Subgroups — C₃₁×#34500 #21491259

Order 7936.

2240 conjugacy classes of subgroups.

The subgroup lattice has 1004 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#559	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2379	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2376	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2367	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2367	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2366	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#2359	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2355	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#2352	C₂	1	2	yes	maximal, complemented
10	C₃₁×#2351	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#2349	C₂	1	1	yes	maximal, characteristic, complemented
12	D₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#2337	C₂	1	2	yes	maximal, complemented
14	C₃₁×#2335	C₂	1	2	yes	maximal, complemented
15	C₃₁×#2328	C₂	1	1	yes	maximal, characteristic
16	C₃₁×#2326	C₂	1	1	yes	maximal, characteristic, complemented
17	C₃₁×#2322	C₂	1	2	yes	maximal, complemented
18	C₃₁×#2320	C₂	1	1	yes	maximal, characteristic, complemented
19	C₃₁×#2319	C₂	1	1	yes	maximal, characteristic, complemented
20	C₃₁×#2318	C₂	1	2	yes	maximal, complemented
21	C₄×C₃₁×#124	C₂	1	1	yes	maximal, characteristic, complemented
22	C₃₁×#2295	C₂	1	1	yes	maximal, characteristic, complemented
23	C₃₁×#2293	C₂	1	1	yes	maximal, characteristic, complemented
24	C₃₁×#559	C₄	1	16	yes	complemented
25	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×#124	C₂²	1	2	yes	—
28	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#123	C₂²	1	2	yes	—
30	C₂×C₃₁×#123	C₂²	1	2	yes	—
31	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
32	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
33	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
34	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
35	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
36	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
37	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
38	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
39	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
40	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
41	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
42	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
43	C₂×C₃₁×#117	C₂²	1	2	yes	—
44	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
45	C₃₁×#391	C₂²	1	1	yes	characteristic
46	C₃₁×#391	C₂²	1	1	yes	characteristic
47	C₃₁×#391	C₂²	1	1	yes	characteristic
48	C₃₁×#391	C₂²	1	2	yes	—
49	C₃₁×#391	C₂²	1	1	yes	characteristic
50	C₃₁×#390	C₂²	1	1	yes	characteristic
51	C₃₁×#390	C₂²	1	1	yes	characteristic
52	C₃₁×#390	C₂²	1	1	yes	characteristic
53	C₃₁×#390	C₂²	1	2	yes	—
54	C₃₁×#390	C₂²	1	1	yes	characteristic
55	C₃₁×#389	C₂²	1	1	yes	characteristic
56	C₃₁×#389	C₂²	1	1	yes	characteristic
57	C₃₁×#389	C₂²	1	1	yes	characteristic
58	C₃₁×#389	C₂²	1	1	yes	characteristic
59	C₃₁×#389	C₂²	1	2	yes	—
60	C₃₁×#389	C₂²	1	2	yes	—
61	C₃₁×#389	C₂²	1	1	yes	characteristic
62	C₃₁×#389	C₂²	1	1	yes	characteristic
63	C₃₁×#388	C₂²	1	2	yes	—
64	C₃₁×#388	C₂²	1	2	yes	—
65	C₃₁×#388	C₂²	1	2	yes	—
66	C₃₁×#388	C₂²	1	2	yes	—
67	C₃₁×#388	C₂²	1	2	yes	—
68	C₃₁×#388	C₂²	1	2	yes	—
69	C₃₁×#388	C₂²	1	2	yes	—
70	C₃₁×#388	C₂²	1	2	yes	—
71	C₃₁×#388	C₂²	1	2	yes	—
72	C₃₁×#388	C₂²	1	2	yes	—
73	C₃₁×#388	C₂²	1	2	yes	—
74	C₃₁×#388	C₂²	1	2	yes	—
75	C₃₁×#387	C₂²	1	1	yes	characteristic
76	C₃₁×#387	C₂²	1	2	yes	—
77	C₃₁×#387	C₂²	1	2	yes	—
78	C₃₁×#387	C₂²	1	2	yes	—
79	C₃₁×#387	C₂²	1	2	yes	—
80	C₃₁×#387	C₂²	1	1	yes	characteristic
81	C₃₁×#387	C₂²	1	1	yes	characteristic
82	C₃₁×#387	C₂²	1	1	yes	characteristic
83	C₃₁×#386	C₂²	1	2	yes	—
84	C₃₁×#386	C₂²	1	2	yes	—
85	C₃₁×#386	C₂²	1	2	yes	—
86	C₃₁×#386	C₂²	1	2	yes	—
87	C₃₁×#386	C₂²	1	2	yes	—
88	C₃₁×#386	C₂²	1	2	yes	—
89	C₃₁×#385	C₂²	1	2	yes	—
90	C₃₁×#385	C₂²	1	2	yes	—
91	C₃₁×#385	C₂²	1	2	yes	—
92	C₃₁×#385	C₂²	1	2	yes	—
93	C₃₁×#385	C₂²	1	2	yes	—
94	C₃₁×#385	C₂²	1	2	yes	—
95	C₃₁×#385	C₂²	1	2	yes	—
96	C₃₁×#385	C₂²	1	2	yes	—
97	C₃₁×#384	C₂²	1	1	yes	characteristic
98	C₃₁×#383	C₂²	1	2	yes	—
99	C₃₁×#382	C₂²	1	2	yes	—
100	C₃₁×#382	C₂²	1	2	yes	—
101	C₃₁×#382	C₂²	1	1	yes	characteristic
102	C₃₁×#382	C₂²	1	1	yes	characteristic
103	C₃₁×#381	C₂²	1	2	yes	—
104	C₃₁×#380	C₂²	1	2	yes	—
105	C₃₁×#380	C₂²	1	2	yes	—
106	C₃₁×#380	C₂²	1	2	yes	—
107	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
108	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
109	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
110	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
111	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
112	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
113	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
114	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
115	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
116	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
117	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
118	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
119	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
120	C₃₁×#376	C₂²	1	1	yes	characteristic
121	C₃₁×#376	C₂²	1	1	yes	characteristic
122	C₂×C₃₁×#95	C₂²	1	2	yes	—
123	C₄³×C₃₁	C₂²	1	1	yes	characteristic
124	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
125	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
126	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
127	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
128	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
129	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
130	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
131	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
132	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
133	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
134	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
135	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
136	C₃₁×#127	C₂×C₄	1	8	yes	—
137	C₃₁×#124	C₂×C₄	1	8	yes	complemented
138	C₃₁×#124	C₂×C₄	1	8	yes	complemented
139	C₃₁×#124	C₂×C₄	1	16	yes	complemented
140	C₃₁×#123	C₂×C₄	1	16	yes	complemented
141	C₃₁×#123	C₂×C₄	1	16	yes	complemented
142	C₄×Q₈×C₃₁	—	4	2	no	—
143	C₄×Q₈×C₃₁	—	4	2	no	—
144	C₄×Q₈×C₃₁	C₂×C₄	1	8	yes	complemented
145	C₄×Q₈×C₃₁	C₂×C₄	1	8	yes	complemented
146	C₄×D₄×C₃₁	—	4	2	no	—
147	C₄×D₄×C₃₁	—	4	2	no	—
148	C₄×D₄×C₃₁	—	4	2	no	—
149	C₄×D₄×C₃₁	—	4	2	no	—
150	C₄×D₄×C₃₁	—	4	4	no	—
151	C₄×D₄×C₃₁	C₂×C₄	1	8	yes	complemented
152	C₄×D₄×C₃₁	C₂×C₄	1	8	yes	complemented
153	C₄×D₄×C₃₁	C₂×C₄	1	16	yes	complemented
154	C₃₁×#117	—	4	4	no	—
155	C₂×C₄⋊C₄×C₃₁	—	4	4	no	—
156	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
157	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
158	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
159	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
160	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
161	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
162	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
163	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
164	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
165	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
166	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
167	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
168	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
169	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
170	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
171	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
172	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
173	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
174	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
175	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
176	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
177	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
178	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
179	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
180	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
181	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
182	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
183	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
184	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
185	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
186	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
187	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
188	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
189	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
190	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
191	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
192	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
193	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
194	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
195	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
196	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
197	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
198	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
199	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
200	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
201	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
202	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
203	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
204	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
205	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
206	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
207	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
208	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
209	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
210	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
211	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
212	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
213	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
214	C₃₁×#95	C₂³	1	1	yes	characteristic
215	C₃₁×#95	C₂³	1	2	yes	—
216	C₃₁×#95	C₂³	1	2	yes	—
217	C₃₁×#95	C₂³	1	2	yes	—
218	C₃₁×#95	C₂³	1	1	yes	characteristic
219	C₃₁×#95	C₂³	1	1	yes	characteristic
220	C₃₁×#95	C₂³	1	1	yes	characteristic
221	C₃₁×#95	C₂³	1	1	yes	characteristic
222	C₃₁×#95	C₂³	1	1	yes	characteristic
223	C₃₁×#95	C₂³	1	1	yes	characteristic
224	C₃₁×#95	C₂³	1	1	yes	characteristic
225	C₃₁×#95	C₂³	1	1	yes	characteristic
226	C₃₁×#95	C₂³	1	1	yes	characteristic
227	C₃₁×#95	C₂³	1	1	yes	characteristic
228	C₃₁×#95	C₂³	1	2	yes	—
229	C₃₁×#95	C₂³	1	2	yes	—
230	C₃₁×#95	C₂³	1	2	yes	—
231	C₃₁×#95	C₂³	1	2	yes	—
232	C₃₁×#95	C₂³	1	2	yes	—
233	C₃₁×#95	C₂³	1	2	yes	—
234	C₃₁×#95	C₂³	1	2	yes	—
235	C₃₁×#95	C₂³	1	1	yes	characteristic
236	C₃₁×#95	C₂³	1	2	yes	—
237	C₃₁×#95	C₂³	1	2	yes	—
238	C₃₁×#95	C₂³	1	1	yes	characteristic
239	C₃₁×#95	C₂³	1	2	yes	—
240	C₃₁×#95	C₂³	1	2	yes	—
241	C₃₁×#95	C₂³	1	1	yes	characteristic
242	C₃₁×#95	C₂³	1	2	yes	—
243	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
244	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
245	C₂×Q₈×C₃₁	—	2	2	no	—
246	C₂×Q₈×C₃₁	—	2	2	no	—
247	C₂×Q₈×C₃₁	—	4	1	no	—
248	C₂×Q₈×C₃₁	—	4	1	no	—
249	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
250	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
251	C₂×D₄×C₃₁	—	2	2	no	—
252	C₂×D₄×C₃₁	—	2	2	no	—
253	C₂×D₄×C₃₁	—	2	2	no	—
254	C₂×D₄×C₃₁	—	2	2	no	—
255	C₂×D₄×C₃₁	—	4	2	no	—
256	C₂×D₄×C₃₁	—	4	2	no	—
257	C₂×D₄×C₃₁	—	4	2	no	—
258	C₂×D₄×C₃₁	—	4	2	no	—
259	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
260	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
261	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
262	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
263	C₂²×C₄×C₃₁	—	2	4	no	—
264	C₂²×C₄×C₃₁	—	4	2	no	—
265	C₂²×C₄×C₃₁	—	4	2	no	—
266	C₂²×C₄×C₃₁	—	4	2	no	—
267	C₂²×C₄×C₃₁	—	4	2	no	—
268	C₂²×C₄×C₃₁	—	4	2	no	—
269	C₂²×C₄×C₃₁	—	4	2	no	—
270	C₂²×C₄×C₃₁	—	4	2	no	—
271	C₂²×C₄×C₃₁	—	4	2	no	—
272	C₂²×C₄×C₃₁	—	4	2	no	—
273	C₂²×C₄×C₃₁	—	4	2	no	—
274	C₂²×C₄×C₃₁	—	4	2	no	—
275	C₂²×C₄×C₃₁	—	4	2	no	—
276	C₂²×C₄×C₃₁	—	4	2	no	—
277	C₂²×C₄×C₃₁	—	4	2	no	—
278	C₂²×C₄×C₃₁	—	4	2	no	—
279	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
280	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
281	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
282	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
283	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
284	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
285	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
286	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
287	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
288	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
289	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
290	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
291	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
292	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
293	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
294	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
295	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
296	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
297	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
298	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
299	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
300	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
301	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
302	C₄⋊C₄×C₃₁	—	2	2	no	—
303	C₄⋊C₄×C₃₁	—	2	4	no	—
304	C₄⋊C₄×C₃₁	—	2	2	no	—
305	C₄⋊C₄×C₃₁	—	2	2	no	—
306	C₄⋊C₄×C₃₁	—	2	2	no	—
307	C₄⋊C₄×C₃₁	—	2	2	no	—
308	C₄⋊C₄×C₃₁	—	2	2	no	—
309	C₄⋊C₄×C₃₁	—	4	2	no	—
310	C₄⋊C₄×C₃₁	—	4	2	no	—
311	C₄⋊C₄×C₃₁	—	4	1	no	—
312	C₄⋊C₄×C₃₁	—	4	1	no	—
313	C₄⋊C₄×C₃₁	—	4	2	no	—
314	C₄⋊C₄×C₃₁	—	4	1	no	—
315	C₄⋊C₄×C₃₁	—	4	2	no	—
316	C₄⋊C₄×C₃₁	—	4	1	no	—
317	C₄⋊C₄×C₃₁	—	4	2	no	—
318	C₄⋊C₄×C₃₁	—	4	2	no	—
319	C₄⋊C₄×C₃₁	—	4	2	no	—
320	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
321	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
322	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
323	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
324	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
325	C₂²⋊C₄×C₃₁	—	2	4	no	—
326	C₂²⋊C₄×C₃₁	—	4	2	no	—
327	C₂²⋊C₄×C₃₁	—	4	2	no	—
328	C₂²⋊C₄×C₃₁	—	4	2	no	—
329	C₂²⋊C₄×C₃₁	—	4	2	no	—
330	C₂²⋊C₄×C₃₁	—	4	2	no	—
331	C₂²⋊C₄×C₃₁	—	4	2	no	—
332	C₂²⋊C₄×C₃₁	—	4	2	no	—
333	C₂²⋊C₄×C₃₁	—	4	2	no	—
334	C₂²⋊C₄×C₃₁	—	4	2	no	—
335	C₂²⋊C₄×C₃₁	—	4	2	no	—
336	C₂²⋊C₄×C₃₁	—	4	2	no	—
337	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
338	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
339	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
340	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
341	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
342	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
343	C₄²×C₃₁	—	2	2	no	—
344	C₄²×C₃₁	—	2	2	no	—
345	C₄²×C₃₁	—	2	2	no	—
346	C₄²×C₃₁	—	2	2	no	—
347	C₄²×C₃₁	—	2	2	no	—
348	C₄²×C₃₁	—	4	2	no	—
349	C₄²×C₃₁	—	4	1	no	—
350	C₄²×C₃₁	—	4	1	no	—
351	C₄²×C₃₁	—	4	1	no	—
352	C₄²×C₃₁	—	4	2	no	—
353	C₄²×C₃₁	—	4	2	no	—
354	C₄²×C₃₁	—	4	2	no	—
355	C₄²×C₃₁	—	4	2	no	—
356	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
357	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
358	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
359	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
360	#34500	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
361	C₂³×C₃₁	—	2	2	no	—
362	C₂³×C₃₁	—	2	2	no	—
363	C₂³×C₃₁	—	2	2	no	—
364	C₂³×C₃₁	—	2	2	no	—
365	C₂³×C₃₁	—	2	2	no	—
366	C₂³×C₃₁	—	2	2	no	—
367	C₂³×C₃₁	—	2	2	no	—
368	C₂³×C₃₁	—	2	2	no	—
369	C₂³×C₃₁	—	2	2	no	—
370	C₂³×C₃₁	—	2	2	no	—
371	C₂³×C₃₁	—	2	2	no	—
372	C₂³×C₃₁	—	2	2	no	—
373	C₂³×C₃₁	C₂³×C₄	1	4	yes	—
374	C₂³×C₃₁	C₂³×C₄	1	4	yes	—
375	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
376	Q₈×C₃₁	—	2	8	no	—
377	Q₈×C₃₁	—	4	4	no	—
378	D₄×C₃₁	—	2	8	no	—
379	D₄×C₃₁	—	2	8	no	—
380	D₄×C₃₁	—	4	8	no	—
381	D₄×C₃₁	—	4	8	no	—
382	C₂×C₄×C₃₁	—	2	1	no	—
383	C₂×C₄×C₃₁	—	2	1	no	—
384	C₂×C₄×C₃₁	—	2	1	no	—
385	C₂×C₄×C₃₁	—	2	2	no	—
386	C₂×C₄×C₃₁	—	2	1	no	—
387	C₂×C₄×C₃₁	—	2	2	no	—
388	C₂×C₄×C₃₁	—	2	2	no	—
389	C₂×C₄×C₃₁	—	2	1	no	—
390	C₂×C₄×C₃₁	—	2	1	no	—
391	C₂×C₄×C₃₁	—	2	1	no	—
392	C₂×C₄×C₃₁	—	2	1	no	—
393	C₂×C₄×C₃₁	—	2	1	no	—
394	C₂×C₄×C₃₁	—	2	1	no	—
395	C₂×C₄×C₃₁	—	2	1	no	—
396	C₂×C₄×C₃₁	—	2	2	no	—
397	C₂×C₄×C₃₁	—	2	1	no	—
398	C₂×C₄×C₃₁	—	2	1	no	—
399	C₂×C₄×C₃₁	—	2	1	no	—
400	C₂×C₄×C₃₁	—	2	1	no	—
401	C₂×C₄×C₃₁	—	2	1	no	—
402	C₂×C₄×C₃₁	—	2	1	no	—
403	C₂×C₄×C₃₁	—	2	1	no	—
404	C₂×C₄×C₃₁	—	2	1	no	—
405	C₂×C₄×C₃₁	—	2	1	no	—
406	C₂×C₄×C₃₁	—	2	2	no	—
407	C₂×C₄×C₃₁	—	2	2	no	—
408	C₂×C₄×C₃₁	—	2	2	no	—
409	C₂×C₄×C₃₁	—	2	2	no	—
410	C₂×C₄×C₃₁	—	2	2	no	—
411	C₂×C₄×C₃₁	—	2	2	no	—
412	C₂×C₄×C₃₁	—	2	1	no	—
413	C₂×C₄×C₃₁	—	2	1	no	—
414	C₂×C₄×C₃₁	—	2	1	no	—
415	C₂×C₄×C₃₁	—	2	1	no	—
416	C₂×C₄×C₃₁	—	2	2	no	—
417	C₂×C₄×C₃₁	—	2	1	no	—
418	C₂×C₄×C₃₁	—	2	2	no	—
419	C₂×C₄×C₃₁	—	2	1	no	—
420	C₂×C₄×C₃₁	—	2	1	no	—
421	C₂×C₄×C₃₁	—	2	1	no	—
422	C₂×C₄×C₃₁	—	2	2	no	—
423	C₂×C₄×C₃₁	—	2	1	no	—
424	C₂×C₄×C₃₁	—	2	1	no	—
425	C₂×C₄×C₃₁	—	2	2	no	—
426	C₂×C₄×C₃₁	—	2	2	no	—
427	C₂×C₄×C₃₁	—	2	1	no	—
428	C₂×C₄×C₃₁	—	2	1	no	—
429	C₂×C₄×C₃₁	—	2	1	no	—
430	C₂×C₄×C₃₁	—	2	2	no	—
431	C₂×C₄×C₃₁	—	2	1	no	—
432	C₂×C₄×C₃₁	—	2	1	no	—
433	C₂×C₄×C₃₁	—	4	8	no	—
434	C₂×C₄×C₃₁	—	4	8	no	—
435	C₂×C₄×C₃₁	—	8	4	no	—
436	C₂×C₄×C₃₁	—	8	4	no	—
437	C₂×C₄×C₃₁	—	8	4	no	—
438	C₂×C₄×C₃₁	—	8	4	no	—
439	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
440	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
441	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
442	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
443	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
444	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
445	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
446	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
447	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
448	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
449	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
450	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
451	C₂×#559	C₂×C₃₁	1	1	yes	characteristic
452	#2379	C₂×C₃₁	1	2	yes	complemented
453	#2376	C₂×C₃₁	1	2	yes	complemented
454	#2367	C₂×C₃₁	1	1	yes	characteristic, complemented
455	#2367	C₂×C₃₁	1	1	yes	characteristic, complemented
456	#2366	C₂×C₃₁	1	1	yes	characteristic
457	#2359	C₂×C₃₁	1	2	yes	complemented
458	#2355	C₂×C₃₁	1	1	yes	characteristic, complemented
459	#2352	C₂×C₃₁	1	2	yes	complemented
460	#2351	C₂×C₃₁	1	1	yes	characteristic, complemented
461	#2349	C₂×C₃₁	1	1	yes	characteristic, complemented
462	D₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
463	#2337	C₂×C₃₁	1	2	yes	complemented
464	#2335	C₂×C₃₁	1	2	yes	complemented
465	#2328	C₂×C₃₁	1	1	yes	characteristic
466	#2326	C₂×C₃₁	1	1	yes	characteristic, complemented
467	#2322	C₂×C₃₁	1	2	yes	complemented
468	#2320	C₂×C₃₁	1	1	yes	characteristic, complemented
469	#2319	C₂×C₃₁	1	1	yes	characteristic, complemented
470	#2318	C₂×C₃₁	1	2	yes	complemented
471	C₄×#124	C₂×C₃₁	1	1	yes	characteristic, complemented
472	#2295	C₂×C₃₁	1	1	yes	characteristic, complemented
473	#2293	C₂×C₃₁	1	1	yes	characteristic, complemented
474	C₂²×C₃₁	—	2	4	no	—
475	C₂²×C₃₁	—	2	4	no	—
476	C₂²×C₃₁	—	2	4	no	—
477	C₂²×C₃₁	—	2	4	no	—
478	C₂²×C₃₁	—	2	4	no	—
479	C₂²×C₃₁	—	2	4	no	—
480	C₂²×C₃₁	—	4	2	no	—
481	C₂²×C₃₁	—	4	2	no	—
482	C₂²×C₃₁	—	4	2	no	—
483	C₂²×C₃₁	—	4	2	no	—
484	C₂²×C₃₁	—	4	2	no	—
485	C₂²×C₃₁	—	4	2	no	—
486	C₂²×C₃₁	—	4	2	no	—
487	C₂²×C₃₁	—	4	2	no	—
488	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
489	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
490	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
491	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
492	C₂²×C₃₁	#585	1	1	yes	characteristic
493	C₂²×C₃₁	#585	1	1	yes	characteristic
494	C₂²×C₃₁	#585	1	1	yes	characteristic
495	C₄×C₃₁	—	2	4	no	—
496	C₄×C₃₁	—	2	2	no	—
497	C₄×C₃₁	—	2	2	no	—
498	C₄×C₃₁	—	2	2	no	—
499	C₄×C₃₁	—	2	2	no	—
500	C₄×C₃₁	—	2	2	no	—
501	C₄×C₃₁	—	2	4	no	—
502	C₄×C₃₁	—	2	2	no	—
503	C₄×C₃₁	—	2	2	no	—
504	C₄×C₃₁	—	4	1	no	—
505	C₄×C₃₁	—	4	2	no	—
506	C₄×C₃₁	—	4	1	no	—
507	C₄×C₃₁	—	4	2	no	—
508	C₄×C₃₁	—	4	1	no	—
509	C₄×C₃₁	—	4	1	no	—
510	C₄×C₃₁	—	4	2	no	—
511	C₄×C₃₁	—	4	1	no	—
512	C₄×C₃₁	—	4	1	no	—
513	C₄×C₃₁	—	4	2	no	—
514	C₄×C₃₁	—	4	1	no	—
515	C₄×C₃₁	#520	1	4	yes	—
516	#559	C₄×C₃₁	1	16	yes	complemented
517	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
518	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
519	C₂×#124	C₂²×C₃₁	1	2	yes	—
520	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
521	C₂×#123	C₂²×C₃₁	1	2	yes	—
522	C₂×#123	C₂²×C₃₁	1	2	yes	—
523	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
524	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
525	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
526	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
527	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
528	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
529	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
530	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
531	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
532	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
533	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
534	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
535	C₂×#117	C₂²×C₃₁	1	2	yes	—
536	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
537	#391	C₂²×C₃₁	1	1	yes	characteristic
538	#391	C₂²×C₃₁	1	1	yes	characteristic
539	#391	C₂²×C₃₁	1	1	yes	characteristic
540	#391	C₂²×C₃₁	1	2	yes	—
541	#391	C₂²×C₃₁	1	1	yes	characteristic
542	#390	C₂²×C₃₁	1	1	yes	characteristic
543	#390	C₂²×C₃₁	1	1	yes	characteristic
544	#390	C₂²×C₃₁	1	1	yes	characteristic
545	#390	C₂²×C₃₁	1	2	yes	—
546	#390	C₂²×C₃₁	1	1	yes	characteristic
547	#389	C₂²×C₃₁	1	1	yes	characteristic
548	#389	C₂²×C₃₁	1	1	yes	characteristic
549	#389	C₂²×C₃₁	1	1	yes	characteristic
550	#389	C₂²×C₃₁	1	1	yes	characteristic
551	#389	C₂²×C₃₁	1	2	yes	—
552	#389	C₂²×C₃₁	1	2	yes	—
553	#389	C₂²×C₃₁	1	1	yes	characteristic
554	#389	C₂²×C₃₁	1	1	yes	characteristic
555	#388	C₂²×C₃₁	1	2	yes	—
556	#388	C₂²×C₃₁	1	2	yes	—
557	#388	C₂²×C₃₁	1	2	yes	—
558	#388	C₂²×C₃₁	1	2	yes	—
559	#388	C₂²×C₃₁	1	2	yes	—
560	#388	C₂²×C₃₁	1	2	yes	—
561	#388	C₂²×C₃₁	1	2	yes	—
562	#388	C₂²×C₃₁	1	2	yes	—
563	#388	C₂²×C₃₁	1	2	yes	—
564	#388	C₂²×C₃₁	1	2	yes	—
565	#388	C₂²×C₃₁	1	2	yes	—
566	#388	C₂²×C₃₁	1	2	yes	—
567	#387	C₂²×C₃₁	1	1	yes	characteristic
568	#387	C₂²×C₃₁	1	2	yes	—
569	#387	C₂²×C₃₁	1	2	yes	—
570	#387	C₂²×C₃₁	1	2	yes	—
571	#387	C₂²×C₃₁	1	2	yes	—
572	#387	C₂²×C₃₁	1	1	yes	characteristic
573	#387	C₂²×C₃₁	1	1	yes	characteristic
574	#387	C₂²×C₃₁	1	1	yes	characteristic
575	#386	C₂²×C₃₁	1	2	yes	—
576	#386	C₂²×C₃₁	1	2	yes	—
577	#386	C₂²×C₃₁	1	2	yes	—
578	#386	C₂²×C₃₁	1	2	yes	—
579	#386	C₂²×C₃₁	1	2	yes	—
580	#386	C₂²×C₃₁	1	2	yes	—
581	#385	C₂²×C₃₁	1	2	yes	—
582	#385	C₂²×C₃₁	1	2	yes	—
583	#385	C₂²×C₃₁	1	2	yes	—
584	#385	C₂²×C₃₁	1	2	yes	—
585	#385	C₂²×C₃₁	1	2	yes	—
586	#385	C₂²×C₃₁	1	2	yes	—
587	#385	C₂²×C₃₁	1	2	yes	—
588	#385	C₂²×C₃₁	1	2	yes	—
589	#384	C₂²×C₃₁	1	1	yes	characteristic
590	#383	C₂²×C₃₁	1	2	yes	—
591	#382	C₂²×C₃₁	1	2	yes	—
592	#382	C₂²×C₃₁	1	2	yes	—
593	#382	C₂²×C₃₁	1	1	yes	characteristic
594	#382	C₂²×C₃₁	1	1	yes	characteristic
595	#381	C₂²×C₃₁	1	2	yes	—
596	#380	C₂²×C₃₁	1	2	yes	—
597	#380	C₂²×C₃₁	1	2	yes	—
598	#380	C₂²×C₃₁	1	2	yes	—
599	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
600	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
601	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
602	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
603	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
604	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
605	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
606	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
607	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
608	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
609	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
610	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
611	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
612	#376	C₂²×C₃₁	1	1	yes	characteristic
613	#376	C₂²×C₃₁	1	1	yes	characteristic
614	C₂×#95	C₂²×C₃₁	1	2	yes	—
615	C₄³	C₂²×C₃₁	1	1	yes	characteristic
616	C₂×C₃₁	—	4	4	no	—
617	C₂×C₃₁	—	4	4	no	—
618	C₂×C₃₁	C₂×#520	1	1	yes	characteristic
619	C₂×C₃₁	#3428	1	1	yes	characteristic
620	C₂×C₃₁	C₄×H(5,𝔽₂)	1	1	yes	characteristic
621	C₂×C₃₁	#3554	1	1	yes	characteristic
622	C₂×C₃₁	#3557	1	1	yes	characteristic
623	C₂×C₃₁	#3558	1	1	yes	characteristic
624	C₂×C₃₁	#3562	1	1	yes	characteristic
625	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
626	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
627	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
628	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
629	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
630	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
631	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
632	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
633	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
634	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
635	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
636	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
637	#127	C₂×C₄×C₃₁	1	8	yes	—
638	#124	C₂×C₄×C₃₁	1	8	yes	complemented
639	#124	C₂×C₄×C₃₁	1	8	yes	complemented
640	#124	C₂×C₄×C₃₁	1	16	yes	complemented
641	#123	C₂×C₄×C₃₁	1	16	yes	complemented
642	#123	C₂×C₄×C₃₁	1	16	yes	complemented
643	C₄×Q₈	—	4	2	no	—
644	C₄×Q₈	—	4	2	no	—
645	C₄×Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
646	C₄×Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
647	C₄×D₄	—	4	2	no	—
648	C₄×D₄	—	4	2	no	—
649	C₄×D₄	—	4	2	no	—
650	C₄×D₄	—	4	2	no	—
651	C₄×D₄	—	4	4	no	—
652	C₄×D₄	C₂×C₄×C₃₁	1	8	yes	complemented
653	C₄×D₄	C₂×C₄×C₃₁	1	8	yes	complemented
654	C₄×D₄	C₂×C₄×C₃₁	1	16	yes	complemented
655	#117	—	4	4	no	—
656	C₂×C₄⋊C₄	—	4	4	no	—
657	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
658	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
659	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
660	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
661	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
662	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
663	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
664	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
665	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
666	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
667	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
668	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
669	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
670	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
671	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
672	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
673	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
674	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
675	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
676	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
677	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
678	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
679	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
680	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
681	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
682	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
683	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
684	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
685	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
686	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
687	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
688	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
689	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
690	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
691	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
692	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
693	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
694	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
695	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
696	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
697	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
698	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
699	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
700	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
701	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
702	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
703	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
704	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
705	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
706	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
707	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
708	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
709	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
710	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
711	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
712	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
713	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
714	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
715	#95	C₂³×C₃₁	1	1	yes	characteristic
716	#95	C₂³×C₃₁	1	2	yes	—
717	#95	C₂³×C₃₁	1	2	yes	—
718	#95	C₂³×C₃₁	1	2	yes	—
719	#95	C₂³×C₃₁	1	1	yes	characteristic
720	#95	C₂³×C₃₁	1	1	yes	characteristic
721	#95	C₂³×C₃₁	1	1	yes	characteristic
722	#95	C₂³×C₃₁	1	1	yes	characteristic
723	#95	C₂³×C₃₁	1	1	yes	characteristic
724	#95	C₂³×C₃₁	1	1	yes	characteristic
725	#95	C₂³×C₃₁	1	1	yes	characteristic
726	#95	C₂³×C₃₁	1	1	yes	characteristic
727	#95	C₂³×C₃₁	1	1	yes	characteristic
728	#95	C₂³×C₃₁	1	1	yes	characteristic
729	#95	C₂³×C₃₁	1	2	yes	—
730	#95	C₂³×C₃₁	1	2	yes	—
731	#95	C₂³×C₃₁	1	2	yes	—
732	#95	C₂³×C₃₁	1	2	yes	—
733	#95	C₂³×C₃₁	1	2	yes	—
734	#95	C₂³×C₃₁	1	2	yes	—
735	#95	C₂³×C₃₁	1	2	yes	—
736	#95	C₂³×C₃₁	1	1	yes	characteristic
737	#95	C₂³×C₃₁	1	2	yes	—
738	#95	C₂³×C₃₁	1	2	yes	—
739	#95	C₂³×C₃₁	1	1	yes	characteristic
740	#95	C₂³×C₃₁	1	2	yes	—
741	#95	C₂³×C₃₁	1	2	yes	—
742	#95	C₂³×C₃₁	1	1	yes	characteristic
743	#95	C₂³×C₃₁	1	2	yes	—
744	C₃₁	#34500	1	1	yes	characteristic, direct product factor, Sylow
745	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
746	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
747	C₂×Q₈	—	2	2	no	—
748	C₂×Q₈	—	2	2	no	—
749	C₂×Q₈	—	4	1	no	—
750	C₂×Q₈	—	4	1	no	—
751	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
752	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
753	C₂×D₄	—	2	2	no	—
754	C₂×D₄	—	2	2	no	—
755	C₂×D₄	—	2	2	no	—
756	C₂×D₄	—	2	2	no	—
757	C₂×D₄	—	4	2	no	—
758	C₂×D₄	—	4	2	no	—
759	C₂×D₄	—	4	2	no	—
760	C₂×D₄	—	4	2	no	—
761	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
762	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
763	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
764	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
765	C₂²×C₄	—	2	4	no	—
766	C₂²×C₄	—	4	2	no	—
767	C₂²×C₄	—	4	2	no	—
768	C₂²×C₄	—	4	2	no	—
769	C₂²×C₄	—	4	2	no	—
770	C₂²×C₄	—	4	2	no	—
771	C₂²×C₄	—	4	2	no	—
772	C₂²×C₄	—	4	2	no	—
773	C₂²×C₄	—	4	2	no	—
774	C₂²×C₄	—	4	2	no	—
775	C₂²×C₄	—	4	2	no	—
776	C₂²×C₄	—	4	2	no	—
777	C₂²×C₄	—	4	2	no	—
778	C₂²×C₄	—	4	2	no	—
779	C₂²×C₄	—	4	2	no	—
780	C₂²×C₄	—	4	2	no	—
781	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
782	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
783	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
784	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
785	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
786	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
787	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
788	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
789	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
790	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
791	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
792	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
793	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
794	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
795	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
796	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
797	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
798	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
799	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
800	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
801	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
802	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
803	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
804	C₄⋊C₄	—	2	2	no	—
805	C₄⋊C₄	—	2	4	no	—
806	C₄⋊C₄	—	2	2	no	—
807	C₄⋊C₄	—	2	2	no	—
808	C₄⋊C₄	—	2	2	no	—
809	C₄⋊C₄	—	2	2	no	—
810	C₄⋊C₄	—	2	2	no	—
811	C₄⋊C₄	—	4	2	no	—
812	C₄⋊C₄	—	4	2	no	—
813	C₄⋊C₄	—	4	1	no	—
814	C₄⋊C₄	—	4	1	no	—
815	C₄⋊C₄	—	4	2	no	—
816	C₄⋊C₄	—	4	1	no	—
817	C₄⋊C₄	—	4	2	no	—
818	C₄⋊C₄	—	4	1	no	—
819	C₄⋊C₄	—	4	2	no	—
820	C₄⋊C₄	—	4	2	no	—
821	C₄⋊C₄	—	4	2	no	—
822	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
823	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
824	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
825	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
826	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
827	C₂²⋊C₄	—	2	4	no	—
828	C₂²⋊C₄	—	4	2	no	—
829	C₂²⋊C₄	—	4	2	no	—
830	C₂²⋊C₄	—	4	2	no	—
831	C₂²⋊C₄	—	4	2	no	—
832	C₂²⋊C₄	—	4	2	no	—
833	C₂²⋊C₄	—	4	2	no	—
834	C₂²⋊C₄	—	4	2	no	—
835	C₂²⋊C₄	—	4	2	no	—
836	C₂²⋊C₄	—	4	2	no	—
837	C₂²⋊C₄	—	4	2	no	—
838	C₂²⋊C₄	—	4	2	no	—
839	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
840	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
841	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
842	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
843	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
844	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
845	C₄²	—	2	2	no	—
846	C₄²	—	2	2	no	—
847	C₄²	—	2	2	no	—
848	C₄²	—	2	2	no	—
849	C₄²	—	2	2	no	—
850	C₄²	—	4	2	no	—
851	C₄²	—	4	1	no	—
852	C₄²	—	4	1	no	—
853	C₄²	—	4	1	no	—
854	C₄²	—	4	2	no	—
855	C₄²	—	4	2	no	—
856	C₄²	—	4	2	no	—
857	C₄²	—	4	2	no	—
858	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
859	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
860	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
861	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
862	C₂³	—	2	2	no	—
863	C₂³	—	2	2	no	—
864	C₂³	—	2	2	no	—
865	C₂³	—	2	2	no	—
866	C₂³	—	2	2	no	—
867	C₂³	—	2	2	no	—
868	C₂³	—	2	2	no	—
869	C₂³	—	2	2	no	—
870	C₂³	—	2	2	no	—
871	C₂³	—	2	2	no	—
872	C₂³	—	2	2	no	—
873	C₂³	—	2	2	no	—
874	C₂³	C₂³×C₄×C₃₁	1	4	yes	—
875	C₂³	C₂³×C₄×C₃₁	1	4	yes	—
876	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
877	Q₈	—	2	8	no	—
878	Q₈	—	4	4	no	—
879	D₄	—	2	8	no	—
880	D₄	—	2	8	no	—
881	D₄	—	4	8	no	—
882	D₄	—	4	8	no	—
883	C₂×C₄	—	2	1	no	—
884	C₂×C₄	—	2	1	no	—
885	C₂×C₄	—	2	1	no	—
886	C₂×C₄	—	2	2	no	—
887	C₂×C₄	—	2	1	no	—
888	C₂×C₄	—	2	2	no	—
889	C₂×C₄	—	2	2	no	—
890	C₂×C₄	—	2	1	no	—
891	C₂×C₄	—	2	1	no	—
892	C₂×C₄	—	2	1	no	—
893	C₂×C₄	—	2	1	no	—
894	C₂×C₄	—	2	1	no	—
895	C₂×C₄	—	2	1	no	—
896	C₂×C₄	—	2	1	no	—
897	C₂×C₄	—	2	2	no	—
898	C₂×C₄	—	2	1	no	—
899	C₂×C₄	—	2	1	no	—
900	C₂×C₄	—	2	1	no	—
901	C₂×C₄	—	2	1	no	—
902	C₂×C₄	—	2	1	no	—
903	C₂×C₄	—	2	1	no	—
904	C₂×C₄	—	2	1	no	—
905	C₂×C₄	—	2	1	no	—
906	C₂×C₄	—	2	1	no	—
907	C₂×C₄	—	2	2	no	—
908	C₂×C₄	—	2	2	no	—
909	C₂×C₄	—	2	2	no	—
910	C₂×C₄	—	2	2	no	—
911	C₂×C₄	—	2	2	no	—
912	C₂×C₄	—	2	2	no	—
913	C₂×C₄	—	2	1	no	—
914	C₂×C₄	—	2	1	no	—
915	C₂×C₄	—	2	1	no	—
916	C₂×C₄	—	2	1	no	—
917	C₂×C₄	—	2	2	no	—
918	C₂×C₄	—	2	1	no	—
919	C₂×C₄	—	2	2	no	—
920	C₂×C₄	—	2	1	no	—
921	C₂×C₄	—	2	1	no	—
922	C₂×C₄	—	2	1	no	—
923	C₂×C₄	—	2	2	no	—
924	C₂×C₄	—	2	1	no	—
925	C₂×C₄	—	2	1	no	—
926	C₂×C₄	—	2	2	no	—
927	C₂×C₄	—	2	2	no	—
928	C₂×C₄	—	2	1	no	—
929	C₂×C₄	—	2	1	no	—
930	C₂×C₄	—	2	1	no	—
931	C₂×C₄	—	2	2	no	—
932	C₂×C₄	—	2	1	no	—
933	C₂×C₄	—	2	1	no	—
934	C₂×C₄	—	4	8	no	—
935	C₂×C₄	—	4	8	no	—
936	C₂×C₄	—	8	4	no	—
937	C₂×C₄	—	8	4	no	—
938	C₂×C₄	—	8	4	no	—
939	C₂×C₄	—	8	4	no	—
940	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
941	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
942	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
943	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
944	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
945	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
946	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
947	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
948	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
949	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
950	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
951	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
952	C₂²	—	2	4	no	—
953	C₂²	—	2	4	no	—
954	C₂²	—	2	4	no	—
955	C₂²	—	2	4	no	—
956	C₂²	—	2	4	no	—
957	C₂²	—	2	4	no	—
958	C₂²	—	4	2	no	—
959	C₂²	—	4	2	no	—
960	C₂²	—	4	2	no	—
961	C₂²	—	4	2	no	—
962	C₂²	—	4	2	no	—
963	C₂²	—	4	2	no	—
964	C₂²	—	4	2	no	—
965	C₂²	—	4	2	no	—
966	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
967	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
968	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
969	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
970	C₂²	C₃₁×#585	1	1	yes	characteristic
971	C₂²	C₃₁×#585	1	1	yes	characteristic
972	C₂²	C₃₁×#585	1	1	yes	characteristic
973	C₄	—	2	4	no	—
974	C₄	—	2	2	no	—
975	C₄	—	2	2	no	—
976	C₄	—	2	2	no	—
977	C₄	—	2	2	no	—
978	C₄	—	2	2	no	—
979	C₄	—	2	4	no	—
980	C₄	—	2	2	no	—
981	C₄	—	2	2	no	—
982	C₄	—	4	1	no	—
983	C₄	—	4	2	no	—
984	C₄	—	4	1	no	—
985	C₄	—	4	1	no	—
986	C₄	—	4	1	no	—
987	C₄	—	4	2	no	—
988	C₄	—	4	2	no	—
989	C₄	—	4	1	no	—
990	C₄	—	4	2	no	—
991	C₄	—	4	1	no	—
992	C₄	—	4	1	no	—
993	C₄	C₃₁×#520	1	4	yes	—
994	C₂	—	4	4	no	—
995	C₂	—	4	4	no	—
996	C₂	C₂×C₃₁×#520	1	1	yes	characteristic
997	C₂	C₃₁×#3428	1	1	yes	characteristic
998	C₂	C₄×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
999	C₂	C₃₁×#3554	1	1	yes	characteristic
1000	C₂	C₃₁×#3557	1	1	yes	characteristic
1001	C₂	C₃₁×#3558	1	1	yes	characteristic
1002	C₂	C₃₁×#3562	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₃₁×#2379	C₂	semidirect
3. C₃₁×#2376	C₂	semidirect
4. C₃₁×#2367	C₂	semidirect
5. C₃₁×#2367	C₂	semidirect
7. C₃₁×#2359	C₂	semidirect
8. C₃₁×#2355	C₂	semidirect
9. C₃₁×#2352	C₂	semidirect
10. C₃₁×#2351	C₂	semidirect
11. C₃₁×#2349	C₂	semidirect
12. D₄×C₄⋊C₄×C₃₁	C₂	semidirect
13. C₃₁×#2337	C₂	semidirect
14. C₃₁×#2335	C₂	semidirect
16. C₃₁×#2326	C₂	semidirect
17. C₃₁×#2322	C₂	semidirect
18. C₃₁×#2320	C₂	semidirect
19. C₃₁×#2319	C₂	semidirect
20. C₃₁×#2318	C₂	semidirect
21. C₄×C₃₁×#124	C₂	semidirect
22. C₃₁×#2295	C₂	semidirect
23. C₃₁×#2293	C₂	semidirect
24. C₃₁×#559	C₄	semidirect
137. C₃₁×#124	C₂×C₄	semidirect
138. C₃₁×#124	C₂×C₄	semidirect
139. C₃₁×#124	C₂×C₄	semidirect
140. C₃₁×#123	C₂×C₄	semidirect
141. C₃₁×#123	C₂×C₄	semidirect
144. C₄×Q₈×C₃₁	C₂×C₄	semidirect
145. C₄×Q₈×C₃₁	C₂×C₄	semidirect
151. C₄×D₄×C₃₁	C₂×C₄	semidirect
152. C₄×D₄×C₃₁	C₂×C₄	semidirect
153. C₄×D₄×C₃₁	C₂×C₄	semidirect
360. #34500	C₃₁	direct (N × H)
452. #2379	C₂×C₃₁	semidirect
453. #2376	C₂×C₃₁	semidirect
454. #2367	C₂×C₃₁	semidirect
455. #2367	C₂×C₃₁	semidirect
457. #2359	C₂×C₃₁	semidirect
458. #2355	C₂×C₃₁	semidirect
459. #2352	C₂×C₃₁	semidirect
460. #2351	C₂×C₃₁	semidirect
461. #2349	C₂×C₃₁	semidirect
462. D₄×C₄⋊C₄	C₂×C₃₁	semidirect
463. #2337	C₂×C₃₁	semidirect
464. #2335	C₂×C₃₁	semidirect
466. #2326	C₂×C₃₁	semidirect
467. #2322	C₂×C₃₁	semidirect
468. #2320	C₂×C₃₁	semidirect
469. #2319	C₂×C₃₁	semidirect
470. #2318	C₂×C₃₁	semidirect
471. C₄×#124	C₂×C₃₁	semidirect
472. #2295	C₂×C₃₁	semidirect
473. #2293	C₂×C₃₁	semidirect
516. #559	C₄×C₃₁	semidirect
638. #124	C₂×C₄×C₃₁	semidirect
639. #124	C₂×C₄×C₃₁	semidirect
640. #124	C₂×C₄×C₃₁	semidirect
641. #123	C₂×C₄×C₃₁	semidirect
642. #123	C₂×C₄×C₃₁	semidirect
645. C₄×Q₈	C₂×C₄×C₃₁	semidirect
646. C₄×Q₈	C₂×C₄×C₃₁	semidirect
652. C₄×D₄	C₂×C₄×C₃₁	semidirect
653. C₄×D₄	C₂×C₄×C₃₁	semidirect
654. C₄×D₄	C₂×C₄×C₃₁	semidirect
744. C₃₁	#34500	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃₁×#559	2
C₃₁×#2379	2
C₃₁×#2376	2
C₃₁×#2367	2
C₃₁×#2366	2
C₃₁×#2359	2
C₃₁×#2355	2
C₃₁×#2352	2
C₃₁×#2351	2
C₃₁×#2349	2
D₄×C₄⋊C₄×C₃₁	2
C₃₁×#2337	2
C₃₁×#2335	2
C₃₁×#2328	2
C₃₁×#2326	2
C₃₁×#2322	2
C₃₁×#2320	2
C₃₁×#2319	2
C₃₁×#2318	2
C₄×C₃₁×#124	2
C₃₁×#2295	2
C₃₁×#2293	2
#34500	31

Subgroups — C31×#34500 #21491259

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34500 #21491259