Subgroups — C₃₁×#34304 #21491063

Order 7936.

2272 conjugacy classes of subgroups.

The subgroup lattice has 986 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#552	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2373	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2370	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2370	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2360	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2360	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2359	C₂	1	2	yes	maximal
8	C₃₁×#2355	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
10	C₃₁×#2347	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#2344	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#2343	C₂	1	2	yes	maximal, complemented
13	C₃₁×#2336	C₂	1	2	yes	maximal, complemented
14	C₃₁×#2334	C₂	1	2	yes	maximal, complemented
15	C₃₁×#2330	C₂	1	1	yes	maximal, characteristic, complemented
16	C₃₁×#2323	C₂	1	1	yes	maximal, characteristic, complemented
17	C₄×C₃₁×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
18	C₄×C₃₁×#123	C₂	1	2	yes	maximal
19	C₃₁×#2296	C₂	1	2	yes	maximal, complemented
20	C₃₁×#2290	C₂	1	1	yes	maximal, characteristic
21	C₃₁×#2276	C₂	1	1	yes	maximal, characteristic, complemented
22	C₃₁×#2274	C₂	1	1	yes	maximal, characteristic, complemented
23	C₂×C₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
24	C₃₁×#552	C₄	1	16	yes	complemented
25	C₂×C₃₁×#126	C₂²	1	2	yes	—
26	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×#125	C₂²	1	2	yes	—
28	C₂×C₃₁×#123	C₂²	1	2	yes	—
29	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
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×D₄×C₃₁	C₂²	1	2	yes	—
35	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
36	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
37	C₂×C₃₁×#117	C₂²	1	2	yes	—
38	C₂×C₃₁×#117	C₂²	1	2	yes	—
39	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
40	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
41	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
42	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
43	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
44	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
45	C₃₁×#391	C₂²	1	1	yes	characteristic
46	C₃₁×#391	C₂²	1	1	yes	characteristic
47	C₃₁×#389	C₂²	1	2	yes	—
48	C₃₁×#389	C₂²	1	1	yes	characteristic
49	C₃₁×#389	C₂²	1	2	yes	—
50	C₃₁×#389	C₂²	1	1	yes	characteristic
51	C₃₁×#389	C₂²	1	1	yes	characteristic
52	C₃₁×#389	C₂²	1	2	yes	—
53	C₃₁×#389	C₂²	1	2	yes	—
54	C₃₁×#389	C₂²	1	2	yes	—
55	C₃₁×#389	C₂²	1	1	yes	characteristic
56	C₃₁×#389	C₂²	1	2	yes	—
57	C₃₁×#389	C₂²	1	2	yes	—
58	C₃₁×#388	C₂²	1	2	yes	—
59	C₃₁×#388	C₂²	1	2	yes	—
60	C₃₁×#388	C₂²	1	1	yes	characteristic
61	C₃₁×#388	C₂²	1	2	yes	—
62	C₃₁×#388	C₂²	1	2	yes	—
63	C₃₁×#388	C₂²	1	1	yes	characteristic
64	C₃₁×#387	C₂²	1	2	yes	—
65	C₃₁×#387	C₂²	1	2	yes	—
66	C₃₁×#387	C₂²	1	2	yes	—
67	C₃₁×#387	C₂²	1	2	yes	—
68	C₃₁×#387	C₂²	1	2	yes	—
69	C₃₁×#387	C₂²	1	2	yes	—
70	C₃₁×#387	C₂²	1	1	yes	characteristic
71	C₃₁×#387	C₂²	1	2	yes	—
72	C₃₁×#387	C₂²	1	2	yes	—
73	C₃₁×#387	C₂²	1	2	yes	—
74	C₃₁×#387	C₂²	1	2	yes	—
75	C₃₁×#387	C₂²	1	2	yes	—
76	C₃₁×#387	C₂²	1	2	yes	—
77	C₃₁×#387	C₂²	1	1	yes	characteristic
78	C₃₁×#386	C₂²	1	1	yes	characteristic
79	C₃₁×#386	C₂²	1	1	yes	characteristic
80	C₃₁×#385	C₂²	1	2	yes	—
81	C₃₁×#385	C₂²	1	2	yes	—
82	C₃₁×#385	C₂²	1	1	yes	characteristic
83	C₃₁×#385	C₂²	1	2	yes	—
84	C₃₁×#385	C₂²	1	2	yes	—
85	C₃₁×#385	C₂²	1	1	yes	characteristic
86	C₃₁×#385	C₂²	1	2	yes	—
87	C₃₁×#384	C₂²	1	1	yes	characteristic
88	C₃₁×#383	C₂²	1	2	yes	—
89	C₃₁×#382	C₂²	1	2	yes	—
90	C₃₁×#381	C₂²	1	2	yes	—
91	C₃₁×#380	C₂²	1	1	yes	characteristic
92	C₃₁×#380	C₂²	1	1	yes	characteristic
93	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
94	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
95	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
96	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
97	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
98	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
99	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
100	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
101	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
102	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
103	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
104	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
105	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
106	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
107	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
108	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
109	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
110	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
111	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
112	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
113	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
114	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
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	1	yes	characteristic
118	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
119	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
120	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
121	C₃₁×#376	C₂²	1	1	yes	characteristic
122	C₃₁×#376	C₂²	1	2	yes	—
123	C₂×C₃₁×#95	C₂²	1	2	yes	—
124	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
125	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
126	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
127	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
128	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
129	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
130	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
131	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
132	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
133	C₃₁×#126	C₂×C₄	1	16	yes	complemented
134	C₃₁×#125	C₂×C₄	1	16	yes	complemented
135	C₃₁×#125	C₂×C₄	1	8	yes	complemented
136	C₃₁×#123	C₂×C₄	1	16	yes	—
137	C₃₁×C₂²⋊Q₈	C₂×C₄	1	8	yes	complemented
138	C₃₁×C₂²⋊Q₈	C₂×C₄	1	8	yes	complemented
139	C₄×Q₈×C₃₁	—	2	4	no	—
140	C₄×Q₈×C₃₁	—	2	4	no	—
141	C₄×Q₈×C₃₁	C₂×C₄	1	8	yes	complemented
142	C₄×D₄×C₃₁	—	4	4	no	—
143	C₄×D₄×C₃₁	C₂×C₄	1	8	yes	—
144	C₃₁×#117	—	2	4	no	—
145	C₃₁×#117	—	4	4	no	—
146	C₃₁×#117	C₂×C₄	1	8	yes	complemented
147	C₃₁×#117	C₂×C₄	1	16	yes	complemented
148	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
149	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
150	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	8	yes	complemented
151	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
152	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
153	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
154	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
155	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
156	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
157	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
158	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
159	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
160	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
161	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
162	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
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	1	yes	characteristic
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	1	yes	characteristic
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	2	yes	—
173	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
174	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
175	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
176	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
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	1	yes	characteristic
180	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
181	C₂×C₂²⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
182	C₂×C₂²⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
183	C₂×C₂²⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
184	C₂×C₂²⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
185	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
186	C₂×C₂²⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
187	C₂×C₂²⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
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	1	yes	characteristic
191	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
192	C₂×C₄²×C₃₁	—	2	4	no	—
193	C₂×C₄²×C₃₁	—	4	4	no	—
194	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
195	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
196	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
197	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
198	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
199	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
200	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
201	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
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	2	yes	—
207	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
208	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
209	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
210	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
211	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
212	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
213	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
214	C₃₁×#95	C₂³	1	2	yes	—
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	2	yes	—
220	C₃₁×#95	C₂³	1	2	yes	—
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	2	yes	—
225	C₃₁×#95	C₂³	1	2	yes	—
226	C₃₁×#95	C₂³	1	2	yes	—
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	1	yes	characteristic
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	1	yes	characteristic
237	C₃₁×#95	C₂³	1	1	yes	characteristic
238	C₃₁×#95	C₂³	1	2	yes	—
239	C₃₁×#95	C₂³	1	2	yes	—
240	C₃₁×#95	C₂³	1	2	yes	—
241	C₃₁×#95	C₂³	1	2	yes	—
242	C₃₁×#95	C₂³	1	1	yes	characteristic
243	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
244	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
245	C₂×Q₈×C₃₁	—	2	2	no	—
246	C₂×Q₈×C₃₁	—	2	2	no	—
247	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
248	C₂×D₄×C₃₁	—	2	2	no	—
249	C₂×D₄×C₃₁	—	2	2	no	—
250	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
251	C₂²×C₄×C₃₁	—	2	2	no	—
252	C₂²×C₄×C₃₁	—	2	4	no	—
253	C₂²×C₄×C₃₁	—	2	4	no	—
254	C₂²×C₄×C₃₁	—	2	2	no	—
255	C₂²×C₄×C₃₁	—	2	2	no	—
256	C₂²×C₄×C₃₁	—	2	2	no	—
257	C₂²×C₄×C₃₁	—	2	2	no	—
258	C₂²×C₄×C₃₁	—	2	2	no	—
259	C₂²×C₄×C₃₁	—	2	2	no	—
260	C₂²×C₄×C₃₁	—	2	4	no	—
261	C₂²×C₄×C₃₁	—	2	2	no	—
262	C₂²×C₄×C₃₁	—	2	2	no	—
263	C₂²×C₄×C₃₁	—	2	4	no	—
264	C₂²×C₄×C₃₁	—	2	4	no	—
265	C₂²×C₄×C₃₁	—	4	1	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	C₂²×C₄	1	8	yes	—
270	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
271	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
272	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
273	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
274	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
275	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
276	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
277	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
278	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
279	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
280	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
281	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
282	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
283	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
284	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
285	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
286	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
287	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
288	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
289	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
290	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
291	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
292	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
293	C₄⋊C₄×C₃₁	—	2	4	no	—
294	C₄⋊C₄×C₃₁	—	2	4	no	—
295	C₄⋊C₄×C₃₁	—	2	2	no	—
296	C₄⋊C₄×C₃₁	—	2	2	no	—
297	C₄⋊C₄×C₃₁	—	2	2	no	—
298	C₄⋊C₄×C₃₁	—	2	2	no	—
299	C₄⋊C₄×C₃₁	—	2	4	no	—
300	C₄⋊C₄×C₃₁	—	2	4	no	—
301	C₄⋊C₄×C₃₁	—	2	2	no	—
302	C₄⋊C₄×C₃₁	—	2	4	no	—
303	C₄⋊C₄×C₃₁	—	2	2	no	—
304	C₄⋊C₄×C₃₁	—	4	2	no	—
305	C₄⋊C₄×C₃₁	—	4	1	no	—
306	C₄⋊C₄×C₃₁	—	4	2	no	—
307	C₄⋊C₄×C₃₁	—	4	2	no	—
308	C₄⋊C₄×C₃₁	—	4	2	no	—
309	C₄⋊C₄×C₃₁	—	4	2	no	—
310	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
311	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
312	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
313	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
314	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
315	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
316	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
317	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
318	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
319	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
320	C₄⋊C₄×C₃₁	#41	1	4	yes	—
321	C₂²⋊C₄×C₃₁	—	2	2	no	—
322	C₂²⋊C₄×C₃₁	—	2	4	no	—
323	C₂²⋊C₄×C₃₁	—	2	2	no	—
324	C₂²⋊C₄×C₃₁	—	2	2	no	—
325	C₂²⋊C₄×C₃₁	—	2	2	no	—
326	C₂²⋊C₄×C₃₁	—	2	2	no	—
327	C₂²⋊C₄×C₃₁	—	4	1	no	—
328	C₂²⋊C₄×C₃₁	—	4	2	no	—
329	C₂²⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
330	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
331	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
332	C₂²⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
333	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
334	C₄²×C₃₁	—	2	2	no	—
335	C₄²×C₃₁	—	2	2	no	—
336	C₄²×C₃₁	—	2	2	no	—
337	C₄²×C₃₁	—	2	2	no	—
338	C₄²×C₃₁	—	2	4	no	—
339	C₄²×C₃₁	—	2	4	no	—
340	C₄²×C₃₁	—	2	2	no	—
341	C₄²×C₃₁	—	4	2	no	—
342	C₄²×C₃₁	—	4	2	no	—
343	C₄²×C₃₁	—	4	2	no	—
344	C₄²×C₃₁	—	4	2	no	—
345	C₄²×C₃₁	—	4	2	no	—
346	C₄²×C₃₁	—	4	2	no	—
347	C₄²×C₃₁	—	4	2	no	—
348	C₄²×C₃₁	—	4	2	no	—
349	C₄²×C₃₁	—	4	2	no	—
350	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
351	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
352	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
353	C₄²×C₃₁	#41	1	4	yes	—
354	#34304	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
355	C₂³×C₃₁	—	2	1	no	—
356	C₂³×C₃₁	—	2	1	no	—
357	C₂³×C₃₁	—	2	1	no	—
358	C₂³×C₃₁	—	2	1	no	—
359	C₂³×C₃₁	—	2	1	no	—
360	C₂³×C₃₁	—	2	1	no	—
361	C₂³×C₃₁	—	2	1	no	—
362	C₂³×C₃₁	—	2	1	no	—
363	C₂³×C₃₁	—	2	1	no	—
364	C₂³×C₃₁	—	2	1	no	—
365	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
366	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
367	C₂³×C₃₁	H(5,𝔽₂)	1	2	yes	—
368	C₂³×C₃₁	2₋¹⁺⁴	1	2	yes	—
369	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
370	Q₈×C₃₁	—	2	8	no	—
371	D₄×C₃₁	—	2	8	no	—
372	C₂×C₄×C₃₁	—	2	8	no	—
373	C₂×C₄×C₃₁	—	2	16	no	—
374	C₂×C₄×C₃₁	—	2	1	no	—
375	C₂×C₄×C₃₁	—	2	1	no	—
376	C₂×C₄×C₃₁	—	2	2	no	—
377	C₂×C₄×C₃₁	—	2	2	no	—
378	C₂×C₄×C₃₁	—	2	1	no	—
379	C₂×C₄×C₃₁	—	2	1	no	—
380	C₂×C₄×C₃₁	—	2	1	no	—
381	C₂×C₄×C₃₁	—	2	2	no	—
382	C₂×C₄×C₃₁	—	2	1	no	—
383	C₂×C₄×C₃₁	—	2	1	no	—
384	C₂×C₄×C₃₁	—	2	2	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	1	no	—
389	C₂×C₄×C₃₁	—	2	2	no	—
390	C₂×C₄×C₃₁	—	2	2	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	2	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	8	no	—
399	C₂×C₄×C₃₁	—	2	8	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	2	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	1	no	—
410	C₂×C₄×C₃₁	—	2	1	no	—
411	C₂×C₄×C₃₁	—	2	2	no	—
412	C₂×C₄×C₃₁	—	2	2	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	1	no	—
417	C₂×C₄×C₃₁	—	2	2	no	—
418	C₂×C₄×C₃₁	—	2	1	no	—
419	C₂×C₄×C₃₁	—	2	1	no	—
420	C₂×C₄×C₃₁	—	2	2	no	—
421	C₂×C₄×C₃₁	—	2	1	no	—
422	C₂×C₄×C₃₁	—	2	1	no	—
423	C₂×C₄×C₃₁	—	2	2	no	—
424	C₂×C₄×C₃₁	—	4	4	no	—
425	C₂×C₄×C₃₁	—	4	8	no	—
426	C₂×C₄×C₃₁	—	8	4	no	—
427	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
428	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
429	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
430	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
431	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
432	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
433	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
434	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
435	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
436	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
437	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
438	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
439	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
440	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
441	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
442	C₂×#552	C₂×C₃₁	1	1	yes	characteristic
443	#2373	C₂×C₃₁	1	2	yes	complemented
444	#2370	C₂×C₃₁	1	1	yes	characteristic, complemented
445	#2370	C₂×C₃₁	1	2	yes	complemented
446	#2360	C₂×C₃₁	1	1	yes	characteristic, complemented
447	#2360	C₂×C₃₁	1	1	yes	characteristic, complemented
448	#2359	C₂×C₃₁	1	2	yes	—
449	#2355	C₂×C₃₁	1	1	yes	characteristic, complemented
450	D₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic
451	#2347	C₂×C₃₁	1	1	yes	characteristic, complemented
452	#2344	C₂×C₃₁	1	1	yes	characteristic, complemented
453	#2343	C₂×C₃₁	1	2	yes	complemented
454	#2336	C₂×C₃₁	1	2	yes	complemented
455	#2334	C₂×C₃₁	1	2	yes	complemented
456	#2330	C₂×C₃₁	1	1	yes	characteristic, complemented
457	#2323	C₂×C₃₁	1	1	yes	characteristic, complemented
458	C₄×C₂²⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
459	C₄×#123	C₂×C₃₁	1	2	yes	—
460	#2296	C₂×C₃₁	1	2	yes	complemented
461	#2290	C₂×C₃₁	1	1	yes	characteristic
462	#2276	C₂×C₃₁	1	1	yes	characteristic, complemented
463	#2274	C₂×C₃₁	1	1	yes	characteristic, complemented
464	C₂×C₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
465	C₂²×C₃₁	—	2	2	no	—
466	C₂²×C₃₁	—	2	2	no	—
467	C₂²×C₃₁	—	2	2	no	—
468	C₂²×C₃₁	—	2	2	no	—
469	C₂²×C₃₁	—	2	2	no	—
470	C₂²×C₃₁	—	2	2	no	—
471	C₂²×C₃₁	—	2	2	no	—
472	C₂²×C₃₁	—	2	2	no	—
473	C₂²×C₃₁	—	2	2	no	—
474	C₂²×C₃₁	—	4	1	no	—
475	C₂²×C₃₁	—	4	1	no	—
476	C₂²×C₃₁	—	4	1	no	—
477	C₂²×C₃₁	—	4	1	no	—
478	C₂²×C₃₁	#520	1	4	yes	—
479	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
480	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
481	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
482	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
483	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
484	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
485	C₂²×C₃₁	#585	1	1	yes	characteristic
486	C₄×C₃₁	—	2	2	no	—
487	C₄×C₃₁	—	2	2	no	—
488	C₄×C₃₁	—	2	4	no	—
489	C₄×C₃₁	—	2	4	no	—
490	C₄×C₃₁	—	2	4	no	—
491	C₄×C₃₁	—	2	2	no	—
492	C₄×C₃₁	—	2	4	no	—
493	C₄×C₃₁	—	2	2	no	—
494	C₄×C₃₁	—	2	2	no	—
495	C₄×C₃₁	—	2	2	no	—
496	C₄×C₃₁	—	2	2	no	—
497	C₄×C₃₁	—	2	4	no	—
498	C₄×C₃₁	—	4	1	no	—
499	C₄×C₃₁	—	4	2	no	—
500	C₄×C₃₁	—	4	1	no	—
501	C₄×C₃₁	—	4	1	no	—
502	C₄×C₃₁	—	4	1	no	—
503	C₄×C₃₁	—	4	2	no	—
504	C₄×C₃₁	—	4	2	no	—
505	C₄×C₃₁	—	4	1	no	—
506	C₄×C₃₁	C₄×#41	1	4	yes	—
507	#552	C₄×C₃₁	1	16	yes	complemented
508	C₂×#126	C₂²×C₃₁	1	2	yes	—
509	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
510	C₂×#125	C₂²×C₃₁	1	2	yes	—
511	C₂×#123	C₂²×C₃₁	1	2	yes	—
512	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
513	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
514	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
515	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
516	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
517	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
518	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
519	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
520	C₂×#117	C₂²×C₃₁	1	2	yes	—
521	C₂×#117	C₂²×C₃₁	1	2	yes	—
522	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
523	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
524	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
525	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
526	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
527	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
528	#391	C₂²×C₃₁	1	1	yes	characteristic
529	#391	C₂²×C₃₁	1	1	yes	characteristic
530	#389	C₂²×C₃₁	1	2	yes	—
531	#389	C₂²×C₃₁	1	1	yes	characteristic
532	#389	C₂²×C₃₁	1	2	yes	—
533	#389	C₂²×C₃₁	1	1	yes	characteristic
534	#389	C₂²×C₃₁	1	1	yes	characteristic
535	#389	C₂²×C₃₁	1	2	yes	—
536	#389	C₂²×C₃₁	1	2	yes	—
537	#389	C₂²×C₃₁	1	2	yes	—
538	#389	C₂²×C₃₁	1	1	yes	characteristic
539	#389	C₂²×C₃₁	1	2	yes	—
540	#389	C₂²×C₃₁	1	2	yes	—
541	#388	C₂²×C₃₁	1	2	yes	—
542	#388	C₂²×C₃₁	1	2	yes	—
543	#388	C₂²×C₃₁	1	1	yes	characteristic
544	#388	C₂²×C₃₁	1	2	yes	—
545	#388	C₂²×C₃₁	1	2	yes	—
546	#388	C₂²×C₃₁	1	1	yes	characteristic
547	#387	C₂²×C₃₁	1	2	yes	—
548	#387	C₂²×C₃₁	1	2	yes	—
549	#387	C₂²×C₃₁	1	2	yes	—
550	#387	C₂²×C₃₁	1	2	yes	—
551	#387	C₂²×C₃₁	1	2	yes	—
552	#387	C₂²×C₃₁	1	2	yes	—
553	#387	C₂²×C₃₁	1	1	yes	characteristic
554	#387	C₂²×C₃₁	1	2	yes	—
555	#387	C₂²×C₃₁	1	2	yes	—
556	#387	C₂²×C₃₁	1	2	yes	—
557	#387	C₂²×C₃₁	1	2	yes	—
558	#387	C₂²×C₃₁	1	2	yes	—
559	#387	C₂²×C₃₁	1	2	yes	—
560	#387	C₂²×C₃₁	1	1	yes	characteristic
561	#386	C₂²×C₃₁	1	1	yes	characteristic
562	#386	C₂²×C₃₁	1	1	yes	characteristic
563	#385	C₂²×C₃₁	1	2	yes	—
564	#385	C₂²×C₃₁	1	2	yes	—
565	#385	C₂²×C₃₁	1	1	yes	characteristic
566	#385	C₂²×C₃₁	1	2	yes	—
567	#385	C₂²×C₃₁	1	2	yes	—
568	#385	C₂²×C₃₁	1	1	yes	characteristic
569	#385	C₂²×C₃₁	1	2	yes	—
570	#384	C₂²×C₃₁	1	1	yes	characteristic
571	#383	C₂²×C₃₁	1	2	yes	—
572	#382	C₂²×C₃₁	1	2	yes	—
573	#381	C₂²×C₃₁	1	2	yes	—
574	#380	C₂²×C₃₁	1	1	yes	characteristic
575	#380	C₂²×C₃₁	1	1	yes	characteristic
576	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
577	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
578	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
579	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
580	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
581	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
582	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
583	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
584	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
585	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
586	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
587	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
588	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
589	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
590	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
591	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
592	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
593	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
594	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
595	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
596	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
597	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
598	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
599	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
600	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
601	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
602	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
603	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
604	#376	C₂²×C₃₁	1	1	yes	characteristic
605	#376	C₂²×C₃₁	1	2	yes	—
606	C₂×#95	C₂²×C₃₁	1	2	yes	—
607	C₂×C₃₁	—	2	4	no	—
608	C₂×C₃₁	—	4	2	no	—
609	C₂×C₃₁	C₂×C₄×#41	1	1	yes	characteristic
610	C₂×C₃₁	C₂×#520	1	1	yes	characteristic
611	C₂×C₃₁	C₄×2₋¹⁺⁴	1	1	yes	characteristic
612	C₂×C₃₁	#3492	1	1	yes	characteristic
613	C₂×C₃₁	#3495	1	1	yes	characteristic
614	C₂×C₃₁	#3513	1	1	yes	characteristic
615	C₂×C₃₁	#3523	1	1	yes	characteristic
616	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
617	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
618	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
619	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
620	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
621	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
622	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
623	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
624	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
625	#126	C₂×C₄×C₃₁	1	16	yes	complemented
626	#125	C₂×C₄×C₃₁	1	16	yes	complemented
627	#125	C₂×C₄×C₃₁	1	8	yes	complemented
628	#123	C₂×C₄×C₃₁	1	16	yes	—
629	C₂²⋊Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
630	C₂²⋊Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
631	C₄×Q₈	—	2	4	no	—
632	C₄×Q₈	—	2	4	no	—
633	C₄×Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
634	C₄×D₄	—	4	4	no	—
635	C₄×D₄	C₂×C₄×C₃₁	1	8	yes	—
636	#117	—	2	4	no	—
637	#117	—	4	4	no	—
638	#117	C₂×C₄×C₃₁	1	8	yes	complemented
639	#117	C₂×C₄×C₃₁	1	16	yes	complemented
640	C₂×C₄⋊C₄	—	2	4	no	—
641	C₂×C₄⋊C₄	—	2	4	no	—
642	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	8	yes	complemented
643	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
644	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
645	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
646	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
647	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
648	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
649	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
650	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
651	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
652	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
653	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
654	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
655	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
656	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
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	2	yes	—
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	2	yes	—
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	1	yes	characteristic
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	2	yes	—
670	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
671	C₂×C₂²⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
672	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
673	C₂×C₂²⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
674	C₂×C₂²⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
675	C₂×C₂²⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
676	C₂×C₂²⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
677	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
678	C₂×C₂²⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
679	C₂×C₂²⋊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	1	yes	characteristic
683	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
684	C₂×C₄²	—	2	4	no	—
685	C₂×C₄²	—	4	4	no	—
686	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
687	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
688	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
689	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
690	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
691	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
692	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
693	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
694	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
695	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
696	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
697	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
698	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
699	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
700	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
701	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
702	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
703	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
704	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
705	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
706	#95	C₂³×C₃₁	1	2	yes	—
707	#95	C₂³×C₃₁	1	2	yes	—
708	#95	C₂³×C₃₁	1	2	yes	—
709	#95	C₂³×C₃₁	1	2	yes	—
710	#95	C₂³×C₃₁	1	1	yes	characteristic
711	#95	C₂³×C₃₁	1	2	yes	—
712	#95	C₂³×C₃₁	1	2	yes	—
713	#95	C₂³×C₃₁	1	1	yes	characteristic
714	#95	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	2	yes	—
721	#95	C₂³×C₃₁	1	2	yes	—
722	#95	C₂³×C₃₁	1	2	yes	—
723	#95	C₂³×C₃₁	1	2	yes	—
724	#95	C₂³×C₃₁	1	1	yes	characteristic
725	#95	C₂³×C₃₁	1	2	yes	—
726	#95	C₂³×C₃₁	1	2	yes	—
727	#95	C₂³×C₃₁	1	1	yes	characteristic
728	#95	C₂³×C₃₁	1	1	yes	characteristic
729	#95	C₂³×C₃₁	1	1	yes	characteristic
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	1	yes	characteristic
735	C₃₁	#34304	1	1	yes	characteristic, direct product factor, Sylow
736	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
737	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
738	C₂×Q₈	—	2	2	no	—
739	C₂×Q₈	—	2	2	no	—
740	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
741	C₂×D₄	—	2	2	no	—
742	C₂×D₄	—	2	2	no	—
743	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
744	C₂²×C₄	—	2	2	no	—
745	C₂²×C₄	—	2	4	no	—
746	C₂²×C₄	—	2	4	no	—
747	C₂²×C₄	—	2	2	no	—
748	C₂²×C₄	—	2	2	no	—
749	C₂²×C₄	—	2	2	no	—
750	C₂²×C₄	—	2	2	no	—
751	C₂²×C₄	—	2	2	no	—
752	C₂²×C₄	—	2	2	no	—
753	C₂²×C₄	—	2	4	no	—
754	C₂²×C₄	—	2	2	no	—
755	C₂²×C₄	—	2	2	no	—
756	C₂²×C₄	—	2	4	no	—
757	C₂²×C₄	—	2	4	no	—
758	C₂²×C₄	—	4	1	no	—
759	C₂²×C₄	—	4	2	no	—
760	C₂²×C₄	—	4	2	no	—
761	C₂²×C₄	—	4	2	no	—
762	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
763	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
764	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
765	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
766	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
767	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
768	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
769	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
770	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
771	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
772	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
773	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
774	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
775	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
776	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
777	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
778	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
779	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
780	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
781	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
782	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
783	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
784	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
785	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
786	C₄⋊C₄	—	2	4	no	—
787	C₄⋊C₄	—	2	4	no	—
788	C₄⋊C₄	—	2	2	no	—
789	C₄⋊C₄	—	2	2	no	—
790	C₄⋊C₄	—	2	2	no	—
791	C₄⋊C₄	—	2	2	no	—
792	C₄⋊C₄	—	2	4	no	—
793	C₄⋊C₄	—	2	4	no	—
794	C₄⋊C₄	—	2	2	no	—
795	C₄⋊C₄	—	2	4	no	—
796	C₄⋊C₄	—	2	2	no	—
797	C₄⋊C₄	—	4	2	no	—
798	C₄⋊C₄	—	4	1	no	—
799	C₄⋊C₄	—	4	2	no	—
800	C₄⋊C₄	—	4	2	no	—
801	C₄⋊C₄	—	4	2	no	—
802	C₄⋊C₄	—	4	2	no	—
803	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
804	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
805	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
806	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
807	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
808	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
809	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
810	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
811	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
812	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
813	C₄⋊C₄	#41×C₃₁	1	4	yes	—
814	C₂²⋊C₄	—	2	2	no	—
815	C₂²⋊C₄	—	2	4	no	—
816	C₂²⋊C₄	—	2	2	no	—
817	C₂²⋊C₄	—	2	2	no	—
818	C₂²⋊C₄	—	2	2	no	—
819	C₂²⋊C₄	—	2	2	no	—
820	C₂²⋊C₄	—	4	1	no	—
821	C₂²⋊C₄	—	4	2	no	—
822	C₂²⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
823	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
824	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
825	C₂²⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
826	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
827	C₄²	—	2	2	no	—
828	C₄²	—	2	2	no	—
829	C₄²	—	2	2	no	—
830	C₄²	—	2	2	no	—
831	C₄²	—	2	4	no	—
832	C₄²	—	2	4	no	—
833	C₄²	—	2	2	no	—
834	C₄²	—	4	2	no	—
835	C₄²	—	4	2	no	—
836	C₄²	—	4	2	no	—
837	C₄²	—	4	2	no	—
838	C₄²	—	4	2	no	—
839	C₄²	—	4	2	no	—
840	C₄²	—	4	2	no	—
841	C₄²	—	4	2	no	—
842	C₄²	—	4	2	no	—
843	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
844	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
845	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
846	C₄²	#41×C₃₁	1	4	yes	—
847	C₂³	—	2	1	no	—
848	C₂³	—	2	1	no	—
849	C₂³	—	2	1	no	—
850	C₂³	—	2	1	no	—
851	C₂³	—	2	1	no	—
852	C₂³	—	2	1	no	—
853	C₂³	—	2	1	no	—
854	C₂³	—	2	1	no	—
855	C₂³	—	2	1	no	—
856	C₂³	—	2	1	no	—
857	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
858	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
859	C₂³	C₃₁×H(5,𝔽₂)	1	2	yes	—
860	C₂³	C₃₁×2₋¹⁺⁴	1	2	yes	—
861	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
862	Q₈	—	2	8	no	—
863	D₄	—	2	8	no	—
864	C₂×C₄	—	2	8	no	—
865	C₂×C₄	—	2	16	no	—
866	C₂×C₄	—	2	1	no	—
867	C₂×C₄	—	2	1	no	—
868	C₂×C₄	—	2	2	no	—
869	C₂×C₄	—	2	2	no	—
870	C₂×C₄	—	2	1	no	—
871	C₂×C₄	—	2	1	no	—
872	C₂×C₄	—	2	1	no	—
873	C₂×C₄	—	2	2	no	—
874	C₂×C₄	—	2	1	no	—
875	C₂×C₄	—	2	1	no	—
876	C₂×C₄	—	2	2	no	—
877	C₂×C₄	—	2	2	no	—
878	C₂×C₄	—	2	1	no	—
879	C₂×C₄	—	2	2	no	—
880	C₂×C₄	—	2	1	no	—
881	C₂×C₄	—	2	2	no	—
882	C₂×C₄	—	2	2	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	1	no	—
890	C₂×C₄	—	2	8	no	—
891	C₂×C₄	—	2	8	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	2	no	—
897	C₂×C₄	—	2	1	no	—
898	C₂×C₄	—	2	2	no	—
899	C₂×C₄	—	2	2	no	—
900	C₂×C₄	—	2	2	no	—
901	C₂×C₄	—	2	1	no	—
902	C₂×C₄	—	2	1	no	—
903	C₂×C₄	—	2	2	no	—
904	C₂×C₄	—	2	2	no	—
905	C₂×C₄	—	2	1	no	—
906	C₂×C₄	—	2	1	no	—
907	C₂×C₄	—	2	1	no	—
908	C₂×C₄	—	2	1	no	—
909	C₂×C₄	—	2	2	no	—
910	C₂×C₄	—	2	1	no	—
911	C₂×C₄	—	2	1	no	—
912	C₂×C₄	—	2	2	no	—
913	C₂×C₄	—	2	1	no	—
914	C₂×C₄	—	2	1	no	—
915	C₂×C₄	—	2	2	no	—
916	C₂×C₄	—	4	4	no	—
917	C₂×C₄	—	4	8	no	—
918	C₂×C₄	—	8	4	no	—
919	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
920	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
921	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
922	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
923	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
924	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
925	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
926	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
927	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
928	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
929	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
930	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
931	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
932	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
933	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
934	C₂²	—	2	2	no	—
935	C₂²	—	2	2	no	—
936	C₂²	—	2	2	no	—
937	C₂²	—	2	2	no	—
938	C₂²	—	2	2	no	—
939	C₂²	—	2	2	no	—
940	C₂²	—	2	2	no	—
941	C₂²	—	2	2	no	—
942	C₂²	—	2	2	no	—
943	C₂²	—	4	1	no	—
944	C₂²	—	4	1	no	—
945	C₂²	—	4	1	no	—
946	C₂²	—	4	1	no	—
947	C₂²	C₃₁×#520	1	4	yes	—
948	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
949	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
950	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
951	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
952	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
953	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
954	C₂²	C₃₁×#585	1	1	yes	characteristic
955	C₄	—	2	2	no	—
956	C₄	—	2	2	no	—
957	C₄	—	2	4	no	—
958	C₄	—	2	4	no	—
959	C₄	—	2	4	no	—
960	C₄	—	2	2	no	—
961	C₄	—	2	4	no	—
962	C₄	—	2	2	no	—
963	C₄	—	2	2	no	—
964	C₄	—	2	2	no	—
965	C₄	—	2	2	no	—
966	C₄	—	2	4	no	—
967	C₄	—	4	1	no	—
968	C₄	—	4	2	no	—
969	C₄	—	4	1	no	—
970	C₄	—	4	1	no	—
971	C₄	—	4	2	no	—
972	C₄	—	4	2	no	—
973	C₄	—	4	1	no	—
974	C₄	—	4	1	no	—
975	C₄	C₄×#41×C₃₁	1	4	yes	—
976	C₂	—	2	4	no	—
977	C₂	—	4	2	no	—
978	C₂	C₂×C₄×#41×C₃₁	1	1	yes	characteristic
979	C₂	C₂×C₃₁×#520	1	1	yes	characteristic
980	C₂	C₄×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
981	C₂	C₃₁×#3492	1	1	yes	characteristic
982	C₂	C₃₁×#3495	1	1	yes	characteristic
983	C₂	C₃₁×#3513	1	1	yes	characteristic
984	C₂	C₃₁×#3523	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂²×D₄×C₃₁	C₂³
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×#2373	C₂	semidirect
3. C₃₁×#2370	C₂	semidirect
4. C₃₁×#2370	C₂	semidirect
5. C₃₁×#2360	C₂	semidirect
6. C₃₁×#2360	C₂	semidirect
8. C₃₁×#2355	C₂	semidirect
10. C₃₁×#2347	C₂	semidirect
11. C₃₁×#2344	C₂	semidirect
12. C₃₁×#2343	C₂	semidirect
13. C₃₁×#2336	C₂	semidirect
14. C₃₁×#2334	C₂	semidirect
15. C₃₁×#2330	C₂	semidirect
16. C₃₁×#2323	C₂	semidirect
17. C₄×C₃₁×C₂²⋊Q₈	C₂	semidirect
19. C₃₁×#2296	C₂	semidirect
21. C₃₁×#2276	C₂	semidirect
22. C₃₁×#2274	C₂	semidirect
23. C₂×C₄×C₄⋊C₄×C₃₁	C₂	semidirect
24. C₃₁×#552	C₄	semidirect
133. C₃₁×#126	C₂×C₄	semidirect
134. C₃₁×#125	C₂×C₄	semidirect
135. C₃₁×#125	C₂×C₄	semidirect
137. C₃₁×C₂²⋊Q₈	C₂×C₄	semidirect
138. C₃₁×C₂²⋊Q₈	C₂×C₄	semidirect
141. C₄×Q₈×C₃₁	C₂×C₄	semidirect
146. C₃₁×#117	C₂×C₄	semidirect
147. C₃₁×#117	C₂×C₄	semidirect
150. C₂×C₄⋊C₄×C₃₁	C₂×C₄	semidirect
354. #34304	C₃₁	direct (N × H)
443. #2373	C₂×C₃₁	semidirect
444. #2370	C₂×C₃₁	semidirect
445. #2370	C₂×C₃₁	semidirect
446. #2360	C₂×C₃₁	semidirect
447. #2360	C₂×C₃₁	semidirect
449. #2355	C₂×C₃₁	semidirect
451. #2347	C₂×C₃₁	semidirect
452. #2344	C₂×C₃₁	semidirect
453. #2343	C₂×C₃₁	semidirect
454. #2336	C₂×C₃₁	semidirect
455. #2334	C₂×C₃₁	semidirect
456. #2330	C₂×C₃₁	semidirect
457. #2323	C₂×C₃₁	semidirect
458. C₄×C₂²⋊Q₈	C₂×C₃₁	semidirect
460. #2296	C₂×C₃₁	semidirect
462. #2276	C₂×C₃₁	semidirect
463. #2274	C₂×C₃₁	semidirect
464. C₂×C₄×C₄⋊C₄	C₂×C₃₁	semidirect
507. #552	C₄×C₃₁	semidirect
625. #126	C₂×C₄×C₃₁	semidirect
626. #125	C₂×C₄×C₃₁	semidirect
627. #125	C₂×C₄×C₃₁	semidirect
629. C₂²⋊Q₈	C₂×C₄×C₃₁	semidirect
630. C₂²⋊Q₈	C₂×C₄×C₃₁	semidirect
633. C₄×Q₈	C₂×C₄×C₃₁	semidirect
638. #117	C₂×C₄×C₃₁	semidirect
639. #117	C₂×C₄×C₃₁	semidirect
642. C₂×C₄⋊C₄	C₂×C₄×C₃₁	semidirect
735. C₃₁	#34304	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×#552	2
C₃₁×#2373	2
C₃₁×#2370	2
C₃₁×#2360	2
C₃₁×#2359	2
C₃₁×#2355	2
D₄×C₄⋊C₄×C₃₁	2
C₃₁×#2347	2
C₃₁×#2344	2
C₃₁×#2343	2
C₃₁×#2336	2
C₃₁×#2334	2
C₃₁×#2330	2
C₃₁×#2323	2
C₄×C₃₁×C₂²⋊Q₈	2
C₄×C₃₁×#123	2
C₃₁×#2296	2
C₃₁×#2290	2
C₃₁×#2276	2
C₃₁×#2274	2
C₂×C₄×C₄⋊C₄×C₃₁	2
#34304	31

Subgroups — C31×#34304 #21491063

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34304 #21491063