Subgroups — C₃₁×#34463 #21491222

Order 7936.

2040 conjugacy classes of subgroups.

The subgroup lattice has 970 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#556	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2383	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2379	C₂	1	2	yes	maximal
4	C₃₁×#2373	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2369	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2362	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2360	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2355	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2352	C₂	1	1	yes	maximal, characteristic
10	C₃₁×#2351	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#2347	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#2346	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#2337	C₂	1	2	yes	maximal
14	C₃₁×#2334	C₂	1	2	yes	maximal, complemented
15	C₃₁×#2334	C₂	1	2	yes	maximal, complemented
16	C₃₁×#2331	C₂	1	1	yes	maximal, characteristic, complemented
17	C₃₁×#2323	C₂	1	1	yes	maximal, characteristic, complemented
18	C₃₁×#2320	C₂	1	1	yes	maximal, characteristic, complemented
19	C₃₁×#2320	C₂	1	2	yes	maximal, complemented
20	C₃₁×#2317	C₂	1	1	yes	maximal, characteristic, complemented
21	C₄×C₃₁×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
22	C₃₁×#2296	C₂	1	1	yes	maximal, characteristic, complemented
23	C₃₁×#2290	C₂	1	1	yes	maximal, characteristic
24	C₃₁×#556	C₄	1	16	yes	complemented
25	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×#126	C₂²	1	2	yes	—
27	C₂×C₃₁×#126	C₂²	1	2	yes	—
28	C₂×C₃₁×#124	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×Q₈×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	1	yes	characteristic
38	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
39	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
40	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
41	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
42	C₃₁×#391	C₂²	1	1	yes	characteristic
43	C₃₁×#391	C₂²	1	2	yes	—
44	C₃₁×#391	C₂²	1	1	yes	characteristic
45	C₃₁×#391	C₂²	1	2	yes	—
46	C₃₁×#391	C₂²	1	2	yes	—
47	C₃₁×#391	C₂²	1	2	yes	—
48	C₃₁×#389	C₂²	1	1	yes	characteristic
49	C₃₁×#389	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
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×#389	C₂²	1	2	yes	—
59	C₃₁×#389	C₂²	1	1	yes	characteristic
60	C₃₁×#389	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×#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	1	yes	characteristic
72	C₃₁×#387	C₂²	1	1	yes	characteristic
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	2	yes	—
78	C₃₁×#387	C₂²	1	2	yes	—
79	C₃₁×#387	C₂²	1	2	yes	—
80	C₃₁×#387	C₂²	1	2	yes	—
81	C₃₁×#387	C₂²	1	2	yes	—
82	C₃₁×#387	C₂²	1	1	yes	characteristic
83	C₃₁×#386	C₂²	1	1	yes	characteristic
84	C₃₁×#386	C₂²	1	2	yes	—
85	C₃₁×#386	C₂²	1	1	yes	characteristic
86	C₃₁×#385	C₂²	1	2	yes	—
87	C₃₁×#385	C₂²	1	2	yes	—
88	C₃₁×#385	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
94	C₃₁×#384	C₂²	1	2	yes	—
95	C₃₁×#383	C₂²	1	2	yes	—
96	C₃₁×#383	C₂²	1	2	yes	—
97	C₃₁×#382	C₂²	1	2	yes	—
98	C₃₁×#382	C₂²	1	2	yes	—
99	C₃₁×#382	C₂²	1	2	yes	—
100	C₃₁×#382	C₂²	1	1	yes	characteristic
101	C₃₁×#382	C₂²	1	2	yes	—
102	C₃₁×#380	C₂²	1	1	yes	characteristic
103	C₃₁×#380	C₂²	1	1	yes	characteristic
104	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
105	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
106	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
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	2	yes	—
112	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
113	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
114	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
115	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
116	C₄×C₄⋊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	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
121	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
122	C₃₁×#376	C₂²	1	1	yes	characteristic
123	C₃₁×#376	C₂²	1	2	yes	—
124	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
125	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
126	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
127	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
128	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
129	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
130	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
131	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
132	C₃₁×C₄⋊Q₈	C₂×C₄	1	8	yes	complemented
133	C₃₁×#126	C₂×C₄	1	16	yes	complemented
134	C₃₁×#126	C₂×C₄	1	16	yes	complemented
135	C₃₁×#124	C₂×C₄	1	16	yes	—
136	C₃₁×C₂²⋊Q₈	C₂×C₄	1	8	yes	complemented
137	C₃₁×C₂²⋊Q₈	C₂×C₄	1	8	yes	complemented
138	C₄×Q₈×C₃₁	—	4	2	no	—
139	C₄×Q₈×C₃₁	—	4	2	no	—
140	C₄×Q₈×C₃₁	C₂×C₄	1	16	yes	complemented
141	C₄×Q₈×C₃₁	C₂×C₄	1	8	yes	complemented
142	C₄×D₄×C₃₁	C₂×C₄	1	8	yes	—
143	C₃₁×#117	—	4	2	no	—
144	C₃₁×#117	—	4	2	no	—
145	C₃₁×#117	C₂×C₄	1	8	yes	complemented
146	C₃₁×#117	C₂×C₄	1	8	yes	complemented
147	C₂×C₄⋊C₄×C₃₁	—	4	2	no	—
148	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
149	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
150	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
151	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
152	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
153	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
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	2	yes	—
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	2	yes	—
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	2	yes	—
166	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
167	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
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	1	yes	characteristic
171	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
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	2	yes	—
175	C₂×C₂²⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
176	C₂×C₂²⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
177	C₂×C₂²⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
178	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
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	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	2	yes	—
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×C₄²×C₃₁	—	4	2	no	—
191	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
192	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
193	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
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	2	yes	—
198	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
199	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
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	1	yes	characteristic
208	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
209	C₃₁×#95	C₂³	1	1	yes	characteristic
210	C₃₁×#95	C₂³	1	2	yes	—
211	C₃₁×#95	C₂³	1	2	yes	—
212	C₃₁×#95	C₂³	1	1	yes	characteristic
213	C₃₁×#95	C₂³	1	1	yes	characteristic
214	C₃₁×#95	C₂³	1	2	yes	—
215	C₃₁×#95	C₂³	1	1	yes	characteristic
216	C₃₁×#95	C₂³	1	1	yes	characteristic
217	C₃₁×#95	C₂³	1	1	yes	characteristic
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	2	yes	—
222	C₃₁×#95	C₂³	1	1	yes	characteristic
223	C₃₁×#95	C₂³	1	2	yes	—
224	C₃₁×#95	C₂³	1	2	yes	—
225	C₃₁×#95	C₂³	1	1	yes	characteristic
226	C₃₁×#95	C₂³	1	2	yes	—
227	C₃₁×#95	C₂³	1	2	yes	—
228	C₃₁×#95	C₂³	1	1	yes	characteristic
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	2	yes	—
236	C₃₁×#95	C₂³	1	2	yes	—
237	C₃₁×#95	C₂³	1	2	yes	—
238	C₃₁×#95	C₂³	1	2	yes	—
239	C₃₁×#95	C₂³	1	2	yes	—
240	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
241	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
242	C₂×Q₈×C₃₁	—	2	2	no	—
243	C₂×Q₈×C₃₁	—	2	2	no	—
244	C₂×Q₈×C₃₁	—	4	2	no	—
245	C₂×Q₈×C₃₁	—	4	2	no	—
246	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
247	C₂×Q₈×C₃₁	C₂²×C₄	1	8	yes	—
248	C₂×D₄×C₃₁	—	4	1	no	—
249	C₂×D₄×C₃₁	—	4	1	no	—
250	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
251	C₂²×C₄×C₃₁	—	2	2	no	—
252	C₂²×C₄×C₃₁	—	2	2	no	—
253	C₂²×C₄×C₃₁	—	2	2	no	—
254	C₂²×C₄×C₃₁	—	2	2	no	—
255	C₂²×C₄×C₃₁	—	2	2	no	—
256	C₂²×C₄×C₃₁	—	4	1	no	—
257	C₂²×C₄×C₃₁	—	4	1	no	—
258	C₂²×C₄×C₃₁	—	4	1	no	—
259	C₂²×C₄×C₃₁	—	4	1	no	—
260	C₂²×C₄×C₃₁	—	4	1	no	—
261	C₂²×C₄×C₃₁	—	4	2	no	—
262	C₂²×C₄×C₃₁	—	4	2	no	—
263	C₂²×C₄×C₃₁	—	4	2	no	—
264	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
265	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
266	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
267	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
268	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
269	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
270	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
271	C₂²×C₄×C₃₁	C₂⁴	1	2	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	2	yes	—
275	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
276	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
277	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
278	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
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	2	yes	—
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	C₄⋊C₄×C₃₁	—	2	2	no	—
288	C₄⋊C₄×C₃₁	—	2	2	no	—
289	C₄⋊C₄×C₃₁	—	2	2	no	—
290	C₄⋊C₄×C₃₁	—	2	4	no	—
291	C₄⋊C₄×C₃₁	—	2	2	no	—
292	C₄⋊C₄×C₃₁	—	2	2	no	—
293	C₄⋊C₄×C₃₁	—	4	2	no	—
294	C₄⋊C₄×C₃₁	—	4	1	no	—
295	C₄⋊C₄×C₃₁	—	4	2	no	—
296	C₄⋊C₄×C₃₁	—	4	2	no	—
297	C₄⋊C₄×C₃₁	—	4	2	no	—
298	C₄⋊C₄×C₃₁	—	4	2	no	—
299	C₄⋊C₄×C₃₁	—	4	2	no	—
300	C₄⋊C₄×C₃₁	—	4	2	no	—
301	C₄⋊C₄×C₃₁	—	4	2	no	—
302	C₄⋊C₄×C₃₁	—	4	1	no	—
303	C₄⋊C₄×C₃₁	—	4	1	no	—
304	C₄⋊C₄×C₃₁	—	4	2	no	—
305	C₄⋊C₄×C₃₁	—	4	2	no	—
306	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
307	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
308	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
309	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
310	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
311	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
312	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
313	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
314	C₂²⋊C₄×C₃₁	—	2	2	no	—
315	C₂²⋊C₄×C₃₁	—	4	1	no	—
316	C₂²⋊C₄×C₃₁	—	4	2	no	—
317	C₂²⋊C₄×C₃₁	—	4	1	no	—
318	C₂²⋊C₄×C₃₁	—	4	2	no	—
319	C₂²⋊C₄×C₃₁	—	4	1	no	—
320	C₂²⋊C₄×C₃₁	—	4	1	no	—
321	C₂²⋊C₄×C₃₁	—	4	1	no	—
322	C₂²⋊C₄×C₃₁	—	4	2	no	—
323	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
324	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
325	C₂²⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
326	C₂²⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
327	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
328	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
329	C₄²×C₃₁	—	2	2	no	—
330	C₄²×C₃₁	—	2	2	no	—
331	C₄²×C₃₁	—	2	2	no	—
332	C₄²×C₃₁	—	2	2	no	—
333	C₄²×C₃₁	—	4	1	no	—
334	C₄²×C₃₁	—	4	1	no	—
335	C₄²×C₃₁	—	4	2	no	—
336	C₄²×C₃₁	—	4	2	no	—
337	C₄²×C₃₁	—	4	2	no	—
338	C₄²×C₃₁	—	4	2	no	—
339	C₄²×C₃₁	—	4	2	no	—
340	C₄²×C₃₁	—	4	2	no	—
341	C₄²×C₃₁	—	4	2	no	—
342	C₄²×C₃₁	—	4	2	no	—
343	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
344	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
345	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
346	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
347	#34463	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
348	C₂³×C₃₁	—	2	1	no	—
349	C₂³×C₃₁	—	2	1	no	—
350	C₂³×C₃₁	—	2	1	no	—
351	C₂³×C₃₁	—	2	1	no	—
352	C₂³×C₃₁	—	2	1	no	—
353	C₂³×C₃₁	—	2	1	no	—
354	C₂³×C₃₁	—	2	1	no	—
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	C₂³×C₄	1	2	yes	—
361	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
362	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
363	Q₈×C₃₁	—	2	8	no	—
364	Q₈×C₃₁	—	4	8	no	—
365	D₄×C₃₁	—	4	4	no	—
366	C₂×C₄×C₃₁	—	2	1	no	—
367	C₂×C₄×C₃₁	—	2	1	no	—
368	C₂×C₄×C₃₁	—	2	2	no	—
369	C₂×C₄×C₃₁	—	2	1	no	—
370	C₂×C₄×C₃₁	—	2	1	no	—
371	C₂×C₄×C₃₁	—	2	2	no	—
372	C₂×C₄×C₃₁	—	2	1	no	—
373	C₂×C₄×C₃₁	—	2	1	no	—
374	C₂×C₄×C₃₁	—	2	2	no	—
375	C₂×C₄×C₃₁	—	2	1	no	—
376	C₂×C₄×C₃₁	—	2	1	no	—
377	C₂×C₄×C₃₁	—	2	2	no	—
378	C₂×C₄×C₃₁	—	2	2	no	—
379	C₂×C₄×C₃₁	—	2	2	no	—
380	C₂×C₄×C₃₁	—	2	2	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	8	no	—
385	C₂×C₄×C₃₁	—	2	8	no	—
386	C₂×C₄×C₃₁	—	2	1	no	—
387	C₂×C₄×C₃₁	—	2	1	no	—
388	C₂×C₄×C₃₁	—	2	1	no	—
389	C₂×C₄×C₃₁	—	2	1	no	—
390	C₂×C₄×C₃₁	—	2	1	no	—
391	C₂×C₄×C₃₁	—	2	2	no	—
392	C₂×C₄×C₃₁	—	2	2	no	—
393	C₂×C₄×C₃₁	—	2	2	no	—
394	C₂×C₄×C₃₁	—	2	1	no	—
395	C₂×C₄×C₃₁	—	2	1	no	—
396	C₂×C₄×C₃₁	—	2	1	no	—
397	C₂×C₄×C₃₁	—	2	1	no	—
398	C₂×C₄×C₃₁	—	2	2	no	—
399	C₂×C₄×C₃₁	—	2	2	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	2	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	1	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	—	4	4	no	—
420	C₂×C₄×C₃₁	—	4	4	no	—
421	C₂×C₄×C₃₁	—	8	4	no	—
422	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
423	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
424	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
425	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
426	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
427	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
428	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
429	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
430	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
431	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
432	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
433	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
434	C₂×#556	C₂×C₃₁	1	1	yes	characteristic
435	#2383	C₂×C₃₁	1	2	yes	complemented
436	#2379	C₂×C₃₁	1	2	yes	—
437	#2373	C₂×C₃₁	1	2	yes	complemented
438	#2369	C₂×C₃₁	1	1	yes	characteristic, complemented
439	#2362	C₂×C₃₁	1	1	yes	characteristic, complemented
440	#2360	C₂×C₃₁	1	1	yes	characteristic, complemented
441	#2355	C₂×C₃₁	1	2	yes	complemented
442	#2352	C₂×C₃₁	1	1	yes	characteristic
443	#2351	C₂×C₃₁	1	1	yes	characteristic, complemented
444	#2347	C₂×C₃₁	1	1	yes	characteristic, complemented
445	#2346	C₂×C₃₁	1	1	yes	characteristic, complemented
446	#2337	C₂×C₃₁	1	2	yes	—
447	#2334	C₂×C₃₁	1	2	yes	complemented
448	#2334	C₂×C₃₁	1	2	yes	complemented
449	#2331	C₂×C₃₁	1	1	yes	characteristic, complemented
450	#2323	C₂×C₃₁	1	1	yes	characteristic, complemented
451	#2320	C₂×C₃₁	1	1	yes	characteristic, complemented
452	#2320	C₂×C₃₁	1	2	yes	complemented
453	#2317	C₂×C₃₁	1	1	yes	characteristic, complemented
454	C₄×C₂²⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
455	#2296	C₂×C₃₁	1	1	yes	characteristic, complemented
456	#2290	C₂×C₃₁	1	1	yes	characteristic
457	C₂²×C₃₁	—	2	2	no	—
458	C₂²×C₃₁	—	2	2	no	—
459	C₂²×C₃₁	—	2	2	no	—
460	C₂²×C₃₁	—	2	2	no	—
461	C₂²×C₃₁	—	2	2	no	—
462	C₂²×C₃₁	—	2	2	no	—
463	C₂²×C₃₁	—	4	1	no	—
464	C₂²×C₃₁	—	4	1	no	—
465	C₂²×C₃₁	—	4	1	no	—
466	C₂²×C₃₁	—	4	1	no	—
467	C₂²×C₃₁	—	4	1	no	—
468	C₂²×C₃₁	—	4	1	no	—
469	C₂²×C₃₁	—	4	1	no	—
470	C₂²×C₃₁	—	4	1	no	—
471	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
472	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
473	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
474	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
475	C₂²×C₃₁	#585	1	1	yes	characteristic
476	C₂²×C₃₁	#585	1	1	yes	characteristic
477	C₂²×C₃₁	#585	1	1	yes	characteristic
478	C₄×C₃₁	—	2	2	no	—
479	C₄×C₃₁	—	2	2	no	—
480	C₄×C₃₁	—	2	4	no	—
481	C₄×C₃₁	—	2	2	no	—
482	C₄×C₃₁	—	2	4	no	—
483	C₄×C₃₁	—	2	4	no	—
484	C₄×C₃₁	—	2	2	no	—
485	C₄×C₃₁	—	2	4	no	—
486	C₄×C₃₁	—	2	2	no	—
487	C₄×C₃₁	—	4	1	no	—
488	C₄×C₃₁	—	4	1	no	—
489	C₄×C₃₁	—	4	1	no	—
490	C₄×C₃₁	—	4	2	no	—
491	C₄×C₃₁	—	4	2	no	—
492	C₄×C₃₁	—	4	1	no	—
493	C₄×C₃₁	—	4	2	no	—
494	C₄×C₃₁	—	4	1	no	—
495	C₄×C₃₁	—	4	1	no	—
496	C₄×C₃₁	—	4	1	no	—
497	C₄×C₃₁	—	4	2	no	—
498	C₄×C₃₁	#520	1	4	yes	—
499	#556	C₄×C₃₁	1	16	yes	complemented
500	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
501	C₂×#126	C₂²×C₃₁	1	2	yes	—
502	C₂×#126	C₂²×C₃₁	1	2	yes	—
503	C₂×#124	C₂²×C₃₁	1	2	yes	—
504	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
505	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
506	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
507	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
508	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
509	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
510	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
511	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
512	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
513	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
514	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
515	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
516	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
517	#391	C₂²×C₃₁	1	1	yes	characteristic
518	#391	C₂²×C₃₁	1	2	yes	—
519	#391	C₂²×C₃₁	1	1	yes	characteristic
520	#391	C₂²×C₃₁	1	2	yes	—
521	#391	C₂²×C₃₁	1	2	yes	—
522	#391	C₂²×C₃₁	1	2	yes	—
523	#389	C₂²×C₃₁	1	1	yes	characteristic
524	#389	C₂²×C₃₁	1	1	yes	characteristic
525	#389	C₂²×C₃₁	1	1	yes	characteristic
526	#389	C₂²×C₃₁	1	1	yes	characteristic
527	#389	C₂²×C₃₁	1	2	yes	—
528	#389	C₂²×C₃₁	1	1	yes	characteristic
529	#389	C₂²×C₃₁	1	2	yes	—
530	#389	C₂²×C₃₁	1	1	yes	characteristic
531	#389	C₂²×C₃₁	1	2	yes	—
532	#389	C₂²×C₃₁	1	2	yes	—
533	#389	C₂²×C₃₁	1	2	yes	—
534	#389	C₂²×C₃₁	1	1	yes	characteristic
535	#389	C₂²×C₃₁	1	1	yes	characteristic
536	#388	C₂²×C₃₁	1	2	yes	—
537	#388	C₂²×C₃₁	1	2	yes	—
538	#388	C₂²×C₃₁	1	1	yes	characteristic
539	#388	C₂²×C₃₁	1	2	yes	—
540	#388	C₂²×C₃₁	1	2	yes	—
541	#388	C₂²×C₃₁	1	2	yes	—
542	#388	C₂²×C₃₁	1	2	yes	—
543	#388	C₂²×C₃₁	1	2	yes	—
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	1	yes	characteristic
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	2	yes	—
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	1	yes	characteristic
558	#386	C₂²×C₃₁	1	1	yes	characteristic
559	#386	C₂²×C₃₁	1	2	yes	—
560	#386	C₂²×C₃₁	1	1	yes	characteristic
561	#385	C₂²×C₃₁	1	2	yes	—
562	#385	C₂²×C₃₁	1	2	yes	—
563	#385	C₂²×C₃₁	1	1	yes	characteristic
564	#385	C₂²×C₃₁	1	2	yes	—
565	#385	C₂²×C₃₁	1	2	yes	—
566	#385	C₂²×C₃₁	1	2	yes	—
567	#385	C₂²×C₃₁	1	2	yes	—
568	#385	C₂²×C₃₁	1	1	yes	characteristic
569	#384	C₂²×C₃₁	1	2	yes	—
570	#383	C₂²×C₃₁	1	2	yes	—
571	#383	C₂²×C₃₁	1	2	yes	—
572	#382	C₂²×C₃₁	1	2	yes	—
573	#382	C₂²×C₃₁	1	2	yes	—
574	#382	C₂²×C₃₁	1	2	yes	—
575	#382	C₂²×C₃₁	1	1	yes	characteristic
576	#382	C₂²×C₃₁	1	2	yes	—
577	#380	C₂²×C₃₁	1	1	yes	characteristic
578	#380	C₂²×C₃₁	1	1	yes	characteristic
579	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
580	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
581	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
582	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
583	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
584	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
585	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
586	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
587	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
588	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
589	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
590	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
591	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
592	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
593	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
594	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
595	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
596	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
597	#376	C₂²×C₃₁	1	1	yes	characteristic
598	#376	C₂²×C₃₁	1	2	yes	—
599	C₂×C₃₁	—	4	2	no	—
600	C₂×C₃₁	—	4	2	no	—
601	C₂×C₃₁	C₂×#520	1	1	yes	characteristic
602	C₂×C₃₁	#3428	1	1	yes	characteristic
603	C₂×C₃₁	C₄×2₋¹⁺⁴	1	1	yes	characteristic
604	C₂×C₃₁	#3540	1	1	yes	characteristic
605	C₂×C₃₁	#3541	1	1	yes	characteristic
606	C₂×C₃₁	#3555	1	1	yes	characteristic
607	C₂×C₃₁	#3565	1	1	yes	characteristic
608	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
609	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
610	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
611	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
612	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
613	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
614	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
615	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
616	C₄⋊Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
617	#126	C₂×C₄×C₃₁	1	16	yes	complemented
618	#126	C₂×C₄×C₃₁	1	16	yes	complemented
619	#124	C₂×C₄×C₃₁	1	16	yes	—
620	C₂²⋊Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
621	C₂²⋊Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
622	C₄×Q₈	—	4	2	no	—
623	C₄×Q₈	—	4	2	no	—
624	C₄×Q₈	C₂×C₄×C₃₁	1	16	yes	complemented
625	C₄×Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
626	C₄×D₄	C₂×C₄×C₃₁	1	8	yes	—
627	#117	—	4	2	no	—
628	#117	—	4	2	no	—
629	#117	C₂×C₄×C₃₁	1	8	yes	complemented
630	#117	C₂×C₄×C₃₁	1	8	yes	complemented
631	C₂×C₄⋊C₄	—	4	2	no	—
632	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
633	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
634	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
635	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
636	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
637	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
638	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
639	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
640	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
641	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
642	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
643	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
644	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
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	2	yes	—
648	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
649	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
650	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
651	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
652	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
653	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
654	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
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	2	yes	—
658	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
659	C₂×C₂²⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
660	C₂×C₂²⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
661	C₂×C₂²⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
662	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
663	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
664	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
665	C₂×C₂²⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
666	C₂×C₂²⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
667	C₂×C₂²⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
668	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
669	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
670	C₂×C₂²⋊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	2	yes	—
674	C₂×C₄²	—	4	2	no	—
675	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
676	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
677	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
678	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
679	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
680	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
681	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
682	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
683	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
684	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
685	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
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	2	yes	—
691	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
692	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
693	#95	C₂³×C₃₁	1	1	yes	characteristic
694	#95	C₂³×C₃₁	1	2	yes	—
695	#95	C₂³×C₃₁	1	2	yes	—
696	#95	C₂³×C₃₁	1	1	yes	characteristic
697	#95	C₂³×C₃₁	1	1	yes	characteristic
698	#95	C₂³×C₃₁	1	2	yes	—
699	#95	C₂³×C₃₁	1	1	yes	characteristic
700	#95	C₂³×C₃₁	1	1	yes	characteristic
701	#95	C₂³×C₃₁	1	1	yes	characteristic
702	#95	C₂³×C₃₁	1	1	yes	characteristic
703	#95	C₂³×C₃₁	1	2	yes	—
704	#95	C₂³×C₃₁	1	2	yes	—
705	#95	C₂³×C₃₁	1	2	yes	—
706	#95	C₂³×C₃₁	1	1	yes	characteristic
707	#95	C₂³×C₃₁	1	2	yes	—
708	#95	C₂³×C₃₁	1	2	yes	—
709	#95	C₂³×C₃₁	1	1	yes	characteristic
710	#95	C₂³×C₃₁	1	2	yes	—
711	#95	C₂³×C₃₁	1	2	yes	—
712	#95	C₂³×C₃₁	1	1	yes	characteristic
713	#95	C₂³×C₃₁	1	2	yes	—
714	#95	C₂³×C₃₁	1	2	yes	—
715	#95	C₂³×C₃₁	1	2	yes	—
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	2	yes	—
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	C₃₁	#34463	1	1	yes	characteristic, direct product factor, Sylow
725	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
726	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
727	C₂×Q₈	—	2	2	no	—
728	C₂×Q₈	—	2	2	no	—
729	C₂×Q₈	—	4	2	no	—
730	C₂×Q₈	—	4	2	no	—
731	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
732	C₂×Q₈	C₂²×C₄×C₃₁	1	8	yes	—
733	C₂×D₄	—	4	1	no	—
734	C₂×D₄	—	4	1	no	—
735	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
736	C₂²×C₄	—	2	2	no	—
737	C₂²×C₄	—	2	2	no	—
738	C₂²×C₄	—	2	2	no	—
739	C₂²×C₄	—	2	2	no	—
740	C₂²×C₄	—	2	2	no	—
741	C₂²×C₄	—	4	1	no	—
742	C₂²×C₄	—	4	1	no	—
743	C₂²×C₄	—	4	1	no	—
744	C₂²×C₄	—	4	1	no	—
745	C₂²×C₄	—	4	1	no	—
746	C₂²×C₄	—	4	2	no	—
747	C₂²×C₄	—	4	2	no	—
748	C₂²×C₄	—	4	2	no	—
749	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
750	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
751	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
752	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
753	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
754	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
755	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
756	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
757	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
758	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
759	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
760	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
761	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
762	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
763	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
764	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
765	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
766	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
767	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
768	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
769	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
770	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
771	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
772	C₄⋊C₄	—	2	2	no	—
773	C₄⋊C₄	—	2	2	no	—
774	C₄⋊C₄	—	2	2	no	—
775	C₄⋊C₄	—	2	4	no	—
776	C₄⋊C₄	—	2	2	no	—
777	C₄⋊C₄	—	2	2	no	—
778	C₄⋊C₄	—	4	2	no	—
779	C₄⋊C₄	—	4	1	no	—
780	C₄⋊C₄	—	4	2	no	—
781	C₄⋊C₄	—	4	2	no	—
782	C₄⋊C₄	—	4	2	no	—
783	C₄⋊C₄	—	4	2	no	—
784	C₄⋊C₄	—	4	2	no	—
785	C₄⋊C₄	—	4	2	no	—
786	C₄⋊C₄	—	4	2	no	—
787	C₄⋊C₄	—	4	1	no	—
788	C₄⋊C₄	—	4	1	no	—
789	C₄⋊C₄	—	4	2	no	—
790	C₄⋊C₄	—	4	2	no	—
791	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
792	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
793	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
794	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
795	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
796	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
797	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
798	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
799	C₂²⋊C₄	—	2	2	no	—
800	C₂²⋊C₄	—	4	1	no	—
801	C₂²⋊C₄	—	4	2	no	—
802	C₂²⋊C₄	—	4	1	no	—
803	C₂²⋊C₄	—	4	2	no	—
804	C₂²⋊C₄	—	4	1	no	—
805	C₂²⋊C₄	—	4	1	no	—
806	C₂²⋊C₄	—	4	1	no	—
807	C₂²⋊C₄	—	4	2	no	—
808	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
809	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
810	C₂²⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
811	C₂²⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
812	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
813	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
814	C₄²	—	2	2	no	—
815	C₄²	—	2	2	no	—
816	C₄²	—	2	2	no	—
817	C₄²	—	2	2	no	—
818	C₄²	—	4	1	no	—
819	C₄²	—	4	1	no	—
820	C₄²	—	4	2	no	—
821	C₄²	—	4	2	no	—
822	C₄²	—	4	2	no	—
823	C₄²	—	4	2	no	—
824	C₄²	—	4	2	no	—
825	C₄²	—	4	2	no	—
826	C₄²	—	4	2	no	—
827	C₄²	—	4	2	no	—
828	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
829	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
830	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
831	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
832	C₂³	—	2	1	no	—
833	C₂³	—	2	1	no	—
834	C₂³	—	2	1	no	—
835	C₂³	—	2	1	no	—
836	C₂³	—	2	1	no	—
837	C₂³	—	2	1	no	—
838	C₂³	—	2	1	no	—
839	C₂³	—	2	1	no	—
840	C₂³	—	2	1	no	—
841	C₂³	—	2	1	no	—
842	C₂³	—	2	1	no	—
843	C₂³	—	2	1	no	—
844	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
845	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
846	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
847	Q₈	—	2	8	no	—
848	Q₈	—	4	8	no	—
849	D₄	—	4	4	no	—
850	C₂×C₄	—	2	1	no	—
851	C₂×C₄	—	2	1	no	—
852	C₂×C₄	—	2	2	no	—
853	C₂×C₄	—	2	1	no	—
854	C₂×C₄	—	2	1	no	—
855	C₂×C₄	—	2	2	no	—
856	C₂×C₄	—	2	1	no	—
857	C₂×C₄	—	2	1	no	—
858	C₂×C₄	—	2	2	no	—
859	C₂×C₄	—	2	1	no	—
860	C₂×C₄	—	2	1	no	—
861	C₂×C₄	—	2	2	no	—
862	C₂×C₄	—	2	2	no	—
863	C₂×C₄	—	2	2	no	—
864	C₂×C₄	—	2	2	no	—
865	C₂×C₄	—	2	2	no	—
866	C₂×C₄	—	2	1	no	—
867	C₂×C₄	—	2	1	no	—
868	C₂×C₄	—	2	8	no	—
869	C₂×C₄	—	2	8	no	—
870	C₂×C₄	—	2	1	no	—
871	C₂×C₄	—	2	1	no	—
872	C₂×C₄	—	2	1	no	—
873	C₂×C₄	—	2	1	no	—
874	C₂×C₄	—	2	1	no	—
875	C₂×C₄	—	2	2	no	—
876	C₂×C₄	—	2	2	no	—
877	C₂×C₄	—	2	2	no	—
878	C₂×C₄	—	2	1	no	—
879	C₂×C₄	—	2	1	no	—
880	C₂×C₄	—	2	1	no	—
881	C₂×C₄	—	2	1	no	—
882	C₂×C₄	—	2	2	no	—
883	C₂×C₄	—	2	2	no	—
884	C₂×C₄	—	2	1	no	—
885	C₂×C₄	—	2	1	no	—
886	C₂×C₄	—	2	1	no	—
887	C₂×C₄	—	2	1	no	—
888	C₂×C₄	—	2	2	no	—
889	C₂×C₄	—	2	2	no	—
890	C₂×C₄	—	2	2	no	—
891	C₂×C₄	—	2	2	no	—
892	C₂×C₄	—	2	2	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	1	no	—
899	C₂×C₄	—	2	1	no	—
900	C₂×C₄	—	2	1	no	—
901	C₂×C₄	—	2	2	no	—
902	C₂×C₄	—	2	1	no	—
903	C₂×C₄	—	4	4	no	—
904	C₂×C₄	—	4	4	no	—
905	C₂×C₄	—	8	4	no	—
906	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
907	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
908	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
909	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
910	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
911	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
912	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
913	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
914	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
915	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
916	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
917	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
918	C₂²	—	2	2	no	—
919	C₂²	—	2	2	no	—
920	C₂²	—	2	2	no	—
921	C₂²	—	2	2	no	—
922	C₂²	—	2	2	no	—
923	C₂²	—	2	2	no	—
924	C₂²	—	4	1	no	—
925	C₂²	—	4	1	no	—
926	C₂²	—	4	1	no	—
927	C₂²	—	4	1	no	—
928	C₂²	—	4	1	no	—
929	C₂²	—	4	1	no	—
930	C₂²	—	4	1	no	—
931	C₂²	—	4	1	no	—
932	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
933	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
934	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
935	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
936	C₂²	C₃₁×#585	1	1	yes	characteristic
937	C₂²	C₃₁×#585	1	1	yes	characteristic
938	C₂²	C₃₁×#585	1	1	yes	characteristic
939	C₄	—	2	2	no	—
940	C₄	—	2	2	no	—
941	C₄	—	2	4	no	—
942	C₄	—	2	2	no	—
943	C₄	—	2	4	no	—
944	C₄	—	2	4	no	—
945	C₄	—	2	2	no	—
946	C₄	—	2	4	no	—
947	C₄	—	2	2	no	—
948	C₄	—	4	1	no	—
949	C₄	—	4	1	no	—
950	C₄	—	4	2	no	—
951	C₄	—	4	2	no	—
952	C₄	—	4	1	no	—
953	C₄	—	4	1	no	—
954	C₄	—	4	1	no	—
955	C₄	—	4	2	no	—
956	C₄	—	4	1	no	—
957	C₄	—	4	1	no	—
958	C₄	—	4	2	no	—
959	C₄	C₃₁×#520	1	4	yes	—
960	C₂	—	4	2	no	—
961	C₂	—	4	2	no	—
962	C₂	C₂×C₃₁×#520	1	1	yes	characteristic
963	C₂	C₃₁×#3428	1	1	yes	characteristic
964	C₂	C₄×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
965	C₂	C₃₁×#3540	1	1	yes	characteristic
966	C₂	C₃₁×#3541	1	1	yes	characteristic
967	C₂	C₃₁×#3555	1	1	yes	characteristic
968	C₂	C₃₁×#3565	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×#2383	C₂	semidirect
4. C₃₁×#2373	C₂	semidirect
5. C₃₁×#2369	C₂	semidirect
6. C₃₁×#2362	C₂	semidirect
7. C₃₁×#2360	C₂	semidirect
8. C₃₁×#2355	C₂	semidirect
10. C₃₁×#2351	C₂	semidirect
11. C₃₁×#2347	C₂	semidirect
12. C₃₁×#2346	C₂	semidirect
14. C₃₁×#2334	C₂	semidirect
15. C₃₁×#2334	C₂	semidirect
16. C₃₁×#2331	C₂	semidirect
17. C₃₁×#2323	C₂	semidirect
18. C₃₁×#2320	C₂	semidirect
19. C₃₁×#2320	C₂	semidirect
20. C₃₁×#2317	C₂	semidirect
21. C₄×C₃₁×C₂²⋊Q₈	C₂	semidirect
22. C₃₁×#2296	C₂	semidirect
24. C₃₁×#556	C₄	semidirect
132. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
133. C₃₁×#126	C₂×C₄	semidirect
134. C₃₁×#126	C₂×C₄	semidirect
136. C₃₁×C₂²⋊Q₈	C₂×C₄	semidirect
137. C₃₁×C₂²⋊Q₈	C₂×C₄	semidirect
140. C₄×Q₈×C₃₁	C₂×C₄	semidirect
141. C₄×Q₈×C₃₁	C₂×C₄	semidirect
145. C₃₁×#117	C₂×C₄	semidirect
146. C₃₁×#117	C₂×C₄	semidirect
347. #34463	C₃₁	direct (N × H)
435. #2383	C₂×C₃₁	semidirect
437. #2373	C₂×C₃₁	semidirect
438. #2369	C₂×C₃₁	semidirect
439. #2362	C₂×C₃₁	semidirect
440. #2360	C₂×C₃₁	semidirect
441. #2355	C₂×C₃₁	semidirect
443. #2351	C₂×C₃₁	semidirect
444. #2347	C₂×C₃₁	semidirect
445. #2346	C₂×C₃₁	semidirect
447. #2334	C₂×C₃₁	semidirect
448. #2334	C₂×C₃₁	semidirect
449. #2331	C₂×C₃₁	semidirect
450. #2323	C₂×C₃₁	semidirect
451. #2320	C₂×C₃₁	semidirect
452. #2320	C₂×C₃₁	semidirect
453. #2317	C₂×C₃₁	semidirect
454. C₄×C₂²⋊Q₈	C₂×C₃₁	semidirect
455. #2296	C₂×C₃₁	semidirect
499. #556	C₄×C₃₁	semidirect
616. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
617. #126	C₂×C₄×C₃₁	semidirect
618. #126	C₂×C₄×C₃₁	semidirect
620. C₂²⋊Q₈	C₂×C₄×C₃₁	semidirect
621. C₂²⋊Q₈	C₂×C₄×C₃₁	semidirect
624. C₄×Q₈	C₂×C₄×C₃₁	semidirect
625. C₄×Q₈	C₂×C₄×C₃₁	semidirect
629. #117	C₂×C₄×C₃₁	semidirect
630. #117	C₂×C₄×C₃₁	semidirect
724. C₃₁	#34463	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×#556	2
C₃₁×#2383	2
C₃₁×#2379	2
C₃₁×#2373	2
C₃₁×#2369	2
C₃₁×#2362	2
C₃₁×#2360	2
C₃₁×#2355	2
C₃₁×#2352	2
C₃₁×#2351	2
C₃₁×#2347	2
C₃₁×#2346	2
C₃₁×#2337	2
C₃₁×#2334	2
C₃₁×#2331	2
C₃₁×#2323	2
C₃₁×#2320	2
C₃₁×#2317	2
C₄×C₃₁×C₂²⋊Q₈	2
C₃₁×#2296	2
C₃₁×#2290	2
#34463	31

Subgroups — C31×#34463 #21491222

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34463 #21491222