Subgroups — C₃₁×#23736 #21480495

Order 7936.

1050 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#428	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#427	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#419	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₃₁×#103	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#2317	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#2310	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#2097	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2097	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2096	C₂	1	2	yes	maximal, complemented
10	C₃₁×#2095	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1729	C₂	1	1	yes	maximal, characteristic
12	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
18	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#428	—	2	4	no	—
20	C₃₁×#427	—	2	4	no	—
21	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#107	C₂²	1	2	yes	—
23	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#106	C₂²	1	2	yes	—
26	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
27	C₃₁×#419	—	2	4	no	—
28	C₂×C₃₁×#103	—	2	4	no	—
29	C₂×C₃₁×#103	C₂²	1	2	yes	—
30	C₂×C₃₁×#103	C₂²	1	2	yes	—
31	C₂×C₃₁×#103	C₂²	1	2	yes	—
32	C₂×C₃₁×#103	C₂²	1	2	yes	—
33	C₃₁×#391	—	2	2	no	—
34	C₃₁×#391	C₂²	1	2	yes	—
35	C₃₁×#391	C₂²	1	2	yes	—
36	C₃₁×#389	—	2	2	no	—
37	C₃₁×#385	—	2	2	no	—
38	C₃₁×#384	C₂²	1	2	yes	—
39	C₃₁×#382	C₂²	1	2	yes	—
40	C₃₁×#381	—	2	2	no	—
41	C₃₁×#336	C₂²	1	2	yes	—
42	C₃₁×#336	C₂²	1	2	yes	—
43	C₂²×Q₈×C₃₁	—	2	1	no	—
44	C₂²×Q₈×C₃₁	—	2	2	no	—
45	C₂²×Q₈×C₃₁	—	2	2	no	—
46	C₂²×Q₈×C₃₁	—	2	2	no	—
47	C₂²×Q₈×C₃₁	—	2	2	no	—
48	C₂²×Q₈×C₃₁	—	2	2	no	—
49	C₂²×Q₈×C₃₁	Q₈	1	4	yes	—
50	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
51	C₂³×C₄×C₃₁	—	2	1	no	—
52	C₂³×C₄×C₃₁	—	2	1	no	—
53	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
54	C₂×M₁₆×C₃₁	—	2	2	no	—
55	C₂×M₁₆×C₃₁	—	2	2	no	—
56	C₂×M₁₆×C₃₁	D₄	1	4	yes	—
57	C₂²×C₈×C₃₁	—	2	2	no	—
58	C₂²×C₈×C₃₁	—	2	2	no	—
59	C₂²×C₈×C₃₁	D₄	1	4	yes	—
60	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
62	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
63	C₃₁×#117	—	2	2	no	—
64	C₃₁×#117	—	2	2	no	—
65	C₃₁×#117	—	2	2	no	—
66	C₃₁×#117	—	2	2	no	—
67	C₃₁×#117	—	2	2	no	—
68	C₃₁×#117	—	2	2	no	—
69	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
70	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
71	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
72	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
73	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
74	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
75	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
76	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
77	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
79	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
80	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
81	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
82	C₂×C₄²×C₃₁	—	2	1	no	—
83	C₂×C₄²×C₃₁	—	2	1	no	—
84	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
85	C₃₁×#107	—	2	2	no	—
86	C₃₁×#107	—	2	2	no	—
87	C₃₁×#107	—	2	4	no	—
88	C₃₁×#106	—	2	2	no	—
89	C₃₁×#106	—	2	2	no	—
90	C₃₁×#106	—	2	4	no	—
91	C₃₁×#103	—	2	4	no	—
92	C₃₁×#103	—	2	4	no	—
93	C₃₁×#103	—	2	4	no	—
94	C₃₁×#103	—	2	4	no	—
95	C₃₁×#95	—	2	4	no	—
96	C₃₁×#95	—	2	4	no	—
97	C₃₁×#95	—	2	4	no	—
98	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
99	C₂×Q₈×C₃₁	—	2	4	no	—
100	C₂×Q₈×C₃₁	—	2	2	no	—
101	C₂×Q₈×C₃₁	—	2	8	no	—
102	C₂×Q₈×C₃₁	—	2	8	no	—
103	C₂×Q₈×C₃₁	—	2	2	no	—
104	C₂×Q₈×C₃₁	—	2	2	no	—
105	C₂×Q₈×C₃₁	—	2	2	no	—
106	C₂×Q₈×C₃₁	—	2	2	no	—
107	C₂×Q₈×C₃₁	—	4	2	no	—
108	C₂×Q₈×C₃₁	—	4	4	no	—
109	C₂×Q₈×C₃₁	—	4	4	no	—
110	C₂×Q₈×C₃₁	—	4	2	no	—
111	C₂×Q₈×C₃₁	C₂×Q₈	1	4	yes	—
112	C₂²×C₄×C₃₁	—	2	4	no	—
113	C₂²×C₄×C₃₁	—	2	2	no	—
114	C₂²×C₄×C₃₁	—	2	1	no	—
115	C₂²×C₄×C₃₁	—	2	1	no	—
116	C₂²×C₄×C₃₁	—	2	1	no	—
117	C₂²×C₄×C₃₁	—	2	2	no	—
118	C₂²×C₄×C₃₁	—	2	1	no	—
119	C₂²×C₄×C₃₁	—	4	2	no	—
120	C₂²×C₄×C₃₁	—	4	2	no	—
121	C₂²×C₄×C₃₁	—	4	2	no	—
122	C₂²×C₄×C₃₁	—	8	2	no	—
123	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
124	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
126	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
127	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
128	C₂²×C₄×C₃₁	#41	1	2	yes	—
129	C₂²×C₄×C₃₁	#41	1	2	yes	—
130	C₂²×C₄×C₃₁	#41	1	2	yes	—
131	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
132	M₁₆×C₃₁	—	2	8	no	—
133	C₂×C₈×C₃₁	—	2	8	no	—
134	C₂×C₈×C₃₁	—	2	1	no	—
135	C₂×C₈×C₃₁	—	2	1	no	—
136	C₂×C₈×C₃₁	—	2	1	no	—
137	C₂×C₈×C₃₁	—	2	1	no	—
138	C₂×C₈×C₃₁	—	2	2	no	—
139	C₂×C₈×C₃₁	—	2	2	no	—
140	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
141	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
142	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
143	C₄⋊C₄×C₃₁	—	2	4	no	—
144	C₄⋊C₄×C₃₁	—	2	2	no	—
145	C₄⋊C₄×C₃₁	—	2	1	no	—
146	C₄⋊C₄×C₃₁	—	2	1	no	—
147	C₄⋊C₄×C₃₁	—	2	1	no	—
148	C₄⋊C₄×C₃₁	—	2	2	no	—
149	C₄⋊C₄×C₃₁	—	2	1	no	—
150	C₄⋊C₄×C₃₁	—	2	1	no	—
151	C₄⋊C₄×C₃₁	—	2	1	no	—
152	C₄⋊C₄×C₃₁	—	2	1	no	—
153	C₄⋊C₄×C₃₁	—	2	1	no	—
154	C₄⋊C₄×C₃₁	—	4	2	no	—
155	C₄⋊C₄×C₃₁	—	8	4	no	—
156	C₂²⋊C₄×C₃₁	—	2	4	no	—
157	C₂²⋊C₄×C₃₁	—	4	2	no	—
158	C₂²⋊C₄×C₃₁	—	4	2	no	—
159	C₄²×C₃₁	—	2	2	no	—
160	C₄²×C₃₁	—	2	2	no	—
161	C₄²×C₃₁	—	4	2	no	—
162	C₄²×C₃₁	—	4	2	no	—
163	#23736	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
164	C₂³×C₃₁	—	2	2	no	—
165	C₂³×C₃₁	—	2	2	no	—
166	C₂³×C₃₁	C₂²⋊Q₈	1	2	yes	—
167	C₂³×C₃₁	#123	1	2	yes	—
168	C₂³×C₃₁	#127	1	2	yes	—
169	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
170	Q₈×C₃₁	—	2	4	no	—
171	Q₈×C₃₁	—	2	4	no	—
172	Q₈×C₃₁	—	4	2	no	—
173	Q₈×C₃₁	—	4	2	no	—
174	Q₈×C₃₁	—	4	8	no	—
175	C₂×C₄×C₃₁	—	2	4	no	—
176	C₂×C₄×C₃₁	—	2	4	no	—
177	C₂×C₄×C₃₁	—	2	4	no	—
178	C₂×C₄×C₃₁	—	2	4	no	—
179	C₂×C₄×C₃₁	—	4	2	no	—
180	C₂×C₄×C₃₁	—	4	2	no	—
181	C₂×C₄×C₃₁	—	4	8	no	—
182	C₂×C₄×C₃₁	—	4	1	no	—
183	C₂×C₄×C₃₁	—	4	1	no	—
184	C₂×C₄×C₃₁	—	4	1	no	—
185	C₂×C₄×C₃₁	—	4	1	no	—
186	C₂×C₄×C₃₁	—	4	2	no	—
187	C₂×C₄×C₃₁	—	4	2	no	—
188	C₂×C₄×C₃₁	—	4	4	no	—
189	C₂×C₄×C₃₁	—	4	2	no	—
190	C₂×C₄×C₃₁	—	4	2	no	—
191	C₂×C₄×C₃₁	—	4	2	no	—
192	C₂×C₄×C₃₁	—	8	4	no	—
193	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
194	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
195	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
196	C₂×C₄×C₃₁	#123	1	4	yes	—
197	C₂×C₄×C₃₁	#123	1	4	yes	—
198	C₂×C₄×C₃₁	#123	1	2	yes	—
199	C₂×C₄×C₃₁	#127	1	2	yes	—
200	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
201	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
202	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
203	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
204	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
205	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
206	C₈×C₃₁	—	2	2	no	—
207	C₈×C₃₁	—	2	2	no	—
208	C₈×C₃₁	—	2	4	no	—
209	C₂×#428	C₂×C₃₁	1	1	yes	characteristic
210	C₂×#427	C₂×C₃₁	1	1	yes	characteristic
211	C₂×#419	C₂×C₃₁	1	1	yes	characteristic
212	C₂²×#103	C₂×C₃₁	1	1	yes	characteristic
213	#2317	C₂×C₃₁	1	1	yes	characteristic
214	#2310	C₂×C₃₁	1	1	yes	characteristic
215	#2097	C₂×C₃₁	1	2	yes	complemented
216	#2097	C₂×C₃₁	1	2	yes	complemented
217	#2096	C₂×C₃₁	1	2	yes	complemented
218	#2095	C₂×C₃₁	1	2	yes	complemented
219	#1729	C₂×C₃₁	1	1	yes	characteristic
220	C₂²×C₃₁	—	2	4	no	—
221	C₂²×C₃₁	—	2	4	no	—
222	C₂²×C₃₁	—	2	2	no	—
223	C₂²×C₃₁	—	2	2	no	—
224	C₂²×C₃₁	#399	1	4	yes	—
225	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
226	C₂²×C₃₁	C₂×#123	1	1	yes	characteristic
227	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
228	C₂²×C₃₁	#577	1	2	yes	—
229	C₂²×C₃₁	#578	1	2	yes	—
230	C₄×C₃₁	—	4	4	no	—
231	C₄×C₃₁	—	4	4	no	—
232	C₄×C₃₁	—	8	1	no	—
233	C₄×C₃₁	—	8	1	no	—
234	C₄×C₃₁	—	8	2	no	—
235	C₄×C₃₁	—	8	1	no	—
236	C₄×C₃₁	—	8	1	no	—
237	C₄×C₃₁	#399	1	4	yes	—
238	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
239	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
240	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
241	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
242	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic
243	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
244	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
245	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
246	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
247	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
248	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
249	#428	—	2	4	no	—
250	#427	—	2	4	no	—
251	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
252	C₂×#107	C₂²×C₃₁	1	2	yes	—
253	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
254	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
255	C₂×#106	C₂²×C₃₁	1	2	yes	—
256	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
257	#419	—	2	4	no	—
258	C₂×#103	—	2	4	no	—
259	C₂×#103	C₂²×C₃₁	1	2	yes	—
260	C₂×#103	C₂²×C₃₁	1	2	yes	—
261	C₂×#103	C₂²×C₃₁	1	2	yes	—
262	C₂×#103	C₂²×C₃₁	1	2	yes	—
263	#391	—	2	2	no	—
264	#391	C₂²×C₃₁	1	2	yes	—
265	#391	C₂²×C₃₁	1	2	yes	—
266	#389	—	2	2	no	—
267	#385	—	2	2	no	—
268	#384	C₂²×C₃₁	1	2	yes	—
269	#382	C₂²×C₃₁	1	2	yes	—
270	#381	—	2	2	no	—
271	#336	C₂²×C₃₁	1	2	yes	—
272	#336	C₂²×C₃₁	1	2	yes	—
273	C₂×C₃₁	—	2	4	no	—
274	C₂×C₃₁	C₂×#399	1	1	yes	characteristic
275	C₂×C₃₁	#3082	1	2	yes	—
276	C₂×C₃₁	#3097	1	2	yes	—
277	C₂×C₃₁	#3154	1	1	yes	characteristic
278	C₂×C₃₁	#3156	1	1	yes	characteristic
279	C₂²×Q₈	—	2	1	no	—
280	C₂²×Q₈	—	2	2	no	—
281	C₂²×Q₈	—	2	2	no	—
282	C₂²×Q₈	—	2	2	no	—
283	C₂²×Q₈	—	2	2	no	—
284	C₂²×Q₈	—	2	2	no	—
285	C₂²×Q₈	Q₈×C₃₁	1	4	yes	—
286	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
287	C₂³×C₄	—	2	1	no	—
288	C₂³×C₄	—	2	1	no	—
289	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
290	C₂×M₁₆	—	2	2	no	—
291	C₂×M₁₆	—	2	2	no	—
292	C₂×M₁₆	D₄×C₃₁	1	4	yes	—
293	C₂²×C₈	—	2	2	no	—
294	C₂²×C₈	—	2	2	no	—
295	C₂²×C₈	D₄×C₃₁	1	4	yes	—
296	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
297	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
298	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
299	#117	—	2	2	no	—
300	#117	—	2	2	no	—
301	#117	—	2	2	no	—
302	#117	—	2	2	no	—
303	#117	—	2	2	no	—
304	#117	—	2	2	no	—
305	C₂×C₄⋊C₄	—	2	2	no	—
306	C₂×C₄⋊C₄	—	2	2	no	—
307	C₂×C₄⋊C₄	—	2	1	no	—
308	C₂×C₄⋊C₄	—	2	4	no	—
309	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
310	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
311	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
312	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
313	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
314	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
315	C₂×C₂²⋊C₄	—	2	1	no	—
316	C₂×C₂²⋊C₄	—	2	1	no	—
317	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
318	C₂×C₄²	—	2	1	no	—
319	C₂×C₄²	—	2	1	no	—
320	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
321	#107	—	2	2	no	—
322	#107	—	2	2	no	—
323	#107	—	2	4	no	—
324	#106	—	2	2	no	—
325	#106	—	2	2	no	—
326	#106	—	2	4	no	—
327	#103	—	2	4	no	—
328	#103	—	2	4	no	—
329	#103	—	2	4	no	—
330	#103	—	2	4	no	—
331	#95	—	2	4	no	—
332	#95	—	2	4	no	—
333	#95	—	2	4	no	—
334	C₃₁	#23736	1	1	yes	characteristic, direct product factor, Sylow
335	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
336	C₂×Q₈	—	2	4	no	—
337	C₂×Q₈	—	2	2	no	—
338	C₂×Q₈	—	2	8	no	—
339	C₂×Q₈	—	2	8	no	—
340	C₂×Q₈	—	2	2	no	—
341	C₂×Q₈	—	2	2	no	—
342	C₂×Q₈	—	2	2	no	—
343	C₂×Q₈	—	2	2	no	—
344	C₂×Q₈	—	4	4	no	—
345	C₂×Q₈	—	4	4	no	—
346	C₂×Q₈	—	4	2	no	—
347	C₂×Q₈	—	4	2	no	—
348	C₂×Q₈	C₂×Q₈×C₃₁	1	4	yes	—
349	C₂²×C₄	—	2	4	no	—
350	C₂²×C₄	—	2	2	no	—
351	C₂²×C₄	—	2	1	no	—
352	C₂²×C₄	—	2	1	no	—
353	C₂²×C₄	—	2	1	no	—
354	C₂²×C₄	—	2	2	no	—
355	C₂²×C₄	—	2	1	no	—
356	C₂²×C₄	—	4	2	no	—
357	C₂²×C₄	—	4	2	no	—
358	C₂²×C₄	—	4	2	no	—
359	C₂²×C₄	—	8	2	no	—
360	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
361	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
362	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
363	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
364	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
365	C₂²×C₄	#41×C₃₁	1	2	yes	—
366	C₂²×C₄	#41×C₃₁	1	2	yes	—
367	C₂²×C₄	#41×C₃₁	1	2	yes	—
368	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
369	M₁₆	—	2	8	no	—
370	C₂×C₈	—	2	8	no	—
371	C₂×C₈	—	2	1	no	—
372	C₂×C₈	—	2	1	no	—
373	C₂×C₈	—	2	1	no	—
374	C₂×C₈	—	2	1	no	—
375	C₂×C₈	—	2	2	no	—
376	C₂×C₈	—	2	2	no	—
377	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
378	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
379	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
380	C₄⋊C₄	—	2	4	no	—
381	C₄⋊C₄	—	2	2	no	—
382	C₄⋊C₄	—	2	1	no	—
383	C₄⋊C₄	—	2	1	no	—
384	C₄⋊C₄	—	2	1	no	—
385	C₄⋊C₄	—	2	2	no	—
386	C₄⋊C₄	—	2	1	no	—
387	C₄⋊C₄	—	2	1	no	—
388	C₄⋊C₄	—	2	1	no	—
389	C₄⋊C₄	—	2	1	no	—
390	C₄⋊C₄	—	2	1	no	—
391	C₄⋊C₄	—	4	2	no	—
392	C₄⋊C₄	—	8	4	no	—
393	C₂²⋊C₄	—	2	4	no	—
394	C₂²⋊C₄	—	4	2	no	—
395	C₂²⋊C₄	—	4	2	no	—
396	C₄²	—	2	2	no	—
397	C₄²	—	2	2	no	—
398	C₄²	—	4	2	no	—
399	C₄²	—	4	2	no	—
400	C₂³	—	2	2	no	—
401	C₂³	—	2	2	no	—
402	C₂³	C₃₁×C₂²⋊Q₈	1	2	yes	—
403	C₂³	C₃₁×#123	1	2	yes	—
404	C₂³	C₃₁×#127	1	2	yes	—
405	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
406	Q₈	—	2	4	no	—
407	Q₈	—	2	4	no	—
408	Q₈	—	4	8	no	—
409	Q₈	—	4	2	no	—
410	Q₈	—	4	2	no	—
411	C₂×C₄	—	2	4	no	—
412	C₂×C₄	—	2	4	no	—
413	C₂×C₄	—	2	4	no	—
414	C₂×C₄	—	2	4	no	—
415	C₂×C₄	—	4	8	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	1	no	—
419	C₂×C₄	—	4	1	no	—
420	C₂×C₄	—	4	1	no	—
421	C₂×C₄	—	4	1	no	—
422	C₂×C₄	—	4	2	no	—
423	C₂×C₄	—	4	2	no	—
424	C₂×C₄	—	4	4	no	—
425	C₂×C₄	—	4	2	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	4	2	no	—
428	C₂×C₄	—	8	4	no	—
429	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
430	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
431	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
432	C₂×C₄	C₃₁×#123	1	2	yes	—
433	C₂×C₄	C₃₁×#123	1	4	yes	—
434	C₂×C₄	C₃₁×#123	1	4	yes	—
435	C₂×C₄	C₃₁×#127	1	2	yes	—
436	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
437	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
438	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
439	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
440	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
441	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
442	C₈	—	2	2	no	—
443	C₈	—	2	2	no	—
444	C₈	—	2	4	no	—
445	C₂²	—	2	4	no	—
446	C₂²	—	2	4	no	—
447	C₂²	—	2	2	no	—
448	C₂²	—	2	2	no	—
449	C₂²	C₃₁×#399	1	4	yes	—
450	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
451	C₂²	C₂×C₃₁×#123	1	1	yes	characteristic
452	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
453	C₂²	C₃₁×#577	1	2	yes	—
454	C₂²	C₃₁×#578	1	2	yes	—
455	C₄	—	4	4	no	—
456	C₄	—	4	4	no	—
457	C₄	—	8	2	no	—
458	C₄	—	8	1	no	—
459	C₄	—	8	1	no	—
460	C₄	—	8	1	no	—
461	C₄	—	8	1	no	—
462	C₄	C₃₁×#399	1	4	yes	—
463	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
464	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
465	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
466	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
467	C₂	—	2	4	no	—
468	C₂	C₂×C₃₁×#399	1	1	yes	characteristic
469	C₂	C₃₁×#3082	1	2	yes	—
470	C₂	C₃₁×#3097	1	2	yes	—
471	C₂	C₃₁×#3154	1	1	yes	characteristic
472	C₂	C₃₁×#3156	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
7. C₃₁×#2097	C₂	semidirect
8. C₃₁×#2097	C₂	semidirect
9. C₃₁×#2096	C₂	semidirect
10. C₃₁×#2095	C₂	semidirect
163. #23736	C₃₁	direct (N × H)
215. #2097	C₂×C₃₁	semidirect
216. #2097	C₂×C₃₁	semidirect
217. #2096	C₂×C₃₁	semidirect
218. #2095	C₂×C₃₁	semidirect
334. C₃₁	#23736	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×#428	2
C₂×C₃₁×#427	2
C₂×C₃₁×#419	2
C₂²×C₃₁×#103	2
C₃₁×#2317	2
C₃₁×#2310	2
C₃₁×#2097	2
C₃₁×#2096	2
C₃₁×#2095	2
C₃₁×#1729	2
#23736	31

Subgroups — C31×#23736 #21480495

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23736 #21480495