Subgroups — C₃₁×#61932 #21518691

Order 7936.

1866 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3548	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3508	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3399	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3391	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3366	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3361	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3220	C₂	1	4	yes	maximal, complemented
8	C₃₁×#3205	C₂	1	2	yes	maximal, complemented
9	C₃₁×#3122	C₂	1	4	yes	maximal, complemented
10	C₃₁×#3106	C₂	1	2	yes	maximal, complemented
11	C₃₁×#2986	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#562	—	2	2	no	—
13	C₃₁×#559	C₂²	1	4	yes	—
14	C₃₁×#554	C₂²	1	4	yes	complemented
15	C₃₁×#550	C₂²	1	4	yes	—
16	C₃₁×#550	C₂²	1	4	yes	complemented
17	C₃₁×#547	—	2	2	no	—
18	C₃₁×#546	—	2	4	no	—
19	C₃₁×#540	—	2	2	no	—
20	C₃₁×#540	C₂²	1	4	yes	—
21	C₃₁×#537	—	2	4	no	—
22	C₃₁×#535	C₂²	1	2	yes	—
23	C₃₁×#533	C₂²	1	1	yes	characteristic
24	C₃₁×#532	—	2	2	no	—
25	C₃₁×#532	C₂²	1	2	yes	—
26	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
27	C₃₁×#518	C₂²	1	2	yes	—
28	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#496	C₂²	1	4	yes	complemented
30	C₃₁×#493	C₂²	1	4	yes	—
31	C₃₁×#493	C₂²	1	4	yes	complemented
32	C₃₁×#490	C₂²	1	4	yes	complemented
33	C₃₁×#488	C₂²	1	4	yes	complemented
34	C₃₁×#486	C₂²	1	4	yes	—
35	C₃₁×#482	C₂²	1	4	yes	—
36	C₃₁×#478	C₂²	1	8	yes	complemented
37	C₃₁×#477	C₂²	1	2	yes	complemented
38	C₃₁×#476	C₂²	1	2	yes	complemented
39	C₃₁×#463	C₂²	1	8	yes	complemented
40	C₃₁×#460	C₂²	1	2	yes	complemented
41	C₃₁×#459	C₂²	1	4	yes	—
42	C₃₁×#459	C₂²	1	4	yes	—
43	C₃₁×#459	C₂²	1	2	yes	complemented
44	C₃₁×#459	C₂²	1	8	yes	complemented
45	C₃₁×#459	C₂²	1	8	yes	complemented
46	C₃₁×#449	C₂²	1	4	yes	—
47	C₃₁×#447	C₂²	1	8	yes	—
48	C₃₁×#446	C₂²	1	4	yes	complemented
49	C₈×Q₈×C₃₁	C₂²	1	4	yes	complemented
50	C₃₁×#442	C₂²	1	8	yes	complemented
51	C₄×SD₁₆×C₃₁	C₂²	1	4	yes	complemented
52	C₄×D₈×C₃₁	C₂²	1	8	yes	complemented
53	C₄×D₈×C₃₁	C₂²	1	4	yes	complemented
54	C₃₁×#433	C₂²	1	4	yes	—
55	C₃₁×#431	C₂²	1	2	yes	—
56	C₂×#41×C₃₁	—	2	4	no	—
57	C₂×#41×C₃₁	—	2	4	no	—
58	C₂×#41×C₃₁	—	2	2	no	—
59	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
60	C₂²×D₄×C₃₁	—	2	2	no	—
61	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
62	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
63	C₂×D₈×C₃₁	C₂³	1	8	yes	—
64	C₂×D₈×C₃₁	C₂³	1	4	yes	—
65	C₂×D₈×C₃₁	C₂³	1	8	yes	—
66	C₃₁×C₄⋊Q₈	D₄	1	2	yes	—
67	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
68	C₃₁×#127	—	2	1	no	—
69	C₃₁×#127	—	2	1	no	—
70	C₃₁×#127	C₂³	1	4	yes	—
71	C₃₁×#127	C₂³	1	4	yes	—
72	C₃₁×#127	C₂³	1	4	yes	—
73	C₃₁×#127	C₂³	1	2	yes	—
74	C₃₁×#125	D₄	1	2	yes	—
75	C₃₁×#125	C₂³	1	4	yes	—
76	C₃₁×#124	—	2	4	no	—
77	C₃₁×#124	—	2	2	no	—
78	C₃₁×#124	C₂³	1	4	yes	—
79	C₃₁×#123	—	2	8	no	—
80	C₃₁×#123	—	2	8	no	—
81	C₃₁×C₂²⋊Q₈	—	2	4	no	—
82	C₃₁×C₂²⋊Q₈	—	2	8	no	—
83	C₃₁×C₂²⋊Q₈	D₄	1	4	yes	—
84	C₃₁×#121	—	2	2	no	—
85	C₃₁×#121	—	2	4	no	—
86	C₃₁×#121	—	2	4	no	—
87	C₃₁×#121	—	2	4	no	—
88	C₃₁×#121	—	2	8	no	—
89	C₃₁×#121	—	2	8	no	—
90	C₃₁×#121	—	2	8	no	—
91	C₃₁×#121	C₂³	1	4	yes	—
92	C₃₁×#120	—	2	4	no	—
93	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
94	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
95	C₄×D₄×C₃₁	—	2	2	no	—
96	C₄×D₄×C₃₁	—	2	4	no	—
97	C₄×D₄×C₃₁	—	2	4	no	—
98	C₄×D₄×C₃₁	—	2	2	no	—
99	C₄×D₄×C₃₁	C₂³	1	2	yes	—
100	C₄×D₄×C₃₁	C₂³	1	4	yes	—
101	C₄×D₄×C₃₁	C₂³	1	8	yes	—
102	C₄×D₄×C₃₁	C₂³	1	8	yes	—
103	C₃₁×#117	—	2	1	no	—
104	C₃₁×#117	C₂³	1	1	yes	characteristic
105	C₃₁×#117	C₂³	1	2	yes	—
106	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
107	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
108	C₂×C₄²×C₃₁	—	2	1	no	—
109	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
110	C₃₁×#107	C₂³	1	4	yes	—
111	C₃₁×#106	C₂³	1	4	yes	—
112	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
113	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
114	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
115	C₃₁×#103	C₂³	1	4	yes	—
116	C₃₁×#102	C₂³	1	4	yes	—
117	C₃₁×#102	C₂³	1	4	yes	—
118	C₃₁×#102	C₂³	1	8	yes	—
119	C₃₁×#102	C₂³	1	4	yes	—
120	C₃₁×#102	C₂³	1	8	yes	—
121	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
122	C₃₁×#97	C₂³	1	4	yes	—
123	C₄×C₈×C₃₁	C₂³	1	4	yes	—
124	C₄×C₈×C₃₁	C₂³	1	4	yes	—
125	C₂⁴×C₃₁	—	2	2	no	—
126	#41×C₃₁	—	4	2	no	—
127	#41×C₃₁	—	8	4	no	—
128	#41×C₃₁	—	8	8	no	—
129	#41×C₃₁	—	8	8	no	—
130	C₂×Q₈×C₃₁	—	2	2	no	—
131	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
132	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
133	C₂×D₄×C₃₁	—	2	1	no	—
134	C₂×D₄×C₃₁	—	2	4	no	—
135	C₂×D₄×C₃₁	—	2	2	no	—
136	C₂×D₄×C₃₁	—	2	2	no	—
137	C₂×D₄×C₃₁	—	2	4	no	—
138	C₂×D₄×C₃₁	—	2	8	no	—
139	C₂×D₄×C₃₁	—	2	8	no	—
140	C₂×D₄×C₃₁	—	2	2	no	—
141	C₂×D₄×C₃₁	—	2	4	no	—
142	C₂×D₄×C₃₁	—	4	4	no	—
143	C₂×D₄×C₃₁	—	8	4	no	—
144	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
145	C₂×D₄×C₃₁	C₂⁴	1	1	yes	characteristic
146	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
147	C₂²×C₄×C₃₁	—	2	1	no	—
148	C₂²×C₄×C₃₁	—	2	2	no	—
149	C₂²×C₄×C₃₁	—	2	4	no	—
150	C₂²×C₄×C₃₁	—	2	8	no	—
151	C₂²×C₄×C₃₁	—	2	8	no	—
152	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
153	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
154	SD₁₆×C₃₁	—	4	4	no	—
155	D₈×C₃₁	—	4	4	no	—
156	D₈×C₃₁	—	4	8	no	—
157	D₈×C₃₁	—	4	8	no	—
158	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
159	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
160	C₄⋊C₄×C₃₁	—	2	2	no	—
161	C₄⋊C₄×C₃₁	—	2	8	no	—
162	C₄⋊C₄×C₃₁	—	2	2	no	—
163	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
164	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
165	C₄⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
166	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
167	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
168	C₂²⋊C₄×C₃₁	—	2	1	no	—
169	C₂²⋊C₄×C₃₁	—	2	8	no	—
170	C₂²⋊C₄×C₃₁	—	2	2	no	—
171	C₂²⋊C₄×C₃₁	—	2	8	no	—
172	C₂²⋊C₄×C₃₁	—	2	4	no	—
173	C₂²⋊C₄×C₃₁	—	2	8	no	—
174	C₂²⋊C₄×C₃₁	—	2	4	no	—
175	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
176	C₄²×C₃₁	—	2	1	no	—
177	C₄²×C₃₁	—	2	2	no	—
178	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
179	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
180	C₄²×C₃₁	C₂⁴	1	2	yes	—
181	C₄²×C₃₁	C₂⁴	1	4	yes	—
182	#61932	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
183	C₂³×C₃₁	—	2	4	no	—
184	C₂³×C₃₁	—	2	4	no	—
185	C₂³×C₃₁	—	2	1	no	—
186	C₂³×C₃₁	—	4	4	no	—
187	C₂³×C₃₁	—	8	4	no	—
188	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
189	Q₈×C₃₁	—	2	8	no	—
190	Q₈×C₃₁	—	2	2	no	—
191	Q₈×C₃₁	—	8	2	no	—
192	D₄×C₃₁	—	2	4	no	—
193	D₄×C₃₁	—	2	4	no	—
194	D₄×C₃₁	—	2	8	no	—
195	D₄×C₃₁	—	2	2	no	—
196	D₄×C₃₁	—	4	4	no	—
197	D₄×C₃₁	—	4	2	no	—
198	D₄×C₃₁	—	4	8	no	—
199	D₄×C₃₁	—	4	4	no	—
200	D₄×C₃₁	—	8	4	no	—
201	D₄×C₃₁	—	8	2	no	—
202	D₄×C₃₁	—	8	8	no	—
203	D₄×C₃₁	—	8	8	no	—
204	D₄×C₃₁	—	8	4	no	—
205	C₂×C₄×C₃₁	—	2	4	no	—
206	C₂×C₄×C₃₁	—	2	2	no	—
207	C₂×C₄×C₃₁	—	2	1	no	—
208	C₂×C₄×C₃₁	—	2	2	no	—
209	C₂×C₄×C₃₁	—	2	4	no	—
210	C₂×C₄×C₃₁	—	4	2	no	—
211	C₂×C₄×C₃₁	—	4	4	no	—
212	C₂×C₄×C₃₁	—	4	8	no	—
213	C₂×C₄×C₃₁	—	8	8	no	—
214	C₂×C₄×C₃₁	—	8	8	no	—
215	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
216	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
217	C₂×C₄×C₃₁	C₂²×D₄	1	8	yes	—
218	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
219	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
220	C₈×C₃₁	—	2	4	no	—
221	C₈×C₃₁	—	2	4	no	—
222	#3548	C₂×C₃₁	1	1	yes	characteristic, complemented
223	#3508	C₂×C₃₁	1	1	yes	characteristic, complemented
224	#3399	C₂×C₃₁	1	4	yes	complemented
225	#3391	C₂×C₃₁	1	4	yes	complemented
226	#3366	C₂×C₃₁	1	4	yes	complemented
227	#3361	C₂×C₃₁	1	4	yes	complemented
228	#3220	C₂×C₃₁	1	4	yes	complemented
229	#3205	C₂×C₃₁	1	2	yes	complemented
230	#3122	C₂×C₃₁	1	4	yes	complemented
231	#3106	C₂×C₃₁	1	2	yes	complemented
232	#2986	C₂×C₃₁	1	1	yes	characteristic, complemented
233	C₂²×C₃₁	—	2	1	no	—
234	C₂²×C₃₁	—	2	2	no	—
235	C₂²×C₃₁	—	4	1	no	—
236	C₂²×C₃₁	—	4	4	no	—
237	C₂²×C₃₁	—	4	4	no	—
238	C₂²×C₃₁	—	4	2	no	—
239	C₂²×C₃₁	—	4	4	no	—
240	C₂²×C₃₁	—	4	4	no	—
241	C₂²×C₃₁	—	4	8	no	—
242	C₂²×C₃₁	—	16	4	no	—
243	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
244	C₄×C₃₁	—	2	1	no	—
245	C₄×C₃₁	—	2	2	no	—
246	C₄×C₃₁	—	2	8	no	—
247	C₄×C₃₁	—	4	1	no	—
248	C₄×C₃₁	—	4	2	no	—
249	C₄×C₃₁	—	4	4	no	—
250	C₄×C₃₁	#576	1	4	yes	—
251	C₄×C₃₁	#585	1	2	yes	—
252	#562	—	2	2	no	—
253	#559	C₂²×C₃₁	1	4	yes	—
254	#554	C₂²×C₃₁	1	4	yes	complemented
255	#550	C₂²×C₃₁	1	4	yes	—
256	#550	C₂²×C₃₁	1	4	yes	complemented
257	#547	—	2	2	no	—
258	#546	—	2	4	no	—
259	#540	—	2	2	no	—
260	#540	C₂²×C₃₁	1	4	yes	—
261	#537	—	2	4	no	—
262	#535	C₂²×C₃₁	1	2	yes	—
263	#533	C₂²×C₃₁	1	1	yes	characteristic
264	#532	—	2	2	no	—
265	#532	C₂²×C₃₁	1	2	yes	—
266	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
267	#518	C₂²×C₃₁	1	2	yes	—
268	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
269	#496	C₂²×C₃₁	1	4	yes	complemented
270	#493	C₂²×C₃₁	1	4	yes	—
271	#493	C₂²×C₃₁	1	4	yes	complemented
272	#490	C₂²×C₃₁	1	4	yes	complemented
273	#488	C₂²×C₃₁	1	4	yes	complemented
274	#486	C₂²×C₃₁	1	4	yes	—
275	#482	C₂²×C₃₁	1	4	yes	—
276	#478	C₂²×C₃₁	1	8	yes	complemented
277	#477	C₂²×C₃₁	1	2	yes	complemented
278	#476	C₂²×C₃₁	1	2	yes	complemented
279	#463	C₂²×C₃₁	1	8	yes	complemented
280	#460	C₂²×C₃₁	1	2	yes	complemented
281	#459	C₂²×C₃₁	1	4	yes	—
282	#459	C₂²×C₃₁	1	4	yes	—
283	#459	C₂²×C₃₁	1	2	yes	complemented
284	#459	C₂²×C₃₁	1	8	yes	complemented
285	#459	C₂²×C₃₁	1	8	yes	complemented
286	#449	C₂²×C₃₁	1	4	yes	—
287	#447	C₂²×C₃₁	1	8	yes	—
288	#446	C₂²×C₃₁	1	4	yes	complemented
289	C₈×Q₈	C₂²×C₃₁	1	4	yes	complemented
290	#442	C₂²×C₃₁	1	8	yes	complemented
291	C₄×SD₁₆	C₂²×C₃₁	1	4	yes	complemented
292	C₄×D₈	C₂²×C₃₁	1	8	yes	complemented
293	C₄×D₈	C₂²×C₃₁	1	4	yes	complemented
294	#433	C₂²×C₃₁	1	4	yes	—
295	#431	C₂²×C₃₁	1	2	yes	—
296	C₂×C₃₁	—	4	1	no	—
297	C₂×C₃₁	—	8	1	no	—
298	C₂×C₃₁	—	8	4	no	—
299	C₂×C₃₁	—	8	4	no	—
300	C₂×C₃₁	#3487	1	1	yes	characteristic
301	C₂×C₃₁	C₂×#576	1	2	yes	—
302	C₂×#41	—	2	4	no	—
303	C₂×#41	—	2	4	no	—
304	C₂×#41	—	2	2	no	—
305	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
306	C₂²×D₄	—	2	2	no	—
307	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
308	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
309	C₂×D₈	C₂³×C₃₁	1	8	yes	—
310	C₂×D₈	C₂³×C₃₁	1	4	yes	—
311	C₂×D₈	C₂³×C₃₁	1	8	yes	—
312	C₄⋊Q₈	D₄×C₃₁	1	2	yes	—
313	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
314	#127	—	2	1	no	—
315	#127	—	2	1	no	—
316	#127	C₂³×C₃₁	1	4	yes	—
317	#127	C₂³×C₃₁	1	2	yes	—
318	#127	C₂³×C₃₁	1	4	yes	—
319	#127	C₂³×C₃₁	1	4	yes	—
320	#125	D₄×C₃₁	1	2	yes	—
321	#125	C₂³×C₃₁	1	4	yes	—
322	#124	—	2	4	no	—
323	#124	—	2	2	no	—
324	#124	C₂³×C₃₁	1	4	yes	—
325	#123	—	2	8	no	—
326	#123	—	2	8	no	—
327	C₂²⋊Q₈	—	2	4	no	—
328	C₂²⋊Q₈	—	2	8	no	—
329	C₂²⋊Q₈	D₄×C₃₁	1	4	yes	—
330	#121	—	2	2	no	—
331	#121	—	2	4	no	—
332	#121	—	2	4	no	—
333	#121	—	2	4	no	—
334	#121	—	2	8	no	—
335	#121	—	2	8	no	—
336	#121	—	2	8	no	—
337	#121	C₂³×C₃₁	1	4	yes	—
338	#120	—	2	4	no	—
339	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
340	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
341	C₄×D₄	—	2	2	no	—
342	C₄×D₄	—	2	4	no	—
343	C₄×D₄	—	2	4	no	—
344	C₄×D₄	—	2	2	no	—
345	C₄×D₄	C₂³×C₃₁	1	2	yes	—
346	C₄×D₄	C₂³×C₃₁	1	8	yes	—
347	C₄×D₄	C₂³×C₃₁	1	4	yes	—
348	C₄×D₄	C₂³×C₃₁	1	8	yes	—
349	#117	—	2	1	no	—
350	#117	C₂³×C₃₁	1	1	yes	characteristic
351	#117	C₂³×C₃₁	1	2	yes	—
352	C₂×C₄⋊C₄	—	2	4	no	—
353	C₂×C₂²⋊C₄	—	2	4	no	—
354	C₂×C₄²	—	2	1	no	—
355	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
356	#107	C₂³×C₃₁	1	4	yes	—
357	#106	C₂³×C₃₁	1	4	yes	—
358	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
359	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
360	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
361	#103	C₂³×C₃₁	1	4	yes	—
362	#102	C₂³×C₃₁	1	4	yes	—
363	#102	C₂³×C₃₁	1	4	yes	—
364	#102	C₂³×C₃₁	1	8	yes	—
365	#102	C₂³×C₃₁	1	4	yes	—
366	#102	C₂³×C₃₁	1	8	yes	—
367	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
368	#97	C₂³×C₃₁	1	4	yes	—
369	C₄×C₈	C₂³×C₃₁	1	4	yes	—
370	C₄×C₈	C₂³×C₃₁	1	4	yes	—
371	C₃₁	#61932	1	1	yes	characteristic, direct product factor, Sylow
372	C₂⁴	—	2	2	no	—
373	#41	—	4	2	no	—
374	#41	—	8	4	no	—
375	#41	—	8	8	no	—
376	#41	—	8	8	no	—
377	C₂×Q₈	—	2	2	no	—
378	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
379	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
380	C₂×D₄	—	2	1	no	—
381	C₂×D₄	—	2	4	no	—
382	C₂×D₄	—	2	2	no	—
383	C₂×D₄	—	2	2	no	—
384	C₂×D₄	—	2	4	no	—
385	C₂×D₄	—	2	8	no	—
386	C₂×D₄	—	2	8	no	—
387	C₂×D₄	—	2	2	no	—
388	C₂×D₄	—	2	4	no	—
389	C₂×D₄	—	4	4	no	—
390	C₂×D₄	—	8	4	no	—
391	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
392	C₂×D₄	C₂⁴×C₃₁	1	1	yes	characteristic
393	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
394	C₂²×C₄	—	2	1	no	—
395	C₂²×C₄	—	2	2	no	—
396	C₂²×C₄	—	2	4	no	—
397	C₂²×C₄	—	2	8	no	—
398	C₂²×C₄	—	2	8	no	—
399	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
400	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
401	SD₁₆	—	4	4	no	—
402	D₈	—	4	4	no	—
403	D₈	—	4	8	no	—
404	D₈	—	4	8	no	—
405	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
406	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
407	C₄⋊C₄	—	2	2	no	—
408	C₄⋊C₄	—	2	8	no	—
409	C₄⋊C₄	—	2	2	no	—
410	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
411	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
412	C₄⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
413	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
414	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
415	C₂²⋊C₄	—	2	1	no	—
416	C₂²⋊C₄	—	2	8	no	—
417	C₂²⋊C₄	—	2	2	no	—
418	C₂²⋊C₄	—	2	8	no	—
419	C₂²⋊C₄	—	2	4	no	—
420	C₂²⋊C₄	—	2	8	no	—
421	C₂²⋊C₄	—	2	4	no	—
422	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
423	C₄²	—	2	1	no	—
424	C₄²	—	2	2	no	—
425	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
426	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
427	C₄²	C₂⁴×C₃₁	1	2	yes	—
428	C₄²	C₂⁴×C₃₁	1	4	yes	—
429	C₂³	—	2	4	no	—
430	C₂³	—	2	4	no	—
431	C₂³	—	2	1	no	—
432	C₂³	—	4	4	no	—
433	C₂³	—	8	4	no	—
434	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
435	Q₈	—	2	8	no	—
436	Q₈	—	2	2	no	—
437	Q₈	—	8	2	no	—
438	D₄	—	2	4	no	—
439	D₄	—	2	4	no	—
440	D₄	—	2	8	no	—
441	D₄	—	2	2	no	—
442	D₄	—	4	4	no	—
443	D₄	—	4	2	no	—
444	D₄	—	4	8	no	—
445	D₄	—	4	4	no	—
446	D₄	—	8	4	no	—
447	D₄	—	8	2	no	—
448	D₄	—	8	8	no	—
449	D₄	—	8	8	no	—
450	D₄	—	8	4	no	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	1	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	4	no	—
456	C₂×C₄	—	4	2	no	—
457	C₂×C₄	—	4	4	no	—
458	C₂×C₄	—	4	8	no	—
459	C₂×C₄	—	8	8	no	—
460	C₂×C₄	—	8	8	no	—
461	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
462	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
463	C₂×C₄	C₂²×D₄×C₃₁	1	8	yes	—
464	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
465	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
466	C₈	—	2	4	no	—
467	C₈	—	2	4	no	—
468	C₂²	—	2	1	no	—
469	C₂²	—	2	2	no	—
470	C₂²	—	4	1	no	—
471	C₂²	—	4	4	no	—
472	C₂²	—	4	4	no	—
473	C₂²	—	4	2	no	—
474	C₂²	—	4	4	no	—
475	C₂²	—	4	4	no	—
476	C₂²	—	4	8	no	—
477	C₂²	—	16	4	no	—
478	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
479	C₄	—	2	1	no	—
480	C₄	—	2	2	no	—
481	C₄	—	2	8	no	—
482	C₄	—	4	1	no	—
483	C₄	—	4	2	no	—
484	C₄	—	4	4	no	—
485	C₄	C₃₁×#576	1	4	yes	—
486	C₄	C₃₁×#585	1	2	yes	—
487	C₂	—	4	1	no	—
488	C₂	—	8	4	no	—
489	C₂	—	8	1	no	—
490	C₂	—	8	4	no	—
491	C₂	C₃₁×#3487	1	1	yes	characteristic
492	C₂	C₂×C₃₁×#576	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₃₁×#533 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3548	C₂	semidirect
2. C₃₁×#3508	C₂	semidirect
3. C₃₁×#3399	C₂	semidirect
4. C₃₁×#3391	C₂	semidirect
5. C₃₁×#3366	C₂	semidirect
6. C₃₁×#3361	C₂	semidirect
7. C₃₁×#3220	C₂	semidirect
8. C₃₁×#3205	C₂	semidirect
9. C₃₁×#3122	C₂	semidirect
10. C₃₁×#3106	C₂	semidirect
11. C₃₁×#2986	C₂	semidirect
14. C₃₁×#554	C₂²	semidirect
16. C₃₁×#550	C₂²	semidirect
29. C₃₁×#496	C₂²	semidirect
31. C₃₁×#493	C₂²	semidirect
32. C₃₁×#490	C₂²	semidirect
33. C₃₁×#488	C₂²	semidirect
36. C₃₁×#478	C₂²	semidirect
37. C₃₁×#477	C₂²	semidirect
38. C₃₁×#476	C₂²	semidirect
39. C₃₁×#463	C₂²	semidirect
40. C₃₁×#460	C₂²	semidirect
43. C₃₁×#459	C₂²	semidirect
44. C₃₁×#459	C₂²	semidirect
45. C₃₁×#459	C₂²	semidirect
48. C₃₁×#446	C₂²	semidirect
49. C₈×Q₈×C₃₁	C₂²	semidirect
50. C₃₁×#442	C₂²	semidirect
51. C₄×SD₁₆×C₃₁	C₂²	semidirect
52. C₄×D₈×C₃₁	C₂²	semidirect
53. C₄×D₈×C₃₁	C₂²	semidirect
182. #61932	C₃₁	direct (N × H)
222. #3548	C₂×C₃₁	semidirect
223. #3508	C₂×C₃₁	semidirect
224. #3399	C₂×C₃₁	semidirect
225. #3391	C₂×C₃₁	semidirect
226. #3366	C₂×C₃₁	semidirect
227. #3361	C₂×C₃₁	semidirect
228. #3220	C₂×C₃₁	semidirect
229. #3205	C₂×C₃₁	semidirect
230. #3122	C₂×C₃₁	semidirect
231. #3106	C₂×C₃₁	semidirect
232. #2986	C₂×C₃₁	semidirect
254. #554	C₂²×C₃₁	semidirect
256. #550	C₂²×C₃₁	semidirect
269. #496	C₂²×C₃₁	semidirect
271. #493	C₂²×C₃₁	semidirect
272. #490	C₂²×C₃₁	semidirect
273. #488	C₂²×C₃₁	semidirect
276. #478	C₂²×C₃₁	semidirect
277. #477	C₂²×C₃₁	semidirect
278. #476	C₂²×C₃₁	semidirect
279. #463	C₂²×C₃₁	semidirect
280. #460	C₂²×C₃₁	semidirect
283. #459	C₂²×C₃₁	semidirect
284. #459	C₂²×C₃₁	semidirect
285. #459	C₂²×C₃₁	semidirect
288. #446	C₂²×C₃₁	semidirect
289. C₈×Q₈	C₂²×C₃₁	semidirect
290. #442	C₂²×C₃₁	semidirect
291. C₄×SD₁₆	C₂²×C₃₁	semidirect
292. C₄×D₈	C₂²×C₃₁	semidirect
293. C₄×D₈	C₂²×C₃₁	semidirect
371. C₃₁	#61932	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3548	2
C₃₁×#3508	2
C₃₁×#3399	2
C₃₁×#3391	2
C₃₁×#3366	2
C₃₁×#3361	2
C₃₁×#3220	2
C₃₁×#3205	2
C₃₁×#3122	2
C₃₁×#3106	2
C₃₁×#2986	2
#61932	31

Subgroups — C31×#61932 #21518691

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61932 #21518691