Subgroups — C₃₁×#33817 #21490576

Order 7936.

534 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3169	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3168	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2228	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2225	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2224	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2220	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2218	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2210	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2207	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#2206	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#2150	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#532	—	2	1	no	—
13	C₃₁×#529	—	2	1	no	—
14	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	—
15	C₂×C₃₁×D₁₆	C₂²	1	2	yes	—
16	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
17	C₃₁×#500	C₂²	1	1	yes	characteristic
18	C₃₁×#499	C₂²	1	1	yes	characteristic
19	C₃₁×#495	C₂²	1	1	yes	characteristic
20	C₃₁×#493	C₂²	1	1	yes	characteristic
21	C₃₁×#488	—	2	1	no	—
22	C₃₁×#486	—	2	1	no	—
23	C₃₁×#467	C₂²	1	1	yes	characteristic
24	C₃₁×#466	—	2	1	no	—
25	C₃₁×#466	C₂²	1	1	yes	characteristic
26	C₃₁×#466	C₂²	1	2	yes	—
27	C₃₁×#465	—	2	1	no	—
28	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
29	C₄×SD₁₆×C₃₁	—	2	1	no	—
30	C₄×D₈×C₃₁	—	2	1	no	—
31	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
32	C₄×D₈×C₃₁	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
34	C₃₁×#367	C₂²	1	2	yes	—
35	C₃₁×#366	C₂²	1	2	yes	—
36	C₃₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
37	C₃₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
38	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
39	C₃₁×#358	C₂²	1	2	yes	—
40	C₃₁×#357	C₂²	1	2	yes	—
41	C₃₁×#357	C₂²	1	2	yes	—
42	C₃₁×#357	C₂²	1	2	yes	—
43	C₂×#41×C₃₁	—	4	1	no	—
44	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
45	C₂×SD₁₆×C₃₁	—	2	1	no	—
46	C₂×D₈×C₃₁	—	2	1	no	—
47	C₂×D₈×C₃₁	C₂³	1	2	yes	—
48	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
51	C₃₁×C₄⋊Q₈	—	2	1	no	—
52	C₃₁×#127	—	2	1	no	—
53	C₃₁×#126	—	4	1	no	—
54	C₃₁×#125	—	2	1	no	—
55	C₃₁×#124	—	2	1	no	—
56	C₃₁×#124	—	4	1	no	—
57	C₃₁×#123	—	4	1	no	—
58	C₃₁×C₂²⋊Q₈	—	2	1	no	—
59	C₃₁×#121	—	2	1	no	—
60	C₃₁×#121	—	2	2	no	—
61	C₃₁×#121	—	4	1	no	—
62	C₄×Q₈×C₃₁	—	2	1	no	—
63	C₄×D₄×C₃₁	—	2	2	no	—
64	C₄×D₄×C₃₁	—	2	1	no	—
65	C₄×D₄×C₃₁	—	4	1	no	—
66	C₄×D₄×C₃₁	—	4	1	no	—
67	C₃₁×#117	—	4	1	no	—
68	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
69	C₃₁×SD₃₂	—	4	2	no	—
70	C₃₁×D₁₆	—	4	2	no	—
71	C₃₁×M₃₂	—	4	2	no	—
72	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
73	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
74	C₃₁×#107	—	2	1	no	—
75	C₃₁×#107	C₂³	1	2	yes	—
76	C₃₁×#107	C₂³	1	1	yes	characteristic
77	C₃₁×#107	C₂³	1	1	yes	characteristic
78	C₃₁×#106	—	2	1	no	—
79	C₃₁×#103	—	2	1	no	—
80	C₃₁×#103	—	2	1	no	—
81	C₃₁×#102	—	2	2	no	—
82	C₃₁×#102	—	2	1	no	—
83	C₃₁×#102	—	2	2	no	—
84	C₃₁×#102	—	2	1	no	—
85	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
86	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
87	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
88	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
89	#41×C₃₁	—	8	4	no	—
90	C₂×Q₈×C₃₁	—	2	1	no	—
91	C₂×Q₈×C₃₁	—	4	1	no	—
92	C₂×D₄×C₃₁	—	2	1	no	—
93	C₂×D₄×C₃₁	—	2	2	no	—
94	C₂×D₄×C₃₁	—	4	1	no	—
95	C₂×D₄×C₃₁	—	4	2	no	—
96	C₂×D₄×C₃₁	—	4	1	no	—
97	C₂×D₄×C₃₁	—	4	1	no	—
98	C₂²×C₄×C₃₁	—	4	1	no	—
99	C₂²×C₄×C₃₁	—	4	2	no	—
100	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
101	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
102	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
103	Q₁₆×C₃₁	—	2	1	no	—
104	Q₁₆×C₃₁	—	2	1	no	—
105	SD₁₆×C₃₁	—	4	2	no	—
106	D₈×C₃₁	—	2	1	no	—
107	D₈×C₃₁	—	2	2	no	—
108	D₈×C₃₁	—	2	2	no	—
109	D₈×C₃₁	—	2	1	no	—
110	D₈×C₃₁	—	4	2	no	—
111	C₂×C₈×C₃₁	—	2	1	no	—
112	C₂×C₈×C₃₁	—	2	1	no	—
113	C₂×C₈×C₃₁	—	4	1	no	—
114	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
118	C₄⋊C₄×C₃₁	—	2	2	no	—
119	C₄⋊C₄×C₃₁	—	2	1	no	—
120	C₄⋊C₄×C₃₁	—	2	1	no	—
121	C₄⋊C₄×C₃₁	—	4	1	no	—
122	C₄⋊C₄×C₃₁	—	4	1	no	—
123	C₄⋊C₄×C₃₁	—	4	1	no	—
124	C₄⋊C₄×C₃₁	—	4	1	no	—
125	C₄⋊C₄×C₃₁	—	4	1	no	—
126	C₂²⋊C₄×C₃₁	—	4	1	no	—
127	C₂²⋊C₄×C₃₁	—	4	1	no	—
128	C₂²⋊C₄×C₃₁	—	4	1	no	—
129	C₂²⋊C₄×C₃₁	—	4	2	no	—
130	C₂²⋊C₄×C₃₁	—	4	1	no	—
131	C₂²⋊C₄×C₃₁	—	4	2	no	—
132	C₂²⋊C₄×C₃₁	—	4	1	no	—
133	C₄²×C₃₁	—	4	1	no	—
134	C₄²×C₃₁	D₈	1	1	yes	characteristic
135	C₄²×C₃₁	D₈	1	1	yes	characteristic
136	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
137	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
138	C₁₆×C₃₁	—	2	2	no	—
139	C₁₆×C₃₁	—	2	2	no	—
140	#33817	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂³×C₃₁	—	4	1	no	—
142	C₂³×C₃₁	—	4	2	no	—
143	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
144	Q₈×C₃₁	—	4	1	no	—
145	Q₈×C₃₁	—	4	1	no	—
146	Q₈×C₃₁	—	8	2	no	—
147	D₄×C₃₁	—	4	1	no	—
148	D₄×C₃₁	—	4	2	no	—
149	D₄×C₃₁	—	4	2	no	—
150	D₄×C₃₁	—	4	1	no	—
151	D₄×C₃₁	—	8	2	no	—
152	D₄×C₃₁	—	8	2	no	—
153	D₄×C₃₁	—	8	4	no	—
154	D₄×C₃₁	—	8	2	no	—
155	C₂×C₄×C₃₁	—	2	1	no	—
156	C₂×C₄×C₃₁	—	2	1	no	—
157	C₂×C₄×C₃₁	—	2	2	no	—
158	C₂×C₄×C₃₁	—	4	1	no	—
159	C₂×C₄×C₃₁	—	4	1	no	—
160	C₂×C₄×C₃₁	—	4	1	no	—
161	C₂×C₄×C₃₁	—	4	1	no	—
162	C₂×C₄×C₃₁	—	4	2	no	—
163	C₂×C₄×C₃₁	—	8	4	no	—
164	C₂×C₄×C₃₁	—	8	2	no	—
165	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
166	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
167	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
168	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
169	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
170	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
171	C₈×C₃₁	—	2	1	no	—
172	C₈×C₃₁	—	2	1	no	—
173	C₈×C₃₁	—	2	1	no	—
174	C₈×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
175	C₈×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
176	#3169	C₂×C₃₁	1	1	yes	characteristic, complemented
177	#3168	C₂×C₃₁	1	1	yes	characteristic, complemented
178	#2228	C₂×C₃₁	1	2	yes	complemented
179	#2225	C₂×C₃₁	1	1	yes	characteristic, complemented
180	#2224	C₂×C₃₁	1	1	yes	characteristic, complemented
181	#2220	C₂×C₃₁	1	2	yes	complemented
182	#2218	C₂×C₃₁	1	2	yes	complemented
183	#2210	C₂×C₃₁	1	2	yes	complemented
184	#2207	C₂×C₃₁	1	1	yes	characteristic, complemented
185	#2206	C₂×C₃₁	1	1	yes	characteristic, complemented
186	#2150	C₂×C₃₁	1	1	yes	characteristic, complemented
187	C₂²×C₃₁	—	2	1	no	—
188	C₂²×C₃₁	—	2	1	no	—
189	C₂²×C₃₁	—	2	1	no	—
190	C₂²×C₃₁	—	8	2	no	—
191	C₂²×C₃₁	—	8	1	no	—
192	C₂²×C₃₁	—	8	2	no	—
193	C₂²×C₃₁	—	8	1	no	—
194	C₂²×C₃₁	—	8	2	no	—
195	C₂²×C₃₁	—	8	1	no	—
196	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
197	C₄×C₃₁	—	2	1	no	—
198	C₄×C₃₁	—	2	1	no	—
199	C₄×C₃₁	—	2	1	no	—
200	C₄×C₃₁	—	2	1	no	—
201	C₄×C₃₁	—	8	1	no	—
202	C₄×C₃₁	—	8	2	no	—
203	C₄×C₃₁	—	8	1	no	—
204	C₄×C₃₁	—	8	1	no	—
205	C₄×C₃₁	#509	1	2	yes	—
206	C₄×C₃₁	#537	1	1	yes	characteristic
207	C₄×C₃₁	#575	1	1	yes	characteristic
208	#532	—	2	1	no	—
209	#529	—	2	1	no	—
210	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	—
211	C₂×D₁₆	C₂²×C₃₁	1	2	yes	—
212	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
213	#500	C₂²×C₃₁	1	1	yes	characteristic
214	#499	C₂²×C₃₁	1	1	yes	characteristic
215	#495	C₂²×C₃₁	1	1	yes	characteristic
216	#493	C₂²×C₃₁	1	1	yes	characteristic
217	#488	—	2	1	no	—
218	#486	—	2	1	no	—
219	#467	C₂²×C₃₁	1	1	yes	characteristic
220	#466	—	2	1	no	—
221	#466	C₂²×C₃₁	1	1	yes	characteristic
222	#466	C₂²×C₃₁	1	2	yes	—
223	#465	—	2	1	no	—
224	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
225	C₄×SD₁₆	—	2	1	no	—
226	C₄×D₈	—	2	1	no	—
227	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
228	C₄×D₈	C₂²×C₃₁	1	2	yes	—
229	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
230	#367	C₂²×C₃₁	1	2	yes	—
231	#366	C₂²×C₃₁	1	2	yes	—
232	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
233	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
234	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
235	#358	C₂²×C₃₁	1	2	yes	—
236	#357	C₂²×C₃₁	1	2	yes	—
237	#357	C₂²×C₃₁	1	2	yes	—
238	#357	C₂²×C₃₁	1	2	yes	—
239	C₂×C₃₁	—	4	1	no	—
240	C₂×C₃₁	—	16	2	no	—
241	C₂×C₃₁	—	16	1	no	—
242	C₂×C₃₁	#3245	1	1	yes	characteristic
243	C₂×C₃₁	C₂×#509	1	1	yes	characteristic
244	C₂×C₃₁	#3414	1	1	yes	characteristic
245	C₂×#41	—	4	1	no	—
246	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
247	C₂×SD₁₆	—	2	1	no	—
248	C₂×D₈	—	2	1	no	—
249	C₂×D₈	C₂³×C₃₁	1	2	yes	—
250	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
251	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
252	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
253	C₄⋊Q₈	—	2	1	no	—
254	#127	—	2	1	no	—
255	#126	—	4	1	no	—
256	#125	—	2	1	no	—
257	#124	—	2	1	no	—
258	#124	—	4	1	no	—
259	#123	—	4	1	no	—
260	C₂²⋊Q₈	—	2	1	no	—
261	#121	—	2	1	no	—
262	#121	—	2	2	no	—
263	#121	—	4	1	no	—
264	C₄×Q₈	—	2	1	no	—
265	C₄×D₄	—	2	1	no	—
266	C₄×D₄	—	2	2	no	—
267	C₄×D₄	—	4	1	no	—
268	C₄×D₄	—	4	1	no	—
269	#117	—	4	1	no	—
270	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
271	SD₃₂	—	4	2	no	—
272	D₁₆	—	4	2	no	—
273	M₃₂	—	4	2	no	—
274	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
275	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
276	#107	—	2	1	no	—
277	#107	C₂³×C₃₁	1	1	yes	characteristic
278	#107	C₂³×C₃₁	1	1	yes	characteristic
279	#107	C₂³×C₃₁	1	2	yes	—
280	#106	—	2	1	no	—
281	#103	—	2	1	no	—
282	#103	—	2	1	no	—
283	#102	—	2	2	no	—
284	#102	—	2	1	no	—
285	#102	—	2	2	no	—
286	#102	—	2	1	no	—
287	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
288	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
289	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
290	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
291	C₃₁	#33817	1	1	yes	characteristic, direct product factor, Sylow
292	#41	—	8	4	no	—
293	C₂×Q₈	—	2	1	no	—
294	C₂×Q₈	—	4	1	no	—
295	C₂×D₄	—	2	1	no	—
296	C₂×D₄	—	2	2	no	—
297	C₂×D₄	—	4	1	no	—
298	C₂×D₄	—	4	2	no	—
299	C₂×D₄	—	4	1	no	—
300	C₂×D₄	—	4	1	no	—
301	C₂²×C₄	—	4	1	no	—
302	C₂²×C₄	—	4	2	no	—
303	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
304	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
305	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
306	Q₁₆	—	2	1	no	—
307	Q₁₆	—	2	1	no	—
308	SD₁₆	—	4	2	no	—
309	D₈	—	2	1	no	—
310	D₈	—	2	2	no	—
311	D₈	—	2	2	no	—
312	D₈	—	2	1	no	—
313	D₈	—	4	2	no	—
314	C₂×C₈	—	2	1	no	—
315	C₂×C₈	—	2	1	no	—
316	C₂×C₈	—	4	1	no	—
317	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
318	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
319	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
320	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	2	1	no	—
323	C₄⋊C₄	—	2	1	no	—
324	C₄⋊C₄	—	4	1	no	—
325	C₄⋊C₄	—	4	1	no	—
326	C₄⋊C₄	—	4	1	no	—
327	C₄⋊C₄	—	4	1	no	—
328	C₄⋊C₄	—	4	1	no	—
329	C₂²⋊C₄	—	4	1	no	—
330	C₂²⋊C₄	—	4	1	no	—
331	C₂²⋊C₄	—	4	1	no	—
332	C₂²⋊C₄	—	4	2	no	—
333	C₂²⋊C₄	—	4	1	no	—
334	C₂²⋊C₄	—	4	2	no	—
335	C₂²⋊C₄	—	4	1	no	—
336	C₄²	—	4	1	no	—
337	C₄²	D₈×C₃₁	1	1	yes	characteristic
338	C₄²	D₈×C₃₁	1	1	yes	characteristic
339	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
340	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
341	C₁₆	—	2	2	no	—
342	C₁₆	—	2	2	no	—
343	C₂³	—	4	1	no	—
344	C₂³	—	4	2	no	—
345	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
346	Q₈	—	4	1	no	—
347	Q₈	—	4	1	no	—
348	Q₈	—	8	2	no	—
349	D₄	—	4	1	no	—
350	D₄	—	4	2	no	—
351	D₄	—	4	2	no	—
352	D₄	—	4	1	no	—
353	D₄	—	8	2	no	—
354	D₄	—	8	2	no	—
355	D₄	—	8	4	no	—
356	D₄	—	8	2	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	4	1	no	—
361	C₂×C₄	—	4	1	no	—
362	C₂×C₄	—	4	1	no	—
363	C₂×C₄	—	4	1	no	—
364	C₂×C₄	—	4	2	no	—
365	C₂×C₄	—	8	4	no	—
366	C₂×C₄	—	8	2	no	—
367	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
368	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
369	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
370	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
371	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
372	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
373	C₈	—	2	1	no	—
374	C₈	—	2	1	no	—
375	C₈	—	2	1	no	—
376	C₈	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
377	C₈	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
378	C₂²	—	2	1	no	—
379	C₂²	—	2	1	no	—
380	C₂²	—	2	1	no	—
381	C₂²	—	8	2	no	—
382	C₂²	—	8	1	no	—
383	C₂²	—	8	2	no	—
384	C₂²	—	8	1	no	—
385	C₂²	—	8	2	no	—
386	C₂²	—	8	1	no	—
387	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
388	C₄	—	2	1	no	—
389	C₄	—	2	1	no	—
390	C₄	—	2	1	no	—
391	C₄	—	2	1	no	—
392	C₄	—	8	2	no	—
393	C₄	—	8	1	no	—
394	C₄	—	8	1	no	—
395	C₄	—	8	1	no	—
396	C₄	C₃₁×#509	1	2	yes	—
397	C₄	C₃₁×#537	1	1	yes	characteristic
398	C₄	C₃₁×#575	1	1	yes	characteristic
399	C₂	—	4	1	no	—
400	C₂	—	16	2	no	—
401	C₂	—	16	1	no	—
402	C₂	C₃₁×#3245	1	1	yes	characteristic
403	C₂	C₂×C₃₁×#509	1	1	yes	characteristic
404	C₂	C₃₁×#3414	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3169	C₂	semidirect
2. C₃₁×#3168	C₂	semidirect
3. C₃₁×#2228	C₂	semidirect
4. C₃₁×#2225	C₂	semidirect
5. C₃₁×#2224	C₂	semidirect
6. C₃₁×#2220	C₂	semidirect
7. C₃₁×#2218	C₂	semidirect
8. C₃₁×#2210	C₂	semidirect
9. C₃₁×#2207	C₂	semidirect
10. C₃₁×#2206	C₂	semidirect
11. C₃₁×#2150	C₂	semidirect
140. #33817	C₃₁	direct (N × H)
176. #3169	C₂×C₃₁	semidirect
177. #3168	C₂×C₃₁	semidirect
178. #2228	C₂×C₃₁	semidirect
179. #2225	C₂×C₃₁	semidirect
180. #2224	C₂×C₃₁	semidirect
181. #2220	C₂×C₃₁	semidirect
182. #2218	C₂×C₃₁	semidirect
183. #2210	C₂×C₃₁	semidirect
184. #2207	C₂×C₃₁	semidirect
185. #2206	C₂×C₃₁	semidirect
186. #2150	C₂×C₃₁	semidirect
291. C₃₁	#33817	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×#3169	2
C₃₁×#3168	2
C₃₁×#2228	2
C₃₁×#2225	2
C₃₁×#2224	2
C₃₁×#2220	2
C₃₁×#2218	2
C₃₁×#2210	2
C₃₁×#2207	2
C₃₁×#2206	2
C₃₁×#2150	2
#33817	31

Subgroups — C31×#33817 #21490576

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33817 #21490576