Subgroups — C₃₁×#33683 #21490442

Order 7936.

556 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3177	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#2948	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2237	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2228	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2208	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2187	C₂	1	2	yes	maximal
7	C₃₁×#2175	C₂	1	2	yes	maximal, complemented
8	C₄×C₃₁×SD₃₂	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2164	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#541	—	2	1	no	—
12	C₃₁×#518	—	2	1	no	—
13	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	—
14	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	—
15	C₃₁×#499	C₂²	1	1	yes	characteristic, complemented
16	C₃₁×#494	C₂²	1	1	yes	characteristic, complemented
17	C₃₁×#491	—	2	1	no	—
18	C₃₁×#470	—	2	1	no	—
19	C₃₁×#467	C₂²	1	2	yes	—
20	C₃₁×#442	—	2	1	no	—
21	C₃₁×#442	—	2	1	no	—
22	C₃₁×#441	—	2	1	no	—
23	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	complemented
24	C₄×D₈×C₃₁	C₂²	1	2	yes	—
25	C₄×D₈×C₃₁	C₂²	1	2	yes	—
26	C₃₁×#427	C₂²	1	1	yes	characteristic
27	C₃₁×#418	—	2	1	no	—
28	C₃₁×#405	C₂²	1	1	yes	characteristic
29	C₃₁×#367	C₂²	1	2	yes	complemented
30	C₃₁×#366	C₂²	1	2	yes	complemented
31	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	complemented
32	C₃₁×#358	C₂²	1	2	yes	complemented
33	C₃₁×#358	C₂²	1	2	yes	complemented
34	C₃₁×#357	C₂²	1	2	yes	—
35	C₃₁×#357	C₂²	1	2	yes	—
36	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
37	C₃₁×#346	C₂²	1	1	yes	characteristic, complemented
38	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
39	C₂²×D₄×C₃₁	—	2	1	no	—
40	C₂×Q₁₆×C₃₁	—	2	1	no	—
41	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
42	C₂×D₈×C₃₁	—	2	4	no	—
43	C₂×D₈×C₃₁	—	2	1	no	—
44	C₂×D₈×C₃₁	C₂³	1	2	yes	—
45	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₃₁×C₄⋊Q₈	—	2	1	no	—
48	C₃₁×#126	—	4	2	no	—
49	C₃₁×#125	—	2	1	no	—
50	C₃₁×#125	—	4	2	no	—
51	C₃₁×C₂²⋊Q₈	—	2	2	no	—
52	C₄×Q₈×C₃₁	—	2	2	no	—
53	C₄×D₄×C₃₁	—	2	2	no	—
54	C₄×D₄×C₃₁	—	2	2	no	—
55	C₄×D₄×C₃₁	—	4	2	no	—
56	C₃₁×#117	—	2	1	no	—
57	C₃₁×#117	D₄	1	1	yes	characteristic
58	C₂×C₂²⋊C₄×C₃₁	—	4	2	no	—
59	C₃₁×SD₃₂	—	2	4	no	—
60	C₃₁×SD₃₂	—	2	4	no	—
61	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
62	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
63	C₃₁×#107	C₂³	1	2	yes	—
64	C₃₁×#107	C₂³	1	2	yes	—
65	C₃₁×#106	—	2	1	no	—
66	C₃₁×#106	—	2	1	no	—
67	C₃₁×#103	—	2	2	no	—
68	C₃₁×#103	—	2	2	no	—
69	C₃₁×#102	—	2	2	no	—
70	C₃₁×#102	—	2	2	no	—
71	C₃₁×#97	D₄	1	1	yes	characteristic
72	C₃₁×#97	D₄	1	1	yes	characteristic
73	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
75	C₂⁴×C₃₁	—	4	1	no	—
76	C₂×Q₈×C₃₁	—	2	2	no	—
77	C₂×D₄×C₃₁	—	2	4	no	—
78	C₂×D₄×C₃₁	—	2	2	no	—
79	C₂×D₄×C₃₁	—	4	2	no	—
80	C₂×D₄×C₃₁	—	4	1	no	—
81	C₂²×C₄×C₃₁	—	4	2	no	—
82	C₂²×C₄×C₃₁	—	4	2	no	—
83	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
84	Q₁₆×C₃₁	—	2	2	no	—
85	Q₁₆×C₃₁	—	2	2	no	—
86	Q₁₆×C₃₁	—	4	2	no	—
87	D₈×C₃₁	—	2	4	no	—
88	D₈×C₃₁	—	2	2	no	—
89	D₈×C₃₁	#41	1	4	yes	—
90	M₁₆×C₃₁	—	4	1	no	—
91	C₂×C₈×C₃₁	—	2	2	no	—
92	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄×C₃₁	—	2	2	no	—
97	C₄⋊C₄×C₃₁	—	2	2	no	—
98	C₄⋊C₄×C₃₁	—	4	2	no	—
99	C₄⋊C₄×C₃₁	—	4	2	no	—
100	C₄⋊C₄×C₃₁	—	4	2	no	—
101	C₄⋊C₄×C₃₁	—	4	2	no	—
102	C₄⋊C₄×C₃₁	D₈	1	2	yes	—
103	C₂²⋊C₄×C₃₁	—	4	1	no	—
104	C₂²⋊C₄×C₃₁	—	4	2	no	—
105	C₂²⋊C₄×C₃₁	—	4	2	no	—
106	C₂²⋊C₄×C₃₁	—	4	2	no	—
107	C₂²⋊C₄×C₃₁	D₈	1	2	yes	—
108	C₄²×C₃₁	—	4	1	no	—
109	C₄²×C₃₁	—	4	2	no	—
110	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
112	C₁₆×C₃₁	—	2	2	no	—
113	C₁₆×C₃₁	—	2	2	no	—
114	#33683	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂³×C₃₁	—	4	4	no	—
116	C₂³×C₃₁	—	4	2	no	—
117	C₂³×C₃₁	—	16	2	no	—
118	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
119	Q₈×C₃₁	—	4	2	no	—
120	Q₈×C₃₁	—	4	2	no	—
121	D₄×C₃₁	—	2	4	no	—
122	D₄×C₃₁	—	4	4	no	—
123	D₄×C₃₁	—	4	2	no	—
124	C₂×C₄×C₃₁	—	2	1	no	—
125	C₂×C₄×C₃₁	—	2	1	no	—
126	C₂×C₄×C₃₁	—	4	2	no	—
127	C₂×C₄×C₃₁	—	4	2	no	—
128	C₂×C₄×C₃₁	—	4	2	no	—
129	C₂×C₄×C₃₁	—	16	2	no	—
130	C₂×C₄×C₃₁	—	16	2	no	—
131	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
132	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
134	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
135	C₈×C₃₁	—	2	1	no	—
136	C₈×C₃₁	—	2	1	no	—
137	C₈×C₃₁	—	2	1	no	—
138	C₈×C₃₁	C₂×#41	1	2	yes	—
139	#3177	C₂×C₃₁	1	1	yes	characteristic, complemented
140	#2948	C₂×C₃₁	1	1	yes	characteristic
141	#2237	C₂×C₃₁	1	2	yes	complemented
142	#2228	C₂×C₃₁	1	2	yes	complemented
143	#2208	C₂×C₃₁	1	2	yes	complemented
144	#2187	C₂×C₃₁	1	2	yes	—
145	#2175	C₂×C₃₁	1	2	yes	complemented
146	C₄×SD₃₂	C₂×C₃₁	1	2	yes	complemented
147	#2164	C₂×C₃₁	1	1	yes	characteristic, complemented
148	C₂²×C₃₁	—	2	1	no	—
149	C₂²×C₃₁	—	2	2	no	—
150	C₂²×C₃₁	—	4	4	no	—
151	C₂²×C₃₁	—	8	2	no	—
152	C₂²×C₃₁	—	8	2	no	—
153	C₂²×C₃₁	—	16	4	no	—
154	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
155	C₄×C₃₁	—	2	1	no	—
156	C₄×C₃₁	—	2	2	no	—
157	C₄×C₃₁	—	2	2	no	—
158	C₄×C₃₁	—	8	2	no	—
159	C₄×C₃₁	—	8	2	no	—
160	C₄×C₃₁	—	8	2	no	—
161	C₄×C₃₁	#525	1	1	yes	characteristic
162	C₄×C₃₁	#575	1	1	yes	characteristic
163	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
164	#541	—	2	1	no	—
165	#518	—	2	1	no	—
166	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	—
167	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	—
168	#499	C₂²×C₃₁	1	1	yes	characteristic, complemented
169	#494	C₂²×C₃₁	1	1	yes	characteristic, complemented
170	#491	—	2	1	no	—
171	#470	—	2	1	no	—
172	#467	C₂²×C₃₁	1	2	yes	—
173	#442	—	2	1	no	—
174	#442	—	2	1	no	—
175	#441	—	2	1	no	—
176	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	complemented
177	C₄×D₈	C₂²×C₃₁	1	2	yes	—
178	C₄×D₈	C₂²×C₃₁	1	2	yes	—
179	#427	C₂²×C₃₁	1	1	yes	characteristic
180	#418	—	2	1	no	—
181	#405	C₂²×C₃₁	1	1	yes	characteristic
182	#367	C₂²×C₃₁	1	2	yes	complemented
183	#366	C₂²×C₃₁	1	2	yes	complemented
184	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	complemented
185	#358	C₂²×C₃₁	1	2	yes	complemented
186	#358	C₂²×C₃₁	1	2	yes	complemented
187	#357	C₂²×C₃₁	1	2	yes	—
188	#357	C₂²×C₃₁	1	2	yes	—
189	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
190	#346	C₂²×C₃₁	1	1	yes	characteristic, complemented
191	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
192	C₂×C₃₁	—	4	1	no	—
193	C₂×C₃₁	—	8	4	no	—
194	C₂×C₃₁	#3100	1	1	yes	characteristic
195	C₂×C₃₁	#3416	1	2	yes	—
196	C₂²×D₄	—	2	1	no	—
197	C₂×Q₁₆	—	2	1	no	—
198	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
199	C₂×D₈	—	2	4	no	—
200	C₂×D₈	—	2	1	no	—
201	C₂×D₈	C₂³×C₃₁	1	2	yes	—
202	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
203	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
204	C₄⋊Q₈	—	2	1	no	—
205	#126	—	4	2	no	—
206	#125	—	2	1	no	—
207	#125	—	4	2	no	—
208	C₂²⋊Q₈	—	2	2	no	—
209	C₄×Q₈	—	2	2	no	—
210	C₄×D₄	—	2	2	no	—
211	C₄×D₄	—	2	2	no	—
212	C₄×D₄	—	4	2	no	—
213	#117	—	2	1	no	—
214	#117	D₄×C₃₁	1	1	yes	characteristic
215	C₂×C₂²⋊C₄	—	4	2	no	—
216	SD₃₂	—	2	4	no	—
217	SD₃₂	—	2	4	no	—
218	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
219	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
220	#107	C₂³×C₃₁	1	2	yes	—
221	#107	C₂³×C₃₁	1	2	yes	—
222	#106	—	2	1	no	—
223	#106	—	2	1	no	—
224	#103	—	2	2	no	—
225	#103	—	2	2	no	—
226	#102	—	2	2	no	—
227	#102	—	2	2	no	—
228	#97	D₄×C₃₁	1	1	yes	characteristic
229	#97	D₄×C₃₁	1	1	yes	characteristic
230	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
231	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
232	C₃₁	#33683	1	1	yes	characteristic, direct product factor, Sylow
233	C₂⁴	—	4	1	no	—
234	C₂×Q₈	—	2	2	no	—
235	C₂×D₄	—	2	4	no	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	—	4	2	no	—
238	C₂×D₄	—	4	1	no	—
239	C₂²×C₄	—	4	2	no	—
240	C₂²×C₄	—	4	2	no	—
241	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
242	Q₁₆	—	2	2	no	—
243	Q₁₆	—	2	2	no	—
244	Q₁₆	—	4	2	no	—
245	D₈	—	2	4	no	—
246	D₈	—	2	2	no	—
247	D₈	#41×C₃₁	1	4	yes	—
248	M₁₆	—	4	1	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
251	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
252	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
253	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	D₈×C₃₁	1	2	yes	—
261	C₂²⋊C₄	—	4	1	no	—
262	C₂²⋊C₄	—	4	2	no	—
263	C₂²⋊C₄	—	4	2	no	—
264	C₂²⋊C₄	—	4	2	no	—
265	C₂²⋊C₄	D₈×C₃₁	1	2	yes	—
266	C₄²	—	4	1	no	—
267	C₄²	—	4	2	no	—
268	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
269	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
270	C₁₆	—	2	2	no	—
271	C₁₆	—	2	2	no	—
272	C₂³	—	4	4	no	—
273	C₂³	—	4	2	no	—
274	C₂³	—	16	2	no	—
275	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
276	Q₈	—	4	2	no	—
277	Q₈	—	4	2	no	—
278	D₄	—	2	4	no	—
279	D₄	—	4	4	no	—
280	D₄	—	4	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	16	2	no	—
287	C₂×C₄	—	16	2	no	—
288	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
289	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
290	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
291	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₈	—	2	1	no	—
295	C₈	C₂×#41×C₃₁	1	2	yes	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	4	4	no	—
299	C₂²	—	8	2	no	—
300	C₂²	—	8	2	no	—
301	C₂²	—	16	4	no	—
302	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
303	C₄	—	2	1	no	—
304	C₄	—	2	2	no	—
305	C₄	—	2	2	no	—
306	C₄	—	8	2	no	—
307	C₄	—	8	2	no	—
308	C₄	—	8	2	no	—
309	C₄	C₃₁×#525	1	1	yes	characteristic
310	C₄	C₃₁×#575	1	1	yes	characteristic
311	C₂	—	4	1	no	—
312	C₂	—	8	4	no	—
313	C₂	C₃₁×#3100	1	1	yes	characteristic
314	C₂	C₃₁×#3416	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₂²×D₈×C₃₁	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₃₁×#117 ◁ C₃₁×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3177	C₂	semidirect
3. C₃₁×#2237	C₂	semidirect
4. C₃₁×#2228	C₂	semidirect
5. C₃₁×#2208	C₂	semidirect
7. C₃₁×#2175	C₂	semidirect
8. C₄×C₃₁×SD₃₂	C₂	semidirect
9. C₃₁×#2164	C₂	semidirect
15. C₃₁×#499	C₂²	semidirect
16. C₃₁×#494	C₂²	semidirect
23. C₄×Q₁₆×C₃₁	C₂²	semidirect
29. C₃₁×#367	C₂²	semidirect
30. C₃₁×#366	C₂²	semidirect
31. C₃₁×C₄⋊C₁₆	C₂²	semidirect
32. C₃₁×#358	C₂²	semidirect
33. C₃₁×#358	C₂²	semidirect
37. C₃₁×#346	C₂²	semidirect
38. C₄×C₁₆×C₃₁	C₂²	semidirect
114. #33683	C₃₁	direct (N × H)
139. #3177	C₂×C₃₁	semidirect
141. #2237	C₂×C₃₁	semidirect
142. #2228	C₂×C₃₁	semidirect
143. #2208	C₂×C₃₁	semidirect
145. #2175	C₂×C₃₁	semidirect
146. C₄×SD₃₂	C₂×C₃₁	semidirect
147. #2164	C₂×C₃₁	semidirect
168. #499	C₂²×C₃₁	semidirect
169. #494	C₂²×C₃₁	semidirect
176. C₄×Q₁₆	C₂²×C₃₁	semidirect
182. #367	C₂²×C₃₁	semidirect
183. #366	C₂²×C₃₁	semidirect
184. C₄⋊C₁₆	C₂²×C₃₁	semidirect
185. #358	C₂²×C₃₁	semidirect
186. #358	C₂²×C₃₁	semidirect
190. #346	C₂²×C₃₁	semidirect
191. C₄×C₁₆	C₂²×C₃₁	semidirect
232. C₃₁	#33683	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×#3177	2
C₃₁×#2948	2
C₃₁×#2237	2
C₃₁×#2228	2
C₃₁×#2208	2
C₃₁×#2187	2
C₃₁×#2175	2
C₄×C₃₁×SD₃₂	2
C₃₁×#2164	2
#33683	31

Subgroups — C31×#33683 #21490442

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33683 #21490442