Subgroups — C₃₁×#61881 #21518640

Order 7936.

2038 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3502	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#550	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3392	C₂	1	8	yes	maximal, complemented
4	C₃₁×#3362	C₂	1	8	yes	maximal, complemented
5	C₃₁×#3208	C₂	1	6	yes	maximal, complemented
6	C₃₁×#3101	C₂	1	6	yes	maximal, complemented
7	C₃₁×#2982	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#41×C₃₁	—	2	1	no	—
9	Q₈²×C₃₁	C₂²	1	4	yes	complemented
10	C₃₁×#557	—	2	4	no	—
11	C₃₁×#557	C₂²	1	4	yes	complemented
12	C₃₁×#550	—	2	4	no	—
13	C₃₁×#550	C₂²	1	8	yes	—
14	C₃₁×#544	—	2	3	no	—
15	C₃₁×#533	C₂²	1	6	yes	—
16	C₂×C₃₁×#127	C₂²	1	3	yes	—
17	C₂×C₃₁×#121	—	2	3	no	—
18	C₃₁×#519	—	2	1	no	—
19	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₃₁	C₂²	1	3	yes	—
21	C₃₁×#492	C₂²	1	12	yes	complemented
22	C₃₁×#486	C₂²	1	12	yes	complemented
23	C₃₁×#483	C₂²	1	4	yes	—
24	C₃₁×#476	C₂²	1	12	yes	complemented
25	C₃₁×#461	C₂²	1	12	yes	complemented
26	C₃₁×#460	C₂²	1	24	yes	complemented
27	C₃₁×#450	C₂²	1	12	yes	—
28	C₈×Q₈×C₃₁	C₂²	1	8	yes	complemented
29	C₄×SD₁₆×C₃₁	C₂²	1	24	yes	complemented
30	C₃₁×#431	C₂²	1	6	yes	—
31	C₂×#41×C₃₁	—	2	8	no	—
32	C₂×#41×C₃₁	—	4	8	no	—
33	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
34	C₂²×D₄×C₃₁	—	2	3	no	—
35	C₂²×D₄×C₃₁	—	2	3	no	—
36	C₂²×D₄×C₃₁	C₂³	1	3	yes	—
37	C₂³×C₄×C₃₁	—	2	3	no	—
38	C₂×SD₁₆×C₃₁	C₂³	1	24	yes	—
39	C₃₁×C₄⋊Q₈	—	2	12	no	—
40	C₃₁×C₄⋊Q₈	C₂³	1	6	yes	—
41	C₃₁×C₄⋊Q₈	C₂³	1	6	yes	—
42	C₃₁×#127	—	2	6	no	—
43	C₃₁×#127	C₂³	1	12	yes	—
44	C₃₁×#125	—	2	12	no	—
45	C₃₁×#125	—	2	6	no	—
46	C₃₁×C₂²⋊Q₈	—	2	6	no	—
47	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
48	C₃₁×#121	—	2	24	no	—
49	C₄×Q₈×C₃₁	—	2	4	no	—
50	C₄×Q₈×C₃₁	—	2	4	no	—
51	C₄×Q₈×C₃₁	—	2	6	no	—
52	C₄×Q₈×C₃₁	D₄	1	4	yes	—
53	C₄×Q₈×C₃₁	C₂³	1	12	yes	—
54	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
55	C₄×D₄×C₃₁	—	2	6	no	—
56	C₄×D₄×C₃₁	C₂³	1	12	yes	—
57	C₃₁×#117	—	2	6	no	—
58	C₂×C₄⋊C₄×C₃₁	D₄	1	3	yes	—
59	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
60	C₂×C₂²⋊C₄×C₃₁	—	2	3	no	—
61	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
62	C₃₁×#106	C₂³	1	8	yes	—
63	C₃₁×C₄⋊C₈	C₂³	1	12	yes	—
64	C₃₁×#103	C₂³	1	8	yes	—
65	C₃₁×#102	C₂³	1	24	yes	—
66	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
67	C₄×C₈×C₃₁	C₂³	1	12	yes	—
68	C₂⁴×C₃₁	—	2	3	no	—
69	#41×C₃₁	—	4	32	no	—
70	C₂×Q₈×C₃₁	—	2	6	no	—
71	C₂×Q₈×C₃₁	SD₁₆	1	8	yes	complemented
72	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
73	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
74	C₂×D₄×C₃₁	—	2	3	no	—
75	C₂×D₄×C₃₁	—	2	12	no	—
76	C₂×D₄×C₃₁	—	2	12	no	—
77	C₂×D₄×C₃₁	—	4	6	no	—
78	C₂×D₄×C₃₁	—	4	12	no	—
79	C₂×D₄×C₃₁	—	8	6	no	—
80	C₂×D₄×C₃₁	C₂⁴	1	6	yes	—
81	C₂²×C₄×C₃₁	—	2	1	no	—
82	C₂²×C₄×C₃₁	—	2	12	no	—
83	C₂²×C₄×C₃₁	—	4	12	no	—
84	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
85	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
86	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
87	SD₁₆×C₃₁	—	4	24	no	—
88	C₂×C₈×C₃₁	C₂⁴	1	8	yes	—
89	C₄⋊C₄×C₃₁	—	2	1	no	—
90	C₄⋊C₄×C₃₁	—	2	6	no	—
91	C₄⋊C₄×C₃₁	—	2	6	no	—
92	C₄⋊C₄×C₃₁	—	2	4	no	—
93	C₄⋊C₄×C₃₁	—	2	6	no	—
94	C₄⋊C₄×C₃₁	—	2	12	no	—
95	C₄⋊C₄×C₃₁	C₂×D₄	1	12	yes	—
96	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
97	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
98	C₂²⋊C₄×C₃₁	—	2	4	no	—
99	C₂²⋊C₄×C₃₁	—	2	12	no	—
100	C₄²×C₃₁	—	2	12	no	—
101	C₄²×C₃₁	C₂×D₄	1	6	yes	—
102	C₄²×C₃₁	C₂⁴	1	6	yes	—
103	#61881	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂³×C₃₁	—	2	6	no	—
105	C₂³×C₃₁	—	4	6	no	—
106	C₂³×C₃₁	—	4	6	no	—
107	C₂³×C₃₁	—	4	6	no	—
108	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
109	Q₈×C₃₁	—	2	8	no	—
110	Q₈×C₃₁	—	8	6	no	—
111	Q₈×C₃₁	C₂×SD₁₆	1	16	yes	—
112	D₄×C₃₁	—	2	12	no	—
113	D₄×C₃₁	—	4	6	no	—
114	D₄×C₃₁	—	4	24	no	—
115	D₄×C₃₁	—	8	12	no	—
116	C₂×C₄×C₃₁	—	2	2	no	—
117	C₂×C₄×C₃₁	—	2	4	no	—
118	C₂×C₄×C₃₁	—	2	4	no	—
119	C₂×C₄×C₃₁	—	2	2	no	—
120	C₂×C₄×C₃₁	—	4	3	no	—
121	C₂×C₄×C₃₁	—	4	2	no	—
122	C₂×C₄×C₃₁	—	4	24	no	—
123	C₂×C₄×C₃₁	C₂×SD₁₆	1	12	yes	—
124	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₃₁	C₂²×D₄	1	6	yes	—
126	C₂×C₄×C₃₁	C₂²×D₄	1	6	yes	—
127	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
128	C₈×C₃₁	—	2	8	no	—
129	#3502	C₂×C₃₁	1	1	yes	characteristic, complemented
130	C₂×#550	C₂×C₃₁	1	1	yes	characteristic
131	#3392	C₂×C₃₁	1	8	yes	complemented
132	#3362	C₂×C₃₁	1	8	yes	complemented
133	#3208	C₂×C₃₁	1	6	yes	complemented
134	#3101	C₂×C₃₁	1	6	yes	complemented
135	#2982	C₂×C₃₁	1	1	yes	characteristic, complemented
136	C₂²×C₃₁	—	2	1	no	—
137	C₂²×C₃₁	—	2	1	no	—
138	C₂²×C₃₁	—	4	6	no	—
139	C₂²×C₃₁	—	4	6	no	—
140	C₂²×C₃₁	—	4	6	no	—
141	C₂²×C₃₁	—	8	12	no	—
142	C₂²×C₃₁	C₂²×SD₁₆	1	2	yes	—
143	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
144	C₄×C₃₁	—	2	6	no	—
145	C₄×C₃₁	—	2	1	no	—
146	C₄×C₃₁	—	4	4	no	—
147	C₄×C₃₁	—	4	4	no	—
148	C₄×C₃₁	—	4	2	no	—
149	C₄×C₃₁	C₂²×SD₁₆	1	12	yes	—
150	C₄×C₃₁	#585	1	2	yes	—
151	C₂²×#41	—	2	1	no	—
152	Q₈²	C₂²×C₃₁	1	4	yes	complemented
153	#557	—	2	4	no	—
154	#557	C₂²×C₃₁	1	4	yes	complemented
155	#550	—	2	4	no	—
156	#550	C₂²×C₃₁	1	8	yes	—
157	#544	—	2	3	no	—
158	#533	C₂²×C₃₁	1	6	yes	—
159	C₂×#127	C₂²×C₃₁	1	3	yes	—
160	C₂×#121	—	2	3	no	—
161	#519	—	2	1	no	—
162	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
163	C₂×C₄×D₄	C₂²×C₃₁	1	3	yes	—
164	#492	C₂²×C₃₁	1	12	yes	complemented
165	#486	C₂²×C₃₁	1	12	yes	complemented
166	#483	C₂²×C₃₁	1	4	yes	—
167	#476	C₂²×C₃₁	1	12	yes	complemented
168	#461	C₂²×C₃₁	1	12	yes	complemented
169	#460	C₂²×C₃₁	1	24	yes	complemented
170	#450	C₂²×C₃₁	1	12	yes	—
171	C₈×Q₈	C₂²×C₃₁	1	8	yes	complemented
172	C₄×SD₁₆	C₂²×C₃₁	1	24	yes	complemented
173	#431	C₂²×C₃₁	1	6	yes	—
174	C₂×C₃₁	—	2	2	no	—
175	C₂×C₃₁	—	8	6	no	—
176	C₂×C₃₁	#3487	1	1	yes	characteristic
177	C₂×C₃₁	C₂³×SD₁₆	1	1	yes	characteristic
178	C₂×C₃₁	#3585	1	1	yes	characteristic
179	C₂×#41	—	2	8	no	—
180	C₂×#41	—	4	8	no	—
181	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
182	C₂²×D₄	—	2	3	no	—
183	C₂²×D₄	—	2	3	no	—
184	C₂²×D₄	C₂³×C₃₁	1	3	yes	—
185	C₂³×C₄	—	2	3	no	—
186	C₂×SD₁₆	C₂³×C₃₁	1	24	yes	—
187	C₄⋊Q₈	—	2	12	no	—
188	C₄⋊Q₈	C₂³×C₃₁	1	6	yes	—
189	C₄⋊Q₈	C₂³×C₃₁	1	6	yes	—
190	#127	—	2	6	no	—
191	#127	C₂³×C₃₁	1	12	yes	—
192	#125	—	2	12	no	—
193	#125	—	2	6	no	—
194	C₂²⋊Q₈	—	2	6	no	—
195	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
196	#121	—	2	24	no	—
197	C₄×Q₈	—	2	4	no	—
198	C₄×Q₈	—	2	4	no	—
199	C₄×Q₈	—	2	6	no	—
200	C₄×Q₈	D₄×C₃₁	1	4	yes	—
201	C₄×Q₈	C₂³×C₃₁	1	12	yes	—
202	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
203	C₄×D₄	—	2	6	no	—
204	C₄×D₄	C₂³×C₃₁	1	12	yes	—
205	#117	—	2	6	no	—
206	C₂×C₄⋊C₄	D₄×C₃₁	1	3	yes	—
207	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
208	C₂×C₂²⋊C₄	—	2	3	no	—
209	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
210	#106	C₂³×C₃₁	1	8	yes	—
211	C₄⋊C₈	C₂³×C₃₁	1	12	yes	—
212	#103	C₂³×C₃₁	1	8	yes	—
213	#102	C₂³×C₃₁	1	24	yes	—
214	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
215	C₄×C₈	C₂³×C₃₁	1	12	yes	—
216	C₃₁	#61881	1	1	yes	characteristic, direct product factor, Sylow
217	C₂⁴	—	2	3	no	—
218	#41	—	4	32	no	—
219	C₂×Q₈	—	2	6	no	—
220	C₂×Q₈	SD₁₆×C₃₁	1	8	yes	complemented
221	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
222	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
223	C₂×D₄	—	2	3	no	—
224	C₂×D₄	—	2	12	no	—
225	C₂×D₄	—	2	12	no	—
226	C₂×D₄	—	4	6	no	—
227	C₂×D₄	—	4	12	no	—
228	C₂×D₄	—	8	6	no	—
229	C₂×D₄	C₂⁴×C₃₁	1	6	yes	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	2	12	no	—
232	C₂²×C₄	—	4	12	no	—
233	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
234	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
235	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
236	SD₁₆	—	4	24	no	—
237	C₂×C₈	C₂⁴×C₃₁	1	8	yes	—
238	C₄⋊C₄	—	2	1	no	—
239	C₄⋊C₄	—	2	6	no	—
240	C₄⋊C₄	—	2	6	no	—
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	—	2	6	no	—
243	C₄⋊C₄	—	2	12	no	—
244	C₄⋊C₄	C₂×D₄×C₃₁	1	12	yes	—
245	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
246	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
247	C₂²⋊C₄	—	2	4	no	—
248	C₂²⋊C₄	—	2	12	no	—
249	C₄²	—	2	12	no	—
250	C₄²	C₂×D₄×C₃₁	1	6	yes	—
251	C₄²	C₂⁴×C₃₁	1	6	yes	—
252	C₂³	—	2	6	no	—
253	C₂³	—	4	6	no	—
254	C₂³	—	4	6	no	—
255	C₂³	—	4	6	no	—
256	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
257	Q₈	—	2	8	no	—
258	Q₈	—	8	6	no	—
259	Q₈	C₂×SD₁₆×C₃₁	1	16	yes	—
260	D₄	—	2	12	no	—
261	D₄	—	4	6	no	—
262	D₄	—	4	24	no	—
263	D₄	—	8	12	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	4	3	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	24	no	—
271	C₂×C₄	C₂×SD₁₆×C₃₁	1	12	yes	—
272	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
273	C₂×C₄	C₂²×D₄×C₃₁	1	6	yes	—
274	C₂×C₄	C₂²×D₄×C₃₁	1	6	yes	—
275	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
276	C₈	—	2	8	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	4	6	no	—
280	C₂²	—	4	6	no	—
281	C₂²	—	4	6	no	—
282	C₂²	—	8	12	no	—
283	C₂²	C₂²×SD₁₆×C₃₁	1	2	yes	—
284	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
285	C₄	—	2	1	no	—
286	C₄	—	2	6	no	—
287	C₄	—	4	4	no	—
288	C₄	—	4	2	no	—
289	C₄	—	4	4	no	—
290	C₄	C₂²×SD₁₆×C₃₁	1	12	yes	—
291	C₄	C₃₁×#585	1	2	yes	—
292	C₂	—	2	2	no	—
293	C₂	—	8	6	no	—
294	C₂	C₃₁×#3487	1	1	yes	characteristic
295	C₂	C₂³×SD₁₆×C₃₁	1	1	yes	characteristic
296	C₂	C₃₁×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₂×C₃₁×#550	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3502	C₂	semidirect
3. C₃₁×#3392	C₂	semidirect
4. C₃₁×#3362	C₂	semidirect
5. C₃₁×#3208	C₂	semidirect
6. C₃₁×#3101	C₂	semidirect
7. C₃₁×#2982	C₂	semidirect
9. Q₈²×C₃₁	C₂²	semidirect
11. C₃₁×#557	C₂²	semidirect
21. C₃₁×#492	C₂²	semidirect
22. C₃₁×#486	C₂²	semidirect
24. C₃₁×#476	C₂²	semidirect
25. C₃₁×#461	C₂²	semidirect
26. C₃₁×#460	C₂²	semidirect
28. C₈×Q₈×C₃₁	C₂²	semidirect
29. C₄×SD₁₆×C₃₁	C₂²	semidirect
71. C₂×Q₈×C₃₁	SD₁₆	semidirect
103. #61881	C₃₁	direct (N × H)
129. #3502	C₂×C₃₁	semidirect
131. #3392	C₂×C₃₁	semidirect
132. #3362	C₂×C₃₁	semidirect
133. #3208	C₂×C₃₁	semidirect
134. #3101	C₂×C₃₁	semidirect
135. #2982	C₂×C₃₁	semidirect
152. Q₈²	C₂²×C₃₁	semidirect
154. #557	C₂²×C₃₁	semidirect
164. #492	C₂²×C₃₁	semidirect
165. #486	C₂²×C₃₁	semidirect
167. #476	C₂²×C₃₁	semidirect
168. #461	C₂²×C₃₁	semidirect
169. #460	C₂²×C₃₁	semidirect
171. C₈×Q₈	C₂²×C₃₁	semidirect
172. C₄×SD₁₆	C₂²×C₃₁	semidirect
216. C₃₁	#61881	direct (N × H)
220. C₂×Q₈	SD₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3502	2
C₂×C₃₁×#550	2
C₃₁×#3392	2
C₃₁×#3362	2
C₃₁×#3208	2
C₃₁×#3101	2
C₃₁×#2982	2
#61881	31

Subgroups — C31×#61881 #21518640

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61881 #21518640