Subgroups — C₃₁×#19717 #21476476

Order 7936.

720 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3231	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3208	C₂	1	2	yes	maximal
3	C₃₁×#3185	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3138	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1720	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1719	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1714	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1708	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1453	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#533	C₂²	1	1	yes	characteristic
11	C₃₁×#533	C₂²	1	2	yes	—
12	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
13	C₃₁×#500	C₂²	1	2	yes	complemented
14	C₃₁×#498	C₂²	1	1	yes	characteristic, complemented
15	C₃₁×#498	C₂²	1	2	yes	complemented
16	C₃₁×#494	C₂²	1	1	yes	characteristic, complemented
17	C₃₁×#492	C₂²	1	2	yes	—
18	C₃₁×#492	C₂²	1	2	yes	—
19	C₃₁×#490	—	2	2	no	—
20	C₃₁×#487	C₂²	1	2	yes	complemented
21	C₃₁×#487	C₂²	1	2	yes	complemented
22	C₃₁×#487	C₂²	1	2	yes	complemented
23	C₃₁×#486	C₂²	1	2	yes	—
24	C₃₁×#486	C₂²	1	2	yes	—
25	C₃₁×#486	C₂²	1	2	yes	—
26	C₃₁×#486	C₂²	1	2	yes	—
27	C₃₁×#485	—	2	4	no	—
28	C₃₁×#479	—	2	4	no	—
29	C₃₁×#470	—	2	2	no	—
30	C₃₁×#460	—	2	4	no	—
31	C₃₁×#450	—	2	4	no	—
32	C₃₁×#441	—	2	2	no	—
33	C₃₁×#431	C₂²	1	2	yes	—
34	C₃₁×#418	—	2	2	no	—
35	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
36	C₃₁×#322	C₂²	1	2	yes	complemented
37	C₃₁×#322	C₂²	1	1	yes	characteristic, complemented
38	C₈²×C₃₁	C₂²	1	1	yes	characteristic, complemented
39	C₂²×D₄×C₃₁	—	2	1	no	—
40	C₂²×D₄×C₃₁	—	2	2	no	—
41	C₂×Q₁₆×C₃₁	—	2	2	no	—
42	C₂×SD₁₆×C₃₁	—	2	4	no	—
43	C₂×SD₁₆×C₃₁	—	2	4	no	—
44	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
45	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
46	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
47	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
48	C₃₁×#127	—	2	2	no	—
49	C₃₁×#127	—	2	1	no	—
50	C₃₁×#127	C₂³	1	2	yes	—
51	C₃₁×#125	—	2	2	no	—
52	C₃₁×#125	—	2	1	no	—
53	C₃₁×C₂²⋊Q₈	—	2	2	no	—
54	C₃₁×C₂²⋊Q₈	—	2	2	no	—
55	C₃₁×C₂²⋊Q₈	—	2	2	no	—
56	C₄×Q₈×C₃₁	—	2	2	no	—
57	C₄×Q₈×C₃₁	—	2	2	no	—
58	C₄×Q₈×C₃₁	—	2	2	no	—
59	C₃₁×#117	—	2	1	no	—
60	C₃₁×#117	—	2	2	no	—
61	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
62	C₃₁×#107	—	2	4	no	—
63	C₃₁×#106	—	2	2	no	—
64	C₃₁×#106	—	2	4	no	—
65	C₃₁×C₄⋊C₈	D₄	1	4	yes	—
66	C₃₁×#103	—	2	4	no	—
67	C₃₁×#103	—	2	4	no	—
68	C₃₁×#103	—	2	4	no	—
69	C₃₁×#102	—	2	4	no	—
70	C₃₁×#102	—	2	4	no	—
71	C₃₁×#102	—	2	4	no	—
72	C₃₁×#102	—	2	4	no	—
73	C₃₁×C₂²⋊C₈	D₄	1	4	yes	—
74	C₃₁×#97	D₄	1	2	yes	—
75	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
76	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₃₁	C₂³	1	2	yes	—
78	C₄×C₈×C₃₁	C₂³	1	2	yes	—
79	C₂⁴×C₃₁	—	4	1	no	—
80	C₂×Q₈×C₃₁	—	2	2	no	—
81	C₂×Q₈×C₃₁	—	2	2	no	—
82	C₂×Q₈×C₃₁	—	2	2	no	—
83	C₂×D₄×C₃₁	—	2	2	no	—
84	C₂×D₄×C₃₁	—	2	2	no	—
85	C₂×D₄×C₃₁	—	2	2	no	—
86	C₂×D₄×C₃₁	—	4	4	no	—
87	C₂×D₄×C₃₁	—	4	1	no	—
88	C₂×D₄×C₃₁	—	4	2	no	—
89	C₂×D₄×C₃₁	—	8	4	no	—
90	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
92	Q₁₆×C₃₁	—	4	4	no	—
93	SD₁₆×C₃₁	—	8	4	no	—
94	SD₁₆×C₃₁	—	8	4	no	—
95	M₁₆×C₃₁	—	4	2	no	—
96	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
99	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
100	C₄⋊C₄×C₃₁	—	2	2	no	—
101	C₄⋊C₄×C₃₁	—	2	2	no	—
102	C₄⋊C₄×C₃₁	—	2	2	no	—
103	C₄⋊C₄×C₃₁	—	2	2	no	—
104	C₄⋊C₄×C₃₁	—	2	2	no	—
105	C₄⋊C₄×C₃₁	—	2	2	no	—
106	C₄⋊C₄×C₃₁	—	4	2	no	—
107	C₄⋊C₄×C₃₁	—	4	2	no	—
108	C₄⋊C₄×C₃₁	—	4	2	no	—
109	C₂²⋊C₄×C₃₁	—	4	1	no	—
110	C₂²⋊C₄×C₃₁	—	4	2	no	—
111	C₄²×C₃₁	—	4	1	no	—
112	C₄²×C₃₁	—	4	2	no	—
113	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₃₁	C₂×D₄	1	2	yes	—
115	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
116	#19717	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂³×C₃₁	—	4	2	no	—
118	C₂³×C₃₁	—	8	2	no	—
119	C₂³×C₃₁	—	8	4	no	—
120	C₂³×C₃₁	#127	1	1	yes	characteristic
121	Q₈×C₃₁	—	4	4	no	—
122	Q₈×C₃₁	—	4	4	no	—
123	Q₈×C₃₁	—	4	4	no	—
124	D₄×C₃₁	—	4	4	no	—
125	D₄×C₃₁	—	4	4	no	—
126	D₄×C₃₁	—	4	4	no	—
127	D₄×C₃₁	—	8	2	no	—
128	D₄×C₃₁	—	8	4	no	—
129	C₂×C₄×C₃₁	—	2	2	no	—
130	C₂×C₄×C₃₁	—	2	4	no	—
131	C₂×C₄×C₃₁	—	4	2	no	—
132	C₂×C₄×C₃₁	—	4	2	no	—
133	C₂×C₄×C₃₁	—	4	2	no	—
134	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
135	C₂×C₄×C₃₁	#127	1	2	yes	—
136	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
138	C₈×C₃₁	—	2	2	no	—
139	C₈×C₃₁	—	2	2	no	—
140	C₈×C₃₁	—	2	4	no	—
141	C₈×C₃₁	—	2	4	no	—
142	#3231	C₂×C₃₁	1	2	yes	complemented
143	#3208	C₂×C₃₁	1	2	yes	—
144	#3185	C₂×C₃₁	1	1	yes	characteristic, complemented
145	#3138	C₂×C₃₁	1	1	yes	characteristic
146	#1720	C₂×C₃₁	1	2	yes	complemented
147	#1719	C₂×C₃₁	1	2	yes	complemented
148	#1714	C₂×C₃₁	1	2	yes	complemented
149	#1708	C₂×C₃₁	1	2	yes	complemented
150	#1453	C₂×C₃₁	1	1	yes	characteristic, complemented
151	C₂²×C₃₁	—	2	1	no	—
152	C₂²×C₃₁	—	2	2	no	—
153	C₂²×C₃₁	—	8	2	no	—
154	C₂²×C₃₁	—	8	2	no	—
155	C₂²×C₃₁	—	8	2	no	—
156	C₂²×C₃₁	—	16	4	no	—
157	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
158	C₄×C₃₁	—	2	1	no	—
159	C₄×C₃₁	—	2	2	no	—
160	C₄×C₃₁	—	8	2	no	—
161	C₄×C₃₁	—	8	2	no	—
162	C₄×C₃₁	—	8	2	no	—
163	C₄×C₃₁	#575	1	2	yes	—
164	C₄×C₃₁	#577	1	4	yes	—
165	#533	C₂²×C₃₁	1	1	yes	characteristic
166	#533	C₂²×C₃₁	1	2	yes	—
167	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
168	#500	C₂²×C₃₁	1	2	yes	complemented
169	#498	C₂²×C₃₁	1	1	yes	characteristic, complemented
170	#498	C₂²×C₃₁	1	2	yes	complemented
171	#494	C₂²×C₃₁	1	1	yes	characteristic, complemented
172	#492	C₂²×C₃₁	1	2	yes	—
173	#492	C₂²×C₃₁	1	2	yes	—
174	#490	—	2	2	no	—
175	#487	C₂²×C₃₁	1	2	yes	complemented
176	#487	C₂²×C₃₁	1	2	yes	complemented
177	#487	C₂²×C₃₁	1	2	yes	complemented
178	#486	C₂²×C₃₁	1	2	yes	—
179	#486	C₂²×C₃₁	1	2	yes	—
180	#486	C₂²×C₃₁	1	2	yes	—
181	#486	C₂²×C₃₁	1	2	yes	—
182	#485	—	2	4	no	—
183	#479	—	2	4	no	—
184	#470	—	2	2	no	—
185	#460	—	2	4	no	—
186	#450	—	2	4	no	—
187	#441	—	2	2	no	—
188	#431	C₂²×C₃₁	1	2	yes	—
189	#418	—	2	2	no	—
190	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
191	#322	C₂²×C₃₁	1	2	yes	complemented
192	#322	C₂²×C₃₁	1	1	yes	characteristic, complemented
193	C₈²	C₂²×C₃₁	1	1	yes	characteristic, complemented
194	C₂×C₃₁	—	4	1	no	—
195	C₂×C₃₁	—	16	2	no	—
196	C₂×C₃₁	#3153	1	1	yes	characteristic
197	C₂×C₃₁	#3154	1	2	yes	—
198	C₂²×D₄	—	2	1	no	—
199	C₂²×D₄	—	2	2	no	—
200	C₂×Q₁₆	—	2	2	no	—
201	C₂×SD₁₆	—	2	4	no	—
202	C₂×SD₁₆	—	2	4	no	—
203	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
204	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
205	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
206	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
207	#127	—	2	2	no	—
208	#127	—	2	1	no	—
209	#127	C₂³×C₃₁	1	2	yes	—
210	#125	—	2	2	no	—
211	#125	—	2	1	no	—
212	C₂²⋊Q₈	—	2	2	no	—
213	C₂²⋊Q₈	—	2	2	no	—
214	C₂²⋊Q₈	—	2	2	no	—
215	C₄×Q₈	—	2	2	no	—
216	C₄×Q₈	—	2	2	no	—
217	C₄×Q₈	—	2	2	no	—
218	#117	—	2	1	no	—
219	#117	—	2	2	no	—
220	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
221	#107	—	2	4	no	—
222	#106	—	2	2	no	—
223	#106	—	2	4	no	—
224	C₄⋊C₈	D₄×C₃₁	1	4	yes	—
225	#103	—	2	4	no	—
226	#103	—	2	4	no	—
227	#103	—	2	4	no	—
228	#102	—	2	4	no	—
229	#102	—	2	4	no	—
230	#102	—	2	4	no	—
231	#102	—	2	4	no	—
232	C₂²⋊C₈	D₄×C₃₁	1	4	yes	—
233	#97	D₄×C₃₁	1	2	yes	—
234	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
235	C₄×C₈	C₂³×C₃₁	1	2	yes	—
236	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
237	C₄×C₈	C₂³×C₃₁	1	2	yes	—
238	C₃₁	#19717	1	1	yes	characteristic, direct product factor, Sylow
239	C₂⁴	—	4	1	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×D₄	—	2	2	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	—	4	4	no	—
247	C₂×D₄	—	4	1	no	—
248	C₂×D₄	—	4	2	no	—
249	C₂×D₄	—	8	4	no	—
250	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
251	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
252	Q₁₆	—	4	4	no	—
253	SD₁₆	—	8	4	no	—
254	SD₁₆	—	8	4	no	—
255	M₁₆	—	4	2	no	—
256	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
257	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
258	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
259	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₂²⋊C₄	—	4	1	no	—
270	C₂²⋊C₄	—	4	2	no	—
271	C₄²	—	4	1	no	—
272	C₄²	—	4	2	no	—
273	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
274	C₄²	C₂×D₄×C₃₁	1	2	yes	—
275	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
276	C₂³	—	4	2	no	—
277	C₂³	—	8	2	no	—
278	C₂³	—	8	4	no	—
279	C₂³	C₃₁×#127	1	1	yes	characteristic
280	Q₈	—	4	4	no	—
281	Q₈	—	4	4	no	—
282	Q₈	—	4	4	no	—
283	D₄	—	4	4	no	—
284	D₄	—	4	4	no	—
285	D₄	—	4	4	no	—
286	D₄	—	8	2	no	—
287	D₄	—	8	4	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
294	C₂×C₄	C₃₁×#127	1	2	yes	—
295	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
296	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
297	C₈	—	2	2	no	—
298	C₈	—	2	2	no	—
299	C₈	—	2	4	no	—
300	C₈	—	2	4	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	8	2	no	—
304	C₂²	—	8	2	no	—
305	C₂²	—	8	2	no	—
306	C₂²	—	16	4	no	—
307	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	2	2	no	—
310	C₄	—	8	2	no	—
311	C₄	—	8	2	no	—
312	C₄	—	8	2	no	—
313	C₄	C₃₁×#575	1	2	yes	—
314	C₄	C₃₁×#577	1	4	yes	—
315	C₂	—	4	1	no	—
316	C₂	—	16	2	no	—
317	C₂	C₃₁×#3153	1	1	yes	characteristic
318	C₂	C₃₁×#3154	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3231	C₂	semidirect
3. C₃₁×#3185	C₂	semidirect
5. C₃₁×#1720	C₂	semidirect
6. C₃₁×#1719	C₂	semidirect
7. C₃₁×#1714	C₂	semidirect
8. C₃₁×#1708	C₂	semidirect
9. C₃₁×#1453	C₂	semidirect
13. C₃₁×#500	C₂²	semidirect
14. C₃₁×#498	C₂²	semidirect
15. C₃₁×#498	C₂²	semidirect
16. C₃₁×#494	C₂²	semidirect
20. C₃₁×#487	C₂²	semidirect
21. C₃₁×#487	C₂²	semidirect
22. C₃₁×#487	C₂²	semidirect
36. C₃₁×#322	C₂²	semidirect
37. C₃₁×#322	C₂²	semidirect
38. C₈²×C₃₁	C₂²	semidirect
116. #19717	C₃₁	direct (N × H)
142. #3231	C₂×C₃₁	semidirect
144. #3185	C₂×C₃₁	semidirect
146. #1720	C₂×C₃₁	semidirect
147. #1719	C₂×C₃₁	semidirect
148. #1714	C₂×C₃₁	semidirect
149. #1708	C₂×C₃₁	semidirect
150. #1453	C₂×C₃₁	semidirect
168. #500	C₂²×C₃₁	semidirect
169. #498	C₂²×C₃₁	semidirect
170. #498	C₂²×C₃₁	semidirect
171. #494	C₂²×C₃₁	semidirect
175. #487	C₂²×C₃₁	semidirect
176. #487	C₂²×C₃₁	semidirect
177. #487	C₂²×C₃₁	semidirect
191. #322	C₂²×C₃₁	semidirect
192. #322	C₂²×C₃₁	semidirect
193. C₈²	C₂²×C₃₁	semidirect
238. C₃₁	#19717	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×#3231	2
C₃₁×#3208	2
C₃₁×#3185	2
C₃₁×#3138	2
C₃₁×#1720	2
C₃₁×#1719	2
C₃₁×#1714	2
C₃₁×#1708	2
C₃₁×#1453	2
#19717	31

Subgroups — C31×#19717 #21476476

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19717 #21476476