Subgroups — C₃₁×#19660 #21476419

Order 7936.

914 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3226	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3205	C₂	1	2	yes	maximal, complemented
3	C₃₁×#3147	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×#492	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1722	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1721	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1706	C₂	1	4	yes	maximal, complemented
8	C₃₁×#1450	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#135	—	2	2	no	—
10	C₂²×SD₁₆×C₃₁	—	2	2	no	—
11	C₃₁×#532	C₂²	1	2	yes	—
12	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
14	C₃₁×#500	C₂²	1	4	yes	complemented
15	C₃₁×#495	—	2	2	no	—
16	C₃₁×#494	C₂²	1	1	yes	characteristic, complemented
17	C₃₁×#493	C₂²	1	4	yes	—
18	C₃₁×#493	C₂²	1	1	yes	characteristic, complemented
19	C₃₁×#492	—	2	2	no	—
20	C₃₁×#492	C₂²	1	2	yes	—
21	C₃₁×#492	C₂²	1	2	yes	—
22	C₃₁×#492	C₂²	1	2	yes	complemented
23	C₃₁×#486	C₂²	1	4	yes	—
24	C₃₁×#486	C₂²	1	4	yes	complemented
25	C₃₁×#480	—	2	4	no	—
26	C₃₁×#474	—	2	4	no	—
27	C₃₁×#459	—	2	4	no	—
28	C₃₁×#447	—	2	4	no	—
29	C₃₁×#431	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
31	C₃₁×#322	C₂²	1	2	yes	complemented
32	C₈²×C₃₁	C₂²	1	2	yes	complemented
33	C₂×#41×C₃₁	—	2	2	no	—
34	C₂²×Q₈×C₃₁	—	2	1	no	—
35	C₂²×D₄×C₃₁	—	2	1	no	—
36	C₂²×D₄×C₃₁	—	2	2	no	—
37	C₃₁×#135	—	4	8	no	—
38	C₂×Q₁₆×C₃₁	—	2	2	no	—
39	C₂×SD₁₆×C₃₁	—	2	4	no	—
40	C₂×SD₁₆×C₃₁	—	2	4	no	—
41	C₂×SD₁₆×C₃₁	—	2	4	no	—
42	C₂×SD₁₆×C₃₁	—	4	8	no	—
43	C₂×D₈×C₃₁	—	2	2	no	—
44	C₂×D₈×C₃₁	—	2	8	no	—
45	C₂²×C₈×C₃₁	D₄	1	2	yes	—
46	C₃₁×C₄⋊Q₈	—	2	2	no	—
47	C₃₁×C₄⋊Q₈	—	2	1	no	—
48	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
49	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
50	C₃₁×#127	—	2	2	no	—
51	C₃₁×#127	—	2	1	no	—
52	C₃₁×#127	C₂³	1	2	yes	—
53	C₃₁×#127	C₂³	1	2	yes	—
54	C₃₁×#124	—	2	2	no	—
55	C₃₁×#121	—	2	2	no	—
56	C₃₁×#121	—	2	2	no	—
57	C₄×D₄×C₃₁	—	2	2	no	—
58	C₄×D₄×C₃₁	—	2	2	no	—
59	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
60	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
61	C₃₁×#107	—	2	8	no	—
62	C₃₁×C₄⋊C₈	D₄	1	4	yes	—
63	C₃₁×#102	—	2	8	no	—
64	C₃₁×#102	—	2	8	no	—
65	C₃₁×C₂²⋊C₈	D₄	1	4	yes	—
66	C₄×C₈×C₃₁	D₄	1	2	yes	—
67	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₃₁	C₂³	1	4	yes	—
70	C₂⁴×C₃₁	—	4	1	no	—
71	#41×C₃₁	—	4	8	no	—
72	C₂×Q₈×C₃₁	—	2	2	no	—
73	C₂×Q₈×C₃₁	—	2	2	no	—
74	C₂×Q₈×C₃₁	—	4	1	no	—
75	C₂×Q₈×C₃₁	—	4	4	no	—
76	C₂×D₄×C₃₁	—	2	2	no	—
77	C₂×D₄×C₃₁	—	2	4	no	—
78	C₂×D₄×C₃₁	—	2	2	no	—
79	C₂×D₄×C₃₁	—	2	2	no	—
80	C₂×D₄×C₃₁	—	2	2	no	—
81	C₂×D₄×C₃₁	—	4	2	no	—
82	C₂×D₄×C₃₁	—	4	2	no	—
83	C₂×D₄×C₃₁	—	4	1	no	—
84	C₂×D₄×C₃₁	—	4	4	no	—
85	C₂×D₄×C₃₁	—	8	4	no	—
86	C₂²×C₄×C₃₁	—	4	1	no	—
87	C₂²×C₄×C₃₁	—	4	2	no	—
88	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
90	Q₁₆×C₃₁	—	4	4	no	—
91	SD₁₆×C₃₁	—	4	8	no	—
92	SD₁₆×C₃₁	—	4	8	no	—
93	SD₁₆×C₃₁	—	4	8	no	—
94	D₈×C₃₁	—	4	4	no	—
95	D₈×C₃₁	—	8	8	no	—
96	C₂×C₈×C₃₁	D₈	1	8	yes	complemented
97	C₂×C₈×C₃₁	C₂×D₄	1	8	yes	—
98	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
100	C₄⋊C₄×C₃₁	—	2	2	no	—
101	C₄⋊C₄×C₃₁	—	2	4	no	—
102	C₄⋊C₄×C₃₁	—	2	2	no	—
103	C₄⋊C₄×C₃₁	—	4	2	no	—
104	C₂²⋊C₄×C₃₁	—	4	2	no	—
105	C₂²⋊C₄×C₃₁	—	4	2	no	—
106	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₃₁	C₂×D₄	1	2	yes	—
108	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
109	#19660	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂³×C₃₁	—	4	2	no	—
111	C₂³×C₃₁	—	4	2	no	—
112	C₂³×C₃₁	—	8	4	no	—
113	C₂³×C₃₁	—	8	2	no	—
114	C₂³×C₃₁	#127	1	1	yes	characteristic
115	Q₈×C₃₁	—	4	4	no	—
116	Q₈×C₃₁	—	4	4	no	—
117	Q₈×C₃₁	—	8	2	no	—
118	D₄×C₃₁	—	4	4	no	—
119	D₄×C₃₁	—	4	4	no	—
120	D₄×C₃₁	—	4	4	no	—
121	D₄×C₃₁	—	4	8	no	—
122	D₄×C₃₁	—	4	4	no	—
123	D₄×C₃₁	—	8	4	no	—
124	D₄×C₃₁	—	8	4	no	—
125	D₄×C₃₁	—	8	2	no	—
126	C₂×C₄×C₃₁	—	2	4	no	—
127	C₂×C₄×C₃₁	—	4	2	no	—
128	C₂×C₄×C₃₁	—	4	2	no	—
129	C₂×C₄×C₃₁	—	8	2	no	—
130	C₂×C₄×C₃₁	—	8	4	no	—
131	C₂×C₄×C₃₁	#127	1	2	yes	—
132	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
133	C₂×C₄×C₃₁	C₂×D₈	1	4	yes	—
134	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
136	C₈×C₃₁	—	2	8	no	—
137	C₈×C₃₁	C₂×D₈	1	4	yes	—
138	C₈×C₃₁	C₂×D₈	1	4	yes	—
139	#3226	C₂×C₃₁	1	2	yes	complemented
140	#3205	C₂×C₃₁	1	2	yes	complemented
141	#3147	C₂×C₃₁	1	1	yes	characteristic, complemented
142	C₂×#492	C₂×C₃₁	1	1	yes	characteristic, complemented
143	#1722	C₂×C₃₁	1	2	yes	complemented
144	#1721	C₂×C₃₁	1	2	yes	complemented
145	#1706	C₂×C₃₁	1	4	yes	complemented
146	#1450	C₂×C₃₁	1	1	yes	characteristic, complemented
147	C₂²×C₃₁	—	2	2	no	—
148	C₂²×C₃₁	—	8	2	no	—
149	C₂²×C₃₁	—	8	2	no	—
150	C₂²×C₃₁	—	8	4	no	—
151	C₂²×C₃₁	—	8	2	no	—
152	C₂²×C₃₁	—	8	2	no	—
153	C₂²×C₃₁	—	16	4	no	—
154	C₂²×C₃₁	#493	1	2	yes	—
155	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
156	C₄×C₃₁	—	2	2	no	—
157	C₄×C₃₁	—	8	2	no	—
158	C₄×C₃₁	—	8	2	no	—
159	C₄×C₃₁	#493	1	2	yes	—
160	C₄×C₃₁	C₂²×D₈	1	2	yes	—
161	C₄×C₃₁	#577	1	4	yes	—
162	C₂×#135	—	2	2	no	—
163	C₂²×SD₁₆	—	2	2	no	—
164	#532	C₂²×C₃₁	1	2	yes	—
165	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
166	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
167	#500	C₂²×C₃₁	1	4	yes	complemented
168	#495	—	2	2	no	—
169	#494	C₂²×C₃₁	1	1	yes	characteristic, complemented
170	#493	C₂²×C₃₁	1	4	yes	—
171	#493	C₂²×C₃₁	1	1	yes	characteristic, complemented
172	#492	—	2	2	no	—
173	#492	C₂²×C₃₁	1	2	yes	—
174	#492	C₂²×C₃₁	1	2	yes	—
175	#492	C₂²×C₃₁	1	2	yes	complemented
176	#486	C₂²×C₃₁	1	4	yes	—
177	#486	C₂²×C₃₁	1	4	yes	complemented
178	#480	—	2	4	no	—
179	#474	—	2	4	no	—
180	#459	—	2	4	no	—
181	#447	—	2	4	no	—
182	#431	C₂²×C₃₁	1	2	yes	—
183	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
184	#322	C₂²×C₃₁	1	2	yes	complemented
185	C₈²	C₂²×C₃₁	1	2	yes	complemented
186	C₂×C₃₁	—	2	2	no	—
187	C₂×C₃₁	—	16	2	no	—
188	C₂×C₃₁	—	16	2	no	—
189	C₂×C₃₁	C₂×#493	1	1	yes	characteristic
190	C₂×C₃₁	#3154	1	2	yes	—
191	C₂×#41	—	2	2	no	—
192	C₂²×Q₈	—	2	1	no	—
193	C₂²×D₄	—	2	1	no	—
194	C₂²×D₄	—	2	2	no	—
195	#135	—	4	8	no	—
196	C₂×Q₁₆	—	2	2	no	—
197	C₂×SD₁₆	—	2	4	no	—
198	C₂×SD₁₆	—	2	4	no	—
199	C₂×SD₁₆	—	2	4	no	—
200	C₂×SD₁₆	—	4	8	no	—
201	C₂×D₈	—	2	2	no	—
202	C₂×D₈	—	2	8	no	—
203	C₂²×C₈	D₄×C₃₁	1	2	yes	—
204	C₄⋊Q₈	—	2	2	no	—
205	C₄⋊Q₈	—	2	1	no	—
206	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
207	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
208	#127	—	2	2	no	—
209	#127	—	2	1	no	—
210	#127	C₂³×C₃₁	1	2	yes	—
211	#127	C₂³×C₃₁	1	2	yes	—
212	#124	—	2	2	no	—
213	#121	—	2	2	no	—
214	#121	—	2	2	no	—
215	C₄×D₄	—	2	2	no	—
216	C₄×D₄	—	2	2	no	—
217	C₂×C₄⋊C₄	—	2	2	no	—
218	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
219	#107	—	2	8	no	—
220	C₄⋊C₈	D₄×C₃₁	1	4	yes	—
221	#102	—	2	8	no	—
222	#102	—	2	8	no	—
223	C₂²⋊C₈	D₄×C₃₁	1	4	yes	—
224	C₄×C₈	D₄×C₃₁	1	2	yes	—
225	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
226	C₄×C₈	C₂³×C₃₁	1	4	yes	—
227	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
228	C₃₁	#19660	1	1	yes	characteristic, direct product factor, Sylow
229	C₂⁴	—	4	1	no	—
230	#41	—	4	8	no	—
231	C₂×Q₈	—	2	2	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	4	1	no	—
234	C₂×Q₈	—	4	4	no	—
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	—	2	4	no	—
237	C₂×D₄	—	2	2	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	4	2	no	—
241	C₂×D₄	—	4	2	no	—
242	C₂×D₄	—	4	1	no	—
243	C₂×D₄	—	4	4	no	—
244	C₂×D₄	—	8	4	no	—
245	C₂²×C₄	—	4	1	no	—
246	C₂²×C₄	—	4	2	no	—
247	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
248	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
249	Q₁₆	—	4	4	no	—
250	SD₁₆	—	4	8	no	—
251	SD₁₆	—	4	8	no	—
252	SD₁₆	—	4	8	no	—
253	D₈	—	4	4	no	—
254	D₈	—	8	8	no	—
255	C₂×C₈	D₈×C₃₁	1	8	yes	complemented
256	C₂×C₈	C₂×D₄×C₃₁	1	8	yes	—
257	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
258	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₂²⋊C₄	—	4	2	no	—
264	C₂²⋊C₄	—	4	2	no	—
265	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
266	C₄²	C₂×D₄×C₃₁	1	2	yes	—
267	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
268	C₂³	—	4	2	no	—
269	C₂³	—	4	2	no	—
270	C₂³	—	8	4	no	—
271	C₂³	—	8	2	no	—
272	C₂³	C₃₁×#127	1	1	yes	characteristic
273	Q₈	—	4	4	no	—
274	Q₈	—	4	4	no	—
275	Q₈	—	8	2	no	—
276	D₄	—	4	4	no	—
277	D₄	—	4	4	no	—
278	D₄	—	4	4	no	—
279	D₄	—	4	8	no	—
280	D₄	—	4	4	no	—
281	D₄	—	8	4	no	—
282	D₄	—	8	4	no	—
283	D₄	—	8	2	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	8	2	no	—
288	C₂×C₄	—	8	4	no	—
289	C₂×C₄	C₃₁×#127	1	2	yes	—
290	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
291	C₂×C₄	C₂×D₈×C₃₁	1	4	yes	—
292	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
293	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
294	C₈	—	2	8	no	—
295	C₈	C₂×D₈×C₃₁	1	4	yes	—
296	C₈	C₂×D₈×C₃₁	1	4	yes	—
297	C₂²	—	2	2	no	—
298	C₂²	—	8	2	no	—
299	C₂²	—	8	2	no	—
300	C₂²	—	8	4	no	—
301	C₂²	—	8	2	no	—
302	C₂²	—	8	2	no	—
303	C₂²	—	16	4	no	—
304	C₂²	C₃₁×#493	1	2	yes	—
305	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
306	C₄	—	2	2	no	—
307	C₄	—	8	2	no	—
308	C₄	—	8	2	no	—
309	C₄	C₃₁×#493	1	2	yes	—
310	C₄	C₂²×D₈×C₃₁	1	2	yes	—
311	C₄	C₃₁×#577	1	4	yes	—
312	C₂	—	2	2	no	—
313	C₂	—	16	2	no	—
314	C₂	—	16	2	no	—
315	C₂	C₂×C₃₁×#493	1	1	yes	characteristic
316	C₂	C₃₁×#3154	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
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×#3226	C₂	semidirect
2. C₃₁×#3205	C₂	semidirect
3. C₃₁×#3147	C₂	semidirect
4. C₂×C₃₁×#492	C₂	semidirect
5. C₃₁×#1722	C₂	semidirect
6. C₃₁×#1721	C₂	semidirect
7. C₃₁×#1706	C₂	semidirect
8. C₃₁×#1450	C₂	semidirect
14. C₃₁×#500	C₂²	semidirect
16. C₃₁×#494	C₂²	semidirect
18. C₃₁×#493	C₂²	semidirect
22. C₃₁×#492	C₂²	semidirect
24. C₃₁×#486	C₂²	semidirect
31. C₃₁×#322	C₂²	semidirect
32. C₈²×C₃₁	C₂²	semidirect
96. C₂×C₈×C₃₁	D₈	semidirect
109. #19660	C₃₁	direct (N × H)
139. #3226	C₂×C₃₁	semidirect
140. #3205	C₂×C₃₁	semidirect
141. #3147	C₂×C₃₁	semidirect
142. C₂×#492	C₂×C₃₁	semidirect
143. #1722	C₂×C₃₁	semidirect
144. #1721	C₂×C₃₁	semidirect
145. #1706	C₂×C₃₁	semidirect
146. #1450	C₂×C₃₁	semidirect
167. #500	C₂²×C₃₁	semidirect
169. #494	C₂²×C₃₁	semidirect
171. #493	C₂²×C₃₁	semidirect
175. #492	C₂²×C₃₁	semidirect
177. #486	C₂²×C₃₁	semidirect
184. #322	C₂²×C₃₁	semidirect
185. C₈²	C₂²×C₃₁	semidirect
228. C₃₁	#19660	direct (N × H)
255. C₂×C₈	D₈×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3226	2
C₃₁×#3205	2
C₃₁×#3147	2
C₂×C₃₁×#492	2
C₃₁×#1722	2
C₃₁×#1721	2
C₃₁×#1706	2
C₃₁×#1450	2
#19660	31

Subgroups — C31×#19660 #21476419

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19660 #21476419