Subgroups — C₃₁×#29293 #21486052

Order 7936.

646 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×#2331	C₂	1	2	yes	maximal
2	C₃₁×#2088	C₂	1	2	yes	maximal
3	C₃₁×#2087	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2086	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1988	C₂	1	2	yes	maximal
6	C₃₁×#1987	C₂	1	2	yes	maximal
7	C₃₁×#1949	C₂	1	2	yes	maximal
8	C₃₁×#1949	C₂	1	2	yes	maximal
9	C₃₁×#1843	C₂	1	1	yes	maximal, characteristic
10	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#106	C₂²	1	2	yes	—
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₃₁×#391	—	2	2	no	—
19	C₃₁×#391	—	2	2	no	—
20	C₃₁×#389	C₂²	1	2	yes	—
21	C₃₁×#389	C₂²	1	2	yes	—
22	C₃₁×#387	—	2	2	no	—
23	C₃₁×#387	—	2	2	no	—
24	C₃₁×#382	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
26	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
27	C₃₁×#376	C₂²	1	1	yes	characteristic
28	C₃₁×#340	C₂²	1	2	yes	—
29	C₃₁×#340	C₂²	1	1	yes	characteristic
30	C₃₁×#340	C₂²	1	2	yes	—
31	C₃₁×#340	C₂²	1	1	yes	characteristic
32	C₃₁×#340	C₂²	1	2	yes	—
33	C₃₁×#340	C₂²	1	2	yes	—
34	C₃₁×#340	C₂²	1	2	yes	—
35	C₂²×Q₈×C₃₁	—	2	2	no	—
36	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
37	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
39	C₃₁×C₄⋊Q₈	—	2	4	no	—
40	C₃₁×C₄⋊Q₈	Q₈	1	8	yes	complemented
41	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
44	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
48	C₂×C₄²×C₃₁	—	2	2	no	—
49	C₂×C₄²×C₃₁	—	2	2	no	—
50	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
52	C₃₁×#107	—	4	1	no	—
53	C₃₁×#107	—	4	1	no	—
54	C₃₁×#106	—	4	2	no	—
55	C₃₁×C₄⋊C₈	—	2	4	no	—
56	C₃₁×C₄⋊C₈	—	2	4	no	—
57	C₃₁×C₄⋊C₈	—	4	1	no	—
58	C₃₁×C₄⋊C₈	—	4	1	no	—
59	C₃₁×#95	—	2	2	no	—
60	C₃₁×#95	—	2	2	no	—
61	C₃₁×#95	—	2	2	no	—
62	C₃₁×#95	—	2	2	no	—
63	C₃₁×#95	—	2	2	no	—
64	C₃₁×#95	—	2	2	no	—
65	C₃₁×#95	D₄	1	1	yes	characteristic
66	C₃₁×#95	D₄	1	2	yes	—
67	C₃₁×#95	D₄	1	1	yes	characteristic
68	C₂×Q₈×C₃₁	—	2	8	no	—
69	C₂×Q₈×C₃₁	—	8	2	no	—
70	C₂×Q₈×C₃₁	—	8	2	no	—
71	C₂²×C₄×C₃₁	—	2	2	no	—
72	C₂²×C₄×C₃₁	—	2	2	no	—
73	C₂²×C₄×C₃₁	—	2	2	no	—
74	C₂²×C₄×C₃₁	—	2	2	no	—
75	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
80	C₂×C₈×C₃₁	—	2	2	no	—
81	C₂×C₈×C₃₁	—	2	2	no	—
82	C₂×C₈×C₃₁	—	2	2	no	—
83	C₂×C₈×C₃₁	—	2	2	no	—
84	C₂×C₈×C₃₁	—	2	2	no	—
85	C₂×C₈×C₃₁	—	2	1	no	—
86	C₂×C₈×C₃₁	—	2	1	no	—
87	C₄⋊C₄×C₃₁	—	2	8	no	—
88	C₄⋊C₄×C₃₁	—	2	2	no	—
89	C₄⋊C₄×C₃₁	—	2	2	no	—
90	C₄⋊C₄×C₃₁	—	2	2	no	—
91	C₄⋊C₄×C₃₁	—	2	2	no	—
92	C₄⋊C₄×C₃₁	—	2	2	no	—
93	C₄⋊C₄×C₃₁	—	2	2	no	—
94	C₄⋊C₄×C₃₁	—	8	2	no	—
95	C₄⋊C₄×C₃₁	—	8	2	no	—
96	C₄⋊C₄×C₃₁	C₂×Q₈	1	4	yes	—
97	C₄⋊C₄×C₃₁	C₂×Q₈	1	4	yes	—
98	C₄²×C₃₁	—	2	1	no	—
99	C₄²×C₃₁	—	2	1	no	—
100	C₄²×C₃₁	—	8	2	no	—
101	C₄²×C₃₁	—	8	2	no	—
102	C₄²×C₃₁	C₂×Q₈	1	4	yes	—
103	C₄²×C₃₁	#41	1	4	yes	—
104	#29293	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
106	Q₈×C₃₁	—	8	8	no	—
107	C₂×C₄×C₃₁	—	2	2	no	—
108	C₂×C₄×C₃₁	—	2	2	no	—
109	C₂×C₄×C₃₁	—	2	2	no	—
110	C₂×C₄×C₃₁	—	2	2	no	—
111	C₂×C₄×C₃₁	—	2	4	no	—
112	C₂×C₄×C₃₁	—	2	4	no	—
113	C₂×C₄×C₃₁	—	2	1	no	—
114	C₂×C₄×C₃₁	—	2	1	no	—
115	C₂×C₄×C₃₁	—	2	1	no	—
116	C₂×C₄×C₃₁	—	2	2	no	—
117	C₂×C₄×C₃₁	—	2	1	no	—
118	C₂×C₄×C₃₁	—	4	2	no	—
119	C₂×C₄×C₃₁	—	4	2	no	—
120	C₂×C₄×C₃₁	—	4	2	no	—
121	C₂×C₄×C₃₁	—	4	2	no	—
122	C₂×C₄×C₃₁	—	4	2	no	—
123	C₂×C₄×C₃₁	—	4	2	no	—
124	C₂×C₄×C₃₁	—	4	2	no	—
125	C₂×C₄×C₃₁	—	4	2	no	—
126	C₂×C₄×C₃₁	—	4	2	no	—
127	C₂×C₄×C₃₁	—	4	2	no	—
128	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
129	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
130	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
131	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
132	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
133	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
134	C₈×C₃₁	—	4	2	no	—
135	C₈×C₃₁	—	4	1	no	—
136	C₈×C₃₁	—	4	1	no	—
137	#2331	C₂×C₃₁	1	2	yes	—
138	#2088	C₂×C₃₁	1	2	yes	—
139	#2087	C₂×C₃₁	1	1	yes	characteristic
140	#2086	C₂×C₃₁	1	1	yes	characteristic
141	#1988	C₂×C₃₁	1	2	yes	—
142	#1987	C₂×C₃₁	1	2	yes	—
143	#1949	C₂×C₃₁	1	2	yes	—
144	#1949	C₂×C₃₁	1	2	yes	—
145	#1843	C₂×C₃₁	1	1	yes	characteristic
146	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
147	C₂²×C₃₁	#537	1	1	yes	characteristic
148	C₂²×C₃₁	#576	1	1	yes	characteristic
149	C₂²×C₃₁	#577	1	2	yes	—
150	C₂²×C₃₁	#578	1	1	yes	characteristic
151	C₄×C₃₁	—	2	4	no	—
152	C₄×C₃₁	—	2	4	no	—
153	C₄×C₃₁	—	4	4	no	—
154	C₄×C₃₁	—	4	4	no	—
155	C₄×C₃₁	—	4	1	no	—
156	C₄×C₃₁	—	4	1	no	—
157	C₄×C₃₁	—	8	2	no	—
158	C₄×C₃₁	—	8	2	no	—
159	C₄×C₃₁	#557	1	2	yes	—
160	C₄×C₃₁	Q₈²	1	1	yes	characteristic
161	C₄×C₃₁	#559	1	1	yes	characteristic
162	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
163	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
164	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
165	C₂×#106	C₂²×C₃₁	1	2	yes	—
166	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
167	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
168	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
169	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
170	#391	—	2	2	no	—
171	#391	—	2	2	no	—
172	#389	C₂²×C₃₁	1	2	yes	—
173	#389	C₂²×C₃₁	1	2	yes	—
174	#387	—	2	2	no	—
175	#387	—	2	2	no	—
176	#382	C₂²×C₃₁	1	2	yes	—
177	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
178	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
179	#376	C₂²×C₃₁	1	1	yes	characteristic
180	#340	C₂²×C₃₁	1	2	yes	—
181	#340	C₂²×C₃₁	1	1	yes	characteristic
182	#340	C₂²×C₃₁	1	2	yes	—
183	#340	C₂²×C₃₁	1	1	yes	characteristic
184	#340	C₂²×C₃₁	1	2	yes	—
185	#340	C₂²×C₃₁	1	2	yes	—
186	#340	C₂²×C₃₁	1	2	yes	—
187	C₂×C₃₁	#2582	1	1	yes	characteristic
188	C₂×C₃₁	#3081	1	2	yes	—
189	C₂×C₃₁	#3082	1	2	yes	—
190	C₂×C₃₁	#3262	1	1	yes	characteristic
191	C₂×C₃₁	#3263	1	1	yes	characteristic
192	C₂²×Q₈	—	2	2	no	—
193	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
194	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
195	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
196	C₄⋊Q₈	—	2	4	no	—
197	C₄⋊Q₈	Q₈×C₃₁	1	8	yes	complemented
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
202	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
203	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
204	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	—	2	2	no	—
207	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
208	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
209	#107	—	4	1	no	—
210	#107	—	4	1	no	—
211	#106	—	4	2	no	—
212	C₄⋊C₈	—	2	4	no	—
213	C₄⋊C₈	—	2	4	no	—
214	C₄⋊C₈	—	4	1	no	—
215	C₄⋊C₈	—	4	1	no	—
216	#95	—	2	2	no	—
217	#95	—	2	2	no	—
218	#95	—	2	2	no	—
219	#95	—	2	2	no	—
220	#95	—	2	2	no	—
221	#95	—	2	2	no	—
222	#95	D₄×C₃₁	1	1	yes	characteristic
223	#95	D₄×C₃₁	1	2	yes	—
224	#95	D₄×C₃₁	1	1	yes	characteristic
225	C₃₁	#29293	1	1	yes	characteristic, direct product factor, Sylow
226	C₂×Q₈	—	2	8	no	—
227	C₂×Q₈	—	8	2	no	—
228	C₂×Q₈	—	8	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
234	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
235	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
236	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
237	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	1	no	—
245	C₄⋊C₄	—	2	8	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	8	2	no	—
253	C₄⋊C₄	—	8	2	no	—
254	C₄⋊C₄	C₂×Q₈×C₃₁	1	4	yes	—
255	C₄⋊C₄	C₂×Q₈×C₃₁	1	4	yes	—
256	C₄²	—	2	1	no	—
257	C₄²	—	2	1	no	—
258	C₄²	—	8	2	no	—
259	C₄²	—	8	2	no	—
260	C₄²	C₂×Q₈×C₃₁	1	4	yes	—
261	C₄²	#41×C₃₁	1	4	yes	—
262	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
263	Q₈	—	8	8	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
286	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
287	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
288	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
289	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
290	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
291	C₈	—	4	2	no	—
292	C₈	—	4	1	no	—
293	C₈	—	4	1	no	—
294	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
295	C₂²	C₃₁×#537	1	1	yes	characteristic
296	C₂²	C₃₁×#576	1	1	yes	characteristic
297	C₂²	C₃₁×#577	1	2	yes	—
298	C₂²	C₃₁×#578	1	1	yes	characteristic
299	C₄	—	2	4	no	—
300	C₄	—	2	4	no	—
301	C₄	—	4	4	no	—
302	C₄	—	4	4	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	8	2	no	—
306	C₄	—	8	2	no	—
307	C₄	C₃₁×#557	1	2	yes	—
308	C₄	Q₈²×C₃₁	1	1	yes	characteristic
309	C₄	C₃₁×#559	1	1	yes	characteristic
310	C₂	C₃₁×#2582	1	1	yes	characteristic
311	C₂	C₃₁×#3081	1	2	yes	—
312	C₂	C₃₁×#3082	1	2	yes	—
313	C₂	C₃₁×#3262	1	1	yes	characteristic
314	C₂	C₃₁×#3263	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₃₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
40. C₃₁×C₄⋊Q₈	Q₈	semidirect
104. #29293	C₃₁	direct (N × H)
197. C₄⋊Q₈	Q₈×C₃₁	semidirect
225. C₃₁	#29293	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×#2331	2
C₃₁×#2088	2
C₃₁×#2087	2
C₃₁×#2086	2
C₃₁×#1988	2
C₃₁×#1987	2
C₃₁×#1949	2
C₃₁×#1843	2
#29293	31

Subgroups — C31×#29293 #21486052

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#29293 #21486052