Subgroups — C₃₁×#33952 #21490711

Order 7936.

464 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	Q₈×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3380	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2255	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2252	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2229	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2227	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2226	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#2224	C₂	1	2	yes	maximal
9	C₃₁×#2183	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#2175	C₂	1	2	yes	maximal
11	C₄×C₃₁×SD₃₂	C₂	1	1	yes	maximal, characteristic
12	Q₈²×C₃₁	—	2	1	no	—
13	C₃₁×#554	—	2	1	no	—
14	C₂×C₃₁×SD₃₂	C₂²	1	1	yes	characteristic
15	C₃₁×#500	C₂²	1	1	yes	characteristic
16	C₃₁×#500	C₂²	1	2	yes	—
17	C₃₁×#500	C₂²	1	1	yes	characteristic
18	C₃₁×#499	C₂²	1	2	yes	—
19	C₃₁×#478	—	2	2	no	—
20	C₃₁×#477	—	2	1	no	—
21	C₃₁×#477	—	2	2	no	—
22	C₃₁×#476	—	2	1	no	—
23	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
25	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
26	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
27	C₄×D₈×C₃₁	C₂²	1	2	yes	—
28	C₃₁×#367	C₂²	1	1	yes	characteristic
29	C₃₁×#367	C₂²	1	2	yes	—
30	C₃₁×#366	C₂²	1	2	yes	—
31	C₃₁×#366	C₂²	1	2	yes	—
32	C₃₁×#366	C₂²	1	2	yes	—
33	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
34	C₃₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
35	C₃₁×#358	C₂²	1	1	yes	characteristic
36	C₃₁×#358	C₂²	1	2	yes	—
37	C₃₁×#357	C₂²	1	1	yes	characteristic
38	C₃₁×#357	C₂²	1	2	yes	—
39	C₃₁×#346	C₂²	1	2	yes	—
40	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
41	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
42	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
43	C₃₁×C₄⋊Q₈	—	2	1	no	—
44	C₃₁×C₄⋊Q₈	—	2	2	no	—
45	C₃₁×C₄⋊Q₈	—	2	1	no	—
46	C₃₁×C₄⋊Q₈	—	4	1	no	—
47	C₃₁×C₄⋊Q₈	—	4	2	no	—
48	C₃₁×#125	—	2	2	no	—
49	C₃₁×C₂²⋊Q₈	—	4	1	no	—
50	C₃₁×C₂²⋊Q₈	—	4	2	no	—
51	C₄×Q₈×C₃₁	—	2	2	no	—
52	C₄×Q₈×C₃₁	—	2	1	no	—
53	C₄×Q₈×C₃₁	—	4	1	no	—
54	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
55	C₄×D₄×C₃₁	—	2	1	no	—
56	C₄×D₄×C₃₁	—	2	2	no	—
57	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
58	C₃₁×SD₃₂	Q₈	1	4	yes	complemented
59	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
60	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
61	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
62	C₃₁×#107	—	2	2	no	—
63	C₃₁×#107	—	2	1	no	—
64	C₃₁×#107	—	2	1	no	—
65	C₃₁×#107	C₂³	1	1	yes	characteristic
66	C₃₁×#107	C₂³	1	2	yes	—
67	C₃₁×#107	C₂³	1	1	yes	characteristic
68	C₃₁×#107	C₂³	1	2	yes	—
69	C₃₁×#106	—	2	2	no	—
70	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
71	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
72	C₃₁×#103	—	2	2	no	—
73	C₃₁×#103	—	2	1	no	—
74	C₃₁×#102	—	2	2	no	—
75	C₃₁×#102	—	2	1	no	—
76	C₄×C₈×C₃₁	C₂³	1	2	yes	—
77	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
78	C₂×Q₈×C₃₁	—	2	1	no	—
79	C₂×Q₈×C₃₁	—	4	1	no	—
80	C₂×Q₈×C₃₁	—	4	1	no	—
81	C₂×Q₈×C₃₁	—	4	2	no	—
82	C₂×Q₈×C₃₁	D₈	1	1	yes	characteristic
83	C₂×D₄×C₃₁	—	2	1	no	—
84	C₂²×C₄×C₃₁	—	4	1	no	—
85	C₂²×C₄×C₃₁	—	4	2	no	—
86	Q₁₆×C₃₁	—	2	1	no	—
87	Q₁₆×C₃₁	C₂×Q₈	1	2	yes	—
88	D₈×C₃₁	—	2	1	no	—
89	D₈×C₃₁	C₂×Q₈	1	2	yes	—
90	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₃₁	—	2	1	no	—
94	C₄⋊C₄×C₃₁	—	2	2	no	—
95	C₄⋊C₄×C₃₁	—	2	2	no	—
96	C₄⋊C₄×C₃₁	—	2	1	no	—
97	C₄⋊C₄×C₃₁	—	4	2	no	—
98	C₄⋊C₄×C₃₁	—	4	1	no	—
99	C₄⋊C₄×C₃₁	—	4	1	no	—
100	C₄⋊C₄×C₃₁	—	4	2	no	—
101	C₄⋊C₄×C₃₁	—	4	2	no	—
102	C₄⋊C₄×C₃₁	—	4	1	no	—
103	C₄⋊C₄×C₃₁	—	4	2	no	—
104	C₄⋊C₄×C₃₁	D₈	1	1	yes	characteristic
105	C₄⋊C₄×C₃₁	D₈	1	2	yes	—
106	C₂²⋊C₄×C₃₁	—	4	1	no	—
107	C₂²⋊C₄×C₃₁	—	4	2	no	—
108	C₄²×C₃₁	—	4	2	no	—
109	C₄²×C₃₁	—	4	1	no	—
110	C₄²×C₃₁	C₂×D₄	1	2	yes	—
111	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
112	C₁₆×C₃₁	—	2	1	no	—
113	C₁₆×C₃₁	—	2	2	no	—
114	C₁₆×C₃₁	C₂×Q₈	1	2	yes	—
115	#33952	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂³×C₃₁	—	4	1	no	—
117	Q₈×C₃₁	—	2	2	no	—
118	Q₈×C₃₁	—	2	2	no	—
119	Q₈×C₃₁	—	4	1	no	—
120	Q₈×C₃₁	—	16	2	no	—
121	Q₈×C₃₁	—	16	1	no	—
122	Q₈×C₃₁	—	16	1	no	—
123	D₄×C₃₁	—	2	2	no	—
124	D₄×C₃₁	—	4	1	no	—
125	C₂×C₄×C₃₁	—	4	1	no	—
126	C₂×C₄×C₃₁	—	4	2	no	—
127	C₂×C₄×C₃₁	—	4	1	no	—
128	C₂×C₄×C₃₁	—	4	1	no	—
129	C₂×C₄×C₃₁	—	4	2	no	—
130	C₂×C₄×C₃₁	—	8	2	no	—
131	C₂×C₄×C₃₁	—	16	2	no	—
132	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
133	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
137	C₈×C₃₁	—	2	2	no	—
138	C₈×C₃₁	—	2	1	no	—
139	C₈×C₃₁	C₂²×Q₈	1	1	yes	characteristic
140	C₈×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
141	Q₈×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
142	#3380	C₂×C₃₁	1	1	yes	characteristic
143	#2255	C₂×C₃₁	1	2	yes	complemented
144	#2252	C₂×C₃₁	1	1	yes	characteristic, complemented
145	#2229	C₂×C₃₁	1	2	yes	complemented
146	#2227	C₂×C₃₁	1	1	yes	characteristic, complemented
147	#2226	C₂×C₃₁	1	1	yes	characteristic
148	#2224	C₂×C₃₁	1	2	yes	—
149	#2183	C₂×C₃₁	1	1	yes	characteristic, complemented
150	#2175	C₂×C₃₁	1	2	yes	—
151	C₄×SD₃₂	C₂×C₃₁	1	1	yes	characteristic
152	C₂²×C₃₁	—	4	2	no	—
153	C₂²×C₃₁	—	8	1	no	—
154	C₂²×C₃₁	—	8	1	no	—
155	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
156	C₄×C₃₁	—	2	2	no	—
157	C₄×C₃₁	—	2	2	no	—
158	C₄×C₃₁	—	2	1	no	—
159	C₄×C₃₁	—	4	2	no	—
160	C₄×C₃₁	—	8	1	no	—
161	C₄×C₃₁	—	8	2	no	—
162	C₄×C₃₁	—	8	1	no	—
163	C₄×C₃₁	—	8	2	no	—
164	C₄×C₃₁	#510	1	2	yes	—
165	C₄×C₃₁	D₄×Q₈	1	1	yes	characteristic
166	C₄×C₃₁	#578	1	1	yes	characteristic
167	Q₈²	—	2	1	no	—
168	#554	—	2	1	no	—
169	C₂×SD₃₂	C₂²×C₃₁	1	1	yes	characteristic
170	#500	C₂²×C₃₁	1	1	yes	characteristic
171	#500	C₂²×C₃₁	1	2	yes	—
172	#500	C₂²×C₃₁	1	1	yes	characteristic
173	#499	C₂²×C₃₁	1	2	yes	—
174	#478	—	2	2	no	—
175	#477	—	2	1	no	—
176	#477	—	2	2	no	—
177	#476	—	2	1	no	—
178	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
179	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
180	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
181	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
182	C₄×D₈	C₂²×C₃₁	1	2	yes	—
183	#367	C₂²×C₃₁	1	1	yes	characteristic
184	#367	C₂²×C₃₁	1	2	yes	—
185	#366	C₂²×C₃₁	1	2	yes	—
186	#366	C₂²×C₃₁	1	2	yes	—
187	#366	C₂²×C₃₁	1	2	yes	—
188	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
189	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
190	#358	C₂²×C₃₁	1	1	yes	characteristic
191	#358	C₂²×C₃₁	1	2	yes	—
192	#357	C₂²×C₃₁	1	1	yes	characteristic
193	#357	C₂²×C₃₁	1	2	yes	—
194	#346	C₂²×C₃₁	1	2	yes	—
195	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
196	C₂×C₃₁	—	8	2	no	—
197	C₂×C₃₁	Q₈×D₈	1	1	yes	characteristic
198	C₂×C₃₁	C₂×#510	1	1	yes	characteristic
199	C₂×C₃₁	#3415	1	1	yes	characteristic
200	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
201	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
202	C₄⋊Q₈	—	2	1	no	—
203	C₄⋊Q₈	—	2	2	no	—
204	C₄⋊Q₈	—	2	1	no	—
205	C₄⋊Q₈	—	4	1	no	—
206	C₄⋊Q₈	—	4	2	no	—
207	#125	—	2	2	no	—
208	C₂²⋊Q₈	—	4	1	no	—
209	C₂²⋊Q₈	—	4	2	no	—
210	C₄×Q₈	—	2	2	no	—
211	C₄×Q₈	—	2	1	no	—
212	C₄×Q₈	—	4	1	no	—
213	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
214	C₄×D₄	—	2	1	no	—
215	C₄×D₄	—	2	2	no	—
216	C₂×C₄⋊C₄	—	4	1	no	—
217	SD₃₂	Q₈×C₃₁	1	4	yes	complemented
218	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
219	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
220	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
221	#107	—	2	2	no	—
222	#107	—	2	1	no	—
223	#107	—	2	1	no	—
224	#107	C₂³×C₃₁	1	1	yes	characteristic
225	#107	C₂³×C₃₁	1	2	yes	—
226	#107	C₂³×C₃₁	1	2	yes	—
227	#107	C₂³×C₃₁	1	1	yes	characteristic
228	#106	—	2	2	no	—
229	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
230	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
231	#103	—	2	2	no	—
232	#103	—	2	1	no	—
233	#102	—	2	2	no	—
234	#102	—	2	1	no	—
235	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
236	C₄×C₈	C₂³×C₃₁	1	2	yes	—
237	C₃₁	#33952	1	1	yes	characteristic, direct product factor, Sylow
238	C₂×Q₈	—	2	1	no	—
239	C₂×Q₈	—	4	1	no	—
240	C₂×Q₈	—	4	1	no	—
241	C₂×Q₈	—	4	2	no	—
242	C₂×Q₈	D₈×C₃₁	1	1	yes	characteristic
243	C₂×D₄	—	2	1	no	—
244	C₂²×C₄	—	4	1	no	—
245	C₂²×C₄	—	4	2	no	—
246	Q₁₆	—	2	1	no	—
247	Q₁₆	C₂×Q₈×C₃₁	1	2	yes	—
248	D₈	—	2	1	no	—
249	D₈	C₂×Q₈×C₃₁	1	2	yes	—
250	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
251	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
252	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	1	no	—
259	C₄⋊C₄	—	4	1	no	—
260	C₄⋊C₄	—	4	2	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
265	C₄⋊C₄	D₈×C₃₁	1	2	yes	—
266	C₂²⋊C₄	—	4	1	no	—
267	C₂²⋊C₄	—	4	2	no	—
268	C₄²	—	4	2	no	—
269	C₄²	—	4	1	no	—
270	C₄²	C₂×D₄×C₃₁	1	2	yes	—
271	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
272	C₁₆	—	2	1	no	—
273	C₁₆	—	2	2	no	—
274	C₁₆	C₂×Q₈×C₃₁	1	2	yes	—
275	C₂³	—	4	1	no	—
276	Q₈	—	2	2	no	—
277	Q₈	—	2	2	no	—
278	Q₈	—	4	1	no	—
279	Q₈	—	16	2	no	—
280	Q₈	—	16	1	no	—
281	Q₈	—	16	1	no	—
282	D₄	—	2	2	no	—
283	D₄	—	4	1	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	8	2	no	—
290	C₂×C₄	—	16	2	no	—
291	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
292	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
293	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
294	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
295	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
296	C₈	—	2	2	no	—
297	C₈	—	2	1	no	—
298	C₈	C₂²×Q₈×C₃₁	1	1	yes	characteristic
299	C₈	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
300	C₂²	—	4	2	no	—
301	C₂²	—	8	1	no	—
302	C₂²	—	8	1	no	—
303	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	1	no	—
307	C₄	—	4	2	no	—
308	C₄	—	8	1	no	—
309	C₄	—	8	2	no	—
310	C₄	—	8	1	no	—
311	C₄	—	8	2	no	—
312	C₄	C₃₁×#510	1	2	yes	—
313	C₄	D₄×Q₈×C₃₁	1	1	yes	characteristic
314	C₄	C₃₁×#578	1	1	yes	characteristic
315	C₂	—	8	2	no	—
316	C₂	Q₈×D₈×C₃₁	1	1	yes	characteristic
317	C₂	C₂×C₃₁×#510	1	1	yes	characteristic
318	C₂	C₃₁×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₂×D₈×C₃₁	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×Q₈×C₃₁ ◁ C₈×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. Q₈×Q₁₆×C₃₁	C₂	semidirect
3. C₃₁×#2255	C₂	semidirect
4. C₃₁×#2252	C₂	semidirect
5. C₃₁×#2229	C₂	semidirect
6. C₃₁×#2227	C₂	semidirect
9. C₃₁×#2183	C₂	semidirect
58. C₃₁×SD₃₂	Q₈	semidirect
115. #33952	C₃₁	direct (N × H)
141. Q₈×Q₁₆	C₂×C₃₁	semidirect
143. #2255	C₂×C₃₁	semidirect
144. #2252	C₂×C₃₁	semidirect
145. #2229	C₂×C₃₁	semidirect
146. #2227	C₂×C₃₁	semidirect
149. #2183	C₂×C₃₁	semidirect
217. SD₃₂	Q₈×C₃₁	semidirect
237. C₃₁	#33952	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×Q₁₆×C₃₁	2
C₃₁×#3380	2
C₃₁×#2255	2
C₃₁×#2252	2
C₃₁×#2229	2
C₃₁×#2227	2
C₃₁×#2226	2
C₃₁×#2224	2
C₃₁×#2183	2
C₃₁×#2175	2
C₄×C₃₁×SD₃₂	2
#33952	31

Subgroups — C31×#33952 #21490711

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33952 #21490711