Subgroups — C₃₁×#34632 #21491391

Order 7936.

5346 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	C₂³×#41×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#525	C₂	1	3	yes	maximal, complemented
3	C₃₁×#2616	C₂	1	4	yes	maximal, complemented
4	C₃₁×#2615	C₂	1	4	yes	maximal, complemented
5	C₃₁×#2614	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2613	C₂	1	6	yes	maximal, complemented
7	C₃₁×#2388	C₂	1	8	yes	maximal, complemented
8	C₃₁×#2280	C₂	1	3	yes	maximal, complemented
9	C₂²×#41×C₃₁	—	2	24	no	—
10	C₂²×#41×C₃₁	C₂²	1	4	yes	—
11	C₂²×#41×C₃₁	C₂²	1	4	yes	—
12	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
13	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂³×D₄×C₃₁	C₂²	1	3	yes	—
15	C₂⁴×C₄×C₃₁	C₂²	1	3	yes	—
16	C₃₁×#525	—	2	24	no	—
17	C₂×C₃₁×#123	C₂²	1	6	yes	—
18	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	6	yes	complemented
19	C₂×C₃₁×#121	C₂²	1	6	yes	complemented
20	C₂×C₃₁×#120	C₂²	1	6	yes	—
21	C₂×C₄×D₄×C₃₁	C₂²	1	12	yes	complemented
22	C₂×C₃₁×#117	C₂²	1	3	yes	—
23	C₃₁×#395	C₂²	1	8	yes	complemented
24	C₃₁×#394	C₂²	1	24	yes	complemented
25	C₃₁×#393	C₂²	1	8	yes	complemented
26	C₃₁×#392	C₂²	1	24	yes	complemented
27	C₃₁×#381	C₂²	1	6	yes	—
28	C₃₁×#380	C₂²	1	12	yes	complemented
29	C₃₁×#379	C₂²	1	6	yes	—
30	C₄×C₂²⋊C₄×C₃₁	C₂²	1	12	yes	complemented
31	C₂⁵×C₃₁	C₂³	1	3	yes	—
32	C₂×#41×C₃₁	—	2	128	no	—
33	C₂×#41×C₃₁	—	2	48	no	—
34	C₂×#41×C₃₁	—	2	48	no	—
35	C₂×#41×C₃₁	—	2	12	no	—
36	C₂²×Q₈×C₃₁	—	2	12	no	—
37	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
38	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
39	C₂²×D₄×C₃₁	—	2	24	no	—
40	C₂²×D₄×C₃₁	—	2	12	no	—
41	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
42	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
43	C₂²×D₄×C₃₁	C₂³	1	12	yes	—
44	C₂²×D₄×C₃₁	C₂³	1	12	yes	—
45	C₂³×C₄×C₃₁	—	2	24	no	—
46	C₂³×C₄×C₃₁	D₄	1	24	yes	complemented
47	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
48	C₂³×C₄×C₃₁	C₂³	1	6	yes	—
49	C₂³×C₄×C₃₁	C₂³	1	3	yes	—
50	C₂³×C₄×C₃₁	C₂³	1	6	yes	—
51	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
52	C₃₁×#123	—	2	24	no	—
53	C₃₁×C₂²⋊Q₈	—	2	24	no	—
54	C₃₁×#121	—	2	24	no	—
55	C₃₁×#120	—	2	24	no	—
56	C₄×D₄×C₃₁	—	2	48	no	—
57	C₃₁×#117	—	2	12	no	—
58	C₂×C₄⋊C₄×C₃₁	C₂³	1	12	yes	—
59	C₂×C₄⋊C₄×C₃₁	C₂³	1	6	yes	—
60	C₂×C₂²⋊C₄×C₃₁	C₂³	1	6	yes	—
61	C₂×C₂²⋊C₄×C₃₁	C₂³	1	12	yes	—
62	C₂×C₂²⋊C₄×C₃₁	C₂³	1	12	yes	—
63	C₂×C₂²⋊C₄×C₃₁	C₂³	1	24	yes	—
64	C₂×C₄²×C₃₁	C₂³	1	6	yes	—
65	C₃₁×#95	C₂³	1	24	yes	—
66	C₂⁴×C₃₁	—	2	12	no	—
67	C₂⁴×C₃₁	—	4	12	no	—
68	C₂⁴×C₃₁	C₂×D₄	1	12	yes	—
69	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
70	C₂⁴×C₃₁	C₂⁴	1	6	yes	—
71	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
72	#41×C₃₁	—	2	128	no	—
73	#41×C₃₁	—	2	128	no	—
74	#41×C₃₁	—	4	16	no	—
75	C₂×Q₈×C₃₁	—	2	32	no	—
76	C₂×Q₈×C₃₁	—	2	24	no	—
77	C₂×Q₈×C₃₁	—	2	6	no	—
78	C₂×D₄×C₃₁	—	2	96	no	—
79	C₂×D₄×C₃₁	—	2	48	no	—
80	C₂×D₄×C₃₁	—	2	24	no	—
81	C₂×D₄×C₃₁	—	2	12	no	—
82	C₂×D₄×C₃₁	—	2	24	no	—
83	C₂×D₄×C₃₁	—	2	6	no	—
84	C₂×D₄×C₃₁	—	4	24	no	—
85	C₂²×C₄×C₃₁	—	2	12	no	—
86	C₂²×C₄×C₃₁	—	2	96	no	—
87	C₂²×C₄×C₃₁	—	2	24	no	—
88	C₂²×C₄×C₃₁	—	2	24	no	—
89	C₂²×C₄×C₃₁	—	2	12	no	—
90	C₂²×C₄×C₃₁	C₂×D₄	1	24	yes	—
91	C₂²×C₄×C₃₁	C₂×D₄	1	24	yes	—
92	C₂²×C₄×C₃₁	C₂×D₄	1	12	yes	—
93	C₂²×C₄×C₃₁	C₂×D₄	1	12	yes	—
94	C₂²×C₄×C₃₁	C₂⁴	1	12	yes	—
95	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
96	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
97	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
98	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
99	C₄⋊C₄×C₃₁	—	2	12	no	—
100	C₄⋊C₄×C₃₁	—	2	24	no	—
101	C₂²⋊C₄×C₃₁	—	2	12	no	—
102	C₂²⋊C₄×C₃₁	—	2	24	no	—
103	C₂²⋊C₄×C₃₁	—	2	24	no	—
104	C₂²⋊C₄×C₃₁	—	4	24	no	—
105	C₄²×C₃₁	—	2	12	no	—
106	#34632	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂³×C₃₁	—	2	4	no	—
108	C₂³×C₃₁	—	2	24	no	—
109	C₂³×C₃₁	—	2	12	no	—
110	C₂³×C₃₁	—	2	12	no	—
111	C₂³×C₃₁	—	2	12	no	—
112	C₂³×C₃₁	—	2	6	no	—
113	C₂³×C₃₁	—	2	8	no	—
114	C₂³×C₃₁	—	4	24	no	—
115	C₂³×C₃₁	—	4	48	no	—
116	C₂³×C₃₁	C₂²×D₄	1	12	yes	—
117	C₂³×C₃₁	C₂²×D₄	1	6	yes	—
118	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
119	Q₈×C₃₁	—	2	32	no	—
120	Q₈×C₃₁	—	4	4	no	—
121	D₄×C₃₁	—	2	96	no	—
122	D₄×C₃₁	—	4	48	no	—
123	D₄×C₃₁	—	4	8	no	—
124	D₄×C₃₁	—	4	4	no	—
125	C₂×C₄×C₃₁	—	2	24	no	—
126	C₂×C₄×C₃₁	—	2	12	no	—
127	C₂×C₄×C₃₁	—	2	48	no	—
128	C₂×C₄×C₃₁	—	2	48	no	—
129	C₂×C₄×C₃₁	—	2	12	no	—
130	C₂×C₄×C₃₁	—	2	6	no	—
131	C₂×C₄×C₃₁	—	2	24	no	—
132	C₂×C₄×C₃₁	—	4	8	no	—
133	C₂×C₄×C₃₁	#120	1	16	yes	—
134	C₂×C₄×C₃₁	C₂²×D₄	1	12	yes	—
135	C₂×C₄×C₃₁	C₂²×D₄	1	6	yes	—
136	C₂×C₄×C₃₁	C₂²×D₄	1	6	yes	—
137	C₂³×#41	C₂×C₃₁	1	1	yes	characteristic, complemented
138	C₂×#525	C₂×C₃₁	1	3	yes	complemented
139	#2616	C₂×C₃₁	1	4	yes	complemented
140	#2615	C₂×C₃₁	1	4	yes	complemented
141	#2614	C₂×C₃₁	1	2	yes	complemented
142	#2613	C₂×C₃₁	1	6	yes	complemented
143	#2388	C₂×C₃₁	1	8	yes	complemented
144	#2280	C₂×C₃₁	1	3	yes	complemented
145	C₂²×C₃₁	—	2	12	no	—
146	C₂²×C₃₁	—	2	12	no	—
147	C₂²×C₃₁	—	2	12	no	—
148	C₂²×C₃₁	—	2	12	no	—
149	C₂²×C₃₁	—	4	48	no	—
150	C₂²×C₃₁	—	4	24	no	—
151	C₂²×C₃₁	—	4	2	no	—
152	C₂²×C₃₁	—	4	12	no	—
153	C₂²×C₃₁	C₂×#120	1	4	yes	—
154	C₂²×C₃₁	C₂³×D₄	1	3	yes	—
155	C₂²×C₃₁	#585	1	4	yes	—
156	C₄×C₃₁	—	2	12	no	—
157	C₄×C₃₁	—	4	2	no	—
158	C₄×C₃₁	—	4	12	no	—
159	C₄×C₃₁	C₂×#120	1	4	yes	—
160	C₄×C₃₁	C₂×#120	1	4	yes	—
161	C₂²×#41	—	2	24	no	—
162	C₂²×#41	C₂²×C₃₁	1	4	yes	—
163	C₂²×#41	C₂²×C₃₁	1	4	yes	—
164	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
165	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic
166	C₂³×D₄	C₂²×C₃₁	1	3	yes	—
167	C₂⁴×C₄	C₂²×C₃₁	1	3	yes	—
168	#525	—	2	24	no	—
169	C₂×#123	C₂²×C₃₁	1	6	yes	—
170	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	6	yes	complemented
171	C₂×#121	C₂²×C₃₁	1	6	yes	complemented
172	C₂×#120	C₂²×C₃₁	1	6	yes	—
173	C₂×C₄×D₄	C₂²×C₃₁	1	12	yes	complemented
174	C₂×#117	C₂²×C₃₁	1	3	yes	—
175	#395	C₂²×C₃₁	1	8	yes	complemented
176	#394	C₂²×C₃₁	1	24	yes	complemented
177	#393	C₂²×C₃₁	1	8	yes	complemented
178	#392	C₂²×C₃₁	1	24	yes	complemented
179	#381	C₂²×C₃₁	1	6	yes	—
180	#380	C₂²×C₃₁	1	12	yes	complemented
181	#379	C₂²×C₃₁	1	6	yes	—
182	C₄×C₂²⋊C₄	C₂²×C₃₁	1	12	yes	complemented
183	C₂×C₃₁	—	2	4	no	—
184	C₂×C₃₁	—	4	12	no	—
185	C₂×C₃₁	—	8	2	no	—
186	C₂×C₃₁	C₂²×#120	1	1	yes	characteristic
187	C₂×C₃₁	#3449	1	6	yes	—
188	C₂⁵	C₂³×C₃₁	1	3	yes	—
189	C₂×#41	—	2	128	no	—
190	C₂×#41	—	2	48	no	—
191	C₂×#41	—	2	48	no	—
192	C₂×#41	—	2	12	no	—
193	C₂²×Q₈	—	2	12	no	—
194	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
195	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
196	C₂²×D₄	—	2	24	no	—
197	C₂²×D₄	—	2	12	no	—
198	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
199	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
200	C₂²×D₄	C₂³×C₃₁	1	12	yes	—
201	C₂²×D₄	C₂³×C₃₁	1	12	yes	—
202	C₂³×C₄	—	2	24	no	—
203	C₂³×C₄	D₄×C₃₁	1	24	yes	complemented
204	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
205	C₂³×C₄	C₂³×C₃₁	1	6	yes	—
206	C₂³×C₄	C₂³×C₃₁	1	6	yes	—
207	C₂³×C₄	C₂³×C₃₁	1	3	yes	—
208	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
209	#123	—	2	24	no	—
210	C₂²⋊Q₈	—	2	24	no	—
211	#121	—	2	24	no	—
212	#120	—	2	24	no	—
213	C₄×D₄	—	2	48	no	—
214	#117	—	2	12	no	—
215	C₂×C₄⋊C₄	C₂³×C₃₁	1	6	yes	—
216	C₂×C₄⋊C₄	C₂³×C₃₁	1	12	yes	—
217	C₂×C₂²⋊C₄	C₂³×C₃₁	1	6	yes	—
218	C₂×C₂²⋊C₄	C₂³×C₃₁	1	12	yes	—
219	C₂×C₂²⋊C₄	C₂³×C₃₁	1	12	yes	—
220	C₂×C₂²⋊C₄	C₂³×C₃₁	1	24	yes	—
221	C₂×C₄²	C₂³×C₃₁	1	6	yes	—
222	#95	C₂³×C₃₁	1	24	yes	—
223	C₃₁	#34632	1	1	yes	characteristic, direct product factor, Sylow
224	C₂⁴	—	2	12	no	—
225	C₂⁴	—	4	12	no	—
226	C₂⁴	C₂×D₄×C₃₁	1	12	yes	—
227	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
228	C₂⁴	C₂⁴×C₃₁	1	6	yes	—
229	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
230	#41	—	2	128	no	—
231	#41	—	2	128	no	—
232	#41	—	4	16	no	—
233	C₂×Q₈	—	2	32	no	—
234	C₂×Q₈	—	2	24	no	—
235	C₂×Q₈	—	2	6	no	—
236	C₂×D₄	—	2	96	no	—
237	C₂×D₄	—	2	48	no	—
238	C₂×D₄	—	2	24	no	—
239	C₂×D₄	—	2	12	no	—
240	C₂×D₄	—	2	24	no	—
241	C₂×D₄	—	2	6	no	—
242	C₂×D₄	—	4	24	no	—
243	C₂²×C₄	—	2	12	no	—
244	C₂²×C₄	—	2	96	no	—
245	C₂²×C₄	—	2	24	no	—
246	C₂²×C₄	—	2	24	no	—
247	C₂²×C₄	—	2	12	no	—
248	C₂²×C₄	C₂×D₄×C₃₁	1	24	yes	—
249	C₂²×C₄	C₂×D₄×C₃₁	1	24	yes	—
250	C₂²×C₄	C₂×D₄×C₃₁	1	12	yes	—
251	C₂²×C₄	C₂×D₄×C₃₁	1	12	yes	—
252	C₂²×C₄	C₂⁴×C₃₁	1	12	yes	—
253	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
254	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
255	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
256	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
257	C₄⋊C₄	—	2	12	no	—
258	C₄⋊C₄	—	2	24	no	—
259	C₂²⋊C₄	—	2	12	no	—
260	C₂²⋊C₄	—	2	24	no	—
261	C₂²⋊C₄	—	2	24	no	—
262	C₂²⋊C₄	—	4	24	no	—
263	C₄²	—	2	12	no	—
264	C₂³	—	2	4	no	—
265	C₂³	—	2	24	no	—
266	C₂³	—	2	12	no	—
267	C₂³	—	2	12	no	—
268	C₂³	—	2	12	no	—
269	C₂³	—	2	6	no	—
270	C₂³	—	2	8	no	—
271	C₂³	—	4	24	no	—
272	C₂³	—	4	48	no	—
273	C₂³	C₂²×D₄×C₃₁	1	12	yes	—
274	C₂³	C₂²×D₄×C₃₁	1	6	yes	—
275	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
276	Q₈	—	2	32	no	—
277	Q₈	—	4	4	no	—
278	D₄	—	2	96	no	—
279	D₄	—	4	48	no	—
280	D₄	—	4	8	no	—
281	D₄	—	4	4	no	—
282	C₂×C₄	—	2	24	no	—
283	C₂×C₄	—	2	12	no	—
284	C₂×C₄	—	2	48	no	—
285	C₂×C₄	—	2	48	no	—
286	C₂×C₄	—	2	12	no	—
287	C₂×C₄	—	2	6	no	—
288	C₂×C₄	—	2	24	no	—
289	C₂×C₄	—	4	8	no	—
290	C₂×C₄	C₃₁×#120	1	16	yes	—
291	C₂×C₄	C₂²×D₄×C₃₁	1	12	yes	—
292	C₂×C₄	C₂²×D₄×C₃₁	1	6	yes	—
293	C₂×C₄	C₂²×D₄×C₃₁	1	6	yes	—
294	C₂²	—	2	12	no	—
295	C₂²	—	2	12	no	—
296	C₂²	—	2	12	no	—
297	C₂²	—	2	12	no	—
298	C₂²	—	4	48	no	—
299	C₂²	—	4	24	no	—
300	C₂²	—	4	2	no	—
301	C₂²	—	4	12	no	—
302	C₂²	C₂×C₃₁×#120	1	4	yes	—
303	C₂²	C₂³×D₄×C₃₁	1	3	yes	—
304	C₂²	C₃₁×#585	1	4	yes	—
305	C₄	—	2	12	no	—
306	C₄	—	4	2	no	—
307	C₄	—	4	12	no	—
308	C₄	C₂×C₃₁×#120	1	4	yes	—
309	C₄	C₂×C₃₁×#120	1	4	yes	—
310	C₂	—	2	4	no	—
311	C₂	—	4	12	no	—
312	C₂	—	8	2	no	—
313	C₂	C₂²×C₃₁×#120	1	1	yes	characteristic
314	C₂	C₃₁×#3449	1	6	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃₁
socle	C₂³×C₃₁	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×#41×C₃₁	C₂	semidirect
2. C₂×C₃₁×#525	C₂	semidirect
3. C₃₁×#2616	C₂	semidirect
4. C₃₁×#2615	C₂	semidirect
5. C₃₁×#2614	C₂	semidirect
6. C₃₁×#2613	C₂	semidirect
7. C₃₁×#2388	C₂	semidirect
8. C₃₁×#2280	C₂	semidirect
18. C₂×C₃₁×C₂²⋊Q₈	C₂²	semidirect
19. C₂×C₃₁×#121	C₂²	semidirect
21. C₂×C₄×D₄×C₃₁	C₂²	semidirect
23. C₃₁×#395	C₂²	semidirect
24. C₃₁×#394	C₂²	semidirect
25. C₃₁×#393	C₂²	semidirect
26. C₃₁×#392	C₂²	semidirect
28. C₃₁×#380	C₂²	semidirect
30. C₄×C₂²⋊C₄×C₃₁	C₂²	semidirect
46. C₂³×C₄×C₃₁	D₄	semidirect
106. #34632	C₃₁	direct (N × H)
137. C₂³×#41	C₂×C₃₁	semidirect
138. C₂×#525	C₂×C₃₁	semidirect
139. #2616	C₂×C₃₁	semidirect
140. #2615	C₂×C₃₁	semidirect
141. #2614	C₂×C₃₁	semidirect
142. #2613	C₂×C₃₁	semidirect
143. #2388	C₂×C₃₁	semidirect
144. #2280	C₂×C₃₁	semidirect
170. C₂×C₂²⋊Q₈	C₂²×C₃₁	semidirect
171. C₂×#121	C₂²×C₃₁	semidirect
173. C₂×C₄×D₄	C₂²×C₃₁	semidirect
175. #395	C₂²×C₃₁	semidirect
176. #394	C₂²×C₃₁	semidirect
177. #393	C₂²×C₃₁	semidirect
178. #392	C₂²×C₃₁	semidirect
180. #380	C₂²×C₃₁	semidirect
182. C₄×C₂²⋊C₄	C₂²×C₃₁	semidirect
203. C₂³×C₄	D₄×C₃₁	semidirect
223. C₃₁	#34632	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#41×C₃₁	2
C₂×C₃₁×#525	2
C₃₁×#2616	2
C₃₁×#2615	2
C₃₁×#2614	2
C₃₁×#2613	2
C₃₁×#2388	2
C₃₁×#2280	2
#34632	31

Subgroups — C31×#34632 #21491391

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34632 #21491391