Subgroups — C₃₁×#8025 #21464784

Order 7936.

654 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#431	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2283	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#383	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1294	C₂	1	4	yes	maximal
5	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
6	C₃₁×#431	—	2	2	no	—
7	C₃₁×#431	C₄	1	4	yes	complemented
8	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
9	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₃₁	—	2	1	no	—
11	C₃₁×#383	—	2	2	no	—
12	C₃₁×#383	—	2	1	no	—
13	C₃₁×#383	C₄	1	4	yes	complemented
14	C₃₁×#382	C₄	1	2	yes	complemented
15	C₃₁×#381	—	2	1	no	—
16	C₃₁×#380	C₂²	1	2	yes	—
17	C₄×C₄⋊C₄×C₃₁	C₄	1	2	yes	complemented
18	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
19	C₃₁×#376	—	2	1	no	—
20	C₂×C₃₁×#95	—	2	1	no	—
21	C₂×C₃₁×#95	—	2	1	no	—
22	C₂×C₃₁×#95	C₂²	1	2	yes	—
23	C₂³×C₄×C₃₁	—	2	2	no	—
24	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₃₁	—	2	2	no	—
28	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
29	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
30	C₂×C₄²×C₃₁	—	2	2	no	—
31	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₃₁	D₄	1	2	yes	—
38	C₂×C₄²×C₃₁	Q₈	1	2	yes	—
39	C₃₁×C₄⋊C₈	—	2	4	no	—
40	C₃₁×C₂²⋊C₈	—	2	2	no	—
41	C₃₁×C₂²⋊C₈	C₂×C₄	1	4	yes	—
42	C₄×C₈×C₃₁	—	2	4	no	—
43	C₃₁×#95	—	2	4	no	—
44	C₃₁×#95	—	2	4	no	—
45	C₃₁×#95	—	2	2	no	—
46	C₃₁×#95	—	2	2	no	—
47	C₃₁×#95	—	2	2	no	—
48	C₃₁×#95	—	2	2	no	—
49	C₃₁×#95	—	2	2	no	—
50	C₃₁×#95	C₂×C₄	1	4	yes	—
51	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₃₁	—	2	4	no	—
53	C₂²×C₄×C₃₁	—	2	4	no	—
54	C₂²×C₄×C₃₁	—	2	4	no	—
55	C₂²×C₄×C₃₁	—	4	2	no	—
56	C₂²×C₄×C₃₁	—	4	2	no	—
57	C₂²×C₄×C₃₁	—	4	4	no	—
58	C₂²×C₄×C₃₁	—	4	4	no	—
59	C₂²×C₄×C₃₁	—	4	4	no	—
60	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
74	C₂×C₈×C₃₁	—	2	4	no	—
75	C₂×C₈×C₃₁	—	4	4	no	—
76	C₄⋊C₄×C₃₁	—	4	4	no	—
77	C₂²⋊C₄×C₃₁	—	4	4	no	—
78	C₄²×C₃₁	—	4	4	no	—
79	C₄²×C₃₁	C₄²	1	2	yes	—
80	C₄²×C₃₁	C₄²	1	2	yes	—
81	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
83	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
84	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
85	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
86	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
87	#8025	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂³×C₃₁	—	2	2	no	—
89	C₂³×C₃₁	—	2	2	no	—
90	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂³×C₃₁	#135	1	4	yes	—
94	C₂×C₄×C₃₁	—	2	2	no	—
95	C₂×C₄×C₃₁	—	2	2	no	—
96	C₂×C₄×C₃₁	—	2	4	no	—
97	C₂×C₄×C₃₁	—	2	2	no	—
98	C₂×C₄×C₃₁	—	2	2	no	—
99	C₂×C₄×C₃₁	—	2	2	no	—
100	C₂×C₄×C₃₁	—	2	2	no	—
101	C₂×C₄×C₃₁	—	2	4	no	—
102	C₂×C₄×C₃₁	—	4	2	no	—
103	C₂×C₄×C₃₁	—	4	2	no	—
104	C₂×C₄×C₃₁	—	4	1	no	—
105	C₂×C₄×C₃₁	—	4	1	no	—
106	C₂×C₄×C₃₁	—	4	1	no	—
107	C₂×C₄×C₃₁	—	4	1	no	—
108	C₂×C₄×C₃₁	—	4	4	no	—
109	C₂×C₄×C₃₁	—	4	4	no	—
110	C₂×C₄×C₃₁	—	4	4	no	—
111	C₂×C₄×C₃₁	—	4	4	no	—
112	C₂×C₄×C₃₁	—	4	4	no	—
113	C₂×C₄×C₃₁	—	4	4	no	—
114	C₂×C₄×C₃₁	—	8	8	no	—
115	C₂×C₄×C₃₁	#95	1	2	yes	—
116	C₂×C₄×C₃₁	#95	1	2	yes	—
117	C₂×C₄×C₃₁	#95	1	2	yes	—
118	C₂×C₄×C₃₁	#95	1	2	yes	—
119	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
120	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₈×C₃₁	—	4	4	no	—
124	C₂×#431	C₂×C₃₁	1	1	yes	characteristic
125	#2283	C₂×C₃₁	1	1	yes	characteristic
126	C₂×#383	C₂×C₃₁	1	1	yes	characteristic
127	#1294	C₂×C₃₁	1	4	yes	—
128	C₂²×C₃₁	—	2	1	no	—
129	C₂²×C₃₁	—	2	1	no	—
130	C₂²×C₃₁	—	2	1	no	—
131	C₂²×C₃₁	—	2	1	no	—
132	C₂²×C₃₁	—	2	8	no	—
133	C₂²×C₃₁	C₂×#95	1	1	yes	characteristic
134	C₂²×C₃₁	#410	1	2	yes	—
135	C₂²×C₃₁	#416	1	2	yes	—
136	C₂²×C₃₁	#416	1	2	yes	—
137	C₂²×C₃₁	#421	1	2	yes	—
138	C₂²×C₃₁	#427	1	2	yes	—
139	C₄×C₃₁	—	2	2	no	—
140	C₄×C₃₁	—	2	2	no	—
141	C₄×C₃₁	—	2	2	no	—
142	C₄×C₃₁	—	2	2	no	—
143	C₄×C₃₁	—	2	1	no	—
144	C₄×C₃₁	—	2	1	no	—
145	C₄×C₃₁	—	2	1	no	—
146	C₄×C₃₁	—	2	1	no	—
147	C₄×C₃₁	—	8	4	no	—
148	C₄×C₃₁	—	8	4	no	—
149	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
150	#431	—	2	2	no	—
151	#431	C₄×C₃₁	1	4	yes	complemented
152	C₂×C₄⋊C₈	—	2	1	no	—
153	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
154	C₂×C₄×C₈	—	2	1	no	—
155	#383	—	2	2	no	—
156	#383	—	2	1	no	—
157	#383	C₄×C₃₁	1	4	yes	complemented
158	#382	C₄×C₃₁	1	2	yes	complemented
159	#381	—	2	1	no	—
160	#380	C₂²×C₃₁	1	2	yes	—
161	C₄×C₄⋊C₄	C₄×C₃₁	1	2	yes	complemented
162	C₄×C₂²⋊C₄	—	2	1	no	—
163	#376	—	2	1	no	—
164	C₂×#95	—	2	1	no	—
165	C₂×#95	—	2	1	no	—
166	C₂×#95	C₂²×C₃₁	1	2	yes	—
167	C₂×C₃₁	—	2	2	no	—
168	C₂×C₃₁	—	2	2	no	—
169	C₂×C₃₁	#1521	1	4	yes	—
170	C₂×C₃₁	#1733	1	1	yes	characteristic
171	C₂×C₃₁	#1735	1	1	yes	characteristic
172	C₂×C₃₁	#1740	1	1	yes	characteristic
173	C₂³×C₄	—	2	2	no	—
174	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
175	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
176	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
177	C₂²×C₈	—	2	2	no	—
178	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
179	C₂×C₂²⋊C₄	—	2	2	no	—
180	C₂×C₄²	—	2	2	no	—
181	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
182	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
183	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
184	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
185	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
186	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
187	C₂×C₄²	D₄×C₃₁	1	2	yes	—
188	C₂×C₄²	Q₈×C₃₁	1	2	yes	—
189	C₄⋊C₈	—	2	4	no	—
190	C₂²⋊C₈	—	2	2	no	—
191	C₂²⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
192	C₄×C₈	—	2	4	no	—
193	#95	—	2	4	no	—
194	#95	—	2	4	no	—
195	#95	—	2	2	no	—
196	#95	—	2	2	no	—
197	#95	—	2	2	no	—
198	#95	—	2	2	no	—
199	#95	—	2	2	no	—
200	#95	C₂×C₄×C₃₁	1	4	yes	—
201	C₃₁	#8025	1	1	yes	characteristic, direct product factor, Sylow
202	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	4	2	no	—
207	C₂²×C₄	—	4	2	no	—
208	C₂²×C₄	—	4	4	no	—
209	C₂²×C₄	—	4	4	no	—
210	C₂²×C₄	—	4	4	no	—
211	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
212	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
213	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
214	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
215	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
216	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
217	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
218	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
219	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
220	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
221	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
224	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	4	4	no	—
227	C₄⋊C₄	—	4	4	no	—
228	C₂²⋊C₄	—	4	4	no	—
229	C₄²	—	4	4	no	—
230	C₄²	C₄²×C₃₁	1	2	yes	—
231	C₄²	C₄²×C₃₁	1	2	yes	—
232	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
233	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
234	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
235	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
236	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
237	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
238	C₂³	—	2	2	no	—
239	C₂³	—	2	2	no	—
240	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
241	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
242	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
243	C₂³	C₃₁×#135	1	4	yes	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	8	8	no	—
265	C₂×C₄	C₃₁×#95	1	2	yes	—
266	C₂×C₄	C₃₁×#95	1	2	yes	—
267	C₂×C₄	C₃₁×#95	1	2	yes	—
268	C₂×C₄	C₃₁×#95	1	2	yes	—
269	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
270	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
271	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
272	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
273	C₈	—	4	4	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	8	no	—
279	C₂²	C₂×C₃₁×#95	1	1	yes	characteristic
280	C₂²	C₃₁×#410	1	2	yes	—
281	C₂²	C₃₁×#416	1	2	yes	—
282	C₂²	C₃₁×#416	1	2	yes	—
283	C₂²	C₃₁×#421	1	2	yes	—
284	C₂²	C₃₁×#427	1	2	yes	—
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	8	4	no	—
294	C₄	—	8	4	no	—
295	C₂	—	2	2	no	—
296	C₂	—	2	2	no	—
297	C₂	C₃₁×#1521	1	4	yes	—
298	C₂	C₃₁×#1733	1	1	yes	characteristic
299	C₂	C₃₁×#1735	1	1	yes	characteristic
300	C₂	C₃₁×#1740	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₃₁
prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
7. C₃₁×#431	C₄	semidirect
13. C₃₁×#383	C₄	semidirect
14. C₃₁×#382	C₄	semidirect
17. C₄×C₄⋊C₄×C₃₁	C₄	semidirect
87. #8025	C₃₁	direct (N × H)
151. #431	C₄×C₃₁	semidirect
157. #383	C₄×C₃₁	semidirect
158. #382	C₄×C₃₁	semidirect
161. C₄×C₄⋊C₄	C₄×C₃₁	semidirect
201. C₃₁	#8025	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#431	2
C₃₁×#2283	2
C₂×C₃₁×#383	2
C₃₁×#1294	2
#8025	31

Subgroups — C31×#8025 #21464784

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8025 #21464784