Subgroups — C₃₁×#10701 #21467460

Order 7936.

998 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#376	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#339	C₂	1	4	yes	maximal
5	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
6	C₂×C₃₁×#117	C₂²	1	2	yes	—
7	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
8	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
9	C₄×M₁₆×C₃₁	—	2	2	no	—
10	C₄×M₁₆×C₃₁	C₄	1	4	yes	complemented
11	C₂×C₃₁×#97	—	2	1	no	—
12	C₂×C₄×C₈×C₃₁	—	2	1	no	—
13	C₄×C₄⋊C₄×C₃₁	C₄	1	4	yes	complemented
14	C₄×C₂²⋊C₄×C₃₁	—	2	2	no	—
15	C₃₁×#376	—	2	2	no	—
16	C₃₁×#376	—	2	1	no	—
17	C₃₁×#376	C₄	1	4	yes	complemented
18	C₂×C₃₁×#95	—	2	2	no	—
19	C₄³×C₃₁	—	2	1	no	—
20	C₃₁×#339	C₄	1	16	yes	complemented
21	C₂³×C₄×C₃₁	—	2	2	no	—
22	C₂³×C₄×C₃₁	D₄	1	2	yes	—
23	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
24	C₂×M₁₆×C₃₁	—	2	2	no	—
25	C₂×M₁₆×C₃₁	—	2	4	no	—
26	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
27	C₂×M₁₆×C₃₁	C₂×C₄	1	8	yes	—
28	C₂²×C₈×C₃₁	—	2	2	no	—
29	C₃₁×#117	—	2	4	no	—
30	C₃₁×#117	C₂×C₄	1	8	yes	—
31	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
32	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
33	C₂×C₄²×C₃₁	—	2	2	no	—
34	C₂×C₄²×C₃₁	—	2	2	no	—
35	C₂×C₄²×C₃₁	—	2	4	no	—
36	C₂×C₄²×C₃₁	—	2	4	no	—
37	C₂×C₄²×C₃₁	C₂×C₄	1	8	yes	—
38	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₃₁	D₄	1	2	yes	—
43	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₃₁	Q₈	1	2	yes	—
46	C₃₁×#97	—	2	4	no	—
47	C₄×C₈×C₃₁	—	2	4	no	—
48	C₃₁×#95	—	2	8	no	—
49	C₃₁×#95	—	2	4	no	—
50	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₃₁	—	2	4	no	—
52	C₂²×C₄×C₃₁	—	2	4	no	—
53	C₂²×C₄×C₃₁	—	2	8	no	—
54	C₂²×C₄×C₃₁	—	4	4	no	—
55	C₂²×C₄×C₃₁	—	4	8	no	—
56	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
59	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
60	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
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	4	yes	—
64	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₃₁	#41	1	4	yes	—
68	M₁₆×C₃₁	—	2	4	no	—
69	M₁₆×C₃₁	—	2	8	no	—
70	M₁₆×C₃₁	C₄²	1	8	yes	complemented
71	C₂×C₈×C₃₁	—	2	4	no	—
72	C₂×C₈×C₃₁	—	2	8	no	—
73	C₄⋊C₄×C₃₁	—	2	4	no	—
74	C₄⋊C₄×C₃₁	C₄²	1	8	yes	—
75	C₂²⋊C₄×C₃₁	—	2	8	no	—
76	C₄²×C₃₁	—	2	4	no	—
77	C₄²×C₃₁	—	2	8	no	—
78	C₄²×C₃₁	—	2	8	no	—
79	C₄²×C₃₁	—	4	8	no	—
80	C₄²×C₃₁	C₄²	1	4	yes	—
81	C₄²×C₃₁	C₄²	1	8	yes	complemented
82	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
83	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
84	C₄²×C₃₁	C₄⋊C₄	1	4	yes	—
85	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
86	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
87	#10701	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂³×C₃₁	—	2	4	no	—
89	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂³×C₃₁	C₄×D₄	1	4	yes	—
93	C₂×C₄×C₃₁	—	2	4	no	—
94	C₂×C₄×C₃₁	—	2	4	no	—
95	C₂×C₄×C₃₁	—	2	8	no	—
96	C₂×C₄×C₃₁	—	2	8	no	—
97	C₂×C₄×C₃₁	—	4	4	no	—
98	C₂×C₄×C₃₁	—	4	4	no	—
99	C₂×C₄×C₃₁	—	4	8	no	—
100	C₂×C₄×C₃₁	—	4	16	no	—
101	C₂×C₄×C₃₁	—	4	8	no	—
102	C₂×C₄×C₃₁	#95	1	4	yes	—
103	C₂×C₄×C₃₁	#95	1	4	yes	—
104	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
105	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
106	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
107	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
108	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₃₁	#117	1	2	yes	—
112	C₂×C₄×C₃₁	#117	1	2	yes	—
113	C₂×C₄×C₃₁	#117	1	2	yes	—
114	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
115	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
116	C₂×C₄×C₃₁	C₄×Q₈	1	4	yes	—
117	C₈×C₃₁	—	2	8	no	—
118	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic
119	C₄×#117	C₂×C₃₁	1	1	yes	characteristic
120	C₂×#376	C₂×C₃₁	1	1	yes	characteristic
121	C₂×#339	C₂×C₃₁	1	4	yes	—
122	C₂²×C₃₁	—	2	2	no	—
123	C₂²×C₃₁	—	2	2	no	—
124	C₂²×C₃₁	—	2	8	no	—
125	C₂²×C₃₁	C₂×#95	1	1	yes	characteristic
126	C₂²×C₃₁	C₄×C₂²⋊C₄	1	2	yes	—
127	C₂²×C₃₁	C₄×C₂²⋊C₄	1	2	yes	—
128	C₂²×C₃₁	C₄×C₄⋊C₄	1	2	yes	—
129	C₂²×C₃₁	#421	1	4	yes	—
130	C₄×C₃₁	—	2	4	no	—
131	C₄×C₃₁	—	2	4	no	—
132	C₄×C₃₁	—	4	8	no	—
133	C₄×C₃₁	—	4	8	no	—
134	C₄×C₃₁	C₄³	1	1	yes	characteristic
135	C₄×C₃₁	#376	1	1	yes	characteristic
136	C₄×C₃₁	C₄×C₂²⋊C₄	1	2	yes	—
137	C₄×C₃₁	C₄×C₄⋊C₄	1	2	yes	—
138	C₄×C₃₁	C₄×C₄⋊C₄	1	2	yes	—
139	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
140	C₂×#117	C₂²×C₃₁	1	2	yes	—
141	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
142	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
143	C₄×M₁₆	—	2	2	no	—
144	C₄×M₁₆	C₄×C₃₁	1	4	yes	complemented
145	C₂×#97	—	2	1	no	—
146	C₂×C₄×C₈	—	2	1	no	—
147	C₄×C₄⋊C₄	C₄×C₃₁	1	4	yes	complemented
148	C₄×C₂²⋊C₄	—	2	2	no	—
149	#376	—	2	2	no	—
150	#376	—	2	1	no	—
151	#376	C₄×C₃₁	1	4	yes	complemented
152	C₂×#95	—	2	2	no	—
153	C₄³	—	2	1	no	—
154	#339	C₄×C₃₁	1	16	yes	complemented
155	C₂×C₃₁	—	2	2	no	—
156	C₂×C₃₁	—	2	2	no	—
157	C₂×C₃₁	C₄×#95	1	1	yes	characteristic
158	C₂×C₃₁	#1733	1	2	yes	—
159	C₂×C₃₁	#1759	1	4	yes	—
160	C₂³×C₄	—	2	2	no	—
161	C₂³×C₄	D₄×C₃₁	1	2	yes	—
162	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	—	2	4	no	—
165	C₂×M₁₆	C₂×C₄×C₃₁	1	8	yes	—
166	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
167	C₂²×C₈	—	2	2	no	—
168	#117	—	2	4	no	—
169	#117	C₂×C₄×C₃₁	1	8	yes	—
170	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
171	C₂×C₂²⋊C₄	—	2	2	no	—
172	C₂×C₄²	—	2	2	no	—
173	C₂×C₄²	—	2	2	no	—
174	C₂×C₄²	—	2	4	no	—
175	C₂×C₄²	—	2	4	no	—
176	C₂×C₄²	C₂×C₄×C₃₁	1	8	yes	—
177	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
178	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
179	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
180	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
181	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
182	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
183	C₂×C₄²	D₄×C₃₁	1	2	yes	—
184	C₂×C₄²	Q₈×C₃₁	1	2	yes	—
185	#97	—	2	4	no	—
186	C₄×C₈	—	2	4	no	—
187	#95	—	2	8	no	—
188	#95	—	2	4	no	—
189	C₃₁	#10701	1	1	yes	characteristic, direct product factor, Sylow
190	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	2	8	no	—
194	C₂²×C₄	—	4	4	no	—
195	C₂²×C₄	—	4	8	no	—
196	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
197	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
198	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
199	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
200	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
201	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
202	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
203	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
204	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
206	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
207	C₂²×C₄	#41×C₃₁	1	4	yes	—
208	M₁₆	—	2	4	no	—
209	M₁₆	—	2	8	no	—
210	M₁₆	C₄²×C₃₁	1	8	yes	complemented
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	8	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	C₄²×C₃₁	1	8	yes	—
215	C₂²⋊C₄	—	2	8	no	—
216	C₄²	—	2	4	no	—
217	C₄²	—	2	8	no	—
218	C₄²	—	2	8	no	—
219	C₄²	—	4	8	no	—
220	C₄²	C₄²×C₃₁	1	4	yes	—
221	C₄²	C₄²×C₃₁	1	8	yes	complemented
222	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
223	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
224	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
225	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
226	C₄²	C₄⋊C₄×C₃₁	1	4	yes	—
227	C₂³	—	2	4	no	—
228	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
229	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
230	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
231	C₂³	C₄×D₄×C₃₁	1	4	yes	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	8	no	—
235	C₂×C₄	—	2	8	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	8	no	—
239	C₂×C₄	—	4	16	no	—
240	C₂×C₄	—	4	8	no	—
241	C₂×C₄	C₃₁×#95	1	4	yes	—
242	C₂×C₄	C₃₁×#95	1	4	yes	—
243	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
244	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
245	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
246	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
247	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
248	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
249	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
250	C₂×C₄	C₃₁×#117	1	2	yes	—
251	C₂×C₄	C₃₁×#117	1	2	yes	—
252	C₂×C₄	C₃₁×#117	1	2	yes	—
253	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
254	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
255	C₂×C₄	C₄×Q₈×C₃₁	1	4	yes	—
256	C₈	—	2	8	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	8	no	—
260	C₂²	C₂×C₃₁×#95	1	1	yes	characteristic
261	C₂²	C₄×C₂²⋊C₄×C₃₁	1	2	yes	—
262	C₂²	C₄×C₂²⋊C₄×C₃₁	1	2	yes	—
263	C₂²	C₄×C₄⋊C₄×C₃₁	1	2	yes	—
264	C₂²	C₃₁×#421	1	4	yes	—
265	C₄	—	2	4	no	—
266	C₄	—	2	4	no	—
267	C₄	—	4	8	no	—
268	C₄	—	4	8	no	—
269	C₄	C₄³×C₃₁	1	1	yes	characteristic
270	C₄	C₃₁×#376	1	1	yes	characteristic
271	C₄	C₄×C₂²⋊C₄×C₃₁	1	2	yes	—
272	C₄	C₄×C₄⋊C₄×C₃₁	1	2	yes	—
273	C₄	C₄×C₄⋊C₄×C₃₁	1	2	yes	—
274	C₂	—	2	2	no	—
275	C₂	—	2	2	no	—
276	C₂	C₄×C₃₁×#95	1	1	yes	characteristic
277	C₂	C₃₁×#1733	1	2	yes	—
278	C₂	C₃₁×#1759	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator	C₄	C₄³×C₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₃₁
socle	C₂³×C₃₁	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₄×M₁₆×C₃₁	C₄	semidirect
13. C₄×C₄⋊C₄×C₃₁	C₄	semidirect
17. C₃₁×#376	C₄	semidirect
20. C₃₁×#339	C₄	semidirect
70. M₁₆×C₃₁	C₄²	semidirect
81. C₄²×C₃₁	C₄²	semidirect
87. #10701	C₃₁	direct (N × H)
144. C₄×M₁₆	C₄×C₃₁	semidirect
147. C₄×C₄⋊C₄	C₄×C₃₁	semidirect
151. #376	C₄×C₃₁	semidirect
154. #339	C₄×C₃₁	semidirect
189. C₃₁	#10701	direct (N × H)
210. M₁₆	C₄²×C₃₁	semidirect
221. C₄²	C₄²×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×M₁₆×C₃₁	2
C₄×C₃₁×#117	2
C₂×C₃₁×#376	2
C₂×C₃₁×#339	2
#10701	31

Subgroups — C31×#10701 #21467460

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#10701 #21467460