Subgroups — C₃₁×#23810 #21480569

Order 7936.

1150 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#442	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₈×C₃₁	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2328	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2092	C₂	1	4	yes	maximal, complemented
5	C₃₁×#2011	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1746	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
8	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
9	C₂×C₃₁×#127	C₂²	1	2	yes	—
10	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
11	C₃₁×#442	—	2	8	no	—
12	C₄×D₈×C₃₁	—	2	8	no	—
13	C₂×C₃₁×#107	C₂²	1	2	yes	—
14	C₂×C₃₁×#106	C₂²	1	2	yes	—
15	C₂×C₃₁×#102	C₂²	1	4	yes	—
16	C₂×C₃₁×#102	C₂²	1	4	yes	—
17	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#390	C₂²	1	4	yes	—
20	C₃₁×#386	—	2	8	no	—
21	C₃₁×#382	C₂²	1	2	yes	—
22	C₃₁×#376	C₂²	1	1	yes	characteristic
23	C₃₁×#336	C₂²	1	4	yes	—
24	C₂²×D₄×C₃₁	—	2	4	no	—
25	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
26	C₂³×C₄×C₃₁	—	2	4	no	—
27	C₂×D₈×C₃₁	—	2	8	no	—
28	C₂×D₈×C₃₁	—	2	8	no	—
29	C₂×D₈×C₃₁	—	2	4	no	—
30	C₂×D₈×C₃₁	—	2	4	no	—
31	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
32	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
33	C₃₁×#127	—	2	8	no	—
34	C₄×D₄×C₃₁	—	2	8	no	—
35	C₄×D₄×C₃₁	—	2	8	no	—
36	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
37	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
38	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
39	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
40	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
41	C₃₁×#107	—	2	4	no	—
42	C₃₁×#106	—	2	4	no	—
43	C₃₁×#102	—	2	8	no	—
44	C₃₁×#102	—	2	8	no	—
45	C₃₁×#97	D₄	1	4	yes	complemented
46	C₄×C₈×C₃₁	D₄	1	4	yes	complemented
47	C₃₁×#95	—	2	4	no	—
48	C₃₁×#95	D₄	1	4	yes	—
49	C₂⁴×C₃₁	—	2	4	no	—
50	C₂×D₄×C₃₁	—	2	16	no	—
51	C₂×D₄×C₃₁	—	2	4	no	—
52	C₂×D₄×C₃₁	—	2	4	no	—
53	C₂×D₄×C₃₁	—	2	4	no	—
54	C₂×D₄×C₃₁	—	2	4	no	—
55	C₂×D₄×C₃₁	—	2	4	no	—
56	C₂×D₄×C₃₁	—	8	8	no	—
57	C₂×D₄×C₃₁	#41	1	8	yes	—
58	C₂²×C₄×C₃₁	—	2	4	no	—
59	C₂²×C₄×C₃₁	—	4	8	no	—
60	C₂²×C₄×C₃₁	—	8	8	no	—
61	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
62	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
64	D₈×C₃₁	—	2	16	no	—
65	D₈×C₃₁	—	2	16	no	—
66	C₂×C₈×C₃₁	—	2	4	no	—
67	C₂×C₈×C₃₁	—	2	4	no	—
68	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
69	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
70	C₄⋊C₄×C₃₁	—	2	4	no	—
71	C₄⋊C₄×C₃₁	—	2	4	no	—
72	C₂²⋊C₄×C₃₁	—	4	8	no	—
73	C₂²⋊C₄×C₃₁	—	8	8	no	—
74	C₄²×C₃₁	—	2	4	no	—
75	C₄²×C₃₁	C₂×D₄	1	2	yes	—
76	C₄²×C₃₁	C₂×D₄	1	2	yes	—
77	#23810	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂³×C₃₁	—	2	8	no	—
79	C₂³×C₃₁	—	4	4	no	—
80	C₂³×C₃₁	—	4	4	no	—
81	C₂³×C₃₁	—	4	8	no	—
82	C₂³×C₃₁	—	4	8	no	—
83	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
84	D₄×C₃₁	—	2	8	no	—
85	D₄×C₃₁	—	2	8	no	—
86	D₄×C₃₁	—	4	4	no	—
87	D₄×C₃₁	—	4	4	no	—
88	D₄×C₃₁	—	8	16	no	—
89	C₂×C₄×C₃₁	—	2	4	no	—
90	C₂×C₄×C₃₁	—	2	4	no	—
91	C₂×C₄×C₃₁	—	2	8	no	—
92	C₂×C₄×C₃₁	—	4	4	no	—
93	C₂×C₄×C₃₁	—	4	8	no	—
94	C₂×C₄×C₃₁	—	8	16	no	—
95	C₂×C₄×C₃₁	#127	1	4	yes	—
96	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
97	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
99	C₈×C₃₁	—	2	4	no	—
100	C₈×C₃₁	—	2	4	no	—
101	C₂×#442	C₂×C₃₁	1	2	yes	complemented
102	C₂×C₄×D₈	C₂×C₃₁	1	2	yes	complemented
103	#2328	C₂×C₃₁	1	2	yes	complemented
104	#2092	C₂×C₃₁	1	4	yes	complemented
105	#2011	C₂×C₃₁	1	4	yes	complemented
106	#1746	C₂×C₃₁	1	1	yes	characteristic, complemented
107	C₂²×C₃₁	—	4	16	no	—
108	C₂²×C₃₁	—	4	8	no	—
109	C₂²×C₃₁	—	8	8	no	—
110	C₂²×C₃₁	—	8	4	no	—
111	C₂²×C₃₁	—	8	4	no	—
112	C₂²×C₃₁	#525	1	2	yes	—
113	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
114	C₂²×C₃₁	#576	1	4	yes	—
115	C₄×C₃₁	—	2	4	no	—
116	C₄×C₃₁	—	4	4	no	—
117	C₄×C₃₁	—	8	4	no	—
118	C₄×C₃₁	#525	1	2	yes	—
119	C₄×C₃₁	#525	1	2	yes	—
120	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
121	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
122	C₂×#127	C₂²×C₃₁	1	2	yes	—
123	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
124	#442	—	2	8	no	—
125	C₄×D₈	—	2	8	no	—
126	C₂×#107	C₂²×C₃₁	1	2	yes	—
127	C₂×#106	C₂²×C₃₁	1	2	yes	—
128	C₂×#102	C₂²×C₃₁	1	4	yes	—
129	C₂×#102	C₂²×C₃₁	1	4	yes	—
130	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
131	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
132	#390	C₂²×C₃₁	1	4	yes	—
133	#386	—	2	8	no	—
134	#382	C₂²×C₃₁	1	2	yes	—
135	#376	C₂²×C₃₁	1	1	yes	characteristic
136	#336	C₂²×C₃₁	1	4	yes	—
137	C₂×C₃₁	—	8	8	no	—
138	C₂×C₃₁	#2397	1	1	yes	characteristic
139	C₂×C₃₁	#3122	1	4	yes	—
140	C₂×C₃₁	#3155	1	2	yes	—
141	C₂²×D₄	—	2	4	no	—
142	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
143	C₂³×C₄	—	2	4	no	—
144	C₂×D₈	—	2	8	no	—
145	C₂×D₈	—	2	8	no	—
146	C₂×D₈	—	2	4	no	—
147	C₂×D₈	—	2	4	no	—
148	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
149	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
150	#127	—	2	8	no	—
151	C₄×D₄	—	2	8	no	—
152	C₄×D₄	—	2	8	no	—
153	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
154	C₂×C₂²⋊C₄	—	2	4	no	—
155	C₂×C₂²⋊C₄	—	2	8	no	—
156	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
157	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
158	#107	—	2	4	no	—
159	#106	—	2	4	no	—
160	#102	—	2	8	no	—
161	#102	—	2	8	no	—
162	#97	D₄×C₃₁	1	4	yes	complemented
163	C₄×C₈	D₄×C₃₁	1	4	yes	complemented
164	#95	—	2	4	no	—
165	#95	D₄×C₃₁	1	4	yes	—
166	C₃₁	#23810	1	1	yes	characteristic, direct product factor, Sylow
167	C₂⁴	—	2	4	no	—
168	C₂×D₄	—	2	16	no	—
169	C₂×D₄	—	2	4	no	—
170	C₂×D₄	—	2	4	no	—
171	C₂×D₄	—	2	4	no	—
172	C₂×D₄	—	2	4	no	—
173	C₂×D₄	—	2	4	no	—
174	C₂×D₄	—	8	8	no	—
175	C₂×D₄	#41×C₃₁	1	8	yes	—
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	4	8	no	—
178	C₂²×C₄	—	8	8	no	—
179	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
180	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
181	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
182	D₈	—	2	16	no	—
183	D₈	—	2	16	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
187	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	2	4	no	—
190	C₂²⋊C₄	—	4	8	no	—
191	C₂²⋊C₄	—	8	8	no	—
192	C₄²	—	2	4	no	—
193	C₄²	C₂×D₄×C₃₁	1	2	yes	—
194	C₄²	C₂×D₄×C₃₁	1	2	yes	—
195	C₂³	—	2	8	no	—
196	C₂³	—	4	4	no	—
197	C₂³	—	4	4	no	—
198	C₂³	—	4	8	no	—
199	C₂³	—	4	8	no	—
200	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
201	D₄	—	2	8	no	—
202	D₄	—	2	8	no	—
203	D₄	—	4	4	no	—
204	D₄	—	4	4	no	—
205	D₄	—	8	16	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	8	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	4	8	no	—
211	C₂×C₄	—	8	16	no	—
212	C₂×C₄	C₃₁×#127	1	4	yes	—
213	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
214	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
215	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
216	C₈	—	2	4	no	—
217	C₈	—	2	4	no	—
218	C₂²	—	4	16	no	—
219	C₂²	—	4	8	no	—
220	C₂²	—	8	8	no	—
221	C₂²	—	8	4	no	—
222	C₂²	—	8	4	no	—
223	C₂²	C₃₁×#525	1	2	yes	—
224	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
225	C₂²	C₃₁×#576	1	4	yes	—
226	C₄	—	2	4	no	—
227	C₄	—	4	4	no	—
228	C₄	—	8	4	no	—
229	C₄	C₃₁×#525	1	2	yes	—
230	C₄	C₃₁×#525	1	2	yes	—
231	C₂	—	8	8	no	—
232	C₂	C₃₁×#2397	1	1	yes	characteristic
233	C₂	C₃₁×#3122	1	4	yes	—
234	C₂	C₃₁×#3155	1	2	yes	—

	subgroup	G/H
center, socle	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
1. C₂×C₃₁×#442	C₂	semidirect
2. C₂×C₄×D₈×C₃₁	C₂	semidirect
3. C₃₁×#2328	C₂	semidirect
4. C₃₁×#2092	C₂	semidirect
5. C₃₁×#2011	C₂	semidirect
6. C₃₁×#1746	C₂	semidirect
45. C₃₁×#97	D₄	semidirect
46. C₄×C₈×C₃₁	D₄	semidirect
77. #23810	C₃₁	direct (N × H)
101. C₂×#442	C₂×C₃₁	semidirect
102. C₂×C₄×D₈	C₂×C₃₁	semidirect
103. #2328	C₂×C₃₁	semidirect
104. #2092	C₂×C₃₁	semidirect
105. #2011	C₂×C₃₁	semidirect
106. #1746	C₂×C₃₁	semidirect
162. #97	D₄×C₃₁	semidirect
163. C₄×C₈	D₄×C₃₁	semidirect
166. C₃₁	#23810	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×#442	2
C₂×C₄×D₈×C₃₁	2
C₃₁×#2328	2
C₃₁×#2092	2
C₃₁×#2011	2
C₃₁×#1746	2
#23810	31

Subgroups — C31×#23810 #21480569

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23810 #21480569