Subgroups — C₃₁×#8275 #21465034

Order 7936.

446 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×C₄⋊C₈	C₂	1	2	yes	maximal
2	C₄×C₃₁×#97	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#335	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#334	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1277	C₂	1	2	yes	maximal
6	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
7	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₃₁×#97	C₄	1	2	yes	—
9	C₂×C₃₁×#97	C₄	1	2	yes	—
10	C₂×C₄×C₈×C₃₁	—	2	2	no	—
11	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₄³×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#336	—	2	2	no	—
15	C₃₁×#335	C₄	1	4	yes	complemented
16	C₃₁×#334	C₄	1	4	yes	complemented
17	C₂²×C₈×C₃₁	—	2	2	no	—
18	C₂²×C₈×C₃₁	—	2	2	no	—
19	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
21	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₃₁	D₄	1	2	yes	—
24	C₂×C₄²×C₃₁	Q₈	1	2	yes	—
25	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
26	C₃₁×C₄⋊C₈	—	2	2	no	—
27	C₃₁×C₄⋊C₈	—	2	2	no	—
28	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
29	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
30	C₃₁×#97	—	2	4	no	—
31	C₃₁×#97	C₈	1	8	yes	complemented
32	C₄×C₈×C₃₁	—	2	1	no	—
33	C₄×C₈×C₃₁	—	2	1	no	—
34	C₄×C₈×C₃₁	—	4	4	no	—
35	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
38	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₈×C₃₁	—	2	2	no	—
42	C₂×C₈×C₃₁	—	2	2	no	—
43	C₂×C₈×C₃₁	—	2	2	no	—
44	C₂×C₈×C₃₁	—	2	4	no	—
45	C₂×C₈×C₃₁	—	2	4	no	—
46	C₂×C₈×C₃₁	—	4	2	no	—
47	C₂×C₈×C₃₁	—	4	2	no	—
48	C₂×C₈×C₃₁	—	4	2	no	—
49	C₂×C₈×C₃₁	—	4	2	no	—
50	C₂×C₈×C₃₁	C₄²	1	2	yes	—
51	C₂×C₈×C₃₁	C₄²	1	2	yes	—
52	C₂×C₈×C₃₁	C₂×C₈	1	4	yes	—
53	C₂×C₈×C₃₁	C₂×C₈	1	4	yes	—
54	C₄²×C₃₁	—	2	2	no	—
55	C₄²×C₃₁	—	2	2	no	—
56	C₄²×C₃₁	—	4	2	no	—
57	C₄²×C₃₁	C₂×C₈	1	4	yes	—
58	C₄²×C₃₁	M₁₆	1	4	yes	—
59	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
61	C₄²×C₃₁	#41	1	1	yes	characteristic
62	C₄²×C₃₁	#41	1	1	yes	characteristic
63	#8275	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₃₁	—	2	4	no	—
66	C₂×C₄×C₃₁	—	2	4	no	—
67	C₂×C₄×C₃₁	—	2	2	no	—
68	C₂×C₄×C₃₁	C₄⋊C₈	1	8	yes	—
69	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
70	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
71	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
73	C₂×C₄×C₃₁	C₄×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₃₁	C₄×Q₈	1	1	yes	characteristic
75	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
76	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
77	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
78	C₂×C₄×C₃₁	#137	1	2	yes	—
79	C₈×C₃₁	—	2	2	no	—
80	C₈×C₃₁	—	2	4	no	—
81	C₈×C₃₁	—	4	4	no	—
82	C₈×C₃₁	—	4	4	no	—
83	C₈×C₃₁	C₄×C₈	1	4	yes	complemented
84	C₄×C₄⋊C₈	C₂×C₃₁	1	2	yes	—
85	C₄×#97	C₂×C₃₁	1	1	yes	characteristic
86	C₂×#335	C₂×C₃₁	1	1	yes	characteristic
87	C₂×#334	C₂×C₃₁	1	1	yes	characteristic
88	#1277	C₂×C₃₁	1	2	yes	—
89	C₂²×C₃₁	C₄×C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₃₁	C₂×C₄⋊C₈	1	2	yes	—
91	C₂²×C₃₁	#428	1	2	yes	—
92	C₂²×C₃₁	#430	1	2	yes	—
93	C₄×C₃₁	—	2	8	no	—
94	C₄×C₃₁	—	2	1	no	—
95	C₄×C₃₁	—	2	1	no	—
96	C₄×C₃₁	C₂×C₄×C₈	1	1	yes	characteristic
97	C₄×C₃₁	C₄×M₁₆	1	1	yes	characteristic
98	C₄×C₃₁	#431	1	2	yes	—
99	C₄×C₃₁	#440	1	2	yes	—
100	C₄×C₃₁	#441	1	1	yes	characteristic
101	C₄×C₃₁	#442	1	1	yes	characteristic
102	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
103	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
104	C₂×#97	C₄×C₃₁	1	2	yes	—
105	C₂×#97	C₄×C₃₁	1	2	yes	—
106	C₂×C₄×C₈	—	2	2	no	—
107	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
108	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
109	C₄³	C₂²×C₃₁	1	1	yes	characteristic
110	#336	—	2	2	no	—
111	#335	C₄×C₃₁	1	4	yes	complemented
112	#334	C₄×C₃₁	1	4	yes	complemented
113	C₂×C₃₁	#1565	1	4	yes	—
114	C₂×C₃₁	C₄×C₄⋊C₈	1	1	yes	characteristic
115	C₂×C₃₁	#1776	1	1	yes	characteristic
116	C₂×C₃₁	#1778	1	1	yes	characteristic
117	C₂²×C₈	—	2	2	no	—
118	C₂²×C₈	—	2	2	no	—
119	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
120	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
121	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
122	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
123	C₂×C₄²	D₄×C₃₁	1	2	yes	—
124	C₂×C₄²	Q₈×C₃₁	1	2	yes	—
125	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
126	C₄⋊C₈	—	2	2	no	—
127	C₄⋊C₈	—	2	2	no	—
128	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
129	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
130	#97	—	2	4	no	—
131	#97	C₈×C₃₁	1	8	yes	complemented
132	C₄×C₈	—	2	1	no	—
133	C₄×C₈	—	2	1	no	—
134	C₄×C₈	—	4	4	no	—
135	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
136	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
137	C₃₁	#8275	1	1	yes	characteristic, direct product factor, Sylow
138	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
139	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
140	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
141	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	2	4	no	—
146	C₂×C₈	—	2	4	no	—
147	C₂×C₈	—	4	2	no	—
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	—	4	2	no	—
151	C₂×C₈	C₄²×C₃₁	1	2	yes	—
152	C₂×C₈	C₄²×C₃₁	1	2	yes	—
153	C₂×C₈	C₂×C₈×C₃₁	1	4	yes	—
154	C₂×C₈	C₂×C₈×C₃₁	1	4	yes	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	2	no	—
157	C₄²	—	4	2	no	—
158	C₄²	C₂×C₈×C₃₁	1	4	yes	—
159	C₄²	M₁₆×C₃₁	1	4	yes	—
160	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
161	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
162	C₄²	#41×C₃₁	1	1	yes	characteristic
163	C₄²	#41×C₃₁	1	1	yes	characteristic
164	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	C₃₁×C₄⋊C₈	1	8	yes	—
169	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
170	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
171	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
172	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
173	C₂×C₄	C₄×Q₈×C₃₁	1	1	yes	characteristic
174	C₂×C₄	C₄×Q₈×C₃₁	1	1	yes	characteristic
175	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
176	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
177	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
178	C₂×C₄	C₃₁×#137	1	2	yes	—
179	C₈	—	2	2	no	—
180	C₈	—	2	4	no	—
181	C₈	—	4	4	no	—
182	C₈	—	4	4	no	—
183	C₈	C₄×C₈×C₃₁	1	4	yes	complemented
184	C₂²	C₄×C₄⋊C₄×C₃₁	1	1	yes	characteristic
185	C₂²	C₂×C₃₁×C₄⋊C₈	1	2	yes	—
186	C₂²	C₃₁×#428	1	2	yes	—
187	C₂²	C₃₁×#430	1	2	yes	—
188	C₄	—	2	8	no	—
189	C₄	—	2	1	no	—
190	C₄	—	2	1	no	—
191	C₄	C₂×C₄×C₈×C₃₁	1	1	yes	characteristic
192	C₄	C₄×M₁₆×C₃₁	1	1	yes	characteristic
193	C₄	C₃₁×#431	1	2	yes	—
194	C₄	C₃₁×#440	1	2	yes	—
195	C₄	C₃₁×#441	1	1	yes	characteristic
196	C₄	C₃₁×#442	1	1	yes	characteristic
197	C₂	C₃₁×#1565	1	4	yes	—
198	C₂	C₄×C₃₁×C₄⋊C₈	1	1	yes	characteristic
199	C₂	C₃₁×#1776	1	1	yes	characteristic
200	C₂	C₃₁×#1778	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator	C₄	C₂×C₄×C₈×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle, prime radical	C₂³×C₃₁	C₂×C₄⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
15. C₃₁×#335	C₄	semidirect
16. C₃₁×#334	C₄	semidirect
31. C₃₁×#97	C₈	semidirect
63. #8275	C₃₁	direct (N × H)
83. C₈×C₃₁	C₄×C₈	semidirect
111. #335	C₄×C₃₁	semidirect
112. #334	C₄×C₃₁	semidirect
131. #97	C₈×C₃₁	semidirect
137. C₃₁	#8275	direct (N × H)
183. C₈	C₄×C₈×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₃₁×C₄⋊C₈	2
C₄×C₃₁×#97	2
C₂×C₃₁×#335	2
C₂×C₃₁×#334	2
C₃₁×#1277	2
#8275	31

Subgroups — C31×#8275 #21465034

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8275 #21465034