Subgroups — C₈×C₃₁×#95 #21467454

Order 7936.

1174 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₈×C₃₁	C₂	1	3	yes	maximal
2	C₂×C₃₁×#336	C₂	1	3	yes	maximal
3	C₄×C₃₁×#95	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₈×C₃₁	C₂²	1	3	yes	—
5	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
6	C₂²×C₄²×C₃₁	C₂²	1	3	yes	—
7	C₂×C₄×C₈×C₃₁	—	2	12	no	—
8	C₂×C₄×C₈×C₃₁	C₄	1	12	yes	complemented
9	C₂×C₃₁×#95	C₄	1	4	yes	—
10	C₃₁×#336	C₄	1	12	yes	complemented
11	C₂³×C₄×C₃₁	C₂×C₄	1	6	yes	—
12	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
13	C₂²×C₈×C₃₁	—	2	6	no	—
14	C₂²×C₈×C₃₁	—	2	3	no	—
15	C₂²×C₈×C₃₁	—	2	3	no	—
16	C₂²×C₈×C₃₁	C₂×C₄	1	6	yes	—
17	C₂²×C₈×C₃₁	C₂×C₄	1	12	yes	—
18	C₂²×C₈×C₃₁	C₂×C₄	1	6	yes	—
19	C₂²×C₈×C₃₁	D₄	1	6	yes	—
20	C₂²×C₈×C₃₁	Q₈	1	2	yes	—
21	C₂×C₄²×C₃₁	—	2	6	no	—
22	C₂×C₄²×C₃₁	—	2	6	no	—
23	C₂×C₄²×C₃₁	C₂×C₄	1	6	yes	—
24	C₂×C₄²×C₃₁	C₂×C₄	1	6	yes	—
25	C₄×C₈×C₃₁	—	2	24	no	—
26	C₃₁×#95	C₈	1	16	yes	direct product factor
27	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₃₁	—	2	12	no	—
29	C₂²×C₄×C₃₁	—	2	12	no	—
30	C₂²×C₄×C₃₁	C₄²	1	12	yes	—
31	C₂²×C₄×C₃₁	C₂×C₈	1	24	yes	—
32	C₂²×C₄×C₃₁	C₂²×C₄	1	3	yes	—
33	C₂²×C₄×C₃₁	C₂²×C₄	1	3	yes	—
34	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
35	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
36	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
37	C₂²×C₄×C₃₁	#41	1	3	yes	—
38	C₂×C₈×C₃₁	—	2	6	no	—
39	C₂×C₈×C₃₁	—	2	6	no	—
40	C₂×C₈×C₃₁	—	2	12	no	—
41	C₂×C₈×C₃₁	—	2	12	no	—
42	C₂×C₈×C₃₁	C₄²	1	12	yes	—
43	C₂×C₈×C₃₁	C₄²	1	4	yes	—
44	C₂×C₈×C₃₁	C₂²⋊C₄	1	12	yes	—
45	C₂×C₈×C₃₁	C₄⋊C₄	1	12	yes	—
46	C₄²×C₃₁	—	2	6	no	—
47	C₄²×C₃₁	—	2	6	no	—
48	C₄²×C₃₁	—	2	12	no	—
49	C₈×#95	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂³×C₃₁	C₂×C₄²	1	3	yes	—
51	C₂³×C₃₁	C₄×D₄	1	3	yes	—
52	C₂³×C₃₁	C₄×Q₈	1	1	yes	characteristic
53	C₂³×C₃₁	C₂²×C₈	1	4	yes	—
54	C₂³×C₃₁	#131	1	4	yes	—
55	C₂×C₄×C₃₁	—	2	12	no	—
56	C₂×C₄×C₃₁	—	2	12	no	—
57	C₂×C₄×C₃₁	—	2	48	no	—
58	C₂×C₄×C₃₁	C₄×C₈	1	24	yes	complemented
59	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
60	C₂×C₄×C₃₁	C₂×C₄²	1	3	yes	—
61	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	3	yes	—
62	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	3	yes	—
63	C₂×C₄×C₃₁	#117	1	3	yes	—
64	C₂×C₄×C₃₁	#117	1	3	yes	—
65	C₂×C₄×C₃₁	C₄×D₄	1	3	yes	—
66	C₂×C₄×C₃₁	C₄×D₄	1	6	yes	—
67	C₂×C₄×C₃₁	C₄×Q₈	1	3	yes	—
68	C₈×C₃₁	—	2	12	no	—
69	C₈×C₃₁	#95	1	8	yes	direct product factor
70	C₂²×C₄×C₈	C₂×C₃₁	1	3	yes	—
71	C₂×#336	C₂×C₃₁	1	3	yes	—
72	C₄×#95	C₂×C₃₁	1	1	yes	characteristic
73	C₂²×C₃₁	C₄³	1	1	yes	characteristic
74	C₂²×C₃₁	C₄×C₂²⋊C₄	1	3	yes	—
75	C₂²×C₃₁	C₄×C₄⋊C₄	1	3	yes	—
76	C₂²×C₃₁	C₂×C₄×C₈	1	6	yes	—
77	C₂²×C₃₁	#405	1	6	yes	—
78	C₂²×C₃₁	C₈×D₄	1	12	yes	—
79	C₂²×C₃₁	C₈×Q₈	1	4	yes	—
80	C₄×C₃₁	—	2	24	no	—
81	C₄×C₃₁	C₂×#95	1	1	yes	characteristic
82	C₄×C₃₁	#376	1	1	yes	characteristic
83	C₄×C₃₁	C₄×C₂²⋊C₄	1	3	yes	—
84	C₄×C₃₁	C₄×C₄⋊C₄	1	3	yes	—
85	C₂³×C₈	C₂²×C₃₁	1	3	yes	—
86	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
87	C₂²×C₄²	C₂²×C₃₁	1	3	yes	—
88	C₂×C₄×C₈	—	2	12	no	—
89	C₂×C₄×C₈	C₄×C₃₁	1	12	yes	complemented
90	C₂×#95	C₄×C₃₁	1	4	yes	—
91	#336	C₄×C₃₁	1	12	yes	complemented
92	C₂×C₃₁	C₄×#95	1	1	yes	characteristic
93	C₂×C₃₁	C₄²×C₈	1	1	yes	characteristic
94	C₂×C₃₁	#1726	1	1	yes	characteristic
95	C₂×C₃₁	C₈×C₂²⋊C₄	1	6	yes	—
96	C₂×C₃₁	C₈×C₄⋊C₄	1	6	yes	—
97	C₂³×C₄	C₂×C₄×C₃₁	1	6	yes	—
98	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
99	C₂²×C₈	—	2	6	no	—
100	C₂²×C₈	—	2	3	no	—
101	C₂²×C₈	—	2	3	no	—
102	C₂²×C₈	C₂×C₄×C₃₁	1	12	yes	—
103	C₂²×C₈	C₂×C₄×C₃₁	1	6	yes	—
104	C₂²×C₈	C₂×C₄×C₃₁	1	6	yes	—
105	C₂²×C₈	D₄×C₃₁	1	6	yes	—
106	C₂²×C₈	Q₈×C₃₁	1	2	yes	—
107	C₂×C₄²	—	2	6	no	—
108	C₂×C₄²	—	2	6	no	—
109	C₂×C₄²	C₂×C₄×C₃₁	1	6	yes	—
110	C₂×C₄²	C₂×C₄×C₃₁	1	6	yes	—
111	C₄×C₈	—	2	24	no	—
112	#95	C₈×C₃₁	1	16	yes	direct product factor
113	C₃₁	C₈×#95	1	1	yes	characteristic, direct product factor, Sylow
114	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
115	C₂²×C₄	—	2	12	no	—
116	C₂²×C₄	—	2	12	no	—
117	C₂²×C₄	C₄²×C₃₁	1	12	yes	—
118	C₂²×C₄	C₂×C₈×C₃₁	1	24	yes	—
119	C₂²×C₄	C₂²×C₄×C₃₁	1	3	yes	—
120	C₂²×C₄	C₂²×C₄×C₃₁	1	3	yes	—
121	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
122	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
123	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
124	C₂²×C₄	#41×C₃₁	1	3	yes	—
125	C₂×C₈	—	2	6	no	—
126	C₂×C₈	—	2	6	no	—
127	C₂×C₈	—	2	12	no	—
128	C₂×C₈	—	2	12	no	—
129	C₂×C₈	C₄²×C₃₁	1	12	yes	—
130	C₂×C₈	C₄²×C₃₁	1	4	yes	—
131	C₂×C₈	C₂²⋊C₄×C₃₁	1	12	yes	—
132	C₂×C₈	C₄⋊C₄×C₃₁	1	12	yes	—
133	C₄²	—	2	6	no	—
134	C₄²	—	2	6	no	—
135	C₄²	—	2	12	no	—
136	C₂³	C₂×C₄²×C₃₁	1	3	yes	—
137	C₂³	C₄×D₄×C₃₁	1	3	yes	—
138	C₂³	C₄×Q₈×C₃₁	1	1	yes	characteristic
139	C₂³	C₂²×C₈×C₃₁	1	4	yes	—
140	C₂³	C₃₁×#131	1	4	yes	—
141	C₂×C₄	—	2	12	no	—
142	C₂×C₄	—	2	48	no	—
143	C₂×C₄	—	2	12	no	—
144	C₂×C₄	C₄×C₈×C₃₁	1	24	yes	complemented
145	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
146	C₂×C₄	C₂×C₄²×C₃₁	1	3	yes	—
147	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	3	yes	—
148	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	3	yes	—
149	C₂×C₄	C₃₁×#117	1	3	yes	—
150	C₂×C₄	C₃₁×#117	1	3	yes	—
151	C₂×C₄	C₄×D₄×C₃₁	1	3	yes	—
152	C₂×C₄	C₄×D₄×C₃₁	1	6	yes	—
153	C₂×C₄	C₄×Q₈×C₃₁	1	3	yes	—
154	C₈	—	2	12	no	—
155	C₈	C₃₁×#95	1	8	yes	direct product factor
156	C₂²	C₄³×C₃₁	1	1	yes	characteristic
157	C₂²	C₄×C₂²⋊C₄×C₃₁	1	3	yes	—
158	C₂²	C₄×C₄⋊C₄×C₃₁	1	3	yes	—
159	C₂²	C₂×C₄×C₈×C₃₁	1	6	yes	—
160	C₂²	C₃₁×#405	1	6	yes	—
161	C₂²	C₈×D₄×C₃₁	1	12	yes	—
162	C₂²	C₈×Q₈×C₃₁	1	4	yes	—
163	C₄	—	2	24	no	—
164	C₄	C₂×C₃₁×#95	1	1	yes	characteristic
165	C₄	C₃₁×#376	1	1	yes	characteristic
166	C₄	C₄×C₂²⋊C₄×C₃₁	1	3	yes	—
167	C₄	C₄×C₄⋊C₄×C₃₁	1	3	yes	—
168	C₂	C₄×C₃₁×#95	1	1	yes	characteristic
169	C₂	C₄²×C₈×C₃₁	1	1	yes	characteristic
170	C₂	C₃₁×#1726	1	1	yes	characteristic
171	C₂	C₈×C₂²⋊C₄×C₃₁	1	6	yes	—
172	C₂	C₈×C₄⋊C₄×C₃₁	1	6	yes	—

	subgroup	G/H
center	C₂³×C₈×C₃₁	C₂²
commutator	C₂	C₄²×C₈×C₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₃₁
socle, prime radical	C₂⁴×C₃₁	C₂²×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₂×C₄×C₈×C₃₁	C₄	semidirect
10. C₃₁×#336	C₄	semidirect
26. C₃₁×#95	C₈	direct (N × H)
49. C₈×#95	C₃₁	direct (N × H)
58. C₂×C₄×C₃₁	C₄×C₈	semidirect
69. C₈×C₃₁	#95	direct (N × H)
89. C₂×C₄×C₈	C₄×C₃₁	semidirect
91. #336	C₄×C₃₁	semidirect
112. #95	C₈×C₃₁	direct (N × H)
113. C₃₁	C₈×#95	direct (N × H)
144. C₂×C₄	C₄×C₈×C₃₁	semidirect
155. C₈	C₃₁×#95	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₈×C₃₁	2
C₂×C₃₁×#336	2
C₄×C₃₁×#95	2
C₈×#95	31

Subgroups — C8×C31×#95 #21467454

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×C₃₁×#95 #21467454