Subgroups — C₂²×C₄×C₃₁⋊C₈ #16854112

Order 3968.

760 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃₁⋊C₈	C₂	1	2	yes	maximal
2	C₂×C₄×C₃₁⋊C₈	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₄²×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₄×C₃₁	C₄	1	2	yes	—
5	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
6	C₂×C₄²×C₃₁	C₄	1	6	yes	—
7	C₂×C₄²×C₃₁	C₂²	1	6	yes	—
8	C₂²×C₃₁⋊C₈	C₄	1	16	yes	direct product factor
9	C₂²×C₃₁⋊C₈	C₂²	1	12	yes	—
10	C₄×C₃₁⋊C₈	C₂²	1	16	yes	direct product factor
11	C₂⁴×C₃₁	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₄×C₃₁	C₈	1	16	yes	complemented
13	C₂²×C₄×C₃₁	C₂×C₄	1	3	yes	—
14	C₂²×C₄×C₃₁	C₂×C₄	1	8	yes	—
15	C₂²×C₄×C₃₁	C₂×C₄	1	12	yes	—
16	C₂²×C₄×C₃₁	C₂³	1	3	yes	—
17	C₄²×C₃₁	C₂×C₄	1	12	yes	—
18	C₄²×C₃₁	C₂³	1	4	yes	—
19	C₂×C₃₁⋊C₈	C₂×C₄	1	48	yes	direct product factor
20	C₂×C₃₁⋊C₈	C₂³	1	8	yes	—
21	C₂³×C₃₁	C₄²	1	4	yes	—
22	C₂³×C₃₁	C₂×C₈	1	8	yes	—
23	C₂³×C₃₁	C₂²×C₄	1	3	yes	—
24	C₂×C₄×C₃₁	C₄²	1	12	yes	—
25	C₂×C₄×C₃₁	C₂×C₈	1	48	yes	complemented
26	C₂×C₄×C₃₁	C₂²×C₄	1	12	yes	—
27	C₂×C₄×C₃₁	C₂²×C₄	1	3	yes	—
28	C₂×C₄×C₃₁	C₂²×C₄	1	8	yes	—
29	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
30	C₃₁⋊C₈	C₂²×C₄	1	16	yes	direct product factor
31	C₂²×C₄×C₈	—	31	1	no	maximal, Sylow
32	C₂²×C₃₁	C₄×C₈	1	16	yes	complemented
33	C₂²×C₃₁	C₂×C₄²	1	6	yes	—
34	C₂²×C₃₁	C₂²×C₈	1	12	yes	—
35	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
36	C₄×C₃₁	C₂×C₄²	1	6	yes	—
37	C₄×C₃₁	C₂²×C₈	1	16	yes	complemented
38	C₄×C₃₁	C₂³×C₄	1	2	yes	—
39	C₂³×C₈	—	31	2	no	—
40	C₂²×C₄²	D₃₁	1	1	yes	characteristic
41	C₂×C₄×C₈	—	31	12	no	—
42	C₂×C₃₁	C₂×C₄×C₈	1	12	yes	complemented
43	C₂×C₃₁	C₂²×C₄²	1	1	yes	characteristic
44	C₂×C₃₁	C₂³×C₈	1	2	yes	—
45	C₂³×C₄	Dic₃₁	1	2	yes	—
46	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
47	C₂²×C₈	—	31	16	no	—
48	C₂²×C₈	—	31	12	no	—
49	C₂×C₄²	Dic₃₁	1	6	yes	—
50	C₂×C₄²	C₂×D₃₁	1	6	yes	—
51	C₄×C₈	—	31	16	no	—
52	C₃₁	C₂²×C₄×C₈	1	1	yes	characteristic, complemented, Sylow
53	C₂⁴	C₂×Dic₃₁	1	1	yes	characteristic
54	C₂²×C₄	C₃₁⋊C₈	1	16	yes	direct product factor
55	C₂²×C₄	C₄×D₃₁	1	8	yes	—
56	C₂²×C₄	C₂×Dic₃₁	1	12	yes	—
57	C₂²×C₄	C₂×Dic₃₁	1	3	yes	—
58	C₂²×C₄	C₂²×D₃₁	1	3	yes	—
59	C₂×C₈	—	31	48	no	—
60	C₂×C₈	—	31	8	no	—
61	C₄²	C₂×Dic₃₁	1	12	yes	—
62	C₄²	C₂²×D₃₁	1	4	yes	—
63	C₂³	C₂×C₃₁⋊C₈	1	8	yes	—
64	C₂³	C₄×Dic₃₁	1	4	yes	—
65	C₂³	C₂²×Dic₃₁	1	3	yes	—
66	C₂×C₄	C₂×C₃₁⋊C₈	1	48	yes	direct product factor
67	C₂×C₄	C₄×Dic₃₁	1	12	yes	—
68	C₂×C₄	C₂×C₄×D₃₁	1	12	yes	—
69	C₂×C₄	C₂²×Dic₃₁	1	8	yes	—
70	C₂×C₄	C₂²×Dic₃₁	1	3	yes	—
71	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
72	C₈	—	31	16	no	—
73	C₂²	C₄×C₃₁⋊C₈	1	16	yes	direct product factor
74	C₂²	C₂²×C₃₁⋊C₈	1	12	yes	—
75	C₂²	C₂×C₄×Dic₃₁	1	6	yes	—
76	C₂²	C₂³×Dic₃₁	1	1	yes	characteristic
77	C₄	C₂²×C₃₁⋊C₈	1	16	yes	direct product factor
78	C₄	C₂×C₄×Dic₃₁	1	6	yes	—
79	C₄	C₂²×C₄×D₃₁	1	2	yes	—
80	C₂	C₂×C₄×C₃₁⋊C₈	1	12	yes	direct product factor
81	C₂	C₂²×C₄×Dic₃₁	1	1	yes	characteristic
82	C₂	C₂³×C₃₁⋊C₈	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂²×C₄²	D₃₁
commutator, hypocenter	C₃₁	C₂²×C₄×C₈
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₂²×C₄²×C₃₁	C₂
socle, prime radical	C₂⁴×C₃₁	C₂×C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
LCS	G ▷ C₃₁
UCS	C₁ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×C₃₁⋊C₈	C₂	direct (N × H)
8. C₂²×C₃₁⋊C₈	C₄	direct (N × H)
10. C₄×C₃₁⋊C₈	C₂²	direct (N × H)
12. C₂²×C₄×C₃₁	C₈	semidirect
19. C₂×C₃₁⋊C₈	C₂×C₄	direct (N × H)
25. C₂×C₄×C₃₁	C₂×C₈	semidirect
30. C₃₁⋊C₈	C₂²×C₄	direct (N × H)
32. C₂²×C₃₁	C₄×C₈	semidirect
37. C₄×C₃₁	C₂²×C₈	semidirect
42. C₂×C₃₁	C₂×C₄×C₈	semidirect
52. C₃₁	C₂²×C₄×C₈	semidirect
54. C₂²×C₄	C₃₁⋊C₈	direct (N × H)
66. C₂×C₄	C₂×C₃₁⋊C₈	direct (N × H)
73. C₂²	C₄×C₃₁⋊C₈	direct (N × H)
77. C₄	C₂²×C₃₁⋊C₈	direct (N × H)
80. C₂	C₂×C₄×C₃₁⋊C₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃₁⋊C₈	2
C₂×C₄×C₃₁⋊C₈	2
C₂²×C₄²×C₃₁	2
C₂²×C₄×C₈	31

Subgroups — C22×C4×C31⋊C8 #16854112

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×C₃₁⋊C₈ #16854112