Subgroups — C₂³×Q₈×D₃₁ #16860505

Order 3968.

6264 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₆₂	C₂	1	3	yes	maximal, complemented
2	C₂²×Q₈×D₃₁	C₂	1	56	yes	maximal, direct product factor
3	C₂³×C₄×D₃₁	C₂	1	3	yes	maximal
4	C₂³×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×D₃₁	C₂²	1	1	yes	characteristic
6	C₂²×Q₈×C₃₁	C₂²	1	28	yes	complemented
7	C₂³×C₄×C₃₁	C₂²	1	3	yes	—
8	C₂³×Dic₃₁	C₂²	1	3	yes	—
9	C₂×Q₈×D₃₁	C₂²	1	448	yes	direct product factor
10	C₂²×C₄×D₃₁	C₂²	1	84	yes	—
11	C₂²×Dic₆₂	C₂²	1	84	yes	complemented
12	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
13	C₂³×D₃₁	Q₈	1	16	yes	direct product factor
14	C₂³×D₃₁	C₂³	1	14	yes	—
15	C₂×Q₈×C₃₁	C₂³	1	112	yes	complemented
16	C₂²×C₄×C₃₁	C₂³	1	42	yes	—
17	C₂²×Dic₃₁	C₂³	1	42	yes	—
18	Q₈×D₃₁	C₂³	1	512	yes	direct product factor
19	C₂×C₄×D₃₁	C₂³	1	336	yes	—
20	C₂×Dic₆₂	C₂³	1	336	yes	complemented
21	C₂³×C₃₁	C₂×Q₈	1	8	yes	complemented
22	C₂³×C₃₁	C₂⁴	1	7	yes	—
23	C₂²×D₃₁	C₂×Q₈	1	112	yes	direct product factor
24	C₂²×D₃₁	C₂⁴	1	28	yes	—
25	Q₈×C₃₁	C₂⁴	1	64	yes	complemented
26	C₂×C₄×C₃₁	C₂⁴	1	84	yes	—
27	C₂×Dic₃₁	C₂⁴	1	84	yes	—
28	C₄×D₃₁	C₂⁴	1	192	yes	—
29	Dic₆₂	C₂⁴	1	192	yes	complemented
30	C₂⁴×Q₈	—	31	1	no	maximal, Sylow
31	C₂²×C₃₁	C₂²×Q₈	1	28	yes	complemented
32	C₂²×C₃₁	C₂⁵	1	7	yes	—
33	C₂×D₃₁	C₂²×Q₈	1	112	yes	direct product factor
34	C₂×D₃₁	C₂⁵	1	8	yes	—
35	C₄×C₃₁	C₂⁵	1	24	yes	—
36	Dic₃₁	C₂⁵	1	24	yes	—
37	C₂³×Q₈	—	31	56	no	—
38	C₂³×Q₈	—	31	3	no	—
39	C₂³×Q₈	D₃₁	1	1	yes	characteristic, direct product factor
40	C₂⁴×C₄	—	31	3	no	—
41	C₂×C₃₁	C₂³×Q₈	1	14	yes	complemented
42	C₂×C₃₁	C₂⁶	1	1	yes	characteristic
43	D₃₁	C₂³×Q₈	1	16	yes	direct product factor
44	C₂⁵	—	31	1	no	—
45	C₂²×Q₈	—	31	448	no	—
46	C₂²×Q₈	—	31	84	no	—
47	C₂²×Q₈	C₂×D₃₁	1	28	yes	direct product factor
48	C₂³×C₄	—	31	3	no	—
49	C₂³×C₄	—	31	84	no	—
50	C₂³×C₄	C₂×D₃₁	1	3	yes	—
51	C₃₁	C₂⁴×Q₈	1	1	yes	characteristic, complemented, Sylow
52	C₂⁴	—	31	16	no	—
53	C₂⁴	—	31	14	no	—
54	C₂⁴	C₂²×D₃₁	1	1	yes	characteristic
55	C₂×Q₈	—	31	512	no	—
56	C₂×Q₈	—	31	336	no	—
57	C₂×Q₈	C₂²×D₃₁	1	112	yes	direct product factor
58	C₂²×C₄	—	31	42	no	—
59	C₂²×C₄	—	31	336	no	—
60	C₂²×C₄	C₂²×D₃₁	1	42	yes	—
61	C₂³	—	31	112	no	—
62	C₂³	—	31	28	no	—
63	C₂³	Q₈×D₃₁	1	8	yes	direct product factor
64	C₂³	C₂³×D₃₁	1	7	yes	—
65	Q₈	—	31	192	no	—
66	Q₈	C₂³×D₃₁	1	64	yes	direct product factor
67	C₂×C₄	—	31	84	no	—
68	C₂×C₄	—	31	192	no	—
69	C₂×C₄	C₂³×D₃₁	1	84	yes	—
70	C₂²	—	31	112	no	—
71	C₂²	—	31	8	no	—
72	C₂²	C₂×Q₈×D₃₁	1	28	yes	direct product factor
73	C₂²	C₂⁴×D₃₁	1	7	yes	—
74	C₄	—	31	24	no	—
75	C₄	C₂⁴×D₃₁	1	24	yes	—
76	C₂	—	31	16	no	—
77	C₂	C₂²×Q₈×D₃₁	1	14	yes	direct product factor
78	C₂	C₂⁵×D₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×D₃₁
commutator, cosocle	C₂×C₃₁	C₂⁶
Frattini	C₂	C₂⁵×D₃₁
Fitting	C₂³×Q₈×C₃₁	C₂
socle	C₂⁴×C₃₁	C₂³
prime radical	C₂⁴×D₃₁	C₂²
hypercenter	C₂³×Q₈	D₃₁
hypocenter	C₃₁	C₂⁴×Q₈
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂³×Dic₆₂	C₂	semidirect
2. C₂²×Q₈×D₃₁	C₂	direct (N × H)
4. C₂³×Q₈×C₃₁	C₂	semidirect
6. C₂²×Q₈×C₃₁	C₂²	semidirect
9. C₂×Q₈×D₃₁	C₂²	direct (N × H)
11. C₂²×Dic₆₂	C₂²	semidirect
13. C₂³×D₃₁	Q₈	direct (N × H)
15. C₂×Q₈×C₃₁	C₂³	semidirect
18. Q₈×D₃₁	C₂³	direct (N × H)
20. C₂×Dic₆₂	C₂³	semidirect
21. C₂³×C₃₁	C₂×Q₈	semidirect
23. C₂²×D₃₁	C₂×Q₈	direct (N × H)
25. Q₈×C₃₁	C₂⁴	semidirect
29. Dic₆₂	C₂⁴	semidirect
31. C₂²×C₃₁	C₂²×Q₈	semidirect
33. C₂×D₃₁	C₂²×Q₈	direct (N × H)
39. C₂³×Q₈	D₃₁	direct (N × H)
41. C₂×C₃₁	C₂³×Q₈	semidirect
43. D₃₁	C₂³×Q₈	direct (N × H)
47. C₂²×Q₈	C₂×D₃₁	direct (N × H)
51. C₃₁	C₂⁴×Q₈	semidirect
57. C₂×Q₈	C₂²×D₃₁	direct (N × H)
63. C₂³	Q₈×D₃₁	direct (N × H)
66. Q₈	C₂³×D₃₁	direct (N × H)
72. C₂²	C₂×Q₈×D₃₁	direct (N × H)
77. C₂	C₂²×Q₈×D₃₁	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×Dic₆₂	2
C₂²×Q₈×D₃₁	2
C₂³×C₄×D₃₁	2
C₂³×Q₈×C₃₁	2
C₂⁴×Q₈	31

Subgroups — C23×Q8×D31 #16860505

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×Q₈×D₃₁ #16860505