Subgroups — C₂³×#92499 #16860510

Order 3968.

6520 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#92499	C₂	1	56	yes	maximal, direct product factor
2	C₂³×D₁₂₄	C₂	1	3	yes	maximal, complemented
3	C₂³×C₄×D₃₁	C₂	1	3	yes	maximal, complemented
4	C₂³×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×D₃₁	C₂²	1	3	yes	—
6	C₂²×Q₈×C₃₁	C₂²	1	28	yes	complemented
7	C₂³×C₄×C₃₁	C₂²	1	3	yes	—
8	C₂³×Dic₃₁	C₂²	1	1	yes	characteristic
9	C₂×#92499	C₂²	1	448	yes	direct product factor
10	C₂²×D₁₂₄	C₂²	1	84	yes	complemented
11	C₂²×C₄×D₃₁	C₂²	1	84	yes	complemented
12	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
13	C₂³×D₃₁	—	2	24	no	—
14	C₂³×D₃₁	C₂³	1	42	yes	—
15	C₂×Q₈×C₃₁	C₂³	1	112	yes	complemented
16	C₂²×C₄×C₃₁	C₂³	1	42	yes	—
17	C₂²×Dic₃₁	C₂³	1	14	yes	—
18	#92499	C₂³	1	512	yes	direct product factor
19	C₂×D₁₂₄	C₂³	1	336	yes	complemented
20	C₂×C₄×D₃₁	C₂³	1	336	yes	complemented
21	C₂³×C₃₁	#41	1	8	yes	complemented
22	C₂³×C₃₁	C₂⁴	1	7	yes	—
23	C₂²×D₃₁	—	2	168	no	—
24	C₂²×D₃₁	C₂⁴	1	84	yes	—
25	Q₈×C₃₁	C₂⁴	1	64	yes	complemented
26	C₂×C₄×C₃₁	C₂⁴	1	84	yes	—
27	C₂×Dic₃₁	C₂⁴	1	28	yes	—
28	D₁₂₄	C₂⁴	1	192	yes	complemented
29	C₄×D₃₁	C₂⁴	1	192	yes	complemented
30	C₂³×#41	—	31	1	no	maximal, Sylow
31	C₂²×C₃₁	C₂×#41	1	28	yes	complemented
32	C₂²×C₃₁	C₂⁵	1	7	yes	—
33	C₂×D₃₁	—	2	168	no	—
34	C₂×D₃₁	C₂⁵	1	24	yes	—
35	C₄×C₃₁	C₂⁵	1	24	yes	—
36	Dic₃₁	C₂⁵	1	8	yes	—
37	C₂²×#41	—	31	56	no	—
38	C₂³×Q₈	D₃₁	1	1	yes	characteristic, complemented
39	C₂³×D₄	—	31	3	no	—
40	C₂⁴×C₄	—	31	3	no	—
41	C₂×C₃₁	C₂²×#41	1	14	yes	complemented
42	C₂×C₃₁	C₂⁶	1	1	yes	characteristic
43	D₃₁	—	2	24	no	—
44	C₂⁵	—	31	3	no	—
45	C₂×#41	—	31	448	no	—
46	C₂²×Q₈	C₂×D₃₁	1	28	yes	complemented
47	C₂²×D₄	—	31	84	no	—
48	C₂³×C₄	—	31	1	no	—
49	C₂³×C₄	—	31	84	no	—
50	C₂³×C₄	C₂×D₃₁	1	3	yes	—
51	C₃₁	C₂³×#41	1	1	yes	characteristic, complemented, Sylow
52	C₂⁴	—	31	42	no	—
53	C₂⁴	—	62	24	no	—
54	C₂⁴	C₂²×D₃₁	1	1	yes	characteristic
55	#41	—	31	512	no	—
56	C₂×Q₈	C₂²×D₃₁	1	112	yes	complemented
57	C₂×D₄	—	31	336	no	—
58	C₂²×C₄	—	31	14	no	—
59	C₂²×C₄	—	31	336	no	—
60	C₂²×C₄	C₂²×D₃₁	1	42	yes	—
61	C₂³	—	31	84	no	—
62	C₂³	—	62	168	no	—
63	C₂³	#92499	1	8	yes	direct product factor
64	C₂³	C₂³×D₃₁	1	7	yes	—
65	Q₈	C₂³×D₃₁	1	64	yes	complemented
66	D₄	—	31	192	no	—
67	C₂×C₄	—	31	28	no	—
68	C₂×C₄	—	31	192	no	—
69	C₂×C₄	C₂³×D₃₁	1	84	yes	—
70	C₂²	—	31	24	no	—
71	C₂²	—	62	168	no	—
72	C₂²	C₂×#92499	1	28	yes	direct product factor
73	C₂²	C₂⁴×D₃₁	1	7	yes	—
74	C₄	—	31	8	no	—
75	C₄	C₂⁴×D₃₁	1	24	yes	—
76	C₂	—	62	24	no	—
77	C₂	C₂²×#92499	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
hypercenter	C₂³×Q₈	D₃₁
hypocenter	C₃₁	C₂³×#41
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. C₂²×#92499	C₂	direct (N × H)
2. C₂³×D₁₂₄	C₂	semidirect
3. C₂³×C₄×D₃₁	C₂	semidirect
4. C₂³×Q₈×C₃₁	C₂	semidirect
6. C₂²×Q₈×C₃₁	C₂²	semidirect
9. C₂×#92499	C₂²	direct (N × H)
10. C₂²×D₁₂₄	C₂²	semidirect
11. C₂²×C₄×D₃₁	C₂²	semidirect
15. C₂×Q₈×C₃₁	C₂³	semidirect
18. #92499	C₂³	direct (N × H)
19. C₂×D₁₂₄	C₂³	semidirect
20. C₂×C₄×D₃₁	C₂³	semidirect
21. C₂³×C₃₁	#41	semidirect
25. Q₈×C₃₁	C₂⁴	semidirect
28. D₁₂₄	C₂⁴	semidirect
29. C₄×D₃₁	C₂⁴	semidirect
31. C₂²×C₃₁	C₂×#41	semidirect
38. C₂³×Q₈	D₃₁	semidirect
41. C₂×C₃₁	C₂²×#41	semidirect
46. C₂²×Q₈	C₂×D₃₁	semidirect
51. C₃₁	C₂³×#41	semidirect
56. C₂×Q₈	C₂²×D₃₁	semidirect
63. C₂³	#92499	direct (N × H)
65. Q₈	C₂³×D₃₁	semidirect
72. C₂²	C₂×#92499	direct (N × H)
77. C₂	C₂²×#92499	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×#92499	2
C₂³×D₁₂₄	2
C₂³×C₄×D₃₁	2
C₂³×Q₈×C₃₁	2
C₂³×#41	31

Subgroups — C23×#92499 #16860510

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#92499 #16860510