Subgroups — C₂⁴×D₅₆₈ #32911227

Order 18176.

15454 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₂₈₄	C₂	1	2	yes	maximal, complemented
2	C₂³×D₅₆₈	C₂	1	60	yes	maximal, direct product factor
3	C₂⁴×C₈×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁵×D₇₁	—	2	2	no	—
5	C₂³×D₂₈₄	—	2	30	no	—
6	C₂³×D₂₈₄	C₂²	1	60	yes	complemented
7	C₂²×D₅₆₈	C₂²	1	560	yes	direct product factor
8	C₂⁴×C₄×C₇₁	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₇₁	C₂²	1	30	yes	complemented
10	C₂⁴×D₇₁	—	2	60	no	—
11	C₂⁴×D₇₁	—	4	32	no	—
12	C₂²×D₂₈₄	—	2	420	no	—
13	C₂²×D₂₈₄	C₂³	1	280	yes	complemented
14	C₂×D₅₆₈	C₂³	1	960	yes	direct product factor
15	C₂⁵×C₇₁	D₄	1	1	yes	characteristic
16	C₂³×C₄×C₇₁	D₄	1	15	yes	—
17	C₂³×C₄×C₇₁	C₂³	1	15	yes	—
18	C₂²×C₈×C₇₁	C₂³	1	140	yes	complemented
19	C₂³×D₇₁	—	2	280	no	—
20	C₂³×D₇₁	—	4	480	no	—
21	C₂×D₂₈₄	—	2	840	no	—
22	C₂×D₂₈₄	C₂⁴	1	240	yes	complemented
23	D₅₆₈	C₂⁴	1	256	yes	direct product factor
24	C₂⁴×C₇₁	D₈	1	16	yes	complemented
25	C₂⁴×C₇₁	C₂×D₄	1	15	yes	—
26	C₂²×C₄×C₇₁	C₂×D₄	1	105	yes	—
27	C₂²×C₄×C₇₁	C₂⁴	1	35	yes	—
28	C₂×C₈×C₇₁	C₂⁴	1	120	yes	complemented
29	C₂³×C₇₁	C₂×D₈	1	120	yes	complemented
30	C₂³×C₇₁	C₂²×D₄	1	35	yes	—
31	C₂²×D₇₁	—	2	240	no	—
32	C₂²×D₇₁	—	4	1120	no	—
33	C₂×C₄×C₇₁	C₂²×D₄	1	105	yes	—
34	C₂×C₄×C₇₁	C₂⁵	1	15	yes	—
35	D₂₈₄	—	2	240	no	—
36	D₂₈₄	C₂⁵	1	32	yes	complemented
37	C₈×C₇₁	C₂⁵	1	16	yes	complemented
38	C₂²×C₇₁	C₂²×D₈	1	140	yes	complemented
39	C₂²×C₇₁	C₂³×D₄	1	15	yes	—
40	C₂×D₇₁	—	2	32	no	—
41	C₂×D₇₁	—	4	480	no	—
42	C₄×C₇₁	C₂³×D₄	1	15	yes	—
43	C₄×C₇₁	C₂⁶	1	1	yes	characteristic
44	C₂⁴×D₈	—	71	1	no	maximal, Sylow
45	C₂×C₇₁	C₂³×D₈	1	30	yes	complemented
46	C₂×C₇₁	C₂⁴×D₄	1	1	yes	characteristic
47	D₇₁	—	4	32	no	—
48	C₂⁴×D₄	—	71	2	no	—
49	C₂³×D₈	—	71	60	no	—
50	C₂⁴×C₈	D₇₁	1	1	yes	characteristic, complemented
51	C₇₁	C₂⁴×D₈	1	1	yes	characteristic, complemented, Sylow
52	C₂⁶	—	142	2	no	—
53	C₂³×D₄	—	71	60	no	—
54	C₂³×D₄	—	142	30	no	—
55	C₂⁴×C₄	C₂×D₇₁	1	1	yes	characteristic
56	C₂²×D₈	—	71	560	no	—
57	C₂³×C₈	C₂×D₇₁	1	30	yes	complemented
58	C₂⁵	—	284	32	no	—
59	C₂⁵	—	142	60	no	—
60	C₂⁵	D₂₈₄	1	1	yes	characteristic
61	C₂²×D₄	—	71	280	no	—
62	C₂²×D₄	—	142	420	no	—
63	C₂³×C₄	D₂₈₄	1	15	yes	—
64	C₂³×C₄	C₂²×D₇₁	1	15	yes	—
65	C₂×D₈	—	71	960	no	—
66	C₂²×C₈	C₂²×D₇₁	1	140	yes	complemented
67	C₂⁴	—	284	480	no	—
68	C₂⁴	—	142	280	no	—
69	C₂⁴	D₅₆₈	1	16	yes	direct product factor
70	C₂⁴	C₂×D₂₈₄	1	15	yes	—
71	C₂×D₄	—	71	240	no	—
72	C₂×D₄	—	142	840	no	—
73	C₂²×C₄	C₂×D₂₈₄	1	105	yes	—
74	C₂²×C₄	C₂³×D₇₁	1	35	yes	—
75	D₈	—	71	256	no	—
76	C₂×C₈	C₂³×D₇₁	1	120	yes	complemented
77	C₂³	—	284	1120	no	—
78	C₂³	—	142	240	no	—
79	C₂³	C₂×D₅₆₈	1	120	yes	direct product factor
80	C₂³	C₂²×D₂₈₄	1	35	yes	—
81	D₄	—	71	32	no	—
82	D₄	—	142	240	no	—
83	C₂×C₄	C₂²×D₂₈₄	1	105	yes	—
84	C₂×C₄	C₂⁴×D₇₁	1	15	yes	—
85	C₈	C₂⁴×D₇₁	1	16	yes	complemented
86	C₂²	—	284	480	no	—
87	C₂²	—	142	32	no	—
88	C₂²	C₂²×D₅₆₈	1	140	yes	direct product factor
89	C₂²	C₂³×D₂₈₄	1	15	yes	—
90	C₄	C₂³×D₂₈₄	1	15	yes	—
91	C₄	C₂⁵×D₇₁	1	1	yes	characteristic
92	C₂	—	284	32	no	—
93	C₂	C₂³×D₅₆₈	1	30	yes	direct product factor
94	C₂	C₂⁴×D₂₈₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	D₂₈₄
commutator, cosocle	C₄×C₇₁	C₂⁶
Frattini	C₄	C₂⁵×D₇₁
Fitting	C₂⁴×C₈×C₇₁	C₂
socle	C₂⁵×C₇₁	D₄
hypercenter	C₂⁴×C₈	D₇₁
hypocenter	C₇₁	C₂⁴×D₈
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
1. C₂⁴×D₂₈₄	C₂	semidirect
2. C₂³×D₅₆₈	C₂	direct (N × H)
3. C₂⁴×C₈×C₇₁	C₂	semidirect
6. C₂³×D₂₈₄	C₂²	semidirect
7. C₂²×D₅₆₈	C₂²	direct (N × H)
9. C₂³×C₈×C₇₁	C₂²	semidirect
13. C₂²×D₂₈₄	C₂³	semidirect
14. C₂×D₅₆₈	C₂³	direct (N × H)
18. C₂²×C₈×C₇₁	C₂³	semidirect
22. C₂×D₂₈₄	C₂⁴	semidirect
23. D₅₆₈	C₂⁴	direct (N × H)
24. C₂⁴×C₇₁	D₈	semidirect
28. C₂×C₈×C₇₁	C₂⁴	semidirect
29. C₂³×C₇₁	C₂×D₈	semidirect
36. D₂₈₄	C₂⁵	semidirect
37. C₈×C₇₁	C₂⁵	semidirect
38. C₂²×C₇₁	C₂²×D₈	semidirect
45. C₂×C₇₁	C₂³×D₈	semidirect
50. C₂⁴×C₈	D₇₁	semidirect
51. C₇₁	C₂⁴×D₈	semidirect
57. C₂³×C₈	C₂×D₇₁	semidirect
66. C₂²×C₈	C₂²×D₇₁	semidirect
69. C₂⁴	D₅₆₈	direct (N × H)
76. C₂×C₈	C₂³×D₇₁	semidirect
79. C₂³	C₂×D₅₆₈	direct (N × H)
85. C₈	C₂⁴×D₇₁	semidirect
88. C₂²	C₂²×D₅₆₈	direct (N × H)
93. C₂	C₂³×D₅₆₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×D₂₈₄	2
C₂³×D₅₆₈	2
C₂⁴×C₈×C₇₁	2
C₂⁴×D₈	71

Subgroups — C24×D568 #32911227

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×D₅₆₈ #32911227