Subgroups — C₂³×D₈×C₆₇ #22558005

Order 8576.

1976 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₄×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₂²×D₈×C₆₇	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁵×C₆₇	—	2	2	no	—
5	C₂²×D₄×C₆₇	—	2	14	no	—
6	C₂²×D₄×C₆₇	C₂²	1	28	yes	complemented
7	C₂³×C₄×C₆₇	C₂²	1	1	yes	characteristic
8	C₂×D₈×C₆₇	C₂²	1	112	yes	direct product factor
9	C₂²×C₈×C₆₇	C₂²	1	14	yes	complemented
10	C₂⁴×C₆₇	—	2	28	no	—
11	C₂⁴×C₆₇	—	4	16	no	—
12	C₂⁴×C₆₇	D₄	1	1	yes	characteristic
13	C₂×D₄×C₆₇	—	2	84	no	—
14	C₂×D₄×C₆₇	C₂³	1	56	yes	complemented
15	C₂²×C₄×C₆₇	D₄	1	7	yes	—
16	C₂²×C₄×C₆₇	C₂³	1	7	yes	—
17	D₈×C₆₇	C₂³	1	64	yes	direct product factor
18	C₂×C₈×C₆₇	C₂³	1	28	yes	complemented
19	C₂³×C₆₇	—	2	56	no	—
20	C₂³×C₆₇	—	4	112	no	—
21	C₂³×C₆₇	D₈	1	8	yes	direct product factor
22	C₂³×C₆₇	C₂×D₄	1	7	yes	—
23	D₄×C₆₇	—	2	56	no	—
24	D₄×C₆₇	C₂⁴	1	16	yes	complemented
25	C₂×C₄×C₆₇	C₂×D₄	1	21	yes	—
26	C₂×C₄×C₆₇	C₂⁴	1	7	yes	—
27	C₈×C₆₇	C₂⁴	1	8	yes	complemented
28	C₂²×C₆₇	—	2	16	no	—
29	C₂²×C₆₇	—	4	112	no	—
30	C₂²×C₆₇	C₂×D₈	1	28	yes	direct product factor
31	C₂²×C₆₇	C₂²×D₄	1	7	yes	—
32	C₄×C₆₇	C₂²×D₄	1	7	yes	—
33	C₄×C₆₇	C₂⁵	1	1	yes	characteristic
34	C₂×C₆₇	—	4	16	no	—
35	C₂×C₆₇	C₂²×D₈	1	14	yes	direct product factor
36	C₂×C₆₇	C₂³×D₄	1	1	yes	characteristic
37	C₂³×D₈	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₆₇	C₂³×D₈	1	1	yes	characteristic, direct product factor, Sylow
39	C₂³×D₄	C₂×C₆₇	1	2	yes	complemented
40	C₂²×D₈	C₂×C₆₇	1	28	yes	direct product factor
41	C₂³×C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
42	C₂⁵	—	2	2	no	—
43	C₂²×D₄	—	2	14	no	—
44	C₂²×D₄	C₂²×C₆₇	1	28	yes	complemented
45	C₂³×C₄	C₂²×C₆₇	1	1	yes	characteristic
46	C₂×D₈	C₂²×C₆₇	1	112	yes	direct product factor
47	C₂²×C₈	C₂²×C₆₇	1	14	yes	complemented
48	C₂⁴	—	2	28	no	—
49	C₂⁴	—	4	16	no	—
50	C₂⁴	D₄×C₆₇	1	1	yes	characteristic
51	C₂×D₄	—	2	84	no	—
52	C₂×D₄	C₂³×C₆₇	1	56	yes	complemented
53	C₂²×C₄	D₄×C₆₇	1	7	yes	—
54	C₂²×C₄	C₂³×C₆₇	1	7	yes	—
55	D₈	C₂³×C₆₇	1	64	yes	direct product factor
56	C₂×C₈	C₂³×C₆₇	1	28	yes	complemented
57	C₂³	—	2	56	no	—
58	C₂³	—	4	112	no	—
59	C₂³	D₈×C₆₇	1	8	yes	direct product factor
60	C₂³	C₂×D₄×C₆₇	1	7	yes	—
61	D₄	—	2	56	no	—
62	D₄	C₂⁴×C₆₇	1	16	yes	complemented
63	C₂×C₄	C₂×D₄×C₆₇	1	21	yes	—
64	C₂×C₄	C₂⁴×C₆₇	1	7	yes	—
65	C₈	C₂⁴×C₆₇	1	8	yes	complemented
66	C₂²	—	2	16	no	—
67	C₂²	—	4	112	no	—
68	C₂²	C₂×D₈×C₆₇	1	28	yes	direct product factor
69	C₂²	C₂²×D₄×C₆₇	1	7	yes	—
70	C₄	C₂²×D₄×C₆₇	1	7	yes	—
71	C₄	C₂⁵×C₆₇	1	1	yes	characteristic
72	C₂	—	4	16	no	—
73	C₂	C₂²×D₈×C₆₇	1	14	yes	direct product factor
74	C₂	C₂³×D₄×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₆₇	D₄
commutator, Frattini, cosocle	C₄	C₂⁵×C₆₇
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×D₄×C₆₇	C₂	semidirect
2. C₂²×D₈×C₆₇	C₂	direct (N × H)
3. C₂³×C₈×C₆₇	C₂	semidirect
6. C₂²×D₄×C₆₇	C₂²	semidirect
8. C₂×D₈×C₆₇	C₂²	direct (N × H)
9. C₂²×C₈×C₆₇	C₂²	semidirect
14. C₂×D₄×C₆₇	C₂³	semidirect
17. D₈×C₆₇	C₂³	direct (N × H)
18. C₂×C₈×C₆₇	C₂³	semidirect
21. C₂³×C₆₇	D₈	direct (N × H)
24. D₄×C₆₇	C₂⁴	semidirect
27. C₈×C₆₇	C₂⁴	semidirect
30. C₂²×C₆₇	C₂×D₈	direct (N × H)
35. C₂×C₆₇	C₂²×D₈	direct (N × H)
37. C₂³×D₈	C₆₇	direct (N × H)
38. C₆₇	C₂³×D₈	direct (N × H)
39. C₂³×D₄	C₂×C₆₇	semidirect
40. C₂²×D₈	C₂×C₆₇	direct (N × H)
41. C₂³×C₈	C₂×C₆₇	semidirect
44. C₂²×D₄	C₂²×C₆₇	semidirect
46. C₂×D₈	C₂²×C₆₇	direct (N × H)
47. C₂²×C₈	C₂²×C₆₇	semidirect
52. C₂×D₄	C₂³×C₆₇	semidirect
55. D₈	C₂³×C₆₇	direct (N × H)
56. C₂×C₈	C₂³×C₆₇	semidirect
59. C₂³	D₈×C₆₇	direct (N × H)
62. D₄	C₂⁴×C₆₇	semidirect
65. C₈	C₂⁴×C₆₇	semidirect
68. C₂²	C₂×D₈×C₆₇	direct (N × H)
73. C₂	C₂²×D₈×C₆₇	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×D₄×C₆₇	2
C₂²×D₈×C₆₇	2
C₂³×C₈×C₆₇	2
C₂³×D₈	67

Subgroups — C23×D8×C67 #22558005

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×D₈×C₆₇ #22558005