Subgroups — C₂×C₃×C₇×D₈ #10621455

Order 672.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₇×D₄	C₂	1	2	yes	maximal, complemented
2	C₃×C₇×D₈	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₃×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₇×D₈	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂³×C₃×C₇	—	2	2	no	—
6	C₃×C₇×D₄	—	2	2	no	—
7	C₃×C₇×D₄	C₂²	1	4	yes	complemented
8	C₂×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₃×C₇×C₈	C₂²	1	2	yes	complemented
10	C₂×C₇×D₄	C₂×C₃	1	2	yes	complemented
11	C₇×D₈	C₂×C₃	1	4	yes	direct product factor
12	C₂×C₇×C₈	C₂×C₃	1	1	yes	characteristic, complemented
13	C₂×C₃×D₈	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂²×C₃×C₇	—	2	4	no	—
15	C₂²×C₃×C₇	—	4	4	no	—
16	C₂²×C₃×C₇	D₄	1	1	yes	characteristic
17	C₃×C₄×C₇	D₄	1	1	yes	characteristic
18	C₃×C₄×C₇	C₂³	1	1	yes	characteristic
19	C₂³×C₇	—	2	2	no	—
20	C₇×D₄	—	2	2	no	—
21	C₇×D₄	C₂²×C₃	1	4	yes	complemented
22	C₂×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
23	C₇×C₈	C₂²×C₃	1	2	yes	complemented
24	C₂×C₃×D₄	C₂×C₇	1	2	yes	complemented
25	C₃×D₈	C₂×C₇	1	4	yes	direct product factor
26	C₂×C₃×C₈	C₂×C₇	1	1	yes	characteristic, complemented
27	C₂×C₃×C₇	—	4	4	no	—
28	C₂×C₃×C₇	D₈	1	2	yes	direct product factor
29	C₂×C₃×C₇	C₂×D₄	1	1	yes	characteristic
30	C₂×D₈	C₃×C₇	1	1	yes	characteristic, direct product factor, Sylow
31	C₂²×C₇	—	2	4	no	—
32	C₂²×C₇	—	4	4	no	—
33	C₂²×C₇	C₃×D₄	1	1	yes	characteristic
34	C₄×C₇	C₃×D₄	1	1	yes	characteristic
35	C₄×C₇	C₂³×C₃	1	1	yes	characteristic
36	C₂³×C₃	—	2	2	no	—
37	C₃×D₄	—	2	2	no	—
38	C₃×D₄	C₂²×C₇	1	4	yes	complemented
39	C₂×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
40	C₃×C₈	C₂²×C₇	1	2	yes	complemented
41	C₃×C₇	C₂×D₈	1	1	yes	characteristic, direct product factor, Hall
42	C₂×D₄	C₂×C₃×C₇	1	2	yes	complemented
43	D₈	C₂×C₃×C₇	1	4	yes	direct product factor
44	C₂×C₈	C₂×C₃×C₇	1	1	yes	characteristic, complemented
45	C₂×C₇	—	4	4	no	—
46	C₂×C₇	C₃×D₈	1	2	yes	direct product factor
47	C₂×C₇	C₂×C₃×D₄	1	1	yes	characteristic
48	C₂²×C₃	—	2	4	no	—
49	C₂²×C₃	—	4	4	no	—
50	C₂²×C₃	C₇×D₄	1	1	yes	characteristic
51	C₃×C₄	C₇×D₄	1	1	yes	characteristic
52	C₃×C₄	C₂³×C₇	1	1	yes	characteristic
53	C₂³	—	2	2	no	—
54	D₄	—	2	2	no	—
55	D₄	C₂²×C₃×C₇	1	4	yes	complemented
56	C₂×C₄	C₂²×C₃×C₇	1	1	yes	characteristic
57	C₈	C₂²×C₃×C₇	1	2	yes	complemented
58	C₇	C₂×C₃×D₈	1	1	yes	characteristic, direct product factor, Sylow
59	C₂×C₃	—	4	4	no	—
60	C₂×C₃	C₇×D₈	1	2	yes	direct product factor
61	C₂×C₃	C₂×C₇×D₄	1	1	yes	characteristic
62	C₂²	—	2	4	no	—
63	C₂²	—	4	4	no	—
64	C₂²	C₃×C₇×D₄	1	1	yes	characteristic
65	C₄	C₃×C₇×D₄	1	1	yes	characteristic
66	C₄	C₂³×C₃×C₇	1	1	yes	characteristic
67	C₃	C₂×C₇×D₈	1	1	yes	characteristic, direct product factor, Sylow
68	C₂	—	4	4	no	—
69	C₂	C₃×C₇×D₈	1	2	yes	direct product factor
70	C₂	C₂×C₃×C₇×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₇	D₄
commutator, Frattini, cosocle	C₄	C₂³×C₃×C₇
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₇ ◁ C₂×C₃×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₇×D₄	C₂	semidirect
2. C₃×C₇×D₈	C₂	direct (N × H)
3. C₂×C₃×C₇×C₈	C₂	semidirect
4. C₂×C₇×D₈	C₃	direct (N × H)
7. C₃×C₇×D₄	C₂²	semidirect
9. C₃×C₇×C₈	C₂²	semidirect
10. C₂×C₇×D₄	C₂×C₃	semidirect
11. C₇×D₈	C₂×C₃	direct (N × H)
12. C₂×C₇×C₈	C₂×C₃	semidirect
13. C₂×C₃×D₈	C₇	direct (N × H)
21. C₇×D₄	C₂²×C₃	semidirect
23. C₇×C₈	C₂²×C₃	semidirect
24. C₂×C₃×D₄	C₂×C₇	semidirect
25. C₃×D₈	C₂×C₇	direct (N × H)
26. C₂×C₃×C₈	C₂×C₇	semidirect
28. C₂×C₃×C₇	D₈	direct (N × H)
30. C₂×D₈	C₃×C₇	direct (N × H)
38. C₃×D₄	C₂²×C₇	semidirect
40. C₃×C₈	C₂²×C₇	semidirect
41. C₃×C₇	C₂×D₈	direct (N × H)
42. C₂×D₄	C₂×C₃×C₇	semidirect
43. D₈	C₂×C₃×C₇	direct (N × H)
44. C₂×C₈	C₂×C₃×C₇	semidirect
46. C₂×C₇	C₃×D₈	direct (N × H)
55. D₄	C₂²×C₃×C₇	semidirect
57. C₈	C₂²×C₃×C₇	semidirect
58. C₇	C₂×C₃×D₈	direct (N × H)
60. C₂×C₃	C₇×D₈	direct (N × H)
67. C₃	C₂×C₇×D₈	direct (N × H)
69. C₂	C₃×C₇×D₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₇×D₄	2
C₃×C₇×D₈	2
C₂×C₃×C₇×C₈	2
C₂×C₇×D₈	3
C₂×C₃×D₈	7

Subgroups — C2×C3×C7×D8 #10621455

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₇×D₈ #10621455