Subgroups — C₇×C₈×D₄ #90080

Order 448.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×C₈	C₂	1	2	yes	maximal, complemented
2	C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic
3	C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×C₂²⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₇×D₄	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₇	C₂²	1	2	yes	—
8	C₂×C₇×C₈	—	2	4	no	—
9	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₇×C₈	C₂²	1	2	yes	—
11	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₇×C₄⋊C₄	C₄	1	1	yes	characteristic
13	C₇×C₂²⋊C₄	C₄	1	2	yes	—
14	C₄²×C₇	C₂²	1	1	yes	characteristic
15	C₈×D₄	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂³×C₇	C₂×C₄	1	2	yes	—
17	C₇×D₄	C₈	1	4	yes	direct product factor
18	C₂×C₄×C₇	—	2	2	no	—
19	C₂×C₄×C₇	—	2	2	no	—
20	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
24	C₇×C₈	—	2	2	no	—
25	C₇×C₈	—	2	1	no	—
26	C₇×C₈	D₄	1	2	yes	direct product factor
27	C₂²×C₈	C₂×C₇	1	2	yes	complemented
28	C₄×D₄	C₂×C₇	1	1	yes	characteristic
29	C₄⋊C₈	C₂×C₇	1	1	yes	characteristic, complemented
30	C₂²⋊C₈	C₂×C₇	1	2	yes	complemented
31	C₄×C₈	C₂×C₇	1	1	yes	characteristic, complemented
32	C₂²×C₇	—	2	2	no	—
33	C₂²×C₇	—	2	2	no	—
34	C₂²×C₇	C₂×C₈	1	4	yes	complemented
35	C₂²×C₇	C₂²×C₄	1	1	yes	characteristic
36	C₄×C₇	—	2	2	no	—
37	C₄×C₇	—	2	1	no	—
38	C₄×C₇	C₂×C₈	1	2	yes	complemented
39	C₄×C₇	C₂×D₄	1	1	yes	characteristic
40	C₄×C₇	#41	1	1	yes	characteristic
41	C₂×D₄	C₄×C₇	1	1	yes	characteristic
42	C₂²×C₄	C₂²×C₇	1	2	yes	—
43	C₂×C₈	—	2	4	no	—
44	C₂×C₈	C₂²×C₇	1	1	yes	characteristic
45	C₂×C₈	C₂²×C₇	1	2	yes	—
46	C₂×C₈	C₂²×C₇	1	1	yes	characteristic
47	C₄⋊C₄	C₄×C₇	1	1	yes	characteristic
48	C₂²⋊C₄	C₄×C₇	1	2	yes	—
49	C₄²	C₂²×C₇	1	1	yes	characteristic
50	C₂×C₇	—	2	4	no	—
51	C₂×C₇	C₄×D₄	1	1	yes	characteristic
52	C₂×C₇	C₂²×C₈	1	1	yes	characteristic
53	C₂×C₇	#131	1	1	yes	characteristic
54	C₂³	C₂×C₄×C₇	1	2	yes	—
55	D₄	C₇×C₈	1	4	yes	direct product factor
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	C₂×C₄×C₇	1	1	yes	characteristic
59	C₂×C₄	C₂×C₄×C₇	1	2	yes	—
60	C₂×C₄	C₂×C₄×C₇	1	1	yes	characteristic
61	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
62	C₈	—	2	2	no	—
63	C₈	—	2	1	no	—
64	C₈	C₇×D₄	1	2	yes	direct product factor
65	C₇	C₈×D₄	1	1	yes	characteristic, direct product factor, Sylow
66	C₂²	—	2	2	no	—
67	C₂²	—	2	2	no	—
68	C₂²	C₂×C₇×C₈	1	4	yes	complemented
69	C₂²	C₂²×C₄×C₇	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	2	1	no	—
72	C₄	C₂×C₇×C₈	1	2	yes	complemented
73	C₄	C₂×C₇×D₄	1	1	yes	characteristic
74	C₄	C₇×#41	1	1	yes	characteristic
75	C₂	—	2	4	no	—
76	C₂	C₄×C₇×D₄	1	1	yes	characteristic
77	C₂	C₂²×C₇×C₈	1	1	yes	characteristic
78	C₂	C₇×#131	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₇×C₈	C₂²
commutator	C₂	C₂²×C₇×C₈
Frattini, cosocle	C₂×C₄	C₂³×C₇
socle	C₂²×C₇	C₂²×C₄
prime radical	C₂×C₇×D₄	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₇×C₈	C₂	semidirect
3. C₇×C₄⋊C₈	C₂	semidirect
4. C₇×C₂²⋊C₈	C₂	semidirect
5. C₄×C₇×C₈	C₂	semidirect
15. C₈×D₄	C₇	direct (N × H)
17. C₇×D₄	C₈	direct (N × H)
26. C₇×C₈	D₄	direct (N × H)
27. C₂²×C₈	C₂×C₇	semidirect
29. C₄⋊C₈	C₂×C₇	semidirect
30. C₂²⋊C₈	C₂×C₇	semidirect
31. C₄×C₈	C₂×C₇	semidirect
34. C₂²×C₇	C₂×C₈	semidirect
38. C₄×C₇	C₂×C₈	semidirect
55. D₄	C₇×C₈	direct (N × H)
64. C₈	C₇×D₄	direct (N × H)
65. C₇	C₈×D₄	direct (N × H)
68. C₂²	C₂×C₇×C₈	semidirect
72. C₄	C₂×C₇×C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₇×C₈	2
C₄×C₇×D₄	2
C₇×C₄⋊C₈	2
C₇×C₂²⋊C₈	2
C₄×C₇×C₈	2
C₈×D₄	7

Subgroups — C7×C8×D4 #90080

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×C₈×D₄ #90080