Subgroups — C₂×C₄⋊(C₇⋊C₈) #89695

Order 448.

138 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄²×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₇⋊C₈	C₂	1	2	yes	maximal
3	C₄⋊(C₇⋊C₈)	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₄×C₇	C₄	1	1	yes	characteristic
5	C₂²×C₄×C₇	C₄	1	1	yes	characteristic
6	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
7	C₄²×C₇	C₄	1	2	yes	—
8	C₄²×C₇	C₂²	1	2	yes	—
9	C₂×C₇⋊C₈	—	2	4	no	—
10	C₂×C₇⋊C₈	C₂²	1	4	yes	—
11	C₂×C₄⋊C₈	—	7	1	no	maximal, Sylow
12	C₂³×C₇	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×C₇	—	2	2	no	—
14	C₂×C₄×C₇	C₈	1	4	yes	complemented
15	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
16	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
18	C₂×C₄×C₇	D₄	1	2	yes	—
19	C₂×C₄×C₇	Q₈	1	2	yes	—
20	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
21	C₇⋊C₈	—	2	4	no	—
22	C₂²×C₈	—	7	2	no	—
23	C₂×C₄²	D₇	1	1	yes	characteristic
24	C₄⋊C₈	—	7	4	no	—
25	C₂²×C₇	C₄⋊C₄	1	2	yes	—
26	C₂²×C₇	C₂×C₈	1	2	yes	—
27	C₂²×C₇	M₁₆	1	2	yes	—
28	C₂²×C₇	C₂²×C₄	1	1	yes	characteristic
29	C₄×C₇	—	2	2	no	—
30	C₄×C₇	C₄⋊C₄	1	2	yes	—
31	C₄×C₇	C₂×C₈	1	4	yes	complemented
32	C₄×C₇	C₂×D₄	1	1	yes	characteristic
33	C₄×C₇	C₂×Q₈	1	1	yes	characteristic
34	C₂²×C₄	Dic₇	1	1	yes	characteristic
35	C₂²×C₄	Dic₇	1	1	yes	characteristic
36	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
37	C₂×C₈	—	7	4	no	—
38	C₂×C₈	—	14	4	no	—
39	C₄²	Dic₇	1	2	yes	—
40	C₄²	C₂×D₇	1	2	yes	—
41	C₂×C₇	C₄⋊C₈	1	4	yes	complemented
42	C₂×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₇	C₂²×C₈	1	1	yes	characteristic
44	C₂×C₇	C₂×M₁₆	1	1	yes	characteristic
45	C₂³	C₂×Dic₇	1	1	yes	characteristic
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	C₇⋊C₈	1	4	yes	complemented
48	C₂×C₄	Dic₁₄	1	2	yes	—
49	C₂×C₄	D₂₈	1	2	yes	—
50	C₂×C₄	C₂×Dic₇	1	2	yes	—
51	C₂×C₄	C₂×Dic₇	1	2	yes	—
52	C₂×C₄	C₂×Dic₇	1	1	yes	characteristic
53	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
54	C₈	—	14	4	no	—
55	C₇	C₂×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
56	C₂²	C₂×C₇⋊C₈	1	2	yes	—
57	C₂²	#1136	1	2	yes	—
58	C₂²	C₄⋊Dic₇	1	2	yes	—
59	C₂²	C₂²×Dic₇	1	1	yes	characteristic
60	C₄	—	2	2	no	—
61	C₄	C₂×C₇⋊C₈	1	4	yes	complemented
62	C₄	C₄⋊Dic₇	1	2	yes	—
63	C₄	C₂×Dic₁₄	1	1	yes	characteristic
64	C₄	C₂×D₂₈	1	1	yes	characteristic
65	C₂	C₄⋊(C₇⋊C₈)	1	4	yes	direct product factor
66	C₂	C₂²×C₇⋊C₈	1	1	yes	characteristic
67	C₂	C₂×#1136	1	1	yes	characteristic
68	C₂	C₂×C₄⋊Dic₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₇
commutator	C₂×C₇	C₂²×C₈
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₂×C₄²×C₇	C₂
socle, prime radical	C₂³×C₇	C₂×C₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	C₂×C₄²	D₇
hypocenter	C₇	C₂×C₄⋊C₈
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
3. C₄⋊(C₇⋊C₈)	C₂	direct (N × H)
14. C₂×C₄×C₇	C₈	semidirect
31. C₄×C₇	C₂×C₈	semidirect
41. C₂×C₇	C₄⋊C₈	semidirect
47. C₂×C₄	C₇⋊C₈	semidirect
55. C₇	C₂×C₄⋊C₈	semidirect
61. C₄	C₂×C₇⋊C₈	semidirect
65. C₂	C₄⋊(C₇⋊C₈)	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×C₄²×C₇	2
C₂²×C₇⋊C₈	2
C₄⋊(C₇⋊C₈)	2
C₂×C₄⋊C₈	7

Subgroups — C2×C4⋊(C7⋊C8) #89695

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄⋊(C₇⋊C₈) #89695