Subgroups — C₂×C₇×C₄⋊C₈ #90068

Order 448.

138 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
2	C₂×C₄²×C₇	C₂	1	1	yes	maximal, characteristic
3	C₇×C₄⋊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	C₂×C₇×C₈	—	2	4	no	—
8	C₂×C₇×C₈	C₂²	1	4	yes	—
9	C₄²×C₇	C₄	1	2	yes	—
10	C₄²×C₇	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈	C₇	1	1	yes	maximal, characteristic, direct product factor, 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	C₂×C₇	1	2	yes	—
23	C₂×C₄²	C₂×C₇	1	1	yes	characteristic
24	C₄⋊C₈	C₂×C₇	1	4	yes	direct product factor
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	C₄×C₇	1	1	yes	characteristic
35	C₂²×C₄	C₄×C₇	1	1	yes	characteristic
36	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic
37	C₂×C₈	—	2	4	no	—
38	C₂×C₈	C₂²×C₇	1	4	yes	—
39	C₄²	C₄×C₇	1	2	yes	—
40	C₄²	C₂²×C₇	1	2	yes	—
41	C₂×C₇	C₄⋊C₈	1	4	yes	direct product factor
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×C₄×C₇	1	1	yes	characteristic
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	C₇×C₈	1	4	yes	complemented
48	C₂×C₄	C₂×C₄×C₇	1	2	yes	—
49	C₂×C₄	C₂×C₄×C₇	1	1	yes	characteristic
50	C₂×C₄	C₂×C₄×C₇	1	2	yes	—
51	C₂×C₄	C₇×D₄	1	2	yes	—
52	C₂×C₄	C₇×Q₈	1	2	yes	—
53	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
54	C₈	—	2	4	no	—
55	C₇	C₂×C₄⋊C₈	1	1	yes	characteristic, direct product factor, Sylow
56	C₂²	C₇×C₄⋊C₄	1	2	yes	—
57	C₂²	C₂×C₇×C₈	1	2	yes	—
58	C₂²	C₇×M₁₆	1	2	yes	—
59	C₂²	C₂²×C₄×C₇	1	1	yes	characteristic
60	C₄	—	2	2	no	—
61	C₄	C₇×C₄⋊C₄	1	2	yes	—
62	C₄	C₂×C₇×C₈	1	4	yes	complemented
63	C₄	C₂×C₇×D₄	1	1	yes	characteristic
64	C₄	C₂×C₇×Q₈	1	1	yes	characteristic
65	C₂	C₇×C₄⋊C₈	1	4	yes	direct product factor
66	C₂	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
67	C₂	C₂²×C₇×C₈	1	1	yes	characteristic
68	C₂	C₂×C₇×M₁₆	1	1	yes	characteristic

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

Complements

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

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

Subgroups — C2×C7×C4⋊C8 #90068

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×C₄⋊C₈ #90068