Subgroups — C₂×C₈×D₇ #6462

Order 224.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×D₇	C₂	1	4	yes	maximal, direct product factor
5	C₂²×D₇	C₄	1	1	yes	characteristic
6	C₂×C₄×C₇	C₂²	1	1	yes	characteristic
7	C₂×Dic₇	C₄	1	1	yes	characteristic
8	C₄×D₇	C₄	1	2	yes	—
9	C₄×D₇	C₂²	1	2	yes	—
10	C₇×C₈	C₂²	1	2	yes	complemented
11	C₇⋊C₈	C₂²	1	2	yes	complemented
12	C₂²×C₈	—	7	1	no	maximal, Sylow
13	C₂²×C₇	C₂×C₄	1	1	yes	characteristic
14	C₂×D₇	C₈	1	4	yes	direct product factor
15	C₂×D₇	C₂×C₄	1	2	yes	—
16	C₄×C₇	C₂×C₄	1	1	yes	characteristic
17	C₄×C₇	C₂³	1	1	yes	characteristic
18	Dic₇	C₂×C₄	1	2	yes	—
19	C₂²×C₄	—	7	1	no	—
20	C₂×C₈	—	7	1	no	—
21	C₂×C₈	—	7	4	no	—
22	C₂×C₈	D₇	1	1	yes	characteristic, direct product factor
23	C₂×C₇	C₂×C₈	1	2	yes	complemented
24	C₂×C₇	C₂²×C₄	1	1	yes	characteristic
25	D₇	C₂×C₈	1	4	yes	direct product factor
26	C₂³	—	7	1	no	—
27	C₂×C₄	—	7	1	no	—
28	C₂×C₄	—	7	2	no	—
29	C₂×C₄	—	7	2	no	—
30	C₂×C₄	C₂×D₇	1	1	yes	characteristic
31	C₈	—	7	2	no	—
32	C₈	C₂×D₇	1	2	yes	direct product factor
33	C₇	C₂²×C₈	1	1	yes	characteristic, complemented, Sylow
34	C₂²	—	7	4	no	—
35	C₂²	—	7	2	no	—
36	C₂²	C₄×D₇	1	1	yes	characteristic
37	C₄	—	7	2	no	—
38	C₄	C₄×D₇	1	1	yes	characteristic
39	C₄	C₂²×D₇	1	1	yes	characteristic
40	C₂	—	7	4	no	—
41	C₂	C₈×D₇	1	2	yes	direct product factor
42	C₂	C₂×C₄×D₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×C₈	C₂	semidirect
3. C₂×C₇⋊C₈	C₂	semidirect
4. C₈×D₇	C₂	direct (N × H)
10. C₇×C₈	C₂²	semidirect
11. C₇⋊C₈	C₂²	semidirect
14. C₂×D₇	C₈	direct (N × H)
22. C₂×C₈	D₇	direct (N × H)
23. C₂×C₇	C₂×C₈	semidirect
25. D₇	C₂×C₈	direct (N × H)
32. C₈	C₂×D₇	direct (N × H)
33. C₇	C₂²×C₈	semidirect
41. C₂	C₈×D₇	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×C₄×D₇	2
C₂×C₇×C₈	2
C₂×C₇⋊C₈	2
C₈×D₇	2
C₂²×C₈	7

Subgroups — C2×C8×D7 #6462

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈×D₇ #6462