Subgroups — C₂×C₇×C₂²⋊C₄ #6518

Order 224.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₇	C₂	1	2	yes	maximal, complemented
3	C₇×C₂²⋊C₄	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₇	—	2	4	no	—
5	C₂³×C₇	C₄	1	4	yes	complemented
6	C₂³×C₇	C₂²	1	2	yes	—
7	C₂³×C₇	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₇	—	2	4	no	—
9	C₂×C₄×C₇	C₂²	1	4	yes	complemented
10	C₂×C₂²⋊C₄	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂²×C₇	—	2	8	no	—
12	C₂²×C₇	—	2	4	no	—
13	C₂²×C₇	C₂×C₄	1	2	yes	—
14	C₂²×C₇	C₂×C₄	1	4	yes	complemented
15	C₂²×C₇	D₄	1	4	yes	—
16	C₂²×C₇	C₂³	1	1	yes	characteristic
17	C₄×C₇	—	2	4	no	—
18	C₂⁴	C₂×C₇	1	1	yes	characteristic
19	C₂²×C₄	C₂×C₇	1	2	yes	complemented
20	C₂²⋊C₄	C₂×C₇	1	4	yes	direct product factor
21	C₂×C₇	—	2	4	no	—
22	C₂×C₇	C₂²⋊C₄	1	4	yes	direct product factor
23	C₂×C₇	C₂²×C₄	1	1	yes	characteristic
24	C₂×C₇	C₂×D₄	1	2	yes	—
25	C₂³	—	2	4	no	—
26	C₂³	C₄×C₇	1	4	yes	complemented
27	C₂³	C₂²×C₇	1	2	yes	—
28	C₂³	C₂²×C₇	1	1	yes	characteristic
29	C₂×C₄	—	2	4	no	—
30	C₂×C₄	C₂²×C₇	1	4	yes	complemented
31	C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
32	C₂²	—	2	8	no	—
33	C₂²	—	2	4	no	—
34	C₂²	C₂×C₄×C₇	1	2	yes	—
35	C₂²	C₂×C₄×C₇	1	4	yes	complemented
36	C₂²	C₇×D₄	1	4	yes	—
37	C₂²	C₂³×C₇	1	1	yes	characteristic
38	C₄	—	2	4	no	—
39	C₂	—	2	4	no	—
40	C₂	C₇×C₂²⋊C₄	1	4	yes	direct product factor
41	C₂	C₂²×C₄×C₇	1	1	yes	characteristic
42	C₂	C₂×C₇×D₄	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₇	C₂²
commutator	C₂	C₂²×C₄×C₇
Frattini, cosocle	C₂²	C₂³×C₇
prime radical	C₂⁴×C₇	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×C₇	C₂	semidirect
3. C₇×C₂²⋊C₄	C₂	direct (N × H)
5. C₂³×C₇	C₄	semidirect
9. C₂×C₄×C₇	C₂²	semidirect
10. C₂×C₂²⋊C₄	C₇	direct (N × H)
14. C₂²×C₇	C₂×C₄	semidirect
19. C₂²×C₄	C₂×C₇	semidirect
20. C₂²⋊C₄	C₂×C₇	direct (N × H)
22. C₂×C₇	C₂²⋊C₄	direct (N × H)
26. C₂³	C₄×C₇	semidirect
30. C₂×C₄	C₂²×C₇	semidirect
31. C₇	C₂×C₂²⋊C₄	direct (N × H)
35. C₂²	C₂×C₄×C₇	semidirect
40. C₂	C₇×C₂²⋊C₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₇	2
C₂²×C₄×C₇	2
C₇×C₂²⋊C₄	2
C₂×C₂²⋊C₄	7

Subgroups — C2×C7×C22⋊C4 #6518

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×C₂²⋊C₄ #6518