Subgroups — C₂²×C₂²⋊D₇ #6556

Order 224.

236 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, complemented
2	C₂³×D₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂²⋊D₇	C₂	1	12	yes	maximal, direct product factor
4	C₂²×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×C₇	—	2	4	no	—
6	C₂³×C₇	C₂²	1	1	yes	characteristic
7	C₂³×C₇	C₂²	1	6	yes	complemented
8	C₂²×D₇	—	2	4	no	—
9	C₂²×D₇	C₂²	1	6	yes	complemented
10	C₂²⋊D₇	C₂²	1	16	yes	direct product factor
11	C₂×Dic₇	C₂²	1	6	yes	complemented
12	C₂²×D₄	—	7	1	no	maximal, Sylow
13	C₂²×C₇	—	2	12	no	—
14	C₂²×C₇	D₄	1	4	yes	complemented
15	C₂²×C₇	C₂³	1	3	yes	—
16	C₂²×C₇	C₂³	1	4	yes	complemented
17	C₂×D₇	—	2	12	no	—
18	C₂×D₇	C₂³	1	4	yes	complemented
19	Dic₇	C₂³	1	4	yes	complemented
20	C₂⁴	—	7	1	no	—
21	C₂⁴	D₇	1	1	yes	characteristic, complemented
22	C₂×D₄	—	7	12	no	—
23	C₂²×C₄	—	7	1	no	—
24	C₂×C₇	—	2	4	no	—
25	C₂×C₇	C₂×D₄	1	6	yes	complemented
26	C₂×C₇	C₂⁴	1	1	yes	characteristic
27	D₇	—	2	4	no	—
28	C₂³	—	2	4	no	—
29	C₂³	—	7	6	no	—
30	C₂³	—	14	4	no	—
31	C₂³	C₂×D₇	1	1	yes	characteristic
32	C₂³	C₂×D₇	1	6	yes	complemented
33	D₄	—	7	16	no	—
34	C₂×C₄	—	7	6	no	—
35	C₇	C₂²×D₄	1	1	yes	characteristic, complemented, Sylow
36	C₂²	—	2	12	no	—
37	C₂²	—	7	4	no	—
38	C₂²	—	14	12	no	—
39	C₂²	C₂²⋊D₇	1	4	yes	direct product factor
40	C₂²	C₂²×D₇	1	3	yes	—
41	C₂²	C₂²×D₇	1	4	yes	complemented
42	C₄	—	7	4	no	—
43	C₂	—	2	4	no	—
44	C₂	—	14	4	no	—
45	C₂	C₂×C₂²⋊D₇	1	6	yes	direct product factor
46	C₂	C₂³×D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₇
commutator, cosocle	C₂×C₇	C₂⁴
Frattini	C₂	C₂³×D₇
Fitting	C₂⁴×C₇	C₂
socle	C₂³×C₇	C₂²
hypercenter	C₂⁴	D₇
hypocenter	C₇	C₂²×D₄
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₇	C₂	semidirect
2. C₂³×D₇	C₂	semidirect
3. C₂×C₂²⋊D₇	C₂	direct (N × H)
4. C₂²×Dic₇	C₂	semidirect
7. C₂³×C₇	C₂²	semidirect
9. C₂²×D₇	C₂²	semidirect
10. C₂²⋊D₇	C₂²	direct (N × H)
11. C₂×Dic₇	C₂²	semidirect
14. C₂²×C₇	D₄	semidirect
16. C₂²×C₇	C₂³	semidirect
18. C₂×D₇	C₂³	semidirect
19. Dic₇	C₂³	semidirect
21. C₂⁴	D₇	semidirect
25. C₂×C₇	C₂×D₄	semidirect
32. C₂³	C₂×D₇	semidirect
35. C₇	C₂²×D₄	semidirect
39. C₂²	C₂²⋊D₇	direct (N × H)
41. C₂²	C₂²×D₇	semidirect
45. C₂	C₂×C₂²⋊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
C₂⁴×C₇	2
C₂³×D₇	2
C₂×C₂²⋊D₇	2
C₂²×Dic₇	2
C₂²×D₄	7

Subgroups — C22×C22⋊D7 #6556

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂²⋊D₇ #6556