Subgroups — C₇×#121 #6524

Order 224.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×D₄	C₂	1	2	yes	maximal, complemented
2	C₂×C₇×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₄×C₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₂³×C₇	C₂²	1	2	yes	—
7	C₂³×C₇	C₂²	1	1	yes	characteristic
8	C₇×D₄	—	2	4	no	—
9	C₇×D₄	—	2	2	no	—
10	C₂×C₄×C₇	—	2	2	no	—
11	C₂×C₄×C₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₇	C₂²	1	2	yes	—
13	C₂×C₄×C₇	C₂²	1	1	yes	characteristic
14	#121	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
15	C₂²×C₇	—	2	1	no	—
16	C₂²×C₇	—	2	1	no	—
17	C₂²×C₇	—	2	2	no	—
18	C₂²×C₇	—	2	2	no	—
19	C₂²×C₇	—	2	2	no	—
20	C₂²×C₇	D₄	1	2	yes	complemented
21	C₂²×C₇	C₂³	1	1	yes	characteristic
22	C₄×C₇	—	2	1	no	—
23	C₄×C₇	—	2	2	no	—
24	C₄×C₇	D₄	1	2	yes	complemented
25	C₂×D₄	C₂×C₇	1	2	yes	complemented
26	C₂×D₄	C₂×C₇	1	1	yes	characteristic, complemented
27	C₂²×C₄	C₂×C₇	1	1	yes	characteristic, complemented
28	C₄⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
29	C₂²⋊C₄	C₂×C₇	1	2	yes	complemented
30	C₂×C₇	—	2	2	no	—
31	C₂×C₇	—	4	2	no	—
32	C₂×C₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₇	#41	1	1	yes	characteristic
35	C₂³	C₂²×C₇	1	2	yes	—
36	C₂³	C₂²×C₇	1	1	yes	characteristic
37	D₄	—	2	4	no	—
38	D₄	—	2	2	no	—
39	C₂×C₄	—	2	2	no	—
40	C₂×C₄	C₂²×C₇	1	1	yes	characteristic
41	C₂×C₄	C₂²×C₇	1	2	yes	—
42	C₂×C₄	C₂²×C₇	1	1	yes	characteristic
43	C₇	#121	1	1	yes	characteristic, direct product factor, Sylow
44	C₂²	—	2	1	no	—
45	C₂²	—	2	1	no	—
46	C₂²	—	2	2	no	—
47	C₂²	—	2	2	no	—
48	C₂²	—	2	2	no	—
49	C₂²	C₇×D₄	1	2	yes	complemented
50	C₂²	C₂³×C₇	1	1	yes	characteristic
51	C₄	—	2	1	no	—
52	C₄	—	2	2	no	—
53	C₄	C₇×D₄	1	2	yes	complemented
54	C₂	—	2	2	no	—
55	C₂	—	4	2	no	—
56	C₂	C₂×C₇×D₄	1	1	yes	characteristic
57	C₂	C₂×C₇×D₄	1	1	yes	characteristic
58	C₂	C₇×#41	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂³
commutator, Frattini, cosocle	C₂²	C₂³×C₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×D₄	C₂	semidirect
2. C₂×C₇×D₄	C₂	semidirect
3. C₂²×C₄×C₇	C₂	semidirect
4. C₇×C₄⋊C₄	C₂	semidirect
5. C₇×C₂²⋊C₄	C₂	semidirect
14. #121	C₇	direct (N × H)
20. C₂²×C₇	D₄	semidirect
24. C₄×C₇	D₄	semidirect
25. C₂×D₄	C₂×C₇	semidirect
26. C₂×D₄	C₂×C₇	semidirect
27. C₂²×C₄	C₂×C₇	semidirect
28. C₄⋊C₄	C₂×C₇	semidirect
29. C₂²⋊C₄	C₂×C₇	semidirect
43. C₇	#121	direct (N × H)
49. C₂²	C₇×D₄	semidirect
53. C₄	C₇×D₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×D₄	2
C₂²×C₄×C₇	2
C₇×C₄⋊C₄	2
C₇×C₂²⋊C₄	2
#121	7

Subgroups — C7×#121 #6524

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#121 #6524