Subgroups — #6493

Order 224.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊D₇	C₂	1	2	yes	maximal, complemented
3	C₂×D₂₈	C₂	1	1	yes	maximal, characteristic, complemented
4	(C₂×D₇)⋊C₄	C₂	1	2	yes	maximal, complemented
5	C₄⋊Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₇	C₂²	1	1	yes	characteristic
7	C₂²×D₇	C₂²	1	2	yes	—
8	C₂×C₄×C₇	—	2	2	no	—
9	C₂×C₄×C₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₇	C₂²	1	1	yes	characteristic
11	C₂²⋊D₇	—	2	4	no	—
12	C₂×Dic₇	C₂²	1	2	yes	—
13	D₂₈	—	2	2	no	—
14	#121	—	7	1	no	maximal, Sylow
15	C₂²×C₇	—	2	1	no	—
16	C₂²×C₇	—	2	1	no	—
17	C₂²×C₇	D₄	1	2	yes	complemented
18	C₂²×C₇	C₂³	1	1	yes	characteristic
19	C₂×D₇	—	2	2	no	—
20	C₂×D₇	—	2	2	no	—
21	C₂×D₇	—	2	2	no	—
22	C₄×C₇	—	2	1	no	—
23	C₄×C₇	D₄	1	2	yes	complemented
24	Dic₇	—	2	2	no	—
25	C₂×D₄	—	7	2	no	—
26	C₂×D₄	—	7	1	no	—
27	C₂²×C₄	D₇	1	1	yes	characteristic, complemented
28	C₄⋊C₄	—	7	1	no	—
29	C₂²⋊C₄	—	7	2	no	—
30	C₂×C₇	—	2	2	no	—
31	C₂×C₇	C₂×D₄	1	1	yes	characteristic
32	C₂×C₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₇	#41	1	1	yes	characteristic
34	D₇	—	4	2	no	—
35	C₂³	—	7	2	no	—
36	C₂³	C₂×D₇	1	1	yes	characteristic
37	D₄	—	14	4	no	—
38	D₄	—	14	2	no	—
39	C₂×C₄	—	2	2	no	—
40	C₂×C₄	—	7	2	no	—
41	C₂×C₄	C₂×D₇	1	1	yes	characteristic
42	C₂×C₄	C₂×D₇	1	1	yes	characteristic
43	C₇	#121	1	1	yes	characteristic, complemented, Sylow
44	C₂²	—	2	1	no	—
45	C₂²	—	2	1	no	—
46	C₂²	—	14	2	no	—
47	C₂²	—	14	2	no	—
48	C₂²	—	14	2	no	—
49	C₂²	D₂₈	1	2	yes	complemented
50	C₂²	C₂²×D₇	1	1	yes	characteristic
51	C₄	—	2	1	no	—
52	C₄	—	14	2	no	—
53	C₄	C₂²⋊D₇	1	2	yes	complemented
54	C₂	—	2	2	no	—
55	C₂	—	28	2	no	—
56	C₂	C₂×D₂₈	1	1	yes	characteristic
57	C₂	#1157	1	1	yes	characteristic
58	C₂	C₂×C₂²⋊D₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₇
commutator, socle, cosocle	C₂²×C₇	C₂³
Fitting	C₂²×C₄×C₇	C₂
hypercenter	C₂²×C₄	D₇
hypocenter	C₇	#121
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₄×C₇	C₂	semidirect
2. C₂×C₂²⋊D₇	C₂	semidirect
3. C₂×D₂₈	C₂	semidirect
4. (C₂×D₇)⋊C₄	C₂	semidirect
5. C₄⋊Dic₇	C₂	semidirect
17. C₂²×C₇	D₄	semidirect
23. C₄×C₇	D₄	semidirect
27. C₂²×C₄	D₇	semidirect
43. C₇	#121	semidirect
49. C₂²	D₂₈	semidirect
53. C₄	C₂²⋊D₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₇	2
C₂×C₂²⋊D₇	2
C₂×D₂₈	2
(C₂×D₇)⋊C₄	2
C₄⋊Dic₇	2
#121	7