Subgroups — #6492

Order 224.

78 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	1	yes	maximal, characteristic
3	C₂²⋊Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
4	(C₂×D₇)⋊C₄	C₂	1	2	yes	maximal, complemented
5	Dic₇⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₂³×C₇	C₂²	1	1	yes	characteristic
7	C₂²×D₇	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₇	—	2	2	no	—
9	C₂×C₄×C₇	C₂²	1	2	yes	—
10	C₂²⋊D₇	—	2	2	no	—
11	C₂×Dic₇	C₂²	1	1	yes	characteristic
12	C₂×Dic₇	C₂²	1	2	yes	—
13	#123	—	7	1	no	maximal, Sylow
14	C₂²×C₇	—	2	2	no	—
15	C₂²×C₇	D₄	1	2	yes	—
16	C₂²×C₇	C₂³	1	1	yes	characteristic
17	C₂×D₇	—	2	1	no	—
18	C₂×D₇	—	2	2	no	—
19	C₄×C₇	—	2	2	no	—
20	Dic₇	—	2	1	no	—
21	Dic₇	—	2	2	no	—
22	C₂×D₄	—	7	1	no	—
23	C₂²×C₄	D₇	1	1	yes	characteristic, complemented
24	C₄⋊C₄	—	7	2	no	—
25	C₂²⋊C₄	—	7	1	no	—
26	C₂²⋊C₄	—	7	2	no	—
27	C₂×C₇	—	2	2	no	—
28	C₂×C₇	C₂×D₄	1	1	yes	characteristic
29	C₂×C₇	#41	1	2	yes	—
30	D₇	—	4	1	no	—
31	C₂³	—	7	1	no	—
32	C₂³	C₂×D₇	1	1	yes	characteristic
33	D₄	—	14	2	no	—
34	C₂×C₄	—	2	2	no	—
35	C₂×C₄	—	7	2	no	—
36	C₂×C₄	—	7	1	no	—
37	C₂×C₄	C₂×D₇	1	2	yes	—
38	C₇	#123	1	1	yes	characteristic, complemented, Sylow
39	C₂²	—	2	2	no	—
40	C₂²	—	14	2	no	—
41	C₂²	—	14	1	no	—
42	C₂²	C₂²⋊D₇	1	2	yes	—
43	C₂²	C₂²×D₇	1	1	yes	characteristic
44	C₄	—	2	2	no	—
45	C₄	—	14	2	no	—
46	C₄	—	14	1	no	—
47	C₂	—	2	2	no	—
48	C₂	—	28	1	no	—
49	C₂	#1157	1	2	yes	—
50	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
prime radical	C₂×C₂²⋊D₇	C₂
hypercenter	C₂²×C₄	D₇
hypocenter	C₇	#123
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
3. C₂²⋊Dic₇	C₂	semidirect
4. (C₂×D₇)⋊C₄	C₂	semidirect
5. Dic₇⋊C₄	C₂	semidirect
23. C₂²×C₄	D₇	semidirect
38. C₇	#123	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
C₂²⋊Dic₇	2
(C₂×D₇)⋊C₄	2
Dic₇⋊C₄	2
#123	7