Subgroups — #6455

Order 224.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₇	C₂	1	1	yes	maximal, characteristic
2	C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	(C₂×D₇)⋊C₄	C₂	1	2	yes	maximal
4	C₄⋊Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×Dic₇	C₂	1	2	yes	maximal, complemented
6	C₂²×D₇	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₇	C₂²	1	2	yes	—
8	C₂×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₂×Dic₇	C₂²	1	2	yes	—
10	C₂×Dic₇	C₂²	1	1	yes	characteristic
11	C₄×D₇	C₄	1	4	yes	complemented
12	#117	—	7	1	no	maximal, Sylow
13	C₂²×C₇	C₂³	1	1	yes	characteristic
14	C₂×D₇	—	2	1	no	—
15	C₂×D₇	—	2	1	no	—
16	C₂×D₇	C₂×C₄	1	2	yes	—
17	C₄×C₇	—	2	2	no	—
18	C₄×C₇	C₂×C₄	1	2	yes	—
19	Dic₇	—	2	2	no	—
20	Dic₇	C₂×C₄	1	2	yes	—
21	C₂²×C₄	—	7	1	no	—
22	C₄⋊C₄	—	7	1	no	—
23	C₄⋊C₄	D₇	1	1	yes	characteristic, complemented
24	C₂²⋊C₄	—	7	2	no	—
25	C₄²	—	7	2	no	—
26	C₂×C₇	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₇	#41	1	1	yes	characteristic
28	C₂×C₇	#41	1	1	yes	characteristic
29	D₇	—	2	2	no	—
30	C₂³	—	7	1	no	—
31	C₂×C₄	—	7	1	no	—
32	C₂×C₄	—	7	4	no	—
33	C₂×C₄	—	7	2	no	—
34	C₂×C₄	C₂×D₇	1	2	yes	—
35	C₂×C₄	C₂×D₇	1	1	yes	characteristic
36	C₇	#117	1	1	yes	characteristic, complemented, Sylow
37	C₂²	—	7	2	no	—
38	C₂²	—	14	1	no	—
39	C₂²	—	14	1	no	—
40	C₂²	C₂²×D₇	1	1	yes	characteristic
41	C₄	—	2	2	no	—
42	C₄	—	7	2	no	—
43	C₄	—	14	2	no	—
44	C₄	C₄×D₇	1	2	yes	—
45	C₂	—	14	2	no	—
46	C₂	C₂×C₄×D₇	1	1	yes	characteristic
47	C₂	#1159	1	1	yes	characteristic
48	C₂	#1161	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₇
commutator	C₂×C₇	C₂²×C₄
Fitting	C₇×C₄⋊C₄	C₂
socle, cosocle	C₂²×C₇	C₂³
prime radical	C₂²×D₇	C₂²
hypercenter	C₄⋊C₄	D₇
hypocenter	C₇	#117
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₇×C₄⋊C₄	C₂	semidirect
4. C₄⋊Dic₇	C₂	semidirect
5. C₄×Dic₇	C₂	semidirect
11. C₄×D₇	C₄	semidirect
23. C₄⋊C₄	D₇	semidirect
36. C₇	#117	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₇	2
C₇×C₄⋊C₄	2
(C₂×D₇)⋊C₄	2
C₄⋊Dic₇	2
C₄×Dic₇	2
#117	7