Subgroups — #6452

Order 224.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₄⋊Dic₇	C₂	1	1	yes	maximal, characteristic
3	Dic₇⋊C₄	C₂	1	1	yes	maximal, characteristic
4	Dic₇⋊C₄	C₂	1	1	yes	maximal, characteristic
5	Dic₇⋊C₄	C₂	1	1	yes	maximal, characteristic
6	Dic₇⋊C₄	C₂	1	1	yes	maximal, characteristic
7	C₄×Dic₇	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₇	C₂²	1	1	yes	characteristic
11	C₂×Dic₇	C₂²	1	1	yes	characteristic
12	C₂×Dic₇	C₂²	1	1	yes	characteristic
13	C₂×Dic₇	C₂²	1	1	yes	characteristic
14	C₂×Dic₇	C₂²	1	1	yes	characteristic
15	#125	—	7	1	no	maximal, Sylow
16	C₂²×C₇	C₂³	1	1	yes	characteristic
17	C₄×C₇	—	2	1	no	—
18	C₄×C₇	—	2	1	no	—
19	C₄×C₇	—	2	1	no	—
20	Dic₇	—	2	1	no	—
21	Dic₇	—	2	1	no	—
22	Dic₇	—	2	1	no	—
23	Dic₇	Q₈	1	2	yes	—
24	C₄⋊C₄	—	7	1	no	—
25	C₄⋊C₄	—	7	1	no	—
26	C₄⋊C₄	—	7	1	no	—
27	C₄⋊C₄	—	7	1	no	—
28	C₄⋊C₄	—	7	1	no	—
29	C₄⋊C₄	D₇	1	1	yes	characteristic
30	C₄²	—	7	1	no	—
31	C₂×C₇	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₇	#41	1	1	yes	characteristic
33	C₂×C₇	#41	1	1	yes	characteristic
34	C₂×C₄	—	7	1	no	—
35	C₂×C₄	—	7	1	no	—
36	C₂×C₄	—	7	1	no	—
37	C₂×C₄	—	7	1	no	—
38	C₂×C₄	C₂×D₇	1	1	yes	characteristic
39	C₂×C₄	C₂×D₇	1	1	yes	characteristic
40	C₂×C₄	C₂×D₇	1	1	yes	characteristic
41	C₇	#125	1	1	yes	characteristic, complemented, Sylow
42	C₂²	C₂²×D₇	1	1	yes	characteristic
43	C₄	—	2	1	no	—
44	C₄	—	2	1	no	—
45	C₄	—	2	1	no	—
46	C₄	—	7	2	no	—
47	C₄	—	14	1	no	—
48	C₄	—	14	1	no	—
49	C₄	—	14	1	no	—
50	C₂	#1157	1	1	yes	characteristic
51	C₂	#1159	1	1	yes	characteristic
52	C₂	Q₈×D₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₇
commutator, socle, prime radical, cosocle	C₂²×C₇	C₂³
Fitting	C₇×C₄⋊C₄	C₂
hypercenter	C₄⋊C₄	D₇
hypocenter	C₇	#125
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
41. C₇	#125	semidirect

Maximal subgroups

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

subgroup	index
C₇×C₄⋊C₄	2
C₄⋊Dic₇	2
Dic₇⋊C₄	2
C₄×Dic₇	2
#125	7