Subgroups — #6433

Order 224.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₇	C₂	1	1	yes	maximal, characteristic
2	C₄⋊Dic₇	C₂	1	2	yes	maximal
3	Dic₇⋊C₄	C₂	1	4	yes	maximal
4	C₂×C₄×C₇	C₂²	1	2	yes	—
5	C₂×C₄×C₇	C₂²	1	1	yes	characteristic
6	C₂×Dic₇	C₂²	1	4	yes	—
7	#125	—	7	1	no	maximal, Sylow
8	C₂²×C₇	C₂³	1	1	yes	characteristic
9	C₄×C₇	—	2	2	no	—
10	C₄×C₇	Q₈	1	2	yes	—
11	Dic₇	—	2	4	no	—
12	C₄⋊C₄	—	7	4	no	—
13	C₄⋊C₄	—	7	2	no	—
14	C₄²	D₇	1	1	yes	characteristic
15	C₂×C₇	C₂×Q₈	1	1	yes	characteristic
16	C₂×C₇	#41	1	2	yes	—
17	C₂×C₄	—	7	4	no	—
18	C₂×C₄	C₂×D₇	1	2	yes	—
19	C₂×C₄	C₂×D₇	1	1	yes	characteristic
20	C₇	#125	1	1	yes	characteristic, complemented, Sylow
21	C₂²	C₂²×D₇	1	1	yes	characteristic
22	C₄	—	2	2	no	—
23	C₄	—	14	4	no	—
24	C₄	Dic₁₄	1	2	yes	—
25	C₂	C₂×Dic₁₄	1	1	yes	characteristic
26	C₂	#1157	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₇
commutator, socle, prime radical, cosocle	C₂²×C₇	C₂³
Fitting	C₄²×C₇	C₂
hypercenter	C₄²	D₇
hypocenter	C₇	#125
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄²

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

N	H ≅ G/N	type
20. C₇	#125	semidirect

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