Subgroups — #6507

Order 224.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₁₄	C₂	1	1	yes	maximal, characteristic
3	Dic₇⋊C₄	C₂	1	4	yes	maximal
4	C₄×Dic₇	C₂	1	1	yes	maximal, characteristic
5	C₇×Q₈	—	2	2	no	—
6	C₂×C₄×C₇	C₂²	1	2	yes	—
7	C₂×C₄×C₇	C₂²	1	1	yes	characteristic
8	C₂×Dic₇	C₂²	1	2	yes	—
9	C₂×Dic₇	C₂²	1	2	yes	—
10	Dic₁₄	—	2	2	no	—
11	C₄⋊Q₈	—	7	1	no	maximal, Sylow
12	C₂²×C₇	C₂³	1	1	yes	characteristic
13	C₄×C₇	—	2	2	no	—
14	C₄×C₇	D₄	1	2	yes	—
15	Dic₇	—	2	2	no	—
16	Dic₇	Q₈	1	4	yes	complemented
17	C₂×Q₈	—	7	1	no	—
18	C₂×Q₈	D₇	1	1	yes	characteristic
19	C₄⋊C₄	—	7	4	no	—
20	C₄²	—	7	1	no	—
21	C₂×C₇	C₂×D₄	1	1	yes	characteristic
22	C₂×C₇	C₂×Q₈	1	2	yes	—
23	Q₈	—	2	2	no	—
24	Q₈	—	14	2	no	—
25	C₂×C₄	—	7	2	no	—
26	C₂×C₄	—	7	2	no	—
27	C₂×C₄	C₂×D₇	1	2	yes	—
28	C₂×C₄	C₂×D₇	1	1	yes	characteristic
29	C₇	C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
30	C₂²	C₂²×D₇	1	1	yes	characteristic
31	C₄	—	2	2	no	—
32	C₄	—	7	4	no	—
33	C₄	—	14	2	no	—
34	C₄	C₂²⋊D₇	1	2	yes	—
35	C₂	Q₈×D₇	1	2	yes	—
36	C₂	C₂×C₂²⋊D₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₇
commutator, socle, prime radical, cosocle	C₂²×C₇	C₂³
Fitting	C₂×C₇×Q₈	C₂
hypercenter	C₂×Q₈	D₇
hypocenter	C₇	C₄⋊Q₈
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

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

N	H ≅ G/N	type
16. Dic₇	Q₈	semidirect
29. C₇	C₄⋊Q₈	semidirect

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