Subgroups — #6432

Order 224.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₄	C₂	1	2	yes	maximal
2	C₄²×C₇	C₂	1	1	yes	maximal, characteristic
3	C₄⋊Dic₇	C₂	1	4	yes	maximal
4	C₂×C₄×C₇	C₂²	1	1	yes	characteristic
5	C₂×C₄×C₇	C₂²	1	2	yes	—
6	C₂×Dic₇	C₂²	1	4	yes	—
7	Dic₁₄	—	2	4	no	—
8	C₄⋊Q₈	—	7	1	no	maximal, Sylow
9	C₂²×C₇	C₂³	1	1	yes	characteristic
10	C₄×C₇	D₄	1	2	yes	—
11	C₄×C₇	Q₈	1	4	yes	complemented
12	Dic₇	—	2	4	no	—
13	C₂×Q₈	—	7	2	no	—
14	C₄⋊C₄	—	7	4	no	—
15	C₄²	D₇	1	1	yes	characteristic
16	C₂×C₇	C₂×D₄	1	1	yes	characteristic
17	C₂×C₇	C₂×Q₈	1	2	yes	—
18	Q₈	—	14	4	no	—
19	C₂×C₄	—	7	4	no	—
20	C₂×C₄	C₂×D₇	1	2	yes	—
21	C₂×C₄	C₂×D₇	1	1	yes	characteristic
22	C₇	C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
23	C₂²	C₂²×D₇	1	1	yes	characteristic
24	C₄	—	14	4	no	—
25	C₄	Dic₁₄	1	4	yes	complemented
26	C₄	D₂₈	1	2	yes	—
27	C₂	C₂×Dic₁₄	1	2	yes	—
28	C₂	C₂×D₂₈	1	1	yes	characteristic

	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	C₄⋊Q₈
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄²

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

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