Subgroups — #6451

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	C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₄⋊Dic₇	C₂	1	1	yes	maximal, characteristic
4	Dic₇⋊C₄	C₂	1	2	yes	maximal
5	C₄×Dic₇	C₂	1	1	yes	maximal, characteristic
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	1	yes	characteristic
10	C₂×Dic₇	C₂²	1	1	yes	characteristic
11	Dic₁₄	—	2	4	no	—
12	C₄⋊Q₈	—	7	1	no	maximal, Sylow
13	C₂²×C₇	C₂³	1	1	yes	characteristic
14	C₄×C₇	—	2	2	no	—
15	C₄×C₇	Q₈	1	2	yes	complemented
16	Dic₇	—	2	2	no	—
17	Dic₇	D₄	1	2	yes	—
18	Dic₇	Q₈	1	2	yes	complemented
19	C₂×Q₈	—	7	2	no	—
20	C₄⋊C₄	—	7	2	no	—
21	C₄⋊C₄	—	7	1	no	—
22	C₄⋊C₄	D₇	1	1	yes	characteristic
23	C₄²	—	7	1	no	—
24	C₂×C₇	C₂×D₄	1	1	yes	characteristic
25	C₂×C₇	C₂×Q₈	1	1	yes	characteristic
26	C₂×C₇	C₂×Q₈	1	1	yes	characteristic
27	Q₈	—	14	4	no	—
28	C₂×C₄	—	7	1	no	—
29	C₂×C₄	—	7	1	no	—
30	C₂×C₄	—	7	2	no	—
31	C₂×C₄	C₂×D₇	1	2	yes	—
32	C₂×C₄	C₂×D₇	1	1	yes	characteristic
33	C₇	C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
34	C₂²	C₂²×D₇	1	1	yes	characteristic
35	C₄	—	2	2	no	—
36	C₄	—	7	2	no	—
37	C₄	—	7	2	no	—
38	C₄	—	14	2	no	—
39	C₄	Dic₁₄	1	2	yes	complemented
40	C₂	C₂×Dic₁₄	1	1	yes	characteristic
41	C₂	D₄×D₇	1	1	yes	characteristic
42	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	C₄⋊Q₈
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

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.