Subgroups — C₇×C₄⋊Q₈ #6531

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	2	yes	maximal
2	C₇×C₄⋊C₄	C₂	1	4	yes	maximal
3	C₄²×C₇	C₂	1	1	yes	maximal, characteristic
4	C₇×Q₈	—	2	4	no	—
5	C₂×C₄×C₇	C₂²	1	4	yes	—
6	C₂×C₄×C₇	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₇	C₂²	1	2	yes	—
8	C₄⋊Q₈	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂²×C₇	C₂³	1	1	yes	characteristic
10	C₄×C₇	—	2	4	no	—
11	C₄×C₇	D₄	1	2	yes	—
12	C₄×C₇	Q₈	1	4	yes	complemented
13	C₂×Q₈	C₂×C₇	1	2	yes	—
14	C₄⋊C₄	C₂×C₇	1	4	yes	—
15	C₄²	C₂×C₇	1	1	yes	characteristic
16	C₂×C₇	C₂×D₄	1	1	yes	characteristic
17	C₂×C₇	C₂×Q₈	1	2	yes	—
18	Q₈	—	2	4	no	—
19	C₂×C₄	C₂²×C₇	1	4	yes	—
20	C₂×C₄	C₂²×C₇	1	1	yes	characteristic
21	C₂×C₄	C₂²×C₇	1	2	yes	—
22	C₇	C₄⋊Q₈	1	1	yes	characteristic, direct product factor, Sylow
23	C₂²	C₂³×C₇	1	1	yes	characteristic
24	C₄	—	2	4	no	—
25	C₄	C₇×D₄	1	2	yes	—
26	C₄	C₇×Q₈	1	4	yes	complemented
27	C₂	C₂×C₇×D₄	1	1	yes	characteristic
28	C₂	C₂×C₇×Q₈	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²×C₇	C₂³
commutator, Frattini, cosocle	C₂²	C₂³×C₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ G

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.