Subgroups — C₂²×C₇×Q₈ #6559

Order 224.

156 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×Q₈	C₂	1	12	yes	maximal, direct product factor
2	C₂²×C₄×C₇	C₂	1	3	yes	maximal
3	C₂³×C₇	C₂²	1	1	yes	characteristic
4	C₇×Q₈	C₂²	1	16	yes	direct product factor
5	C₂×C₄×C₇	C₂²	1	18	yes	—
6	C₂²×Q₈	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂²×C₇	Q₈	1	4	yes	direct product factor
8	C₂²×C₇	C₂³	1	3	yes	—
9	C₄×C₇	C₂³	1	12	yes	—
10	C₂×Q₈	C₂×C₇	1	12	yes	direct product factor
11	C₂²×C₄	C₂×C₇	1	3	yes	—
12	C₂×C₇	C₂×Q₈	1	6	yes	direct product factor
13	C₂×C₇	C₂⁴	1	1	yes	characteristic
14	C₂³	C₂²×C₇	1	1	yes	characteristic
15	Q₈	C₂²×C₇	1	16	yes	direct product factor
16	C₂×C₄	C₂²×C₇	1	18	yes	—
17	C₇	C₂²×Q₈	1	1	yes	characteristic, direct product factor, Sylow
18	C₂²	C₇×Q₈	1	4	yes	direct product factor
19	C₂²	C₂³×C₇	1	3	yes	—
20	C₄	C₂³×C₇	1	12	yes	—
21	C₂	C₂×C₇×Q₈	1	6	yes	direct product factor
22	C₂	C₂⁴×C₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	C₂²
commutator, Frattini, cosocle	C₂	C₂⁴×C₇
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×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.