Subgroups — C₂²×Q₈×C₇₂₇ #37551449

Order 23264.

156 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×C₇₂₇	C₂	1	12	yes	maximal, direct product factor
2	C₂²×C₄×C₇₂₇	C₂	1	3	yes	maximal
3	C₂³×C₇₂₇	C₂²	1	1	yes	characteristic
4	Q₈×C₇₂₇	C₂²	1	16	yes	direct product factor
5	C₂×C₄×C₇₂₇	C₂²	1	18	yes	—
6	C₂²×C₇₂₇	Q₈	1	4	yes	direct product factor
7	C₂²×C₇₂₇	C₂³	1	3	yes	—
8	C₄×C₇₂₇	C₂³	1	12	yes	—
9	C₂×C₇₂₇	C₂×Q₈	1	6	yes	direct product factor
10	C₂×C₇₂₇	C₂⁴	1	1	yes	characteristic
11	C₇₂₇	C₂²×Q₈	1	1	yes	characteristic, direct product factor, Sylow
12	C₂²×Q₈	C₇₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×Q₈	C₂×C₇₂₇	1	12	yes	direct product factor
14	C₂²×C₄	C₂×C₇₂₇	1	3	yes	—
15	C₂³	C₂²×C₇₂₇	1	1	yes	characteristic
16	Q₈	C₂²×C₇₂₇	1	16	yes	direct product factor
17	C₂×C₄	C₂²×C₇₂₇	1	18	yes	—
18	C₂²	Q₈×C₇₂₇	1	4	yes	direct product factor
19	C₂²	C₂³×C₇₂₇	1	3	yes	—
20	C₄	C₂³×C₇₂₇	1	12	yes	—
21	C₂	C₂×Q₈×C₇₂₇	1	6	yes	direct product factor
22	C₂	C₂⁴×C₇₂₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₇₂₇	C₂²
commutator, Frattini, cosocle	C₂	C₂⁴×C₇₂₇
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇₂₇ ◁ 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.