Subgroups — C₂×D₄×C₂₂₆₇ #49136708

Order 36272.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₂₂₆₇	C₂	1	2	yes	maximal, complemented
2	D₄×C₂₂₆₇	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×C₂₂₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₂₂₆₇	—	2	4	no	—
5	C₂²×C₂₂₆₇	C₂²	1	1	yes	characteristic
6	C₂²×C₂₂₆₇	C₂²	1	4	yes	complemented
7	C₄×C₂₂₆₇	C₂²	1	2	yes	complemented
8	C₂×C₂₂₆₇	—	2	4	no	—
9	C₂×C₂₂₆₇	D₄	1	2	yes	direct product factor
10	C₂×C₂₂₆₇	C₂³	1	1	yes	characteristic
11	C₂₂₆₇	C₂×D₄	1	1	yes	characteristic, direct product factor, Sylow
12	C₂×D₄	C₂₂₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂³	C₂×C₂₂₆₇	1	2	yes	complemented
14	D₄	C₂×C₂₂₆₇	1	4	yes	direct product factor
15	C₂×C₄	C₂×C₂₂₆₇	1	1	yes	characteristic, complemented
16	C₂²	—	2	4	no	—
17	C₂²	C₂²×C₂₂₆₇	1	1	yes	characteristic
18	C₂²	C₂²×C₂₂₆₇	1	4	yes	complemented
19	C₄	C₂²×C₂₂₆₇	1	2	yes	complemented
20	C₂	—	2	4	no	—
21	C₂	D₄×C₂₂₆₇	1	2	yes	direct product factor
22	C₂	C₂³×C₂₂₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₂₆₇	C₂²
commutator, Frattini, cosocle	C₂	C₂³×C₂₂₆₇
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₂₆₇ ◁ 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.