Subgroups — H(5,𝔽₂)×C₃₃₇ #24924862

Order 10784.

166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×C₃₃₇	C₂	1	6	yes	maximal, complemented
2	C₂×D₄×C₃₃₇	C₂	1	9	yes	maximal, complemented
3	C₂³×C₃₃₇	C₂²	1	6	yes	complemented
4	Q₈×C₃₃₇	C₂²	1	2	yes	complemented
5	D₄×C₃₃₇	C₂²	1	18	yes	complemented
6	C₂×C₄×C₃₃₇	C₂²	1	9	yes	complemented
7	C₂²×C₃₃₇	—	4	6	no	—
8	C₂²×C₃₃₇	C₂³	1	9	yes	—
9	C₄×C₃₃₇	C₂³	1	6	yes	—
10	C₂×C₃₃₇	—	2	9	no	—
11	C₂×C₃₃₇	C₂⁴	1	1	yes	characteristic
12	C₃₃₇	H(5,𝔽₂)	1	1	yes	characteristic, direct product factor, Sylow
13	H(5,𝔽₂)	C₃₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
14	#41	C₂×C₃₃₇	1	6	yes	complemented
15	C₂×D₄	C₂×C₃₃₇	1	9	yes	complemented
16	C₂³	C₂²×C₃₃₇	1	6	yes	complemented
17	Q₈	C₂²×C₃₃₇	1	2	yes	complemented
18	D₄	C₂²×C₃₃₇	1	18	yes	complemented
19	C₂×C₄	C₂²×C₃₃₇	1	9	yes	complemented
20	C₂²	—	4	6	no	—
21	C₂²	C₂³×C₃₃₇	1	9	yes	—
22	C₄	C₂³×C₃₃₇	1	6	yes	—
23	C₂	—	2	9	no	—
24	C₂	C₂⁴×C₃₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₃₇	C₂⁴
commutator, Frattini, cosocle	C₂	C₂⁴×C₃₃₇
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₃₇ ◁ 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.