Subgroups — H(5,𝔽₂)×C₇₆₉ #37589802

Order 24608.

166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×C₇₆₉	C₂	1	6	yes	maximal, complemented
2	C₂×D₄×C₇₆₉	C₂	1	9	yes	maximal, complemented
3	C₂³×C₇₆₉	C₂²	1	6	yes	complemented
4	Q₈×C₇₆₉	C₂²	1	2	yes	complemented
5	D₄×C₇₆₉	C₂²	1	18	yes	complemented
6	C₂×C₄×C₇₆₉	C₂²	1	9	yes	complemented
7	C₂²×C₇₆₉	—	4	6	no	—
8	C₂²×C₇₆₉	C₂³	1	9	yes	—
9	C₄×C₇₆₉	C₂³	1	6	yes	—
10	C₂×C₇₆₉	—	2	9	no	—
11	C₂×C₇₆₉	C₂⁴	1	1	yes	characteristic
12	C₇₆₉	H(5,𝔽₂)	1	1	yes	characteristic, direct product factor, Sylow
13	H(5,𝔽₂)	C₇₆₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
14	#41	C₂×C₇₆₉	1	6	yes	complemented
15	C₂×D₄	C₂×C₇₆₉	1	9	yes	complemented
16	C₂³	C₂²×C₇₆₉	1	6	yes	complemented
17	Q₈	C₂²×C₇₆₉	1	2	yes	complemented
18	D₄	C₂²×C₇₆₉	1	18	yes	complemented
19	C₂×C₄	C₂²×C₇₆₉	1	9	yes	complemented
20	C₂²	—	4	6	no	—
21	C₂²	C₂³×C₇₆₉	1	9	yes	—
22	C₄	C₂³×C₇₆₉	1	6	yes	—
23	C₂	—	2	9	no	—
24	C₂	C₂⁴×C₇₆₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₇₆₉	C₂⁴
commutator, Frattini, cosocle	C₂	C₂⁴×C₇₆₉
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₆₉ ◁ 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.