Subgroups — C₇×H(5,𝔽₂) #6561

Order 224.

166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#41	C₂	1	6	yes	maximal, complemented
2	C₂×C₇×D₄	C₂	1	9	yes	maximal, complemented
3	C₂³×C₇	C₂²	1	6	yes	complemented
4	C₇×Q₈	C₂²	1	2	yes	complemented
5	C₇×D₄	C₂²	1	18	yes	complemented
6	C₂×C₄×C₇	C₂²	1	9	yes	complemented
7	H(5,𝔽₂)	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂²×C₇	—	4	6	no	—
9	C₂²×C₇	C₂³	1	9	yes	—
10	C₄×C₇	C₂³	1	6	yes	—
11	#41	C₂×C₇	1	6	yes	complemented
12	C₂×D₄	C₂×C₇	1	9	yes	complemented
13	C₂×C₇	—	2	9	no	—
14	C₂×C₇	C₂⁴	1	1	yes	characteristic
15	C₂³	C₂²×C₇	1	6	yes	complemented
16	Q₈	C₂²×C₇	1	2	yes	complemented
17	D₄	C₂²×C₇	1	18	yes	complemented
18	C₂×C₄	C₂²×C₇	1	9	yes	complemented
19	C₇	H(5,𝔽₂)	1	1	yes	characteristic, direct product factor, Sylow
20	C₂²	—	4	6	no	—
21	C₂²	C₂³×C₇	1	9	yes	—
22	C₄	C₂³×C₇	1	6	yes	—
23	C₂	—	2	9	no	—
24	C₂	C₂⁴×C₇	1	1	yes	characteristic

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