Subgroups — C₇×#125 #6528

Order 224.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₄⋊C₄	C₂	1	4	yes	maximal
2	C₇×C₄⋊C₄	C₂	1	2	yes	maximal
3	C₄²×C₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₇	C₂²	1	2	yes	—
5	C₂×C₄×C₇	C₂²	1	4	yes	—
6	C₂×C₄×C₇	C₂²	1	1	yes	characteristic
7	#125	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂²×C₇	C₂³	1	1	yes	characteristic
9	C₄×C₇	—	2	2	no	—
10	C₄×C₇	—	2	4	no	—
11	C₄×C₇	Q₈	1	2	yes	—
12	C₄⋊C₄	C₂×C₇	1	4	yes	—
13	C₄⋊C₄	C₂×C₇	1	2	yes	—
14	C₄²	C₂×C₇	1	1	yes	characteristic
15	C₂×C₇	C₂×Q₈	1	1	yes	characteristic
16	C₂×C₇	#41	1	2	yes	—
17	C₂×C₄	C₂²×C₇	1	2	yes	—
18	C₂×C₄	C₂²×C₇	1	4	yes	—
19	C₂×C₄	C₂²×C₇	1	1	yes	characteristic
20	C₇	#125	1	1	yes	characteristic, direct product factor, Sylow
21	C₂²	C₂³×C₇	1	1	yes	characteristic
22	C₄	—	2	2	no	—
23	C₄	—	2	4	no	—
24	C₄	C₇×Q₈	1	2	yes	—
25	C₂	C₂×C₇×Q₈	1	1	yes	characteristic
26	C₂	C₇×#41	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²×C₇	C₂³
commutator, Frattini, cosocle	C₂²	C₂³×C₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ G

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
7. #125	C₇	direct (N × H)
20. C₇	#125	direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.