Subgroups — C₇×#120 #6523

Order 224.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×D₄	C₂	1	3	yes	maximal, complemented
3	C₇×C₂²⋊C₄	C₂	1	3	yes	maximal, complemented
4	C₂³×C₇	—	2	6	no	—
5	C₂³×C₇	C₂²	1	3	yes	—
6	C₂³×C₇	C₂²	1	1	yes	characteristic, complemented
7	C₇×D₄	—	2	6	no	—
8	C₂×C₄×C₇	C₂²	1	3	yes	complemented
9	#120	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂²×C₇	—	2	8	no	—
11	C₂²×C₇	—	2	6	no	—
12	C₂²×C₇	—	2	3	no	—
13	C₂²×C₇	D₄	1	6	yes	complemented
14	C₂²×C₇	C₂³	1	1	yes	characteristic
15	C₄×C₇	—	2	3	no	—
16	C₂⁴	C₂×C₇	1	1	yes	characteristic, complemented
17	C₂×D₄	C₂×C₇	1	3	yes	complemented
18	C₂²⋊C₄	C₂×C₇	1	3	yes	complemented
19	C₂×C₇	—	2	6	no	—
20	C₂×C₇	—	4	1	no	—
21	C₂×C₇	C₂×D₄	1	3	yes	—
22	C₂³	—	2	6	no	—
23	C₂³	C₂²×C₇	1	3	yes	—
24	C₂³	C₂²×C₇	1	1	yes	characteristic, complemented
25	D₄	—	2	6	no	—
26	C₂×C₄	C₂²×C₇	1	3	yes	complemented
27	C₇	#120	1	1	yes	characteristic, direct product factor, Sylow
28	C₂²	—	2	8	no	—
29	C₂²	—	2	6	no	—
30	C₂²	—	2	3	no	—
31	C₂²	C₇×D₄	1	6	yes	complemented
32	C₂²	C₂³×C₇	1	1	yes	characteristic
33	C₄	—	2	3	no	—
34	C₂	—	2	6	no	—
35	C₂	—	4	1	no	—
36	C₂	C₂×C₇×D₄	1	3	yes	—

	subgroup	G/H
center, socle	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.

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