Subgroups — C₇×#124 #6527

Order 224.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×D₄	C₂	1	1	yes	maximal, characteristic
3	C₇×C₂²⋊C₄	C₂	1	4	yes	maximal, complemented
4	C₄²×C₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×C₇	C₂²	1	2	yes	—
6	C₇×Q₈	—	2	2	no	—
7	C₇×D₄	—	2	2	no	—
8	C₂×C₄×C₇	C₂²	1	2	yes	—
9	C₂×C₄×C₇	C₂²	1	2	yes	—
10	C₂×C₄×C₇	C₂²	1	1	yes	characteristic
11	#124	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
12	C₂²×C₇	—	2	2	no	—
13	C₂²×C₇	—	2	4	no	—
14	C₂²×C₇	C₂³	1	1	yes	characteristic
15	C₄×C₇	—	2	2	no	—
16	C₄×C₇	—	2	2	no	—
17	C₄×C₇	D₄	1	2	yes	—
18	C₂×Q₈	C₂×C₇	1	1	yes	characteristic, complemented
19	C₂×D₄	C₂×C₇	1	1	yes	characteristic
20	C₂²⋊C₄	C₂×C₇	1	4	yes	complemented
21	C₄²	C₂×C₇	1	1	yes	characteristic, complemented
22	C₂×C₇	—	4	2	no	—
23	C₂×C₇	C₂×D₄	1	1	yes	characteristic
24	C₂×C₇	#41	1	2	yes	—
25	C₂³	C₂²×C₇	1	2	yes	—
26	Q₈	—	2	2	no	—
27	D₄	—	2	2	no	—
28	C₂×C₄	C₂²×C₇	1	2	yes	—
29	C₂×C₄	C₂²×C₇	1	2	yes	—
30	C₂×C₄	C₂²×C₇	1	1	yes	characteristic
31	C₇	#124	1	1	yes	characteristic, direct product factor, Sylow
32	C₂²	—	2	2	no	—
33	C₂²	—	2	4	no	—
34	C₂²	C₂³×C₇	1	1	yes	characteristic
35	C₄	—	2	2	no	—
36	C₄	—	2	2	no	—
37	C₄	C₇×D₄	1	2	yes	—
38	C₂	—	4	2	no	—
39	C₂	C₂×C₇×D₄	1	1	yes	characteristic
40	C₂	C₇×#41	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂³
commutator, Frattini, cosocle	C₂²	C₂³×C₇
prime radical	C₂×C₇×D₄	C₂
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.