Subgroups — C₂²×C₇×D₄ #6558

Order 224.

236 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, socle	C₂³×C₇	C₂²
commutator, Frattini, cosocle	C₂	C₂⁴×C₇
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×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.