Subgroups — C₂×C₇×#41 #6560

Order 224.

164 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#41	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₇×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×D₄	C₂	1	3	yes	maximal, complemented
4	C₂²×C₄×C₇	C₂	1	3	yes	maximal, complemented
5	C₂³×C₇	C₂²	1	3	yes	—
6	C₇×Q₈	C₂²	1	4	yes	complemented
7	C₇×D₄	C₂²	1	12	yes	complemented
8	C₂×C₄×C₇	C₂²	1	3	yes	—
9	C₂×C₄×C₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₇	C₂²	1	12	yes	complemented
11	C₂×#41	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
12	C₂²×C₇	—	2	6	no	—
13	C₂²×C₇	C₂³	1	6	yes	—
14	C₂²×C₇	C₂³	1	1	yes	characteristic
15	C₄×C₇	C₂³	1	6	yes	—
16	C₄×C₇	C₂³	1	2	yes	—
17	#41	C₂×C₇	1	8	yes	direct product factor
18	C₂×Q₈	C₂×C₇	1	1	yes	characteristic, complemented
19	C₂×D₄	C₂×C₇	1	3	yes	complemented
20	C₂²×C₄	C₂×C₇	1	3	yes	complemented
21	C₂×C₇	—	2	6	no	—
22	C₂×C₇	#41	1	2	yes	direct product factor
23	C₂×C₇	C₂⁴	1	1	yes	characteristic
24	C₂³	C₂²×C₇	1	3	yes	—
25	Q₈	C₂²×C₇	1	4	yes	complemented
26	D₄	C₂²×C₇	1	12	yes	complemented
27	C₂×C₄	C₂²×C₇	1	3	yes	—
28	C₂×C₄	C₂²×C₇	1	1	yes	characteristic
29	C₂×C₄	C₂²×C₇	1	12	yes	complemented
30	C₇	C₂×#41	1	1	yes	characteristic, direct product factor, Sylow
31	C₂²	—	2	6	no	—
32	C₂²	C₂³×C₇	1	6	yes	—
33	C₂²	C₂³×C₇	1	1	yes	characteristic
34	C₄	C₂³×C₇	1	2	yes	—
35	C₄	C₂³×C₇	1	6	yes	—
36	C₂	—	2	6	no	—
37	C₂	C₇×#41	1	2	yes	direct product factor
38	C₂	C₂⁴×C₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂²
commutator, Frattini, cosocle	C₂	C₂⁴×C₇
socle	C₂²×C₇	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×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.