Subgroups — #10621502

Order 672.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#41⋊C₃	C₂	1	1	yes	maximal, characteristic
2	C₇⋊#226	C₂	1	2	yes	maximal, complemented
3	#6514	—	3	1	no	maximal, Hall
4	Dic₄₂	—	4	1	no	maximal
5	C₇×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
6	C₇×#41	D₃	1	1	yes	characteristic
7	C₂×Dic₁₄	—	3	1	no	—
8	#1144	—	3	2	no	—
9	#1142	—	3	2	no	—
10	#1136	—	3	1	no	—
11	#984	—	7	1	no	maximal, Hall
12	C₃×C₄×C₇	—	4	1	no	—
13	Dic₂₁	—	4	2	no	—
14	C₇×Q₈	C₂×D₃	1	1	yes	characteristic
15	C₇×D₄	—	3	1	no	—
16	C₂×C₄×C₇	—	3	1	no	—
17	C₂×Dic₇	—	6	1	no	—
18	Dic₁₄	—	3	2	no	—
19	Dic₁₄	—	6	1	no	—
20	C₇⋊C₈	—	3	2	no	—
21	#41⋊C₃	D₇	1	1	yes	characteristic
22	#226	—	7	2	no	—
23	C₂×C₃×C₇	—	4	1	no	—
24	#137	—	21	1	no	Sylow
25	C₂²×C₇	—	3	1	no	—
26	C₄×C₇	—	3	1	no	—
27	C₄×C₇	S₄	1	1	yes	characteristic
28	Dic₇	—	6	2	no	—
29	Dic₆	—	28	1	no	—
30	SL(2,𝔽₃)	C₂×D₇	1	1	yes	characteristic
31	C₃×C₇	—	4	1	no	Hall
32	#41	D₂₁	1	1	yes	characteristic
33	C₂×Q₈	—	21	1	no	—
34	Q₁₆	—	21	2	no	—
35	SD₁₆	—	21	2	no	—
36	M₁₆	—	21	1	no	—
37	C₂×C₇	—	6	1	no	—
38	C₂×C₇	C₂×S₄	1	1	yes	characteristic
39	C₃×C₄	—	4	1	no	—
40	Dic₃	—	28	2	no	—
41	Q₈	—	21	2	no	—
42	Q₈	—	42	1	no	—
43	Q₈	C₂×D₂₁	1	1	yes	characteristic
44	D₄	—	3	1	no	—
45	C₂×C₄	—	3	1	no	—
46	C₂×C₄	—	42	1	no	—
47	C₈	—	21	2	no	—
48	C₇	#984	1	1	yes	characteristic, complemented, Sylow
49	C₂×C₃	—	4	1	no	—
50	C₂²	—	3	1	no	—
51	C₄	—	3	1	no	—
52	C₄	—	42	2	no	—
53	C₄	C₇⋊S₄	1	1	yes	characteristic
54	C₃	—	4	1	no	Sylow
55	C₂	—	6	1	no	—
56	C₂	C₂×C₇⋊S₄	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×C₇⋊S₄
commutator, cosocle, hypocenter	C₇×SL(2,𝔽₃)	C₂²
Fitting	C₇×#41	D₃
socle	C₂×C₇	C₂×S₄
prime radical	C₇×#41⋊C₃	C₂
hypercenter	C₄	C₇⋊S₄
LCS	G ▷ C₇×SL(2,𝔽₃)
UCS	C₁ ◁ C₂ ◁ C₄

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

N	H ≅ G/N	type
2. C₇⋊#226	C₂	semidirect
48. C₇	#984	semidirect

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