Subgroups — #10621501

Order 672.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇⋊GL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇⋊#226	C₂	1	1	yes	maximal, characteristic, complemented
4	#6505	—	3	1	no	maximal, Hall
5	#4268	—	4	1	no	maximal
6	C₇×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
7	C₂×C₇×Q₈	D₃	1	1	yes	characteristic, complemented
8	#1157	—	3	1	no	—
9	#1144	—	3	1	no	—
10	#1144	—	3	1	no	—
11	#1143	—	3	1	no	—
12	#1143	—	3	1	no	—
13	#1136	—	3	1	no	—
14	#983	—	7	1	no	maximal, Hall
15	C₂²×C₃×C₇	—	4	1	no	—
16	C₂×D₂₁	—	4	1	no	—
17	Dic₂₁	—	4	1	no	—
18	C₇×Q₈	—	3	1	no	—
19	C₇×Q₈	C₂×D₃	1	1	yes	characteristic
20	C₂×C₄×C₇	—	3	1	no	—
21	C₂²⋊D₇	—	6	1	no	—
22	D₂₈	—	3	1	no	—
23	C₄×D₇	—	6	1	no	—
24	Dic₁₄	—	3	1	no	—
25	C₇⋊C₈	—	3	1	no	—
26	C₇⋊C₈	—	3	1	no	—
27	C₂×SL(2,𝔽₃)	D₇	1	1	yes	characteristic, complemented
28	GL(2,𝔽₃)	—	7	1	no	—
29	#226	—	7	1	no	—
30	C₂×C₃×C₇	—	4	1	no	—
31	C₂×C₃×C₇	—	8	1	no	—
32	D₂₁	—	8	1	no	—
33	#137	—	21	1	no	Sylow
34	C₂²×C₇	S₄	1	1	yes	characteristic
35	C₂×D₇	—	6	1	no	—
36	C₄×C₇	—	3	1	no	—
37	C₄×C₇	—	3	1	no	—
38	Dic₇	—	6	1	no	—
39	#67	—	28	1	no	—
40	SL(2,𝔽₃)	C₂×D₇	1	1	yes	characteristic
41	C₃×C₇	—	4	1	no	Hall
42	#41	—	21	1	no	—
43	C₂×Q₈	D₂₁	1	1	yes	characteristic, complemented
44	Q₁₆	—	21	1	no	—
45	Q₁₆	—	21	1	no	—
46	SD₁₆	—	21	1	no	—
47	SD₁₆	—	21	1	no	—
48	M₁₆	—	21	1	no	—
49	C₂×C₇	—	2	1	no	—
50	C₂×C₇	C₂×S₄	1	1	yes	characteristic
51	D₇	—	12	1	no	—
52	C₂²×C₃	—	4	1	no	—
53	C₂×D₃	—	28	1	no	—
54	Dic₃	—	28	1	no	—
55	Q₈	—	3	1	no	—
56	Q₈	—	21	1	no	—
57	Q₈	C₂×D₂₁	1	1	yes	characteristic
58	D₄	—	21	1	no	—
59	D₄	—	42	1	no	—
60	C₂×C₄	—	3	1	no	—
61	C₂×C₄	—	42	1	no	—
62	C₈	—	21	1	no	—
63	C₈	—	21	1	no	—
64	C₇	#983	1	1	yes	characteristic, complemented, Sylow
65	C₂×C₃	—	4	1	no	—
66	C₂×C₃	—	8	1	no	—
67	D₃	—	56	1	no	—
68	C₂²	—	42	1	no	—
69	C₂²	C₇⋊S₄	1	1	yes	characteristic
70	C₄	—	3	1	no	—
71	C₄	—	3	1	no	—
72	C₄	—	42	1	no	—
73	C₃	—	4	1	no	Sylow
74	C₂	—	2	1	no	—
75	C₂	—	84	1	no	—
76	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	C₂×C₇×Q₈	D₃
socle	C₂×C₇	C₂×S₄
hypercenter	C₂²	C₇⋊S₄
LCS	G ▷ C₇×SL(2,𝔽₃)
UCS	C₁ ◁ C₂ ◁ C₂²

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×SL(2,𝔽₃)	C₂	semidirect
2. C₇⋊GL(2,𝔽₃)	C₂	semidirect
3. C₇⋊#226	C₂	semidirect
7. C₂×C₇×Q₈	D₃	semidirect
27. C₂×SL(2,𝔽₃)	D₇	semidirect
43. C₂×Q₈	D₂₁	semidirect
64. C₇	#983	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×SL(2,𝔽₃)	2
C₇⋊GL(2,𝔽₃)	2
C₇⋊#226	2
#6505	3
#4268	4
#983	7