Subgroups — #10621530

Order 672.

92 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, complemented
2	D₇×SL(2,𝔽₃)	C₂	1	2	yes	maximal, complemented
3	#6554	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₃×D₂₈	—	4	1	no	maximal
5	C₇×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
6	C₇×#41	C₂×C₃	1	1	yes	characteristic, complemented
7	Q₈×D₇	C₂×C₃	1	2	yes	complemented
8	#1159	—	3	2	no	—
9	#1157	—	3	1	no	—
10	C₂×Dic₁₄	—	3	1	no	—
11	2₋¹⁺⁴⋊C₃	—	7	1	no	maximal, Hall
12	C₂×C₃×D₇	—	4	2	no	—
13	C₃×C₄×C₇	—	4	1	no	—
14	C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
15	C₇×D₄	—	3	1	no	—
16	C₂×C₄×C₇	—	3	1	no	—
17	C₂²⋊D₇	—	3	2	no	—
18	C₂×Dic₇	—	3	2	no	—
19	D₂₈	A₄	1	1	yes	characteristic
20	C₄×D₇	—	3	2	no	—
21	Dic₁₄	—	3	1	no	—
22	Dic₁₄	—	3	2	no	—
23	#41⋊C₃	D₇	1	1	yes	characteristic, complemented
24	C₂×SL(2,𝔽₃)	—	7	2	no	—
25	C₂×C₃×C₇	—	4	1	no	—
26	C₃×D₇	—	8	2	no	—
27	2₋¹⁺⁴	—	7	1	no	Sylow
28	C₂²×C₇	—	3	1	no	—
29	C₂×D₇	C₂×A₄	1	2	yes	—
30	C₄×C₇	—	3	1	no	—
31	C₄×C₇	C₂×A₄	1	1	yes	characteristic
32	Dic₇	—	3	2	no	—
33	C₃×D₄	—	28	1	no	—
34	SL(2,𝔽₃)	C₂×D₇	1	1	yes	characteristic
35	C₃×C₇	—	4	1	no	Hall
36	#41	—	21	1	no	—
37	#41	—	21	2	no	—
38	#41	C₃×D₇	1	1	yes	characteristic, complemented
39	C₂×Q₈	—	7	2	no	—
40	C₂×Q₈	—	21	1	no	—
41	C₂×C₇	—	6	1	no	—
42	C₂×C₇	C₂²×A₄	1	1	yes	characteristic
43	D₇	—	2	2	no	—
44	C₂²×C₃	—	28	2	no	—
45	C₃×C₄	—	4	1	no	—
46	Q₈	—	21	2	no	—
47	Q₈	—	21	1	no	—
48	Q₈	C₂×C₃×D₇	1	1	yes	characteristic
49	D₄	—	3	1	no	—
50	D₄	—	7	1	no	—
51	D₄	—	21	2	no	—
52	C₂×C₄	—	3	1	no	—
53	C₂×C₄	—	21	2	no	—
54	C₂×C₄	—	21	2	no	—
55	C₇	2₋¹⁺⁴⋊C₃	1	1	yes	characteristic, complemented, Sylow
56	C₂×C₃	—	4	1	no	—
57	C₂×C₃	—	56	2	no	—
58	C₂²	—	3	1	no	—
59	C₂²	—	7	2	no	—
60	C₄	—	3	1	no	—
61	C₄	—	21	2	no	—
62	C₄	A₄×D₇	1	1	yes	characteristic
63	C₃	—	4	1	no	Sylow
64	C₂	—	6	1	no	—
65	C₂	—	14	2	no	—
66	C₂	C₂×A₄×D₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×A₄×D₇
commutator, cosocle, hypocenter	C₇×Q₈	C₂²×C₃
Fitting	C₇×#41	C₂×C₃
socle	C₂×C₇	C₂²×A₄
hypercenter	C₄	A₄×D₇
LCS	G ▷ C₇×Q₈
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
1. C₇×#41⋊C₃	C₂	semidirect
2. D₇×SL(2,𝔽₃)	C₂	semidirect
3. #6554	C₃	semidirect
6. C₇×#41	C₂×C₃	semidirect
7. Q₈×D₇	C₂×C₃	semidirect
23. #41⋊C₃	D₇	semidirect
38. #41	C₃×D₇	semidirect
55. C₇	2₋¹⁺⁴⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₇×#41⋊C₃	2
D₇×SL(2,𝔽₃)	2
#6554	3
C₃×D₂₈	4
2₋¹⁺⁴⋊C₃	7