Subgroups — #10621527

Order 672.

92 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	D₇×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
3	#66662	C₂	1	1	yes	maximal, characteristic, complemented
4	#6551	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₃×C₂²⋊D₇	—	4	1	no	maximal
6	C₇×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
7	C₂×C₇×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
8	#1161	—	3	1	no	—
9	#1161	C₂×C₃	1	1	yes	characteristic, complemented
10	Q₈×D₇	—	3	1	no	—
11	Q₈×D₇	C₂×C₃	1	1	yes	characteristic, complemented
12	#1157	—	3	1	no	—
13	#1157	—	3	1	no	—
14	2₋¹⁺⁴⋊C₃	—	7	1	no	maximal, Hall
15	C₂²×C₃×C₇	—	4	1	no	—
16	C₂×C₃×D₇	—	4	1	no	—
17	C₃×Dic₇	—	4	1	no	—
18	C₇×Q₈	—	3	1	no	—
19	C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
20	C₂×C₄×C₇	—	3	1	no	—
21	C₂²⋊D₇	—	3	1	no	—
22	C₂²⋊D₇	A₄	1	1	yes	characteristic
23	D₂₈	—	3	1	no	—
24	D₂₈	—	3	1	no	—
25	C₄×D₇	—	3	2	no	—
26	C₄×D₇	—	3	1	no	—
27	C₄×D₇	—	3	1	no	—
28	Dic₁₄	—	3	1	no	—
29	Dic₁₄	—	3	1	no	—
30	#41⋊C₃	—	7	1	no	—
31	C₂×SL(2,𝔽₃)	—	7	1	no	—
32	C₂×SL(2,𝔽₃)	D₇	1	1	yes	characteristic, complemented
33	C₂×C₃×C₇	—	4	1	no	—
34	C₂×C₃×C₇	—	8	1	no	—
35	C₃×D₇	—	8	1	no	—
36	2₋¹⁺⁴	—	7	1	no	Sylow
37	C₂²×C₇	C₂×A₄	1	1	yes	characteristic
38	C₂×D₇	—	3	1	no	—
39	C₂×D₇	C₂×A₄	1	1	yes	characteristic
40	C₄×C₇	—	3	1	no	—
41	C₄×C₇	—	3	1	no	—
42	Dic₇	—	3	1	no	—
43	Dic₇	C₂×A₄	1	1	yes	characteristic
44	C₃×D₄	—	28	1	no	—
45	SL(2,𝔽₃)	C₂×D₇	1	1	yes	characteristic
46	C₃×C₇	—	4	1	no	Hall
47	#41	—	7	1	no	—
48	#41	—	21	1	no	—
49	#41	—	21	1	no	—
50	#41	—	21	1	no	—
51	C₂×Q₈	—	7	1	no	—
52	C₂×Q₈	—	21	1	no	—
53	C₂×Q₈	C₃×D₇	1	1	yes	characteristic, complemented
54	C₂×C₇	—	2	1	no	—
55	C₂×C₇	C₂²×A₄	1	1	yes	characteristic
56	D₇	—	2	1	no	—
57	D₇	—	6	1	no	—
58	C₂²×C₃	—	4	1	no	—
59	C₂²×C₃	—	28	1	no	—
60	C₃×C₄	—	28	1	no	—
61	Q₈	—	3	1	no	—
62	Q₈	—	21	1	no	—
63	Q₈	—	21	1	no	—
64	Q₈	C₂×C₃×D₇	1	1	yes	characteristic
65	D₄	—	7	1	no	—
66	D₄	—	21	1	no	—
67	D₄	—	21	1	no	—
68	D₄	—	21	1	no	—
69	C₂×C₄	—	3	1	no	—
70	C₂×C₄	—	21	1	no	—
71	C₂×C₄	—	21	1	no	—
72	C₂×C₄	—	21	2	no	—
73	C₇	2₋¹⁺⁴⋊C₃	1	1	yes	characteristic, complemented, Sylow
74	C₂×C₃	—	4	1	no	—
75	C₂×C₃	—	8	1	no	—
76	C₂×C₃	—	56	1	no	—
77	C₂²	—	7	1	no	—
78	C₂²	—	21	1	no	—
79	C₂²	A₄×D₇	1	1	yes	characteristic
80	C₄	—	3	1	no	—
81	C₄	—	3	1	no	—
82	C₄	—	7	1	no	—
83	C₄	—	21	1	no	—
84	C₃	—	4	1	no	Sylow
85	C₂	—	2	1	no	—
86	C₂	—	14	1	no	—
87	C₂	—	42	1	no	—
88	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	C₂×C₇×Q₈	C₂×C₃
socle	C₂×C₇	C₂²×A₄
hypercenter	C₂²	A₄×D₇
LCS	G ▷ C₇×Q₈
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. D₇×SL(2,𝔽₃)	C₂	semidirect
3. #66662	C₂	semidirect
4. #6551	C₃	semidirect
7. C₂×C₇×Q₈	C₂×C₃	semidirect
9. #1161	C₂×C₃	semidirect
11. Q₈×D₇	C₂×C₃	semidirect
32. C₂×SL(2,𝔽₃)	D₇	semidirect
53. C₂×Q₈	C₃×D₇	semidirect
73. 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
C₂×C₇×SL(2,𝔽₃)	2
D₇×SL(2,𝔽₃)	2
#66662	2
#6551	3
C₃×C₂²⋊D₇	4
2₋¹⁺⁴⋊C₃	7