Subgroups — #10621390

Order 672.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#66728	C₂	1	1	yes	maximal, characteristic
2	C₃×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#66622	C₂	1	2	yes	maximal, complemented
5	C₈×D₂₁	C₂	1	2	yes	maximal, complemented
6	#6470	—	3	1	no	maximal, Hall
7	C₂×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
8	#4268	C₄	1	2	yes	—
9	D₈₄	C₄	1	1	yes	characteristic
10	C₄×D₂₁	C₂²	1	2	yes	—
11	Dic₄₂	C₄	1	1	yes	characteristic
12	C₃×C₇×C₈	C₂²	1	2	yes	—
13	C₃⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₃⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
15	#1157	—	3	1	no	—
16	C₇×M₁₆	D₃	1	1	yes	characteristic, complemented
17	C₂×C₇⋊C₈	—	3	1	no	—
18	#1131	—	3	2	no	—
19	C₈×D₇	—	3	2	no	—
20	#907	—	7	1	no	maximal, Hall
21	C₂²×C₃×C₇	C₂×C₄	1	1	yes	characteristic
22	C₂×D₂₁	C₂×C₄	1	2	yes	—
23	C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
24	C₃×C₄×C₇	C₂³	1	1	yes	characteristic
25	Dic₂₁	C₂×C₄	1	2	yes	—
26	C₂×C₄×C₇	C₂×D₃	1	1	yes	characteristic
27	C₂²⋊D₇	—	3	2	no	—
28	D₂₈	—	3	1	no	—
29	C₄×D₇	—	3	2	no	—
30	Dic₁₄	—	3	1	no	—
31	C₇×C₈	C₂×D₃	1	2	yes	—
32	C₇⋊C₈	—	3	1	no	—
33	C₇⋊C₈	—	3	1	no	—
34	#235	—	7	1	no	—
35	C₃×M₁₆	D₇	1	1	yes	characteristic, complemented
36	C₂×C₃⋊C₈	—	7	1	no	—
37	ΓL(1,𝔽₂₅)	—	7	2	no	—
38	D₃×C₈	—	7	2	no	—
39	C₂×C₃×C₇	—	2	1	no	—
40	C₂×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
41	D₂₁	—	2	2	no	—
42	#131	—	21	1	no	Sylow
43	C₂²×C₇	C₄×D₃	1	1	yes	characteristic
44	C₂×D₇	—	3	2	no	—
45	C₄×C₇	C₄×D₃	1	1	yes	characteristic
46	C₄×C₇	C₂²×D₃	1	1	yes	characteristic
47	Dic₇	—	3	2	no	—
48	C₂×C₃×C₄	C₂×D₇	1	1	yes	characteristic
49	#67	—	7	2	no	—
50	D₁₂	—	7	1	no	—
51	C₄×D₃	—	7	2	no	—
52	Dic₆	—	7	1	no	—
53	C₃×C₈	C₂×D₇	1	2	yes	—
54	C₃⋊C₈	—	7	1	no	—
55	C₃⋊C₈	—	7	1	no	—
56	C₃×C₇	#131	1	1	yes	characteristic, complemented, Hall
57	#41	—	21	1	no	—
58	M₁₆	—	21	2	no	—
59	M₁₆	D₂₁	1	1	yes	characteristic, complemented
60	C₂×C₈	—	21	1	no	—
61	C₂×C₈	—	21	2	no	—
62	C₂×C₇	—	2	1	no	—
63	C₂×C₇	C₂×C₄×D₃	1	1	yes	characteristic
64	D₇	—	6	2	no	—
65	C₂²×C₃	C₄×D₇	1	1	yes	characteristic
66	C₂×D₃	—	7	2	no	—
67	C₃×C₄	C₄×D₇	1	1	yes	characteristic
68	C₃×C₄	C₂²×D₇	1	1	yes	characteristic
69	Dic₃	—	7	2	no	—
70	Q₈	—	21	1	no	—
71	D₄	—	21	2	no	—
72	D₄	—	21	1	no	—
73	C₂×C₄	—	21	2	no	—
74	C₂×C₄	C₂×D₂₁	1	1	yes	characteristic
75	C₈	—	21	1	no	—
76	C₈	—	21	1	no	—
77	C₈	C₂×D₂₁	1	2	yes	—
78	C₇	#907	1	1	yes	characteristic, complemented, Sylow
79	C₂×C₃	—	2	1	no	—
80	C₂×C₃	C₂×C₄×D₇	1	1	yes	characteristic
81	D₃	—	14	2	no	—
82	C₂²	—	21	2	no	—
83	C₂²	C₄×D₂₁	1	1	yes	characteristic
84	C₄	—	21	2	no	—
85	C₄	C₄×D₂₁	1	1	yes	characteristic
86	C₄	C₂²×D₂₁	1	1	yes	characteristic
87	C₃	#6470	1	1	yes	characteristic, complemented, Sylow
88	C₂	—	2	1	no	—
89	C₂	—	42	2	no	—
90	C₂	C₂×C₄×D₂₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂²×D₂₁
commutator, socle	C₂×C₃×C₇	C₂²×C₄
Fitting	C₃×C₇×M₁₆	C₂
prime radical	#66728	C₂
cosocle	C₃×C₄×C₇	C₂³
hypercenter	M₁₆	D₂₁
hypocenter	C₃×C₇	#131
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
2. C₃×C₇×M₁₆	C₂	semidirect
3. C₂×C₃⋊(C₇⋊C₈)	C₂	semidirect
4. #66622	C₂	semidirect
5. C₈×D₂₁	C₂	semidirect
16. C₇×M₁₆	D₃	semidirect
35. C₃×M₁₆	D₇	semidirect
56. C₃×C₇	#131	semidirect
59. M₁₆	D₂₁	semidirect
78. C₇	#907	semidirect
87. C₃	#6470	semidirect

Maximal subgroups

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

subgroup	index
#66728	2
C₃×C₇×M₁₆	2
C₂×C₃⋊(C₇⋊C₈)	2
#66622	2
C₈×D₂₁	2
#6470	3
#907	7