Subgroups — #10621663

Order 672.

292 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×D₂₁	C₂	1	2	yes	maximal, complemented
3	#66730	C₂	1	6	yes	maximal, complemented
4	#66728	C₂	1	3	yes	maximal, complemented
5	C₂×Dic₄₂	C₂	1	3	yes	maximal, complemented
6	#6554	—	3	1	no	maximal, Hall
7	C₃×C₇×Q₈	C₂²	1	1	yes	characteristic
8	C₃×C₇×D₄	C₂²	1	3	yes	—
9	C₂×C₃×C₄×C₇	C₂²	1	3	yes	—
10	#4268	C₂²	1	6	yes	—
11	C₂×Dic₂₁	C₂²	1	6	yes	—
12	D₈₄	C₂²	1	1	yes	characteristic
13	C₄×D₂₁	C₂²	1	6	yes	—
14	Dic₄₂	C₂²	1	3	yes	—
15	Dic₄₂	C₂²	1	6	yes	—
16	C₇×#41	D₃	1	1	yes	characteristic, complemented
17	Q₈×D₇	—	3	2	no	—
18	#1159	—	3	6	no	—
19	#1157	—	3	3	no	—
20	C₂×Dic₁₄	—	3	3	no	—
21	#1010	—	7	1	no	maximal, Hall
22	C₂²×C₃×C₇	C₂³	1	3	yes	—
23	C₂×D₂₁	C₂³	1	2	yes	—
24	C₃×C₄×C₇	C₂³	1	1	yes	characteristic
25	C₃×C₄×C₇	C₂³	1	3	yes	—
26	Dic₂₁	C₂³	1	6	yes	—
27	C₇×Q₈	C₂×D₃	1	1	yes	characteristic
28	C₇×D₄	C₂×D₃	1	3	yes	—
29	C₂×C₄×C₇	C₂×D₃	1	3	yes	—
30	C₂²⋊D₇	—	3	6	no	—
31	C₂×Dic₇	—	3	6	no	—
32	D₂₈	—	3	1	no	—
33	C₄×D₇	—	3	6	no	—
34	Dic₁₄	—	3	3	no	—
35	Dic₁₄	—	3	6	no	—
36	C₃×#41	D₇	1	1	yes	characteristic, complemented
37	D₃×Q₈	—	7	2	no	—
38	#237	—	7	6	no	—
39	#235	—	7	3	no	—
40	C₂×Dic₆	—	7	3	no	—
41	C₂×C₃×C₇	—	2	3	no	—
42	C₂×C₃×C₇	C₂⁴	1	1	yes	characteristic
43	D₂₁	—	2	2	no	—
44	2₋¹⁺⁴	—	21	1	no	Sylow
45	C₂²×C₇	C₂²×D₃	1	3	yes	—
46	C₂×D₇	—	3	2	no	—
47	C₄×C₇	C₂²×D₃	1	1	yes	characteristic
48	C₄×C₇	C₂²×D₃	1	3	yes	—
49	Dic₇	—	3	6	no	—
50	C₃×Q₈	C₂×D₇	1	1	yes	characteristic
51	C₃×D₄	C₂×D₇	1	3	yes	—
52	C₂×C₃×C₄	C₂×D₇	1	3	yes	—
53	#67	—	7	6	no	—
54	C₂×Dic₃	—	7	6	no	—
55	D₁₂	—	7	1	no	—
56	C₄×D₃	—	7	6	no	—
57	Dic₆	—	7	3	no	—
58	Dic₆	—	7	6	no	—
59	C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic, complemented, Hall
60	#41	—	21	6	no	—
61	#41	—	21	3	no	—
62	#41	D₂₁	1	1	yes	characteristic, complemented
63	C₂×Q₈	—	21	2	no	—
64	C₂×Q₈	—	21	3	no	—
65	C₂×C₇	—	2	3	no	—
66	C₂×C₇	C₂³×D₃	1	1	yes	characteristic
67	D₇	—	6	2	no	—
68	C₂²×C₃	C₂²×D₇	1	3	yes	—
69	C₂×D₃	—	7	2	no	—
70	C₃×C₄	C₂²×D₇	1	1	yes	characteristic
71	C₃×C₄	C₂²×D₇	1	3	yes	—
72	Dic₃	—	7	6	no	—
73	Q₈	—	21	6	no	—
74	Q₈	—	21	3	no	—
75	Q₈	C₂×D₂₁	1	1	yes	characteristic
76	D₄	—	21	6	no	—
77	D₄	—	21	1	no	—
78	D₄	C₂×D₂₁	1	3	yes	—
79	C₂×C₄	—	21	6	no	—
80	C₂×C₄	—	21	6	no	—
81	C₂×C₄	C₂×D₂₁	1	3	yes	—
82	C₇	#1010	1	1	yes	characteristic, complemented, Sylow
83	C₂×C₃	—	2	3	no	—
84	C₂×C₃	C₂³×D₇	1	1	yes	characteristic
85	D₃	—	14	2	no	—
86	C₂²	—	21	2	no	—
87	C₂²	C₂²×D₂₁	1	3	yes	—
88	C₄	—	21	6	no	—
89	C₄	C₂²×D₂₁	1	3	yes	—
90	C₄	C₂²×D₂₁	1	1	yes	characteristic
91	C₃	#6554	1	1	yes	characteristic, complemented, Sylow
92	C₂	—	2	3	no	—
93	C₂	—	42	2	no	—
94	C₂	C₂³×D₂₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×D₂₁
commutator, socle, cosocle	C₂×C₃×C₇	C₂⁴
Fitting	C₃×C₇×#41	C₂
hypercenter	#41	D₂₁
hypocenter	C₃×C₇	2₋¹⁺⁴
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂ ◁ #41

Complements

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

N	H ≅ G/N	type
1. C₃×C₇×#41	C₂	semidirect
2. Q₈×D₂₁	C₂	semidirect
3. #66730	C₂	semidirect
4. #66728	C₂	semidirect
5. C₂×Dic₄₂	C₂	semidirect
16. C₇×#41	D₃	semidirect
36. C₃×#41	D₇	semidirect
59. C₃×C₇	2₋¹⁺⁴	semidirect
62. #41	D₂₁	semidirect
82. C₇	#1010	semidirect
91. C₃	#6554	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₇×#41	2
Q₈×D₂₁	2
#66730	2
#66728	2
C₂×Dic₄₂	2
#6554	3
#1010	7