Subgroups — #10621402

Order 672.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#66732	C₂	1	2	yes	maximal, complemented
2	C₃×C₇×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#66634	C₂	1	2	yes	maximal, complemented
4	D₁₆₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×D₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#6482	—	3	1	no	maximal, Hall
7	C₃×C₇×Q₈	C₂²	1	2	yes	—
8	D₈₄	—	2	2	no	—
9	D₈₄	C₂²	1	2	yes	—
10	C₄×D₂₁	—	2	2	no	—
11	C₄×D₂₁	C₂²	1	1	yes	characteristic
12	C₃×C₇×C₈	C₂²	1	1	yes	characteristic
13	C₃⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
14	#1161	—	3	2	no	—
15	C₇×Q₁₆	D₃	1	1	yes	characteristic, complemented
16	#1143	—	3	2	no	—
17	D₅₆	—	3	1	no	—
18	C₈×D₇	—	3	1	no	—
19	#919	—	7	1	no	maximal, Hall
20	C₂×D₂₁	—	2	2	no	—
21	C₂×D₂₁	D₄	1	1	yes	characteristic
22	C₃×C₄×C₇	—	2	2	no	—
23	C₃×C₄×C₇	C₂³	1	1	yes	characteristic
24	Dic₂₁	D₄	1	1	yes	characteristic
25	C₇×Q₈	C₂×D₃	1	2	yes	—
26	D₂₈	—	3	2	no	—
27	D₂₈	—	6	2	no	—
28	C₄×D₇	—	3	1	no	—
29	C₄×D₇	—	6	2	no	—
30	C₇×C₈	C₂×D₃	1	1	yes	characteristic
31	C₇⋊C₈	—	3	1	no	—
32	#239	—	7	2	no	—
33	C₃×Q₁₆	D₇	1	1	yes	characteristic, complemented
34	#215	—	7	2	no	—
35	D₂₄	—	7	1	no	—
36	D₃×C₈	—	7	1	no	—
37	C₂×C₃×C₇	C₂×D₄	1	1	yes	characteristic
38	D₂₁	—	2	1	no	—
39	D₂₁	—	4	2	no	—
40	#135	—	21	1	no	Sylow
41	C₂×D₇	—	3	1	no	—
42	C₂×D₇	—	6	2	no	—
43	C₄×C₇	—	2	2	no	—
44	C₄×C₇	C₂²×D₃	1	1	yes	characteristic
45	Dic₇	—	3	1	no	—
46	C₃×Q₈	C₂×D₇	1	2	yes	—
47	D₁₂	—	7	2	no	—
48	D₁₂	—	14	2	no	—
49	C₄×D₃	—	7	1	no	—
50	C₄×D₃	—	14	2	no	—
51	C₃×C₈	C₂×D₇	1	1	yes	characteristic
52	C₃⋊C₈	—	7	1	no	—
53	C₃×C₇	#135	1	1	yes	characteristic, complemented, Hall
54	#41	—	21	2	no	—
55	Q₁₆	D₂₁	1	1	yes	characteristic, complemented
56	SD₁₆	—	21	2	no	—
57	D₈	—	21	1	no	—
58	C₂×C₈	—	21	1	no	—
59	C₂×C₇	D₃×D₄	1	1	yes	characteristic
60	D₇	—	6	1	no	—
61	D₇	—	12	2	no	—
62	C₂×D₃	—	7	1	no	—
63	C₂×D₃	—	14	2	no	—
64	C₃×C₄	—	2	2	no	—
65	C₃×C₄	C₂²×D₇	1	1	yes	characteristic
66	Dic₃	—	7	1	no	—
67	Q₈	C₂×D₂₁	1	2	yes	—
68	D₄	—	21	2	no	—
69	D₄	—	42	2	no	—
70	C₂×C₄	—	21	1	no	—
71	C₂×C₄	—	42	2	no	—
72	C₈	—	21	1	no	—
73	C₈	C₂×D₂₁	1	1	yes	characteristic
74	C₇	#919	1	1	yes	characteristic, complemented, Sylow
75	C₂×C₃	D₄×D₇	1	1	yes	characteristic
76	D₃	—	14	1	no	—
77	D₃	—	28	2	no	—
78	C₂²	—	21	1	no	—
79	C₂²	—	42	2	no	—
80	C₄	—	2	2	no	—
81	C₄	—	21	1	no	—
82	C₄	C₂²×D₂₁	1	1	yes	characteristic
83	C₃	#6482	1	1	yes	characteristic, complemented, Sylow
84	C₂	—	42	1	no	—
85	C₂	—	84	2	no	—
86	C₂	D₄×D₂₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₂₁
commutator, cosocle	C₃×C₄×C₇	C₂³
Frattini	C₄	C₂²×D₂₁
Fitting	C₃×C₇×Q₁₆	C₂
socle	C₂×C₃×C₇	C₂×D₄
hypercenter	Q₁₆	D₂₁
hypocenter	C₃×C₇	#135
LCS	G ▷ C₃×C₄×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
1. #66732	C₂	semidirect
2. C₃×C₇×Q₁₆	C₂	semidirect
3. #66634	C₂	semidirect
4. D₁₆₈	C₂	semidirect
5. C₈×D₂₁	C₂	semidirect
15. C₇×Q₁₆	D₃	semidirect
33. C₃×Q₁₆	D₇	semidirect
53. C₃×C₇	#135	semidirect
55. Q₁₆	D₂₁	semidirect
74. C₇	#919	semidirect
83. C₃	#6482	semidirect

Maximal subgroups

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

subgroup	index
#66732	2
C₃×C₇×Q₁₆	2
#66634	2
D₁₆₈	2
C₈×D₂₁	2
#6482	3
#919	7