Subgroups — #10621419

Order 672.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₇×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₄₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#66636	C₂	1	4	yes	maximal, complemented
4	C₄×Dic₂₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#6499	—	3	1	no	maximal, Hall
6	C₂³×C₃×C₇	C₂²	1	2	yes	—
7	C₃×C₇×D₄	—	2	2	no	—
8	C₂×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₂×Dic₂₁	C₂²	1	2	yes	—
10	C₂×Dic₂₁	C₂²	1	2	yes	—
11	Dic₄₂	—	2	2	no	—
12	C₂×C₇×D₄	D₃	1	1	yes	characteristic
13	C₂×Dic₁₄	—	3	1	no	—
14	C₂²⋊Dic₇	—	3	4	no	—
15	C₄×Dic₇	—	3	1	no	—
16	#936	—	7	1	no	maximal, Hall
17	C₂²×C₃×C₇	—	2	2	no	—
18	C₂²×C₃×C₇	—	2	4	no	—
19	C₂²×C₃×C₇	C₂³	1	1	yes	characteristic
20	C₃×C₄×C₇	D₄	1	2	yes	—
21	Dic₂₁	—	2	2	no	—
22	Dic₂₁	—	2	2	no	—
23	C₂³×C₇	C₂×D₃	1	2	yes	—
24	C₇×D₄	—	2	2	no	—
25	C₂×C₄×C₇	C₂×D₃	1	1	yes	characteristic
26	C₂×Dic₇	—	3	2	no	—
27	C₂×Dic₇	—	3	2	no	—
28	Dic₁₄	—	6	2	no	—
29	C₂×C₃×D₄	D₇	1	1	yes	characteristic
30	C₂×Dic₆	—	7	1	no	—
31	#217	—	7	4	no	—
32	C₄×Dic₃	—	7	1	no	—
33	C₂×C₃×C₇	—	4	2	no	—
34	C₂×C₃×C₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₃×C₇	#41	1	2	yes	—
36	#124	—	21	1	no	Sylow
37	C₂²×C₇	—	2	4	no	—
38	C₂²×C₇	—	2	2	no	—
39	C₂²×C₇	C₂²×D₃	1	1	yes	characteristic
40	C₄×C₇	#67	1	2	yes	—
41	Dic₇	—	6	2	no	—
42	Dic₇	—	6	2	no	—
43	C₂³×C₃	C₂×D₇	1	2	yes	—
44	C₃×D₄	—	2	2	no	—
45	C₂×C₃×C₄	C₂×D₇	1	1	yes	characteristic
46	C₂×Dic₃	—	7	2	no	—
47	C₂×Dic₃	—	7	2	no	—
48	Dic₆	—	14	2	no	—
49	C₃×C₇	#124	1	1	yes	characteristic, complemented, Hall
50	C₂×Q₈	—	21	1	no	—
51	C₂×D₄	D₂₁	1	1	yes	characteristic
52	C₂²⋊C₄	—	21	4	no	—
53	C₄²	—	21	1	no	—
54	C₂×C₇	—	4	2	no	—
55	C₂×C₇	#237	1	2	yes	—
56	C₂×C₇	C₂×#67	1	1	yes	characteristic
57	C₂²×C₃	—	2	4	no	—
58	C₂²×C₃	—	2	2	no	—
59	C₂²×C₃	C₂²×D₇	1	1	yes	characteristic
60	C₃×C₄	C₂²⋊D₇	1	2	yes	—
61	Dic₃	—	14	2	no	—
62	Dic₃	—	14	2	no	—
63	C₂³	C₂×D₂₁	1	2	yes	—
64	Q₈	—	42	2	no	—
65	D₄	—	2	2	no	—
66	C₂×C₄	—	21	2	no	—
67	C₂×C₄	—	21	2	no	—
68	C₂×C₄	C₂×D₂₁	1	1	yes	characteristic
69	C₇	#936	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₃	—	4	2	no	—
71	C₂×C₃	#1159	1	2	yes	—
72	C₂×C₃	C₂×C₂²⋊D₇	1	1	yes	characteristic
73	C₂²	—	2	4	no	—
74	C₂²	—	2	2	no	—
75	C₂²	C₂²×D₂₁	1	1	yes	characteristic
76	C₄	—	42	2	no	—
77	C₄	—	42	2	no	—
78	C₄	#4268	1	2	yes	—
79	C₃	#6499	1	1	yes	characteristic, complemented, Sylow
80	C₂	—	4	2	no	—
81	C₂	#66730	1	2	yes	—
82	C₂	C₂×#4268	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₂₁
commutator, socle, cosocle	C₂²×C₃×C₇	C₂³
Fitting, prime radical	C₂×C₃×C₇×D₄	C₂
hypercenter	C₂×D₄	D₂₁
hypocenter	C₃×C₇	#124
LCS	G ▷ C₂²×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₄₂	C₂	semidirect
3. #66636	C₂	semidirect
4. C₄×Dic₂₁	C₂	semidirect
49. C₃×C₇	#124	semidirect
69. C₇	#936	semidirect
79. C₃	#6499	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₇×D₄	2
C₂×Dic₄₂	2
#66636	2
C₄×Dic₂₁	2
#6499	3
#936	7