Group #20291132 — subgroups

Subgroups — #20291132

Order 6784.

126 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811544	C₂	1	1	yes	maximal, characteristic
2	#16811544	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11719910	C₂	1	1	yes	maximal, characteristic
4	#16811512	C₂	1	1	yes	maximal, characteristic
5	#16811511	C₂	1	1	yes	maximal, characteristic
6	#16811509	C₂	1	1	yes	maximal, characteristic
7	#16810747	C₂	1	1	yes	maximal, characteristic
8	C₂×#11719910	C₂²	1	1	yes	characteristic
9	C₂×#11719910	C₂²	1	1	yes	characteristic
10	C₂×#11719910	C₂²	1	1	yes	characteristic
11	C₂×#11719910	C₂²	1	1	yes	characteristic
12	C₄×Dic₁₀₆	C₄	1	1	yes	characteristic
13	#13133146	C₄	1	1	yes	characteristic
14	#13133146	C₄	1	1	yes	characteristic
15	C₄×#88322	C₂²	1	1	yes	characteristic
16	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
17	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
18	C₄⋊C₈×C₅₃	C₄	1	1	yes	characteristic
19	C₂×Dic₁₀₆	C₈	1	1	yes	characteristic
20	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₄⋊Dic₅₃	C₈	1	1	yes	characteristic
24	#11719919	C₈	1	1	yes	characteristic
25	#11719919	C₈	1	1	yes	characteristic
26	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×#88322	C₂×C₄	1	1	yes	characteristic
29	C₂×#88322	C₂³	1	1	yes	characteristic
30	#11719910	—	2	1	no	—
31	#11719910	—	2	1	no	—
32	#11719910	—	2	1	no	—
33	#11719910	Q₈	1	2	yes	complemented
34	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
36	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×Dic₅₃	C₂×C₈	1	1	yes	characteristic
38	C₂×Dic₅₃	C₂×C₈	1	1	yes	characteristic
39	C₂×Dic₅₃	C₂×C₈	1	1	yes	characteristic
40	C₂×Dic₅₃	C₂×C₈	1	1	yes	characteristic
41	Dic₁₀₆	—	2	2	no	—
42	C₈×C₅₃	—	2	1	no	—
43	C₈×C₅₃	—	2	1	no	—
44	#88322	—	2	1	no	—
45	#88322	C₂×Q₈	1	1	yes	characteristic
46	#88322	#41	1	1	yes	characteristic
47	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
48	C₄×C₅₃	—	2	1	no	—
49	C₄×C₅₃	—	2	1	no	—
50	C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
51	C₄×C₅₃	#131	1	1	yes	characteristic
52	Dic₅₃	—	2	1	no	—
53	Dic₅₃	—	2	1	no	—
54	Dic₅₃	—	2	1	no	—
55	Dic₅₃	M₃₂	1	2	yes	—
56	#2183	—	53	1	no	maximal, Sylow
57	C₂×C₅₃	C₈×Q₈	1	1	yes	characteristic
58	C₂×C₅₃	C₂×M₃₂	1	1	yes	characteristic
59	C₂×C₅₃	#504	1	1	yes	characteristic
60	C₈×Q₈	—	53	1	no	—
61	C₄⋊C₁₆	—	53	1	no	—
62	C₄⋊C₁₆	—	53	1	no	—
63	C₄⋊C₁₆	—	53	1	no	—
64	#346	—	53	1	no	—
65	#346	—	53	1	no	—
66	C₄×C₁₆	—	53	1	no	—
67	C₅₃	#2183	1	1	yes	characteristic, complemented, Sylow
68	C₄×Q₈	—	53	1	no	—
69	C₂×C₁₆	—	53	1	no	—
70	C₂×C₁₆	—	53	1	no	—
71	C₂×C₁₆	—	53	1	no	—
72	C₂×C₁₆	—	53	1	no	—
73	C₄⋊C₈	—	53	1	no	—
74	C₄⋊C₈	—	53	1	no	—
75	C₄⋊C₈	#6157	1	1	yes	characteristic
76	C₄×C₈	—	53	1	no	—
77	C₄×C₈	—	53	1	no	—
78	C₄×C₈	—	53	1	no	—
79	C₂×Q₈	—	53	1	no	—
80	C₂×C₈	—	53	1	no	—
81	C₂×C₈	—	53	1	no	—
82	C₂×C₈	C₂×#6157	1	1	yes	characteristic
83	C₂×C₈	C₂×#6157	1	1	yes	characteristic
84	C₄⋊C₄	—	53	1	no	—
85	C₄⋊C₄	—	53	1	no	—
86	C₄⋊C₄	—	53	1	no	—
87	C₄²	—	53	1	no	—
88	C₄²	—	53	1	no	—
89	C₄²	C₂×#6157	1	1	yes	characteristic
90	C₁₆	—	53	2	no	—
91	C₁₆	—	106	1	no	—
92	C₁₆	—	106	1	no	—
93	C₁₆	—	106	1	no	—
94	Q₈	—	106	2	no	—
95	C₂×C₄	—	53	1	no	—
96	C₂×C₄	—	53	1	no	—
97	C₂×C₄	—	53	1	no	—
98	C₂×C₄	—	53	1	no	—
99	C₂×C₄	D₅₃⋊C₈	1	1	yes	characteristic
100	C₂×C₄	D₅₃⋊C₈	1	1	yes	characteristic
101	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
102	C₈	—	2	1	no	—
103	C₈	—	2	1	no	—
104	C₈	—	53	1	no	—
105	C₈	—	53	1	no	—
106	C₈	—	106	1	no	—
107	C₂²	C₂×D₅₃⋊C₈	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	53	2	no	—
111	C₄	—	106	1	no	—
112	C₄	—	106	1	no	—
113	C₄	—	106	1	no	—
114	C₄	Q₈×#6157	1	1	yes	characteristic
115	C₄	#13133304	1	1	yes	characteristic
116	C₂	#16811672	1	1	yes	characteristic
117	C₂	D₅₃⋊M₃₂	1	1	yes	characteristic
118	C₂	#16811682	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811544

C₂

yes

maximal, characteristic

#16811544

C₂

yes

maximal, characteristic

C₄⋊#11719910

C₂

yes

maximal, characteristic

#16811512

C₂

yes

maximal, characteristic

#16811511

C₂

yes

maximal, characteristic

#16811509

C₂

yes

maximal, characteristic

#16810747

C₂

yes

maximal, characteristic

C₂×#11719910

C₂²

yes

characteristic

C₂×#11719910

C₂²

yes

characteristic

C₂×#11719910

C₂²

yes

characteristic

C₂×#11719910

C₂²

yes

characteristic

C₄×Dic₁₀₆

C₄

yes

characteristic

#13133146

C₄

yes

characteristic

#13133146

C₄

yes

characteristic

C₄×#88322

C₂²

yes

characteristic

C₈×Dic₅₃

C₂²

yes

characteristic

C₈×Dic₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₄

yes

characteristic

C₂×Dic₁₀₆

C₈

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄⋊Dic₅₃

C₈

yes

characteristic

#11719919

C₈

yes

characteristic

#11719919

C₈

yes

characteristic

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₂×#88322

C₂³

yes

—

—

—

Q₈

yes

complemented

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

C₂×C₈

yes

characteristic

C₂×Dic₅₃

C₂×C₈

yes

characteristic

C₂×Dic₅₃

C₂×C₈

yes

characteristic

C₂×Dic₅₃

C₂×C₈

yes

characteristic

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

C₂×Q₈

yes

yes

C₂²×C₅₃

C₂²×C₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×Q₈

yes

characteristic

C₄×C₅₃

#131

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

M₃₂

yes

—

#2183

—

maximal, Sylow

C₂×C₅₃

C₈×Q₈

yes

characteristic

C₂×C₅₃

C₂×M₃₂

yes

characteristic

C₂×C₅₃

#504

yes

characteristic

C₈×Q₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#346

—

#346

—

C₄×C₁₆

—

C₅₃

#2183

yes

characteristic, complemented, Sylow

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

#6157

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂×#6157

yes

characteristic

C₁₆

—

C₁₆

—

106

—

C₁₆

—

106

—

C₁₆

—

106

—

Q₈

—

106

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₅₃⋊C₈

yes

characteristic

100

C₂×C₄

D₅₃⋊C₈

yes

characteristic

101

C₂×C₄

C₂²×#6157

yes

characteristic

102

C₈

—

103

C₈

—

104

C₈

—

105

C₈

—

106

C₈

—

106

—

107

C₂²

C₂×D₅₃⋊C₈

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

106

—

112

C₄

—

106

—

113

C₄

—

106

—

114

C₄

Q₈×#6157

yes

characteristic

115

C₄

#13133304

yes

characteristic

116

C₂

#16811672

yes

characteristic

117

C₂

D₅₃⋊M₃₂

yes

characteristic

118

C₂

#16811682

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#6157
commutator, socle	C₂²×C₅₃	C₂²×C₈
Fitting	C₄⋊C₈×C₅₃	C₄
hypercenter	C₄⋊C₈	#6157
hypocenter	C₅₃	#2183
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#6157

commutator, socle

C₂²×C₅₃

C₂²×C₈

Fitting

C₄⋊C₈×C₅₃

C₄

C₄⋊C₈

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

N	H ≅ G/N	type
33. #11719910	Q₈	semidirect
67. C₅₃	#2183	semidirect

H ≅ G/N

type

33. #11719910

Q₈

semidirect

67. C₅₃

#2183

semidirect

Subgroups — #20291132

Conjugacy classes of subgroups

Complements