Group #20274172 — subgroups

Subgroups — #20274172

Order 6784.

144 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 51 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	#16810973	C₂	1	1	yes	maximal, characteristic
2	#16810919	C₂	1	1	yes	maximal, characteristic
3	#16810905	C₂	1	1	yes	maximal, characteristic
4	#16810747	C₂	1	1	yes	maximal, characteristic
5	#16810375	C₂	1	1	yes	maximal, characteristic
6	#16810371	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#332	C₂	1	1	yes	maximal, characteristic
8	#13133217	C₂²	1	1	yes	characteristic
9	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
10	#13133151	—	2	1	no	—
11	#13133148	—	2	1	no	—
12	C₄⋊#88322	C₂²	1	1	yes	characteristic
13	#13133146	—	2	1	no	—
14	#13133142	—	2	1	no	—
15	#13133141	—	2	1	no	—
16	#13133141	—	2	1	no	—
17	#13133140	—	2	1	no	—
18	C₄×#88322	C₂²	1	1	yes	characteristic
19	C₈×Dic₅₃	—	2	1	no	—
20	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₅₃	—	2	1	no	—
24	C₂×Dic₁₀₆	—	2	1	no	—
25	C₂×Dic₁₀₆	D₄	1	1	yes	characteristic
26	C₂×C₈×C₅₃	—	2	1	no	—
27	C₂×C₈×C₅₃	—	2	1	no	—
28	C₄⋊C₄×C₅₃	—	2	1	no	—
29	C₄⋊C₄×C₅₃	—	2	1	no	—
30	C₄²×C₅₃	C₂³	1	1	yes	characteristic
31	C₄⋊Dic₅₃	—	2	1	no	—
32	C₄⋊Dic₅₃	—	2	1	no	—
33	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
34	#11719919	—	2	1	no	—
35	C₄×Dic₅₃	—	2	1	no	—
36	C₂×#88322	—	2	1	no	—
37	C₂×#88322	D₄	1	1	yes	characteristic
38	C₂×#88322	D₄	1	1	yes	characteristic
39	Q₈×C₅₃	—	4	1	no	—
40	Q₈×C₅₃	—	4	1	no	—
41	C₂×C₄×C₅₃	—	4	1	no	—
42	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
45	C₂×Dic₅₃	—	2	1	no	—
46	C₂×Dic₅₃	—	2	1	no	—
47	C₂×Dic₅₃	—	4	1	no	—
48	Dic₁₀₆	—	2	1	no	—
49	Dic₁₀₆	—	8	1	no	—
50	Dic₁₀₆	Q₁₆	1	2	yes	—
51	C₈×C₅₃	—	4	1	no	—
52	C₈×C₅₃	—	4	1	no	—
53	#88322	—	2	1	no	—
54	#88322	—	2	1	no	—
55	#88322	—	4	1	no	—
56	C₂²×C₅₃	#121	1	1	yes	characteristic
57	C₄×C₅₃	—	2	1	no	—
58	C₄×C₅₃	—	8	1	no	—
59	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
60	C₄×C₅₃	#135	1	1	yes	characteristic
61	C₄×C₅₃	#135	1	1	yes	characteristic
62	C₄×C₅₃	#137	1	1	yes	characteristic
63	Dic₅₃	—	2	2	no	—
64	Dic₅₃	—	4	1	no	—
65	Dic₅₃	—	8	1	no	—
66	#1684	—	53	1	no	maximal, Sylow
67	C₂×C₅₃	#464	1	1	yes	characteristic
68	C₂×C₅₃	#467	1	1	yes	characteristic
69	C₂×C₅₃	#471	1	1	yes	characteristic
70	#487	—	53	1	no	—
71	#477	—	53	1	no	—
72	#475	—	53	1	no	—
73	C₈×Q₈	—	53	1	no	—
74	#335	—	53	1	no	—
75	#333	—	53	1	no	—
76	#332	D₅₃	1	1	yes	characteristic
77	C₅₃	#1684	1	1	yes	characteristic, complemented, Sylow
78	C₄⋊Q₈	—	53	1	no	—
79	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
80	C₄×Q₈	—	53	1	no	—
81	#107	—	106	1	no	—
82	#106	—	106	1	no	—
83	C₄⋊C₈	—	53	1	no	—
84	C₄⋊C₈	—	106	1	no	—
85	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
86	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
87	#103	—	106	1	no	—
88	#103	—	106	1	no	—
89	#103	—	106	1	no	—
90	#103	—	106	1	no	—
91	C₄×C₈	—	53	1	no	—
92	C₄×C₈	—	106	1	no	—
93	C₂×Q₈	—	2	1	no	—
94	C₂×Q₈	—	53	1	no	—
95	C₂×Q₈	—	106	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	53	1	no	—
99	C₂×C₈	—	53	1	no	—
100	C₂×C₈	—	106	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	53	1	no	—
104	C₄⋊C₄	—	106	1	no	—
105	C₄⋊C₄	—	106	1	no	—
106	C₄⋊C₄	—	106	1	no	—
107	C₄²	—	106	1	no	—
108	C₄²	C₂²×D₅₃	1	1	yes	characteristic
109	Q₈	—	4	1	no	—
110	Q₈	—	4	1	no	—
111	Q₈	—	53	2	no	—
112	Q₈	—	106	1	no	—
113	Q₈	—	424	1	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	106	1	no	—
116	C₂×C₄	—	106	1	no	—
117	C₂×C₄	—	212	1	no	—
118	C₂×C₄	#11719945	1	1	yes	characteristic
119	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
120	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
121	C₈	—	4	1	no	—
122	C₈	—	4	1	no	—
123	C₈	—	106	1	no	—
124	C₈	—	106	1	no	—
125	C₈	—	212	1	no	—
126	C₂²	#13133190	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	8	1	no	—
129	C₄	—	106	2	no	—
130	C₄	—	424	1	no	—
131	C₄	—	212	1	no	—
132	C₄	Q₁₆×D₅₃	1	1	yes	characteristic
133	C₄	#13133236	1	1	yes	characteristic
134	C₄	#13133238	1	1	yes	characteristic
135	C₄	#13133246	1	1	yes	characteristic
136	C₂	#16810844	1	1	yes	characteristic
137	C₂	#16810863	1	1	yes	characteristic
138	C₂	#16810889	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810973

C₂

yes

maximal, characteristic

#16810919

C₂

yes

maximal, characteristic

#16810905

C₂

yes

maximal, characteristic

#16810747

C₂

yes

maximal, characteristic

#16810375

C₂

yes

maximal, characteristic

#16810371

C₂

yes

maximal, characteristic

C₅₃×#332

C₂

yes

maximal, characteristic

#13133217

C₂²

yes

characteristic

C₄×Dic₁₀₆

C₂²

yes

—

—

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

C₈×Dic₅₃

—

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

D₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

characteristic

#11719919

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

D₄

yes

characteristic

C₂×#88322

D₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

Q₁₆

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

—

#88322

—

C₂²×C₅₃

#121

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×Q₁₆

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

—

C₈×Q₈

—

#335

—

#333

—

#332

D₅₃

yes

characteristic

C₅₃

#1684

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

C₄×Q₈

—

#107

—

106

—

#106

—

106

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₄⋊C₈

C₂×D₅₃

yes

characteristic

C₄⋊C₈

C₂×D₅₃

yes

characteristic

#103

—

106

—

#103

—

106

—

#103

—

106

—

#103

—

106

—

C₄×C₈

—

C₄×C₈

—

106

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

106

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

106

—

105

C₄⋊C₄

—

106

—

106

C₄⋊C₄

—

106

—

107

C₄²

—

106

—

108

C₄²

C₂²×D₅₃

yes

characteristic

109

Q₈

—

110

Q₈

—

111

Q₈

—

112

Q₈

—

106

—

113

Q₈

—

424

—

114

C₂×C₄

—

115

C₂×C₄

—

106

—

116

C₂×C₄

—

106

—

117

C₂×C₄

—

212

—

118

C₂×C₄

#11719945

yes

characteristic

119

C₂×C₄

D₄×D₅₃

yes

characteristic

120

C₂×C₄

D₄×D₅₃

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

106

—

124

C₈

—

106

—

125

C₈

—

212

—

126

C₂²

#13133190

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

106

—

130

C₄

—

424

—

131

C₄

—

212

—

132

C₄

Q₁₆×D₅₃

yes

characteristic

133

C₄

#13133236

yes

characteristic

134

C₄

#13133238

yes

characteristic

135

C₄

#13133246

yes

characteristic

136

C₂

#16810844

yes

characteristic

137

C₂

#16810863

yes

characteristic

138

C₂

#16810889

yes

characteristic

	subgroup	G/H
center	C₂²	#13133190
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#332	C₂
socle	C₂²×C₅₃	#121
hypercenter	#332	D₅₃
hypocenter	C₅₃	#1684
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

#13133190

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#332

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
77. C₅₃	#1684	semidirect

H ≅ G/N

type

77. C₅₃

#1684

semidirect

Subgroups — #20274172

Conjugacy classes of subgroups

Complements