Group #20275438 — subgroups

Subgroups — #20275438

Order 6784.

248 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 113 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	C₂×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133146	C₂	1	2	yes	maximal
3	C₄×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	#16810377	C₂	1	2	yes	maximal
5	C₂×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#88322	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₅₃	C₄	1	2	yes	—
8	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
9	C₄⋊#88322	—	2	2	no	—
10	#13133146	—	2	4	no	—
11	#13133124	C₄	1	1	yes	characteristic
12	#13133124	C₄	1	1	yes	characteristic
13	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	#97×C₅₃	—	2	2	no	—
17	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
20	C₂²×Dic₅₃	C₂×C₄	1	2	yes	—
21	C₂²×Dic₅₃	C₂×C₄	1	2	yes	—
22	C₂×C₈×C₅₃	—	2	1	no	—
23	C₂×C₈×C₅₃	—	2	1	no	—
24	C₂×C₈×C₅₃	—	2	1	no	—
25	C₂×C₈×C₅₃	—	2	1	no	—
26	C₂×C₈×C₅₃	D₄	1	2	yes	—
27	C₂×C₈×C₅₃	Q₈	1	2	yes	—
28	C₄²×C₅₃	D₄	1	2	yes	—
29	C₄²×C₅₃	Q₈	1	2	yes	—
30	C₄⋊Dic₅₃	—	2	4	no	—
31	C₄×Dic₅₃	—	2	2	no	—
32	C₄×Dic₅₃	—	2	2	no	—
33	C₂×#88322	—	2	2	no	—
34	C₂×#88322	—	2	2	no	—
35	C₂×#88322	—	2	2	no	—
36	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₅₃	—	2	2	no	—
38	C₂×C₄×C₅₃	—	2	2	no	—
39	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
40	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
41	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
46	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
47	C₂×Dic₅₃	—	2	2	no	—
48	C₂×Dic₅₃	—	2	2	no	—
49	C₂×Dic₅₃	—	2	2	no	—
50	C₂×Dic₅₃	—	2	2	no	—
51	C₂×Dic₅₃	—	2	2	no	—
52	C₂×Dic₅₃	M₁₆	1	4	yes	—
53	C₈×C₅₃	—	2	2	no	—
54	C₈×C₅₃	—	2	2	no	—
55	#88322	—	4	2	no	—
56	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
58	C₂²×C₅₃	C₄×Q₈	1	1	yes	characteristic
59	C₂²×C₅₃	C₂×M₁₆	1	2	yes	—
60	C₂²×C₅₃	#131	1	2	yes	—
61	C₄×C₅₃	—	2	2	no	—
62	C₄×C₅₃	—	2	2	no	—
63	C₄×C₅₃	#121	1	1	yes	characteristic
64	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
65	C₄×C₅₃	#125	1	1	yes	characteristic
66	C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
67	Dic₅₃	—	2	4	no	—
68	Dic₅₃	—	4	2	no	—
69	#1940	—	53	1	no	maximal, Sylow
70	C₂×C₅₃	#389	1	1	yes	characteristic
71	C₂×C₅₃	#423	1	1	yes	characteristic
72	C₂×C₅₃	#424	1	1	yes	characteristic
73	C₂×C₅₃	#435	1	2	yes	—
74	C₂×C₅₃	#446	1	2	yes	—
75	C₂×C₄⋊C₈	—	53	2	no	—
76	C₂×C₄⋊C₈	—	53	1	no	—
77	C₂×#97	D₅₃	1	1	yes	characteristic
78	C₄×C₄⋊C₄	—	53	1	no	—
79	#336	—	53	2	no	—
80	C₅₃	#1940	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₈	—	53	2	no	—
82	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
83	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
84	C₂×C₄⋊C₄	—	53	2	no	—
85	C₂×C₄²	—	53	2	no	—
86	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
87	C₄⋊C₈	—	106	4	no	—
88	C₄⋊C₈	—	106	2	no	—
89	#97	—	2	2	no	—
90	#95	—	53	1	no	—
91	#95	—	53	1	no	—
92	C₂²×C₄	—	53	2	no	—
93	C₂²×C₄	—	53	2	no	—
94	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
95	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
96	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	106	2	no	—
102	C₂×C₈	—	106	2	no	—
103	C₂×C₈	—	106	2	no	—
104	C₂×C₈	Dic₁₀₆	1	2	yes	—
105	C₂×C₈	D₂₁₂	1	2	yes	—
106	C₄⋊C₄	—	106	4	no	—
107	C₄²	—	106	2	no	—
108	C₄²	—	106	2	no	—
109	C₄²	Dic₁₀₆	1	2	yes	—
110	C₄²	C₂²⋊D₅₃	1	2	yes	—
111	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	53	4	no	—
115	C₂×C₄	—	106	2	no	—
116	C₂×C₄	—	106	2	no	—
117	C₂×C₄	—	106	2	no	—
118	C₂×C₄	—	106	2	no	—
119	C₂×C₄	—	106	2	no	—
120	C₂×C₄	#11719919	1	2	yes	—
121	C₂×C₄	#11719919	1	2	yes	—
122	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
123	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
124	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
125	C₂×C₄	#11719945	1	1	yes	characteristic
126	C₂×C₄	#11719945	1	1	yes	characteristic
127	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
128	C₈	—	2	2	no	—
129	C₈	—	2	2	no	—
130	C₈	—	212	2	no	—
131	C₂²	C₂×#11719919	1	1	yes	characteristic
132	C₂²	C₄×D₂₁₂	1	1	yes	characteristic
133	C₂²	C₄×Dic₁₀₆	1	1	yes	characteristic
134	C₂²	M₁₆×D₅₃	1	2	yes	—
135	C₂²	#13133225	1	2	yes	—
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	106	4	no	—
139	C₄	—	212	2	no	—
140	C₄	#13133191	1	1	yes	characteristic
141	C₄	#13133196	1	1	yes	characteristic
142	C₄	#13133209	1	1	yes	characteristic
143	C₄	#13133217	1	1	yes	characteristic
144	C₂	#16810521	1	1	yes	characteristic
145	C₂	#16810646	1	1	yes	characteristic
146	C₂	#16810651	1	1	yes	characteristic
147	C₂	#16810696	1	2	yes	—
148	C₂	#16810751	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#88322

C₂

yes

maximal, characteristic

C₂×#13133146

C₂

yes

maximal

C₄×C₄⋊Dic₅₃

C₂

yes

maximal, characteristic

#16810377

C₂

yes

maximal

C₂×#97×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₄

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

C₄⋊#88322

—

#13133146

—

#13133124

C₄

yes

characteristic

#13133124

C₄

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#97×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

—

C₂²×Dic₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

Q₈

yes

—

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

Q₈

yes

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

M₁₆

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₃

C₄×D₄

yes

characteristic

C₂²×C₅₃

C₄×Q₈

yes

characteristic

C₂²×C₅₃

C₂×M₁₆

yes

—

C₂²×C₅₃

#131

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#121

yes

characteristic

C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₄×C₅₃

#125

yes

characteristic

C₄×C₅₃

C₄⋊Q₈

yes

characteristic

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#446

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#97

D₅₃

yes

characteristic

C₄×C₄⋊C₄

—

#336

—

C₅₃

#1940

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

106

—

C₄⋊C₈

—

106

—

#97

—

#95

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

106

—

102

C₂×C₈

—

106

—

103

C₂×C₈

—

106

—

104

C₂×C₈

Dic₁₀₆

yes

—

105

C₂×C₈

D₂₁₂

yes

—

106

C₄⋊C₄

—

106

—

107

C₄²

—

106

—

108

C₄²

—

106

—

109

C₄²

Dic₁₀₆

yes

—

110

C₄²

C₂²⋊D₅₃

yes

—

111

C₂³

C₂×C₄×D₅₃

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

106

—

116

C₂×C₄

—

106

—

117

C₂×C₄

—

106

—

118

C₂×C₄

—

106

—

119

C₂×C₄

—

106

—

120

C₂×C₄

#11719919

yes

—

121

C₂×C₄

#11719919

yes

—

122

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

123

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

124

C₂×C₄

C₂×D₂₁₂

yes

characteristic

125

C₂×C₄

#11719945

yes

characteristic

126

C₂×C₄

#11719945

yes

characteristic

127

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

212

—

131

C₂²

C₂×#11719919

yes

characteristic

132

C₂²

C₄×D₂₁₂

yes

characteristic

133

C₂²

C₄×Dic₁₀₆

yes

characteristic

134

C₂²

M₁₆×D₅₃

yes

—

135

C₂²

#13133225

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

106

—

139

C₄

—

212

—

140

C₄

#13133191

yes

characteristic

141

C₄

#13133196

yes

characteristic

142

C₄

#13133209

yes

characteristic

143

C₄

#13133217

yes

characteristic

144

C₂

#16810521

yes

characteristic

145

C₂

#16810646

yes

characteristic

146

C₂

#16810651

yes

characteristic

147

C₂

#16810696

yes

—

148

C₂

#16810751

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₅₃
commutator, socle	C₂³×C₅₃	C₂²×C₄
Fitting	C₂×#97×C₅₃	C₂
hypercenter	C₂×#97	D₅₃
hypocenter	C₅₃	#1940
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₅₃

commutator, socle

C₂³×C₅₃

C₂²×C₄

Fitting

C₂×#97×C₅₃

C₂

hypercenter

C₂×#97

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂²×C₄ ◁ C₂×#97

N	H ≅ G/N	type
80. C₅₃	#1940	semidirect

H ≅ G/N

type

80. C₅₃

#1940

semidirect

Subgroups — #20275438

Conjugacy classes of subgroups

Complements