Group #20285255 — subgroups

Subgroups — #20285255

Order 6784.

340 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 149 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	#16811251	C₂	1	2	yes	maximal
2	#16811030	C₂	1	4	yes	maximal, complemented
3	#16811027	C₂	1	2	yes	maximal, complemented
4	#16811024	C₂	1	2	yes	maximal, complemented
5	#16810666	C₂	1	2	yes	maximal
6	C₂×#13133151	C₂	1	1	yes	maximal, characteristic
7	#16810578	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#427	C₂	1	1	yes	maximal, characteristic
9	C₂×#11719917	C₂²	1	1	yes	characteristic
10	#13133217	—	2	2	no	—
11	#13133216	C₂²	1	4	yes	—
12	#13133209	—	2	2	no	—
13	#13133208	C₂²	1	4	yes	—
14	#13133194	—	2	4	no	—
15	#13133176	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
17	#13133151	C₂²	1	2	yes	—
18	#13133151	C₂²	1	2	yes	—
19	#13133150	C₂²	1	4	yes	—
20	#13133148	C₂²	1	4	yes	—
21	#13133127	C₂²	1	2	yes	—
22	C₈×Dic₅₃	C₂²	1	2	yes	—
23	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	#117×C₅₃	C₂²	1	2	yes	—
25	#107×C₅₃	C₂²	1	1	yes	characteristic
26	#107×C₅₃	C₂²	1	1	yes	characteristic
27	#106×C₅₃	C₂²	1	2	yes	—
28	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
29	C₂²×Dic₅₃	—	2	2	no	—
30	C₂×Dic₁₀₆	—	2	4	no	—
31	C₂×C₈×C₅₃	Q₈	1	4	yes	complemented
32	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
35	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
36	C₂²⋊C₄×C₅₃	—	2	2	no	—
37	C₄²×C₅₃	—	2	2	no	—
38	C₂²⋊Dic₅₃	D₄	1	2	yes	—
39	C₄⋊Dic₅₃	—	2	2	no	—
40	C₄⋊Dic₅₃	—	2	4	no	—
41	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
42	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
43	C₄⋊Dic₅₃	C₂³	1	4	yes	—
44	#11719919	—	2	4	no	—
45	#11719919	—	2	4	no	—
46	C₄×Dic₅₃	C₂³	1	2	yes	—
47	#11719917	Q₈	1	4	yes	complemented
48	C₂×#88322	C₂³	1	2	yes	—
49	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₅₃	—	2	2	no	—
51	C₂×C₄×C₅₃	—	2	2	no	—
52	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
54	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
55	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
56	C₂×Dic₅₃	—	2	4	no	—
57	C₂×Dic₅₃	—	8	2	no	—
58	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
59	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
60	Dic₁₀₆	—	8	4	no	—
61	C₈×C₅₃	C₂×Q₈	1	2	yes	—
62	C₈×C₅₃	C₂×Q₈	1	2	yes	—
63	#88322	C₂×Q₈	1	4	yes	—
64	C₂²×C₅₃	—	2	1	no	—
65	C₂²×C₅₃	—	2	1	no	—
66	C₂²×C₅₃	C₄⋊Q₈	1	2	yes	—
67	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
68	C₄×C₅₃	—	4	2	no	—
69	C₄×C₅₃	—	4	2	no	—
70	C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
71	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
72	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
73	Dic₅₃	—	2	2	no	—
74	Dic₅₃	—	2	2	no	—
75	Dic₅₃	—	4	4	no	—
76	#3164	—	53	1	no	maximal, Sylow
77	C₂×C₅₃	—	2	2	no	—
78	C₂×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
79	C₂×C₅₃	#576	1	1	yes	characteristic
80	C₂×C₅₃	#578	1	1	yes	characteristic
81	#544	—	53	2	no	—
82	#501	—	53	4	no	—
83	#500	—	53	2	no	—
84	#499	—	53	2	no	—
85	#428	—	53	2	no	—
86	#427	D₅₃	1	1	yes	characteristic
87	C₂×#107	—	53	1	no	—
88	#405	—	53	1	no	—
89	C₅₃	#3164	1	1	yes	characteristic, complemented, Sylow
90	C₂×M₁₆	—	53	1	no	—
91	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
92	C₄⋊Q₈	—	53	4	no	—
93	C₄⋊Q₈	—	106	2	no	—
94	#125	—	53	4	no	—
95	#125	—	106	2	no	—
96	C₂²⋊Q₈	—	106	4	no	—
97	#117	—	53	1	no	—
98	#117	C₂×D₅₃	1	2	yes	—
99	C₂×C₄⋊C₄	—	53	2	no	—
100	#107	—	53	2	no	—
101	#107	—	53	2	no	—
102	#107	—	53	4	no	—
103	#107	C₂×D₅₃	1	1	yes	characteristic
104	#107	C₂×D₅₃	1	1	yes	characteristic
105	#106	—	53	4	no	—
106	#106	C₂×D₅₃	1	2	yes	—
107	#97	—	53	2	no	—
108	C₄×C₈	—	53	2	no	—
109	C₂×Q₈	—	106	4	no	—
110	C₂²×C₄	—	106	2	no	—
111	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
112	M₁₆	—	53	4	no	—
113	C₂×C₈	—	53	2	no	—
114	C₂×C₈	Dic₁₀₆	1	4	yes	complemented
115	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
116	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
117	C₄⋊C₄	—	53	1	no	—
118	C₄⋊C₄	—	53	1	no	—
119	C₄⋊C₄	—	53	4	no	—
120	C₄⋊C₄	—	106	4	no	—
121	C₄⋊C₄	—	106	4	no	—
122	C₄⋊C₄	—	106	4	no	—
123	C₄⋊C₄	—	106	2	no	—
124	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
125	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
126	C₂²⋊C₄	—	2	2	no	—
127	C₂²⋊C₄	—	53	2	no	—
128	C₄²	—	2	2	no	—
129	C₄²	—	53	2	no	—
130	C₂³	D₄×D₅₃	1	1	yes	characteristic
131	Q₈	—	424	4	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	53	2	no	—
135	C₂×C₄	—	53	2	no	—
136	C₂×C₄	—	106	4	no	—
137	C₂×C₄	—	424	2	no	—
138	C₂×C₄	C₂×Dic₁₀₆	1	2	yes	—
139	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
140	C₂×C₄	Q₈×D₅₃	1	2	yes	—
141	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
142	C₈	—	53	4	no	—
143	C₈	C₂×Dic₁₀₆	1	2	yes	—
144	C₈	C₂×Dic₁₀₆	1	2	yes	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	#13133216	1	2	yes	—
148	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
149	C₄	—	4	2	no	—
150	C₄	—	4	2	no	—
151	C₄	—	106	2	no	—
152	C₄	—	106	2	no	—
153	C₄	—	212	4	no	—
154	C₄	#13133216	1	2	yes	—
155	C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
156	C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
157	C₂	—	2	2	no	—
158	C₂	C₂×#13133216	1	1	yes	characteristic
159	C₂	#16811425	1	1	yes	characteristic
160	C₂	#16811437	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811251

C₂

yes

maximal

#16811030

C₂

yes

maximal, complemented

#16811027

C₂

yes

maximal, complemented

#16811024

C₂

yes

maximal, complemented

#16810666

C₂

yes

maximal

C₂×#13133151

C₂

yes

maximal, characteristic

#16810578

C₂

yes

maximal, characteristic

C₅₃×#427

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

characteristic

#13133217

—

#13133216

C₂²

yes

—

#13133209

—

#13133208

C₂²

yes

—

#13133194

—

#13133176

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

—

#13133151

C₂²

yes

—

#13133151

C₂²

yes

—

#13133150

C₂²

yes

—

#13133148

C₂²

yes

—

#13133127

C₂²

yes

—

C₈×Dic₅₃

C₂²

yes

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

#117×C₅₃

C₂²

yes

—

#107×C₅₃

C₂²

yes

characteristic

#107×C₅₃

C₂²

yes

characteristic

#106×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

Q₈

yes

complemented

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₂²⋊Dic₅₃

D₄

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

characteristic

C₄⋊Dic₅₃

D₄

yes

characteristic

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

#11719919

—

C₄×Dic₅₃

C₂³

yes

—

#11719917

Q₈

yes

complemented

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₄

yes

—

C₂×Dic₅₃

C₂×D₄

yes

—

Dic₁₀₆

—

C₈×C₅₃

C₂×Q₈

yes

—

C₈×C₅₃

C₂×Q₈

yes

—

#88322

C₂×Q₈

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₄⋊Q₈

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄⋊Q₈

yes

—

C₄×C₅₃

C₂²×Q₈

yes

characteristic

C₄×C₅₃

C₂²×Q₈

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

#3164

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×C₄⋊Q₈

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

—

—

—

D₅₃

yes

characteristic

C₂×#107

—

#405

—

C₅₃

#3164

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

106

—

#125

—

#125

—

106

—

C₂²⋊Q₈

—

106

—

#117

—

#117

C₂×D₅₃

yes

—

C₂×C₄⋊C₄

—

100

#107

—

101

#107

—

102

#107

—

103

#107

C₂×D₅₃

yes

characteristic

104

#107

C₂×D₅₃

yes

characteristic

105

#106

—

106

#106

C₂×D₅₃

yes

—

107

#97

—

108

C₄×C₈

—

109

C₂×Q₈

—

106

—

110

C₂²×C₄

—

106

—

111

C₂²×C₄

C₂²×D₅₃

yes

characteristic

112

M₁₆

—

113

C₂×C₈

—

114

C₂×C₈

Dic₁₀₆

yes

complemented

115

C₂×C₈

C₂²×D₅₃

yes

characteristic

116

C₂×C₈

C₂²×D₅₃

yes

characteristic

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

106

—

121

C₄⋊C₄

—

106

—

122

C₄⋊C₄

—

106

—

123

C₄⋊C₄

—

106

—

124

C₄⋊C₄

C₂²×D₅₃

yes

—

125

C₄⋊C₄

C₂²×D₅₃

yes

—

126

C₂²⋊C₄

—

127

C₂²⋊C₄

—

128

C₄²

—

129

C₄²

—

130

C₂³

D₄×D₅₃

yes

characteristic

131

Q₈

—

424

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

106

—

137

C₂×C₄

—

424

—

138

C₂×C₄

C₂×Dic₁₀₆

yes

—

139

C₂×C₄

D₄×D₅₃

yes

characteristic

140

C₂×C₄

Q₈×D₅₃

yes

—

141

C₂×C₄

C₂³×D₅₃

yes

characteristic

142

C₈

—

143

C₈

C₂×Dic₁₀₆

yes

—

144

C₈

C₂×Dic₁₀₆

yes

—

145

C₂²

—

146

C₂²

—

147

C₂²

#13133216

yes

—

148

C₂²

C₂×D₄×D₅₃

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

106

—

152

C₄

—

106

—

153

C₄

—

212

—

154

C₄

#13133216

yes

—

155

C₄

C₂×Q₈×D₅₃

yes

characteristic

156

C₄

C₂²×Dic₁₀₆

yes

characteristic

157

C₂

—

158

C₂

C₂×#13133216

yes

characteristic

159

C₂

#16811425

yes

characteristic

160

C₂

#16811437

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#427	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#427	D₅₃
hypocenter	C₅₃	#3164
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂²

C₂×D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#427

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
2. #16811030	C₂	semidirect
3. #16811027	C₂	semidirect
4. #16811024	C₂	semidirect
31. C₂×C₈×C₅₃	Q₈	semidirect
47. #11719917	Q₈	semidirect
89. C₅₃	#3164	semidirect
114. C₂×C₈	Dic₁₀₆	semidirect

H ≅ G/N

type

2. #16811030

C₂

semidirect

3. #16811027

C₂

semidirect

4. #16811024

C₂

semidirect

31. C₂×C₈×C₅₃

Q₈

semidirect

47. #11719917

Q₈

semidirect

89. C₅₃

#3164

semidirect

114. C₂×C₈

Dic₁₀₆

semidirect

Subgroups — #20285255

Conjugacy classes of subgroups

Complements