Group #20274177 — subgroups

Subgroups — #20274177

Order 6784.

182 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 55 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	#16810900	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810815	C₂	1	1	yes	maximal, characteristic
3	#16810702	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810376	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810369	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810367	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#492	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11719922	—	2	1	no	—
9	C₂×#11719912	—	2	1	no	—
10	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
11	#13133150	—	2	1	no	—
12	C₄⋊#88322	C₂²	1	1	yes	characteristic
13	C₄⋊#88322	C₂²	1	1	yes	characteristic
14	C₄⋊#88322	C₂²	1	1	yes	characteristic
15	#13133138	—	2	1	no	—
16	#13133137	—	2	1	no	—
17	#13133129	—	2	1	no	—
18	C₂×SD₁₆×C₅₃	—	2	1	no	—
19	C₂×SD₁₆×C₅₃	—	2	1	no	—
20	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
21	#127×C₅₃	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₅₃	—	2	1	no	—
24	C₂×D₄×C₅₃	—	2	1	no	—
25	C₂×D₄×C₅₃	—	2	1	no	—
26	C₂×D₄×C₅₃	—	2	1	no	—
27	C₂×D₂₁₂	D₄	1	1	yes	characteristic
28	C₂×C₄×D₅₃	—	2	1	no	—
29	SD₁₆×C₅₃	—	4	2	no	—
30	SD₁₆×C₅₃	—	4	2	no	—
31	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
33	C₄⋊C₄×C₅₃	—	2	1	no	—
34	C₄⋊C₄×C₅₃	—	2	1	no	—
35	C₄²×C₅₃	C₂³	1	1	yes	characteristic
36	#11719922	—	4	2	no	—
37	#11719921	—	2	1	no	—
38	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
39	C₂×#88322	—	2	1	no	—
40	C₂×#88322	—	2	1	no	—
41	C₂×#88322	—	2	1	no	—
42	#11719912	—	4	2	no	—
43	C₂³×C₅₃	—	4	1	no	—
44	C₂²×D₅₃	—	2	1	no	—
45	Q₈×C₅₃	—	4	1	no	—
46	Q₈×C₅₃	—	4	1	no	—
47	D₄×C₅₃	—	4	1	no	—
48	D₄×C₅₃	—	4	1	no	—
49	D₄×C₅₃	—	4	1	no	—
50	D₄×C₅₃	—	4	1	no	—
51	D₄×C₅₃	—	8	1	no	—
52	C₂×C₄×C₅₃	—	4	1	no	—
53	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
56	C₂×Dic₅₃	—	2	1	no	—
57	D₂₁₂	—	2	1	no	—
58	D₂₁₂	—	2	1	no	—
59	C₄×D₅₃	—	4	1	no	—
60	C₄×D₅₃	—	4	1	no	—
61	C₈×C₅₃	—	2	1	no	—
62	C₈×C₅₃	D₈	1	2	yes	complemented
63	#88322	—	4	1	no	—
64	#88322	—	4	1	no	—
65	#88322	—	4	1	no	—
66	C₂²×C₅₃	—	8	1	no	—
67	C₂²×C₅₃	—	8	1	no	—
68	C₂²×C₅₃	—	8	1	no	—
69	C₂²×C₅₃	#121	1	1	yes	characteristic
70	C₂×D₅₃	—	4	1	no	—
71	C₂×D₅₃	—	4	1	no	—
72	C₂×D₅₃	—	4	1	no	—
73	C₄×C₅₃	—	2	1	no	—
74	C₄×C₅₃	—	8	1	no	—
75	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
76	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₄×C₅₃	#137	1	1	yes	characteristic
79	Dic₅₃	—	4	1	no	—
80	#1685	—	53	1	no	maximal, Sylow
81	C₂×C₅₃	—	16	1	no	—
82	C₂×C₅₃	#459	1	1	yes	characteristic
83	C₂×C₅₃	#468	1	1	yes	characteristic
84	C₂×C₅₃	#471	1	1	yes	characteristic
85	D₅₃	—	8	1	no	—
86	#492	D₅₃	1	1	yes	characteristic, complemented
87	#474	—	53	1	no	—
88	#459	—	53	1	no	—
89	#436	—	53	1	no	—
90	#335	—	53	1	no	—
91	#332	—	53	1	no	—
92	#331	—	53	1	no	—
93	C₅₃	#1685	1	1	yes	characteristic, complemented, Sylow
94	C₂×SD₁₆	—	2	1	no	—
95	C₂×SD₁₆	—	2	1	no	—
96	C₂×D₈	—	106	1	no	—
97	C₂×M₁₆	—	106	1	no	—
98	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
99	#127	C₂×D₅₃	1	1	yes	characteristic
100	C₄×D₄	—	53	1	no	—
101	#107	—	106	1	no	—
102	C₄⋊C₈	—	53	1	no	—
103	C₄⋊C₈	—	53	1	no	—
104	C₄⋊C₈	—	53	1	no	—
105	#102	—	106	1	no	—
106	#102	—	106	1	no	—
107	C₂²⋊C₈	—	106	1	no	—
108	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
109	C₂×Q₈	—	2	1	no	—
110	C₂×D₄	—	2	1	no	—
111	C₂×D₄	—	2	1	no	—
112	C₂×D₄	—	2	1	no	—
113	C₂×D₄	—	53	1	no	—
114	C₂²×C₄	—	106	1	no	—
115	SD₁₆	—	4	2	no	—
116	SD₁₆	—	4	2	no	—
117	D₈	—	212	2	no	—
118	M₁₆	—	212	2	no	—
119	C₂×C₈	—	106	1	no	—
120	C₂×C₈	—	106	1	no	—
121	C₂×C₈	—	106	1	no	—
122	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
123	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
124	C₄⋊C₄	—	2	1	no	—
125	C₄⋊C₄	—	2	1	no	—
126	C₄⋊C₄	—	53	1	no	—
127	C₂²⋊C₄	—	106	1	no	—
128	C₄²	C₂²×D₅₃	1	1	yes	characteristic
129	C₂³	—	4	1	no	—
130	C₂³	—	106	1	no	—
131	Q₈	—	4	1	no	—
132	Q₈	—	4	1	no	—
133	D₄	—	4	1	no	—
134	D₄	—	4	1	no	—
135	D₄	—	4	1	no	—
136	D₄	—	4	1	no	—
137	D₄	—	8	1	no	—
138	D₄	—	106	1	no	—
139	D₄	—	106	1	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	106	1	no	—
142	C₂×C₄	—	212	1	no	—
143	C₂×C₄	—	212	1	no	—
144	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
145	C₂×C₄	#11719947	1	1	yes	characteristic
146	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
147	C₈	—	2	1	no	—
148	C₈	—	212	1	no	—
149	C₈	—	212	1	no	—
150	C₈	—	212	1	no	—
151	C₈	#11719922	1	2	yes	complemented
152	C₂²	—	8	1	no	—
153	C₂²	—	8	1	no	—
154	C₂²	—	8	1	no	—
155	C₂²	—	212	1	no	—
156	C₂²	—	212	1	no	—
157	C₂²	—	212	1	no	—
158	C₂²	#13133188	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	8	1	no	—
161	C₄	—	212	1	no	—
162	C₄	C₂×#11719922	1	1	yes	characteristic
163	C₄	#13133243	1	1	yes	characteristic
164	C₄	#13133244	1	1	yes	characteristic
165	C₄	#13133246	1	1	yes	characteristic
166	C₂	—	16	1	no	—
167	C₂	—	424	1	no	—
168	C₂	#16810815	1	1	yes	characteristic
169	C₂	#16810866	1	1	yes	characteristic
170	C₂	#16810883	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810900

C₂

yes

maximal, characteristic, complemented

#16810815

C₂

yes

maximal, characteristic

#16810702

C₂

yes

maximal, characteristic, complemented

#16810376

C₂

yes

maximal, characteristic, complemented

#16810369

C₂

yes

maximal, characteristic, complemented

#16810367

C₂

yes

maximal, characteristic, complemented

C₅₃×#492

C₂

yes

maximal, characteristic, complemented

C₂×#11719922

—

C₂×#11719912

—

C₄×D₂₁₂

C₂²

yes

characteristic

#13133150

—

C₄⋊#88322

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

—

—

—

C₂×SD₁₆×C₅₃

—

C₂×SD₁₆×C₅₃

—

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

#127×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₂₁₂

D₄

yes

characteristic

C₂×C₄×D₅₃

—

SD₁₆×C₅₃

—

SD₁₆×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

#11719922

—

#11719921

—

C₄⋊Dic₅₃

D₄

yes

—

—

—

—

C₂³×C₅₃

—

C₂²×D₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×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

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

C₈×C₅₃

D₈

yes

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#121

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×D₈

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

#1685

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

#492

D₅₃

yes

characteristic, complemented

—

—

—

—

—

—

C₅₃

#1685

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

106

—

C₂×M₁₆

—

106

—

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

#127

C₂×D₅₃

yes

characteristic

100

C₄×D₄

—

101

#107

—

106

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

#102

—

106

—

106

#102

—

106

—

107

C₂²⋊C₈

—

106

—

108

C₄×C₈

C₂×D₅₃

yes

characteristic

109

C₂×Q₈

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂²×C₄

—

106

—

115

SD₁₆

—

116

SD₁₆

—

117

D₈

—

212

—

118

M₁₆

—

212

—

119

C₂×C₈

—

106

—

120

C₂×C₈

—

106

—

121

C₂×C₈

—

106

—

122

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

123

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₂²⋊C₄

—

106

—

128

C₄²

C₂²×D₅₃

yes

characteristic

129

C₂³

—

130

C₂³

—

106

—

131

Q₈

—

132

Q₈

—

133

D₄

—

134

D₄

—

135

D₄

—

136

D₄

—

137

D₄

—

138

D₄

—

106

—

139

D₄

—

106

—

140

C₂×C₄

—

141

C₂×C₄

—

106

—

142

C₂×C₄

—

212

—

143

C₂×C₄

—

212

—

144

C₂×C₄

D₄×D₅₃

yes

characteristic

145

C₂×C₄

#11719947

yes

characteristic

146

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

147

C₈

—

148

C₈

—

212

—

149

C₈

—

212

—

150

C₈

—

212

—

151

C₈

#11719922

yes

complemented

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

212

—

156

C₂²

—

212

—

157

C₂²

—

212

—

158

C₂²

#13133188

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

212

—

162

C₄

C₂×#11719922

yes

characteristic

163

C₄

#13133243

yes

characteristic

164

C₄

#13133244

yes

characteristic

165

C₄

#13133246

yes

characteristic

166

C₂

—

167

C₂

—

424

—

168

C₂

#16810815

yes

characteristic

169

C₂

#16810866

yes

characteristic

170

C₂

#16810883

yes

characteristic

	subgroup	G/H
center	C₂²	#13133188
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#492	C₂
socle	C₂²×C₅₃	#121
hypercenter	#492	D₅₃
hypocenter	C₅₃	#1685
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#13133188

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#492

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #16810900	C₂	semidirect
3. #16810702	C₂	semidirect
4. #16810376	C₂	semidirect
5. #16810369	C₂	semidirect
6. #16810367	C₂	semidirect
7. C₅₃×#492	C₂	semidirect
62. C₈×C₅₃	D₈	semidirect
86. #492	D₅₃	semidirect
93. C₅₃	#1685	semidirect
151. C₈	#11719922	semidirect

H ≅ G/N

type

1. #16810900

C₂

semidirect

3. #16810702

C₂

semidirect

4. #16810376

C₂

semidirect

5. #16810369

C₂

semidirect

6. #16810367

C₂

semidirect

7. C₅₃×#492

C₂

semidirect

62. C₈×C₅₃

D₈

semidirect

86. #492

D₅₃

semidirect

93. C₅₃

#1685

semidirect

151. C₈

#11719922

semidirect

Subgroups — #20274177

Conjugacy classes of subgroups

Complements