Group #20274178 — subgroups

Subgroups — #20274178

Order 6784.

182 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 53 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	#16810995	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810903	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810701	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810375	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810368	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810367	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#459	C₂	1	1	yes	maximal, characteristic
8	C₂×#11719923	—	2	1	no	—
9	C₂×#11719923	—	2	1	no	—
10	C₂×#11719917	—	2	1	no	—
11	#13133217	C₂²	1	1	yes	characteristic
12	#13133150	—	2	1	no	—
13	C₄⋊#88322	C₂²	1	1	yes	characteristic
14	C₄⋊#88322	C₂²	1	1	yes	characteristic
15	#13133138	—	2	1	no	—
16	C₂²⋊#88322	—	2	1	no	—
17	C₄×#88322	C₂²	1	1	yes	characteristic
18	C₂×D₈×C₅₃	—	2	1	no	—
19	#127×C₅₃	C₂²	1	1	yes	characteristic
20	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
22	#102×C₅₃	—	2	1	no	—
23	C₂×D₄×C₅₃	—	2	1	no	—
24	C₂×D₄×C₅₃	—	2	1	no	—
25	C₂×D₄×C₅₃	—	2	1	no	—
26	C₂×D₄×C₅₃	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₅₃	—	2	1	no	—
28	C₂×Dic₁₀₆	—	2	1	no	—
29	D₈×C₅₃	—	4	2	no	—
30	C₂×C₈×C₅₃	—	2	1	no	—
31	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
32	C₂²⋊C₄×C₅₃	—	2	1	no	—
33	C₄²×C₅₃	C₂³	1	1	yes	characteristic
34	#11719923	—	4	2	no	—
35	#11719923	—	4	2	no	—
36	C₄⋊Dic₅₃	—	2	1	no	—
37	C₄⋊Dic₅₃	—	2	1	no	—
38	#11719917	—	4	2	no	—
39	C₂×#88322	—	2	1	no	—
40	C₂×#88322	—	2	1	no	—
41	C₂×#88322	D₄	1	1	yes	characteristic
42	C₂×#88322	D₄	1	1	yes	characteristic
43	C₂³×C₅₃	—	2	1	no	—
44	C₂³×C₅₃	—	4	1	no	—
45	D₄×C₅₃	—	2	1	no	—
46	D₄×C₅₃	—	2	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	—	2	1	no	—
53	C₂×C₄×C₅₃	—	4	1	no	—
54	C₂×C₄×C₅₃	—	4	1	no	—
55	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
58	C₂×Dic₅₃	—	4	1	no	—
59	Dic₁₀₆	—	4	1	no	—
60	Dic₁₀₆	—	4	1	no	—
61	C₈×C₅₃	—	4	1	no	—
62	#88322	—	2	1	no	—
63	#88322	—	4	1	no	—
64	#88322	—	4	1	no	—
65	#88322	D₈	1	2	yes	complemented
66	C₂²×C₅₃	—	4	1	no	—
67	C₂²×C₅₃	—	4	1	no	—
68	C₂²×C₅₃	—	4	1	no	—
69	C₂²×C₅₃	—	8	1	no	—
70	C₂²×C₅₃	—	8	1	no	—
71	C₂²×C₅₃	—	8	1	no	—
72	C₂²×C₅₃	#121	1	1	yes	characteristic
73	C₄×C₅₃	—	2	1	no	—
74	C₄×C₅₃	—	4	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	—	8	1	no	—
80	#1685	—	53	1	no	maximal, Sylow
81	C₂×C₅₃	—	8	1	no	—
82	C₂×C₅₃	—	16	1	no	—
83	C₂×C₅₃	#459	1	1	yes	characteristic
84	C₂×C₅₃	#468	1	1	yes	characteristic
85	C₂×C₅₃	#471	1	1	yes	characteristic
86	#492	—	53	1	no	—
87	#474	—	53	1	no	—
88	#459	D₅₃	1	1	yes	characteristic
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	—	106	1	no	—
95	C₂×SD₁₆	—	106	1	no	—
96	C₂×D₈	—	2	1	no	—
97	C₂×M₁₆	—	106	1	no	—
98	C₄⋊Q₈	—	53	1	no	—
99	#127	C₂×D₅₃	1	1	yes	characteristic
100	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
101	#107	—	106	1	no	—
102	C₄⋊C₈	—	53	1	no	—
103	C₄⋊C₈	—	53	1	no	—
104	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
105	#102	—	2	1	no	—
106	#102	—	106	1	no	—
107	C₂²⋊C₈	—	106	1	no	—
108	C₄×C₈	—	53	1	no	—
109	C₂×Q₈	—	106	1	no	—
110	C₂×D₄	—	2	1	no	—
111	C₂×D₄	—	2	1	no	—
112	C₂×D₄	—	2	1	no	—
113	C₂×D₄	C₂²⋊D₅₃	1	1	yes	characteristic
114	C₂²×C₄	—	2	1	no	—
115	SD₁₆	—	212	2	no	—
116	SD₁₆	—	212	2	no	—
117	D₈	—	4	2	no	—
118	M₁₆	—	212	2	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	53	1	no	—
121	C₂×C₈	—	53	1	no	—
122	C₂×C₈	—	106	1	no	—
123	C₂×C₈	—	106	1	no	—
124	C₄⋊C₄	—	106	1	no	—
125	C₄⋊C₄	—	106	1	no	—
126	C₄⋊C₄	C₂²⋊D₅₃	1	1	yes	characteristic
127	C₂²⋊C₄	—	2	1	no	—
128	C₄²	C₂²×D₅₃	1	1	yes	characteristic
129	C₂³	—	2	1	no	—
130	C₂³	—	4	1	no	—
131	Q₈	—	212	1	no	—
132	Q₈	—	212	1	no	—
133	D₄	—	2	1	no	—
134	D₄	—	2	1	no	—
135	D₄	—	4	1	no	—
136	D₄	—	4	1	no	—
137	D₄	—	4	1	no	—
138	D₄	—	4	1	no	—
139	D₄	—	8	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	4	1	no	—
142	C₂×C₄	—	4	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	—	4	1	no	—
148	C₈	—	53	2	no	—
149	C₈	—	106	1	no	—
150	C₈	—	212	1	no	—
151	C₈	—	212	1	no	—
152	C₂²	—	4	1	no	—
153	C₂²	—	4	1	no	—
154	C₂²	—	4	1	no	—
155	C₂²	—	8	1	no	—
156	C₂²	—	8	1	no	—
157	C₂²	—	8	1	no	—
158	C₂²	#13133189	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	4	1	no	—
161	C₄	—	424	1	no	—
162	C₄	D₈×D₅₃	1	1	yes	characteristic
163	C₄	#13133241	1	1	yes	characteristic
164	C₄	#13133243	1	1	yes	characteristic
165	C₄	#13133247	1	1	yes	characteristic
166	C₂	—	8	1	no	—
167	C₂	—	16	1	no	—
168	C₂	#16810816	1	1	yes	characteristic
169	C₂	#16810867	1	1	yes	characteristic
170	C₂	#16810884	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810995

C₂

yes

maximal, characteristic, complemented

#16810903

C₂

yes

maximal, characteristic, complemented

#16810701

C₂

yes

maximal, characteristic, complemented

#16810375

C₂

yes

maximal, characteristic, complemented

#16810368

C₂

yes

maximal, characteristic, complemented

#16810367

C₂

yes

maximal, characteristic, complemented

C₅₃×#459

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

characteristic

#13133150

—

C₄⋊#88322

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

—

—

C₂²

yes

characteristic

C₂×D₈×C₅₃

—

#127×C₅₃

C₂²

yes

characteristic

C₄×D₄×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

#102×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂×Dic₁₀₆

—

D₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

#11719923

—

#11719923

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

—

—

—

D₄

yes

characteristic

C₂×#88322

D₄

yes

characteristic

C₂³×C₅₃

—

C₂³×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×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

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

—

—

—

D₈

yes

complemented

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#121

yes

characteristic

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

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

D₅₃

yes

—

—

—

—

C₅₃

#1685

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

106

—

C₂×SD₁₆

—

106

—

C₂×D₈

—

C₂×M₁₆

—

106

—

C₄⋊Q₈

—

#127

C₂×D₅₃

yes

characteristic

100

C₄×D₄

C₂×D₅₃

yes

characteristic

101

#107

—

106

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

104

C₄⋊C₈

C₂×D₅₃

yes

characteristic

105

#102

—

106

#102

—

106

—

107

C₂²⋊C₈

—

106

—

108

C₄×C₈

—

109

C₂×Q₈

—

106

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

C₂²⋊D₅₃

yes

characteristic

114

C₂²×C₄

—

115

SD₁₆

—

212

—

116

SD₁₆

—

212

—

117

D₈

—

118

M₁₆

—

212

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

106

—

123

C₂×C₈

—

106

—

124

C₄⋊C₄

—

106

—

125

C₄⋊C₄

—

106

—

126

C₄⋊C₄

C₂²⋊D₅₃

yes

characteristic

127

C₂²⋊C₄

—

128

C₄²

C₂²×D₅₃

yes

characteristic

129

C₂³

—

130

C₂³

—

131

Q₈

—

212

—

132

Q₈

—

212

—

133

D₄

—

134

D₄

—

135

D₄

—

136

D₄

—

137

D₄

—

138

D₄

—

139

D₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

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

—

149

C₈

—

106

—

150

C₈

—

212

—

151

C₈

—

212

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

#13133189

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

424

—

162

C₄

D₈×D₅₃

yes

characteristic

163

C₄

#13133241

yes

characteristic

164

C₄

#13133243

yes

characteristic

165

C₄

#13133247

yes

characteristic

166

C₂

—

167

C₂

—

168

C₂

#16810816

yes

characteristic

169

C₂

#16810867

yes

characteristic

170

C₂

#16810884

yes

characteristic

	subgroup	G/H
center	C₂²	#13133189
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#459	C₂
socle	C₂²×C₅₃	#121
hypercenter	#459	D₅₃
hypocenter	C₅₃	#1685
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #459

subgroup

G/H

center

C₂²

#13133189

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#459

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #459

N	H ≅ G/N	type
1. #16810995	C₂	semidirect
2. #16810903	C₂	semidirect
3. #16810701	C₂	semidirect
4. #16810375	C₂	semidirect
5. #16810368	C₂	semidirect
6. #16810367	C₂	semidirect
65. #88322	D₈	semidirect
93. C₅₃	#1685	semidirect

H ≅ G/N

type

1. #16810995

C₂

semidirect

2. #16810903

C₂

semidirect

3. #16810701

C₂

semidirect

4. #16810375

C₂

semidirect

5. #16810368

C₂

semidirect

6. #16810367

C₂

semidirect

65. #88322

D₈

semidirect

93. C₅₃

#1685

semidirect

Subgroups — #20274178

Conjugacy classes of subgroups

Complements