Group #20274197 — subgroups

Subgroups — #20274197

Order 6784.

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

normal subgroups only

This group has 184 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	#16810999	C₂	1	1	yes	maximal, characteristic
2	#16810823	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810754	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810373	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810367	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810366	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#460	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11719924	—	2	1	no	—
9	C₂×#11719922	—	2	1	no	—
10	C₂×#11719922	—	2	1	no	—
11	#13133214	C₂²	1	1	yes	characteristic
12	C₄⋊#88322	—	2	1	no	—
13	C₄⋊#88322	C₂²	1	1	yes	characteristic
14	C₄⋊#88322	C₂²	1	1	yes	characteristic
15	#13133137	—	2	1	no	—
16	#13133128	—	2	1	no	—
17	C₄×#88322	C₂²	1	1	yes	characteristic
18	C₂×SD₁₆×C₅₃	—	2	1	no	—
19	#127×C₅₃	C₂²	1	1	yes	characteristic
20	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
22	#102×C₅₃	—	2	1	no	—
23	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
24	C₂×D₄×C₅₃	—	2	1	no	—
25	C₂×D₄×C₅₃	—	2	1	no	—
26	C₂×D₄×C₅₃	—	2	1	no	—
27	C₂×D₂₁₂	—	2	1	no	—
28	C₂×D₂₁₂	—	2	1	no	—
29	C₂×D₂₁₂	—	2	1	no	—
30	SD₁₆×C₅₃	—	4	2	no	—
31	C₂×C₈×C₅₃	—	2	1	no	—
32	C₄⋊C₄×C₅₃	—	2	1	no	—
33	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
34	C₄²×C₅₃	—	2	1	no	—
35	C₄²×C₅₃	C₂³	1	1	yes	characteristic
36	#11719924	—	4	2	no	—
37	#11719922	—	4	2	no	—
38	#11719922	—	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	—	4	1	no	—
44	C₂²×D₅₃	—	4	1	no	—
45	Q₈×C₅₃	—	2	1	no	—
46	Q₈×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	—	2	1	no	—
54	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
57	D₂₁₂	—	4	1	no	—
58	D₂₁₂	—	4	1	no	—
59	D₂₁₂	—	4	1	no	—
60	D₂₁₂	—	4	1	no	—
61	D₂₁₂	—	8	1	no	—
62	C₈×C₅₃	—	4	1	no	—
63	#88322	—	2	1	no	—
64	#88322	—	4	1	no	—
65	#88322	—	4	1	no	—
66	#88322	SD₁₆	1	2	yes	complemented
67	C₂²×C₅₃	—	8	1	no	—
68	C₂²×C₅₃	—	8	1	no	—
69	C₂²×C₅₃	—	8	1	no	—
70	C₂²×C₅₃	#121	1	1	yes	characteristic
71	C₂×D₅₃	—	8	1	no	—
72	C₂×D₅₃	—	8	1	no	—
73	C₂×D₅₃	—	8	1	no	—
74	C₄×C₅₃	—	2	1	no	—
75	C₄×C₅₃	—	4	1	no	—
76	C₄×C₅₃	—	4	1	no	—
77	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
78	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
80	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
81	#1688	—	53	1	no	maximal, Sylow
82	C₂×C₅₃	—	16	1	no	—
83	C₂×C₅₃	#460	1	1	yes	characteristic
84	C₂×C₅₃	#469	1	1	yes	characteristic
85	C₂×C₅₃	#472	1	1	yes	characteristic
86	D₅₃	—	16	1	no	—
87	#493	—	53	1	no	—
88	#460	—	53	1	no	—
89	#460	D₅₃	1	1	yes	characteristic, complemented
90	#446	—	53	1	no	—
91	#334	—	53	1	no	—
92	#331	—	53	1	no	—
93	#331	—	53	1	no	—
94	C₅₃	#1688	1	1	yes	characteristic, complemented, Sylow
95	C₂×SD₁₆	—	2	1	no	—
96	C₂×SD₁₆	—	106	1	no	—
97	C₂×D₈	—	106	1	no	—
98	C₂×D₈	—	106	1	no	—
99	#127	—	53	1	no	—
100	#127	C₂×D₅₃	1	1	yes	characteristic
101	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
102	C₄⋊C₈	—	53	1	no	—
103	C₄⋊C₈	—	53	1	no	—
104	C₄⋊C₈	—	106	1	no	—
105	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
106	#102	—	2	1	no	—
107	#102	—	106	1	no	—
108	#97	—	106	1	no	—
109	C₄×C₈	—	53	1	no	—
110	C₂×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
111	C₂×D₄	—	2	1	no	—
112	C₂×D₄	—	2	1	no	—
113	C₂×D₄	—	2	1	no	—
114	C₂×D₄	—	106	1	no	—
115	C₂×D₄	—	106	1	no	—
116	C₂×D₄	—	106	1	no	—
117	SD₁₆	—	4	2	no	—
118	SD₁₆	—	212	2	no	—
119	D₈	—	212	2	no	—
120	D₈	—	212	2	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	53	1	no	—
123	C₂×C₈	—	53	1	no	—
124	C₂×C₈	—	106	1	no	—
125	C₂×C₈	—	106	1	no	—
126	C₄⋊C₄	—	2	1	no	—
127	C₄⋊C₄	C₂²⋊D₅₃	1	1	yes	characteristic
128	C₄²	—	2	1	no	—
129	C₄²	C₂²×D₅₃	1	1	yes	characteristic
130	C₂³	—	4	1	no	—
131	C₂³	—	212	1	no	—
132	Q₈	—	2	1	no	—
133	Q₈	—	2	1	no	—
134	D₄	—	4	1	no	—
135	D₄	—	4	1	no	—
136	D₄	—	4	1	no	—
137	D₄	—	4	1	no	—
138	D₄	—	8	1	no	—
139	D₄	—	424	1	no	—
140	D₄	—	212	1	no	—
141	D₄	—	212	1	no	—
142	D₄	—	212	1	no	—
143	D₄	—	212	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
147	C₂×C₄	#11719947	1	1	yes	characteristic
148	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
149	C₈	—	4	1	no	—
150	C₈	—	53	2	no	—
151	C₈	—	106	1	no	—
152	C₈	—	212	1	no	—
153	C₈	—	212	1	no	—
154	C₂²	—	8	1	no	—
155	C₂²	—	8	1	no	—
156	C₂²	—	8	1	no	—
157	C₂²	—	424	1	no	—
158	C₂²	—	424	1	no	—
159	C₂²	—	424	1	no	—
160	C₂²	#13133189	1	1	yes	characteristic
161	C₄	—	2	1	no	—
162	C₄	—	4	1	no	—
163	C₄	—	4	1	no	—
164	C₄	SD₁₆×D₅₃	1	1	yes	characteristic
165	C₄	#13133242	1	1	yes	characteristic
166	C₄	#13133243	1	1	yes	characteristic
167	C₄	#13133244	1	1	yes	characteristic
168	C₂	—	16	1	no	—
169	C₂	—	848	1	no	—
170	C₂	#16810821	1	1	yes	characteristic
171	C₂	#16810874	1	1	yes	characteristic
172	C₂	#16810891	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810999

C₂

yes

maximal, characteristic

#16810823

C₂

yes

maximal, characteristic, complemented

#16810754

C₂

yes

maximal, characteristic, complemented

#16810373

C₂

yes

maximal, characteristic, complemented

#16810367

C₂

yes

maximal, characteristic, complemented

#16810366

C₂

yes

maximal, characteristic, complemented

C₅₃×#460

C₂

yes

maximal, characteristic, complemented

—

—

—

C₂²

yes

characteristic

C₄⋊#88322

—

C₄⋊#88322

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

—

—

C₂²

yes

characteristic

C₂×SD₁₆×C₅₃

—

#127×C₅₃

C₂²

yes

characteristic

C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

#102×C₅₃

—

C₂×Q₈×C₅₃

D₄

yes

characteristic

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

SD₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

—

C₄²×C₅₃

C₂³

yes

—

—

—

—

—

D₄

yes

characteristic

C₂×#88322

D₄

yes

characteristic

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×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₈×C₅₃

—

—

—

—

SD₁₆

yes

complemented

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#121

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×SD₁₆

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

#493

—

#460

—

#460

D₅₃

yes

characteristic, complemented

—

—

—

—

C₅₃

#1688

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂×SD₁₆

—

106

—

C₂×D₈

—

106

—

C₂×D₈

—

106

—

#127

—

100

#127

C₂×D₅₃

yes

characteristic

101

C₄×Q₈

C₂×D₅₃

yes

characteristic

102

C₄⋊C₈

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

106

—

105

C₄⋊C₈

C₂×D₅₃

yes

characteristic

106

#102

—

107

#102

—

106

—

108

#97

—

106

—

109

C₄×C₈

—

110

C₂×Q₈

C₂²⋊D₅₃

yes

characteristic

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂×D₄

—

106

—

115

C₂×D₄

—

106

—

116

C₂×D₄

—

106

—

117

SD₁₆

—

118

SD₁₆

—

212

—

119

D₈

—

212

—

120

D₈

—

212

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

106

—

125

C₂×C₈

—

106

—

126

C₄⋊C₄

—

127

C₄⋊C₄

C₂²⋊D₅₃

yes

characteristic

128

C₄²

—

129

C₄²

C₂²×D₅₃

yes

characteristic

130

C₂³

—

131

C₂³

—

212

—

132

Q₈

—

133

Q₈

—

134

D₄

—

135

D₄

—

136

D₄

—

137

D₄

—

138

D₄

—

139

D₄

—

424

—

140

D₄

—

212

—

141

D₄

—

212

—

142

D₄

—

212

—

143

D₄

—

212

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

D₄×D₅₃

yes

characteristic

147

C₂×C₄

#11719947

yes

characteristic

148

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

149

C₈

—

150

C₈

—

151

C₈

—

106

—

152

C₈

—

212

—

153

C₈

—

212

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

424

—

158

C₂²

—

424

—

159

C₂²

—

424

—

160

C₂²

#13133189

yes

characteristic

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

SD₁₆×D₅₃

yes

characteristic

165

C₄

#13133242

yes

characteristic

166

C₄

#13133243

yes

characteristic

167

C₄

#13133244

yes

characteristic

168

C₂

—

169

C₂

—

848

—

170

C₂

#16810821

yes

characteristic

171

C₂

#16810874

yes

characteristic

172

C₂

#16810891

yes

characteristic

	subgroup	G/H
center	C₂²	#13133189
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#460	C₂
socle	C₂²×C₅₃	#121
hypercenter	#460	D₅₃
hypocenter	C₅₃	#1688
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #460

subgroup

G/H

center

C₂²

#13133189

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#460

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #460

N	H ≅ G/N	type
2. #16810823	C₂	semidirect
3. #16810754	C₂	semidirect
4. #16810373	C₂	semidirect
5. #16810367	C₂	semidirect
6. #16810366	C₂	semidirect
7. C₅₃×#460	C₂	semidirect
66. #88322	SD₁₆	semidirect
89. #460	D₅₃	semidirect
94. C₅₃	#1688	semidirect

H ≅ G/N

type

2. #16810823

C₂

semidirect

3. #16810754

C₂

semidirect

4. #16810373

C₂

semidirect

5. #16810367

C₂

semidirect

6. #16810366

C₂

semidirect

7. C₅₃×#460

C₂

semidirect

66. #88322

SD₁₆

semidirect

89. #460

D₅₃

semidirect

94. C₅₃

#1688

semidirect

Subgroups — #20274197

Conjugacy classes of subgroups

Complements