Group #20274278 — subgroups

Subgroups — #20274278

Order 6784.

176 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 51 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	#16810971	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810818	C₂	1	1	yes	maximal, characteristic
3	#16810701	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810373	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810369	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810366	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#461	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11719922	—	2	1	no	—
9	C₂×#11719917	—	2	1	no	—
10	#13133214	C₂²	1	1	yes	characteristic
11	C₄⋊#88322	C₂²	1	1	yes	characteristic
12	C₄⋊#88322	C₂²	1	1	yes	characteristic
13	#13133137	—	2	1	no	—
14	#13133137	—	2	1	no	—
15	#13133137	—	2	1	no	—
16	C₂²⋊#88322	—	2	1	no	—
17	C₄×#88322	C₂²	1	1	yes	characteristic
18	C₂×SD₁₆×C₅₃	—	2	1	no	—
19	C₄⋊Q₈×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	#103×C₅₃	—	2	1	no	—
23	C₂×Q₈×C₅₃	—	2	1	no	—
24	C₂×D₄×C₅₃	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	—	2	1	no	—
26	C₂×D₂₁₂	—	2	1	no	—
27	C₂×D₂₁₂	—	2	1	no	—
28	C₂×D₂₁₂	—	2	1	no	—
29	SD₁₆×C₅₃	—	4	2	no	—
30	C₂×C₈×C₅₃	—	2	1	no	—
31	C₄⋊C₄×C₅₃	—	2	1	no	—
32	C₄⋊C₄×C₅₃	—	2	1	no	—
33	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
34	C₂²⋊C₄×C₅₃	—	2	1	no	—
35	C₄²×C₅₃	C₂³	1	1	yes	characteristic
36	#11719922	—	4	2	no	—
37	#11719917	—	4	2	no	—
38	C₂×#88322	—	2	1	no	—
39	C₂×#88322	—	2	1	no	—
40	C₂×#88322	D₄	1	1	yes	characteristic
41	C₂×#88322	D₄	1	1	yes	characteristic
42	C₂³×C₅₃	—	2	1	no	—
43	C₂²×D₅₃	—	4	1	no	—
44	Q₈×C₅₃	—	4	1	no	—
45	Q₈×C₅₃	—	4	1	no	—
46	D₄×C₅₃	—	2	1	no	—
47	D₄×C₅₃	—	2	1	no	—
48	C₂×C₄×C₅₃	—	2	1	no	—
49	C₂×C₄×C₅₃	—	4	1	no	—
50	C₂×C₄×C₅₃	—	4	1	no	—
51	C₂×C₄×C₅₃	—	4	1	no	—
52	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
55	D₂₁₂	—	4	1	no	—
56	D₂₁₂	—	4	1	no	—
57	D₂₁₂	—	4	1	no	—
58	D₂₁₂	—	4	1	no	—
59	D₂₁₂	—	8	1	no	—
60	C₈×C₅₃	—	4	1	no	—
61	#88322	—	2	1	no	—
62	#88322	—	2	1	no	—
63	#88322	—	4	1	no	—
64	#88322	—	4	1	no	—
65	C₂²×C₅₃	—	4	1	no	—
66	C₂²×C₅₃	—	4	1	no	—
67	C₂²×C₅₃	—	4	1	no	—
68	C₂²×C₅₃	#121	1	1	yes	characteristic
69	C₂×D₅₃	—	8	1	no	—
70	C₂×D₅₃	—	8	1	no	—
71	C₂×D₅₃	—	8	1	no	—
72	C₄×C₅₃	—	2	1	no	—
73	C₄×C₅₃	—	4	1	no	—
74	C₄×C₅₃	—	8	1	no	—
75	C₄×C₅₃	#135	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	AGL(1,ℤ/8)	1	1	yes	characteristic
79	#1701	—	53	1	no	maximal, Sylow
80	C₂×C₅₃	—	8	1	no	—
81	C₂×C₅₃	#463	1	1	yes	characteristic
82	C₂×C₅₃	#469	1	1	yes	characteristic
83	C₂×C₅₃	#471	1	1	yes	characteristic
84	D₅₃	—	16	1	no	—
85	#486	—	53	1	no	—
86	#461	D₅₃	1	1	yes	characteristic, complemented
87	#459	—	53	1	no	—
88	#436	—	53	1	no	—
89	#334	—	53	1	no	—
90	#332	—	53	1	no	—
91	#331	—	53	1	no	—
92	C₅₃	#1701	1	1	yes	characteristic, complemented, Sylow
93	C₂×SD₁₆	—	2	1	no	—
94	C₂×D₈	—	106	1	no	—
95	C₂×M₁₆	—	106	1	no	—
96	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
97	#127	—	53	1	no	—
98	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
99	C₄⋊C₈	—	53	1	no	—
100	C₄⋊C₈	—	53	1	no	—
101	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
102	#103	—	2	1	no	—
103	#102	—	106	1	no	—
104	#102	—	106	1	no	—
105	#102	—	106	1	no	—
106	C₂²⋊C₈	—	106	1	no	—
107	C₄×C₈	—	53	1	no	—
108	C₂×Q₈	—	2	1	no	—
109	C₂×D₄	—	106	1	no	—
110	C₂×D₄	—	106	1	no	—
111	C₂×D₄	—	106	1	no	—
112	C₂×D₄	C₂²⋊D₅₃	1	1	yes	characteristic
113	C₂²×C₄	—	2	1	no	—
114	SD₁₆	—	4	2	no	—
115	D₈	—	212	2	no	—
116	M₁₆	—	212	2	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	53	1	no	—
119	C₂×C₈	—	53	1	no	—
120	C₂×C₈	—	106	1	no	—
121	C₂×C₈	—	106	1	no	—
122	C₄⋊C₄	—	2	1	no	—
123	C₄⋊C₄	—	2	1	no	—
124	C₄⋊C₄	C₂²⋊D₅₃	1	1	yes	characteristic
125	C₂²⋊C₄	—	2	1	no	—
126	C₄²	C₂²×D₅₃	1	1	yes	characteristic
127	C₂³	—	2	1	no	—
128	C₂³	—	212	1	no	—
129	Q₈	—	4	1	no	—
130	Q₈	—	4	1	no	—
131	D₄	—	2	1	no	—
132	D₄	—	2	1	no	—
133	D₄	—	424	1	no	—
134	D₄	—	212	1	no	—
135	D₄	—	212	1	no	—
136	D₄	—	212	1	no	—
137	D₄	—	212	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	4	1	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	4	1	no	—
142	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
143	C₂×C₄	#11719947	1	1	yes	characteristic
144	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
145	C₈	—	4	1	no	—
146	C₈	—	106	1	no	—
147	C₈	—	106	1	no	—
148	C₈	—	212	1	no	—
149	C₈	—	212	1	no	—
150	C₂²	—	4	1	no	—
151	C₂²	—	4	1	no	—
152	C₂²	—	4	1	no	—
153	C₂²	—	424	1	no	—
154	C₂²	—	424	1	no	—
155	C₂²	—	424	1	no	—
156	C₂²	#13133189	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	4	1	no	—
159	C₄	—	8	1	no	—
160	C₄	#13133238	1	1	yes	characteristic
161	C₄	#13133242	1	1	yes	characteristic
162	C₄	#13133243	1	1	yes	characteristic
163	C₄	#13133244	1	1	yes	characteristic
164	C₂	—	8	1	no	—
165	C₂	—	848	1	no	—
166	C₂	#16810840	1	1	yes	characteristic
167	C₂	#16810874	1	1	yes	characteristic
168	C₂	#16810888	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810971

C₂

yes

maximal, characteristic, complemented

#16810818

C₂

yes

maximal, characteristic

#16810701

C₂

yes

maximal, characteristic, complemented

#16810373

C₂

yes

maximal, characteristic, complemented

#16810369

C₂

yes

maximal, characteristic, complemented

#16810366

C₂

yes

maximal, characteristic, complemented

C₅₃×#461

C₂

yes

maximal, characteristic, complemented

C₂×#11719922

—

C₂×#11719917

—

#13133214

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₂×SD₁₆×C₅₃

—

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₄×D₄×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

#103×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

SD₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₂²⋊C₄×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

—

C₂×C₄×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

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₈×C₅₃

—

—

—

—

—

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

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

#486

—

#461

D₅₃

yes

characteristic, complemented

—

—

—

—

—

C₅₃

#1701

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂×D₈

—

106

—

C₂×M₁₆

—

106

—

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

#127

—

C₄×D₄

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

100

C₄⋊C₈

—

101

C₄⋊C₈

C₂×D₅₃

yes

characteristic

102

#103

—

103

#102

—

106

—

104

#102

—

106

—

105

#102

—

106

—

106

C₂²⋊C₈

—

106

—

107

C₄×C₈

—

108

C₂×Q₈

—

109

C₂×D₄

—

106

—

110

C₂×D₄

—

106

—

111

C₂×D₄

—

106

—

112

C₂×D₄

C₂²⋊D₅₃

yes

characteristic

113

C₂²×C₄

—

114

SD₁₆

—

115

D₈

—

212

—

116

M₁₆

—

212

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

106

—

121

C₂×C₈

—

106

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

C₂²⋊D₅₃

yes

characteristic

125

C₂²⋊C₄

—

126

C₄²

C₂²×D₅₃

yes

characteristic

127

C₂³

—

128

C₂³

—

212

—

129

Q₈

—

130

Q₈

—

131

D₄

—

132

D₄

—

133

D₄

—

424

—

134

D₄

—

212

—

135

D₄

—

212

—

136

D₄

—

212

—

137

D₄

—

212

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

D₄×D₅₃

yes

characteristic

143

C₂×C₄

#11719947

yes

characteristic

144

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

145

C₈

—

146

C₈

—

106

—

147

C₈

—

106

—

148

C₈

—

212

—

149

C₈

—

212

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

424

—

154

C₂²

—

424

—

155

C₂²

—

424

—

156

C₂²

#13133189

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

#13133238

yes

characteristic

161

C₄

#13133242

yes

characteristic

162

C₄

#13133243

yes

characteristic

163

C₄

#13133244

yes

characteristic

164

C₂

—

165

C₂

—

848

—

166

C₂

#16810840

yes

characteristic

167

C₂

#16810874

yes

characteristic

168

C₂

#16810888

yes

characteristic

	subgroup	G/H
center	C₂²	#13133189
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#461	C₂
socle	C₂²×C₅₃	#121
hypercenter	#461	D₅₃
hypocenter	C₅₃	#1701
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #461

subgroup

G/H

center

C₂²

#13133189

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#461

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #461

N	H ≅ G/N	type
1. #16810971	C₂	semidirect
3. #16810701	C₂	semidirect
4. #16810373	C₂	semidirect
5. #16810369	C₂	semidirect
6. #16810366	C₂	semidirect
7. C₅₃×#461	C₂	semidirect
86. #461	D₅₃	semidirect
92. C₅₃	#1701	semidirect

H ≅ G/N

type

1. #16810971

C₂

semidirect

3. #16810701

C₂

semidirect

4. #16810373

C₂

semidirect

5. #16810369

C₂

semidirect

6. #16810366

C₂

semidirect

7. C₅₃×#461

C₂

semidirect

86. #461

D₅₃

semidirect

92. C₅₃

#1701

semidirect

Subgroups — #20274278

Conjugacy classes of subgroups

Complements