Group #20274900 — subgroups

Subgroups — #20274900

Order 6784.

324 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 121 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	C₂²×#11719917	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#88322	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊#88322	C₂	1	1	yes	maximal, characteristic
4	#16810378	C₂	1	2	yes	maximal, complemented
5	C₄×C₂²⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	—	2	4	no	—
7	C₂×#11719917	—	2	2	no	—
8	C₂²×#88322	C₂²	1	1	yes	characteristic
9	C₂²×#88322	C₂²	1	1	yes	characteristic
10	C₂²×#88322	C₂²	1	2	yes	—
11	C₄⋊#88322	—	2	4	no	—
12	C₂²⋊#88322	—	2	2	no	—
13	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊C₄×C₅₃	C₄	1	2	yes	—
15	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
16	#95×C₅₃	C₄	1	1	yes	characteristic
17	#95×C₅₃	C₄	1	1	yes	characteristic
18	C₂⁴×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₅₃	—	2	2	no	—
20	C₂²×C₄×C₅₃	—	2	2	no	—
21	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₅₃	D₄	1	2	yes	—
25	C₂²×C₄×C₅₃	Q₈	1	2	yes	—
26	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
27	C₂²⋊C₄×C₅₃	—	2	4	no	—
28	C₄²×C₅₃	—	2	2	no	—
29	C₄²×C₅₃	—	2	2	no	—
30	#11719917	—	2	8	no	—
31	C₂×#88322	—	2	2	no	—
32	C₂×#88322	—	2	2	no	—
33	C₂×#88322	—	2	1	no	—
34	C₂×#88322	—	2	1	no	—
35	C₂×#88322	—	2	2	no	—
36	C₂×#88322	—	2	2	no	—
37	C₂×#88322	D₄	1	4	yes	—
38	C₂³×C₅₃	—	2	2	no	—
39	C₂³×C₅₃	—	2	2	no	—
40	C₂³×C₅₃	—	2	1	no	—
41	C₂³×C₅₃	—	2	1	no	—
42	C₂³×C₅₃	C₄⋊C₄	1	2	yes	—
43	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	—	2	2	no	—
45	C₂×C₄×C₅₃	—	2	2	no	—
46	C₂×C₄×C₅₃	—	2	2	no	—
47	C₂×C₄×C₅₃	—	2	2	no	—
48	C₂×C₄×C₅₃	—	2	4	no	—
49	C₂×C₄×C₅₃	—	2	2	no	—
50	C₂×C₄×C₅₃	—	2	2	no	—
51	C₂×C₄×C₅₃	—	2	2	no	—
52	C₂×C₄×C₅₃	—	2	2	no	—
53	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
54	C₂×C₄×C₅₃	M₁₆	1	4	yes	complemented
55	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₄×C₅₃	#41	1	2	yes	—
59	#88322	—	2	4	no	—
60	#88322	—	4	1	no	—
61	#88322	—	4	1	no	—
62	C₂²×C₅₃	—	2	2	no	—
63	C₂²×C₅₃	—	2	2	no	—
64	C₂²×C₅₃	—	2	2	no	—
65	C₂²×C₅₃	—	4	2	no	—
66	C₂²×C₅₃	—	4	2	no	—
67	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₅₃	C₄×D₄	1	2	yes	—
69	C₂²×C₅₃	C₂×M₁₆	1	2	yes	—
70	C₂²×C₅₃	#131	1	1	yes	characteristic
71	C₂²×C₅₃	#131	1	1	yes	characteristic
72	C₄×C₅₃	—	2	2	no	—
73	C₄×C₅₃	—	2	4	no	—
74	C₄×C₅₃	—	4	2	no	—
75	C₄×C₅₃	#120	1	1	yes	characteristic
76	C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
77	C₄×C₅₃	#123	1	1	yes	characteristic
78	#1851	—	53	1	no	maximal, Sylow
79	C₂×C₅₃	—	4	2	no	—
80	C₂×C₅₃	#385	1	1	yes	characteristic
81	C₂×C₅₃	#423	1	1	yes	characteristic
82	C₂×C₅₃	#424	1	1	yes	characteristic
83	C₂×C₅₃	#435	1	2	yes	—
84	C₂×C₅₃	#436	1	2	yes	—
85	C₂²×M₁₆	—	53	1	no	—
86	C₂×C₄⋊C₈	—	53	2	no	—
87	C₂×C₂²⋊C₈	—	53	1	no	—
88	C₄×C₂²⋊C₄	D₅₃	1	1	yes	characteristic
89	#336	—	53	2	no	—
90	C₅₃	#1851	1	1	yes	characteristic, complemented, Sylow
91	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
92	C₂×M₁₆	—	106	4	no	—
93	C₂×M₁₆	—	106	2	no	—
94	C₂²×C₈	—	53	2	no	—
95	C₂²×C₈	—	53	1	no	—
96	C₂²×C₈	—	53	1	no	—
97	C₂×C₂²⋊C₄	Dic₅₃	1	2	yes	—
98	C₂×C₄²	C₂×D₅₃	1	2	yes	—
99	C₄⋊C₈	—	106	4	no	—
100	C₂²⋊C₈	—	106	2	no	—
101	#95	Dic₅₃	1	1	yes	characteristic
102	#95	Dic₅₃	1	1	yes	characteristic
103	C₂⁴	C₂×Dic₅₃	1	1	yes	characteristic
104	C₂²×C₄	—	2	2	no	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	Dic₁₀₆	1	2	yes	—
107	C₂²×C₄	D₂₁₂	1	2	yes	—
108	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
109	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
110	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
111	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
112	M₁₆	—	106	8	no	—
113	C₂×C₈	—	53	4	no	—
114	C₂×C₈	—	106	2	no	—
115	C₂×C₈	—	106	2	no	—
116	C₂×C₈	—	106	1	no	—
117	C₂×C₈	—	106	1	no	—
118	C₂×C₈	—	106	2	no	—
119	C₂×C₈	—	106	2	no	—
120	C₂²⋊C₄	—	2	4	no	—
121	C₄²	—	2	2	no	—
122	C₄²	—	2	2	no	—
123	C₂³	—	2	2	no	—
124	C₂³	—	2	2	no	—
125	C₂³	—	2	1	no	—
126	C₂³	—	2	1	no	—
127	C₂³	C₄⋊Dic₅₃	1	2	yes	—
128	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	#11719917	1	4	yes	complemented
139	C₂×C₄	C₄⋊Dic₅₃	1	2	yes	—
140	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
141	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
142	C₂×C₄	D₄×D₅₃	1	2	yes	—
143	C₂×C₄	#11719947	1	2	yes	—
144	C₈	—	106	4	no	—
145	C₈	—	212	1	no	—
146	C₈	—	212	1	no	—
147	C₂²	—	2	2	no	—
148	C₂²	—	2	2	no	—
149	C₂²	—	2	2	no	—
150	C₂²	—	4	2	no	—
151	C₂²	—	4	2	no	—
152	C₂²	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
153	C₂²	D₄×Dic₅₃	1	2	yes	—
154	C₂²	C₂×#11719917	1	2	yes	—
155	C₂²	#13133224	1	1	yes	characteristic
156	C₂²	#13133224	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	2	4	no	—
159	C₄	—	4	2	no	—
160	C₄	#13133187	1	1	yes	characteristic
161	C₄	#13133194	1	2	yes	—
162	C₄	#13133201	1	1	yes	characteristic
163	C₂	—	4	2	no	—
164	C₂	#16810498	1	1	yes	characteristic
165	C₂	#16810647	1	1	yes	characteristic
166	C₂	#16810650	1	1	yes	characteristic
167	C₂	#16810695	1	2	yes	—
168	C₂	#16810701	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11719917

C₂

yes

maximal, characteristic

C₂×C₄⋊#88322

C₂

yes

maximal, complemented

C₂×C₂²⋊#88322

C₂

yes

maximal, characteristic

#16810378

C₂

yes

maximal, complemented

C₄×C₂²⋊C₄×C₅₃

C₂

yes

maximal, characteristic

C₂×#11719917

—

C₂×#11719917

—

C₂²×#88322

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

—

C₄⋊#88322

—

C₂²⋊#88322

—

C₂³×C₄×C₅₃

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₅₃

C₄

yes

—

C₂×C₄²×C₅₃

C₂²

yes

—

#95×C₅₃

C₄

yes

characteristic

#95×C₅₃

C₄

yes

characteristic

C₂⁴×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

—

—

—

—

—

—

—

D₄

yes

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₄⋊C₄

yes

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

M₁₆

yes

complemented

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

#88322

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₃

C₄×D₄

yes

—

C₂²×C₅₃

C₂×M₁₆

yes

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#120

yes

characteristic

C₄×C₅₃

C₂²⋊Q₈

yes

—

C₄×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#436

yes

—

C₂²×M₁₆

—

C₂×C₄⋊C₈

—

C₂×C₂²⋊C₈

—

C₄×C₂²⋊C₄

D₅₃

yes

characteristic

#336

—

C₅₃

#1851

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₅₃

yes

characteristic

C₂×M₁₆

—

106

—

C₂×M₁₆

—

106

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₂²⋊C₄

Dic₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

—

C₄⋊C₈

—

106

—

100

C₂²⋊C₈

—

106

—

101

#95

Dic₅₃

yes

characteristic

102

#95

Dic₅₃

yes

characteristic

103

C₂⁴

C₂×Dic₅₃

yes

characteristic

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

Dic₁₀₆

yes

—

107

C₂²×C₄

D₂₁₂

yes

—

108

C₂²×C₄

C₂×Dic₅₃

yes

—

109

C₂²×C₄

C₂×Dic₅₃

yes

—

110

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

111

C₂²×C₄

C₂²×D₅₃

yes

characteristic

112

M₁₆

—

106

—

113

C₂×C₈

—

114

C₂×C₈

—

106

—

115

C₂×C₈

—

106

—

116

C₂×C₈

—

106

—

117

C₂×C₈

—

106

—

118

C₂×C₈

—

106

—

119

C₂×C₈

—

106

—

120

C₂²⋊C₄

—

121

C₄²

—

122

C₄²

—

123

C₂³

—

124

C₂³

—

125

C₂³

—

126

C₂³

—

127

C₂³

C₄⋊Dic₅₃

yes

—

128

C₂³

C₂²×Dic₅₃

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

#11719917

yes

complemented

139

C₂×C₄

C₄⋊Dic₅₃

yes

—

140

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

141

C₂×C₄

C₂×D₂₁₂

yes

characteristic

142

C₂×C₄

D₄×D₅₃

yes

—

143

C₂×C₄

#11719947

yes

—

144

C₈

—

106

—

145

C₈

—

212

—

146

C₈

—

212

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

C₂×C₄⋊Dic₅₃

yes

characteristic

153

C₂²

D₄×Dic₅₃

yes

—

154

C₂²

C₂×#11719917

yes

—

155

C₂²

#13133224

yes

characteristic

156

C₂²

#13133224

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

#13133187

yes

characteristic

161

C₄

#13133194

yes

—

162

C₄

#13133201

yes

characteristic

163

C₂

—

164

C₂

#16810498

yes

characteristic

165

C₂

#16810647

yes

characteristic

166

C₂

#16810650

yes

characteristic

167

C₂

#16810695

yes

—

168

C₂

#16810701

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₅₃
commutator, socle	C₂³×C₅₃	C₂²×C₄
Fitting	C₄×C₂²⋊C₄×C₅₃	C₂
hypercenter	C₄×C₂²⋊C₄	D₅₃
hypocenter	C₅₃	#1851
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₄

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₅₃

commutator, socle

C₂³×C₅₃

C₂²×C₄

Fitting

C₄×C₂²⋊C₄×C₅₃

C₂

hypercenter

C₄×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₄

N	H ≅ G/N	type
2. C₂×C₄⋊#88322	C₂	semidirect
4. #16810378	C₂	semidirect
54. C₂×C₄×C₅₃	M₁₆	semidirect
90. C₅₃	#1851	semidirect
138. C₂×C₄	#11719917	semidirect

H ≅ G/N

type

2. C₂×C₄⋊#88322

C₂

semidirect

4. #16810378

C₂

semidirect

54. C₂×C₄×C₅₃

M₁₆

semidirect

90. C₅₃

#1851

semidirect

138. C₂×C₄

#11719917

semidirect

Subgroups — #20274900

Conjugacy classes of subgroups

Complements