Group #20274526 — subgroups

Subgroups — #20274526

Order 6784.

320 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 113 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	C₄×#11719945	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133145	C₂	1	2	yes	maximal
3	C₄×#11719917	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810383	C₂	1	2	yes	maximal, complemented
5	C₅₃×#376	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11719945	C₂²	1	1	yes	characteristic
7	C₂×#11719917	C₂²	1	2	yes	—
8	C₄×Dic₁₀₆	C₄	1	2	yes	complemented
9	C₄×D₂₁₂	C₄	1	2	yes	complemented
10	C₄×C₂²⋊D₅₃	—	2	2	no	—
11	#13133176	C₂²	1	1	yes	characteristic
12	#13133174	—	2	1	no	—
13	C₄²×D₅₃	—	2	1	no	—
14	#13133145	C₄	1	8	yes	complemented
15	#13133128	—	2	1	no	—
16	C₄×#88322	—	2	1	no	—
17	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
19	#95×C₅₃	—	2	2	no	—
20	C₂²×C₄×C₅₃	—	2	2	no	—
21	C₂²×C₄×C₅₃	D₄	1	2	yes	—
22	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
23	C₂×C₂²⋊D₅₃	—	2	1	no	—
24	#11719945	—	2	2	no	—
25	#11719945	C₂×C₄	1	4	yes	—
26	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×D₅₃	—	2	1	no	—
28	C₂×C₄×D₅₃	—	2	2	no	—
29	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₅₃	—	2	2	no	—
31	C₄²×C₅₃	C₂×C₄	1	2	yes	—
32	C₄²×C₅₃	C₂×C₄	1	4	yes	complemented
33	C₄²×C₅₃	D₄	1	1	yes	characteristic
34	C₄²×C₅₃	D₄	1	1	yes	characteristic
35	C₂²⋊Dic₅₃	—	2	1	no	—
36	#11719921	—	2	2	no	—
37	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
38	#11719919	—	2	2	no	—
39	C₄×Dic₅₃	—	2	2	no	—
40	C₄×Dic₅₃	—	2	1	no	—
41	#11719917	—	2	2	no	—
42	#11719917	C₂×C₄	1	4	yes	complemented
43	C₂×#88322	—	2	2	no	—
44	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×D₅₃	—	2	1	no	—
46	C₂×C₄×C₅₃	—	2	4	no	—
47	C₂×C₄×C₅₃	—	4	2	no	—
48	C₂×C₄×C₅₃	—	4	4	no	—
49	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
50	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
51	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
53	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₅₃	#41	1	2	yes	—
55	C₂²⋊D₅₃	—	2	4	no	—
56	C₂×Dic₅₃	—	2	1	no	—
57	C₂×Dic₅₃	—	2	2	no	—
58	D₂₁₂	—	2	1	no	—
59	D₂₁₂	C₄²	1	2	yes	—
60	C₄×D₅₃	—	2	4	no	—
61	C₄×D₅₃	—	4	4	no	—
62	Dic₁₀₆	—	2	1	no	—
63	Dic₁₀₆	C₄²	1	2	yes	—
64	#88322	—	2	4	no	—
65	C₂²×C₅₃	—	2	2	no	—
66	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₅₃	C₄×D₄	1	2	yes	—
68	C₂×D₅₃	—	2	2	no	—
69	C₂×D₅₃	—	4	2	no	—
70	C₄×C₅₃	—	2	2	no	—
71	C₄×C₅₃	—	2	2	no	—
72	C₄×C₅₃	—	4	4	no	—
73	C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
74	C₄×C₅₃	#117	1	1	yes	characteristic
75	C₄×C₅₃	C₄×D₄	1	2	yes	—
76	Dic₅₃	—	2	2	no	—
77	Dic₅₃	—	4	2	no	—
78	#1759	—	53	1	no	maximal, Sylow
79	C₂×C₅₃	—	2	2	no	—
80	C₂×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₅₃	#421	1	2	yes	—
82	D₅₃	—	4	2	no	—
83	C₄×#41	—	53	1	no	—
84	C₂×#104	—	53	2	no	—
85	C₄×M₁₆	—	53	1	no	—
86	#376	D₅₃	1	1	yes	characteristic, complemented
87	#339	—	53	2	no	—
88	C₅₃	#1759	1	1	yes	characteristic, complemented, Sylow
89	C₂×#41	—	53	1	no	—
90	C₂×M₁₆	—	53	2	no	—
91	C₄×Q₈	—	53	2	no	—
92	C₄×D₄	—	53	2	no	—
93	C₄×D₄	—	106	2	no	—
94	#117	—	53	1	no	—
95	#117	—	106	1	no	—
96	C₂×C₄²	—	106	1	no	—
97	C₂×C₄²	C₂×D₅₃	1	2	yes	—
98	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
99	#104	—	53	8	no	—
100	#97	—	106	1	no	—
101	C₄×C₈	—	106	1	no	—
102	#95	—	2	2	no	—
103	#41	—	53	4	no	—
104	#41	—	106	2	no	—
105	C₂×Q₈	—	53	1	no	—
106	C₂×D₄	—	53	1	no	—
107	C₂×D₄	—	106	1	no	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	—	106	1	no	—
110	C₂²×C₄	—	106	2	no	—
111	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
112	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
113	M₁₆	—	53	4	no	—
114	M₁₆	—	106	2	no	—
115	C₂×C₈	—	106	2	no	—
116	C₄⋊C₄	—	53	2	no	—
117	C₄⋊C₄	—	106	2	no	—
118	C₂²⋊C₄	—	106	1	no	—
119	C₂²⋊C₄	—	106	2	no	—
120	C₄²	—	2	2	no	—
121	C₄²	—	106	1	no	—
122	C₄²	—	106	2	no	—
123	C₄²	C₄×D₅₃	1	2	yes	—
124	C₄²	C₄×D₅₃	1	4	yes	complemented
125	C₄²	D₂₁₂	1	1	yes	characteristic
126	C₄²	D₂₁₂	1	1	yes	characteristic
127	C₂³	—	106	1	no	—
128	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
129	Q₈	—	53	2	no	—
130	Q₈	—	106	1	no	—
131	D₄	—	53	2	no	—
132	D₄	—	106	4	no	—
133	D₄	—	106	1	no	—
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	4	2	no	—
136	C₂×C₄	—	4	4	no	—
137	C₂×C₄	—	106	1	no	—
138	C₂×C₄	—	106	4	no	—
139	C₂×C₄	—	106	2	no	—
140	C₂×C₄	—	212	4	no	—
141	C₂×C₄	#11719921	1	2	yes	—
142	C₂×C₄	#11719921	1	2	yes	—
143	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
144	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
145	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
146	C₂×C₄	#11719945	1	2	yes	—
147	C₈	—	106	4	no	—
148	C₂²	—	2	2	no	—
149	C₂²	—	106	2	no	—
150	C₂²	—	212	2	no	—
151	C₂²	C₂×#11719921	1	1	yes	characteristic
152	C₂²	C₄×C₂²⋊D₅₃	1	2	yes	—
153	C₄	—	2	2	no	—
154	C₄	—	2	2	no	—
155	C₄	—	4	4	no	—
156	C₄	—	106	2	no	—
157	C₄	—	212	2	no	—
158	C₄	C₄²×D₅₃	1	1	yes	characteristic
159	C₄	#13133174	1	1	yes	characteristic
160	C₄	C₄×D₂₁₂	1	2	yes	—
161	C₂	—	2	2	no	—
162	C₂	—	212	2	no	—
163	C₂	C₄×#11719921	1	1	yes	characteristic
164	C₂	#16810641	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#11719945

C₂

yes

maximal, characteristic

C₂×#13133145

C₂

yes

maximal

C₄×#11719917

C₂

yes

maximal, characteristic, complemented

#16810383

C₂

yes

maximal, complemented

C₅₃×#376

C₂

yes

maximal, characteristic, complemented

C₂×#11719945

C₂²

yes

characteristic

C₂×#11719917

C₂²

yes

—

C₄×Dic₁₀₆

C₄

yes

complemented

C₄×D₂₁₂

C₄

yes

complemented

C₄×C₂²⋊D₅₃

—

#13133176

C₂²

yes

characteristic

#13133174

—

C₄²×D₅₃

—

#13133145

C₄

yes

complemented

#13133128

—

C₄×#88322

—

C₂×C₄²×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#95×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₂²⋊D₅₃

—

#11719945

—

#11719945

C₂×C₄

yes

—

C₂×D₂₁₂

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×Dic₁₀₆

C₂×C₄

yes

characteristic

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

complemented

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₂²⋊Dic₅₃

—

#11719921

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

#11719917

—

#11719917

C₂×C₄

yes

complemented

C₂×#88322

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂²×D₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

C₄²

yes

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

Dic₁₀₆

C₄²

yes

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

C₄×D₄

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄²

yes

characteristic

C₄×C₅₃

#117

yes

characteristic

C₄×C₅₃

C₄×D₄

yes

—

Dic₅₃

—

Dic₅₃

—

#1759

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₄×C₂²⋊C₄

yes

characteristic

C₂×C₅₃

#421

yes

—

D₅₃

—

C₄×#41

—

C₂×#104

—

C₄×M₁₆

—

#376

D₅₃

yes

characteristic, complemented

#339

—

C₅₃

#1759

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

106

—

#117

—

#117

—

106

—

C₂×C₄²

—

106

—

C₂×C₄²

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#104

—

100

#97

—

106

—

101

C₄×C₈

—

106

—

102

#95

—

103

#41

—

104

#41

—

106

—

105

C₂×Q₈

—

106

C₂×D₄

—

107

C₂×D₄

—

106

—

108

C₂²×C₄

—

109

C₂²×C₄

—

106

—

110

C₂²×C₄

—

106

—

111

C₂²×C₄

C₂²⋊D₅₃

yes

—

112

C₂²×C₄

C₂²×D₅₃

yes

characteristic

113

M₁₆

—

114

M₁₆

—

106

—

115

C₂×C₈

—

106

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

106

—

118

C₂²⋊C₄

—

106

—

119

C₂²⋊C₄

—

106

—

120

C₄²

—

121

C₄²

—

106

—

122

C₄²

—

106

—

123

C₄²

C₄×D₅₃

yes

—

124

C₄²

C₄×D₅₃

yes

complemented

125

C₄²

D₂₁₂

yes

characteristic

126

C₄²

D₂₁₂

yes

characteristic

127

C₂³

—

106

—

128

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

129

Q₈

—

130

Q₈

—

106

—

131

D₄

—

132

D₄

—

106

—

133

D₄

—

106

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

106

—

138

C₂×C₄

—

106

—

139

C₂×C₄

—

106

—

140

C₂×C₄

—

212

—

141

C₂×C₄

#11719921

yes

—

142

C₂×C₄

#11719921

yes

—

143

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

144

C₂×C₄

C₂×C₄×D₅₃

yes

—

145

C₂×C₄

C₂×D₂₁₂

yes

characteristic

146

C₂×C₄

#11719945

yes

—

147

C₈

—

106

—

148

C₂²

—

149

C₂²

—

106

—

150

C₂²

—

212

—

151

C₂²

C₂×#11719921

yes

characteristic

152

C₂²

C₄×C₂²⋊D₅₃

yes

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

106

—

157

C₄

—

212

—

158

C₄

C₄²×D₅₃

yes

characteristic

159

C₄

#13133174

yes

characteristic

160

C₄

C₄×D₂₁₂

yes

—

161

C₂

—

162

C₂

—

212

—

163

C₂

C₄×#11719921

yes

characteristic

164

C₂

#16810641

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₂₁₂
commutator	C₄×C₅₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#376	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	#376	D₅₃
hypocenter	C₅₃	#1759
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

subgroup

G/H

center

C₂×C₄

C₂×D₂₁₂

commutator

C₄×C₅₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#376

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

N	H ≅ G/N	type
3. C₄×#11719917	C₂	semidirect
4. #16810383	C₂	semidirect
5. C₅₃×#376	C₂	semidirect
8. C₄×Dic₁₀₆	C₄	semidirect
9. C₄×D₂₁₂	C₄	semidirect
14. #13133145	C₄	semidirect
32. C₄²×C₅₃	C₂×C₄	semidirect
42. #11719917	C₂×C₄	semidirect
86. #376	D₅₃	semidirect
88. C₅₃	#1759	semidirect
124. C₄²	C₄×D₅₃	semidirect

H ≅ G/N

type

3. C₄×#11719917

C₂

semidirect

4. #16810383

C₂

semidirect

5. C₅₃×#376

C₂

semidirect

8. C₄×Dic₁₀₆

C₄

semidirect

9. C₄×D₂₁₂

C₄

semidirect

14. #13133145

C₄

semidirect

32. C₄²×C₅₃

C₂×C₄

semidirect

42. #11719917

C₂×C₄

semidirect

86. #376

D₅₃

semidirect

88. C₅₃

#1759

semidirect

124. C₄²

C₄×D₅₃

semidirect

Subgroups — #20274526

Conjugacy classes of subgroups

Complements