Group #20285374 — subgroups

Subgroups — #20285374

Order 6784.

338 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 125 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	#16811235	C₂	1	2	yes	maximal
2	#16811025	C₂	1	1	yes	maximal, characteristic, complemented
3	#16811003	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810992	C₂	1	2	yes	maximal, complemented
5	#16810881	C₂	1	2	yes	maximal
6	#16810864	C₂	1	2	yes	maximal
7	#16810731	C₂	1	2	yes	maximal, complemented
8	C₄×Dic₂₁₂	C₂	1	2	yes	maximal, complemented
9	C₅₃×#405	C₂	1	1	yes	maximal, characteristic
10	C₂×Dic₂₁₂	C₂²	1	2	yes	—
11	C₂×Dic₂₁₂	C₂²	1	2	yes	—
12	#13133217	C₂²	1	2	yes	—
13	#13133210	—	2	4	no	—
14	#13133209	C₂²	1	2	yes	—
15	#13133207	—	2	4	no	—
16	#13133196	C₂²	1	4	yes	—
17	C₄×Dic₁₀₆	C₂²	1	4	yes	—
18	#13133151	C₂²	1	2	yes	—
19	#13133149	C₂²	1	2	yes	—
20	#13133142	C₂²	1	4	yes	—
21	#13133142	C₂²	1	4	yes	—
22	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
23	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	#117×C₅₃	C₂²	1	1	yes	characteristic
25	#97×C₅₃	C₂²	1	1	yes	characteristic
26	#97×C₅₃	C₂²	1	1	yes	characteristic
27	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
30	C₂×Dic₁₀₆	C₂³	1	4	yes	—
31	M₁₆×C₅₃	—	2	2	no	—
32	C₂×C₈×C₅₃	—	2	2	no	—
33	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
36	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
37	C₄⋊C₄×C₅₃	D₄	1	2	yes	—
38	C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
39	C₄²×C₅₃	C₂³	1	1	yes	characteristic
40	C₄²×C₅₃	C₂³	1	1	yes	characteristic
41	C₂²⋊Dic₅₃	—	2	4	no	—
42	C₄⋊Dic₅₃	—	2	4	no	—
43	C₄⋊Dic₅₃	C₂³	1	4	yes	—
44	#11719919	—	2	4	no	—
45	#11719919	—	2	4	no	—
46	#11719919	—	2	4	no	—
47	C₄×Dic₅₃	—	2	4	no	—
48	Dic₂₁₂	—	2	4	no	—
49	Dic₂₁₂	—	2	4	no	—
50	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₅₃	—	2	1	no	—
52	C₂×C₄×C₅₃	—	2	1	no	—
53	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
57	C₂×Dic₅₃	—	2	4	no	—
58	C₂×Dic₅₃	—	2	4	no	—
59	Dic₁₀₆	—	2	4	no	—
60	Dic₁₀₆	—	2	4	no	—
61	C₈×C₅₃	—	2	1	no	—
62	C₈×C₅₃	—	2	1	no	—
63	C₈×C₅₃	#41	1	2	yes	—
64	C₈×C₅₃	#41	1	2	yes	—
65	C₂²×C₅₃	—	2	1	no	—
66	C₂²×C₅₃	—	2	2	no	—
67	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
68	C₄×C₅₃	—	2	1	no	—
69	C₄×C₅₃	—	2	2	no	—
70	C₄×C₅₃	—	2	2	no	—
71	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
72	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
73	Dic₅₃	—	4	4	no	—
74	Dic₅₃	—	4	4	no	—
75	#3177	—	53	1	no	maximal, Sylow
76	C₂×C₅₃	—	4	1	no	—
77	C₂×C₅₃	#532	1	1	yes	characteristic
78	C₂×C₅₃	#578	1	2	yes	—
79	#541	—	53	2	no	—
80	#499	—	53	1	no	—
81	#494	—	53	1	no	—
82	#491	—	53	2	no	—
83	#470	—	53	2	no	—
84	#467	—	53	2	no	—
85	#441	—	53	2	no	—
86	C₄×Q₁₆	—	53	2	no	—
87	#405	D₅₃	1	1	yes	characteristic
88	C₅₃	#3177	1	1	yes	characteristic, complemented, Sylow
89	C₂×Q₁₆	—	53	2	no	—
90	C₂×Q₁₆	—	53	2	no	—
91	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
92	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
93	C₄⋊Q₈	—	53	2	no	—
94	#126	—	106	4	no	—
95	#125	—	53	2	no	—
96	#125	—	106	4	no	—
97	C₂²⋊Q₈	—	53	4	no	—
98	C₄×Q₈	—	53	4	no	—
99	#117	C₂×D₅₃	1	1	yes	characteristic
100	#107	—	53	2	no	—
101	#106	—	53	2	no	—
102	#103	—	53	4	no	—
103	#103	—	53	4	no	—
104	#97	C₂×D₅₃	1	1	yes	characteristic
105	#97	C₂×D₅₃	1	1	yes	characteristic
106	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
107	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
108	C₂×Q₈	—	53	4	no	—
109	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
110	Q₁₆	—	106	4	no	—
111	Q₁₆	—	106	4	no	—
112	M₁₆	—	2	2	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
115	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
116	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
117	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
118	C₄⋊C₄	—	53	4	no	—
119	C₄⋊C₄	—	106	4	no	—
120	C₄⋊C₄	—	106	4	no	—
121	C₄⋊C₄	—	106	4	no	—
122	C₄⋊C₄	—	106	4	no	—
123	C₄⋊C₄	D₂₁₂	1	2	yes	—
124	C₂²⋊C₄	—	106	4	no	—
125	C₂²⋊C₄	D₂₁₂	1	2	yes	—
126	C₄²	—	106	4	no	—
127	C₄²	C₂²×D₅₃	1	1	yes	characteristic
128	C₄²	C₂²×D₅₃	1	1	yes	characteristic
129	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
130	Q₈	—	106	4	no	—
131	Q₈	—	106	4	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	106	4	no	—
135	C₂×C₄	—	106	4	no	—
136	C₂×C₄	C₂×D₂₁₂	1	2	yes	—
137	C₂×C₄	C₂×D₂₁₂	1	2	yes	—
138	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
139	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
140	C₈	—	2	1	no	—
141	C₈	—	2	1	no	—
142	C₈	#11719945	1	2	yes	—
143	C₈	#11719945	1	2	yes	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	2	no	—
146	C₂²	C₂²×D₂₁₂	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	2	2	no	—
149	C₄	—	2	2	no	—
150	C₄	—	212	4	no	—
151	C₄	—	212	4	no	—
152	C₄	C₂×#11719945	1	1	yes	characteristic
153	C₄	C₂×#11719945	1	1	yes	characteristic
154	C₂	—	4	1	no	—
155	C₂	#16811173	1	1	yes	characteristic
156	C₂	#16811440	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16811235

C₂

yes

maximal

#16811025

C₂

yes

maximal, characteristic, complemented

#16811003

C₂

yes

maximal, characteristic, complemented

#16810992

C₂

yes

maximal, complemented

#16810881

C₂

yes

maximal

#16810864

C₂

yes

maximal

#16810731

C₂

yes

maximal, complemented

C₄×Dic₂₁₂

C₂

yes

maximal, complemented

C₅₃×#405

C₂

yes

maximal, characteristic

C₂×Dic₂₁₂

C₂²

yes

—

C₂×Dic₂₁₂

C₂²

yes

—

#13133217

C₂²

yes

—

#13133210

—

#13133209

C₂²

yes

—

#13133207

—

#13133196

C₂²

yes

—

C₄×Dic₁₀₆

C₂²

yes

—

#13133151

C₂²

yes

—

#13133149

C₂²

yes

—

#13133142

C₂²

yes

—

#13133142

C₂²

yes

—

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

#117×C₅₃

C₂²

yes

characteristic

#97×C₅₃

C₂²

yes

characteristic

#97×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

C₂³

yes

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

D₄

yes

—

C₂²⋊C₄×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

#11719919

—

#11719919

—

C₄×Dic₅₃

—

Dic₂₁₂

—

Dic₂₁₂

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#41

yes

—

C₈×C₅₃

#41

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×#41

yes

characteristic

C₄×C₅₃

C₂×#41

yes

characteristic

Dic₅₃

—

Dic₅₃

—

#3177

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#532

yes

characteristic

C₂×C₅₃

#578

yes

—

—

—

—

—

—

—

—

C₄×Q₁₆

—

#405

D₅₃

yes

characteristic

C₅₃

#3177

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

characteristic

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₄⋊Q₈

—

#126

—

106

—

#125

—

#125

—

106

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₅₃

yes

characteristic

100

#107

—

101

#106

—

102

#103

—

103

#103

—

104

#97

C₂×D₅₃

yes

characteristic

105

#97

C₂×D₅₃

yes

characteristic

106

C₄×C₈

C₂×D₅₃

yes

characteristic

107

C₄×C₈

C₂×D₅₃

yes

characteristic

108

C₂×Q₈

—

109

C₂²×C₄

C₂²×D₅₃

yes

characteristic

110

Q₁₆

—

106

—

111

Q₁₆

—

106

—

112

M₁₆

—

113

C₂×C₈

—

114

C₂×C₈

C₂²×D₅₃

yes

characteristic

115

C₂×C₈

C₂²×D₅₃

yes

characteristic

116

C₂×C₈

C₂²×D₅₃

yes

characteristic

117

C₂×C₈

C₂²×D₅₃

yes

characteristic

118

C₄⋊C₄

—

119

C₄⋊C₄

—

106

—

120

C₄⋊C₄

—

106

—

121

C₄⋊C₄

—

106

—

122

C₄⋊C₄

—

106

—

123

C₄⋊C₄

D₂₁₂

yes

—

124

C₂²⋊C₄

—

106

—

125

C₂²⋊C₄

D₂₁₂

yes

—

126

C₄²

—

106

—

127

C₄²

C₂²×D₅₃

yes

characteristic

128

C₄²

C₂²×D₅₃

yes

characteristic

129

C₂³

C₂×D₂₁₂

yes

characteristic

130

Q₈

—

106

—

131

Q₈

—

106

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

106

—

135

C₂×C₄

—

106

—

136

C₂×C₄

C₂×D₂₁₂

yes

—

137

C₂×C₄

C₂×D₂₁₂

yes

—

138

C₂×C₄

C₂×D₂₁₂

yes

characteristic

139

C₂×C₄

C₂³×D₅₃

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

#11719945

yes

—

143

C₈

#11719945

yes

—

144

C₂²

—

145

C₂²

—

146

C₂²

C₂²×D₂₁₂

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

212

—

151

C₄

—

212

—

152

C₄

C₂×#11719945

yes

characteristic

153

C₄

C₂×#11719945

yes

characteristic

154

C₂

—

155

C₂

#16811173

yes

characteristic

156

C₂

#16811440

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#405	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#405	D₅₃
hypocenter	C₅₃	#3177
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

subgroup

G/H

center

C₂²

C₂²×D₂₁₂

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#405

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ #117 ◁ #405

N	H ≅ G/N	type
2. #16811025	C₂	semidirect
3. #16811003	C₂	semidirect
4. #16810992	C₂	semidirect
7. #16810731	C₂	semidirect
8. C₄×Dic₂₁₂	C₂	semidirect
88. C₅₃	#3177	semidirect

H ≅ G/N

type

2. #16811025

C₂

semidirect

3. #16811003

C₂

semidirect

4. #16810992

C₂

semidirect

7. #16810731

C₂

semidirect

8. C₄×Dic₂₁₂

C₂

semidirect

88. C₅₃

#3177

semidirect

Subgroups — #20285374

Conjugacy classes of subgroups

Complements