Group #20273262 — subgroups

Subgroups — #20273262

Order 6784.

252 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 77 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	#16810680	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810491	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810357	C₂	1	4	yes	maximal, complemented
4	C₅₃×#532	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#88322	—	2	1	no	—
6	C₂²⋊#88322	C₂²	1	2	yes	—
7	#13133128	—	2	1	no	—
8	#13133124	—	2	1	no	—
9	#13133124	—	2	1	no	—
10	#13133124	C₂²	1	2	yes	—
11	C₂×#41×C₅₃	—	2	1	no	—
12	C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic, complemented
13	#127×C₅₃	C₄	1	1	yes	characteristic, complemented
14	#124×C₅₃	—	2	1	no	—
15	#121×C₅₃	—	2	1	no	—
16	#121×C₅₃	C₂²	1	2	yes	—
17	C₄×D₄×C₅₃	—	2	1	no	—
18	C₄×D₄×C₅₃	C₄	1	2	yes	complemented
19	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
20	#41×C₅₃	—	4	4	no	—
21	C₂×Q₈×C₅₃	—	2	1	no	—
22	C₂×D₄×C₅₃	—	2	2	no	—
23	C₂×D₄×C₅₃	—	2	1	no	—
24	C₂×D₄×C₅₃	—	2	1	no	—
25	C₂×D₄×C₅₃	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₅₃	—	2	2	no	—
27	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
30	C₂²×Dic₅₃	—	2	2	no	—
31	C₄⋊C₄×C₅₃	—	2	1	no	—
32	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
33	C₂²⋊C₄×C₅₃	—	2	2	no	—
34	C₂²⋊C₄×C₅₃	—	2	2	no	—
35	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₅₃	D₄	1	1	yes	characteristic
38	C₄²×C₅₃	D₄	1	1	yes	characteristic
39	C₂×#88322	—	2	2	no	—
40	C₂³×C₅₃	—	2	2	no	—
41	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
42	Q₈×C₅₃	—	4	2	no	—
43	D₄×C₅₃	—	4	2	no	—
44	D₄×C₅₃	—	4	2	no	—
45	D₄×C₅₃	—	4	2	no	—
46	D₄×C₅₃	—	4	4	no	—
47	C₂×C₄×C₅₃	—	2	2	no	—
48	C₂×C₄×C₅₃	—	2	2	no	—
49	C₂×C₄×C₅₃	—	4	1	no	—
50	C₂×C₄×C₅₃	—	4	1	no	—
51	C₂×C₄×C₅₃	—	4	4	no	—
52	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂×Dic₅₃	—	4	2	no	—
59	C₂×Dic₅₃	—	4	2	no	—
60	C₂×Dic₅₃	—	4	2	no	—
61	#88322	—	4	2	no	—
62	C₂²×C₅₃	—	2	1	no	—
63	C₂²×C₅₃	—	2	1	no	—
64	C₂²×C₅₃	—	4	2	no	—
65	C₂²×C₅₃	—	4	2	no	—
66	C₂²×C₅₃	—	4	2	no	—
67	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
69	C₄×C₅₃	—	2	1	no	—
70	C₄×C₅₃	—	2	1	no	—
71	C₄×C₅₃	—	2	1	no	—
72	C₄×C₅₃	—	2	1	no	—
73	C₄×C₅₃	—	2	1	no	—
74	C₄×C₅₃	—	2	1	no	—
75	C₄×C₅₃	—	4	2	no	—
76	Dic₅₃	—	8	2	no	—
77	#1514	—	53	1	no	maximal, Sylow
78	C₂×C₅₃	—	2	2	no	—
79	C₂×C₅₃	—	8	2	no	—
80	C₂×C₅₃	#410	1	1	yes	characteristic
81	C₂×C₅₃	#418	1	1	yes	characteristic
82	C₂×C₅₃	#421	1	1	yes	characteristic
83	#532	D₅₃	1	1	yes	characteristic
84	#432	—	53	1	no	—
85	#383	—	53	1	no	—
86	#327	—	53	4	no	—
87	C₅₃	#1514	1	1	yes	characteristic, complemented, Sylow
88	C₂×#41	—	2	1	no	—
89	C₄⋊Q₈	Dic₅₃	1	1	yes	characteristic, complemented
90	#127	Dic₅₃	1	1	yes	characteristic, complemented
91	#124	—	2	1	no	—
92	#121	—	2	1	no	—
93	#121	C₂×D₅₃	1	2	yes	—
94	C₄×D₄	—	2	1	no	—
95	C₄×D₄	Dic₅₃	1	2	yes	complemented
96	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
97	C₄⋊C₈	—	106	1	no	—
98	C₂²⋊C₈	—	53	2	no	—
99	#97	—	106	1	no	—
100	#95	—	53	2	no	—
101	#95	—	106	1	no	—
102	#95	—	106	1	no	—
103	#41	—	4	4	no	—
104	C₂×Q₈	—	2	1	no	—
105	C₂×D₄	—	2	2	no	—
106	C₂×D₄	—	2	1	no	—
107	C₂×D₄	—	2	1	no	—
108	C₂×D₄	C₂×Dic₅₃	1	2	yes	—
109	C₂²×C₄	—	2	2	no	—
110	C₂²×C₄	—	106	2	no	—
111	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
112	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
113	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
114	C₂×C₈	—	106	2	no	—
115	C₄⋊C₄	—	2	1	no	—
116	C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
117	C₂²⋊C₄	—	2	2	no	—
118	C₂²⋊C₄	—	2	2	no	—
119	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
120	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
121	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
122	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
123	C₂³	—	2	2	no	—
124	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
125	Q₈	—	4	2	no	—
126	D₄	—	4	2	no	—
127	D₄	—	4	2	no	—
128	D₄	—	4	2	no	—
129	D₄	—	4	4	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	4	no	—
135	C₂×C₄	—	212	2	no	—
136	C₂×C₄	—	212	2	no	—
137	C₂×C₄	—	212	2	no	—
138	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
139	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
140	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
141	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
142	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
143	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
144	C₈	—	212	2	no	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	—	4	2	no	—
148	C₂²	—	4	2	no	—
149	C₂²	—	4	2	no	—
150	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
151	C₂²	#13133243	1	2	yes	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	—	4	2	no	—
159	C₄	—	424	2	no	—
160	C₂	—	2	2	no	—
161	C₂	—	8	2	no	—
162	C₂	#16810597	1	1	yes	characteristic
163	C₂	#16810628	1	1	yes	characteristic
164	C₂	#16810640	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810680

C₂

yes

maximal, characteristic, complemented

#16810491

C₂

yes

maximal, characteristic, complemented

#16810357

C₂

yes

maximal, complemented

C₅₃×#532

C₂

yes

maximal, characteristic

C₄⋊#88322

—

C₂²⋊#88322

C₂²

yes

—

—

—

—

C₂²

yes

—

C₂×#41×C₅₃

—

C₄⋊Q₈×C₅₃

C₄

yes

characteristic, complemented

#127×C₅₃

C₄

yes

characteristic, complemented

#124×C₅₃

—

#121×C₅₃

—

#121×C₅₃

C₂²

yes

—

C₄×D₄×C₅₃

—

C₄×D₄×C₅₃

C₄

yes

complemented

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#41×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₂×#88322

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

AGL(1,ℤ/8)

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

#1514

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

yes

—

—

—

C₅₃

#1514

yes

characteristic, complemented, Sylow

C₂×#41

—

C₄⋊Q₈

Dic₅₃

yes

characteristic, complemented

#127

Dic₅₃

yes

characteristic, complemented

#124

—

#121

—

#121

C₂×D₅₃

yes

—

C₄×D₄

—

C₄×D₄

Dic₅₃

yes

complemented

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

#97

—

106

—

100

#95

—

101

#95

—

106

—

102

#95

—

106

—

103

#41

—

104

C₂×Q₈

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

C₂×Dic₅₃

yes

—

109

C₂²×C₄

—

110

C₂²×C₄

—

106

—

111

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

112

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

113

C₂²×C₄

C₂²×D₅₃

yes

characteristic

114

C₂×C₈

—

106

—

115

C₄⋊C₄

—

116

C₄⋊C₄

C₂×Dic₅₃

yes

—

117

C₂²⋊C₄

—

118

C₂²⋊C₄

—

119

C₄²

C₂×Dic₅₃

yes

characteristic

120

C₄²

C₂×Dic₅₃

yes

characteristic

121

C₄²

C₂²⋊D₅₃

yes

characteristic

122

C₄²

C₂²⋊D₅₃

yes

characteristic

123

C₂³

—

124

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

125

Q₈

—

126

D₄

—

127

D₄

—

128

D₄

—

129

D₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

212

—

136

C₂×C₄

—

212

—

137

C₂×C₄

—

212

—

138

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

139

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

140

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

141

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

142

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

143

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

144

C₈

—

212

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

151

C₂²

#13133243

yes

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

424

—

160

C₂

—

161

C₂

—

162

C₂

#16810597

yes

characteristic

163

C₂

#16810628

yes

characteristic

164

C₂

#16810640

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#532	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	#532	D₅₃
hypocenter	C₅₃	#1514
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#532

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. #16810680	C₂	semidirect
2. #16810491	C₂	semidirect
3. #16810357	C₂	semidirect
12. C₄⋊Q₈×C₅₃	C₄	semidirect
13. #127×C₅₃	C₄	semidirect
18. C₄×D₄×C₅₃	C₄	semidirect
87. C₅₃	#1514	semidirect
89. C₄⋊Q₈	Dic₅₃	semidirect
90. #127	Dic₅₃	semidirect
95. C₄×D₄	Dic₅₃	semidirect

H ≅ G/N

type

1. #16810680

C₂

semidirect

2. #16810491

C₂

semidirect

3. #16810357

C₂

semidirect

12. C₄⋊Q₈×C₅₃

C₄

semidirect

13. #127×C₅₃

C₄

semidirect

18. C₄×D₄×C₅₃

C₄

semidirect

87. C₅₃

#1514

semidirect

89. C₄⋊Q₈

Dic₅₃

semidirect

90. #127

Dic₅₃

semidirect

95. C₄×D₄

Dic₅₃

semidirect

Subgroups — #20273262

Conjugacy classes of subgroups

Complements