Group #27295289 — subgroups

Subgroups — #27295289

Order 13568.

274 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 79 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	#20276527	C₂	1	2	yes	maximal
2	#20272750	C₂	1	4	yes	maximal, complemented
3	C₅₃×#3139	C₂	1	1	yes	maximal, characteristic
4	#16810444	—	2	2	no	—
5	#16810405	—	2	2	no	—
6	#16810401	C₂²	1	4	yes	—
7	C₅₃×#533	C₄	1	1	yes	characteristic
8	C₂×C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
9	C₅₃×#491	C₄	1	2	yes	—
10	C₅₃×#491	C₂²	1	2	yes	—
11	C₅₃×#487	—	2	2	no	—
12	C₅₃×#419	—	2	2	no	—
13	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	#13133157	—	2	4	no	—
15	C₂²×Q₈×C₅₃	—	2	1	no	—
16	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
17	C₄⋊Q₈×C₅₃	—	2	1	no	—
18	C₄⋊Q₈×C₅₃	—	2	1	no	—
19	C₄⋊Q₈×C₅₃	—	4	1	no	—
20	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
21	#125×C₅₃	C₂×C₄	1	2	yes	—
22	C₂²⋊Q₈×C₅₃	—	4	1	no	—
23	C₄×Q₈×C₅₃	—	4	1	no	—
24	#117×C₅₃	—	2	1	no	—
25	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
26	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
27	#103×C₅₃	—	2	4	no	—
28	#103×C₅₃	—	2	4	no	—
29	#97×C₅₃	C₂×C₄	1	1	yes	characteristic
30	#97×C₅₃	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₅₃	D₄	1	2	yes	—
32	C₂×Q₈×C₅₃	—	2	2	no	—
33	C₂×Q₈×C₅₃	—	4	1	no	—
34	C₂×Q₈×C₅₃	—	4	4	no	—
35	C₂×Q₈×C₅₃	—	4	1	no	—
36	C₂²×C₄×C₅₃	—	2	1	no	—
37	C₂²×C₄×C₅₃	—	4	1	no	—
38	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
39	M₁₆×C₅₃	—	4	2	no	—
40	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
41	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
42	C₄⋊C₄×C₅₃	—	2	2	no	—
43	C₄⋊C₄×C₅₃	—	4	2	no	—
44	C₄⋊C₄×C₅₃	—	4	1	no	—
45	C₄⋊C₄×C₅₃	—	4	2	no	—
46	C₄⋊C₄×C₅₃	—	4	2	no	—
47	C₂²⋊C₄×C₅₃	—	4	1	no	—
48	C₄²×C₅₃	—	2	1	no	—
49	C₄²×C₅₃	—	4	1	no	—
50	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	#11719908	—	4	4	no	—
53	C₂³×C₅₃	#100	1	1	yes	characteristic
54	Q₈×C₅₃	—	4	4	no	—
55	Q₈×C₅₃	—	8	2	no	—
56	Q₈×C₅₃	—	8	2	no	—
57	C₂×C₄×C₅₃	—	2	1	no	—
58	C₂×C₄×C₅₃	—	2	2	no	—
59	C₂×C₄×C₅₃	—	2	1	no	—
60	C₂×C₄×C₅₃	—	4	2	no	—
61	C₂×C₄×C₅₃	—	4	2	no	—
62	C₂×C₄×C₅₃	—	4	2	no	—
63	C₂×C₄×C₅₃	—	8	2	no	—
64	C₂×C₄×C₅₃	#100	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₈×C₅₃	—	2	2	no	—
67	C₈×C₅₃	—	2	2	no	—
68	#11916	—	53	1	no	maximal, Sylow
69	C₂²×C₅₃	—	2	1	no	—
70	C₂²×C₅₃	—	4	1	no	—
71	C₂²×C₅₃	C₂×#100	1	1	yes	characteristic
72	C₄×C₅₃	—	2	2	no	—
73	C₄×C₅₃	—	2	1	no	—
74	C₄×C₅₃	—	4	1	no	—
75	C₄×C₅₃	—	8	2	no	—
76	C₄×C₅₃	—	8	2	no	—
77	C₄×C₅₃	#421	1	2	yes	—
78	#3139	D₅₃	1	1	yes	characteristic
79	#2165	—	53	2	no	—
80	#1356	—	53	4	no	—
81	C₂×C₅₃	—	2	1	no	—
82	C₂×C₅₃	—	4	1	no	—
83	C₂×C₅₃	#1533	1	1	yes	characteristic
84	#533	Dic₅₃	1	1	yes	characteristic
85	C₂×C₄⋊Q₈	Dic₅₃	1	1	yes	characteristic
86	#491	Dic₅₃	1	2	yes	—
87	#491	C₂×D₅₃	1	2	yes	—
88	#487	—	2	2	no	—
89	#419	—	2	2	no	—
90	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
91	#364	—	106	2	no	—
92	#349	—	106	2	no	—
93	#347	—	53	4	no	—
94	C₅₃	#11916	1	1	yes	characteristic, complemented, Sylow
95	C₂²×Q₈	—	2	1	no	—
96	C₂×M₁₆	C₂²⋊D₅₃	1	2	yes	—
97	C₄⋊Q₈	—	2	1	no	—
98	C₄⋊Q₈	—	2	1	no	—
99	C₄⋊Q₈	—	4	1	no	—
100	C₄⋊Q₈	C₂×Dic₅₃	1	2	yes	—
101	#125	C₂×Dic₅₃	1	2	yes	—
102	C₂²⋊Q₈	—	4	1	no	—
103	C₄×Q₈	—	4	1	no	—
104	#117	—	2	1	no	—
105	C₂×C₄⋊C₄	—	4	1	no	—
106	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
107	M₃₂	—	106	4	no	—
108	#103	—	2	4	no	—
109	#103	—	2	4	no	—
110	#97	C₂×Dic₅₃	1	1	yes	characteristic
111	#97	C₂²×D₅₃	1	1	yes	characteristic
112	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
113	C₂×Q₈	—	2	2	no	—
114	C₂×Q₈	—	4	1	no	—
115	C₂×Q₈	—	4	4	no	—
116	C₂×Q₈	—	4	1	no	—
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	4	1	no	—
119	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
120	M₁₆	—	4	2	no	—
121	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
122	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
123	C₄⋊C₄	—	2	2	no	—
124	C₄⋊C₄	—	4	2	no	—
125	C₄⋊C₄	—	4	2	no	—
126	C₄⋊C₄	—	4	2	no	—
127	C₄⋊C₄	—	4	1	no	—
128	C₂²⋊C₄	—	4	1	no	—
129	C₄²	—	2	1	no	—
130	C₄²	—	4	1	no	—
131	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
132	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
133	C₁₆	—	212	4	no	—
134	C₂³	#13133134	1	1	yes	characteristic
135	Q₈	—	4	4	no	—
136	Q₈	—	8	2	no	—
137	Q₈	—	8	2	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	4	2	no	—
144	C₂×C₄	—	8	2	no	—
145	C₂×C₄	#13133134	1	1	yes	characteristic
146	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
147	C₈	—	2	2	no	—
148	C₈	—	2	2	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	4	1	no	—
151	C₂²	C₂×#13133134	1	1	yes	characteristic
152	C₄	—	2	2	no	—
153	C₄	—	2	1	no	—
154	C₄	—	4	1	no	—
155	C₄	—	8	2	no	—
156	C₄	—	8	2	no	—
157	C₄	#16810640	1	2	yes	—
158	C₂	—	2	1	no	—
159	C₂	—	4	1	no	—
160	C₂	#20273339	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276527

C₂

yes

maximal

#20272750

C₂

yes

maximal, complemented

C₅₃×#3139

C₂

yes

maximal, characteristic

#16810444

—

#16810405

—

#16810401

C₂²

yes

—

C₅₃×#533

C₄

yes

characteristic

C₂×C₄⋊Q₈×C₅₃

C₄

yes

characteristic

C₅₃×#491

C₄

yes

—

C₅₃×#491

C₂²

yes

—

C₅₃×#487

—

C₅₃×#419

—

C₄×M₁₆×C₅₃

C₂²

yes

characteristic

#13133157

—

C₂²×Q₈×C₅₃

—

C₂×M₁₆×C₅₃

D₄

yes

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

C₂×C₄

yes

—

#125×C₅₃

C₂×C₄

yes

—

C₂²⋊Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

#117×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

#103×C₅₃

—

#103×C₅₃

—

#97×C₅₃

C₂×C₄

yes

characteristic

#97×C₅₃

C₂³

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

M₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂²⋊C₄

yes

characteristic

C₄²×C₅₃

C₂²×C₄

yes

characteristic

#11719908

—

C₂³×C₅₃

#100

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×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

#100

yes

characteristic

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

—

#11916

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×#100

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#421

yes

—

#3139

D₅₃

yes

characteristic

#2165

—

#1356

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#1533

yes

characteristic

#533

Dic₅₃

yes

characteristic

C₂×C₄⋊Q₈

Dic₅₃

yes

characteristic

#491

Dic₅₃

yes

—

#491

C₂×D₅₃

yes

—

#487

—

#419

—

C₄×M₁₆

C₂×D₅₃

yes

characteristic

#364

—

106

—

#349

—

106

—

#347

—

C₅₃

#11916

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×M₁₆

C₂²⋊D₅₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

100

C₄⋊Q₈

C₂×Dic₅₃

yes

—

101

#125

C₂×Dic₅₃

yes

—

102

C₂²⋊Q₈

—

103

C₄×Q₈

—

104

#117

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄²

C₂×Dic₅₃

yes

characteristic

107

M₃₂

—

106

—

108

#103

—

109

#103

—

110

#97

C₂×Dic₅₃

yes

characteristic

111

#97

C₂²×D₅₃

yes

characteristic

112

C₄×C₈

C₂²⋊D₅₃

yes

—

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂×Q₈

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

C₂²⋊Dic₅₃

yes

characteristic

120

M₁₆

—

121

C₂×C₈

C₂²⋊Dic₅₃

yes

—

122

C₂×C₈

C₂×C₂²⋊D₅₃

yes

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₂²⋊C₄

—

129

C₄²

—

130

C₄²

—

131

C₄²

C₂²⋊Dic₅₃

yes

characteristic

132

C₄²

C₂²×Dic₅₃

yes

characteristic

133

C₁₆

—

212

—

134

C₂³

#13133134

yes

characteristic

135

Q₈

—

136

Q₈

—

137

Q₈

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

#13133134

yes

characteristic

146

C₂×C₄

C₂×C₂²⋊Dic₅₃

yes

characteristic

147

C₈

—

148

C₈

—

149

C₂²

—

150

C₂²

—

151

C₂²

C₂×#13133134

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

#16810640

yes

—

158

C₂

—

159

C₂

—

160

C₂

#20273339

yes

characteristic

	subgroup	G/H
center	C₂	#20273339
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	#97	C₂²×D₅₃
Fitting	C₅₃×#3139	C₂
socle	C₂×C₅₃	#1533
prime radical	C₂³×C₅₃	#100
cosocle	#97×C₅₃	C₂³
hypercenter	#3139	D₅₃
hypocenter	C₅₃	#11916
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ #3139

subgroup

G/H

center

C₂

#20273339

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

#97

C₂²×D₅₃

Fitting

C₅₃×#3139

C₂

C₂×C₅₃

C₂³×C₅₃

#97×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ #3139

N	H ≅ G/N	type
2. #20272750	C₂	semidirect
94. C₅₃	#11916	semidirect

H ≅ G/N

type

2. #20272750

C₂

semidirect

94. C₅₃

#11916

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20276527	2
#20272750	2
C₅₃×#3139	2
#11916	53

subgroup

index

#20276527

#20272750

C₅₃×#3139

#11916

Subgroups — #27295289

Conjugacy classes of subgroups

Complements

Maximal subgroups