Group #27284059 — subgroups

Subgroups — #27284059

Order 13568.

406 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 151 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	#20285229	C₂	1	1	yes	maximal, characteristic
2	#20273169	C₂	1	4	yes	maximal
3	C₂×#16810344	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1848	C₂	1	1	yes	maximal, characteristic
5	#16811180	C₂²	1	2	yes	—
6	#16811028	C₄	1	2	yes	—
7	#16811025	—	2	2	no	—
8	#16811021	C₄	1	2	yes	—
9	#16810661	—	2	2	no	—
10	C₂×C₄×#88322	C₂²	1	2	yes	—
11	#16810353	—	2	8	no	—
12	#16810344	—	2	2	no	—
13	C₂×#108×C₅₃	—	2	2	no	—
14	C₅₃×#423	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×#88322	—	2	2	no	—
17	#13133217	C₂×C₄	1	4	yes	—
18	#13133209	—	2	2	no	—
19	#13133196	—	2	4	no	—
20	C₄×Dic₁₀₆	—	2	4	no	—
21	#13133176	—	2	2	no	—
22	#13133151	—	2	4	no	—
23	#13133149	—	2	4	no	—
24	C₄×#88322	—	2	2	no	—
25	C₄×#88322	D₄	1	4	yes	—
26	C₂×M₁₆×C₅₃	—	2	2	no	—
27	C₂²×C₈×C₅₃	D₄	1	2	yes	—
28	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
29	#108×C₅₃	—	4	4	no	—
30	C₄⋊C₈×C₅₃	—	2	4	no	—
31	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₅₃	D₄	1	2	yes	—
34	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	#41	1	2	yes	—
36	C₂×Dic₁₀₆	—	2	4	no	—
37	M₁₆×C₅₃	—	4	4	no	—
38	C₂×C₈×C₅₃	—	4	2	no	—
39	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
40	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
41	C₂×C₈×C₅₃	SD₁₆	1	4	yes	—
42	C₂×C₈×C₅₃	Q₁₆	1	4	yes	—
43	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₄²×C₅₃	C₂×D₄	1	2	yes	—
46	C₂²⋊Dic₅₃	—	4	2	no	—
47	C₄⋊Dic₅₃	—	2	4	no	—
48	C₄⋊Dic₅₃	—	2	4	no	—
49	#11719919	—	4	4	no	—
50	C₄×Dic₅₃	—	4	2	no	—
51	C₂×#88322	—	2	4	no	—
52	C₂×#88322	—	4	4	no	—
53	C₂³×C₅₃	#124	1	1	yes	characteristic
54	C₂×C₄×C₅₃	—	2	4	no	—
55	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
57	C₂×C₄×C₅₃	#121	1	2	yes	—
58	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
60	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
61	C₂×Dic₅₃	—	4	4	no	—
62	Dic₁₀₆	—	4	8	no	—
63	C₈×C₅₃	—	8	2	no	—
64	C₈×C₅₃	#103	1	4	yes	—
65	C₈×C₅₃	#103	1	4	yes	—
66	#88322	—	4	4	no	—
67	#9861	—	53	1	no	maximal, Sylow
68	C₂²×C₅₃	—	2	2	no	—
69	C₂²×C₅₃	#390	1	1	yes	characteristic
70	C₂²×C₅₃	#487	1	2	yes	—
71	C₄×C₅₃	—	2	2	no	—
72	C₄×C₅₃	C₂×#103	1	2	yes	—
73	C₄×C₅₃	#444	1	4	yes	—
74	C₄×C₅₃	#492	1	1	yes	characteristic
75	C₄×C₅₃	#494	1	1	yes	characteristic
76	Dic₅₃	—	8	4	no	—
77	#3160	—	53	1	no	—
78	#1848	D₅₃	1	1	yes	characteristic
79	#1493	—	53	4	no	—
80	C₂×#322	—	53	1	no	—
81	C₂×C₅₃	—	2	2	no	—
82	C₂×C₅₃	#1956	1	1	yes	characteristic
83	C₂×C₅₃	#1975	1	2	yes	—
84	#533	—	53	2	no	—
85	#500	—	53	2	no	—
86	#499	—	106	2	no	—
87	#498	—	53	2	no	—
88	C₂×#108	—	2	2	no	—
89	#427	—	106	2	no	—
90	#423	C₂×D₅₃	1	2	yes	—
91	C₂×C₄×C₈	—	53	2	no	—
92	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
93	Q₈⋊C₈	—	106	8	no	—
94	#322	—	106	2	no	—
95	C₅₃	#9861	1	1	yes	characteristic, complemented, Sylow
96	C₂×M₁₆	—	2	2	no	—
97	C₂²×C₈	—	106	2	no	—
98	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
99	C₄⋊Q₈	—	53	4	no	—
100	#125	—	106	2	no	—
101	C₂²⋊Q₈	—	106	4	no	—
102	C₄×Q₈	—	106	4	no	—
103	#117	—	106	2	no	—
104	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
105	#108	—	4	4	no	—
106	#107	—	106	4	no	—
107	#106	—	106	4	no	—
108	C₄⋊C₈	—	2	4	no	—
109	C₄×C₈	—	53	4	no	—
110	C₄×C₈	—	106	2	no	—
111	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
112	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
113	C₄×C₈	D₂₁₂	1	2	yes	—
114	C₂×Q₈	—	106	4	no	—
115	C₂²×C₄	#11719949	1	2	yes	—
116	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
117	M₁₆	—	4	4	no	—
118	C₂×C₈	—	4	2	no	—
119	C₂×C₈	—	106	4	no	—
120	C₂×C₈	—	212	4	no	—
121	C₂×C₈	#11719921	1	2	yes	—
122	C₂×C₈	#11719921	1	2	yes	—
123	C₂×C₈	#11719923	1	4	yes	—
124	C₂×C₈	#11719925	1	4	yes	—
125	C₄⋊C₄	—	106	4	no	—
126	C₄⋊C₄	—	106	4	no	—
127	C₄⋊C₄	—	212	4	no	—
128	C₂²⋊C₄	—	212	2	no	—
129	C₄²	—	212	2	no	—
130	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
131	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
132	C₄²	D₄×D₅₃	1	2	yes	—
133	C₂³	#13133206	1	1	yes	characteristic
134	Q₈	—	212	8	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	212	4	no	—
137	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
138	C₂×C₄	#13133179	1	2	yes	—
139	C₂×C₄	#13133192	1	2	yes	—
140	C₂×C₄	#13133213	1	1	yes	characteristic
141	C₂×C₄	C₂×#11719923	1	2	yes	—
142	C₂×C₄	C₂×#11719925	1	2	yes	—
143	C₈	—	8	2	no	—
144	C₈	—	212	4	no	—
145	C₈	#13133140	1	4	yes	—
146	C₈	#13133140	1	4	yes	—
147	C₂²	—	2	2	no	—
148	C₂²	#16810524	1	1	yes	characteristic
149	C₂²	#16810975	1	2	yes	—
150	C₄	—	2	2	no	—
151	C₄	—	424	4	no	—
152	C₄	C₂×#13133140	1	2	yes	—
153	C₄	#16810744	1	4	yes	—
154	C₄	#16810995	1	1	yes	characteristic
155	C₄	#16811002	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	#20275516	1	1	yes	characteristic
158	C₂	#20275606	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20285229

C₂

yes

maximal, characteristic

#20273169

C₂

yes

maximal

C₂×#16810344

C₂

yes

maximal, characteristic

C₅₃×#1848

C₂

yes

maximal, characteristic

#16811180

C₂²

yes

—

#16811028

C₄

yes

—

#16811025

—

#16811021

C₄

yes

—

#16810661

—

C₂×C₄×#88322

C₂²

yes

—

#16810353

—

#16810344

—

C₂×#108×C₅₃

—

C₅₃×#423

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂²×#88322

—

#13133217

C₂×C₄

yes

—

#13133209

—

#13133196

—

C₄×Dic₁₀₆

—

—

—

—

—

D₄

yes

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#108×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

C₂×Dic₁₀₆

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₂×C₈×C₅₃

Q₁₆

yes

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

#124

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

#121

yes

—

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

C₂×C₄×C₅₃

C₂×Q₁₆

yes

—

C₂×Dic₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

#103

yes

—

C₈×C₅₃

#103

yes

—

#88322

—

#9861

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

#390

yes

characteristic

C₂²×C₅₃

#487

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×#103

yes

—

C₄×C₅₃

#444

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

#3160

—

#1848

D₅₃

yes

characteristic

#1493

—

C₂×#322

—

C₂×C₅₃

—

C₂×C₅₃

#1956

yes

characteristic

C₂×C₅₃

#1975

yes

—

#533

—

#500

—

#499

—

106

—

#498

—

C₂×#108

—

#427

—

106

—

#423

C₂×D₅₃

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

Q₈⋊C₈

—

106

—

#322

—

106

—

C₅₃

#9861

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

106

—

C₂²×C₈

C₂²⋊D₅₃

yes

—

C₄⋊Q₈

—

100

#125

—

106

—

101

C₂²⋊Q₈

—

106

—

102

C₄×Q₈

—

106

—

103

#117

—

106

—

104

C₂×C₄²

C₂²×D₅₃

yes

characteristic

105

#108

—

106

#107

—

106

—

107

#106

—

106

—

108

C₄⋊C₈

—

109

C₄×C₈

—

110

C₄×C₈

—

106

—

111

C₄×C₈

C₄×D₅₃

yes

characteristic

112

C₄×C₈

C₄×D₅₃

yes

characteristic

113

C₄×C₈

D₂₁₂

yes

—

114

C₂×Q₈

—

106

—

115

C₂²×C₄

#11719949

yes

—

116

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

117

M₁₆

—

118

C₂×C₈

—

119

C₂×C₈

—

106

—

120

C₂×C₈

—

212

—

121

C₂×C₈

#11719921

yes

—

122

C₂×C₈

#11719921

yes

—

123

C₂×C₈

#11719923

yes

—

124

C₂×C₈

#11719925

yes

—

125

C₄⋊C₄

—

106

—

126

C₄⋊C₄

—

106

—

127

C₄⋊C₄

—

212

—

128

C₂²⋊C₄

—

212

—

129

C₄²

—

212

—

130

C₄²

C₂×C₄×D₅₃

yes

characteristic

131

C₄²

C₂×D₂₁₂

yes

characteristic

132

C₄²

D₄×D₅₃

yes

—

133

C₂³

#13133206

yes

characteristic

134

Q₈

—

212

—

135

C₂×C₄

—

136

C₂×C₄

—

212

—

137

C₂×C₄

C₂×#11719921

yes

characteristic

138

C₂×C₄

#13133179

yes

—

139

C₂×C₄

#13133192

yes

—

140

C₂×C₄

#13133213

yes

characteristic

141

C₂×C₄

C₂×#11719923

yes

—

142

C₂×C₄

C₂×#11719925

yes

—

143

C₈

—

144

C₈

—

212

—

145

C₈

#13133140

yes

—

146

C₈

#13133140

yes

—

147

C₂²

—

148

C₂²

#16810524

yes

characteristic

149

C₂²

#16810975

yes

—

150

C₄

—

151

C₄

—

424

—

152

C₄

C₂×#13133140

yes

—

153

C₄

#16810744

yes

—

154

C₄

#16810995

yes

characteristic

155

C₄

#16811002

yes

characteristic

156

C₂

—

157

C₂

#20275516

yes

characteristic

158

C₂

#20275606

yes

—

	subgroup	G/H
center	C₂×C₄	#13133213
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1848	C₂
socle	C₂²×C₅₃	#390
prime radical	C₂³×C₅₃	#124
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1848	D₅₃
hypocenter	C₅₃	#9861
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

subgroup

G/H

center

C₂×C₄

#13133213

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1848

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

N	H ≅ G/N	type
95. C₅₃	#9861	semidirect

H ≅ G/N

type

95. C₅₃

#9861

semidirect

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

subgroup	index
#20285229	2
#20273169	2
C₂×#16810344	2
C₅₃×#1848	2
#9861	53

subgroup

index

C₅₃×#1848

Subgroups — #27284059

Conjugacy classes of subgroups

Complements

Maximal subgroups