Group #27277817 — subgroups

Subgroups — #27277817

Order 13568.

258 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 91 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	#20274464	C₂	1	1	yes	maximal, characteristic
2	#20272559	C₂	1	2	yes	maximal
3	#20272548	C₂	1	1	yes	maximal, characteristic
4	#20272545	C₂	1	2	yes	maximal
5	C₅₃×#1276	C₂	1	1	yes	maximal, characteristic
6	#16810676	C₂²	1	2	yes	—
7	C₂×#13133128	C₂²	1	1	yes	characteristic
8	C₂×#13133128	C₂²	1	1	yes	characteristic
9	#16810378	C₄	1	2	yes	—
10	#16810378	C₄	1	2	yes	—
11	C₅₃×#431	C₂²	1	2	yes	—
12	C₄³×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
14	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
15	C₂²×#88322	C₂×C₄	1	2	yes	—
16	C₄⋊#88322	—	2	2	no	—
17	C₂²⋊#88322	—	2	2	no	—
18	#13133128	—	2	2	no	—
19	#13133128	—	2	1	no	—
20	#13133128	—	2	1	no	—
21	C₄×#88322	—	2	2	no	—
22	C₂×C₄²×C₅₃	—	2	2	no	—
23	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
25	C₄⋊C₈×C₅₃	—	2	2	no	—
26	C₂²⋊C₈×C₅₃	—	2	2	no	—
27	C₄×C₈×C₅₃	—	2	2	no	—
28	C₂²×C₄×C₅₃	—	2	2	no	—
29	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	#41	1	2	yes	—
31	C₂×C₈×C₅₃	—	4	2	no	—
32	C₄²×C₅₃	—	2	2	no	—
33	C₄²×C₅₃	—	2	2	no	—
34	C₄²×C₅₃	—	4	4	no	—
35	C₄²×C₅₃	#41	1	2	yes	—
36	C₄²×C₅₃	#41	1	1	yes	characteristic
37	C₄²×C₅₃	#41	1	1	yes	characteristic
38	C₂×#88322	—	2	1	no	—
39	C₂×#88322	—	2	1	no	—
40	C₂×#88322	—	2	2	no	—
41	C₂×#88322	—	2	2	no	—
42	C₂×#88322	—	2	2	no	—
43	C₂×#88322	—	2	2	no	—
44	C₂×#88322	—	2	2	no	—
45	C₂×#88322	—	4	2	no	—
46	C₂³×C₅₃	#117	1	1	yes	characteristic
47	C₂×C₄×C₅₃	—	2	4	no	—
48	C₂×C₄×C₅₃	—	4	2	no	—
49	C₂×C₄×C₅₃	—	4	2	no	—
50	C₂×C₄×C₅₃	—	4	2	no	—
51	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
52	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
53	C₂×C₄×C₅₃	#126	1	2	yes	—
54	C₂×C₄×C₅₃	#126	1	2	yes	—
55	C₂×C₄×C₅₃	#131	1	2	yes	—
56	C₂×C₄×C₅₃	#131	1	2	yes	—
57	C₈×C₅₃	—	8	2	no	—
58	#88322	—	4	1	no	—
59	#88322	—	4	2	no	—
60	#88322	—	4	1	no	—
61	#88322	—	8	2	no	—
62	#8802	—	53	1	no	maximal, Sylow
63	C₂²×C₅₃	—	2	2	no	—
64	C₂²×C₅₃	#383	1	1	yes	characteristic
65	C₂²×C₅₃	#433	1	2	yes	—
66	C₄×C₅₃	—	2	2	no	—
67	C₄×C₅₃	—	2	2	no	—
68	C₄×C₅₃	—	4	4	no	—
69	C₄×C₅₃	#433	1	1	yes	characteristic
70	C₄×C₅₃	#433	1	2	yes	—
71	C₄×C₅₃	#433	1	1	yes	characteristic
72	#1745	—	53	1	no	—
73	#1281	—	53	2	no	—
74	#1276	—	53	1	no	—
75	#1276	D₅₃	1	1	yes	characteristic
76	#1275	—	53	2	no	—
77	C₂×C₅₃	—	2	2	no	—
78	C₂×C₅₃	#1840	1	1	yes	characteristic
79	C₂×C₅₃	#1841	1	2	yes	—
80	#431	—	53	2	no	—
81	#431	C₂×D₅₃	1	2	yes	—
82	C₂×#97	—	53	1	no	—
83	C₂×#97	—	53	1	no	—
84	C₄³	C₂×D₅₃	1	1	yes	characteristic
85	#336	—	53	2	no	—
86	#336	—	53	2	no	—
87	C₅₃	#8802	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₈	—	53	1	no	—
89	C₂²×C₈	—	53	1	no	—
90	C₂²×C₈	—	53	2	no	—
91	C₂×C₄²	—	2	2	no	—
92	C₂×C₄²	C₄×D₅₃	1	2	yes	—
93	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
94	C₄⋊C₈	—	2	2	no	—
95	C₄⋊C₈	—	106	2	no	—
96	C₂²⋊C₈	—	2	2	no	—
97	C₂²⋊C₈	—	106	2	no	—
98	#97	—	106	2	no	—
99	#97	—	106	1	no	—
100	#97	—	106	1	no	—
101	C₄×C₈	—	2	2	no	—
102	C₄×C₈	—	106	2	no	—
103	C₂²×C₄	—	2	2	no	—
104	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
105	C₂²×C₄	#11719947	1	2	yes	—
106	C₂×C₈	—	4	2	no	—
107	C₂×C₈	—	106	1	no	—
108	C₂×C₈	—	106	1	no	—
109	C₂×C₈	—	106	2	no	—
110	C₂×C₈	—	106	2	no	—
111	C₂×C₈	—	106	2	no	—
112	C₂×C₈	—	106	2	no	—
113	C₂×C₈	—	106	2	no	—
114	C₂×C₈	—	212	2	no	—
115	C₄²	—	2	2	no	—
116	C₄²	—	2	2	no	—
117	C₄²	—	4	4	no	—
118	C₄²	#11719945	1	2	yes	—
119	C₄²	#11719947	1	1	yes	characteristic
120	C₄²	#11719949	1	1	yes	characteristic
121	C₂³	#13133173	1	1	yes	characteristic
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	#13133174	1	1	yes	characteristic
127	C₂×C₄	#13133175	1	1	yes	characteristic
128	C₂×C₄	#13133210	1	2	yes	—
129	C₂×C₄	#13133212	1	2	yes	—
130	C₂×C₄	#13133223	1	2	yes	—
131	C₂×C₄	#13133225	1	2	yes	—
132	C₈	—	8	2	no	—
133	C₈	—	424	2	no	—
134	C₈	—	212	1	no	—
135	C₈	—	212	2	no	—
136	C₈	—	212	1	no	—
137	C₂²	—	2	2	no	—
138	C₂²	#16810490	1	1	yes	characteristic
139	C₂²	#16810681	1	2	yes	—
140	C₄	—	2	2	no	—
141	C₄	—	2	2	no	—
142	C₄	—	4	4	no	—
143	C₄	#16810682	1	1	yes	characteristic
144	C₄	#16810683	1	2	yes	—
145	C₄	#16810685	1	1	yes	characteristic
146	C₂	—	2	2	no	—
147	C₂	#20274859	1	1	yes	characteristic
148	C₂	#20274861	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20274464

C₂

yes

maximal, characteristic

#20272559

C₂

yes

maximal

#20272548

C₂

yes

maximal, characteristic

#20272545

C₂

yes

maximal

C₅₃×#1276

C₂

yes

maximal, characteristic

#16810676

C₂²

yes

—

C₂×#13133128

C₂²

yes

characteristic

C₂×#13133128

C₂²

yes

characteristic

#16810378

C₄

yes

—

#16810378

C₄

yes

—

C₅₃×#431

C₂²

yes

—

C₄³×C₅₃

C₂²

yes

characteristic

C₂²×#88322

C₂×C₄

yes

characteristic

C₂²×#88322

C₂×C₄

yes

characteristic

C₂²×#88322

C₂×C₄

yes

—

—

—

—

—

—

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

#41

yes

—

C₄²×C₅₃

yes

C₄²×C₅₃

yes

—

—

—

—

—

—

—

—

C₂³×C₅₃

#117

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×C₄×C₅₃

#126

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₈×C₅₃

—

—

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

#383

yes

characteristic

C₂²×C₅₃

#433

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#433

yes

characteristic

C₄×C₅₃

#433

yes

—

C₄×C₅₃

yes

—

—

—

D₅₃

yes

characteristic

#1275

—

C₂×C₅₃

—

C₂×C₅₃

#1840

yes

characteristic

C₂×C₅₃

#1841

yes

—

#431

—

#431

C₂×D₅₃

yes

—

C₂×#97

—

C₂×#97

—

C₄³

C₂×D₅₃

yes

characteristic

#336

—

#336

—

C₅₃

#8802

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

C₄×D₅₃

yes

—

C₂×C₄²

C₂²×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

C₂²⋊C₈

—

106

—

#97

—

106

—

#97

—

106

—

100

#97

—

106

—

101

C₄×C₈

—

102

C₄×C₈

—

106

—

103

C₂²×C₄

—

104

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

105

C₂²×C₄

#11719947

yes

—

106

C₂×C₈

—

107

C₂×C₈

—

106

—

108

C₂×C₈

—

106

—

109

C₂×C₈

—

106

—

110

C₂×C₈

—

106

—

111

C₂×C₈

—

106

—

112

C₂×C₈

—

106

—

113

C₂×C₈

—

106

—

114

C₂×C₈

—

212

—

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₄²

#11719945

yes

—

119

C₄²

#11719947

yes

characteristic

120

C₄²

#11719949

yes

characteristic

121

C₂³

#13133173

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

#13133174

yes

characteristic

127

C₂×C₄

#13133175

yes

characteristic

128

C₂×C₄

#13133210

yes

—

129

C₂×C₄

#13133212

yes

—

130

C₂×C₄

#13133223

yes

—

131

C₂×C₄

#13133225

yes

—

132

C₈

—

133

C₈

—

424

—

134

C₈

—

212

—

135

C₈

—

212

—

136

C₈

—

212

—

137

C₂²

—

138

C₂²

#16810490

yes

characteristic

139

C₂²

#16810681

yes

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

#16810682

yes

characteristic

144

C₄

#16810683

yes

—

145

C₄

#16810685

yes

characteristic

146

C₂

—

147

C₂

#20274859

yes

characteristic

148

C₂

#20274861

yes

—

	subgroup	G/H
center	C₂×C₄	#13133175
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1276	C₂
socle	C₂²×C₅₃	#383
prime radical	C₂³×C₅₃	#117
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1276	D₅₃
hypocenter	C₅₃	#8802
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1276

subgroup

G/H

center

C₂×C₄

#13133175

commutator

C₂²×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1276

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1276

N	H ≅ G/N	type
87. C₅₃	#8802	semidirect

H ≅ G/N

type

87. C₅₃

#8802

semidirect

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

subgroup	index
#20274464	2
#20272559	2
#20272548	2
#20272545	2
C₅₃×#1276	2
#8802	53

subgroup

index

C₅₃×#1276

Subgroups — #27277817

Conjugacy classes of subgroups

Complements

Maximal subgroups