Group #27277748 — subgroups

Subgroups — #27277748

Order 13568.

262 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 99 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	#20274467	C₂	1	1	yes	maximal, characteristic
2	#20273461	C₂	1	2	yes	maximal
3	#20272559	C₂	1	2	yes	maximal
4	#20272544	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1274	C₂	1	1	yes	maximal, characteristic
6	#16810676	C₂²	1	2	yes	—
7	C₂×#13133128	C₂²	1	1	yes	characteristic
8	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
9	#16810378	C₄	1	2	yes	—
10	#16810378	C₄	1	2	yes	—
11	#16810375	—	2	2	no	—
12	#16810373	—	2	2	no	—
13	C₅₃×#431	C₂²	1	2	yes	—
14	C₅₃×#376	C₂²	1	1	yes	characteristic
15	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
16	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
17	C₂²×#88322	C₂×C₄	1	2	yes	—
18	C₄⋊#88322	—	2	2	no	—
19	C₂²⋊#88322	—	2	2	no	—
20	#13133128	—	2	2	no	—
21	C₄×#88322	—	2	2	no	—
22	C₄×#88322	D₄	1	2	yes	—
23	C₄×#88322	Q₈	1	2	yes	—
24	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
26	C₄⋊C₈×C₅₃	—	2	2	no	—
27	C₂²⋊C₈×C₅₃	—	2	2	no	—
28	C₄×C₈×C₅₃	—	2	2	no	—
29	#95×C₅₃	—	2	2	no	—
30	C₂²×C₄×C₅₃	—	2	2	no	—
31	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	#41	1	2	yes	—
33	C₂×C₈×C₅₃	—	4	2	no	—
34	C₄²×C₅₃	—	2	2	no	—
35	C₄²×C₅₃	—	2	2	no	—
36	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
38	C₄²×C₅₃	#41	1	2	yes	—
39	C₂×#88322	—	2	2	no	—
40	C₂×#88322	—	2	2	no	—
41	C₂×#88322	—	2	1	no	—
42	C₂×#88322	—	2	1	no	—
43	C₂×#88322	—	2	2	no	—
44	C₂×#88322	—	2	2	no	—
45	C₂×#88322	—	4	2	no	—
46	C₂×#88322	C₄⋊C₄	1	4	yes	—
47	C₂³×C₅₃	#117	1	1	yes	characteristic
48	C₂×C₄×C₅₃	—	4	2	no	—
49	C₂×C₄×C₅₃	—	4	4	no	—
50	C₂×C₄×C₅₃	—	4	2	no	—
51	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
53	C₂×C₄×C₅₃	#124	1	2	yes	—
54	C₂×C₄×C₅₃	#125	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	—	2	4	no	—
59	#88322	—	4	1	no	—
60	#88322	—	4	1	no	—
61	#88322	—	8	2	no	—
62	#8783	—	53	1	no	maximal, Sylow
63	C₂²×C₅₃	—	2	2	no	—
64	C₂²×C₅₃	#382	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	—	8	2	no	—
69	C₄×C₅₃	#424	1	2	yes	—
70	C₄×C₅₃	#433	1	1	yes	characteristic
71	C₄×C₅₃	#433	1	1	yes	characteristic
72	#1746	—	53	1	no	—
73	#1564	—	53	2	no	—
74	#1281	—	53	2	no	—
75	#1274	—	53	1	no	—
76	#1274	D₅₃	1	1	yes	characteristic
77	C₂×C₅₃	—	2	2	no	—
78	C₂×C₅₃	#1832	1	1	yes	characteristic
79	C₂×C₅₃	#1833	1	2	yes	—
80	#431	—	53	2	no	—
81	#431	C₂×D₅₃	1	2	yes	—
82	C₂×#97	—	53	1	no	—
83	C₂×C₄×C₈	—	53	1	no	—
84	#376	C₂×D₅₃	1	1	yes	characteristic
85	#336	—	53	2	no	—
86	#336	—	53	2	no	—
87	#335	—	106	2	no	—
88	#334	—	106	2	no	—
89	C₅₃	#8783	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₈	—	53	2	no	—
91	C₂²×C₈	—	53	1	no	—
92	C₂²×C₈	—	53	1	no	—
93	C₂×C₄²	C₄×D₅₃	1	2	yes	—
94	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
95	C₄⋊C₈	—	2	2	no	—
96	C₄⋊C₈	—	106	2	no	—
97	C₂²⋊C₈	—	2	2	no	—
98	C₂²⋊C₈	—	106	2	no	—
99	#97	—	106	2	no	—
100	C₄×C₈	—	2	2	no	—
101	C₄×C₈	—	53	2	no	—
102	C₄×C₈	—	53	2	no	—
103	C₄×C₈	—	106	2	no	—
104	#95	—	2	2	no	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
107	C₂²×C₄	#11719947	1	2	yes	—
108	C₂×C₈	—	4	2	no	—
109	C₂×C₈	—	53	4	no	—
110	C₂×C₈	—	106	2	no	—
111	C₂×C₈	—	106	2	no	—
112	C₂×C₈	—	106	1	no	—
113	C₂×C₈	—	106	1	no	—
114	C₂×C₈	—	106	2	no	—
115	C₂×C₈	—	106	2	no	—
116	C₂×C₈	—	212	2	no	—
117	C₄²	—	2	2	no	—
118	C₄²	—	2	2	no	—
119	C₄²	#11719945	1	2	yes	—
120	C₄²	D₄×D₅₃	1	1	yes	characteristic
121	C₄²	Q₈×D₅₃	1	1	yes	characteristic
122	C₂³	#13133173	1	1	yes	characteristic
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	4	4	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
127	C₂×C₄	#13133174	1	1	yes	characteristic
128	C₂×C₄	#13133203	1	2	yes	—
129	C₂×C₄	#13133207	1	2	yes	—
130	C₂×C₄	#13133223	1	2	yes	—
131	C₂×C₄	#13133225	1	2	yes	—
132	C₈	—	8	2	no	—
133	C₈	—	106	4	no	—
134	C₈	—	424	2	no	—
135	C₈	—	212	1	no	—
136	C₈	—	212	1	no	—
137	C₂²	—	2	2	no	—
138	C₂²	#16810487	1	1	yes	characteristic
139	C₂²	#16810681	1	2	yes	—
140	C₄	—	2	2	no	—
141	C₄	—	2	2	no	—
142	C₄	—	8	2	no	—
143	C₄	#16810648	1	2	yes	—
144	C₄	#16810682	1	1	yes	characteristic
145	C₄	#16810685	1	1	yes	characteristic
146	C₂	—	2	2	no	—
147	C₂	#20274833	1	1	yes	characteristic
148	C₂	#20274837	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20274467

C₂

yes

maximal, characteristic

#20273461

C₂

yes

maximal

#20272559

C₂

yes

maximal

#20272544

C₂

yes

maximal, characteristic

C₅₃×#1274

C₂

yes

maximal, characteristic

#16810676

C₂²

yes

—

C₂×#13133128

C₂²

yes

characteristic

C₂×C₄×#88322

C₂²

yes

characteristic

#16810378

C₄

yes

—

#16810378

C₄

yes

—

#16810375

—

#16810373

—

C₅₃×#431

C₂²

yes

—

C₅₃×#376

C₂²

yes

characteristic

C₂²×#88322

C₂×C₄

yes

characteristic

C₂²×#88322

C₂×C₄

yes

characteristic

C₂²×#88322

C₂×C₄

yes

—

—

—

—

—

D₄

yes

—

C₄×#88322

Q₈

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

—

#95×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

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×Q₈

yes

characteristic

C₄²×C₅₃

#41

yes

—

—

—

—

—

—

—

—

C₄⋊C₄

yes

—

C₂³×C₅₃

#117

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×C₄×C₅₃

#125

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₈×C₅₃

—

—

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

#382

yes

characteristic

C₂²×C₅₃

#433

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#424

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

—

—

—

—

D₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

#1832

yes

characteristic

C₂×C₅₃

#1833

yes

—

#431

—

#431

C₂×D₅₃

yes

—

C₂×#97

—

C₂×C₄×C₈

—

#376

C₂×D₅₃

yes

—

—

—

106

—

#334

—

106

—

C₅₃

#8783

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₄×D₅₃

yes

—

C₂×C₄²

C₂²×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

C₂²⋊C₈

—

106

—

#97

—

106

—

100

C₄×C₈

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

—

106

—

104

#95

—

105

C₂²×C₄

—

106

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

107

C₂²×C₄

#11719947

yes

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

106

—

111

C₂×C₈

—

106

—

112

C₂×C₈

—

106

—

113

C₂×C₈

—

106

—

114

C₂×C₈

—

106

—

115

C₂×C₈

—

106

—

116

C₂×C₈

—

212

—

117

C₄²

—

118

C₄²

—

119

C₄²

#11719945

yes

—

120

C₄²

D₄×D₅₃

yes

characteristic

121

C₄²

Q₈×D₅₃

yes

characteristic

122

C₂³

#13133173

yes

characteristic

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

C₄⋊C₄×D₅₃

yes

characteristic

127

C₂×C₄

#13133174

yes

characteristic

128

C₂×C₄

#13133203

yes

—

129

C₂×C₄

#13133207

yes

—

130

C₂×C₄

#13133223

yes

—

131

C₂×C₄

#13133225

yes

—

132

C₈

—

133

C₈

—

106

—

134

C₈

—

424

—

135

C₈

—

212

—

136

C₈

—

212

—

137

C₂²

—

138

C₂²

#16810487

yes

characteristic

139

C₂²

#16810681

yes

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

#16810648

yes

—

144

C₄

#16810682

yes

characteristic

145

C₄

#16810685

yes

characteristic

146

C₂

—

147

C₂

#20274833

yes

characteristic

148

C₂

#20274837

yes

—

	subgroup	G/H
center	C₂×C₄	#13133174
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1274	C₂
socle	C₂²×C₅₃	#382
prime radical	C₂³×C₅₃	#117
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1274	D₅₃
hypocenter	C₅₃	#8783
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #376 ◁ #1274

subgroup

G/H

center

C₂×C₄

#13133174

commutator

C₂²×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1274

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ #376 ◁ #1274

N	H ≅ G/N	type
89. C₅₃	#8783	semidirect

H ≅ G/N

type

89. C₅₃

#8783

semidirect

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

subgroup	index
#20274467	2
#20273461	2
#20272559	2
#20272544	2
C₅₃×#1274	2
#8783	53

subgroup

index

C₅₃×#1274

Subgroups — #27277748

Conjugacy classes of subgroups

Complements

Maximal subgroups