Group #27289113 — subgroups

Subgroups — #27289113

Order 13568.

316 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 101 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	#20275696	C₂	1	1	yes	maximal, characteristic
2	#20274869	C₂	1	1	yes	maximal, characteristic
3	#20273490	C₂	1	2	yes	maximal
4	#20273119	C₂	1	2	yes	maximal
5	C₅₃×#1274	C₂	1	1	yes	maximal, characteristic
6	#16811180	C₂²	1	1	yes	characteristic
7	#16810647	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊#88322	—	2	2	no	—
9	C₂×C₄⋊#88322	C₂²	1	2	yes	—
10	#16810383	—	2	2	no	—
11	#16810376	—	2	4	no	—
12	#16810355	—	2	4	no	—
13	C₅₃×#431	C₂²	1	2	yes	—
14	C₅₃×#376	C₂²	1	1	yes	characteristic
15	C₂×#11719917	—	2	1	no	—
16	C₂²×#88322	—	2	2	no	—
17	#13133217	Q₈	1	2	yes	—
18	#13133209	—	2	1	no	—
19	#13133196	—	2	2	no	—
20	C₄×Dic₁₀₆	—	2	2	no	—
21	#13133176	—	2	1	no	—
22	C₄⋊#88322	—	2	2	no	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄⋊#88322	—	2	2	no	—
25	C₄⋊#88322	—	4	4	no	—
26	C₂×C₄²×C₅₃	D₄	1	2	yes	—
27	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
28	C₄⋊C₈×C₅₃	—	2	2	no	—
29	C₂²⋊C₈×C₅₃	—	2	2	no	—
30	C₄×C₈×C₅₃	D₄	1	4	yes	—
31	#95×C₅₃	—	2	2	no	—
32	C₂²×C₄×C₅₃	—	2	2	no	—
33	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	#41	1	2	yes	—
35	C₂×Dic₁₀₆	—	2	2	no	—
36	C₂×C₈×C₅₃	—	2	4	no	—
37	C₄²×C₅₃	—	2	2	no	—
38	C₄²×C₅₃	Q₁₆	1	4	yes	—
39	C₄²×C₅₃	C₂×D₄	1	2	yes	—
40	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
41	C₄²×C₅₃	#41	1	1	yes	characteristic
42	C₂²⋊Dic₅₃	—	4	1	no	—
43	C₄⋊Dic₅₃	—	2	2	no	—
44	C₄⋊Dic₅₃	—	2	2	no	—
45	#11719919	—	4	2	no	—
46	C₄×Dic₅₃	—	4	1	no	—
47	#11719917	—	4	2	no	—
48	C₂×#88322	—	4	2	no	—
49	C₂×#88322	—	4	1	no	—
50	C₂×#88322	—	4	4	no	—
51	C₂³×C₅₃	#123	1	1	yes	characteristic
52	C₂×C₄×C₅₃	—	2	4	no	—
53	C₂×C₄×C₅₃	—	4	2	no	—
54	C₂×C₄×C₅₃	—	4	4	no	—
55	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
56	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
57	C₂×C₄×C₅₃	#123	1	2	yes	—
58	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
60	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
61	C₂×Dic₅₃	—	4	2	no	—
62	Dic₁₀₆	—	4	4	no	—
63	C₈×C₅₃	—	4	4	no	—
64	#88322	—	8	2	no	—
65	#88322	—	8	1	no	—
66	#10668	—	53	1	no	maximal, Sylow
67	C₂²×C₅₃	—	2	2	no	—
68	C₂²×C₅₃	#399	1	1	yes	characteristic
69	C₂²×C₅₃	#484	1	2	yes	—
70	C₄×C₅₃	—	2	2	no	—
71	C₄×C₅₃	—	2	2	no	—
72	C₄×C₅₃	—	4	4	no	—
73	C₄×C₅₃	#477	1	2	yes	—
74	C₄×C₅₃	#494	1	1	yes	characteristic
75	C₄×C₅₃	#496	1	1	yes	characteristic
76	Dic₅₃	—	8	2	no	—
77	#1995	—	53	1	no	—
78	#1843	—	53	1	no	—
79	#1571	—	53	2	no	—
80	#1481	—	53	2	no	—
81	#1274	D₅₃	1	1	yes	characteristic
82	C₂×C₅₃	—	2	2	no	—
83	C₂×C₅₃	#2095	1	1	yes	characteristic
84	C₂×C₅₃	#2098	1	2	yes	—
85	#533	—	53	1	no	—
86	#431	C₂×D₅₃	1	2	yes	—
87	#423	—	53	1	no	—
88	C₂×C₄⋊C₈	—	53	2	no	—
89	C₂×C₄⋊C₈	—	106	2	no	—
90	#376	C₂×D₅₃	1	1	yes	characteristic
91	#339	—	106	2	no	—
92	#335	—	106	4	no	—
93	Q₈⋊C₈	—	106	4	no	—
94	C₅₃	#10668	1	1	yes	characteristic, complemented, Sylow
95	C₂×M₁₆	—	106	1	no	—
96	C₂²×C₈	—	106	2	no	—
97	C₄⋊Q₈	—	53	2	no	—
98	#125	—	106	1	no	—
99	C₂²⋊Q₈	—	106	2	no	—
100	C₄×Q₈	—	106	2	no	—
101	#117	—	106	1	no	—
102	C₂×C₄²	D₂₁₂	1	2	yes	—
103	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	106	2	no	—
106	C₄⋊C₈	—	106	2	no	—
107	C₄⋊C₈	—	106	2	no	—
108	C₄⋊C₈	—	212	4	no	—
109	C₂²⋊C₈	—	2	2	no	—
110	C₄×C₈	D₂₁₂	1	4	yes	—
111	#95	—	2	2	no	—
112	C₂×Q₈	—	106	2	no	—
113	C₂²×C₄	—	2	2	no	—
114	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
115	C₂²×C₄	#11719947	1	2	yes	—
116	M₁₆	—	212	2	no	—
117	C₂×C₈	—	2	4	no	—
118	C₂×C₈	—	212	2	no	—
119	C₂×C₈	—	212	1	no	—
120	C₂×C₈	—	212	4	no	—
121	C₄⋊C₄	—	106	2	no	—
122	C₄⋊C₄	—	106	2	no	—
123	C₄⋊C₄	—	212	2	no	—
124	C₂²⋊C₄	—	212	1	no	—
125	C₄²	—	2	2	no	—
126	C₄²	—	212	1	no	—
127	C₄²	Dic₂₁₂	1	4	yes	—
128	C₄²	C₂×D₂₁₂	1	2	yes	—
129	C₄²	#11719947	1	1	yes	characteristic
130	C₄²	Q₈×D₅₃	1	1	yes	characteristic
131	C₂³	#13133201	1	1	yes	characteristic
132	Q₈	—	212	4	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	4	4	no	—
136	C₂×C₄	—	212	2	no	—
137	C₂×C₄	#13133197	1	2	yes	—
138	C₂×C₄	#13133197	1	1	yes	characteristic
139	C₂×C₄	#13133201	1	2	yes	—
140	C₂×C₄	#13133214	1	1	yes	characteristic
141	C₂×C₄	C₂×Dic₂₁₂	1	2	yes	—
142	C₂×C₄	#13133245	1	2	yes	—
143	C₈	—	4	4	no	—
144	C₈	—	424	2	no	—
145	C₈	—	424	1	no	—
146	C₂²	—	2	2	no	—
147	C₂²	#16810560	1	1	yes	characteristic
148	C₂²	#16810959	1	2	yes	—
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	4	4	no	—
152	C₄	—	424	2	no	—
153	C₄	#16810919	1	2	yes	—
154	C₄	#16811003	1	1	yes	characteristic
155	C₄	#16811012	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	#20276252	1	1	yes	characteristic
158	C₂	#20276269	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20275696

C₂

yes

maximal, characteristic

#20274869

C₂

yes

maximal, characteristic

#20273490

C₂

yes

maximal

#20273119

C₂

yes

maximal

C₅₃×#1274

C₂

yes

maximal, characteristic

#16811180

C₂²

yes

characteristic

#16810647

C₂²

yes

characteristic

C₂×C₄⋊#88322

—

C₂×C₄⋊#88322

C₂²

yes

—

#16810383

—

#16810376

—

#16810355

—

C₅₃×#431

C₂²

yes

—

C₅₃×#376

C₂²

yes

characteristic

C₂×#11719917

—

C₂²×#88322

—

#13133217

Q₈

yes

—

#13133209

—

#13133196

—

C₄×Dic₁₀₆

—

—

—

—

—

—

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

—

#95×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

Q₁₆

yes

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

C₂×Q₈

yes

characteristic

C₄²×C₅₃

#41

yes

characteristic

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

C₄×Dic₅₃

—

—

—

—

—

C₂³×C₅₃

#123

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

#123

yes

—

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₁₆

yes

—

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

#399

yes

characteristic

C₂²×C₅₃

#484

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#477

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

—

—

—

—

D₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

#2095

yes

characteristic

C₂×C₅₃

#2098

yes

—

#533

—

#431

C₂×D₅₃

yes

—

#423

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

106

—

#376

C₂×D₅₃

yes

characteristic

#339

—

106

—

#335

—

106

—

Q₈⋊C₈

—

106

—

C₅₃

#10668

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

106

—

C₂²×C₈

—

106

—

C₄⋊Q₈

—

#125

—

106

—

C₂²⋊Q₈

—

106

—

100

C₄×Q₈

—

106

—

101

#117

—

106

—

102

C₂×C₄²

D₂₁₂

yes

—

103

C₂×C₄²

C₂²×D₅₃

yes

characteristic

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

—

106

C₄⋊C₈

—

106

—

107

C₄⋊C₈

—

106

—

108

C₄⋊C₈

—

212

—

109

C₂²⋊C₈

—

110

C₄×C₈

D₂₁₂

yes

—

111

#95

—

112

C₂×Q₈

—

106

—

113

C₂²×C₄

—

114

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

115

C₂²×C₄

#11719947

yes

—

116

M₁₆

—

212

—

117

C₂×C₈

—

118

C₂×C₈

—

212

—

119

C₂×C₈

—

212

—

120

C₂×C₈

—

212

—

121

C₄⋊C₄

—

106

—

122

C₄⋊C₄

—

106

—

123

C₄⋊C₄

—

212

—

124

C₂²⋊C₄

—

212

—

125

C₄²

—

126

C₄²

—

212

—

127

C₄²

Dic₂₁₂

yes

—

128

C₄²

C₂×D₂₁₂

yes

—

129

C₄²

#11719947

yes

characteristic

130

C₄²

Q₈×D₅₃

yes

characteristic

131

C₂³

#13133201

yes

characteristic

132

Q₈

—

212

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

212

—

137

C₂×C₄

#13133197

yes

—

138

C₂×C₄

#13133197

yes

characteristic

139

C₂×C₄

#13133201

yes

—

140

C₂×C₄

#13133214

yes

characteristic

141

C₂×C₄

C₂×Dic₂₁₂

yes

—

142

C₂×C₄

#13133245

yes

—

143

C₈

—

144

C₈

—

424

—

145

C₈

—

424

—

146

C₂²

—

147

C₂²

#16810560

yes

characteristic

148

C₂²

#16810959

yes

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

424

—

153

C₄

#16810919

yes

—

154

C₄

#16811003

yes

characteristic

155

C₄

#16811012

yes

characteristic

156

C₂

—

157

C₂

#20276252

yes

characteristic

158

C₂

#20276269

yes

—

	subgroup	G/H
center	C₂×C₄	#13133214
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1274	C₂
socle	C₂²×C₅₃	#399
prime radical	C₂³×C₅₃	#123
hypercenter	#1274	D₅₃
hypocenter	C₅₃	#10668
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

subgroup

G/H

center

C₂×C₄

#13133214

commutator, cosocle

C₂×C₄²×C₅₃

C₂³

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1274

C₂

C₂²×C₅₃

C₂³×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

N	H ≅ G/N	type
94. C₅₃	#10668	semidirect

H ≅ G/N

type

94. C₅₃

#10668

semidirect

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

subgroup	index
#20275696	2
#20274869	2
#20273490	2
#20273119	2
C₅₃×#1274	2
#10668	53

subgroup

index

C₅₃×#1274

Subgroups — #27289113

Conjugacy classes of subgroups

Complements

Maximal subgroups