Group #27304227 — subgroups

Subgroups — #27304227

Order 13568.

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

normal subgroups only

This group has 234 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	#20274890	C₂	1	1	yes	maximal, characteristic
2	#20272831	C₂	1	2	yes	maximal
3	#20272828	C₂	1	2	yes	maximal
4	#20272795	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1946	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	C₂²	1	2	yes	—
7	C₂×#13133153	—	2	1	no	—
8	C₂×#13133153	C₂²	1	2	yes	—
9	#16810647	—	2	1	no	—
10	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
11	C₅₃×#427	C₂²	1	1	yes	characteristic
12	C₅₃×#424	—	2	1	no	—
13	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#339	—	2	1	no	—
15	C₂×#11719917	—	2	2	no	—
16	#13133157	—	2	2	no	—
17	#13133157	—	2	2	no	—
18	C₂×#11719908	—	2	2	no	—
19	#13133153	—	2	4	no	—
20	#13133153	—	4	2	no	—
21	C₄⋊#88322	—	4	2	no	—
22	C₂×M₁₆×C₅₃	—	2	1	no	—
23	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
25	#117×C₅₃	—	2	1	no	—
26	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
27	#108×C₅₃	—	2	2	no	—
28	#107×C₅₃	—	2	1	no	—
29	#106×C₅₃	Q₈	1	2	yes	—
30	C₄⋊C₈×C₅₃	—	4	2	no	—
31	#97×C₅₃	—	2	2	no	—
32	C₂²×C₄×C₅₃	—	2	1	no	—
33	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
34	M₁₆×C₅₃	—	4	1	no	—
35	M₁₆×C₅₃	—	4	1	no	—
36	C₂×C₈×C₅₃	—	2	1	no	—
37	C₂×C₈×C₅₃	—	2	1	no	—
38	C₂×C₈×C₅₃	—	4	1	no	—
39	C₂×C₈×C₅₃	D₈	1	2	yes	—
40	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₈×C₅₃	#41	1	2	yes	—
42	C₂×C₈×C₅₃	#41	1	2	yes	—
43	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
44	C₄⋊C₄×C₅₃	—	2	2	no	—
45	C₂²⋊C₄×C₅₃	—	4	1	no	—
46	C₄²×C₅₃	—	2	1	no	—
47	C₄²×C₅₃	—	4	1	no	—
48	C₄²×C₅₃	SD₁₆	1	2	yes	—
49	#11719917	—	4	2	no	—
50	#11719917	—	4	2	no	—
51	C₂×#88322	—	4	2	no	—
52	#11719908	—	4	2	no	—
53	C₂³×C₅₃	#137	1	1	yes	characteristic
54	C₂×C₄×C₅₃	—	2	2	no	—
55	C₂×C₄×C₅₃	—	4	2	no	—
56	C₂×C₄×C₅₃	—	8	1	no	—
57	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
58	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
59	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
63	C₈×C₅₃	—	2	2	no	—
64	C₈×C₅₃	—	8	1	no	—
65	C₈×C₅₃	#125	1	2	yes	—
66	C₈×C₅₃	#126	1	2	yes	—
67	#88322	—	8	2	no	—
68	#13540	—	53	1	no	maximal, Sylow
69	C₂²×C₅₃	—	4	1	no	—
70	C₂²×C₅₃	#474	1	1	yes	characteristic
71	C₂²×C₅₃	#483	1	1	yes	characteristic
72	C₂²×C₅₃	#491	1	1	yes	characteristic
73	C₄×C₅₃	—	4	2	no	—
74	C₄×C₅₃	—	8	2	no	—
75	C₄×C₅₃	#400	1	1	yes	characteristic
76	C₄×C₅₃	#475	1	1	yes	characteristic
77	C₄×C₅₃	#480	1	1	yes	characteristic
78	C₄×C₅₃	#486	1	1	yes	characteristic
79	#1946	D₅₃	1	1	yes	characteristic
80	#1849	—	53	1	no	—
81	#1389	—	53	2	no	—
82	#1388	—	53	2	no	—
83	#1375	—	53	1	no	—
84	C₂×C₅₃	—	2	1	no	—
85	C₂×C₅₃	—	2	1	no	—
86	C₂×C₅₃	—	2	1	no	—
87	C₂×C₅₃	#2099	1	1	yes	characteristic
88	C₂×M₃₂	—	53	2	no	—
89	C₂×#108	—	53	2	no	—
90	C₂×#108	—	106	1	no	—
91	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
92	#427	C₂×D₅₃	1	1	yes	characteristic
93	#424	—	2	1	no	—
94	#423	—	106	1	no	—
95	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
96	#339	—	2	1	no	—
97	C₅₃	#13540	1	1	yes	characteristic, complemented, Sylow
98	C₂×M₁₆	—	2	1	no	—
99	C₂×M₁₆	—	106	2	no	—
100	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
101	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
102	#117	—	2	1	no	—
103	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
104	M₃₂	—	106	2	no	—
105	M₃₂	—	106	2	no	—
106	C₂×C₁₆	—	106	2	no	—
107	#108	—	2	2	no	—
108	#108	—	106	4	no	—
109	#108	—	212	2	no	—
110	#107	—	2	1	no	—
111	#106	Dic₁₀₆	1	2	yes	—
112	C₄⋊C₈	—	4	2	no	—
113	C₄⋊C₈	—	212	2	no	—
114	#97	—	2	2	no	—
115	C₂²×C₄	—	2	1	no	—
116	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
117	M₁₆	—	4	1	no	—
118	M₁₆	—	4	1	no	—
119	M₁₆	—	212	2	no	—
120	M₁₆	—	212	2	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	4	1	no	—
124	C₂×C₈	—	212	2	no	—
125	C₂×C₈	#11719922	1	2	yes	—
126	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
127	C₂×C₈	#11719945	1	1	yes	characteristic
128	C₂×C₈	#11719947	1	2	yes	—
129	C₂×C₈	#11719949	1	2	yes	—
130	C₄⋊C₄	—	2	2	no	—
131	C₂²⋊C₄	—	4	1	no	—
132	C₄²	—	2	1	no	—
133	C₄²	—	4	1	no	—
134	C₄²	#11719924	1	2	yes	—
135	C₁₆	—	212	2	no	—
136	C₂³	#13133247	1	1	yes	characteristic
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	—	8	1	no	—
140	C₂×C₄	#13133196	1	1	yes	characteristic
141	C₂×C₄	#13133198	1	1	yes	characteristic
142	C₂×C₄	#13133206	1	1	yes	characteristic
143	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
144	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
145	C₂×C₄	#13133247	1	1	yes	characteristic
146	C₈	—	2	2	no	—
147	C₈	—	8	1	no	—
148	C₈	—	424	2	no	—
149	C₈	#13133208	1	2	yes	—
150	C₈	#13133212	1	2	yes	—
151	C₂²	—	4	1	no	—
152	C₂²	#16810903	1	1	yes	characteristic
153	C₂²	#16810951	1	1	yes	characteristic
154	C₂²	#16810993	1	1	yes	characteristic
155	C₄	—	4	2	no	—
156	C₄	—	8	2	no	—
157	C₄	#16810562	1	1	yes	characteristic
158	C₄	#16810908	1	1	yes	characteristic
159	C₄	#16810934	1	1	yes	characteristic
160	C₄	#16810971	1	1	yes	characteristic
161	C₂	—	2	1	no	—
162	C₂	—	2	1	no	—
163	C₂	—	2	1	no	—
164	C₂	#20276273	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20274890

C₂

yes

maximal, characteristic

#20272831

C₂

yes

maximal

#20272828

C₂

yes

maximal

#20272795

C₂

yes

maximal, characteristic

C₅₃×#1946

C₂

yes

maximal, characteristic

C₂×#13133157

C₂²

yes

—

C₂×#13133153

—

C₂×#13133153

C₂²

yes

—

#16810647

—

C₂×#108×C₅₃

C₂²

yes

characteristic

C₅₃×#427

C₂²

yes

characteristic

C₅₃×#424

—

C₂×#97×C₅₃

C₂²

yes

characteristic

C₅₃×#339

—

—

—

—

—

—

—

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

#117×C₅₃

—

C₂×C₄²×C₅₃

D₄

yes

characteristic

#108×C₅₃

—

#107×C₅₃

—

#106×C₅₃

Q₈

yes

—

C₄⋊C₈×C₅₃

—

#97×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₃

#41

yes

—

C₂×C₈×C₅₃

#41

yes

—

C₂×C₈×C₅₃

#41

yes

characteristic

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

SD₁₆

yes

—

—

—

—

—

C₂³×C₅₃

#137

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₃

#137

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#125

yes

—

C₈×C₅₃

#126

yes

—

#88322

—

#13540

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

D₅₃

yes

—

—

—

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#2099

yes

characteristic

C₂×M₃₂

—

C₂×#108

—

C₂×#108

—

106

—

C₂×#108

C₂×D₅₃

yes

characteristic

#427

C₂×D₅₃

yes

characteristic

#424

—

#423

—

106

—

C₂×#97

C₂×D₅₃

yes

characteristic

#339

—

C₅₃

#13540

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

106

—

100

C₂²×C₈

C₂²⋊D₅₃

yes

characteristic

101

C₂²×C₈

C₂²×D₅₃

yes

characteristic

102

#117

—

103

C₂×C₄²

C₂²⋊D₅₃

yes

characteristic

104

M₃₂

—

106

—

105

M₃₂

—

106

—

106

C₂×C₁₆

—

106

—

107

#108

—

108

#108

—

106

—

109

#108

—

212

—

110

#107

—

111

#106

Dic₁₀₆

yes

—

112

C₄⋊C₈

—

113

C₄⋊C₈

—

212

—

114

#97

—

115

C₂²×C₄

—

116

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

117

M₁₆

—

118

M₁₆

—

119

M₁₆

—

212

—

120

M₁₆

—

212

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

212

—

125

C₂×C₈

#11719922

yes

—

126

C₂×C₈

C₂×Dic₁₀₆

yes

characteristic

127

C₂×C₈

#11719945

yes

characteristic

128

C₂×C₈

#11719947

yes

—

129

C₂×C₈

#11719949

yes

—

130

C₄⋊C₄

—

131

C₂²⋊C₄

—

132

C₄²

—

133

C₄²

—

134

C₄²

#11719924

yes

—

135

C₁₆

—

212

—

136

C₂³

#13133247

yes

characteristic

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

#13133196

yes

characteristic

141

C₂×C₄

#13133198

yes

characteristic

142

C₂×C₄

#13133206

yes

characteristic

143

C₂×C₄

C₂×#11719922

yes

characteristic

144

C₂×C₄

C₂×#11719924

yes

characteristic

145

C₂×C₄

#13133247

yes

characteristic

146

C₈

—

147

C₈

—

148

C₈

—

424

—

149

C₈

#13133208

yes

—

150

C₈

#13133212

yes

—

151

C₂²

—

152

C₂²

#16810903

yes

characteristic

153

C₂²

#16810951

yes

characteristic

154

C₂²

#16810993

yes

characteristic

155

C₄

—

156

C₄

—

157

C₄

#16810562

yes

characteristic

158

C₄

#16810908

yes

characteristic

159

C₄

#16810934

yes

characteristic

160

C₄

#16810971

yes

characteristic

161

C₂

—

162

C₂

—

163

C₂

—

164

C₂

#20276273

yes

characteristic

	subgroup	G/H
center	C₄	#16810971
commutator, cosocle	C₂²×C₈×C₅₃	C₂³
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1946	C₂
socle	C₂×C₅₃	#2099
prime radical	C₂³×C₅₃	#137
hypercenter	#1946	D₅₃
hypocenter	C₅₃	#13540
LCS	G ▷ C₂²×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1946

subgroup

G/H

center

C₄

#16810971

commutator, cosocle

C₂²×C₈×C₅₃

C₂³

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₅₃×#1946

C₂

C₂×C₅₃

C₂³×C₅₃

D₅₃

C₅₃

G ▷ C₂²×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1946

N	H ≅ G/N	type
97. C₅₃	#13540	semidirect

H ≅ G/N

type

97. C₅₃

#13540

semidirect

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

subgroup	index
#20274890	2
#20272831	2
#20272828	2
#20272795	2
C₅₃×#1946	2
#13540	53

subgroup

index

C₅₃×#1946

Subgroups — #27304227

Conjugacy classes of subgroups

Complements

Maximal subgroups