Group #27299472 — subgroups

Subgroups — #27299472

Order 13568.

308 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 131 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	#20276489	C₂	1	2	yes	maximal
2	M₃₂×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#20274597	C₂	1	2	yes	maximal
4	#20272817	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1385	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	C₂²	1	1	yes	characteristic
7	#16810661	C₂²	1	2	yes	—
8	#16810581	—	2	2	no	—
9	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
10	#16810447	C₄	1	8	yes	complemented
11	#16810398	C₄	1	2	yes	complemented
12	C₁₆×Dic₅₃	C₄	1	2	yes	complemented
13	#16810384	—	2	2	no	—
14	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#108×C₅₃	C₂²	1	2	yes	—
16	C₂²×#88322	D₄	1	1	yes	characteristic
17	#13133176	—	2	2	no	—
18	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
19	#13133157	C₂×C₄	1	4	yes	—
20	C₂×#11719908	C₂×C₄	1	2	yes	—
21	#13133151	—	2	2	no	—
22	#13133149	C₂×C₄	1	4	yes	—
23	#13133127	—	4	2	no	—
24	C₈×Dic₅₃	—	4	2	no	—
25	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
26	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₅₃	Q₈	1	1	yes	characteristic
28	C₈×Dic₅₃	Q₈	1	1	yes	characteristic
29	C₂×M₁₆×C₅₃	—	2	2	no	—
30	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
31	M₃₂×C₅₃	—	2	2	no	—
32	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
33	#108×C₅₃	—	2	2	no	—
34	#108×C₅₃	C₂×C₄	1	4	yes	—
35	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₂²×Dic₅₃	—	2	1	no	—
37	M₁₆×C₅₃	—	2	4	no	—
38	M₁₆×C₅₃	—	4	2	no	—
39	C₂×C₈×C₅₃	—	4	2	no	—
40	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
42	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₈×C₅₃	#41	1	2	yes	—
44	C₂²⋊Dic₅₃	—	4	2	no	—
45	C₄⋊Dic₅₃	—	2	4	no	—
46	C₄×Dic₅₃	—	2	1	no	—
47	C₄×Dic₅₃	—	4	2	no	—
48	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
49	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
51	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
52	C₂×#88322	SD₁₆	1	2	yes	—
53	C₂×#88322	SD₁₆	1	2	yes	—
54	C₁₆×C₅₃	—	2	2	no	—
55	#11719908	C₄²	1	4	yes	—
56	C₂³×C₅₃	#135	1	1	yes	characteristic
57	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
59	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
60	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
61	C₂×Dic₅₃	—	2	1	no	—
62	C₂×Dic₅₃	—	2	1	no	—
63	C₂×Dic₅₃	—	4	2	no	—
64	C₂×Dic₅₃	—	4	4	no	—
65	C₈×C₅₃	—	4	4	no	—
66	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
67	C₈×C₅₃	#117	1	1	yes	characteristic
68	C₈×C₅₃	C₄×Q₈	1	2	yes	—
69	#88322	#106	1	1	yes	characteristic
70	#88322	#106	1	2	yes	—
71	#88322	#106	1	1	yes	characteristic
72	#12668	—	53	1	no	maximal, Sylow
73	C₂²×C₅₃	—	4	1	no	—
74	C₂²×C₅₃	#427	1	1	yes	characteristic
75	C₂²×C₅₃	C₄×SD₁₆	1	2	yes	—
76	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
77	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
78	C₄×C₅₃	C₄×SD₁₆	1	2	yes	—
79	Dic₅₃	—	2	2	no	—
80	Dic₅₃	—	4	1	no	—
81	Dic₅₃	—	8	4	no	—
82	#2155	—	53	2	no	—
83	C₄×M₃₂	—	53	1	no	—
84	#1772	—	53	2	no	—
85	#1385	D₅₃	1	1	yes	characteristic
86	#1384	—	53	1	no	—
87	C₂×C₅₃	—	2	1	no	—
88	C₂×C₅₃	—	2	2	no	—
89	C₂×C₅₃	C₄×#106	1	1	yes	characteristic
90	C₂×M₃₂	—	53	1	no	—
91	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
92	C₂×#108	C₂×D₅₃	1	2	yes	—
93	#427	—	53	2	no	—
94	#405	—	106	2	no	—
95	C₂×C₄×C₈	—	53	1	no	—
96	#365	—	53	8	no	—
97	#346	—	53	2	no	—
98	C₄×C₁₆	—	53	2	no	—
99	#339	—	106	2	no	—
100	C₅₃	#12668	1	1	yes	characteristic, complemented, Sylow
101	C₂×M₁₆	—	2	2	no	—
102	C₂²×C₈	—	53	1	no	—
103	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
104	#117	—	106	2	no	—
105	C₂×C₄²	—	53	1	no	—
106	M₃₂	—	2	2	no	—
107	M₃₂	—	53	4	no	—
108	C₂×C₁₆	—	53	2	no	—
109	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
110	#108	—	2	2	no	—
111	#108	C₄×D₅₃	1	4	yes	—
112	#107	—	106	2	no	—
113	#106	—	53	4	no	—
114	#97	—	212	2	no	—
115	C₄×C₈	—	53	1	no	—
116	C₄×C₈	—	53	1	no	—
117	C₄×C₈	—	53	1	no	—
118	C₄×C₈	—	53	1	no	—
119	C₄×C₈	—	212	2	no	—
120	C₂²×C₄	—	106	1	no	—
121	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
122	M₁₆	—	2	4	no	—
123	M₁₆	—	4	2	no	—
124	C₂×C₈	—	4	2	no	—
125	C₂×C₈	—	53	1	no	—
126	C₂×C₈	—	53	1	no	—
127	C₂×C₈	—	53	2	no	—
128	C₂×C₈	—	53	2	no	—
129	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
130	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
131	C₂×C₈	#11719947	1	2	yes	—
132	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
133	C₄⋊C₄	—	106	4	no	—
134	C₂²⋊C₄	—	212	2	no	—
135	C₄²	—	53	1	no	—
136	C₄²	—	53	1	no	—
137	C₄²	—	106	1	no	—
138	C₄²	—	212	2	no	—
139	C₁₆	—	2	2	no	—
140	C₁₆	—	53	4	no	—
141	C₂³	#13133240	1	1	yes	characteristic
142	C₂×C₄	—	106	1	no	—
143	C₂×C₄	—	106	1	no	—
144	C₂×C₄	—	212	2	no	—
145	C₂×C₄	—	212	4	no	—
146	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
147	C₂×C₄	#13133177	1	2	yes	—
148	C₂×C₄	SD₁₆×D₅₃	1	2	yes	—
149	C₂×C₄	#13133236	1	1	yes	characteristic
150	C₈	—	4	4	no	—
151	C₈	—	53	1	no	—
152	C₈	—	53	2	no	—
153	C₈	—	53	1	no	—
154	C₈	C₄²×D₅₃	1	1	yes	characteristic
155	C₈	#13133173	1	1	yes	characteristic
156	C₈	#13133182	1	2	yes	—
157	C₂²	—	4	1	no	—
158	C₂²	#16810660	1	1	yes	characteristic
159	C₂²	#16810708	1	2	yes	—
160	C₄	—	106	2	no	—
161	C₄	—	424	4	no	—
162	C₄	—	212	1	no	—
163	C₄	#16810474	1	1	yes	characteristic
164	C₄	#106×D₅₃	1	1	yes	characteristic
165	C₄	#16810714	1	2	yes	—
166	C₂	—	2	1	no	—
167	C₂	—	2	2	no	—
168	C₂	#20274603	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276489

C₂

yes

maximal

M₃₂×Dic₅₃

C₂

yes

maximal, characteristic

#20274597

C₂

yes

maximal

#20272817

C₂

yes

maximal, characteristic

C₅₃×#1385

C₂

yes

maximal, characteristic

C₂×#13133157

C₂²

yes

characteristic

#16810661

C₂²

yes

—

#16810581

—

C₂×C₈×Dic₅₃

C₂²

yes

characteristic

#16810447

C₄

yes

complemented

#16810398

C₄

yes

complemented

C₁₆×Dic₅₃

C₄

yes

complemented

#16810384

—

C₂×M₃₂×C₅₃

C₂²

yes

characteristic

C₂×#108×C₅₃

C₂²

yes

—

C₂²×#88322

D₄

yes

characteristic

#13133176

—

C₂×C₄×Dic₅₃

D₄

yes

characteristic

#13133157

C₂×C₄

yes

—

C₂×#11719908

C₂×C₄

yes

—

#13133151

—

#13133149

C₂×C₄

yes

—

#13133127

—

C₈×Dic₅₃

—

C₈×Dic₅₃

C₂×C₄

yes

characteristic

C₈×Dic₅₃

C₂×C₄

yes

characteristic

C₈×Dic₅₃

Q₈

yes

characteristic

C₈×Dic₅₃

Q₈

yes

characteristic

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

C₂³

yes

characteristic

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

#108×C₅₃

—

#108×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×Dic₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂²×C₄

yes

—

C₂×C₈×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₃

#41

yes

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₄⋊C₄

yes

characteristic

C₄×Dic₅₃

C₄⋊C₄

yes

characteristic

C₂×#88322

C₄⋊C₄

yes

characteristic

C₂×#88322

C₄⋊C₄

yes

characteristic

C₂×#88322

SD₁₆

yes

—

C₂×#88322

SD₁₆

yes

—

C₁₆×C₅₃

—

#11719908

C₄²

yes

—

C₂³×C₅₃

#135

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

C₂×C₄×C₅₃

#135

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₂×C₄²

yes

characteristic

C₈×C₅₃

#117

yes

characteristic

C₈×C₅₃

C₄×Q₈

yes

—

yes

yes

—

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

#427

yes

characteristic

C₂²×C₅₃

C₄×SD₁₆

yes

—

C₄×C₅₃

C₄×C₄⋊C₄

yes

characteristic

C₄×C₅₃

C₂×#106

yes

characteristic

C₄×C₅₃

C₄×SD₁₆

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

#2155

—

C₄×M₃₂

—

#1772

—

#1385

D₅₃

yes

characteristic

#1384

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₄×#106

yes

characteristic

C₂×M₃₂

—

C₂×M₃₂

C₂×D₅₃

yes

characteristic

C₂×#108

C₂×D₅₃

yes

—

#427

—

#405

—

106

—

C₂×C₄×C₈

—

#365

—

#346

—

C₄×C₁₆

—

#339

—

106

—

100

C₅₃

#12668

yes

characteristic, complemented, Sylow

101

C₂×M₁₆

—

102

C₂²×C₈

—

103

C₂²×C₈

C₂²×D₅₃

yes

characteristic

104

#117

—

106

—

105

C₂×C₄²

—

106

M₃₂

—

107

M₃₂

—

108

C₂×C₁₆

—

109

C₂×C₁₆

C₄×D₅₃

yes

—

110

#108

—

111

#108

C₄×D₅₃

yes

—

112

#107

—

106

—

113

#106

—

114

#97

—

212

—

115

C₄×C₈

—

116

C₄×C₈

—

117

C₄×C₈

—

118

C₄×C₈

—

119

C₄×C₈

—

212

—

120

C₂²×C₄

—

106

—

121

C₂²×C₄

D₄×D₅₃

yes

characteristic

122

M₁₆

—

123

M₁₆

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

C₂×C₄×D₅₃

yes

characteristic

130

C₂×C₈

C₂×C₄×D₅₃

yes

—

131

C₂×C₈

#11719947

yes

—

132

C₂×C₈

Q₈×D₅₃

yes

characteristic

133

C₄⋊C₄

—

106

—

134

C₂²⋊C₄

—

212

—

135

C₄²

—

136

C₄²

—

137

C₄²

—

106

—

138

C₄²

—

212

—

139

C₁₆

—

140

C₁₆

—

141

C₂³

#13133240

yes

characteristic

142

C₂×C₄

—

106

—

143

C₂×C₄

—

106

—

144

C₂×C₄

—

212

—

145

C₂×C₄

—

212

—

146

C₂×C₄

C₄⋊C₄×D₅₃

yes

characteristic

147

C₂×C₄

#13133177

yes

—

148

C₂×C₄

SD₁₆×D₅₃

yes

—

149

C₂×C₄

#13133236

yes

characteristic

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

C₄²×D₅₃

yes

characteristic

155

C₈

#13133173

yes

characteristic

156

C₈

#13133182

yes

—

157

C₂²

—

158

C₂²

#16810660

yes

characteristic

159

C₂²

#16810708

yes

—

160

C₄

—

106

—

161

C₄

—

424

—

162

C₄

—

212

—

163

C₄

#16810474

yes

characteristic

164

C₄

#106×D₅₃

yes

characteristic

165

C₄

#16810714

yes

—

166

C₂

—

167

C₂

—

168

C₂

#20274603

yes

characteristic

	subgroup	G/H
center	C₄	#106×D₅₃
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1385	C₂
socle	C₂×C₅₃	C₄×#106
prime radical	C₂³×C₅₃	#135
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1385	D₅₃
hypocenter	C₅₃	#12668
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

subgroup

G/H

center

C₄

#106×D₅₃

commutator

C₈×C₅₃

C₂×C₄²

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₅₃×#1385

C₂

C₂×C₅₃

C₂³×C₅₃

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

N	H ≅ G/N	type
10. #16810447	C₄	semidirect
11. #16810398	C₄	semidirect
12. C₁₆×Dic₅₃	C₄	semidirect
100. C₅₃	#12668	semidirect

H ≅ G/N

type

10. #16810447

C₄

semidirect

11. #16810398

C₄

semidirect

12. C₁₆×Dic₅₃

C₄

semidirect

100. C₅₃

#12668

semidirect

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

subgroup	index
#20276489	2
M₃₂×Dic₅₃	2
#20274597	2
#20272817	2
C₅₃×#1385	2
#12668	53

subgroup

index

M₃₂×Dic₅₃

C₅₃×#1385

Subgroups — #27299472

Conjugacy classes of subgroups

Complements

Maximal subgroups