Group #27295726 — subgroups

Subgroups — #27295726

Order 13568.

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

normal subgroups only

This group has 232 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	#20276308	C₂	1	1	yes	maximal, characteristic
2	#20273469	C₂	1	2	yes	maximal
3	#20272781	C₂	1	1	yes	maximal, characteristic, complemented
4	#20272780	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1368	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13133156	C₄	1	1	yes	characteristic
7	C₂×#13133156	C₄	1	1	yes	characteristic
8	#16810645	—	2	2	no	—
9	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
10	#16810399	C₂²	1	1	yes	characteristic
11	C₁₆×Dic₅₃	C₄	1	1	yes	characteristic
12	C₁₆×Dic₅₃	C₄	1	1	yes	characteristic
13	#16810374	C₂²	1	2	yes	—
14	#16810373	—	2	2	no	—
15	C₅₃×#346	C₂²	1	1	yes	characteristic
16	C₅₃×#334	C₂²	1	2	yes	—
17	C₂×#11719912	—	4	2	no	—
18	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×D₅₃	Q₈	1	1	yes	characteristic
20	C₄²×D₅₃	D₄	1	1	yes	characteristic
21	#13133156	—	2	2	no	—
22	#13133156	—	2	2	no	—
23	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
24	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
25	C₄⋊#88322	—	2	2	no	—
26	#13133146	—	4	2	no	—
27	C₄×#88322	D₄	1	1	yes	characteristic
28	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₅₃	Q₈	1	1	yes	characteristic
30	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₄⋊C₈×C₅₃	—	2	2	no	—
33	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×D₅₃	SD₁₆	1	2	yes	—
36	C₂×C₈×C₅₃	—	4	2	no	—
37	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
39	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
41	C₄×Dic₅₃	SD₁₆	1	2	yes	—
42	C₂×#88322	—	4	2	no	—
43	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
44	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
45	#11719912	—	8	4	no	—
46	C₈×D₅₃	—	2	1	no	—
47	C₈×D₅₃	—	2	1	no	—
48	C₈×D₅₃	—	2	2	no	—
49	C₁₆×C₅₃	—	2	1	no	—
50	C₁₆×C₅₃	M₁₆	1	2	yes	complemented
51	#11719908	—	2	1	no	—
52	#11719908	M₁₆	1	2	yes	complemented
53	C₂²×D₅₃	#106	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
56	C₂×Dic₅₃	#106	1	1	yes	characteristic
57	C₂×Dic₅₃	#106	1	2	yes	—
58	C₄×D₅₃	—	2	1	no	—
59	C₄×D₅₃	—	2	1	no	—
60	C₄×D₅₃	—	2	4	no	—
61	C₈×C₅₃	—	2	1	no	—
62	C₈×C₅₃	—	8	2	no	—
63	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
64	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
65	#88322	—	2	1	no	—
66	#88322	—	8	2	no	—
67	#88322	#108	1	2	yes	—
68	#12020	—	53	1	no	maximal, Sylow
69	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
70	C₂×D₅₃	—	2	1	no	—
71	C₂×D₅₃	—	2	2	no	—
72	C₄×C₅₃	—	2	2	no	—
73	C₄×C₅₃	#423	1	1	yes	characteristic
74	C₄×C₅₃	C₂×#108	1	1	yes	characteristic
75	Dic₅₃	—	2	1	no	—
76	Dic₅₃	#365	1	4	yes	—
77	C₄×M₃₂	—	53	1	no	—
78	#1566	—	53	2	no	—
79	#1368	—	53	1	no	—
80	#1368	—	53	2	no	—
81	#1368	D₅₃	1	1	yes	characteristic, complemented
82	C₂×C₅₃	#1566	1	1	yes	characteristic
83	C₂×C₅₃	C₂×#365	1	2	yes	—
84	D₅₃	—	4	1	no	—
85	C₂×M₃₂	—	53	1	no	—
86	C₂×M₃₂	—	53	1	no	—
87	#423	—	106	2	no	—
88	C₂×C₄×C₈	—	53	1	no	—
89	#346	—	53	1	no	—
90	#346	C₂×D₅₃	1	1	yes	characteristic
91	C₄×C₁₆	—	53	1	no	—
92	C₄×C₁₆	—	53	1	no	—
93	#334	—	53	2	no	—
94	#334	—	106	2	no	—
95	#334	C₂×D₅₃	1	2	yes	—
96	C₅₃	#12020	1	1	yes	characteristic, complemented, Sylow
97	C₂×M₁₆	—	212	2	no	—
98	C₂²×C₈	—	53	1	no	—
99	C₂²×C₈	—	53	1	no	—
100	C₂×C₄²	—	53	1	no	—
101	M₃₂	—	106	2	no	—
102	M₃₂	—	106	2	no	—
103	C₂×C₁₆	—	53	1	no	—
104	C₂×C₁₆	—	53	1	no	—
105	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
106	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
107	C₄⋊C₈	—	2	2	no	—
108	C₄⋊C₈	—	106	2	no	—
109	C₄⋊C₈	—	212	2	no	—
110	C₄×C₈	—	53	1	no	—
111	C₄×C₈	—	53	1	no	—
112	C₄×C₈	—	53	1	no	—
113	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
114	C₂²×C₄	—	53	1	no	—
115	C₂²×C₄	—	53	2	no	—
116	M₁₆	—	424	4	no	—
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	53	1	no	—
119	C₂×C₈	—	53	1	no	—
120	C₂×C₈	—	106	2	no	—
121	C₂×C₈	—	106	1	no	—
122	C₂×C₈	—	106	1	no	—
123	C₂×C₈	—	212	2	no	—
124	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
125	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
126	C₄²	—	53	1	no	—
127	C₄²	—	53	2	no	—
128	C₄²	D₄×D₅₃	1	1	yes	characteristic
129	C₁₆	—	2	1	no	—
130	C₁₆	—	53	2	no	—
131	C₁₆	—	106	1	no	—
132	C₁₆	#11719912	1	2	yes	complemented
133	C₂³	—	53	1	no	—
134	C₂×C₄	—	53	2	no	—
135	C₂×C₄	—	53	1	no	—
136	C₂×C₄	—	106	4	no	—
137	C₂×C₄	—	106	1	no	—
138	C₂×C₄	—	106	1	no	—
139	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
140	C₂×C₄	SD₁₆×D₅₃	1	2	yes	—
141	C₈	—	2	1	no	—
142	C₈	—	8	2	no	—
143	C₈	—	53	2	no	—
144	C₈	—	106	1	no	—
145	C₈	—	424	2	no	—
146	C₈	M₁₆×D₅₃	1	1	yes	characteristic
147	C₈	C₂×#11719912	1	1	yes	characteristic
148	C₂²	—	106	2	no	—
149	C₂²	—	106	1	no	—
150	C₂²	#106×D₅₃	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	53	4	no	—
153	C₄	—	106	1	no	—
154	C₄	#16810645	1	1	yes	characteristic
155	C₄	#108×D₅₃	1	1	yes	characteristic
156	C₂	—	212	1	no	—
157	C₂	#20273469	1	1	yes	characteristic
158	C₂	#365×D₅₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20276308

C₂

yes

maximal, characteristic

#20273469

C₂

yes

maximal

#20272781

C₂

yes

maximal, characteristic, complemented

#20272780

C₂

yes

maximal, complemented

C₅₃×#1368

C₂

yes

maximal, characteristic, complemented

C₂×#13133156

C₄

yes

characteristic

C₂×#13133156

C₄

yes

characteristic

#16810645

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

#16810399

C₂²

yes

characteristic

C₁₆×Dic₅₃

C₄

yes

characteristic

C₁₆×Dic₅₃

C₄

yes

characteristic

#16810374

C₂²

yes

—

#16810373

—

C₅₃×#346

C₂²

yes

characteristic

C₅₃×#334

C₂²

yes

—

C₂×#11719912

—

C₂×C₈×D₅₃

C₂×C₄

yes

characteristic

C₂×C₈×D₅₃

Q₈

yes

characteristic

C₄²×D₅₃

D₄

yes

—

—

C₂×C₄

yes

characteristic

C₂×#11719908

C₂×C₄

yes

—

—

D₄

yes

characteristic

C₈×Dic₅₃

C₂×C₄

yes

characteristic

C₈×Dic₅₃

Q₈

yes

characteristic

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₄×D₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×D₅₃

SD₁₆

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×Q₈

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄×Dic₅₃

C₄⋊C₄

yes

characteristic

C₄×Dic₅₃

SD₁₆

yes

—

C₂×#88322

—

C₂×#88322

C₄⋊C₄

yes

characteristic

C₂×#88322

C₄⋊C₄

yes

characteristic

#11719912

—

C₈×D₅₃

—

C₈×D₅₃

—

C₈×D₅₃

—

C₁₆×C₅₃

—

C₁₆×C₅₃

M₁₆

yes

complemented

#11719908

—

#11719908

M₁₆

yes

complemented

C₂²×D₅₃

#106

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

C₂×Dic₅₃

#106

yes

characteristic

C₂×Dic₅₃

#106

yes

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₂×M₁₆

yes

characteristic

C₈×C₅₃

C₂×M₁₆

yes

—

—

yes

—

—

C₂²×C₅₃

yes

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

#365

yes

—

C₄×M₃₂

—

—

—

—

D₅₃

yes

characteristic, complemented

C₂×C₅₃

#1566

yes

characteristic

C₂×C₅₃

C₂×#365

yes

—

D₅₃

—

C₂×M₃₂

—

C₂×M₃₂

—

#423

—

106

—

C₂×C₄×C₈

—

#346

—

#346

C₂×D₅₃

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

—

#334

—

#334

—

106

—

#334

C₂×D₅₃

yes

—

C₅₃

#12020

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

212

—

C₂²×C₈

—

C₂²×C₈

—

100

C₂×C₄²

—

101

M₃₂

—

106

—

102

M₃₂

—

106

—

103

C₂×C₁₆

—

104

C₂×C₁₆

—

105

C₂×C₁₆

C₄×D₅₃

yes

characteristic

106

C₂×C₁₆

C₄×D₅₃

yes

characteristic

107

C₄⋊C₈

—

108

C₄⋊C₈

—

106

—

109

C₄⋊C₈

—

212

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

C₂²×D₅₃

yes

characteristic

114

C₂²×C₄

—

115

C₂²×C₄

—

116

M₁₆

—

424

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

106

—

121

C₂×C₈

—

106

—

122

C₂×C₈

—

106

—

123

C₂×C₈

—

212

—

124

C₂×C₈

C₂×C₄×D₅₃

yes

characteristic

125

C₂×C₈

Q₈×D₅₃

yes

characteristic

126

C₄²

—

127

C₄²

—

128

C₄²

D₄×D₅₃

yes

characteristic

129

C₁₆

—

130

C₁₆

—

131

C₁₆

—

106

—

132

C₁₆

#11719912

yes

complemented

133

C₂³

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

106

—

137

C₂×C₄

—

106

—

138

C₂×C₄

—

106

—

139

C₂×C₄

C₄⋊C₄×D₅₃

yes

characteristic

140

C₂×C₄

SD₁₆×D₅₃

yes

—

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

106

—

145

C₈

—

424

—

146

C₈

M₁₆×D₅₃

yes

characteristic

147

C₈

C₂×#11719912

yes

characteristic

148

C₂²

—

106

—

149

C₂²

—

106

—

150

C₂²

#106×D₅₃

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

106

—

154

C₄

#16810645

yes

characteristic

155

C₄

#108×D₅₃

yes

characteristic

156

C₂

—

212

—

157

C₂

#20273469

yes

characteristic

158

C₂

#365×D₅₃

yes

—

	subgroup	G/H
center	C₂²	#106×D₅₃
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#1368	C₂
socle	C₂²×C₅₃	C₂×#106
prime radical	C₂²×D₅₃	#106
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#1368	D₅₃
hypocenter	C₅₃	#12020
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1368

subgroup

G/H

center

C₂²

#106×D₅₃

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₅₃

Fitting

C₅₃×#1368

C₂

C₂²×C₅₃

C₂²×D₅₃

C₄×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1368

N	H ≅ G/N	type
3. #20272781	C₂	semidirect
4. #20272780	C₂	semidirect
5. C₅₃×#1368	C₂	semidirect
50. C₁₆×C₅₃	M₁₆	semidirect
52. #11719908	M₁₆	semidirect
81. #1368	D₅₃	semidirect
96. C₅₃	#12020	semidirect
132. C₁₆	#11719912	semidirect

H ≅ G/N

type

3. #20272781

C₂

semidirect

4. #20272780

C₂

semidirect

5. C₅₃×#1368

C₂

semidirect

50. C₁₆×C₅₃

M₁₆

semidirect

52. #11719908

M₁₆

semidirect

81. #1368

D₅₃

semidirect

96. C₅₃

#12020

semidirect

132. C₁₆

#11719912

semidirect

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

subgroup	index
#20276308	2
#20273469	2
#20272781	2
#20272780	2
C₅₃×#1368	2
#12020	53

subgroup

index

C₅₃×#1368

Subgroups — #27295726

Conjugacy classes of subgroups

Complements

Maximal subgroups