Group #27299503 — subgroups

Subgroups — #27299503

Order 13568.

314 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 139 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	C₂×#16810452	C₂	1	2	yes	maximal
2	C₂×C₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#108×Dic₅₃	C₂	1	2	yes	maximal
4	#20272809	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1381	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719908	C₂²	1	1	yes	characteristic
7	C₂×#13133153	C₂²	1	2	yes	—
8	M₁₆×Dic₅₃	—	2	2	no	—
9	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
10	#16810452	C₄	1	8	yes	complemented
11	C₁₆×Dic₅₃	C₄	1	4	yes	—
12	#16810387	—	2	2	no	—
13	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#108×C₅₃	C₂²	1	2	yes	—
15	C₂×#11719917	—	2	2	no	—
16	C₂²×#88322	D₄	1	1	yes	characteristic
17	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
18	C₂×#11719908	C₂×C₄	1	2	yes	—
19	C₂×#11719908	C₂×C₄	1	4	yes	—
20	#13133153	—	2	2	no	—
21	#13133153	C₂×C₄	1	4	yes	—
22	#13133127	—	4	2	no	—
23	C₈×Dic₅₃	—	4	2	no	—
24	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
25	C₈×Dic₅₃	Q₈	1	2	yes	—
26	C₂×M₁₆×C₅₃	—	2	2	no	—
27	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
28	C₂×C₁₆×C₅₃	—	2	2	no	—
29	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
30	#108×C₅₃	—	2	2	no	—
31	#108×C₅₃	C₂×C₄	1	4	yes	—
32	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
33	C₂²×Dic₅₃	Q₁₆	1	2	yes	—
34	M₁₆×C₅₃	—	2	4	no	—
35	M₁₆×C₅₃	—	4	2	no	—
36	C₂×C₈×C₅₃	—	4	2	no	—
37	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
39	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₈×C₅₃	#41	1	2	yes	—
41	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
42	C₄×Dic₅₃	D₈	1	1	yes	characteristic
43	C₄×Dic₅₃	D₈	1	1	yes	characteristic
44	#11719917	—	2	4	no	—
45	#11719917	—	4	2	no	—
46	C₂×#88322	—	2	2	no	—
47	C₂×#88322	—	4	2	no	—
48	C₂×#88322	C₄⋊C₄	1	2	yes	—
49	C₁₆×C₅₃	—	2	2	no	—
50	#11719908	C₄²	1	4	yes	—
51	C₂³×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
54	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₅₃	#135	1	2	yes	—
56	C₂×Dic₅₃	—	2	4	no	—
57	C₂×Dic₅₃	#107	1	2	yes	—
58	C₂×Dic₅₃	#107	1	2	yes	—
59	C₈×C₅₃	—	4	4	no	—
60	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
61	C₈×C₅₃	#117	1	1	yes	characteristic
62	C₈×C₅₃	C₄×Q₈	1	2	yes	—
63	#88322	—	2	2	no	—
64	#88322	—	4	4	no	—
65	C₄×#368	—	53	1	no	maximal, Sylow
66	C₂²×C₅₃	—	2	2	no	—
67	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
68	C₂²×C₅₃	C₄×Q₁₆	1	2	yes	—
69	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
70	C₄×C₅₃	#427	1	1	yes	characteristic
71	C₄×C₅₃	C₄×D₈	1	2	yes	—
72	Dic₅₃	—	2	2	no	—
73	Dic₅₃	#368	1	4	yes	complemented
74	C₂×#368	—	53	2	no	—
75	C₂×C₄×C₁₆	—	53	1	no	—
76	C₄×#108	—	53	2	no	—
77	#1381	—	53	1	no	—
78	#1381	D₅₃	1	1	yes	characteristic
79	C₂×C₅₃	—	2	2	no	—
80	C₂×C₅₃	C₄×#107	1	1	yes	characteristic
81	C₂×C₅₃	C₂×#368	1	2	yes	—
82	C₂²×C₁₆	—	53	1	no	—
83	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
84	C₂×#108	—	53	2	no	—
85	C₂×#108	C₂×D₅₃	1	2	yes	—
86	C₄×M₁₆	—	106	2	no	—
87	C₂×C₄×C₈	—	53	1	no	—
88	#368	—	53	8	no	—
89	C₄×C₁₆	—	53	4	no	—
90	#341	—	106	2	no	—
91	C₅₃	C₄×#368	1	1	yes	characteristic, complemented, Sylow
92	C₂×M₁₆	—	2	2	no	—
93	C₂×M₁₆	—	106	2	no	—
94	C₂²×C₈	—	53	1	no	—
95	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
96	C₂×C₄²	—	53	1	no	—
97	C₂×C₁₆	—	2	2	no	—
98	C₂×C₁₆	—	53	2	no	—
99	C₂×C₁₆	—	53	4	no	—
100	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
101	#108	—	2	2	no	—
102	#108	—	53	4	no	—
103	#108	—	106	2	no	—
104	#108	C₄×D₅₃	1	4	yes	—
105	#97	—	212	2	no	—
106	C₄×C₈	—	53	2	no	—
107	C₄×C₈	—	53	2	no	—
108	C₄×C₈	—	212	2	no	—
109	C₂²×C₄	—	53	2	no	—
110	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
111	M₁₆	—	2	4	no	—
112	M₁₆	—	4	2	no	—
113	M₁₆	—	106	4	no	—
114	M₁₆	—	212	2	no	—
115	C₂×C₈	—	4	2	no	—
116	C₂×C₈	—	53	2	no	—
117	C₂×C₈	—	106	2	no	—
118	C₂×C₈	—	212	2	no	—
119	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
120	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
121	C₂×C₈	#11719947	1	2	yes	—
122	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
123	C₄²	—	53	2	no	—
124	C₄²	—	53	1	no	—
125	C₄²	—	53	1	no	—
126	C₁₆	—	2	2	no	—
127	C₁₆	—	53	4	no	—
128	C₂³	Q₁₆×D₅₃	1	1	yes	characteristic
129	C₂×C₄	—	53	2	no	—
130	C₂×C₄	—	53	2	no	—
131	C₂×C₄	—	106	4	no	—
132	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
133	C₂×C₄	#13133177	1	2	yes	—
134	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
135	C₂×C₄	#13133238	1	2	yes	—
136	C₈	—	4	4	no	—
137	C₈	—	106	2	no	—
138	C₈	—	212	4	no	—
139	C₈	C₄²×D₅₃	1	1	yes	characteristic
140	C₈	#13133173	1	1	yes	characteristic
141	C₈	#13133182	1	2	yes	—
142	C₂²	—	2	2	no	—
143	C₂²	#107×D₅₃	1	1	yes	characteristic
144	C₂²	#16810715	1	2	yes	—
145	C₄	—	53	4	no	—
146	C₄	—	106	2	no	—
147	C₄	#16810474	1	1	yes	characteristic
148	C₄	#16810658	1	1	yes	characteristic
149	C₄	#16810703	1	2	yes	—
150	C₂	—	2	2	no	—
151	C₂	#20274607	1	1	yes	characteristic
152	C₂	#368×D₅₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16810452

C₂

yes

maximal

C₂×C₁₆×Dic₅₃

C₂

yes

maximal, characteristic

#108×Dic₅₃

C₂

yes

maximal

#20272809

C₂

yes

maximal, characteristic

C₅₃×#1381

C₂

yes

maximal, characteristic

C₂²×#11719908

C₂²

yes

characteristic

C₂×#13133153

C₂²

yes

—

M₁₆×Dic₅₃

—

C₂×C₈×Dic₅₃

C₂²

yes

characteristic

#16810452

C₄

yes

complemented

C₁₆×Dic₅₃

C₄

yes

—

#16810387

—

C₂²×C₁₆×C₅₃

C₂²

yes

characteristic

C₂×#108×C₅₃

C₂²

yes

—

C₂×#11719917

—

C₂²×#88322

D₄

yes

characteristic

C₂×C₄×Dic₅₃

D₄

yes

characteristic

C₂×#11719908

C₂×C₄

yes

—

C₂×#11719908

C₂×C₄

yes

—

#13133153

—

#13133153

C₂×C₄

yes

—

#13133127

—

C₈×Dic₅₃

—

C₈×Dic₅₃

C₂×C₄

yes

—

C₈×Dic₅₃

Q₈

yes

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

#108×C₅₃

—

#108×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×Dic₅₃

Q₁₆

yes

—

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

—

C₄×Dic₅₃

C₄⋊C₄

yes

—

C₄×Dic₅₃

D₈

yes

characteristic

C₄×Dic₅₃

D₈

yes

—

—

—

—

C₄⋊C₄

yes

—

C₁₆×C₅₃

—

#11719908

C₄²

yes

—

C₂³×C₅₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

#135

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

#107

yes

—

C₂×Dic₅₃

#107

yes

—

C₈×C₅₃

—

C₈×C₅₃

C₂×C₄²

yes

characteristic

C₈×C₅₃

#117

yes

characteristic

C₈×C₅₃

C₄×Q₈

yes

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×#107

yes

characteristic

C₂²×C₅₃

C₄×Q₁₆

yes

—

C₄×C₅₃

C₄×C₄⋊C₄

yes

characteristic

C₄×C₅₃

#427

yes

characteristic

C₄×C₅₃

C₄×D₈

yes

—

Dic₅₃

—

Dic₅₃

#368

yes

complemented

C₂×#368

—

C₂×C₄×C₁₆

—

C₄×#108

—

#1381

—

#1381

D₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

C₄×#107

yes

characteristic

C₂×C₅₃

C₂×#368

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₅₃

yes

characteristic

C₂×#108

—

C₂×#108

C₂×D₅₃

yes

—

C₄×M₁₆

—

106

—

C₂×C₄×C₈

—

#368

—

C₄×C₁₆

—

#341

—

106

—

C₅₃

C₄×#368

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

106

—

C₂²×C₈

—

C₂²×C₈

C₂²×D₅₃

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

100

C₂×C₁₆

C₄×D₅₃

yes

—

101

#108

—

102

#108

—

103

#108

—

106

—

104

#108

C₄×D₅₃

yes

—

105

#97

—

212

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

212

—

109

C₂²×C₄

—

110

C₂²×C₄

D₄×D₅₃

yes

characteristic

111

M₁₆

—

112

M₁₆

—

113

M₁₆

—

106

—

114

M₁₆

—

212

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

106

—

118

C₂×C₈

—

212

—

119

C₂×C₈

C₂×C₄×D₅₃

yes

characteristic

120

C₂×C₈

C₂×C₄×D₅₃

yes

—

121

C₂×C₈

#11719947

yes

—

122

C₂×C₈

Q₈×D₅₃

yes

characteristic

123

C₄²

—

124

C₄²

—

125

C₄²

—

126

C₁₆

—

127

C₁₆

—

128

C₂³

Q₁₆×D₅₃

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

106

—

132

C₂×C₄

C₄⋊C₄×D₅₃

yes

characteristic

133

C₂×C₄

#13133177

yes

—

134

C₂×C₄

D₈×D₅₃

yes

characteristic

135

C₂×C₄

#13133238

yes

—

136

C₈

—

137

C₈

—

106

—

138

C₈

—

212

—

139

C₈

C₄²×D₅₃

yes

characteristic

140

C₈

#13133173

yes

characteristic

141

C₈

#13133182

yes

—

142

C₂²

—

143

C₂²

#107×D₅₃

yes

characteristic

144

C₂²

#16810715

yes

—

145

C₄

—

146

C₄

—

106

—

147

C₄

#16810474

yes

characteristic

148

C₄

#16810658

yes

characteristic

149

C₄

#16810703

yes

—

150

C₂

—

151

C₂

#20274607

yes

characteristic

152

C₂

#368×D₅₃

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₅₃
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1381	C₂
socle	C₂²×C₅₃	C₂×#107
prime radical	C₂³×C₅₃	C₂×Q₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1381	D₅₃
hypocenter	C₅₃	C₄×#368
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

subgroup

G/H

center

C₂×C₄

D₈×D₅₃

commutator

C₈×C₅₃

C₂×C₄²

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₅₃×#1381

C₂

socle

C₂²×C₅₃

C₂×#107

prime radical

C₂³×C₅₃

C₂×Q₁₆

cosocle

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

N	H ≅ G/N	type
10. #16810452	C₄	semidirect
73. Dic₅₃	#368	semidirect
91. C₅₃	C₄×#368	semidirect

H ≅ G/N

type

10. #16810452

C₄

semidirect

73. Dic₅₃

#368

semidirect

91. C₅₃

C₄×#368

semidirect

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

subgroup	index
C₂×#16810452	2
C₂×C₁₆×Dic₅₃	2
#108×Dic₅₃	2
#20272809	2
C₅₃×#1381	2
C₄×#368	53

subgroup

index

C₂×#16810452

C₂×C₁₆×Dic₅₃

#108×Dic₅₃

#20272809

C₅₃×#1381

C₄×#368

Subgroups — #27299503

Conjugacy classes of subgroups

Complements

Maximal subgroups