Group #27270948 — subgroups

Subgroups — #27270948

Order 13568.

302 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 127 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	#20276514	C₂	1	2	yes	maximal
2	C₄²×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11719908	C₄	1	4	yes	complemented
4	C₂²⋊#11719908	—	2	2	no	—
5	C₄×#11719908	—	2	2	no	—
6	C₂×C₄×C₈×C₅₃	—	2	2	no	—
7	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₄³×C₅₃	C₄	1	1	yes	characteristic
10	C₂×#11719908	—	2	4	no	—
11	C₂²×C₈×C₅₃	—	2	1	no	—
12	C₂²×C₈×C₅₃	D₄	1	2	yes	—
13	C₂×C₄²×C₅₃	—	2	1	no	—
14	C₂×C₄²×C₅₃	—	2	1	no	—
15	C₂×C₄²×C₅₃	C₈	1	2	yes	—
16	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₄×C₈×C₅₃	—	2	8	no	—
18	C₄×C₈×C₅₃	—	2	1	no	—
19	C₄×C₈×C₅₃	C₈	1	2	yes	—
20	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
22	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	—	2	2	no	—
24	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
27	C₂×C₈×C₅₃	—	2	4	no	—
28	C₂×C₈×C₅₃	—	2	2	no	—
29	C₂×C₈×C₅₃	—	2	4	no	—
30	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
31	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
32	C₄²×C₅₃	—	2	4	no	—
33	C₄²×C₅₃	—	2	4	no	—
34	C₄²×C₅₃	—	2	2	no	—
35	C₄²×C₅₃	C₁₆	1	4	yes	complemented
36	C₄²×C₅₃	C₄²	1	1	yes	characteristic
37	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
38	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
39	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
40	#11719908	—	4	4	no	—
41	C₂³×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
42	C₂×C₄×C₅₃	—	2	4	no	—
43	C₂×C₄×C₅₃	—	2	4	no	—
44	C₂×C₄×C₅₃	—	2	4	no	—
45	C₂×C₄×C₅₃	—	2	2	no	—
46	C₂×C₄×C₅₃	—	2	2	no	—
47	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
49	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
54	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
55	C₈×C₅₃	—	2	4	no	—
56	C₈×C₅₃	—	2	2	no	—
57	C₈×C₅₃	#104	1	4	yes	—
58	#7148	—	53	1	no	maximal, Sylow
59	C₂²×C₅₃	—	2	1	no	—
60	C₂²×C₅₃	—	2	1	no	—
61	C₂²×C₅₃	#336	1	1	yes	characteristic
62	C₂²×C₅₃	C₂²⋊C₁₆	1	2	yes	—
63	C₄×C₅₃	—	2	8	no	—
64	C₄×C₅₃	—	2	1	no	—
65	C₄×C₅₃	—	2	1	no	—
66	C₄×C₅₃	#339	1	2	yes	—
67	C₄×C₅₃	C₄×C₁₆	1	1	yes	characteristic
68	C₄×C₅₃	#346	1	1	yes	characteristic
69	C₄×C₅₃	C₄⋊C₁₆	1	2	yes	—
70	C₄×C₅₃	#364	1	2	yes	—
71	#2162	—	53	2	no	—
72	C₄²×C₈	D₅₃	1	1	yes	characteristic
73	C₂×C₅₃	—	2	2	no	—
74	C₂×C₅₃	#1276	1	1	yes	characteristic
75	C₂×C₅₃	#1374	1	1	yes	characteristic
76	C₂×C₅₃	#1376	1	1	yes	characteristic
77	C₂×C₄×C₈	—	2	2	no	—
78	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
79	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
80	C₄³	Dic₅₃	1	1	yes	characteristic
81	C₄⋊C₁₆	—	53	4	no	—
82	C₂²⋊C₁₆	—	106	2	no	—
83	C₄×C₁₆	—	106	2	no	—
84	C₅₃	#7148	1	1	yes	characteristic, complemented, Sylow
85	C₂²×C₈	—	2	1	no	—
86	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
87	C₂×C₄²	—	2	1	no	—
88	C₂×C₄²	—	2	1	no	—
89	C₂×C₄²	#88322	1	2	yes	—
90	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
91	C₂×C₁₆	—	106	4	no	—
92	C₄×C₈	—	2	8	no	—
93	C₄×C₈	—	2	1	no	—
94	C₄×C₈	#88322	1	2	yes	—
95	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
96	C₄×C₈	C₄×D₅₃	1	2	yes	—
97	C₄×C₈	D₂₁₂	1	1	yes	characteristic
98	C₂²×C₄	—	2	2	no	—
99	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
100	C₂²×C₄	#11719917	1	1	yes	characteristic
101	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
102	C₂×C₈	—	2	4	no	—
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	2	4	no	—
105	C₂×C₈	#11719919	1	2	yes	—
106	C₂×C₈	#11719921	1	2	yes	—
107	C₄²	—	2	4	no	—
108	C₄²	—	2	4	no	—
109	C₄²	—	2	2	no	—
110	C₄²	#11719908	1	4	yes	complemented
111	C₄²	C₂×#88322	1	1	yes	characteristic
112	C₄²	#11719917	1	1	yes	characteristic
113	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
114	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
115	C₁₆	—	212	4	no	—
116	C₂³	C₂²⋊#88322	1	1	yes	characteristic
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	#13133124	1	1	yes	characteristic
123	C₂×C₄	C₄×#88322	1	1	yes	characteristic
124	C₂×C₄	#13133128	1	1	yes	characteristic
125	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
126	C₂×C₄	C₄⋊#88322	1	1	yes	characteristic
127	C₂×C₄	C₄⋊#88322	1	1	yes	characteristic
128	C₂×C₄	C₂×#11719908	1	2	yes	—
129	C₂×C₄	#13133157	1	2	yes	—
130	C₈	—	2	4	no	—
131	C₈	—	2	2	no	—
132	C₈	#13133145	1	4	yes	—
133	C₂²	—	2	1	no	—
134	C₂²	—	2	1	no	—
135	C₂²	#16810378	1	1	yes	characteristic
136	C₂²	C₂²⋊#11719908	1	2	yes	—
137	C₄	—	2	8	no	—
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	#16810383	1	2	yes	—
141	C₄	C₄×#11719908	1	1	yes	characteristic
142	C₄	#16810399	1	1	yes	characteristic
143	C₄	C₄⋊#11719908	1	2	yes	—
144	C₄	#16810444	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	#20272548	1	1	yes	characteristic
147	C₂	#20272793	1	1	yes	characteristic
148	C₂	#20272797	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276514

C₂

yes

maximal

C₄²×C₈×C₅₃

C₂

yes

maximal, characteristic

C₄⋊#11719908

C₄

yes

complemented

C₂²⋊#11719908

—

C₄×#11719908

—

C₂×C₄×C₈×C₅₃

—

C₂×C₄×C₈×C₅₃

C₄

yes

characteristic

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₄³×C₅₃

C₄

yes

characteristic

C₂×#11719908

—

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₈

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

C₈

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

Q₈

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₅₃

M₁₆

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₁₆

yes

complemented

C₄²×C₅₃

C₄²

yes

characteristic

C₄²×C₅₃

C₄⋊C₄

yes

characteristic

C₄²×C₅₃

C₂×C₈

yes

characteristic

C₄²×C₅₃

M₁₆

yes

characteristic

#11719908

—

C₂³×C₅₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#95

yes

characteristic

C₂×C₄×C₅₃

C₄×C₈

yes

characteristic

C₂×C₄×C₅₃

#97

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₈

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×C₁₆

yes

—

C₂×C₄×C₅₃

M₃₂

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#104

yes

—

#7148

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#336

yes

characteristic

C₂²×C₅₃

C₂²⋊C₁₆

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#339

yes

—

C₄×C₅₃

C₄×C₁₆

yes

characteristic

C₄×C₅₃

#346

yes

characteristic

C₄×C₅₃

C₄⋊C₁₆

yes

—

C₄×C₅₃

#364

yes

—

#2162

—

C₄²×C₈

D₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄×C₈

—

C₂×C₄×C₈

Dic₅₃

yes

characteristic

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

C₄³

Dic₅₃

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

106

—

C₄×C₁₆

—

106

—

C₅₃

#7148

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

#88322

yes

—

C₂×C₄²

C₂×Dic₅₃

yes

characteristic

C₂×C₁₆

—

106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

#88322

yes

—

C₄×C₈

Dic₁₀₆

yes

characteristic

C₄×C₈

C₄×D₅₃

yes

—

C₄×C₈

D₂₁₂

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂×#88322

yes

characteristic

100

C₂²×C₄

#11719917

yes

characteristic

101

C₂²×C₄

C₂²⋊Dic₅₃

yes

characteristic

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

#11719919

yes

—

106

C₂×C₈

#11719921

yes

—

107

C₄²

—

108

C₄²

—

109

C₄²

—

110

C₄²

#11719908

yes

complemented

111

C₄²

C₂×#88322

yes

characteristic

112

C₄²

#11719917

yes

characteristic

113

C₄²

C₄×Dic₅₃

yes

characteristic

114

C₄²

C₄⋊Dic₅₃

yes

characteristic

115

C₁₆

—

212

—

116

C₂³

C₂²⋊#88322

yes

characteristic

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

#13133124

yes

characteristic

123

C₂×C₄

C₄×#88322

yes

characteristic

124

C₂×C₄

#13133128

yes

characteristic

125

C₂×C₄

C₂²⋊#88322

yes

characteristic

126

C₂×C₄

C₄⋊#88322

yes

characteristic

127

C₂×C₄

C₄⋊#88322

yes

characteristic

128

C₂×C₄

C₂×#11719908

yes

—

129

C₂×C₄

#13133157

yes

—

130

C₈

—

131

C₈

—

132

C₈

#13133145

yes

—

133

C₂²

—

134

C₂²

—

135

C₂²

#16810378

yes

characteristic

136

C₂²

C₂²⋊#11719908

yes

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

#16810383

yes

—

141

C₄

C₄×#11719908

yes

characteristic

142

C₄

#16810399

yes

characteristic

143

C₄

C₄⋊#11719908

yes

—

144

C₄

#16810444

yes

—

145

C₂

—

146

C₂

#20272548

yes

characteristic

147

C₂

#20272793

yes

characteristic

148

C₂

#20272797

yes

characteristic

	subgroup	G/H
center	C₄×C₈	D₂₁₂
commutator	C₄×C₅₃	C₄×C₁₆
Frattini	C₂×C₄×C₈	C₂×D₅₃
Fitting	C₄²×C₈×C₅₃	C₂
socle	C₂²×C₅₃	#336
prime radical	C₂³×C₅₃	C₂²⋊C₈
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	C₄²×C₈	D₅₃
hypocenter	C₅₃	#7148
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×C₈

subgroup

G/H

center

C₄×C₈

D₂₁₂

commutator

C₄×C₅₃

C₄×C₁₆

Frattini

C₂×C₄×C₈

C₂×D₅₃

Fitting

C₄²×C₈×C₅₃

C₂

socle

C₂²×C₅₃

#336

prime radical

C₂³×C₅₃

C₂²⋊C₈

cosocle

C₂×C₄×C₈×C₅₃

C₂²

hypercenter

C₄²×C₈

D₅₃

hypocenter

C₅₃

#7148

LCS

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×C₈

N	H ≅ G/N	type
3. C₄⋊#11719908	C₄	semidirect
35. C₄²×C₅₃	C₁₆	semidirect
84. C₅₃	#7148	semidirect
110. C₄²	#11719908	semidirect

H ≅ G/N

type

3. C₄⋊#11719908

C₄

semidirect

35. C₄²×C₅₃

C₁₆

semidirect

84. C₅₃

#7148

semidirect

110. C₄²

#11719908

semidirect

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

subgroup	index
#20276514	2
C₄²×C₈×C₅₃	2
#7148	53

subgroup

index

#20276514

C₄²×C₈×C₅₃

#7148

Subgroups — #27270948

Conjugacy classes of subgroups

Complements

Maximal subgroups