Group #27349840 — subgroups

Subgroups — #27349840

Order 13568.

712 conjugacy classes of subgroups; only the 323 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 323 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	#20283346	C₂	1	3	yes	maximal
2	#20273018	C₂	1	3	yes	maximal, complemented
3	#20273016	C₂	1	6	yes	maximal, complemented
4	#20273014	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1455	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
7	#16810678	C₄	1	12	yes	—
8	#16810674	C₂²	1	6	yes	—
9	#16810674	C₂²	1	6	yes	—
10	#16810646	C₄	1	6	yes	—
11	#16810586	C₄	1	6	yes	—
12	C₄×#11719917	C₂²	1	3	yes	—
13	#16810345	C₂²	1	6	yes	complemented
14	#16810344	C₂²	1	3	yes	complemented
15	C₈×#88322	C₂²	1	3	yes	complemented
16	C₄×M₁₆×C₅₃	C₂²	1	3	yes	—
17	C₅₃×#322	C₂²	1	3	yes	complemented
18	C₈²×C₅₃	C₂²	1	1	yes	characteristic, complemented
19	C₂²×C₄×D₅₃	C₂×C₄	1	3	yes	—
20	C₂×#11719917	C₂×C₄	1	6	yes	—
21	C₂×C₄×Dic₅₃	C₂×C₄	1	3	yes	—
22	C₄²×D₅₃	C₂×C₄	1	6	yes	—
23	C₄²×D₅₃	C₂³	1	2	yes	—
24	#13133146	C₂×C₄	1	12	yes	—
25	#13133146	C₂×C₄	1	12	yes	—
26	#13133129	C₂×C₄	1	12	yes	—
27	#13133129	C₂×C₄	1	12	yes	—
28	#13133128	C₂×C₄	1	6	yes	—
29	C₄×#88322	C₂³	1	6	yes	—
30	C₂×M₁₆×C₅₃	C₂×C₄	1	6	yes	—
31	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
32	#97×C₅₃	C₂×C₄	1	6	yes	—
33	C₄×C₈×C₅₃	C₂³	1	3	yes	—
34	C₄×C₈×C₅₃	C₂³	1	3	yes	—
35	C₂³×D₅₃	C₄²	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
37	C₂²×Dic₅₃	C₄²	1	3	yes	—
38	C₂×C₄×D₅₃	—	4	6	no	—
39	C₂×C₄×D₅₃	C₄²	1	6	yes	—
40	C₂×C₄×D₅₃	C₂²×C₄	1	6	yes	—
41	M₁₆×C₅₃	—	4	6	no	—
42	C₂×C₈×C₅₃	C₂²×C₄	1	6	yes	—
43	C₂×C₈×C₅₃	C₂²×C₄	1	6	yes	—
44	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
45	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
46	C₄×Dic₅₃	C₄²	1	6	yes	—
47	C₄×Dic₅₃	C₂²×C₄	1	6	yes	—
48	#11719917	—	4	6	no	—
49	C₂×#88322	C₂²×C₄	1	12	yes	—
50	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
51	C₂²×D₅₃	—	4	3	no	—
52	C₂²×D₅₃	C₂×C₄²	1	2	yes	—
53	C₂×C₄×C₅₃	—	2	6	no	—
54	C₂×C₄×C₅₃	C₂×C₄²	1	3	yes	—
55	C₂×C₄×C₅₃	C₂³×C₄	1	3	yes	—
56	C₂×Dic₅₃	—	4	3	no	—
57	C₂×Dic₅₃	C₂×C₄²	1	6	yes	—
58	C₄×D₅₃	—	2	12	no	—
59	C₄×D₅₃	—	4	6	no	—
60	C₈×C₅₃	—	2	6	no	—
61	C₈×C₅₃	—	2	6	no	—
62	#88322	—	2	12	no	—
63	#17339	—	53	1	no	maximal, Sylow
64	C₂²×C₅₃	—	2	3	no	—
65	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
66	C₂×D₅₃	—	2	6	no	—
67	C₂×D₅₃	—	4	4	no	—
68	C₄×C₅₃	—	2	3	no	—
69	C₄×C₅₃	#568	1	6	yes	—
70	Dic₅₃	—	2	6	no	—
71	#2920	—	53	3	no	—
72	#1455	—	53	3	no	—
73	#1455	D₅₃	1	1	yes	characteristic, complemented
74	#1454	—	53	6	no	—
75	#1454	—	53	2	no	—
76	C₂×C₅₃	—	4	1	no	—
77	C₂×C₅₃	#2873	1	3	yes	—
78	D₅₃	—	4	2	no	—
79	C₂²×C₄²	—	53	1	no	—
80	#432	—	53	12	no	—
81	#431	—	53	6	no	—
82	#431	—	53	6	no	—
83	#423	—	53	6	no	—
84	#407	—	53	6	no	—
85	C₄×M₁₆	—	53	3	no	—
86	C₄×M₁₆	C₂×D₅₃	1	3	yes	—
87	#322	—	53	3	no	—
88	#322	—	53	6	no	—
89	#322	C₂×D₅₃	1	3	yes	complemented
90	C₈²	—	53	3	no	—
91	C₈²	C₂×D₅₃	1	1	yes	characteristic, complemented
92	C₅₃	#17339	1	1	yes	characteristic, complemented, Sylow
93	C₂³×C₄	—	53	3	no	—
94	C₂×M₁₆	—	53	6	no	—
95	C₂×M₁₆	C₄×D₅₃	1	6	yes	—
96	C₂×C₄²	—	53	3	no	—
97	C₂×C₄²	—	53	6	no	—
98	C₂×C₄²	—	53	2	no	—
99	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
100	C₄⋊C₈	—	53	12	no	—
101	C₄⋊C₈	—	53	12	no	—
102	C₂²⋊C₈	—	53	12	no	—
103	C₂²⋊C₈	—	53	12	no	—
104	#97	—	53	6	no	—
105	#97	C₄×D₅₃	1	6	yes	—
106	C₄×C₈	—	53	6	no	—
107	C₄×C₈	C₂²×D₅₃	1	3	yes	—
108	C₄×C₈	C₂²×D₅₃	1	3	yes	—
109	C₂⁴	—	53	1	no	—
110	C₂²×C₄	—	53	3	no	—
111	C₂²×C₄	—	53	6	no	—
112	C₂²×C₄	—	53	6	no	—
113	C₂²×C₄	—	212	6	no	—
114	C₂²×C₄	C₂×C₄×D₅₃	1	3	yes	—
115	M₁₆	—	4	6	no	—
116	M₁₆	—	212	6	no	—
117	C₂×C₈	—	53	12	no	—
118	C₂×C₈	C₂×C₄×D₅₃	1	6	yes	—
119	C₂×C₈	C₂×C₄×D₅₃	1	6	yes	—
120	C₄²	—	53	6	no	—
121	C₄²	—	53	6	no	—
122	C₄²	C₂×C₄×D₅₃	1	3	yes	—
123	C₄²	C₂³×D₅₃	1	1	yes	characteristic
124	C₂³	—	53	2	no	—
125	C₂³	—	212	3	no	—
126	C₂³	C₄²×D₅₃	1	1	yes	characteristic
127	C₂×C₄	—	2	6	no	—
128	C₂×C₄	—	53	6	no	—
129	C₂×C₄	—	106	12	no	—
130	C₂×C₄	—	212	6	no	—
131	C₂×C₄	—	212	3	no	—
132	C₂×C₄	C₄²×D₅₃	1	3	yes	—
133	C₂×C₄	C₂²×C₄×D₅₃	1	3	yes	—
134	C₈	—	2	6	no	—
135	C₈	—	2	6	no	—
136	C₈	—	106	12	no	—
137	C₂²	—	2	3	no	—
138	C₂²	—	106	6	no	—
139	C₂²	—	212	4	no	—
140	C₂²	C₂×C₄²×D₅₃	1	1	yes	characteristic
141	C₄	—	2	3	no	—
142	C₄	—	106	6	no	—
143	C₄	#16811389	1	6	yes	—
144	C₂	—	4	1	no	—
145	C₂	—	212	2	no	—
146	C₂	#20283132	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20283346

C₂

yes

maximal

#20273018

C₂

yes

maximal, complemented

#20273016

C₂

yes

maximal, complemented

#20273014

C₂

yes

maximal, complemented

C₅₃×#1455

C₂

yes

maximal, characteristic, complemented

C₂×C₄²×D₅₃

C₂²

yes

characteristic

#16810678

C₄

yes

—

#16810674

C₂²

yes

—

#16810674

C₂²

yes

—

#16810646

C₄

yes

—

#16810586

C₄

yes

—

C₄×#11719917

C₂²

yes

—

#16810345

C₂²

yes

complemented

#16810344

C₂²

yes

complemented

C₈×#88322

C₂²

yes

complemented

C₄×M₁₆×C₅₃

C₂²

yes

—

C₅₃×#322

C₂²

yes

complemented

C₈²×C₅₃

C₂²

yes

characteristic, complemented

C₂²×C₄×D₅₃

C₂×C₄

yes

—

C₂×#11719917

C₂×C₄

yes

—

C₂×C₄×Dic₅₃

C₂×C₄

yes

—

C₄²×D₅₃

C₂×C₄

yes

—

C₄²×D₅₃

C₂³

yes

—

#13133146

C₂×C₄

yes

—

#13133146

C₂×C₄

yes

—

#13133129

C₂×C₄

yes

—

#13133129

C₂×C₄

yes

—

#13133128

C₂×C₄

yes

—

C₄×#88322

C₂³

yes

—

C₂×M₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#97×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂³

yes

—

C₄×C₈×C₅₃

C₂³

yes

—

C₂³×D₅₃

C₄²

yes

characteristic

C₂²×C₄×C₅₃

C₂²×C₄

yes

—

C₂²×Dic₅₃

C₄²

yes

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

C₄²

yes

—

C₂×C₄×D₅₃

C₂²×C₄

yes

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂²×C₄

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

—

C₄²×C₅₃

C₂²×C₄

yes

—

C₄²×C₅₃

C₂⁴

yes

characteristic

C₄×Dic₅₃

C₄²

yes

—

C₄×Dic₅₃

C₂²×C₄

yes

—

#11719917

—

C₂×#88322

C₂²×C₄

yes

—

C₂³×C₅₃

C₂×C₄²

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

C₂×C₄²

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄²

yes

—

C₂×C₄×C₅₃

C₂³×C₄

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×C₄²

yes

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#17339

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

C₂²×C₄²

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#568

yes

—

Dic₅₃

—

#2920

—

#1455

—

#1455

D₅₃

yes

characteristic, complemented

#1454

—

#1454

—

C₂×C₅₃

—

C₂×C₅₃

#2873

yes

—

D₅₃

—

C₂²×C₄²

—

—

—

—

—

—

C₄×M₁₆

—

C₄×M₁₆

C₂×D₅₃

yes

—

#322

—

#322

—

#322

C₂×D₅₃

yes

complemented

C₈²

—

C₈²

C₂×D₅₃

yes

characteristic, complemented

C₅₃

#17339

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂×M₁₆

—

C₂×M₁₆

C₄×D₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₅₃

yes

characteristic

100

C₄⋊C₈

—

101

C₄⋊C₈

—

102

C₂²⋊C₈

—

103

C₂²⋊C₈

—

104

#97

—

105

#97

C₄×D₅₃

yes

—

106

C₄×C₈

—

107

C₄×C₈

C₂²×D₅₃

yes

—

108

C₄×C₈

C₂²×D₅₃

yes

—

109

C₂⁴

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

212

—

114

C₂²×C₄

C₂×C₄×D₅₃

yes

—

115

M₁₆

—

116

M₁₆

—

212

—

117

C₂×C₈

—

118

C₂×C₈

C₂×C₄×D₅₃

yes

—

119

C₂×C₈

C₂×C₄×D₅₃

yes

—

120

C₄²

—

121

C₄²

—

122

C₄²

C₂×C₄×D₅₃

yes

—

123

C₄²

C₂³×D₅₃

yes

characteristic

124

C₂³

—

125

C₂³

—

212

—

126

C₂³

C₄²×D₅₃

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

106

—

130

C₂×C₄

—

212

—

131

C₂×C₄

—

212

—

132

C₂×C₄

C₄²×D₅₃

yes

—

133

C₂×C₄

C₂²×C₄×D₅₃

yes

—

134

C₈

—

135

C₈

—

136

C₈

—

106

—

137

C₂²

—

138

C₂²

—

106

—

139

C₂²

—

212

—

140

C₂²

C₂×C₄²×D₅₃

yes

characteristic

141

C₄

—

142

C₄

—

106

—

143

C₄

#16811389

yes

—

144

C₂

—

145

C₂

—

212

—

146

C₂

#20283132

yes

—

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂²×C₄²
Fitting	C₅₃×#1455	C₂
prime radical	C₂³×D₅₃	C₄²
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1455	D₅₃
hypocenter	C₅₃	#17339
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ #1455

subgroup

G/H

center, Frattini

C₄²

C₂³×D₅₃

commutator, socle

C₂²×C₅₃

C₂²×C₄²

Fitting

C₅₃×#1455

C₂

prime radical

C₂³×D₅₃

C₄²

cosocle

C₄²×C₅₃

C₂⁴

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₄² ◁ #1455

N	H ≅ G/N	type
2. #20273018	C₂	semidirect
3. #20273016	C₂	semidirect
4. #20273014	C₂	semidirect
5. C₅₃×#1455	C₂	semidirect
13. #16810345	C₂²	semidirect
14. #16810344	C₂²	semidirect
15. C₈×#88322	C₂²	semidirect
17. C₅₃×#322	C₂²	semidirect
18. C₈²×C₅₃	C₂²	semidirect
73. #1455	D₅₃	semidirect
89. #322	C₂×D₅₃	semidirect
91. C₈²	C₂×D₅₃	semidirect
92. C₅₃	#17339	semidirect

H ≅ G/N

type

2. #20273018

C₂

semidirect

3. #20273016

C₂

semidirect

4. #20273014

C₂

semidirect

5. C₅₃×#1455

C₂

semidirect

13. #16810345

C₂²

semidirect

14. #16810344

C₂²

semidirect

15. C₈×#88322

C₂²

semidirect

17. C₅₃×#322

C₂²

semidirect

18. C₈²×C₅₃

C₂²

semidirect

73. #1455

D₅₃

semidirect

89. #322

C₂×D₅₃

semidirect

91. C₈²

C₂×D₅₃

semidirect

92. C₅₃

#17339

semidirect

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

subgroup	index
#20283346	2
#20273018	2
#20273016	2
#20273014	2
C₅₃×#1455	2
#17339	53

subgroup

index

C₅₃×#1455

Subgroups — #27349840

Conjugacy classes of subgroups

Complements

Maximal subgroups