Group #27270721 — subgroups

Subgroups — #27270721

Order 13568.

194 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 51 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	#20272639	C₂	1	2	yes	maximal
2	C₅₃×#1818	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊#11719908	—	2	2	no	—
4	C₂×#117×C₅₃	C₄	1	1	yes	characteristic
5	C₂×C₂²⋊C₈×C₅₃	C₄	1	1	yes	characteristic
6	C₂×C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₅₃×#336	—	2	2	no	—
8	C₂×#11719908	—	4	2	no	—
9	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
10	C₂²×C₈×C₅₃	—	2	1	no	—
11	C₂²×C₈×C₅₃	D₄	1	2	yes	—
12	#117×C₅₃	—	4	1	no	—
13	#117×C₅₃	—	4	1	no	—
14	C₂×C₄⋊C₄×C₅₃	C₈	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₅₃	C₈	1	1	yes	characteristic
16	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
17	C₂×C₄²×C₅₃	—	2	1	no	—
18	C₂²⋊C₈×C₅₃	—	4	1	no	—
19	C₂²⋊C₈×C₅₃	—	4	1	no	—
20	C₂⁴×C₅₃	M₁₆	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	—	2	1	no	—
22	C₂²×C₄×C₅₃	—	2	1	no	—
23	C₂²×C₄×C₅₃	—	2	1	no	—
24	C₂²×C₄×C₅₃	—	2	1	no	—
25	C₂²×C₄×C₅₃	—	4	2	no	—
26	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
28	C₂×C₈×C₅₃	—	2	2	no	—
29	C₂×C₈×C₅₃	—	4	2	no	—
30	C₂×C₈×C₅₃	—	4	2	no	—
31	C₂×C₈×C₅₃	—	4	1	no	—
32	C₄⋊C₄×C₅₃	—	4	1	no	—
33	C₄⋊C₄×C₅₃	—	4	1	no	—
34	C₂²⋊C₄×C₅₃	—	4	2	no	—
35	C₄²×C₅₃	—	4	1	no	—
36	C₄²×C₅₃	—	4	1	no	—
37	#11719908	—	8	2	no	—
38	C₂³×C₅₃	—	2	1	no	—
39	C₂³×C₅₃	—	2	2	no	—
40	C₂³×C₅₃	—	4	1	no	—
41	C₂³×C₅₃	—	4	1	no	—
42	C₂³×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
43	C₂×C₄×C₅₃	—	2	2	no	—
44	C₂×C₄×C₅₃	—	2	1	no	—
45	C₂×C₄×C₅₃	—	4	2	no	—
46	C₂×C₄×C₅₃	—	4	2	no	—
47	C₂×C₄×C₅₃	—	4	1	no	—
48	C₂×C₄×C₅₃	—	4	2	no	—
49	C₂×C₄×C₅₃	—	4	1	no	—
50	C₂×C₄×C₅₃	—	4	1	no	—
51	C₂×C₄×C₅₃	—	4	2	no	—
52	C₂×C₄×C₅₃	#99	1	1	yes	characteristic
53	C₂×C₄×C₅₃	#100	1	1	yes	characteristic
54	C₈×C₅₃	—	4	2	no	—
55	C₈×C₅₃	—	8	1	no	—
56	#7059	—	53	1	no	maximal, Sylow
57	C₂²×C₅₃	—	2	1	no	—
58	C₂²×C₅₃	—	2	1	no	—
59	C₂²×C₅₃	—	4	1	no	—
60	C₂²×C₅₃	—	4	1	no	—
61	C₂²×C₅₃	—	4	1	no	—
62	C₂²×C₅₃	—	8	2	no	—
63	C₂²×C₅₃	#323	1	1	yes	characteristic
64	C₂²×C₅₃	#350	1	2	yes	—
65	C₄×C₅₃	—	2	1	no	—
66	C₄×C₅₃	—	4	1	no	—
67	C₄×C₅₃	—	8	1	no	—
68	C₄×C₅₃	—	8	1	no	—
69	C₄×C₅₃	#352	1	2	yes	—
70	#1818	D₅₃	1	1	yes	characteristic
71	#1314	—	53	2	no	—
72	C₂×C₅₃	—	2	2	no	—
73	C₂×C₅₃	—	8	1	no	—
74	C₂×C₅₃	#1315	1	1	yes	characteristic
75	C₂×C₅₃	#1317	1	1	yes	characteristic
76	C₂×C₅₃	#1321	1	1	yes	characteristic
77	C₂×#117	Dic₅₃	1	1	yes	characteristic
78	C₂×C₂²⋊C₈	Dic₅₃	1	1	yes	characteristic
79	C₂×C₂²⋊C₈	C₂×D₅₃	1	1	yes	characteristic
80	C₂²⋊C₁₆	—	106	2	no	—
81	#336	—	2	2	no	—
82	C₅₃	#7059	1	1	yes	characteristic, complemented, Sylow
83	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
84	C₂²×C₈	—	2	1	no	—
85	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
86	#117	—	4	1	no	—
87	#117	—	4	1	no	—
88	C₂×C₄⋊C₄	#88322	1	1	yes	characteristic
89	C₂×C₄⋊C₄	#88322	1	1	yes	characteristic
90	C₂×C₂²⋊C₄	—	2	1	no	—
91	C₂×C₄²	—	2	1	no	—
92	C₂×C₁₆	—	212	2	no	—
93	C₂²⋊C₈	—	4	1	no	—
94	C₂²⋊C₈	—	4	1	no	—
95	C₂⁴	#11719917	1	1	yes	characteristic
96	C₂²×C₄	—	2	1	no	—
97	C₂²×C₄	—	2	1	no	—
98	C₂²×C₄	—	2	1	no	—
99	C₂²×C₄	—	2	1	no	—
100	C₂²×C₄	—	4	2	no	—
101	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
102	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	4	2	no	—
105	C₂×C₈	—	4	2	no	—
106	C₂×C₈	—	4	1	no	—
107	C₄⋊C₄	—	4	1	no	—
108	C₄⋊C₄	—	4	1	no	—
109	C₂²⋊C₄	—	4	2	no	—
110	C₄²	—	4	1	no	—
111	C₄²	—	4	1	no	—
112	C₁₆	—	424	2	no	—
113	C₂³	—	2	1	no	—
114	C₂³	—	2	2	no	—
115	C₂³	—	4	1	no	—
116	C₂³	—	4	1	no	—
117	C₂³	C₂²⋊#88322	1	1	yes	characteristic
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	#13133132	1	1	yes	characteristic
128	C₂×C₄	#13133134	1	1	yes	characteristic
129	C₈	—	4	2	no	—
130	C₈	—	8	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	—	2	1	no	—
133	C₂²	—	4	1	no	—
134	C₂²	—	4	1	no	—
135	C₂²	—	4	1	no	—
136	C₂²	—	8	2	no	—
137	C₂²	#16810347	1	1	yes	characteristic
138	C₂²	#16810407	1	2	yes	—
139	C₄	—	2	1	no	—
140	C₄	—	4	1	no	—
141	C₄	—	8	1	no	—
142	C₄	—	8	1	no	—
143	C₄	#16810413	1	2	yes	—
144	C₂	—	2	2	no	—
145	C₂	—	8	1	no	—
146	C₂	#20272641	1	1	yes	characteristic
147	C₂	#20272645	1	1	yes	characteristic
148	C₂	#20272653	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20272639

C₂

yes

maximal

C₅₃×#1818

C₂

yes

maximal, characteristic

C₂²⋊#11719908

—

C₂×#117×C₅₃

C₄

yes

characteristic

C₂×C₂²⋊C₈×C₅₃

C₄

yes

characteristic

C₂×C₂²⋊C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#336

—

C₂×#11719908

—

C₂³×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

—

#117×C₅₃

—

#117×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₈

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₈

yes

characteristic

C₂×C₂²⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₂⁴×C₅₃

M₁₆

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

#11719908

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₈×C₅₃

—

C₈×C₅₃

—

#7059

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#323

yes

characteristic

C₂²×C₅₃

#350

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#352

yes

—

#1818

D₅₃

yes

characteristic

#1314

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

Dic₅₃

yes

characteristic

C₂×C₂²⋊C₈

Dic₅₃

yes

characteristic

C₂×C₂²⋊C₈

C₂×D₅₃

yes

characteristic

C₂²⋊C₁₆

—

106

—

#336

—

C₅₃

#7059

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₅₃

yes

—

#117

—

#117

—

C₂×C₄⋊C₄

yes

C₂×C₄⋊C₄

yes

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₁₆

—

212

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂⁴

#11719917

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

C₂×#88322

yes

characteristic

102

C₂²×C₄

C₂²⋊Dic₅₃

yes

characteristic

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₂²⋊C₄

—

110

C₄²

—

111

C₄²

—

112

C₁₆

—

424

—

113

C₂³

—

114

C₂³

—

115

C₂³

—

116

C₂³

—

117

C₂³

C₂²⋊#88322

yes

characteristic

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

#13133132

yes

characteristic

128

C₂×C₄

#13133134

yes

characteristic

129

C₈

—

130

C₈

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

#16810347

yes

characteristic

138

C₂²

#16810407

yes

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

#16810413

yes

—

144

C₂

—

145

C₂

—

146

C₂

#20272641

yes

characteristic

147

C₂

#20272645

yes

characteristic

148

C₂

#20272653

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13133132
commutator	C₂²×C₄×C₅₃	C₂×C₈
Frattini	C₂×C₂²⋊C₈	C₂×D₅₃
Fitting	C₅₃×#1818	C₂
socle	C₂²×C₅₃	#323
prime radical	C₂⁴×C₅₃	M₁₆
cosocle	C₂×C₂²⋊C₈×C₅₃	C₂²
hypercenter	#1818	D₅₃
hypocenter	C₅₃	#7059
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈ ◁ #1818

subgroup

G/H

center

C₂×C₄

#13133132

commutator

C₂²×C₄×C₅₃

C₂×C₈

Frattini

C₂×C₂²⋊C₈

C₂×D₅₃

Fitting

C₅₃×#1818

C₂

socle

C₂²×C₅₃

#323

prime radical

C₂⁴×C₅₃

M₁₆

cosocle

C₂×C₂²⋊C₈×C₅₃

C₂²

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈ ◁ #1818

N	H ≅ G/N	type
82. C₅₃	#7059	semidirect

H ≅ G/N

type

82. C₅₃

#7059

semidirect

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

subgroup	index
#20272639	2
C₅₃×#1818	2
#7059	53

subgroup

index

#20272639

C₅₃×#1818

#7059

Subgroups — #27270721

Conjugacy classes of subgroups

Complements

Maximal subgroups