Group #27300455 — subgroups

Subgroups — #27300455

Order 13568.

302 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 135 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	#20276471	C₂	1	1	yes	maximal, characteristic
2	#20274401	C₂	1	1	yes	maximal, characteristic
3	#20272796	C₂	1	4	yes	maximal
4	C₅₃×#2150	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	C₂²	1	2	yes	—
6	C₂×#13133128	C₄	1	1	yes	characteristic
7	C₂×#13133127	C₄	1	2	yes	—
8	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
9	#16810469	C₄	1	1	yes	characteristic
10	C₄⋊#11719908	—	2	2	no	—
11	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×C₄⋊C₁₆	—	2	2	no	—
14	C₂²×#88322	C₂×C₄	1	2	yes	—
15	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
16	#13133157	—	2	4	no	—
17	C₂×#11719908	—	2	2	no	—
18	#13133128	C₈	1	4	yes	—
19	#13133127	—	2	4	no	—
20	C₈×Dic₅₃	—	2	2	no	—
21	C₈×Dic₅₃	—	2	2	no	—
22	#13133124	—	2	2	no	—
23	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
26	M₃₂×C₅₃	—	2	4	no	—
27	C₂×C₁₆×C₅₃	—	2	2	no	—
28	C₄×C₈×C₅₃	D₄	1	2	yes	—
29	C₄×C₈×C₅₃	Q₈	1	2	yes	—
30	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
31	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
32	C₂²×Dic₅₃	—	2	2	no	—
33	C₂×C₈×C₅₃	—	2	2	no	—
34	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
35	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₈×C₅₃	#41	1	2	yes	—
38	C₂×C₈×C₅₃	#41	1	2	yes	—
39	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
40	C₄²×C₅₃	C₂×C₈	1	2	yes	—
41	C₄×Dic₅₃	—	2	2	no	—
42	C₄×Dic₅₃	M₁₆	1	4	yes	—
43	C₂×#88322	—	2	2	no	—
44	C₂×#88322	—	2	2	no	—
45	C₂×#88322	C₂×C₈	1	4	yes	—
46	C₁₆×C₅₃	—	4	2	no	—
47	#11719908	—	4	2	no	—
48	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
49	C₂×C₄×C₅₃	—	4	2	no	—
50	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
51	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	#117	1	2	yes	—
53	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
55	C₂×Dic₅₃	—	2	4	no	—
56	C₂×Dic₅₃	—	4	4	no	—
57	C₈×C₅₃	—	2	2	no	—
58	C₈×C₅₃	#124	1	1	yes	characteristic
59	C₈×C₅₃	#124	1	1	yes	characteristic
60	C₈×C₅₃	#125	1	2	yes	—
61	#88322	—	2	4	no	—
62	#12884	—	53	1	no	maximal, Sylow
63	C₂²×C₅₃	—	2	2	no	—
64	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
65	C₂²×C₅₃	#432	1	2	yes	—
66	C₄×C₅₃	—	2	4	no	—
67	C₄×C₅₃	#382	1	1	yes	characteristic
68	C₄×C₅₃	#423	1	1	yes	characteristic
69	C₄×C₅₃	#431	1	2	yes	—
70	Dic₅₃	—	4	4	no	—
71	#2150	—	53	1	no	—
72	#2150	D₅₃	1	1	yes	characteristic
73	#1726	—	53	1	no	—
74	#1376	—	53	4	no	—
75	C₂×C₅₃	—	2	2	no	—
76	C₂×C₅₃	#1831	1	1	yes	characteristic
77	C₂×C₅₃	#2153	1	2	yes	—
78	C₂×M₃₂	—	53	2	no	—
79	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
80	C₂×#97	—	53	1	no	—
81	C₂×#97	—	53	2	no	—
82	C₂×C₄×C₈	—	53	2	no	—
83	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
84	#376	—	53	1	no	—
85	C₄⋊C₁₆	—	2	2	no	—
86	C₄⋊C₁₆	—	106	2	no	—
87	C₅₃	#12884	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₈	—	53	2	no	—
89	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
90	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
91	C₂×C₄²	—	53	2	no	—
92	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
93	M₃₂	—	2	4	no	—
94	M₃₂	—	106	4	no	—
95	C₂×C₁₆	—	2	2	no	—
96	C₂×C₁₆	—	106	2	no	—
97	#97	—	53	4	no	—
98	#97	—	106	4	no	—
99	C₄×C₈	—	106	2	no	—
100	C₄×C₈	—	106	2	no	—
101	C₄×C₈	Dic₁₀₆	1	2	yes	—
102	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
103	#95	—	106	2	no	—
104	C₂²×C₄	—	106	2	no	—
105	C₂²×C₄	#11719912	1	2	yes	—
106	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	53	4	no	—
109	C₂×C₈	—	106	2	no	—
110	C₂×C₈	—	106	2	no	—
111	C₂×C₈	#11719919	1	2	yes	—
112	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
113	C₂×C₈	#11719947	1	2	yes	—
114	C₂×C₈	#11719949	1	2	yes	—
115	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
116	C₄²	—	53	4	no	—
117	C₄²	—	106	2	no	—
118	C₄²	C₈×D₅₃	1	2	yes	—
119	C₄²	#11719919	1	2	yes	—
120	C₁₆	—	4	2	no	—
121	C₁₆	—	212	2	no	—
122	C₂³	C₂×#11719912	1	1	yes	characteristic
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	106	4	no	—
125	C₂×C₄	—	212	4	no	—
126	C₂×C₄	#13133146	1	4	yes	—
127	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
128	C₂×C₄	#13133175	1	2	yes	—
129	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
130	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
131	C₈	—	2	2	no	—
132	C₈	—	106	4	no	—
133	C₈	#13133204	1	1	yes	characteristic
134	C₈	#13133206	1	1	yes	characteristic
135	C₈	#13133208	1	2	yes	—
136	C₂²	—	2	2	no	—
137	C₂²	C₂×#13133146	1	1	yes	characteristic
138	C₂²	#16810679	1	2	yes	—
139	C₄	—	2	4	no	—
140	C₄	—	212	4	no	—
141	C₄	#16810488	1	1	yes	characteristic
142	C₄	#16810646	1	1	yes	characteristic
143	C₄	#16810675	1	2	yes	—
144	C₂	—	2	2	no	—
145	C₂	#20274831	1	1	yes	characteristic
146	C₂	#20276480	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20276471

C₂

yes

maximal, characteristic

#20274401

C₂

yes

maximal, characteristic

#20272796

C₂

yes

maximal

C₅₃×#2150

C₂

yes

maximal, characteristic

C₂×#13133157

C₂²

yes

—

C₂×#13133128

C₄

yes

characteristic

C₂×#13133127

C₄

yes

—

C₂×C₈×Dic₅₃

C₂²

yes

—

#16810469

C₄

yes

characteristic

C₄⋊#11719908

—

C₂×M₃₂×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×C₄⋊C₁₆

—

C₂²×#88322

C₂×C₄

yes

—

C₂×C₄×Dic₅₃

C₂×C₄

yes

—

#13133157

—

C₂×#11719908

—

#13133128

C₈

yes

—

#13133127

—

C₈×Dic₅₃

—

C₈×Dic₅₃

—

#13133124

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

M₁₆

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₃

#41

yes

—

C₂×C₈×C₅₃

#41

yes

—

C₄²×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

C₂×C₈

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

M₁₆

yes

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

C₂×C₈

yes

—

C₁₆×C₅₃

—

#11719908

—

C₂³×C₅₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₈

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

yes

C₈×C₅₃

yes

C₈×C₅₃

yes

—

#88322

—

#12884

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₅₃

#432

yes

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

Dic₅₃

—

#2150

—

#2150

D₅₃

yes

characteristic

#1726

—

#1376

—

C₂×C₅₃

—

C₂×C₅₃

#1831

yes

characteristic

C₂×C₅₃

#2153

yes

—

C₂×M₃₂

—

C₂×M₃₂

C₂×D₅₃

yes

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

#376

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

106

—

C₅₃

#12884

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₄×D₅₃

yes

characteristic

C₂²×C₈

C₂²×D₅₃

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₄×D₅₃

yes

characteristic

M₃₂

—

M₃₂

—

106

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

#97

—

#97

—

106

—

C₄×C₈

—

106

—

100

C₄×C₈

—

106

—

101

C₄×C₈

Dic₁₀₆

yes

—

102

C₄×C₈

C₂²⋊D₅₃

yes

—

103

#95

—

106

—

104

C₂²×C₄

—

106

—

105

C₂²×C₄

#11719912

yes

—

106

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

106

—

110

C₂×C₈

—

106

—

111

C₂×C₈

#11719919

yes

—

112

C₂×C₈

C₂×Dic₁₀₆

yes

characteristic

113

C₂×C₈

#11719947

yes

—

114

C₂×C₈

#11719949

yes

—

115

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

116

C₄²

—

117

C₄²

—

106

—

118

C₄²

C₈×D₅₃

yes

—

119

C₄²

#11719919

yes

—

120

C₁₆

—

121

C₁₆

—

212

—

122

C₂³

C₂×#11719912

yes

characteristic

123

C₂×C₄

—

124

C₂×C₄

—

106

—

125

C₂×C₄

—

212

—

126

C₂×C₄

#13133146

yes

—

127

C₂×C₄

C₂×#11719919

yes

characteristic

128

C₂×C₄

#13133175

yes

—

129

C₂×C₄

C₂×C₈×D₅₃

yes

characteristic

130

C₂×C₄

M₁₆×D₅₃

yes

—

131

C₈

—

132

C₈

—

106

—

133

C₈

#13133204

yes

characteristic

134

C₈

#13133206

yes

characteristic

135

C₈

#13133208

yes

—

136

C₂²

—

137

C₂²

C₂×#13133146

yes

characteristic

138

C₂²

#16810679

yes

—

139

C₄

—

140

C₄

—

212

—

141

C₄

#16810488

yes

characteristic

142

C₄

#16810646

yes

characteristic

143

C₄

#16810675

yes

—

144

C₂

—

145

C₂

#20274831

yes

characteristic

146

C₂

#20276480

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#2150	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₈
prime radical	C₂³×C₅₃	C₂×M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#2150	D₅₃
hypocenter	C₅₃	#12884
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

subgroup

G/H

center

C₂×C₈

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₅₃×#2150

C₂

socle

C₂²×C₅₃

C₂×C₄⋊C₈

prime radical

C₂³×C₅₃

C₂×M₁₆

cosocle

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

N	H ≅ G/N	type
87. C₅₃	#12884	semidirect

H ≅ G/N

type

87. C₅₃

#12884

semidirect

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

subgroup	index
#20276471	2
#20274401	2
#20272796	2
C₅₃×#2150	2
#12884	53

subgroup

index

C₅₃×#2150

Subgroups — #27300455

Conjugacy classes of subgroups

Complements

Maximal subgroups