Group #27300511 — subgroups

Subgroups — #27300511

Order 13568.

342 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 159 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×#16810443	C₂	1	4	yes	maximal
2	#20276471	C₂	1	1	yes	maximal, characteristic
3	#20274401	C₂	1	1	yes	maximal, characteristic
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	#16810443	—	2	8	no	—
11	C₄⋊#11719908	—	2	2	no	—
12	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×C₄⋊C₁₆	—	2	2	no	—
15	C₂²×#88322	C₂×C₄	1	2	yes	—
16	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
17	#13133157	—	2	4	no	—
18	C₂×#11719908	—	2	2	no	—
19	#13133128	C₈	1	4	yes	—
20	#13133127	—	2	4	no	—
21	C₈×Dic₅₃	D₄	1	4	yes	—
22	C₈×Dic₅₃	Q₈	1	4	yes	—
23	#13133124	—	2	2	no	—
24	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
27	M₃₂×C₅₃	—	2	4	no	—
28	C₂×C₁₆×C₅₃	—	2	2	no	—
29	C₄×C₈×C₅₃	D₄	1	2	yes	—
30	C₄×C₈×C₅₃	Q₈	1	2	yes	—
31	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
32	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
33	C₂²×Dic₅₃	—	2	2	no	—
34	C₂×C₈×C₅₃	—	2	2	no	—
35	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
36	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
37	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
40	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
41	C₄²×C₅₃	C₂×C₈	1	2	yes	—
42	C₄×Dic₅₃	—	2	2	no	—
43	C₄×Dic₅₃	C₄⋊C₄	1	4	yes	—
44	C₂×#88322	C₄⋊C₄	1	4	yes	—
45	C₂×#88322	C₂×C₈	1	4	yes	—
46	C₂×#88322	M₁₆	1	4	yes	—
47	C₁₆×C₅₃	—	4	2	no	—
48	#11719908	—	4	2	no	—
49	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
50	C₂×C₄×C₅₃	—	4	2	no	—
51	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
52	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
53	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
56	C₂×Dic₅₃	—	2	4	no	—
57	C₂×Dic₅₃	—	4	4	no	—
58	C₈×C₅₃	—	2	2	no	—
59	C₈×C₅₃	#127	1	1	yes	characteristic
60	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
61	C₈×C₅₃	C₄⋊Q₈	1	2	yes	—
62	#88322	C₄⋊C₈	1	8	yes	—
63	#12901	—	53	1	no	maximal, Sylow
64	C₂²×C₅₃	—	2	2	no	—
65	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
66	C₂²×C₅₃	#423	1	2	yes	—
67	C₄×C₅₃	—	2	4	no	—
68	C₄×C₅₃	#384	1	1	yes	characteristic
69	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
70	C₄×C₅₃	#423	1	1	yes	characteristic
71	Dic₅₃	—	4	4	no	—
72	C₂×#364	—	53	4	no	—
73	#2150	—	53	1	no	—
74	#2150	D₅₃	1	1	yes	characteristic
75	#1726	—	53	1	no	—
76	C₂×C₅₃	—	2	2	no	—
77	C₂×C₅₃	#1842	1	1	yes	characteristic
78	C₂×C₅₃	#2153	1	2	yes	—
79	C₂×M₃₂	—	53	2	no	—
80	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
81	C₂×#97	—	53	1	no	—
82	C₂×#97	—	53	2	no	—
83	C₂×C₄×C₈	—	53	2	no	—
84	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
85	#376	—	53	1	no	—
86	#364	—	106	8	no	—
87	C₄⋊C₁₆	—	2	2	no	—
88	C₄⋊C₁₆	—	106	2	no	—
89	C₅₃	#12901	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₈	—	53	2	no	—
91	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
92	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
93	C₂×C₄²	—	53	2	no	—
94	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
95	M₃₂	—	2	4	no	—
96	M₃₂	—	106	4	no	—
97	C₂×C₁₆	—	2	2	no	—
98	C₂×C₁₆	—	106	2	no	—
99	#97	—	53	4	no	—
100	#97	—	106	4	no	—
101	C₄×C₈	—	53	4	no	—
102	C₄×C₈	—	53	4	no	—
103	C₄×C₈	Dic₁₀₆	1	2	yes	—
104	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
105	#95	—	106	2	no	—
106	C₂²×C₄	—	106	2	no	—
107	C₂²×C₄	#11719912	1	2	yes	—
108	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	53	4	no	—
111	C₂×C₈	—	53	4	no	—
112	C₂×C₈	—	53	4	no	—
113	C₂×C₈	#11719919	1	2	yes	—
114	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
115	C₂×C₈	D₄×D₅₃	1	2	yes	—
116	C₂×C₈	Q₈×D₅₃	1	2	yes	—
117	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
118	C₄²	—	53	4	no	—
119	C₄²	—	106	2	no	—
120	C₄²	C₈×D₅₃	1	2	yes	—
121	C₄²	#11719919	1	2	yes	—
122	C₁₆	—	4	2	no	—
123	C₁₆	—	212	2	no	—
124	C₂³	C₂×#11719912	1	1	yes	characteristic
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	106	4	no	—
127	C₂×C₄	—	212	4	no	—
128	C₂×C₄	#13133146	1	4	yes	—
129	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
130	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
131	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
132	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
133	C₈	—	2	2	no	—
134	C₈	—	53	8	no	—
135	C₈	#13133213	1	1	yes	characteristic
136	C₈	#13133215	1	1	yes	characteristic
137	C₈	#13133216	1	2	yes	—
138	C₂²	—	2	2	no	—
139	C₂²	C₂×#13133146	1	1	yes	characteristic
140	C₂²	#16810645	1	2	yes	—
141	C₄	—	2	4	no	—
142	C₄	—	212	4	no	—
143	C₄	#16810492	1	1	yes	characteristic
144	C₄	C₄⋊C₈×D₅₃	1	2	yes	—
145	C₄	#16810646	1	1	yes	characteristic
146	C₂	—	2	2	no	—
147	C₂	#20274866	1	1	yes	characteristic
148	C₂	#20276480	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16810443

C₂

yes

maximal

#20276471

C₂

yes

maximal, characteristic

#20274401

C₂

yes

maximal, characteristic

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

#16810443

—

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

D₄

yes

—

C₈×Dic₅₃

Q₈

yes

—

#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

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₃

C₂×Q₈

yes

—

C₄²×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

C₂×C₈

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₄⋊C₄

yes

—

C₂×#88322

C₄⋊C₄

yes

—

C₂×#88322

C₂×C₈

yes

—

C₂×#88322

M₁₆

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

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#127

yes

characteristic

C₈×C₅₃

C₄⋊Q₈

yes

characteristic

C₈×C₅₃

C₄⋊Q₈

yes

—

#88322

C₄⋊C₈

yes

—

#12901

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₅₃

#423

yes

—

C₄×C₅₃

—

C₄×C₅₃

#384

yes

characteristic

C₄×C₅₃

C₂×C₄⋊C₈

yes

—

C₄×C₅₃

#423

yes

characteristic

Dic₅₃

—

C₂×#364

—

#2150

—

#2150

D₅₃

yes

characteristic

#1726

—

C₂×C₅₃

—

C₂×C₅₃

#1842

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

—

#364

—

106

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

106

—

C₅₃

#12901

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

—

100

#97

—

106

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

Dic₁₀₆

yes

—

104

C₄×C₈

C₂²⋊D₅₃

yes

—

105

#95

—

106

—

106

C₂²×C₄

—

106

—

107

C₂²×C₄

#11719912

yes

—

108

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

#11719919

yes

—

114

C₂×C₈

C₂×Dic₁₀₆

yes

characteristic

115

C₂×C₈

D₄×D₅₃

yes

—

116

C₂×C₈

Q₈×D₅₃

yes

—

117

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

118

C₄²

—

119

C₄²

—

106

—

120

C₄²

C₈×D₅₃

yes

—

121

C₄²

#11719919

yes

—

122

C₁₆

—

123

C₁₆

—

212

—

124

C₂³

C₂×#11719912

yes

characteristic

125

C₂×C₄

—

126

C₂×C₄

—

106

—

127

C₂×C₄

—

212

—

128

C₂×C₄

#13133146

yes

—

129

C₂×C₄

C₄⋊C₄×D₅₃

yes

—

130

C₂×C₄

C₂×#11719919

yes

characteristic

131

C₂×C₄

C₂×C₈×D₅₃

yes

characteristic

132

C₂×C₄

M₁₆×D₅₃

yes

—

133

C₈

—

134

C₈

—

135

C₈

#13133213

yes

characteristic

136

C₈

#13133215

yes

characteristic

137

C₈

#13133216

yes

—

138

C₂²

—

139

C₂²

C₂×#13133146

yes

characteristic

140

C₂²

#16810645

yes

—

141

C₄

—

142

C₄

—

212

—

143

C₄

#16810492

yes

characteristic

144

C₄

C₄⋊C₈×D₅₃

yes

—

145

C₄

#16810646

yes

characteristic

146

C₂

—

147

C₂

#20274866

yes

characteristic

148

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	#12901
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
89. C₅₃	#12901	semidirect

H ≅ G/N

type

89. C₅₃

#12901

semidirect

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

subgroup	index
C₂×#16810443	2
#20276471	2
#20274401	2
C₅₃×#2150	2
#12901	53

subgroup

index

C₅₃×#2150

Subgroups — #27300511

Conjugacy classes of subgroups

Complements

Maximal subgroups