Group #27277909 — subgroups

Subgroups — #27277909

Order 13568.

306 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 131 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	#20274852	C₂	1	2	yes	maximal
2	#20273474	C₂	1	1	yes	maximal, characteristic
3	#20273470	C₂	1	2	yes	maximal
4	C₂×#16810345	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1567	C₂	1	1	yes	maximal, characteristic
6	#16810647	C₂²	1	2	yes	—
7	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
8	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
9	#16810387	—	2	4	no	—
10	#16810376	—	2	2	no	—
11	#16810373	—	2	4	no	—
12	#16810345	C₄	1	4	yes	—
13	C₅₃×#423	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#335	—	2	2	no	—
16	C₂×#11719917	—	2	2	no	—
17	C₂²×#88322	D₄	1	2	yes	—
18	C₂²×#88322	Q₈	1	2	yes	—
19	C₄⋊#88322	—	2	2	no	—
20	C₄⋊#88322	—	2	2	no	—
21	C₄×#88322	C₂×C₄	1	2	yes	—
22	C₄×#88322	C₂×C₄	1	2	yes	—
23	C₄×#88322	D₄	1	2	yes	—
24	C₄×#88322	Q₈	1	2	yes	—
25	C₂×M₁₆×C₅₃	—	2	2	no	—
26	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
27	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
28	C₄⋊C₈×C₅₃	—	2	2	no	—
29	C₄⋊C₈×C₅₃	—	2	2	no	—
30	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₅₃	D₄	1	2	yes	—
32	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
35	M₁₆×C₅₃	—	4	4	no	—
36	C₂×C₈×C₅₃	—	4	2	no	—
37	C₂×C₈×C₅₃	—	4	2	no	—
38	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
39	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
43	#11719917	—	4	4	no	—
44	C₂×#88322	—	2	2	no	—
45	C₂×#88322	—	2	2	no	—
46	C₂×#88322	—	4	2	no	—
47	C₂×#88322	—	4	2	no	—
48	C₂×#88322	C₄⋊C₄	1	4	yes	—
49	C₂×#88322	C₄⋊C₄	1	4	yes	—
50	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
51	C₂×C₄×C₅₃	—	2	2	no	—
52	C₂×C₄×C₅₃	—	2	2	no	—
53	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
57	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
58	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
59	C₂×C₄×C₅₃	#135	1	2	yes	—
60	C₂×C₄×C₅₃	#135	1	2	yes	—
61	C₈×C₅₃	—	2	4	no	—
62	C₈×C₅₃	—	8	2	no	—
63	#88322	—	2	4	no	—
64	#88322	—	2	4	no	—
65	#88322	—	8	2	no	—
66	#8827	—	53	1	no	maximal, Sylow
67	C₂²×C₅₃	—	2	1	no	—
68	C₂²×C₅₃	—	2	1	no	—
69	C₂²×C₅₃	#384	1	1	yes	characteristic
70	C₂²×C₅₃	#499	1	2	yes	—
71	C₄×C₅₃	—	2	1	no	—
72	C₄×C₅₃	—	2	1	no	—
73	C₄×C₅₃	#427	1	2	yes	—
74	C₄×C₅₃	#430	1	2	yes	—
75	C₄×C₅₃	#430	1	2	yes	—
76	C₄×C₅₃	#495	1	1	yes	characteristic
77	C₄×C₅₃	#495	1	1	yes	characteristic
78	#1837	—	53	2	no	—
79	#1567	—	53	1	no	—
80	#1567	D₅₃	1	1	yes	characteristic
81	#1566	—	53	2	no	—
82	C₂×#322	—	53	1	no	—
83	C₂×C₅₃	—	2	2	no	—
84	C₂×C₅₃	#1846	1	1	yes	characteristic
85	C₂×C₅₃	#1849	1	1	yes	characteristic
86	C₂×C₅₃	#1849	1	1	yes	characteristic
87	#423	—	53	2	no	—
88	#423	C₂×D₅₃	1	2	yes	—
89	C₂×C₄×C₈	—	53	1	no	—
90	C₂×C₄×C₈	—	53	1	no	—
91	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
92	#341	—	106	4	no	—
93	#335	—	2	2	no	—
94	#335	—	106	2	no	—
95	#334	—	106	4	no	—
96	#322	—	53	4	no	—
97	C₅₃	#8827	1	1	yes	characteristic, complemented, Sylow
98	C₂×M₁₆	—	2	2	no	—
99	C₂×M₁₆	—	106	2	no	—
100	C₂²×C₈	—	53	2	no	—
101	C₂²×C₈	—	53	2	no	—
102	C₂²×C₈	Dic₁₀₆	1	2	yes	—
103	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	2	2	no	—
106	C₄⋊C₈	—	106	2	no	—
107	C₄⋊C₈	—	106	2	no	—
108	C₄×C₈	—	53	2	no	—
109	C₄×C₈	—	53	2	no	—
110	C₄×C₈	—	53	2	no	—
111	C₄×C₈	—	53	2	no	—
112	C₄×C₈	C₄×D₅₃	1	2	yes	—
113	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
114	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
115	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
116	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
117	M₁₆	—	4	4	no	—
118	M₁₆	—	212	4	no	—
119	C₂×C₈	—	4	2	no	—
120	C₂×C₈	—	4	2	no	—
121	C₂×C₈	—	53	4	no	—
122	C₂×C₈	—	53	4	no	—
123	C₂×C₈	—	106	2	no	—
124	C₂×C₈	—	106	2	no	—
125	C₂×C₈	—	212	2	no	—
126	C₂×C₈	—	212	2	no	—
127	C₂×C₈	#11719919	1	4	yes	—
128	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
129	C₄²	D₄×D₅₃	1	1	yes	characteristic
130	C₄²	Q₈×D₅₃	1	1	yes	characteristic
131	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
132	C₂³	#13133216	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
136	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
137	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
138	C₂×C₄	#13133213	1	1	yes	characteristic
139	C₂×C₄	#13133215	1	1	yes	characteristic
140	C₂×C₄	#13133216	1	1	yes	characteristic
141	C₂×C₄	#13133236	1	2	yes	—
142	C₂×C₄	#13133240	1	2	yes	—
143	C₈	—	2	4	no	—
144	C₈	—	8	2	no	—
145	C₈	—	106	4	no	—
146	C₈	—	106	4	no	—
147	C₈	—	424	2	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	#16810492	1	1	yes	characteristic
151	C₂²	#16811024	1	2	yes	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	#16810660	1	2	yes	—
155	C₄	#16810670	1	2	yes	—
156	C₄	#16810671	1	2	yes	—
157	C₄	#16811006	1	1	yes	characteristic
158	C₄	#16811008	1	1	yes	characteristic
159	C₂	—	2	2	no	—
160	C₂	#20274878	1	1	yes	characteristic
161	C₂	#20274888	1	1	yes	characteristic
162	C₂	#20274889	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20274852

C₂

yes

maximal

#20273474

C₂

yes

maximal, characteristic

#20273470

C₂

yes

maximal

C₂×#16810345

C₂

yes

maximal, characteristic

C₅₃×#1567

C₂

yes

maximal, characteristic

#16810647

C₂²

yes

—

C₂×C₄×#88322

C₂²

yes

characteristic

C₂×C₄×#88322

C₂²

yes

—

—

—

C₄

yes

—

C₅₃×#423

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#335

—

C₂×#11719917

—

C₂²×#88322

D₄

yes

—

C₂²×#88322

Q₈

yes

—

C₄⋊#88322

—

C₄⋊#88322

—

C₄×#88322

C₂×C₄

yes

—

C₄×#88322

C₂×C₄

yes

—

C₄×#88322

D₄

yes

—

C₄×#88322

Q₈

yes

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

Q₈

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₅₃

C₂×Q₈

yes

characteristic

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×Q₈

yes

—

—

—

—

—

C₄⋊C₄

yes

—

C₂×#88322

C₄⋊C₄

yes

—

C₂³×C₅₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

#135

yes

—

C₂×C₄×C₅₃

#135

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#384

yes

characteristic

C₂²×C₅₃

#499

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#427

yes

—

C₄×C₅₃

#430

yes

—

C₄×C₅₃

#430

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

—

—

D₅₃

yes

characteristic

#1566

—

C₂×#322

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×D₅₃

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

#341

—

106

—

#335

—

#335

—

106

—

#334

—

106

—

#322

—

C₅₃

#8827

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

106

—

100

C₂²×C₈

—

101

C₂²×C₈

—

102

C₂²×C₈

Dic₁₀₆

yes

—

103

C₂×C₄²

C₂²×D₅₃

yes

characteristic

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

C₄⋊C₈

—

106

—

107

C₄⋊C₈

—

106

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

C₄×D₅₃

yes

—

113

C₄×C₈

C₂²⋊D₅₃

yes

—

114

C₂²×C₄

C₂×Dic₁₀₆

yes

characteristic

115

C₂²×C₄

D₄×D₅₃

yes

characteristic

116

C₂²×C₄

Q₈×D₅₃

yes

characteristic

117

M₁₆

—

118

M₁₆

—

212

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

106

—

124

C₂×C₈

—

106

—

125

C₂×C₈

—

212

—

126

C₂×C₈

—

212

—

127

C₂×C₈

#11719919

yes

—

128

C₄²

C₂×C₄×D₅₃

yes

characteristic

129

C₄²

D₄×D₅₃

yes

characteristic

130

C₄²

Q₈×D₅₃

yes

characteristic

131

C₄²

C₂×C₂²⋊D₅₃

yes

characteristic

132

C₂³

#13133216

yes

characteristic

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

C₄⋊C₄×D₅₃

yes

characteristic

136

C₂×C₄

C₄⋊C₄×D₅₃

yes

characteristic

137

C₂×C₄

C₂×#11719919

yes

characteristic

138

C₂×C₄

#13133213

yes

characteristic

139

C₂×C₄

#13133215

yes

characteristic

140

C₂×C₄

#13133216

yes

characteristic

141

C₂×C₄

#13133236

yes

—

142

C₂×C₄

#13133240

yes

—

143

C₈

—

144

C₈

—

145

C₈

—

106

—

146

C₈

—

106

—

147

C₈

—

424

—

148

C₂²

—

149

C₂²

—

150

C₂²

#16810492

yes

characteristic

151

C₂²

#16811024

yes

—

152

C₄

—

153

C₄

—

154

C₄

#16810660

yes

—

155

C₄

#16810670

yes

—

156

C₄

#16810671

yes

—

157

C₄

#16811006

yes

characteristic

158

C₄

#16811008

yes

characteristic

159

C₂

—

160

C₂

#20274878

yes

characteristic

161

C₂

#20274888

yes

characteristic

162

C₂

#20274889

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13133213
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1567	C₂
socle	C₂²×C₅₃	#384
prime radical	C₂³×C₅₃	C₄⋊Q₈
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1567	D₅₃
hypocenter	C₅₃	#8827
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

subgroup

G/H

center

C₂×C₄

#13133213

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1567

C₂

socle

C₂²×C₅₃

#384

prime radical

C₂³×C₅₃

C₄⋊Q₈

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

N	H ≅ G/N	type
97. C₅₃	#8827	semidirect

H ≅ G/N

type

97. C₅₃

#8827

semidirect

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

subgroup	index
#20274852	2
#20273474	2
#20273470	2
C₂×#16810345	2
C₅₃×#1567	2
#8827	53

subgroup

index

C₅₃×#1567

Subgroups — #27277909

Conjugacy classes of subgroups

Complements

Maximal subgroups