Group #27277847 — subgroups

Subgroups — #27277847

Order 13568.

402 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 187 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	#20274867	C₂	1	1	yes	maximal, characteristic
2	#20274461	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810375	C₂	1	4	yes	maximal
4	C₅₃×#1842	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#88322	—	2	2	no	—
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	C₂×C₄×#88322	C₂²	1	2	yes	—
8	#16810378	C₄	1	2	yes	—
9	#16810378	C₄	1	2	yes	—
10	#16810375	—	2	8	no	—
11	C₂×C₄⋊C₈×C₅₃	—	2	2	no	—
12	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
13	C₄³×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×#88322	—	2	2	no	—
15	C₂²×#88322	C₂×C₄	1	4	yes	—
16	C₄⋊#88322	—	2	4	no	—
17	C₄⋊#88322	—	4	4	no	—
18	C₄×#88322	D₄	1	4	yes	—
19	C₄×#88322	Q₈	1	4	yes	—
20	C₂²×C₈×C₅₃	—	2	2	no	—
21	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₅₃	D₄	1	2	yes	—
24	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
25	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
26	C₄⋊C₈×C₅₃	—	2	4	no	—
27	C₄⋊C₈×C₅₃	—	4	4	no	—
28	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
29	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
30	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₈×C₅₃	—	4	2	no	—
34	C₂×C₈×C₅₃	—	4	4	no	—
35	C₄²×C₅₃	—	2	2	no	—
36	C₄²×C₅₃	—	2	2	no	—
37	C₄²×C₅₃	—	4	2	no	—
38	C₄²×C₅₃	D₈	1	4	yes	—
39	C₄²×C₅₃	Q₁₆	1	4	yes	—
40	C₄²×C₅₃	C₂×D₄	1	2	yes	—
41	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
42	C₂×#88322	—	2	4	no	—
43	C₂×#88322	—	2	4	no	—
44	C₂×#88322	—	4	2	no	—
45	C₂×#88322	—	4	4	no	—
46	C₂×#88322	C₄⋊C₄	1	8	yes	—
47	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₅₃	—	2	2	no	—
49	C₂×C₄×C₅₃	—	2	2	no	—
50	C₂×C₄×C₅₃	—	2	2	no	—
51	C₂×C₄×C₅₃	—	2	2	no	—
52	C₂×C₄×C₅₃	—	2	2	no	—
53	C₂×C₄×C₅₃	#107	1	4	yes	—
54	C₂×C₄×C₅₃	#107	1	4	yes	—
55	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
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	2	yes	—
59	C₂×C₄×C₅₃	#131	1	2	yes	—
60	C₂×C₄×C₅₃	#131	1	2	yes	—
61	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
62	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
63	C₈×C₅₃	—	8	2	no	—
64	#88322	—	2	8	no	—
65	#88322	—	8	2	no	—
66	#8810	—	53	1	no	maximal, Sylow
67	C₂²×C₅₃	#384	1	1	yes	characteristic
68	C₂²×C₅₃	#424	1	1	yes	characteristic
69	C₂²×C₅₃	#424	1	1	yes	characteristic
70	C₂²×C₅₃	C₂×#107	1	2	yes	—
71	C₂²×C₅₃	#430	1	2	yes	—
72	C₄×C₅₃	—	2	4	no	—
73	C₄×C₅₃	—	2	4	no	—
74	C₄×C₅₃	—	2	2	no	—
75	C₄×C₅₃	#424	1	4	yes	—
76	C₄×C₅₃	#493	1	1	yes	characteristic
77	C₄×C₅₃	#494	1	1	yes	characteristic
78	C₄×C₅₃	#500	1	2	yes	—
79	#1842	—	53	1	no	—
80	#1842	D₅₃	1	1	yes	characteristic
81	#1744	—	53	1	no	—
82	C₂×#335	—	53	4	no	—
83	C₂×C₅₃	#1571	1	2	yes	—
84	C₂×C₅₃	#1571	1	2	yes	—
85	C₂×C₅₃	#1843	1	1	yes	characteristic
86	C₂×C₅₃	#1845	1	1	yes	characteristic
87	C₂×C₅₃	#1849	1	1	yes	characteristic
88	C₂×C₄⋊C₈	—	2	2	no	—
89	C₂×C₄⋊C₈	—	53	2	no	—
90	C₂×C₄⋊C₈	—	106	2	no	—
91	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
92	C₂×C₄×C₈	—	53	2	no	—
93	C₄³	C₂×D₅₃	1	1	yes	characteristic
94	#336	—	53	2	no	—
95	#336	—	53	2	no	—
96	#335	—	106	8	no	—
97	C₅₃	#8810	1	1	yes	characteristic, complemented, Sylow
98	C₂²×C₈	—	2	2	no	—
99	C₂²×C₈	—	53	4	no	—
100	C₂²×C₈	—	106	2	no	—
101	C₂×C₄²	Dic₁₀₆	1	2	yes	—
102	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
103	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
104	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
105	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
106	C₄⋊C₈	—	2	4	no	—
107	C₄⋊C₈	—	4	4	no	—
108	C₄⋊C₈	—	106	4	no	—
109	C₄⋊C₈	—	212	4	no	—
110	C₄×C₈	—	53	4	no	—
111	C₄×C₈	—	53	4	no	—
112	C₂²×C₄	#11719919	1	2	yes	—
113	C₂²×C₄	#11719919	1	2	yes	—
114	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
115	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
116	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	4	4	no	—
119	C₂×C₈	—	53	8	no	—
120	C₂×C₈	—	106	4	no	—
121	C₂×C₈	—	106	4	no	—
122	C₂×C₈	—	212	2	no	—
123	C₂×C₈	—	212	4	no	—
124	C₄²	—	2	2	no	—
125	C₄²	—	2	2	no	—
126	C₄²	—	4	2	no	—
127	C₄²	#11719922	1	4	yes	—
128	C₄²	#11719925	1	4	yes	—
129	C₄²	D₄×D₅₃	1	2	yes	—
130	C₄²	Q₈×D₅₃	1	2	yes	—
131	C₂³	C₂×#11719919	1	1	yes	characteristic
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	#13133150	1	4	yes	—
138	C₂×C₄	#13133150	1	4	yes	—
139	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
140	C₂×C₄	#13133213	1	1	yes	characteristic
141	C₂×C₄	#13133215	1	1	yes	characteristic
142	C₂×C₄	#13133216	1	2	yes	—
143	C₂×C₄	#13133223	1	2	yes	—
144	C₂×C₄	#13133225	1	2	yes	—
145	C₂×C₄	C₂×#11719922	1	2	yes	—
146	C₂×C₄	C₂×#11719925	1	2	yes	—
147	C₈	—	8	2	no	—
148	C₈	—	106	8	no	—
149	C₈	—	424	2	no	—
150	C₂²	#16810492	1	1	yes	characteristic
151	C₂²	#16810649	1	1	yes	characteristic
152	C₂²	#16810651	1	1	yes	characteristic
153	C₂²	C₂×#13133150	1	2	yes	—
154	C₂²	#16810671	1	2	yes	—
155	C₄	—	2	4	no	—
156	C₄	—	2	4	no	—
157	C₄	—	2	2	no	—
158	C₄	#16810648	1	4	yes	—
159	C₄	#16810999	1	1	yes	characteristic
160	C₄	#16811002	1	1	yes	characteristic
161	C₄	#16811026	1	2	yes	—
162	C₂	#20273489	1	2	yes	—
163	C₂	#20273491	1	2	yes	—
164	C₂	#20274868	1	1	yes	characteristic
165	C₂	#20274874	1	1	yes	characteristic
166	C₂	#20274889	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20274867

C₂

yes

maximal, characteristic

#20274461

C₂

yes

maximal, characteristic

C₂×#16810375

C₂

yes

maximal

C₅₃×#1842

C₂

yes

maximal, characteristic

C₂×C₄⋊#88322

—

C₂×C₄⋊#88322

C₂²

yes

—

C₂×C₄×#88322

C₂²

yes

—

#16810378

C₄

yes

—

#16810378

C₄

yes

—

#16810375

—

C₂×C₄⋊C₈×C₅₃

—

C₂×C₄⋊C₈×C₅₃

C₂²

yes

—

C₄³×C₅₃

C₂²

yes

characteristic

C₂²×#88322

—

C₂²×#88322

C₂×C₄

yes

—

C₄⋊#88322

—

C₄⋊#88322

—

C₄×#88322

D₄

yes

—

C₄×#88322

Q₈

yes

—

C₂²×C₈×C₅₃

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

Q₈

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

D₈

yes

—

C₄²×C₅₃

Q₁₆

yes

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

C₂×Q₈

yes

—

—

—

—

—

C₄⋊C₄

yes

—

C₂³×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#107

yes

—

C₂×C₄×C₅₃

#107

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

C₄⋊Q₈

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₂×C₄×C₅₃

C₂×D₈

yes

—

C₂×C₄×C₅₃

C₂×Q₁₆

yes

—

C₈×C₅₃

—

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#430

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#424

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

#1842

—

#1842

D₅₃

yes

characteristic

#1744

—

C₂×#335

—

C₂×C₅₃

#1571

yes

—

C₂×C₅₃

#1571

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

106

—

C₂×C₄⋊C₈

C₂×D₅₃

yes

—

C₂×C₄×C₈

—

C₄³

C₂×D₅₃

yes

—

—

—

106

—

C₅₃

#8810

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

100

C₂²×C₈

—

106

—

101

C₂×C₄²

Dic₁₀₆

yes

—

102

C₂×C₄²

C₄×D₅₃

yes

characteristic

103

C₂×C₄²

C₄×D₅₃

yes

characteristic

104

C₂×C₄²

C₂²⋊D₅₃

yes

—

105

C₂×C₄²

C₂²×D₅₃

yes

characteristic

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

106

—

109

C₄⋊C₈

—

212

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₂²×C₄

#11719919

yes

—

113

C₂²×C₄

#11719919

yes

—

114

C₂²×C₄

C₂×Dic₁₀₆

yes

characteristic

115

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

116

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

106

—

121

C₂×C₈

—

106

—

122

C₂×C₈

—

212

—

123

C₂×C₈

—

212

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

#11719922

yes

—

128

C₄²

#11719925

yes

—

129

C₄²

D₄×D₅₃

yes

—

130

C₄²

Q₈×D₅₃

yes

—

131

C₂³

C₂×#11719919

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#13133150

yes

—

138

C₂×C₄

#13133150

yes

—

139

C₂×C₄

C₄⋊C₄×D₅₃

yes

—

140

C₂×C₄

#13133213

yes

characteristic

141

C₂×C₄

#13133215

yes

characteristic

142

C₂×C₄

#13133216

yes

—

143

C₂×C₄

#13133223

yes

—

144

C₂×C₄

#13133225

yes

—

145

C₂×C₄

C₂×#11719922

yes

—

146

C₂×C₄

C₂×#11719925

yes

—

147

C₈

—

148

C₈

—

106

—

149

C₈

—

424

—

150

C₂²

#16810492

yes

characteristic

151

C₂²

#16810649

yes

characteristic

152

C₂²

#16810651

yes

characteristic

153

C₂²

C₂×#13133150

yes

—

154

C₂²

#16810671

yes

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

#16810648

yes

—

159

C₄

#16810999

yes

characteristic

160

C₄

#16811002

yes

characteristic

161

C₄

#16811026

yes

—

162

C₂

#20273489

yes

—

163

C₂

#20273491

yes

—

164

C₂

#20274868

yes

characteristic

165

C₂

#20274874

yes

characteristic

166

C₂

#20274889

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1842	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1842	D₅₃
hypocenter	C₅₃	#8810
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

subgroup

G/H

center

C₂²×C₄

C₂×C₂²⋊D₅₃

commutator

C₂²×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1842

C₂

socle, prime radical

C₂³×C₅₃

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

N	H ≅ G/N	type
97. C₅₃	#8810	semidirect

H ≅ G/N

type

97. C₅₃

#8810

semidirect

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

subgroup	index
#20274867	2
#20274461	2
C₂×#16810375	2
C₅₃×#1842	2
#8810	53

subgroup

index

C₅₃×#1842

Subgroups — #27277847

Conjugacy classes of subgroups

Complements

Maximal subgroups