Group #27300474 — subgroups

Subgroups — #27300474

Order 13568.

310 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 155 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×C₄×#11719908	C₂	1	1	yes	maximal, characteristic
2	#20274879	C₂	1	1	yes	maximal, characteristic
3	#20272806	C₂	1	4	yes	maximal
4	C₅₃×#1846	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719908	C₂²	1	2	yes	—
6	C₂×#13133151	C₂²	1	2	yes	—
7	C₂×#13133149	—	2	1	no	—
8	#16810489	—	2	1	no	—
9	C₄×#11719908	C₄	1	4	yes	complemented
10	C₂×#107×C₅₃	C₂²	1	2	yes	—
11	C₂×#106×C₅₃	—	2	1	no	—
12	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#382	—	2	1	no	—
14	C₂×C₄⋊Dic₅₃	—	2	2	no	—
15	C₂×#11719908	—	2	2	no	—
16	C₂×#11719908	—	2	2	no	—
17	C₂×#11719908	C₂×C₄	1	4	yes	—
18	#13133151	—	2	4	no	—
19	#13133149	—	4	2	no	—
20	#13133124	—	4	2	no	—
21	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
24	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
25	#107×C₅₃	—	2	4	no	—
26	#106×C₅₃	—	4	2	no	—
27	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₅₃	Q₈	1	2	yes	—
29	#95×C₅₃	—	4	2	no	—
30	C₂²×C₄×C₅₃	—	4	2	no	—
31	C₂²×C₄×C₅₃	D₈	1	2	yes	—
32	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
33	C₂²×Dic₅₃	—	4	2	no	—
34	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
35	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
36	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
37	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈×C₅₃	#41	1	2	yes	—
40	C₂×C₈×C₅₃	#41	1	2	yes	—
41	C₄⋊C₄×C₅₃	—	4	2	no	—
42	C₄⋊C₄×C₅₃	—	4	2	no	—
43	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
44	C₄²×C₅₃	Q₁₆	1	2	yes	—
45	C₄⋊Dic₅₃	—	4	2	no	—
46	C₄⋊Dic₅₃	—	4	2	no	—
47	#11719908	—	2	4	no	—
48	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
49	C₂×C₄×C₅₃	—	2	4	no	—
50	C₂×C₄×C₅₃	—	8	2	no	—
51	C₂×C₄×C₅₃	—	8	4	no	—
52	C₂×C₄×C₅₃	#107	1	4	yes	—
53	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
55	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
57	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
58	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
59	C₂×Dic₅₃	—	8	2	no	—
60	C₂×Dic₅₃	—	8	4	no	—
61	C₈×C₅₃	#106	1	4	yes	—
62	C₈×C₅₃	#117	1	2	yes	—
63	C₈×C₅₃	#125	1	2	yes	—
64	#12889	—	53	1	no	maximal, Sylow
65	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
66	C₂²×C₅₃	#486	1	1	yes	characteristic
67	C₂²×C₅₃	#486	1	1	yes	characteristic
68	C₂²×C₅₃	#508	1	4	yes	—
69	C₄×C₅₃	—	2	4	no	—
70	C₄×C₅₃	—	16	2	no	—
71	C₄×C₅₃	#382	1	1	yes	characteristic
72	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
73	C₄×C₅₃	#487	1	1	yes	characteristic
74	C₄×C₅₃	#487	1	1	yes	characteristic
75	Dic₅₃	—	16	2	no	—
76	C₂×C₄×C₁₆	—	53	1	no	—
77	#1846	—	53	1	no	—
78	#1846	D₅₃	1	1	yes	characteristic
79	#1380	—	53	4	no	—
80	C₂×C₅₃	#1834	1	1	yes	characteristic
81	C₂×C₅₃	#2158	1	2	yes	—
82	C₂×C₅₃	#2242	1	2	yes	—
83	C₂×C₅₃	#2242	1	2	yes	—
84	C₂²×C₁₆	—	53	2	no	—
85	C₂×#107	—	53	2	no	—
86	C₂×#107	C₂×D₅₃	1	2	yes	—
87	C₂×#106	—	2	1	no	—
88	C₂×#106	—	106	1	no	—
89	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
90	#382	—	2	1	no	—
91	#382	—	106	1	no	—
92	C₄×C₁₆	—	53	4	no	—
93	C₅₃	#12889	1	1	yes	characteristic, complemented, Sylow
94	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
95	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
96	C₂×C₄⋊C₄	—	2	2	no	—
97	C₂×C₄⋊C₄	—	106	2	no	—
98	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
99	C₂×C₁₆	—	53	4	no	—
100	C₂×C₁₆	—	106	2	no	—
101	C₂×C₁₆	—	106	2	no	—
102	#107	—	2	4	no	—
103	#107	—	106	4	no	—
104	#106	—	4	2	no	—
105	#106	—	212	2	no	—
106	C₄×C₈	Dic₁₀₆	1	2	yes	—
107	C₄×C₈	C₄×D₅₃	1	2	yes	—
108	#95	—	4	2	no	—
109	#95	—	212	2	no	—
110	C₂²×C₄	—	4	2	no	—
111	C₂²×C₄	—	212	2	no	—
112	C₂²×C₄	#11719922	1	2	yes	—
113	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
114	C₂×C₈	#11719919	1	2	yes	—
115	C₂×C₈	#11719923	1	2	yes	—
116	C₂×C₈	#11719924	1	2	yes	—
117	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
118	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
119	C₂×C₈	#11719947	1	2	yes	—
120	C₂×C₈	#11719949	1	2	yes	—
121	C₄⋊C₄	—	4	2	no	—
122	C₄⋊C₄	—	4	2	no	—
123	C₄⋊C₄	—	212	2	no	—
124	C₄⋊C₄	—	212	2	no	—
125	C₄²	#11719919	1	2	yes	—
126	C₄²	#11719925	1	2	yes	—
127	C₁₆	—	106	4	no	—
128	C₂³	C₂×#11719922	1	1	yes	characteristic
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	8	4	no	—
131	C₂×C₄	—	8	2	no	—
132	C₂×C₄	—	424	4	no	—
133	C₂×C₄	—	424	2	no	—
134	C₂×C₄	#13133150	1	4	yes	—
135	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
136	C₂×C₄	#13133204	1	1	yes	characteristic
137	C₂×C₄	#13133206	1	1	yes	characteristic
138	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
139	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
140	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
141	C₈	#13133148	1	4	yes	—
142	C₈	#13133175	1	2	yes	—
143	C₈	#13133208	1	2	yes	—
144	C₂²	C₂×#13133150	1	1	yes	characteristic
145	C₂²	#16810969	1	1	yes	characteristic
146	C₂²	#16810971	1	1	yes	characteristic
147	C₂²	#16811053	1	4	yes	—
148	C₄	—	2	4	no	—
149	C₄	—	16	2	no	—
150	C₄	—	848	2	no	—
151	C₄	#16810488	1	1	yes	characteristic
152	C₄	C₂×#13133148	1	1	yes	characteristic
153	C₄	#16810973	1	1	yes	characteristic
154	C₄	#16810975	1	1	yes	characteristic
155	C₂	#20274842	1	1	yes	characteristic
156	C₂	#20276497	1	2	yes	—
157	C₂	#20276946	1	2	yes	—
158	C₂	#20276949	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×#11719908

C₂

yes

maximal, characteristic

#20274879

C₂

yes

maximal, characteristic

#20272806

C₂

yes

maximal

C₅₃×#1846

C₂

yes

maximal, characteristic

C₂²×#11719908

C₂²

yes

—

C₂×#13133151

C₂²

yes

—

C₂×#13133149

—

#16810489

—

C₄×#11719908

C₄

yes

complemented

C₂×#107×C₅₃

C₂²

yes

—

C₂×#106×C₅₃

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#382

—

C₂×C₄⋊Dic₅₃

—

C₂×#11719908

—

C₂×#11719908

—

C₂×#11719908

C₂×C₄

yes

—

#13133151

—

#13133149

—

#13133124

—

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

D₄

yes

characteristic

#107×C₅₃

—

#106×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

Q₈

yes

—

#95×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₈

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₃

#41

yes

—

C₂×C₈×C₅₃

#41

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

Q₁₆

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719908

—

C₂³×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#107

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₁₆

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

#106

yes

—

C₈×C₅₃

#117

yes

—

C₈×C₅₃

#125

yes

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

C₂×C₄×C₁₆

—

#1846

—

#1846

D₅₃

yes

characteristic

#1380

—

C₂×C₅₃

#1834

yes

characteristic

C₂×C₅₃

#2158

yes

—

C₂×C₅₃

#2242

yes

—

C₂×C₅₃

#2242

yes

—

C₂²×C₁₆

—

C₂×#107

—

C₂×#107

C₂×D₅₃

yes

—

C₂×#106

—

C₂×#106

—

106

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

#382

—

#382

—

106

—

C₄×C₁₆

—

C₅₃

#12889

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂²⋊D₅₃

yes

characteristic

C₂²×C₈

C₂²×D₅₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

C₂²⋊D₅₃

yes

characteristic

C₂×C₁₆

—

100

C₂×C₁₆

—

106

—

101

C₂×C₁₆

—

106

—

102

#107

—

103

#107

—

106

—

104

#106

—

105

#106

—

212

—

106

C₄×C₈

Dic₁₀₆

yes

—

107

C₄×C₈

C₄×D₅₃

yes

—

108

#95

—

109

#95

—

212

—

110

C₂²×C₄

—

111

C₂²×C₄

—

212

—

112

C₂²×C₄

#11719922

yes

—

113

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

114

C₂×C₈

#11719919

yes

—

115

C₂×C₈

#11719923

yes

—

116

C₂×C₈

#11719924

yes

—

117

C₂×C₈

C₂×Dic₁₀₆

yes

characteristic

118

C₂×C₈

C₂×C₄×D₅₃

yes

characteristic

119

C₂×C₈

#11719947

yes

—

120

C₂×C₈

#11719949

yes

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

212

—

124

C₄⋊C₄

—

212

—

125

C₄²

#11719919

yes

—

126

C₄²

#11719925

yes

—

127

C₁₆

—

106

—

128

C₂³

C₂×#11719922

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

424

—

133

C₂×C₄

—

424

—

134

C₂×C₄

#13133150

yes

—

135

C₂×C₄

C₂×#11719919

yes

characteristic

136

C₂×C₄

#13133204

yes

characteristic

137

C₂×C₄

#13133206

yes

characteristic

138

C₂×C₄

C₂×#11719924

yes

characteristic

139

C₂×C₄

C₂×#11719923

yes

characteristic

140

C₂×C₄

C₂×#11719925

yes

characteristic

141

C₈

#13133148

yes

—

142

C₈

#13133175

yes

—

143

C₈

#13133208

yes

—

144

C₂²

C₂×#13133150

yes

characteristic

145

C₂²

#16810969

yes

characteristic

146

C₂²

#16810971

yes

characteristic

147

C₂²

#16811053

yes

—

148

C₄

—

149

C₄

—

150

C₄

—

848

—

151

C₄

#16810488

yes

characteristic

152

C₄

C₂×#13133148

yes

characteristic

153

C₄

#16810973

yes

characteristic

154

C₄

#16810975

yes

characteristic

155

C₂

#20274842

yes

characteristic

156

C₂

#20276497

yes

—

157

C₂

#20276946

yes

—

158

C₂

#20276949

yes

—

	subgroup	G/H
center	C₂³	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1846	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1846	D₅₃
hypocenter	C₅₃	#12889
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

subgroup

G/H

center

C₂³

C₂×#11719922

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₅₃×#1846

C₂

socle, prime radical

C₂³×C₅₃

C₂×D₈

cosocle

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

N	H ≅ G/N	type
9. C₄×#11719908	C₄	semidirect
93. C₅₃	#12889	semidirect

H ≅ G/N

type

9. C₄×#11719908

C₄

semidirect

93. C₅₃

#12889

semidirect

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

subgroup	index
C₂×C₄×#11719908	2
#20274879	2
#20272806	2
C₅₃×#1846	2
#12889	53

subgroup

index

C₅₃×#1846

Subgroups — #27300474

Conjugacy classes of subgroups

Complements

Maximal subgroups