Group #27298422 — subgroups

Subgroups — #27298422

Order 13568.

212 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 75 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	#20276990	C₂	1	1	yes	maximal, characteristic, complemented
2	#20274114	C₂	1	2	yes	maximal
3	#20272787	C₂	1	1	yes	maximal, characteristic, complemented
4	#20272772	C₂	1	2	yes	maximal, complemented
5	D₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
6	#16811028	C₂²	1	2	yes	—
7	#16810913	—	2	2	no	—
8	#16810449	—	2	1	no	—
9	#16810449	—	2	1	no	—
10	#16810376	C₂²	1	2	yes	—
11	#16810371	—	2	2	no	—
12	C₂²×C₁₆×C₅₃	—	2	1	no	—
13	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₅₃×C₂²⋊C₁₆	—	2	1	no	—
16	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	#13133217	—	2	2	no	—
18	#13133151	—	2	2	no	—
19	#13133151	—	2	2	no	—
20	#13133148	—	4	2	no	—
21	C₄⋊#88322	—	2	2	no	—
22	#13133138	—	4	2	no	—
23	C₂²×C₈×C₅₃	—	2	1	no	—
24	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
25	C₂×C₁₆×C₅₃	—	2	1	no	—
26	C₂×C₁₆×C₅₃	—	4	2	no	—
27	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
29	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
30	C₂²⋊C₈×C₅₃	—	2	1	no	—
31	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
32	C₂×D₄×C₅₃	D₈	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	—	2	1	no	—
34	C₂×Dic₁₀₆	—	4	2	no	—
35	C₂×C₈×C₅₃	—	2	1	no	—
36	C₂×C₈×C₅₃	—	4	1	no	—
37	C₂×C₈×C₅₃	—	4	1	no	—
38	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
40	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
41	C₂²⋊C₄×C₅₃	—	2	1	no	—
42	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₄⋊Dic₅₃	—	4	2	no	—
44	C₄⋊Dic₅₃	—	4	2	no	—
45	C₂×#88322	—	4	2	no	—
46	C₁₆×C₅₃	—	2	1	no	—
47	C₁₆×C₅₃	—	4	1	no	—
48	C₁₆×C₅₃	SD₁₆	1	2	yes	—
49	C₂³×C₅₃	—	2	1	no	—
50	D₄×C₅₃	—	2	1	no	—
51	D₄×C₅₃	Q₃₂	1	2	yes	—
52	C₂×C₄×C₅₃	—	2	1	no	—
53	C₂×C₄×C₅₃	—	4	1	no	—
54	C₂×C₄×C₅₃	—	4	1	no	—
55	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
58	C₂×Dic₅₃	—	8	2	no	—
59	Dic₁₀₆	—	16	2	no	—
60	C₈×C₅₃	—	2	1	no	—
61	C₈×C₅₃	—	4	1	no	—
62	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
63	C₈×C₅₃	#137	1	1	yes	characteristic
64	#88322	—	8	2	no	—
65	#12452	—	53	1	no	maximal, Sylow
66	C₂²×C₅₃	—	2	2	no	—
67	C₂²×C₅₃	—	4	1	no	—
68	C₂²×C₅₃	—	4	1	no	—
69	C₂²×C₅₃	#466	1	1	yes	characteristic
70	C₄×C₅₃	—	2	1	no	—
71	C₄×C₅₃	—	4	1	no	—
72	C₄×C₅₃	#461	1	1	yes	characteristic
73	C₄×C₅₃	#473	1	1	yes	characteristic
74	C₄×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
75	C₄×C₅₃	#508	1	1	yes	characteristic
76	Dic₅₃	—	16	2	no	—
77	#2252	—	53	1	no	—
78	D₄×C₁₆	D₅₃	1	1	yes	characteristic
79	#1675	—	53	2	no	—
80	#1371	—	53	1	no	—
81	#1365	—	53	2	no	—
82	C₂×C₅₃	—	4	2	no	—
83	C₂×C₅₃	#1673	1	1	yes	characteristic
84	C₂×C₅₃	#2216	1	1	yes	characteristic
85	C₂×C₅₃	#2221	1	1	yes	characteristic
86	C₂²×C₁₆	—	2	1	no	—
87	#500	—	53	2	no	—
88	#476	—	106	2	no	—
89	C₈×D₄	C₂×D₅₃	1	1	yes	characteristic
90	#366	—	106	1	no	—
91	#366	—	106	1	no	—
92	C₄⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
93	C₂²⋊C₁₆	—	2	1	no	—
94	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
95	#335	—	53	2	no	—
96	#333	—	106	2	no	—
97	C₅₃	#12452	1	1	yes	characteristic, complemented, Sylow
98	C₂²×C₈	—	2	1	no	—
99	C₄⋊Q₈	—	106	2	no	—
100	C₄×D₄	D₂₁₂	1	1	yes	characteristic
101	C₂×C₁₆	—	2	1	no	—
102	C₂×C₁₆	—	4	2	no	—
103	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
104	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
105	#107	—	106	2	no	—
106	#107	—	106	2	no	—
107	#106	—	212	2	no	—
108	C₄⋊C₈	—	106	2	no	—
109	C₄⋊C₈	D₂₁₂	1	1	yes	characteristic
110	#102	—	212	2	no	—
111	C₂²⋊C₈	—	2	1	no	—
112	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
113	C₂×Q₈	—	212	2	no	—
114	C₂×D₄	D₄₂₄	1	1	yes	characteristic
115	C₂²×C₄	—	2	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	4	1	no	—
118	C₂×C₈	—	4	1	no	—
119	C₂×C₈	—	212	2	no	—
120	C₂×C₈	#11719945	1	1	yes	characteristic
121	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
122	C₄⋊C₄	—	212	2	no	—
123	C₄⋊C₄	—	212	2	no	—
124	C₄⋊C₄	D₄₂₄	1	1	yes	characteristic
125	C₂²⋊C₄	—	2	1	no	—
126	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
127	C₁₆	—	2	1	no	—
128	C₁₆	—	4	1	no	—
129	C₁₆	#11719923	1	2	yes	—
130	C₂³	—	2	1	no	—
131	Q₈	—	848	2	no	—
132	D₄	—	2	1	no	—
133	D₄	Dic₄₂₄	1	2	yes	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	424	2	no	—
138	C₂×C₄	#13133191	1	1	yes	characteristic
139	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
140	C₂×C₄	#13133239	1	1	yes	characteristic
141	C₈	—	2	1	no	—
142	C₈	—	4	1	no	—
143	C₈	—	424	2	no	—
144	C₈	C₂×#11719923	1	1	yes	characteristic
145	C₈	#13133247	1	1	yes	characteristic
146	C₂²	—	2	2	no	—
147	C₂²	—	4	1	no	—
148	C₂²	—	4	1	no	—
149	C₂²	#16810859	1	1	yes	characteristic
150	C₄	—	2	1	no	—
151	C₄	—	4	1	no	—
152	C₄	—	848	2	no	—
153	C₄	#16810828	1	1	yes	characteristic
154	C₄	#16810898	1	1	yes	characteristic
155	C₄	C₂×Dic₄₂₄	1	1	yes	characteristic
156	C₄	#16811055	1	1	yes	characteristic
157	C₂	—	4	2	no	—
158	C₂	#20274104	1	1	yes	characteristic
159	C₂	#20276806	1	1	yes	characteristic
160	C₂	#20276835	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276990

C₂

yes

maximal, characteristic, complemented

#20274114

C₂

yes

maximal

#20272787

C₂

yes

maximal, characteristic, complemented

#20272772

C₂

yes

maximal, complemented

D₄×C₁₆×C₅₃

C₂

yes

maximal, characteristic

#16811028

C₂²

yes

—

—

—

—

C₂²

yes

—

#16810371

—

C₂²×C₁₆×C₅₃

—

C₈×D₄×C₅₃

C₂²

yes

characteristic

C₅₃×C₄⋊C₁₆

C₂²

yes

characteristic

C₅₃×C₂²⋊C₁₆

—

C₄×C₁₆×C₅₃

C₂²

yes

—

—

—

—

—

—

C₂²×C₈×C₅₃

—

C₄×D₄×C₅₃

D₄

yes

characteristic

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

D₄

yes

characteristic

C₂×C₁₆×C₅₃

D₄

yes

characteristic

C₄⋊C₈×C₅₃

D₄

yes

characteristic

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂×D₄×C₅₃

D₈

yes

characteristic

C₂²×C₄×C₅₃

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

#41

yes

characteristic

C₄⋊C₄×C₅₃

D₈

yes

characteristic

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₁₆×C₅₃

—

C₁₆×C₅₃

—

C₁₆×C₅₃

SD₁₆

yes

—

C₂³×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

Q₃₂

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#121

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

#135

yes

characteristic

C₂×Dic₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₂×SD₁₆

yes

C₈×C₅₃

yes

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#466

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

C₂×Q₃₂

yes

characteristic

C₄×C₅₃

#508

yes

characteristic

Dic₅₃

—

#2252

—

D₄×C₁₆

D₅₃

yes

—

—

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂²×C₁₆

—

#500

—

#476

—

106

—

C₈×D₄

C₂×D₅₃

yes

characteristic

#366

—

106

—

#366

—

106

—

C₄⋊C₁₆

C₂×D₅₃

yes

characteristic

C₂²⋊C₁₆

—

C₄×C₁₆

C₂×D₅₃

yes

characteristic

#335

—

#333

—

106

—

C₅₃

#12452

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₄⋊Q₈

—

106

—

100

C₄×D₄

D₂₁₂

yes

characteristic

101

C₂×C₁₆

—

102

C₂×C₁₆

—

103

C₂×C₁₆

C₂²⋊D₅₃

yes

characteristic

104

C₂×C₁₆

C₂²⋊D₅₃

yes

characteristic

105

#107

—

106

—

106

#107

—

106

—

107

#106

—

212

—

108

C₄⋊C₈

—

106

—

109

C₄⋊C₈

D₂₁₂

yes

characteristic

110

#102

—

212

—

111

C₂²⋊C₈

—

112

C₄×C₈

C₂²×D₅₃

yes

characteristic

113

C₂×Q₈

—

212

—

114

C₂×D₄

D₄₂₄

yes

characteristic

115

C₂²×C₄

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

212

—

120

C₂×C₈

#11719945

yes

characteristic

121

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

122

C₄⋊C₄

—

212

—

123

C₄⋊C₄

—

212

—

124

C₄⋊C₄

D₄₂₄

yes

characteristic

125

C₂²⋊C₄

—

126

C₄²

C₂×D₂₁₂

yes

characteristic

127

C₁₆

—

128

C₁₆

—

129

C₁₆

#11719923

yes

—

130

C₂³

—

131

Q₈

—

848

—

132

D₄

—

133

D₄

Dic₄₂₄

yes

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

424

—

138

C₂×C₄

#13133191

yes

characteristic

139

C₂×C₄

C₂×D₄₂₄

yes

characteristic

140

C₂×C₄

#13133239

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

—

424

—

144

C₈

C₂×#11719923

yes

characteristic

145

C₈

#13133247

yes

characteristic

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

#16810859

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

848

—

153

C₄

#16810828

yes

characteristic

154

C₄

#16810898

yes

characteristic

155

C₄

C₂×Dic₄₂₄

yes

characteristic

156

C₄

#16811055

yes

characteristic

157

C₂

—

158

C₂

#20274104

yes

characteristic

159

C₂

#20276806

yes

characteristic

160

C₂

#20276835

yes

characteristic

	subgroup	G/H
center	C₂²	#16810859
commutator, cosocle	C₄×C₈×C₅₃	C₂³
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	D₄×C₁₆×C₅₃	C₂
socle	C₂²×C₅₃	#466
prime radical	C₂×D₄×C₅₃	D₈
hypercenter	D₄×C₁₆	D₅₃
hypocenter	C₅₃	#12452
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄ ◁ D₄×C₁₆

subgroup

G/H

center

C₂²

#16810859

commutator, cosocle

C₄×C₈×C₅₃

C₂³

Frattini

C₄×C₈

C₂²×D₅₃

Fitting

D₄×C₁₆×C₅₃

C₂

socle

C₂²×C₅₃

#466

prime radical

C₂×D₄×C₅₃

D₈

hypercenter

D₄×C₁₆

D₅₃

hypocenter

C₅₃

#12452

LCS

G ▷ C₄×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄ ◁ D₄×C₁₆

N	H ≅ G/N	type
1. #20276990	C₂	semidirect
3. #20272787	C₂	semidirect
4. #20272772	C₂	semidirect
97. C₅₃	#12452	semidirect

H ≅ G/N

type

1. #20276990

C₂

semidirect

3. #20272787

C₂

semidirect

4. #20272772

C₂

semidirect

97. C₅₃

#12452

semidirect

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

subgroup	index
#20276990	2
#20274114	2
#20272787	2
#20272772	2
D₄×C₁₆×C₅₃	2
#12452	53

subgroup

index

D₄×C₁₆×C₅₃

Subgroups — #27298422

Conjugacy classes of subgroups

Complements

Maximal subgroups