Group #27298280 — subgroups

Subgroups — #27298280

Order 13568.

230 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 71 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	#20276766	C₂	1	2	yes	maximal
2	#20274333	C₂	1	1	yes	maximal, characteristic
3	C₈×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
4	#20272765	C₂	1	2	yes	maximal
5	C₅₃×#1370	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₂₄	—	2	2	no	—
7	#16811028	—	2	1	no	—
8	#16811028	—	2	1	no	—
9	#16811003	C₂²	1	2	yes	—
10	C₈×Dic₁₀₆	—	2	1	no	—
11	C₄×Dic₂₁₂	C₂²	1	1	yes	characteristic
12	#16810428	—	2	2	no	—
13	#16810376	C₂²	1	1	yes	characteristic
14	#16810355	—	2	1	no	—
15	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	—
16	C₈²×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₁₂	—	2	2	no	—
18	C₂×Dic₂₁₂	—	2	2	no	—
19	C₂×Dic₂₁₂	D₄	1	1	yes	characteristic
20	#13133217	—	2	2	no	—
21	C₄×Dic₁₀₆	—	2	1	no	—
22	Dic₄₂₄	—	4	4	no	—
23	#13133151	—	4	2	no	—
24	#13133151	—	4	1	no	—
25	#13133151	—	4	1	no	—
26	#13133151	D₄	1	1	yes	characteristic
27	C₄⋊#88322	—	2	1	no	—
28	#13133146	—	4	1	no	—
29	#13133142	—	2	1	no	—
30	C₈×Dic₅₃	—	4	1	no	—
31	C₂×C₁₆×C₅₃	—	2	2	no	—
32	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
33	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
34	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	C₂×Dic₁₀₆	—	2	1	no	—
36	C₂×Dic₁₀₆	—	4	2	no	—
37	C₂×C₈×C₅₃	—	2	1	no	—
38	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
39	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
40	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
42	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₄⋊Dic₅₃	—	2	1	no	—
44	C₄⋊Dic₅₃	—	4	2	no	—
45	C₄⋊Dic₅₃	—	4	2	no	—
46	#11719919	—	4	1	no	—
47	C₄×Dic₅₃	—	4	1	no	—
48	C₂×#88322	—	4	1	no	—
49	Dic₂₁₂	—	2	1	no	—
50	Dic₂₁₂	—	4	2	no	—
51	Dic₂₁₂	—	4	2	no	—
52	Dic₂₁₂	—	8	2	no	—
53	Dic₂₁₂	D₈	1	2	yes	—
54	C₁₆×C₅₃	—	4	2	no	—
55	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
58	C₂×Dic₅₃	—	4	1	no	—
59	C₂×Dic₅₃	—	4	1	no	—
60	C₂×Dic₅₃	—	8	2	no	—
61	Dic₁₀₆	—	2	2	no	—
62	Dic₁₀₆	—	4	1	no	—
63	Dic₁₀₆	—	8	2	no	—
64	Dic₁₀₆	—	8	2	no	—
65	C₈×C₅₃	—	2	2	no	—
66	C₈×C₅₃	—	2	1	no	—
67	C₈×C₅₃	—	2	1	no	—
68	C₈×C₅₃	Q₃₂	1	2	yes	complemented
69	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
70	C₈×C₅₃	#135	1	1	yes	characteristic
71	#88322	—	8	1	no	—
72	#12427	—	53	1	no	maximal, Sylow
73	C₂²×C₅₃	#459	1	1	yes	characteristic
74	C₄×C₅₃	—	2	1	no	—
75	C₄×C₅₃	#466	1	1	yes	characteristic
76	C₄×C₅₃	#472	1	1	yes	characteristic
77	C₄×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
78	C₄×C₅₃	#510	1	1	yes	characteristic
79	Dic₅₃	—	4	2	no	—
80	Dic₅₃	—	8	1	no	—
81	Dic₅₃	—	16	2	no	—
82	#2209	—	53	2	no	—
83	#1713	—	53	1	no	—
84	C₈×Q₁₆	—	53	1	no	—
85	#1370	D₅₃	1	1	yes	characteristic
86	#1362	—	53	2	no	—
87	C₂×C₅₃	#1667	1	1	yes	characteristic
88	C₂×C₅₃	#2209	1	1	yes	characteristic
89	C₂×C₅₃	#2214	1	1	yes	characteristic
90	C₂×Q₃₂	—	106	2	no	—
91	#500	—	106	1	no	—
92	#500	—	106	1	no	—
93	#494	—	53	2	no	—
94	C₈×Q₈	—	106	1	no	—
95	C₄×Q₁₆	—	53	1	no	—
96	C₄⋊C₁₆	C₂×D₅₃	1	2	yes	—
97	#358	—	106	2	no	—
98	#335	—	53	1	no	—
99	Q₈⋊C₈	—	106	1	no	—
100	C₈²	C₂×D₅₃	1	1	yes	characteristic
101	C₅₃	#12427	1	1	yes	characteristic, complemented, Sylow
102	C₂×Q₁₆	—	53	1	no	—
103	C₂×Q₁₆	—	106	2	no	—
104	C₂×Q₁₆	—	106	2	no	—
105	C₄⋊Q₈	—	106	2	no	—
106	C₄×Q₈	—	106	1	no	—
107	Q₃₂	—	212	4	no	—
108	C₂×C₁₆	—	2	2	no	—
109	#107	—	53	1	no	—
110	#107	—	212	2	no	—
111	#107	—	212	1	no	—
112	#107	—	212	1	no	—
113	C₄⋊C₈	—	106	1	no	—
114	C₄⋊C₈	—	212	1	no	—
115	#103	—	106	1	no	—
116	C₄×C₈	—	212	1	no	—
117	C₄×C₈	D₂₁₂	1	1	yes	characteristic
118	C₄×C₈	D₂₁₂	1	1	yes	characteristic
119	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
120	C₂×Q₈	—	106	1	no	—
121	C₂×Q₈	—	212	2	no	—
122	Q₁₆	—	53	2	no	—
123	Q₁₆	—	106	1	no	—
124	Q₁₆	—	424	2	no	—
125	Q₁₆	—	212	2	no	—
126	Q₁₆	—	212	2	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	212	1	no	—
129	C₂×C₈	D₄₂₄	1	1	yes	characteristic
130	C₂×C₈	D₄₂₄	1	1	yes	characteristic
131	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
132	C₂×C₈	#11719949	1	1	yes	characteristic
133	C₄⋊C₄	—	106	1	no	—
134	C₄⋊C₄	—	212	2	no	—
135	C₄⋊C₄	—	212	2	no	—
136	C₄⋊C₄	—	212	1	no	—
137	C₄²	—	212	1	no	—
138	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
139	C₁₆	—	4	2	no	—
140	Q₈	—	106	2	no	—
141	Q₈	—	424	2	no	—
142	Q₈	—	424	2	no	—
143	Q₈	—	212	1	no	—
144	C₂×C₄	—	424	2	no	—
145	C₂×C₄	—	212	1	no	—
146	C₂×C₄	—	212	1	no	—
147	C₂×C₄	#13133192	1	1	yes	characteristic
148	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
149	C₂×C₄	#13133245	1	1	yes	characteristic
150	C₈	—	2	2	no	—
151	C₈	—	2	1	no	—
152	C₈	—	2	1	no	—
153	C₈	—	424	1	no	—
154	C₈	Dic₄₂₄	1	2	yes	complemented
155	C₈	D₈×D₅₃	1	1	yes	characteristic
156	C₈	#13133240	1	1	yes	characteristic
157	C₂²	#16810819	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	848	2	no	—
160	C₄	—	424	1	no	—
161	C₄	—	212	2	no	—
162	C₄	#16810860	1	1	yes	characteristic
163	C₄	#16810894	1	1	yes	characteristic
164	C₄	C₂×Dic₄₂₄	1	1	yes	characteristic
165	C₄	#16811066	1	1	yes	characteristic
166	C₂	#20274071	1	1	yes	characteristic
167	C₂	#20276766	1	1	yes	characteristic
168	C₂	#20276797	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276766

C₂

yes

maximal

#20274333

C₂

yes

maximal, characteristic

C₈×Dic₂₁₂

C₂

yes

maximal, characteristic

#20272765

C₂

yes

maximal

C₅₃×#1370

C₂

yes

maximal, characteristic

C₂×Dic₄₂₄

—

#16811028

—

#16811028

—

#16811003

C₂²

yes

—

C₈×Dic₁₀₆

—

C₄×Dic₂₁₂

C₂²

yes

characteristic

#16810428

—

#16810376

C₂²

yes

characteristic

#16810355

—

C₅₃×C₄⋊C₁₆

C₂²

yes

—

C₈²×C₅₃

C₂²

yes

characteristic

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

D₄

yes

characteristic

#13133217

—

C₄×Dic₁₀₆

—

Dic₄₂₄

—

—

—

—

D₄

yes

—

—

—

C₈×Dic₅₃

—

C₂×C₁₆×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₈

yes

characteristic

C₂×C₈×C₅₃

D₈

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

#41

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

C₄×Dic₅₃

—

C₂×#88322

—

Dic₂₁₂

—

Dic₂₁₂

—

Dic₂₁₂

—

Dic₂₁₂

—

Dic₂₁₂

D₈

yes

—

C₁₆×C₅₃

—

C₂×C₄×C₅₃

#121

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

Q₃₂

yes

complemented

C₈×C₅₃

C₂×D₈

yes

C₈×C₅₃

yes

—

—

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

C₂×Q₃₂

yes

characteristic

C₄×C₅₃

#510

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

#2209

—

#1713

—

C₈×Q₁₆

—

#1370

D₅₃

yes

characteristic

#1362

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×Q₃₂

—

106

—

#500

—

106

—

#500

—

106

—

#494

—

C₈×Q₈

—

106

—

C₄×Q₁₆

—

C₄⋊C₁₆

C₂×D₅₃

yes

—

#358

—

106

—

#335

—

Q₈⋊C₈

—

106

—

100

C₈²

C₂×D₅₃

yes

characteristic

101

C₅₃

#12427

yes

characteristic, complemented, Sylow

102

C₂×Q₁₆

—

103

C₂×Q₁₆

—

106

—

104

C₂×Q₁₆

—

106

—

105

C₄⋊Q₈

—

106

—

106

C₄×Q₈

—

106

—

107

Q₃₂

—

212

—

108

C₂×C₁₆

—

109

#107

—

110

#107

—

212

—

111

#107

—

212

—

112

#107

—

212

—

113

C₄⋊C₈

—

106

—

114

C₄⋊C₈

—

212

—

115

#103

—

106

—

116

C₄×C₈

—

212

—

117

C₄×C₈

D₂₁₂

yes

characteristic

118

C₄×C₈

D₂₁₂

yes

characteristic

119

C₄×C₈

C₂²×D₅₃

yes

characteristic

120

C₂×Q₈

—

106

—

121

C₂×Q₈

—

212

—

122

Q₁₆

—

123

Q₁₆

—

106

—

124

Q₁₆

—

424

—

125

Q₁₆

—

212

—

126

Q₁₆

—

212

—

127

C₂×C₈

—

128

C₂×C₈

—

212

—

129

C₂×C₈

D₄₂₄

yes

characteristic

130

C₂×C₈

D₄₂₄

yes

characteristic

131

C₂×C₈

D₄×D₅₃

yes

characteristic

132

C₂×C₈

#11719949

yes

characteristic

133

C₄⋊C₄

—

106

—

134

C₄⋊C₄

—

212

—

135

C₄⋊C₄

—

212

—

136

C₄⋊C₄

—

212

—

137

C₄²

—

212

—

138

C₄²

C₂×D₂₁₂

yes

characteristic

139

C₁₆

—

140

Q₈

—

106

—

141

Q₈

—

424

—

142

Q₈

—

424

—

143

Q₈

—

212

—

144

C₂×C₄

—

424

—

145

C₂×C₄

—

212

—

146

C₂×C₄

—

212

—

147

C₂×C₄

#13133192

yes

characteristic

148

C₂×C₄

C₂×D₄₂₄

yes

characteristic

149

C₂×C₄

#13133245

yes

characteristic

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

—

424

—

154

C₈

Dic₄₂₄

yes

complemented

155

C₈

D₈×D₅₃

yes

characteristic

156

C₈

#13133240

yes

characteristic

157

C₂²

#16810819

yes

characteristic

158

C₄

—

159

C₄

—

848

—

160

C₄

—

424

—

161

C₄

—

212

—

162

C₄

#16810860

yes

characteristic

163

C₄

#16810894

yes

characteristic

164

C₄

C₂×Dic₄₂₄

yes

characteristic

165

C₄

#16811066

yes

characteristic

166

C₂

#20274071

yes

characteristic

167

C₂

#20276766

yes

characteristic

168

C₂

#20276797

yes

characteristic

	subgroup	G/H
center	C₂²	#16810819
commutator, cosocle	C₄×C₈×C₅₃	C₂³
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#1370	C₂
socle, prime radical	C₂²×C₅₃	#459
hypercenter	#1370	D₅₃
hypocenter	C₅₃	#12427
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1370

subgroup

G/H

center

C₂²

#16810819

commutator, cosocle

C₄×C₈×C₅₃

C₂³

Frattini

C₄×C₈

C₂²×D₅₃

Fitting

C₅₃×#1370

C₂

C₂²×C₅₃

D₅₃

C₅₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1370

N	H ≅ G/N	type
68. C₈×C₅₃	Q₃₂	semidirect
101. C₅₃	#12427	semidirect
154. C₈	Dic₄₂₄	semidirect

H ≅ G/N

type

68. C₈×C₅₃

Q₃₂

semidirect

101. C₅₃

#12427

semidirect

154. C₈

Dic₄₂₄

semidirect

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

subgroup	index
#20276766	2
#20274333	2
C₈×Dic₂₁₂	2
#20272765	2
C₅₃×#1370	2
#12427	53

subgroup

index

C₈×Dic₂₁₂

C₅₃×#1370

Subgroups — #27298280

Conjugacy classes of subgroups

Complements

Maximal subgroups