Group #27298399 — subgroups

Subgroups — #27298399

Order 13568.

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

normal subgroups only

This group has 218 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	#20276868	C₂	1	2	yes	maximal
2	#20274333	C₂	1	1	yes	maximal, characteristic
3	C₈×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
4	#20272772	C₂	1	2	yes	maximal
5	C₅₃×#1370	C₂	1	1	yes	maximal, characteristic
6	#16811028	—	2	1	no	—
7	#16811028	C₂²	1	2	yes	—
8	#16811003	—	2	1	no	—
9	C₈×Dic₁₀₆	—	2	1	no	—
10	C₄×Dic₂₁₂	C₂²	1	1	yes	characteristic
11	#16810449	—	2	2	no	—
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	—	4	2	no	—
18	C₂×Dic₂₁₂	D₄	1	1	yes	characteristic
19	#13133217	—	2	2	no	—
20	C₄×Dic₁₀₆	—	2	1	no	—
21	#13133151	—	2	2	no	—
22	#13133151	—	2	2	no	—
23	#13133151	—	4	1	no	—
24	#13133151	—	4	1	no	—
25	#13133151	D₄	1	1	yes	characteristic
26	C₄⋊#88322	—	2	1	no	—
27	#13133146	—	4	1	no	—
28	#13133142	—	2	1	no	—
29	C₈×Dic₅₃	—	4	1	no	—
30	C₂×C₁₆×C₅₃	—	2	2	no	—
31	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
33	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₂×Dic₁₀₆	—	2	1	no	—
35	C₂×Dic₁₀₆	—	4	2	no	—
36	C₂×C₈×C₅₃	—	2	1	no	—
37	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
38	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
39	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
41	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₄⋊Dic₅₃	—	2	1	no	—
43	C₄⋊Dic₅₃	—	4	2	no	—
44	C₄⋊Dic₅₃	—	4	2	no	—
45	#11719919	—	4	1	no	—
46	C₄×Dic₅₃	—	4	1	no	—
47	C₂×#88322	—	4	1	no	—
48	Dic₂₁₂	—	2	1	no	—
49	Dic₂₁₂	—	8	4	no	—
50	Dic₂₁₂	Q₁₆	1	2	yes	complemented
51	C₁₆×C₅₃	—	4	2	no	—
52	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
54	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
55	C₂×Dic₅₃	—	4	1	no	—
56	C₂×Dic₅₃	—	4	1	no	—
57	C₂×Dic₅₃	—	8	2	no	—
58	Dic₁₀₆	—	2	2	no	—
59	Dic₁₀₆	—	4	1	no	—
60	Dic₁₀₆	—	8	4	no	—
61	C₈×C₅₃	—	2	2	no	—
62	C₈×C₅₃	—	2	1	no	—
63	C₈×C₅₃	—	2	1	no	—
64	C₈×C₅₃	Q₃₂	1	2	yes	—
65	C₈×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
66	C₈×C₅₃	#135	1	1	yes	characteristic
67	#88322	—	8	1	no	—
68	#12448	—	53	1	no	maximal, Sylow
69	C₂²×C₅₃	#459	1	1	yes	characteristic
70	C₄×C₅₃	—	2	1	no	—
71	C₄×C₅₃	#467	1	1	yes	characteristic
72	C₄×C₅₃	#473	1	1	yes	characteristic
73	C₄×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
74	C₄×C₅₃	#510	1	1	yes	characteristic
75	Dic₅₃	—	4	2	no	—
76	Dic₅₃	—	8	1	no	—
77	Dic₅₃	—	16	2	no	—
78	#2227	—	53	2	no	—
79	#1713	—	53	1	no	—
80	C₈×Q₁₆	—	53	1	no	—
81	#1370	D₅₃	1	1	yes	characteristic
82	#1365	—	53	2	no	—
83	C₂×C₅₃	#1669	1	1	yes	characteristic
84	C₂×C₅₃	#2209	1	1	yes	characteristic
85	C₂×C₅₃	#2213	1	1	yes	characteristic
86	#500	—	53	2	no	—
87	#500	—	106	1	no	—
88	#494	—	106	1	no	—
89	C₈×Q₈	—	106	1	no	—
90	C₄×Q₁₆	—	53	1	no	—
91	#366	—	106	2	no	—
92	C₄⋊C₁₆	C₂×D₅₃	1	2	yes	—
93	#358	—	106	2	no	—
94	#335	—	53	1	no	—
95	Q₈⋊C₈	—	106	1	no	—
96	C₈²	C₂×D₅₃	1	1	yes	characteristic
97	C₅₃	#12448	1	1	yes	characteristic, complemented, Sylow
98	C₂×Q₁₆	—	53	1	no	—
99	C₂×Q₁₆	—	212	2	no	—
100	C₄⋊Q₈	—	106	2	no	—
101	C₄×Q₈	—	106	1	no	—
102	C₂×C₁₆	—	2	2	no	—
103	#107	—	53	1	no	—
104	#107	—	106	2	no	—
105	#107	—	106	2	no	—
106	#107	—	212	1	no	—
107	#107	—	212	1	no	—
108	C₄⋊C₈	—	106	1	no	—
109	C₄⋊C₈	—	212	1	no	—
110	#103	—	106	1	no	—
111	C₄×C₈	—	212	1	no	—
112	C₄×C₈	D₂₁₂	1	1	yes	characteristic
113	C₄×C₈	D₂₁₂	1	1	yes	characteristic
114	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
115	C₂×Q₈	—	106	1	no	—
116	C₂×Q₈	—	212	2	no	—
117	Q₁₆	—	53	2	no	—
118	Q₁₆	—	106	1	no	—
119	Q₁₆	—	424	4	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	212	1	no	—
122	C₂×C₈	D₄₂₄	1	1	yes	characteristic
123	C₂×C₈	D₄₂₄	1	1	yes	characteristic
124	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
125	C₂×C₈	#11719949	1	1	yes	characteristic
126	C₄⋊C₄	—	106	1	no	—
127	C₄⋊C₄	—	212	2	no	—
128	C₄⋊C₄	—	212	2	no	—
129	C₄⋊C₄	—	212	1	no	—
130	C₄²	—	212	1	no	—
131	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
132	C₁₆	—	4	2	no	—
133	Q₈	—	106	2	no	—
134	Q₈	—	424	4	no	—
135	Q₈	—	212	1	no	—
136	C₂×C₄	—	424	2	no	—
137	C₂×C₄	—	212	1	no	—
138	C₂×C₄	—	212	1	no	—
139	C₂×C₄	#13133192	1	1	yes	characteristic
140	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
141	C₂×C₄	#13133245	1	1	yes	characteristic
142	C₈	—	2	2	no	—
143	C₈	—	2	1	no	—
144	C₈	—	2	1	no	—
145	C₈	—	424	1	no	—
146	C₈	Dic₄₂₄	1	2	yes	—
147	C₈	Q₁₆×D₅₃	1	1	yes	characteristic
148	C₈	#13133236	1	1	yes	characteristic
149	C₂²	#16810819	1	1	yes	characteristic
150	C₄	—	2	1	no	—
151	C₄	—	848	2	no	—
152	C₄	—	424	1	no	—
153	C₄	—	212	2	no	—
154	C₄	#16810865	1	1	yes	characteristic
155	C₄	#16810899	1	1	yes	characteristic
156	C₄	C₂×Dic₄₂₄	1	1	yes	characteristic
157	C₄	#16811066	1	1	yes	characteristic
158	C₂	#20274081	1	1	yes	characteristic
159	C₂	#20276766	1	1	yes	characteristic
160	C₂	#20276792	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276868

C₂

yes

maximal

#20274333

C₂

yes

maximal, characteristic

C₈×Dic₂₁₂

C₂

yes

maximal, characteristic

#20272772

C₂

yes

maximal

C₅₃×#1370

C₂

yes

maximal, characteristic

#16811028

—

#16811028

C₂²

yes

—

#16811003

—

C₈×Dic₁₀₆

—

C₄×Dic₂₁₂

C₂²

yes

—

—

C₂²

yes

characteristic

#16810355

—

C₅₃×C₄⋊C₁₆

C₂²

yes

—

C₈²×C₅₃

C₂²

yes

characteristic

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

D₄

yes

characteristic

#13133217

—

C₄×Dic₁₀₆

—

—

—

—

—

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

Q₁₆

yes

complemented

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

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

Q₃₂

yes

—

C₈×C₅₃

C₂×Q₁₆

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

—

#2227

—

#1713

—

C₈×Q₁₆

—

#1370

D₅₃

yes

characteristic

#1365

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

106

—

#494

—

106

—

C₈×Q₈

—

106

—

C₄×Q₁₆

—

#366

—

106

—

C₄⋊C₁₆

C₂×D₅₃

yes

—

#358

—

106

—

#335

—

Q₈⋊C₈

—

106

—

C₈²

C₂×D₅₃

yes

characteristic

C₅₃

#12448

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

212

—

100

C₄⋊Q₈

—

106

—

101

C₄×Q₈

—

106

—

102

C₂×C₁₆

—

103

#107

—

104

#107

—

106

—

105

#107

—

106

—

106

#107

—

212

—

107

#107

—

212

—

108

C₄⋊C₈

—

106

—

109

C₄⋊C₈

—

212

—

110

#103

—

106

—

111

C₄×C₈

—

212

—

112

C₄×C₈

D₂₁₂

yes

characteristic

113

C₄×C₈

D₂₁₂

yes

characteristic

114

C₄×C₈

C₂²×D₅₃

yes

characteristic

115

C₂×Q₈

—

106

—

116

C₂×Q₈

—

212

—

117

Q₁₆

—

118

Q₁₆

—

106

—

119

Q₁₆

—

424

—

120

C₂×C₈

—

121

C₂×C₈

—

212

—

122

C₂×C₈

D₄₂₄

yes

characteristic

123

C₂×C₈

D₄₂₄

yes

characteristic

124

C₂×C₈

D₄×D₅₃

yes

characteristic

125

C₂×C₈

#11719949

yes

characteristic

126

C₄⋊C₄

—

106

—

127

C₄⋊C₄

—

212

—

128

C₄⋊C₄

—

212

—

129

C₄⋊C₄

—

212

—

130

C₄²

—

212

—

131

C₄²

C₂×D₂₁₂

yes

characteristic

132

C₁₆

—

133

Q₈

—

106

—

134

Q₈

—

424

—

135

Q₈

—

212

—

136

C₂×C₄

—

424

—

137

C₂×C₄

—

212

—

138

C₂×C₄

—

212

—

139

C₂×C₄

#13133192

yes

characteristic

140

C₂×C₄

C₂×D₄₂₄

yes

characteristic

141

C₂×C₄

#13133245

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

424

—

146

C₈

Dic₄₂₄

yes

—

147

C₈

Q₁₆×D₅₃

yes

characteristic

148

C₈

#13133236

yes

characteristic

149

C₂²

#16810819

yes

characteristic

150

C₄

—

151

C₄

—

848

—

152

C₄

—

424

—

153

C₄

—

212

—

154

C₄

#16810865

yes

characteristic

155

C₄

#16810899

yes

characteristic

156

C₄

C₂×Dic₄₂₄

yes

characteristic

157

C₄

#16811066

yes

characteristic

158

C₂

#20274081

yes

characteristic

159

C₂

#20276766

yes

characteristic

160

C₂

#20276792

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	#12448
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
50. Dic₂₁₂	Q₁₆	semidirect
97. C₅₃	#12448	semidirect

H ≅ G/N

type

50. Dic₂₁₂

Q₁₆

semidirect

97. C₅₃

#12448

semidirect

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

subgroup	index
#20276868	2
#20274333	2
C₈×Dic₂₁₂	2
#20272772	2
C₅₃×#1370	2
#12448	53

subgroup

index

C₈×Dic₂₁₂

C₅₃×#1370

Subgroups — #27298399

Conjugacy classes of subgroups

Complements

Maximal subgroups