Group #27298371 — subgroups

Subgroups — #27298371

Order 13568.

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

normal subgroups only

This group has 212 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	#20276968	C₂	1	1	yes	maximal, characteristic
2	#20274085	C₂	1	2	yes	maximal
3	#20272789	C₂	1	1	yes	maximal, characteristic
4	#20272765	C₂	1	2	yes	maximal
5	C₅₃×#2183	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₂₄	—	2	1	no	—
7	C₂×Dic₄₂₄	—	2	1	no	—
8	#16811003	C₂²	1	2	yes	—
9	#16810908	—	2	2	no	—
10	#16810376	C₂²	1	2	yes	—
11	#16810370	—	2	2	no	—
12	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×C₄⋊C₁₆	—	2	1	no	—
14	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₅₃×#346	—	2	1	no	—
16	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₁₂	—	2	2	no	—
18	C₂×Dic₂₁₂	—	2	2	no	—
19	#13133217	—	2	2	no	—
20	Dic₄₂₄	—	4	2	no	—
21	Dic₄₂₄	—	4	2	no	—
22	#13133148	—	4	2	no	—
23	C₄⋊#88322	—	2	2	no	—
24	#13133141	—	4	2	no	—
25	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₅₃	—	2	1	no	—
27	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
29	C₄⋊C₈×C₅₃	—	2	1	no	—
30	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
31	C₄×C₈×C₅₃	—	2	1	no	—
32	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
33	C₂×Q₈×C₅₃	D₈	1	1	yes	characteristic
34	C₂×Dic₁₀₆	—	4	2	no	—
35	C₂×C₈×C₅₃	—	2	1	no	—
36	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
38	C₄⋊C₄×C₅₃	—	2	1	no	—
39	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
40	C₄²×C₅₃	—	2	1	no	—
41	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₄⋊Dic₅₃	—	4	2	no	—
43	C₄⋊Dic₅₃	—	4	2	no	—
44	C₂×#88322	—	4	2	no	—
45	Dic₂₁₂	—	4	2	no	—
46	Dic₂₁₂	—	4	2	no	—
47	Dic₂₁₂	—	8	2	no	—
48	C₁₆×C₅₃	—	2	1	no	—
49	C₁₆×C₅₃	—	4	1	no	—
50	C₁₆×C₅₃	SD₁₆	1	2	yes	—
51	Q₈×C₅₃	—	2	2	no	—
52	C₂×C₄×C₅₃	—	2	1	no	—
53	C₂×C₄×C₅₃	—	2	1	no	—
54	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
57	C₂×Dic₅₃	—	8	2	no	—
58	Dic₁₀₆	—	8	2	no	—
59	Dic₁₀₆	—	8	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	AGL(1,ℤ/8)	1	1	yes	characteristic
64	#88322	—	8	2	no	—
65	#12443	—	53	1	no	maximal, Sylow
66	C₂²×C₅₃	#466	1	1	yes	characteristic
67	C₄×C₅₃	—	2	1	no	—
68	C₄×C₅₃	—	4	1	no	—
69	C₄×C₅₃	—	4	1	no	—
70	C₄×C₅₃	#460	1	1	yes	characteristic
71	C₄×C₅₃	#472	1	1	yes	characteristic
72	C₄×C₅₃	#510	1	2	yes	—
73	Dic₅₃	—	16	2	no	—
74	#2247	—	53	1	no	—
75	#2183	D₅₃	1	1	yes	characteristic
76	#1670	—	53	2	no	—
77	#1372	—	53	1	no	—
78	#1362	—	53	2	no	—
79	C₂×C₅₃	#1668	1	1	yes	characteristic
80	C₂×C₅₃	#2219	1	1	yes	characteristic
81	C₂×C₅₃	#2222	1	1	yes	characteristic
82	C₂×Q₃₂	—	106	1	no	—
83	C₂×Q₃₂	—	106	1	no	—
84	#494	—	53	2	no	—
85	#475	—	106	2	no	—
86	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
87	C₄⋊C₁₆	—	2	1	no	—
88	C₄⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
89	#346	—	2	1	no	—
90	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
91	#335	—	53	2	no	—
92	#332	—	106	2	no	—
93	C₅₃	#12443	1	1	yes	characteristic, complemented, Sylow
94	C₂×Q₁₆	—	106	2	no	—
95	C₂×Q₁₆	—	106	2	no	—
96	C₄⋊Q₈	—	106	2	no	—
97	C₄×Q₈	D₂₁₂	1	1	yes	characteristic
98	Q₃₂	—	212	2	no	—
99	Q₃₂	—	212	2	no	—
100	C₂×C₁₆	—	2	1	no	—
101	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
102	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
103	#106	—	212	2	no	—
104	C₄⋊C₈	—	2	1	no	—
105	C₄⋊C₈	—	106	2	no	—
106	C₄⋊C₈	D₂₁₂	1	1	yes	characteristic
107	#103	—	212	2	no	—
108	C₄×C₈	—	2	1	no	—
109	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
110	C₂×Q₈	—	212	2	no	—
111	C₂×Q₈	D₄₂₄	1	1	yes	characteristic
112	Q₁₆	—	424	2	no	—
113	Q₁₆	—	212	2	no	—
114	Q₁₆	—	212	2	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	212	2	no	—
117	C₂×C₈	#11719945	1	1	yes	characteristic
118	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	212	2	no	—
121	C₄⋊C₄	—	212	2	no	—
122	C₄⋊C₄	D₄₂₄	1	1	yes	characteristic
123	C₄²	—	2	1	no	—
124	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
125	C₁₆	—	2	1	no	—
126	C₁₆	—	4	1	no	—
127	C₁₆	#11719924	1	2	yes	—
128	Q₈	—	2	2	no	—
129	Q₈	—	424	2	no	—
130	Q₈	—	424	2	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	424	2	no	—
134	C₂×C₄	#13133191	1	1	yes	characteristic
135	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
136	C₂×C₄	#13133239	1	1	yes	characteristic
137	C₈	—	2	1	no	—
138	C₈	—	4	1	no	—
139	C₈	—	424	2	no	—
140	C₈	C₂×#11719924	1	1	yes	characteristic
141	C₈	#13133242	1	1	yes	characteristic
142	C₂²	#16810859	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	4	1	no	—
145	C₄	—	4	1	no	—
146	C₄	—	848	2	no	—
147	C₄	#16810823	1	1	yes	characteristic
148	C₄	#16810893	1	1	yes	characteristic
149	C₄	#16811066	1	2	yes	—
150	C₂	#20274075	1	1	yes	characteristic
151	C₂	#20276823	1	1	yes	characteristic
152	C₂	#20276840	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276968

C₂

yes

maximal, characteristic

#20274085

C₂

yes

maximal

#20272789

C₂

yes

maximal, characteristic

#20272765

C₂

yes

maximal

C₅₃×#2183

C₂

yes

maximal, characteristic

C₂×Dic₄₂₄

—

C₂×Dic₄₂₄

—

#16811003

C₂²

yes

—

#16810908

—

#16810376

C₂²

yes

—

#16810370

—

C₈×Q₈×C₅₃

C₂²

yes

characteristic

C₅₃×C₄⋊C₁₆

—

C₅₃×C₄⋊C₁₆

C₂²

yes

characteristic

C₅₃×#346

—

C₄×C₁₆×C₅₃

C₂²

yes

characteristic

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

#13133217

—

Dic₄₂₄

—

Dic₄₂₄

—

#13133148

—

C₄⋊#88322

—

#13133141

—

C₄×Q₈×C₅₃

D₄

yes

characteristic

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

D₄

yes

characteristic

C₂×C₁₆×C₅₃

D₄

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂×Q₈×C₅₃

D₈

yes

characteristic

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

#41

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₈

yes

characteristic

C₄²×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

Dic₂₁₂

—

Dic₂₁₂

—

Dic₂₁₂

—

C₁₆×C₅₃

—

C₁₆×C₅₃

—

C₁₆×C₅₃

SD₁₆

yes

—

Q₈×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

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₂×SD₁₆

yes

C₈×C₅₃

yes

—

—

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

Dic₅₃

—

#2247

—

#2183

D₅₃

yes

—

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×Q₃₂

—

106

—

C₂×Q₃₂

—

106

—

#494

—

#475

—

106

—

C₈×Q₈

C₂×D₅₃

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×D₅₃

yes

characteristic

#346

—

C₄×C₁₆

C₂×D₅₃

yes

characteristic

#335

—

#332

—

106

—

C₅₃

#12443

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

106

—

C₂×Q₁₆

—

106

—

C₄⋊Q₈

—

106

—

C₄×Q₈

D₂₁₂

yes

characteristic

Q₃₂

—

212

—

Q₃₂

—

212

—

100

C₂×C₁₆

—

101

C₂×C₁₆

C₂²⋊D₅₃

yes

characteristic

102

C₂×C₁₆

C₂²⋊D₅₃

yes

characteristic

103

#106

—

212

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

—

106

C₄⋊C₈

D₂₁₂

yes

characteristic

107

#103

—

212

—

108

C₄×C₈

—

109

C₄×C₈

C₂²×D₅₃

yes

characteristic

110

C₂×Q₈

—

212

—

111

C₂×Q₈

D₄₂₄

yes

characteristic

112

Q₁₆

—

424

—

113

Q₁₆

—

212

—

114

Q₁₆

—

212

—

115

C₂×C₈

—

116

C₂×C₈

—

212

—

117

C₂×C₈

#11719945

yes

characteristic

118

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

119

C₄⋊C₄

—

120

C₄⋊C₄

—

212

—

121

C₄⋊C₄

—

212

—

122

C₄⋊C₄

D₄₂₄

yes

characteristic

123

C₄²

—

124

C₄²

C₂×D₂₁₂

yes

characteristic

125

C₁₆

—

126

C₁₆

—

127

C₁₆

#11719924

yes

—

128

Q₈

—

129

Q₈

—

424

—

130

Q₈

—

424

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

424

—

134

C₂×C₄

#13133191

yes

characteristic

135

C₂×C₄

C₂×D₄₂₄

yes

characteristic

136

C₂×C₄

#13133239

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

—

424

—

140

C₈

C₂×#11719924

yes

characteristic

141

C₈

#13133242

yes

characteristic

142

C₂²

#16810859

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

848

—

147

C₄

#16810823

yes

characteristic

148

C₄

#16810893

yes

characteristic

149

C₄

#16811066

yes

—

150

C₂

#20274075

yes

characteristic

151

C₂

#20276823

yes

characteristic

152

C₂

#20276840

yes

characteristic

	subgroup	G/H
center	C₂²	#16810859
commutator, cosocle	C₄×C₈×C₅₃	C₂³
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#2183	C₂
socle, prime radical	C₂²×C₅₃	#466
hypercenter	#2183	D₅₃
hypocenter	C₅₃	#12443
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ #2183

subgroup

G/H

center

C₂²

#16810859

commutator, cosocle

C₄×C₈×C₅₃

C₂³

Frattini

C₄×C₈

C₂²×D₅₃

Fitting

C₅₃×#2183

C₂

C₂²×C₅₃

D₅₃

C₅₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ #2183

N	H ≅ G/N	type
93. C₅₃	#12443	semidirect

H ≅ G/N

type

93. C₅₃

#12443

semidirect

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

subgroup	index
#20276968	2
#20274085	2
#20272789	2
#20272765	2
C₅₃×#2183	2
#12443	53

subgroup

index

C₅₃×#2183

Subgroups — #27298371

Conjugacy classes of subgroups

Complements

Maximal subgroups