Group #27298355 — subgroups

Subgroups — #27298355

Order 13568.

216 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₁₆×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
3	#20274086	C₂	1	2	yes	maximal
4	#20272763	C₂	1	2	yes	maximal
5	C₅₃×#1371	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	#16810909	—	2	2	no	—
10	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic
11	C₄⋊#11719908	C₂²	1	1	yes	characteristic
12	#16810441	—	2	1	no	—
13	C₁₆×Dic₅₃	—	2	1	no	—
14	#16810368	—	2	2	no	—
15	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#335	C₂²	1	2	yes	—
17	C₂×Dic₂₁₂	—	2	2	no	—
18	C₂×Dic₂₁₂	—	2	2	no	—
19	#13133217	—	2	2	no	—
20	C₄×Dic₁₀₆	D₄	1	1	yes	characteristic
21	Dic₄₂₄	—	4	2	no	—
22	Dic₄₂₄	—	4	2	no	—
23	C₂×#11719908	—	2	1	no	—
24	#13133149	—	4	2	no	—
25	C₄⋊#88322	D₄	1	1	yes	characteristic
26	#13133146	—	2	1	no	—
27	#13133142	—	4	2	no	—
28	C₈×Dic₅₃	—	2	1	no	—
29	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
30	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
31	C₄⋊C₈×C₅₃	—	2	2	no	—
32	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
33	C₂×Dic₁₀₆	—	4	2	no	—
34	C₂×Dic₁₀₆	D₈	1	1	yes	characteristic
35	C₂×C₈×C₅₃	—	4	2	no	—
36	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
38	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₄⋊Dic₅₃	—	4	2	no	—
40	C₄⋊Dic₅₃	—	4	2	no	—
41	C₄⋊Dic₅₃	D₈	1	1	yes	characteristic
42	#11719919	—	2	1	no	—
43	C₄×Dic₅₃	—	2	1	no	—
44	C₂×#88322	—	2	1	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	SD₁₆	1	2	yes	—
50	#11719908	—	4	1	no	—
51	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
54	C₂×Dic₅₃	—	2	1	no	—
55	C₂×Dic₅₃	—	2	1	no	—
56	C₂×Dic₅₃	—	8	2	no	—
57	Dic₁₀₆	—	2	1	no	—
58	Dic₁₀₆	—	8	2	no	—
59	Dic₁₀₆	—	8	2	no	—
60	Dic₁₀₆	Q₃₂	1	2	yes	complemented
61	C₈×C₅₃	—	2	1	no	—
62	C₈×C₅₃	—	8	2	no	—
63	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
64	C₈×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
65	#88322	—	4	1	no	—
66	#12440	—	53	1	no	maximal, Sylow
67	C₂²×C₅₃	#466	1	1	yes	characteristic
68	C₄×C₅₃	—	2	1	no	—
69	C₄×C₅₃	#460	1	1	yes	characteristic
70	C₄×C₅₃	#472	1	1	yes	characteristic
71	C₄×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
72	C₄×C₅₃	#508	1	1	yes	characteristic
73	Dic₅₃	—	2	2	no	—
74	Dic₅₃	—	4	1	no	—
75	Dic₅₃	—	16	2	no	—
76	#2247	—	53	1	no	—
77	Q₈×C₁₆	—	53	1	no	—
78	#1670	—	53	2	no	—
79	#1371	D₅₃	1	1	yes	characteristic
80	#1362	—	53	2	no	—
81	C₂×C₅₃	#1668	1	1	yes	characteristic
82	C₂×C₅₃	#2216	1	1	yes	characteristic
83	C₂×C₅₃	#2222	1	1	yes	characteristic
84	C₂×Q₃₂	—	106	1	no	—
85	C₂×Q₃₂	—	106	1	no	—
86	#494	—	53	2	no	—
87	#475	—	106	2	no	—
88	C₈×Q₈	—	53	1	no	—
89	C₄⋊C₁₆	—	53	1	no	—
90	C₄⋊C₁₆	—	106	1	no	—
91	C₄×C₁₆	—	106	1	no	—
92	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
93	#335	C₂×D₅₃	1	2	yes	—
94	#332	—	106	2	no	—
95	C₅₃	#12440	1	1	yes	characteristic, complemented, Sylow
96	C₂×Q₁₆	—	106	2	no	—
97	C₂×Q₁₆	—	106	2	no	—
98	C₄⋊Q₈	—	106	2	no	—
99	C₄×Q₈	—	53	1	no	—
100	Q₃₂	—	212	2	no	—
101	Q₃₂	—	212	2	no	—
102	C₂×C₁₆	—	106	1	no	—
103	C₂×C₁₆	D₂₁₂	1	1	yes	characteristic
104	C₂×C₁₆	D₂₁₂	1	1	yes	characteristic
105	#106	—	212	2	no	—
106	C₄⋊C₈	—	2	2	no	—
107	C₄⋊C₈	—	53	1	no	—
108	C₄⋊C₈	—	106	1	no	—
109	#103	—	212	2	no	—
110	C₄×C₈	—	106	1	no	—
111	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
112	C₂×Q₈	—	53	1	no	—
113	C₂×Q₈	—	212	2	no	—
114	Q₁₆	—	424	2	no	—
115	Q₁₆	—	212	2	no	—
116	Q₁₆	—	212	2	no	—
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	106	1	no	—
119	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
120	C₂×C₈	#11719949	1	1	yes	characteristic
121	C₄⋊C₄	—	53	1	no	—
122	C₄⋊C₄	—	106	1	no	—
123	C₄⋊C₄	—	212	2	no	—
124	C₄⋊C₄	—	212	2	no	—
125	C₄²	—	106	1	no	—
126	C₄²	D₄×D₅₃	1	1	yes	characteristic
127	C₁₆	—	2	1	no	—
128	C₁₆	—	212	1	no	—
129	C₁₆	#11719913	1	2	yes	—
130	Q₈	—	53	2	no	—
131	Q₈	—	106	1	no	—
132	Q₈	—	424	2	no	—
133	Q₈	—	424	2	no	—
134	C₂×C₄	—	106	1	no	—
135	C₂×C₄	—	106	1	no	—
136	C₂×C₄	—	424	2	no	—
137	C₂×C₄	#13133192	1	1	yes	characteristic
138	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
139	C₂×C₄	#13133240	1	1	yes	characteristic
140	C₈	—	2	1	no	—
141	C₈	—	8	2	no	—
142	C₈	—	212	1	no	—
143	C₈	C₂×#11719913	1	1	yes	characteristic
144	C₈	#13133245	1	1	yes	characteristic
145	C₂²	#16810860	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	848	2	no	—
148	C₄	—	106	2	no	—
149	C₄	—	212	1	no	—
150	C₄	#16810824	1	1	yes	characteristic
151	C₄	#16810894	1	1	yes	characteristic
152	C₄	Q₃₂×D₅₃	1	1	yes	characteristic
153	C₄	#16811052	1	1	yes	characteristic
154	C₂	#20274076	1	1	yes	characteristic
155	C₂	#20276807	1	1	yes	characteristic
156	C₂	#20276841	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276968

C₂

yes

maximal, characteristic

C₁₆×Dic₁₀₆

C₂

yes

maximal, characteristic

#20274086

C₂

yes

maximal

#20272763

C₂

yes

maximal

C₅₃×#1371

C₂

yes

maximal, characteristic

C₂×Dic₄₂₄

—

C₂×Dic₄₂₄

—

#16811003

C₂²

yes

—

#16810909

—

C₈×Dic₁₀₆

C₂²

yes

characteristic

C₄⋊#11719908

C₂²

yes

characteristic

#16810441

—

C₁₆×Dic₅₃

—

#16810368

—

C₄×C₁₆×C₅₃

C₂²

yes

characteristic

C₅₃×#335

C₂²

yes

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

#13133217

—

C₄×Dic₁₀₆

D₄

yes

characteristic

Dic₄₂₄

—

Dic₄₂₄

—

C₂×#11719908

—

#13133149

—

C₄⋊#88322

D₄

yes

characteristic

#13133146

—

#13133142

—

C₈×Dic₅₃

—

C₂×C₁₆×C₅₃

D₄

yes

characteristic

C₂×C₁₆×C₅₃

D₄

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

D₈

yes

characteristic

C₂×C₈×C₅₃

—

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

D₈

yes

characteristic

#11719919

—

C₄×Dic₅₃

—

C₂×#88322

—

Dic₂₁₂

—

Dic₂₁₂

—

Dic₂₁₂

—

C₁₆×C₅₃

—

C₁₆×C₅₃

SD₁₆

yes

—

#11719908

—

C₂×C₄×C₅₃

#121

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

#135

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

Q₃₂

yes

complemented

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₂×SD₁₆

yes

C₈×C₅₃

yes

—

—

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

C₂×Q₃₂

yes

characteristic

C₄×C₅₃

#508

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

#2247

—

Q₈×C₁₆

—

#1670

—

#1371

D₅₃

yes

characteristic

#1362

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×Q₃₂

—

106

—

C₂×Q₃₂

—

106

—

#494

—

#475

—

106

—

C₈×Q₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

106

—

C₄×C₁₆

—

106

—

C₄×C₁₆

C₂×D₅₃

yes

characteristic

#335

C₂×D₅₃

yes

—

#332

—

106

—

C₅₃

#12440

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

106

—

C₂×Q₁₆

—

106

—

C₄⋊Q₈

—

106

—

C₄×Q₈

—

100

Q₃₂

—

212

—

101

Q₃₂

—

212

—

102

C₂×C₁₆

—

106

—

103

C₂×C₁₆

D₂₁₂

yes

characteristic

104

C₂×C₁₆

D₂₁₂

yes

characteristic

105

#106

—

212

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

106

—

109

#103

—

212

—

110

C₄×C₈

—

106

—

111

C₄×C₈

C₂²×D₅₃

yes

characteristic

112

C₂×Q₈

—

113

C₂×Q₈

—

212

—

114

Q₁₆

—

424

—

115

Q₁₆

—

212

—

116

Q₁₆

—

212

—

117

C₂×C₈

—

118

C₂×C₈

—

106

—

119

C₂×C₈

C₂×D₂₁₂

yes

characteristic

120

C₂×C₈

#11719949

yes

characteristic

121

C₄⋊C₄

—

122

C₄⋊C₄

—

106

—

123

C₄⋊C₄

—

212

—

124

C₄⋊C₄

—

212

—

125

C₄²

—

106

—

126

C₄²

D₄×D₅₃

yes

characteristic

127

C₁₆

—

128

C₁₆

—

212

—

129

C₁₆

#11719913

yes

—

130

Q₈

—

131

Q₈

—

106

—

132

Q₈

—

424

—

133

Q₈

—

424

—

134

C₂×C₄

—

106

—

135

C₂×C₄

—

106

—

136

C₂×C₄

—

424

—

137

C₂×C₄

#13133192

yes

characteristic

138

C₂×C₄

D₈×D₅₃

yes

characteristic

139

C₂×C₄

#13133240

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

—

212

—

143

C₈

C₂×#11719913

yes

characteristic

144

C₈

#13133245

yes

characteristic

145

C₂²

#16810860

yes

characteristic

146

C₄

—

147

C₄

—

848

—

148

C₄

—

106

—

149

C₄

—

212

—

150

C₄

#16810824

yes

characteristic

151

C₄

#16810894

yes

characteristic

152

C₄

Q₃₂×D₅₃

yes

characteristic

153

C₄

#16811052

yes

characteristic

154

C₂

#20274076

yes

characteristic

155

C₂

#20276807

yes

characteristic

156

C₂

#20276841

yes

characteristic

	subgroup	G/H
center	C₂²	#16810860
commutator, cosocle	C₄×C₈×C₅₃	C₂³
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#1371	C₂
socle, prime radical	C₂²×C₅₃	#466
hypercenter	#1371	D₅₃
hypocenter	C₅₃	#12440
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1371

subgroup

G/H

center

C₂²

#16810860

commutator, cosocle

C₄×C₈×C₅₃

C₂³

Frattini

C₄×C₈

C₂²×D₅₃

Fitting

C₅₃×#1371

C₂

C₂²×C₅₃

D₅₃

C₅₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1371

N	H ≅ G/N	type
60. Dic₁₀₆	Q₃₂	semidirect
95. C₅₃	#12440	semidirect

H ≅ G/N

type

60. Dic₁₀₆

Q₃₂

semidirect

95. C₅₃

#12440

semidirect

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

subgroup	index
#20276968	2
C₁₆×Dic₁₀₆	2
#20274086	2
#20272763	2
C₅₃×#1371	2
#12440	53

subgroup

index

C₁₆×Dic₁₀₆

C₅₃×#1371

Subgroups — #27298355

Conjugacy classes of subgroups

Complements

Maximal subgroups