Group #27298457 — subgroups

Subgroups — #27298457

Order 13568.

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

normal subgroups only

This group has 192 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	#20276990	C₂	1	1	yes	maximal, characteristic
2	C₁₆×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
3	#20274120	C₂	1	2	yes	maximal
4	#20272774	C₂	1	2	yes	maximal
5	C₅₃×#1371	C₂	1	1	yes	maximal, characteristic
6	#16811028	C₂²	1	2	yes	—
7	#16810919	—	2	2	no	—
8	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic
9	#16810449	—	2	1	no	—
10	#16810449	—	2	1	no	—
11	C₄⋊#11719908	C₂²	1	1	yes	characteristic
12	#16810441	—	2	1	no	—
13	C₁₆×Dic₅₃	—	2	1	no	—
14	#16810371	—	2	2	no	—
15	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#335	C₂²	1	2	yes	—
17	#13133217	—	2	2	no	—
18	C₄×Dic₁₀₆	D₄	1	1	yes	characteristic
19	C₂×#11719908	—	2	1	no	—
20	#13133151	—	2	2	no	—
21	#13133151	—	2	2	no	—
22	#13133151	—	4	2	no	—
23	C₄⋊#88322	D₄	1	1	yes	characteristic
24	#13133146	—	2	1	no	—
25	#13133142	—	4	2	no	—
26	C₈×Dic₅₃	—	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	2	no	—
30	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
31	C₂×Dic₁₀₆	—	4	2	no	—
32	C₂×Dic₁₀₆	D₈	1	1	yes	characteristic
33	C₂×C₈×C₅₃	—	4	2	no	—
34	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
36	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₄⋊Dic₅₃	—	4	2	no	—
38	C₄⋊Dic₅₃	—	4	2	no	—
39	C₄⋊Dic₅₃	D₈	1	1	yes	characteristic
40	#11719919	—	2	1	no	—
41	C₄×Dic₅₃	—	2	1	no	—
42	C₂×#88322	—	2	1	no	—
43	C₁₆×C₅₃	—	2	1	no	—
44	C₁₆×C₅₃	Q₁₆	1	2	yes	—
45	#11719908	—	4	1	no	—
46	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
48	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
49	C₂×Dic₅₃	—	2	1	no	—
50	C₂×Dic₅₃	—	2	1	no	—
51	C₂×Dic₅₃	—	8	2	no	—
52	Dic₁₀₆	—	2	1	no	—
53	Dic₁₀₆	—	16	2	no	—
54	Dic₁₀₆	Q₃₂	1	2	yes	—
55	C₈×C₅₃	—	2	1	no	—
56	C₈×C₅₃	—	8	2	no	—
57	C₈×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
58	C₈×C₅₃	#137	1	1	yes	characteristic
59	#88322	—	4	1	no	—
60	#12458	—	53	1	no	maximal, Sylow
61	C₂²×C₅₃	#466	1	1	yes	characteristic
62	C₄×C₅₃	—	2	1	no	—
63	C₄×C₅₃	#462	1	1	yes	characteristic
64	C₄×C₅₃	#473	1	1	yes	characteristic
65	C₄×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
66	C₄×C₅₃	#508	1	1	yes	characteristic
67	Dic₅₃	—	2	2	no	—
68	Dic₅₃	—	4	1	no	—
69	Dic₅₃	—	16	2	no	—
70	#2252	—	53	1	no	—
71	Q₈×C₁₆	—	53	1	no	—
72	#1676	—	53	2	no	—
73	#1371	D₅₃	1	1	yes	characteristic
74	#1365	—	53	2	no	—
75	C₂×C₅₃	#1674	1	1	yes	characteristic
76	C₂×C₅₃	#2216	1	1	yes	characteristic
77	C₂×C₅₃	#2221	1	1	yes	characteristic
78	#500	—	53	2	no	—
79	#477	—	106	2	no	—
80	C₈×Q₈	—	53	1	no	—
81	#366	—	106	1	no	—
82	#366	—	106	1	no	—
83	C₄⋊C₁₆	—	53	1	no	—
84	C₄⋊C₁₆	—	106	1	no	—
85	C₄×C₁₆	—	106	1	no	—
86	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
87	#335	C₂×D₅₃	1	2	yes	—
88	#333	—	106	2	no	—
89	C₅₃	#12458	1	1	yes	characteristic, complemented, Sylow
90	C₄⋊Q₈	—	106	2	no	—
91	C₄×Q₈	—	53	1	no	—
92	C₂×C₁₆	—	106	1	no	—
93	C₂×C₁₆	D₂₁₂	1	1	yes	characteristic
94	C₂×C₁₆	D₂₁₂	1	1	yes	characteristic
95	#107	—	106	2	no	—
96	#107	—	106	2	no	—
97	#107	—	212	2	no	—
98	C₄⋊C₈	—	2	2	no	—
99	C₄⋊C₈	—	53	1	no	—
100	C₄⋊C₈	—	106	1	no	—
101	#103	—	212	2	no	—
102	C₄×C₈	—	106	1	no	—
103	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
104	C₂×Q₈	—	53	1	no	—
105	C₂×Q₈	—	212	2	no	—
106	C₂×C₈	—	4	2	no	—
107	C₂×C₈	—	106	1	no	—
108	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
109	C₂×C₈	#11719949	1	1	yes	characteristic
110	C₄⋊C₄	—	53	1	no	—
111	C₄⋊C₄	—	106	1	no	—
112	C₄⋊C₄	—	212	2	no	—
113	C₄⋊C₄	—	212	2	no	—
114	C₄²	—	106	1	no	—
115	C₄²	D₄×D₅₃	1	1	yes	characteristic
116	C₁₆	—	2	1	no	—
117	C₁₆	—	212	1	no	—
118	C₁₆	Dic₂₁₂	1	2	yes	—
119	Q₈	—	53	2	no	—
120	Q₈	—	848	2	no	—
121	Q₈	—	106	1	no	—
122	C₂×C₄	—	106	1	no	—
123	C₂×C₄	—	106	1	no	—
124	C₂×C₄	—	424	2	no	—
125	C₂×C₄	#13133192	1	1	yes	characteristic
126	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
127	C₂×C₄	#13133240	1	1	yes	characteristic
128	C₈	—	2	1	no	—
129	C₈	—	8	2	no	—
130	C₈	—	212	1	no	—
131	C₈	C₂×Dic₂₁₂	1	1	yes	characteristic
132	C₈	#13133250	1	1	yes	characteristic
133	C₂²	#16810860	1	1	yes	characteristic
134	C₄	—	2	1	no	—
135	C₄	—	848	2	no	—
136	C₄	—	106	2	no	—
137	C₄	—	212	1	no	—
138	C₄	#16810834	1	1	yes	characteristic
139	C₄	#16810899	1	1	yes	characteristic
140	C₄	Q₃₂×D₅₃	1	1	yes	characteristic
141	C₄	#16811052	1	1	yes	characteristic
142	C₂	#20274110	1	1	yes	characteristic
143	C₂	#20276807	1	1	yes	characteristic
144	C₂	#20276836	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276990

C₂

yes

maximal, characteristic

C₁₆×Dic₁₀₆

C₂

yes

maximal, characteristic

#20274120

C₂

yes

maximal

#20272774

C₂

yes

maximal

C₅₃×#1371

C₂

yes

maximal, characteristic

#16811028

C₂²

yes

—

#16810919

—

C₈×Dic₁₀₆

C₂²

yes

—

—

C₂²

yes

characteristic

#16810441

—

C₁₆×Dic₅₃

—

#16810371

—

C₄×C₁₆×C₅₃

C₂²

yes

characteristic

C₅₃×#335

C₂²

yes

—

#13133217

—

C₄×Dic₁₀₆

D₄

yes

—

—

—

—

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

—

C₁₆×C₅₃

—

C₁₆×C₅₃

Q₁₆

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

Q₃₂

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

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

#508

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

#2252

—

Q₈×C₁₆

—

#1676

—

#1371

D₅₃

yes

characteristic

#1365

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

106

—

C₈×Q₈

—

#366

—

106

—

#366

—

106

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

106

—

C₄×C₁₆

—

106

—

C₄×C₁₆

C₂×D₅₃

yes

characteristic

#335

C₂×D₅₃

yes

—

#333

—

106

—

C₅₃

#12458

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

106

—

C₄×Q₈

—

C₂×C₁₆

—

106

—

C₂×C₁₆

D₂₁₂

yes

characteristic

C₂×C₁₆

D₂₁₂

yes

characteristic

#107

—

106

—

#107

—

106

—

#107

—

212

—

C₄⋊C₈

—

C₄⋊C₈

—

100

C₄⋊C₈

—

106

—

101

#103

—

212

—

102

C₄×C₈

—

106

—

103

C₄×C₈

C₂²×D₅₃

yes

characteristic

104

C₂×Q₈

—

105

C₂×Q₈

—

212

—

106

C₂×C₈

—

107

C₂×C₈

—

106

—

108

C₂×C₈

C₂×D₂₁₂

yes

characteristic

109

C₂×C₈

#11719949

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

106

—

112

C₄⋊C₄

—

212

—

113

C₄⋊C₄

—

212

—

114

C₄²

—

106

—

115

C₄²

D₄×D₅₃

yes

characteristic

116

C₁₆

—

117

C₁₆

—

212

—

118

C₁₆

Dic₂₁₂

yes

—

119

Q₈

—

120

Q₈

—

848

—

121

Q₈

—

106

—

122

C₂×C₄

—

106

—

123

C₂×C₄

—

106

—

124

C₂×C₄

—

424

—

125

C₂×C₄

#13133192

yes

characteristic

126

C₂×C₄

D₈×D₅₃

yes

characteristic

127

C₂×C₄

#13133240

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

212

—

131

C₈

C₂×Dic₂₁₂

yes

characteristic

132

C₈

#13133250

yes

characteristic

133

C₂²

#16810860

yes

characteristic

134

C₄

—

135

C₄

—

848

—

136

C₄

—

106

—

137

C₄

—

212

—

138

C₄

#16810834

yes

characteristic

139

C₄

#16810899

yes

characteristic

140

C₄

Q₃₂×D₅₃

yes

characteristic

141

C₄

#16811052

yes

characteristic

142

C₂

#20274110

yes

characteristic

143

C₂

#20276807

yes

characteristic

144

C₂

#20276836

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	#12458
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
89. C₅₃	#12458	semidirect

H ≅ G/N

type

89. C₅₃

#12458

semidirect

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

subgroup	index
#20276990	2
C₁₆×Dic₁₀₆	2
#20274120	2
#20272774	2
C₅₃×#1371	2
#12458	53

subgroup

index

C₁₆×Dic₁₀₆

C₅₃×#1371

Subgroups — #27298457

Conjugacy classes of subgroups

Complements

Maximal subgroups