Group #27298451 — subgroups

Subgroups — #27298451

Order 13568.

204 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 75 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	#20274109	C₂	1	2	yes	maximal
3	#20272787	C₂	1	1	yes	maximal, characteristic
4	#20272769	C₂	1	2	yes	maximal
5	Q₈×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
6	#16811028	C₂²	1	2	yes	—
7	#16810833	—	2	2	no	—
8	#16810449	—	2	1	no	—
9	#16810449	—	2	1	no	—
10	#16810376	C₂²	1	2	yes	—
11	#16810368	—	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	C₄×C₁₆×C₅₃	—	2	1	no	—
16	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#11719925	—	4	2	no	—
18	#13133217	—	2	2	no	—
19	#13133151	—	2	2	no	—
20	#13133151	—	2	2	no	—
21	C₄⋊#88322	—	2	2	no	—
22	#13133140	—	4	2	no	—
23	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₅₃	—	2	1	no	—
25	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
27	C₄⋊C₈×C₅₃	—	2	1	no	—
28	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
29	C₄×C₈×C₅₃	—	2	1	no	—
30	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
31	C₂×Q₈×C₅₃	D₈	1	1	yes	characteristic
32	C₂×Dic₁₀₆	—	4	2	no	—
33	C₂×C₈×C₅₃	—	2	1	no	—
34	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
36	C₄⋊C₄×C₅₃	—	2	1	no	—
37	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
38	C₄²×C₅₃	—	2	1	no	—
39	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
40	#11719925	—	8	4	no	—
41	C₄⋊Dic₅₃	—	4	2	no	—
42	C₄⋊Dic₅₃	—	4	2	no	—
43	C₂×#88322	—	4	2	no	—
44	C₁₆×C₅₃	—	2	1	no	—
45	C₁₆×C₅₃	—	4	1	no	—
46	C₁₆×C₅₃	Q₁₆	1	2	yes	complemented
47	Q₈×C₅₃	—	2	1	no	—
48	Q₈×C₅₃	D₁₆	1	2	yes	—
49	C₂×C₄×C₅₃	—	2	1	no	—
50	C₂×C₄×C₅₃	—	2	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	—	8	2	no	—
55	Dic₁₀₆	—	8	2	no	—
56	Dic₁₀₆	—	8	2	no	—
57	C₈×C₅₃	—	2	1	no	—
58	C₈×C₅₃	—	4	1	no	—
59	C₈×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
60	C₈×C₅₃	#137	1	1	yes	characteristic
61	#88322	—	8	2	no	—
62	#12457	—	53	1	no	maximal, Sylow
63	C₂²×C₅₃	#466	1	1	yes	characteristic
64	C₄×C₅₃	—	2	2	no	—
65	C₄×C₅₃	—	2	1	no	—
66	C₄×C₅₃	—	4	1	no	—
67	C₄×C₅₃	#462	1	1	yes	characteristic
68	C₄×C₅₃	#473	1	1	yes	characteristic
69	C₄×C₅₃	C₂×D₁₆	1	1	yes	characteristic
70	C₄×C₅₃	#508	1	1	yes	characteristic
71	Dic₅₃	—	16	2	no	—
72	#2252	—	53	1	no	—
73	Q₈×C₁₆	D₅₃	1	1	yes	characteristic
74	#1674	—	53	2	no	—
75	#1371	—	53	1	no	—
76	#1364	—	53	2	no	—
77	C₂×C₅₃	#1674	1	1	yes	characteristic
78	C₂×C₅₃	#2215	1	1	yes	characteristic
79	C₂×C₅₃	#2222	1	1	yes	characteristic
80	#500	—	53	2	no	—
81	#462	—	106	2	no	—
82	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
83	#366	—	106	1	no	—
84	#366	—	106	1	no	—
85	C₄⋊C₁₆	—	2	1	no	—
86	C₄⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
87	C₄×C₁₆	—	2	1	no	—
88	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
89	#335	—	53	2	no	—
90	#332	—	106	2	no	—
91	C₅₃	#12457	1	1	yes	characteristic, complemented, Sylow
92	C₂×Q₁₆	—	212	2	no	—
93	C₄⋊Q₈	—	106	2	no	—
94	C₄×Q₈	D₂₁₂	1	1	yes	characteristic
95	C₂×C₁₆	—	2	1	no	—
96	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
97	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
98	#107	—	106	2	no	—
99	#107	—	106	2	no	—
100	C₄⋊C₈	—	2	1	no	—
101	C₄⋊C₈	—	106	2	no	—
102	C₄⋊C₈	D₂₁₂	1	1	yes	characteristic
103	#103	—	212	2	no	—
104	C₄×C₈	—	2	1	no	—
105	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
106	C₂×Q₈	—	212	2	no	—
107	C₂×Q₈	D₄₂₄	1	1	yes	characteristic
108	Q₁₆	—	424	4	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	212	2	no	—
111	C₂×C₈	#11719945	1	1	yes	characteristic
112	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	212	2	no	—
115	C₄⋊C₄	—	212	2	no	—
116	C₄⋊C₄	D₄₂₄	1	1	yes	characteristic
117	C₄²	—	2	1	no	—
118	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
119	C₁₆	—	2	1	no	—
120	C₁₆	—	4	1	no	—
121	C₁₆	#11719925	1	2	yes	complemented
122	Q₈	—	2	1	no	—
123	Q₈	—	424	2	no	—
124	Q₈	—	424	2	no	—
125	Q₈	D₈₄₈	1	2	yes	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	424	2	no	—
129	C₂×C₄	#13133191	1	1	yes	characteristic
130	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
131	C₂×C₄	#13133239	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	4	1	no	—
134	C₈	—	424	2	no	—
135	C₈	C₂×#11719925	1	1	yes	characteristic
136	C₈	#13133247	1	1	yes	characteristic
137	C₂²	#16810859	1	1	yes	characteristic
138	C₄	—	2	2	no	—
139	C₄	—	2	1	no	—
140	C₄	—	4	1	no	—
141	C₄	—	848	2	no	—
142	C₄	#16810833	1	1	yes	characteristic
143	C₄	#16810898	1	1	yes	characteristic
144	C₄	C₂×D₈₄₈	1	1	yes	characteristic
145	C₄	#16811055	1	1	yes	characteristic
146	C₂	#20274109	1	1	yes	characteristic
147	C₂	#20276801	1	1	yes	characteristic
148	C₂	#20276840	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276990

C₂

yes

maximal, characteristic

#20274109

C₂

yes

maximal

#20272787

C₂

yes

maximal, characteristic

#20272769

C₂

yes

maximal

Q₈×C₁₆×C₅₃

C₂

yes

maximal, characteristic

#16811028

C₂²

yes

—

—

—

—

C₂²

yes

—

#16810368

—

C₈×Q₈×C₅₃

C₂²

yes

characteristic

C₅₃×C₄⋊C₁₆

—

C₅₃×C₄⋊C₁₆

C₂²

yes

characteristic

C₄×C₁₆×C₅₃

—

C₄×C₁₆×C₅₃

C₂²

yes

—

—

—

—

—

—

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

#11719925

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₁₆×C₅₃

—

C₁₆×C₅₃

—

C₁₆×C₅₃

Q₁₆

yes

complemented

Q₈×C₅₃

—

Q₈×C₅₃

D₁₆

yes

—

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×Q₁₆

yes

C₈×C₅₃

yes

—

—

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

C₂×D₁₆

yes

characteristic

C₄×C₅₃

#508

yes

characteristic

Dic₅₃

—

#2252

—

Q₈×C₁₆

D₅₃

yes

—

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

106

—

C₈×Q₈

C₂×D₅₃

yes

characteristic

#366

—

106

—

#366

—

106

—

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×D₅₃

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

C₂×D₅₃

yes

characteristic

#335

—

#332

—

106

—

C₅₃

#12457

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

212

—

C₄⋊Q₈

—

106

—

C₄×Q₈

D₂₁₂

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₅₃

yes

characteristic

C₂×C₁₆

C₂²⋊D₅₃

yes

characteristic

#107

—

106

—

#107

—

106

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

106

—

102

C₄⋊C₈

D₂₁₂

yes

characteristic

103

#103

—

212

—

104

C₄×C₈

—

105

C₄×C₈

C₂²×D₅₃

yes

characteristic

106

C₂×Q₈

—

212

—

107

C₂×Q₈

D₄₂₄

yes

characteristic

108

Q₁₆

—

424

—

109

C₂×C₈

—

110

C₂×C₈

—

212

—

111

C₂×C₈

#11719945

yes

characteristic

112

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

—

212

—

115

C₄⋊C₄

—

212

—

116

C₄⋊C₄

D₄₂₄

yes

characteristic

117

C₄²

—

118

C₄²

C₂×D₂₁₂

yes

characteristic

119

C₁₆

—

120

C₁₆

—

121

C₁₆

#11719925

yes

complemented

122

Q₈

—

123

Q₈

—

424

—

124

Q₈

—

424

—

125

Q₈

D₈₄₈

yes

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

424

—

129

C₂×C₄

#13133191

yes

characteristic

130

C₂×C₄

C₂×D₄₂₄

yes

characteristic

131

C₂×C₄

#13133239

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

424

—

135

C₈

C₂×#11719925

yes

characteristic

136

C₈

#13133247

yes

characteristic

137

C₂²

#16810859

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

848

—

142

C₄

#16810833

yes

characteristic

143

C₄

#16810898

yes

characteristic

144

C₄

C₂×D₈₄₈

yes

characteristic

145

C₄

#16811055

yes

characteristic

146

C₂

#20274109

yes

characteristic

147

C₂

#20276801

yes

characteristic

148

C₂

#20276840

yes

characteristic

	subgroup	G/H
center	C₂²	#16810859
commutator, cosocle	C₄×C₈×C₅₃	C₂³
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	Q₈×C₁₆×C₅₃	C₂
socle, prime radical	C₂²×C₅₃	#466
hypercenter	Q₈×C₁₆	D₅₃
hypocenter	C₅₃	#12457
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ Q₈×C₁₆

subgroup

G/H

center

C₂²

#16810859

commutator, cosocle

C₄×C₈×C₅₃

C₂³

Frattini

C₄×C₈

C₂²×D₅₃

Fitting

Q₈×C₁₆×C₅₃

C₂

socle, prime radical

C₂²×C₅₃

#466

hypercenter

Q₈×C₁₆

D₅₃

hypocenter

C₅₃

#12457

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ Q₈×C₁₆

N	H ≅ G/N	type
46. C₁₆×C₅₃	Q₁₆	semidirect
91. C₅₃	#12457	semidirect
121. C₁₆	#11719925	semidirect

H ≅ G/N

type

46. C₁₆×C₅₃

Q₁₆

semidirect

91. C₅₃

#12457

semidirect

121. C₁₆

#11719925

semidirect

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

subgroup	index
#20276990	2
#20274109	2
#20272787	2
#20272769	2
Q₈×C₁₆×C₅₃	2
#12457	53

subgroup

index

Q₈×C₁₆×C₅₃

Subgroups — #27298451

Conjugacy classes of subgroups

Complements

Maximal subgroups