Group #27298790 — subgroups

Subgroups — #27298790

Order 13568.

268 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 79 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	#20276942	C₂	1	2	yes	maximal, complemented
2	#20276938	C₂	1	2	yes	maximal
3	#20274333	C₂	1	1	yes	maximal, characteristic, complemented
4	#20272632	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1709	C₂	1	1	yes	maximal, characteristic
6	#16811028	—	2	1	no	—
7	#16811028	C₂²	1	2	yes	—
8	#16811003	—	2	1	no	—
9	#16810427	—	2	4	no	—
10	#16810424	—	2	4	no	—
11	C₄×#11719908	C₂²	1	2	yes	—
12	C₅₃×#494	C₂²	1	1	yes	characteristic
13	C₅₃×#493	C₂²	1	1	yes	characteristic
14	C₅₃×#486	—	2	1	no	—
15	C₅₃×#486	—	2	1	no	—
16	C₈²×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₁₂	—	4	2	no	—
18	#13133217	—	2	2	no	—
19	C₂×#11719908	D₄	1	4	yes	—
20	#13133151	—	2	2	no	—
21	#13133151	—	2	2	no	—
22	#13133151	—	4	2	no	—
23	C₂×Q₁₆×C₅₃	—	2	2	no	—
24	C₂×D₈×C₅₃	—	2	2	no	—
25	C₄⋊Q₈×C₅₃	—	2	1	no	—
26	#127×C₅₃	—	2	1	no	—
27	#102×C₅₃	—	4	2	no	—
28	#102×C₅₃	—	4	2	no	—
29	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
30	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
31	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
32	C₂×Q₈×C₅₃	—	4	1	no	—
33	C₂×D₄×C₅₃	—	4	1	no	—
34	C₂×D₄×C₅₃	—	4	1	no	—
35	C₂×D₄×C₅₃	—	4	1	no	—
36	C₂×Dic₁₀₆	—	4	2	no	—
37	Q₁₆×C₅₃	—	4	4	no	—
38	D₈×C₅₃	—	4	4	no	—
39	C₂×C₈×C₅₃	D₈	1	2	yes	—
40	C₂×C₈×C₅₃	D₈	1	2	yes	—
41	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
42	C₄⋊C₄×C₅₃	—	4	1	no	—
43	C₄⋊C₄×C₅₃	—	4	1	no	—
44	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₄⋊Dic₅₃	—	4	2	no	—
46	C₄⋊Dic₅₃	—	4	2	no	—
47	Dic₂₁₂	—	8	4	no	—
48	#11719908	—	2	4	no	—
49	C₂³×C₅₃	—	8	1	no	—
50	Q₈×C₅₃	—	8	2	no	—
51	D₄×C₅₃	—	8	2	no	—
52	D₄×C₅₃	—	8	2	no	—
53	D₄×C₅₃	—	8	2	no	—
54	C₂×C₄×C₅₃	—	8	1	no	—
55	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
58	C₂×Dic₅₃	—	8	2	no	—
59	Dic₁₀₆	—	8	4	no	—
60	C₈×C₅₃	—	2	2	no	—
61	C₈×C₅₃	—	2	2	no	—
62	C₈×C₅₃	#135	1	4	yes	—
63	#12512	—	53	1	no	maximal, Sylow
64	C₂²×C₅₃	—	16	1	no	—
65	C₂²×C₅₃	—	16	1	no	—
66	C₂²×C₅₃	—	16	1	no	—
67	C₂²×C₅₃	#493	1	1	yes	characteristic
68	C₄×C₅₃	—	16	1	no	—
69	C₄×C₅₃	#495	1	2	yes	—
70	C₄×C₅₃	#509	1	2	yes	—
71	C₄×C₅₃	#510	1	2	yes	—
72	Dic₅₃	—	16	2	no	—
73	#2241	—	53	2	no	—
74	#2240	—	53	2	no	—
75	#1713	—	53	1	no	—
76	#1709	D₅₃	1	1	yes	characteristic
77	#1311	—	53	1	no	—
78	C₂×C₅₃	—	32	1	no	—
79	C₂×C₅₃	#1709	1	1	yes	characteristic
80	C₂×C₅₃	#2250	1	1	yes	characteristic
81	C₂×C₅₃	#2251	1	1	yes	characteristic
82	#500	—	53	2	no	—
83	#500	—	106	1	no	—
84	#494	—	106	1	no	—
85	#494	C₂×D₅₃	1	1	yes	characteristic
86	#493	C₂×D₅₃	1	1	yes	characteristic
87	#486	—	2	1	no	—
88	#486	—	2	1	no	—
89	#358	—	106	4	no	—
90	#357	—	106	4	no	—
91	C₄×C₁₆	—	53	2	no	—
92	C₈²	C₂×D₅₃	1	1	yes	characteristic
93	C₅₃	#12512	1	1	yes	characteristic, complemented, Sylow
94	C₂×Q₁₆	—	2	2	no	—
95	C₂×Q₁₆	—	212	2	no	—
96	C₂×D₈	—	2	2	no	—
97	C₄⋊Q₈	—	2	1	no	—
98	C₄⋊Q₈	—	106	2	no	—
99	#127	—	2	1	no	—
100	C₂×C₁₆	—	53	4	no	—
101	#107	—	106	2	no	—
102	#107	—	106	2	no	—
103	#107	—	212	2	no	—
104	#102	—	4	2	no	—
105	#102	—	4	2	no	—
106	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
107	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
108	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
109	C₂×Q₈	—	4	1	no	—
110	C₂×Q₈	—	212	2	no	—
111	C₂×D₄	—	4	1	no	—
112	C₂×D₄	—	4	1	no	—
113	C₂×D₄	—	4	1	no	—
114	Q₁₆	—	4	4	no	—
115	Q₁₆	—	424	4	no	—
116	D₈	—	4	4	no	—
117	C₂×C₈	#11719922	1	2	yes	—
118	C₂×C₈	#11719922	1	2	yes	—
119	C₂×C₈	D₄×D₅₃	1	2	yes	—
120	C₄⋊C₄	—	4	1	no	—
121	C₄⋊C₄	—	4	1	no	—
122	C₄⋊C₄	—	212	2	no	—
123	C₄⋊C₄	—	212	2	no	—
124	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
125	C₁₆	—	106	4	no	—
126	C₂³	—	8	1	no	—
127	Q₈	—	8	2	no	—
128	Q₈	—	424	4	no	—
129	D₄	—	8	2	no	—
130	D₄	—	8	2	no	—
131	D₄	—	8	2	no	—
132	C₂×C₄	—	8	1	no	—
133	C₂×C₄	—	424	2	no	—
134	C₂×C₄	#13133213	1	1	yes	characteristic
135	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
136	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	—	2	2	no	—
139	C₈	#13133236	1	4	yes	—
140	C₂²	—	16	1	no	—
141	C₂²	—	16	1	no	—
142	C₂²	—	16	1	no	—
143	C₂²	#16810999	1	1	yes	characteristic
144	C₄	—	16	1	no	—
145	C₄	—	848	2	no	—
146	C₄	#16811006	1	2	yes	—
147	C₄	#16811058	1	2	yes	—
148	C₄	#16811063	1	2	yes	—
149	C₂	—	32	1	no	—
150	C₂	#20274320	1	1	yes	characteristic
151	C₂	#20276982	1	1	yes	characteristic
152	C₂	#20276987	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276942

C₂

yes

maximal, complemented

#20276938

C₂

yes

maximal

#20274333

C₂

yes

maximal, characteristic, complemented

#20272632

C₂

yes

maximal, characteristic, complemented

C₅₃×#1709

C₂

yes

maximal, characteristic

#16811028

—

#16811028

C₂²

yes

—

—

—

—

C₂²

yes

—

C₅₃×#494

C₂²

yes

characteristic

C₅₃×#493

C₂²

yes

characteristic

C₅₃×#486

—

C₅₃×#486

—

C₈²×C₅₃

C₂²

yes

characteristic

C₂×Dic₂₁₂

—

#13133217

—

C₂×#11719908

D₄

yes

—

#13133151

—

#13133151

—

#13133151

—

C₂×Q₁₆×C₅₃

—

C₂×D₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

#127×C₅₃

—

#102×C₅₃

—

#102×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×Dic₁₀₆

—

Q₁₆×C₅₃

—

D₈×C₅₃

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

Dic₂₁₂

—

#11719908

—

C₂³×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×Dic₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#135

yes

—

#12512

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#493

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

#495

yes

—

C₄×C₅₃

#509

yes

—

C₄×C₅₃

#510

yes

—

Dic₅₃

—

—

—

—

D₅₃

yes

characteristic

#1311

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

106

—

#494

—

106

—

#494

C₂×D₅₃

yes

characteristic

#493

C₂×D₅₃

yes

—

—

—

106

—

#357

—

106

—

C₄×C₁₆

—

C₈²

C₂×D₅₃

yes

characteristic

C₅₃

#12512

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

212

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

106

—

#127

—

100

C₂×C₁₆

—

101

#107

—

106

—

102

#107

—

106

—

103

#107

—

212

—

104

#102

—

105

#102

—

106

C₄×C₈

C₂²⋊D₅₃

yes

characteristic

107

C₄×C₈

C₂²⋊D₅₃

yes

characteristic

108

C₄×C₈

C₂²×D₅₃

yes

characteristic

109

C₂×Q₈

—

110

C₂×Q₈

—

212

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

Q₁₆

—

115

Q₁₆

—

424

—

116

D₈

—

117

C₂×C₈

#11719922

yes

—

118

C₂×C₈

#11719922

yes

—

119

C₂×C₈

D₄×D₅₃

yes

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

212

—

123

C₄⋊C₄

—

212

—

124

C₄²

C₂×C₂²⋊D₅₃

yes

characteristic

125

C₁₆

—

106

—

126

C₂³

—

127

Q₈

—

128

Q₈

—

424

—

129

D₄

—

130

D₄

—

131

D₄

—

132

C₂×C₄

—

133

C₂×C₄

—

424

—

134

C₂×C₄

#13133213

yes

characteristic

135

C₂×C₄

C₂×#11719922

yes

characteristic

136

C₂×C₄

C₂×#11719922

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

#13133236

yes

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

#16810999

yes

characteristic

144

C₄

—

145

C₄

—

848

—

146

C₄

#16811006

yes

—

147

C₄

#16811058

yes

—

148

C₄

#16811063

yes

—

149

C₂

—

150

C₂

#20274320

yes

characteristic

151

C₂

#20276982

yes

characteristic

152

C₂

#20276987

yes

characteristic

	subgroup	G/H
center	C₂²	#16810999
commutator, cosocle	C₄×C₈×C₅₃	C₂³
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#1709	C₂
socle	C₂²×C₅₃	#493
prime radical	C₅₃×#493	C₂²
hypercenter	#1709	D₅₃
hypocenter	C₅₃	#12512
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

subgroup

G/H

center

C₂²

#16810999

commutator, cosocle

C₄×C₈×C₅₃

C₂³

Frattini

C₄×C₈

C₂²×D₅₃

Fitting

C₅₃×#1709

C₂

socle

C₂²×C₅₃

#493

prime radical

C₅₃×#493

C₂²

D₅₃

C₅₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

N	H ≅ G/N	type
1. #20276942	C₂	semidirect
3. #20274333	C₂	semidirect
4. #20272632	C₂	semidirect
93. C₅₃	#12512	semidirect

H ≅ G/N

type

1. #20276942

C₂

semidirect

3. #20274333

C₂

semidirect

4. #20272632

C₂

semidirect

93. C₅₃

#12512

semidirect

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

subgroup	index
#20276942	2
#20276938	2
#20274333	2
#20272632	2
C₅₃×#1709	2
#12512	53

subgroup

index

C₅₃×#1709

Subgroups — #27298790

Conjugacy classes of subgroups

Complements

Maximal subgroups