Group #27299098 — subgroups

Subgroups — #27299098

Order 13568.

358 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 195 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	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic
2	#20274401	C₂	1	1	yes	maximal, characteristic
3	#20272792	C₂	1	4	yes	maximal
4	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719908	C₂²	1	2	yes	—
6	C₂×#13133128	C₄	1	1	yes	characteristic
7	C₂×#13133127	C₄	1	2	yes	—
8	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
9	#16810469	C₄	1	1	yes	characteristic
10	C₄×#11719908	C₄	1	4	yes	complemented
11	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#346	C₄	1	4	yes	complemented
14	C₂²×#88322	C₂×C₄	1	2	yes	—
15	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
16	C₂×#11719908	—	2	4	no	—
17	C₂×#11719908	C₂×C₄	1	4	yes	—
18	#13133128	C₈	1	4	yes	—
19	#13133127	—	2	4	no	—
20	C₈×Dic₅₃	—	2	4	no	—
21	#13133124	C₈	1	4	yes	—
22	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₅₃	—	2	4	no	—
26	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
27	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
30	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
31	C₂²×Dic₅₃	C₂×C₈	1	4	yes	—
32	C₂×C₈×C₅₃	C₄²	1	2	yes	—
33	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
34	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈×C₅₃	#41	1	4	yes	—
37	C₄²×C₅₃	C₄²	1	2	yes	—
38	C₄²×C₅₃	C₂×C₈	1	2	yes	—
39	C₄×Dic₅₃	—	2	2	no	—
40	C₄×Dic₅₃	—	2	2	no	—
41	C₂×#88322	—	2	2	no	—
42	C₂×#88322	—	2	2	no	—
43	C₂×#88322	C₂×C₈	1	4	yes	—
44	C₁₆×C₅₃	—	2	4	no	—
45	#11719908	—	2	4	no	—
46	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
47	C₂×C₄×C₅₃	—	2	4	no	—
48	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
49	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
50	C₂×C₄×C₅₃	#117	1	2	yes	—
51	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
53	C₂×Dic₅₃	—	2	4	no	—
54	C₂×Dic₅₃	—	2	8	no	—
55	C₈×C₅₃	#97	1	4	yes	—
56	C₈×C₅₃	#117	1	2	yes	—
57	C₈×C₅₃	#117	1	2	yes	—
58	#88322	—	2	4	no	—
59	#12592	—	53	1	no	maximal, Sylow
60	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
61	C₂²×C₅₃	#431	1	2	yes	—
62	C₂²×C₅₃	#504	1	4	yes	—
63	C₄×C₅₃	—	2	4	no	—
64	C₄×C₅₃	#376	1	1	yes	characteristic
65	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
66	C₄×C₅₃	#431	1	2	yes	—
67	Dic₅₃	—	4	4	no	—
68	C₂×#346	D₅₃	1	1	yes	characteristic
69	C₂×C₄×C₁₆	—	53	1	no	—
70	#1726	—	53	1	no	—
71	#1374	—	53	4	no	—
72	C₂×C₅₃	#1744	1	1	yes	characteristic
73	C₂×C₅₃	#2108	1	2	yes	—
74	C₂×C₅₃	#2164	1	4	yes	—
75	C₂²×C₁₆	—	53	2	no	—
76	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
77	C₂×#97	—	53	1	no	—
78	C₂×#97	—	53	2	no	—
79	C₂×C₄×C₈	—	53	2	no	—
80	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
81	#376	—	53	1	no	—
82	#346	Dic₅₃	1	4	yes	complemented
83	C₄×C₁₆	—	53	4	no	—
84	C₅₃	#12592	1	1	yes	characteristic, complemented, Sylow
85	C₂²×C₈	—	53	2	no	—
86	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
87	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
88	C₂×C₄²	—	53	2	no	—
89	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
90	C₂×C₁₆	—	2	4	no	—
91	C₂×C₁₆	—	53	4	no	—
92	C₂×C₁₆	—	106	4	no	—
93	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
94	#97	—	53	4	no	—
95	#97	—	106	4	no	—
96	C₄×C₈	—	106	4	no	—
97	C₄×C₈	C₄×D₅₃	1	2	yes	—
98	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
99	#95	—	53	4	no	—
100	C₂²×C₄	—	53	4	no	—
101	C₂²×C₄	C₈×D₅₃	1	2	yes	—
102	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
103	C₂×C₈	—	53	4	no	—
104	C₂×C₈	—	106	2	no	—
105	C₂×C₈	—	106	2	no	—
106	C₂×C₈	#11719912	1	4	yes	—
107	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
108	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
109	C₂×C₈	#11719945	1	4	yes	—
110	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
111	C₄²	—	106	2	no	—
112	C₄²	—	106	2	no	—
113	C₄²	C₈×D₅₃	1	2	yes	—
114	C₄²	C₄×Dic₅₃	1	2	yes	—
115	C₁₆	—	2	4	no	—
116	C₁₆	—	106	4	no	—
117	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	106	8	no	—
120	C₂×C₄	—	106	4	no	—
121	C₂×C₄	C₈×Dic₅₃	1	4	yes	—
122	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
123	C₂×C₄	#13133174	1	2	yes	—
124	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
125	C₂×C₄	C₂×#11719912	1	2	yes	—
126	C₈	—	106	4	no	—
127	C₈	#13133127	1	4	yes	—
128	C₈	#13133174	1	2	yes	—
129	C₈	#13133176	1	2	yes	—
130	C₂²	C₂×C₈×Dic₅₃	1	1	yes	characteristic
131	C₂²	#16810674	1	2	yes	—
132	C₂²	#16811041	1	4	yes	—
133	C₄	—	2	4	no	—
134	C₄	—	212	4	no	—
135	C₄	#16810469	1	1	yes	characteristic
136	C₄	C₂×#13133127	1	1	yes	characteristic
137	C₄	#16810674	1	2	yes	—
138	C₂	#20274460	1	1	yes	characteristic
139	C₂	#20276314	1	2	yes	—
140	C₂	#20276523	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×#11719908

C₂

yes

maximal, characteristic

#20274401

C₂

yes

maximal, characteristic

#20272792

C₂

yes

maximal

C₂×C₅₃×#346

C₂

yes

maximal, characteristic

C₂²×#11719908

C₂²

yes

—

C₂×#13133128

C₄

yes

characteristic

C₂×#13133127

C₄

yes

—

C₂×C₈×Dic₅₃

C₂²

yes

—

#16810469

C₄

yes

characteristic

C₄×#11719908

C₄

yes

complemented

C₂²×C₁₆×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#346

C₄

yes

complemented

C₂²×#88322

C₂×C₄

yes

—

C₂×C₄×Dic₅₃

C₂×C₄

yes

—

C₂×#11719908

—

C₂×#11719908

C₂×C₄

yes

—

#13133128

C₈

yes

—

#13133127

—

C₈×Dic₅₃

—

#13133124

C₈

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₈

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×Dic₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

C₄²

yes

—

C₂×C₈×C₅₃

M₁₆

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

#41

yes

—

C₄²×C₅₃

C₄²

yes

—

C₄²×C₅₃

C₂×C₈

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

C₂×C₈

yes

—

C₁₆×C₅₃

—

#11719908

—

C₂³×C₅₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄×C₈

yes

—

C₂×C₄×C₅₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

#97

yes

—

C₈×C₅₃

#117

yes

—

C₈×C₅₃

#117

yes

—

#88322

—

#12592

—

maximal, Sylow

C₂²×C₅₃

C₂×C₄×C₈

yes

characteristic

C₂²×C₅₃

#431

yes

—

C₂²×C₅₃

#504

yes

—

C₄×C₅₃

—

C₄×C₅₃

#376

yes

characteristic

C₄×C₅₃

C₂×#97

yes

characteristic

C₄×C₅₃

#431

yes

—

Dic₅₃

—

C₂×#346

D₅₃

yes

characteristic

C₂×C₄×C₁₆

—

#1726

—

#1374

—

C₂×C₅₃

#1744

yes

characteristic

C₂×C₅₃

#2108

yes

—

C₂×C₅₃

#2164

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₅₃

yes

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

#376

—

#346

Dic₅₃

yes

complemented

C₄×C₁₆

—

C₅₃

#12592

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₄×D₅₃

yes

characteristic

C₂²×C₈

C₂²×D₅₃

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₄×D₅₃

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

C₂×C₁₆

C₂×Dic₅₃

yes

—

#97

—

#97

—

106

—

C₄×C₈

—

106

—

C₄×C₈

C₄×D₅₃

yes

—

C₄×C₈

C₂×Dic₅₃

yes

—

#95

—

100

C₂²×C₄

—

101

C₂²×C₄

C₈×D₅₃

yes

—

102

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

103

C₂×C₈

—

104

C₂×C₈

—

106

—

105

C₂×C₈

—

106

—

106

C₂×C₈

#11719912

yes

—

107

C₂×C₈

C₄×Dic₅₃

yes

—

108

C₂×C₈

C₂×C₄×D₅₃

yes

characteristic

109

C₂×C₈

#11719945

yes

—

110

C₂×C₈

C₂²×Dic₅₃

yes

characteristic

111

C₄²

—

106

—

112

C₄²

—

106

—

113

C₄²

C₈×D₅₃

yes

—

114

C₄²

C₄×Dic₅₃

yes

—

115

C₁₆

—

116

C₁₆

—

106

—

117

C₂³

C₂×C₈×D₅₃

yes

characteristic

118

C₂×C₄

—

119

C₂×C₄

—

106

—

120

C₂×C₄

—

106

—

121

C₂×C₄

C₈×Dic₅₃

yes

—

122

C₂×C₄

C₂×C₄×Dic₅₃

yes

characteristic

123

C₂×C₄

#13133174

yes

—

124

C₂×C₄

C₂×C₈×D₅₃

yes

characteristic

125

C₂×C₄

C₂×#11719912

yes

—

126

C₈

—

106

—

127

C₈

#13133127

yes

—

128

C₈

#13133174

yes

—

129

C₈

#13133176

yes

—

130

C₂²

C₂×C₈×Dic₅₃

yes

characteristic

131

C₂²

#16810674

yes

—

132

C₂²

#16811041

yes

—

133

C₄

—

134

C₄

—

212

—

135

C₄

#16810469

yes

characteristic

136

C₄

C₂×#13133127

yes

characteristic

137

C₄

#16810674

yes

—

138

C₂

#20274460

yes

characteristic

139

C₂

#20276314

yes

—

140

C₂

#20276523

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₂×C₅₃×#346	C₂
socle, prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#346	D₅₃
hypocenter	C₅₃	#12592
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#346

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₅₃

commutator

C₂²×C₅₃

C₂×C₄×C₈

Fitting

C₂×C₅₃×#346

C₂

socle, prime radical

C₂³×C₅₃

C₂²×C₈

cosocle

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂²×C₈ ◁ C₂×#346

N	H ≅ G/N	type
10. C₄×#11719908	C₄	semidirect
13. C₅₃×#346	C₄	semidirect
82. #346	Dic₅₃	semidirect
84. C₅₃	#12592	semidirect

H ≅ G/N

type

10. C₄×#11719908

C₄

semidirect

13. C₅₃×#346

C₄

semidirect

82. #346

Dic₅₃

semidirect

84. C₅₃

#12592

semidirect

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

subgroup	index
C₂×C₄×#11719908	2
#20274401	2
#20272792	2
C₂×C₅₃×#346	2
#12592	53

subgroup

index

C₂×C₅₃×#346

Subgroups — #27299098

Conjugacy classes of subgroups

Complements

Maximal subgroups