Group #28307783 — subgroups

Subgroups — #28307783

Order 13568.

302 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 140 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	#20290916	C₂	1	1	yes	maximal, characteristic
2	#20289909	C₂	1	1	yes	maximal, characteristic
3	#20289634	C₂	1	4	yes	maximal, complemented
4	#346×D₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×D₅₃⋊C₁₆	C₄	1	2	yes	—
6	C₄×D₅₃⋊C₈	C₄	1	2	yes	—
7	#16811512	C₂²	1	2	yes	—
8	#16811511	C₄	1	2	yes	—
9	#16811464	C₄	1	2	yes	—
10	#16811464	C₂²	1	2	yes	—
11	C₂×C₁₆×D₅₃	C₄	1	2	yes	—
12	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
13	#16810399	C₄	1	1	yes	characteristic
14	#16810398	C₂²	1	2	yes	—
15	C₅₃×#346	C₄	1	1	yes	characteristic
16	C₂×D₅₃⋊C₈	—	2	2	no	—
17	D₅₃⋊C₁₆	C₈	1	8	yes	complemented
18	C₂×#11719910	C₂×C₄	1	4	yes	—
19	C₄×#88324	—	2	2	no	—
20	C₄×#88324	C₂×C₄	1	4	yes	—
21	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
23	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×#11719908	C₂×C₄	1	2	yes	—
25	C₁₆×D₅₃	—	2	4	no	—
26	C₄×#88322	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
28	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
29	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
32	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
33	C₂×#88324	—	2	4	no	—
34	D₅₃⋊C₈	—	4	4	no	—
35	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₄²×C₅₃	C₄²	1	1	yes	characteristic
38	C₄×Dic₅₃	M₁₆	1	2	yes	—
39	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×#88322	C₄²	1	1	yes	characteristic
41	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
42	C₈×D₅₃	C₂×C₈	1	4	yes	—
43	C₈×D₅₃	M₁₆	1	4	yes	—
44	#11719910	C₂×C₈	1	8	yes	complemented
45	C₁₆×C₅₃	—	2	2	no	—
46	#11719908	—	2	2	no	—
47	C₂²×D₅₃	#97	1	1	yes	characteristic
48	C₂×C₄×C₅₃	#97	1	2	yes	—
49	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
50	C₂×Dic₅₃	#97	1	1	yes	characteristic
51	C₂×Dic₅₃	C₂×M₁₆	1	2	yes	—
52	C₄×D₅₃	—	2	4	no	—
53	C₄×D₅₃	C₄×C₈	1	2	yes	—
54	C₄×D₅₃	#97	1	2	yes	—
55	#88324	—	4	4	no	—
56	C₈×C₅₃	C₄×C₈	1	2	yes	—
57	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
58	#88322	#97	1	2	yes	—
59	#88322	C₂²×C₈	1	2	yes	—
60	C₂×#1313	—	53	1	no	maximal, Sylow
61	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
62	C₂×D₅₃	#322	1	2	yes	—
63	C₂×D₅₃	#347	1	4	yes	—
64	C₄×C₅₃	—	2	2	no	—
65	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
66	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
67	Dic₅₃	—	2	2	no	—
68	Dic₅₃	#322	1	2	yes	—
69	C₂×#346	—	53	1	no	—
70	C₂×#346	—	53	1	no	—
71	C₂×#322	—	53	1	no	—
72	#1313	—	53	4	no	—
73	C₂×C₅₃	C₂×#322	1	1	yes	characteristic
74	C₂×C₅₃	C₂×#347	1	2	yes	—
75	D₅₃	#1313	1	4	yes	complemented
76	C₂²×C₁₆	—	53	2	no	—
77	C₂²×C₁₆	—	53	2	no	—
78	C₂×C₄×C₈	—	53	1	no	—
79	C₂×C₄×C₈	—	53	2	no	—
80	#346	—	53	2	no	—
81	#346	—	53	2	no	—
82	#346	—	53	2	no	—
83	#346	—	53	1	no	—
84	#346	#6157	1	1	yes	characteristic
85	#322	—	53	2	no	—
86	#322	—	53	2	no	—
87	C₅₃	C₂×#1313	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₈	—	53	1	no	—
89	C₂²×C₈	—	53	1	no	—
90	C₂²×C₈	—	106	2	no	—
91	C₂×C₄²	—	53	1	no	—
92	C₂×C₁₆	—	53	4	no	—
93	C₂×C₁₆	—	53	8	no	—
94	C₂×C₁₆	—	53	2	no	—
95	C₂×C₁₆	—	106	4	no	—
96	C₂×C₁₆	C₂×#6157	1	2	yes	—
97	C₄×C₈	—	53	1	no	—
98	C₄×C₈	—	53	1	no	—
99	C₄×C₈	—	53	4	no	—
100	C₄×C₈	—	53	1	no	—
101	C₄×C₈	—	106	2	no	—
102	C₄×C₈	C₂×#6157	1	1	yes	characteristic
103	C₂²×C₄	—	53	1	no	—
104	C₂²×C₄	—	53	2	no	—
105	C₂×C₈	—	53	1	no	—
106	C₂×C₈	—	53	1	no	—
107	C₂×C₈	—	53	4	no	—
108	C₂×C₈	—	53	4	no	—
109	C₂×C₈	—	106	4	no	—
110	C₂×C₈	—	212	4	no	—
111	C₂×C₈	C₄×#6157	1	1	yes	characteristic
112	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
113	C₄²	—	53	2	no	—
114	C₄²	—	53	1	no	—
115	C₄²	C₄×#6157	1	1	yes	characteristic
116	C₁₆	—	2	2	no	—
117	C₁₆	—	53	8	no	—
118	C₁₆	—	106	2	no	—
119	C₂³	—	53	1	no	—
120	C₂×C₄	—	53	1	no	—
121	C₂×C₄	—	53	2	no	—
122	C₂×C₄	—	53	2	no	—
123	C₂×C₄	—	53	2	no	—
124	C₂×C₄	—	106	4	no	—
125	C₂×C₄	#13133271	1	2	yes	—
126	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
127	C₈	—	53	2	no	—
128	C₈	—	53	2	no	—
129	C₈	—	212	4	no	—
130	C₈	C₈×#6157	1	2	yes	—
131	C₈	D₅₃⋊M₁₆	1	2	yes	—
132	C₂²	—	53	4	no	—
133	C₂²	—	53	2	no	—
134	C₂²	C₂×#13133271	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	—	53	2	no	—
137	C₄	—	106	2	no	—
138	C₄	C₂×C₈×#6157	1	1	yes	characteristic
139	C₄	#16811582	1	1	yes	characteristic
140	C₂	—	53	4	no	—
141	C₂	#20289911	1	1	yes	characteristic
142	C₂	#20290930	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20290916

C₂

yes

maximal, characteristic

#20289909

C₂

yes

maximal, characteristic

#20289634

C₂

yes

maximal, complemented

#346×D₅₃

C₂

yes

maximal, characteristic

C₂×D₅₃⋊C₁₆

C₄

yes

—

C₄×D₅₃⋊C₈

C₄

yes

—

#16811512

C₂²

yes

—

#16811511

C₄

yes

—

#16811464

C₄

yes

—

#16811464

C₂²

yes

—

C₂×C₁₆×D₅₃

C₄

yes

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

#16810399

C₄

yes

characteristic

#16810398

C₂²

yes

—

C₅₃×#346

C₄

yes

characteristic

C₂×D₅₃⋊C₈

—

D₅₃⋊C₁₆

C₈

yes

complemented

C₂×#11719910

C₂×C₄

yes

—

C₄×#88324

—

C₄×#88324

C₂×C₄

yes

—

C₂×C₈×D₅₃

C₂×C₄

yes

characteristic

C₂×C₈×D₅₃

C₂×C₄

yes

characteristic

C₄²×D₅₃

C₂×C₄

yes

characteristic

C₂×#11719908

C₂×C₄

yes

—

C₁₆×D₅₃

—

C₄×#88322

C₂×C₄

yes

characteristic

C₈×Dic₅₃

C₂×C₄

yes

characteristic

C₈×Dic₅₃

C₂³

yes

characteristic

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

C₄²

yes

characteristic

C₂×C₄×D₅₃

M₁₆

yes

—

C₂×#88324

—

D₅₃⋊C₈

—

C₂×C₈×C₅₃

C₄²

yes

characteristic

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₄²

yes

characteristic

C₄×Dic₅₃

M₁₆

yes

—

C₄×Dic₅₃

C₂²×C₄

yes

characteristic

C₂×#88322

C₄²

yes

characteristic

C₂×#88322

C₂²×C₄

yes

characteristic

C₈×D₅₃

C₂×C₈

yes

—

C₈×D₅₃

M₁₆

yes

—

#11719910

C₂×C₈

yes

complemented

C₁₆×C₅₃

—

#11719908

—

C₂²×D₅₃

#97

yes

characteristic

C₂×C₄×C₅₃

#97

yes

—

C₂×C₄×C₅₃

C₂×C₄²

yes

characteristic

C₂×Dic₅₃

#97

yes

characteristic

C₂×Dic₅₃

C₂×M₁₆

yes

—

C₄×D₅₃

—

C₄×D₅₃

C₄×C₈

yes

—

C₄×D₅₃

#97

yes

—

#88324

—

C₈×C₅₃

C₄×C₈

yes

—

C₈×C₅₃

C₂×M₁₆

yes

—

#88322

#97

yes

—

#88322

C₂²×C₈

yes

—

C₂×#1313

—

maximal, Sylow

C₂²×C₅₃

C₂×#97

yes

characteristic

C₂×D₅₃

#322

yes

—

C₂×D₅₃

#347

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄×C₈

yes

characteristic

C₄×C₅₃

C₂×#97

yes

characteristic

Dic₅₃

—

Dic₅₃

#322

yes

—

—

—

—

—

C₂×C₅₃

C₂×#322

yes

characteristic

C₂×C₅₃

C₂×#347

yes

—

D₅₃

#1313

yes

complemented

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

—

—

—

—

yes

—

—

C₅₃

C₂×#1313

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

106

—

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

C₂×C₁₆

C₂×#6157

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

—

106

—

102

C₄×C₈

C₂×#6157

yes

characteristic

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

106

—

110

C₂×C₈

—

212

—

111

C₂×C₈

C₄×#6157

yes

characteristic

112

C₂×C₈

C₂²×#6157

yes

characteristic

113

C₄²

—

114

C₄²

—

115

C₄²

C₄×#6157

yes

characteristic

116

C₁₆

—

117

C₁₆

—

118

C₁₆

—

106

—

119

C₂³

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

106

—

125

C₂×C₄

#13133271

yes

—

126

C₂×C₄

C₂×C₄×#6157

yes

characteristic

127

C₈

—

128

C₈

—

129

C₈

—

212

—

130

C₈

C₈×#6157

yes

—

131

C₈

D₅₃⋊M₁₆

yes

—

132

C₂²

—

133

C₂²

—

134

C₂²

C₂×#13133271

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

106

—

138

C₄

C₂×C₈×#6157

yes

characteristic

139

C₄

#16811582

yes

characteristic

140

C₂

—

141

C₂

#20289911

yes

characteristic

142

C₂

#20290930

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#6157
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₅₃×#346	C₄
socle	C₂²×C₅₃	C₂×#97
prime radical	C₂²×D₅₃	#97
cosocle	C₈×Dic₅₃	C₂³
hypercenter	#346	#6157
hypocenter	C₅₃	C₂×#1313
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #346

subgroup

G/H

center

C₂×C₄

C₂×C₄×#6157

commutator

C₄×C₅₃

C₂×C₄×C₈

Frattini

C₂×C₈

C₂²×#6157

Fitting

C₅₃×#346

C₄

socle

C₂²×C₅₃

C₂×#97

prime radical

C₂²×D₅₃

#97

cosocle

C₈×Dic₅₃

C₂³

C₅₃

G ▷ C₄×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ #346

N	H ≅ G/N	type
3. #20289634	C₂	semidirect
17. D₅₃⋊C₁₆	C₈	semidirect
44. #11719910	C₂×C₈	semidirect
75. D₅₃	#1313	semidirect
87. C₅₃	C₂×#1313	semidirect

H ≅ G/N

type

3. #20289634

C₂

semidirect

17. D₅₃⋊C₁₆

C₈

semidirect

44. #11719910

C₂×C₈

semidirect

75. D₅₃

#1313

semidirect

87. C₅₃

C₂×#1313

semidirect

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

subgroup	index
#20290916	2
#20289909	2
#20289634	2
#346×D₅₃	2
C₂×#1313	53

subgroup

index

#346×D₅₃

Subgroups — #28307783

Conjugacy classes of subgroups

Complements

Maximal subgroups