Group #28307801 — subgroups

Subgroups — #28307801

Order 13568.

282 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 144 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	#20290921	C₂	1	1	yes	maximal, characteristic
2	#20289918	C₂	1	1	yes	maximal, characteristic
3	#20289634	C₂	1	4	yes	maximal, complemented
4	C₂×#16810398	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133291	C₄	1	2	yes	—
6	#16811639	C₄	1	2	yes	—
7	#16811512	C₂²	1	2	yes	—
8	#16811509	C₄	1	2	yes	—
9	#16811464	C₂²	1	2	yes	—
10	C₈×#88324	C₄	1	2	yes	—
11	C₂²×#11719908	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
13	#16810398	C₄	1	2	yes	—
14	#16810398	C₂²	1	2	yes	—
15	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
16	#13133291	C₈	1	8	yes	complemented
17	C₂×#11719910	C₂×C₄	1	4	yes	—
18	C₄⋊#88324	—	2	2	no	—
19	C₂²⋊#88324	—	2	2	no	—
20	C₄×#88324	C₂×C₄	1	4	yes	—
21	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×#11719908	—	2	2	no	—
24	C₂×#11719908	C₂×C₄	1	2	yes	—
25	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
26	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
28	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆×C₅₃	—	2	2	no	—
31	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
33	C₂²×Dic₅₃	M₁₆	1	2	yes	—
34	C₂×#88324	—	2	4	no	—
35	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
37	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
38	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
40	C₄×Dic₅₃	M₁₆	1	2	yes	—
41	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×#88322	C₄²	1	1	yes	characteristic
43	C₂×#88322	C₂×C₈	1	4	yes	—
44	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
45	#11719910	C₂×C₈	1	8	yes	—
46	C₁₆×C₅₃	—	2	2	no	—
47	#11719908	—	2	2	no	—
48	C₂³×C₅₃	#97	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
50	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
51	C₂×C₄×C₅₃	#97	1	2	yes	—
52	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
53	C₂×Dic₅₃	—	4	2	no	—
54	C₂×Dic₅₃	#97	1	2	yes	—
55	C₂×Dic₅₃	C₂×M₁₆	1	2	yes	—
56	#88324	—	4	4	no	—
57	C₈×C₅₃	#97	1	2	yes	—
58	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
59	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
60	#88322	C₄×C₈	1	2	yes	—
61	#88322	C₂²×C₈	1	2	yes	—
62	#11406	—	53	1	no	maximal, Sylow
63	C₂²×C₅₃	—	2	2	no	—
64	C₂²×C₅₃	#322	1	2	yes	—
65	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
66	C₄×C₅₃	#322	1	2	yes	—
67	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
68	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
69	Dic₅₃	—	2	2	no	—
70	Dic₅₃	—	2	2	no	—
71	C₄×M₃₂	—	53	1	no	—
72	C₂×#346	—	53	1	no	—
73	#1450	—	53	1	no	—
74	#1313	—	53	4	no	—
75	C₂×C₅₃	—	2	2	no	—
76	C₂×C₅₃	C₂×#322	1	1	yes	characteristic
77	C₂×C₅₃	#2110	1	2	yes	—
78	C₂×M₃₂	—	53	2	no	—
79	C₂²×C₁₆	—	53	1	no	—
80	C₂²×C₁₆	#6157	1	1	yes	characteristic
81	#431	—	53	2	no	—
82	C₂×C₄×C₈	—	53	1	no	—
83	#346	—	53	2	no	—
84	#346	—	53	2	no	—
85	#346	—	53	2	no	—
86	C₄×C₁₆	—	53	2	no	—
87	#322	—	53	2	no	—
88	C₈²	—	53	2	no	—
89	C₅₃	#11406	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₈	—	53	1	no	—
91	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
92	C₂×C₄²	—	53	1	no	—
93	M₃₂	—	53	8	no	—
94	C₂×C₁₆	—	2	2	no	—
95	C₂×C₁₆	—	53	4	no	—
96	C₂×C₁₆	—	53	2	no	—
97	C₂×C₁₆	—	106	2	no	—
98	C₂×C₁₆	C₂×#6157	1	2	yes	—
99	C₄⋊C₈	—	106	2	no	—
100	C₂²⋊C₈	—	106	2	no	—
101	C₄×C₈	—	53	1	no	—
102	C₄×C₈	—	53	1	no	—
103	C₄×C₈	—	53	1	no	—
104	C₄×C₈	—	53	1	no	—
105	C₄×C₈	—	53	4	no	—
106	C₂²×C₄	—	53	2	no	—
107	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
108	C₂×C₈	—	53	4	no	—
109	C₂×C₈	—	53	1	no	—
110	C₂×C₈	—	53	1	no	—
111	C₂×C₈	—	106	4	no	—
112	C₂×C₈	#11719936	1	2	yes	—
113	C₂×C₈	C₄×#6157	1	1	yes	characteristic
114	C₂×C₈	#11719940	1	2	yes	—
115	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
116	C₄²	—	53	2	no	—
117	C₄²	—	53	1	no	—
118	C₄²	—	53	1	no	—
119	C₁₆	—	2	2	no	—
120	C₁₆	—	53	8	no	—
121	C₁₆	—	106	2	no	—
122	C₂³	#13133271	1	1	yes	characteristic
123	C₂×C₄	—	53	2	no	—
124	C₂×C₄	—	53	2	no	—
125	C₂×C₄	—	212	2	no	—
126	C₂×C₄	C₈×#6157	1	2	yes	—
127	C₂×C₄	#13133271	1	1	yes	characteristic
128	C₂×C₄	#13133272	1	2	yes	—
129	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
130	C₈	—	53	2	no	—
131	C₈	—	53	2	no	—
132	C₈	—	212	4	no	—
133	C₈	#13133272	1	2	yes	—
134	C₈	C₂×#11719940	1	1	yes	characteristic
135	C₈	C₂×#11719936	1	1	yes	characteristic
136	C₂²	—	2	2	no	—
137	C₂²	#88324⋊C₈	1	2	yes	—
138	C₂²	C₂×#13133271	1	1	yes	characteristic
139	C₄	—	106	2	no	—
140	C₄	—	106	2	no	—
141	C₄	#88324⋊C₈	1	2	yes	—
142	C₄	C₂×C₈×#6157	1	1	yes	characteristic
143	C₄	C₂×#13133272	1	1	yes	characteristic
144	C₂	—	2	2	no	—
145	C₂	C₂×#88324⋊C₈	1	1	yes	characteristic
146	C₂	#20290934	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20290921

C₂

yes

maximal, characteristic

#20289918

C₂

yes

maximal, characteristic

#20289634

C₂

yes

maximal, complemented

C₂×#16810398

C₂

yes

maximal, characteristic

C₂×#13133291

C₄

yes

—

#16811639

C₄

yes

—

#16811512

C₂²

yes

—

#16811509

C₄

yes

—

#16811464

C₂²

yes

—

C₈×#88324

C₄

yes

—

C₂²×#11719908

C₄

yes

characteristic

C₂×C₈×Dic₅₃

C₂²

yes

characteristic

#16810398

C₄

yes

—

#16810398

C₂²

yes

—

C₂²×C₁₆×C₅₃

C₄

yes

characteristic

#13133291

C₈

yes

complemented

C₂×#11719910

C₂×C₄

yes

—

C₄⋊#88324

—

C₂²⋊#88324

—

C₄×#88324

C₂×C₄

yes

—

C₂²×#88322

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₂×#11719908

—

C₂×#11719908

C₂×C₄

yes

—

C₈×Dic₅₃

C₂×C₄

yes

characteristic

C₈×Dic₅₃

C₂×C₄

yes

characteristic

C₈×Dic₅₃

C₂×C₄

yes

characteristic

C₈×Dic₅₃

C₂³

yes

characteristic

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₄²

yes

characteristic

C₂²×Dic₅₃

M₁₆

yes

—

C₂×#88324

—

C₂×C₈×C₅₃

C₄²

yes

characteristic

C₂×C₈×C₅₃

M₁₆

yes

—

C₂×C₈×C₅₃

M₁₆

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₄×Dic₅₃

C₄²

yes

characteristic

C₄×Dic₅₃

M₁₆

yes

—

C₄×Dic₅₃

C₂²×C₄

yes

characteristic

C₂×#88322

C₄²

yes

characteristic

C₂×#88322

C₂×C₈

yes

—

C₂×#88322

C₂²×C₄

yes

characteristic

#11719910

C₂×C₈

yes

—

C₁₆×C₅₃

—

#11719908

—

C₂³×C₅₃

#97

yes

characteristic

C₂×C₄×C₅₃

C₄×C₈

yes

—

C₂×C₄×C₅₃

#97

yes

characteristic

C₂×C₄×C₅₃

#97

yes

—

C₂×C₄×C₅₃

C₂×C₄²

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

#97

yes

—

C₂×Dic₅₃

C₂×M₁₆

yes

—

#88324

—

C₈×C₅₃

#97

yes

—

C₈×C₅₃

C₂×M₁₆

yes

characteristic

C₈×C₅₃

C₂×M₁₆

yes

characteristic

#88322

C₄×C₈

yes

—

#88322

C₂²×C₈

yes

—

#11406

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

#322

yes

—

C₂²×C₅₃

C₂×#97

yes

characteristic

C₄×C₅₃

#322

yes

—

C₄×C₅₃

C₂×C₄×C₈

yes

characteristic

C₄×C₅₃

C₂×#97

yes

characteristic

Dic₅₃

—

Dic₅₃

—

C₄×M₃₂

—

C₂×#346

—

#1450

—

#1313

—

C₂×C₅₃

—

C₂×C₅₃

C₂×#322

yes

characteristic

C₂×C₅₃

#2110

yes

—

C₂×M₃₂

—

C₂²×C₁₆

—

C₂²×C₁₆

#6157

yes

characteristic

#431

—

C₂×C₄×C₈

—

#346

—

#346

—

#346

—

C₄×C₁₆

—

#322

—

C₈²

—

C₅₃

#11406

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×#6157

yes

characteristic

C₂×C₄²

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

C₂×C₁₆

C₂×#6157

yes

—

C₄⋊C₈

—

106

—

100

C₂²⋊C₈

—

106

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

—

104

C₄×C₈

—

105

C₄×C₈

—

106

C₂²×C₄

—

107

C₂²×C₄

C₄×#6157

yes

characteristic

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

106

—

112

C₂×C₈

#11719936

yes

—

113

C₂×C₈

C₄×#6157

yes

characteristic

114

C₂×C₈

#11719940

yes

—

115

C₂×C₈

C₂²×#6157

yes

characteristic

116

C₄²

—

117

C₄²

—

118

C₄²

—

119

C₁₆

—

120

C₁₆

—

121

C₁₆

—

106

—

122

C₂³

#13133271

yes

characteristic

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

212

—

126

C₂×C₄

C₈×#6157

yes

—

127

C₂×C₄

#13133271

yes

characteristic

128

C₂×C₄

#13133272

yes

—

129

C₂×C₄

C₂×C₄×#6157

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

—

212

—

133

C₈

#13133272

yes

—

134

C₈

C₂×#11719940

yes

characteristic

135

C₈

C₂×#11719936

yes

characteristic

136

C₂²

—

137

C₂²

#88324⋊C₈

yes

—

138

C₂²

C₂×#13133271

yes

characteristic

139

C₄

—

106

—

140

C₄

—

106

—

141

C₄

#88324⋊C₈

yes

—

142

C₄

C₂×C₈×#6157

yes

characteristic

143

C₄

C₂×#13133272

yes

characteristic

144

C₂

—

145

C₂

C₂×#88324⋊C₈

yes

characteristic

146

C₂

#20290934

yes

—

	subgroup	G/H
center	C₂×C₈	C₄×#6157
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₂²×C₁₆×C₅₃	C₄
socle	C₂²×C₅₃	C₂×#97
prime radical	C₂³×C₅₃	#97
cosocle	C₈×Dic₅₃	C₂³
hypercenter	C₂²×C₁₆	#6157
hypocenter	C₅₃	#11406
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂×C₈

C₄×#6157

commutator

C₄×C₅₃

C₂×C₄×C₈

Frattini

C₂×C₈

C₂²×#6157

Fitting

C₂²×C₁₆×C₅₃

C₄

socle

C₂²×C₅₃

C₂×#97

prime radical

C₂³×C₅₃

#97

cosocle

C₈×Dic₅₃

C₂³

C₂²×C₁₆

C₅₃

G ▷ C₄×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
3. #20289634	C₂	semidirect
16. #13133291	C₈	semidirect
89. C₅₃	#11406	semidirect

H ≅ G/N

type

3. #20289634

C₂

semidirect

16. #13133291

C₈

semidirect

89. C₅₃

#11406

semidirect

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

subgroup	index
#20290921	2
#20289918	2
#20289634	2
C₂×#16810398	2
#11406	53

subgroup

index

Subgroups — #28307801

Conjugacy classes of subgroups

Complements

Maximal subgroups