Group #28308144 — subgroups

Subgroups — #28308144

Order 13568.

296 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 98 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	#20291072	C₂	1	2	yes	maximal
2	#20289654	C₂	1	4	yes	maximal
3	#20283645	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#11719910	—	2	2	no	—
5	C₂²⋊#11719910	C₂²	1	4	yes	—
6	C₄×#11719910	—	2	2	no	—
7	#16811180	C₄	1	1	yes	characteristic
8	#16810750	—	2	2	no	—
9	#16810747	—	2	2	no	—
10	#16810651	C₂²	1	2	yes	—
11	#16810581	C₄	1	2	yes	—
12	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
13	C₅₃×#423	C₄	1	1	yes	characteristic
14	C₂×#11719910	—	2	4	no	—
15	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
16	C₂²×#88322	C₂³	1	1	yes	characteristic
17	#13133217	C₈	1	2	yes	—
18	#13133209	C₈	1	2	yes	—
19	#13133196	—	2	2	no	—
20	C₄×Dic₁₀₆	—	2	2	no	—
21	#13133176	C₂×C₄	1	2	yes	—
22	#13133146	—	2	4	no	—
23	#13133127	—	2	2	no	—
24	C₄×#88322	D₄	1	2	yes	—
25	C₄×#88322	D₄	1	2	yes	—
26	C₈×Dic₅₃	—	2	2	no	—
27	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₄⋊C₈×C₅₃	—	2	2	no	—
30	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
31	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
32	C₂×Dic₁₀₆	—	2	2	no	—
33	M₁₆×C₅₃	—	4	2	no	—
34	C₂×C₈×C₅₃	—	2	2	no	—
35	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
36	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
37	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
38	C₂²⋊Dic₅₃	—	2	2	no	—
39	C₄⋊Dic₅₃	—	2	2	no	—
40	C₄⋊Dic₅₃	C₂×C₈	1	4	yes	—
41	#11719919	—	2	4	no	—
42	C₄×Dic₅₃	—	2	2	no	—
43	C₂×#88322	—	2	2	no	—
44	C₂×#88322	—	4	1	no	—
45	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×#88322	C₂×D₄	1	2	yes	—
48	#11719910	—	4	4	no	—
49	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
50	C₂×C₄×C₅₃	—	2	4	no	—
51	C₂×C₄×C₅₃	—	2	1	no	—
52	C₂×C₄×C₅₃	—	2	1	no	—
53	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
54	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
55	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
57	C₂×Dic₅₃	—	2	4	no	—
58	Dic₁₀₆	—	4	4	no	—
59	C₈×C₅₃	—	4	2	no	—
60	#88322	—	2	2	no	—
61	#88322	—	2	1	no	—
62	#88322	#101	1	2	yes	—
63	#11496	—	53	1	no	maximal, Sylow
64	C₂²×C₅₃	—	2	2	no	—
65	C₂²×C₅₃	—	2	1	no	—
66	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
67	C₄×C₅₃	—	2	2	no	—
68	C₄×C₅₃	—	2	1	no	—
69	C₄×C₅₃	#350	1	4	yes	—
70	C₄×C₅₃	#410	1	1	yes	characteristic
71	C₄×C₅₃	C₂×#101	1	1	yes	characteristic
72	Dic₅₃	—	4	4	no	—
73	#2960	—	53	1	no	—
74	#2162	—	53	2	no	—
75	#1323	—	53	4	no	—
76	C₂×C₅₃	—	4	1	no	—
77	C₂×C₅₃	#1461	1	1	yes	characteristic
78	C₂×C₅₃	C₂×#350	1	2	yes	—
79	#533	—	53	1	no	—
80	#446	—	106	2	no	—
81	C₈×Q₈	—	106	2	no	—
82	#424	—	53	2	no	—
83	#423	#6157	1	1	yes	characteristic
84	#405	—	53	2	no	—
85	C₂×C₄×C₈	—	53	1	no	—
86	C₄⋊C₁₆	—	106	2	no	—
87	C₂²⋊C₁₆	—	53	4	no	—
88	C₄×C₁₆	—	106	2	no	—
89	C₅₃	#11496	1	1	yes	characteristic, complemented, Sylow
90	C₂×M₁₆	C₂×#6157	1	2	yes	—
91	C₂²×C₈	—	53	1	no	—
92	C₂²×C₈	—	53	1	no	—
93	C₄⋊Q₈	—	53	2	no	—
94	#125	—	53	2	no	—
95	C₂²⋊Q₈	—	106	2	no	—
96	C₄×Q₈	—	106	2	no	—
97	#117	—	53	2	no	—
98	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
99	C₂×C₁₆	—	106	4	no	—
100	C₄⋊C₈	—	2	2	no	—
101	C₄⋊C₈	—	106	4	no	—
102	#97	—	106	2	no	—
103	C₄×C₈	—	53	2	no	—
104	C₄×C₈	—	53	2	no	—
105	C₄×C₈	—	106	2	no	—
106	C₂×Q₈	—	106	2	no	—
107	C₂²×C₄	#11719936	1	2	yes	—
108	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
109	M₁₆	—	4	2	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	53	1	no	—
112	C₂×C₈	—	53	1	no	—
113	C₂×C₈	—	53	2	no	—
114	C₂×C₈	—	106	2	no	—
115	C₂×C₈	—	212	1	no	—
116	C₄⋊C₄	—	53	4	no	—
117	C₄⋊C₄	—	106	4	no	—
118	C₄⋊C₄	—	106	2	no	—
119	C₂²⋊C₄	—	106	2	no	—
120	C₄²	—	106	2	no	—
121	C₄²	D₅₃⋊C₈	1	1	yes	characteristic
122	C₄²	D₅₃⋊C₈	1	1	yes	characteristic
123	C₄²	C₂²⋊#6157	1	2	yes	—
124	C₁₆	—	212	4	no	—
125	C₂³	C₂×#11719936	1	1	yes	characteristic
126	Q₈	—	212	4	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	106	4	no	—
131	C₂×C₄	#13133274	1	2	yes	—
132	C₂×C₄	#13133274	1	2	yes	—
133	C₂×C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
134	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
135	C₈	—	4	2	no	—
136	C₈	—	53	2	no	—
137	C₈	—	106	2	no	—
138	C₈	—	106	1	no	—
139	C₂²	—	2	2	no	—
140	C₂²	—	2	1	no	—
141	C₂²	C₂×#13133274	1	1	yes	characteristic
142	C₄	—	2	2	no	—
143	C₄	—	2	1	no	—
144	C₄	—	212	4	no	—
145	C₄	#16811522	1	4	yes	—
146	C₄	#16811603	1	1	yes	characteristic
147	C₄	D₅₃⋊#101	1	1	yes	characteristic
148	C₂	—	4	1	no	—
149	C₂	#20289959	1	1	yes	characteristic
150	C₂	C₂×#16811522	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20291072

C₂

yes

maximal

#20289654

C₂

yes

maximal

#20283645

C₂

yes

maximal, characteristic

C₄⋊#11719910

—

C₂²⋊#11719910

C₂²

yes

—

C₄×#11719910

—

#16811180

C₄

yes

—

—

C₂²

yes

—

#16810581

C₄

yes

—

C₂×C₄×#88322

C₂²

yes

characteristic

C₅₃×#423

C₄

yes

characteristic

C₂×#11719910

—

C₂²×#88322

C₂×C₄

yes

characteristic

C₂²×#88322

C₂³

yes

characteristic

#13133217

C₈

yes

—

#13133209

C₈

yes

—

#13133196

—

C₄×Dic₁₀₆

—

#13133176

C₂×C₄

yes

—

#13133146

—

#13133127

—

C₄×#88322

D₄

yes

—

C₄×#88322

D₄

yes

—

C₈×Dic₅₃

—

C₂×M₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²×C₄×C₅₃

M₁₆

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₆

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂²⋊C₄

yes

—

C₄²×C₅₃

C₂×C₈

yes

characteristic

C₄²×C₅₃

C₂×C₈

yes

characteristic

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₈

yes

—

#11719919

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

C₂²⋊C₄

yes

characteristic

C₂×#88322

C₂²⋊C₄

yes

characteristic

C₂×#88322

C₂×D₄

yes

—

#11719910

—

C₂³×C₅₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₈

yes

—

C₂×C₄×C₅₃

C₂²⋊C₈

yes

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₂×Dic₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

—

—

yes

—

#11496

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#350

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

#2960

—

#2162

—

#1323

—

C₂×C₅₃

—

C₂×C₅₃

#1461

yes

characteristic

C₂×C₅₃

C₂×#350

yes

—

#533

—

#446

—

106

—

C₈×Q₈

—

106

—

—

yes

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

106

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

106

—

C₅₃

#11496

yes

characteristic, complemented, Sylow

C₂×M₁₆

C₂×#6157

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

106

—

C₄×Q₈

—

106

—

#117

—

C₂×C₄²

C₂×#6157

yes

characteristic

C₂×C₁₆

—

106

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

106

—

102

#97

—

106

—

103

C₄×C₈

—

104

C₄×C₈

—

105

C₄×C₈

—

106

—

106

C₂×Q₈

—

106

—

107

C₂²×C₄

#11719936

yes

—

108

C₂²×C₄

C₂²×#6157

yes

characteristic

109

M₁₆

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

106

—

115

C₂×C₈

—

212

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

106

—

118

C₄⋊C₄

—

106

—

119

C₂²⋊C₄

—

106

—

120

C₄²

—

106

—

121

C₄²

D₅₃⋊C₈

yes

characteristic

122

C₄²

D₅₃⋊C₈

yes

characteristic

123

C₄²

C₂²⋊#6157

yes

—

124

C₁₆

—

212

—

125

C₂³

C₂×#11719936

yes

characteristic

126

Q₈

—

212

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

106

—

131

C₂×C₄

#13133274

yes

—

132

C₂×C₄

#13133274

yes

—

133

C₂×C₄

C₂×C₂²⋊#6157

yes

characteristic

134

C₂×C₄

C₂×D₅₃⋊C₈

yes

characteristic

135

C₈

—

136

C₈

—

137

C₈

—

106

—

138

C₈

—

106

—

139

C₂²

—

140

C₂²

—

141

C₂²

C₂×#13133274

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

212

—

145

C₄

#16811522

yes

—

146

C₄

#16811603

yes

characteristic

147

C₄

D₅₃⋊#101

yes

characteristic

148

C₂

—

149

C₂

#20289959

yes

characteristic

150

C₂

C₂×#16811522

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊#6157
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₅₃×#423	C₄
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂³×C₅₃	C₂×M₁₆
cosocle	C₂²×#88322	C₂³
hypercenter	#423	#6157
hypocenter	C₅₃	#11496
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊#6157

commutator

C₂×C₄×C₅₃

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#6157

Fitting

C₅₃×#423

C₄

socle

C₂²×C₅₃

C₂×C₂²⋊C₈

prime radical

C₂³×C₅₃

C₂×M₁₆

cosocle

C₂²×#88322

C₂³

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
89. C₅₃	#11496	semidirect

H ≅ G/N

type

89. C₅₃

#11496

semidirect

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

subgroup	index
#20291072	2
#20289654	2
#20283645	2
#11496	53

subgroup

index

Subgroups — #28308144

Conjugacy classes of subgroups

Complements

Maximal subgroups