Group #28312586 — subgroups

Subgroups — #28312586

Order 13568.

206 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 100 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×#16811556	C₂	1	1	yes	maximal, characteristic
2	C₂×D₅₃⋊C₃₂	C₂	1	1	yes	maximal, characteristic
3	#20289873	C₂	1	2	yes	maximal, complemented
4	C₄⋊#13133267	C₂	1	2	yes	maximal, complemented
5	#20276313	C₂	1	1	yes	maximal, characteristic
6	#16811556	—	2	2	no	—
7	D₅₃⋊C₃₂	—	2	2	no	—
8	C₂×#13133267	C₂²	1	2	yes	—
9	C₂×#13133267	C₂²	1	2	yes	—
10	C₂×#13133156	C₄	1	1	yes	characteristic
11	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
12	#16810577	C₄	1	1	yes	characteristic
13	#16810398	C₄	1	1	yes	characteristic
14	C₄×#11719908	C₂²	1	1	yes	characteristic
15	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#346	C₄	1	1	yes	characteristic
17	#13133267	—	2	2	no	—
18	#13133267	—	2	2	no	—
19	C₂×#11719912	C₈	1	1	yes	characteristic
20	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
21	#13133174	C₈	1	1	yes	characteristic
22	#13133156	—	2	2	no	—
23	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
24	C₂×#11719908	C₂³	1	1	yes	characteristic
25	C₁₆×D₅₃	D₄	1	2	yes	—
26	C₁₆×D₅₃	Q₈	1	2	yes	—
27	#13133128	C₈	1	1	yes	characteristic
28	#13133127	C₈	1	1	yes	characteristic
29	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
37	#11719921	C₁₆	1	2	yes	—
38	#11719919	C₁₆	1	2	yes	—
39	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
40	C₂×#88322	C₂×C₈	1	1	yes	characteristic
41	C₂×#88322	C₂×C₈	1	1	yes	characteristic
42	#11719912	—	2	2	no	—
43	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
44	C₈×D₅₃	M₁₆	1	2	yes	—
45	C₁₆×C₅₃	—	2	1	no	—
46	C₁₆×C₅₃	C₄⋊C₄	1	2	yes	—
47	#11719908	—	2	1	no	—
48	#11719908	C₂×D₄	1	1	yes	characteristic
49	#11719908	C₂×Q₈	1	1	yes	characteristic
50	C₂²×D₅₃	C₂×C₁₆	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
52	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
53	C₂×Dic₅₃	C₂×C₁₆	1	2	yes	—
54	C₂×Dic₅₃	C₂×C₁₆	1	1	yes	characteristic
55	C₄×D₅₃	C₄⋊C₈	1	2	yes	—
56	C₄×D₅₃	M₃₂	1	2	yes	—
57	C₈×C₅₃	—	2	1	no	—
58	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
60	#88322	—	2	1	no	—
61	#88322	C₄⋊C₈	1	2	yes	—
62	#13628	—	53	1	no	maximal, Sylow
63	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
64	C₂×D₅₃	—	2	2	no	—
65	C₂×D₅₃	C₄⋊C₁₆	1	2	yes	—
66	C₄×C₅₃	—	2	2	no	—
67	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
68	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
69	Dic₅₃	—	2	2	no	—
70	Dic₅₃	C₄⋊C₁₆	1	2	yes	—
71	C₂×M₆₄	—	53	1	no	—
72	C₂²×C₃₂	—	53	1	no	—
73	#2108	—	53	1	no	—
74	C₄⋊C₃₂	—	53	2	no	—
75	C₄⋊C₃₂	—	53	2	no	—
76	C₂×C₅₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
77	C₂×C₅₃	#2258	1	2	yes	—
78	D₅₃	—	2	2	no	—
79	C₂×M₃₂	—	53	1	no	—
80	C₂²×C₁₆	—	53	1	no	—
81	#405	—	53	1	no	—
82	M₆₄	—	106	2	no	—
83	C₂×C₃₂	—	53	2	no	—
84	C₂×C₃₂	—	53	2	no	—
85	C₂×C₃₂	—	106	2	no	—
86	#346	—	53	1	no	—
87	#346	#6157	1	1	yes	characteristic
88	C₄×C₁₆	—	53	1	no	—
89	C₄×C₁₆	—	53	1	no	—
90	C₅₃	#13628	1	1	yes	characteristic, complemented, Sylow
91	C₂×M₁₆	—	53	1	no	—
92	C₂²×C₈	—	53	1	no	—
93	#117	—	53	1	no	—
94	M₃₂	—	106	2	no	—
95	C₂×C₁₆	—	53	1	no	—
96	C₂×C₁₆	—	53	2	no	—
97	C₂×C₁₆	—	53	2	no	—
98	C₂×C₁₆	—	53	1	no	—
99	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
100	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
101	#97	—	53	1	no	—
102	#97	—	53	1	no	—
103	C₄×C₈	—	53	1	no	—
104	C₄×C₈	C₂×#6157	1	1	yes	characteristic
105	C₃₂	—	106	2	no	—
106	C₃₂	—	106	2	no	—
107	C₂²×C₄	—	53	1	no	—
108	M₁₆	—	106	2	no	—
109	C₂×C₈	—	53	1	no	—
110	C₂×C₈	—	53	2	no	—
111	C₂×C₈	—	53	2	no	—
112	C₂×C₈	—	53	1	no	—
113	C₂×C₈	D₅₃⋊C₈	1	1	yes	characteristic
114	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
115	C₄⋊C₄	—	53	2	no	—
116	C₂²⋊C₄	—	53	2	no	—
117	C₄²	—	53	1	no	—
118	C₄²	D₅₃⋊C₈	1	1	yes	characteristic
119	C₁₆	—	2	1	no	—
120	C₁₆	—	53	1	no	—
121	C₁₆	—	53	1	no	—
122	C₁₆	—	106	1	no	—
123	C₁₆	C₄⋊#6157	1	2	yes	—
124	C₂³	—	53	1	no	—
125	C₂×C₄	—	53	1	no	—
126	C₂×C₄	—	53	2	no	—
127	C₂×C₄	—	53	2	no	—
128	C₂×C₄	—	53	2	no	—
129	C₂×C₄	D₅₃⋊C₁₆	1	2	yes	—
130	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
131	C₈	—	2	1	no	—
132	C₈	—	53	2	no	—
133	C₈	—	106	1	no	—
134	C₈	C₂×C₄⋊#6157	1	1	yes	characteristic
135	C₈	D₅₃⋊M₁₆	1	1	yes	characteristic
136	C₂²	—	53	2	no	—
137	C₂²	—	106	2	no	—
138	C₂²	C₂×D₅₃⋊C₁₆	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	53	2	no	—
141	C₄	—	106	2	no	—
142	C₄	D₅₃⋊(C₄⋊C₈)	1	1	yes	characteristic
143	C₄	D₅₃⋊M₃₂	1	1	yes	characteristic
144	C₂	—	106	2	no	—
145	C₂	D₅₃⋊(C₄⋊C₁₆)	1	1	yes	characteristic
146	C₂	#20291139	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16811556

C₂

yes

maximal, characteristic

C₂×D₅₃⋊C₃₂

C₂

yes

maximal, characteristic

#20289873

C₂

yes

maximal, complemented

C₄⋊#13133267

C₂

yes

maximal, complemented

#20276313

C₂

yes

maximal, characteristic

#16811556

—

D₅₃⋊C₃₂

—

C₂×#13133267

C₂²

yes

—

C₂×#13133267

C₂²

yes

—

C₂×#13133156

C₄

yes

characteristic

C₂×C₁₆×D₅₃

C₂²

yes

characteristic

#16810577

C₄

yes

characteristic

#16810398

C₄

yes

characteristic

C₄×#11719908

C₂²

yes

characteristic

C₁₆×Dic₅₃

C₂²

yes

characteristic

C₅₃×#346

C₄

yes

—

—

C₈

yes

characteristic

C₂×C₈×D₅₃

C₂×C₄

yes

characteristic

#13133174

C₈

yes

characteristic

#13133156

—

C₂×#11719908

C₂×C₄

yes

characteristic

C₂×#11719908

C₂³

yes

characteristic

C₁₆×D₅₃

D₄

yes

—

C₁₆×D₅₃

Q₈

yes

—

#13133128

C₈

yes

characteristic

#13133127

C₈

yes

characteristic

C₈×Dic₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

C₂×C₈

yes

characteristic

C₂×C₈×C₅₃

C₂×C₈

yes

characteristic

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₈

yes

characteristic

#11719921

C₁₆

yes

—

#11719919

C₁₆

yes

—

C₄×Dic₅₃

C₂×C₈

yes

characteristic

C₂×#88322

C₂×C₈

yes

characteristic

C₂×#88322

C₂×C₈

yes

characteristic

#11719912

—

C₈×D₅₃

C₄⋊C₄

yes

—

C₈×D₅₃

M₁₆

yes

—

C₁₆×C₅₃

—

C₁₆×C₅₃

C₄⋊C₄

yes

—

#11719908

—

#11719908

C₂×D₄

yes

characteristic

#11719908

C₂×Q₈

yes

characteristic

C₂²×D₅₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×C₁₆

yes

—

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₂×Dic₅₃

C₂×C₁₆

yes

—

C₂×Dic₅₃

C₂×C₁₆

yes

characteristic

C₄×D₅₃

C₄⋊C₈

yes

—

C₄×D₅₃

M₃₂

yes

—

C₈×C₅₃

—

C₈×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₈×C₅₃

C₂×M₁₆

yes

characteristic

#88322

—

#88322

C₄⋊C₈

yes

—

#13628

—

maximal, Sylow

C₂²×C₅₃

C₂²×C₁₆

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

C₄⋊C₁₆

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄⋊C₈

yes

characteristic

C₄×C₅₃

C₂×M₃₂

yes

characteristic

Dic₅₃

—

Dic₅₃

C₄⋊C₁₆

yes

—

C₂×M₆₄

—

C₂²×C₃₂

—

#2108

—

C₄⋊C₃₂

—

C₄⋊C₃₂

—

C₂×C₅₃

C₂×C₄⋊C₁₆

yes

characteristic

C₂×C₅₃

#2258

yes

—

D₅₃

—

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

M₆₄

—

106

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

106

—

—

yes

C₄×C₁₆

—

C₄×C₁₆

—

C₅₃

#13628

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

#117

—

M₃₂

—

106

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×#6157

yes

characteristic

100

C₂×C₁₆

C₂×#6157

yes

characteristic

101

#97

—

102

#97

—

103

C₄×C₈

—

104

C₄×C₈

C₂×#6157

yes

characteristic

105

C₃₂

—

106

—

106

C₃₂

—

106

—

107

C₂²×C₄

—

108

M₁₆

—

106

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

D₅₃⋊C₈

yes

characteristic

114

C₂×C₈

C₂²×#6157

yes

characteristic

115

C₄⋊C₄

—

116

C₂²⋊C₄

—

117

C₄²

—

118

C₄²

D₅₃⋊C₈

yes

characteristic

119

C₁₆

—

120

C₁₆

—

121

C₁₆

—

122

C₁₆

—

106

—

123

C₁₆

C₄⋊#6157

yes

—

124

C₂³

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

D₅₃⋊C₁₆

yes

—

130

C₂×C₄

C₂×D₅₃⋊C₈

yes

characteristic

131

C₈

—

132

C₈

—

133

C₈

—

106

—

134

C₈

C₂×C₄⋊#6157

yes

characteristic

135

C₈

D₅₃⋊M₁₆

yes

characteristic

136

C₂²

—

137

C₂²

—

106

—

138

C₂²

C₂×D₅₃⋊C₁₆

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

106

—

142

C₄

D₅₃⋊(C₄⋊C₈)

yes

characteristic

143

C₄

D₅₃⋊M₃₂

yes

characteristic

144

C₂

—

106

—

145

C₂

D₅₃⋊(C₄⋊C₁₆)

yes

characteristic

146

C₂

#20291139

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#6157
commutator, socle	C₂²×C₅₃	C₂²×C₁₆
Fitting	C₅₃×#346	C₄
prime radical	C₂²×D₅₃	C₂×C₁₆
cosocle	C₂×#11719908	C₂³
hypercenter	#346	#6157
hypocenter	C₅₃	#13628
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #346

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×#6157

commutator, socle

C₂²×C₅₃

C₂²×C₁₆

Fitting

C₅₃×#346

C₄

prime radical

C₂²×D₅₃

C₂×C₁₆

C₂³

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₈ ◁ #346

N	H ≅ G/N	type
3. #20289873	C₂	semidirect
4. C₄⋊#13133267	C₂	semidirect
90. C₅₃	#13628	semidirect

H ≅ G/N

type

3. #20289873

C₂

semidirect

4. C₄⋊#13133267

C₂

semidirect

90. C₅₃

#13628

semidirect

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

subgroup	index
C₂×#16811556	2
C₂×D₅₃⋊C₃₂	2
#20289873	2
C₄⋊#13133267	2
#20276313	2
#13628	53

subgroup

index

C₂×D₅₃⋊C₃₂

Subgroups — #28312586

Conjugacy classes of subgroups

Complements

Maximal subgroups