Group #28312617 — subgroups

Subgroups — #28312617

Order 13568.

180 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 84 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	#20289873	C₂	1	2	yes	maximal, complemented
2	#20289829	C₂	1	2	yes	maximal
3	#20289822	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#13133267	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276313	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133267	C₂²	1	2	yes	—
7	C₂×#13133267	C₂²	1	2	yes	—
8	C₂×#13133156	C₄	1	1	yes	characteristic
9	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
10	#16810577	C₄	1	1	yes	characteristic
11	#16810398	C₄	1	1	yes	characteristic
12	C₄×#11719908	C₂²	1	1	yes	characteristic
13	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#346	C₄	1	1	yes	characteristic
15	#13133267	—	2	2	no	—
16	#13133267	—	2	2	no	—
17	C₂×#11719912	C₈	1	1	yes	characteristic
18	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
19	#13133174	C₈	1	1	yes	characteristic
20	#13133156	—	2	2	no	—
21	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
22	C₂×#11719908	C₂³	1	1	yes	characteristic
23	C₁₆×D₅₃	—	2	2	no	—
24	#13133128	C₈	1	1	yes	characteristic
25	#13133127	C₈	1	1	yes	characteristic
26	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
31	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
33	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
34	#11719921	C₁₆	1	2	yes	—
35	#11719919	C₁₆	1	2	yes	—
36	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
37	C₂×#88322	C₂×C₈	1	1	yes	characteristic
38	C₂×#88322	C₂×C₈	1	1	yes	characteristic
39	#11719912	—	2	2	no	—
40	C₈×D₅₃	—	2	1	no	—
41	C₈×D₅₃	—	2	1	no	—
42	C₁₆×C₅₃	—	2	1	no	—
43	C₁₆×C₅₃	—	2	1	no	—
44	#11719908	M₁₆	1	2	yes	—
45	#11719908	#41	1	2	yes	—
46	C₂²×D₅₃	C₂×C₁₆	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
48	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
49	C₂×Dic₅₃	C₂×C₁₆	1	1	yes	characteristic
50	C₂×Dic₅₃	C₂×C₁₆	1	2	yes	—
51	C₄×D₅₃	—	2	1	no	—
52	C₄×D₅₃	—	2	1	no	—
53	C₈×C₅₃	M₃₂	1	2	yes	—
54	C₈×C₅₃	#117	1	1	yes	characteristic
55	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
56	#88322	—	2	1	no	—
57	#88322	—	2	1	no	—
58	#13641	—	53	1	no	maximal, Sylow
59	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
60	C₂×D₅₃	—	2	1	no	—
61	C₂×D₅₃	—	2	2	no	—
62	C₄×C₅₃	—	2	2	no	—
63	C₄×C₅₃	#431	1	1	yes	characteristic
64	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
65	Dic₅₃	—	2	1	no	—
66	Dic₅₃	—	2	2	no	—
67	#2108	—	53	1	no	—
68	C₄⋊C₃₂	—	53	2	no	—
69	C₂²⋊C₃₂	—	53	2	no	—
70	#1397	—	53	1	no	—
71	C₄×C₃₂	—	53	1	no	—
72	C₂×C₅₃	#2162	1	1	yes	characteristic
73	C₂×C₅₃	#2258	1	2	yes	—
74	D₅₃	—	4	1	no	—
75	C₂×M₃₂	—	53	1	no	—
76	C₂²×C₁₆	—	53	1	no	—
77	#405	—	53	1	no	—
78	C₂×C₃₂	—	53	2	no	—
79	C₂×C₃₂	—	53	2	no	—
80	#346	—	53	1	no	—
81	#346	#6157	1	1	yes	characteristic
82	C₄×C₁₆	—	53	1	no	—
83	C₄×C₁₆	—	53	1	no	—
84	C₅₃	#13641	1	1	yes	characteristic, complemented, Sylow
85	C₂×M₁₆	—	53	1	no	—
86	C₂²×C₈	—	53	1	no	—
87	#117	—	53	1	no	—
88	M₃₂	—	106	2	no	—
89	C₂×C₁₆	—	53	1	no	—
90	C₂×C₁₆	—	53	1	no	—
91	C₂×C₁₆	—	106	2	no	—
92	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
93	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
94	#97	—	53	1	no	—
95	#97	—	53	1	no	—
96	C₄×C₈	—	53	1	no	—
97	C₄×C₈	C₂×#6157	1	1	yes	characteristic
98	C₃₂	—	106	2	no	—
99	C₃₂	—	106	2	no	—
100	C₂²×C₄	—	53	1	no	—
101	M₁₆	—	106	2	no	—
102	C₂×C₈	—	53	1	no	—
103	C₂×C₈	—	53	1	no	—
104	C₂×C₈	—	106	1	no	—
105	C₂×C₈	—	106	1	no	—
106	C₂×C₈	D₅₃⋊C₈	1	1	yes	characteristic
107	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
108	C₄⋊C₄	—	53	2	no	—
109	C₂²⋊C₄	—	53	2	no	—
110	C₄²	—	53	1	no	—
111	C₄²	D₅₃⋊C₈	1	1	yes	characteristic
112	C₁₆	—	2	1	no	—
113	C₁₆	—	2	1	no	—
114	C₁₆	—	53	2	no	—
115	C₁₆	—	53	2	no	—
116	C₂³	—	53	1	no	—
117	C₂×C₄	—	53	2	no	—
118	C₂×C₄	—	53	1	no	—
119	C₂×C₄	—	106	1	no	—
120	C₂×C₄	—	106	1	no	—
121	C₂×C₄	D₅₃⋊C₁₆	1	2	yes	—
122	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
123	C₈	—	106	1	no	—
124	C₈	—	106	1	no	—
125	C₈	#13133292	1	2	yes	—
126	C₈	#13133296	1	1	yes	characteristic
127	C₈	D₅₃⋊M₁₆	1	1	yes	characteristic
128	C₂²	—	106	1	no	—
129	C₂²	—	106	2	no	—
130	C₂²	C₂×D₅₃⋊C₁₆	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	106	1	no	—
133	C₄	—	106	2	no	—
134	C₄	#16811641	1	1	yes	characteristic
135	C₄	C₂×#13133292	1	1	yes	characteristic
136	C₂	—	212	1	no	—
137	C₂	#20291074	1	1	yes	characteristic
138	C₂	#20291139	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20289873

C₂

yes

maximal, complemented

#20289829

C₂

yes

maximal

#20289822

C₂

yes

maximal, characteristic, complemented

C₄×#13133267

C₂

yes

maximal, characteristic, complemented

#20276313

C₂

yes

maximal, characteristic

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

—

#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

—

C₈×D₅₃

—

C₁₆×C₅₃

—

C₁₆×C₅₃

—

#11719908

M₁₆

yes

—

#11719908

#41

yes

—

C₂²×D₅₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×C₁₆

yes

—

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₂×Dic₅₃

C₂×C₁₆

yes

characteristic

C₂×Dic₅₃

C₂×C₁₆

yes

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

M₃₂

yes

—

C₈×C₅₃

#117

yes

characteristic

C₈×C₅₃

C₂×M₁₆

yes

—

—

—

C₂²×C₅₃

C₂²×C₁₆

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#431

yes

characteristic

C₄×C₅₃

C₂×M₃₂

yes

characteristic

Dic₅₃

—

Dic₅₃

—

#2108

—

C₄⋊C₃₂

—

C₂²⋊C₃₂

—

#1397

—

C₄×C₃₂

—

C₂×C₅₃

#2162

yes

characteristic

C₂×C₅₃

#2258

yes

—

D₅₃

—

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

C₂×C₃₂

—

C₂×C₃₂

—

—

yes

C₄×C₁₆

—

C₄×C₁₆

—

C₅₃

#13641

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

#117

—

M₃₂

—

106

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

C₂×C₁₆

yes

C₂×C₁₆

yes

—

—

C₄×C₈

—

C₄×C₈

C₂×#6157

yes

characteristic

C₃₂

—

106

—

C₃₂

—

106

—

100

C₂²×C₄

—

101

M₁₆

—

106

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

106

—

105

C₂×C₈

—

106

—

106

C₂×C₈

D₅₃⋊C₈

yes

characteristic

107

C₂×C₈

C₂²×#6157

yes

characteristic

108

C₄⋊C₄

—

109

C₂²⋊C₄

—

110

C₄²

—

111

C₄²

D₅₃⋊C₈

yes

characteristic

112

C₁₆

—

113

C₁₆

—

114

C₁₆

—

115

C₁₆

—

116

C₂³

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

106

—

120

C₂×C₄

—

106

—

121

C₂×C₄

D₅₃⋊C₁₆

yes

—

122

C₂×C₄

C₂×D₅₃⋊C₈

yes

characteristic

123

C₈

—

106

—

124

C₈

—

106

—

125

C₈

#13133292

yes

—

126

C₈

#13133296

yes

characteristic

127

C₈

D₅₃⋊M₁₆

yes

characteristic

128

C₂²

—

106

—

129

C₂²

—

106

—

130

C₂²

C₂×D₅₃⋊C₁₆

yes

characteristic

131

C₄

—

132

C₄

—

106

—

133

C₄

—

106

—

134

C₄

#16811641

yes

characteristic

135

C₄

C₂×#13133292

yes

characteristic

136

C₂

—

212

—

137

C₂

#20291074

yes

characteristic

138

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	#13641
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
1. #20289873	C₂	semidirect
3. #20289822	C₂	semidirect
4. C₄×#13133267	C₂	semidirect
84. C₅₃	#13641	semidirect

H ≅ G/N

type

1. #20289873

C₂

semidirect

3. #20289822

C₂

semidirect

4. C₄×#13133267

C₂

semidirect

84. C₅₃

#13641

semidirect

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

subgroup	index
#20289873	2
#20289829	2
#20289822	2
C₄×#13133267	2
#20276313	2
#13641	53

subgroup

index

Subgroups — #28312617

Conjugacy classes of subgroups

Complements

Maximal subgroups