Group #28312621 — subgroups

Subgroups — #28312621

Order 13568.

170 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 68 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	#20289875	C₂	1	2	yes	maximal, complemented
2	#20289831	C₂	1	2	yes	maximal
3	#20289826	C₂	1	2	yes	maximal, complemented
4	#20276308	C₂	1	1	yes	maximal, characteristic
5	#16811555	C₂²	1	4	yes	—
6	C₂×#13133156	C₄	1	1	yes	characteristic
7	C₂×#13133156	C₂²	1	1	yes	characteristic
8	C₄×C₈×D₅₃	C₄	1	1	yes	characteristic
9	#16810399	C₂²	1	1	yes	characteristic
10	C₁₆×Dic₅₃	C₄	1	1	yes	characteristic
11	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#346	C₄	1	1	yes	characteristic
13	#13133267	—	2	4	no	—
14	C₂×C₈×D₅₃	C₈	1	1	yes	characteristic
15	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
16	C₄²×D₅₃	C₈	1	1	yes	characteristic
17	#13133156	—	2	2	no	—
18	#13133156	—	2	2	no	—
19	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
20	C₂×#11719908	C₂³	1	1	yes	characteristic
21	C₄×#88322	C₈	1	1	yes	characteristic
22	C₈×Dic₅₃	C₈	1	1	yes	characteristic
23	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×D₅₃	—	2	1	no	—
28	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
29	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
30	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
31	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
32	C₄×Dic₅₃	—	2	1	no	—
33	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
34	C₂×#88322	C₂×C₈	1	1	yes	characteristic
35	C₂×#88322	C₂×C₈	1	1	yes	characteristic
36	C₈×D₅₃	—	2	1	no	—
37	C₈×D₅₃	—	2	1	no	—
38	C₈×D₅₃	—	2	2	no	—
39	C₁₆×C₅₃	—	2	1	no	—
40	C₁₆×C₅₃	—	2	1	no	—
41	#11719908	M₁₆	1	2	yes	—
42	#11719908	#41	1	2	yes	—
43	C₂²×D₅₃	M₃₂	1	1	yes	characteristic
44	C₂×C₄×C₅₃	—	2	1	no	—
45	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
46	C₂×Dic₅₃	—	2	1	no	—
47	C₂×Dic₅₃	M₃₂	1	1	yes	characteristic
48	C₄×D₅₃	—	2	1	no	—
49	C₄×D₅₃	—	2	1	no	—
50	C₄×D₅₃	—	4	2	no	—
51	C₈×C₅₃	—	2	1	no	—
52	C₈×C₅₃	#117	1	1	yes	characteristic
53	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
54	#88322	—	2	1	no	—
55	#88322	M₃₂	1	2	yes	—
56	#13642	—	53	1	no	maximal, Sylow
57	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
58	C₂×D₅₃	—	2	1	no	—
59	C₂×D₅₃	—	4	1	no	—
60	C₄×C₅₃	—	4	1	no	—
61	C₄×C₅₃	#431	1	1	yes	characteristic
62	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
63	Dic₅₃	—	2	2	no	—
64	Dic₅₃	—	2	1	no	—
65	C₄×M₃₂	—	53	1	no	—
66	#1422	—	53	2	no	—
67	#1400	—	53	2	no	—
68	#1398	—	53	2	no	—
69	C₂×C₅₃	—	2	1	no	—
70	C₂×C₅₃	#2163	1	1	yes	characteristic
71	D₅₃	—	4	1	no	—
72	C₂×M₃₂	—	53	1	no	—
73	C₂×M₃₂	—	53	1	no	—
74	C₂×C₄×C₈	—	53	1	no	—
75	M₆₄	—	53	4	no	—
76	#346	—	53	1	no	—
77	#346	#6157	1	1	yes	characteristic
78	C₄×C₁₆	—	53	1	no	—
79	C₄×C₁₆	—	53	1	no	—
80	C₅₃	#13642	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₈	—	53	1	no	—
82	C₂²×C₈	—	53	1	no	—
83	C₂×C₄²	—	53	1	no	—
84	M₃₂	—	106	2	no	—
85	M₃₂	—	106	2	no	—
86	C₂×C₁₆	—	53	1	no	—
87	C₂×C₁₆	—	53	1	no	—
88	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
89	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
90	C₄×C₈	—	53	1	no	—
91	C₄×C₈	—	53	1	no	—
92	C₄×C₈	—	53	1	no	—
93	C₄×C₈	C₂×#6157	1	1	yes	characteristic
94	C₃₂	—	106	4	no	—
95	C₂²×C₄	—	53	1	no	—
96	C₂²×C₄	—	106	1	no	—
97	C₂×C₈	—	53	1	no	—
98	C₂×C₈	—	53	1	no	—
99	C₂×C₈	—	106	2	no	—
100	C₂×C₈	—	106	1	no	—
101	C₂×C₈	—	106	1	no	—
102	C₂×C₈	D₅₃⋊C₈	1	1	yes	characteristic
103	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
104	C₄²	—	53	1	no	—
105	C₄²	—	106	1	no	—
106	C₄²	D₅₃⋊C₈	1	1	yes	characteristic
107	C₁₆	—	2	1	no	—
108	C₁₆	—	2	1	no	—
109	C₁₆	—	53	2	no	—
110	C₁₆	—	53	2	no	—
111	C₂³	—	53	1	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	53	1	no	—
114	C₂×C₄	—	106	1	no	—
115	C₂×C₄	—	106	1	no	—
116	C₂×C₄	—	106	1	no	—
117	C₂×C₄	—	212	2	no	—
118	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
119	C₈	—	2	1	no	—
120	C₈	—	53	2	no	—
121	C₈	—	106	1	no	—
122	C₈	#13133296	1	1	yes	characteristic
123	C₈	D₅₃⋊M₁₆	1	1	yes	characteristic
124	C₂²	—	106	1	no	—
125	C₂²	—	212	1	no	—
126	C₂²	D₅₃⋊M₃₂	1	1	yes	characteristic
127	C₄	—	4	1	no	—
128	C₄	—	106	2	no	—
129	C₄	—	106	1	no	—
130	C₄	#16811641	1	1	yes	characteristic
131	C₄	D₅₃⋊M₃₂	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	212	1	no	—
134	C₂	#20291078	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20289875

C₂

yes

maximal, complemented

#20289831

C₂

yes

maximal

#20289826

C₂

yes

maximal, complemented

#20276308

C₂

yes

maximal, characteristic

#16811555

C₂²

yes

—

C₂×#13133156

C₄

yes

characteristic

C₂×#13133156

C₂²

yes

characteristic

C₄×C₈×D₅₃

C₄

yes

characteristic

#16810399

C₂²

yes

characteristic

C₁₆×Dic₅₃

C₄

yes

characteristic

C₁₆×Dic₅₃

C₂²

yes

characteristic

C₅₃×#346

C₄

yes

characteristic

#13133267

—

C₂×C₈×D₅₃

C₈

yes

characteristic

C₂×C₈×D₅₃

C₂×C₄

yes

characteristic

C₄²×D₅₃

C₈

yes

—

—

C₂×C₄

yes

characteristic

C₂×#11719908

C₂³

yes

characteristic

C₄×#88322

C₈

yes

characteristic

C₈×Dic₅₃

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×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

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂×C₈

yes

characteristic

C₂×#88322

C₂×C₈

yes

characteristic

C₂×#88322

C₂×C₈

yes

characteristic

C₈×D₅₃

—

C₈×D₅₃

—

C₈×D₅₃

—

C₁₆×C₅₃

—

C₁₆×C₅₃

—

#11719908

M₁₆

yes

—

#11719908

#41

yes

—

C₂²×D₅₃

M₃₂

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

M₃₂

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#117

yes

characteristic

C₈×C₅₃

C₂×M₁₆

yes

characteristic

#88322

—

#88322

M₃₂

yes

—

#13642

—

maximal, Sylow

C₂²×C₅₃

C₂×M₃₂

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#431

yes

characteristic

C₄×C₅₃

C₂×M₃₂

yes

characteristic

Dic₅₃

—

Dic₅₃

—

C₄×M₃₂

—

#1422

—

#1400

—

#1398

—

C₂×C₅₃

—

C₂×C₅₃

#2163

yes

characteristic

D₅₃

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂×C₄×C₈

—

M₆₄

—

—

yes

C₄×C₁₆

—

C₄×C₁₆

—

C₅₃

#13642

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

106

—

M₃₂

—

106

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

yes

C₂×C₁₆

yes

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×#6157

yes

characteristic

C₃₂

—

106

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

100

C₂×C₈

—

106

—

101

C₂×C₈

—

106

—

102

C₂×C₈

D₅₃⋊C₈

yes

characteristic

103

C₂×C₈

C₂²×#6157

yes

characteristic

104

C₄²

—

105

C₄²

—

106

—

106

C₄²

D₅₃⋊C₈

yes

characteristic

107

C₁₆

—

108

C₁₆

—

109

C₁₆

—

110

C₁₆

—

111

C₂³

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

106

—

115

C₂×C₄

—

106

—

116

C₂×C₄

—

106

—

117

C₂×C₄

—

212

—

118

C₂×C₄

C₂×D₅₃⋊C₈

yes

characteristic

119

C₈

—

120

C₈

—

121

C₈

—

106

—

122

C₈

#13133296

yes

characteristic

123

C₈

D₅₃⋊M₁₆

yes

characteristic

124

C₂²

—

106

—

125

C₂²

—

212

—

126

C₂²

D₅₃⋊M₃₂

yes

characteristic

127

C₄

—

128

C₄

—

106

—

129

C₄

—

106

—

130

C₄

#16811641

yes

characteristic

131

C₄

D₅₃⋊M₃₂

yes

characteristic

132

C₂

—

133

C₂

—

212

—

134

C₂

#20291078

yes

characteristic

	subgroup	G/H
center	C₈	#13133296
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₅₃×#346	C₄
socle	C₂×C₅₃	#2163
prime radical	C₂²×D₅₃	M₃₂
cosocle	C₂×#11719908	C₂³
hypercenter	#346	#6157
hypocenter	C₅₃	#13642
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₄×C₈ ◁ #346

subgroup

G/H

center

C₈

#13133296

commutator

C₂×C₄×C₅₃

C₂²×C₈

Frattini

C₂×C₈

C₂²×#6157

Fitting

C₅₃×#346

C₄

socle

C₂×C₅₃

#2163

prime radical

C₂²×D₅₃

M₃₂

C₂³

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₈ ◁ C₄×C₈ ◁ #346

N	H ≅ G/N	type
1. #20289875	C₂	semidirect
3. #20289826	C₂	semidirect
80. C₅₃	#13642	semidirect

H ≅ G/N

type

1. #20289875

C₂

semidirect

3. #20289826

C₂

semidirect

80. C₅₃

#13642

semidirect

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

subgroup	index
#20289875	2
#20289831	2
#20289826	2
#20276308	2
#13642	53

subgroup

index

Subgroups — #28312621

Conjugacy classes of subgroups

Complements

Maximal subgroups