Group #28301460 — subgroups

Subgroups — #28301460

Order 13568.

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

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 180 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	#20290429	C₂	1	2	yes	maximal
2	#20289537	C₂	1	4	yes	maximal
3	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×#88324	C₂²	1	4	yes	—
5	#16811489	C₄	1	8	yes	—
6	C₂×C₄×#88322	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
8	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
9	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
11	C₂²×#88324	C₂×C₄	1	8	yes	—
12	C₄×#88324	—	2	8	no	—
13	C₂²×#88322	C₂×C₄	1	2	yes	—
14	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×Dic₅₃	C₂×C₄	1	4	yes	—
16	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
17	C₄×#88322	—	2	2	no	—
18	C₈×Dic₅₃	—	2	4	no	—
19	C₈×Dic₅₃	C₈	1	8	yes	complemented
20	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₅₃	—	2	2	no	—
23	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
24	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
25	C₂²×Dic₅₃	C₄²	1	2	yes	—
26	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
27	C₂×#88324	—	2	8	no	—
28	C₂×#88324	—	2	16	no	—
29	C₂×C₈×C₅₃	—	2	2	no	—
30	C₂×C₈×C₅₃	—	2	2	no	—
31	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
32	C₄²×C₅₃	M₁₆	1	2	yes	—
33	C₄²×C₅₃	M₁₆	1	2	yes	—
34	C₄×Dic₅₃	—	2	8	no	—
35	C₄×Dic₅₃	C₂×C₈	1	4	yes	—
36	C₄×Dic₅₃	#41	1	4	yes	—
37	C₂×#88322	—	2	2	no	—
38	C₂×#88322	—	2	2	no	—
39	C₂×#88322	C₂×C₈	1	4	yes	—
40	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
41	C₂×C₄×C₅₃	—	2	4	no	—
42	C₂×C₄×C₅₃	#97	1	4	yes	—
43	C₂×C₄×C₅₃	#117	1	2	yes	—
44	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
45	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
47	C₂×Dic₅₃	—	2	4	no	—
48	C₂×Dic₅₃	C₄×C₈	1	4	yes	—
49	C₂×Dic₅₃	#117	1	4	yes	—
50	C₂×Dic₅₃	#131	1	8	yes	—
51	#88324	—	4	8	no	—
52	C₈×C₅₃	—	2	4	no	—
53	#88322	—	2	4	no	—
54	#8133	—	53	1	no	maximal, Sylow
55	C₂²×C₅₃	#376	1	1	yes	characteristic
56	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
57	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
58	C₂²×C₅₃	#405	1	4	yes	—
59	C₄×C₅₃	—	2	4	no	—
60	C₄×C₅₃	#431	1	2	yes	—
61	C₄×C₅₃	#431	1	2	yes	—
62	Dic₅₃	—	2	4	no	—
63	Dic₅₃	#433	1	8	yes	—
64	#1744	—	53	2	no	—
65	C₄²×C₈	—	53	1	no	—
66	#1273	—	53	4	no	—
67	C₂×C₅₃	#1450	1	2	yes	—
68	C₂×C₅₃	#1452	1	2	yes	—
69	C₂×C₅₃	#1744	1	1	yes	characteristic
70	C₂×C₅₃	#1746	1	2	yes	—
71	C₂×C₄×C₈	—	53	2	no	—
72	C₂×C₄×C₈	—	53	4	no	—
73	C₂×C₄×C₈	—	53	2	no	—
74	C₂×C₄×C₈	—	53	1	no	—
75	C₂×C₄×C₈	#6157	1	1	yes	characteristic
76	C₄³	—	53	1	no	—
77	#336	—	53	8	no	—
78	C₅₃	#8133	1	1	yes	characteristic, complemented, Sylow
79	C₂²×C₈	—	53	2	no	—
80	C₂²×C₈	—	53	8	no	—
81	C₂²×C₈	C₂×#6157	1	2	yes	—
82	C₂×C₄²	—	53	1	no	—
83	C₂×C₄²	—	53	1	no	—
84	C₂×C₄²	—	53	4	no	—
85	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
86	C₄×C₈	—	2	2	no	—
87	C₄×C₈	—	53	8	no	—
88	C₄×C₈	—	106	4	no	—
89	C₄×C₈	—	106	8	no	—
90	C₄×C₈	—	106	2	no	—
91	C₂²×C₄	—	53	2	no	—
92	C₂²×C₄	—	53	2	no	—
93	C₂²×C₄	C₄×#6157	1	2	yes	—
94	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	53	4	no	—
98	C₂×C₈	—	106	2	no	—
99	C₂×C₈	—	106	2	no	—
100	C₂×C₈	—	106	8	no	—
101	C₂×C₈	—	106	16	no	—
102	C₂×C₈	D₅₃⋊C₈	1	4	yes	—
103	C₄²	—	53	4	no	—
104	C₄²	—	53	4	no	—
105	C₄²	—	106	8	no	—
106	C₄²	#11719936	1	2	yes	—
107	C₄²	#11719940	1	2	yes	—
108	C₂³	C₂×C₄×#6157	1	1	yes	characteristic
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	53	4	no	—
111	C₂×C₄	—	53	8	no	—
112	C₂×C₄	—	53	4	no	—
113	C₂×C₄	—	106	4	no	—
114	C₂×C₄	#13133272	1	4	yes	—
115	C₂×C₄	#13133296	1	2	yes	—
116	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
117	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
118	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
119	C₈	—	2	4	no	—
120	C₈	—	106	4	no	—
121	C₈	—	212	8	no	—
122	C₂²	#16811559	1	1	yes	characteristic
123	C₂²	C₄×D₅₃⋊C₈	1	1	yes	characteristic
124	C₂²	C₂×#13133272	1	1	yes	characteristic
125	C₂²	#16811586	1	4	yes	—
126	C₄	—	2	4	no	—
127	C₄	—	53	8	no	—
128	C₄	—	106	4	no	—
129	C₄	#16811640	1	2	yes	—
130	C₄	#16811642	1	2	yes	—
131	C₂	#20289919	1	2	yes	—
132	C₂	#20289928	1	2	yes	—
133	C₂	#20290430	1	1	yes	characteristic
134	C₂	#20290433	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20290429

C₂

yes

maximal

#20289537

C₂

yes

maximal

C₄×C₈×Dic₅₃

C₂

yes

maximal, characteristic

C₂×C₄×#88324

C₂²

yes

—

#16811489

C₄

yes

—

C₂×C₄×#88322

C₄

yes

characteristic

C₂×C₈×Dic₅₃

C₄

yes

—

C₂×C₈×Dic₅₃

C₂²

yes

—

C₄²×Dic₅₃

C₂²

yes

characteristic

C₂×C₄×C₈×C₅₃

C₄

yes

characteristic

C₂²×#88324

C₂×C₄

yes

—

C₄×#88324

—

C₂²×#88322

C₂×C₄

yes

—

C₂×C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅₃

C₂×C₄

yes

—

C₂×C₄×Dic₅₃

C₂³

yes

characteristic

C₄×#88322

—

C₈×Dic₅₃

—

C₈×Dic₅₃

C₈

yes

complemented

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

—

C₂²×C₄×C₅₃

C₄²

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×Dic₅₃

C₄²

yes

—

C₂²×Dic₅₃

C₂²×C₄

yes

—

C₂×#88324

—

C₂×#88324

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₈

yes

—

C₄²×C₅₃

M₁₆

yes

—

C₄²×C₅₃

M₁₆

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂×C₈

yes

—

C₄×Dic₅₃

#41

yes

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

C₂×C₈

yes

—

C₂³×C₅₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#97

yes

—

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

C₂²×C₈

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

C₄×C₈

yes

—

C₂×Dic₅₃

#117

yes

—

C₂×Dic₅₃

#131

yes

—

#88324

—

C₈×C₅₃

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

C₂×C₄×C₈

yes

characteristic

C₂²×C₅₃

C₂×#97

yes

characteristic

C₂²×C₅₃

#405

yes

—

C₄×C₅₃

—

C₄×C₅₃

#431

yes

—

C₄×C₅₃

#431

yes

—

Dic₅₃

—

Dic₅₃

#433

yes

—

#1744

—

C₄²×C₈

—

#1273

—

C₂×C₅₃

#1450

yes

—

C₂×C₅₃

#1452

yes

—

C₂×C₅₃

#1744

yes

characteristic

C₂×C₅₃

#1746

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

#6157

yes

characteristic

C₄³

—

#336

—

C₅₃

#8133

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×#6157

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×#6157

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

106

—

C₄×C₈

—

106

—

C₄×C₈

—

106

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×#6157

yes

—

C₂²×C₄

C₂²×#6157

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

100

C₂×C₈

—

106

—

101

C₂×C₈

—

106

—

102

C₂×C₈

D₅₃⋊C₈

yes

—

103

C₄²

—

104

C₄²

—

105

C₄²

—

106

—

106

C₄²

#11719936

yes

—

107

C₄²

#11719940

yes

—

108

C₂³

C₂×C₄×#6157

yes

characteristic

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

106

—

114

C₂×C₄

#13133272

yes

—

115

C₂×C₄

#13133296

yes

—

116

C₂×C₄

C₂×D₅₃⋊C₈

yes

—

117

C₂×C₄

C₂×#11719940

yes

characteristic

118

C₂×C₄

C₂×#11719936

yes

characteristic

119

C₈

—

120

C₈

—

106

—

121

C₈

—

212

—

122

C₂²

#16811559

yes

characteristic

123

C₂²

C₄×D₅₃⋊C₈

yes

characteristic

124

C₂²

C₂×#13133272

yes

characteristic

125

C₂²

#16811586

yes

—

126

C₄

—

127

C₄

—

128

C₄

—

106

—

129

C₄

#16811640

yes

—

130

C₄

#16811642

yes

—

131

C₂

#20289919

yes

—

132

C₂

#20289928

yes

—

133

C₂

#20290430

yes

characteristic

134

C₂

#20290433

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6157
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₂×C₄²
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	#8133
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×#6157

commutator

C₂²×C₅₃

C₂×C₄×C₈

Fitting

C₂×C₄×C₈×C₅₃

C₄

socle, prime radical

C₂³×C₅₃

C₂×C₄²

cosocle

C₂×C₄×Dic₅₃

C₂³

C₂×C₄×C₈

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
19. C₈×Dic₅₃	C₈	semidirect
78. C₅₃	#8133	semidirect

H ≅ G/N

type

19. C₈×Dic₅₃

C₈

semidirect

78. C₅₃

#8133

semidirect

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

subgroup	index
#20290429	2
#20289537	2
C₄×C₈×Dic₅₃	2
#8133	53

subgroup

index

#20290429

#20289537

C₄×C₈×Dic₅₃

#8133

Subgroups — #28301460

Conjugacy classes of subgroups

Complements

Maximal subgroups