Group #27300482 — subgroups

Subgroups — #27300482

Order 13568.

282 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 155 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	#20274887	C₂	1	2	yes	maximal
2	#20272809	C₂	1	4	yes	maximal
3	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133153	—	2	2	no	—
5	C₂×#13133153	C₂²	1	4	yes	—
6	#16810647	—	2	2	no	—
7	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₄×C₁₆×C₅₃	C₄	1	2	yes	—
10	C₄×C₁₆×C₅₃	C₄	1	2	yes	—
11	C₂×#11719917	—	2	4	no	—
12	#13133153	—	2	8	no	—
13	#13133153	—	4	4	no	—
14	C₄⋊#88322	—	4	4	no	—
15	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
17	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₅₃	—	2	4	no	—
19	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
20	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
23	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
25	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
26	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
27	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₈×C₅₃	SD₁₆	1	4	yes	—
29	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
31	C₂×C₈×C₅₃	#41	1	4	yes	—
32	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
33	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
34	C₄²×C₅₃	D₈	1	2	yes	—
35	#11719917	—	4	4	no	—
36	#11719917	—	4	4	no	—
37	C₂×#88322	—	4	4	no	—
38	C₁₆×C₅₃	—	2	4	no	—
39	C₂³×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
40	C₂×C₄×C₅₃	—	2	4	no	—
41	C₂×C₄×C₅₃	#107	1	2	yes	—
42	C₂×C₄×C₅₃	#107	1	2	yes	—
43	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	#124	1	2	yes	—
45	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
47	C₈×C₅₃	#106	1	2	yes	—
48	C₈×C₅₃	#106	1	2	yes	—
49	C₈×C₅₃	#117	1	2	yes	—
50	C₈×C₅₃	#125	1	2	yes	—
51	#88322	—	8	4	no	—
52	#12891	—	53	1	no	maximal, Sylow
53	C₂²×C₅₃	—	2	2	no	—
54	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
55	C₂²×C₅₃	#487	1	2	yes	—
56	C₄×C₅₃	—	2	2	no	—
57	C₄×C₅₃	#368	1	4	yes	—
58	C₄×C₅₃	#382	1	1	yes	characteristic
59	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
60	C₄×C₅₃	#486	1	2	yes	—
61	C₂×C₄×C₁₆	D₅₃	1	1	yes	characteristic
62	#1848	—	53	2	no	—
63	#1381	—	53	4	no	—
64	C₂×C₅₃	—	2	2	no	—
65	C₂×C₅₃	#1834	1	1	yes	characteristic
66	C₂×C₅₃	C₂×#368	1	2	yes	—
67	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
68	C₂×#108	—	53	4	no	—
69	C₂×#108	—	106	2	no	—
70	#423	—	106	2	no	—
71	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
72	C₄×C₁₆	Dic₅₃	1	2	yes	—
73	C₄×C₁₆	Dic₅₃	1	2	yes	—
74	C₅₃	#12891	1	1	yes	characteristic, complemented, Sylow
75	C₂×M₁₆	—	106	4	no	—
76	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
77	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
78	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
79	C₂×C₁₆	—	2	4	no	—
80	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
81	#108	—	106	8	no	—
82	#108	—	212	4	no	—
83	C₄⋊C₈	—	212	4	no	—
84	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
85	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
86	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
87	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
88	C₂²×C₄	Dic₂₁₂	1	2	yes	—
89	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
90	M₁₆	—	212	4	no	—
91	M₁₆	—	212	4	no	—
92	C₂×C₈	—	212	4	no	—
93	C₂×C₈	#11719913	1	4	yes	—
94	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
95	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
96	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
97	C₂×C₈	#11719945	1	4	yes	—
98	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
99	C₄²	D₄₂₄	1	2	yes	—
100	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
101	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
102	C₁₆	—	2	4	no	—
103	C₂³	C₂×Dic₂₁₂	1	1	yes	characteristic
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	#13133151	1	2	yes	—
106	C₂×C₄	#13133151	1	2	yes	—
107	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
108	C₂×C₄	#13133205	1	2	yes	—
109	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
110	C₂×C₄	C₂×#11719913	1	2	yes	—
111	C₈	—	424	4	no	—
112	C₈	#13133149	1	2	yes	—
113	C₈	#13133149	1	2	yes	—
114	C₈	#13133176	1	2	yes	—
115	C₈	#13133209	1	2	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	C₂×#13133151	1	1	yes	characteristic
118	C₂²	#16810974	1	2	yes	—
119	C₄	—	2	2	no	—
120	C₄	#16810453	1	4	yes	—
121	C₄	#16810489	1	1	yes	characteristic
122	C₄	C₂×#13133149	1	1	yes	characteristic
123	C₄	#16810970	1	2	yes	—
124	C₂	—	2	2	no	—
125	C₂	#20274843	1	1	yes	characteristic
126	C₂	C₂×#16810453	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20274887

C₂

yes

maximal

#20272809

C₂

yes

maximal

C₂×C₄×C₁₆×C₅₃

C₂

yes

maximal, characteristic

C₂×#13133153

—

C₂×#13133153

C₂²

yes

—

#16810647

—

C₂²×C₁₆×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₄×C₁₆×C₅₃

C₄

yes

—

C₄×C₁₆×C₅₃

C₄

yes

—

—

—

—

—

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

characteristic

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

Q₈

yes

characteristic

C₄×C₈×C₅₃

Q₈

yes

characteristic

C₂²×C₄×C₅₃

Q₁₆

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

SD₁₆

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₃

#41

yes

—

C₄²×C₅₃

C₄⋊C₄

yes

characteristic

C₄²×C₅₃

C₄⋊C₄

yes

characteristic

C₄²×C₅₃

D₈

yes

—

#11719917

—

#11719917

—

C₂×#88322

—

C₁₆×C₅₃

—

C₂³×C₅₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#107

yes

—

C₂×C₄×C₅₃

#107

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#124

yes

—

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

C₈×C₅₃

#106

yes

—

C₈×C₅₃

#106

yes

—

C₈×C₅₃

#117

yes

—

C₈×C₅₃

#125

yes

—

#88322

—

#12891

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

C₂×#107

yes

characteristic

C₂²×C₅₃

#487

yes

—

C₄×C₅₃

—

C₄×C₅₃

#368

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₂×C₄×C₁₆

D₅₃

yes

characteristic

#1848

—

#1381

—

C₂×C₅₃

—

C₂×C₅₃

#1834

yes

characteristic

C₂×C₅₃

C₂×#368

yes

—

C₂²×C₁₆

C₂×D₅₃

yes

—

C₂×#108

—

C₂×#108

—

106

—

#423

—

106

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

C₄×C₁₆

Dic₅₃

yes

—

C₄×C₁₆

Dic₅₃

yes

—

C₅₃

#12891

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

106

—

C₂²×C₈

D₂₁₂

yes

characteristic

C₂²×C₈

C₂²×D₅₃

yes

characteristic

C₂×C₄²

D₂₁₂

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₅₃

yes

—

#108

—

106

—

#108

—

212

—

C₄⋊C₈

—

212

—

C₄×C₈

Dic₁₀₆

yes

characteristic

C₄×C₈

Dic₁₀₆

yes

characteristic

C₄×C₈

C₂×Dic₅₃

yes

characteristic

C₄×C₈

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

Dic₂₁₂

yes

—

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

M₁₆

—

212

—

M₁₆

—

212

—

C₂×C₈

—

212

—

C₂×C₈

#11719913

yes

—

C₂×C₈

C₄⋊Dic₅₃

yes

characteristic

C₂×C₈

C₄⋊Dic₅₃

yes

characteristic

C₂×C₈

C₂×Dic₁₀₆

yes

characteristic

C₂×C₈

#11719945

yes

—

C₂×C₈

C₂²×Dic₅₃

yes

characteristic

C₄²

D₄₂₄

yes

—

100

C₄²

C₄⋊Dic₅₃

yes

characteristic

101

C₄²

C₄⋊Dic₅₃

yes

characteristic

102

C₁₆

—

103

C₂³

C₂×Dic₂₁₂

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

#13133151

yes

—

106

C₂×C₄

#13133151

yes

—

107

C₂×C₄

C₂×C₄⋊Dic₅₃

yes

characteristic

108

C₂×C₄

#13133205

yes

—

109

C₂×C₄

C₂×D₄₂₄

yes

characteristic

110

C₂×C₄

C₂×#11719913

yes

—

111

C₈

—

424

—

112

C₈

#13133149

yes

—

113

C₈

#13133149

yes

—

114

C₈

#13133176

yes

—

115

C₈

#13133209

yes

—

116

C₂²

—

117

C₂²

C₂×#13133151

yes

characteristic

118

C₂²

#16810974

yes

—

119

C₄

—

120

C₄

#16810453

yes

—

121

C₄

#16810489

yes

characteristic

122

C₄

C₂×#13133149

yes

characteristic

123

C₄

#16810970

yes

—

124

C₂

—

125

C₂

#20274843

yes

characteristic

126

C₂

C₂×#16810453

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₄₂₄
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₄×C₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#107
prime radical	C₂³×C₅₃	C₂×Q₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₅₃
hypocenter	C₅₃	#12891
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

subgroup

G/H

center

C₂×C₄

C₂×D₄₂₄

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₂×C₄×C₁₆×C₅₃

C₂

socle

C₂²×C₅₃

C₂×#107

prime radical

C₂³×C₅₃

C₂×Q₁₆

cosocle

C₂²×C₈×C₅₃

C₂³

hypercenter

C₂×C₄×C₁₆

D₅₃

hypocenter

C₅₃

#12891

LCS

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

N	H ≅ G/N	type
74. C₅₃	#12891	semidirect

H ≅ G/N

type

74. C₅₃

#12891

semidirect

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

subgroup	index
#20274887	2
#20272809	2
C₂×C₄×C₁₆×C₅₃	2
#12891	53

subgroup

index

#20274887

#20272809

C₂×C₄×C₁₆×C₅₃

#12891

Subgroups — #27300482

Conjugacy classes of subgroups

Complements

Maximal subgroups