Group #27295647 — subgroups

Subgroups — #27295647

Order 13568.

234 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 147 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	#20273463	C₂	1	2	yes	maximal
2	#20272781	C₂	1	4	yes	maximal, complemented
3	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic
4	#16810676	—	2	2	no	—
5	#16810376	—	2	2	no	—
6	#16810374	C₂²	1	4	yes	—
7	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
8	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₅₃×#346	C₄	1	2	yes	—
11	C₅₃×#346	C₂²	1	2	yes	—
12	C₄⋊#88322	—	2	4	no	—
13	C₂²⋊#88322	—	4	2	no	—
14	C₄×#88322	—	4	2	no	—
15	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
17	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₅₃	—	2	2	no	—
19	C₂×C₁₆×C₅₃	C₈	1	4	yes	complemented
20	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
21	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
25	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
28	C₂×C₈×C₅₃	—	2	2	no	—
29	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
32	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
33	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
35	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
36	C₄²×C₅₃	SD₁₆	1	2	yes	—
37	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂×#88322	—	4	4	no	—
39	C₁₆×C₅₃	—	2	2	no	—
40	C₁₆×C₅₃	C₂×C₈	1	4	yes	—
41	C₂³×C₅₃	#106	1	1	yes	characteristic
42	C₂×C₄×C₅₃	—	4	2	no	—
43	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
44	C₂×C₄×C₅₃	#106	1	2	yes	—
45	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
46	C₂×C₄×C₅₃	#108	1	2	yes	—
47	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
49	C₈×C₅₃	—	2	1	no	—
50	C₈×C₅₃	—	2	1	no	—
51	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
52	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
53	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
54	#88322	—	8	4	no	—
55	#11999	—	53	1	no	maximal, Sylow
56	C₂²×C₅₃	—	2	2	no	—
57	C₂²×C₅₃	#334	1	2	yes	—
58	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
59	C₄×C₅₃	—	2	2	no	—
60	C₄×C₅₃	—	2	2	no	—
61	C₄×C₅₃	#334	1	2	yes	—
62	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
63	C₄×C₅₃	C₂×#108	1	1	yes	characteristic
64	C₂×#346	D₅₃	1	1	yes	characteristic
65	#1564	—	53	2	no	—
66	#1368	—	53	4	no	—
67	C₂×C₅₃	—	2	2	no	—
68	C₂×C₅₃	C₂×#334	1	1	yes	characteristic
69	C₂×C₅₃	#2155	1	2	yes	—
70	C₂²×C₁₆	Dic₅₃	1	1	yes	characteristic
71	C₂²×C₁₆	Dic₅₃	1	1	yes	characteristic
72	#431	—	106	2	no	—
73	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
74	#346	Dic₅₃	1	2	yes	—
75	#346	C₂×D₅₃	1	2	yes	—
76	#335	—	106	2	no	—
77	#334	—	53	4	no	—
78	C₅₃	#11999	1	1	yes	characteristic, complemented, Sylow
79	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
80	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
81	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
82	C₂×C₁₆	—	2	2	no	—
83	C₂×C₁₆	#88322	1	4	yes	complemented
84	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
85	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
86	C₄⋊C₈	—	106	4	no	—
87	C₂²⋊C₈	—	212	2	no	—
88	C₄×C₈	—	212	2	no	—
89	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
90	C₄×C₈	D₂₁₂	1	1	yes	characteristic
91	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
92	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
93	C₂²×C₄	#11719913	1	2	yes	—
94	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	212	4	no	—
97	C₂×C₈	C₂×#88322	1	2	yes	—
98	C₂×C₈	#11719917	1	2	yes	—
99	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
100	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
101	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
102	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
103	C₄²	#11719913	1	2	yes	—
104	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
105	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
106	C₁₆	—	2	2	no	—
107	C₁₆	C₂×#88322	1	4	yes	—
108	C₂³	#13133149	1	1	yes	characteristic
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	C₄⋊#88322	1	2	yes	—
111	C₂×C₄	#13133149	1	2	yes	—
112	C₂×C₄	#13133149	1	1	yes	characteristic
113	C₂×C₄	#13133153	1	2	yes	—
114	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
115	C₂×C₄	C₂×#11719913	1	2	yes	—
116	C₈	—	2	1	no	—
117	C₈	—	2	1	no	—
118	C₈	—	424	4	no	—
119	C₈	C₄⋊#88322	1	2	yes	—
120	C₈	C₂²×#88322	1	1	yes	characteristic
121	C₈	C₂×#11719917	1	1	yes	characteristic
122	C₂²	—	2	2	no	—
123	C₂²	#16810374	1	2	yes	—
124	C₂²	C₂×#13133149	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	—	2	2	no	—
127	C₄	#16810374	1	2	yes	—
128	C₄	C₂×C₄⋊#88322	1	1	yes	characteristic
129	C₄	C₂×#13133153	1	1	yes	characteristic
130	C₂	—	2	2	no	—
131	C₂	C₂×#16810374	1	1	yes	characteristic
132	C₂	#20276488	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20273463

C₂

yes

maximal

#20272781

C₂

yes

maximal, complemented

C₂×C₅₃×#346

C₂

yes

maximal, characteristic

#16810676

—

#16810376

—

#16810374

C₂²

yes

—

C₂²×C₁₆×C₅₃

C₄

yes

characteristic

C₂²×C₁₆×C₅₃

C₄

yes

characteristic

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#346

C₄

yes

—

C₅₃×#346

C₂²

yes

—

C₄⋊#88322

—

C₂²⋊#88322

—

C₄×#88322

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

Q₈

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

characteristic

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

C₈

yes

complemented

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

Q₈

yes

characteristic

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₅₃

SD₁₆

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

M₁₆

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×Q₈

yes

characteristic

C₄²×C₅₃

C₄⋊C₄

yes

characteristic

C₄²×C₅₃

SD₁₆

yes

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₂×#88322

—

C₁₆×C₅₃

—

C₁₆×C₅₃

C₂×C₈

yes

—

C₂³×C₅₃

#106

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₈

yes

—

C₂×C₄×C₅₃

#106

yes

—

C₂×C₄×C₅₃

#106

yes

characteristic

C₂×C₄×C₅₃

#108

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₄⋊C₈

yes

—

C₈×C₅₃

C₂²×C₈

yes

characteristic

C₈×C₅₃

C₂×M₁₆

yes

—

—

C₂²×C₅₃

—

C₂²×C₅₃

#334

yes

—

C₂²×C₅₃

C₂×#106

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#334

yes

—

C₄×C₅₃

C₂×C₄⋊C₈

yes

C₄×C₅₃

yes

D₅₃

yes

characteristic

#1564

—

#1368

—

C₂×C₅₃

—

C₂×C₅₃

C₂×#334

yes

characteristic

C₂×C₅₃

#2155

yes

—

C₂²×C₁₆

Dic₅₃

yes

characteristic

C₂²×C₁₆

Dic₅₃

yes

characteristic

#431

—

106

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

#346

Dic₅₃

yes

—

#346

C₂×D₅₃

yes

—

#335

—

106

—

#334

—

C₅₃

#11999

yes

characteristic, complemented, Sylow

C₂²×C₈

Dic₁₀₆

yes

characteristic

C₂²×C₈

C₂×Dic₅₃

yes

characteristic

C₂×C₄²

D₂₁₂

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

#88322

yes

complemented

C₂×C₁₆

C₂×Dic₅₃

yes

—

C₂×C₁₆

C₂×Dic₅₃

yes

—

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

212

—

C₄×C₈

—

212

—

C₄×C₈

Dic₁₀₆

yes

characteristic

C₄×C₈

D₂₁₂

yes

characteristic

C₄×C₈

C₂×Dic₅₃

yes

characteristic

C₄×C₈

C₂²×D₅₃

yes

characteristic

C₂²×C₄

#11719913

yes

—

C₂²×C₄

C₄⋊Dic₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

212

—

C₂×C₈

C₂×#88322

yes

—

C₂×C₈

#11719917

yes

—

C₂×C₈

C₄⋊Dic₅₃

yes

characteristic

100

C₂×C₈

C₄⋊Dic₅₃

yes

characteristic

101

C₂×C₈

C₂×Dic₁₀₆

yes

characteristic

102

C₂×C₈

C₂²×Dic₅₃

yes

characteristic

103

C₄²

#11719913

yes

—

104

C₄²

C₄⋊Dic₅₃

yes

characteristic

105

C₄²

C₂×D₂₁₂

yes

characteristic

106

C₁₆

—

107

C₁₆

C₂×#88322

yes

—

108

C₂³

#13133149

yes

characteristic

109

C₂×C₄

—

110

C₂×C₄

C₄⋊#88322

yes

—

111

C₂×C₄

#13133149

yes

—

112

C₂×C₄

#13133149

yes

characteristic

113

C₂×C₄

#13133153

yes

—

114

C₂×C₄

C₂×C₄⋊Dic₅₃

yes

characteristic

115

C₂×C₄

C₂×#11719913

yes

—

116

C₈

—

117

C₈

—

118

C₈

—

424

—

119

C₈

C₄⋊#88322

yes

—

120

C₈

C₂²×#88322

yes

characteristic

121

C₈

C₂×#11719917

yes

characteristic

122

C₂²

—

123

C₂²

#16810374

yes

—

124

C₂²

C₂×#13133149

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

#16810374

yes

—

128

C₄

C₂×C₄⋊#88322

yes

characteristic

129

C₄

C₂×#13133153

yes

characteristic

130

C₂

—

131

C₂

C₂×#16810374

yes

characteristic

132

C₂

#20276488

yes

—

	subgroup	G/H
center	C₂×C₄	#13133149
commutator	C₈×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#346	C₂
socle	C₂²×C₅₃	C₂×#106
prime radical	C₂³×C₅₃	#106
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₂×#346	D₅₃
hypocenter	C₅₃	#11999
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₄

#13133149

commutator

C₈×C₅₃

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₅₃

Fitting

C₂×C₅₃×#346

C₂

C₂²×C₅₃

C₂³×C₅₃

C₄×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
2. #20272781	C₂	semidirect
19. C₂×C₁₆×C₅₃	C₈	semidirect
78. C₅₃	#11999	semidirect
83. C₂×C₁₆	#88322	semidirect

H ≅ G/N

type

2. #20272781

C₂

semidirect

19. C₂×C₁₆×C₅₃

C₈

semidirect

78. C₅₃

#11999

semidirect

83. C₂×C₁₆

#88322

semidirect

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

subgroup	index
#20273463	2
#20272781	2
C₂×C₅₃×#346	2
#11999	53

subgroup

index

#20273463

#20272781

C₂×C₅₃×#346

#11999

Subgroups — #27295647

Conjugacy classes of subgroups

Complements

Maximal subgroups