Group #27295728 — subgroups

Subgroups — #27295728

Order 13568.

232 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 123 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	#20273471	C₂	1	2	yes	maximal
2	#20272781	C₂	1	4	yes	maximal, complemented
3	C₄×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
4	#16810647	—	2	2	no	—
5	#16810374	—	2	2	no	—
6	#16810374	C₂²	1	4	yes	—
7	C₂×M₃₂×C₅₃	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#346	C₂²	1	2	yes	—
10	C₄×C₁₆×C₅₃	C₄	1	1	yes	characteristic
11	C₄×C₁₆×C₅₃	C₄	1	1	yes	characteristic
12	C₂×#11719917	—	4	2	no	—
13	C₄⋊#88322	—	2	4	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	M₃₂×C₅₃	—	2	4	no	—
19	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
20	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
23	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
24	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
27	C₂×C₈×C₅₃	—	2	2	no	—
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	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
33	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
34	C₄²×C₅₃	SD₁₆	1	2	yes	—
35	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
36	#11719917	—	8	4	no	—
37	C₂×#88322	—	4	4	no	—
38	C₁₆×C₅₃	—	2	2	no	—
39	C₁₆×C₅₃	M₁₆	1	4	yes	complemented
40	C₂³×C₅₃	#106	1	1	yes	characteristic
41	C₂×C₄×C₅₃	—	2	4	no	—
42	C₂×C₄×C₅₃	—	2	1	no	—
43	C₂×C₄×C₅₃	—	2	1	no	—
44	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
45	C₂×C₄×C₅₃	#106	1	2	yes	—
46	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
48	C₈×C₅₃	—	2	1	no	—
49	C₈×C₅₃	—	2	1	no	—
50	C₈×C₅₃	#108	1	2	yes	—
51	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
52	#88322	—	8	4	no	—
53	#12020	—	53	1	no	maximal, Sylow
54	C₂²×C₅₃	—	2	1	no	—
55	C₂²×C₅₃	—	2	2	no	—
56	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
57	C₄×C₅₃	—	2	1	no	—
58	C₄×C₅₃	—	2	2	no	—
59	C₄×C₅₃	#365	1	4	yes	—
60	C₄×C₅₃	#423	1	1	yes	characteristic
61	C₄×C₅₃	C₂×#108	1	1	yes	characteristic
62	C₄×M₃₂	D₅₃	1	1	yes	characteristic
63	#1566	—	53	2	no	—
64	#1368	—	53	4	no	—
65	C₂×C₅₃	—	4	1	no	—
66	C₂×C₅₃	#1566	1	1	yes	characteristic
67	C₂×C₅₃	C₂×#365	1	2	yes	—
68	C₂×M₃₂	Dic₅₃	1	2	yes	—
69	#423	—	106	2	no	—
70	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
71	#346	C₂×D₅₃	1	2	yes	—
72	C₄×C₁₆	Dic₅₃	1	1	yes	characteristic
73	C₄×C₁₆	Dic₅₃	1	1	yes	characteristic
74	#334	—	53	4	no	—
75	#334	—	106	2	no	—
76	C₅₃	#12020	1	1	yes	characteristic, complemented, Sylow
77	C₂×M₁₆	—	212	2	no	—
78	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
79	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
80	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
81	M₃₂	—	2	4	no	—
82	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
83	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
84	C₄⋊C₈	—	106	4	no	—
85	C₄⋊C₈	—	212	2	no	—
86	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
87	C₄×C₈	D₂₁₂	1	1	yes	characteristic
88	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
89	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
90	C₂²×C₄	#11719913	1	2	yes	—
91	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
92	M₁₆	—	424	4	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	212	4	no	—
96	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
97	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
98	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
99	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
100	C₄²	#11719913	1	2	yes	—
101	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
102	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
103	C₁₆	—	2	2	no	—
104	C₁₆	#11719917	1	4	yes	complemented
105	C₂³	#13133149	1	1	yes	characteristic
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	#13133149	1	1	yes	characteristic
110	C₂×C₄	#13133149	1	2	yes	—
111	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
112	C₂×C₄	C₂×#11719913	1	2	yes	—
113	C₈	—	2	1	no	—
114	C₈	—	2	1	no	—
115	C₈	—	424	4	no	—
116	C₈	#13133153	1	2	yes	—
117	C₈	C₂×#11719917	1	2	yes	—
118	C₂²	—	2	2	no	—
119	C₂²	—	2	1	no	—
120	C₂²	C₂×#13133149	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	2	1	no	—
123	C₄	#16810446	1	4	yes	—
124	C₄	#16810647	1	1	yes	characteristic
125	C₄	C₂×#13133153	1	1	yes	characteristic
126	C₂	—	4	1	no	—
127	C₂	#20273471	1	1	yes	characteristic
128	C₂	C₂×#16810446	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20273471

C₂

yes

maximal

#20272781

C₂

yes

maximal, complemented

C₄×M₃₂×C₅₃

C₂

yes

maximal, characteristic

#16810647

—

#16810374

—

#16810374

C₂²

yes

—

C₂×M₃₂×C₅₃

C₄

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#346

C₂²

yes

—

C₄×C₁₆×C₅₃

C₄

yes

characteristic

C₄×C₁₆×C₅₃

C₄

yes

—

—

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

Q₈

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

characteristic

M₃₂×C₅₃

—

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

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

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

#11719917

—

C₂×#88322

—

C₁₆×C₅₃

—

C₁₆×C₅₃

M₁₆

yes

complemented

C₂³×C₅₃

#106

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#106

yes

characteristic

C₂×C₄×C₅₃

#106

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#108

yes

—

C₈×C₅₃

C₂×M₁₆

yes

—

#88322

—

#12020

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×#106

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#365

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×M₃₂

D₅₃

yes

characteristic

#1566

—

#1368

—

C₂×C₅₃

—

C₂×C₅₃

#1566

yes

characteristic

C₂×C₅₃

C₂×#365

yes

—

C₂×M₃₂

Dic₅₃

yes

—

#423

—

106

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

#346

C₂×D₅₃

yes

—

C₄×C₁₆

Dic₅₃

yes

characteristic

C₄×C₁₆

Dic₅₃

yes

characteristic

#334

—

#334

—

106

—

C₅₃

#12020

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

212

—

C₂²×C₈

Dic₁₀₆

yes

characteristic

C₂²×C₈

C₂×Dic₅₃

yes

characteristic

C₂×C₄²

D₂₁₂

yes

characteristic

M₃₂

—

C₂×C₁₆

C₂×Dic₅₃

yes

—

C₂×C₁₆

C₂×Dic₅₃

yes

—

C₄⋊C₈

—

106

—

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

M₁₆

—

424

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

212

—

C₂×C₈

C₄⋊Dic₅₃

yes

characteristic

C₂×C₈

C₄⋊Dic₅₃

yes

characteristic

C₂×C₈

C₂×Dic₁₀₆

yes

characteristic

C₂×C₈

C₂²×Dic₅₃

yes

characteristic

100

C₄²

#11719913

yes

—

101

C₄²

C₄⋊Dic₅₃

yes

characteristic

102

C₄²

C₂×D₂₁₂

yes

characteristic

103

C₁₆

—

104

C₁₆

#11719917

yes

complemented

105

C₂³

#13133149

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

#13133149

yes

characteristic

110

C₂×C₄

#13133149

yes

—

111

C₂×C₄

C₂×C₄⋊Dic₅₃

yes

characteristic

112

C₂×C₄

C₂×#11719913

yes

—

113

C₈

—

114

C₈

—

115

C₈

—

424

—

116

C₈

#13133153

yes

—

117

C₈

C₂×#11719917

yes

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂×#13133149

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

#16810446

yes

—

124

C₄

#16810647

yes

characteristic

125

C₄

C₂×#13133153

yes

characteristic

126

C₂

—

127

C₂

#20273471

yes

characteristic

128

C₂

C₂×#16810446

yes

—

	subgroup	G/H
center	C₂²	C₂×#13133149
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₄×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#106
prime radical	C₂³×C₅₃	#106
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₄×M₃₂	D₅₃
hypocenter	C₅₃	#12020
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

subgroup

G/H

center

C₂²

C₂×#13133149

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₅₃

Fitting

C₄×M₃₂×C₅₃

C₂

C₂²×C₅₃

C₂³×C₅₃

C₄×C₈×C₅₃

C₂³

hypercenter

C₄×M₃₂

D₅₃

hypocenter

C₅₃

#12020

LCS

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

N	H ≅ G/N	type
2. #20272781	C₂	semidirect
39. C₁₆×C₅₃	M₁₆	semidirect
76. C₅₃	#12020	semidirect
104. C₁₆	#11719917	semidirect

H ≅ G/N

type

2. #20272781

C₂

semidirect

39. C₁₆×C₅₃

M₁₆

semidirect

76. C₅₃

#12020

semidirect

104. C₁₆

#11719917

semidirect

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

subgroup	index
#20273471	2
#20272781	2
C₄×M₃₂×C₅₃	2
#12020	53

subgroup

index

#20273471

#20272781

C₄×M₃₂×C₅₃

#12020

Subgroups — #27295728

Conjugacy classes of subgroups

Complements

Maximal subgroups