Group #27295981 — subgroups

Subgroups — #27295981

Order 13568.

204 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 73 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	#20276602	C₂	1	1	yes	maximal, characteristic
2	#20276556	C₂	1	2	yes	maximal, complemented
3	#20272690	C₂	1	2	yes	maximal, complemented
4	#20272635	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1369	C₂	1	1	yes	maximal, characteristic, complemented
6	#16811055	C₄	1	1	yes	characteristic
7	#16811041	—	2	2	no	—
8	#16810741	C₂²	1	2	yes	—
9	#16810453	C₄	1	1	yes	characteristic
10	#16810431	C₄	1	2	yes	—
11	#16810408	—	2	2	no	—
12	#16810399	C₂²	1	1	yes	characteristic
13	#16810344	C₂²	1	1	yes	characteristic
14	C₅₃×#364	C₂²	1	2	yes	—
15	C₅₃×#346	C₂²	1	1	yes	characteristic
16	#13133239	C₂×C₄	1	2	yes	—
17	#13133223	—	2	2	no	—
18	Dic₄₂₄	C₈	1	1	yes	characteristic, complemented
19	#13133162	C₈	1	2	yes	complemented
20	D₈₄₈	C₈	1	1	yes	characteristic, complemented
21	#13133156	—	4	2	no	—
22	C₂×#11719908	D₄	1	2	yes	—
23	C₁₆×D₅₃	—	4	2	no	—
24	#13133153	C₂×C₄	1	2	yes	—
25	#13133145	—	2	2	no	—
26	C₄×#88322	—	2	1	no	—
27	M₃₂×C₅₃	—	2	2	no	—
28	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
31	#11719945	—	2	2	no	—
32	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
34	C₄²×C₅₃	#41	1	1	yes	characteristic
35	#11719917	—	2	2	no	—
36	C₂×#88322	—	4	1	no	—
37	Dic₂₁₂	C₂×C₈	1	2	yes	—
38	D₄₂₄	C₂×C₈	1	2	yes	—
39	#11719913	—	2	2	no	—
40	#11719912	—	4	2	no	—
41	C₈×D₅₃	—	4	2	no	—
42	C₁₆×C₅₃	—	2	1	no	—
43	C₁₆×C₅₃	—	4	2	no	—
44	C₁₆×C₅₃	C₂×C₈	1	1	yes	characteristic
45	C₁₆×C₅₃	C₂×C₈	1	1	yes	characteristic
46	#11719908	—	2	2	no	—
47	C₂×C₄×C₅₃	—	2	1	no	—
48	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
49	C₂²⋊D₅₃	—	4	2	no	—
50	D₂₁₂	—	2	2	no	—
51	C₄×D₅₃	—	4	2	no	—
52	Dic₁₀₆	—	2	2	no	—
53	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
54	C₈×C₅₃	#131	1	1	yes	characteristic
55	C₈×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
56	#88322	—	4	2	no	—
57	#88322	—	8	1	no	—
58	#12068	—	53	1	no	maximal, Sylow
59	C₂²×C₅₃	#444	1	1	yes	characteristic
60	C₂×D₅₃	—	4	2	no	—
61	C₄×C₅₃	—	4	1	no	—
62	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
63	C₄×C₅₃	#442	1	1	yes	characteristic
64	Dic₅₃	—	4	2	no	—
65	#2179	—	53	1	no	—
66	#2171	—	53	2	no	—
67	#1369	D₅₃	1	1	yes	characteristic, complemented
68	#1336	—	53	2	no	—
69	#1313	—	53	1	no	—
70	C₂×C₅₃	—	2	1	no	—
71	C₂×C₅₃	#1580	1	1	yes	characteristic
72	D₅₃	—	8	2	no	—
73	#508	—	53	1	no	—
74	#504	—	106	2	no	—
75	#443	—	53	2	no	—
76	#368	—	53	1	no	—
77	#364	C₂×D₅₃	1	2	yes	—
78	#359	—	53	2	no	—
79	#350	—	106	2	no	—
80	#346	—	53	1	no	—
81	#346	C₂×D₅₃	1	1	yes	characteristic
82	#322	—	53	1	no	—
83	C₅₃	#12068	1	1	yes	characteristic, complemented, Sylow
84	#135	—	53	2	no	—
85	#131	—	106	2	no	—
86	Q₃₂	—	53	1	no	—
87	SD₃₂	—	53	2	no	—
88	D₁₆	—	53	1	no	—
89	M₃₂	—	2	2	no	—
90	M₃₂	—	212	2	no	—
91	C₂×C₁₆	—	53	2	no	—
92	C₂×C₁₆	—	212	2	no	—
93	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
94	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
95	#108	—	53	2	no	—
96	#104	—	106	2	no	—
97	C₄×C₈	—	106	1	no	—
98	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
99	#41	—	106	2	no	—
100	Q₁₆	—	53	2	no	—
101	SD₁₆	—	106	2	no	—
102	D₈	—	53	2	no	—
103	M₁₆	—	106	2	no	—
104	M₁₆	—	212	2	no	—
105	C₂×C₈	—	212	1	no	—
106	C₂×C₈	—	212	2	no	—
107	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
108	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
109	C₄²	#11719949	1	1	yes	characteristic
110	C₁₆	—	2	1	no	—
111	C₁₆	—	4	2	no	—
112	C₁₆	—	106	2	no	—
113	C₁₆	C₈×D₅₃	1	1	yes	characteristic
114	C₁₆	C₈×D₅₃	1	1	yes	characteristic
115	Q₈	—	106	2	no	—
116	D₄	—	106	2	no	—
117	D₄	—	212	2	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	212	2	no	—
120	C₂×C₄	#13133179	1	1	yes	characteristic
121	C₈	—	424	1	no	—
122	C₈	—	212	2	no	—
123	C₈	C₂×C₈×D₅₃	1	1	yes	characteristic
124	C₈	#13133225	1	1	yes	characteristic
125	C₈	#13133244	1	2	yes	—
126	C₂²	—	212	2	no	—
127	C₂²	#16810744	1	1	yes	characteristic
128	C₄	—	4	1	no	—
129	C₄	—	212	2	no	—
130	C₄	#16810688	1	1	yes	characteristic
131	C₄	#16810734	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	424	2	no	—
134	C₂	#20273538	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276602

C₂

yes

maximal, characteristic

#20276556

C₂

yes

maximal, complemented

#20272690

C₂

yes

maximal, complemented

#20272635

C₂

yes

maximal, characteristic, complemented

C₅₃×#1369

C₂

yes

maximal, characteristic, complemented

#16811055

C₄

yes

characteristic

#16811041

—

#16810741

C₂²

yes

—

#16810453

C₄

yes

characteristic

#16810431

C₄

yes

—

#16810408

—

#16810399

C₂²

yes

characteristic

#16810344

C₂²

yes

characteristic

C₅₃×#364

C₂²

yes

—

C₅₃×#346

C₂²

yes

characteristic

#13133239

C₂×C₄

yes

—

#13133223

—

Dic₄₂₄

C₈

yes

characteristic, complemented

#13133162

C₈

yes

complemented

D₈₄₈

C₈

yes

characteristic, complemented

#13133156

—

C₂×#11719908

D₄

yes

—

C₁₆×D₅₃

—

#13133153

C₂×C₄

yes

—

#13133145

—

C₄×#88322

—

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂³

yes

characteristic

#11719945

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

C₄²×C₅₃

yes

—

—

Dic₂₁₂

C₂×C₈

yes

—

D₄₂₄

C₂×C₈

yes

—

#11719913

—

#11719912

—

C₈×D₅₃

—

C₁₆×C₅₃

—

C₁₆×C₅₃

—

C₁₆×C₅₃

C₂×C₈

yes

characteristic

C₁₆×C₅₃

C₂×C₈

yes

characteristic

#11719908

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄×D₄

yes

characteristic

C₂²⋊D₅₃

—

D₂₁₂

—

C₄×D₅₃

—

Dic₁₀₆

—

C₈×C₅₃

C₂²×C₈

yes

C₈×C₅₃

yes

C₈×C₅₃

yes

—

—

—

—

C₂²×C₅₃

yes

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₈×D₄

yes

characteristic

C₄×C₅₃

#442

yes

characteristic

Dic₅₃

—

#2179

—

#2171

—

#1369

D₅₃

yes

characteristic, complemented

#1336

—

#1313

—

C₂×C₅₃

—

C₂×C₅₃

#1580

yes

characteristic

D₅₃

—

#508

—

#504

—

106

—

#443

—

#368

—

#364

C₂×D₅₃

yes

—

#359

—

#350

—

106

—

#346

—

#346

C₂×D₅₃

yes

characteristic

#322

—

C₅₃

#12068

yes

characteristic, complemented, Sylow

#135

—

#131

—

106

—

Q₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

—

M₃₂

—

212

—

C₂×C₁₆

—

C₂×C₁₆

—

212

—

C₂×C₁₆

C₄×D₅₃

yes

characteristic

C₂×C₁₆

C₄×D₅₃

yes

characteristic

#108

—

#104

—

106

—

C₄×C₈

—

106

—

C₄×C₈

C₂²×D₅₃

yes

characteristic

#41

—

106

—

100

Q₁₆

—

101

SD₁₆

—

106

—

102

D₈

—

103

M₁₆

—

106

—

104

M₁₆

—

212

—

105

C₂×C₈

—

212

—

106

C₂×C₈

—

212

—

107

C₂×C₈

C₂×C₄×D₅₃

yes

characteristic

108

C₂×C₈

D₄×D₅₃

yes

characteristic

109

C₄²

#11719949

yes

characteristic

110

C₁₆

—

111

C₁₆

—

112

C₁₆

—

106

—

113

C₁₆

C₈×D₅₃

yes

characteristic

114

C₁₆

C₈×D₅₃

yes

characteristic

115

Q₈

—

106

—

116

D₄

—

106

—

117

D₄

—

212

—

118

C₂×C₄

—

119

C₂×C₄

—

212

—

120

C₂×C₄

#13133179

yes

characteristic

121

C₈

—

424

—

122

C₈

—

212

—

123

C₈

C₂×C₈×D₅₃

yes

characteristic

124

C₈

#13133225

yes

characteristic

125

C₈

#13133244

yes

—

126

C₂²

—

212

—

127

C₂²

#16810744

yes

characteristic

128

C₄

—

129

C₄

—

212

—

130

C₄

#16810688

yes

characteristic

131

C₄

#16810734

yes

characteristic

132

C₂

—

133

C₂

—

424

—

134

C₂

#20273538

yes

characteristic

	subgroup	G/H
center	C₄	#16810734
commutator	C₈×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#1369	C₂
socle	C₂×C₅₃	#1580
prime radical	#16811055	C₄
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#1369	D₅₃
hypocenter	C₅₃	#12068
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1369

subgroup

G/H

center

C₄

#16810734

commutator

C₈×C₅₃

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₅₃

Fitting

C₅₃×#1369

C₂

C₂×C₅₃

C₄

C₄×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1369

N	H ≅ G/N	type
2. #20276556	C₂	semidirect
3. #20272690	C₂	semidirect
4. #20272635	C₂	semidirect
5. C₅₃×#1369	C₂	semidirect
18. Dic₄₂₄	C₈	semidirect
19. #13133162	C₈	semidirect
20. D₈₄₈	C₈	semidirect
67. #1369	D₅₃	semidirect
83. C₅₃	#12068	semidirect

H ≅ G/N

type

2. #20276556

C₂

semidirect

3. #20272690

C₂

semidirect

4. #20272635

C₂

semidirect

5. C₅₃×#1369

C₂

semidirect

18. Dic₄₂₄

C₈

semidirect

19. #13133162

C₈

semidirect

20. D₈₄₈

C₈

semidirect

67. #1369

D₅₃

semidirect

83. C₅₃

#12068

semidirect

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

subgroup	index
#20276602	2
#20276556	2
#20272690	2
#20272635	2
C₅₃×#1369	2
#12068	53

subgroup

index

C₅₃×#1369

Subgroups — #27295981

Conjugacy classes of subgroups

Complements

Maximal subgroups