Group #27295973 — subgroups

Subgroups — #27295973

Order 13568.

214 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 79 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	#20276587	C₂	1	1	yes	maximal, characteristic
2	#20273535	C₂	1	2	yes	maximal
3	#20272781	C₂	1	1	yes	maximal, characteristic
4	#20272686	C₂	1	2	yes	maximal
5	C₅₃×#1313	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₂₄	C₄	1	1	yes	characteristic
7	C₈×Dic₁₀₆	—	2	2	no	—
8	C₄×Dic₂₁₂	C₂²	1	2	yes	—
9	#16810451	C₄	1	1	yes	characteristic
10	#16810428	C₄	1	2	yes	—
11	#16810374	C₂²	1	2	yes	—
12	#16810355	—	2	2	no	—
13	C₅₃×#346	C₂²	1	1	yes	characteristic
14	C₅₃×#346	C₂²	1	1	yes	characteristic
15	C₅₃×#322	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₁₂	C₂×C₄	1	2	yes	—
17	C₄×Dic₁₀₆	—	2	2	no	—
18	Dic₄₂₄	C₈	1	4	yes	complemented
19	#13133151	C₂×C₄	1	2	yes	—
20	C₄⋊#88322	—	2	2	no	—
21	#13133146	—	4	2	no	—
22	#13133142	—	2	2	no	—
23	C₈×Dic₅₃	—	4	2	no	—
24	C₂×C₁₆×C₅₃	—	2	1	no	—
25	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₅₃	D₄	1	2	yes	—
28	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₀₆	—	2	2	no	—
30	C₂×C₈×C₅₃	—	2	2	no	—
31	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
33	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
34	C₄⋊Dic₅₃	—	2	2	no	—
35	#11719919	—	4	2	no	—
36	C₄×Dic₅₃	—	4	2	no	—
37	C₂×#88322	—	4	2	no	—
38	Dic₂₁₂	—	2	2	no	—
39	Dic₂₁₂	C₂×C₈	1	4	yes	—
40	C₁₆×C₅₃	—	2	1	no	—
41	C₁₆×C₅₃	—	4	1	no	—
42	C₁₆×C₅₃	C₂×C₈	1	2	yes	—
43	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
45	C₂×Dic₅₃	—	4	2	no	—
46	C₂×Dic₅₃	—	4	2	no	—
47	Dic₁₀₆	—	2	4	no	—
48	Dic₁₀₆	—	4	2	no	—
49	C₈×C₅₃	—	2	1	no	—
50	C₈×C₅₃	—	4	2	no	—
51	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
52	C₈×C₅₃	#131	1	1	yes	characteristic
53	#88322	—	8	2	no	—
54	#12066	—	53	1	no	maximal, Sylow
55	C₂²×C₅₃	#442	1	1	yes	characteristic
56	C₄×C₅₃	—	2	2	no	—
57	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
58	C₄×C₅₃	#444	1	1	yes	characteristic
59	Dic₅₃	—	4	4	no	—
60	Dic₅₃	—	8	2	no	—
61	#2176	—	53	1	no	—
62	#1579	—	53	2	no	—
63	#1368	—	53	1	no	—
64	#1334	—	53	2	no	—
65	#1313	D₅₃	1	1	yes	characteristic
66	C₂×C₅₃	#1580	1	1	yes	characteristic
67	C₂×C₅₃	#2180	1	2	yes	—
68	C₂×Q₃₂	—	53	1	no	—
69	C₈×Q₈	—	106	2	no	—
70	C₄×Q₁₆	—	53	2	no	—
71	#367	—	53	1	no	—
72	#358	—	53	2	no	—
73	#346	C₂×D₅₃	1	1	yes	characteristic
74	#346	C₂×D₅₃	1	1	yes	characteristic
75	#334	—	53	2	no	—
76	Q₈⋊C₈	—	106	2	no	—
77	#322	C₂×D₅₃	1	1	yes	characteristic
78	C₅₃	#12066	1	1	yes	characteristic, complemented, Sylow
79	C₂×Q₁₆	—	53	2	no	—
80	C₄×Q₈	—	106	2	no	—
81	Q₃₂	—	53	4	no	—
82	C₂×C₁₆	—	2	1	no	—
83	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
84	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
85	#107	—	53	2	no	—
86	C₄⋊C₈	—	106	2	no	—
87	C₄⋊C₈	—	212	2	no	—
88	#103	—	106	2	no	—
89	C₄×C₈	—	212	2	no	—
90	C₄×C₈	D₂₁₂	1	2	yes	—
91	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
92	C₂×Q₈	—	106	2	no	—
93	Q₁₆	—	53	4	no	—
94	Q₁₆	—	106	2	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	212	2	no	—
97	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
98	C₂×C₈	#11719945	1	1	yes	characteristic
99	C₄⋊C₄	—	106	2	no	—
100	C₄⋊C₄	—	212	2	no	—
101	C₄²	—	212	2	no	—
102	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
103	C₁₆	—	2	1	no	—
104	C₁₆	—	4	1	no	—
105	C₁₆	C₈×D₅₃	1	2	yes	—
106	Q₈	—	106	4	no	—
107	Q₈	—	212	2	no	—
108	C₂×C₄	—	212	2	no	—
109	C₂×C₄	—	212	2	no	—
110	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
111	C₂×C₄	#13133245	1	2	yes	—
112	C₈	—	2	1	no	—
113	C₈	—	4	2	no	—
114	C₈	—	424	2	no	—
115	C₈	C₂×C₈×D₅₃	1	1	yes	characteristic
116	C₈	#13133223	1	1	yes	characteristic
117	C₂²	#16810736	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	424	2	no	—
120	C₄	—	212	4	no	—
121	C₄	C₈×D₂₁₂	1	1	yes	characteristic
122	C₄	#16810746	1	1	yes	characteristic
123	C₂	#20273540	1	1	yes	characteristic
124	C₂	#20276608	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20276587

C₂

yes

maximal, characteristic

#20273535

C₂

yes

maximal

#20272781

C₂

yes

maximal, characteristic

#20272686

C₂

yes

maximal

C₅₃×#1313

C₂

yes

maximal, characteristic

C₂×Dic₄₂₄

C₄

yes

characteristic

C₈×Dic₁₀₆

—

C₄×Dic₂₁₂

C₂²

yes

—

#16810451

C₄

yes

characteristic

#16810428

C₄

yes

—

#16810374

C₂²

yes

—

#16810355

—

C₅₃×#346

C₂²

yes

characteristic

C₅₃×#346

C₂²

yes

characteristic

C₅₃×#322

C₂²

yes

characteristic

C₂×Dic₂₁₂

C₂×C₄

yes

—

C₄×Dic₁₀₆

—

Dic₄₂₄

C₈

yes

complemented

#13133151

C₂×C₄

yes

—

C₄⋊#88322

—

#13133146

—

#13133142

—

C₈×Dic₅₃

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

#41

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄⋊Dic₅₃

—

#11719919

—

C₄×Dic₅₃

—

C₂×#88322

—

Dic₂₁₂

—

Dic₂₁₂

C₂×C₈

yes

—

C₁₆×C₅₃

—

C₁₆×C₅₃

—

C₁₆×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

C₄×D₄

yes

characteristic

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₂²×C₈

yes

C₈×C₅₃

yes

—

—

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

C₈×D₄

yes

characteristic

C₄×C₅₃

#444

yes

characteristic

Dic₅₃

—

Dic₅₃

—

—

—

—

—

D₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×Q₃₂

—

C₈×Q₈

—

106

—

C₄×Q₁₆

—

#367

—

#358

—

#346

C₂×D₅₃

yes

characteristic

#346

C₂×D₅₃

yes

characteristic

#334

—

Q₈⋊C₈

—

106

—

#322

C₂×D₅₃

yes

characteristic

C₅₃

#12066

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

106

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₅₃

yes

characteristic

C₂×C₁₆

C₄×D₅₃

yes

characteristic

#107

—

C₄⋊C₈

—

106

—

C₄⋊C₈

—

212

—

#103

—

106

—

C₄×C₈

—

212

—

C₄×C₈

D₂₁₂

yes

—

C₄×C₈

C₂²×D₅₃

yes

characteristic

C₂×Q₈

—

106

—

Q₁₆

—

Q₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

212

—

C₂×C₈

C₂×C₄×D₅₃

yes

characteristic

C₂×C₈

#11719945

yes

characteristic

C₄⋊C₄

—

106

—

100

C₄⋊C₄

—

212

—

101

C₄²

—

212

—

102

C₄²

C₂×D₂₁₂

yes

characteristic

103

C₁₆

—

104

C₁₆

—

105

C₁₆

C₈×D₅₃

yes

—

106

Q₈

—

106

—

107

Q₈

—

212

—

108

C₂×C₄

—

212

—

109

C₂×C₄

—

212

—

110

C₂×C₄

C₄×D₂₁₂

yes

characteristic

111

C₂×C₄

#13133245

yes

—

112

C₈

—

113

C₈

—

114

C₈

—

424

—

115

C₈

C₂×C₈×D₅₃

yes

characteristic

116

C₈

#13133223

yes

characteristic

117

C₂²

#16810736

yes

characteristic

118

C₄

—

119

C₄

—

424

—

120

C₄

—

212

—

121

C₄

C₈×D₂₁₂

yes

characteristic

122

C₄

#16810746

yes

characteristic

123

C₂

#20273540

yes

characteristic

124

C₂

#20276608

yes

—

	subgroup	G/H
center	C₂²	#16810736
commutator	C₈×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#1313	C₂
socle, prime radical	C₂²×C₅₃	#442
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#1313	D₅₃
hypocenter	C₅₃	#12066
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322 ◁ #1313

subgroup

G/H

center

C₂²

#16810736

commutator

C₈×C₅₃

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₅₃

Fitting

C₅₃×#1313

C₂

socle, prime radical

C₂²×C₅₃

#442

cosocle

C₄×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322 ◁ #1313

N	H ≅ G/N	type
18. Dic₄₂₄	C₈	semidirect
78. C₅₃	#12066	semidirect

H ≅ G/N

type

18. Dic₄₂₄

C₈

semidirect

78. C₅₃

#12066

semidirect

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

subgroup	index
#20276587	2
#20273535	2
#20272781	2
#20272686	2
C₅₃×#1313	2
#12066	53

subgroup

index

C₅₃×#1313

Subgroups — #27295973

Conjugacy classes of subgroups

Complements

Maximal subgroups