Group #27296111 — subgroups

Subgroups — #27296111

Order 13568.

198 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 69 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	#20273623	C₂	1	2	yes	maximal
3	#20272684	C₂	1	2	yes	maximal
4	#20272635	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1368	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₂₄	C₄	1	1	yes	characteristic
7	#16810750	—	2	2	no	—
8	C₄×Dic₂₁₂	C₂²	1	2	yes	—
9	#16810451	C₄	1	1	yes	characteristic
10	#16810428	C₄	1	2	yes	—
11	#16810399	C₂²	1	1	yes	characteristic
12	#16810353	—	2	2	no	—
13	#16810344	C₂²	1	1	yes	characteristic
14	C₅₃×#346	C₂²	1	1	yes	characteristic
15	C₅₃×#334	C₂²	1	2	yes	—
16	C₂×Dic₂₁₂	C₂×C₄	1	2	yes	—
17	C₄×Dic₁₀₆	—	2	2	no	—
18	Dic₄₂₄	—	2	2	no	—
19	C₂×#11719908	—	2	1	no	—
20	#13133151	C₂×C₄	1	2	yes	—
21	#13133146	—	4	2	no	—
22	#13133142	—	2	2	no	—
23	#13133127	—	4	2	no	—
24	C₄×#88322	D₄	1	2	yes	—
25	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₄⋊C₈×C₅₃	—	2	2	no	—
28	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₀₆	—	2	2	no	—
30	C₂×C₈×C₅₃	—	4	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	—	2	2	no	—
38	Dic₂₁₂	—	2	2	no	—
39	Dic₂₁₂	—	2	2	no	—
40	C₁₆×C₅₃	—	2	1	no	—
41	C₁₆×C₅₃	M₁₆	1	2	yes	—
42	#11719908	—	4	1	no	—
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	—	4	2	no	—
48	Dic₁₀₆	—	4	2	no	—
49	C₈×C₅₃	—	2	1	no	—
50	C₈×C₅₃	—	8	2	no	—
51	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
52	C₈×C₅₃	#131	1	1	yes	characteristic
53	#88322	—	4	2	no	—
54	#12088	—	53	1	no	maximal, Sylow
55	C₂²×C₅₃	#442	1	1	yes	characteristic
56	C₄×C₅₃	—	2	2	no	—
57	C₄×C₅₃	#436	1	1	yes	characteristic
58	C₄×C₅₃	#444	1	1	yes	characteristic
59	Dic₅₃	—	8	2	no	—
60	Dic₅₃	—	8	2	no	—
61	#2176	—	53	1	no	—
62	#1593	—	53	2	no	—
63	#1368	D₅₃	1	1	yes	characteristic
64	#1334	—	53	2	no	—
65	#1313	—	53	1	no	—
66	C₂×C₅₃	#1594	1	1	yes	characteristic
67	C₂×C₅₃	#2180	1	2	yes	—
68	C₂×Q₃₂	—	53	1	no	—
69	#446	—	106	2	no	—
70	C₄×Q₁₆	—	53	2	no	—
71	#367	—	53	1	no	—
72	#358	—	53	2	no	—
73	#346	—	53	1	no	—
74	#346	C₂×D₅₃	1	1	yes	characteristic
75	#334	C₂×D₅₃	1	2	yes	—
76	Q₈⋊C₈	—	106	2	no	—
77	#322	—	53	1	no	—
78	C₅₃	#12088	1	1	yes	characteristic, complemented, Sylow
79	C₂×Q₁₆	—	53	2	no	—
80	C₄×Q₈	—	106	2	no	—
81	Q₃₂	—	106	2	no	—
82	C₂×C₁₆	—	106	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	—	2	2	no	—
87	C₄⋊C₈	—	212	2	no	—
88	#103	—	106	2	no	—
89	#97	—	212	2	no	—
90	C₄×C₈	—	53	2	no	—
91	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
92	C₂×Q₈	—	106	2	no	—
93	Q₁₆	—	106	2	no	—
94	Q₁₆	—	106	2	no	—
95	C₂×C₈	—	4	2	no	—
96	C₂×C₈	—	106	2	no	—
97	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
98	C₂×C₈	#11719949	1	1	yes	characteristic
99	C₄⋊C₄	—	106	2	no	—
100	C₄⋊C₄	—	212	2	no	—
101	C₄²	—	212	2	no	—
102	C₄²	D₄×D₅₃	1	1	yes	characteristic
103	C₁₆	—	2	1	no	—
104	C₁₆	—	212	1	no	—
105	C₁₆	#11719912	1	2	yes	—
106	Q₈	—	212	2	no	—
107	Q₈	—	212	2	no	—
108	C₂×C₄	—	212	2	no	—
109	C₂×C₄	—	212	2	no	—
110	C₂×C₄	#13133179	1	1	yes	characteristic
111	C₂×C₄	#13133244	1	2	yes	—
112	C₈	—	2	1	no	—
113	C₈	—	8	2	no	—
114	C₈	—	212	2	no	—
115	C₈	C₂×#11719912	1	1	yes	characteristic
116	C₈	#13133225	1	1	yes	characteristic
117	C₂²	#16810734	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	424	2	no	—
120	C₄	—	424	2	no	—
121	C₄	#16810700	1	1	yes	characteristic
122	C₄	#16810744	1	1	yes	characteristic
123	C₂	#20273628	1	1	yes	characteristic
124	C₂	#20276605	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20276587

C₂

yes

maximal, characteristic

#20273623

C₂

yes

maximal

#20272684

C₂

yes

maximal

#20272635

C₂

yes

maximal, characteristic

C₅₃×#1368

C₂

yes

maximal, characteristic

C₂×Dic₄₂₄

C₄

yes

characteristic

#16810750

—

C₄×Dic₂₁₂

C₂²

yes

—

#16810451

C₄

yes

characteristic

#16810428

C₄

yes

—

#16810399

C₂²

yes

characteristic

#16810353

—

#16810344

C₂²

yes

characteristic

C₅₃×#346

C₂²

yes

characteristic

C₅₃×#334

C₂²

yes

—

C₂×Dic₂₁₂

C₂×C₄

yes

—

C₄×Dic₁₀₆

—

Dic₄₂₄

—

C₂×#11719908

—

#13133151

C₂×C₄

yes

—

—

—

—

D₄

yes

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₄⋊C₈×C₅₃

—

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

—

C₁₆×C₅₃

—

C₁₆×C₅₃

M₁₆

yes

—

#11719908

—

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

C₂×M₁₆

yes

C₈×C₅₃

yes

—

—

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

#2176

—

#1593

—

#1368

D₅₃

yes

characteristic

#1334

—

#1313

—

C₂×C₅₃

#1594

yes

characteristic

C₂×C₅₃

#2180

yes

—

C₂×Q₃₂

—

#446

—

106

—

C₄×Q₁₆

—

—

—

—

C₂×D₅₃

yes

characteristic

#334

C₂×D₅₃

yes

—

Q₈⋊C₈

—

106

—

#322

—

C₅₃

#12088

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

106

—

Q₃₂

—

106

—

C₂×C₁₆

—

106

—

C₂×C₁₆

C₄×D₅₃

yes

characteristic

C₂×C₁₆

C₄×D₅₃

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

212

—

#103

—

106

—

#97

—

212

—

C₄×C₈

—

C₄×C₈

C₂²×D₅₃

yes

characteristic

C₂×Q₈

—

106

—

Q₁₆

—

106

—

Q₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

C₂×C₄×D₅₃

yes

characteristic

C₂×C₈

#11719949

yes

characteristic

C₄⋊C₄

—

106

—

100

C₄⋊C₄

—

212

—

101

C₄²

—

212

—

102

C₄²

D₄×D₅₃

yes

characteristic

103

C₁₆

—

104

C₁₆

—

212

—

105

C₁₆

#11719912

yes

—

106

Q₈

—

212

—

107

Q₈

—

212

—

108

C₂×C₄

—

212

—

109

C₂×C₄

—

212

—

110

C₂×C₄

#13133179

yes

characteristic

111

C₂×C₄

#13133244

yes

—

112

C₈

—

113

C₈

—

114

C₈

—

212

—

115

C₈

C₂×#11719912

yes

characteristic

116

C₈

#13133225

yes

characteristic

117

C₂²

#16810734

yes

characteristic

118

C₄

—

119

C₄

—

424

—

120

C₄

—

424

—

121

C₄

#16810700

yes

characteristic

122

C₄

#16810744

yes

characteristic

123

C₂

#20273628

yes

characteristic

124

C₂

#20276605

yes

—

	subgroup	G/H
center	C₂²	#16810734
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#1368	C₂
socle, prime radical	C₂²×C₅₃	#442
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#1368	D₅₃
hypocenter	C₅₃	#12088
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1368

subgroup

G/H

center

C₂²

#16810734

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₅₃

Fitting

C₅₃×#1368

C₂

socle, prime radical

C₂²×C₅₃

#442

cosocle

C₄×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1368

N	H ≅ G/N	type
78. C₅₃	#12088	semidirect

H ≅ G/N

type

78. C₅₃

#12088

semidirect

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

subgroup	index
#20276587	2
#20273623	2
#20272684	2
#20272635	2
C₅₃×#1368	2
#12088	53

subgroup

index

C₅₃×#1368

Subgroups — #27296111

Conjugacy classes of subgroups

Complements

Maximal subgroups