Group #27298788 — subgroups

Subgroups — #27298788

Order 13568.

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

normal subgroups only

This group has 260 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	#20276944	C₂	1	2	yes	maximal, complemented
2	#20276940	C₂	1	2	yes	maximal
3	#20274321	C₂	1	1	yes	maximal, characteristic
4	C₈×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1710	C₂	1	1	yes	maximal, characteristic, complemented
6	#16811003	C₂²	1	1	yes	characteristic
7	#16811000	C₂²	1	1	yes	characteristic
8	#16810970	—	2	2	no	—
9	#16810426	—	2	4	no	—
10	#16810423	—	2	4	no	—
11	C₄×#11719908	C₂²	1	2	yes	—
12	C₅₃×#500	C₂²	1	2	yes	—
13	C₅₃×#487	—	2	2	no	—
14	C₈²×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₁₂	—	2	2	no	—
16	C₂×D₄₂₄	—	2	2	no	—
17	#13133217	—	2	1	no	—
18	#13133214	—	2	1	no	—
19	C₂×#11719908	D₄	1	4	yes	—
20	#13133139	—	4	4	no	—
21	C₄⋊Q₈×C₅₃	—	2	2	no	—
22	#107×C₅₃	—	2	2	no	—
23	#107×C₅₃	—	2	2	no	—
24	#103×C₅₃	—	4	4	no	—
25	C₄×C₈×C₅₃	D₄	1	2	yes	—
26	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
27	C₂×Q₈×C₅₃	—	4	2	no	—
28	C₂×D₂₁₂	—	4	1	no	—
29	C₂×D₂₁₂	—	4	2	no	—
30	C₂×Dic₁₀₆	—	4	1	no	—
31	C₂×C₈×C₅₃	D₈	1	4	yes	—
32	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
33	C₄⋊C₄×C₅₃	—	4	2	no	—
34	C₄⋊C₄×C₅₃	—	4	2	no	—
35	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₄⋊Dic₅₃	—	4	2	no	—
37	Dic₂₁₂	—	4	4	no	—
38	D₄₂₄	—	4	4	no	—
39	#11719908	—	2	4	no	—
40	C₂²×D₅₃	—	8	1	no	—
41	Q₈×C₅₃	—	8	4	no	—
42	C₂×C₄×C₅₃	—	8	2	no	—
43	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
45	C₂×Dic₅₃	—	8	1	no	—
46	D₂₁₂	—	8	2	no	—
47	D₂₁₂	—	8	4	no	—
48	Dic₁₀₆	—	8	2	no	—
49	C₈×C₅₃	—	2	4	no	—
50	C₈×C₅₃	#135	1	4	yes	—
51	#12511	—	53	1	no	maximal, Sylow
52	C₂²×C₅₃	#493	1	1	yes	characteristic
53	C₂×D₅₃	—	16	1	no	—
54	C₂×D₅₃	—	16	2	no	—
55	C₄×C₅₃	—	16	2	no	—
56	C₄×C₅₃	#495	1	2	yes	—
57	C₄×C₅₃	#508	1	4	yes	—
58	Dic₅₃	—	16	1	no	—
59	#2241	—	53	2	no	—
60	#2240	—	53	2	no	—
61	#1710	D₅₃	1	1	yes	characteristic, complemented
62	#1709	—	53	1	no	—
63	C₈×C₁₆	—	53	1	no	—
64	C₂×C₅₃	#1709	1	1	yes	characteristic
65	C₂×C₅₃	#2249	1	2	yes	—
66	D₅₃	—	32	1	no	—
67	#500	C₂×D₅₃	1	2	yes	—
68	#494	—	53	1	no	—
69	#493	—	53	1	no	—
70	#487	—	2	2	no	—
71	#486	—	106	2	no	—
72	#358	—	106	4	no	—
73	#357	—	106	4	no	—
74	C₄×C₁₆	—	53	2	no	—
75	C₈²	C₂×D₅₃	1	1	yes	characteristic
76	C₅₃	#12511	1	1	yes	characteristic, complemented, Sylow
77	C₂×Q₁₆	—	106	2	no	—
78	C₂×D₈	—	106	2	no	—
79	C₄⋊Q₈	—	2	2	no	—
80	C₄⋊Q₈	—	106	1	no	—
81	#127	—	106	1	no	—
82	C₂×C₁₆	—	53	4	no	—
83	#107	—	2	2	no	—
84	#107	—	2	2	no	—
85	#103	—	4	4	no	—
86	#102	—	212	4	no	—
87	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
88	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
89	C₂×Q₈	—	4	2	no	—
90	C₂×Q₈	—	212	1	no	—
91	C₂×D₄	—	212	1	no	—
92	C₂×D₄	—	212	2	no	—
93	Q₁₆	—	212	4	no	—
94	D₈	—	212	4	no	—
95	C₂×C₈	#11719922	1	4	yes	—
96	C₂×C₈	D₄×D₅₃	1	2	yes	—
97	C₄⋊C₄	—	4	2	no	—
98	C₄⋊C₄	—	4	2	no	—
99	C₄⋊C₄	—	212	2	no	—
100	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
101	C₁₆	—	106	4	no	—
102	C₂³	—	424	1	no	—
103	Q₈	—	8	4	no	—
104	Q₈	—	424	2	no	—
105	D₄	—	424	2	no	—
106	D₄	—	424	4	no	—
107	C₂×C₄	—	8	2	no	—
108	C₂×C₄	—	424	1	no	—
109	C₂×C₄	#13133213	1	1	yes	characteristic
110	C₂×C₄	C₂×#11719922	1	2	yes	—
111	C₈	—	2	4	no	—
112	C₈	#13133240	1	4	yes	—
113	C₂²	—	848	1	no	—
114	C₂²	—	848	2	no	—
115	C₂²	#16810999	1	1	yes	characteristic
116	C₄	—	16	2	no	—
117	C₄	—	848	1	no	—
118	C₄	#16811008	1	2	yes	—
119	C₄	#16811053	1	4	yes	—
120	C₂	—	1696	1	no	—
121	C₂	#20274322	1	1	yes	characteristic
122	C₂	#20276974	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20276944

C₂

yes

maximal, complemented

#20276940

C₂

yes

maximal

#20274321

C₂

yes

maximal, characteristic

C₈×#11719908

C₂

yes

maximal, characteristic, complemented

C₅₃×#1710

C₂

yes

maximal, characteristic, complemented

#16811003

C₂²

yes

characteristic

#16811000

C₂²

yes

—

—

—

C₂²

yes

—

C₅₃×#500

C₂²

yes

—

C₅₃×#487

—

C₈²×C₅₃

C₂²

yes

characteristic

C₂×Dic₂₁₂

—

C₂×D₄₂₄

—

#13133217

—

#13133214

—

C₂×#11719908

D₄

yes

—

#13133139

—

C₄⋊Q₈×C₅₃

—

#107×C₅₃

—

#107×C₅₃

—

#103×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄⋊Dic₅₃

—

Dic₂₁₂

—

D₄₂₄

—

#11719908

—

C₂²×D₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

#135

yes

—

—

C₂²×C₅₃

yes

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#495

yes

—

C₄×C₅₃

#508

yes

—

Dic₅₃

—

#2241

—

#2240

—

#1710

D₅₃

yes

characteristic, complemented

#1709

—

C₈×C₁₆

—

C₂×C₅₃

#1709

yes

characteristic

C₂×C₅₃

#2249

yes

—

D₅₃

—

#500

C₂×D₅₃

yes

—

—

—

—

—

106

—

#358

—

106

—

#357

—

106

—

C₄×C₁₆

—

C₈²

C₂×D₅₃

yes

characteristic

C₅₃

#12511

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

106

—

C₂×D₈

—

106

—

C₄⋊Q₈

—

C₄⋊Q₈

—

106

—

#127

—

106

—

C₂×C₁₆

—

—

—

—

—

212

—

C₄×C₈

C₂²⋊D₅₃

yes

—

C₄×C₈

C₂²×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

212

—

C₂×D₄

—

212

—

C₂×D₄

—

212

—

Q₁₆

—

212

—

D₈

—

212

—

C₂×C₈

#11719922

yes

—

C₂×C₈

D₄×D₅₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

212

—

100

C₄²

C₂×C₂²⋊D₅₃

yes

characteristic

101

C₁₆

—

106

—

102

C₂³

—

424

—

103

Q₈

—

104

Q₈

—

424

—

105

D₄

—

424

—

106

D₄

—

424

—

107

C₂×C₄

—

108

C₂×C₄

—

424

—

109

C₂×C₄

#13133213

yes

characteristic

110

C₂×C₄

C₂×#11719922

yes

—

111

C₈

—

112

C₈

#13133240

yes

—

113

C₂²

—

848

—

114

C₂²

—

848

—

115

C₂²

#16810999

yes

characteristic

116

C₄

—

117

C₄

—

848

—

118

C₄

#16811008

yes

—

119

C₄

#16811053

yes

—

120

C₂

—

1696

—

121

C₂

#20274322

yes

characteristic

122

C₂

#20276974

yes

—

	subgroup	G/H
center	C₂²	#16810999
commutator, cosocle	C₄×C₈×C₅₃	C₂³
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#1710	C₂
socle	C₂²×C₅₃	#493
prime radical	#16811000	C₂²
hypercenter	#1710	D₅₃
hypocenter	C₅₃	#12511
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

subgroup

G/H

center

C₂²

#16810999

commutator, cosocle

C₄×C₈×C₅₃

C₂³

Frattini

C₄×C₈

C₂²×D₅₃

Fitting

C₅₃×#1710

C₂

C₂²×C₅₃

C₂²

D₅₃

C₅₃

G ▷ C₄×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

N	H ≅ G/N	type
1. #20276944	C₂	semidirect
4. C₈×#11719908	C₂	semidirect
5. C₅₃×#1710	C₂	semidirect
61. #1710	D₅₃	semidirect
76. C₅₃	#12511	semidirect

H ≅ G/N

type

1. #20276944

C₂

semidirect

4. C₈×#11719908

C₂

semidirect

5. C₅₃×#1710

C₂

semidirect

61. #1710

D₅₃

semidirect

76. C₅₃

#12511

semidirect

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

subgroup	index
#20276944	2
#20276940	2
#20274321	2
C₈×#11719908	2
C₅₃×#1710	2
#12511	53

subgroup

index

C₅₃×#1710

Subgroups — #27298788

Conjugacy classes of subgroups

Complements

Maximal subgroups