Group #27298785 — subgroups

Subgroups — #27298785

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	#20276942	C₂	1	2	yes	maximal, complemented
2	#20276938	C₂	1	2	yes	maximal
3	#20274325	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1709	C₂	1	1	yes	maximal, characteristic
6	#16811028	C₂²	1	2	yes	—
7	#16810974	—	2	2	no	—
8	#16810427	—	2	4	no	—
9	#16810424	—	2	4	no	—
10	C₄×#11719908	C₂²	1	2	yes	—
11	C₅₃×#494	C₂²	1	1	yes	characteristic
12	C₅₃×#493	C₂²	1	1	yes	characteristic
13	C₅₃×#486	—	2	2	no	—
14	C₈²×C₅₃	C₂²	1	1	yes	characteristic
15	#13133217	—	2	2	no	—
16	C₂×#11719908	D₄	1	4	yes	—
17	#13133151	—	2	2	no	—
18	#13133151	—	2	2	no	—
19	#13133142	—	4	4	no	—
20	C₂×Q₁₆×C₅₃	—	2	2	no	—
21	C₂×D₈×C₅₃	—	2	2	no	—
22	C₄⋊Q₈×C₅₃	—	2	1	no	—
23	#127×C₅₃	—	2	1	no	—
24	#102×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	1	no	—
28	C₂×D₄×C₅₃	—	4	1	no	—
29	C₂×D₄×C₅₃	—	4	2	no	—
30	C₂×Dic₁₀₆	—	4	2	no	—
31	Q₁₆×C₅₃	—	4	4	no	—
32	D₈×C₅₃	—	4	4	no	—
33	C₂×C₈×C₅₃	D₈	1	4	yes	—
34	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
35	C₄⋊C₄×C₅₃	—	4	2	no	—
36	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₄⋊Dic₅₃	—	4	2	no	—
38	C₄⋊Dic₅₃	—	4	2	no	—
39	#11719908	—	2	4	no	—
40	C₂³×C₅₃	—	8	1	no	—
41	Q₈×C₅₃	—	8	2	no	—
42	D₄×C₅₃	—	8	2	no	—
43	D₄×C₅₃	—	8	4	no	—
44	C₂×C₄×C₅₃	—	8	1	no	—
45	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
47	C₂×Dic₅₃	—	8	2	no	—
48	Dic₁₀₆	—	8	4	no	—
49	C₈×C₅₃	—	2	4	no	—
50	C₈×C₅₃	#135	1	4	yes	—
51	#12511	—	53	1	no	maximal, Sylow
52	C₂²×C₅₃	—	16	1	no	—
53	C₂²×C₅₃	—	16	2	no	—
54	C₂²×C₅₃	#493	1	1	yes	characteristic
55	C₄×C₅₃	—	16	1	no	—
56	C₄×C₅₃	#495	1	2	yes	—
57	C₄×C₅₃	#508	1	4	yes	—
58	Dic₅₃	—	16	2	no	—
59	#2241	—	53	2	no	—
60	#2240	—	53	2	no	—
61	#1710	—	53	1	no	—
62	#1709	D₅₃	1	1	yes	characteristic
63	C₈×C₁₆	—	53	1	no	—
64	C₂×C₅₃	—	32	1	no	—
65	C₂×C₅₃	#1709	1	1	yes	characteristic
66	C₂×C₅₃	#2249	1	2	yes	—
67	#500	—	53	2	no	—
68	#494	C₂×D₅₃	1	1	yes	characteristic
69	#493	C₂×D₅₃	1	1	yes	characteristic
70	#487	—	106	2	no	—
71	#486	—	2	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	—	2	2	no	—
78	C₂×D₈	—	2	2	no	—
79	C₄⋊Q₈	—	2	1	no	—
80	C₄⋊Q₈	—	106	2	no	—
81	#127	—	2	1	no	—
82	C₂×C₁₆	—	53	4	no	—
83	#107	—	106	2	no	—
84	#107	—	106	2	no	—
85	#103	—	212	4	no	—
86	#102	—	4	4	no	—
87	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
88	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
89	C₂×Q₈	—	4	1	no	—
90	C₂×Q₈	—	212	2	no	—
91	C₂×D₄	—	4	1	no	—
92	C₂×D₄	—	4	2	no	—
93	Q₁₆	—	4	4	no	—
94	D₈	—	4	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	—	212	2	no	—
99	C₄⋊C₄	—	212	2	no	—
100	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
101	C₁₆	—	106	4	no	—
102	C₂³	—	8	1	no	—
103	Q₈	—	8	2	no	—
104	Q₈	—	424	4	no	—
105	D₄	—	8	4	no	—
106	D₄	—	8	2	no	—
107	C₂×C₄	—	8	1	no	—
108	C₂×C₄	—	424	2	no	—
109	C₂×C₄	#13133213	1	1	yes	characteristic
110	C₂×C₄	C₂×#11719922	1	2	yes	—
111	C₈	—	2	4	no	—
112	C₈	#13133236	1	4	yes	—
113	C₂²	—	16	1	no	—
114	C₂²	—	16	2	no	—
115	C₂²	#16810999	1	1	yes	characteristic
116	C₄	—	16	1	no	—
117	C₄	—	848	2	no	—
118	C₄	#16811006	1	2	yes	—
119	C₄	#16811053	1	4	yes	—
120	C₂	—	32	1	no	—
121	C₂	#20274320	1	1	yes	characteristic
122	C₂	#20276974	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20276942

C₂

yes

maximal, complemented

#20276938

C₂

yes

maximal

#20274325

C₂

yes

maximal, characteristic, complemented

C₈×#11719908

C₂

yes

maximal, characteristic, complemented

C₅₃×#1709

C₂

yes

maximal, characteristic

#16811028

C₂²

yes

—

—

—

—

C₂²

yes

—

C₅₃×#494

C₂²

yes

characteristic

C₅₃×#493

C₂²

yes

characteristic

C₅₃×#486

—

C₈²×C₅₃

C₂²

yes

characteristic

#13133217

—

C₂×#11719908

D₄

yes

—

#13133151

—

#13133151

—

#13133142

—

C₂×Q₁₆×C₅₃

—

C₂×D₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

#127×C₅₃

—

#102×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×Dic₁₀₆

—

Q₁₆×C₅₃

—

D₈×C₅₃

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719908

—

C₂³×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

—

C₂×Dic₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

#135

yes

—

#12511

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#493

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

#495

yes

—

C₄×C₅₃

#508

yes

—

Dic₅₃

—

—

—

—

D₅₃

yes

characteristic

C₈×C₁₆

—

C₂×C₅₃

—

C₂×C₅₃

#1709

yes

characteristic

C₂×C₅₃

#2249

yes

—

#500

—

#494

C₂×D₅₃

yes

characteristic

#493

C₂×D₅₃

yes

characteristic

#487

—

106

—

#486

—

#358

—

106

—

#357

—

106

—

C₄×C₁₆

—

C₈²

C₂×D₅₃

yes

characteristic

C₅₃

#12511

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

106

—

#127

—

C₂×C₁₆

—

#107

—

106

—

#107

—

106

—

#103

—

212

—

#102

—

C₄×C₈

C₂²⋊D₅₃

yes

—

C₄×C₈

C₂²×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

212

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

D₈

—

C₂×C₈

#11719922

yes

—

C₂×C₈

D₄×D₅₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

212

—

C₄⋊C₄

—

212

—

100

C₄²

C₂×C₂²⋊D₅₃

yes

characteristic

101

C₁₆

—

106

—

102

C₂³

—

103

Q₈

—

104

Q₈

—

424

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

424

—

109

C₂×C₄

#13133213

yes

characteristic

110

C₂×C₄

C₂×#11719922

yes

—

111

C₈

—

112

C₈

#13133236

yes

—

113

C₂²

—

114

C₂²

—

115

C₂²

#16810999

yes

characteristic

116

C₄

—

117

C₄

—

848

—

118

C₄

#16811006

yes

—

119

C₄

#16811053

yes

—

120

C₂

—

121

C₂

#20274320

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×#1709	C₂
socle	C₂²×C₅₃	#493
prime radical	C₅₃×#493	C₂²
hypercenter	#1709	D₅₃
hypocenter	C₅₃	#12511
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

subgroup

G/H

center

C₂²

#16810999

commutator, cosocle

C₄×C₈×C₅₃

C₂³

Frattini

C₄×C₈

C₂²×D₅₃

Fitting

C₅₃×#1709

C₂

socle

C₂²×C₅₃

#493

prime radical

C₅₃×#493

C₂²

D₅₃

C₅₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

N	H ≅ G/N	type
1. #20276942	C₂	semidirect
3. #20274325	C₂	semidirect
4. C₈×#11719908	C₂	semidirect
76. C₅₃	#12511	semidirect

H ≅ G/N

type

1. #20276942

C₂

semidirect

3. #20274325

C₂

semidirect

4. C₈×#11719908

C₂

semidirect

76. C₅₃

#12511

semidirect

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

subgroup	index
#20276942	2
#20276938	2
#20274325	2
C₈×#11719908	2
C₅₃×#1709	2
#12511	53

subgroup

index

C₅₃×#1709

Subgroups — #27298785

Conjugacy classes of subgroups

Complements

Maximal subgroups