Group #27298831 — subgroups

Subgroups — #27298831

Order 13568.

238 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 87 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	#20276995	C₂	1	2	yes	maximal
2	#20276990	C₂	1	1	yes	maximal, characteristic
3	#20274324	C₂	1	2	yes	maximal
4	C₁₆×#88322	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#2252	C₂	1	1	yes	maximal, characteristic
6	#16811028	C₂²	1	2	yes	—
7	#16810975	—	2	2	no	—
8	#16810973	—	2	2	no	—
9	#16810450	—	2	4	no	—
10	#16810449	—	2	2	no	—
11	C₄×#11719908	C₂²	1	1	yes	characteristic
12	C₈×#88322	C₂²	1	1	yes	characteristic
13	C₅₃×#500	C₂²	1	2	yes	—
14	C₅₃×#366	—	2	2	no	—
15	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	#13133217	—	2	2	no	—
17	C₂×#11719908	D₄	1	2	yes	—
18	#13133151	—	2	2	no	—
19	#13133151	—	2	2	no	—
20	#13133141	—	4	4	no	—
21	#13133140	—	4	4	no	—
22	C₄×#88322	D₄	1	1	yes	characteristic
23	C₄×#88322	D₄	1	1	yes	characteristic
24	C₄⋊Q₈×C₅₃	—	2	2	no	—
25	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
26	#107×C₅₃	—	2	2	no	—
27	#107×C₅₃	—	2	2	no	—
28	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
29	C₂×Q₈×C₅₃	—	4	2	no	—
30	C₂×Dic₁₀₆	—	4	2	no	—
31	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
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	C₄⋊Dic₅₃	—	4	2	no	—
38	C₂×#88322	D₈	1	2	yes	—
39	C₂×#88322	D₈	1	2	yes	—
40	C₁₆×C₅₃	Q₁₆	1	4	yes	—
41	#11719908	Q₁₆	1	4	yes	—
42	Q₈×C₅₃	—	8	4	no	—
43	C₂×C₄×C₅₃	—	8	2	no	—
44	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
47	C₂×Dic₅₃	—	8	2	no	—
48	Dic₁₀₆	—	8	4	no	—
49	C₈×C₅₃	C₂×Q₁₆	1	2	yes	—
50	C₈×C₅₃	C₂×Q₁₆	1	2	yes	—
51	#88322	—	2	2	no	—
52	#88322	—	2	2	no	—
53	#12522	—	53	1	no	maximal, Sylow
54	C₂²×C₅₃	#493	1	1	yes	characteristic
55	C₄×C₅₃	—	16	2	no	—
56	C₄×C₅₃	#494	1	1	yes	characteristic
57	C₄×C₅₃	#494	1	1	yes	characteristic
58	C₄×C₅₃	#508	1	2	yes	—
59	C₄×C₅₃	#508	1	2	yes	—
60	Dic₅₃	—	16	2	no	—
61	#2254	—	53	2	no	—
62	#2252	—	53	1	no	—
63	#2252	D₅₃	1	1	yes	characteristic
64	#1710	—	53	2	no	—
65	C₈×C₁₆	—	53	1	no	—
66	C₂×C₅₃	#1713	1	1	yes	characteristic
67	C₂×C₅₃	#2249	1	1	yes	characteristic
68	C₂×C₅₃	#2249	1	1	yes	characteristic
69	#500	—	53	2	no	—
70	#500	C₂×D₅₃	1	2	yes	—
71	#487	—	106	2	no	—
72	#487	—	106	2	no	—
73	#367	—	106	4	no	—
74	#366	—	2	2	no	—
75	#366	—	106	2	no	—
76	C₄×C₁₆	—	53	1	no	—
77	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
78	C₈²	—	53	1	no	—
79	C₅₃	#12522	1	1	yes	characteristic, complemented, Sylow
80	C₄⋊Q₈	—	2	2	no	—
81	C₄⋊Q₈	—	106	2	no	—
82	C₂×C₁₆	—	53	2	no	—
83	C₂×C₁₆	C₂²⋊D₅₃	1	2	yes	—
84	#107	—	2	2	no	—
85	#107	—	2	2	no	—
86	#107	—	106	2	no	—
87	#107	—	106	2	no	—
88	#103	—	212	4	no	—
89	#103	—	212	4	no	—
90	C₄×C₈	—	53	1	no	—
91	C₄×C₈	—	53	1	no	—
92	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
93	C₂×Q₈	—	4	2	no	—
94	C₂×Q₈	—	212	2	no	—
95	C₂×C₈	—	53	2	no	—
96	C₂×C₈	—	53	2	no	—
97	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
98	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
99	C₄⋊C₄	—	4	2	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄⋊C₄	—	212	2	no	—
102	C₄⋊C₄	—	212	2	no	—
103	C₄²	D₄×D₅₃	1	1	yes	characteristic
104	C₁₆	—	53	4	no	—
105	C₁₆	#11719925	1	4	yes	—
106	Q₈	—	8	4	no	—
107	Q₈	—	424	4	no	—
108	C₂×C₄	—	8	2	no	—
109	C₂×C₄	—	424	2	no	—
110	C₂×C₄	#13133213	1	1	yes	characteristic
111	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
112	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
113	C₈	—	106	2	no	—
114	C₈	—	106	2	no	—
115	C₈	Q₁₆×D₅₃	1	2	yes	—
116	C₈	C₂×#11719925	1	2	yes	—
117	C₂²	#16810998	1	1	yes	characteristic
118	C₄	—	16	2	no	—
119	C₄	—	848	2	no	—
120	C₄	#16811001	1	1	yes	characteristic
121	C₄	#16811002	1	1	yes	characteristic
122	C₄	#16811052	1	2	yes	—
123	C₄	#16811056	1	2	yes	—
124	C₂	#20274332	1	1	yes	characteristic
125	C₂	#20276973	1	1	yes	characteristic
126	C₂	#20276977	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276995

C₂

yes

maximal

#20276990

C₂

yes

maximal, characteristic

#20274324

C₂

yes

maximal

C₁₆×#88322

C₂

yes

maximal, characteristic

C₅₃×#2252

C₂

yes

maximal, characteristic

#16811028

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

C₈×#88322

C₂²

yes

characteristic

C₅₃×#500

C₂²

yes

—

C₅₃×#366

—

C₄×C₁₆×C₅₃

C₂²

yes

characteristic

#13133217

—

C₂×#11719908

D₄

yes

—

—

—

—

—

D₄

yes

characteristic

C₄×#88322

D₄

yes

characteristic

C₄⋊Q₈×C₅₃

—

C₂×C₁₆×C₅₃

D₄

yes

—

#107×C₅₃

—

#107×C₅₃

—

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

D₈

yes

—

C₂×#88322

D₈

yes

—

C₁₆×C₅₃

Q₁₆

yes

—

#11719908

Q₁₆

yes

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×Dic₅₃

—

Dic₁₀₆

—

C₈×C₅₃

C₂×Q₁₆

yes

—

C₈×C₅₃

C₂×Q₁₆

yes

—

—

—

—

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₄×C₅₃

#508

yes

—

Dic₅₃

—

#2254

—

#2252

—

#2252

D₅₃

yes

characteristic

#1710

—

C₈×C₁₆

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×D₅₃

yes

—

#487

—

106

—

#487

—

106

—

#367

—

106

—

#366

—

#366

—

106

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₅₃

yes

characteristic

C₈²

—

C₅₃

#12522

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

—

106

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₅₃

yes

—

#107

—

#107

—

#107

—

106

—

#107

—

106

—

#103

—

212

—

#103

—

212

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

212

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×D₅₃

yes

characteristic

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

212

—

102

C₄⋊C₄

—

212

—

103

C₄²

D₄×D₅₃

yes

characteristic

104

C₁₆

—

105

C₁₆

#11719925

yes

—

106

Q₈

—

107

Q₈

—

424

—

108

C₂×C₄

—

109

C₂×C₄

—

424

—

110

C₂×C₄

#13133213

yes

characteristic

111

C₂×C₄

D₈×D₅₃

yes

characteristic

112

C₂×C₄

D₈×D₅₃

yes

characteristic

113

C₈

—

106

—

114

C₈

—

106

—

115

C₈

Q₁₆×D₅₃

yes

—

116

C₈

C₂×#11719925

yes

—

117

C₂²

#16810998

yes

characteristic

118

C₄

—

119

C₄

—

848

—

120

C₄

#16811001

yes

characteristic

121

C₄

#16811002

yes

characteristic

122

C₄

#16811052

yes

—

123

C₄

#16811056

yes

—

124

C₂

#20274332

yes

characteristic

125

C₂

#20276973

yes

characteristic

126

C₂

#20276977

yes

characteristic

	subgroup	G/H
center	C₂²	#16810998
commutator, cosocle	C₄×C₈×C₅₃	C₂³
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#2252	C₂
socle, prime radical	C₂²×C₅₃	#493
hypercenter	#2252	D₅₃
hypocenter	C₅₃	#12522
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

subgroup

G/H

center

C₂²

#16810998

commutator, cosocle

C₄×C₈×C₅₃

C₂³

Frattini

C₄×C₈

C₂²×D₅₃

Fitting

C₅₃×#2252

C₂

C₂²×C₅₃

D₅₃

C₅₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

N	H ≅ G/N	type
79. C₅₃	#12522	semidirect

H ≅ G/N

type

79. C₅₃

#12522

semidirect

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

subgroup	index
#20276995	2
#20276990	2
#20274324	2
C₁₆×#88322	2
C₅₃×#2252	2
#12522	53

subgroup

index

#20276995

#20276990

#20274324

C₁₆×#88322

C₅₃×#2252

#12522

Subgroups — #27298831

Conjugacy classes of subgroups

Complements

Maximal subgroups