Group #27287169 — subgroups

Subgroups — #27287169

Order 13568.

476 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 93 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	#20275538	C₂	1	3	yes	maximal
2	#20272596	C₂	1	3	yes	maximal, complemented
3	C₅₃×#2845	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13133214	C₂²	1	1	yes	characteristic
5	C₂×#13133137	—	2	6	no	—
6	C₂×#13133128	C₂²	1	3	yes	—
7	C₅₃×#398	—	2	6	no	—
8	C₅₃×#382	C₂²	1	3	yes	—
9	C₂²×D₂₁₂	—	2	3	no	—
10	C₂²×#88322	—	2	3	no	—
11	#13133214	—	2	3	no	—
12	#13133214	Q₈	1	2	yes	—
13	#13133137	—	4	12	no	—
14	#13133128	—	2	6	no	—
15	C₂×C₄⋊C₄×C₅₃	—	2	6	no	—
16	C₂×C₄⋊C₄×C₅₃	D₄	1	6	yes	—
17	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
18	#95×C₅₃	—	2	2	no	—
19	#95×C₅₃	—	2	6	no	—
20	C₂³×D₅₃	—	4	1	no	—
21	C₂²×C₄×C₅₃	—	2	6	no	—
22	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
23	C₂×D₂₁₂	—	2	6	no	—
24	C₂×D₂₁₂	—	4	6	no	—
25	C₂×D₂₁₂	—	4	6	no	—
26	C₂×D₂₁₂	—	4	3	no	—
27	C₄⋊C₄×C₅₃	—	4	6	no	—
28	C₄⋊C₄×C₅₃	—	8	6	no	—
29	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
30	C₄²×C₅₃	#41	1	3	yes	—
31	C₂×#88322	—	2	6	no	—
32	C₂×#88322	—	4	3	no	—
33	C₂×#88322	—	4	3	no	—
34	C₂³×C₅₃	#120	1	1	yes	characteristic
35	C₂²×D₅₃	—	4	2	no	—
36	C₂²×D₅₃	—	8	3	no	—
37	C₂²×D₅₃	—	8	3	no	—
38	C₂×C₄×C₅₃	—	2	3	no	—
39	C₂×C₄×C₅₃	—	2	3	no	—
40	C₂×C₄×C₅₃	—	4	6	no	—
41	C₂×C₄×C₅₃	—	4	6	no	—
42	C₂×C₄×C₅₃	—	4	6	no	—
43	C₂×C₄×C₅₃	C₂²⋊Q₈	1	3	yes	—
44	C₂×C₄×C₅₃	#124	1	3	yes	—
45	D₂₁₂	—	4	12	no	—
46	D₂₁₂	—	8	6	no	—
47	#88322	—	4	6	no	—
48	#10360	—	53	1	no	maximal, Sylow
49	C₂²×C₅₃	#393	1	1	yes	characteristic
50	C₂²×C₅₃	#453	1	3	yes	—
51	C₂²×C₅₃	#454	1	3	yes	—
52	C₂×D₅₃	—	8	6	no	—
53	C₂×D₅₃	—	16	3	no	—
54	C₂×D₅₃	—	16	1	no	—
55	C₄×C₅₃	—	2	3	no	—
56	C₄×C₅₃	—	2	3	no	—
57	C₄×C₅₃	—	8	6	no	—
58	#2845	D₅₃	1	1	yes	characteristic, complemented
59	#1961	—	53	3	no	—
60	#1295	—	53	3	no	—
61	C₂×C₅₃	#1656	1	3	yes	—
62	C₂×C₅₃	#1658	1	1	yes	characteristic
63	C₂×C₅₃	#2027	1	3	yes	—
64	D₅₃	—	16	2	no	—
65	C₂×#127	—	53	1	no	—
66	C₂×#102	—	106	6	no	—
67	C₂×#97	—	53	3	no	—
68	#398	—	2	6	no	—
69	#382	C₂×D₅₃	1	3	yes	—
70	C₅₃	#10360	1	1	yes	characteristic, complemented, Sylow
71	C₂²×D₄	—	106	3	no	—
72	C₂²×C₈	—	106	3	no	—
73	#127	—	53	2	no	—
74	#127	—	106	3	no	—
75	C₂×C₄⋊C₄	—	2	6	no	—
76	C₂×C₄⋊C₄	C₂²⋊D₅₃	1	6	yes	—
77	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
78	#102	—	212	12	no	—
79	#97	—	106	6	no	—
80	#95	—	2	2	no	—
81	#95	—	2	6	no	—
82	C₂⁴	—	212	1	no	—
83	C₂×D₄	—	106	6	no	—
84	C₂×D₄	—	212	6	no	—
85	C₂×D₄	—	212	6	no	—
86	C₂×D₄	—	212	3	no	—
87	C₂²×C₄	—	2	6	no	—
88	C₂²×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
89	C₂×C₈	—	106	6	no	—
90	C₂×C₈	—	212	3	no	—
91	C₂×C₈	—	212	3	no	—
92	C₄⋊C₄	—	4	6	no	—
93	C₄⋊C₄	—	8	6	no	—
94	C₄²	Q₈×D₅₃	1	1	yes	characteristic
95	C₄²	#11719949	1	3	yes	—
96	C₂³	—	424	3	no	—
97	C₂³	—	424	3	no	—
98	C₂³	—	212	2	no	—
99	C₂³	#13133186	1	1	yes	characteristic
100	D₄	—	424	6	no	—
101	D₄	—	212	12	no	—
102	C₂×C₄	—	2	3	no	—
103	C₂×C₄	—	2	3	no	—
104	C₂×C₄	—	4	6	no	—
105	C₂×C₄	—	4	6	no	—
106	C₂×C₄	—	4	6	no	—
107	C₂×C₄	#13133193	1	3	yes	—
108	C₂×C₄	#13133206	1	3	yes	—
109	C₈	—	212	6	no	—
110	C₂²	—	848	3	no	—
111	C₂²	—	848	1	no	—
112	C₂²	—	424	6	no	—
113	C₂²	#16810534	1	1	yes	characteristic
114	C₂²	#16810791	1	3	yes	—
115	C₂²	#16810796	1	3	yes	—
116	C₄	—	2	3	no	—
117	C₄	—	2	3	no	—
118	C₄	—	8	6	no	—
119	C₂	—	848	2	no	—
120	C₂	#20274008	1	3	yes	—
121	C₂	#20274017	1	1	yes	characteristic
122	C₂	#20275868	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20275538

C₂

yes

maximal

#20272596

C₂

yes

maximal, complemented

C₅₃×#2845

C₂

yes

maximal, characteristic, complemented

C₂×#13133214

C₂²

yes

characteristic

C₂×#13133137

—

C₂×#13133128

C₂²

yes

—

C₅₃×#398

—

C₅₃×#382

C₂²

yes

—

C₂²×D₂₁₂

—

C₂²×#88322

—

#13133214

—

#13133214

Q₈

yes

—

#13133137

—

#13133128

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#95×C₅₃

—

#95×C₅₃

—

C₂³×D₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×Q₈

yes

characteristic

C₄²×C₅₃

#41

yes

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

#120

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₃

#124

yes

—

D₂₁₂

—

D₂₁₂

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#454

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

#2845

D₅₃

yes

characteristic, complemented

#1961

—

#1295

—

C₂×C₅₃

#1656

yes

—

C₂×C₅₃

#1658

yes

characteristic

C₂×C₅₃

#2027

yes

—

D₅₃

—

C₂×#127

—

C₂×#102

—

106

—

C₂×#97

—

#398

—

#382

C₂×D₅₃

yes

—

C₅₃

#10360

yes

characteristic, complemented, Sylow

C₂²×D₄

—

106

—

C₂²×C₈

—

106

—

#127

—

#127

—

106

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²⋊D₅₃

yes

—

C₂×C₄²

C₂²×D₅₃

yes

characteristic

#102

—

212

—

#97

—

106

—

#95

—

#95

—

C₂⁴

—

212

—

C₂×D₄

—

106

—

C₂×D₄

—

212

—

C₂×D₄

—

212

—

C₂×D₄

—

212

—

C₂²×C₄

—

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

—

C₂×C₈

—

106

—

C₂×C₈

—

212

—

C₂×C₈

—

212

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

Q₈×D₅₃

yes

characteristic

C₄²

#11719949

yes

—

C₂³

—

424

—

C₂³

—

424

—

C₂³

—

212

—

C₂³

#13133186

yes

characteristic

100

D₄

—

424

—

101

D₄

—

212

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

#13133193

yes

—

108

C₂×C₄

#13133206

yes

—

109

C₈

—

212

—

110

C₂²

—

848

—

111

C₂²

—

848

—

112

C₂²

—

424

—

113

C₂²

#16810534

yes

characteristic

114

C₂²

#16810791

yes

—

115

C₂²

#16810796

yes

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₂

—

848

—

120

C₂

#20274008

yes

—

121

C₂

#20274017

yes

characteristic

122

C₂

#20275868

yes

—

	subgroup	G/H
center	C₂³	#13133186
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#2845	C₂
socle	C₂³×C₅₃	#120
prime radical	C₂×#13133214	C₂²
hypercenter	#2845	D₅₃
hypocenter	C₅₃	#10360
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2845

subgroup

G/H

center

C₂³

#13133186

commutator, cosocle

C₂×C₄²×C₅₃

C₂³

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#2845

C₂

C₂³×C₅₃

C₂²

D₅₃

C₅₃

G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2845

N	H ≅ G/N	type
2. #20272596	C₂	semidirect
3. C₅₃×#2845	C₂	semidirect
58. #2845	D₅₃	semidirect
70. C₅₃	#10360	semidirect

H ≅ G/N

type

2. #20272596

C₂

semidirect

3. C₅₃×#2845

C₂

semidirect

58. #2845

D₅₃

semidirect

70. C₅₃

#10360

semidirect

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

subgroup	index
#20275538	2
#20272596	2
C₅₃×#2845	2
#10360	53

subgroup

index

#20275538

#20272596

C₅₃×#2845

#10360

Subgroups — #27287169

Conjugacy classes of subgroups

Complements

Maximal subgroups