Group #27288588 — subgroups

Subgroups — #27288588

Order 13568.

406 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 97 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	#20274884	C₂	1	3	yes	maximal
2	#20272596	C₂	1	3	yes	maximal
3	C₅₃×#2844	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133150	—	2	3	no	—
5	C₂×#13133148	—	2	3	no	—
6	C₂×#13133128	C₂²	1	3	yes	—
7	#16810493	C₂²	1	1	yes	characteristic
8	C₅₃×#400	—	2	3	no	—
9	C₅₃×#395	—	2	3	no	—
10	C₅₃×#382	C₂²	1	3	yes	—
11	C₂²×#88322	—	2	3	no	—
12	C₂×C₄⋊Dic₅₃	—	2	3	no	—
13	#13133150	—	4	6	no	—
14	#13133148	—	4	6	no	—
15	#13133128	—	2	3	no	—
16	#13133128	Q₈	1	6	yes	complemented
17	C₂²×Q₈×C₅₃	—	2	1	no	—
18	C₂×C₄⋊C₄×C₅₃	—	2	3	no	—
19	C₂×C₄⋊C₄×C₅₃	D₄	1	3	yes	—
20	C₂×C₄⋊C₄×C₅₃	D₄	1	3	yes	—
21	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
22	#95×C₅₃	—	2	3	no	—
23	#95×C₅₃	—	2	1	no	—
24	#95×C₅₃	—	2	6	no	—
25	C₂×Q₈×C₅₃	—	8	3	no	—
26	C₂²×C₄×C₅₃	—	2	3	no	—
27	C₂²×C₄×C₅₃	—	2	3	no	—
28	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
29	C₂²×Dic₅₃	—	4	1	no	—
30	C₄⋊C₄×C₅₃	—	4	3	no	—
31	C₄⋊C₄×C₅₃	—	4	3	no	—
32	C₄⋊C₄×C₅₃	—	8	3	no	—
33	C₄²×C₅₃	C₂×Q₈	1	3	yes	—
34	C₄²×C₅₃	#41	1	1	yes	characteristic
35	C₄⋊Dic₅₃	—	4	3	no	—
36	C₄⋊Dic₅₃	—	4	3	no	—
37	C₂×#88322	—	2	6	no	—
38	C₂×#88322	—	2	6	no	—
39	C₂×#88322	—	2	6	no	—
40	C₂³×C₅₃	#120	1	1	yes	characteristic
41	Q₈×C₅₃	—	8	4	no	—
42	C₂×C₄×C₅₃	—	2	3	no	—
43	C₂×C₄×C₅₃	—	2	3	no	—
44	C₂×C₄×C₅₃	—	4	3	no	—
45	C₂×C₄×C₅₃	—	4	6	no	—
46	C₂×C₄×C₅₃	—	4	3	no	—
47	C₂×C₄×C₅₃	—	4	6	no	—
48	C₂×C₄×C₅₃	C₂²⋊Q₈	1	3	yes	—
49	C₂×C₄×C₅₃	C₄⋊Q₈	1	3	yes	—
50	C₂×Dic₅₃	—	8	3	no	—
51	#88322	—	2	12	no	—
52	#10583	—	53	1	no	maximal, Sylow
53	C₂²×C₅₃	#395	1	1	yes	characteristic
54	C₂²×C₅₃	#453	1	3	yes	—
55	C₂²×C₅₃	#456	1	3	yes	—
56	C₄×C₅₃	—	2	3	no	—
57	C₄×C₅₃	—	2	3	no	—
58	C₄×C₅₃	—	8	3	no	—
59	C₄×C₅₃	—	8	3	no	—
60	Dic₅₃	—	16	1	no	—
61	#2844	D₅₃	1	1	yes	characteristic
62	#1847	—	53	3	no	—
63	#1295	—	53	3	no	—
64	C₂×C₅₃	#1655	1	1	yes	characteristic
65	C₂×C₅₃	#1660	1	3	yes	—
66	C₂×C₅₃	#2060	1	3	yes	—
67	C₂×#107	—	106	3	no	—
68	C₂×#106	—	106	3	no	—
69	C₂×#97	—	53	3	no	—
70	#400	—	2	3	no	—
71	#395	—	2	3	no	—
72	#384	—	53	1	no	—
73	#382	C₂×D₅₃	1	3	yes	—
74	C₅₃	#10583	1	1	yes	characteristic, complemented, Sylow
75	C₂²×Q₈	—	2	1	no	—
76	C₂²×C₈	—	106	3	no	—
77	C₂×C₄⋊C₄	—	2	3	no	—
78	C₂×C₄⋊C₄	—	106	3	no	—
79	C₂×C₄⋊C₄	C₂²⋊D₅₃	1	3	yes	—
80	C₂×C₄⋊C₄	C₂²⋊D₅₃	1	3	yes	—
81	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
82	#107	—	212	6	no	—
83	#106	—	212	6	no	—
84	#97	—	53	6	no	—
85	#97	—	106	3	no	—
86	#95	—	2	3	no	—
87	#95	—	2	1	no	—
88	#95	—	2	6	no	—
89	C₂×Q₈	—	8	3	no	—
90	C₂²×C₄	—	2	3	no	—
91	C₂²×C₄	—	2	3	no	—
92	C₂²×C₄	—	212	1	no	—
93	C₂²×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
94	C₂×C₈	—	106	6	no	—
95	C₂×C₈	—	106	6	no	—
96	C₂×C₈	—	106	6	no	—
97	C₄⋊C₄	—	4	3	no	—
98	C₄⋊C₄	—	4	3	no	—
99	C₄⋊C₄	—	8	3	no	—
100	C₄⋊C₄	—	212	3	no	—
101	C₄⋊C₄	—	212	3	no	—
102	C₄²	Q₈×D₅₃	1	3	yes	—
103	C₄²	#11719949	1	1	yes	characteristic
104	C₂³	#13133186	1	1	yes	characteristic
105	Q₈	—	8	4	no	—
106	C₂×C₄	—	2	3	no	—
107	C₂×C₄	—	2	3	no	—
108	C₂×C₄	—	4	3	no	—
109	C₂×C₄	—	4	3	no	—
110	C₂×C₄	—	4	6	no	—
111	C₂×C₄	—	4	6	no	—
112	C₂×C₄	—	424	3	no	—
113	C₂×C₄	#13133193	1	3	yes	—
114	C₂×C₄	#13133215	1	3	yes	—
115	C₈	—	106	12	no	—
116	C₂²	#16810542	1	1	yes	characteristic
117	C₂²	#16810791	1	3	yes	—
118	C₂²	#16810806	1	3	yes	—
119	C₄	—	2	3	no	—
120	C₄	—	2	3	no	—
121	C₄	—	8	3	no	—
122	C₄	—	8	3	no	—
123	C₄	—	848	1	no	—
124	C₂	#20274004	1	1	yes	characteristic
125	C₂	#20274026	1	3	yes	—
126	C₂	#20276057	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20274884

C₂

yes

maximal

#20272596

C₂

yes

maximal

C₅₃×#2844

C₂

yes

maximal, characteristic

C₂×#13133150

—

C₂×#13133148

—

C₂×#13133128

C₂²

yes

—

#16810493

C₂²

yes

characteristic

C₅₃×#400

—

C₅₃×#395

—

C₅₃×#382

C₂²

yes

—

C₂²×#88322

—

C₂×C₄⋊Dic₅₃

—

—

—

—

Q₈

yes

complemented

C₂²×Q₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

D₄

yes

—

C₂×C₄⋊C₄×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#95×C₅₃

—

#95×C₅₃

—

#95×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×Dic₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×Q₈

yes

—

C₄²×C₅₃

#41

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

#120

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₃

C₄⋊Q₈

yes

—

C₂×Dic₅₃

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#456

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

#2844

D₅₃

yes

characteristic

#1847

—

#1295

—

C₂×C₅₃

#1655

yes

characteristic

C₂×C₅₃

#1660

yes

—

C₂×C₅₃

#2060

yes

—

C₂×#107

—

106

—

C₂×#106

—

106

—

C₂×#97

—

—

—

—

C₂×D₅₃

yes

—

C₅₃

#10583

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

106

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

C₂²⋊D₅₃

yes

—

C₂×C₄⋊C₄

C₂²⋊D₅₃

yes

—

C₂×C₄²

C₂²×D₅₃

yes

characteristic

#107

—

212

—

#106

—

212

—

#97

—

#97

—

106

—

#95

—

#95

—

#95

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

212

—

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

212

—

101

C₄⋊C₄

—

212

—

102

C₄²

Q₈×D₅₃

yes

—

103

C₄²

#11719949

yes

characteristic

104

C₂³

#13133186

yes

characteristic

105

Q₈

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

424

—

113

C₂×C₄

#13133193

yes

—

114

C₂×C₄

#13133215

yes

—

115

C₈

—

106

—

116

C₂²

#16810542

yes

characteristic

117

C₂²

#16810791

yes

—

118

C₂²

#16810806

yes

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

848

—

124

C₂

#20274004

yes

characteristic

125

C₂

#20274026

yes

—

126

C₂

#20276057

yes

—

	subgroup	G/H
center	C₂³	#13133186
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#2844	C₂
socle, prime radical	C₂³×C₅₃	#120
hypercenter	#2844	D₅₃
hypocenter	C₅₃	#10583
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2844

subgroup

G/H

center

C₂³

#13133186

commutator, cosocle

C₂×C₄²×C₅₃

C₂³

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#2844

C₂

C₂³×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2844

N	H ≅ G/N	type
16. #13133128	Q₈	semidirect
74. C₅₃	#10583	semidirect

H ≅ G/N

type

16. #13133128

Q₈

semidirect

74. C₅₃

#10583

semidirect

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

subgroup	index
#20274884	2
#20272596	2
C₅₃×#2844	2
#10583	53

subgroup

index

#20274884

#20272596

C₅₃×#2844

#10583

Subgroups — #27288588

Conjugacy classes of subgroups

Complements

Maximal subgroups