Group #27311326 — subgroups

Subgroups — #27311326

Order 13568.

1118 conjugacy classes of subgroups; only the 411 normal subgroups are shown.

normal subgroups only

This group has 1118 conjugacy classes of subgroups — too many to draw. Only its 411 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	#20272977	C₂	1	8	yes	maximal
2	#20272945	C₂	1	6	yes	maximal
3	C₂×C₅₃×#376	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄⋊Dic₅₃	C₂²	1	4	yes	—
5	C₂×#13133124	C₂²	1	24	yes	—
6	C₂²×C₄²×C₅₃	C₂²	1	3	yes	—
7	C₅₃×#376	C₄	1	8	yes	complemented
8	C₂×#95×C₅₃	C₂²	1	4	yes	—
9	C₂³×Dic₅₃	C₂³	1	8	yes	—
10	C₂×C₄⋊Dic₅₃	—	2	24	no	—
11	#13133124	—	4	24	no	—
12	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
13	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
14	C₂×C₄²×C₅₃	—	2	12	no	—
15	C₂×C₄²×C₅₃	C₂×C₄	1	12	yes	—
16	#95×C₅₃	C₂×C₄	1	16	yes	—
17	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	—	2	6	no	—
19	C₂²×C₄×C₅₃	—	2	6	no	—
20	C₂²×C₄×C₅₃	—	2	12	no	—
21	C₂²×C₄×C₅₃	—	2	12	no	—
22	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
23	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
24	C₂²×C₄×C₅₃	#41	1	12	yes	—
25	C₂²×Dic₅₃	—	2	24	no	—
26	C₂²×Dic₅₃	—	2	24	no	—
27	C₂²×Dic₅₃	—	2	8	no	—
28	C₄²×C₅₃	—	2	24	no	—
29	C₄⋊Dic₅₃	—	4	16	no	—
30	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
31	C₂³×C₅₃	C₂×#41	1	6	yes	—
32	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
33	C₂³×C₅₃	2₋¹⁺⁴	1	4	yes	—
34	C₂×C₄×C₅₃	—	2	12	no	—
35	C₂×C₄×C₅₃	—	2	24	no	—
36	C₂×C₄×C₅₃	—	4	6	no	—
37	C₂×C₄×C₅₃	—	4	6	no	—
38	C₂×C₄×C₅₃	—	4	12	no	—
39	C₂×C₄×C₅₃	#117	1	12	yes	—
40	C₂×C₄×C₅₃	#126	1	16	yes	—
41	C₂×Dic₅₃	—	4	8	no	—
42	C₂×Dic₅₃	—	4	24	no	—
43	C₂×Dic₅₃	—	4	24	no	—
44	#14350	—	53	1	no	maximal, Sylow
45	C₂²×C₅₃	C₂×#117	1	3	yes	—
46	C₂²×C₅₃	#520	1	4	yes	—
47	C₂²×C₅₃	C₂×#126	1	4	yes	—
48	C₂²×C₅₃	#540	1	6	yes	—
49	C₂²×C₅₃	#541	1	6	yes	—
50	C₂²×C₅₃	#542	1	12	yes	—
51	C₄×C₅₃	—	4	12	no	—
52	C₄×C₅₃	#383	1	8	yes	—
53	Dic₅₃	—	8	8	no	—
54	C₂×#376	D₅₃	1	1	yes	characteristic
55	#1443	—	53	8	no	—
56	C₄⋊#95	—	53	6	no	—
57	C₂×C₅₃	C₂×#383	1	1	yes	characteristic
58	C₂×C₅₃	#2314	1	6	yes	—
59	C₂×C₅₃	#2644	1	4	yes	—
60	C₂×C₅₃	#2645	1	4	yes	—
61	C₂²×C₄⋊C₄	—	53	4	no	—
62	C₂²×C₄²	C₂×D₅₃	1	3	yes	—
63	#376	Dic₅₃	1	8	yes	complemented
64	C₂×#95	—	53	24	no	—
65	C₂×#95	C₂×D₅₃	1	4	yes	—
66	C₅₃	#14350	1	1	yes	characteristic, complemented, Sylow
67	C₂³×C₄	—	53	8	no	—
68	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
69	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
70	C₂×C₄⋊C₄	—	106	24	no	—
71	C₂×C₄²	—	2	12	no	—
72	C₂×C₄²	C₂×Dic₅₃	1	12	yes	—
73	#95	—	212	24	no	—
74	#95	C₂×Dic₅₃	1	16	yes	—
75	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
76	C₂²×C₄	—	2	6	no	—
77	C₂²×C₄	—	2	6	no	—
78	C₂²×C₄	—	2	12	no	—
79	C₂²×C₄	—	2	12	no	—
80	C₂²×C₄	—	106	24	no	—
81	C₂²×C₄	—	106	24	no	—
82	C₂²×C₄	—	106	8	no	—
83	C₂²×C₄	#11719945	1	12	yes	—
84	C₂²×C₄	C₂²×Dic₅₃	1	12	yes	—
85	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
86	C₄⋊C₄	—	212	16	no	—
87	C₄²	—	2	24	no	—
88	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
89	C₂³	C₂×#11719945	1	6	yes	—
90	C₂³	#13133263	1	4	yes	—
91	C₂³	#13133264	1	4	yes	—
92	C₂×C₄	—	2	12	no	—
93	C₂×C₄	—	2	24	no	—
94	C₂×C₄	—	4	6	no	—
95	C₂×C₄	—	4	6	no	—
96	C₂×C₄	—	4	12	no	—
97	C₂×C₄	—	212	8	no	—
98	C₂×C₄	—	212	24	no	—
99	C₂×C₄	—	212	24	no	—
100	C₂×C₄	#13133176	1	12	yes	—
101	C₂×C₄	#13133211	1	16	yes	—
102	C₂²	C₂×#13133176	1	3	yes	—
103	C₂²	#16811106	1	4	yes	—
104	C₂²	C₂×#13133211	1	4	yes	—
105	C₂²	#16811228	1	6	yes	—
106	C₂²	#16811235	1	6	yes	—
107	C₂²	#16811244	1	12	yes	—
108	C₄	—	4	12	no	—
109	C₄	—	424	8	no	—
110	C₄	#16810491	1	8	yes	—
111	C₂	C₂×#16810491	1	1	yes	characteristic
112	C₂	#20277324	1	6	yes	—
113	C₂	#20280675	1	4	yes	—
114	C₂	#20280680	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20272977

C₂

yes

maximal

#20272945

C₂

yes

maximal

C₂×C₅₃×#376

C₂

yes

maximal, characteristic

C₂²×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#13133124

C₂²

yes

—

C₂²×C₄²×C₅₃

C₂²

yes

—

C₅₃×#376

C₄

yes

complemented

C₂×#95×C₅₃

C₂²

yes

—

C₂³×Dic₅₃

C₂³

yes

—

C₂×C₄⋊Dic₅₃

—

#13133124

—

C₂³×C₄×C₅₃

C₂³

yes

—

C₂³×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₂×C₄

yes

—

#95×C₅₃

C₂×C₄

yes

—

C₂⁴×C₅₃

C₂⁴

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

—

C₂²×C₄×C₅₃

#41

yes

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₄²×C₅₃

—

C₄⋊Dic₅₃

—

C₂³×C₅₃

C₂³×C₄

yes

characteristic

C₂³×C₅₃

C₂×#41

yes

—

C₂³×C₅₃

H(5,𝔽₂)

yes

—

C₂³×C₅₃

2₋¹⁺⁴

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

#126

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

#14350

—

maximal, Sylow

C₂²×C₅₃

C₂×#117

yes

—

C₂²×C₅₃

#520

yes

—

C₂²×C₅₃

C₂×#126

yes

—

C₂²×C₅₃

#540

yes

—

C₂²×C₅₃

#541

yes

—

C₂²×C₅₃

#542

yes

—

C₄×C₅₃

—

C₄×C₅₃

#383

yes

—

Dic₅₃

—

C₂×#376

D₅₃

yes

characteristic

#1443

—

C₄⋊#95

—

C₂×C₅₃

C₂×#383

yes

characteristic

C₂×C₅₃

#2314

yes

—

C₂×C₅₃

#2644

yes

—

C₂×C₅₃

#2645

yes

—

C₂²×C₄⋊C₄

—

C₂²×C₄²

C₂×D₅₃

yes

—

#376

Dic₅₃

yes

complemented

C₂×#95

—

C₂×#95

C₂×D₅₃

yes

—

C₅₃

#14350

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂³×C₄

C₂²×D₅₃

yes

—

C₂³×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

—

C₂×C₄²

C₂×Dic₅₃

yes

—

#95

—

212

—

#95

C₂×Dic₅₃

yes

—

C₂⁴

C₂³×D₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

#11719945

yes

—

C₂²×C₄

C₂²×Dic₅₃

yes

—

C₂²×C₄

C₂²×Dic₅₃

yes

—

C₄⋊C₄

—

212

—

C₄²

—

C₂³

C₂³×Dic₅₃

yes

characteristic

C₂³

C₂×#11719945

yes

—

C₂³

#13133263

yes

—

C₂³

#13133264

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

100

C₂×C₄

#13133176

yes

—

101

C₂×C₄

#13133211

yes

—

102

C₂²

C₂×#13133176

yes

—

103

C₂²

#16811106

yes

—

104

C₂²

C₂×#13133211

yes

—

105

C₂²

#16811228

yes

—

106

C₂²

#16811235

yes

—

107

C₂²

#16811244

yes

—

108

C₄

—

109

C₄

—

424

—

110

C₄

#16810491

yes

—

111

C₂

C₂×#16810491

yes

characteristic

112

C₂

#20277324

yes

—

113

C₂

#20280675

yes

—

114

C₂

#20280680

yes

—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₅₃
commutator	C₂³×C₅₃	C₂³×C₄
Fitting	C₂×C₅₃×#376	C₂
socle, prime radical, cosocle	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂×#376	D₅₃
hypocenter	C₅₃	#14350
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376

subgroup

G/H

center, Frattini

C₂⁴

C₂³×D₅₃

commutator

C₂³×C₅₃

C₂³×C₄

Fitting

C₂×C₅₃×#376

C₂

socle, prime radical, cosocle

C₂⁴×C₅₃

C₂⁴

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂⁴ ◁ C₂×#376

N	H ≅ G/N	type
7. C₅₃×#376	C₄	semidirect
63. #376	Dic₅₃	semidirect
66. C₅₃	#14350	semidirect

H ≅ G/N

type

7. C₅₃×#376

C₄

semidirect

63. #376

Dic₅₃

semidirect

66. C₅₃

#14350

semidirect

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

subgroup	index
#20272977	2
#20272945	2
C₂×C₅₃×#376	2
#14350	53

subgroup

index

#20272977

#20272945

C₂×C₅₃×#376

#14350

Subgroups — #27311326

Conjugacy classes of subgroups

Complements

Maximal subgroups