Group #27300485 — subgroups

Subgroups — #27300485

Order 13568.

274 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 147 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	#20274887	C₂	1	2	yes	maximal
2	#20272809	C₂	1	4	yes	maximal
3	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133153	—	2	2	no	—
5	C₂×#13133153	C₂²	1	4	yes	—
6	#16810647	—	2	2	no	—
7	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#346	C₄	1	4	yes	—
10	C₂×#11719917	—	2	4	no	—
11	#13133153	—	2	8	no	—
12	#13133153	—	4	4	no	—
13	C₄⋊#88322	—	4	4	no	—
14	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
15	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
17	C₂×C₁₆×C₅₃	—	2	4	no	—
18	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
19	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
20	C₄×C₈×C₅₃	Q₈	1	2	yes	—
21	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
22	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
23	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
24	C₂×C₈×C₅₃	SD₁₆	1	4	yes	—
25	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
26	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
27	C₂×C₈×C₅₃	#41	1	4	yes	—
28	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
29	C₄²×C₅₃	D₈	1	2	yes	—
30	#11719917	—	4	4	no	—
31	#11719917	—	4	4	no	—
32	C₂×#88322	—	4	4	no	—
33	C₁₆×C₅₃	—	2	4	no	—
34	C₂³×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
35	C₂×C₄×C₅₃	—	4	2	no	—
36	C₂×C₄×C₅₃	#107	1	4	yes	—
37	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₅₃	#124	1	2	yes	—
39	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
40	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
41	C₈×C₅₃	#106	1	4	yes	—
42	C₈×C₅₃	#117	1	2	yes	—
43	C₈×C₅₃	#125	1	2	yes	—
44	#88322	—	8	4	no	—
45	#12892	—	53	1	no	maximal, Sylow
46	C₂²×C₅₃	—	2	2	no	—
47	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
48	C₂²×C₅₃	#487	1	2	yes	—
49	C₄×C₅₃	—	2	4	no	—
50	C₄×C₅₃	#382	1	1	yes	characteristic
51	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
52	C₄×C₅₃	#486	1	2	yes	—
53	C₂×#346	D₅₃	1	1	yes	characteristic
54	#1848	—	53	2	no	—
55	#1381	—	53	4	no	—
56	C₂×C₅₃	—	2	2	no	—
57	C₂×C₅₃	#1834	1	1	yes	characteristic
58	C₂×C₅₃	#2161	1	2	yes	—
59	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
60	C₂×#108	—	53	4	no	—
61	C₂×#108	—	106	2	no	—
62	#423	—	106	2	no	—
63	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
64	#346	Dic₅₃	1	4	yes	—
65	C₅₃	#12892	1	1	yes	characteristic, complemented, Sylow
66	C₂×M₁₆	—	106	4	no	—
67	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
68	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
69	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
70	C₂×C₁₆	—	2	4	no	—
71	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
72	#108	—	106	8	no	—
73	#108	—	212	4	no	—
74	C₄⋊C₈	—	212	4	no	—
75	C₄×C₈	Dic₁₀₆	1	2	yes	—
76	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
77	C₂²×C₄	Dic₂₁₂	1	2	yes	—
78	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
79	M₁₆	—	212	4	no	—
80	M₁₆	—	212	4	no	—
81	C₂×C₈	—	212	4	no	—
82	C₂×C₈	#11719913	1	4	yes	—
83	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
84	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
85	C₂×C₈	#11719945	1	4	yes	—
86	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
87	C₄²	D₄₂₄	1	2	yes	—
88	C₄²	C₄⋊Dic₅₃	1	2	yes	—
89	C₁₆	—	2	4	no	—
90	C₂³	C₂×Dic₂₁₂	1	1	yes	characteristic
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	#13133151	1	4	yes	—
93	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
94	C₂×C₄	#13133205	1	2	yes	—
95	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
96	C₂×C₄	C₂×#11719913	1	2	yes	—
97	C₈	—	424	4	no	—
98	C₈	#13133149	1	4	yes	—
99	C₈	#13133176	1	2	yes	—
100	C₈	#13133209	1	2	yes	—
101	C₂²	—	2	2	no	—
102	C₂²	C₂×#13133151	1	1	yes	characteristic
103	C₂²	#16810974	1	2	yes	—
104	C₄	—	2	4	no	—
105	C₄	#16810489	1	1	yes	characteristic
106	C₄	C₂×#13133149	1	1	yes	characteristic
107	C₄	#16810970	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	#20274843	1	1	yes	characteristic
110	C₂	#20276510	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20274887

C₂

yes

maximal

#20272809

C₂

yes

maximal

C₂×C₅₃×#346

C₂

yes

maximal, characteristic

C₂×#13133153

—

C₂×#13133153

C₂²

yes

—

#16810647

—

C₂²×C₁₆×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#346

C₄

yes

—

—

—

—

—

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

characteristic

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

Q₁₆

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₃

#41

yes

—

C₄²×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

D₈

yes

—

#11719917

—

#11719917

—

C₂×#88322

—

C₁₆×C₅₃

—

C₂³×C₅₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#107

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#124

yes

—

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

C₈×C₅₃

#106

yes

—

C₈×C₅₃

#117

yes

—

C₈×C₅₃

#125

yes

—

#88322

—

#12892

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

C₂×#107

yes

characteristic

C₂²×C₅₃

#487

yes

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₂×#346

D₅₃

yes

characteristic

#1848

—

#1381

—

C₂×C₅₃

—

C₂×C₅₃

#1834

yes

characteristic

C₂×C₅₃

#2161

yes

—

C₂²×C₁₆

C₂×D₅₃

yes

—

C₂×#108

—

C₂×#108

—

106

—

#423

—

106

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

#346

Dic₅₃

yes

—

C₅₃

#12892

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

106

—

C₂²×C₈

D₂₁₂

yes

characteristic

C₂²×C₈

C₂²×D₅₃

yes

characteristic

C₂×C₄²

D₂₁₂

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₅₃

yes

—

#108

—

106

—

#108

—

212

—

C₄⋊C₈

—

212

—

C₄×C₈

Dic₁₀₆

yes

—

C₄×C₈

C₂×Dic₅₃

yes

—

C₂²×C₄

Dic₂₁₂

yes

—

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

M₁₆

—

212

—

M₁₆

—

212

—

C₂×C₈

—

212

—

C₂×C₈

#11719913

yes

—

C₂×C₈

C₄⋊Dic₅₃

yes

—

C₂×C₈

C₂×Dic₁₀₆

yes

characteristic

C₂×C₈

#11719945

yes

—

C₂×C₈

C₂²×Dic₅₃

yes

characteristic

C₄²

D₄₂₄

yes

—

C₄²

C₄⋊Dic₅₃

yes

—

C₁₆

—

C₂³

C₂×Dic₂₁₂

yes

characteristic

C₂×C₄

—

C₂×C₄

#13133151

yes

—

C₂×C₄

C₂×C₄⋊Dic₅₃

yes

characteristic

C₂×C₄

#13133205

yes

—

C₂×C₄

C₂×D₄₂₄

yes

characteristic

C₂×C₄

C₂×#11719913

yes

—

C₈

—

424

—

C₈

#13133149

yes

—

C₈

#13133176

yes

—

100

C₈

#13133209

yes

—

101

C₂²

—

102

C₂²

C₂×#13133151

yes

characteristic

103

C₂²

#16810974

yes

—

104

C₄

—

105

C₄

#16810489

yes

characteristic

106

C₄

C₂×#13133149

yes

characteristic

107

C₄

#16810970

yes

—

108

C₂

—

109

C₂

#20274843

yes

characteristic

110

C₂

#20276510

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₄₂₄
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#346	C₂
socle	C₂²×C₅₃	C₂×#107
prime radical	C₂³×C₅₃	C₂×Q₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#346	D₅₃
hypocenter	C₅₃	#12892
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₄

C₂×D₄₂₄

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₂×C₅₃×#346

C₂

socle

C₂²×C₅₃

C₂×#107

prime radical

C₂³×C₅₃

C₂×Q₁₆

cosocle

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
65. C₅₃	#12892	semidirect

H ≅ G/N

type

65. C₅₃

#12892

semidirect

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

subgroup	index
#20274887	2
#20272809	2
C₂×C₅₃×#346	2
#12892	53

subgroup

index

#20274887

#20272809

C₂×C₅₃×#346

#12892

Subgroups — #27300485

Conjugacy classes of subgroups

Complements

Maximal subgroups