Group #27277906 — subgroups

Subgroups — #27277906

Order 13568.

338 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 187 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	#20273474	C₂	1	4	yes	maximal
3	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133153	—	2	4	no	—
5	#16810647	C₂²	1	4	yes	—
6	#16810376	—	2	8	no	—
7	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#322	C₄	1	4	yes	—
10	C₂×#11719917	—	2	4	no	—
11	#13133153	—	4	8	no	—
12	C₄⋊#88322	—	2	8	no	—
13	C₂²×C₈×C₅₃	D₄	1	2	yes	—
14	C₂²×C₈×C₅₃	Q₈	1	4	yes	—
15	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
16	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
17	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₅₃	D₄	1	4	yes	—
19	C₄×C₈×C₅₃	Q₈	1	2	yes	—
20	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
22	C₂×C₈×C₅₃	—	4	4	no	—
23	C₂×C₈×C₅₃	C₄⋊C₄	1	8	yes	—
24	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
25	C₂×C₈×C₅₃	D₈	1	4	yes	—
26	C₂×C₈×C₅₃	Q₁₆	1	4	yes	—
27	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
28	C₄²×C₅₃	C₂×D₄	1	2	yes	—
29	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
30	#11719917	—	4	8	no	—
31	C₂×#88322	—	4	4	no	—
32	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
33	C₂×C₄×C₅₃	—	2	4	no	—
34	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
36	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
37	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
38	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
39	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
40	C₈×C₅₃	—	2	8	no	—
41	C₈×C₅₃	#107	1	8	yes	—
42	#88322	—	8	4	no	—
43	#8826	—	53	1	no	maximal, Sylow
44	C₂²×C₅₃	—	2	2	no	—
45	C₂²×C₅₃	#384	1	1	yes	characteristic
46	C₂²×C₅₃	#500	1	2	yes	—
47	C₄×C₅₃	—	2	2	no	—
48	C₄×C₅₃	C₂×#107	1	2	yes	—
49	C₄×C₅₃	#430	1	4	yes	—
50	C₄×C₅₃	#493	1	1	yes	characteristic
51	C₄×C₅₃	#494	1	1	yes	characteristic
52	#1848	—	53	2	no	—
53	#1567	—	53	4	no	—
54	C₂×#322	D₅₃	1	1	yes	characteristic
55	C₂×C₅₃	—	2	2	no	—
56	C₂×C₅₃	#1845	1	1	yes	characteristic
57	C₂×C₅₃	#1849	1	2	yes	—
58	C₂×#108	—	106	4	no	—
59	#423	—	53	4	no	—
60	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
61	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
62	#335	—	106	8	no	—
63	#322	Dic₅₃	1	4	yes	—
64	C₅₃	#8826	1	1	yes	characteristic, complemented, Sylow
65	C₂×M₁₆	—	106	4	no	—
66	C₂²×C₈	Dic₁₀₆	1	4	yes	—
67	C₂²×C₈	D₂₁₂	1	2	yes	—
68	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
69	#108	—	212	8	no	—
70	C₄⋊C₈	—	106	8	no	—
71	C₄×C₈	Dic₁₀₆	1	2	yes	—
72	C₄×C₈	D₂₁₂	1	4	yes	—
73	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
74	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
75	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
76	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
77	M₁₆	—	212	8	no	—
78	C₂×C₈	—	4	4	no	—
79	C₂×C₈	—	212	4	no	—
80	C₂×C₈	D₄₂₄	1	4	yes	—
81	C₂×C₈	Dic₂₁₂	1	4	yes	—
82	C₂×C₈	C₄⋊Dic₅₃	1	8	yes	—
83	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
84	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
85	C₄²	C₂×D₂₁₂	1	2	yes	—
86	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
87	C₂³	#13133217	1	1	yes	characteristic
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
90	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
91	C₂×C₄	#13133214	1	1	yes	characteristic
92	C₂×C₄	#13133217	1	2	yes	—
93	C₂×C₄	C₂×D₄₂₄	1	2	yes	—
94	C₂×C₄	C₂×Dic₂₁₂	1	2	yes	—
95	C₈	—	2	8	no	—
96	C₈	—	424	4	no	—
97	C₈	#13133151	1	8	yes	—
98	C₂²	—	2	2	no	—
99	C₂²	#16810493	1	1	yes	characteristic
100	C₂²	#16811028	1	2	yes	—
101	C₄	—	2	2	no	—
102	C₄	C₂×#13133151	1	2	yes	—
103	C₄	#16810672	1	4	yes	—
104	C₄	#16811000	1	1	yes	characteristic
105	C₄	#16811003	1	1	yes	characteristic
106	C₂	—	2	2	no	—
107	C₂	#20274875	1	1	yes	characteristic
108	C₂	#20274890	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20274887

C₂

yes

maximal

#20273474

C₂

yes

maximal

C₂×C₅₃×#322

C₂

yes

maximal, characteristic

C₂×#13133153

—

#16810647

C₂²

yes

—

#16810376

—

C₂×C₄×C₈×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#322

C₄

yes

—

C₂×#11719917

—

#13133153

—

C₄⋊#88322

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂²×C₈×C₅₃

Q₈

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

Q₁₆

yes

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

C₂×Q₈

yes

characteristic

#11719917

—

C₂×#88322

—

C₂³×C₅₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₄⋊Q₈

yes

—

C₂×C₄×C₅₃

C₂×D₈

yes

—

C₂×C₄×C₅₃

C₂×Q₁₆

yes

—

C₈×C₅₃

—

C₈×C₅₃

#107

yes

—

#88322

—

#8826

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

#384

yes

characteristic

C₂²×C₅₃

#500

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×#107

yes

—

C₄×C₅₃

#430

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

—

—

D₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

#1845

yes

characteristic

C₂×C₅₃

#1849

yes

—

C₂×#108

—

106

—

#423

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×C₄×C₈

C₂×D₅₃

yes

—

#335

—

106

—

#322

Dic₅₃

yes

—

C₅₃

#8826

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

106

—

C₂²×C₈

Dic₁₀₆

yes

—

C₂²×C₈

D₂₁₂

yes

—

C₂×C₄²

C₂²×D₅₃

yes

characteristic

#108

—

212

—

C₄⋊C₈

—

106

—

C₄×C₈

Dic₁₀₆

yes

—

C₄×C₈

D₂₁₂

yes

—

C₄×C₈

C₂×Dic₅₃

yes

—

C₄×C₈

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₁₀₆

yes

—

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

M₁₆

—

212

—

C₂×C₈

—

C₂×C₈

—

212

—

C₂×C₈

D₄₂₄

yes

—

C₂×C₈

Dic₂₁₂

yes

—

C₂×C₈

C₄⋊Dic₅₃

yes

—

C₂×C₈

C₄⋊Dic₅₃

yes

—

C₄²

C₂×Dic₁₀₆

yes

characteristic

C₄²

C₂×D₂₁₂

yes

—

C₄²

C₂²×Dic₅₃

yes

characteristic

C₂³

#13133217

yes

characteristic

C₂×C₄

—

C₂×C₄

C₂×C₄⋊Dic₅₃

yes

characteristic

C₂×C₄

C₂×C₄⋊Dic₅₃

yes

—

C₂×C₄

#13133214

yes

characteristic

C₂×C₄

#13133217

yes

—

C₂×C₄

C₂×D₄₂₄

yes

—

C₂×C₄

C₂×Dic₂₁₂

yes

—

C₈

—

C₈

—

424

—

C₈

#13133151

yes

—

C₂²

—

C₂²

#16810493

yes

characteristic

100

C₂²

#16811028

yes

—

101

C₄

—

102

C₄

C₂×#13133151

yes

—

103

C₄

#16810672

yes

—

104

C₄

#16811000

yes

characteristic

105

C₄

#16811003

yes

characteristic

106

C₂

—

107

C₂

#20274875

yes

characteristic

108

C₂

#20274890

yes

—

	subgroup	G/H
center	C₂×C₄	#13133214
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#322	C₂
socle	C₂²×C₅₃	#384
prime radical	C₂³×C₅₃	C₄⋊Q₈
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#322	D₅₃
hypocenter	C₅₃	#8826
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂×C₄

#13133214

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₂×C₅₃×#322

C₂

socle

C₂²×C₅₃

#384

prime radical

C₂³×C₅₃

C₄⋊Q₈

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

N	H ≅ G/N	type
64. C₅₃	#8826	semidirect

H ≅ G/N

type

64. C₅₃

#8826

semidirect

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

subgroup	index
#20274887	2
#20273474	2
C₂×C₅₃×#322	2
#8826	53

subgroup

index

#20274887

#20273474

C₂×C₅₃×#322

#8826

Subgroups — #27277906

Conjugacy classes of subgroups

Complements

Maximal subgroups