Group #27304690 — subgroups

Subgroups — #27304690

Order 13568.

188 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 73 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	Q₃₂×Dic₅₃	C₂	1	2	yes	maximal
2	#20272917	C₂	1	1	yes	maximal, characteristic
3	#20272896	C₂	1	2	yes	maximal
4	#20272848	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₃×Q₆₄	C₂	1	1	yes	maximal, characteristic
6	Q₁₆×Dic₅₃	—	2	2	no	—
7	C₂×#13133123	C₂²	1	1	yes	characteristic
8	#16810449	C₂²	1	2	yes	—
9	#16810427	—	2	2	no	—
10	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂×Q₃₂×C₅₃	C₂²	1	2	yes	—
12	C₅₃×Q₆₄	C₄	1	4	yes	complemented
13	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
14	Q₈×Dic₅₃	—	4	2	no	—
15	C₂×#11719908	D₄	1	1	yes	characteristic
16	#13133151	—	2	2	no	—
17	#13133141	—	4	2	no	—
18	C₈×Dic₅₃	D₄	1	1	yes	characteristic
19	#13133123	—	2	1	no	—
20	C₂×Q₁₆×C₅₃	—	2	2	no	—
21	Q₃₂×C₅₃	—	2	2	no	—
22	Q₃₂×C₅₃	C₂×C₄	1	4	yes	—
23	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
24	C₃₂×C₅₃	C₂×C₄	1	2	yes	—
25	C₂×Q₈×C₅₃	—	4	2	no	—
26	Q₁₆×C₅₃	—	2	4	no	—
27	Q₁₆×C₅₃	—	4	2	no	—
28	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
29	C₄⋊Dic₅₃	—	4	2	no	—
30	C₄⋊Dic₅₃	—	8	2	no	—
31	C₄×Dic₅₃	—	8	2	no	—
32	C₄×Dic₅₃	D₈	1	1	yes	characteristic
33	C₂×#88322	D₈	1	1	yes	characteristic
34	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₁₆×C₅₃	#41	1	1	yes	characteristic
36	#11719908	—	2	1	no	—
37	Q₈×C₅₃	—	4	4	no	—
38	Q₈×C₅₃	—	8	2	no	—
39	C₂×C₄×C₅₃	—	8	2	no	—
40	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
41	C₂×Dic₅₃	—	8	2	no	—
42	C₂×Dic₅₃	D₁₆	1	2	yes	—
43	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
44	C₈×C₅₃	#135	1	1	yes	characteristic
45	#88322	—	2	1	no	—
46	#13653	—	53	1	no	maximal, Sylow
47	C₂²×C₅₃	C₂×D₁₆	1	1	yes	characteristic
48	C₄×C₅₃	—	8	4	no	—
49	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
50	C₄×C₅₃	#508	1	1	yes	characteristic
51	Dic₅₃	—	2	2	no	—
52	Dic₅₃	—	16	2	no	—
53	C₂×Q₆₄	D₅₃	1	1	yes	characteristic
54	C₄×Q₃₂	—	53	2	no	—
55	#1424	—	53	1	no	—
56	#1416	—	53	2	no	—
57	#1397	—	53	1	no	—
58	C₂×C₅₃	C₄×D₁₆	1	1	yes	characteristic
59	C₂×C₅₃	#2264	1	2	yes	—
60	C₂×Q₃₂	C₂×D₅₃	1	2	yes	—
61	C₄×Q₁₆	—	106	2	no	—
62	Q₆₄	Dic₅₃	1	4	yes	complemented
63	C₂×C₃₂	—	53	1	no	—
64	C₂×C₃₂	C₂×D₅₃	1	1	yes	characteristic
65	#366	—	53	2	no	—
66	#358	—	106	2	no	—
67	C₄×C₁₆	—	53	1	no	—
68	C₅₃	#13653	1	1	yes	characteristic, complemented, Sylow
69	C₂×Q₁₆	—	2	2	no	—
70	C₄×Q₈	—	212	2	no	—
71	Q₃₂	—	2	2	no	—
72	Q₃₂	C₂×Dic₅₃	1	4	yes	—
73	C₂×C₁₆	—	53	1	no	—
74	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
75	#107	—	106	2	no	—
76	#103	—	212	2	no	—
77	C₄×C₈	—	53	1	no	—
78	C₃₂	—	106	1	no	—
79	C₃₂	C₂×Dic₅₃	1	2	yes	—
80	C₂×Q₈	—	4	2	no	—
81	Q₁₆	—	2	4	no	—
82	Q₁₆	—	4	2	no	—
83	C₂×C₈	—	53	1	no	—
84	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
85	C₄⋊C₄	—	424	2	no	—
86	C₄⋊C₄	—	212	2	no	—
87	C₄²	—	53	1	no	—
88	C₄²	—	424	2	no	—
89	C₁₆	—	106	1	no	—
90	C₁₆	#11719947	1	1	yes	characteristic
91	C₁₆	C₂²×Dic₅₃	1	1	yes	characteristic
92	Q₈	—	4	4	no	—
93	Q₈	—	8	2	no	—
94	C₂×C₄	—	8	2	no	—
95	C₂×C₄	—	53	2	no	—
96	C₂×C₄	—	424	2	no	—
97	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
98	C₈	—	106	1	no	—
99	C₈	D₄×Dic₅₃	1	1	yes	characteristic
100	C₈	#13133236	1	1	yes	characteristic
101	C₂²	D₁₆×D₅₃	1	1	yes	characteristic
102	C₄	—	8	4	no	—
103	C₄	—	848	2	no	—
104	C₄	—	106	2	no	—
105	C₄	D₈×Dic₅₃	1	1	yes	characteristic
106	C₄	#16811052	1	1	yes	characteristic
107	C₂	D₁₆×Dic₅₃	1	1	yes	characteristic
108	C₂	#20277034	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

Q₃₂×Dic₅₃

C₂

yes

maximal

#20272917

C₂

yes

maximal, characteristic

#20272896

C₂

yes

maximal

#20272848

C₂

yes

maximal, characteristic

C₂×C₅₃×Q₆₄

C₂

yes

maximal, characteristic

Q₁₆×Dic₅₃

—

C₂×#13133123

C₂²

yes

characteristic

#16810449

C₂²

yes

—

#16810427

—

C₁₆×Dic₅₃

C₂²

yes

characteristic

C₂×Q₃₂×C₅₃

C₂²

yes

—

C₅₃×Q₆₄

C₄

yes

complemented

C₂×C₃₂×C₅₃

C₂²

yes

characteristic

Q₈×Dic₅₃

—

C₂×#11719908

D₄

yes

characteristic

#13133151

—

#13133141

—

C₈×Dic₅₃

D₄

yes

characteristic

#13133123

—

C₂×Q₁₆×C₅₃

—

Q₃₂×C₅₃

—

Q₃₂×C₅₃

C₂×C₄

yes

—

C₂×C₁₆×C₅₃

C₂³

yes

characteristic

C₃₂×C₅₃

C₂×C₄

yes

—

C₂×Q₈×C₅₃

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

D₈

yes

characteristic

C₂×#88322

D₈

yes

characteristic

C₁₆×C₅₃

C₂²×C₄

yes

C₁₆×C₅₃

yes

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

D₁₆

yes

—

C₈×C₅₃

C₄×D₄

yes

C₈×C₅₃

yes

—

—

C₂²×C₅₃

C₂×D₁₆

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₄×D₈

yes

characteristic

C₄×C₅₃

#508

yes

characteristic

Dic₅₃

—

Dic₅₃

—

C₂×Q₆₄

D₅₃

yes

characteristic

C₄×Q₃₂

—

#1424

—

#1416

—

#1397

—

C₂×C₅₃

C₄×D₁₆

yes

characteristic

C₂×C₅₃

#2264

yes

—

C₂×Q₃₂

C₂×D₅₃

yes

—

C₄×Q₁₆

—

106

—

Q₆₄

Dic₅₃

yes

complemented

C₂×C₃₂

—

C₂×C₃₂

C₂×D₅₃

yes

characteristic

#366

—

#358

—

106

—

C₄×C₁₆

—

C₅₃

#13653

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

212

—

Q₃₂

—

Q₃₂

C₂×Dic₅₃

yes

—

C₂×C₁₆

—

C₂×C₁₆

C₂²×D₅₃

yes

characteristic

#107

—

106

—

#103

—

212

—

C₄×C₈

—

C₃₂

—

106

—

C₃₂

C₂×Dic₅₃

yes

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

D₄×D₅₃

yes

characteristic

C₄⋊C₄

—

424

—

C₄⋊C₄

—

212

—

C₄²

—

C₄²

—

424

—

C₁₆

—

106

—

C₁₆

#11719947

yes

characteristic

C₁₆

C₂²×Dic₅₃

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

424

—

C₂×C₄

D₈×D₅₃

yes

characteristic

C₈

—

106

—

C₈

D₄×Dic₅₃

yes

characteristic

100

C₈

#13133236

yes

characteristic

101

C₂²

D₁₆×D₅₃

yes

characteristic

102

C₄

—

103

C₄

—

848

—

104

C₄

—

106

—

105

C₄

D₈×Dic₅₃

yes

characteristic

106

C₄

#16811052

yes

characteristic

107

C₂

D₁₆×Dic₅₃

yes

characteristic

108

C₂

#20277034

yes

—

	subgroup	G/H
center	C₂²	D₁₆×D₅₃
commutator	C₁₆×C₅₃	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₂×C₅₃×Q₆₄	C₂
socle, prime radical	C₂²×C₅₃	C₂×D₁₆
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	C₂×Q₆₄	D₅₃
hypocenter	C₅₃	#13653
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×Q₆₄

subgroup

G/H

center

C₂²

D₁₆×D₅₃

commutator

C₁₆×C₅₃

C₂²×C₄

Frattini

C₂×C₁₆

C₂²×D₅₃

Fitting

C₂×C₅₃×Q₆₄

C₂

socle, prime radical

C₂²×C₅₃

C₂×D₁₆

cosocle

C₂×C₁₆×C₅₃

C₂³

hypercenter

C₂×Q₆₄

D₅₃

hypocenter

C₅₃

#13653

LCS

G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×Q₆₄

N	H ≅ G/N	type
12. C₅₃×Q₆₄	C₄	semidirect
62. Q₆₄	Dic₅₃	semidirect
68. C₅₃	#13653	semidirect

H ≅ G/N

type

12. C₅₃×Q₆₄

C₄

semidirect

62. Q₆₄

Dic₅₃

semidirect

68. C₅₃

#13653

semidirect

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

subgroup	index
Q₃₂×Dic₅₃	2
#20272917	2
#20272896	2
#20272848	2
C₂×C₅₃×Q₆₄	2
#13653	53

subgroup

index

Q₃₂×Dic₅₃

C₂×C₅₃×Q₆₄

Subgroups — #27304690

Conjugacy classes of subgroups

Complements

Maximal subgroups