Group #27536613 — subgroups

Subgroups — #27536613

Order 13568.

1422 conjugacy classes of subgroups; only the 395 normal subgroups are shown.

normal subgroups only

This group has 1422 conjugacy classes of subgroups — too many to draw. Only its 395 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	C₂×#16811002	C₂	1	12	yes	maximal
2	#20283113	C₂	1	1	yes	maximal, characteristic
3	C₄²×#88322	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#2868	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719925	C₂²	1	16	yes	—
6	C₂×#13133217	C₂²	1	6	yes	—
7	#16811002	—	2	48	no	—
8	C₂×C₄×#88322	C₂²	1	6	yes	—
9	#16810493	—	2	4	no	—
10	C₂×C₄⋊Q₈×C₅₃	C₂²	1	6	yes	—
11	C₅₃×#384	—	2	4	no	—
12	C₄³×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×Dic₁₀₆	C₂³	1	4	yes	—
14	C₂×#11719925	—	2	96	no	—
15	C₂²×#88322	C₂³	1	4	yes	—
16	#13133217	—	2	24	no	—
17	C₂×C₄⋊Dic₅₃	—	2	12	no	—
18	C₄×#88322	D₄	1	24	yes	—
19	C₂²×Q₈×C₅₃	C₂³	1	4	yes	—
20	C₄⋊Q₈×C₅₃	—	2	24	no	—
21	C₂×C₄⋊C₄×C₅₃	—	2	12	no	—
22	C₂×C₄²×C₅₃	D₄	1	4	yes	—
23	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
24	C₂×Q₈×C₅₃	—	2	24	no	—
25	C₂²×C₄×C₅₃	—	2	4	no	—
26	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
27	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
28	C₂²×Dic₅₃	—	2	4	no	—
29	C₂×Dic₁₀₆	—	2	24	no	—
30	C₄⋊C₄×C₅₃	—	4	24	no	—
31	C₄²×C₅₃	Q₁₆	1	16	yes	complemented
32	C₄²×C₅₃	C₂×D₄	1	12	yes	—
33	#11719925	—	4	64	no	—
34	C₄⋊Dic₅₃	—	4	24	no	—
35	C₂×#88322	C₂×D₄	1	24	yes	—
36	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
37	Q₈×C₅₃	—	4	16	no	—
38	C₂×C₄×C₅₃	—	4	12	no	—
39	C₂×C₄×C₅₃	#127	1	12	yes	—
40	C₂×C₄×C₅₃	C₂×Q₁₆	1	24	yes	—
41	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
42	C₂×Dic₅₃	—	4	12	no	—
43	Dic₁₀₆	—	4	16	no	—
44	#88322	#127	1	16	yes	—
45	#33302	—	53	1	no	maximal, Sylow
46	C₂²×C₅₃	C₂×#127	1	3	yes	—
47	C₂²×C₅₃	C₂²×Q₁₆	1	4	yes	—
48	C₄×C₅₃	—	8	4	no	—
49	C₄×C₅₃	#494	1	24	yes	complemented
50	C₄×C₅₃	C₂×#127	1	4	yes	—
51	Dic₅₃	—	8	4	no	—
52	C₂×#494	—	53	12	no	—
53	#2868	—	53	1	no	—
54	#2868	D₅₃	1	1	yes	characteristic
55	C₄²×C₈	—	53	1	no	—
56	C₂×C₅₃	#2867	1	1	yes	characteristic
57	C₂×C₅₃	C₂×#494	1	6	yes	—
58	C₂²×Q₁₆	—	53	16	no	—
59	C₂×C₄⋊Q₈	—	53	6	no	—
60	C₂×C₄⋊Q₈	C₂×D₅₃	1	6	yes	—
61	#494	—	106	48	no	—
62	C₂×C₄×C₈	—	53	6	no	—
63	#384	—	2	4	no	—
64	#384	—	106	4	no	—
65	C₄³	C₂×D₅₃	1	1	yes	characteristic
66	C₅₃	#33302	1	1	yes	characteristic, complemented, Sylow
67	C₂²×Q₈	—	53	4	no	—
68	C₂²×Q₈	C₂²×D₅₃	1	4	yes	—
69	C₂×Q₁₆	—	106	96	no	—
70	C₂²×C₈	—	53	4	no	—
71	C₄⋊Q₈	—	2	24	no	—
72	C₄⋊Q₈	—	106	24	no	—
73	C₂×C₄⋊C₄	—	2	12	no	—
74	C₂×C₄⋊C₄	—	106	12	no	—
75	C₂×C₄²	C₂²⋊D₅₃	1	4	yes	—
76	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
77	C₄×C₈	—	53	24	no	—
78	C₂×Q₈	—	2	24	no	—
79	C₂×Q₈	—	106	24	no	—
80	C₂²×C₄	—	2	4	no	—
81	C₂²×C₄	—	106	4	no	—
82	C₂²×C₄	C₂×C₂²⋊D₅₃	1	6	yes	—
83	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
84	Q₁₆	—	212	64	no	—
85	C₂×C₈	—	53	24	no	—
86	C₄⋊C₄	—	4	24	no	—
87	C₄⋊C₄	—	212	24	no	—
88	C₄²	#11719925	1	16	yes	complemented
89	C₄²	D₄×D₅₃	1	12	yes	—
90	C₂³	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
91	Q₈	—	4	16	no	—
92	Q₈	—	212	16	no	—
93	C₂×C₄	—	4	12	no	—
94	C₂×C₄	—	212	12	no	—
95	C₂×C₄	#13133213	1	12	yes	—
96	C₂×C₄	C₂×#11719925	1	24	yes	—
97	C₂×C₄	C₂×D₄×D₅₃	1	6	yes	—
98	C₈	—	53	16	no	—
99	C₂²	C₂×#13133213	1	3	yes	—
100	C₂²	C₂²×#11719925	1	4	yes	—
101	C₄	—	8	4	no	—
102	C₄	—	424	4	no	—
103	C₄	#16811002	1	24	yes	complemented
104	C₄	#127×D₅₃	1	4	yes	—
105	C₂	#20283109	1	1	yes	characteristic
106	C₂	C₂×#16811002	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16811002

C₂

yes

maximal

#20283113

C₂

yes

maximal, characteristic

C₄²×#88322

C₂

yes

maximal, characteristic

C₅₃×#2868

C₂

yes

maximal, characteristic

C₂²×#11719925

C₂²

yes

—

C₂×#13133217

C₂²

yes

—

#16811002

—

C₂×C₄×#88322

C₂²

yes

—

#16810493

—

C₂×C₄⋊Q₈×C₅₃

C₂²

yes

—

C₅₃×#384

—

C₄³×C₅₃

C₂²

yes

characteristic

C₂²×Dic₁₀₆

C₂³

yes

—

C₂×#11719925

—

C₂²×#88322

C₂³

yes

—

#13133217

—

C₂×C₄⋊Dic₅₃

—

C₄×#88322

D₄

yes

—

C₂²×Q₈×C₅₃

C₂³

yes

—

C₄⋊Q₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×C₄×C₅₃

C₂⁴

yes

characteristic

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

Q₁₆

yes

complemented

C₄²×C₅₃

C₂×D₄

yes

—

#11719925

—

C₄⋊Dic₅₃

—

C₂×#88322

C₂×D₄

yes

—

C₂³×C₅₃

C₂²×D₄

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#127

yes

—

C₂×C₄×C₅₃

C₂×Q₁₆

yes

—

C₂×C₄×C₅₃

C₂²×D₄

yes

—

C₂×Dic₅₃

—

Dic₁₀₆

—

#88322

#127

yes

—

#33302

—

maximal, Sylow

C₂²×C₅₃

C₂×#127

yes

—

C₂²×C₅₃

C₂²×Q₁₆

yes

—

C₄×C₅₃

—

C₄×C₅₃

#494

yes

complemented

C₄×C₅₃

C₂×#127

yes

—

Dic₅₃

—

C₂×#494

—

#2868

—

#2868

D₅₃

yes

characteristic

C₄²×C₈

—

C₂×C₅₃

#2867

yes

characteristic

C₂×C₅₃

C₂×#494

yes

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

C₂×D₅₃

yes

—

#494

—

106

—

C₂×C₄×C₈

—

#384

—

#384

—

106

—

C₄³

C₂×D₅₃

yes

characteristic

C₅₃

#33302

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

C₂²×D₅₃

yes

—

C₂×Q₁₆

—

106

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

106

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

C₂²⋊D₅₃

yes

—

C₂×C₄²

C₂²×D₅₃

yes

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

—

C₂²×C₄

C₂³×D₅₃

yes

characteristic

Q₁₆

—

212

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

212

—

C₄²

#11719925

yes

complemented

C₄²

D₄×D₅₃

yes

—

C₂³

C₂²×C₂²⋊D₅₃

yes

characteristic

Q₈

—

Q₈

—

212

—

C₂×C₄

—

C₂×C₄

—

212

—

C₂×C₄

#13133213

yes

—

C₂×C₄

C₂×#11719925

yes

—

C₂×C₄

C₂×D₄×D₅₃

yes

—

C₈

—

C₂²

C₂×#13133213

yes

—

100

C₂²

C₂²×#11719925

yes

—

101

C₄

—

102

C₄

—

424

—

103

C₄

#16811002

yes

complemented

104

C₄

#127×D₅₃

yes

—

105

C₂

#20283109

yes

characteristic

106

C₂

C₂×#16811002

yes

—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2868	C₂
socle, prime radical	C₂³×C₅₃	C₂²×D₄
hypercenter	#2868	D₅₃
hypocenter	C₅₃	#33302
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

subgroup

G/H

center

C₂³

C₂²×C₂²⋊D₅₃

commutator, cosocle

C₂²×C₄×C₅₃

C₂⁴

Frattini

C₂²×C₄

C₂³×D₅₃

Fitting

C₅₃×#2868

C₂

socle, prime radical

C₂³×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

N	H ≅ G/N	type
31. C₄²×C₅₃	Q₁₆	semidirect
49. C₄×C₅₃	#494	semidirect
66. C₅₃	#33302	semidirect
88. C₄²	#11719925	semidirect
103. C₄	#16811002	semidirect

H ≅ G/N

type

31. C₄²×C₅₃

Q₁₆

semidirect

49. C₄×C₅₃

#494

semidirect

66. C₅₃

#33302

semidirect

88. C₄²

#11719925

semidirect

103. C₄

#16811002

semidirect

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

subgroup	index
C₂×#16811002	2
#20283113	2
C₄²×#88322	2
C₅₃×#2868	2
#33302	53

subgroup

index

C₂×#16811002

#20283113

C₄²×#88322

C₅₃×#2868

#33302

Subgroups — #27536613

Conjugacy classes of subgroups

Complements

Maximal subgroups