Group #27305084 — subgroups

Subgroups — #27305084

Order 13568.

174 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 67 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	#20276968	C₂	1	1	yes	maximal, characteristic
2	#20272895	C₂	1	4	yes	maximal
3	#20272849	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#2253	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₄₂₄	—	2	1	no	—
6	C₂×Dic₄₂₄	C₂²	1	2	yes	—
7	#16811003	—	2	1	no	—
8	C₂×#13133123	C₂²	1	2	yes	—
9	C₅₃×#499	—	2	1	no	—
10	C₅₃×#367	—	2	1	no	—
11	C₅₃×#366	C₂²	1	2	yes	—
12	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×Dic₂₁₂	—	2	2	no	—
14	C₂×Dic₂₁₂	—	4	1	no	—
15	#13133217	—	4	1	no	—
16	Dic₄₂₄	—	2	4	no	—
17	Dic₄₂₄	—	4	2	no	—
18	#13133123	—	2	2	no	—
19	#125×C₅₃	—	4	1	no	—
20	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
21	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
22	#107×C₅₃	—	2	2	no	—
23	#106×C₅₃	—	4	1	no	—
24	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
25	C₂×Dic₁₀₆	—	4	2	no	—
26	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
27	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
28	C₄⋊C₄×C₅₃	—	4	2	no	—
29	C₄⋊C₄×C₅₃	—	8	2	no	—
30	C₄²×C₅₃	D₈	1	1	yes	characteristic
31	C₄⋊Dic₅₃	—	8	2	no	—
32	Dic₂₁₂	—	4	2	no	—
33	Dic₂₁₂	—	4	2	no	—
34	Dic₂₁₂	—	8	2	no	—
35	C₁₆×C₅₃	SD₁₆	1	2	yes	—
36	C₁₆×C₅₃	#41	1	2	yes	—
37	C₂×C₄×C₅₃	—	8	2	no	—
38	C₂×C₄×C₅₃	D₁₆	1	2	yes	—
39	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
40	C₂×Dic₅₃	—	8	2	no	—
41	Dic₁₀₆	—	8	2	no	—
42	Dic₁₀₆	—	8	2	no	—
43	C₈×C₅₃	SD₃₂	1	2	yes	—
44	C₈×C₅₃	#124	1	1	yes	characteristic
45	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
46	#13724	—	53	1	no	maximal, Sylow
47	C₂²×C₅₃	C₂×D₁₆	1	1	yes	characteristic
48	C₄×C₅₃	—	2	2	no	—
49	C₄×C₅₃	—	16	2	no	—
50	C₄×C₅₃	#486	1	1	yes	characteristic
51	C₄×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
52	Dic₅₃	—	16	2	no	—
53	#2253	D₅₃	1	1	yes	characteristic
54	#2247	—	53	1	no	—
55	#1416	—	53	4	no	—
56	#1397	—	53	1	no	—
57	C₂×C₅₃	#2240	1	1	yes	characteristic
58	C₂×C₅₃	#2264	1	2	yes	—
59	C₂×Q₃₂	—	53	2	no	—
60	C₂×Q₃₂	—	106	1	no	—
61	#499	—	2	1	no	—
62	#494	—	106	1	no	—
63	C₂×C₃₂	—	53	2	no	—
64	#367	—	2	1	no	—
65	#366	C₂×D₅₃	1	2	yes	—
66	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
67	C₅₃	#13724	1	1	yes	characteristic, complemented, Sylow
68	C₂×Q₁₆	—	106	2	no	—
69	C₂×Q₁₆	—	212	1	no	—
70	C₄⋊Q₈	—	212	1	no	—
71	#125	—	4	1	no	—
72	Q₃₂	—	106	4	no	—
73	Q₃₂	—	212	2	no	—
74	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
75	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
76	#107	—	2	2	no	—
77	#106	—	4	1	no	—
78	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
79	C₃₂	—	106	2	no	—
80	C₂×Q₈	—	212	2	no	—
81	Q₁₆	—	424	2	no	—
82	Q₁₆	—	212	2	no	—
83	Q₁₆	—	212	2	no	—
84	C₂×C₈	#11719922	1	1	yes	characteristic
85	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
86	C₄⋊C₄	—	4	2	no	—
87	C₄⋊C₄	—	8	2	no	—
88	C₄⋊C₄	—	424	2	no	—
89	C₄²	#11719922	1	1	yes	characteristic
90	C₁₆	#11719924	1	2	yes	—
91	C₁₆	#11719949	1	2	yes	—
92	Q₈	—	424	2	no	—
93	Q₈	—	424	2	no	—
94	C₂×C₄	—	8	2	no	—
95	C₂×C₄	—	424	2	no	—
96	C₂×C₄	#13133158	1	2	yes	—
97	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
98	C₈	#13133160	1	2	yes	—
99	C₈	#13133206	1	1	yes	characteristic
100	C₈	C₂×#11719924	1	1	yes	characteristic
101	C₂²	C₂×#13133158	1	1	yes	characteristic
102	C₄	—	2	2	no	—
103	C₄	—	16	2	no	—
104	C₄	—	848	2	no	—
105	C₄	#16810971	1	1	yes	characteristic
106	C₄	C₂×#13133160	1	1	yes	characteristic
107	C₂	#20276940	1	1	yes	characteristic
108	C₂	#20277032	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20276968

C₂

yes

maximal, characteristic

#20272895

C₂

yes

maximal

#20272849

C₂

yes

maximal, characteristic

C₅₃×#2253

C₂

yes

maximal, characteristic

C₂×Dic₄₂₄

—

C₂×Dic₄₂₄

C₂²

yes

—

#16811003

—

C₂×#13133123

C₂²

yes

—

C₅₃×#499

—

C₅₃×#367

—

C₅₃×#366

C₂²

yes

—

C₄×C₁₆×C₅₃

C₂²

yes

characteristic

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

#13133217

—

Dic₄₂₄

—

Dic₄₂₄

—

#13133123

—

#125×C₅₃

—

C₂×C₁₆×C₅₃

D₄

yes

characteristic

C₂×C₁₆×C₅₃

C₂³

yes

characteristic

#107×C₅₃

—

#106×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

characteristic

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

D₈

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

D₈

yes

characteristic

C₄⋊Dic₅₃

—

Dic₂₁₂

—

Dic₂₁₂

—

Dic₂₁₂

—

C₁₆×C₅₃

SD₁₆

yes

—

C₁₆×C₅₃

#41

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

D₁₆

yes

—

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

SD₃₂

yes

—

C₈×C₅₃

#124

yes

characteristic

C₈×C₅₃

C₂×SD₁₆

yes

characteristic

#13724

—

maximal, Sylow

C₂²×C₅₃

C₂×D₁₆

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#486

yes

characteristic

C₄×C₅₃

C₂×SD₃₂

yes

characteristic

Dic₅₃

—

#2253

D₅₃

yes

—

—

—

C₂×C₅₃

#2240

yes

characteristic

C₂×C₅₃

#2264

yes

—

C₂×Q₃₂

—

C₂×Q₃₂

—

106

—

#499

—

#494

—

106

—

C₂×C₃₂

—

#367

—

#366

C₂×D₅₃

yes

—

C₄×C₁₆

C₂×D₅₃

yes

characteristic

C₅₃

#13724

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

106

—

C₂×Q₁₆

—

212

—

C₄⋊Q₈

—

212

—

#125

—

Q₃₂

—

106

—

Q₃₂

—

212

—

C₂×C₁₆

C₂²⋊D₅₃

yes

characteristic

C₂×C₁₆

C₂²×D₅₃

yes

characteristic

#107

—

#106

—

C₄×C₈

C₂²⋊D₅₃

yes

characteristic

C₃₂

—

106

—

C₂×Q₈

—

212

—

Q₁₆

—

424

—

Q₁₆

—

212

—

Q₁₆

—

212

—

C₂×C₈

#11719922

yes

characteristic

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

424

—

C₄²

#11719922

yes

characteristic

C₁₆

#11719924

yes

—

C₁₆

#11719949

yes

—

Q₈

—

424

—

Q₈

—

424

—

C₂×C₄

—

C₂×C₄

—

424

—

C₂×C₄

#13133158

yes

—

C₂×C₄

C₂×#11719922

yes

characteristic

C₈

#13133160

yes

—

C₈

#13133206

yes

characteristic

100

C₈

C₂×#11719924

yes

characteristic

101

C₂²

C₂×#13133158

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

848

—

105

C₄

#16810971

yes

characteristic

106

C₄

C₂×#13133160

yes

characteristic

107

C₂

#20276940

yes

characteristic

108

C₂

#20277032

yes

—

	subgroup	G/H
center	C₂²	C₂×#13133158
commutator, cosocle	C₂×C₁₆×C₅₃	C₂³
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₅₃×#2253	C₂
socle, prime radical	C₂²×C₅₃	C₂×D₁₆
hypercenter	#2253	D₅₃
hypocenter	C₅₃	#13724
LCS	G ▷ C₂×C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2253

subgroup

G/H

center

C₂²

C₂×#13133158

commutator, cosocle

C₂×C₁₆×C₅₃

C₂³

Frattini

C₂×C₁₆

C₂²×D₅₃

Fitting

C₅₃×#2253

C₂

socle, prime radical

C₂²×C₅₃

C₂×D₁₆

D₅₃

C₅₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2253

N	H ≅ G/N	type
67. C₅₃	#13724	semidirect

H ≅ G/N

type

67. C₅₃

#13724

semidirect

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

subgroup	index
#20276968	2
#20272895	2
#20272849	2
C₅₃×#2253	2
#13724	53

subgroup

index

C₅₃×#2253

Subgroups — #27305084

Conjugacy classes of subgroups

Complements

Maximal subgroups