Group #27300475 — subgroups

Subgroups — #27300475

Order 13568.

310 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 163 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	#20274879	C₂	1	2	yes	maximal
2	#20272807	C₂	1	4	yes	maximal
3	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133151	C₂²	1	4	yes	—
5	C₂×#13133149	—	2	2	no	—
6	#16810489	—	2	2	no	—
7	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₄×C₁₆×C₅₃	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₅₃	—	2	4	no	—
11	#13133151	—	2	8	no	—
12	#13133149	—	4	4	no	—
13	#13133124	—	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	D₈	1	2	yes	—
22	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
23	C₂²×Dic₅₃	—	4	4	no	—
24	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
25	C₂×C₈×C₅₃	SD₁₆	1	4	yes	—
26	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₈×C₅₃	#41	1	4	yes	—
29	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
30	C₄²×C₅₃	Q₁₆	1	2	yes	—
31	C₄⋊Dic₅₃	—	4	4	no	—
32	C₄⋊Dic₅₃	—	4	4	no	—
33	C₁₆×C₅₃	—	2	4	no	—
34	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
35	C₂×C₄×C₅₃	—	2	4	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×SD₁₆	1	2	yes	—
40	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
41	C₂×Dic₅₃	—	8	4	no	—
42	C₂×Dic₅₃	—	8	8	no	—
43	C₈×C₅₃	#106	1	4	yes	—
44	C₈×C₅₃	#117	1	2	yes	—
45	C₈×C₅₃	#125	1	2	yes	—
46	#12889	—	53	1	no	maximal, Sylow
47	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
48	C₂²×C₅₃	#486	1	2	yes	—
49	C₂²×C₅₃	#508	1	4	yes	—
50	C₄×C₅₃	—	2	4	no	—
51	C₄×C₅₃	#382	1	1	yes	characteristic
52	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
53	C₄×C₅₃	#487	1	2	yes	—
54	Dic₅₃	—	16	4	no	—
55	C₂×C₄×C₁₆	D₅₃	1	1	yes	characteristic
56	#1846	—	53	2	no	—
57	#1380	—	53	4	no	—
58	C₂×C₅₃	#1834	1	1	yes	characteristic
59	C₂×C₅₃	#2158	1	2	yes	—
60	C₂×C₅₃	#2242	1	4	yes	—
61	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
62	C₂×#107	—	53	4	no	—
63	C₂×#106	—	106	2	no	—
64	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
65	#382	—	106	2	no	—
66	C₄×C₁₆	Dic₅₃	1	4	yes	complemented
67	C₅₃	#12889	1	1	yes	characteristic, complemented, Sylow
68	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
69	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
70	C₂×C₄⋊C₄	—	106	4	no	—
71	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
72	C₂×C₁₆	—	2	4	no	—
73	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
74	#107	—	106	8	no	—
75	#106	—	212	4	no	—
76	C₄×C₈	Dic₁₀₆	1	2	yes	—
77	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
78	#95	—	212	4	no	—
79	C₂²×C₄	—	212	4	no	—
80	C₂²×C₄	D₄₂₄	1	2	yes	—
81	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
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	C₄⋊C₄	—	212	4	no	—
88	C₄⋊C₄	—	212	4	no	—
89	C₄²	Dic₂₁₂	1	2	yes	—
90	C₄²	C₄⋊Dic₅₃	1	2	yes	—
91	C₁₆	—	2	4	no	—
92	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	424	8	no	—
95	C₂×C₄	—	424	4	no	—
96	C₂×C₄	#13133151	1	4	yes	—
97	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
98	C₂×C₄	#13133205	1	2	yes	—
99	C₂×C₄	C₂×#11719913	1	2	yes	—
100	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
101	C₈	#13133149	1	4	yes	—
102	C₈	#13133176	1	2	yes	—
103	C₈	#13133209	1	2	yes	—
104	C₂²	C₂×#13133151	1	1	yes	characteristic
105	C₂²	#16810970	1	2	yes	—
106	C₂²	#16811055	1	4	yes	—
107	C₄	—	2	4	no	—
108	C₄	—	848	4	no	—
109	C₄	#16810489	1	1	yes	characteristic
110	C₄	C₂×#13133149	1	1	yes	characteristic
111	C₄	#16810974	1	2	yes	—
112	C₂	#20274843	1	1	yes	characteristic
113	C₂	#20276499	1	2	yes	—
114	C₂	#20276947	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20274879

C₂

yes

maximal

#20272807

C₂

yes

maximal

C₂×C₄×C₁₆×C₅₃

C₂

yes

maximal, characteristic

C₂×#13133151

C₂²

yes

—

C₂×#13133149

—

#16810489

—

C₂²×C₁₆×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₄×C₁₆×C₅₃

C₄

yes

complemented

C₂×C₄⋊Dic₅₃

—

#13133151

—

#13133149

—

#13133124

—

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

D₈

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×Dic₅₃

—

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

Q₁₆

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₁₆×C₅₃

—

C₂³×C₅₃

C₂×D₈

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×SD₁₆

yes

—

C₂×C₄×C₅₃

C₂×Q₁₆

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

#106

yes

—

C₈×C₅₃

#117

yes

—

C₈×C₅₃

#125

yes

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#508

yes

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

Dic₅₃

—

C₂×C₄×C₁₆

D₅₃

yes

characteristic

#1846

—

#1380

—

C₂×C₅₃

#1834

yes

characteristic

C₂×C₅₃

#2158

yes

—

C₂×C₅₃

#2242

yes

—

C₂²×C₁₆

C₂×D₅₃

yes

—

C₂×#107

—

C₂×#106

—

106

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

#382

—

106

—

C₄×C₁₆

Dic₅₃

yes

complemented

C₅₃

#12889

yes

characteristic, complemented, Sylow

C₂²×C₈

D₂₁₂

yes

characteristic

C₂²×C₈

C₂²×D₅₃

yes

characteristic

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

D₂₁₂

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₅₃

yes

—

#107

—

106

—

#106

—

212

—

C₄×C₈

Dic₁₀₆

yes

—

C₄×C₈

C₂×Dic₅₃

yes

—

#95

—

212

—

C₂²×C₄

—

212

—

C₂²×C₄

D₄₂₄

yes

—

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

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

C₄⋊C₄

—

212

—

C₄⋊C₄

—

212

—

C₄²

Dic₂₁₂

yes

—

C₄²

C₄⋊Dic₅₃

yes

—

C₁₆

—

C₂³

C₂×D₄₂₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

424

—

C₂×C₄

—

424

—

C₂×C₄

#13133151

yes

—

C₂×C₄

C₂×C₄⋊Dic₅₃

yes

characteristic

C₂×C₄

#13133205

yes

—

C₂×C₄

C₂×#11719913

yes

—

100

C₂×C₄

C₂×Dic₂₁₂

yes

characteristic

101

C₈

#13133149

yes

—

102

C₈

#13133176

yes

—

103

C₈

#13133209

yes

—

104

C₂²

C₂×#13133151

yes

characteristic

105

C₂²

#16810970

yes

—

106

C₂²

#16811055

yes

—

107

C₄

—

108

C₄

—

848

—

109

C₄

#16810489

yes

characteristic

110

C₄

C₂×#13133149

yes

characteristic

111

C₄

#16810974

yes

—

112

C₂

#20274843

yes

characteristic

113

C₂

#20276499

yes

—

114

C₂

#20276947

yes

—

	subgroup	G/H
center	C₂³	C₂×D₄₂₄
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₄×C₁₆×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₅₃
hypocenter	C₅₃	#12889
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

subgroup

G/H

center

C₂³

C₂×D₄₂₄

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₂×C₄×C₁₆×C₅₃

C₂

socle, prime radical

C₂³×C₅₃

C₂×D₈

cosocle

C₂²×C₈×C₅₃

C₂³

hypercenter

C₂×C₄×C₁₆

D₅₃

hypocenter

C₅₃

#12889

LCS

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

N	H ≅ G/N	type
9. C₄×C₁₆×C₅₃	C₄	semidirect
66. C₄×C₁₆	Dic₅₃	semidirect
67. C₅₃	#12889	semidirect

H ≅ G/N

type

9. C₄×C₁₆×C₅₃

C₄

semidirect

66. C₄×C₁₆

Dic₅₃

semidirect

67. C₅₃

#12889

semidirect

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

subgroup	index
#20274879	2
#20272807	2
C₂×C₄×C₁₆×C₅₃	2
#12889	53

subgroup

index

#20274879

#20272807

C₂×C₄×C₁₆×C₅₃

#12889

Subgroups — #27300475

Conjugacy classes of subgroups

Complements

Maximal subgroups