Group #27289198 — subgroups

Subgroups — #27289198

Order 13568.

526 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 165 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	#20275521	C₂	1	3	yes	maximal
2	#20274872	C₂	1	3	yes	maximal, complemented
3	C₅₃×#1273	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13133214	C₂²	1	1	yes	characteristic
5	C₂×#13133149	—	2	6	no	—
6	C₂×#13133139	—	2	6	no	—
7	#16810493	C₂²	1	3	yes	—
8	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	—
9	C₂²×D₂₁₂	—	2	3	no	—
10	#13133214	—	2	3	no	—
11	#13133214	Q₈	1	2	yes	—
12	C₂×C₄⋊Dic₅₃	—	2	3	no	—
13	C₂×C₄⋊Dic₅₃	—	2	6	no	—
14	#13133149	—	4	12	no	—
15	#13133139	—	4	12	no	—
16	C₂²×C₈×C₅₃	D₄	1	6	yes	—
17	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
18	C₄×C₈×C₅₃	—	2	6	no	—
19	C₂³×D₅₃	—	4	1	no	—
20	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
21	C₂²×Dic₅₃	—	4	3	no	—
22	C₂×D₂₁₂	—	2	6	no	—
23	C₂×D₂₁₂	—	4	6	no	—
24	C₂×D₂₁₂	—	4	6	no	—
25	C₂×D₂₁₂	—	4	3	no	—
26	C₂×C₈×C₅₃	—	2	6	no	—
27	C₂×C₈×C₅₃	—	2	6	no	—
28	C₂×C₈×C₅₃	SD₁₆	1	12	yes	—
29	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
30	C₄²×C₅₃	#41	1	3	yes	—
31	C₄⋊Dic₅₃	—	4	6	no	—
32	C₄⋊Dic₅₃	—	4	12	no	—
33	C₂³×C₅₃	#127	1	1	yes	characteristic
34	C₂²×D₅₃	—	4	2	no	—
35	C₂²×D₅₃	—	8	3	no	—
36	C₂²×D₅₃	—	8	3	no	—
37	C₂×C₄×C₅₃	C₂²⋊Q₈	1	3	yes	—
38	C₂×C₄×C₅₃	#123	1	3	yes	—
39	C₂×C₄×C₅₃	C₂×SD₁₆	1	6	yes	—
40	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	6	yes	—
41	C₂×Dic₅₃	—	8	3	no	—
42	C₂×Dic₅₃	—	8	6	no	—
43	D₂₁₂	—	4	12	no	—
44	D₂₁₂	—	8	6	no	—
45	C₈×C₅₃	—	2	12	no	—
46	#10683	—	53	1	no	maximal, Sylow
47	C₂²×C₅₃	#399	1	1	yes	characteristic
48	C₂²×C₅₃	#492	1	3	yes	—
49	C₂²×C₅₃	#496	1	3	yes	—
50	C₂×D₅₃	—	8	6	no	—
51	C₂×D₅₃	—	16	1	no	—
52	C₂×D₅₃	—	16	3	no	—
53	C₄×C₅₃	#476	1	6	yes	—
54	C₄×C₅₃	#481	1	6	yes	—
55	Dic₅₃	—	16	3	no	—
56	#1957	—	53	3	no	—
57	#1844	—	53	3	no	—
58	#1273	D₅₃	1	1	yes	characteristic, complemented
59	C₂×C₅₃	#1705	1	1	yes	characteristic
60	C₂×C₅₃	#1715	1	3	yes	—
61	C₂×C₅₃	#2094	1	3	yes	—
62	D₅₃	—	16	2	no	—
63	C₂×#127	—	53	1	no	—
64	C₂×#106	—	106	6	no	—
65	C₂×#102	—	106	6	no	—
66	C₂×C₄×C₈	C₂×D₅₃	1	3	yes	—
67	#384	—	53	3	no	—
68	C₅₃	#10683	1	1	yes	characteristic, complemented, Sylow
69	C₂²×D₄	—	106	3	no	—
70	C₂²×C₈	D₂₁₂	1	6	yes	—
71	#127	—	53	2	no	—
72	#127	—	106	3	no	—
73	C₂×C₄⋊C₄	—	106	3	no	—
74	C₂×C₄⋊C₄	—	106	6	no	—
75	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
76	#106	—	212	12	no	—
77	#102	—	212	12	no	—
78	C₄×C₈	—	2	6	no	—
79	C₂⁴	—	212	1	no	—
80	C₂×D₄	—	106	6	no	—
81	C₂×D₄	—	212	3	no	—
82	C₂×D₄	—	212	6	no	—
83	C₂×D₄	—	212	6	no	—
84	C₂²×C₄	—	212	3	no	—
85	C₂²×C₄	C₂×D₂₁₂	1	3	yes	—
86	C₂×C₈	—	2	6	no	—
87	C₂×C₈	—	2	6	no	—
88	C₂×C₈	#11719913	1	12	yes	—
89	C₄⋊C₄	—	212	6	no	—
90	C₄⋊C₄	—	212	12	no	—
91	C₄²	#11719947	1	3	yes	—
92	C₄²	Q₈×D₅₃	1	1	yes	characteristic
93	C₂³	—	424	3	no	—
94	C₂³	—	424	3	no	—
95	C₂³	—	212	2	no	—
96	C₂³	#13133214	1	1	yes	characteristic
97	D₄	—	424	6	no	—
98	D₄	—	212	12	no	—
99	C₂×C₄	—	424	6	no	—
100	C₂×C₄	—	424	3	no	—
101	C₂×C₄	#13133197	1	3	yes	—
102	C₂×C₄	#13133201	1	3	yes	—
103	C₂×C₄	C₂×#11719913	1	6	yes	—
104	C₂×C₄	#13133245	1	6	yes	—
105	C₈	—	2	12	no	—
106	C₂²	—	848	1	no	—
107	C₂²	—	848	3	no	—
108	C₂²	—	424	6	no	—
109	C₂²	#16810560	1	1	yes	characteristic
110	C₂²	#16810996	1	3	yes	—
111	C₂²	#16811012	1	3	yes	—
112	C₄	—	848	3	no	—
113	C₄	#16810914	1	6	yes	—
114	C₄	#16810942	1	6	yes	—
115	C₂	—	848	2	no	—
116	C₂	#20274302	1	1	yes	characteristic
117	C₂	#20274343	1	3	yes	—
118	C₂	#20276247	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20275521

C₂

yes

maximal

#20274872

C₂

yes

maximal, complemented

C₅₃×#1273

C₂

yes

maximal, characteristic, complemented

C₂×#13133214

C₂²

yes

—

—

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

—

C₂²×D₂₁₂

—

#13133214

—

#13133214

Q₈

yes

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

#13133149

—

#13133139

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄×C₈×C₅₃

—

C₂³×D₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×Dic₅₃

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₄²×C₅₃

C₂×Q₈

yes

characteristic

C₄²×C₅₃

#41

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂³×C₅₃

#127

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₃

#123

yes

—

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

C₈×C₅₃

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#496

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

#476

yes

—

C₄×C₅₃

#481

yes

—

Dic₅₃

—

#1957

—

#1844

—

#1273

D₅₃

yes

characteristic, complemented

C₂×C₅₃

#1705

yes

characteristic

C₂×C₅₃

#1715

yes

—

C₂×C₅₃

#2094

yes

—

D₅₃

—

C₂×#127

—

C₂×#106

—

106

—

C₂×#102

—

106

—

C₂×C₄×C₈

C₂×D₅₃

yes

—

#384

—

C₅₃

#10683

yes

characteristic, complemented, Sylow

C₂²×D₄

—

106

—

C₂²×C₈

D₂₁₂

yes

—

#127

—

#127

—

106

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

C₂²×D₅₃

yes

characteristic

#106

—

212

—

#102

—

212

—

C₄×C₈

—

C₂⁴

—

212

—

C₂×D₄

—

106

—

C₂×D₄

—

212

—

C₂×D₄

—

212

—

C₂×D₄

—

212

—

C₂²×C₄

—

212

—

C₂²×C₄

C₂×D₂₁₂

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#11719913

yes

—

C₄⋊C₄

—

212

—

C₄⋊C₄

—

212

—

C₄²

#11719947

yes

—

C₄²

Q₈×D₅₃

yes

characteristic

C₂³

—

424

—

C₂³

—

424

—

C₂³

—

212

—

C₂³

#13133214

yes

characteristic

D₄

—

424

—

D₄

—

212

—

C₂×C₄

—

424

—

100

C₂×C₄

—

424

—

101

C₂×C₄

#13133197

yes

—

102

C₂×C₄

#13133201

yes

—

103

C₂×C₄

C₂×#11719913

yes

—

104

C₂×C₄

#13133245

yes

—

105

C₈

—

106

C₂²

—

848

—

107

C₂²

—

848

—

108

C₂²

—

424

—

109

C₂²

#16810560

yes

characteristic

110

C₂²

#16810996

yes

—

111

C₂²

#16811012

yes

—

112

C₄

—

848

—

113

C₄

#16810914

yes

—

114

C₄

#16810942

yes

—

115

C₂

—

848

—

116

C₂

#20274302

yes

characteristic

117

C₂

#20274343

yes

—

118

C₂

#20276247

yes

—

	subgroup	G/H
center	C₂³	#13133214
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1273	C₂
socle	C₂³×C₅₃	#127
prime radical	C₂×#13133214	C₂²
hypercenter	#1273	D₅₃
hypocenter	C₅₃	#10683
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

subgroup

G/H

center

C₂³

#13133214

commutator, cosocle

C₂×C₄²×C₅₃

C₂³

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1273

C₂

C₂³×C₅₃

C₂²

D₅₃

C₅₃

G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

N	H ≅ G/N	type
2. #20274872	C₂	semidirect
3. C₅₃×#1273	C₂	semidirect
58. #1273	D₅₃	semidirect
68. C₅₃	#10683	semidirect

H ≅ G/N

type

2. #20274872

C₂

semidirect

3. C₅₃×#1273

C₂

semidirect

58. #1273

D₅₃

semidirect

68. C₅₃

#10683

semidirect

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

subgroup	index
#20275521	2
#20274872	2
C₅₃×#1273	2
#10683	53

subgroup

index

#20275521

#20274872

C₅₃×#1273

#10683

Subgroups — #27289198

Conjugacy classes of subgroups

Complements

Maximal subgroups