Group #27288774 — subgroups

Subgroups — #27288774

Order 13568.

328 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 79 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	#20273506	C₂	1	3	yes	maximal, complemented
2	#20273384	C₂	1	3	yes	maximal
3	C₅₃×#1309	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13133214	C₂²	1	1	yes	characteristic
5	#16810676	C₂²	1	3	yes	—
6	#16810375	—	2	3	no	—
7	#16810373	—	2	3	no	—
8	#16810366	—	2	3	no	—
9	#16810366	—	2	3	no	—
10	C₅₃×#423	C₂²	1	3	yes	—
11	C₂²×D₂₁₂	—	2	3	no	—
12	#13133214	—	2	3	no	—
13	#13133214	Q₈	1	2	yes	—
14	C₄⋊#88322	D₄	1	3	yes	—
15	C₄⋊#88322	D₄	1	3	yes	—
16	C₂²⋊#88322	—	2	3	no	—
17	C₄×#88322	—	2	3	no	—
18	C₂×M₁₆×C₅₃	—	2	3	no	—
19	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
20	C₄⋊C₈×C₅₃	—	2	3	no	—
21	C₄⋊C₈×C₅₃	—	2	3	no	—
22	C₂³×D₅₃	—	4	1	no	—
23	C₂²×C₄×C₅₃	#41	1	3	yes	—
24	C₂×D₂₁₂	—	2	6	no	—
25	C₂×D₂₁₂	—	4	3	no	—
26	C₂×D₂₁₂	—	8	3	no	—
27	C₂×D₂₁₂	—	8	3	no	—
28	M₁₆×C₅₃	—	8	3	no	—
29	C₂×C₈×C₅₃	—	4	3	no	—
30	C₄²×C₅₃	C₂×D₄	1	3	yes	—
31	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
32	C₂×#88322	—	2	3	no	—
33	C₂×#88322	—	2	3	no	—
34	C₂³×C₅₃	#126	1	1	yes	characteristic
35	C₂²×D₅₃	—	4	2	no	—
36	C₂²×D₅₃	—	8	6	no	—
37	C₂×C₄×C₅₃	—	2	3	no	—
38	C₂×C₄×C₅₃	—	2	3	no	—
39	C₂×C₄×C₅₃	#121	1	3	yes	—
40	C₂×C₄×C₅₃	C₂²⋊Q₈	1	3	yes	—
41	D₂₁₂	—	8	6	no	—
42	D₂₁₂	—	8	3	no	—
43	D₂₁₂	—	8	3	no	—
44	C₈×C₅₃	—	8	3	no	—
45	#88322	—	4	3	no	—
46	#88322	—	4	3	no	—
47	#10611	—	53	1	no	maximal, Sylow
48	C₂²×C₅₃	—	2	3	no	—
49	C₂²×C₅₃	#396	1	1	yes	characteristic
50	C₂×D₅₃	—	8	6	no	—
51	C₂×D₅₃	—	16	4	no	—
52	C₄×C₅₃	—	2	3	no	—
53	C₄×C₅₃	#471	1	3	yes	—
54	C₄×C₅₃	#472	1	3	yes	—
55	#1574	—	53	3	no	—
56	#1545	—	53	3	no	—
57	#1309	D₅₃	1	1	yes	characteristic, complemented
58	C₂×C₅₃	—	4	1	no	—
59	C₂×C₅₃	#2063	1	3	yes	—
60	D₅₃	—	16	2	no	—
61	C₂×#127	—	53	1	no	—
62	#431	—	53	3	no	—
63	#423	C₂×D₅₃	1	3	yes	—
64	#335	—	106	3	no	—
65	#334	—	106	3	no	—
66	#331	—	106	3	no	—
67	#331	—	106	3	no	—
68	C₅₃	#10611	1	1	yes	characteristic, complemented, Sylow
69	C₂²×D₄	—	106	3	no	—
70	C₂×M₁₆	—	2	3	no	—
71	#127	—	53	2	no	—
72	#127	—	106	3	no	—
73	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
74	C₄⋊C₈	—	2	3	no	—
75	C₄⋊C₈	—	2	3	no	—
76	C₄⋊C₈	—	53	3	no	—
77	C₄⋊C₈	—	53	3	no	—
78	C₂²⋊C₈	—	106	3	no	—
79	C₄×C₈	—	106	3	no	—
80	C₂⁴	—	212	1	no	—
81	C₂×D₄	—	106	6	no	—
82	C₂×D₄	—	424	3	no	—
83	C₂×D₄	—	424	3	no	—
84	C₂×D₄	—	212	3	no	—
85	C₂²×C₄	#11719945	1	3	yes	—
86	M₁₆	—	8	3	no	—
87	C₂×C₈	—	4	3	no	—
88	C₂×C₈	—	106	3	no	—
89	C₂×C₈	—	106	3	no	—
90	C₄²	D₄×D₅₃	1	3	yes	—
91	C₄²	Q₈×D₅₃	1	1	yes	characteristic
92	C₂³	—	424	6	no	—
93	C₂³	—	212	2	no	—
94	C₂³	#13133211	1	1	yes	characteristic
95	D₄	—	424	6	no	—
96	D₄	—	424	3	no	—
97	D₄	—	424	3	no	—
98	C₂×C₄	—	2	3	no	—
99	C₂×C₄	—	2	3	no	—
100	C₂×C₄	#13133190	1	3	yes	—
101	C₂×C₄	#13133195	1	3	yes	—
102	C₈	—	8	3	no	—
103	C₈	—	212	3	no	—
104	C₈	—	212	3	no	—
105	C₂²	—	2	3	no	—
106	C₂²	—	848	4	no	—
107	C₂²	—	424	6	no	—
108	C₂²	#16810547	1	1	yes	characteristic
109	C₄	—	2	3	no	—
110	C₄	#16810889	1	3	yes	—
111	C₄	#16810892	1	3	yes	—
112	C₂	—	4	1	no	—
113	C₂	—	848	2	no	—
114	C₂	#20276076	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20273506

C₂

yes

maximal, complemented

#20273384

C₂

yes

maximal

C₅₃×#1309

C₂

yes

maximal, characteristic, complemented

C₂×#13133214

C₂²

yes

characteristic

#16810676

C₂²

yes

—

—

—

—

—

C₅₃×#423

C₂²

yes

—

C₂²×D₂₁₂

—

#13133214

—

#13133214

Q₈

yes

—

C₄⋊#88322

D₄

yes

—

C₄⋊#88322

D₄

yes

—

C₂²⋊#88322

—

C₄×#88322

—

C₂×M₁₆×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₂³×D₅₃

—

C₂²×C₄×C₅₃

#41

yes

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

C₂×Q₈

yes

characteristic

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

#126

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#121

yes

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₈×C₅₃

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

#396

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#471

yes

—

C₄×C₅₃

#472

yes

—

#1574

—

#1545

—

#1309

D₅₃

yes

characteristic, complemented

C₂×C₅₃

—

C₂×C₅₃

#2063

yes

—

D₅₃

—

C₂×#127

—

#431

—

#423

C₂×D₅₃

yes

—

#335

—

106

—

#334

—

106

—

#331

—

106

—

#331

—

106

—

C₅₃

#10611

yes

characteristic, complemented, Sylow

C₂²×D₄

—

106

—

C₂×M₁₆

—

#127

—

#127

—

106

—

C₂×C₄²

C₂²×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

106

—

C₄×C₈

—

106

—

C₂⁴

—

212

—

C₂×D₄

—

106

—

C₂×D₄

—

424

—

C₂×D₄

—

424

—

C₂×D₄

—

212

—

C₂²×C₄

#11719945

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄²

D₄×D₅₃

yes

—

C₄²

Q₈×D₅₃

yes

characteristic

C₂³

—

424

—

C₂³

—

212

—

C₂³

#13133211

yes

characteristic

D₄

—

424

—

D₄

—

424

—

D₄

—

424

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

#13133190

yes

—

101

C₂×C₄

#13133195

yes

—

102

C₈

—

103

C₈

—

212

—

104

C₈

—

212

—

105

C₂²

—

106

C₂²

—

848

—

107

C₂²

—

424

—

108

C₂²

#16810547

yes

characteristic

109

C₄

—

110

C₄

#16810889

yes

—

111

C₄

#16810892

yes

—

112

C₂

—

113

C₂

—

848

—

114

C₂

#20276076

yes

—

	subgroup	G/H
center	C₂²	#16810547
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1309	C₂
socle	C₂²×C₅₃	#396
prime radical	C₂×#13133214	C₂²
hypercenter	#1309	D₅₃
hypocenter	C₅₃	#10611
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

subgroup

G/H

center

C₂²

#16810547

commutator, cosocle

C₂×C₄²×C₅₃

C₂³

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1309

C₂

C₂²×C₅₃

C₂²

D₅₃

C₅₃

G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

N	H ≅ G/N	type
1. #20273506	C₂	semidirect
3. C₅₃×#1309	C₂	semidirect
57. #1309	D₅₃	semidirect
68. C₅₃	#10611	semidirect

H ≅ G/N

type

1. #20273506

C₂

semidirect

3. C₅₃×#1309

C₂

semidirect

57. #1309

D₅₃

semidirect

68. C₅₃

#10611

semidirect

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

subgroup	index
#20273506	2
#20273384	2
C₅₃×#1309	2
#10611	53

subgroup

index

#20273506

#20273384

C₅₃×#1309

#10611

Subgroups — #27288774

Conjugacy classes of subgroups

Complements

Maximal subgroups