Group #27298783 — subgroups

Subgroups — #27298783

Order 13568.

262 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 95 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	#20276944	C₂	1	4	yes	maximal
2	#20274333	C₂	1	1	yes	maximal, characteristic
3	C₈×#11719908	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1713	C₂	1	1	yes	maximal, characteristic
5	#16811028	—	2	1	no	—
6	#16811028	—	2	1	no	—
7	#16811003	C₂²	1	2	yes	—
8	#16810426	—	2	8	no	—
9	C₄×#11719908	C₂²	1	2	yes	—
10	C₅₃×#500	—	2	1	no	—
11	C₅₃×#500	C₂²	1	2	yes	—
12	C₅₃×#494	—	2	1	no	—
13	C₈²×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×Dic₂₁₂	—	2	4	no	—
15	#13133217	—	2	2	no	—
16	C₂×#11719908	D₄	1	4	yes	—
17	#13133151	—	4	2	no	—
18	#13133151	—	4	2	no	—
19	C₂×Q₁₆×C₅₃	—	4	2	no	—
20	C₄⋊Q₈×C₅₃	—	2	2	no	—
21	#107×C₅₃	—	2	4	no	—
22	#107×C₅₃	—	4	2	no	—
23	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
25	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
26	C₂×Q₈×C₅₃	—	4	2	no	—
27	C₂×Dic₁₀₆	—	4	2	no	—
28	Q₁₆×C₅₃	—	8	4	no	—
29	C₂×C₈×C₅₃	D₈	1	2	yes	—
30	C₂×C₈×C₅₃	D₈	1	2	yes	—
31	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
32	C₄⋊C₄×C₅₃	—	4	2	no	—
33	C₄⋊C₄×C₅₃	—	4	2	no	—
34	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₄⋊Dic₅₃	—	4	2	no	—
36	C₄⋊Dic₅₃	—	4	2	no	—
37	Dic₂₁₂	—	4	8	no	—
38	#11719908	—	2	4	no	—
39	Q₈×C₅₃	—	8	4	no	—
40	C₂×C₄×C₅₃	—	8	2	no	—
41	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
44	C₂×Dic₅₃	—	8	2	no	—
45	Dic₁₀₆	—	8	4	no	—
46	C₈×C₅₃	SD₃₂	1	4	yes	—
47	C₈×C₅₃	Q₃₂	1	4	yes	—
48	C₈×C₅₃	#135	1	4	yes	—
49	#12510	—	53	1	no	maximal, Sylow
50	C₂²×C₅₃	#493	1	1	yes	characteristic
51	C₄×C₅₃	—	16	2	no	—
52	C₄×C₅₃	#495	1	2	yes	—
53	C₄×C₅₃	C₂×SD₃₂	1	2	yes	—
54	C₄×C₅₃	C₂×Q₃₂	1	2	yes	—
55	Dic₅₃	—	16	2	no	—
56	#2241	—	53	4	no	—
57	#1713	—	53	1	no	—
58	#1713	D₅₃	1	1	yes	characteristic
59	C₈×C₁₆	—	53	1	no	—
60	C₂×C₅₃	#1709	1	1	yes	characteristic
61	C₂×C₅₃	#2247	1	1	yes	characteristic
62	C₂×C₅₃	#2248	1	1	yes	characteristic
63	#500	—	2	1	no	—
64	#500	—	106	1	no	—
65	#500	—	106	1	no	—
66	#500	C₂×D₅₃	1	2	yes	—
67	#494	—	2	1	no	—
68	#494	—	53	2	no	—
69	#358	—	106	8	no	—
70	C₄×C₁₆	—	53	2	no	—
71	C₈²	C₂×D₅₃	1	1	yes	characteristic
72	C₅₃	#12510	1	1	yes	characteristic, complemented, Sylow
73	C₂×Q₁₆	—	4	2	no	—
74	C₂×Q₁₆	—	106	4	no	—
75	C₄⋊Q₈	—	2	2	no	—
76	C₄⋊Q₈	—	106	2	no	—
77	C₂×C₁₆	—	53	4	no	—
78	#107	—	2	4	no	—
79	#107	—	4	2	no	—
80	#107	—	212	2	no	—
81	#107	—	212	2	no	—
82	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
83	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
84	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
85	C₂×Q₈	—	4	2	no	—
86	C₂×Q₈	—	212	2	no	—
87	Q₁₆	—	8	4	no	—
88	Q₁₆	—	212	8	no	—
89	C₂×C₈	#11719922	1	2	yes	—
90	C₂×C₈	#11719922	1	2	yes	—
91	C₂×C₈	D₄×D₅₃	1	2	yes	—
92	C₄⋊C₄	—	4	2	no	—
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	212	2	no	—
95	C₄⋊C₄	—	212	2	no	—
96	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
97	C₁₆	—	106	4	no	—
98	Q₈	—	8	4	no	—
99	Q₈	—	424	4	no	—
100	C₂×C₄	—	8	2	no	—
101	C₂×C₄	—	424	2	no	—
102	C₂×C₄	#13133213	1	1	yes	characteristic
103	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
104	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
105	C₈	#13133161	1	4	yes	—
106	C₈	#13133163	1	4	yes	—
107	C₈	#13133240	1	4	yes	—
108	C₂²	#16810999	1	1	yes	characteristic
109	C₄	—	16	2	no	—
110	C₄	—	848	2	no	—
111	C₄	#16811008	1	2	yes	—
112	C₄	C₂×#13133161	1	2	yes	—
113	C₄	C₂×#13133163	1	2	yes	—
114	C₂	#20274322	1	1	yes	characteristic
115	C₂	#20276967	1	1	yes	characteristic
116	C₂	#20276970	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276944

C₂

yes

maximal

#20274333

C₂

yes

maximal, characteristic

C₈×#11719908

C₂

yes

maximal, characteristic

C₅₃×#1713

C₂

yes

maximal, characteristic

#16811028

—

#16811028

—

#16811003

C₂²

yes

—

#16810426

—

C₄×#11719908

C₂²

yes

—

C₅₃×#500

—

C₅₃×#500

C₂²

yes

—

C₅₃×#494

—

C₈²×C₅₃

C₂²

yes

characteristic

C₂×Dic₂₁₂

—

#13133217

—

C₂×#11719908

D₄

yes

—

#13133151

—

#13133151

—

C₂×Q₁₆×C₅₃

—

C₄⋊Q₈×C₅₃

—

#107×C₅₃

—

#107×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Dic₁₀₆

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

Dic₂₁₂

—

#11719908

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×Dic₅₃

—

Dic₁₀₆

—

C₈×C₅₃

SD₃₂

yes

—

C₈×C₅₃

Q₃₂

yes

—

C₈×C₅₃

#135

yes

—

—

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

#495

yes

—

C₄×C₅₃

C₂×SD₃₂

yes

—

C₄×C₅₃

C₂×Q₃₂

yes

—

Dic₅₃

—

#2241

—

#1713

—

#1713

D₅₃

yes

characteristic

C₈×C₁₆

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

106

—

#500

—

106

—

#500

C₂×D₅₃

yes

—

#494

—

#494

—

#358

—

106

—

C₄×C₁₆

—

C₈²

C₂×D₅₃

yes

characteristic

C₅₃

#12510

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

106

—

C₄⋊Q₈

—

C₄⋊Q₈

—

106

—

C₂×C₁₆

—

#107

—

#107

—

#107

—

212

—

#107

—

212

—

C₄×C₈

C₂²⋊D₅₃

yes

characteristic

C₄×C₈

C₂²⋊D₅₃

yes

characteristic

C₄×C₈

C₂²×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

212

—

Q₁₆

—

Q₁₆

—

212

—

C₂×C₈

#11719922

yes

—

C₂×C₈

#11719922

yes

—

C₂×C₈

D₄×D₅₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

212

—

C₄⋊C₄

—

212

—

C₄²

C₂×C₂²⋊D₅₃

yes

characteristic

C₁₆

—

106

—

Q₈

—

Q₈

—

424

—

100

C₂×C₄

—

101

C₂×C₄

—

424

—

102

C₂×C₄

#13133213

yes

characteristic

103

C₂×C₄

C₂×#11719922

yes

characteristic

104

C₂×C₄

C₂×#11719922

yes

characteristic

105

C₈

#13133161

yes

—

106

C₈

#13133163

yes

—

107

C₈

#13133240

yes

—

108

C₂²

#16810999

yes

characteristic

109

C₄

—

110

C₄

—

848

—

111

C₄

#16811008

yes

—

112

C₄

C₂×#13133161

yes

—

113

C₄

C₂×#13133163

yes

—

114

C₂

#20274322

yes

characteristic

115

C₂

#20276967

yes

characteristic

116

C₂

#20276970

yes

characteristic

	subgroup	G/H
center	C₂²	#16810999
commutator, cosocle	C₄×C₈×C₅₃	C₂³
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#1713	C₂
socle, prime radical	C₂²×C₅₃	#493
hypercenter	#1713	D₅₃
hypocenter	C₅₃	#12510
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

subgroup

G/H

center

C₂²

#16810999

commutator, cosocle

C₄×C₈×C₅₃

C₂³

Frattini

C₄×C₈

C₂²×D₅₃

Fitting

C₅₃×#1713

C₂

C₂²×C₅₃

D₅₃

C₅₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

N	H ≅ G/N	type
72. C₅₃	#12510	semidirect

H ≅ G/N

type

72. C₅₃

#12510

semidirect

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

subgroup	index
#20276944	2
#20274333	2
C₈×#11719908	2
C₅₃×#1713	2
#12510	53

subgroup

index

C₅₃×#1713

Subgroups — #27298783

Conjugacy classes of subgroups

Complements

Maximal subgroups