Group #28298612 — subgroups

Subgroups — #28298612

Order 13568.

184 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 62 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	#20290277	C₂	1	2	yes	maximal
2	#20276470	C₂	1	1	yes	maximal, characteristic
3	#16811644	—	2	2	no	—
4	#16811486	—	2	2	no	—
5	#16811485	C₄	1	4	yes	—
6	C₂×#13133157	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
8	#16810441	—	2	2	no	—
9	C₂×M₃₂×C₅₃	C₄	1	1	yes	characteristic
10	C₄⋊#88324	—	2	4	no	—
11	C₂²⋊#88324	—	4	2	no	—
12	#13133272	—	4	2	no	—
13	C₂²×#88322	Q₈	1	1	yes	characteristic
14	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
15	#13133157	—	4	1	no	—
16	C₂×#11719908	—	2	1	no	—
17	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
18	C₈×Dic₅₃	D₄	1	2	yes	—
19	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	M₃₂×C₅₃	—	4	1	no	—
21	C₂×C₁₆×C₅₃	—	2	1	no	—
22	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
23	C₂²×Dic₅₃	SD₁₆	1	2	yes	—
24	C₂×#88324	—	4	4	no	—
25	C₂×C₈×C₅₃	—	2	1	no	—
26	C₂×C₈×C₅₃	—	2	1	no	—
27	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
28	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₄×Dic₅₃	C₂²⋊C₄	1	2	yes	—
30	C₄×Dic₅₃	D₈	1	1	yes	characteristic
31	C₄×Dic₅₃	Q₁₆	1	1	yes	characteristic
32	C₂×#88322	—	4	1	no	—
33	C₂×#88322	C₄⋊C₄	1	2	yes	—
34	C₁₆×C₅₃	—	4	1	no	—
35	#11719908	—	4	1	no	—
36	C₂³×C₅₃	#106	1	1	yes	characteristic
37	C₂×C₄×C₅₃	—	2	1	no	—
38	C₂×C₄×C₅₃	—	2	1	no	—
39	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
40	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
41	C₂×Dic₅₃	—	4	2	no	—
42	C₂×Dic₅₃	#102	1	2	yes	—
43	C₂×Dic₅₃	#103	1	2	yes	—
44	#88324	—	8	4	no	—
45	C₈×C₅₃	—	2	1	no	—
46	C₈×C₅₃	#100	1	1	yes	characteristic
47	C₈×C₅₃	#101	1	1	yes	characteristic
48	#88322	—	2	2	no	—
49	#7246	—	53	1	no	maximal, Sylow
50	C₂²×C₅₃	—	2	1	no	—
51	C₂²×C₅₃	—	2	2	no	—
52	C₂²×C₅₃	#340	1	1	yes	characteristic
53	C₄×C₅₃	—	2	1	no	—
54	C₄×C₅₃	#338	1	1	yes	characteristic
55	C₄×C₅₃	#343	1	1	yes	characteristic
56	Dic₅₃	—	2	2	no	—
57	Dic₅₃	—	2	2	no	—
58	#2150	—	53	1	no	—
59	#1568	—	53	2	no	—
60	C₂×C₅₃	—	4	1	no	—
61	C₂×C₅₃	#1292	1	1	yes	characteristic
62	C₂×C₅₃	#1389	1	2	yes	—
63	C₂×M₃₂	—	53	1	no	—
64	C₂×M₃₂	#6157	1	1	yes	characteristic
65	#432	—	106	2	no	—
66	C₂×C₄×C₈	—	53	1	no	—
67	C₄⋊C₁₆	—	106	2	no	—
68	#335	—	106	2	no	—
69	#334	—	53	4	no	—
70	C₅₃	#7246	1	1	yes	characteristic, complemented, Sylow
71	C₂²×C₈	—	53	1	no	—
72	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
73	C₂×C₄²	—	53	1	no	—
74	M₃₂	—	4	1	no	—
75	M₃₂	—	212	1	no	—
76	C₂×C₁₆	—	2	1	no	—
77	C₂×C₁₆	—	106	1	no	—
78	C₄⋊C₈	—	106	4	no	—
79	C₂²⋊C₈	—	212	2	no	—
80	#97	—	212	2	no	—
81	C₄×C₈	—	53	2	no	—
82	C₄×C₈	—	53	2	no	—
83	C₂²×C₄	—	53	2	no	—
84	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	2	1	no	—
87	C₂×C₈	—	53	2	no	—
88	C₂×C₈	—	212	1	no	—
89	C₂×C₈	—	212	4	no	—
90	C₂×C₈	C₄×#6157	1	1	yes	characteristic
91	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
92	C₄²	—	53	2	no	—
93	C₄²	—	53	1	no	—
94	C₄²	—	53	1	no	—
95	C₁₆	—	4	1	no	—
96	C₁₆	—	212	1	no	—
97	C₂³	#13133285	1	1	yes	characteristic
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	53	2	no	—
101	C₂×C₄	—	53	2	no	—
102	C₂×C₄	—	212	2	no	—
103	C₂×C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
104	C₂×C₄	#13133286	1	1	yes	characteristic
105	C₈	—	2	1	no	—
106	C₈	—	106	2	no	—
107	C₈	—	424	4	no	—
108	C₈	#13133277	1	1	yes	characteristic
109	C₈	#13133278	1	1	yes	characteristic
110	C₂²	—	2	2	no	—
111	C₂²	—	2	1	no	—
112	C₂²	#16811497	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	106	2	no	—
115	C₄	—	106	2	no	—
116	C₄	#16811492	1	1	yes	characteristic
117	C₄	#16811503	1	1	yes	characteristic
118	C₂	—	4	1	no	—
119	C₂	#20289585	1	1	yes	characteristic
120	C₂	#20289796	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20290277

C₂

yes

maximal

#20276470

C₂

yes

maximal, characteristic

#16811644

—

#16811486

—

#16811485

C₄

yes

—

C₂×#13133157

C₄

yes

characteristic

C₂×C₈×Dic₅₃

C₂²

yes

characteristic

#16810441

—

C₂×M₃₂×C₅₃

C₄

yes

—

—

—

C₂²×#88322

Q₈

yes

characteristic

C₂×C₄×Dic₅₃

D₄

yes

characteristic

#13133157

—

C₂×#11719908

—

C₈×Dic₅₃

C₂×C₄

yes

—

C₈×Dic₅₃

D₄

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂²×Dic₅₃

SD₁₆

yes

—

C₂×#88324

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄²

yes

characteristic

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₅₃

C₂²⋊C₄

yes

—

C₄×Dic₅₃

D₈

yes

characteristic

C₄×Dic₅₃

Q₁₆

yes

characteristic

C₂×#88322

—

C₂×#88322

C₄⋊C₄

yes

—

C₁₆×C₅₃

—

#11719908

—

C₂³×C₅₃

#106

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

#102

yes

—

C₂×Dic₅₃

#103

yes

—

#88324

—

C₈×C₅₃

—

C₈×C₅₃

yes

C₈×C₅₃

yes

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#340

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

#2150

—

#1568

—

C₂×C₅₃

—

C₂×C₅₃

#1292

yes

characteristic

C₂×C₅₃

#1389

yes

—

C₂×M₃₂

—

C₂×M₃₂

#6157

yes

characteristic

#432

—

106

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

106

—

#335

—

106

—

#334

—

C₅₃

#7246

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×#6157

yes

characteristic

C₂×C₄²

—

M₃₂

—

M₃₂

—

212

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

212

—

#97

—

212

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

C₄⋊#6157

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

212

—

C₂×C₈

—

212

—

C₂×C₈

C₄×#6157

yes

characteristic

C₂×C₈

C₂²⋊#6157

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

C₁₆

—

C₁₆

—

212

—

C₂³

#13133285

yes

characteristic

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

212

—

103

C₂×C₄

(C₂×#6157)⋊C₄

yes

characteristic

104

C₂×C₄

#13133286

yes

characteristic

105

C₈

—

106

C₈

—

106

—

107

C₈

—

424

—

108

C₈

#13133277

yes

characteristic

109

C₈

#13133278

yes

characteristic

110

C₂²

—

111

C₂²

—

112

C₂²

#16811497

yes

characteristic

113

C₄

—

114

C₄

—

106

—

115

C₄

—

106

—

116

C₄

#16811492

yes

characteristic

117

C₄

#16811503

yes

characteristic

118

C₂

—

119

C₂

#20289585

yes

characteristic

120

C₂

#20289796

yes

—

	subgroup	G/H
center	C₂²	#16811497
commutator	C₂×C₈×C₅₃	C₄²
Frattini	C₂²×C₈	C₂×#6157
Fitting	C₂×M₃₂×C₅₃	C₄
socle	C₂²×C₅₃	#340
prime radical	C₂³×C₅₃	#106
cosocle	C₂×C₈×Dic₅₃	C₂²
hypercenter	C₂×M₃₂	#6157
hypocenter	C₅₃	#7246
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₂²

#16811497

commutator

C₂×C₈×C₅₃

C₄²

Frattini

C₂²×C₈

C₂×#6157

Fitting

C₂×M₃₂×C₅₃

C₄

C₂²×C₅₃

C₂³×C₅₃

C₂×C₈×Dic₅₃

C₂²

C₂×M₃₂

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
70. C₅₃	#7246	semidirect

H ≅ G/N

type

70. C₅₃

#7246

semidirect

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

subgroup	index
#20290277	2
#20276470	2
#7246	53

subgroup

index

#20290277

#20276470

#7246

Subgroups — #28298612

Conjugacy classes of subgroups

Complements

Maximal subgroups