Group #28298610 — subgroups

Subgroups — #28298610

Order 13568.

200 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 66 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	#20290272	C₂	1	2	yes	maximal
2	#20276470	C₂	1	1	yes	maximal, characteristic
3	#16811626	—	2	2	no	—
4	#16811487	C₄	1	4	yes	—
5	#16811486	—	2	2	no	—
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	C₂×#11719940	—	4	2	no	—
11	#13133284	—	4	2	no	—
12	C₄⋊#88324	—	2	4	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	#11719940	—	8	4	no	—
25	C₂×#88324	—	4	4	no	—
26	C₂×C₈×C₅₃	—	2	1	no	—
27	C₂×C₈×C₅₃	—	2	1	no	—
28	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
29	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
30	C₄×Dic₅₃	C₂²⋊C₄	1	2	yes	—
31	C₄×Dic₅₃	D₈	1	1	yes	characteristic
32	C₄×Dic₅₃	Q₁₆	1	1	yes	characteristic
33	C₂×#88322	—	4	1	no	—
34	C₂×#88322	C₄⋊C₄	1	2	yes	—
35	C₁₆×C₅₃	—	4	1	no	—
36	#11719908	—	4	1	no	—
37	C₂³×C₅₃	#106	1	1	yes	characteristic
38	C₂×C₄×C₅₃	—	2	1	no	—
39	C₂×C₄×C₅₃	—	2	1	no	—
40	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
41	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
42	C₂×Dic₅₃	—	2	2	no	—
43	C₂×Dic₅₃	—	2	2	no	—
44	C₂×Dic₅₃	#102	1	2	yes	—
45	C₂×Dic₅₃	#103	1	2	yes	—
46	#88324	—	8	4	no	—
47	C₈×C₅₃	—	2	1	no	—
48	C₈×C₅₃	#100	1	1	yes	characteristic
49	C₈×C₅₃	#101	1	1	yes	characteristic
50	#88322	—	2	2	no	—
51	#7245	—	53	1	no	maximal, Sylow
52	C₂²×C₅₃	—	2	1	no	—
53	C₂²×C₅₃	—	2	2	no	—
54	C₂²×C₅₃	#340	1	1	yes	characteristic
55	C₄×C₅₃	—	2	1	no	—
56	C₄×C₅₃	#338	1	1	yes	characteristic
57	C₄×C₅₃	#343	1	1	yes	characteristic
58	Dic₅₃	—	2	2	no	—
59	Dic₅₃	#361	1	2	yes	—
60	Dic₅₃	#362	1	2	yes	—
61	#2150	—	53	1	no	—
62	#1567	—	53	2	no	—
63	C₂×C₅₃	—	4	1	no	—
64	C₂×C₅₃	#1292	1	1	yes	characteristic
65	C₂×C₅₃	#1387	1	2	yes	—
66	C₂×M₃₂	—	53	1	no	—
67	C₂×M₃₂	#6157	1	1	yes	characteristic
68	#423	—	106	2	no	—
69	C₂×C₄×C₈	—	53	1	no	—
70	C₄⋊C₁₆	—	106	2	no	—
71	#335	—	53	4	no	—
72	#335	—	106	2	no	—
73	C₅₃	#7245	1	1	yes	characteristic, complemented, Sylow
74	C₂×M₁₆	—	212	2	no	—
75	C₂²×C₈	—	53	1	no	—
76	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
77	C₂×C₄²	—	53	1	no	—
78	M₃₂	—	4	1	no	—
79	M₃₂	—	212	1	no	—
80	C₂×C₁₆	—	2	1	no	—
81	C₂×C₁₆	—	106	1	no	—
82	C₄⋊C₈	—	106	4	no	—
83	C₄⋊C₈	—	212	2	no	—
84	C₄×C₈	—	53	2	no	—
85	C₄×C₈	—	53	2	no	—
86	C₂²×C₄	—	53	2	no	—
87	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
88	M₁₆	—	424	4	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	53	2	no	—
92	C₂×C₈	—	212	1	no	—
93	C₂×C₈	—	212	4	no	—
94	C₂×C₈	C₄×#6157	1	1	yes	characteristic
95	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
96	C₄²	—	53	2	no	—
97	C₄²	—	53	1	no	—
98	C₄²	—	53	1	no	—
99	C₁₆	—	4	1	no	—
100	C₁₆	—	212	1	no	—
101	C₂³	#13133285	1	1	yes	characteristic
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	53	2	no	—
105	C₂×C₄	—	53	2	no	—
106	C₂×C₄	—	106	2	no	—
107	C₂×C₄	—	106	2	no	—
108	C₂×C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
109	C₂×C₄	#13133286	1	1	yes	characteristic
110	C₈	—	2	1	no	—
111	C₈	—	106	2	no	—
112	C₈	—	424	4	no	—
113	C₈	#13133277	1	1	yes	characteristic
114	C₈	#13133278	1	1	yes	characteristic
115	C₂²	—	2	1	no	—
116	C₂²	—	2	2	no	—
117	C₂²	#16811497	1	1	yes	characteristic
118	C₄	—	2	1	no	—
119	C₄	—	53	2	no	—
120	C₄	—	53	2	no	—
121	C₄	—	106	2	no	—
122	C₄	#16811492	1	1	yes	characteristic
123	C₄	#16811503	1	1	yes	characteristic
124	C₂	—	4	1	no	—
125	C₂	#20289585	1	1	yes	characteristic
126	C₂	#20289790	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20290272

C₂

yes

maximal

#20276470

C₂

yes

maximal, characteristic

#16811626

—

#16811487

C₄

yes

—

#16811486

—

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

—

#11719940

—

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

—

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

#361

yes

—

Dic₅₃

#362

yes

—

#2150

—

#1567

—

C₂×C₅₃

—

C₂×C₅₃

#1292

yes

characteristic

C₂×C₅₃

#1387

yes

—

C₂×M₃₂

—

C₂×M₃₂

#6157

yes

characteristic

#423

—

106

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

106

—

#335

—

#335

—

106

—

C₅₃

#7245

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

212

—

C₂²×C₈

—

C₂²×C₈

C₂×#6157

yes

characteristic

C₂×C₄²

—

M₃₂

—

M₃₂

—

212

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

C₄⋊C₈

—

106

—

C₄⋊C₈

—

212

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

C₄⋊#6157

yes

characteristic

M₁₆

—

424

—

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

—

100

C₁₆

—

212

—

101

C₂³

#13133285

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

106

—

107

C₂×C₄

—

106

—

108

C₂×C₄

(C₂×#6157)⋊C₄

yes

characteristic

109

C₂×C₄

#13133286

yes

characteristic

110

C₈

—

111

C₈

—

106

—

112

C₈

—

424

—

113

C₈

#13133277

yes

characteristic

114

C₈

#13133278

yes

characteristic

115

C₂²

—

116

C₂²

—

117

C₂²

#16811497

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

106

—

122

C₄

#16811492

yes

characteristic

123

C₄

#16811503

yes

characteristic

124

C₂

—

125

C₂

#20289585

yes

characteristic

126

C₂

#20289790

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	#7245
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
73. C₅₃	#7245	semidirect

H ≅ G/N

type

73. C₅₃

#7245

semidirect

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

subgroup	index
#20290272	2
#20276470	2
#7245	53

subgroup

index

#20290272

#20276470

#7245

Subgroups — #28298610

Conjugacy classes of subgroups

Complements

Maximal subgroups