Group #29570448 — subgroups

Subgroups — #29570448

Order 15616.

382 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 179 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	#21445565	C₂	1	1	yes	maximal, characteristic
2	#21445565	C₂	1	1	yes	maximal, characteristic
3	#21443646	C₂	1	4	yes	maximal
4	C₆₁×#1744	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×#92418	C₂²	1	2	yes	—
6	C₂×C₄×#92418	C₂²	1	2	yes	—
7	#16839517	C₄	1	4	yes	—
8	#16839517	C₄	1	4	yes	—
9	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
10	C₄³×C₆₁	C₂²	1	1	yes	characteristic
11	C₆₁×#336	C₄	1	4	yes	—
12	C₂²×#92418	C₂×C₄	1	4	yes	—
13	C₂²×#92418	C₂×C₄	1	4	yes	—
14	C₄×#92418	—	2	4	no	—
15	C₄×#92418	—	2	4	no	—
16	C₂²×C₈×C₆₁	C₂×C₄	1	4	yes	—
17	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
21	C₄×C₈×C₆₁	—	2	4	no	—
22	C₂²×C₄×C₆₁	C₄²	1	4	yes	—
23	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
26	C₂×C₈×C₆₁	—	2	4	no	—
27	C₂×C₈×C₆₁	—	2	8	no	—
28	C₄²×C₆₁	—	2	4	no	—
29	C₄²×C₆₁	—	2	4	no	—
30	C₄²×C₆₁	—	2	4	no	—
31	C₄²×C₆₁	#41	1	4	yes	—
32	C₂×#92418	—	2	4	no	—
33	C₂×#92418	—	2	4	no	—
34	C₂×#92418	—	2	8	no	—
35	C₂×#92418	—	2	8	no	—
36	C₂³×C₆₁	C₂×C₄²	1	1	yes	characteristic
37	C₂×C₄×C₆₁	—	2	4	no	—
38	C₂×C₄×C₆₁	—	2	4	no	—
39	C₂×C₄×C₆₁	—	2	4	no	—
40	C₂×C₄×C₆₁	#117	1	2	yes	—
41	C₂×C₄×C₆₁	#117	1	2	yes	—
42	C₂×C₄×C₆₁	#117	1	2	yes	—
43	C₂×C₄×C₆₁	#131	1	4	yes	—
44	C₂×C₄×C₆₁	#131	1	4	yes	—
45	C₂×C₄×C₆₁	#131	1	4	yes	—
46	C₈×C₆₁	—	4	4	no	—
47	#92418	—	4	4	no	—
48	#92418	—	4	4	no	—
49	#8135	—	61	1	no	maximal, Sylow
50	C₂²×C₆₁	#376	1	1	yes	characteristic
51	C₂²×C₆₁	#405	1	2	yes	—
52	C₂²×C₆₁	#405	1	2	yes	—
53	C₂²×C₆₁	#405	1	2	yes	—
54	C₄×C₆₁	—	4	4	no	—
55	C₄×C₆₁	#433	1	4	yes	—
56	C₄×C₆₁	#433	1	4	yes	—
57	C₄×C₆₁	#433	1	4	yes	—
58	#1744	—	61	1	no	—
59	#1744	—	61	1	no	—
60	#1744	D₆₁	1	1	yes	characteristic
61	#1273	—	61	4	no	—
62	C₂×C₆₁	#1454	1	4	yes	—
63	C₂×C₆₁	#1746	1	1	yes	characteristic
64	C₂×C₆₁	#1746	1	1	yes	characteristic
65	C₂×C₆₁	#1746	1	1	yes	characteristic
66	C₂×C₄×C₈	—	61	2	no	—
67	C₂×C₄×C₈	—	61	2	no	—
68	C₂×C₄×C₈	C₂×D₆₁	1	2	yes	—
69	C₄³	C₂×D₆₁	1	1	yes	characteristic
70	#336	—	61	4	no	—
71	#336	—	61	4	no	—
72	#336	Dic₆₁	1	4	yes	—
73	C₆₁	#8135	1	1	yes	characteristic, complemented, Sylow
74	C₂²×C₈	—	61	4	no	—
75	C₂²×C₈	—	61	4	no	—
76	C₂²×C₈	C₂×Dic₆₁	1	4	yes	—
77	C₂×C₄²	C₄×D₆₁	1	2	yes	—
78	C₂×C₄²	C₄×D₆₁	1	2	yes	—
79	C₂×C₄²	C₂×Dic₆₁	1	2	yes	—
80	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
81	C₄×C₈	—	2	4	no	—
82	C₄×C₈	—	122	4	no	—
83	C₄×C₈	—	122	4	no	—
84	C₂²×C₄	C₄×Dic₆₁	1	4	yes	—
85	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
86	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
87	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
88	C₂×C₈	—	2	4	no	—
89	C₂×C₈	—	2	8	no	—
90	C₂×C₈	—	122	4	no	—
91	C₂×C₈	—	122	4	no	—
92	C₂×C₈	—	122	8	no	—
93	C₂×C₈	—	122	8	no	—
94	C₄²	—	2	4	no	—
95	C₄²	—	2	4	no	—
96	C₄²	—	2	4	no	—
97	C₄²	#11758222	1	4	yes	—
98	C₂³	C₂×C₄×Dic₆₁	1	1	yes	characteristic
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	#14519734	1	2	yes	—
103	C₂×C₄	#14519734	1	2	yes	—
104	C₂×C₄	#14519736	1	2	yes	—
105	C₂×C₄	#14519783	1	4	yes	—
106	C₂×C₄	#14519784	1	4	yes	—
107	C₂×C₄	#14519785	1	4	yes	—
108	C₈	—	4	4	no	—
109	C₈	—	244	4	no	—
110	C₈	—	244	4	no	—
111	C₂²	#16839608	1	1	yes	characteristic
112	C₂²	#16839717	1	2	yes	—
113	C₂²	#16839718	1	2	yes	—
114	C₂²	#16839720	1	2	yes	—
115	C₄	—	4	4	no	—
116	C₄	#16839821	1	4	yes	—
117	C₄	#16839823	1	4	yes	—
118	C₄	#16839824	1	4	yes	—
119	C₂	#21444119	1	4	yes	—
120	C₂	#21445570	1	1	yes	characteristic
121	C₂	#21445571	1	1	yes	characteristic
122	C₂	#21445572	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21445565

C₂

yes

maximal, characteristic

#21445565

C₂

yes

maximal, characteristic

#21443646

C₂

yes

maximal

C₆₁×#1744

C₂

yes

maximal, characteristic

C₂×C₄×#92418

C₂²

yes

—

C₂×C₄×#92418

C₂²

yes

—

#16839517

C₄

yes

—

#16839517

C₄

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

—

C₄³×C₆₁

C₂²

yes

characteristic

C₆₁×#336

C₄

yes

—

C₂²×#92418

C₂×C₄

yes

—

C₂²×#92418

C₂×C₄

yes

—

C₄×#92418

—

C₄×#92418

—

C₂²×C₈×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄×C₈×C₆₁

—

C₂²×C₄×C₆₁

C₄²

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

#41

yes

—

—

—

—

—

C₂³×C₆₁

C₂×C₄²

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#117

yes

—

C₂×C₄×C₆₁

#117

yes

—

C₂×C₄×C₆₁

#117

yes

—

C₂×C₄×C₆₁

#131

yes

—

C₂×C₄×C₆₁

#131

yes

—

C₂×C₄×C₆₁

#131

yes

—

C₈×C₆₁

—

—

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

#405

yes

—

C₂²×C₆₁

#405

yes

—

C₄×C₆₁

—

C₄×C₆₁

#433

yes

—

C₄×C₆₁

#433

yes

—

C₄×C₆₁

#433

yes

—

#1744

—

#1744

—

#1744

D₆₁

yes

characteristic

#1273

—

C₂×C₆₁

#1454

yes

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₆₁

yes

—

C₄³

C₂×D₆₁

yes

—

—

Dic₆₁

yes

—

C₆₁

#8135

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×Dic₆₁

yes

—

C₂×C₄²

C₄×D₆₁

yes

—

C₂×C₄²

C₄×D₆₁

yes

—

C₂×C₄²

C₂×Dic₆₁

yes

—

C₂×C₄²

C₂²×D₆₁

yes

characteristic

C₄×C₈

—

C₄×C₈

—

122

—

C₄×C₈

—

122

—

C₂²×C₄

C₄×Dic₆₁

yes

—

C₂²×C₄

C₂×C₄×D₆₁

yes

characteristic

C₂²×C₄

C₂×C₄×D₆₁

yes

characteristic

C₂²×C₄

C₂²×Dic₆₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

—

122

—

C₂×C₈

—

122

—

C₂×C₈

—

122

—

C₄²

—

C₄²

—

C₄²

—

C₄²

#11758222

yes

—

C₂³

C₂×C₄×Dic₆₁

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

#14519734

yes

—

103

C₂×C₄

#14519734

yes

—

104

C₂×C₄

#14519736

yes

—

105

C₂×C₄

#14519783

yes

—

106

C₂×C₄

#14519784

yes

—

107

C₂×C₄

#14519785

yes

—

108

C₈

—

109

C₈

—

244

—

110

C₈

—

244

—

111

C₂²

#16839608

yes

characteristic

112

C₂²

#16839717

yes

—

113

C₂²

#16839718

yes

—

114

C₂²

#16839720

yes

—

115

C₄

—

116

C₄

#16839821

yes

—

117

C₄

#16839823

yes

—

118

C₄

#16839824

yes

—

119

C₂

#21444119

yes

—

120

C₂

#21445570

yes

characteristic

121

C₂

#21445571

yes

characteristic

122

C₂

#21445572

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₆₁
commutator, socle, prime radical	C₂³×C₆₁	C₂×C₄²
Fitting	C₆₁×#1744	C₂
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1744	D₆₁
hypocenter	C₆₁	#8135
LCS	G ▷ C₂³×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄² ◁ #1744

subgroup

G/H

center, Frattini

C₂×C₄²

C₂²×D₆₁

commutator, socle, prime radical

C₂³×C₆₁

C₂×C₄²

Fitting

C₆₁×#1744

C₂

cosocle

C₂×C₄²×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂³×C₆₁ ▷ C₆₁

C₁ ◁ C₂×C₄² ◁ #1744

N	H ≅ G/N	type
73. C₆₁	#8135	semidirect

H ≅ G/N

type

73. C₆₁

#8135

semidirect

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

subgroup	index
#21445565	2
#21443646	2
C₆₁×#1744	2
#8135	61

subgroup

index

#21445565

#21443646

C₆₁×#1744

#8135

Subgroups — #29570448

Conjugacy classes of subgroups

Complements

Maximal subgroups