Group #29589050 — subgroups

Subgroups — #29589050

Order 15616.

260 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 163 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	#21447618	C₂	1	2	yes	maximal
2	#21443737	C₂	1	2	yes	maximal, complemented
3	#21443736	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×#11758185	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1450	C₂	1	1	yes	maximal, characteristic
6	C₄⋊#11758185	C₄	1	4	yes	—
7	C₂²⋊#11758185	C₄	1	4	yes	—
8	C₄×#11758185	C₂²	1	2	yes	—
9	C₄×#11758185	C₂²	1	2	yes	—
10	C₆₁×#431	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
12	C₆₁×#322	C₄	1	2	yes	—
13	C₈²×C₆₁	C₂²	1	1	yes	characteristic
14	C₈²×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×#11758185	C₂×C₄	1	4	yes	—
16	C₂×#11758185	C₂×C₄	1	4	yes	—
17	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
19	C₄⋊C₈×C₆₁	C₈	1	4	yes	—
20	C₂²⋊C₈×C₆₁	C₈	1	4	yes	—
21	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
24	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
27	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
28	C₂×C₈×C₆₁	—	2	4	no	—
29	C₂×C₈×C₆₁	C₄²	1	2	yes	—
30	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
31	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
32	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
33	C₄²×C₆₁	C₄²	1	1	yes	characteristic
34	C₄²×C₆₁	C₂×C₈	1	2	yes	—
35	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
36	#11758185	—	2	4	no	—
37	#11758185	—	2	4	no	—
38	C₂³×C₆₁	C₄×C₈	1	1	yes	characteristic
39	C₂×C₄×C₆₁	—	2	2	no	—
40	C₂×C₄×C₆₁	—	2	4	no	—
41	C₂×C₄×C₆₁	C₄×C₈	1	2	yes	—
42	C₂×C₄×C₆₁	C₄×C₈	1	1	yes	characteristic
43	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
44	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
45	C₈×C₆₁	—	2	2	no	—
46	C₈×C₆₁	—	2	4	no	—
47	C₈×C₆₁	—	2	4	no	—
48	C₈×C₆₁	#131	1	4	yes	—
49	#11412	—	61	1	no	maximal, Sylow
50	C₂²×C₆₁	—	2	2	no	—
51	C₂²×C₆₁	—	2	1	no	—
52	C₂²×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
53	C₄×C₆₁	—	2	2	no	—
54	C₄×C₆₁	—	2	1	no	—
55	C₄×C₆₁	#405	1	2	yes	—
56	C₄×C₆₁	#504	1	4	yes	—
57	#2162	—	61	2	no	—
58	#1450	D₆₁	1	1	yes	characteristic
59	#1312	—	61	2	no	—
60	#1311	—	61	1	no	—
61	C₈×C₁₆	—	61	1	no	—
62	C₂×C₆₁	—	4	1	no	—
63	C₂×C₆₁	#1450	1	1	yes	characteristic
64	C₂×C₆₁	#2108	1	2	yes	—
65	#431	Dic₆₁	1	2	yes	—
66	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
67	C₄⋊C₁₆	—	61	4	no	—
68	C₂²⋊C₁₆	—	61	4	no	—
69	C₄×C₁₆	—	61	2	no	—
70	C₄×C₁₆	—	61	2	no	—
71	#322	Dic₆₁	1	2	yes	—
72	C₈²	C₂×D₆₁	1	1	yes	characteristic
73	C₈²	C₂×D₆₁	1	1	yes	characteristic
74	C₆₁	#11412	1	1	yes	characteristic, complemented, Sylow
75	C₂²×C₈	C₄×D₆₁	1	2	yes	—
76	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
77	C₂×C₁₆	—	61	4	no	—
78	C₂×C₁₆	—	61	4	no	—
79	C₄⋊C₈	#92418	1	4	yes	—
80	C₂²⋊C₈	#92418	1	4	yes	—
81	C₄×C₈	C₄×D₆₁	1	2	yes	—
82	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
83	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
84	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
85	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
86	C₂²×C₄	C₂×#92418	1	2	yes	—
87	C₂²×C₄	C₄×Dic₆₁	1	1	yes	characteristic
88	C₂×C₈	—	2	4	no	—
89	C₂×C₈	C₂×#92418	1	4	yes	—
90	C₂×C₈	C₂×#92418	1	4	yes	—
91	C₂×C₈	C₄×Dic₆₁	1	2	yes	—
92	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
93	C₄²	C₂×#92418	1	2	yes	—
94	C₄²	C₄×Dic₆₁	1	1	yes	characteristic
95	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
96	C₁₆	—	122	4	no	—
97	C₁₆	—	122	4	no	—
98	C₂³	C₄×#92418	1	1	yes	characteristic
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	C₄×#92418	1	1	yes	characteristic
102	C₂×C₄	C₄×#92418	1	2	yes	—
103	C₂×C₄	C₂×C₄×Dic₆₁	1	1	yes	characteristic
104	C₂×C₄	C₂²×#92418	1	2	yes	—
105	C₈	—	2	2	no	—
106	C₈	—	2	4	no	—
107	C₈	—	2	4	no	—
108	C₈	#14519783	1	4	yes	—
109	C₂²	—	2	1	no	—
110	C₂²	—	2	2	no	—
111	C₂²	C₂×C₄×#92418	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	2	2	no	—
114	C₄	#16839717	1	2	yes	—
115	C₄	#16840179	1	4	yes	—
116	C₂	—	4	1	no	—
117	C₂	#21444102	1	1	yes	characteristic
118	C₂	#21447416	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21447618

C₂

yes

maximal

#21443737

C₂

yes

maximal, complemented

#21443736

C₂

yes

maximal, characteristic, complemented

C₈×#11758185

C₂

yes

maximal, characteristic, complemented

C₆₁×#1450

C₂

yes

maximal, characteristic

C₄⋊#11758185

C₄

yes

—

C₂²⋊#11758185

C₄

yes

—

C₄×#11758185

C₂²

yes

—

C₄×#11758185

C₂²

yes

—

C₆₁×#431

C₄

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₆₁×#322

C₄

yes

—

C₈²×C₆₁

C₂²

yes

characteristic

C₈²×C₆₁

C₂²

yes

characteristic

C₂×#11758185

C₂×C₄

yes

—

C₂×#11758185

C₂×C₄

yes

—

C₂²×C₈×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄⋊C₈×C₆₁

C₈

yes

—

C₂²⋊C₈×C₆₁

C₈

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

C₂³

yes

characteristic

C₂²×C₄×C₆₁

C₄²

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₈

yes

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₄²

yes

—

C₂×C₈×C₆₁

C₂×C₈

yes

—

C₂×C₈×C₆₁

C₂×C₈

yes

—

C₂×C₈×C₆₁

C₂²×C₄

yes

—

C₄²×C₆₁

C₄²

yes

characteristic

C₄²×C₆₁

C₂×C₈

yes

—

C₄²×C₆₁

C₂²×C₄

yes

characteristic

#11758185

—

#11758185

—

C₂³×C₆₁

C₄×C₈

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₄×C₈

yes

—

C₂×C₄×C₆₁

C₄×C₈

yes

characteristic

C₂×C₄×C₆₁

C₂×C₄²

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₈

yes

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

#131

yes

—

#11412

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×C₄×C₈

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#405

yes

—

C₄×C₆₁

#504

yes

—

#2162

—

#1450

D₆₁

yes

characteristic

#1312

—

#1311

—

C₈×C₁₆

—

C₂×C₆₁

—

C₂×C₆₁

#1450

yes

characteristic

C₂×C₆₁

#2108

yes

—

#431

Dic₆₁

yes

—

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

#322

Dic₆₁

yes

—

C₈²

C₂×D₆₁

yes

characteristic

C₈²

C₂×D₆₁

yes

characteristic

C₆₁

#11412

yes

characteristic, complemented, Sylow

C₂²×C₈

C₄×D₆₁

yes

—

C₂×C₄²

C₂×Dic₆₁

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

#92418

yes

—

C₂²⋊C₈

#92418

yes

—

C₄×C₈

C₄×D₆₁

yes

—

C₄×C₈

C₂×Dic₆₁

yes

—

C₄×C₈

C₂×Dic₆₁

yes

—

C₄×C₈

C₂×Dic₆₁

yes

characteristic

C₄×C₈

C₂²×D₆₁

yes

characteristic

C₂²×C₄

C₂×#92418

yes

—

C₂²×C₄

C₄×Dic₆₁

yes

characteristic

C₂×C₈

—

C₂×C₈

C₂×#92418

yes

—

C₂×C₈

C₂×#92418

yes

—

C₂×C₈

C₄×Dic₆₁

yes

—

C₂×C₈

C₂×C₄×D₆₁

yes

—

C₄²

C₂×#92418

yes

—

C₄²

C₄×Dic₆₁

yes

characteristic

C₄²

C₂²×Dic₆₁

yes

characteristic

C₁₆

—

122

—

C₁₆

—

122

—

C₂³

C₄×#92418

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₄×#92418

yes

characteristic

102

C₂×C₄

C₄×#92418

yes

—

103

C₂×C₄

C₂×C₄×Dic₆₁

yes

characteristic

104

C₂×C₄

C₂²×#92418

yes

—

105

C₈

—

106

C₈

—

107

C₈

—

108

C₈

#14519783

yes

—

109

C₂²

—

110

C₂²

—

111

C₂²

C₂×C₄×#92418

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

#16839717

yes

—

115

C₄

#16840179

yes

—

116

C₂

—

117

C₂

#21444102

yes

characteristic

118

C₂

#21447416

yes

—

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₆₁
commutator, socle	C₂²×C₆₁	C₂×C₄×C₈
Fitting	C₆₁×#1450	C₂
prime radical	C₂³×C₆₁	C₄×C₈
cosocle	C₄×C₈×C₆₁	C₂³
hypercenter	#1450	D₆₁
hypocenter	C₆₁	#11412
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄×C₈ ◁ #1450

subgroup

G/H

center, Frattini

C₄×C₈

C₂²×D₆₁

commutator, socle

C₂²×C₆₁

C₂×C₄×C₈

Fitting

C₆₁×#1450

C₂

prime radical

C₂³×C₆₁

C₄×C₈

cosocle

C₄×C₈×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₄×C₈ ◁ #1450

N	H ≅ G/N	type
2. #21443737	C₂	semidirect
3. #21443736	C₂	semidirect
4. C₈×#11758185	C₂	semidirect
74. C₆₁	#11412	semidirect

H ≅ G/N

type

2. #21443737

C₂

semidirect

3. #21443736

C₂

semidirect

4. C₈×#11758185

C₂

semidirect

74. C₆₁

#11412

semidirect

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

subgroup	index
#21447618	2
#21443737	2
#21443736	2
C₈×#11758185	2
C₆₁×#1450	2
#11412	61

subgroup

index

C₆₁×#1450

Subgroups — #29589050

Conjugacy classes of subgroups

Complements

Maximal subgroups