Group #29589068 — subgroups

Subgroups — #29589068

Order 15616.

228 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 123 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	#21447622	C₂	1	2	yes	maximal
2	#21443740	C₂	1	4	yes	maximal, complemented
3	C₆₁×#1450	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#11758185	C₄	1	4	yes	—
5	C₂²⋊#11758185	C₄	1	4	yes	—
6	#16839538	C₂²	1	4	yes	—
7	C₆₁×#431	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
9	C₆₁×#322	C₂²	1	2	yes	—
10	C₈²×C₆₁	C₄	1	2	yes	—
11	C₂×#11758185	C₂×C₄	1	8	yes	—
12	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
13	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
14	C₄⋊C₈×C₆₁	—	2	2	no	—
15	C₂²⋊C₈×C₆₁	—	2	2	no	—
16	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
17	C₄×C₈×C₆₁	C₂×C₄	1	4	yes	—
18	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
19	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
20	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
21	C₂²×C₄×C₆₁	M₁₆	1	2	yes	—
22	C₂×C₈×C₆₁	—	2	4	no	—
23	C₂×C₈×C₆₁	—	2	4	no	—
24	C₂×C₈×C₆₁	C₄²	1	2	yes	—
25	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
26	C₄²×C₆₁	C₄²	1	1	yes	characteristic
27	C₄²×C₆₁	M₁₆	1	2	yes	—
28	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
29	#11758185	—	2	8	no	—
30	C₂³×C₆₁	#97	1	1	yes	characteristic
31	C₂×C₄×C₆₁	—	2	2	no	—
32	C₂×C₄×C₆₁	—	4	2	no	—
33	C₂×C₄×C₆₁	#97	1	2	yes	—
34	C₂×C₄×C₆₁	#97	1	1	yes	characteristic
35	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
36	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
37	C₈×C₆₁	—	2	2	no	—
38	C₈×C₆₁	—	4	4	no	—
39	C₈×C₆₁	#131	1	4	yes	—
40	#11416	—	61	1	no	maximal, Sylow
41	C₂²×C₆₁	—	2	2	no	—
42	C₂²×C₆₁	—	2	1	no	—
43	C₂²×C₆₁	C₂×#97	1	1	yes	characteristic
44	C₄×C₆₁	—	2	2	no	—
45	C₄×C₆₁	—	2	2	no	—
46	C₄×C₆₁	—	2	1	no	—
47	C₄×C₆₁	#405	1	2	yes	—
48	#2163	—	61	2	no	—
49	#1450	D₆₁	1	1	yes	characteristic
50	#1313	—	61	4	no	—
51	C₂×C₆₁	—	4	1	no	—
52	C₂×C₆₁	#1452	1	1	yes	characteristic
53	C₂×C₆₁	#2110	1	2	yes	—
54	#431	Dic₆₁	1	2	yes	—
55	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
56	C₄⋊C₁₆	—	61	4	no	—
57	C₂²⋊C₁₆	—	61	4	no	—
58	#346	—	61	4	no	—
59	#322	C₂×D₆₁	1	2	yes	—
60	C₈²	Dic₆₁	1	2	yes	—
61	C₆₁	#11416	1	1	yes	characteristic, complemented, Sylow
62	C₂²×C₈	C₄×D₆₁	1	2	yes	—
63	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
64	C₂×C₁₆	—	61	8	no	—
65	C₄⋊C₈	—	2	2	no	—
66	C₂²⋊C₈	—	2	2	no	—
67	C₄×C₈	C₄×D₆₁	1	2	yes	—
68	C₄×C₈	C₂×Dic₆₁	1	4	yes	—
69	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
70	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
71	C₂²×C₄	#11758194	1	2	yes	—
72	C₂²×C₄	C₄×Dic₆₁	1	1	yes	characteristic
73	C₂×C₈	—	2	4	no	—
74	C₂×C₈	—	2	4	no	—
75	C₂×C₈	C₄×Dic₆₁	1	2	yes	—
76	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
77	C₄²	#11758194	1	2	yes	—
78	C₄²	C₄×Dic₆₁	1	1	yes	characteristic
79	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
80	C₁₆	—	122	8	no	—
81	C₂³	#14519688	1	1	yes	characteristic
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	4	2	no	—
84	C₂×C₄	#14519688	1	1	yes	characteristic
85	C₂×C₄	#14519688	1	2	yes	—
86	C₂×C₄	C₂×C₄×Dic₆₁	1	1	yes	characteristic
87	C₂×C₄	C₂×#11758194	1	2	yes	—
88	C₈	—	2	2	no	—
89	C₈	—	4	4	no	—
90	C₈	#14519783	1	4	yes	—
91	C₂²	—	2	2	no	—
92	C₂²	—	2	1	no	—
93	C₂²	C₂×#14519688	1	1	yes	characteristic
94	C₄	—	2	2	no	—
95	C₄	—	2	2	no	—
96	C₄	—	2	1	no	—
97	C₄	#16839717	1	2	yes	—
98	C₂	—	4	1	no	—
99	C₂	#21444111	1	1	yes	characteristic
100	C₂	#21447424	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21447622

C₂

yes

maximal

#21443740

C₂

yes

maximal, complemented

C₆₁×#1450

C₂

yes

maximal, characteristic

C₄⋊#11758185

C₄

yes

—

C₂²⋊#11758185

C₄

yes

—

#16839538

C₂²

yes

—

C₆₁×#431

C₄

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₆₁×#322

C₂²

yes

—

C₈²×C₆₁

C₄

yes

—

C₂×#11758185

C₂×C₄

yes

—

C₂²×C₈×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄⋊C₈×C₆₁

—

C₂²⋊C₈×C₆₁

—

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

C₂³

yes

characteristic

C₂²×C₄×C₆₁

C₄²

yes

characteristic

C₂²×C₄×C₆₁

M₁₆

yes

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₄²

yes

—

C₂×C₈×C₆₁

C₂²×C₄

yes

—

C₄²×C₆₁

C₄²

yes

characteristic

C₄²×C₆₁

M₁₆

yes

—

C₄²×C₆₁

C₂²×C₄

yes

characteristic

#11758185

—

C₂³×C₆₁

#97

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#97

yes

—

C₂×C₄×C₆₁

#97

yes

characteristic

C₂×C₄×C₆₁

C₂×C₄²

yes

characteristic

C₂×C₄×C₆₁

C₂×M₁₆

yes

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

#131

yes

—

#11416

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×#97

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#405

yes

—

#2163

—

#1450

D₆₁

yes

characteristic

#1313

—

C₂×C₆₁

—

C₂×C₆₁

#1452

yes

characteristic

C₂×C₆₁

#2110

yes

—

#431

Dic₆₁

yes

—

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

#322

C₂×D₆₁

yes

—

C₈²

Dic₆₁

yes

—

C₆₁

#11416

yes

characteristic, complemented, Sylow

C₂²×C₈

C₄×D₆₁

yes

—

C₂×C₄²

C₂×Dic₆₁

yes

characteristic

C₂×C₁₆

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

C₄×D₆₁

yes

—

C₄×C₈

C₂×Dic₆₁

yes

—

C₄×C₈

C₂×Dic₆₁

yes

characteristic

C₄×C₈

C₂²×D₆₁

yes

characteristic

C₂²×C₄

#11758194

yes

—

C₂²×C₄

C₄×Dic₆₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×Dic₆₁

yes

—

C₂×C₈

C₂×C₄×D₆₁

yes

—

C₄²

#11758194

yes

—

C₄²

C₄×Dic₆₁

yes

characteristic

C₄²

C₂²×Dic₆₁

yes

characteristic

C₁₆

—

122

—

C₂³

#14519688

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#14519688

yes

characteristic

C₂×C₄

#14519688

yes

—

C₂×C₄

C₂×C₄×Dic₆₁

yes

characteristic

C₂×C₄

C₂×#11758194

yes

—

C₈

—

C₈

—

C₈

#14519783

yes

—

C₂²

—

C₂²

—

C₂²

C₂×#14519688

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#16839717

yes

—

C₂

—

C₂

#21444111

yes

characteristic

100

C₂

#21447424

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₆₁
commutator	C₂×C₄×C₆₁	C₂×C₄²
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₆₁×#1450	C₂
socle	C₂²×C₆₁	C₂×#97
prime radical	C₂³×C₆₁	#97
cosocle	C₄×C₈×C₆₁	C₂³
hypercenter	#1450	D₆₁
hypocenter	C₆₁	#11416
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ #1450

subgroup

G/H

center

C₂×C₄

C₂×C₄×Dic₆₁

commutator

C₂×C₄×C₆₁

C₂×C₄²

Frattini

C₄×C₈

C₂²×D₆₁

Fitting

C₆₁×#1450

C₂

socle

C₂²×C₆₁

C₂×#97

prime radical

C₂³×C₆₁

#97

cosocle

C₄×C₈×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₆₁

C₁ ◁ C₂×C₄ ◁ #1450

N	H ≅ G/N	type
2. #21443740	C₂	semidirect
61. C₆₁	#11416	semidirect

H ≅ G/N

type

2. #21443740

C₂

semidirect

61. C₆₁

#11416

semidirect

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

subgroup	index
#21447622	2
#21443740	2
C₆₁×#1450	2
#11416	61

subgroup

index

#21447622

#21443740

C₆₁×#1450

#11416

Subgroups — #29589068

Conjugacy classes of subgroups

Complements

Maximal subgroups