Group #29570447 — subgroups

Subgroups — #29570447

Order 15616.

382 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 167 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	#21445564	C₂	1	3	yes	maximal
2	#21443646	C₂	1	3	yes	maximal
3	C₆₁×#1273	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×#92418	C₂²	1	3	yes	—
5	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
6	#16839516	C₄	1	6	yes	—
7	#16839516	C₄	1	6	yes	—
8	C₂×C₄×C₈×C₆₁	C₂²	1	3	yes	—
9	C₂²×#92418	C₂×C₄	1	6	yes	—
10	C₂×C₄×Dic₆₁	C₂×C₄	1	3	yes	—
11	C₂×C₄×Dic₆₁	C₂×C₄	1	3	yes	—
12	C₄×#92418	—	2	3	no	—
13	C₄×#92418	—	2	3	no	—
14	C₂²×C₈×C₆₁	C₂×C₄	1	6	yes	—
15	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
16	C₄×C₈×C₆₁	—	2	6	no	—
17	C₂²×C₄×C₆₁	C₂²×C₄	1	3	yes	—
18	C₂²×Dic₆₁	C₄²	1	1	yes	characteristic
19	C₂²×Dic₆₁	C₄²	1	3	yes	—
20	C₂×C₈×C₆₁	—	2	6	no	—
21	C₂×C₈×C₆₁	—	2	6	no	—
22	C₂×C₈×C₆₁	—	2	6	no	—
23	C₄²×C₆₁	#41	1	3	yes	—
24	C₄²×C₆₁	#41	1	1	yes	characteristic
25	C₄×Dic₆₁	—	2	6	no	—
26	C₄×Dic₆₁	—	2	6	no	—
27	C₂×#92418	—	2	6	no	—
28	C₂×#92418	—	2	6	no	—
29	C₂×#92418	—	2	6	no	—
30	C₂³×C₆₁	C₂×C₄²	1	1	yes	characteristic
31	C₂×C₄×C₆₁	#117	1	3	yes	—
32	C₂×C₄×C₆₁	#117	1	3	yes	—
33	C₂×C₄×C₆₁	#131	1	6	yes	—
34	C₂×C₄×C₆₁	#131	1	6	yes	—
35	C₂×Dic₆₁	—	2	3	no	—
36	C₂×Dic₆₁	—	2	3	no	—
37	C₂×Dic₆₁	—	2	6	no	—
38	C₈×C₆₁	—	4	6	no	—
39	#92418	—	4	6	no	—
40	#8135	—	61	1	no	maximal, Sylow
41	C₂²×C₆₁	#376	1	1	yes	characteristic
42	C₂²×C₆₁	#405	1	3	yes	—
43	C₂²×C₆₁	#405	1	3	yes	—
44	C₄×C₆₁	#433	1	6	yes	—
45	C₄×C₆₁	#433	1	6	yes	—
46	Dic₆₁	—	4	1	no	—
47	Dic₆₁	—	4	3	no	—
48	#1744	—	61	3	no	—
49	#1273	—	61	3	no	—
50	#1273	D₆₁	1	1	yes	characteristic
51	C₂×C₆₁	#1454	1	1	yes	characteristic
52	C₂×C₆₁	#1454	1	3	yes	—
53	C₂×C₆₁	#1746	1	3	yes	—
54	C₂×C₄×C₈	—	61	3	no	—
55	C₂×C₄×C₈	C₂×D₆₁	1	3	yes	—
56	C₄³	—	61	1	no	—
57	#336	—	61	6	no	—
58	#336	—	61	6	no	—
59	C₆₁	#8135	1	1	yes	characteristic, complemented, Sylow
60	C₂²×C₈	—	61	6	no	—
61	C₂²×C₈	C₄×D₆₁	1	6	yes	—
62	C₂×C₄²	—	61	3	no	—
63	C₂×C₄²	—	61	3	no	—
64	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
65	C₄×C₈	—	2	6	no	—
66	C₄×C₈	—	122	3	no	—
67	C₄×C₈	—	122	3	no	—
68	C₂²×C₄	—	61	1	no	—
69	C₂²×C₄	—	61	3	no	—
70	C₂²×C₄	C₂×C₄×D₆₁	1	3	yes	—
71	C₂×C₈	—	2	6	no	—
72	C₂×C₈	—	2	6	no	—
73	C₂×C₈	—	2	6	no	—
74	C₂×C₈	—	122	6	no	—
75	C₂×C₈	—	122	6	no	—
76	C₂×C₈	—	122	6	no	—
77	C₄²	—	122	6	no	—
78	C₄²	—	122	6	no	—
79	C₄²	#11758224	1	3	yes	—
80	C₄²	#11758226	1	1	yes	characteristic
81	C₂³	C₄²×D₆₁	1	1	yes	characteristic
82	C₂×C₄	—	122	3	no	—
83	C₂×C₄	—	122	3	no	—
84	C₂×C₄	—	122	6	no	—
85	C₂×C₄	#14519733	1	3	yes	—
86	C₂×C₄	#14519735	1	3	yes	—
87	C₂×C₄	#14519783	1	6	yes	—
88	C₂×C₄	#14519785	1	6	yes	—
89	C₈	—	4	6	no	—
90	C₈	—	244	6	no	—
91	C₂²	#16839607	1	1	yes	characteristic
92	C₂²	#16839716	1	3	yes	—
93	C₂²	#16839719	1	3	yes	—
94	C₄	—	244	1	no	—
95	C₄	—	244	3	no	—
96	C₄	#16839820	1	6	yes	—
97	C₄	#16839822	1	6	yes	—
98	C₂	#21444118	1	1	yes	characteristic
99	C₂	#21444120	1	3	yes	—
100	C₂	#21445569	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21445564

C₂

yes

maximal

#21443646

C₂

yes

maximal

C₆₁×#1273

C₂

yes

maximal, characteristic

C₂×C₄×#92418

C₂²

yes

—

C₄²×Dic₆₁

C₂²

yes

characteristic

#16839516

C₄

yes

—

#16839516

C₄

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

—

C₂²×#92418

C₂×C₄

yes

—

C₂×C₄×Dic₆₁

C₂×C₄

yes

—

C₂×C₄×Dic₆₁

C₂×C₄

yes

—

C₄×#92418

—

C₄×#92418

—

C₂²×C₈×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄×C₈×C₆₁

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

—

C₂²×Dic₆₁

C₄²

yes

characteristic

C₂²×Dic₆₁

C₄²

yes

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

#41

yes

—

C₄²×C₆₁

#41

yes

characteristic

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

—

C₂³×C₆₁

C₂×C₄²

yes

characteristic

C₂×C₄×C₆₁

#117

yes

—

C₂×C₄×C₆₁

#117

yes

—

C₂×C₄×C₆₁

#131

yes

—

C₂×C₄×C₆₁

#131

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₈×C₆₁

—

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

#405

yes

—

C₄×C₆₁

#433

yes

—

C₄×C₆₁

#433

yes

—

Dic₆₁

—

Dic₆₁

—

#1744

—

#1273

—

#1273

D₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

C₂×C₆₁

#1746

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₆₁

yes

—

C₄³

—

#336

—

#336

—

C₆₁

#8135

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₄×D₆₁

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₆₁

yes

characteristic

C₄×C₈

—

C₄×C₈

—

122

—

C₄×C₈

—

122

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄×D₆₁

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

—

122

—

C₂×C₈

—

122

—

C₄²

—

122

—

C₄²

—

122

—

C₄²

#11758224

yes

—

C₄²

#11758226

yes

characteristic

C₂³

C₄²×D₆₁

yes

characteristic

C₂×C₄

—

122

—

C₂×C₄

—

122

—

C₂×C₄

—

122

—

C₂×C₄

#14519733

yes

—

C₂×C₄

#14519735

yes

—

C₂×C₄

#14519783

yes

—

C₂×C₄

#14519785

yes

—

C₈

—

C₈

—

244

—

C₂²

#16839607

yes

characteristic

C₂²

#16839716

yes

—

C₂²

#16839719

yes

—

C₄

—

244

—

C₄

—

244

—

C₄

#16839820

yes

—

C₄

#16839822

yes

—

C₂

#21444118

yes

characteristic

C₂

#21444120

yes

—

100

C₂

#21445569

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₆₁
commutator, socle, prime radical	C₂³×C₆₁	C₂×C₄²
Fitting	C₆₁×#1273	C₂
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1273	D₆₁
hypocenter	C₆₁	#8135
LCS	G ▷ C₂³×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄² ◁ #1273

subgroup

G/H

center, Frattini

C₂×C₄²

C₂²×D₆₁

commutator, socle, prime radical

C₂³×C₆₁

C₂×C₄²

Fitting

C₆₁×#1273

C₂

cosocle

C₂×C₄²×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂³×C₆₁ ▷ C₆₁

C₁ ◁ C₂×C₄² ◁ #1273

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

H ≅ G/N

type

59. C₆₁

#8135

semidirect

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

subgroup	index
#21445564	2
#21443646	2
C₆₁×#1273	2
#8135	61

subgroup

index

#21445564

#21443646

C₆₁×#1273

#8135

Subgroups — #29570447

Conjugacy classes of subgroups

Complements

Maximal subgroups