Group #29584522 — subgroups

Subgroups — #29584522

Order 15616.

350 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 97 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	#21445952	C₂	1	3	yes	maximal
2	#21443700	C₂	1	3	yes	maximal
3	C₆₁×#2845	C₂	1	1	yes	maximal, characteristic
4	C₂×#14519688	C₂²	1	3	yes	—
5	#16839632	C₂²	1	1	yes	characteristic
6	#16839524	—	2	6	no	—
7	C₆₁×#398	—	2	6	no	—
8	C₆₁×#382	C₂²	1	3	yes	—
9	C₂²×#92418	—	2	3	no	—
10	C₂×C₄⋊Dic₆₁	—	2	3	no	—
11	#14519688	—	2	3	no	—
12	#14519688	Q₈	1	6	yes	—
13	C₂×C₄⋊C₄×C₆₁	—	2	6	no	—
14	C₂×C₄⋊C₄×C₆₁	D₄	1	6	yes	—
15	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
16	#95×C₆₁	—	2	2	no	—
17	#95×C₆₁	—	2	6	no	—
18	C₂²×C₄×C₆₁	—	2	6	no	—
19	C₂²×C₄×C₆₁	C₂×D₄	1	3	yes	—
20	C₂²×Dic₆₁	—	4	1	no	—
21	C₄⋊C₄×C₆₁	—	4	6	no	—
22	C₄⋊C₄×C₆₁	—	8	6	no	—
23	C₄²×C₆₁	C₂×Q₈	1	3	yes	—
24	C₄²×C₆₁	#41	1	1	yes	characteristic
25	C₄⋊Dic₆₁	—	4	6	no	—
26	C₂×#92418	—	2	6	no	—
27	C₂×#92418	—	4	3	no	—
28	C₂×#92418	—	4	3	no	—
29	C₂³×C₆₁	#120	1	1	yes	characteristic
30	C₂×C₄×C₆₁	—	2	3	no	—
31	C₂×C₄×C₆₁	—	2	3	no	—
32	C₂×C₄×C₆₁	—	4	6	no	—
33	C₂×C₄×C₆₁	—	4	6	no	—
34	C₂×C₄×C₆₁	—	4	6	no	—
35	C₂×C₄×C₆₁	C₂²⋊Q₈	1	3	yes	—
36	C₂×C₄×C₆₁	C₄⋊Q₈	1	3	yes	—
37	C₂×Dic₆₁	—	8	3	no	—
38	#92418	—	4	6	no	—
39	#10586	—	61	1	no	maximal, Sylow
40	C₂²×C₆₁	#395	1	1	yes	characteristic
41	C₂²×C₆₁	#454	1	3	yes	—
42	C₂²×C₆₁	#455	1	3	yes	—
43	C₄×C₆₁	—	2	3	no	—
44	C₄×C₆₁	—	2	3	no	—
45	C₄×C₆₁	—	8	6	no	—
46	Dic₆₁	—	16	1	no	—
47	#2845	D₆₁	1	1	yes	characteristic
48	#1836	—	61	3	no	—
49	#1295	—	61	3	no	—
50	C₂×C₆₁	#1658	1	1	yes	characteristic
51	C₂×C₆₁	#1662	1	3	yes	—
52	C₂×C₆₁	#2061	1	3	yes	—
53	C₂×#97	—	61	3	no	—
54	#398	—	2	6	no	—
55	#384	—	61	1	no	—
56	#382	C₂×D₆₁	1	3	yes	—
57	#340	—	122	6	no	—
58	C₆₁	#10586	1	1	yes	characteristic, complemented, Sylow
59	C₂²×C₈	—	122	3	no	—
60	C₂×C₄⋊C₄	—	2	6	no	—
61	C₂×C₄⋊C₄	—	122	3	no	—
62	C₂×C₄⋊C₄	C₂²⋊D₆₁	1	6	yes	—
63	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
64	#97	—	61	6	no	—
65	#97	—	122	3	no	—
66	#95	—	2	2	no	—
67	#95	—	2	6	no	—
68	C₂²×C₄	—	2	6	no	—
69	C₂²×C₄	—	244	1	no	—
70	C₂²×C₄	C₂×C₂²⋊D₆₁	1	3	yes	—
71	C₂×C₈	—	122	6	no	—
72	C₂×C₈	—	244	3	no	—
73	C₂×C₈	—	244	3	no	—
74	C₄⋊C₄	—	4	6	no	—
75	C₄⋊C₄	—	8	6	no	—
76	C₄⋊C₄	—	244	6	no	—
77	C₄²	Q₈×D₆₁	1	3	yes	—
78	C₄²	#11758226	1	1	yes	characteristic
79	C₂³	#14519746	1	1	yes	characteristic
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	—	2	3	no	—
82	C₂×C₄	—	4	6	no	—
83	C₂×C₄	—	4	6	no	—
84	C₂×C₄	—	4	6	no	—
85	C₂×C₄	—	488	3	no	—
86	C₂×C₄	#14519753	1	3	yes	—
87	C₂×C₄	#14519775	1	3	yes	—
88	C₈	—	244	6	no	—
89	C₂²	#16839681	1	1	yes	characteristic
90	C₂²	#16839935	1	3	yes	—
91	C₂²	#16839940	1	3	yes	—
92	C₄	—	2	3	no	—
93	C₄	—	2	3	no	—
94	C₄	—	8	6	no	—
95	C₄	—	976	1	no	—
96	C₂	#21445121	1	1	yes	characteristic
97	C₂	#21445142	1	3	yes	—
98	C₂	#21447166	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21445952

C₂

yes

maximal

#21443700

C₂

yes

maximal

C₆₁×#2845

C₂

yes

maximal, characteristic

C₂×#14519688

C₂²

yes

—

#16839632

C₂²

yes

characteristic

#16839524

—

C₆₁×#398

—

C₆₁×#382

C₂²

yes

—

C₂²×#92418

—

C₂×C₄⋊Dic₆₁

—

#14519688

—

#14519688

Q₈

yes

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

D₄

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

#95×C₆₁

—

#95×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

—

C₂²×Dic₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂×Q₈

yes

—

C₄²×C₆₁

#41

yes

characteristic

C₄⋊Dic₆₁

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

—

C₂³×C₆₁

#120

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊Q₈

yes

—

C₂×C₄×C₆₁

C₄⋊Q₈

yes

—

C₂×Dic₆₁

—

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

#455

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

Dic₆₁

—

#2845

D₆₁

yes

characteristic

#1836

—

#1295

—

C₂×C₆₁

#1658

yes

characteristic

C₂×C₆₁

#1662

yes

—

C₂×C₆₁

#2061

yes

—

C₂×#97

—

#398

—

#384

—

#382

C₂×D₆₁

yes

—

#340

—

122

—

C₆₁

#10586

yes

characteristic, complemented, Sylow

C₂²×C₈

—

122

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄⋊C₄

C₂²⋊D₆₁

yes

—

C₂×C₄²

C₂²×D₆₁

yes

characteristic

#97

—

#97

—

122

—

#95

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

244

—

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

—

C₂×C₈

—

122

—

C₂×C₈

—

244

—

C₂×C₈

—

244

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

244

—

C₄²

Q₈×D₆₁

yes

—

C₄²

yes

C₂³

yes

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

488

—

C₂×C₄

#14519753

yes

—

C₂×C₄

#14519775

yes

—

C₈

—

244

—

C₂²

#16839681

yes

characteristic

C₂²

#16839935

yes

—

C₂²

#16839940

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

976

—

C₂

#21445121

yes

characteristic

C₂

#21445142

yes

—

C₂

#21447166

yes

—

	subgroup	G/H
center	C₂³	#14519746
commutator, cosocle	C₂×C₄²×C₆₁	C₂³
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#2845	C₂
socle, prime radical	C₂³×C₆₁	#120
hypercenter	#2845	D₆₁
hypocenter	C₆₁	#10586
LCS	G ▷ C₂×C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2845

subgroup

G/H

center

C₂³

#14519746

commutator, cosocle

C₂×C₄²×C₆₁

C₂³

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#2845

C₂

C₂³×C₆₁

D₆₁

C₆₁

G ▷ C₂×C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2845

N	H ≅ G/N	type
58. C₆₁	#10586	semidirect

H ≅ G/N

type

58. C₆₁

#10586

semidirect

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

subgroup	index
#21445952	2
#21443700	2
C₆₁×#2845	2
#10586	61

subgroup

index

#21445952

#21443700

C₆₁×#2845

#10586

Subgroups — #29584522

Conjugacy classes of subgroups

Complements

Maximal subgroups