Group #29573687 — subgroups

Subgroups — #29573687

Order 15616.

322 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 147 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	#21445936	C₂	1	2	yes	maximal
2	#21443654	C₂	1	4	yes	maximal
3	C₆₁×#1744	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#92418	C₂²	1	4	yes	—
5	#16839517	—	2	4	no	—
6	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
7	C₄³×C₆₁	C₂²	1	1	yes	characteristic
8	C₆₁×#336	C₄	1	4	yes	—
9	C₂²×#92418	—	2	4	no	—
10	C₄⋊#92418	—	2	8	no	—
11	C₂²×C₈×C₆₁	C₂×C₄	1	4	yes	—
12	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
13	C₂×C₄²×C₆₁	D₄	1	2	yes	—
14	C₂×C₄²×C₆₁	Q₈	1	2	yes	—
15	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
16	C₄×C₈×C₆₁	—	2	4	no	—
17	C₂²×C₄×C₆₁	C₄⋊C₄	1	4	yes	—
18	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
20	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
21	C₂×C₈×C₆₁	—	2	4	no	—
22	C₂×C₈×C₆₁	—	2	8	no	—
23	C₄²×C₆₁	—	2	4	no	—
24	C₄²×C₆₁	—	2	4	no	—
25	C₄²×C₆₁	—	4	2	no	—
26	C₄²×C₆₁	#41	1	4	yes	—
27	C₂×#92418	—	4	4	no	—
28	C₂×#92418	—	4	8	no	—
29	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₆₁	—	2	2	no	—
31	C₂×C₄×C₆₁	—	2	4	no	—
32	C₂×C₄×C₆₁	—	2	4	no	—
33	C₂×C₄×C₆₁	—	2	4	no	—
34	C₂×C₄×C₆₁	#117	1	2	yes	—
35	C₂×C₄×C₆₁	#124	1	2	yes	—
36	C₂×C₄×C₆₁	#125	1	2	yes	—
37	C₂×C₄×C₆₁	#131	1	4	yes	—
38	C₂×C₄×C₆₁	#135	1	4	yes	—
39	C₈×C₆₁	—	4	4	no	—
40	#92418	—	8	4	no	—
41	#8788	—	61	1	no	maximal, Sylow
42	C₂²×C₆₁	#382	1	1	yes	characteristic
43	C₂²×C₆₁	#424	1	2	yes	—
44	C₂²×C₆₁	#427	1	2	yes	—
45	C₂²×C₆₁	#430	1	2	yes	—
46	C₄×C₆₁	—	2	2	no	—
47	C₄×C₆₁	—	4	4	no	—
48	C₄×C₆₁	#433	1	4	yes	—
49	C₄×C₆₁	#488	1	4	yes	—
50	#1831	—	61	2	no	—
51	#1744	D₆₁	1	1	yes	characteristic
52	#1277	—	61	4	no	—
53	C₂×C₆₁	#1572	1	4	yes	—
54	C₂×C₆₁	#1832	1	1	yes	characteristic
55	C₂×C₆₁	#1835	1	1	yes	characteristic
56	C₂×C₆₁	#1838	1	1	yes	characteristic
57	C₂×C₄⋊C₈	—	61	4	no	—
58	C₂×C₄×C₈	C₂×D₆₁	1	2	yes	—
59	C₄³	C₂×D₆₁	1	1	yes	characteristic
60	#336	—	122	4	no	—
61	#336	Dic₆₁	1	4	yes	—
62	C₆₁	#8788	1	1	yes	characteristic, complemented, Sylow
63	C₂²×C₈	—	122	4	no	—
64	C₂²×C₈	C₂×Dic₆₁	1	4	yes	—
65	C₂×C₄²	Dic₁₂₂	1	2	yes	—
66	C₂×C₄²	D₂₄₄	1	2	yes	—
67	C₂×C₄²	C₂×Dic₆₁	1	2	yes	—
68	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
69	C₄⋊C₈	—	122	8	no	—
70	C₄×C₈	—	2	4	no	—
71	C₂²×C₄	C₄⋊Dic₆₁	1	4	yes	—
72	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
73	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
74	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
75	C₂×C₈	—	2	4	no	—
76	C₂×C₈	—	2	8	no	—
77	C₂×C₈	—	244	4	no	—
78	C₂×C₈	—	244	8	no	—
79	C₄²	—	2	4	no	—
80	C₄²	—	2	4	no	—
81	C₄²	—	4	2	no	—
82	C₄²	#11758222	1	4	yes	—
83	C₂³	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	4	no	—
86	C₂×C₄	—	2	4	no	—
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	#14519736	1	2	yes	—
89	C₂×C₄	#14519765	1	2	yes	—
90	C₂×C₄	#14519769	1	2	yes	—
91	C₂×C₄	#14519784	1	4	yes	—
92	C₂×C₄	#14519799	1	4	yes	—
93	C₈	—	4	4	no	—
94	C₈	—	488	4	no	—
95	C₂²	#16839628	1	1	yes	characteristic
96	C₂²	#16839789	1	2	yes	—
97	C₂²	#16839800	1	2	yes	—
98	C₂²	#16839811	1	2	yes	—
99	C₄	—	2	2	no	—
100	C₄	—	4	4	no	—
101	C₄	#16839823	1	4	yes	—
102	C₄	#16840117	1	4	yes	—
103	C₂	#21444599	1	4	yes	—
104	C₂	#21445939	1	1	yes	characteristic
105	C₂	#21445950	1	1	yes	characteristic
106	C₂	#21445960	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21445936

C₂

yes

maximal

#21443654

C₂

yes

maximal

C₆₁×#1744

C₂

yes

maximal, characteristic

C₂×C₄⋊#92418

C₂²

yes

—

#16839517

—

C₂×C₄×C₈×C₆₁

C₂²

yes

—

C₄³×C₆₁

C₂²

yes

characteristic

C₆₁×#336

C₄

yes

—

C₂²×#92418

—

C₄⋊#92418

—

C₂²×C₈×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

D₄

yes

—

C₂×C₄²×C₆₁

Q₈

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄×C₈×C₆₁

—

C₂²×C₄×C₆₁

C₄⋊C₄

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

#41

yes

—

C₂×#92418

—

C₂×#92418

—

C₂³×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#117

yes

—

C₂×C₄×C₆₁

#124

yes

—

C₂×C₄×C₆₁

#125

yes

—

C₂×C₄×C₆₁

#131

yes

—

C₂×C₄×C₆₁

#135

yes

—

C₈×C₆₁

—

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

#427

yes

—

C₂²×C₆₁

#430

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#433

yes

—

C₄×C₆₁

#488

yes

—

#1831

—

#1744

D₆₁

yes

characteristic

#1277

—

C₂×C₆₁

#1572

yes

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×D₆₁

yes

—

C₄³

C₂×D₆₁

yes

characteristic

#336

—

122

—

#336

Dic₆₁

yes

—

C₆₁

#8788

yes

characteristic, complemented, Sylow

C₂²×C₈

—

122

—

C₂²×C₈

C₂×Dic₆₁

yes

—

C₂×C₄²

Dic₁₂₂

yes

—

C₂×C₄²

D₂₄₄

yes

—

C₂×C₄²

C₂×Dic₆₁

yes

—

C₂×C₄²

C₂²×D₆₁

yes

characteristic

C₄⋊C₈

—

122

—

C₄×C₈

—

C₂²×C₄

C₄⋊Dic₆₁

yes

—

C₂²×C₄

C₂×Dic₁₂₂

yes

characteristic

C₂²×C₄

C₂×D₂₄₄

yes

characteristic

C₂²×C₄

C₂²×Dic₆₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

244

—

C₂×C₈

—

244

—

C₄²

—

C₄²

—

C₄²

—

C₄²

#11758222

yes

—

C₂³

C₂×C₄⋊Dic₆₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#14519736

yes

—

C₂×C₄

#14519765

yes

—

C₂×C₄

#14519769

yes

—

C₂×C₄

#14519784

yes

—

C₂×C₄

#14519799

yes

—

C₈

—

C₈

—

488

—

C₂²

#16839628

yes

characteristic

C₂²

#16839789

yes

—

C₂²

#16839800

yes

—

C₂²

#16839811

yes

—

C₄

—

100

C₄

—

101

C₄

#16839823

yes

—

102

C₄

#16840117

yes

—

103

C₂

#21444599

yes

—

104

C₂

#21445939

yes

characteristic

105

C₂

#21445950

yes

characteristic

106

C₂

#21445960

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₄₄
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1744	C₂
socle, prime radical	C₂³×C₆₁	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1744	D₆₁
hypocenter	C₆₁	#8788
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

subgroup

G/H

center

C₂²×C₄

C₂×D₂₄₄

commutator

C₂²×C₄×C₆₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#1744

C₂

socle, prime radical

C₂³×C₆₁

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

N	H ≅ G/N	type
62. C₆₁	#8788	semidirect

H ≅ G/N

type

62. C₆₁

#8788

semidirect

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

subgroup	index
#21445936	2
#21443654	2
C₆₁×#1744	2
#8788	61

subgroup

index

#21445936

#21443654

C₆₁×#1744

#8788

Subgroups — #29573687

Conjugacy classes of subgroups

Complements

Maximal subgroups