Group #29594718 — subgroups

Subgroups — #29594718

Order 15616.

290 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 79 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	#21448044	C₂	1	4	yes	maximal, complemented
2	#21445434	C₂	1	1	yes	maximal, characteristic
3	#21443736	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#1710	C₂	1	1	yes	maximal, characteristic, complemented
5	#16840139	—	2	2	no	—
6	#16840139	C₂²	1	2	yes	—
7	#16839562	—	2	8	no	—
8	C₄×#11758185	C₂²	1	2	yes	—
9	C₆₁×#500	C₂²	1	2	yes	—
10	C₆₁×#487	—	2	2	no	—
11	C₈²×C₆₁	C₂²	1	1	yes	characteristic
12	C₂×D₄₈₈	—	2	4	no	—
13	C₂×D₄₈₈	—	4	4	no	—
14	#14519774	—	2	2	no	—
15	C₂×#11758185	D₄	1	4	yes	—
16	C₄⋊Q₈×C₆₁	—	2	2	no	—
17	#107×C₆₁	—	2	4	no	—
18	#103×C₆₁	—	4	4	no	—
19	C₄×C₈×C₆₁	D₄	1	2	yes	—
20	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
21	C₂×Q₈×C₆₁	—	4	2	no	—
22	C₂×D₂₄₄	—	4	2	no	—
23	C₂×D₂₄₄	—	4	4	no	—
24	C₂×C₈×C₆₁	D₈	1	4	yes	—
25	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
26	C₄⋊C₄×C₆₁	—	4	2	no	—
27	C₄⋊C₄×C₆₁	—	4	2	no	—
28	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
29	D₄₈₈	—	4	8	no	—
30	D₄₈₈	—	8	8	no	—
31	#11758185	—	2	4	no	—
32	C₂²×D₆₁	—	8	2	no	—
33	Q₈×C₆₁	—	8	4	no	—
34	C₂×C₄×C₆₁	—	8	2	no	—
35	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
36	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
37	D₂₄₄	—	8	4	no	—
38	D₂₄₄	—	8	8	no	—
39	C₈×C₆₁	—	2	4	no	—
40	C₈×C₆₁	#135	1	4	yes	—
41	#12513	—	61	1	no	maximal, Sylow
42	C₂²×C₆₁	#493	1	1	yes	characteristic
43	C₂×D₆₁	—	16	2	no	—
44	C₂×D₆₁	—	16	4	no	—
45	C₄×C₆₁	—	16	2	no	—
46	C₄×C₆₁	#495	1	2	yes	—
47	C₄×C₆₁	#509	1	4	yes	—
48	#2240	—	61	4	no	—
49	#1712	—	61	1	no	—
50	#1710	D₆₁	1	1	yes	characteristic, complemented
51	#1311	—	61	1	no	—
52	C₂×C₆₁	#1709	1	1	yes	characteristic
53	C₂×C₆₁	#2250	1	2	yes	—
54	D₆₁	—	32	2	no	—
55	#500	C₂×D₆₁	1	2	yes	—
56	#493	—	61	2	no	—
57	#493	—	122	2	no	—
58	#487	—	2	2	no	—
59	#357	—	122	8	no	—
60	C₄×C₁₆	—	61	2	no	—
61	C₈²	C₂×D₆₁	1	1	yes	characteristic
62	C₆₁	#12513	1	1	yes	characteristic, complemented, Sylow
63	C₂×D₈	—	122	4	no	—
64	C₂×D₈	—	244	4	no	—
65	C₄⋊Q₈	—	2	2	no	—
66	#127	—	122	2	no	—
67	C₂×C₁₆	—	61	4	no	—
68	#107	—	2	4	no	—
69	#103	—	4	4	no	—
70	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
71	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
72	C₂×Q₈	—	4	2	no	—
73	C₂×D₄	—	244	2	no	—
74	C₂×D₄	—	244	4	no	—
75	D₈	—	488	8	no	—
76	D₈	—	244	8	no	—
77	C₂×C₈	#11758199	1	4	yes	—
78	C₂×C₈	D₄×D₆₁	1	2	yes	—
79	C₄⋊C₄	—	4	2	no	—
80	C₄⋊C₄	—	4	2	no	—
81	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
82	C₁₆	—	122	4	no	—
83	C₂³	—	488	2	no	—
84	Q₈	—	8	4	no	—
85	D₄	—	488	8	no	—
86	D₄	—	488	4	no	—
87	C₂×C₄	—	8	2	no	—
88	C₂×C₄	#14519773	1	1	yes	characteristic
89	C₂×C₄	C₂×#11758199	1	2	yes	—
90	C₈	—	2	4	no	—
91	C₈	#14519800	1	4	yes	—
92	C₂²	—	976	2	no	—
93	C₂²	—	976	4	no	—
94	C₂²	#16840138	1	1	yes	characteristic
95	C₄	—	16	2	no	—
96	C₄	#16840147	1	2	yes	—
97	C₄	#16840197	1	4	yes	—
98	C₂	—	1952	2	no	—
99	C₂	#21445426	1	1	yes	characteristic
100	C₂	#21448086	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21448044

C₂

yes

maximal, complemented

#21445434

C₂

yes

maximal, characteristic

#21443736

C₂

yes

maximal, characteristic, complemented

C₆₁×#1710

C₂

yes

maximal, characteristic, complemented

#16840139

—

#16840139

C₂²

yes

—

#16839562

—

C₄×#11758185

C₂²

yes

—

C₆₁×#500

C₂²

yes

—

C₆₁×#487

—

C₈²×C₆₁

C₂²

yes

characteristic

C₂×D₄₈₈

—

C₂×D₄₈₈

—

#14519774

—

C₂×#11758185

D₄

yes

—

C₄⋊Q₈×C₆₁

—

#107×C₆₁

—

#103×C₆₁

—

C₄×C₈×C₆₁

D₄

yes

—

C₄×C₈×C₆₁

C₂³

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×C₈×C₆₁

D₈

yes

—

C₂×C₈×C₆₁

C₂×D₄

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂×D₄

yes

characteristic

D₄₈₈

—

D₄₈₈

—

#11758185

—

C₂²×D₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#127

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

—

D₂₄₄

—

D₂₄₄

—

C₈×C₆₁

—

C₈×C₆₁

#135

yes

—

—

C₂²×C₆₁

yes

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#495

yes

—

C₄×C₆₁

#509

yes

—

#2240

—

#1712

—

#1710

D₆₁

yes

characteristic, complemented

#1311

—

C₂×C₆₁

#1709

yes

characteristic

C₂×C₆₁

#2250

yes

—

D₆₁

—

#500

C₂×D₆₁

yes

—

#493

—

#493

—

122

—

#487

—

#357

—

122

—

C₄×C₁₆

—

C₈²

C₂×D₆₁

yes

characteristic

C₆₁

#12513

yes

characteristic, complemented, Sylow

C₂×D₈

—

122

—

C₂×D₈

—

244

—

C₄⋊Q₈

—

#127

—

122

—

C₂×C₁₆

—

#107

—

#103

—

C₄×C₈

C₂²⋊D₆₁

yes

—

C₄×C₈

C₂²×D₆₁

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

244

—

C₂×D₄

—

244

—

D₈

—

488

—

D₈

—

244

—

C₂×C₈

#11758199

yes

—

C₂×C₈

D₄×D₆₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₂²⋊D₆₁

yes

characteristic

C₁₆

—

122

—

C₂³

—

488

—

Q₈

—

D₄

—

488

—

D₄

—

488

—

C₂×C₄

—

C₂×C₄

#14519773

yes

characteristic

C₂×C₄

C₂×#11758199

yes

—

C₈

—

C₈

#14519800

yes

—

C₂²

—

976

—

C₂²

—

976

—

C₂²

#16840138

yes

characteristic

C₄

—

C₄

#16840147

yes

—

C₄

#16840197

yes

—

C₂

—

1952

—

C₂

#21445426

yes

characteristic

100

C₂

#21448086

yes

—

	subgroup	G/H
center	C₂²	#16840138
commutator, cosocle	C₄×C₈×C₆₁	C₂³
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₆₁×#1710	C₂
socle	C₂²×C₆₁	#493
prime radical	#21445434	C₂
hypercenter	#1710	D₆₁
hypocenter	C₆₁	#12513
LCS	G ▷ C₄×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

subgroup

G/H

center

C₂²

#16840138

commutator, cosocle

C₄×C₈×C₆₁

C₂³

Frattini

C₄×C₈

C₂²×D₆₁

Fitting

C₆₁×#1710

C₂

C₂²×C₆₁

C₂

D₆₁

C₆₁

G ▷ C₄×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

N	H ≅ G/N	type
1. #21448044	C₂	semidirect
3. #21443736	C₂	semidirect
4. C₆₁×#1710	C₂	semidirect
50. #1710	D₆₁	semidirect
62. C₆₁	#12513	semidirect

H ≅ G/N

type

1. #21448044

C₂

semidirect

3. #21443736

C₂

semidirect

4. C₆₁×#1710

C₂

semidirect

50. #1710

D₆₁

semidirect

62. C₆₁

#12513

semidirect

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

subgroup	index
#21448044	2
#21445434	2
#21443736	2
C₆₁×#1710	2
#12513	61

subgroup

index

C₆₁×#1710

Subgroups — #29594718

Conjugacy classes of subgroups

Complements

Maximal subgroups