Group #29594717 — subgroups

Subgroups — #29594717

Order 15616.

290 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 83 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	#21448043	C₂	1	4	yes	maximal, complemented
2	#21445434	C₂	1	1	yes	maximal, characteristic
3	#21445429	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#1311	C₂	1	1	yes	maximal, characteristic, complemented
5	#16840167	C₂²	1	2	yes	—
6	#16840139	—	2	2	no	—
7	#16840139	C₂²	1	2	yes	—
8	#16840113	—	2	2	no	—
9	#16839564	—	2	8	no	—
10	C₄×C₁₆×C₆₁	C₂²	1	2	yes	—
11	C₈²×C₆₁	C₂²	1	1	yes	characteristic
12	C₂×D₄₈₈	—	2	4	no	—
13	C₂×D₄₈₈	—	4	4	no	—
14	#14519777	—	2	2	no	—
15	#14519774	—	2	2	no	—
16	#14519711	—	2	4	no	—
17	#14519702	—	4	4	no	—
18	C₂×C₁₆×C₆₁	D₄	1	4	yes	—
19	C₄×C₈×C₆₁	D₄	1	2	yes	—
20	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
21	C₂×D₂₄₄	—	4	2	no	—
22	C₂×D₂₄₄	—	4	4	no	—
23	C₂×Dic₁₂₂	—	4	2	no	—
24	C₂×C₈×C₆₁	D₈	1	4	yes	—
25	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
26	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
27	C₄⋊Dic₆₁	—	4	2	no	—
28	C₄⋊Dic₆₁	—	4	2	no	—
29	D₄₈₈	—	4	8	no	—
30	D₄₈₈	—	8	8	no	—
31	C₁₆×C₆₁	—	2	4	no	—
32	C₂²×D₆₁	—	8	2	no	—
33	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
34	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
35	C₂×Dic₆₁	—	8	2	no	—
36	D₂₄₄	—	8	4	no	—
37	D₂₄₄	—	8	8	no	—
38	Dic₁₂₂	—	8	4	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	#495	1	2	yes	—
46	C₄×C₆₁	#509	1	4	yes	—
47	Dic₆₁	—	16	2	no	—
48	#2240	—	61	4	no	—
49	#1712	—	61	1	no	—
50	#1710	—	61	1	no	—
51	#1311	D₆₁	1	1	yes	characteristic, complemented
52	C₂×C₆₁	#1709	1	1	yes	characteristic
53	C₂×C₆₁	#2250	1	2	yes	—
54	D₆₁	—	32	2	no	—
55	#500	—	61	2	no	—
56	#493	—	61	2	no	—
57	#493	—	122	2	no	—
58	#487	—	122	2	no	—
59	#357	—	122	8	no	—
60	C₄×C₁₆	C₂×D₆₁	1	2	yes	—
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	—	122	2	no	—
66	#127	—	122	2	no	—
67	C₂×C₁₆	D₂₄₄	1	4	yes	—
68	#107	—	122	4	no	—
69	#103	—	244	4	no	—
70	C₄×C₈	D₂₄₄	1	2	yes	—
71	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
72	C₂×Q₈	—	244	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	D₄₈₈	1	4	yes	—
78	C₂×C₈	C₂×D₂₄₄	1	2	yes	—
79	C₄⋊C₄	—	244	2	no	—
80	C₄⋊C₄	—	244	2	no	—
81	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
82	C₁₆	—	2	4	no	—
83	C₂³	—	488	2	no	—
84	Q₈	—	488	4	no	—
85	D₄	—	488	8	no	—
86	D₄	—	488	4	no	—
87	C₂×C₄	—	488	2	no	—
88	C₂×C₄	#14519774	1	1	yes	characteristic
89	C₂×C₄	C₂×D₄₈₈	1	2	yes	—
90	C₈	—	2	4	no	—
91	C₈	#14519799	1	4	yes	—
92	C₂²	—	976	2	no	—
93	C₂²	—	976	4	no	—
94	C₂²	#16840139	1	1	yes	characteristic
95	C₄	—	976	2	no	—
96	C₄	#16840146	1	2	yes	—
97	C₄	#16840200	1	4	yes	—
98	C₂	—	1952	2	no	—
99	C₂	#21445425	1	1	yes	characteristic
100	C₂	#21448085	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21448043

C₂

yes

maximal, complemented

#21445434

C₂

yes

maximal, characteristic

#21445429

C₂

yes

maximal, characteristic, complemented

C₆₁×#1311

C₂

yes

maximal, characteristic, complemented

#16840167

C₂²

yes

—

#16840139

—

#16840139

C₂²

yes

—

#16840113

—

#16839564

—

C₄×C₁₆×C₆₁

C₂²

yes

—

C₈²×C₆₁

C₂²

yes

characteristic

C₂×D₄₈₈

—

C₂×D₄₈₈

—

—

—

—

—

C₂×C₁₆×C₆₁

D₄

yes

—

C₄×C₈×C₆₁

D₄

yes

—

C₄×C₈×C₆₁

C₂³

yes

characteristic

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

D₈

yes

—

C₂×C₈×C₆₁

C₂×D₄

yes

—

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

D₄₈₈

—

D₄₈₈

—

C₁₆×C₆₁

—

C₂²×D₆₁

—

C₂×C₄×C₆₁

#127

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

—

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

#135

yes

—

—

C₂²×C₆₁

yes

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

#495

yes

—

C₄×C₆₁

#509

yes

—

Dic₆₁

—

—

—

—

D₆₁

yes

characteristic, complemented

C₂×C₆₁

#1709

yes

characteristic

C₂×C₆₁

#2250

yes

—

D₆₁

—

#500

—

#493

—

#493

—

122

—

#487

—

122

—

#357

—

122

—

C₄×C₁₆

C₂×D₆₁

yes

—

C₈²

C₂×D₆₁

yes

characteristic

C₆₁

#12513

yes

characteristic, complemented, Sylow

C₂×D₈

—

122

—

C₂×D₈

—

244

—

C₄⋊Q₈

—

122

—

#127

—

122

—

C₂×C₁₆

D₂₄₄

yes

—

#107

—

122

—

#103

—

244

—

C₄×C₈

D₂₄₄

yes

—

C₄×C₈

C₂²×D₆₁

yes

characteristic

C₂×Q₈

—

244

—

C₂×D₄

—

244

—

C₂×D₄

—

244

—

D₈

—

488

—

D₈

—

244

—

C₂×C₈

D₄₈₈

yes

—

C₂×C₈

C₂×D₂₄₄

yes

—

C₄⋊C₄

—

244

—

C₄⋊C₄

—

244

—

C₄²

C₂×D₂₄₄

yes

characteristic

C₁₆

—

C₂³

—

488

—

Q₈

—

488

—

D₄

—

488

—

D₄

—

488

—

C₂×C₄

—

488

—

C₂×C₄

#14519774

yes

characteristic

C₂×C₄

C₂×D₄₈₈

yes

—

C₈

—

C₈

#14519799

yes

—

C₂²

—

976

—

C₂²

—

976

—

C₂²

#16840139

yes

characteristic

C₄

—

976

—

C₄

#16840146

yes

—

C₄

#16840200

yes

—

C₂

—

1952

—

C₂

#21445425

yes

characteristic

100

C₂

#21448085

yes

—

	subgroup	G/H
center	C₂²	#16840139
commutator, cosocle	C₄×C₈×C₆₁	C₂³
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₆₁×#1311	C₂
socle	C₂²×C₆₁	#493
prime radical	#21445434	C₂
hypercenter	#1311	D₆₁
hypocenter	C₆₁	#12513
LCS	G ▷ C₄×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂²

#16840139

commutator, cosocle

C₄×C₈×C₆₁

C₂³

Frattini

C₄×C₈

C₂²×D₆₁

Fitting

C₆₁×#1311

C₂

C₂²×C₆₁

C₂

D₆₁

C₆₁

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

N	H ≅ G/N	type
1. #21448043	C₂	semidirect
3. #21445429	C₂	semidirect
4. C₆₁×#1311	C₂	semidirect
51. #1311	D₆₁	semidirect
62. C₆₁	#12513	semidirect

H ≅ G/N

type

1. #21448043

C₂

semidirect

3. #21445429

C₂

semidirect

4. C₆₁×#1311

C₂

semidirect

51. #1311

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
#21448043	2
#21445434	2
#21445429	2
C₆₁×#1311	2
#12513	61

subgroup

index

C₆₁×#1311

Subgroups — #29594717

Conjugacy classes of subgroups

Complements

Maximal subgroups