Group #29594730 — subgroups

Subgroups — #29594730

Order 15616.

354 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 103 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	#21448074	C₂	1	4	yes	maximal, complemented
2	#21445437	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×#11758185	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#1712	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519721	—	2	8	no	—
6	#16840167	—	2	2	no	—
7	#16840142	C₂²	1	2	yes	—
8	C₄×#11758185	C₂²	1	2	yes	—
9	C₆₁×#493	—	2	2	no	—
10	C₆₁×#493	C₂²	1	2	yes	—
11	C₈²×C₆₁	C₂²	1	1	yes	characteristic
12	C₂×Dic₂₄₄	—	2	4	no	—
13	#14519777	—	2	2	no	—
14	#14519721	—	4	16	no	—
15	C₂×#11758185	D₄	1	4	yes	—
16	#14519711	—	4	4	no	—
17	C₂×D₈×C₆₁	—	2	4	no	—
18	C₂×D₈×C₆₁	—	4	4	no	—
19	#127×C₆₁	—	2	2	no	—
20	C₄×C₈×C₆₁	D₄	1	2	yes	—
21	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
22	C₂×D₄×C₆₁	—	4	2	no	—
23	C₂×D₄×C₆₁	—	4	4	no	—
24	C₂×Dic₁₂₂	—	4	2	no	—
25	D₈×C₆₁	—	4	8	no	—
26	D₈×C₆₁	—	8	8	no	—
27	C₂×C₈×C₆₁	D₈	1	4	yes	—
28	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
29	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
30	C₄⋊Dic₆₁	—	4	4	no	—
31	Dic₂₄₄	—	4	8	no	—
32	#11758185	D₈	1	8	yes	complemented
33	C₂³×C₆₁	—	8	2	no	—
34	D₄×C₆₁	—	8	4	no	—
35	D₄×C₆₁	—	8	8	no	—
36	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
37	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
38	C₂×Dic₆₁	—	8	2	no	—
39	Dic₁₂₂	—	8	4	no	—
40	C₈×C₆₁	SD₃₂	1	8	yes	complemented
41	C₈×C₆₁	C₂×D₈	1	4	yes	—
42	#12517	—	61	1	no	maximal, Sylow
43	C₂²×C₆₁	—	16	2	no	—
44	C₂²×C₆₁	—	16	4	no	—
45	C₂²×C₆₁	#493	1	1	yes	characteristic
46	C₄×C₆₁	#493	1	2	yes	—
47	C₄×C₆₁	C₂×SD₃₂	1	4	yes	—
48	Dic₆₁	—	16	2	no	—
49	#2248	—	61	4	no	—
50	#1713	—	61	1	no	—
51	#1712	D₆₁	1	1	yes	characteristic
52	C₈×C₁₆	—	61	1	no	—
53	C₂×C₆₁	—	32	2	no	—
54	C₂×C₆₁	#1712	1	1	yes	characteristic
55	C₂×C₆₁	#2248	1	2	yes	—
56	C₂×SD₃₂	—	122	8	no	—
57	#500	—	122	2	no	—
58	#494	—	61	2	no	—
59	#493	—	2	2	no	—
60	#493	C₂×D₆₁	1	2	yes	—
61	C₄×C₁₆	—	61	2	no	—
62	C₈²	C₂×D₆₁	1	1	yes	characteristic
63	C₆₁	#12517	1	1	yes	characteristic, complemented, Sylow
64	C₂×Q₁₆	—	122	4	no	—
65	C₂×D₈	—	2	4	no	—
66	C₂×D₈	—	4	4	no	—
67	C₄⋊Q₈	—	122	2	no	—
68	#127	—	2	2	no	—
69	SD₃₂	—	244	16	no	—
70	C₂×C₁₆	—	61	4	no	—
71	#107	—	244	4	no	—
72	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
73	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
74	C₂×Q₈	—	244	2	no	—
75	C₂×D₄	—	4	2	no	—
76	C₂×D₄	—	4	4	no	—
77	Q₁₆	—	244	8	no	—
78	D₈	—	4	8	no	—
79	D₈	—	8	8	no	—
80	C₂×C₈	#11758199	1	4	yes	—
81	C₂×C₈	D₄×D₆₁	1	2	yes	—
82	C₄⋊C₄	—	244	4	no	—
83	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
84	C₁₆	—	61	8	no	—
85	C₂³	—	8	2	no	—
86	Q₈	—	488	4	no	—
87	D₄	—	8	4	no	—
88	D₄	—	8	8	no	—
89	C₂×C₄	—	488	2	no	—
90	C₂×C₄	#14519773	1	1	yes	characteristic
91	C₂×C₄	C₂×#11758199	1	2	yes	—
92	C₈	#14519721	1	8	yes	complemented
93	C₈	D₈×D₆₁	1	4	yes	—
94	C₂²	—	16	2	no	—
95	C₂²	—	16	4	no	—
96	C₂²	#16840138	1	1	yes	characteristic
97	C₄	—	976	2	no	—
98	C₄	#16840137	1	2	yes	—
99	C₄	C₂×#14519721	1	4	yes	—
100	C₂	—	32	2	no	—
101	C₂	#21445433	1	1	yes	characteristic
102	C₂	#21448074	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21448074

C₂

yes

maximal, complemented

#21445437

C₂

yes

maximal, characteristic, complemented

C₈×#11758185

C₂

yes

maximal, characteristic, complemented

C₆₁×#1712

C₂

yes

maximal, characteristic

C₂×#14519721

—

#16840167

—

#16840142

C₂²

yes

—

C₄×#11758185

C₂²

yes

—

C₆₁×#493

—

C₆₁×#493

C₂²

yes

—

C₈²×C₆₁

C₂²

yes

characteristic

C₂×Dic₂₄₄

—

#14519777

—

#14519721

—

C₂×#11758185

D₄

yes

—

#14519711

—

C₂×D₈×C₆₁

—

C₂×D₈×C₆₁

—

#127×C₆₁

—

C₄×C₈×C₆₁

D₄

yes

—

C₄×C₈×C₆₁

C₂³

yes

characteristic

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×Dic₁₂₂

—

D₈×C₆₁

—

D₈×C₆₁

—

C₂×C₈×C₆₁

D₈

yes

—

C₂×C₈×C₆₁

C₂×D₄

yes

—

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄⋊Dic₆₁

—

Dic₂₄₄

—

#11758185

D₈

yes

complemented

C₂³×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

#127

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

—

C₂×Dic₆₁

—

Dic₁₂₂

—

C₈×C₆₁

SD₃₂

yes

complemented

C₈×C₆₁

C₂×D₈

yes

—

#12517

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#493

yes

characteristic

C₄×C₆₁

#493

yes

—

C₄×C₆₁

C₂×SD₃₂

yes

—

Dic₆₁

—

#2248

—

#1713

—

#1712

D₆₁

yes

characteristic

C₈×C₁₆

—

C₂×C₆₁

—

C₂×C₆₁

#1712

yes

characteristic

C₂×C₆₁

#2248

yes

—

C₂×SD₃₂

—

122

—

#500

—

122

—

#494

—

#493

—

#493

C₂×D₆₁

yes

—

C₄×C₁₆

—

C₈²

C₂×D₆₁

yes

characteristic

C₆₁

#12517

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

122

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

122

—

#127

—

SD₃₂

—

244

—

C₂×C₁₆

—

#107

—

244

—

C₄×C₈

C₂²⋊D₆₁

yes

—

C₄×C₈

C₂²×D₆₁

yes

characteristic

C₂×Q₈

—

244

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

244

—

D₈

—

D₈

—

C₂×C₈

#11758199

yes

—

C₂×C₈

D₄×D₆₁

yes

—

C₄⋊C₄

—

244

—

C₄²

C₂×C₂²⋊D₆₁

yes

characteristic

C₁₆

—

C₂³

—

Q₈

—

488

—

D₄

—

D₄

—

C₂×C₄

—

488

—

C₂×C₄

#14519773

yes

characteristic

C₂×C₄

C₂×#11758199

yes

—

C₈

#14519721

yes

complemented

C₈

D₈×D₆₁

yes

—

C₂²

—

C₂²

—

C₂²

#16840138

yes

characteristic

C₄

—

976

—

C₄

#16840137

yes

—

C₄

C₂×#14519721

yes

—

100

C₂

—

101

C₂

#21445433

yes

characteristic

102

C₂

#21448074

yes

—

	subgroup	G/H
center	C₂²	#16840138
commutator, cosocle	C₄×C₈×C₆₁	C₂³
Frattini	C₄×C₈	C₂²×D₆₁
Fitting, prime radical	C₆₁×#1712	C₂
socle	C₂²×C₆₁	#493
hypercenter	#1712	D₆₁
hypocenter	C₆₁	#12517
LCS	G ▷ C₄×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

subgroup

G/H

center

C₂²

#16840138

commutator, cosocle

C₄×C₈×C₆₁

C₂³

Frattini

C₄×C₈

C₂²×D₆₁

Fitting, prime radical

C₆₁×#1712

C₂

C₂²×C₆₁

D₆₁

C₆₁

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

N	H ≅ G/N	type
1. #21448074	C₂	semidirect
2. #21445437	C₂	semidirect
3. C₈×#11758185	C₂	semidirect
32. #11758185	D₈	semidirect
40. C₈×C₆₁	SD₃₂	semidirect
63. C₆₁	#12517	semidirect
92. C₈	#14519721	semidirect

H ≅ G/N

type

1. #21448074

C₂

semidirect

2. #21445437

C₂

semidirect

3. C₈×#11758185

C₂

semidirect

32. #11758185

D₈

semidirect

40. C₈×C₆₁

SD₃₂

semidirect

63. C₆₁

#12517

semidirect

92. C₈

#14519721

semidirect

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

subgroup	index
#21448074	2
#21445437	2
C₈×#11758185	2
C₆₁×#1712	2
#12517	61

subgroup

index

C₆₁×#1712

Subgroups — #29594730

Conjugacy classes of subgroups

Complements

Maximal subgroups