Group #29594731 — subgroups

Subgroups — #29594731

Order 15616.

354 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 115 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	#21448075	C₂	1	4	yes	maximal, complemented
2	#21445437	C₂	1	1	yes	maximal, characteristic, complemented
3	#21445434	C₂	1	1	yes	maximal, characteristic
4	C₈×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14519722	—	2	8	no	—
6	#16840167	—	2	2	no	—
7	#16840142	C₂²	1	2	yes	—
8	#16840139	—	2	2	no	—
9	#16840139	C₂²	1	2	yes	—
10	C₄×C₁₆×C₆₁	C₂²	1	2	yes	—
11	C₈²×C₆₁	C₂²	1	1	yes	characteristic
12	C₂×Dic₂₄₄	—	2	4	no	—
13	C₂×D₄₈₈	—	2	4	no	—
14	C₂×D₄₈₈	—	4	4	no	—
15	#14519777	—	2	2	no	—
16	#14519774	—	2	2	no	—
17	#14519722	—	4	16	no	—
18	#14519711	—	4	4	no	—
19	C₂×C₁₆×C₆₁	D₄	1	4	yes	—
20	C₄×C₈×C₆₁	D₄	1	2	yes	—
21	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
22	C₂×D₂₄₄	—	4	2	no	—
23	C₂×D₂₄₄	—	4	4	no	—
24	C₂×Dic₁₂₂	—	4	2	no	—
25	C₂×C₈×C₆₁	D₈	1	4	yes	—
26	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
27	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
28	C₄⋊Dic₆₁	—	4	4	no	—
29	Dic₂₄₄	—	4	8	no	—
30	D₄₈₈	—	4	8	no	—
31	D₄₈₈	—	8	8	no	—
32	C₁₆×C₆₁	D₈	1	8	yes	complemented
33	C₂²×D₆₁	—	8	2	no	—
34	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
35	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
36	C₂×Dic₆₁	—	8	2	no	—
37	D₂₄₄	—	8	4	no	—
38	D₂₄₄	—	8	8	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	#493	1	1	yes	characteristic
44	C₂×D₆₁	—	16	2	no	—
45	C₂×D₆₁	—	16	4	no	—
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	—	61	1	no	—
52	C₈×C₁₆	D₆₁	1	1	yes	characteristic, complemented
53	C₂×C₆₁	#1712	1	1	yes	characteristic
54	C₂×C₆₁	#2248	1	2	yes	—
55	D₆₁	—	32	2	no	—
56	C₂×SD₃₂	—	122	8	no	—
57	#500	—	122	2	no	—
58	#494	—	61	2	no	—
59	#493	—	61	2	no	—
60	#493	—	122	2	no	—
61	C₄×C₁₆	C₂×D₆₁	1	2	yes	—
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	—	122	4	no	—
66	C₂×D₈	—	244	4	no	—
67	C₄⋊Q₈	—	122	2	no	—
68	#127	—	122	2	no	—
69	SD₃₂	—	244	16	no	—
70	C₂×C₁₆	D₂₄₄	1	4	yes	—
71	#107	—	244	4	no	—
72	C₄×C₈	D₂₄₄	1	2	yes	—
73	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
74	C₂×Q₈	—	244	2	no	—
75	C₂×D₄	—	244	2	no	—
76	C₂×D₄	—	244	4	no	—
77	Q₁₆	—	244	8	no	—
78	D₈	—	488	8	no	—
79	D₈	—	244	8	no	—
80	C₂×C₈	D₄₈₈	1	4	yes	—
81	C₂×C₈	C₂×D₂₄₄	1	2	yes	—
82	C₄⋊C₄	—	244	4	no	—
83	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
84	C₁₆	D₄₈₈	1	8	yes	complemented
85	C₂³	—	488	2	no	—
86	Q₈	—	488	4	no	—
87	D₄	—	488	4	no	—
88	D₄	—	488	8	no	—
89	C₂×C₄	—	488	2	no	—
90	C₂×C₄	#14519774	1	1	yes	characteristic
91	C₂×C₄	C₂×D₄₈₈	1	2	yes	—
92	C₈	#14519722	1	8	yes	complemented
93	C₈	C₂×D₄₈₈	1	4	yes	—
94	C₂²	—	976	2	no	—
95	C₂²	—	976	4	no	—
96	C₂²	#16840139	1	1	yes	characteristic
97	C₄	—	976	2	no	—
98	C₄	#16840139	1	2	yes	—
99	C₄	C₂×#14519722	1	4	yes	—
100	C₂	—	1952	2	no	—
101	C₂	#21445434	1	1	yes	characteristic
102	C₂	#21448075	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21448075

C₂

yes

maximal, complemented

#21445437

C₂

yes

maximal, characteristic, complemented

#21445434

C₂

yes

maximal, characteristic

C₈×C₁₆×C₆₁

C₂

yes

maximal, characteristic, complemented

C₂×#14519722

—

#16840167

—

#16840142

C₂²

yes

—

#16840139

—

#16840139

C₂²

yes

—

C₄×C₁₆×C₆₁

C₂²

yes

—

C₈²×C₆₁

C₂²

yes

characteristic

C₂×Dic₂₄₄

—

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

—

Dic₂₄₄

—

D₄₈₈

—

D₄₈₈

—

C₁₆×C₆₁

D₈

yes

complemented

C₂²×D₆₁

—

C₂×C₄×C₆₁

#127

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

—

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

—

Dic₁₂₂

—

C₈×C₆₁

SD₃₂

yes

complemented

C₈×C₆₁

C₂×D₈

yes

—

—

C₂²×C₆₁

yes

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

#493

yes

—

C₄×C₆₁

C₂×SD₃₂

yes

—

Dic₆₁

—

#2248

—

#1713

—

#1712

—

C₈×C₁₆

D₆₁

yes

characteristic, complemented

C₂×C₆₁

#1712

yes

characteristic

C₂×C₆₁

#2248

yes

—

D₆₁

—

C₂×SD₃₂

—

122

—

#500

—

122

—

#494

—

#493

—

#493

—

122

—

C₄×C₁₆

C₂×D₆₁

yes

—

C₈²

C₂×D₆₁

yes

characteristic

C₆₁

#12517

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

122

—

C₂×D₈

—

122

—

C₂×D₈

—

244

—

C₄⋊Q₈

—

122

—

#127

—

122

—

SD₃₂

—

244

—

C₂×C₁₆

D₂₄₄

yes

—

#107

—

244

—

C₄×C₈

D₂₄₄

yes

—

C₄×C₈

C₂²×D₆₁

yes

characteristic

C₂×Q₈

—

244

—

C₂×D₄

—

244

—

C₂×D₄

—

244

—

Q₁₆

—

244

—

D₈

—

488

—

D₈

—

244

—

C₂×C₈

D₄₈₈

yes

—

C₂×C₈

C₂×D₂₄₄

yes

—

C₄⋊C₄

—

244

—

C₄²

C₂×D₂₄₄

yes

characteristic

C₁₆

D₄₈₈

yes

complemented

C₂³

—

488

—

Q₈

—

488

—

D₄

—

488

—

D₄

—

488

—

C₂×C₄

—

488

—

C₂×C₄

#14519774

yes

characteristic

C₂×C₄

C₂×D₄₈₈

yes

—

C₈

#14519722

yes

complemented

C₈

C₂×D₄₈₈

yes

—

C₂²

—

976

—

C₂²

—

976

—

C₂²

#16840139

yes

characteristic

C₄

—

976

—

C₄

#16840139

yes

—

C₄

C₂×#14519722

yes

—

100

C₂

—

1952

—

101

C₂

#21445434

yes

characteristic

102

C₂

#21448075

yes

—

	subgroup	G/H
center	C₂²	#16840139
commutator, cosocle	C₄×C₈×C₆₁	C₂³
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₈×C₁₆×C₆₁	C₂
socle	C₂²×C₆₁	#493
prime radical	#21445434	C₂
hypercenter	C₈×C₁₆	D₆₁
hypocenter	C₆₁	#12517
LCS	G ▷ C₄×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

subgroup

G/H

center

C₂²

#16840139

commutator, cosocle

C₄×C₈×C₆₁

C₂³

Frattini

C₄×C₈

C₂²×D₆₁

Fitting

C₈×C₁₆×C₆₁

C₂

C₂²×C₆₁

C₂

C₈×C₁₆

D₆₁

hypocenter

C₆₁

#12517

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
1. #21448075	C₂	semidirect
2. #21445437	C₂	semidirect
4. C₈×C₁₆×C₆₁	C₂	semidirect
32. C₁₆×C₆₁	D₈	semidirect
40. C₈×C₆₁	SD₃₂	semidirect
52. C₈×C₁₆	D₆₁	semidirect
63. C₆₁	#12517	semidirect
84. C₁₆	D₄₈₈	semidirect
92. C₈	#14519722	semidirect

H ≅ G/N

type

1. #21448075

C₂

semidirect

2. #21445437

C₂

semidirect

4. C₈×C₁₆×C₆₁

C₂

semidirect

32. C₁₆×C₆₁

D₈

semidirect

40. C₈×C₆₁

SD₃₂

semidirect

52. C₈×C₁₆

D₆₁

semidirect

63. C₆₁

#12517

semidirect

84. C₁₆

D₄₈₈

semidirect

92. C₈

#14519722

semidirect

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

subgroup	index
#21448075	2
#21445437	2
#21445434	2
C₈×C₁₆×C₆₁	2
#12517	61

subgroup

index

C₈×C₁₆×C₆₁

Subgroups — #29594731

Conjugacy classes of subgroups

Complements

Maximal subgroups