Group #21458972 — subgroups

Subgroups — #21458972

Order 7808.

350 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 129 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	#16840579	C₂	1	2	yes	maximal, complemented
2	#16840205	C₂	1	6	yes	maximal, complemented
3	#16840194	C₂	1	3	yes	maximal, complemented
4	C₂×Dic₄₈₈	C₂	1	3	yes	maximal, complemented
5	C₆₁×#504	C₂	1	1	yes	maximal, characteristic, complemented
6	#14519824	—	2	2	no	—
7	#14519810	—	2	6	no	—
8	#14519799	C₂²	1	6	yes	—
9	C₂×Dic₂₄₄	C₂²	1	6	yes	—
10	Dic₄₈₈	C₂²	1	3	yes	—
11	Dic₄₈₈	C₂²	1	6	yes	—
12	#14519722	C₂²	1	6	yes	—
13	D₉₇₆	C₂²	1	1	yes	characteristic
14	#131×C₆₁	C₂²	1	1	yes	characteristic
15	M₃₂×C₆₁	C₂²	1	3	yes	—
16	C₂×C₁₆×C₆₁	C₂²	1	3	yes	—
17	#41×C₆₁	D₄	1	1	yes	characteristic
18	Q₈×D₆₁	—	4	2	no	—
19	#11758224	—	4	6	no	—
20	#11758222	—	2	6	no	—
21	C₂×Dic₁₂₂	—	2	6	no	—
22	M₁₆×C₆₁	D₄	1	3	yes	—
23	C₂×C₈×C₆₁	C₂³	1	3	yes	—
24	Dic₂₄₄	—	2	6	no	—
25	Dic₂₄₄	C₂³	1	6	yes	—
26	D₄₈₈	C₂³	1	2	yes	—
27	#11758190	—	2	6	no	—
28	C₁₆×C₆₁	C₂³	1	1	yes	characteristic
29	C₁₆×C₆₁	C₂³	1	3	yes	—
30	Q₈×C₆₁	D₈	1	1	yes	characteristic
31	D₄×C₆₁	D₈	1	3	yes	—
32	C₂×C₄×C₆₁	C₂×D₄	1	3	yes	—
33	C₂²⋊D₆₁	—	4	6	no	—
34	C₂×Dic₆₁	—	4	6	no	—
35	D₂₄₄	—	2	2	no	—
36	C₄×D₆₁	—	4	6	no	—
37	Dic₁₂₂	—	2	6	no	—
38	Dic₁₂₂	—	4	6	no	—
39	C₈×C₆₁	C₂×D₄	1	3	yes	—
40	C₈×C₆₁	C₂⁴	1	1	yes	characteristic
41	C₂²×C₆₁	C₂×D₈	1	3	yes	—
42	C₂×D₆₁	—	4	2	no	—
43	C₄×C₆₁	C₂×D₈	1	3	yes	—
44	C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
45	Dic₆₁	—	4	6	no	—
46	#3417	—	61	1	no	maximal, Sylow
47	C₂×C₆₁	—	2	3	no	—
48	C₂×C₆₁	C₂²×D₈	1	1	yes	characteristic
49	D₆₁	—	8	2	no	—
50	#578	—	61	2	no	—
51	#510	—	61	6	no	—
52	#508	—	61	3	no	—
53	C₂×Q₃₂	—	61	3	no	—
54	#504	D₆₁	1	1	yes	characteristic, complemented
55	C₆₁	#3417	1	1	yes	characteristic, complemented, Sylow
56	2₋¹⁺⁴	—	122	2	no	—
57	#137	—	122	6	no	—
58	#135	—	61	6	no	—
59	C₂×Q₁₆	—	61	6	no	—
60	#131	C₂×D₆₁	1	1	yes	characteristic
61	Q₃₂	—	61	3	no	—
62	Q₃₂	—	61	6	no	—
63	SD₃₂	—	61	6	no	—
64	D₁₆	—	61	1	no	—
65	M₃₂	C₂×D₆₁	1	3	yes	—
66	C₂×C₁₆	C₂×D₆₁	1	3	yes	—
67	#41	—	122	6	no	—
68	#41	—	244	6	no	—
69	#41	D₂₄₄	1	1	yes	characteristic
70	C₂×Q₈	—	122	6	no	—
71	C₂×Q₈	—	244	2	no	—
72	Q₁₆	—	61	6	no	—
73	Q₁₆	—	122	6	no	—
74	SD₁₆	—	122	6	no	—
75	D₈	—	61	2	no	—
76	M₁₆	D₂₄₄	1	3	yes	—
77	C₂×C₈	C₂²×D₆₁	1	3	yes	—
78	C₁₆	C₂²×D₆₁	1	1	yes	characteristic
79	C₁₆	C₂²×D₆₁	1	3	yes	—
80	Q₈	—	122	6	no	—
81	Q₈	—	244	6	no	—
82	Q₈	D₄₈₈	1	1	yes	characteristic
83	D₄	—	122	2	no	—
84	D₄	—	244	6	no	—
85	D₄	D₄₈₈	1	3	yes	—
86	C₂×C₄	—	244	6	no	—
87	C₂×C₄	—	244	6	no	—
88	C₂×C₄	C₂×D₂₄₄	1	3	yes	—
89	C₈	C₂×D₂₄₄	1	3	yes	—
90	C₈	C₂³×D₆₁	1	1	yes	characteristic
91	C₂²	—	244	2	no	—
92	C₂²	C₂×D₄₈₈	1	3	yes	—
93	C₄	—	244	6	no	—
94	C₄	C₂×D₄₈₈	1	3	yes	—
95	C₄	C₂²×D₂₄₄	1	1	yes	characteristic
96	C₂	—	2	3	no	—
97	C₂	—	488	2	no	—
98	C₂	C₂²×D₄₈₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840579

C₂

yes

maximal, complemented

#16840205

C₂

yes

maximal, complemented

#16840194

C₂

yes

maximal, complemented

C₂×Dic₄₈₈

C₂

yes

maximal, complemented

C₆₁×#504

C₂

yes

maximal, characteristic, complemented

#14519824

—

#14519810

—

#14519799

C₂²

yes

—

C₂×Dic₂₄₄

C₂²

yes

—

Dic₄₈₈

C₂²

yes

—

Dic₄₈₈

C₂²

yes

—

#14519722

C₂²

yes

—

D₉₇₆

C₂²

yes

characteristic

#131×C₆₁

C₂²

yes

characteristic

M₃₂×C₆₁

C₂²

yes

—

C₂×C₁₆×C₆₁

C₂²

yes

—

#41×C₆₁

D₄

yes

characteristic

Q₈×D₆₁

—

#11758224

—

#11758222

—

C₂×Dic₁₂₂

—

M₁₆×C₆₁

D₄

yes

—

C₂×C₈×C₆₁

C₂³

yes

—

Dic₂₄₄

—

Dic₂₄₄

C₂³

yes

—

D₄₈₈

C₂³

yes

—

#11758190

—

C₁₆×C₆₁

C₂³

yes

characteristic

C₁₆×C₆₁

C₂³

yes

—

Q₈×C₆₁

D₈

yes

characteristic

D₄×C₆₁

D₈

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂²⋊D₆₁

—

C₂×Dic₆₁

—

D₂₄₄

—

C₄×D₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

C₂×D₄

yes

—

C₈×C₆₁

C₂⁴

yes

characteristic

C₂²×C₆₁

C₂×D₈

yes

—

C₂×D₆₁

—

C₄×C₆₁

C₂×D₈

yes

—

C₄×C₆₁

C₂²×D₄

yes

characteristic

Dic₆₁

—

#3417

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

C₂²×D₈

yes

characteristic

D₆₁

—

#578

—

#510

—

#508

—

C₂×Q₃₂

—

#504

D₆₁

yes

characteristic, complemented

C₆₁

#3417

yes

characteristic, complemented, Sylow

2₋¹⁺⁴

—

122

—

#137

—

122

—

#135

—

C₂×Q₁₆

—

#131

C₂×D₆₁

yes

characteristic

Q₃₂

—

Q₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

C₂×D₆₁

yes

—

C₂×C₁₆

C₂×D₆₁

yes

—

#41

—

122

—

#41

—

244

—

#41

D₂₄₄

yes

characteristic

C₂×Q₈

—

122

—

C₂×Q₈

—

244

—

Q₁₆

—

Q₁₆

—

122

—

SD₁₆

—

122

—

D₈

—

M₁₆

D₂₄₄

yes

—

C₂×C₈

C₂²×D₆₁

yes

—

C₁₆

C₂²×D₆₁

yes

characteristic

C₁₆

C₂²×D₆₁

yes

—

Q₈

—

122

—

Q₈

—

244

—

Q₈

D₄₈₈

yes

characteristic

D₄

—

122

—

D₄

—

244

—

D₄

D₄₈₈

yes

—

C₂×C₄

—

244

—

C₂×C₄

—

244

—

C₂×C₄

C₂×D₂₄₄

yes

—

C₈

C₂×D₂₄₄

yes

—

C₈

C₂³×D₆₁

yes

characteristic

C₂²

—

244

—

C₂²

C₂×D₄₈₈

yes

—

C₄

—

244

—

C₄

C₂×D₄₈₈

yes

—

C₄

C₂²×D₂₄₄

yes

characteristic

C₂

—

C₂

—

488

—

C₂

C₂²×D₄₈₈

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×D₄₈₈
commutator	C₈×C₆₁	C₂⁴
Frattini	C₈	C₂³×D₆₁
Fitting	C₆₁×#504	C₂
socle	C₂×C₆₁	C₂²×D₈
hypercenter	#504	D₆₁
hypocenter	C₆₁	#3417
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #504

subgroup

G/H

center

C₂

C₂²×D₄₈₈

commutator

C₈×C₆₁

C₂⁴

Frattini

C₈

C₂³×D₆₁

Fitting

C₆₁×#504

C₂

socle

C₂×C₆₁

C₂²×D₈

D₆₁

C₆₁

G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #504

N	H ≅ G/N	type
1. #16840579	C₂	semidirect
2. #16840205	C₂	semidirect
3. #16840194	C₂	semidirect
4. C₂×Dic₄₈₈	C₂	semidirect
5. C₆₁×#504	C₂	semidirect
54. #504	D₆₁	semidirect
55. C₆₁	#3417	semidirect

H ≅ G/N

type

1. #16840579

C₂

semidirect

2. #16840205

C₂

semidirect

3. #16840194

C₂

semidirect

4. C₂×Dic₄₈₈

C₂

semidirect

5. C₆₁×#504

C₂

semidirect

54. #504

D₆₁

semidirect

55. C₆₁

#3417

semidirect

Subgroups — #21458972

Conjugacy classes of subgroups

Complements