Group #21447700 — subgroups

Subgroups — #21447700

Order 7808.

146 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 61 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	#16839879	C₂	1	2	yes	maximal, complemented
2	#16839592	C₂	1	1	yes	maximal, characteristic, complemented
3	#16839569	C₂	1	2	yes	maximal, complemented
4	C₁₆×Dic₆₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#508	C₂	1	1	yes	maximal, characteristic
6	#14519784	—	2	2	no	—
7	C₂×#11758185	C₂²	1	1	yes	characteristic
8	#14519713	C₂²	1	2	yes	—
9	#14519704	—	2	2	no	—
10	C₈×Dic₆₁	C₂²	1	1	yes	characteristic
11	#135×C₆₁	C₂²	1	2	yes	—
12	Q₃₂×C₆₁	C₄	1	1	yes	characteristic, complemented
13	SD₃₂×C₆₁	C₄	1	2	yes	complemented
14	D₁₆×C₆₁	C₄	1	1	yes	characteristic, complemented
15	C₂×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
16	#41×C₆₁	—	2	2	no	—
17	Q₁₆×C₆₁	C₂×C₄	1	2	yes	—
18	SD₁₆×C₆₁	—	2	2	no	—
19	D₈×C₆₁	C₂×C₄	1	2	yes	—
20	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
21	C₄×Dic₆₁	D₄	1	1	yes	characteristic
22	#11758194	—	2	2	no	—
23	#11758194	—	4	2	no	—
24	C₂×#92418	—	4	2	no	—
25	C₂×#92418	D₄	1	1	yes	characteristic
26	C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
27	C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
28	#11758185	—	2	1	no	—
29	Q₈×C₆₁	—	2	2	no	—
30	D₄×C₆₁	—	2	2	no	—
31	D₄×C₆₁	—	4	2	no	—
32	C₂×C₄×C₆₁	—	4	2	no	—
33	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
34	C₂×Dic₆₁	—	2	1	no	—
35	C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
36	C₈×C₆₁	#41	1	1	yes	characteristic
37	#92418	—	4	2	no	—
38	#92418	D₈	1	1	yes	characteristic
39	#92418	D₈	1	1	yes	characteristic
40	C₂²×C₆₁	—	4	2	no	—
41	C₂²×C₆₁	#135	1	1	yes	characteristic
42	C₄×C₆₁	—	4	2	no	—
43	C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
44	C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
45	Dic₆₁	—	2	2	no	—
46	#2178	—	61	1	no	maximal, Sylow
47	C₂×C₆₁	—	2	1	no	—
48	C₂×C₆₁	—	8	2	no	—
49	C₂×C₆₁	C₄×D₈	1	1	yes	characteristic
50	#508	D₆₁	1	1	yes	characteristic
51	#443	—	61	2	no	—
52	#368	—	61	1	no	—
53	#359	—	61	2	no	—
54	C₄×C₁₆	—	61	1	no	—
55	C₆₁	#2178	1	1	yes	characteristic, complemented, Sylow
56	#135	C₂×D₆₁	1	2	yes	—
57	#131	—	122	2	no	—
58	Q₃₂	Dic₆₁	1	1	yes	characteristic, complemented
59	SD₃₂	Dic₆₁	1	2	yes	complemented
60	D₁₆	Dic₆₁	1	1	yes	characteristic, complemented
61	C₂×C₁₆	—	61	1	no	—
62	C₂×C₁₆	C₂×D₆₁	1	1	yes	characteristic
63	#108	—	61	2	no	—
64	#104	—	122	2	no	—
65	C₄×C₈	—	61	1	no	—
66	#41	—	2	2	no	—
67	Q₁₆	C₂×Dic₆₁	1	2	yes	—
68	SD₁₆	—	2	2	no	—
69	D₈	C₂×Dic₆₁	1	2	yes	—
70	M₁₆	—	122	2	no	—
71	M₁₆	—	244	2	no	—
72	C₂×C₈	—	61	1	no	—
73	C₂×C₈	—	244	2	no	—
74	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
75	C₄²	—	61	1	no	—
76	C₁₆	—	122	1	no	—
77	C₁₆	C₂×Dic₆₁	1	1	yes	characteristic
78	C₁₆	C₂×Dic₆₁	1	1	yes	characteristic
79	Q₈	—	2	2	no	—
80	D₄	—	2	2	no	—
81	D₄	—	4	2	no	—
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	122	1	no	—
84	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
85	C₈	—	61	1	no	—
86	C₈	—	61	1	no	—
87	C₈	—	244	2	no	—
88	C₈	#11758224	1	1	yes	characteristic
89	C₈	C₂²×Dic₆₁	1	1	yes	characteristic
90	C₂²	—	4	2	no	—
91	C₂²	#14519796	1	1	yes	characteristic
92	C₄	—	4	2	no	—
93	C₄	—	122	2	no	—
94	C₄	D₄×Dic₆₁	1	1	yes	characteristic
95	C₄	D₈×D₆₁	1	1	yes	characteristic
96	C₂	—	2	1	no	—
97	C₂	—	8	2	no	—
98	C₂	D₈×Dic₆₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16839879

C₂

yes

maximal, complemented

#16839592

C₂

yes

maximal, characteristic, complemented

#16839569

C₂

yes

maximal, complemented

C₁₆×Dic₆₁

C₂

yes

maximal, characteristic, complemented

C₆₁×#508

C₂

yes

maximal, characteristic

#14519784

—

C₂×#11758185

C₂²

yes

characteristic

#14519713

C₂²

yes

—

#14519704

—

C₈×Dic₆₁

C₂²

yes

characteristic

#135×C₆₁

C₂²

yes

—

Q₃₂×C₆₁

C₄

yes

characteristic, complemented

SD₃₂×C₆₁

C₄

yes

complemented

D₁₆×C₆₁

C₄

yes

characteristic, complemented

C₂×C₁₆×C₆₁

C₂²

yes

characteristic

#41×C₆₁

—

Q₁₆×C₆₁

C₂×C₄

yes

—

SD₁₆×C₆₁

—

D₈×C₆₁

C₂×C₄

yes

—

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₄×Dic₆₁

D₄

yes

—

—

—

D₄

yes

characteristic

C₁₆×C₆₁

C₂×C₄

yes

characteristic

C₁₆×C₆₁

C₂×C₄

yes

characteristic

#11758185

—

Q₈×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×Dic₆₁

—

C₈×C₆₁

C₂²×C₄

yes

C₈×C₆₁

yes

—

D₈

yes

characteristic

#92418

D₈

yes

characteristic

C₂²×C₆₁

—

C₂²×C₆₁

#135

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

C₄×D₄

yes

characteristic

C₄×C₆₁

C₂×D₈

yes

characteristic

Dic₆₁

—

#2178

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

C₄×D₈

yes

characteristic

#508

D₆₁

yes

—

—

—

C₄×C₁₆

—

C₆₁

#2178

yes

characteristic, complemented, Sylow

#135

C₂×D₆₁

yes

—

#131

—

122

—

Q₃₂

Dic₆₁

yes

characteristic, complemented

SD₃₂

Dic₆₁

yes

complemented

D₁₆

Dic₆₁

yes

characteristic, complemented

C₂×C₁₆

—

C₂×C₁₆

C₂×D₆₁

yes

characteristic

#108

—

#104

—

122

—

C₄×C₈

—

#41

—

Q₁₆

C₂×Dic₆₁

yes

—

SD₁₆

—

D₈

C₂×Dic₆₁

yes

—

M₁₆

—

122

—

M₁₆

—

244

—

C₂×C₈

—

C₂×C₈

—

244

—

C₂×C₈

C₂²×D₆₁

yes

characteristic

C₄²

—

C₁₆

—

122

—

C₁₆

C₂×Dic₆₁

yes

characteristic

C₁₆

C₂×Dic₆₁

yes

characteristic

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

122

—

C₂×C₄

D₄×D₆₁

yes

characteristic

C₈

—

C₈

—

C₈

—

244

—

C₈

#11758224

yes

characteristic

C₈

C₂²×Dic₆₁

yes

characteristic

C₂²

—

C₂²

#14519796

yes

characteristic

C₄

—

C₄

—

122

—

C₄

D₄×Dic₆₁

yes

characteristic

C₄

D₈×D₆₁

yes

characteristic

C₂

—

C₂

—

C₂

D₈×Dic₆₁

yes

characteristic

	subgroup	G/H
center	C₄	D₈×D₆₁
commutator	C₈×C₆₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₆₁
Fitting	C₆₁×#508	C₂
socle	C₂×C₆₁	C₄×D₈
hypercenter	#508	D₆₁
hypocenter	C₆₁	#2178
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #508

subgroup

G/H

center

C₄

D₈×D₆₁

commutator

C₈×C₆₁

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₆₁

Fitting

C₆₁×#508

C₂

socle

C₂×C₆₁

C₄×D₈

D₆₁

C₆₁

G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #508

N	H ≅ G/N	type
1. #16839879	C₂	semidirect
2. #16839592	C₂	semidirect
3. #16839569	C₂	semidirect
4. C₁₆×Dic₆₁	C₂	semidirect
12. Q₃₂×C₆₁	C₄	semidirect
13. SD₃₂×C₆₁	C₄	semidirect
14. D₁₆×C₆₁	C₄	semidirect
55. C₆₁	#2178	semidirect
58. Q₃₂	Dic₆₁	semidirect
59. SD₃₂	Dic₆₁	semidirect
60. D₁₆	Dic₆₁	semidirect

H ≅ G/N

type

1. #16839879

C₂

semidirect

2. #16839592

C₂

semidirect

3. #16839569

C₂

semidirect

4. C₁₆×Dic₆₁

C₂

semidirect

12. Q₃₂×C₆₁

C₄

semidirect

13. SD₃₂×C₆₁

C₄

semidirect

14. D₁₆×C₆₁

C₄

semidirect

55. C₆₁

#2178

semidirect

58. Q₃₂

Dic₆₁

semidirect

59. SD₃₂

Dic₆₁

semidirect

60. D₁₆

Dic₆₁

semidirect

Subgroups — #21447700

Conjugacy classes of subgroups

Complements