Group #21447657 — subgroups

Subgroups — #21447657

Order 7808.

146 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 65 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	#16840178	C₂	1	2	yes	maximal, complemented
2	#16839880	C₂	1	1	yes	maximal, characteristic
3	#16839583	C₂	1	1	yes	maximal, characteristic, complemented
4	#16839545	C₂	1	2	yes	maximal, complemented
5	C₄×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#14519799	C₄	1	1	yes	characteristic
7	#14519783	C₂²	1	2	yes	—
8	#14519717	C₂²	1	2	yes	—
9	#14519716	—	2	2	no	—
10	C₁₆×D₆₁	—	2	2	no	—
11	#14519713	C₄	1	1	yes	characteristic
12	#14519705	C₄	1	2	yes	—
13	C₂×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	#11758222	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
19	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
20	#11758194	C₂×C₄	1	2	yes	—
21	Dic₂₄₄	C₈	1	1	yes	characteristic
22	D₄₈₈	C₈	1	1	yes	characteristic
23	#11758190	C₈	1	2	yes	—
24	#11758189	—	2	2	no	—
25	C₈×D₆₁	—	2	2	no	—
26	C₁₆×C₆₁	—	2	1	no	—
27	C₁₆×C₆₁	D₄	1	1	yes	characteristic
28	C₁₆×C₆₁	D₄	1	1	yes	characteristic
29	#11758185	—	2	2	no	—
30	C₂×C₄×C₆₁	—	2	1	no	—
31	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
32	C₂²⋊D₆₁	—	2	2	no	—
33	D₂₄₄	C₂×C₈	1	2	yes	—
34	C₄×D₆₁	—	2	2	no	—
35	Dic₁₂₂	C₂×C₈	1	2	yes	—
36	C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
37	C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
38	C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
39	C₈×C₆₁	#41	1	1	yes	characteristic
40	#92418	—	2	2	no	—
41	C₂²×C₆₁	#131	1	1	yes	characteristic
42	C₂×D₆₁	—	2	2	no	—
43	C₄×C₆₁	—	2	2	no	—
44	C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
45	C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
46	Dic₆₁	—	2	2	no	—
47	#2170	—	61	1	no	maximal, Sylow
48	C₂×C₆₁	—	2	1	no	—
49	C₂×C₆₁	C₈×D₄	1	1	yes	characteristic
50	D₆₁	—	4	2	no	—
51	#504	—	61	2	no	—
52	#443	—	61	1	no	—
53	#364	—	61	1	no	—
54	#350	—	61	2	no	—
55	C₄×C₁₆	D₆₁	1	1	yes	characteristic, complemented
56	C₆₁	#2170	1	1	yes	characteristic, complemented, Sylow
57	#135	—	61	1	no	—
58	#131	—	61	2	no	—
59	M₃₂	—	61	2	no	—
60	M₃₂	—	122	2	no	—
61	C₂×C₁₆	—	122	2	no	—
62	C₂×C₁₆	C₂×D₆₁	1	1	yes	characteristic
63	C₂×C₁₆	C₂×D₆₁	1	1	yes	characteristic
64	#108	—	61	1	no	—
65	#104	—	61	2	no	—
66	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
67	#41	—	61	2	no	—
68	Q₁₆	—	61	1	no	—
69	SD₁₆	—	61	2	no	—
70	D₈	—	61	1	no	—
71	M₁₆	—	61	2	no	—
72	M₁₆	—	122	2	no	—
73	C₂×C₈	—	122	2	no	—
74	C₂×C₈	C₄×D₆₁	1	1	yes	characteristic
75	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
76	C₄²	C₄×D₆₁	1	1	yes	characteristic
77	C₁₆	—	2	1	no	—
78	C₁₆	—	122	2	no	—
79	C₁₆	D₂₄₄	1	1	yes	characteristic
80	C₁₆	D₂₄₄	1	1	yes	characteristic
81	Q₈	—	61	2	no	—
82	D₄	—	61	2	no	—
83	D₄	—	122	2	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	122	2	no	—
86	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
87	C₈	—	122	2	no	—
88	C₈	C₈×D₆₁	1	1	yes	characteristic
89	C₈	C₈×D₆₁	1	1	yes	characteristic
90	C₈	C₂×D₂₄₄	1	1	yes	characteristic
91	C₈	#11758222	1	1	yes	characteristic
92	C₂²	—	122	2	no	—
93	C₂²	#14519783	1	1	yes	characteristic
94	C₄	—	2	2	no	—
95	C₄	—	122	2	no	—
96	C₄	C₄×D₂₄₄	1	1	yes	characteristic
97	C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
98	C₂	—	2	1	no	—
99	C₂	—	244	2	no	—
100	C₂	C₈×D₂₄₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840178

C₂

yes

maximal, complemented

#16839880

C₂

yes

maximal, characteristic

#16839583

C₂

yes

maximal, characteristic, complemented

#16839545

C₂

yes

maximal, complemented

C₄×C₁₆×C₆₁

C₂

yes

maximal, characteristic, complemented

#14519799

C₄

yes

characteristic

#14519783

C₂²

yes

—

#14519717

C₂²

yes

—

#14519716

—

C₁₆×D₆₁

—

#14519713

C₄

yes

characteristic

#14519705

C₄

yes

—

C₂×C₁₆×C₆₁

C₂²

yes

characteristic

C₂×C₁₆×C₆₁

C₂²

yes

characteristic

C₄×C₈×C₆₁

C₂²

yes

characteristic

#11758222

C₂×C₄

yes

—

C₂×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₄²×C₆₁

C₂×C₄

yes

characteristic

#11758194

C₂×C₄

yes

—

Dic₂₄₄

C₈

yes

characteristic

D₄₈₈

C₈

yes

characteristic

#11758190

C₈

yes

—

#11758189

—

C₈×D₆₁

—

C₁₆×C₆₁

—

C₁₆×C₆₁

D₄

yes

characteristic

C₁₆×C₆₁

D₄

yes

characteristic

#11758185

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²⋊D₆₁

—

D₂₄₄

C₂×C₈

yes

—

C₄×D₆₁

—

Dic₁₂₂

C₂×C₈

yes

—

C₈×C₆₁

C₂×C₈

yes

characteristic

C₈×C₆₁

C₂×C₈

yes

characteristic

C₈×C₆₁

C₂×D₄

yes

C₈×C₆₁

yes

—

C₂²×C₆₁

#131

yes

characteristic

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₄×D₄

yes

characteristic

C₄×C₆₁

C₂²×C₈

yes

characteristic

Dic₆₁

—

#2170

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

C₈×D₄

yes

characteristic

D₆₁

—

—

—

—

—

C₄×C₁₆

D₆₁

yes

characteristic, complemented

C₆₁

#2170

yes

characteristic, complemented, Sylow

#135

—

#131

—

M₃₂

—

M₃₂

—

122

—

C₂×C₁₆

—

122

—

C₂×C₁₆

C₂×D₆₁

yes

characteristic

C₂×C₁₆

C₂×D₆₁

yes

characteristic

#108

—

#104

—

C₄×C₈

C₂×D₆₁

yes

characteristic

#41

—

Q₁₆

—

SD₁₆

—

D₈

—

M₁₆

—

M₁₆

—

122

—

C₂×C₈

—

122

—

C₂×C₈

C₄×D₆₁

yes

characteristic

C₂×C₈

C₂²×D₆₁

yes

characteristic

C₄²

C₄×D₆₁

yes

characteristic

C₁₆

—

C₁₆

—

122

—

C₁₆

D₂₄₄

yes

characteristic

C₁₆

D₂₄₄

yes

characteristic

Q₈

—

D₄

—

D₄

—

122

—

C₂×C₄

—

C₂×C₄

—

122

—

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

C₈

—

122

—

C₈

C₈×D₆₁

yes

characteristic

C₈

C₈×D₆₁

yes

characteristic

C₈

C₂×D₂₄₄

yes

characteristic

C₈

#11758222

yes

characteristic

C₂²

—

122

—

C₂²

#14519783

yes

characteristic

C₄

—

C₄

—

122

—

C₄

C₄×D₂₄₄

yes

characteristic

C₄

C₂×C₈×D₆₁

yes

characteristic

C₂

—

C₂

—

244

—

100

C₂

C₈×D₂₄₄

yes

characteristic

	subgroup	G/H
center	C₁₆	D₂₄₄
commutator	C₄×C₆₁	C₂²×C₈
Frattini	C₂×C₈	C₂²×D₆₁
Fitting	C₄×C₁₆×C₆₁	C₂
socle	C₂×C₆₁	C₈×D₄
hypercenter	C₄×C₁₆	D₆₁
hypocenter	C₆₁	#2170
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ C₄×C₁₆

subgroup

G/H

center

C₁₆

D₂₄₄

commutator

C₄×C₆₁

C₂²×C₈

Frattini

C₂×C₈

C₂²×D₆₁

Fitting

C₄×C₁₆×C₆₁

C₂

socle

C₂×C₆₁

C₈×D₄

hypercenter

C₄×C₁₆

D₆₁

hypocenter

C₆₁

#2170

LCS

G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ C₄×C₁₆

N	H ≅ G/N	type
1. #16840178	C₂	semidirect
3. #16839583	C₂	semidirect
4. #16839545	C₂	semidirect
5. C₄×C₁₆×C₆₁	C₂	semidirect
55. C₄×C₁₆	D₆₁	semidirect
56. C₆₁	#2170	semidirect

H ≅ G/N

type

1. #16840178

C₂

semidirect

3. #16839583

C₂

semidirect

4. #16839545

C₂

semidirect

5. C₄×C₁₆×C₆₁

C₂

semidirect

55. C₄×C₁₆

D₆₁

semidirect

56. C₆₁

#2170

semidirect

Subgroups — #21447657

Conjugacy classes of subgroups

Complements