Group #21447699 — subgroups

Subgroups — #21447699

Order 7808.

146 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 53 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	#16840194	C₂	1	1	yes	maximal, characteristic
2	#16839878	C₂	1	2	yes	maximal, complemented
3	#16839568	C₂	1	2	yes	maximal, complemented
4	C₁₆×Dic₆₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#368	C₂	1	1	yes	maximal, characteristic, complemented
6	#14519799	C₂²	1	2	yes	—
7	#14519785	—	2	2	no	—
8	Dic₄₈₈	C₄	1	1	yes	characteristic, complemented
9	#14519722	C₄	1	2	yes	complemented
10	D₉₇₆	C₄	1	1	yes	characteristic, complemented
11	C₂×#11758185	C₂²	1	1	yes	characteristic
12	#14519703	—	2	2	no	—
13	C₈×Dic₆₁	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
15	#108×C₆₁	C₂²	1	2	yes	—
16	#11758222	—	2	2	no	—
17	M₁₆×C₆₁	—	2	2	no	—
18	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
19	C₄×Dic₆₁	D₄	1	1	yes	characteristic
20	C₂×#92418	D₄	1	1	yes	characteristic
21	Dic₂₄₄	C₂×C₄	1	2	yes	—
22	D₄₈₈	C₂×C₄	1	2	yes	—
23	#11758190	—	2	2	no	—
24	#11758189	—	4	2	no	—
25	C₈×D₆₁	—	4	2	no	—
26	C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
27	C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
28	#11758185	—	2	1	no	—
29	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
30	C₂²⋊D₆₁	—	4	2	no	—
31	C₂×Dic₆₁	—	2	1	no	—
32	D₂₄₄	—	2	2	no	—
33	C₄×D₆₁	—	4	2	no	—
34	Dic₁₂₂	—	2	2	no	—
35	C₈×C₆₁	—	4	2	no	—
36	C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
37	C₈×C₆₁	#41	1	1	yes	characteristic
38	#92418	D₈	1	1	yes	characteristic
39	#92418	D₈	1	1	yes	characteristic
40	C₂²×C₆₁	#135	1	1	yes	characteristic
41	C₂×D₆₁	—	4	2	no	—
42	C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
43	C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
44	Dic₆₁	—	2	2	no	—
45	Dic₆₁	—	4	2	no	—
46	#2178	—	61	1	no	maximal, Sylow
47	C₂×C₆₁	—	2	1	no	—
48	C₂×C₆₁	C₄×D₈	1	1	yes	characteristic
49	D₆₁	—	8	2	no	—
50	#508	—	61	1	no	—
51	#443	—	61	2	no	—
52	#368	D₆₁	1	1	yes	characteristic, complemented
53	#359	—	61	2	no	—
54	C₄×C₁₆	—	61	1	no	—
55	C₆₁	#2178	1	1	yes	characteristic, complemented, Sylow
56	#135	—	61	2	no	—
57	#131	—	122	2	no	—
58	Q₃₂	—	61	1	no	—
59	SD₃₂	—	61	2	no	—
60	D₁₆	—	61	1	no	—
61	C₂×C₁₆	—	61	1	no	—
62	C₂×C₁₆	C₂×D₆₁	1	1	yes	characteristic
63	#108	C₂×D₆₁	1	2	yes	—
64	#104	—	122	2	no	—
65	C₄×C₈	—	61	1	no	—
66	#41	—	122	2	no	—
67	Q₁₆	—	61	2	no	—
68	SD₁₆	—	122	2	no	—
69	D₈	—	61	2	no	—
70	M₁₆	—	2	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	C₄×D₆₁	1	1	yes	characteristic
78	C₁₆	C₄×D₆₁	1	1	yes	characteristic
79	Q₈	—	122	2	no	—
80	D₄	—	122	2	no	—
81	D₄	—	244	2	no	—
82	C₂×C₄	—	122	1	no	—
83	C₂×C₄	—	244	2	no	—
84	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
85	C₈	—	4	2	no	—
86	C₈	—	61	1	no	—
87	C₈	—	61	1	no	—
88	C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
89	C₈	#11758226	1	1	yes	characteristic
90	C₂²	—	244	2	no	—
91	C₂²	#14519800	1	1	yes	characteristic
92	C₄	—	122	2	no	—
93	C₄	—	244	2	no	—
94	C₄	#14519739	1	1	yes	characteristic
95	C₄	D₈×D₆₁	1	1	yes	characteristic
96	C₂	—	2	1	no	—
97	C₂	—	488	2	no	—
98	C₂	#16839844	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840194

C₂

yes

maximal, characteristic

#16839878

C₂

yes

maximal, complemented

#16839568

C₂

yes

maximal, complemented

C₁₆×Dic₆₁

C₂

yes

maximal, characteristic, complemented

C₆₁×#368

C₂

yes

maximal, characteristic, complemented

#14519799

C₂²

yes

—

#14519785

—

Dic₄₈₈

C₄

yes

characteristic, complemented

#14519722

C₄

yes

complemented

D₉₇₆

C₄

yes

characteristic, complemented

C₂×#11758185

C₂²

yes

characteristic

#14519703

—

C₈×Dic₆₁

C₂²

yes

characteristic

C₂×C₁₆×C₆₁

C₂²

yes

characteristic

#108×C₆₁

C₂²

yes

—

#11758222

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₄×Dic₆₁

D₄

yes

characteristic

C₂×#92418

D₄

yes

characteristic

Dic₂₄₄

C₂×C₄

yes

—

D₄₈₈

C₂×C₄

yes

—

#11758190

—

#11758189

—

C₈×D₆₁

—

C₁₆×C₆₁

C₂×C₄

yes

characteristic

C₁₆×C₆₁

C₂×C₄

yes

characteristic

#11758185

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²⋊D₆₁

—

C₂×Dic₆₁

—

D₂₄₄

—

C₄×D₆₁

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

C₂²×C₄

yes

C₈×C₆₁

yes

D₈

yes

characteristic

#92418

D₈

yes

characteristic

C₂²×C₆₁

#135

yes

characteristic

C₂×D₆₁

—

C₄×C₆₁

C₄×D₄

yes

characteristic

C₄×C₆₁

C₂×D₈

yes

characteristic

Dic₆₁

—

Dic₆₁

—

#2178

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

C₄×D₈

yes

characteristic

D₆₁

—

#508

—

#443

—

#368

D₆₁

yes

characteristic, complemented

#359

—

C₄×C₁₆

—

C₆₁

#2178

yes

characteristic, complemented, Sylow

#135

—

#131

—

122

—

Q₃₂

—

SD₃₂

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₆₁

yes

characteristic

#108

C₂×D₆₁

yes

—

#104

—

122

—

C₄×C₈

—

#41

—

122

—

Q₁₆

—

SD₁₆

—

122

—

D₈

—

M₁₆

—

M₁₆

—

244

—

C₂×C₈

—

C₂×C₈

—

244

—

C₂×C₈

C₂²×D₆₁

yes

characteristic

C₄²

—

C₁₆

—

122

—

C₁₆

C₄×D₆₁

yes

characteristic

C₁₆

C₄×D₆₁

yes

characteristic

Q₈

—

122

—

D₄

—

122

—

D₄

—

244

—

C₂×C₄

—

122

—

C₂×C₄

—

244

—

C₂×C₄

D₄×D₆₁

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

C₂×C₄×D₆₁

yes

characteristic

C₈

#11758226

yes

characteristic

C₂²

—

244

—

C₂²

#14519800

yes

characteristic

C₄

—

122

—

C₄

—

244

—

C₄

#14519739

yes

characteristic

C₄

D₈×D₆₁

yes

characteristic

C₂

—

C₂

—

488

—

C₂

#16839844

yes

characteristic

	subgroup	G/H
center	C₄	D₈×D₆₁
commutator	C₈×C₆₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₆₁
Fitting	C₆₁×#368	C₂
socle	C₂×C₆₁	C₄×D₈
hypercenter	#368	D₆₁
hypocenter	C₆₁	#2178
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

subgroup

G/H

center

C₄

D₈×D₆₁

commutator

C₈×C₆₁

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₆₁

Fitting

C₆₁×#368

C₂

socle

C₂×C₆₁

C₄×D₈

D₆₁

C₆₁

G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

N	H ≅ G/N	type
2. #16839878	C₂	semidirect
3. #16839568	C₂	semidirect
4. C₁₆×Dic₆₁	C₂	semidirect
5. C₆₁×#368	C₂	semidirect
8. Dic₄₈₈	C₄	semidirect
9. #14519722	C₄	semidirect
10. D₉₇₆	C₄	semidirect
52. #368	D₆₁	semidirect
55. C₆₁	#2178	semidirect

H ≅ G/N

type

2. #16839878

C₂

semidirect

3. #16839568

C₂

semidirect

4. C₁₆×Dic₆₁

C₂

semidirect

5. C₆₁×#368

C₂

semidirect

8. Dic₄₈₈

C₄

semidirect

9. #14519722

C₄

semidirect

10. D₉₇₆

C₄

semidirect

52. #368

D₆₁

semidirect

55. C₆₁

#2178

semidirect

Subgroups — #21447699

Conjugacy classes of subgroups

Complements