Group #21445573 — subgroups

Subgroups — #21445573

Order 7808.

200 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 93 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	#16839720	C₂	1	1	yes	maximal, characteristic
2	#16839718	C₂	1	2	yes	maximal
3	#16839520	C₂	1	1	yes	maximal, characteristic
4	#16839518	C₂	1	2	yes	maximal
5	C₆₁×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
6	C₂×#11758194	C₂²	1	2	yes	—
7	C₂²×#92418	C₂²	1	1	yes	characteristic
8	#14519736	C₂²	1	1	yes	characteristic
9	#14519688	C₄	1	4	yes	complemented
10	#14519687	C₄	1	2	yes	complemented
11	C₄×#92418	C₄	1	4	yes	complemented
12	C₈×Dic₆₁	C₄	1	2	yes	complemented
13	C₂×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
14	#117×C₆₁	C₂²	1	2	yes	—
15	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
16	M₁₆×C₆₁	—	2	2	no	—
17	C₂×C₈×C₆₁	C₂×C₄	1	2	yes	—
18	C₄⋊C₄×C₆₁	—	2	2	no	—
19	C₂²⋊C₄×C₆₁	—	2	2	no	—
20	C₄²×C₆₁	C₂×C₄	1	4	yes	—
21	C₂²⋊Dic₆₁	—	2	1	no	—
22	C₄⋊Dic₆₁	—	2	1	no	—
23	C₄×Dic₆₁	C₂×C₄	1	2	yes	—
24	#11758194	—	2	2	no	—
25	#11758194	—	2	2	no	—
26	C₂×#92418	C₂×C₄	1	4	yes	—
27	C₂×#92418	C₂×C₄	1	1	yes	characteristic
28	C₂×#92418	C₂×C₄	1	2	yes	—
29	C₂×#92418	C₂×C₄	1	1	yes	characteristic
30	C₂×#92418	C₂×C₄	1	2	yes	—
31	C₂³×C₆₁	#41	1	1	yes	characteristic
32	C₂×C₄×C₆₁	—	2	4	no	—
33	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
34	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₆₁	#41	1	2	yes	—
36	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
37	C₂×Dic₆₁	—	2	2	no	—
38	C₈×C₆₁	—	2	2	no	—
39	#92418	—	2	4	no	—
40	#92418	C₄²	1	1	yes	characteristic
41	#92418	C₄²	1	1	yes	characteristic
42	#92418	C₄²	1	2	yes	—
43	C₂²×C₆₁	—	4	1	no	—
44	C₂²×C₆₁	#117	1	2	yes	—
45	C₂²×C₆₁	#117	1	1	yes	characteristic
46	C₄×C₆₁	—	4	4	no	—
47	C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₆₁	#117	1	1	yes	characteristic
49	C₄×C₆₁	#117	1	2	yes	—
50	Dic₆₁	—	4	2	no	—
51	#1747	—	61	1	no	maximal, Sylow
52	C₂×C₆₁	—	2	1	no	—
53	C₂×C₆₁	—	2	2	no	—
54	C₂×C₆₁	#376	1	1	yes	characteristic
55	#405	—	61	1	no	—
56	#405	—	61	2	no	—
57	#338	—	61	1	no	—
58	H(3,ℤ/4)	—	61	2	no	—
59	H(3,ℤ/4)	D₆₁	1	1	yes	characteristic
60	C₆₁	#1747	1	1	yes	characteristic, complemented, Sylow
61	C₂×M₁₆	—	61	2	no	—
62	C₂×M₁₆	C₂×D₆₁	1	1	yes	characteristic
63	C₂²×C₈	—	61	1	no	—
64	#117	—	61	1	no	—
65	#117	C₂×D₆₁	1	2	yes	—
66	#97	—	61	2	no	—
67	#97	—	61	4	no	—
68	C₄×C₈	—	61	2	no	—
69	C₄×C₈	—	61	4	no	—
70	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
71	M₁₆	—	2	2	no	—
72	M₁₆	—	122	2	no	—
73	M₁₆	—	122	2	no	—
74	C₂×C₈	—	61	1	no	—
75	C₂×C₈	—	61	2	no	—
76	C₂×C₈	—	61	1	no	—
77	C₂×C₈	—	61	2	no	—
78	C₂×C₈	—	61	4	no	—
79	C₂×C₈	C₄×D₆₁	1	2	yes	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	122	1	no	—
82	C₂²⋊C₄	—	2	2	no	—
83	C₂²⋊C₄	—	122	1	no	—
84	C₄²	—	61	2	no	—
85	C₄²	C₄×D₆₁	1	4	yes	—
86	C₂³	#11758224	1	1	yes	characteristic
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	122	2	no	—
89	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
90	C₂×C₄	C₂×C₄×D₆₁	1	2	yes	—
91	C₂×C₄	#11758224	1	2	yes	—
92	C₂×C₄	#11758226	1	1	yes	characteristic
93	C₈	—	2	2	no	—
94	C₈	—	61	1	no	—
95	C₈	—	61	1	no	—
96	C₈	—	61	2	no	—
97	C₈	—	122	4	no	—
98	C₂²	—	4	1	no	—
99	C₂²	#14519733	1	2	yes	—
100	C₂²	#14519735	1	1	yes	characteristic
101	C₄	—	4	4	no	—
102	C₄	—	244	2	no	—
103	C₄	C₄²×D₆₁	1	1	yes	characteristic
104	C₄	#14519733	1	1	yes	characteristic
105	C₄	#14519735	1	2	yes	—
106	C₂	—	2	2	no	—
107	C₂	—	2	1	no	—
108	C₂	#16839607	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16839720

C₂

yes

maximal, characteristic

#16839718

C₂

yes

maximal

#16839520

C₂

yes

maximal, characteristic

#16839518

C₂

yes

maximal

C₆₁×H(3,ℤ/4)

C₂

yes

maximal, characteristic

C₂×#11758194

C₂²

yes

—

C₂²×#92418

C₂²

yes

characteristic

#14519736

C₂²

yes

characteristic

#14519688

C₄

yes

complemented

#14519687

C₄

yes

complemented

C₄×#92418

C₄

yes

complemented

C₈×Dic₆₁

C₄

yes

complemented

C₂×M₁₆×C₆₁

C₂²

yes

characteristic

#117×C₆₁

C₂²

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

M₁₆×C₆₁

—

C₂×C₈×C₆₁

C₂×C₄

yes

—

C₄⋊C₄×C₆₁

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

C₂×C₄

yes

—

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄×Dic₆₁

C₂×C₄

yes

—

#11758194

—

#11758194

—

C₂×#92418

C₂×C₄

yes

—

C₂×#92418

C₂×C₄

yes

characteristic

C₂×#92418

C₂×C₄

yes

—

C₂×#92418

C₂×C₄

yes

characteristic

C₂×#92418

C₂×C₄

yes

—

C₂³×C₆₁

#41

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

#41

yes

—

C₂×C₄×C₆₁

#41

yes

characteristic

C₂×Dic₆₁

—

C₈×C₆₁

—

#92418

—

#92418

C₄²

yes

characteristic

#92418

C₄²

yes

characteristic

#92418

C₄²

yes

—

C₂²×C₆₁

—

C₂²×C₆₁

#117

yes

—

C₂²×C₆₁

#117

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

C₂×C₄²

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

Dic₆₁

—

#1747

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

yes

—

—

—

—

D₆₁

yes

characteristic

C₆₁

#1747

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₂×D₆₁

yes

characteristic

C₂²×C₈

—

#117

—

#117

C₂×D₆₁

yes

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

M₁₆

—

M₁₆

—

122

—

M₁₆

—

122

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₆₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

122

—

C₂²⋊C₄

—

C₂²⋊C₄

—

122

—

C₄²

—

C₄²

C₄×D₆₁

yes

—

C₂³

#11758224

yes

characteristic

C₂×C₄

—

C₂×C₄

—

122

—

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

C₂×C₄

C₂×C₄×D₆₁

yes

—

C₂×C₄

#11758224

yes

—

C₂×C₄

#11758226

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

122

—

C₂²

—

C₂²

#14519733

yes

—

100

C₂²

#14519735

yes

characteristic

101

C₄

—

102

C₄

—

244

—

103

C₄

C₄²×D₆₁

yes

characteristic

104

C₄

#14519733

yes

characteristic

105

C₄

#14519735

yes

—

106

C₂

—

107

C₂

—

108

C₂

#16839607

yes

characteristic

	subgroup	G/H
center	C₄	C₄²×D₆₁
commutator	C₄×C₆₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₆₁×H(3,ℤ/4)	C₂
socle	C₂×C₆₁	#376
hypercenter	H(3,ℤ/4)	D₆₁
hypocenter	C₆₁	#1747
LCS	G ▷ C₄×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

C₄²×D₆₁

commutator

C₄×C₆₁

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₆₁

Fitting

C₆₁×H(3,ℤ/4)

C₂

C₂×C₆₁

D₆₁

C₆₁

G ▷ C₄×C₆₁ ▷ C₆₁

C₁ ◁ C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
9. #14519688	C₄	semidirect
10. #14519687	C₄	semidirect
11. C₄×#92418	C₄	semidirect
12. C₈×Dic₆₁	C₄	semidirect
60. C₆₁	#1747	semidirect

H ≅ G/N

type

9. #14519688

C₄

semidirect

10. #14519687

C₄

semidirect

11. C₄×#92418

C₄

semidirect

12. C₈×Dic₆₁

C₄

semidirect

60. C₆₁

#1747

semidirect

Subgroups — #21445573

Conjugacy classes of subgroups

Complements