Group #21459967 — subgroups

Subgroups — #21459967

Order 7808.

930 conjugacy classes of subgroups; only the 451 normal subgroups are shown.

normal subgroups only

This group has 930 conjugacy classes of subgroups — too many to draw. Only its 451 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	#16840516	C₂	1	10	yes	maximal, complemented
2	#16840510	C₂	1	5	yes	maximal, complemented
3	#16840318	C₂	1	5	yes	maximal, complemented
4	#16840297	C₂	1	10	yes	maximal, complemented
5	C₆₁×#564	C₂	1	1	yes	maximal, characteristic, complemented
6	#14519776	C₂²	1	10	yes	—
7	#14519772	C₂²	1	20	yes	—
8	#14519769	C₂²	1	5	yes	—
9	#14519765	C₂²	1	10	yes	—
10	#14519762	C₂²	1	20	yes	—
11	#14519757	C₂²	1	10	yes	—
12	#14519755	C₂²	1	20	yes	—
13	#14519752	C₂²	1	10	yes	—
14	#14519742	C₂²	1	10	yes	—
15	#14519739	C₂²	1	10	yes	—
16	#14519735	C₂²	1	10	yes	—
17	C₄²×D₆₁	C₂²	1	5	yes	—
18	#125×C₆₁	C₂²	1	5	yes	—
19	#125×C₆₁	C₂²	1	10	yes	—
20	C₂×D₂₄₄	C₂³	1	10	yes	—
21	C₂×C₄×D₆₁	C₂³	1	5	yes	—
22	C₂×C₄×D₆₁	C₂³	1	10	yes	—
23	C₂×Dic₁₂₂	C₂³	1	10	yes	—
24	C₄⋊C₄×C₆₁	C₂³	1	10	yes	—
25	C₄⋊C₄×C₆₁	C₂³	1	10	yes	—
26	C₄⋊C₄×C₆₁	C₂³	1	10	yes	—
27	C₄²×C₆₁	C₂³	1	5	yes	—
28	#11758198	C₂³	1	20	yes	—
29	#11758198	C₂³	1	20	yes	—
30	C₄⋊Dic₆₁	C₂³	1	10	yes	—
31	#11758196	C₂³	1	20	yes	—
32	C₄×Dic₆₁	C₂³	1	10	yes	—
33	C₄×Dic₆₁	C₂³	1	5	yes	—
34	C₂²×D₆₁	C₂⁴	1	5	yes	—
35	C₂×C₄×C₆₁	C₂⁴	1	10	yes	—
36	C₂×C₄×C₆₁	C₂⁴	1	5	yes	—
37	C₂×Dic₆₁	C₂⁴	1	10	yes	—
38	C₂×Dic₆₁	C₂⁴	1	1	yes	characteristic
39	D₂₄₄	—	4	10	no	—
40	C₄×D₆₁	—	2	10	no	—
41	C₄×D₆₁	—	4	10	no	—
42	Dic₁₂₂	—	4	10	no	—
43	C₂²×C₆₁	C₂⁵	1	1	yes	characteristic
44	C₂×D₆₁	—	2	5	no	—
45	C₂×D₆₁	—	2	10	no	—
46	C₄×C₆₁	—	2	5	no	—
47	C₄×C₆₁	—	2	10	no	—
48	Dic₆₁	—	2	10	no	—
49	Dic₆₁	2₋¹⁺⁴	1	2	yes	—
50	#3531	—	61	1	no	maximal, Sylow
51	C₂×C₆₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
52	C₂×C₆₁	#585	1	2	yes	—
53	D₆₁	—	4	5	no	—
54	#564	D₆₁	1	1	yes	characteristic, complemented
55	#563	—	61	10	no	—
56	#562	—	61	5	no	—
57	#533	—	61	5	no	—
58	#529	—	61	10	no	—
59	C₆₁	#3531	1	1	yes	characteristic, complemented, Sylow
60	C₄⋊Q₈	—	61	10	no	—
61	#126	—	61	20	no	—
62	#125	—	61	5	no	—
63	#125	C₂×D₆₁	1	10	yes	—
64	#125	C₂×D₆₁	1	5	yes	—
65	#124	—	61	10	no	—
66	#123	—	61	20	no	—
67	C₂²⋊Q₈	—	61	10	no	—
68	C₂²⋊Q₈	—	61	20	no	—
69	#121	—	61	10	no	—
70	C₄×Q₈	—	61	10	no	—
71	C₄×D₄	—	61	10	no	—
72	#117	—	61	10	no	—
73	C₂×C₄²	—	61	5	no	—
74	C₂×Q₈	—	61	10	no	—
75	C₂×D₄	—	61	10	no	—
76	C₂²×C₄	—	61	5	no	—
77	C₂²×C₄	—	61	10	no	—
78	C₄⋊C₄	—	61	20	no	—
79	C₄⋊C₄	—	61	10	no	—
80	C₄⋊C₄	C₂²×D₆₁	1	10	yes	—
81	C₄⋊C₄	C₂²×D₆₁	1	10	yes	—
82	C₄⋊C₄	C₂²×D₆₁	1	10	yes	—
83	C₂²⋊C₄	—	61	20	no	—
84	C₂²⋊C₄	—	61	20	no	—
85	C₄²	—	61	10	no	—
86	C₄²	—	61	5	no	—
87	C₄²	C₂²×D₆₁	1	5	yes	—
88	C₂³	—	61	5	no	—
89	Q₈	—	244	10	no	—
90	D₄	—	244	10	no	—
91	C₂×C₄	—	61	1	no	—
92	C₂×C₄	—	61	10	no	—
93	C₂×C₄	—	122	10	no	—
94	C₂×C₄	—	244	10	no	—
95	C₂×C₄	C₂³×D₆₁	1	5	yes	—
96	C₂×C₄	C₂³×D₆₁	1	10	yes	—
97	C₂²	—	122	5	no	—
98	C₂²	—	122	10	no	—
99	C₂²	C₂⁴×D₆₁	1	1	yes	characteristic
100	C₄	—	2	5	no	—
101	C₄	—	2	10	no	—
102	C₄	—	61	2	no	—
103	C₄	—	122	10	no	—
104	C₂	—	244	5	no	—
105	C₂	2₋¹⁺⁴×D₆₁	1	1	yes	characteristic
106	C₂	#16840599	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16840516

C₂

yes

maximal, complemented

#16840510

C₂

yes

maximal, complemented

#16840318

C₂

yes

maximal, complemented

#16840297

C₂

yes

maximal, complemented

C₆₁×#564

C₂

yes

maximal, characteristic, complemented

#14519776

C₂²

yes

—

#14519772

C₂²

yes

—

#14519769

C₂²

yes

—

#14519765

C₂²

yes

—

#14519762

C₂²

yes

—

#14519757

C₂²

yes

—

#14519755

C₂²

yes

—

#14519752

C₂²

yes

—

#14519742

C₂²

yes

—

#14519739

C₂²

yes

—

#14519735

C₂²

yes

—

C₄²×D₆₁

C₂²

yes

—

#125×C₆₁

C₂²

yes

—

#125×C₆₁

C₂²

yes

—

C₂×D₂₄₄

C₂³

yes

—

C₂×C₄×D₆₁

C₂³

yes

—

C₂×C₄×D₆₁

C₂³

yes

—

C₂×Dic₁₂₂

C₂³

yes

—

C₄⋊C₄×C₆₁

C₂³

yes

—

C₄⋊C₄×C₆₁

C₂³

yes

—

C₄⋊C₄×C₆₁

C₂³

yes

—

C₄²×C₆₁

C₂³

yes

—

#11758198

C₂³

yes

—

#11758198

C₂³

yes

—

C₄⋊Dic₆₁

C₂³

yes

—

#11758196

C₂³

yes

—

C₄×Dic₆₁

C₂³

yes

—

C₄×Dic₆₁

C₂³

yes

—

C₂²×D₆₁

C₂⁴

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

—

C₂×Dic₆₁

C₂⁴

yes

—

C₂×Dic₆₁

C₂⁴

yes

characteristic

D₂₄₄

—

C₄×D₆₁

—

C₄×D₆₁

—

Dic₁₂₂

—

C₂²×C₆₁

C₂⁵

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

Dic₆₁

—

Dic₆₁

2₋¹⁺⁴

yes

—

#3531

—

maximal, Sylow

C₂×C₆₁

C₂×2₋¹⁺⁴

yes

characteristic

C₂×C₆₁

#585

yes

—

D₆₁

—

#564

D₆₁

yes

characteristic, complemented

—

—

—

—

C₆₁

#3531

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

#126

—

#125

—

#125

C₂×D₆₁

yes

—

#125

C₂×D₆₁

yes

—

#124

—

#123

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

#121

—

C₄×Q₈

—

C₄×D₄

—

#117

—

C₂×C₄²

—

C₂×Q₈

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₆₁

yes

—

C₄⋊C₄

C₂²×D₆₁

yes

—

C₄⋊C₄

C₂²×D₆₁

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²×D₆₁

yes

—

C₂³

—

Q₈

—

244

—

D₄

—

244

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

122

—

C₂×C₄

—

244

—

C₂×C₄

C₂³×D₆₁

yes

—

C₂×C₄

C₂³×D₆₁

yes

—

C₂²

—

122

—

C₂²

—

122

—

C₂²

C₂⁴×D₆₁

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

122

—

104

C₂

—

244

—

105

C₂

2₋¹⁺⁴×D₆₁

yes

characteristic

106

C₂

#16840599

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₆₁
commutator, socle	C₂²×C₆₁	C₂⁵
Fitting	C₆₁×#564	C₂
hypercenter	#564	D₆₁
hypocenter	C₆₁	#3531
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ #564

subgroup

G/H

center, Frattini

C₂²

C₂⁴×D₆₁

commutator, socle

C₂²×C₆₁

C₂⁵

Fitting

C₆₁×#564

C₂

D₆₁

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂² ◁ #564

N	H ≅ G/N	type
1. #16840516	C₂	semidirect
2. #16840510	C₂	semidirect
3. #16840318	C₂	semidirect
4. #16840297	C₂	semidirect
5. C₆₁×#564	C₂	semidirect
54. #564	D₆₁	semidirect
59. C₆₁	#3531	semidirect

H ≅ G/N

type

1. #16840516

C₂

semidirect

2. #16840510

C₂

semidirect

3. #16840318

C₂

semidirect

4. #16840297

C₂

semidirect

5. C₆₁×#564

C₂

semidirect

54. #564

D₆₁

semidirect

59. C₆₁

#3531

semidirect

Subgroups — #21459967

Conjugacy classes of subgroups

Complements