Group #21441861 — subgroups

Subgroups — C₆₁×#1820 #21441861

Order 7808.

300 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 104 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	C₂×#117×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#381	C₂	1	2	yes	maximal
3	C₆₁×#342	C₂	1	4	yes	maximal
4	C₂³×C₄×C₆₁	C₂²	1	1	yes	characteristic
5	#117×C₆₁	—	2	4	no	—
6	C₂×C₄⋊C₄×C₆₁	C₄	1	2	yes	complemented
7	C₂×C₂²⋊C₄×C₆₁	C₂²	1	4	yes	—
8	C₂×C₂²⋊C₄×C₆₁	C₂²	1	2	yes	—
9	C₂×C₄²×C₆₁	C₄	1	2	yes	complemented
10	#95×C₆₁	—	2	4	no	—
11	C₂⁴×C₆₁	C₂³	1	1	yes	characteristic
12	C₂²×C₄×C₆₁	—	2	4	no	—
13	C₂²×C₄×C₆₁	—	2	4	no	—
14	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₆₁	C₂×C₄	1	4	yes	—
17	C₂²×C₄×C₆₁	D₄	1	4	yes	—
18	C₄⋊C₄×C₆₁	—	2	4	no	—
19	C₂²⋊C₄×C₆₁	—	2	4	no	—
20	C₂²⋊C₄×C₆₁	—	2	8	no	—
21	C₄²×C₆₁	—	2	4	no	—
22	C₂³×C₆₁	—	4	2	no	—
23	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
24	C₂³×C₆₁	C₂×D₄	1	2	yes	—
25	C₂³×C₆₁	#41	1	4	yes	—
26	C₂×C₄×C₆₁	—	2	2	no	—
27	C₂×C₄×C₆₁	—	2	2	no	—
28	C₂×C₄×C₆₁	—	2	8	no	—
29	C₂×C₄×C₆₁	—	2	4	no	—
30	C₂×C₄×C₆₁	—	4	4	no	—
31	C₂×C₄×C₆₁	—	4	8	no	—
32	C₂×C₄×C₆₁	—	4	4	no	—
33	C₂×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
34	C₂×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₆₁	—	2	2	no	—
36	C₂²×C₆₁	—	2	4	no	—
37	C₂²×C₆₁	—	4	4	no	—
38	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₆₁	#117	1	2	yes	—
40	C₂²×C₆₁	#123	1	4	yes	—
41	C₄×C₆₁	—	2	2	no	—
42	C₄×C₆₁	—	2	2	no	—
43	C₄×C₆₁	—	4	4	no	—
44	C₄×C₆₁	—	8	4	no	—
45	#1820	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	C₂×C₆₁	—	2	4	no	—
47	C₂×C₆₁	—	2	2	no	—
48	C₂×C₆₁	#381	1	1	yes	characteristic
49	C₂×C₆₁	#410	1	2	yes	—
50	C₂×#117	C₂×C₆₁	1	1	yes	characteristic
51	#381	C₂×C₆₁	1	2	yes	—
52	#342	C₂×C₆₁	1	4	yes	—
53	C₆₁	#1820	1	1	yes	characteristic, direct product factor, Sylow
54	C₂³×C₄	C₂²×C₆₁	1	1	yes	characteristic
55	#117	—	2	4	no	—
56	C₂×C₄⋊C₄	C₄×C₆₁	1	2	yes	complemented
57	C₂×C₂²⋊C₄	C₂²×C₆₁	1	4	yes	—
58	C₂×C₂²⋊C₄	C₂²×C₆₁	1	2	yes	—
59	C₂×C₄²	C₄×C₆₁	1	2	yes	complemented
60	#95	—	2	4	no	—
61	C₂⁴	C₂³×C₆₁	1	1	yes	characteristic
62	C₂²×C₄	—	2	4	no	—
63	C₂²×C₄	—	2	4	no	—
64	C₂²×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
65	C₂²×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
66	C₂²×C₄	C₂×C₄×C₆₁	1	4	yes	—
67	C₂²×C₄	D₄×C₆₁	1	4	yes	—
68	C₄⋊C₄	—	2	4	no	—
69	C₂²⋊C₄	—	2	4	no	—
70	C₂²⋊C₄	—	2	8	no	—
71	C₄²	—	2	4	no	—
72	C₂³	—	4	2	no	—
73	C₂³	C₂²×C₄×C₆₁	1	1	yes	characteristic
74	C₂³	C₂×D₄×C₆₁	1	2	yes	—
75	C₂³	#41×C₆₁	1	4	yes	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	2	2	no	—
78	C₂×C₄	—	2	8	no	—
79	C₂×C₄	—	2	4	no	—
80	C₂×C₄	—	4	4	no	—
81	C₂×C₄	—	4	8	no	—
82	C₂×C₄	—	4	4	no	—
83	C₂×C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
84	C₂×C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
85	C₂²	—	2	2	no	—
86	C₂²	—	2	4	no	—
87	C₂²	—	4	4	no	—
88	C₂²	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
89	C₂²	#117×C₆₁	1	2	yes	—
90	C₂²	#123×C₆₁	1	4	yes	—
91	C₄	—	2	2	no	—
92	C₄	—	2	2	no	—
93	C₄	—	4	4	no	—
94	C₄	—	8	4	no	—
95	C₂	—	2	4	no	—
96	C₂	—	2	2	no	—
97	C₂	C₆₁×#381	1	1	yes	characteristic
98	C₂	C₆₁×#410	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#117×C₆₁

C₂

yes

maximal, characteristic

C₆₁×#381

C₂

yes

maximal

C₆₁×#342

C₂

yes

maximal

C₂³×C₄×C₆₁

C₂²

yes

characteristic

#117×C₆₁

—

C₂×C₄⋊C₄×C₆₁

C₄

yes

complemented

C₂×C₂²⋊C₄×C₆₁

C₂²

yes

—

C₂×C₂²⋊C₄×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₄

yes

complemented

#95×C₆₁

—

C₂⁴×C₆₁

C₂³

yes

characteristic

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

D₄

yes

—

C₄⋊C₄×C₆₁

—

C₂²⋊C₄×C₆₁

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

C₂²×C₄

yes

characteristic

C₂³×C₆₁

C₂×D₄

yes

—

C₂³×C₆₁

#41

yes

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₆₁

#117

yes

—

C₂²×C₆₁

#123

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

#1820

C₆₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

#381

yes

characteristic

C₂×C₆₁

#410

yes

—

C₂×#117

C₂×C₆₁

yes

characteristic

#381

C₂×C₆₁

yes

—

#342

C₂×C₆₁

yes

—

C₆₁

#1820

yes

characteristic, direct product factor, Sylow

C₂³×C₄

C₂²×C₆₁

yes

characteristic

#117

—

C₂×C₄⋊C₄

C₄×C₆₁

yes

complemented

C₂×C₂²⋊C₄

C₂²×C₆₁

yes

—

C₂×C₂²⋊C₄

C₂²×C₆₁

yes

—

C₂×C₄²

C₄×C₆₁

yes

complemented

#95

—

C₂⁴

C₂³×C₆₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄×C₆₁

yes

characteristic

C₂²×C₄

C₂×C₄×C₆₁

yes

characteristic

C₂²×C₄

C₂×C₄×C₆₁

yes

—

C₂²×C₄

D₄×C₆₁

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

C₂³

C₂²×C₄×C₆₁

yes

characteristic

C₂³

C₂×D₄×C₆₁

yes

—

C₂³

#41×C₆₁

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊C₄×C₆₁

yes

—

C₂×C₄

C₂²⋊C₄×C₆₁

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂×C₂²⋊C₄×C₆₁

yes

characteristic

C₂²

#117×C₆₁

yes

—

C₂²

#123×C₆₁

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

C₂

—

C₂

—

C₂

C₆₁×#381

yes

characteristic

C₂

C₆₁×#410

yes

—

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₄×C₆₁
Frattini, cosocle	C₂⁴	C₂³×C₆₁
prime radical	C₂⁴×C₆₁	C₂³
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂³×C₄×C₆₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₆₁

C₂×C₂²⋊C₄

commutator

C₂³

C₂²×C₄×C₆₁

Frattini, cosocle

C₂⁴

C₂³×C₆₁

prime radical

C₂⁴×C₆₁

C₂³

LCS

G ▷ C₂³ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₆₁ ◁ C₂³×C₄×C₆₁ ◁ G

N	H ≅ G/N	type
6. C₂×C₄⋊C₄×C₆₁	C₄	semidirect
9. C₂×C₄²×C₆₁	C₄	semidirect
45. #1820	C₆₁	direct (N × H)
53. C₆₁	#1820	direct (N × H)
56. C₂×C₄⋊C₄	C₄×C₆₁	semidirect
59. C₂×C₄²	C₄×C₆₁	semidirect

H ≅ G/N

type

6. C₂×C₄⋊C₄×C₆₁

C₄

semidirect

9. C₂×C₄²×C₆₁

C₄

semidirect

45. #1820

C₆₁

direct (N × H)

53. C₆₁

#1820

direct (N × H)

56. C₂×C₄⋊C₄

C₄×C₆₁

semidirect

59. C₂×C₄²

C₄×C₆₁

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#117×C₆₁	2
C₆₁×#381	2
C₆₁×#342	2
#1820	61

subgroup

index

C₂×#117×C₆₁

C₆₁×#381

C₆₁×#342

#1820

Subgroups — C₆₁×#1820 #21441861

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C61×#1820 #21441861

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#1820 #21441861