Group #29524205 — subgroups

Subgroups — C₆₁×#20772 #29524205

Order 15616.

1062 conjugacy classes of subgroups; only the 462 normal subgroups are shown.

normal subgroups only

This group has 1062 conjugacy classes of subgroups — too many to draw. Only its 462 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×C₄³×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#1839	C₂	1	8	yes	maximal, complemented
3	C₆₁×#1728	C₂	1	6	yes	maximal
4	C₂²×M₁₆×C₆₁	C₂²	1	4	yes	—
5	C₂²×C₄²×C₆₁	C₄	1	4	yes	—
6	C₂²×C₄²×C₆₁	C₂²	1	3	yes	—
7	C₄³×C₆₁	C₄	1	4	yes	—
8	C₄³×C₆₁	C₂²	1	4	yes	—
9	C₆₁×#336	C₂²	1	24	yes	—
10	C₂³×C₄×C₆₁	C₂×C₄	1	6	yes	—
11	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
12	C₂×M₁₆×C₆₁	—	2	24	no	—
13	C₂²×C₈×C₆₁	C₂³	1	8	yes	—
14	C₂×C₄²×C₆₁	—	2	12	no	—
15	C₂×C₄²×C₆₁	—	2	16	no	—
16	C₂×C₄²×C₆₁	C₂×C₄	1	6	yes	—
17	C₂×C₄²×C₆₁	C₂×C₄	1	12	yes	—
18	C₂×C₄²×C₆₁	C₂×C₄	1	4	yes	—
19	C₂×C₄²×C₆₁	C₂³	1	6	yes	—
20	C₂⁴×C₆₁	C₂²×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₆₁	—	2	12	no	—
22	C₂²×C₄×C₆₁	—	2	24	no	—
23	C₂²×C₄×C₆₁	C₂²×C₄	1	12	yes	—
24	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₆₁	#41	1	12	yes	—
26	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
27	M₁₆×C₆₁	—	4	16	no	—
28	C₂×C₈×C₆₁	—	2	24	no	—
29	C₄²×C₆₁	—	2	12	no	—
30	C₄²×C₆₁	—	2	24	no	—
31	C₄²×C₆₁	—	2	12	no	—
32	C₄²×C₆₁	M₁₆	1	16	yes	complemented
33	C₂³×C₆₁	—	2	4	no	—
34	C₂³×C₆₁	#117	1	6	yes	—
35	C₂³×C₆₁	C₂³×C₄	1	1	yes	characteristic
36	C₂×C₄×C₆₁	—	2	48	no	—
37	C₂×C₄×C₆₁	—	2	12	no	—
38	C₂×C₄×C₆₁	—	2	12	no	—
39	C₂×C₄×C₆₁	#117	1	6	yes	—
40	C₂×C₄×C₆₁	#126	1	16	yes	—
41	C₂×C₄×C₆₁	C₂×M₁₆	1	24	yes	—
42	C₂×C₄×C₆₁	C₂×#41	1	6	yes	—
43	C₈×C₆₁	—	4	8	no	—
44	#20772	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂²×C₆₁	—	2	12	no	—
46	C₂²×C₆₁	#383	1	4	yes	—
47	C₂²×C₆₁	C₂×#117	1	3	yes	—
48	C₂²×C₆₁	C₂²×M₁₆	1	4	yes	—
49	C₄×C₆₁	—	2	24	no	—
50	C₄×C₆₁	#383	1	4	yes	—
51	C₄×C₆₁	C₂×#126	1	4	yes	—
52	C₂×C₄³	C₂×C₆₁	1	1	yes	characteristic
53	#1839	C₂×C₆₁	1	8	yes	complemented
54	#1728	C₂×C₆₁	1	6	yes	—
55	C₂×C₆₁	—	2	4	no	—
56	C₂×C₆₁	C₂×#383	1	1	yes	characteristic
57	C₂×C₆₁	#2920	1	6	yes	—
58	C₂²×M₁₆	C₂²×C₆₁	1	4	yes	—
59	C₂²×C₄²	C₄×C₆₁	1	4	yes	—
60	C₂²×C₄²	C₂²×C₆₁	1	3	yes	—
61	C₄³	C₄×C₆₁	1	4	yes	—
62	C₄³	C₂²×C₆₁	1	4	yes	—
63	#336	C₂²×C₆₁	1	24	yes	—
64	C₆₁	#20772	1	1	yes	characteristic, direct product factor, Sylow
65	C₂³×C₄	C₂×C₄×C₆₁	1	6	yes	—
66	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
67	C₂×M₁₆	—	2	24	no	—
68	C₂²×C₈	C₂³×C₆₁	1	8	yes	—
69	C₂×C₄²	—	2	12	no	—
70	C₂×C₄²	—	2	16	no	—
71	C₂×C₄²	C₂×C₄×C₆₁	1	6	yes	—
72	C₂×C₄²	C₂×C₄×C₆₁	1	12	yes	—
73	C₂×C₄²	C₂×C₄×C₆₁	1	4	yes	—
74	C₂×C₄²	C₂³×C₆₁	1	6	yes	—
75	C₂⁴	C₂²×C₄×C₆₁	1	1	yes	characteristic
76	C₂²×C₄	—	2	12	no	—
77	C₂²×C₄	—	2	24	no	—
78	C₂²×C₄	C₂²×C₄×C₆₁	1	12	yes	—
79	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
80	C₂²×C₄	#41×C₆₁	1	12	yes	—
81	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
82	M₁₆	—	4	16	no	—
83	C₂×C₈	—	2	24	no	—
84	C₄²	—	2	12	no	—
85	C₄²	—	2	24	no	—
86	C₄²	—	2	12	no	—
87	C₄²	M₁₆×C₆₁	1	16	yes	complemented
88	C₂³	—	2	4	no	—
89	C₂³	#117×C₆₁	1	6	yes	—
90	C₂³	C₂³×C₄×C₆₁	1	1	yes	characteristic
91	C₂×C₄	—	2	48	no	—
92	C₂×C₄	—	2	12	no	—
93	C₂×C₄	—	2	12	no	—
94	C₂×C₄	#117×C₆₁	1	6	yes	—
95	C₂×C₄	#126×C₆₁	1	16	yes	—
96	C₂×C₄	C₂×M₁₆×C₆₁	1	24	yes	—
97	C₂×C₄	C₂×#41×C₆₁	1	6	yes	—
98	C₈	—	4	8	no	—
99	C₂²	—	2	12	no	—
100	C₂²	C₆₁×#383	1	4	yes	—
101	C₂²	C₂×#117×C₆₁	1	3	yes	—
102	C₂²	C₂²×M₁₆×C₆₁	1	4	yes	—
103	C₄	—	2	24	no	—
104	C₄	C₆₁×#383	1	4	yes	—
105	C₄	C₂×#126×C₆₁	1	4	yes	—
106	C₂	—	2	4	no	—
107	C₂	C₂×C₆₁×#383	1	1	yes	characteristic
108	C₂	C₆₁×#2920	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄³×C₆₁

C₂

yes

maximal, characteristic

C₆₁×#1839

C₂

yes

maximal, complemented

C₆₁×#1728

C₂

yes

maximal

C₂²×M₁₆×C₆₁

C₂²

yes

—

C₂²×C₄²×C₆₁

C₄

yes

—

C₂²×C₄²×C₆₁

C₂²

yes

—

C₄³×C₆₁

C₄

yes

—

C₄³×C₆₁

C₂²

yes

—

C₆₁×#336

C₂²

yes

—

C₂³×C₄×C₆₁

C₂×C₄

yes

—

C₂³×C₄×C₆₁

C₂³

yes

characteristic

C₂×M₁₆×C₆₁

—

C₂²×C₈×C₆₁

C₂³

yes

—

C₂×C₄²×C₆₁

—

C₂×C₄²×C₆₁

—

C₂×C₄²×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂³

yes

—

C₂⁴×C₆₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₁

#41

yes

—

C₂²×C₄×C₆₁

C₂⁴

yes

characteristic

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

M₁₆

yes

complemented

C₂³×C₆₁

—

C₂³×C₆₁

#117

yes

—

C₂³×C₆₁

C₂³×C₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#117

yes

—

C₂×C₄×C₆₁

#126

yes

—

C₂×C₄×C₆₁

C₂×M₁₆

yes

—

C₂×C₄×C₆₁

C₂×#41

yes

—

C₈×C₆₁

—

#20772

C₆₁

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

#383

yes

—

C₂²×C₆₁

C₂×#117

yes

—

C₂²×C₆₁

C₂²×M₁₆

yes

—

C₄×C₆₁

—

C₄×C₆₁

#383

yes

—

C₄×C₆₁

C₂×#126

yes

—

C₂×C₄³

C₂×C₆₁

yes

characteristic

#1839

C₂×C₆₁

yes

complemented

#1728

C₂×C₆₁

yes

—

C₂×C₆₁

—

C₂×C₆₁

C₂×#383

yes

characteristic

C₂×C₆₁

#2920

yes

—

C₂²×M₁₆

C₂²×C₆₁

yes

—

C₂²×C₄²

C₄×C₆₁

yes

—

C₂²×C₄²

C₂²×C₆₁

yes

—

C₄³

C₄×C₆₁

yes

—

C₄³

C₂²×C₆₁

yes

—

#336

C₂²×C₆₁

yes

—

C₆₁

#20772

yes

characteristic, direct product factor, Sylow

C₂³×C₄

C₂×C₄×C₆₁

yes

—

C₂³×C₄

C₂³×C₆₁

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

C₂³×C₆₁

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×C₄×C₆₁

yes

—

C₂×C₄²

C₂×C₄×C₆₁

yes

—

C₂×C₄²

C₂×C₄×C₆₁

yes

—

C₂×C₄²

C₂³×C₆₁

yes

—

C₂⁴

C₂²×C₄×C₆₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×C₄×C₆₁

yes

—

C₂²×C₄

C₂²×C₄×C₆₁

yes

characteristic

C₂²×C₄

#41×C₆₁

yes

—

C₂²×C₄

C₂⁴×C₆₁

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄²

—

C₄²

—

C₄²

—

C₄²

M₁₆×C₆₁

yes

complemented

C₂³

—

C₂³

#117×C₆₁

yes

—

C₂³

C₂³×C₄×C₆₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#117×C₆₁

yes

—

C₂×C₄

#126×C₆₁

yes

—

C₂×C₄

C₂×M₁₆×C₆₁

yes

—

C₂×C₄

C₂×#41×C₆₁

yes

—

C₈

—

C₂²

—

100

C₂²

C₆₁×#383

yes

—

101

C₂²

C₂×#117×C₆₁

yes

—

102

C₂²

C₂²×M₁₆×C₆₁

yes

—

103

C₄

—

104

C₄

C₆₁×#383

yes

—

105

C₄

C₂×#126×C₆₁

yes

—

106

C₂

—

107

C₂

C₂×C₆₁×#383

yes

characteristic

108

C₂

C₆₁×#2920

yes

—

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

subgroup

G/H

center

C₂²×C₄×C₆₁

C₂⁴

commutator

C₂³

C₂³×C₄×C₆₁

Frattini, cosocle

C₂²×C₄

C₂⁴×C₆₁

socle

C₂³×C₆₁

C₂³×C₄

prime radical

C₂⁴×C₆₁

C₂²×C₄

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂²×C₄×C₆₁ ◁ G

N	H ≅ G/N	type
2. C₆₁×#1839	C₂	semidirect
32. C₄²×C₆₁	M₁₆	semidirect
44. #20772	C₆₁	direct (N × H)
53. #1839	C₂×C₆₁	semidirect
64. C₆₁	#20772	direct (N × H)
87. C₄²	M₁₆×C₆₁	semidirect

H ≅ G/N

type

2. C₆₁×#1839

C₂

semidirect

32. C₄²×C₆₁

M₁₆

semidirect

44. #20772

C₆₁

direct (N × H)

53. #1839

C₂×C₆₁

semidirect

64. C₆₁

#20772

direct (N × H)

87. C₄²

M₁₆×C₆₁

semidirect

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

subgroup	index
C₂×C₄³×C₆₁	2
C₆₁×#1839	2
C₆₁×#1728	2
#20772	61

subgroup

index

C₂×C₄³×C₆₁

C₆₁×#1839

C₆₁×#1728

#20772

Subgroups — C₆₁×#20772 #29524205

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C61×#20772 #29524205

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#20772 #29524205