Group #29524225 — subgroups

Subgroups — C₆₁×#20792 #29524225

Order 15616.

1118 conjugacy classes of subgroups; only the 622 normal subgroups are shown.

normal subgroups only

This group has 1118 conjugacy classes of subgroups — too many to draw. Only its 622 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×C₆₁×#423	C₂	1	6	yes	maximal
2	C₂×C₆₁×#376	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#1843	C₂	1	8	yes	maximal, complemented
4	C₂²×M₁₆×C₆₁	C₂²	1	4	yes	—
5	C₂²×C₄²×C₆₁	C₂²	1	3	yes	—
6	C₆₁×#423	—	2	24	no	—
7	C₂×C₄⋊C₈×C₆₁	C₂²	1	24	yes	—
8	C₆₁×#376	C₄	1	4	yes	—
9	C₆₁×#376	C₂²	1	4	yes	—
10	C₂×#95×C₆₁	C₄	1	4	yes	—
11	C₂³×C₄×C₆₁	C₂×C₄	1	6	yes	—
12	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
13	C₂×M₁₆×C₆₁	—	2	24	no	—
14	C₂²×C₈×C₆₁	C₂³	1	8	yes	—
15	C₂×C₄²×C₆₁	C₂×C₄	1	6	yes	—
16	C₂×C₄²×C₆₁	D₄	1	12	yes	—
17	C₂×C₄²×C₆₁	Q₈	1	12	yes	—
18	C₂×C₄²×C₆₁	C₂³	1	6	yes	—
19	C₄⋊C₈×C₆₁	—	2	48	no	—
20	#95×C₆₁	C₂×C₄	1	16	yes	—
21	C₂⁴×C₆₁	C₂²×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₆₁	—	4	12	no	—
23	C₂²×C₄×C₆₁	C₄⋊C₄	1	24	yes	—
24	C₂²×C₄×C₆₁	C₂²×C₄	1	12	yes	—
25	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₆₁	C₂×D₄	1	6	yes	—
27	C₂²×C₄×C₆₁	C₂×Q₈	1	6	yes	—
28	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
29	M₁₆×C₆₁	—	4	16	no	—
30	C₂×C₈×C₆₁	—	2	24	no	—
31	C₄²×C₆₁	C₄⋊C₄	1	24	yes	—
32	C₄²×C₆₁	C₂×D₄	1	12	yes	—
33	C₄²×C₆₁	C₂×Q₈	1	12	yes	—
34	C₂³×C₆₁	—	2	4	no	—
35	C₂³×C₆₁	C₂×C₄⋊C₄	1	6	yes	—
36	C₂³×C₆₁	C₂³×C₄	1	1	yes	characteristic
37	C₂×C₄×C₆₁	—	2	24	no	—
38	C₂×C₄×C₆₁	—	4	24	no	—
39	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	24	yes	—
40	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	6	yes	—
41	C₂×C₄×C₆₁	#127	1	4	yes	—
42	C₂×C₄×C₆₁	C₄⋊Q₈	1	12	yes	—
43	C₂×C₄×C₆₁	C₂²×D₄	1	3	yes	—
44	C₂×C₄×C₆₁	C₂²×Q₈	1	3	yes	—
45	C₈×C₆₁	—	4	8	no	—
46	#20792	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
47	C₂²×C₆₁	—	2	12	no	—
48	C₂²×C₆₁	#384	1	4	yes	—
49	C₂²×C₆₁	C₂²×C₄⋊C₄	1	3	yes	—
50	C₂²×C₆₁	#568	1	4	yes	—
51	C₄×C₆₁	—	2	24	no	—
52	C₄×C₆₁	#384	1	4	yes	—
53	C₄×C₆₁	C₂×#127	1	1	yes	characteristic
54	C₄×C₆₁	C₂×C₄⋊Q₈	1	3	yes	—
55	C₂×#423	C₂×C₆₁	1	6	yes	—
56	C₂×#376	C₂×C₆₁	1	1	yes	characteristic
57	#1843	C₂×C₆₁	1	8	yes	complemented
58	C₂×C₆₁	—	2	4	no	—
59	C₂×C₆₁	C₂×#384	1	1	yes	characteristic
60	C₂×C₆₁	#2905	1	6	yes	—
61	C₂²×M₁₆	C₂²×C₆₁	1	4	yes	—
62	C₂²×C₄²	C₂²×C₆₁	1	3	yes	—
63	#423	—	2	24	no	—
64	C₂×C₄⋊C₈	C₂²×C₆₁	1	24	yes	—
65	#376	C₄×C₆₁	1	4	yes	—
66	#376	C₂²×C₆₁	1	4	yes	—
67	C₂×#95	C₄×C₆₁	1	4	yes	—
68	C₆₁	#20792	1	1	yes	characteristic, direct product factor, Sylow
69	C₂³×C₄	C₂×C₄×C₆₁	1	6	yes	—
70	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
71	C₂×M₁₆	—	2	24	no	—
72	C₂²×C₈	C₂³×C₆₁	1	8	yes	—
73	C₂×C₄²	C₂×C₄×C₆₁	1	6	yes	—
74	C₂×C₄²	D₄×C₆₁	1	12	yes	—
75	C₂×C₄²	Q₈×C₆₁	1	12	yes	—
76	C₂×C₄²	C₂³×C₆₁	1	6	yes	—
77	C₄⋊C₈	—	2	48	no	—
78	#95	C₂×C₄×C₆₁	1	16	yes	—
79	C₂⁴	C₂²×C₄×C₆₁	1	1	yes	characteristic
80	C₂²×C₄	—	4	12	no	—
81	C₂²×C₄	C₄⋊C₄×C₆₁	1	24	yes	—
82	C₂²×C₄	C₂²×C₄×C₆₁	1	12	yes	—
83	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
84	C₂²×C₄	C₂×D₄×C₆₁	1	6	yes	—
85	C₂²×C₄	C₂×Q₈×C₆₁	1	6	yes	—
86	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
87	M₁₆	—	4	16	no	—
88	C₂×C₈	—	2	24	no	—
89	C₄²	C₄⋊C₄×C₆₁	1	24	yes	—
90	C₄²	C₂×D₄×C₆₁	1	12	yes	—
91	C₄²	C₂×Q₈×C₆₁	1	12	yes	—
92	C₂³	—	2	4	no	—
93	C₂³	C₂×C₄⋊C₄×C₆₁	1	6	yes	—
94	C₂³	C₂³×C₄×C₆₁	1	1	yes	characteristic
95	C₂×C₄	—	2	24	no	—
96	C₂×C₄	—	4	24	no	—
97	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	24	yes	—
98	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	6	yes	—
99	C₂×C₄	#127×C₆₁	1	4	yes	—
100	C₂×C₄	C₄⋊Q₈×C₆₁	1	12	yes	—
101	C₂×C₄	C₂²×D₄×C₆₁	1	3	yes	—
102	C₂×C₄	C₂²×Q₈×C₆₁	1	3	yes	—
103	C₈	—	4	8	no	—
104	C₂²	—	2	12	no	—
105	C₂²	C₆₁×#384	1	4	yes	—
106	C₂²	C₂²×C₄⋊C₄×C₆₁	1	3	yes	—
107	C₂²	C₆₁×#568	1	4	yes	—
108	C₄	—	2	24	no	—
109	C₄	C₆₁×#384	1	4	yes	—
110	C₄	C₂×#127×C₆₁	1	1	yes	characteristic
111	C₄	C₂×C₄⋊Q₈×C₆₁	1	3	yes	—
112	C₂	—	2	4	no	—
113	C₂	C₂×C₆₁×#384	1	1	yes	characteristic
114	C₂	C₆₁×#2905	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₆₁×#423

C₂

yes

maximal

C₂×C₆₁×#376

C₂

yes

maximal, characteristic

C₆₁×#1843

C₂

yes

maximal, complemented

C₂²×M₁₆×C₆₁

C₂²

yes

—

C₂²×C₄²×C₆₁

C₂²

yes

—

C₆₁×#423

—

C₂×C₄⋊C₈×C₆₁

C₂²

yes

—

C₆₁×#376

C₄

yes

—

C₆₁×#376

C₂²

yes

—

C₂×#95×C₆₁

C₄

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×C₄

yes

—

C₂×C₄²×C₆₁

D₄

yes

—

C₂×C₄²×C₆₁

Q₈

yes

—

C₂×C₄²×C₆₁

C₂³

yes

—

C₄⋊C₈×C₆₁

—

#95×C₆₁

C₂×C₄

yes

—

C₂⁴×C₆₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₄⋊C₄

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×D₄

yes

—

C₂²×C₄×C₆₁

C₂×Q₈

yes

—

C₂²×C₄×C₆₁

C₂⁴

yes

characteristic

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

C₄⋊C₄

yes

—

C₄²×C₆₁

C₂×D₄

yes

—

C₄²×C₆₁

C₂×Q₈

yes

—

C₂³×C₆₁

—

C₂³×C₆₁

C₂×C₄⋊C₄

yes

—

C₂³×C₆₁

C₂³×C₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₆₁

#127

yes

—

C₂×C₄×C₆₁

C₄⋊Q₈

yes

—

C₂×C₄×C₆₁

C₂²×D₄

yes

—

C₂×C₄×C₆₁

C₂²×Q₈

yes

—

C₈×C₆₁

—

#20792

C₆₁

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

#384

yes

—

C₂²×C₆₁

C₂²×C₄⋊C₄

yes

—

C₂²×C₆₁

#568

yes

—

C₄×C₆₁

—

C₄×C₆₁

#384

yes

—

C₄×C₆₁

C₂×#127

yes

characteristic

C₄×C₆₁

C₂×C₄⋊Q₈

yes

—

C₂×#423

C₂×C₆₁

yes

—

C₂×#376

C₂×C₆₁

yes

characteristic

#1843

C₂×C₆₁

yes

complemented

C₂×C₆₁

—

C₂×C₆₁

C₂×#384

yes

characteristic

C₂×C₆₁

#2905

yes

—

C₂²×M₁₆

C₂²×C₆₁

yes

—

C₂²×C₄²

C₂²×C₆₁

yes

—

#423

—

C₂×C₄⋊C₈

C₂²×C₆₁

yes

—

#376

C₄×C₆₁

yes

—

#376

C₂²×C₆₁

yes

—

C₂×#95

C₄×C₆₁

yes

—

C₆₁

#20792

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×C₄×C₆₁

yes

—

C₂×C₄²

D₄×C₆₁

yes

—

C₂×C₄²

Q₈×C₆₁

yes

—

C₂×C₄²

C₂³×C₆₁

yes

—

C₄⋊C₈

—

#95

C₂×C₄×C₆₁

yes

—

C₂⁴

C₂²×C₄×C₆₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₄⋊C₄×C₆₁

yes

—

C₂²×C₄

C₂²×C₄×C₆₁

yes

—

C₂²×C₄

C₂²×C₄×C₆₁

yes

characteristic

C₂²×C₄

C₂×D₄×C₆₁

yes

—

C₂²×C₄

C₂×Q₈×C₆₁

yes

—

C₂²×C₄

C₂⁴×C₆₁

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄²

C₄⋊C₄×C₆₁

yes

—

C₄²

C₂×D₄×C₆₁

yes

—

C₄²

C₂×Q₈×C₆₁

yes

—

C₂³

—

C₂³

C₂×C₄⋊C₄×C₆₁

yes

—

C₂³

C₂³×C₄×C₆₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄⋊C₄×C₆₁

yes

—

C₂×C₄

C₂×C₄⋊C₄×C₆₁

yes

—

C₂×C₄

#127×C₆₁

yes

—

100

C₂×C₄

C₄⋊Q₈×C₆₁

yes

—

101

C₂×C₄

C₂²×D₄×C₆₁

yes

—

102

C₂×C₄

C₂²×Q₈×C₆₁

yes

—

103

C₈

—

104

C₂²

—

105

C₂²

C₆₁×#384

yes

—

106

C₂²

C₂²×C₄⋊C₄×C₆₁

yes

—

107

C₂²

C₆₁×#568

yes

—

108

C₄

—

109

C₄

C₆₁×#384

yes

—

110

C₄

C₂×#127×C₆₁

yes

characteristic

111

C₄

C₂×C₄⋊Q₈×C₆₁

yes

—

112

C₂

—

113

C₂

C₂×C₆₁×#384

yes

characteristic

114

C₂

C₆₁×#2905

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
3. C₆₁×#1843	C₂	semidirect
46. #20792	C₆₁	direct (N × H)
57. #1843	C₂×C₆₁	semidirect
68. C₆₁	#20792	direct (N × H)

H ≅ G/N

type

3. C₆₁×#1843

C₂

semidirect

46. #20792

C₆₁

direct (N × H)

57. #1843

C₂×C₆₁

semidirect

68. C₆₁

#20792

direct (N × H)

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

subgroup	index
C₂×C₆₁×#423	2
C₂×C₆₁×#376	2
C₆₁×#1843	2
#20792	61

subgroup

index

C₂×C₆₁×#423

C₂×C₆₁×#376

C₆₁×#1843

#20792

Subgroups — C₆₁×#20792 #29524225

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C61×#20792 #29524225

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#20792 #29524225