Group #21460632 — subgroups

Subgroups — #21460632

Order 7808.

5796 conjugacy classes of subgroups; only the 3201 normal subgroups are shown.

normal subgroups only

This group has 5796 conjugacy classes of subgroups — too many to draw. Only its 3201 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	#16840598	C₂	1	15	yes	maximal, complemented
2	#16840597	C₂	1	20	yes	maximal, complemented
3	C₂×#14519824	C₂	1	15	yes	maximal, complemented
4	2₋¹⁺⁴×D₆₁	C₂	1	12	yes	maximal, complemented
5	C₆₁×#585	C₂	1	1	yes	maximal, characteristic, complemented
6	#14519825	C₂²	1	30	yes	complemented
7	#14519824	C₂²	1	120	yes	complemented
8	#14519824	C₂²	1	60	yes	complemented
9	#14519823	C₂²	1	20	yes	complemented
10	#14519822	C₂²	1	90	yes	complemented
11	C₂×#11758222	C₂²	1	45	yes	complemented
12	C₂×#11758224	C₂²	1	90	yes	complemented
13	#41×D₆₁	C₂²	1	120	yes	complemented
14	C₂²×Dic₁₂₂	C₂²	1	15	yes	complemented
15	C₂×Q₈×D₆₁	C₂²	1	30	yes	complemented
16	2₋¹⁺⁴×C₆₁	C₂²	1	6	yes	complemented
17	H(5,𝔽₂)×C₆₁	C₂²	1	10	yes	complemented
18	C₂×#41×C₆₁	C₂²	1	15	yes	complemented
19	#41×C₆₁	C₂³	1	60	yes	—
20	#41×C₆₁	C₂³	1	20	yes	—
21	C₂×Q₈×C₆₁	C₂³	1	15	yes	—
22	C₂×D₄×C₆₁	C₂³	1	45	yes	—
23	C₂²×C₄×C₆₁	C₂³	1	15	yes	—
24	C₂×C₂²⋊D₆₁	C₂³	1	90	yes	—
25	C₂²×Dic₆₁	C₂³	1	30	yes	—
26	#11758226	C₂³	1	40	yes	—
27	Q₈×D₆₁	C₂³	1	120	yes	—
28	#11758224	C₂³	1	360	yes	—
29	D₄×D₆₁	C₂³	1	120	yes	—
30	#11758222	C₂³	1	180	yes	—
31	#11758222	C₂³	1	60	yes	—
32	C₂×D₂₄₄	C₂³	1	15	yes	—
33	C₂×C₄×D₆₁	C₂³	1	90	yes	—
34	C₂×Dic₁₂₂	C₂³	1	90	yes	—
35	C₂×Dic₁₂₂	C₂³	1	45	yes	—
36	C₂³×C₆₁	C₂⁴	1	15	yes	—
37	C₂²×D₆₁	C₂⁴	1	30	yes	—
38	Q₈×C₆₁	C₂⁴	1	20	yes	—
39	D₄×C₆₁	C₂⁴	1	60	yes	—
40	C₂×C₄×C₆₁	C₂⁴	1	15	yes	—
41	C₂×C₄×C₆₁	C₂⁴	1	45	yes	—
42	C₂²⋊D₆₁	C₂⁴	1	120	yes	—
43	C₂×Dic₆₁	C₂⁴	1	90	yes	—
44	D₂₄₄	C₂⁴	1	6	yes	—
45	D₂₄₄	C₂⁴	1	30	yes	—
46	C₄×D₆₁	C₂⁴	1	120	yes	—
47	Dic₁₂₂	C₂⁴	1	10	yes	—
48	Dic₁₂₂	C₂⁴	1	90	yes	—
49	C₂²×C₆₁	—	4	15	no	—
50	C₂²×C₆₁	C₂⁵	1	15	yes	—
51	C₂×D₆₁	—	4	30	no	—
52	C₂×D₆₁	C₂⁵	1	12	yes	—
53	C₄×C₆₁	C₂⁵	1	1	yes	characteristic
54	C₄×C₆₁	C₂⁵	1	15	yes	—
55	Dic₆₁	C₂⁵	1	20	yes	—
56	2₋¹⁺⁶	—	61	1	no	maximal, Sylow
57	C₂×C₆₁	—	2	15	no	—
58	C₂×C₆₁	C₂⁶	1	1	yes	characteristic
59	D₆₁	—	2	12	no	—
60	#585	—	61	20	no	—
61	#585	—	61	15	no	—
62	#585	D₆₁	1	1	yes	characteristic, complemented
63	C₂×2₋¹⁺⁴	—	61	12	no	—
64	C₂×2₋¹⁺⁴	—	61	15	no	—
65	C₆₁	2₋¹⁺⁶	1	1	yes	characteristic, complemented, Sylow
66	2₋¹⁺⁴	—	61	120	no	—
67	2₋¹⁺⁴	—	61	60	no	—
68	2₋¹⁺⁴	—	61	30	no	—
69	2₋¹⁺⁴	C₂×D₆₁	1	6	yes	complemented
70	H(5,𝔽₂)	—	61	20	no	—
71	H(5,𝔽₂)	—	61	90	no	—
72	H(5,𝔽₂)	C₂×D₆₁	1	10	yes	complemented
73	C₂×#41	—	61	120	no	—
74	C₂×#41	—	61	90	no	—
75	C₂×#41	—	61	45	no	—
76	C₂×#41	C₂×D₆₁	1	15	yes	complemented
77	C₂²×Q₈	—	61	30	no	—
78	C₂²×Q₈	—	61	15	no	—
79	#41	—	61	40	no	—
80	#41	—	61	360	no	—
81	#41	—	61	180	no	—
82	#41	—	61	60	no	—
83	#41	C₂²×D₆₁	1	60	yes	—
84	#41	C₂²×D₆₁	1	20	yes	—
85	C₂×Q₈	—	61	120	no	—
86	C₂×Q₈	—	61	90	no	—
87	C₂×Q₈	—	61	45	no	—
88	C₂×Q₈	C₂²×D₆₁	1	15	yes	—
89	C₂×D₄	—	61	90	no	—
90	C₂×D₄	—	61	120	no	—
91	C₂×D₄	—	61	15	no	—
92	C₂×D₄	C₂²×D₆₁	1	45	yes	—
93	C₂²×C₄	—	61	30	no	—
94	C₂²×C₄	—	61	90	no	—
95	C₂²×C₄	C₂²×D₆₁	1	15	yes	—
96	C₂³	—	61	30	no	—
97	C₂³	C₂³×D₆₁	1	15	yes	—
98	Q₈	—	61	90	no	—
99	Q₈	—	61	10	no	—
100	Q₈	C₂³×D₆₁	1	20	yes	—
101	D₄	—	61	120	no	—
102	D₄	—	61	30	no	—
103	D₄	—	61	6	no	—
104	D₄	C₂³×D₆₁	1	60	yes	—
105	C₂×C₄	—	61	90	no	—
106	C₂×C₄	—	61	120	no	—
107	C₂×C₄	C₂³×D₆₁	1	45	yes	—
108	C₂×C₄	C₂³×D₆₁	1	15	yes	—
109	C₂²	—	4	15	no	—
110	C₂²	—	61	12	no	—
111	C₂²	—	244	30	no	—
112	C₂²	C₂⁴×D₆₁	1	15	yes	—
113	C₄	—	61	20	no	—
114	C₄	C₂⁴×D₆₁	1	15	yes	—
115	C₄	C₂⁴×D₆₁	1	1	yes	characteristic
116	C₂	—	2	15	no	—
117	C₂	—	122	12	no	—
118	C₂	C₂⁵×D₆₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840598

C₂

yes

maximal, complemented

#16840597

C₂

yes

maximal, complemented

C₂×#14519824

C₂

yes

maximal, complemented

2₋¹⁺⁴×D₆₁

C₂

yes

maximal, complemented

C₆₁×#585

C₂

yes

maximal, characteristic, complemented

#14519825

C₂²

yes

complemented

#14519824

C₂²

120

yes

complemented

#14519824

C₂²

yes

complemented

#14519823

C₂²

yes

complemented

#14519822

C₂²

yes

complemented

C₂×#11758222

C₂²

yes

complemented

C₂×#11758224

C₂²

yes

complemented

#41×D₆₁

C₂²

120

yes

complemented

C₂²×Dic₁₂₂

C₂²

yes

complemented

C₂×Q₈×D₆₁

C₂²

yes

complemented

2₋¹⁺⁴×C₆₁

C₂²

yes

complemented

H(5,𝔽₂)×C₆₁

C₂²

yes

complemented

C₂×#41×C₆₁

C₂²

yes

complemented

#41×C₆₁

C₂³

yes

—

#41×C₆₁

C₂³

yes

—

C₂×Q₈×C₆₁

C₂³

yes

—

C₂×D₄×C₆₁

C₂³

yes

—

C₂²×C₄×C₆₁

C₂³

yes

—

C₂×C₂²⋊D₆₁

C₂³

yes

—

C₂²×Dic₆₁

C₂³

yes

—

#11758226

C₂³

yes

—

Q₈×D₆₁

C₂³

120

yes

—

#11758224

C₂³

360

yes

—

D₄×D₆₁

C₂³

120

yes

—

#11758222

C₂³

180

yes

—

#11758222

C₂³

yes

—

C₂×D₂₄₄

C₂³

yes

—

C₂×C₄×D₆₁

C₂³

yes

—

C₂×Dic₁₂₂

C₂³

yes

—

C₂×Dic₁₂₂

C₂³

yes

—

C₂³×C₆₁

C₂⁴

yes

—

C₂²×D₆₁

C₂⁴

yes

—

Q₈×C₆₁

C₂⁴

yes

—

D₄×C₆₁

C₂⁴

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

—

C₂²⋊D₆₁

C₂⁴

120

yes

—

C₂×Dic₆₁

C₂⁴

yes

—

D₂₄₄

C₂⁴

yes

—

D₂₄₄

C₂⁴

yes

—

C₄×D₆₁

C₂⁴

120

yes

—

Dic₁₂₂

C₂⁴

yes

—

Dic₁₂₂

C₂⁴

yes

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂⁵

yes

—

C₂×D₆₁

—

C₂×D₆₁

C₂⁵

yes

—

C₄×C₆₁

C₂⁵

yes

characteristic

C₄×C₆₁

C₂⁵

yes

—

Dic₆₁

C₂⁵

yes

—

2₋¹⁺⁶

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

C₂⁶

yes

characteristic

D₆₁

—

#585

—

#585

—

#585

D₆₁

yes

characteristic, complemented

C₂×2₋¹⁺⁴

—

C₂×2₋¹⁺⁴

—

C₆₁

2₋¹⁺⁶

yes

characteristic, complemented, Sylow

2₋¹⁺⁴

—

120

—

2₋¹⁺⁴

—

2₋¹⁺⁴

—

2₋¹⁺⁴

C₂×D₆₁

yes

—

—

C₂×D₆₁

yes

complemented

C₂×#41

—

120

—

C₂×#41

—

C₂×#41

—

C₂×#41

C₂×D₆₁

yes

complemented

C₂²×Q₈

—

C₂²×Q₈

—

#41

—

#41

—

360

—

#41

—

180

—

#41

—

#41

C₂²×D₆₁

yes

—

#41

C₂²×D₆₁

yes

—

C₂×Q₈

—

120

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²×D₆₁

yes

—

C₂×D₄

—

C₂×D₄

—

120

—

C₂×D₄

—

C₂×D₄

C₂²×D₆₁

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₆₁

yes

—

C₂³

—

C₂³

C₂³×D₆₁

yes

—

Q₈

—

Q₈

—

100

Q₈

C₂³×D₆₁

yes

—

101

D₄

—

120

—

102

D₄

—

103

D₄

—

104

D₄

C₂³×D₆₁

yes

—

105

C₂×C₄

—

106

C₂×C₄

—

120

—

107

C₂×C₄

C₂³×D₆₁

yes

—

108

C₂×C₄

C₂³×D₆₁

yes

—

109

C₂²

—

110

C₂²

—

111

C₂²

—

244

—

112

C₂²

C₂⁴×D₆₁

yes

—

113

C₄

—

114

C₄

C₂⁴×D₆₁

yes

—

115

C₄

C₂⁴×D₆₁

yes

characteristic

116

C₂

—

117

C₂

—

122

—

118

C₂

C₂⁵×D₆₁

yes

characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁵×D₆₁
commutator, socle	C₂×C₆₁	C₂⁶
Fitting	C₆₁×#585	C₂
hypercenter	#585	D₆₁
hypocenter	C₆₁	2₋¹⁺⁶
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ #585

subgroup

G/H

center, Frattini

C₂

C₂⁵×D₆₁

commutator, socle

C₂×C₆₁

C₂⁶

Fitting

C₆₁×#585

C₂

hypercenter

#585

D₆₁

hypocenter

C₆₁

2₋¹⁺⁶

LCS

G ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂ ◁ #585

N	H ≅ G/N	type
1. #16840598	C₂	semidirect
2. #16840597	C₂	semidirect
3. C₂×#14519824	C₂	semidirect
4. 2₋¹⁺⁴×D₆₁	C₂	semidirect
5. C₆₁×#585	C₂	semidirect
6. #14519825	C₂²	semidirect
7. #14519824	C₂²	semidirect
8. #14519824	C₂²	semidirect
9. #14519823	C₂²	semidirect
10. #14519822	C₂²	semidirect
11. C₂×#11758222	C₂²	semidirect
12. C₂×#11758224	C₂²	semidirect
13. #41×D₆₁	C₂²	semidirect
14. C₂²×Dic₁₂₂	C₂²	semidirect
15. C₂×Q₈×D₆₁	C₂²	semidirect
16. 2₋¹⁺⁴×C₆₁	C₂²	semidirect
17. H(5,𝔽₂)×C₆₁	C₂²	semidirect
18. C₂×#41×C₆₁	C₂²	semidirect
62. #585	D₆₁	semidirect
65. C₆₁	2₋¹⁺⁶	semidirect
69. 2₋¹⁺⁴	C₂×D₆₁	semidirect
72. H(5,𝔽₂)	C₂×D₆₁	semidirect
76. C₂×#41	C₂×D₆₁	semidirect

H ≅ G/N

type

1. #16840598

C₂

semidirect

2. #16840597

C₂

semidirect

3. C₂×#14519824

C₂

semidirect

4. 2₋¹⁺⁴×D₆₁

C₂

semidirect

5. C₆₁×#585

C₂

semidirect

6. #14519825

C₂²

semidirect

7. #14519824

C₂²

semidirect

8. #14519824

C₂²

semidirect

9. #14519823

C₂²

semidirect

10. #14519822

C₂²

semidirect

11. C₂×#11758222

C₂²

semidirect

12. C₂×#11758224

C₂²

semidirect

13. #41×D₆₁

C₂²

semidirect

14. C₂²×Dic₁₂₂

C₂²

semidirect

15. C₂×Q₈×D₆₁

C₂²

semidirect

16. 2₋¹⁺⁴×C₆₁

C₂²

semidirect

17. H(5,𝔽₂)×C₆₁

C₂²

semidirect

18. C₂×#41×C₆₁

C₂²

semidirect

62. #585

D₆₁

semidirect

65. C₆₁

2₋¹⁺⁶

semidirect

69. 2₋¹⁺⁴

C₂×D₆₁

semidirect

72. H(5,𝔽₂)

C₂×D₆₁

semidirect

76. C₂×#41

C₂×D₆₁

semidirect

Subgroups — #21460632

Conjugacy classes of subgroups

Complements