Group #30626736 — subgroups

Subgroups — C₂⁴×D₆₁⋊C₈ #30626736

Order 15616.

15454 conjugacy classes of subgroups; only the 8408 normal subgroups are shown.

normal subgroups only

This group has 15454 conjugacy classes of subgroups — too many to draw. Only its 8408 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	C₂³×D₆₁⋊C₈	C₂	1	60	yes	maximal, direct product factor
2	C₂⁴×#92420	C₂	1	2	yes	maximal, complemented
3	C₂⁴×C₄×D₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×D₆₁⋊C₈	C₂²	1	560	yes	direct product factor
5	C₂³×#92420	C₂²	1	60	yes	complemented
6	C₂⁵×D₆₁	C₄	1	1	yes	characteristic
7	C₂⁴×Dic₆₁	C₂²	1	1	yes	characteristic
8	C₂³×C₄×D₆₁	C₄	1	30	yes	—
9	C₂³×C₄×D₆₁	C₂²	1	30	yes	—
10	C₂⁴×C₄×C₆₁	C₄	1	1	yes	characteristic
11	C₂×D₆₁⋊C₈	C₂³	1	960	yes	direct product factor
12	C₂²×#92420	C₂³	1	280	yes	complemented
13	C₂⁴×D₆₁	C₈	1	32	yes	complemented
14	C₂⁴×D₆₁	C₂×C₄	1	30	yes	—
15	C₂³×Dic₆₁	C₂×C₄	1	15	yes	—
16	C₂³×Dic₆₁	C₂³	1	15	yes	—
17	C₂²×C₄×D₆₁	C₂×C₄	1	420	yes	—
18	C₂²×C₄×D₆₁	C₂³	1	140	yes	—
19	C₂⁵×C₆₁	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₆₁	C₂×C₄	1	30	yes	—
21	C₂⁴×C₆₁	C₂×C₈	1	16	yes	complemented
22	C₂⁴×C₆₁	C₂²×C₄	1	15	yes	—
23	C₂³×D₆₁	C₂×C₈	1	480	yes	complemented
24	C₂³×D₆₁	C₂²×C₄	1	140	yes	—
25	C₂²×C₄×C₆₁	C₂²×C₄	1	140	yes	—
26	C₂²×Dic₆₁	C₂²×C₄	1	105	yes	—
27	C₂²×Dic₆₁	C₂⁴	1	35	yes	—
28	C₂×C₄×D₆₁	C₂²×C₄	1	840	yes	—
29	C₂×C₄×D₆₁	C₂⁴	1	120	yes	—
30	C₂×#92420	C₂⁴	1	240	yes	complemented
31	D₆₁⋊C₈	C₂⁴	1	256	yes	direct product factor
32	C₂³×C₆₁	C₂²×C₈	1	120	yes	complemented
33	C₂³×C₆₁	C₂³×C₄	1	35	yes	—
34	C₂²×D₆₁	C₂²×C₈	1	1120	yes	complemented
35	C₂²×D₆₁	C₂³×C₄	1	120	yes	—
36	C₂×C₄×C₆₁	C₂³×C₄	1	120	yes	—
37	C₂×Dic₆₁	C₂³×C₄	1	105	yes	—
38	C₂×Dic₆₁	C₂⁵	1	15	yes	—
39	C₄×D₆₁	C₂³×C₄	1	240	yes	—
40	C₄×D₆₁	C₂⁵	1	16	yes	—
41	#92420	C₂⁵	1	32	yes	complemented
42	C₂⁵×C₈	—	61	1	no	maximal, Sylow
43	C₂²×C₆₁	C₂³×C₈	1	140	yes	complemented
44	C₂²×C₆₁	C₂⁴×C₄	1	15	yes	—
45	C₂×D₆₁	C₂³×C₈	1	480	yes	complemented
46	C₂×D₆₁	C₂⁴×C₄	1	16	yes	—
47	C₄×C₆₁	C₂⁴×C₄	1	16	yes	—
48	Dic₆₁	C₂⁴×C₄	1	15	yes	—
49	Dic₆₁	C₂⁶	1	1	yes	characteristic
50	C₂⁵×C₄	—	61	1	no	—
51	C₂⁴×C₈	—	61	2	no	—
52	C₂⁴×C₈	—	61	60	no	—
53	C₂×C₆₁	C₂⁴×C₈	1	30	yes	complemented
54	C₂×C₆₁	C₂⁵×C₄	1	1	yes	characteristic
55	D₆₁	C₂⁴×C₈	1	32	yes	complemented
56	C₂⁶	—	61	1	no	—
57	C₂⁴×C₄	—	61	1	no	—
58	C₂⁴×C₄	—	61	30	no	—
59	C₂⁴×C₄	—	61	30	no	—
60	C₂⁴×C₄	#6923	1	1	yes	characteristic
61	C₂³×C₈	—	61	60	no	—
62	C₂³×C₈	—	61	560	no	—
63	C₆₁	C₂⁵×C₈	1	1	yes	characteristic, complemented, Sylow
64	C₂⁵	—	61	32	no	—
65	C₂⁵	—	61	30	no	—
66	C₂⁵	C₂×#6923	1	1	yes	characteristic
67	C₂³×C₄	—	61	15	no	—
68	C₂³×C₄	—	61	15	no	—
69	C₂³×C₄	—	61	420	no	—
70	C₂³×C₄	—	61	140	no	—
71	C₂³×C₄	C₂×#6923	1	30	yes	—
72	C₂²×C₈	—	61	280	no	—
73	C₂²×C₈	—	61	960	no	—
74	C₂⁴	—	61	480	no	—
75	C₂⁴	—	61	140	no	—
76	C₂⁴	D₆₁⋊C₈	1	16	yes	direct product factor
77	C₂⁴	C₂²×#6923	1	15	yes	—
78	C₂²×C₄	—	61	35	no	—
79	C₂²×C₄	—	61	105	no	—
80	C₂²×C₄	—	61	120	no	—
81	C₂²×C₄	—	61	840	no	—
82	C₂²×C₄	C₂²×#6923	1	140	yes	—
83	C₂×C₈	—	61	256	no	—
84	C₂×C₈	—	61	240	no	—
85	C₂³	—	61	1120	no	—
86	C₂³	—	61	120	no	—
87	C₂³	C₂×D₆₁⋊C₈	1	120	yes	direct product factor
88	C₂³	C₂³×#6923	1	35	yes	—
89	C₂×C₄	—	61	105	no	—
90	C₂×C₄	—	61	15	no	—
91	C₂×C₄	—	61	240	no	—
92	C₂×C₄	—	61	16	no	—
93	C₂×C₄	C₂³×#6923	1	120	yes	—
94	C₈	—	61	32	no	—
95	C₂²	—	61	480	no	—
96	C₂²	—	61	16	no	—
97	C₂²	C₂²×D₆₁⋊C₈	1	140	yes	direct product factor
98	C₂²	C₂⁴×#6923	1	15	yes	—
99	C₄	—	61	15	no	—
100	C₄	—	61	1	no	—
101	C₄	C₂⁴×#6923	1	16	yes	—
102	C₂	—	61	32	no	—
103	C₂	C₂³×D₆₁⋊C₈	1	30	yes	direct product factor
104	C₂	C₂⁵×#6923	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×D₆₁⋊C₈

C₂

yes

maximal, direct product factor

C₂⁴×#92420

C₂

yes

maximal, complemented

C₂⁴×C₄×D₆₁

C₂

yes

maximal, characteristic

C₂²×D₆₁⋊C₈

C₂²

560

yes

direct product factor

C₂³×#92420

C₂²

yes

complemented

C₂⁵×D₆₁

C₄

yes

characteristic

C₂⁴×Dic₆₁

C₂²

yes

characteristic

C₂³×C₄×D₆₁

C₄

yes

—

C₂³×C₄×D₆₁

C₂²

yes

—

C₂⁴×C₄×C₆₁

C₄

yes

characteristic

C₂×D₆₁⋊C₈

C₂³

960

yes

direct product factor

C₂²×#92420

C₂³

280

yes

complemented

C₂⁴×D₆₁

C₈

yes

complemented

C₂⁴×D₆₁

C₂×C₄

yes

—

C₂³×Dic₆₁

C₂×C₄

yes

—

C₂³×Dic₆₁

C₂³

yes

—

C₂²×C₄×D₆₁

C₂×C₄

420

yes

—

C₂²×C₄×D₆₁

C₂³

140

yes

—

C₂⁵×C₆₁

C₂×C₄

yes

characteristic

C₂³×C₄×C₆₁

C₂×C₄

yes

—

C₂⁴×C₆₁

C₂×C₈

yes

complemented

C₂⁴×C₆₁

C₂²×C₄

yes

—

C₂³×D₆₁

C₂×C₈

480

yes

complemented

C₂³×D₆₁

C₂²×C₄

140

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

140

yes

—

C₂²×Dic₆₁

C₂²×C₄

105

yes

—

C₂²×Dic₆₁

C₂⁴

yes

—

C₂×C₄×D₆₁

C₂²×C₄

840

yes

—

C₂×C₄×D₆₁

C₂⁴

120

yes

—

C₂×#92420

C₂⁴

240

yes

complemented

D₆₁⋊C₈

C₂⁴

256

yes

direct product factor

C₂³×C₆₁

C₂²×C₈

120

yes

complemented

C₂³×C₆₁

C₂³×C₄

yes

—

C₂²×D₆₁

C₂²×C₈

1120

yes

complemented

C₂²×D₆₁

C₂³×C₄

120

yes

—

C₂×C₄×C₆₁

C₂³×C₄

120

yes

—

C₂×Dic₆₁

C₂³×C₄

105

yes

—

C₂×Dic₆₁

C₂⁵

yes

—

C₄×D₆₁

C₂³×C₄

240

yes

—

C₄×D₆₁

C₂⁵

yes

—

#92420

C₂⁵

yes

complemented

C₂⁵×C₈

—

maximal, Sylow

C₂²×C₆₁

C₂³×C₈

140

yes

complemented

C₂²×C₆₁

C₂⁴×C₄

yes

—

C₂×D₆₁

C₂³×C₈

480

yes

complemented

C₂×D₆₁

C₂⁴×C₄

yes

—

C₄×C₆₁

C₂⁴×C₄

yes

—

Dic₆₁

C₂⁴×C₄

yes

—

Dic₆₁

C₂⁶

yes

characteristic

C₂⁵×C₄

—

C₂⁴×C₈

—

C₂⁴×C₈

—

C₂×C₆₁

C₂⁴×C₈

yes

complemented

C₂×C₆₁

C₂⁵×C₄

yes

characteristic

D₆₁

C₂⁴×C₈

yes

complemented

C₂⁶

—

C₂⁴×C₄

—

C₂⁴×C₄

—

C₂⁴×C₄

—

C₂⁴×C₄

#6923

yes

characteristic

C₂³×C₈

—

C₂³×C₈

—

560

—

C₆₁

C₂⁵×C₈

yes

characteristic, complemented, Sylow

C₂⁵

—

C₂⁵

—

C₂⁵

C₂×#6923

yes

characteristic

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

—

420

—

C₂³×C₄

—

140

—

C₂³×C₄

C₂×#6923

yes

—

C₂²×C₈

—

280

—

C₂²×C₈

—

960

—

C₂⁴

—

480

—

C₂⁴

—

140

—

C₂⁴

D₆₁⋊C₈

yes

direct product factor

C₂⁴

C₂²×#6923

yes

—

C₂²×C₄

—

C₂²×C₄

—

105

—

C₂²×C₄

—

120

—

C₂²×C₄

—

840

—

C₂²×C₄

C₂²×#6923

140

yes

—

C₂×C₈

—

256

—

C₂×C₈

—

240

—

C₂³

—

1120

—

C₂³

—

120

—

C₂³

C₂×D₆₁⋊C₈

120

yes

direct product factor

C₂³

C₂³×#6923

yes

—

C₂×C₄

—

105

—

C₂×C₄

—

C₂×C₄

—

240

—

C₂×C₄

—

C₂×C₄

C₂³×#6923

120

yes

—

C₈

—

C₂²

—

480

—

C₂²

—

C₂²

C₂²×D₆₁⋊C₈

140

yes

direct product factor

C₂²

C₂⁴×#6923

yes

—

C₄

—

100

C₄

—

101

C₄

C₂⁴×#6923

yes

—

102

C₂

—

103

C₂

C₂³×D₆₁⋊C₈

yes

direct product factor

104

C₂

C₂⁵×#6923

yes

characteristic

	subgroup	G/H
center, hypercenter	C₂⁴×C₄	#6923
commutator, hypocenter	C₆₁	C₂⁵×C₈
Frattini	C₂	C₂⁵×#6923
Fitting	C₂⁴×C₄×C₆₁	C₄
socle	C₂⁵×C₆₁	C₂×C₄
prime radical	C₂⁵×D₆₁	C₄
cosocle	Dic₆₁	C₂⁶
LCS	G ▷ C₆₁
UCS	C₁ ◁ C₂⁴×C₄

subgroup

G/H

center, hypercenter

C₂⁴×C₄

#6923

commutator, hypocenter

C₆₁

C₂⁵×C₈

Frattini

C₂

C₂⁵×#6923

Fitting

C₂⁴×C₄×C₆₁

C₄

socle

C₂⁵×C₆₁

C₂×C₄

prime radical

C₂⁵×D₆₁

C₄

cosocle

Dic₆₁

C₂⁶

LCS

G ▷ C₆₁

UCS

C₁ ◁ C₂⁴×C₄

N	H ≅ G/N	type
1. C₂³×D₆₁⋊C₈	C₂	direct (N × H)
2. C₂⁴×#92420	C₂	semidirect
4. C₂²×D₆₁⋊C₈	C₂²	direct (N × H)
5. C₂³×#92420	C₂²	semidirect
11. C₂×D₆₁⋊C₈	C₂³	direct (N × H)
12. C₂²×#92420	C₂³	semidirect
13. C₂⁴×D₆₁	C₈	semidirect
21. C₂⁴×C₆₁	C₂×C₈	semidirect
23. C₂³×D₆₁	C₂×C₈	semidirect
30. C₂×#92420	C₂⁴	semidirect
31. D₆₁⋊C₈	C₂⁴	direct (N × H)
32. C₂³×C₆₁	C₂²×C₈	semidirect
34. C₂²×D₆₁	C₂²×C₈	semidirect
41. #92420	C₂⁵	semidirect
43. C₂²×C₆₁	C₂³×C₈	semidirect
45. C₂×D₆₁	C₂³×C₈	semidirect
53. C₂×C₆₁	C₂⁴×C₈	semidirect
55. D₆₁	C₂⁴×C₈	semidirect
63. C₆₁	C₂⁵×C₈	semidirect
76. C₂⁴	D₆₁⋊C₈	direct (N × H)
87. C₂³	C₂×D₆₁⋊C₈	direct (N × H)
97. C₂²	C₂²×D₆₁⋊C₈	direct (N × H)
103. C₂	C₂³×D₆₁⋊C₈	direct (N × H)

H ≅ G/N

type

1. C₂³×D₆₁⋊C₈

C₂

direct (N × H)

2. C₂⁴×#92420

C₂

semidirect

4. C₂²×D₆₁⋊C₈

C₂²

direct (N × H)

5. C₂³×#92420

C₂²

semidirect

11. C₂×D₆₁⋊C₈

C₂³

direct (N × H)

12. C₂²×#92420

C₂³

semidirect

13. C₂⁴×D₆₁

C₈

semidirect

21. C₂⁴×C₆₁

C₂×C₈

semidirect

23. C₂³×D₆₁

C₂×C₈

semidirect

30. C₂×#92420

C₂⁴

semidirect

31. D₆₁⋊C₈

C₂⁴

direct (N × H)

32. C₂³×C₆₁

C₂²×C₈

semidirect

34. C₂²×D₆₁

C₂²×C₈

semidirect

41. #92420

C₂⁵

semidirect

43. C₂²×C₆₁

C₂³×C₈

semidirect

45. C₂×D₆₁

C₂³×C₈

semidirect

53. C₂×C₆₁

C₂⁴×C₈

semidirect

55. D₆₁

C₂⁴×C₈

semidirect

63. C₆₁

C₂⁵×C₈

semidirect

76. C₂⁴

D₆₁⋊C₈

direct (N × H)

87. C₂³

C₂×D₆₁⋊C₈

direct (N × H)

97. C₂²

C₂²×D₆₁⋊C₈

direct (N × H)

103. C₂

C₂³×D₆₁⋊C₈

direct (N × H)

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

subgroup	index
C₂³×D₆₁⋊C₈	2
C₂⁴×#92420	2
C₂⁴×C₄×D₆₁	2
C₂⁵×C₈	61

subgroup

index

C₂³×D₆₁⋊C₈

C₂⁴×#92420

C₂⁴×C₄×D₆₁

C₂⁵×C₈

Subgroups — C₂⁴×D₆₁⋊C₈ #30626736

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C24×D61⋊C8 #30626736

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×D₆₁⋊C₈ #30626736