Group #18068599 — subgroups

Subgroups — C₂⁴×C₄⋊C₄×C₁₉ #18068599

Order 4864.

19742 conjugacy classes of subgroups; only the 15454 normal subgroups are shown.

normal subgroups only

This group has 19742 conjugacy classes of subgroups — too many to draw. Only its 15454 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₄×C₁₉	C₂	1	2	yes	maximal
2	C₂⁵×C₄×C₁₉	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₄⋊C₄×C₁₉	C₂	1	60	yes	maximal, direct product factor
4	C₂⁶×C₁₉	C₂²	1	1	yes	characteristic
5	C₂⁴×C₄×C₁₉	—	2	32	no	—
6	C₂⁴×C₄×C₁₉	C₄	1	32	yes	complemented
7	C₂⁴×C₄×C₁₉	C₂²	1	60	yes	—
8	C₂⁴×C₄×C₁₉	C₂²	1	30	yes	—
9	C₂²×C₄⋊C₄×C₁₉	C₂²	1	560	yes	direct product factor
10	C₂⁵×C₁₉	C₂×C₄	1	16	yes	—
11	C₂⁵×C₁₉	D₄	1	16	yes	—
12	C₂⁵×C₁₉	Q₈	1	16	yes	—
13	C₂⁵×C₁₉	C₂³	1	15	yes	—
14	C₂³×C₄×C₁₉	—	2	480	no	—
15	C₂³×C₄×C₁₉	C₂×C₄	1	480	yes	complemented
16	C₂³×C₄×C₁₉	C₂³	1	280	yes	—
17	C₂³×C₄×C₁₉	C₂³	1	140	yes	—
18	C₂×C₄⋊C₄×C₁₉	C₂³	1	960	yes	direct product factor
19	C₂⁴×C₁₉	C₄⋊C₄	1	256	yes	direct product factor
20	C₂⁴×C₁₉	C₂²×C₄	1	120	yes	—
21	C₂⁴×C₁₉	C₂×D₄	1	120	yes	—
22	C₂⁴×C₁₉	C₂×Q₈	1	120	yes	—
23	C₂⁴×C₁₉	C₂⁴	1	35	yes	—
24	C₂²×C₄×C₁₉	—	2	1120	no	—
25	C₂²×C₄×C₁₉	C₂²×C₄	1	1120	yes	complemented
26	C₂²×C₄×C₁₉	C₂⁴	1	240	yes	—
27	C₂²×C₄×C₁₉	C₂⁴	1	120	yes	—
28	C₄⋊C₄×C₁₉	C₂⁴	1	256	yes	direct product factor
29	C₂⁴×C₄⋊C₄	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂³×C₁₉	C₂×C₄⋊C₄	1	960	yes	direct product factor
31	C₂³×C₁₉	C₂³×C₄	1	140	yes	—
32	C₂³×C₁₉	C₂²×D₄	1	140	yes	—
33	C₂³×C₁₉	C₂²×Q₈	1	140	yes	—
34	C₂³×C₁₉	C₂⁵	1	15	yes	—
35	C₂×C₄×C₁₉	—	2	480	no	—
36	C₂×C₄×C₁₉	C₂³×C₄	1	480	yes	complemented
37	C₂×C₄×C₁₉	C₂⁵	1	32	yes	—
38	C₂×C₄×C₁₉	C₂⁵	1	16	yes	—
39	C₂⁵×C₄	C₂×C₁₉	1	2	yes	—
40	C₂⁵×C₄	C₂×C₁₉	1	1	yes	characteristic
41	C₂³×C₄⋊C₄	C₂×C₁₉	1	60	yes	direct product factor
42	C₂²×C₁₉	C₂²×C₄⋊C₄	1	560	yes	direct product factor
43	C₂²×C₁₉	C₂⁴×C₄	1	30	yes	—
44	C₂²×C₁₉	C₂³×D₄	1	30	yes	—
45	C₂²×C₁₉	C₂³×Q₈	1	30	yes	—
46	C₂²×C₁₉	C₂⁶	1	1	yes	characteristic
47	C₄×C₁₉	—	2	32	no	—
48	C₄×C₁₉	C₂⁴×C₄	1	32	yes	complemented
49	C₂⁶	C₂²×C₁₉	1	1	yes	characteristic
50	C₂⁴×C₄	—	2	32	no	—
51	C₂⁴×C₄	C₄×C₁₉	1	32	yes	complemented
52	C₂⁴×C₄	C₂²×C₁₉	1	60	yes	—
53	C₂⁴×C₄	C₂²×C₁₉	1	30	yes	—
54	C₂²×C₄⋊C₄	C₂²×C₁₉	1	560	yes	direct product factor
55	C₂×C₁₉	C₂³×C₄⋊C₄	1	60	yes	direct product factor
56	C₂×C₁₉	C₂⁵×C₄	1	1	yes	characteristic
57	C₂×C₁₉	C₂⁴×D₄	1	1	yes	characteristic
58	C₂×C₁₉	C₂⁴×Q₈	1	1	yes	characteristic
59	C₂⁵	C₂×C₄×C₁₉	1	16	yes	—
60	C₂⁵	D₄×C₁₉	1	16	yes	—
61	C₂⁵	Q₈×C₁₉	1	16	yes	—
62	C₂⁵	C₂³×C₁₉	1	15	yes	—
63	C₂³×C₄	—	2	480	no	—
64	C₂³×C₄	C₂×C₄×C₁₉	1	480	yes	complemented
65	C₂³×C₄	C₂³×C₁₉	1	280	yes	—
66	C₂³×C₄	C₂³×C₁₉	1	140	yes	—
67	C₂×C₄⋊C₄	C₂³×C₁₉	1	960	yes	direct product factor
68	C₁₉	C₂⁴×C₄⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
69	C₂⁴	C₄⋊C₄×C₁₉	1	256	yes	direct product factor
70	C₂⁴	C₂²×C₄×C₁₉	1	120	yes	—
71	C₂⁴	C₂×D₄×C₁₉	1	120	yes	—
72	C₂⁴	C₂×Q₈×C₁₉	1	120	yes	—
73	C₂⁴	C₂⁴×C₁₉	1	35	yes	—
74	C₂²×C₄	—	2	1120	no	—
75	C₂²×C₄	C₂²×C₄×C₁₉	1	1120	yes	complemented
76	C₂²×C₄	C₂⁴×C₁₉	1	240	yes	—
77	C₂²×C₄	C₂⁴×C₁₉	1	120	yes	—
78	C₄⋊C₄	C₂⁴×C₁₉	1	256	yes	direct product factor
79	C₂³	C₂×C₄⋊C₄×C₁₉	1	960	yes	direct product factor
80	C₂³	C₂³×C₄×C₁₉	1	140	yes	—
81	C₂³	C₂²×D₄×C₁₉	1	140	yes	—
82	C₂³	C₂²×Q₈×C₁₉	1	140	yes	—
83	C₂³	C₂⁵×C₁₉	1	15	yes	—
84	C₂×C₄	—	2	480	no	—
85	C₂×C₄	C₂³×C₄×C₁₉	1	480	yes	complemented
86	C₂×C₄	C₂⁵×C₁₉	1	32	yes	—
87	C₂×C₄	C₂⁵×C₁₉	1	16	yes	—
88	C₂²	C₂²×C₄⋊C₄×C₁₉	1	560	yes	direct product factor
89	C₂²	C₂⁴×C₄×C₁₉	1	30	yes	—
90	C₂²	C₂³×D₄×C₁₉	1	30	yes	—
91	C₂²	C₂³×Q₈×C₁₉	1	30	yes	—
92	C₂²	C₂⁶×C₁₉	1	1	yes	characteristic
93	C₄	—	2	32	no	—
94	C₄	C₂⁴×C₄×C₁₉	1	32	yes	complemented
95	C₂	C₂³×C₄⋊C₄×C₁₉	1	60	yes	direct product factor
96	C₂	C₂⁵×C₄×C₁₉	1	1	yes	characteristic
97	C₂	C₂⁴×D₄×C₁₉	1	1	yes	characteristic
98	C₂	C₂⁴×Q₈×C₁₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂⁵×C₄×C₁₉

C₂

yes

maximal

C₂⁵×C₄×C₁₉

C₂

yes

maximal, characteristic

C₂³×C₄⋊C₄×C₁₉

C₂

yes

maximal, direct product factor

C₂⁶×C₁₉

C₂²

yes

characteristic

C₂⁴×C₄×C₁₉

—

C₂⁴×C₄×C₁₉

C₄

yes

complemented

C₂⁴×C₄×C₁₉

C₂²

yes

—

C₂⁴×C₄×C₁₉

C₂²

yes

—

C₂²×C₄⋊C₄×C₁₉

C₂²

560

yes

direct product factor

C₂⁵×C₁₉

C₂×C₄

yes

—

C₂⁵×C₁₉

D₄

yes

—

C₂⁵×C₁₉

Q₈

yes

—

C₂⁵×C₁₉

C₂³

yes

—

C₂³×C₄×C₁₉

—

480

—

C₂³×C₄×C₁₉

C₂×C₄

480

yes

complemented

C₂³×C₄×C₁₉

C₂³

280

yes

—

C₂³×C₄×C₁₉

C₂³

140

yes

—

C₂×C₄⋊C₄×C₁₉

C₂³

960

yes

direct product factor

C₂⁴×C₁₉

C₄⋊C₄

256

yes

direct product factor

C₂⁴×C₁₉

C₂²×C₄

120

yes

—

C₂⁴×C₁₉

C₂×D₄

120

yes

—

C₂⁴×C₁₉

C₂×Q₈

120

yes

—

C₂⁴×C₁₉

C₂⁴

yes

—

C₂²×C₄×C₁₉

—

1120

—

C₂²×C₄×C₁₉

C₂²×C₄

1120

yes

complemented

C₂²×C₄×C₁₉

C₂⁴

240

yes

—

C₂²×C₄×C₁₉

C₂⁴

120

yes

—

C₄⋊C₄×C₁₉

C₂⁴

256

yes

direct product factor

C₂⁴×C₄⋊C₄

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₉

C₂×C₄⋊C₄

960

yes

direct product factor

C₂³×C₁₉

C₂³×C₄

140

yes

—

C₂³×C₁₉

C₂²×D₄

140

yes

—

C₂³×C₁₉

C₂²×Q₈

140

yes

—

C₂³×C₁₉

C₂⁵

yes

—

C₂×C₄×C₁₉

—

480

—

C₂×C₄×C₁₉

C₂³×C₄

480

yes

complemented

C₂×C₄×C₁₉

C₂⁵

yes

—

C₂×C₄×C₁₉

C₂⁵

yes

—

C₂⁵×C₄

C₂×C₁₉

yes

—

C₂⁵×C₄

C₂×C₁₉

yes

characteristic

C₂³×C₄⋊C₄

C₂×C₁₉

yes

direct product factor

C₂²×C₁₉

C₂²×C₄⋊C₄

560

yes

direct product factor

C₂²×C₁₉

C₂⁴×C₄

yes

—

C₂²×C₁₉

C₂³×D₄

yes

—

C₂²×C₁₉

C₂³×Q₈

yes

—

C₂²×C₁₉

C₂⁶

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

C₂⁴×C₄

yes

complemented

C₂⁶

C₂²×C₁₉

yes

characteristic

C₂⁴×C₄

—

C₂⁴×C₄

C₄×C₁₉

yes

complemented

C₂⁴×C₄

C₂²×C₁₉

yes

—

C₂⁴×C₄

C₂²×C₁₉

yes

—

C₂²×C₄⋊C₄

C₂²×C₁₉

560

yes

direct product factor

C₂×C₁₉

C₂³×C₄⋊C₄

yes

direct product factor

C₂×C₁₉

C₂⁵×C₄

yes

characteristic

C₂×C₁₉

C₂⁴×D₄

yes

characteristic

C₂×C₁₉

C₂⁴×Q₈

yes

characteristic

C₂⁵

C₂×C₄×C₁₉

yes

—

C₂⁵

D₄×C₁₉

yes

—

C₂⁵

Q₈×C₁₉

yes

—

C₂⁵

C₂³×C₁₉

yes

—

C₂³×C₄

—

480

—

C₂³×C₄

C₂×C₄×C₁₉

480

yes

complemented

C₂³×C₄

C₂³×C₁₉

280

yes

—

C₂³×C₄

C₂³×C₁₉

140

yes

—

C₂×C₄⋊C₄

C₂³×C₁₉

960

yes

direct product factor

C₁₉

C₂⁴×C₄⋊C₄

yes

characteristic, direct product factor, Sylow

C₂⁴

C₄⋊C₄×C₁₉

256

yes

direct product factor

C₂⁴

C₂²×C₄×C₁₉

120

yes

—

C₂⁴

C₂×D₄×C₁₉

120

yes

—

C₂⁴

C₂×Q₈×C₁₉

120

yes

—

C₂⁴

C₂⁴×C₁₉

yes

—

C₂²×C₄

—

1120

—

C₂²×C₄

C₂²×C₄×C₁₉

1120

yes

complemented

C₂²×C₄

C₂⁴×C₁₉

240

yes

—

C₂²×C₄

C₂⁴×C₁₉

120

yes

—

C₄⋊C₄

C₂⁴×C₁₉

256

yes

direct product factor

C₂³

C₂×C₄⋊C₄×C₁₉

960

yes

direct product factor

C₂³

C₂³×C₄×C₁₉

140

yes

—

C₂³

C₂²×D₄×C₁₉

140

yes

—

C₂³

C₂²×Q₈×C₁₉

140

yes

—

C₂³

C₂⁵×C₁₉

yes

—

C₂×C₄

—

480

—

C₂×C₄

C₂³×C₄×C₁₉

480

yes

complemented

C₂×C₄

C₂⁵×C₁₉

yes

—

C₂×C₄

C₂⁵×C₁₉

yes

—

C₂²

C₂²×C₄⋊C₄×C₁₉

560

yes

direct product factor

C₂²

C₂⁴×C₄×C₁₉

yes

—

C₂²

C₂³×D₄×C₁₉

yes

—

C₂²

C₂³×Q₈×C₁₉

yes

—

C₂²

C₂⁶×C₁₉

yes

characteristic

C₄

—

C₄

C₂⁴×C₄×C₁₉

yes

complemented

C₂

C₂³×C₄⋊C₄×C₁₉

yes

direct product factor

C₂

C₂⁵×C₄×C₁₉

yes

characteristic

C₂

C₂⁴×D₄×C₁₉

yes

characteristic

C₂

C₂⁴×Q₈×C₁₉

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁶×C₁₉	C₂²
commutator	C₂	C₂⁵×C₄×C₁₉
Frattini, cosocle	C₂²	C₂⁶×C₁₉
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁶×C₁₉ ◁ G

subgroup

G/H

center, socle, prime radical

C₂⁶×C₁₉

C₂²

commutator

C₂

C₂⁵×C₄×C₁₉

Frattini, cosocle

C₂²

C₂⁶×C₁₉

LCS

G ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂⁶×C₁₉ ◁ G

Complements

N	H ≅ G/N	type
3. C₂³×C₄⋊C₄×C₁₉	C₂	direct (N × H)
6. C₂⁴×C₄×C₁₉	C₄	semidirect
9. C₂²×C₄⋊C₄×C₁₉	C₂²	direct (N × H)
15. C₂³×C₄×C₁₉	C₂×C₄	semidirect
18. C₂×C₄⋊C₄×C₁₉	C₂³	direct (N × H)
19. C₂⁴×C₁₉	C₄⋊C₄	direct (N × H)
25. C₂²×C₄×C₁₉	C₂²×C₄	semidirect
28. C₄⋊C₄×C₁₉	C₂⁴	direct (N × H)
29. C₂⁴×C₄⋊C₄	C₁₉	direct (N × H)
30. C₂³×C₁₉	C₂×C₄⋊C₄	direct (N × H)
36. C₂×C₄×C₁₉	C₂³×C₄	semidirect
41. C₂³×C₄⋊C₄	C₂×C₁₉	direct (N × H)
42. C₂²×C₁₉	C₂²×C₄⋊C₄	direct (N × H)
48. C₄×C₁₉	C₂⁴×C₄	semidirect
51. C₂⁴×C₄	C₄×C₁₉	semidirect
54. C₂²×C₄⋊C₄	C₂²×C₁₉	direct (N × H)
55. C₂×C₁₉	C₂³×C₄⋊C₄	direct (N × H)
64. C₂³×C₄	C₂×C₄×C₁₉	semidirect
67. C₂×C₄⋊C₄	C₂³×C₁₉	direct (N × H)
68. C₁₉	C₂⁴×C₄⋊C₄	direct (N × H)
69. C₂⁴	C₄⋊C₄×C₁₉	direct (N × H)
75. C₂²×C₄	C₂²×C₄×C₁₉	semidirect
78. C₄⋊C₄	C₂⁴×C₁₉	direct (N × H)
79. C₂³	C₂×C₄⋊C₄×C₁₉	direct (N × H)
85. C₂×C₄	C₂³×C₄×C₁₉	semidirect
88. C₂²	C₂²×C₄⋊C₄×C₁₉	direct (N × H)
94. C₄	C₂⁴×C₄×C₁₉	semidirect
95. C₂	C₂³×C₄⋊C₄×C₁₉	direct (N × H)

H ≅ G/N

type

3. C₂³×C₄⋊C₄×C₁₉

C₂

direct (N × H)

6. C₂⁴×C₄×C₁₉

C₄

semidirect

9. C₂²×C₄⋊C₄×C₁₉

C₂²

direct (N × H)

15. C₂³×C₄×C₁₉

C₂×C₄

semidirect

18. C₂×C₄⋊C₄×C₁₉

C₂³

direct (N × H)

19. C₂⁴×C₁₉

C₄⋊C₄

direct (N × H)

25. C₂²×C₄×C₁₉

C₂²×C₄

semidirect

28. C₄⋊C₄×C₁₉

C₂⁴

direct (N × H)

29. C₂⁴×C₄⋊C₄

C₁₉

direct (N × H)

30. C₂³×C₁₉

C₂×C₄⋊C₄

direct (N × H)

36. C₂×C₄×C₁₉

C₂³×C₄

semidirect

41. C₂³×C₄⋊C₄

C₂×C₁₉

direct (N × H)

42. C₂²×C₁₉

C₂²×C₄⋊C₄

direct (N × H)

48. C₄×C₁₉

C₂⁴×C₄

semidirect

51. C₂⁴×C₄

C₄×C₁₉

semidirect

54. C₂²×C₄⋊C₄

C₂²×C₁₉

direct (N × H)

55. C₂×C₁₉

C₂³×C₄⋊C₄

direct (N × H)

64. C₂³×C₄

C₂×C₄×C₁₉

semidirect

67. C₂×C₄⋊C₄

C₂³×C₁₉

direct (N × H)

68. C₁₉

C₂⁴×C₄⋊C₄

direct (N × H)

69. C₂⁴

C₄⋊C₄×C₁₉

direct (N × H)

75. C₂²×C₄

C₂²×C₄×C₁₉

semidirect

78. C₄⋊C₄

C₂⁴×C₁₉

direct (N × H)

79. C₂³

C₂×C₄⋊C₄×C₁₉

direct (N × H)

85. C₂×C₄

C₂³×C₄×C₁₉

semidirect

88. C₂²

C₂²×C₄⋊C₄×C₁₉

direct (N × H)

94. C₄

C₂⁴×C₄×C₁₉

semidirect

95. C₂

C₂³×C₄⋊C₄×C₁₉

direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×C₄×C₁₉	2
C₂³×C₄⋊C₄×C₁₉	2
C₂⁴×C₄⋊C₄	19

subgroup

index

C₂⁵×C₄×C₁₉

C₂³×C₄⋊C₄×C₁₉

C₂⁴×C₄⋊C₄

Subgroups — C24×C4⋊C4×C19 #18068599

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₄⋊C₄×C₁₉ #18068599