Group #19210251 — subgroups

Subgroups — C₂³×C₂₃×D₁₆ #19210251

Order 5888.

2590 conjugacy classes of subgroups; only the 1054 normal subgroups are shown.

normal subgroups only

This group has 2590 conjugacy classes of subgroups — too many to draw. Only its 1054 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×D₈×C₂₃	C₂	1	2	yes	maximal, complemented
2	C₂²×C₂₃×D₁₆	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₄×C₂₃	—	2	2	no	—
5	C₂²×D₈×C₂₃	—	2	14	no	—
6	C₂²×D₈×C₂₃	C₂²	1	28	yes	complemented
7	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₂₃×D₁₆	C₂²	1	112	yes	direct product factor
9	C₂²×C₁₆×C₂₃	C₂²	1	14	yes	complemented
10	C₂⁵×C₂₃	—	4	2	no	—
11	C₂²×D₄×C₂₃	—	2	28	no	—
12	C₂²×D₄×C₂₃	—	4	14	no	—
13	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
14	C₂×D₈×C₂₃	—	2	84	no	—
15	C₂×D₈×C₂₃	C₂³	1	56	yes	complemented
16	C₂²×C₈×C₂₃	D₄	1	7	yes	—
17	C₂²×C₈×C₂₃	C₂³	1	7	yes	—
18	C₂₃×D₁₆	C₂³	1	64	yes	direct product factor
19	C₂×C₁₆×C₂₃	C₂³	1	28	yes	complemented
20	C₂⁴×C₂₃	—	4	28	no	—
21	C₂⁴×C₂₃	—	8	16	no	—
22	C₂⁴×C₂₃	D₈	1	1	yes	characteristic
23	C₂×D₄×C₂₃	—	2	56	no	—
24	C₂×D₄×C₂₃	—	4	84	no	—
25	C₂²×C₄×C₂₃	D₈	1	7	yes	—
26	C₂²×C₄×C₂₃	C₂×D₄	1	7	yes	—
27	D₈×C₂₃	—	2	56	no	—
28	D₈×C₂₃	C₂⁴	1	16	yes	complemented
29	C₂×C₈×C₂₃	C₂×D₄	1	21	yes	—
30	C₂×C₈×C₂₃	C₂⁴	1	7	yes	—
31	C₁₆×C₂₃	C₂⁴	1	8	yes	complemented
32	C₂³×D₁₆	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂³×C₂₃	—	4	56	no	—
34	C₂³×C₂₃	—	8	112	no	—
35	C₂³×C₂₃	D₁₆	1	8	yes	direct product factor
36	C₂³×C₂₃	C₂×D₈	1	7	yes	—
37	D₄×C₂₃	—	2	16	no	—
38	D₄×C₂₃	—	4	56	no	—
39	C₂×C₄×C₂₃	C₂×D₈	1	21	yes	—
40	C₂×C₄×C₂₃	C₂²×D₄	1	7	yes	—
41	C₈×C₂₃	C₂²×D₄	1	7	yes	—
42	C₈×C₂₃	C₂⁵	1	1	yes	characteristic
43	C₂³×D₈	C₂×C₂₃	1	2	yes	complemented
44	C₂²×D₁₆	C₂×C₂₃	1	28	yes	direct product factor
45	C₂³×C₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
46	C₂²×C₂₃	—	4	16	no	—
47	C₂²×C₂₃	—	8	112	no	—
48	C₂²×C₂₃	C₂×D₁₆	1	28	yes	direct product factor
49	C₂²×C₂₃	C₂²×D₈	1	7	yes	—
50	C₄×C₂₃	C₂²×D₈	1	7	yes	—
51	C₄×C₂₃	C₂³×D₄	1	1	yes	characteristic
52	C₂³×D₄	—	2	2	no	—
53	C₂²×D₈	—	2	14	no	—
54	C₂²×D₈	C₂²×C₂₃	1	28	yes	complemented
55	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
56	C₂×D₁₆	C₂²×C₂₃	1	112	yes	direct product factor
57	C₂²×C₁₆	C₂²×C₂₃	1	14	yes	complemented
58	C₂×C₂₃	—	8	16	no	—
59	C₂×C₂₃	C₂²×D₁₆	1	14	yes	direct product factor
60	C₂×C₂₃	C₂³×D₈	1	1	yes	characteristic
61	C₂⁵	—	4	2	no	—
62	C₂²×D₄	—	2	28	no	—
63	C₂²×D₄	—	4	14	no	—
64	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
65	C₂×D₈	—	2	84	no	—
66	C₂×D₈	C₂³×C₂₃	1	56	yes	complemented
67	C₂²×C₈	D₄×C₂₃	1	7	yes	—
68	C₂²×C₈	C₂³×C₂₃	1	7	yes	—
69	D₁₆	C₂³×C₂₃	1	64	yes	direct product factor
70	C₂×C₁₆	C₂³×C₂₃	1	28	yes	complemented
71	C₂₃	C₂³×D₁₆	1	1	yes	characteristic, direct product factor, Sylow
72	C₂⁴	—	4	28	no	—
73	C₂⁴	—	8	16	no	—
74	C₂⁴	D₈×C₂₃	1	1	yes	characteristic
75	C₂×D₄	—	2	56	no	—
76	C₂×D₄	—	4	84	no	—
77	C₂²×C₄	D₈×C₂₃	1	7	yes	—
78	C₂²×C₄	C₂×D₄×C₂₃	1	7	yes	—
79	D₈	—	2	56	no	—
80	D₈	C₂⁴×C₂₃	1	16	yes	complemented
81	C₂×C₈	C₂×D₄×C₂₃	1	21	yes	—
82	C₂×C₈	C₂⁴×C₂₃	1	7	yes	—
83	C₁₆	C₂⁴×C₂₃	1	8	yes	complemented
84	C₂³	—	4	56	no	—
85	C₂³	—	8	112	no	—
86	C₂³	C₂₃×D₁₆	1	8	yes	direct product factor
87	C₂³	C₂×D₈×C₂₃	1	7	yes	—
88	D₄	—	2	16	no	—
89	D₄	—	4	56	no	—
90	C₂×C₄	C₂×D₈×C₂₃	1	21	yes	—
91	C₂×C₄	C₂²×D₄×C₂₃	1	7	yes	—
92	C₈	C₂²×D₄×C₂₃	1	7	yes	—
93	C₈	C₂⁵×C₂₃	1	1	yes	characteristic
94	C₂²	—	4	16	no	—
95	C₂²	—	8	112	no	—
96	C₂²	C₂×C₂₃×D₁₆	1	28	yes	direct product factor
97	C₂²	C₂²×D₈×C₂₃	1	7	yes	—
98	C₄	C₂²×D₈×C₂₃	1	7	yes	—
99	C₄	C₂³×D₄×C₂₃	1	1	yes	characteristic
100	C₂	—	8	16	no	—
101	C₂	C₂²×C₂₃×D₁₆	1	14	yes	direct product factor
102	C₂	C₂³×D₈×C₂₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×D₈×C₂₃

C₂

yes

maximal, complemented

C₂²×C₂₃×D₁₆

C₂

yes

maximal, direct product factor

C₂³×C₁₆×C₂₃

C₂

yes

maximal, characteristic, complemented

C₂³×D₄×C₂₃

—

C₂²×D₈×C₂₃

—

C₂²×D₈×C₂₃

C₂²

yes

complemented

C₂³×C₈×C₂₃

C₂²

yes

characteristic

C₂×C₂₃×D₁₆

C₂²

112

yes

direct product factor

C₂²×C₁₆×C₂₃

C₂²

yes

complemented

C₂⁵×C₂₃

—

C₂²×D₄×C₂₃

—

C₂²×D₄×C₂₃

—

C₂³×C₄×C₂₃

D₄

yes

characteristic

C₂×D₈×C₂₃

—

C₂×D₈×C₂₃

C₂³

yes

complemented

C₂²×C₈×C₂₃

D₄

yes

—

C₂²×C₈×C₂₃

C₂³

yes

—

C₂₃×D₁₆

C₂³

yes

direct product factor

C₂×C₁₆×C₂₃

C₂³

yes

complemented

C₂⁴×C₂₃

—

C₂⁴×C₂₃

—

C₂⁴×C₂₃

D₈

yes

characteristic

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂²×C₄×C₂₃

D₈

yes

—

C₂²×C₄×C₂₃

C₂×D₄

yes

—

D₈×C₂₃

—

D₈×C₂₃

C₂⁴

yes

complemented

C₂×C₈×C₂₃

C₂×D₄

yes

—

C₂×C₈×C₂₃

C₂⁴

yes

—

C₁₆×C₂₃

C₂⁴

yes

complemented

C₂³×D₁₆

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

—

C₂³×C₂₃

—

112

—

C₂³×C₂₃

D₁₆

yes

direct product factor

C₂³×C₂₃

C₂×D₈

yes

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₈

yes

—

C₂×C₄×C₂₃

C₂²×D₄

yes

—

C₈×C₂₃

C₂²×D₄

yes

—

C₈×C₂₃

C₂⁵

yes

characteristic

C₂³×D₈

C₂×C₂₃

yes

complemented

C₂²×D₁₆

C₂×C₂₃

yes

direct product factor

C₂³×C₁₆

C₂×C₂₃

yes

characteristic, complemented

C₂²×C₂₃

—

C₂²×C₂₃

—

112

—

C₂²×C₂₃

C₂×D₁₆

yes

direct product factor

C₂²×C₂₃

C₂²×D₈

yes

—

C₄×C₂₃

C₂²×D₈

yes

—

C₄×C₂₃

C₂³×D₄

yes

characteristic

C₂³×D₄

—

C₂²×D₈

—

C₂²×D₈

C₂²×C₂₃

yes

complemented

C₂³×C₈

C₂²×C₂₃

yes

characteristic

C₂×D₁₆

C₂²×C₂₃

112

yes

direct product factor

C₂²×C₁₆

C₂²×C₂₃

yes

complemented

C₂×C₂₃

—

C₂×C₂₃

C₂²×D₁₆

yes

direct product factor

C₂×C₂₃

C₂³×D₈

yes

characteristic

C₂⁵

—

C₂²×D₄

—

C₂²×D₄

—

C₂³×C₄

D₄×C₂₃

yes

characteristic

C₂×D₈

—

C₂×D₈

C₂³×C₂₃

yes

complemented

C₂²×C₈

D₄×C₂₃

yes

—

C₂²×C₈

C₂³×C₂₃

yes

—

D₁₆

C₂³×C₂₃

yes

direct product factor

C₂×C₁₆

C₂³×C₂₃

yes

complemented

C₂₃

C₂³×D₁₆

yes

characteristic, direct product factor, Sylow

C₂⁴

—

C₂⁴

—

C₂⁴

D₈×C₂₃

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

D₈×C₂₃

yes

—

C₂²×C₄

C₂×D₄×C₂₃

yes

—

D₈

—

D₈

C₂⁴×C₂₃

yes

complemented

C₂×C₈

C₂×D₄×C₂₃

yes

—

C₂×C₈

C₂⁴×C₂₃

yes

—

C₁₆

C₂⁴×C₂₃

yes

complemented

C₂³

—

C₂³

—

112

—

C₂³

C₂₃×D₁₆

yes

direct product factor

C₂³

C₂×D₈×C₂₃

yes

—

D₄

—

D₄

—

C₂×C₄

C₂×D₈×C₂₃

yes

—

C₂×C₄

C₂²×D₄×C₂₃

yes

—

C₈

C₂²×D₄×C₂₃

yes

—

C₈

C₂⁵×C₂₃

yes

characteristic

C₂²

—

C₂²

—

112

—

C₂²

C₂×C₂₃×D₁₆

yes

direct product factor

C₂²

C₂²×D₈×C₂₃

yes

—

C₄

C₂²×D₈×C₂₃

yes

—

C₄

C₂³×D₄×C₂₃

yes

characteristic

100

C₂

—

101

C₂

C₂²×C₂₃×D₁₆

yes

direct product factor

102

C₂

C₂³×D₈×C₂₃

yes

characteristic

	subgroup	G/H
center, socle	C₂⁴×C₂₃	D₈
commutator, Frattini, cosocle	C₈	C₂⁵×C₂₃
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ C₂³×C₈×C₂₃ ◁ G

subgroup

G/H

center, socle

C₂⁴×C₂₃

D₈

commutator, Frattini, cosocle

C₈

C₂⁵×C₂₃

LCS

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂⁴×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ C₂³×C₈×C₂₃ ◁ G

N	H ≅ G/N	type
1. C₂³×D₈×C₂₃	C₂	semidirect
2. C₂²×C₂₃×D₁₆	C₂	direct (N × H)
3. C₂³×C₁₆×C₂₃	C₂	semidirect
6. C₂²×D₈×C₂₃	C₂²	semidirect
8. C₂×C₂₃×D₁₆	C₂²	direct (N × H)
9. C₂²×C₁₆×C₂₃	C₂²	semidirect
15. C₂×D₈×C₂₃	C₂³	semidirect
18. C₂₃×D₁₆	C₂³	direct (N × H)
19. C₂×C₁₆×C₂₃	C₂³	semidirect
28. D₈×C₂₃	C₂⁴	semidirect
31. C₁₆×C₂₃	C₂⁴	semidirect
32. C₂³×D₁₆	C₂₃	direct (N × H)
35. C₂³×C₂₃	D₁₆	direct (N × H)
43. C₂³×D₈	C₂×C₂₃	semidirect
44. C₂²×D₁₆	C₂×C₂₃	direct (N × H)
45. C₂³×C₁₆	C₂×C₂₃	semidirect
48. C₂²×C₂₃	C₂×D₁₆	direct (N × H)
54. C₂²×D₈	C₂²×C₂₃	semidirect
56. C₂×D₁₆	C₂²×C₂₃	direct (N × H)
57. C₂²×C₁₆	C₂²×C₂₃	semidirect
59. C₂×C₂₃	C₂²×D₁₆	direct (N × H)
66. C₂×D₈	C₂³×C₂₃	semidirect
69. D₁₆	C₂³×C₂₃	direct (N × H)
70. C₂×C₁₆	C₂³×C₂₃	semidirect
71. C₂₃	C₂³×D₁₆	direct (N × H)
80. D₈	C₂⁴×C₂₃	semidirect
83. C₁₆	C₂⁴×C₂₃	semidirect
86. C₂³	C₂₃×D₁₆	direct (N × H)
96. C₂²	C₂×C₂₃×D₁₆	direct (N × H)
101. C₂	C₂²×C₂₃×D₁₆	direct (N × H)

H ≅ G/N

type

1. C₂³×D₈×C₂₃

C₂

semidirect

2. C₂²×C₂₃×D₁₆

C₂

direct (N × H)

3. C₂³×C₁₆×C₂₃

C₂

semidirect

6. C₂²×D₈×C₂₃

C₂²

semidirect

8. C₂×C₂₃×D₁₆

C₂²

direct (N × H)

9. C₂²×C₁₆×C₂₃

C₂²

semidirect

15. C₂×D₈×C₂₃

C₂³

semidirect

18. C₂₃×D₁₆

C₂³

direct (N × H)

19. C₂×C₁₆×C₂₃

C₂³

semidirect

28. D₈×C₂₃

C₂⁴

semidirect

31. C₁₆×C₂₃

C₂⁴

semidirect

32. C₂³×D₁₆

C₂₃

direct (N × H)

35. C₂³×C₂₃

D₁₆

direct (N × H)

43. C₂³×D₈

C₂×C₂₃

semidirect

44. C₂²×D₁₆

C₂×C₂₃

direct (N × H)

45. C₂³×C₁₆

C₂×C₂₃

semidirect

48. C₂²×C₂₃

C₂×D₁₆

direct (N × H)

54. C₂²×D₈

C₂²×C₂₃

semidirect

56. C₂×D₁₆

C₂²×C₂₃

direct (N × H)

57. C₂²×C₁₆

C₂²×C₂₃

semidirect

59. C₂×C₂₃

C₂²×D₁₆

direct (N × H)

66. C₂×D₈

C₂³×C₂₃

semidirect

69. D₁₆

C₂³×C₂₃

direct (N × H)

70. C₂×C₁₆

C₂³×C₂₃

semidirect

71. C₂₃

C₂³×D₁₆

direct (N × H)

80. D₈

C₂⁴×C₂₃

semidirect

83. C₁₆

C₂⁴×C₂₃

semidirect

86. C₂³

C₂₃×D₁₆

direct (N × H)

96. C₂²

C₂×C₂₃×D₁₆

direct (N × H)

101. C₂

C₂²×C₂₃×D₁₆

direct (N × H)

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

subgroup	index
C₂³×D₈×C₂₃	2
C₂²×C₂₃×D₁₆	2
C₂³×C₁₆×C₂₃	2
C₂³×D₁₆	23

subgroup

index

C₂³×D₈×C₂₃

C₂²×C₂₃×D₁₆

C₂³×C₁₆×C₂₃

C₂³×D₁₆

Subgroups — C₂³×C₂₃×D₁₆ #19210251

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×C23×D16 #19210251

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₂₃×D₁₆ #19210251