Group #22683878 — subgroups

Subgroups — C₂³×D₁₆×C₃₇ #22683878

Order 9472.

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×C₃₇	C₂	1	2	yes	maximal, complemented
2	C₂²×D₁₆×C₃₇	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₁₆×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₄×C₃₇	—	2	2	no	—
5	C₂²×D₈×C₃₇	—	2	14	no	—
6	C₂²×D₈×C₃₇	C₂²	1	28	yes	complemented
7	C₂³×C₈×C₃₇	C₂²	1	1	yes	characteristic
8	C₂×D₁₆×C₃₇	C₂²	1	112	yes	direct product factor
9	C₂²×C₁₆×C₃₇	C₂²	1	14	yes	complemented
10	C₂⁵×C₃₇	—	4	2	no	—
11	C₂²×D₄×C₃₇	—	2	28	no	—
12	C₂²×D₄×C₃₇	—	4	14	no	—
13	C₂³×C₄×C₃₇	D₄	1	1	yes	characteristic
14	C₂×D₈×C₃₇	—	2	84	no	—
15	C₂×D₈×C₃₇	C₂³	1	56	yes	complemented
16	C₂²×C₈×C₃₇	D₄	1	7	yes	—
17	C₂²×C₈×C₃₇	C₂³	1	7	yes	—
18	D₁₆×C₃₇	C₂³	1	64	yes	direct product factor
19	C₂×C₁₆×C₃₇	C₂³	1	28	yes	complemented
20	C₂⁴×C₃₇	—	4	28	no	—
21	C₂⁴×C₃₇	—	8	16	no	—
22	C₂⁴×C₃₇	D₈	1	1	yes	characteristic
23	C₂×D₄×C₃₇	—	2	56	no	—
24	C₂×D₄×C₃₇	—	4	84	no	—
25	C₂²×C₄×C₃₇	D₈	1	7	yes	—
26	C₂²×C₄×C₃₇	C₂×D₄	1	7	yes	—
27	D₈×C₃₇	—	2	56	no	—
28	D₈×C₃₇	C₂⁴	1	16	yes	complemented
29	C₂×C₈×C₃₇	C₂×D₄	1	21	yes	—
30	C₂×C₈×C₃₇	C₂⁴	1	7	yes	—
31	C₁₆×C₃₇	C₂⁴	1	8	yes	complemented
32	C₂³×C₃₇	—	4	56	no	—
33	C₂³×C₃₇	—	8	112	no	—
34	C₂³×C₃₇	D₁₆	1	8	yes	direct product factor
35	C₂³×C₃₇	C₂×D₈	1	7	yes	—
36	D₄×C₃₇	—	2	16	no	—
37	D₄×C₃₇	—	4	56	no	—
38	C₂×C₄×C₃₇	C₂×D₈	1	21	yes	—
39	C₂×C₄×C₃₇	C₂²×D₄	1	7	yes	—
40	C₈×C₃₇	C₂²×D₄	1	7	yes	—
41	C₈×C₃₇	C₂⁵	1	1	yes	characteristic
42	C₂³×D₁₆	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂²×C₃₇	—	4	16	no	—
44	C₂²×C₃₇	—	8	112	no	—
45	C₂²×C₃₇	C₂×D₁₆	1	28	yes	direct product factor
46	C₂²×C₃₇	C₂²×D₈	1	7	yes	—
47	C₄×C₃₇	C₂²×D₈	1	7	yes	—
48	C₄×C₃₇	C₂³×D₄	1	1	yes	characteristic
49	C₂³×D₈	C₂×C₃₇	1	2	yes	complemented
50	C₂²×D₁₆	C₂×C₃₇	1	28	yes	direct product factor
51	C₂³×C₁₆	C₂×C₃₇	1	1	yes	characteristic, complemented
52	C₂×C₃₇	—	8	16	no	—
53	C₂×C₃₇	C₂²×D₁₆	1	14	yes	direct product factor
54	C₂×C₃₇	C₂³×D₈	1	1	yes	characteristic
55	C₂³×D₄	—	2	2	no	—
56	C₂²×D₈	—	2	14	no	—
57	C₂²×D₈	C₂²×C₃₇	1	28	yes	complemented
58	C₂³×C₈	C₂²×C₃₇	1	1	yes	characteristic
59	C₂×D₁₆	C₂²×C₃₇	1	112	yes	direct product factor
60	C₂²×C₁₆	C₂²×C₃₇	1	14	yes	complemented
61	C₃₇	C₂³×D₁₆	1	1	yes	characteristic, direct product factor, Sylow
62	C₂⁵	—	4	2	no	—
63	C₂²×D₄	—	2	28	no	—
64	C₂²×D₄	—	4	14	no	—
65	C₂³×C₄	D₄×C₃₇	1	1	yes	characteristic
66	C₂×D₈	—	2	84	no	—
67	C₂×D₈	C₂³×C₃₇	1	56	yes	complemented
68	C₂²×C₈	D₄×C₃₇	1	7	yes	—
69	C₂²×C₈	C₂³×C₃₇	1	7	yes	—
70	D₁₆	C₂³×C₃₇	1	64	yes	direct product factor
71	C₂×C₁₆	C₂³×C₃₇	1	28	yes	complemented
72	C₂⁴	—	4	28	no	—
73	C₂⁴	—	8	16	no	—
74	C₂⁴	D₈×C₃₇	1	1	yes	characteristic
75	C₂×D₄	—	2	56	no	—
76	C₂×D₄	—	4	84	no	—
77	C₂²×C₄	D₈×C₃₇	1	7	yes	—
78	C₂²×C₄	C₂×D₄×C₃₇	1	7	yes	—
79	D₈	—	2	56	no	—
80	D₈	C₂⁴×C₃₇	1	16	yes	complemented
81	C₂×C₈	C₂×D₄×C₃₇	1	21	yes	—
82	C₂×C₈	C₂⁴×C₃₇	1	7	yes	—
83	C₁₆	C₂⁴×C₃₇	1	8	yes	complemented
84	C₂³	—	4	56	no	—
85	C₂³	—	8	112	no	—
86	C₂³	D₁₆×C₃₇	1	8	yes	direct product factor
87	C₂³	C₂×D₈×C₃₇	1	7	yes	—
88	D₄	—	2	16	no	—
89	D₄	—	4	56	no	—
90	C₂×C₄	C₂×D₈×C₃₇	1	21	yes	—
91	C₂×C₄	C₂²×D₄×C₃₇	1	7	yes	—
92	C₈	C₂²×D₄×C₃₇	1	7	yes	—
93	C₈	C₂⁵×C₃₇	1	1	yes	characteristic
94	C₂²	—	4	16	no	—
95	C₂²	—	8	112	no	—
96	C₂²	C₂×D₁₆×C₃₇	1	28	yes	direct product factor
97	C₂²	C₂²×D₈×C₃₇	1	7	yes	—
98	C₄	C₂²×D₈×C₃₇	1	7	yes	—
99	C₄	C₂³×D₄×C₃₇	1	1	yes	characteristic
100	C₂	—	8	16	no	—
101	C₂	C₂²×D₁₆×C₃₇	1	14	yes	direct product factor
102	C₂	C₂³×D₈×C₃₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×D₈×C₃₇

C₂

yes

maximal, complemented

C₂²×D₁₆×C₃₇

C₂

yes

maximal, direct product factor

C₂³×C₁₆×C₃₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₄×C₃₇

—

C₂²×D₈×C₃₇

—

C₂²×D₈×C₃₇

C₂²

yes

complemented

C₂³×C₈×C₃₇

C₂²

yes

characteristic

C₂×D₁₆×C₃₇

C₂²

112

yes

direct product factor

C₂²×C₁₆×C₃₇

C₂²

yes

complemented

C₂⁵×C₃₇

—

C₂²×D₄×C₃₇

—

C₂²×D₄×C₃₇

—

C₂³×C₄×C₃₇

D₄

yes

characteristic

C₂×D₈×C₃₇

—

C₂×D₈×C₃₇

C₂³

yes

complemented

C₂²×C₈×C₃₇

D₄

yes

—

C₂²×C₈×C₃₇

C₂³

yes

—

D₁₆×C₃₇

C₂³

yes

direct product factor

C₂×C₁₆×C₃₇

C₂³

yes

complemented

C₂⁴×C₃₇

—

C₂⁴×C₃₇

—

C₂⁴×C₃₇

D₈

yes

characteristic

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

—

C₂²×C₄×C₃₇

D₈

yes

—

C₂²×C₄×C₃₇

C₂×D₄

yes

—

D₈×C₃₇

—

D₈×C₃₇

C₂⁴

yes

complemented

C₂×C₈×C₃₇

C₂×D₄

yes

—

C₂×C₈×C₃₇

C₂⁴

yes

—

C₁₆×C₃₇

C₂⁴

yes

complemented

C₂³×C₃₇

—

C₂³×C₃₇

—

112

—

C₂³×C₃₇

D₁₆

yes

direct product factor

C₂³×C₃₇

C₂×D₈

yes

—

D₄×C₃₇

—

D₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₈

yes

—

C₂×C₄×C₃₇

C₂²×D₄

yes

—

C₈×C₃₇

C₂²×D₄

yes

—

C₈×C₃₇

C₂⁵

yes

characteristic

C₂³×D₁₆

C₃₇

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₃₇

—

C₂²×C₃₇

—

112

—

C₂²×C₃₇

C₂×D₁₆

yes

direct product factor

C₂²×C₃₇

C₂²×D₈

yes

—

C₄×C₃₇

C₂²×D₈

yes

—

C₄×C₃₇

C₂³×D₄

yes

characteristic

C₂³×D₈

C₂×C₃₇

yes

complemented

C₂²×D₁₆

C₂×C₃₇

yes

direct product factor

C₂³×C₁₆

C₂×C₃₇

yes

characteristic, complemented

C₂×C₃₇

—

C₂×C₃₇

C₂²×D₁₆

yes

direct product factor

C₂×C₃₇

C₂³×D₈

yes

characteristic

C₂³×D₄

—

C₂²×D₈

—

C₂²×D₈

C₂²×C₃₇

yes

complemented

C₂³×C₈

C₂²×C₃₇

yes

characteristic

C₂×D₁₆

C₂²×C₃₇

112

yes

direct product factor

C₂²×C₁₆

C₂²×C₃₇

yes

complemented

C₃₇

C₂³×D₁₆

yes

characteristic, direct product factor, Sylow

C₂⁵

—

C₂²×D₄

—

C₂²×D₄

—

C₂³×C₄

D₄×C₃₇

yes

characteristic

C₂×D₈

—

C₂×D₈

C₂³×C₃₇

yes

complemented

C₂²×C₈

D₄×C₃₇

yes

—

C₂²×C₈

C₂³×C₃₇

yes

—

D₁₆

C₂³×C₃₇

yes

direct product factor

C₂×C₁₆

C₂³×C₃₇

yes

complemented

C₂⁴

—

C₂⁴

—

C₂⁴

D₈×C₃₇

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

D₈×C₃₇

yes

—

C₂²×C₄

C₂×D₄×C₃₇

yes

—

D₈

—

D₈

C₂⁴×C₃₇

yes

complemented

C₂×C₈

C₂×D₄×C₃₇

yes

—

C₂×C₈

C₂⁴×C₃₇

yes

—

C₁₆

C₂⁴×C₃₇

yes

complemented

C₂³

—

C₂³

—

112

—

C₂³

D₁₆×C₃₇

yes

direct product factor

C₂³

C₂×D₈×C₃₇

yes

—

D₄

—

D₄

—

C₂×C₄

C₂×D₈×C₃₇

yes

—

C₂×C₄

C₂²×D₄×C₃₇

yes

—

C₈

C₂²×D₄×C₃₇

yes

—

C₈

C₂⁵×C₃₇

yes

characteristic

C₂²

—

C₂²

—

112

—

C₂²

C₂×D₁₆×C₃₇

yes

direct product factor

C₂²

C₂²×D₈×C₃₇

yes

—

C₄

C₂²×D₈×C₃₇

yes

—

C₄

C₂³×D₄×C₃₇

yes

characteristic

100

C₂

—

101

C₂

C₂²×D₁₆×C₃₇

yes

direct product factor

102

C₂

C₂³×D₈×C₃₇

yes

characteristic

	subgroup	G/H
center, socle	C₂⁴×C₃₇	D₈
commutator, Frattini, cosocle	C₈	C₂⁵×C₃₇
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₇ ◁ C₂³×C₄×C₃₇ ◁ C₂³×C₈×C₃₇ ◁ G

subgroup

G/H

center, socle

C₂⁴×C₃₇

D₈

commutator, Frattini, cosocle

C₈

C₂⁵×C₃₇

LCS

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂⁴×C₃₇ ◁ C₂³×C₄×C₃₇ ◁ C₂³×C₈×C₃₇ ◁ G

N	H ≅ G/N	type
1. C₂³×D₈×C₃₇	C₂	semidirect
2. C₂²×D₁₆×C₃₇	C₂	direct (N × H)
3. C₂³×C₁₆×C₃₇	C₂	semidirect
6. C₂²×D₈×C₃₇	C₂²	semidirect
8. C₂×D₁₆×C₃₇	C₂²	direct (N × H)
9. C₂²×C₁₆×C₃₇	C₂²	semidirect
15. C₂×D₈×C₃₇	C₂³	semidirect
18. D₁₆×C₃₇	C₂³	direct (N × H)
19. C₂×C₁₆×C₃₇	C₂³	semidirect
28. D₈×C₃₇	C₂⁴	semidirect
31. C₁₆×C₃₇	C₂⁴	semidirect
34. C₂³×C₃₇	D₁₆	direct (N × H)
42. C₂³×D₁₆	C₃₇	direct (N × H)
45. C₂²×C₃₇	C₂×D₁₆	direct (N × H)
49. C₂³×D₈	C₂×C₃₇	semidirect
50. C₂²×D₁₆	C₂×C₃₇	direct (N × H)
51. C₂³×C₁₆	C₂×C₃₇	semidirect
53. C₂×C₃₇	C₂²×D₁₆	direct (N × H)
57. C₂²×D₈	C₂²×C₃₇	semidirect
59. C₂×D₁₆	C₂²×C₃₇	direct (N × H)
60. C₂²×C₁₆	C₂²×C₃₇	semidirect
61. C₃₇	C₂³×D₁₆	direct (N × H)
67. C₂×D₈	C₂³×C₃₇	semidirect
70. D₁₆	C₂³×C₃₇	direct (N × H)
71. C₂×C₁₆	C₂³×C₃₇	semidirect
80. D₈	C₂⁴×C₃₇	semidirect
83. C₁₆	C₂⁴×C₃₇	semidirect
86. C₂³	D₁₆×C₃₇	direct (N × H)
96. C₂²	C₂×D₁₆×C₃₇	direct (N × H)
101. C₂	C₂²×D₁₆×C₃₇	direct (N × H)

H ≅ G/N

type

1. C₂³×D₈×C₃₇

C₂

semidirect

2. C₂²×D₁₆×C₃₇

C₂

direct (N × H)

3. C₂³×C₁₆×C₃₇

C₂

semidirect

6. C₂²×D₈×C₃₇

C₂²

semidirect

8. C₂×D₁₆×C₃₇

C₂²

direct (N × H)

9. C₂²×C₁₆×C₃₇

C₂²

semidirect

15. C₂×D₈×C₃₇

C₂³

semidirect

18. D₁₆×C₃₇

C₂³

direct (N × H)

19. C₂×C₁₆×C₃₇

C₂³

semidirect

28. D₈×C₃₇

C₂⁴

semidirect

31. C₁₆×C₃₇

C₂⁴

semidirect

34. C₂³×C₃₇

D₁₆

direct (N × H)

42. C₂³×D₁₆

C₃₇

direct (N × H)

45. C₂²×C₃₇

C₂×D₁₆

direct (N × H)

49. C₂³×D₈

C₂×C₃₇

semidirect

50. C₂²×D₁₆

C₂×C₃₇

direct (N × H)

51. C₂³×C₁₆

C₂×C₃₇

semidirect

53. C₂×C₃₇

C₂²×D₁₆

direct (N × H)

57. C₂²×D₈

C₂²×C₃₇

semidirect

59. C₂×D₁₆

C₂²×C₃₇

direct (N × H)

60. C₂²×C₁₆

C₂²×C₃₇

semidirect

61. C₃₇

C₂³×D₁₆

direct (N × H)

67. C₂×D₈

C₂³×C₃₇

semidirect

70. D₁₆

C₂³×C₃₇

direct (N × H)

71. C₂×C₁₆

C₂³×C₃₇

semidirect

80. D₈

C₂⁴×C₃₇

semidirect

83. C₁₆

C₂⁴×C₃₇

semidirect

86. C₂³

D₁₆×C₃₇

direct (N × H)

96. C₂²

C₂×D₁₆×C₃₇

direct (N × H)

101. C₂

C₂²×D₁₆×C₃₇

direct (N × H)

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

subgroup	index
C₂³×D₈×C₃₇	2
C₂²×D₁₆×C₃₇	2
C₂³×C₁₆×C₃₇	2
C₂³×D₁₆	37

subgroup

index

C₂³×D₈×C₃₇

C₂²×D₁₆×C₃₇

C₂³×C₁₆×C₃₇

C₂³×D₁₆

Subgroups — C₂³×D₁₆×C₃₇ #22683878

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×D16×C37 #22683878

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×D₁₆×C₃₇ #22683878