Group #17990664 — subgroups

Subgroups — C₃₇×#1846 #17990664

Order 4736.

248 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 152 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	C₂×#107×C₃₇	C₂	1	2	yes	maximal
2	C₂×#106×C₃₇	C₂	1	2	yes	maximal
3	C₂×C₄×C₈×C₃₇	C₂	1	1	yes	maximal, characteristic
4	C₃₇×#382	C₂	1	2	yes	maximal
5	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
6	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₃₇	C₂²	1	4	yes	—
8	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
9	#107×C₃₇	—	2	4	no	—
10	#106×C₃₇	—	2	4	no	—
11	C₄×C₈×C₃₇	C₄	1	4	yes	complemented
12	#95×C₃₇	—	2	4	no	—
13	C₂²×C₄×C₃₇	—	2	4	no	—
14	C₂²×C₄×C₃₇	D₄	1	2	yes	—
15	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
16	C₂×C₈×C₃₇	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₃₇	C₂×C₄	1	2	yes	—
18	C₂×C₈×C₃₇	D₄	1	2	yes	—
19	C₂×C₈×C₃₇	D₄	1	2	yes	—
20	C₂×C₈×C₃₇	Q₈	1	2	yes	—
21	C₂×C₈×C₃₇	Q₈	1	2	yes	—
22	C₄⋊C₄×C₃₇	—	2	4	no	—
23	C₄⋊C₄×C₃₇	—	2	4	no	—
24	C₄²×C₃₇	C₂×C₄	1	2	yes	—
25	C₄²×C₃₇	Q₈	1	2	yes	—
26	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
27	C₂×C₄×C₃₇	—	2	4	no	—
28	C₂×C₄×C₃₇	—	4	4	no	—
29	C₂×C₄×C₃₇	—	4	8	no	—
30	C₂×C₄×C₃₇	C₄⋊C₄	1	4	yes	—
31	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
37	C₈×C₃₇	C₄⋊C₄	1	4	yes	—
38	C₈×C₃₇	C₄⋊C₄	1	4	yes	—
39	C₂²×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₃₇	#127	1	1	yes	characteristic
41	C₂²×C₃₇	C₄⋊Q₈	1	1	yes	characteristic
42	C₂²×C₃₇	#135	1	4	yes	—
43	C₄×C₃₇	—	2	4	no	—
44	C₄×C₃₇	—	8	4	no	—
45	C₄×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₄×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₄×C₃₇	C₄⋊Q₈	1	1	yes	characteristic
48	C₄×C₃₇	C₄⋊Q₈	1	1	yes	characteristic
49	#1846	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂×C₃₇	#384	1	1	yes	characteristic
51	C₂×C₃₇	#427	1	2	yes	—
52	C₂×C₃₇	#495	1	2	yes	—
53	C₂×C₃₇	#499	1	2	yes	—
54	C₂×#107	C₂×C₃₇	1	2	yes	—
55	C₂×#106	C₂×C₃₇	1	2	yes	—
56	C₂×C₄×C₈	C₂×C₃₇	1	1	yes	characteristic
57	#382	C₂×C₃₇	1	2	yes	—
58	C₃₇	#1846	1	1	yes	characteristic, direct product factor, Sylow
59	C₂²×C₈	C₂²×C₃₇	1	1	yes	characteristic
60	C₂²×C₈	C₂²×C₃₇	1	1	yes	characteristic
61	C₂×C₄⋊C₄	C₂²×C₃₇	1	4	yes	—
62	C₂×C₄²	C₂²×C₃₇	1	1	yes	characteristic
63	#107	—	2	4	no	—
64	#106	—	2	4	no	—
65	C₄×C₈	C₄×C₃₇	1	4	yes	complemented
66	#95	—	2	4	no	—
67	C₂²×C₄	—	2	4	no	—
68	C₂²×C₄	D₄×C₃₇	1	2	yes	—
69	C₂²×C₄	C₂³×C₃₇	1	1	yes	characteristic
70	C₂×C₈	C₂×C₄×C₃₇	1	2	yes	—
71	C₂×C₈	C₂×C₄×C₃₇	1	2	yes	—
72	C₂×C₈	D₄×C₃₇	1	2	yes	—
73	C₂×C₈	D₄×C₃₇	1	2	yes	—
74	C₂×C₈	Q₈×C₃₇	1	2	yes	—
75	C₂×C₈	Q₈×C₃₇	1	2	yes	—
76	C₄⋊C₄	—	2	4	no	—
77	C₄⋊C₄	—	2	4	no	—
78	C₄²	C₂×C₄×C₃₇	1	2	yes	—
79	C₄²	Q₈×C₃₇	1	2	yes	—
80	C₂³	C₂×D₄×C₃₇	1	1	yes	characteristic
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	4	4	no	—
83	C₂×C₄	—	4	8	no	—
84	C₂×C₄	C₄⋊C₄×C₃₇	1	4	yes	—
85	C₂×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
86	C₂×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
87	C₂×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
88	C₂×C₄	C₂×Q₈×C₃₇	1	1	yes	characteristic
89	C₂×C₄	C₂×Q₈×C₃₇	1	1	yes	characteristic
90	C₂×C₄	C₂×Q₈×C₃₇	1	1	yes	characteristic
91	C₈	C₄⋊C₄×C₃₇	1	4	yes	—
92	C₈	C₄⋊C₄×C₃₇	1	4	yes	—
93	C₂²	C₂×C₄⋊C₄×C₃₇	1	1	yes	characteristic
94	C₂²	#127×C₃₇	1	1	yes	characteristic
95	C₂²	C₄⋊Q₈×C₃₇	1	1	yes	characteristic
96	C₂²	#135×C₃₇	1	4	yes	—
97	C₄	—	2	4	no	—
98	C₄	—	8	4	no	—
99	C₄	C₂×C₄⋊C₄×C₃₇	1	1	yes	characteristic
100	C₄	C₂×C₄⋊C₄×C₃₇	1	1	yes	characteristic
101	C₄	C₄⋊Q₈×C₃₇	1	1	yes	characteristic
102	C₄	C₄⋊Q₈×C₃₇	1	1	yes	characteristic
103	C₂	C₃₇×#384	1	1	yes	characteristic
104	C₂	C₃₇×#427	1	2	yes	—
105	C₂	C₃₇×#495	1	2	yes	—
106	C₂	C₃₇×#499	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#107×C₃₇

C₂

yes

maximal

C₂×#106×C₃₇

C₂

yes

maximal

C₂×C₄×C₈×C₃₇

C₂

yes

maximal, characteristic

C₃₇×#382

C₂

yes

maximal

C₂²×C₈×C₃₇

C₂²

yes

characteristic

C₂²×C₈×C₃₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₃₇

C₂²

yes

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

#107×C₃₇

—

#106×C₃₇

—

C₄×C₈×C₃₇

C₄

yes

complemented

#95×C₃₇

—

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂×C₈×C₃₇

C₂×C₄

yes

—

C₂×C₈×C₃₇

C₂×C₄

yes

—

C₂×C₈×C₃₇

D₄

yes

—

C₂×C₈×C₃₇

D₄

yes

—

C₂×C₈×C₃₇

Q₈

yes

—

C₂×C₈×C₃₇

Q₈

yes

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄²×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

Q₈

yes

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₄⋊C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₇

C₂×Q₈

yes

characteristic

C₈×C₃₇

C₄⋊C₄

yes

—

C₈×C₃₇

C₄⋊C₄

yes

—

C₂²×C₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₇

#127

yes

characteristic

C₂²×C₃₇

C₄⋊Q₈

yes

characteristic

C₂²×C₃₇

#135

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₃₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₃₇

C₄⋊Q₈

yes

characteristic

C₄×C₃₇

C₄⋊Q₈

yes

characteristic

#1846

C₃₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₃₇

#384

yes

characteristic

C₂×C₃₇

#427

yes

—

C₂×C₃₇

#495

yes

—

C₂×C₃₇

#499

yes

—

C₂×#107

C₂×C₃₇

yes

—

C₂×#106

C₂×C₃₇

yes

—

C₂×C₄×C₈

C₂×C₃₇

yes

characteristic

#382

C₂×C₃₇

yes

—

C₃₇

#1846

yes

characteristic, direct product factor, Sylow

C₂²×C₈

C₂²×C₃₇

yes

characteristic

C₂²×C₈

C₂²×C₃₇

yes

characteristic

C₂×C₄⋊C₄

C₂²×C₃₇

yes

—

C₂×C₄²

C₂²×C₃₇

yes

characteristic

#107

—

#106

—

C₄×C₈

C₄×C₃₇

yes

complemented

#95

—

C₂²×C₄

—

C₂²×C₄

D₄×C₃₇

yes

—

C₂²×C₄

C₂³×C₃₇

yes

characteristic

C₂×C₈

C₂×C₄×C₃₇

yes

—

C₂×C₈

C₂×C₄×C₃₇

yes

—

C₂×C₈

D₄×C₃₇

yes

—

C₂×C₈

D₄×C₃₇

yes

—

C₂×C₈

Q₈×C₃₇

yes

—

C₂×C₈

Q₈×C₃₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₄×C₃₇

yes

—

C₄²

Q₈×C₃₇

yes

—

C₂³

C₂×D₄×C₃₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊C₄×C₃₇

yes

—

C₂×C₄

C₂²×C₄×C₃₇

yes

characteristic

C₂×C₄

C₂×D₄×C₃₇

yes

characteristic

C₂×C₄

C₂×D₄×C₃₇

yes

characteristic

C₂×C₄

C₂×Q₈×C₃₇

yes

characteristic

C₂×C₄

C₂×Q₈×C₃₇

yes

characteristic

C₂×C₄

C₂×Q₈×C₃₇

yes

characteristic

C₈

C₄⋊C₄×C₃₇

yes

—

C₈

C₄⋊C₄×C₃₇

yes

—

C₂²

C₂×C₄⋊C₄×C₃₇

yes

characteristic

C₂²

#127×C₃₇

yes

characteristic

C₂²

C₄⋊Q₈×C₃₇

yes

characteristic

C₂²

#135×C₃₇

yes

—

C₄

—

C₄

—

C₄

C₂×C₄⋊C₄×C₃₇

yes

characteristic

100

C₄

C₂×C₄⋊C₄×C₃₇

yes

characteristic

101

C₄

C₄⋊Q₈×C₃₇

yes

characteristic

102

C₄

C₄⋊Q₈×C₃₇

yes

characteristic

103

C₂

C₃₇×#384

yes

characteristic

104

C₂

C₃₇×#427

yes

—

105

C₂

C₃₇×#495

yes

—

106

C₂

C₃₇×#499

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₇	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₃₇
Frattini, cosocle	C₂²×C₄	C₂³×C₃₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₃₇

C₂×D₄

commutator

C₂×C₄

C₂²×C₄×C₃₇

Frattini, cosocle

C₂²×C₄

C₂³×C₃₇

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ G

N	H ≅ G/N	type
11. C₄×C₈×C₃₇	C₄	semidirect
49. #1846	C₃₇	direct (N × H)
58. C₃₇	#1846	direct (N × H)
65. C₄×C₈	C₄×C₃₇	semidirect

H ≅ G/N

type

11. C₄×C₈×C₃₇

C₄

semidirect

49. #1846

C₃₇

direct (N × H)

58. C₃₇

#1846

direct (N × H)

65. C₄×C₈

C₄×C₃₇

semidirect

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

subgroup	index
C₂×#107×C₃₇	2
C₂×#106×C₃₇	2
C₂×C₄×C₈×C₃₇	2
C₃₇×#382	2
#1846	37

subgroup

index

C₂×#107×C₃₇

C₂×#106×C₃₇

C₂×C₄×C₈×C₃₇

C₃₇×#382

#1846

Subgroups — C₃₇×#1846 #17990664

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C37×#1846 #17990664

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#1846 #17990664