Group #18010045 — subgroups

Subgroups — D₃₇⋊#330 #18010045

Order 4736.

232 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 64 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	#14558492	C₂	1	2	yes	maximal
2	#14558390	C₂	1	4	yes	maximal, complemented
3	#125×D₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×D₃₇⋊C₈	—	2	2	no	—
5	#11924591	—	2	2	no	—
6	#11924591	C₂²	1	4	yes	—
7	#11924528	C₄	1	1	yes	characteristic
8	#11924527	C₂²	1	2	yes	—
9	#11924526	—	2	2	no	—
10	C₄⋊C₄×D₃₇	—	2	2	no	—
11	C₄⋊C₄×D₃₇	C₄	1	2	yes	—
12	C₄²×D₃₇	C₂²	1	1	yes	characteristic
13	#125×C₃₇	C₄	1	1	yes	characteristic
14	C₂×C₄×D₃₇	—	2	2	no	—
15	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×D₃₇	D₄	1	2	yes	—
17	C₂×#64438	—	2	4	no	—
18	D₃₇⋊C₈	—	2	8	no	—
19	C₄⋊C₄×C₃₇	—	2	2	no	—
20	C₄⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
21	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
22	C₄⋊Dic₃₇	—	2	2	no	—
23	C₄⋊Dic₃₇	C₂×C₄	1	2	yes	—
24	#10596936	—	2	2	no	—
25	#10596936	—	2	2	no	—
26	#10596936	—	2	2	no	—
27	C₄×Dic₃₇	D₄	1	2	yes	—
28	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
29	C₂²×D₃₇	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₃₇	—	2	2	no	—
31	C₂×C₄×C₃₇	C₂²⋊C₄	1	2	yes	—
32	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
33	C₂×Dic₃₇	—	2	2	no	—
34	C₂×Dic₃₇	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×Dic₃₇	C₂×D₄	1	2	yes	—
36	C₄×D₃₇	—	2	2	no	—
37	C₄×D₃₇	—	2	4	no	—
38	C₄×D₃₇	—	4	4	no	—
39	#64438	—	2	8	no	—
40	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
41	C₂×D₃₇	#101	1	2	yes	—
42	C₂×D₃₇	#104	1	4	yes	—
43	C₄×C₃₇	—	2	1	no	—
44	C₄×C₃₇	—	2	2	no	—
45	C₄×C₃₇	—	4	2	no	—
46	Dic₃₇	—	2	1	no	—
47	Dic₃₇	—	2	2	no	—
48	Dic₃₇	—	4	2	no	—
49	C₂×#330	—	37	1	no	maximal, Sylow
50	C₂×C₃₇	C₂×#101	1	1	yes	characteristic
51	C₂×C₃₇	C₂×#104	1	2	yes	—
52	D₃₇	#330	1	4	yes	complemented
53	C₂×#125	—	37	1	no	—
54	C₂×#97	—	37	2	no	—
55	#330	—	37	4	no	—
56	C₃₇	C₂×#330	1	1	yes	characteristic, complemented, Sylow
57	C₂²×C₈	—	74	2	no	—
58	#125	—	37	2	no	—
59	#125	—	37	1	no	—
60	#125	—	74	2	no	—
61	#125	#3866	1	1	yes	characteristic
62	C₂×C₄⋊C₄	—	37	2	no	—
63	C₂×C₄⋊C₄	—	74	2	no	—
64	C₂×C₄²	—	37	1	no	—
65	#97	—	37	4	no	—
66	#97	—	74	2	no	—
67	C₂²×C₄	—	37	2	no	—
68	C₂²×C₄	—	37	1	no	—
69	C₂²×C₄	—	74	2	no	—
70	C₂×C₈	—	74	4	no	—
71	C₂×C₈	—	74	8	no	—
72	C₄⋊C₄	—	2	2	no	—
73	C₄⋊C₄	—	37	2	no	—
74	C₄⋊C₄	—	74	2	no	—
75	C₄⋊C₄	—	74	2	no	—
76	C₄⋊C₄	—	74	2	no	—
77	C₄⋊C₄	—	74	2	no	—
78	C₄⋊C₄	C₂×#3866	1	2	yes	—
79	C₄²	—	37	2	no	—
80	C₄²	—	37	1	no	—
81	C₄²	C₂×#3866	1	1	yes	characteristic
82	C₂³	—	37	1	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	37	2	no	—
85	C₂×C₄	—	37	1	no	—
86	C₂×C₄	—	74	4	no	—
87	C₂×C₄	—	74	2	no	—
88	C₂×C₄	—	74	2	no	—
89	C₂×C₄	—	148	4	no	—
90	C₂×C₄	C₂²⋊#3866	1	2	yes	—
91	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
92	C₈	—	74	8	no	—
93	C₂²	—	37	2	no	—
94	C₂²	—	37	4	no	—
95	C₂²	C₂×C₂²⋊#3866	1	1	yes	characteristic
96	C₄	—	2	1	no	—
97	C₄	—	2	2	no	—
98	C₄	—	4	2	no	—
99	C₄	—	74	1	no	—
100	C₄	—	74	2	no	—
101	C₄	—	148	2	no	—
102	C₂	—	37	4	no	—
103	C₂	D₃₇⋊#101	1	1	yes	characteristic
104	C₂	D₃₇⋊#104	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14558492

C₂

yes

maximal

#14558390

C₂

yes

maximal, complemented

#125×D₃₇

C₂

yes

maximal, characteristic

C₂×D₃₇⋊C₈

—

#11924591

—

#11924591

C₂²

yes

—

#11924528

C₄

yes

characteristic

#11924527

C₂²

yes

—

#11924526

—

C₄⋊C₄×D₃₇

—

C₄⋊C₄×D₃₇

C₄

yes

—

C₄²×D₃₇

C₂²

yes

characteristic

#125×C₃₇

C₄

yes

characteristic

C₂×C₄×D₃₇

—

C₂×C₄×D₃₇

C₂×C₄

yes

characteristic

C₂×C₄×D₃₇

D₄

yes

—

C₂×#64438

—

D₃₇⋊C₈

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

C₂×C₄

yes

characteristic

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

C₂×C₄

yes

—

#10596936

—

#10596936

—

#10596936

—

C₄×Dic₃₇

D₄

yes

—

C₄×Dic₃₇

C₂³

yes

characteristic

C₂²×D₃₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₃₇

C₂×D₄

yes

—

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

—

#64438

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₃₇

#101

yes

—

C₂×D₃₇

#104

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

—

D₃₇

#330

yes

complemented

C₂×#125

—

C₂×#97

—

#330

—

C₃₇

C₂×#330

yes

characteristic, complemented, Sylow

C₂²×C₈

—

—

—

—

yes

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

#97

—

#97

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×#3866

yes

—

C₄²

—

C₄²

—

C₄²

C₂×#3866

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

148

—

C₂×C₄

C₂²⋊#3866

yes

—

C₂×C₄

C₂²×#3866

yes

characteristic

C₈

—

C₂²

—

C₂²

—

C₂²

C₂×C₂²⋊#3866

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

100

C₄

—

101

C₄

—

148

—

102

C₂

—

103

C₂

D₃₇⋊#101

yes

characteristic

104

C₂

D₃₇⋊#104

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#3866
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#3866
Fitting	#125×C₃₇	C₄
socle	C₂²×C₃₇	C₂×C₂²⋊C₄
prime radical	C₂²×D₃₇	C₂²⋊C₄
cosocle	C₄×Dic₃₇	C₂³
hypercenter	#125	#3866
hypocenter	C₃₇	C₂×#330
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

subgroup

G/H

center

C₂²

C₂×C₂²⋊#3866

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#3866

Fitting

#125×C₃₇

C₄

socle

C₂²×C₃₇

C₂×C₂²⋊C₄

prime radical

C₂²×D₃₇

C₂²⋊C₄

cosocle

C₄×Dic₃₇

C₂³

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #125

N	H ≅ G/N	type
2. #14558390	C₂	semidirect
52. D₃₇	#330	semidirect
56. C₃₇	C₂×#330	semidirect

H ≅ G/N

type

2. #14558390

C₂

semidirect

52. D₃₇

#330

semidirect

56. C₃₇

C₂×#330

semidirect

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

subgroup	index
#14558492	2
#14558390	2
#125×D₃₇	2
C₂×#330	37

subgroup

index

#14558492

#14558390

#125×D₃₇

C₂×#330

Subgroups — D₃₇⋊#330 #18010045

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D37⋊#330 #18010045

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃₇⋊#330 #18010045