Group #17993899 — subgroups

Subgroups — #17993899

Order 4736.

182 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 45 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	C₃₇⋊#564	C₂	1	1	yes	maximal, characteristic, complemented
2	#14557899	C₂	1	1	yes	maximal, characteristic
3	#14557898	C₂	1	2	yes	maximal
4	#14557274	C₂	1	2	yes	maximal, complemented
5	C₃₇×#330	C₂	1	1	yes	maximal, characteristic, complemented
6	#11924533	C₂²	1	1	yes	characteristic
7	#11924528	C₂²	1	2	yes	—
8	#11924527	—	2	2	no	—
9	#11924526	—	2	2	no	—
10	#11924526	—	2	2	no	—
11	#11924458	—	2	2	no	—
12	#11924458	—	2	2	no	—
13	#11924456	—	2	2	no	—
14	#11924447	C₂²	1	1	yes	characteristic
15	#125×C₃₇	C₂²	1	1	yes	characteristic
16	#97×C₃₇	C₂²	1	2	yes	—
17	C₂×D₁₄₈	—	2	1	no	—
18	C₂×D₁₄₈	—	2	2	no	—
19	C₂×C₈×C₃₇	—	2	2	no	—
20	C₄⋊C₄×C₃₇	—	2	2	no	—
21	C₄⋊C₄×C₃₇	D₄	1	2	yes	—
22	C₄²×C₃₇	C₂³	1	1	yes	characteristic
23	C₄⋊Dic₃₇	—	2	2	no	—
24	C₄⋊Dic₃₇	D₄	1	2	yes	—
25	C₄⋊Dic₃₇	D₄	1	2	yes	—
26	#10596936	—	2	2	no	—
27	#10596936	—	2	2	no	—
28	#10596936	—	2	2	no	—
29	#10596936	—	2	2	no	—
30	C₄×Dic₃₇	—	2	2	no	—
31	C₂×#64436	—	2	1	no	—
32	C₂²×D₃₇	—	4	1	no	—
33	C₂×C₄×C₃₇	—	2	2	no	—
34	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
36	C₂×Dic₃₇	—	2	2	no	—
37	C₂×Dic₃₇	—	2	2	no	—
38	D₁₄₈	—	4	2	no	—
39	D₁₄₈	—	4	4	no	—
40	C₈×C₃₇	—	4	2	no	—
41	#64436	—	4	1	no	—
42	C₂²×C₃₇	#120	1	1	yes	characteristic
43	C₂×D₃₇	—	8	2	no	—
44	C₂×D₃₇	—	8	1	no	—
45	C₄×C₃₇	—	2	1	no	—
46	C₄×C₃₇	—	2	2	no	—
47	C₄×C₃₇	—	4	2	no	—
48	Dic₃₇	—	4	2	no	—
49	Dic₃₇	—	4	2	no	—
50	#1658	—	37	1	no	maximal, Sylow
51	C₂×C₃₇	#454	1	2	yes	—
52	C₂×C₃₇	#454	1	1	yes	characteristic
53	D₃₇	—	16	1	no	—
54	#564	—	37	1	no	—
55	#490	—	37	1	no	—
56	#490	—	37	2	no	—
57	#330	—	37	2	no	—
58	#330	D₃₇	1	1	yes	characteristic, complemented
59	C₃₇	#1658	1	1	yes	characteristic, complemented, Sylow
60	#127	—	37	1	no	—
61	#125	—	37	2	no	—
62	#125	—	74	2	no	—
63	#125	—	74	2	no	—
64	#125	—	74	2	no	—
65	#125	C₂×D₃₇	1	1	yes	characteristic
66	#102	—	74	2	no	—
67	#102	—	74	2	no	—
68	#102	—	74	2	no	—
69	#97	—	37	1	no	—
70	#97	C₂×D₃₇	1	2	yes	—
71	C₂×D₄	—	74	1	no	—
72	C₂×D₄	—	74	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	74	1	no	—
75	C₄⋊C₄	—	2	2	no	—
76	C₄⋊C₄	—	37	2	no	—
77	C₄⋊C₄	—	37	2	no	—
78	C₄⋊C₄	—	74	2	no	—
79	C₄⋊C₄	—	74	2	no	—
80	C₄⋊C₄	—	74	2	no	—
81	C₄⋊C₄	—	74	2	no	—
82	C₄⋊C₄	—	74	2	no	—
83	C₄⋊C₄	D₁₄₈	1	2	yes	—
84	C₄²	—	74	2	no	—
85	C₄²	C₂²×D₃₇	1	1	yes	characteristic
86	C₂³	—	148	1	no	—
87	D₄	—	148	4	no	—
88	D₄	—	148	2	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	74	2	no	—
91	C₂×C₄	—	74	2	no	—
92	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
93	C₂×C₄	D₄×D₃₇	1	2	yes	—
94	C₈	—	4	2	no	—
95	C₈	—	148	1	no	—
96	C₂²	—	296	2	no	—
97	C₂²	—	296	1	no	—
98	C₂²	#11924506	1	1	yes	characteristic
99	C₄	—	2	2	no	—
100	C₄	—	2	1	no	—
101	C₄	—	4	2	no	—
102	C₄	—	148	2	no	—
103	C₄	—	148	2	no	—
104	C₂	—	592	1	no	—
105	C₂	#14557707	1	2	yes	—
106	C₂	#14557708	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃₇⋊#564

C₂

yes

maximal, characteristic, complemented

#14557899

C₂

yes

maximal, characteristic

#14557898

C₂

yes

maximal

#14557274

C₂

yes

maximal, complemented

C₃₇×#330

C₂

yes

maximal, characteristic, complemented

#11924533

C₂²

yes

characteristic

#11924528

C₂²

yes

—

—

—

—

—

—

—

C₂²

yes

characteristic

#125×C₃₇

C₂²

yes

characteristic

#97×C₃₇

C₂²

yes

—

C₂×D₁₄₈

—

C₂×D₁₄₈

—

C₂×C₈×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

D₄

yes

—

C₄²×C₃₇

C₂³

yes

characteristic

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

D₄

yes

—

C₄⋊Dic₃₇

D₄

yes

—

—

—

—

—

C₄×Dic₃₇

—

C₂×#64436

—

C₂²×D₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

D₁₄₈

—

D₁₄₈

—

C₈×C₃₇

—

#64436

—

C₂²×C₃₇

#120

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

Dic₃₇

—

Dic₃₇

—

#1658

—

maximal, Sylow

C₂×C₃₇

#454

yes

—

C₂×C₃₇

#454

yes

characteristic

D₃₇

—

—

—

—

—

D₃₇

yes

characteristic, complemented

C₃₇

#1658

yes

characteristic, complemented, Sylow

—

—

—

—

—

C₂×D₃₇

yes

—

—

—

—

C₂×D₃₇

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₁₄₈

yes

—

C₄²

—

C₄²

C₂²×D₃₇

yes

characteristic

C₂³

—

148

—

D₄

—

148

—

D₄

—

148

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₁₄₈

yes

characteristic

C₂×C₄

D₄×D₃₇

yes

—

C₈

—

C₈

—

148

—

C₂²

—

296

—

C₂²

—

296

—

C₂²

#11924506

yes

characteristic

C₄

—

100

C₄

—

101

C₄

—

102

C₄

—

148

—

103

C₄

—

148

—

104

C₂

—

592

—

105

C₂

#14557707

yes

—

106

C₂

#14557708

yes

characteristic

	subgroup	G/H
center	C₂²	#11924506
commutator, cosocle	C₄²×C₃₇	C₂³
Frattini	C₄²	C₂²×D₃₇
Fitting	C₃₇×#330	C₂
socle	C₂²×C₃₇	#120
prime radical	#11924533	C₂²
hypercenter	#330	D₃₇
hypocenter	C₃₇	#1658
LCS	G ▷ C₄²×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #330

subgroup

G/H

center

C₂²

#11924506

commutator, cosocle

C₄²×C₃₇

C₂³

Frattini

C₄²

C₂²×D₃₇

Fitting

C₃₇×#330

C₂

C₂²×C₃₇

C₂²

D₃₇

C₃₇

G ▷ C₄²×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #330

N	H ≅ G/N	type
1. C₃₇⋊#564	C₂	semidirect
4. #14557274	C₂	semidirect
5. C₃₇×#330	C₂	semidirect
58. #330	D₃₇	semidirect
59. C₃₇	#1658	semidirect

H ≅ G/N

type

1. C₃₇⋊#564

C₂

semidirect

4. #14557274

C₂

semidirect

5. C₃₇×#330

C₂

semidirect

58. #330

D₃₇

semidirect

59. C₃₇

#1658

semidirect

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

subgroup	index
C₃₇⋊#564	2
#14557899	2
#14557898	2
#14557274	2
C₃₇×#330	2
#1658	37

subgroup

index

C₃₇⋊#564

C₃₇×#330

Subgroups — #17993899

Conjugacy classes of subgroups

Complements

Maximal subgroups