Group #17994351 — subgroups

Subgroups — #17994351

Order 4736.

200 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 105 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	#14557488	C₂	1	2	yes	maximal
2	#14557291	C₂	1	1	yes	maximal, characteristic
3	#14557290	C₂	1	1	yes	maximal, characteristic
4	#14557289	C₂	1	2	yes	maximal
5	C₃₇×#405	C₂	1	1	yes	maximal, characteristic
6	C₂×#10596934	C₂²	1	2	yes	—
7	#11924495	C₂²	1	2	yes	—
8	#11924446	C₄	1	4	yes	complemented
9	C₈×Dic₃₇	C₄	1	4	yes	complemented
10	C₂×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
12	#117×C₃₇	C₂²	1	1	yes	characteristic
13	#97×C₃₇	C₄	1	2	yes	complemented
14	C₄×C₈×C₃₇	C₄	1	2	yes	complemented
15	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
16	M₁₆×C₃₇	—	2	2	no	—
17	C₂×C₈×C₃₇	C₂×C₄	1	2	yes	—
18	C₂×C₈×C₃₇	C₂×C₄	1	2	yes	—
19	C₂×C₈×C₃₇	C₂×C₄	1	2	yes	—
20	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
22	C₄⋊C₄×C₃₇	—	2	1	no	—
23	C₂²⋊C₄×C₃₇	—	2	1	no	—
24	C₄²×C₃₇	C₂×C₄	1	2	yes	—
25	C₂²⋊Dic₃₇	—	2	2	no	—
26	C₄⋊Dic₃₇	—	2	2	no	—
27	C₄×Dic₃₇	C₂×C₄	1	4	yes	—
28	#10596934	—	2	2	no	—
29	#10596934	—	2	2	no	—
30	C₂×#64436	C₂×C₄	1	4	yes	—
31	C₂³×C₃₇	#41	1	1	yes	characteristic
32	C₂×C₄×C₃₇	—	2	2	no	—
33	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
34	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₃₇	#41	1	1	yes	characteristic
36	C₂×C₄×C₃₇	#41	1	2	yes	—
37	C₂×Dic₃₇	—	2	4	no	—
38	C₈×C₃₇	—	2	2	no	—
39	C₈×C₃₇	C₄²	1	2	yes	—
40	C₈×C₃₇	C₄²	1	1	yes	characteristic
41	C₈×C₃₇	C₄²	1	1	yes	characteristic
42	#64436	—	2	4	no	—
43	C₂²×C₃₇	—	4	1	no	—
44	C₂²×C₃₇	#117	1	2	yes	—
45	C₂²×C₃₇	#117	1	1	yes	characteristic
46	C₄×C₃₇	—	4	2	no	—
47	C₄×C₃₇	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₃₇	#117	1	2	yes	—
49	C₄×C₃₇	#117	1	1	yes	characteristic
50	Dic₃₇	—	4	4	no	—
51	#1747	—	37	1	no	maximal, Sylow
52	C₂×C₃₇	—	2	2	no	—
53	C₂×C₃₇	—	2	1	no	—
54	C₂×C₃₇	#376	1	1	yes	characteristic
55	#405	—	37	2	no	—
56	#405	D₃₇	1	1	yes	characteristic
57	#338	—	37	1	no	—
58	H(3,ℤ/4)	—	37	1	no	—
59	H(3,ℤ/4)	—	37	2	no	—
60	C₃₇	#1747	1	1	yes	characteristic, complemented, Sylow
61	C₂×M₁₆	—	37	2	no	—
62	C₂×M₁₆	C₂×D₃₇	1	1	yes	characteristic
63	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
64	#117	—	37	2	no	—
65	#117	C₂×D₃₇	1	1	yes	characteristic
66	#97	—	37	4	no	—
67	#97	Dic₃₇	1	2	yes	complemented
68	C₄×C₈	—	37	4	no	—
69	C₄×C₈	Dic₃₇	1	2	yes	complemented
70	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
71	M₁₆	—	2	2	no	—
72	M₁₆	—	74	2	no	—
73	M₁₆	—	74	2	no	—
74	C₂×C₈	—	37	4	no	—
75	C₂×C₈	C₄×D₃₇	1	2	yes	—
76	C₂×C₈	C₄×D₃₇	1	2	yes	—
77	C₂×C₈	C₂×Dic₃₇	1	2	yes	—
78	C₂×C₈	C₂×Dic₃₇	1	1	yes	characteristic
79	C₂×C₈	C₂×Dic₃₇	1	1	yes	characteristic
80	C₄⋊C₄	—	2	1	no	—
81	C₄⋊C₄	—	74	2	no	—
82	C₂²⋊C₄	—	2	1	no	—
83	C₂²⋊C₄	—	74	2	no	—
84	C₄²	—	37	4	no	—
85	C₄²	C₂×Dic₃₇	1	2	yes	—
86	C₂³	#10596962	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	74	4	no	—
89	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
90	C₂×C₄	#10596962	1	1	yes	characteristic
91	C₂×C₄	#10596962	1	2	yes	—
92	C₂×C₄	C₂²×Dic₃₇	1	1	yes	characteristic
93	C₈	—	2	2	no	—
94	C₈	—	74	4	no	—
95	C₈	C₄×Dic₃₇	1	1	yes	characteristic
96	C₈	C₄×Dic₃₇	1	2	yes	—
97	C₈	C₄×Dic₃₇	1	1	yes	characteristic
98	C₂²	—	4	1	no	—
99	C₂²	#11924493	1	2	yes	—
100	C₂²	#11924495	1	1	yes	characteristic
101	C₄	—	4	2	no	—
102	C₄	—	148	4	no	—
103	C₄	C₂×C₄×Dic₃₇	1	1	yes	characteristic
104	C₄	#11924493	1	2	yes	—
105	C₄	#11924495	1	1	yes	characteristic
106	C₂	—	2	1	no	—
107	C₂	—	2	2	no	—
108	C₂	#14557379	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14557488

C₂

yes

maximal

#14557291

C₂

yes

maximal, characteristic

#14557290

C₂

yes

maximal, characteristic

#14557289

C₂

yes

maximal

C₃₇×#405

C₂

yes

maximal, characteristic

C₂×#10596934

C₂²

yes

—

#11924495

C₂²

yes

—

#11924446

C₄

yes

complemented

C₈×Dic₃₇

C₄

yes

complemented

C₂×M₁₆×C₃₇

C₂²

yes

characteristic

C₂²×C₈×C₃₇

C₂²

yes

characteristic

#117×C₃₇

C₂²

yes

characteristic

#97×C₃₇

C₄

yes

complemented

C₄×C₈×C₃₇

C₄

yes

complemented

C₂²×C₄×C₃₇

C₂³

yes

characteristic

M₁₆×C₃₇

—

C₂×C₈×C₃₇

C₂×C₄

yes

—

C₂×C₈×C₃₇

C₂×C₄

yes

—

C₂×C₈×C₃₇

C₂×C₄

yes

—

C₂×C₈×C₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₃₇

C₂×C₄

yes

characteristic

C₄⋊C₄×C₃₇

—

C₂²⋊C₄×C₃₇

—

C₄²×C₃₇

C₂×C₄

yes

—

C₂²⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄×Dic₃₇

C₂×C₄

yes

—

#10596934

—

#10596934

—

C₂×#64436

C₂×C₄

yes

—

C₂³×C₃₇

#41

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

C₂×C₄×C₃₇

yes

C₂×C₄×C₃₇

yes

—

C₂×Dic₃₇

—

C₈×C₃₇

—

C₈×C₃₇

C₄²

yes

—

C₈×C₃₇

C₄²

yes

characteristic

C₈×C₃₇

C₄²

yes

characteristic

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

#117

yes

—

C₂²×C₃₇

#117

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

C₂×C₄²

yes

characteristic

C₄×C₃₇

#117

yes

—

C₄×C₃₇

#117

yes

characteristic

Dic₃₇

—

#1747

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

yes

—

D₃₇

yes

—

—

—

C₃₇

#1747

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃₇

yes

characteristic

C₂²×C₈

C₂×D₃₇

yes

characteristic

#117

—

#117

C₂×D₃₇

yes

characteristic

#97

—

#97

Dic₃₇

yes

complemented

C₄×C₈

—

C₄×C₈

Dic₃₇

yes

complemented

C₂²×C₄

C₂²×D₃₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₄×D₃₇

yes

—

C₂×C₈

C₄×D₃₇

yes

—

C₂×C₈

C₂×Dic₃₇

yes

—

C₂×C₈

C₂×Dic₃₇

yes

characteristic

C₂×C₈

C₂×Dic₃₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×Dic₃₇

yes

—

C₂³

#10596962

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄×D₃₇

yes

—

C₂×C₄

#10596962

yes

characteristic

C₂×C₄

#10596962

yes

—

C₂×C₄

C₂²×Dic₃₇

yes

characteristic

C₈

—

C₈

—

C₈

C₄×Dic₃₇

yes

characteristic

C₈

C₄×Dic₃₇

yes

—

C₈

C₄×Dic₃₇

yes

characteristic

C₂²

—

C₂²

#11924493

yes

—

100

C₂²

#11924495

yes

characteristic

101

C₄

—

102

C₄

—

148

—

103

C₄

C₂×C₄×Dic₃₇

yes

characteristic

104

C₄

#11924493

yes

—

105

C₄

#11924495

yes

characteristic

106

C₂

—

107

C₂

—

108

C₂

#14557379

yes

characteristic

	subgroup	G/H
center	C₈	C₄×Dic₃₇
commutator	C₄×C₃₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₃₇×#405	C₂
socle	C₂×C₃₇	#376
prime radical	C₂³×C₃₇	#41
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	#405	D₃₇
hypocenter	C₃₇	#1747
LCS	G ▷ C₄×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₈ ◁ #405

subgroup

G/H

center

C₈

C₄×Dic₃₇

commutator

C₄×C₃₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₃₇×#405

C₂

C₂×C₃₇

C₂³×C₃₇

C₂²×C₄×C₃₇

C₂³

D₃₇

C₃₇

G ▷ C₄×C₃₇ ▷ C₃₇

C₁ ◁ C₈ ◁ #405

N	H ≅ G/N	type
8. #11924446	C₄	semidirect
9. C₈×Dic₃₇	C₄	semidirect
13. #97×C₃₇	C₄	semidirect
14. C₄×C₈×C₃₇	C₄	semidirect
60. C₃₇	#1747	semidirect
67. #97	Dic₃₇	semidirect
69. C₄×C₈	Dic₃₇	semidirect

H ≅ G/N

type

8. #11924446

C₄

semidirect

9. C₈×Dic₃₇

C₄

semidirect

13. #97×C₃₇

C₄

semidirect

14. C₄×C₈×C₃₇

C₄

semidirect

60. C₃₇

#1747

semidirect

67. #97

Dic₃₇

semidirect

69. C₄×C₈

Dic₃₇

semidirect

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

subgroup	index
#14557488	2
#14557291	2
#14557290	2
#14557289	2
C₃₇×#405	2
#1747	37

subgroup

index

C₃₇×#405

Subgroups — #17994351

Conjugacy classes of subgroups

Complements

Maximal subgroups