Group #18009659 — subgroups

Subgroups — #18009659

Order 4736.

132 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 50 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×D₃₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#14558543	C₂	1	1	yes	maximal, characteristic
3	#108×D₃₇	C₂	1	1	yes	maximal, characteristic
4	#11924615	—	2	1	no	—
5	D₃₇⋊C₁₆	—	2	1	no	—
6	D₃₇⋊C₁₆	C₄	1	2	yes	complemented
7	C₂×#10596927	—	2	1	no	—
8	#11924605	C₄	1	2	yes	—
9	#11924604	—	2	1	no	—
10	M₁₆×D₃₇	—	2	1	no	—
11	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
12	#11924472	C₄	1	1	yes	characteristic, complemented
13	#11924471	—	2	1	no	—
14	#108×C₃₇	C₄	1	1	yes	characteristic, complemented
15	C₂×C₄×D₃₇	D₄	1	1	yes	characteristic
16	C₂²⋊#3866	—	4	1	no	—
17	C₄⋊#3866	—	2	2	no	—
18	C₄×#3866	—	4	1	no	—
19	M₁₆×C₃₇	—	2	1	no	—
20	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
21	#10596934	—	2	1	no	—
22	C₂×#64436	D₄	1	1	yes	characteristic
23	#10596929	—	4	1	no	—
24	C₈×D₃₇	—	4	1	no	—
25	C₈×D₃₇	C₂×C₄	1	1	yes	characteristic
26	C₈×D₃₇	C₂×C₄	1	1	yes	characteristic
27	C₈×D₃₇	D₄	1	1	yes	characteristic
28	C₈×D₃₇	Q₈	1	1	yes	characteristic
29	#10596927	—	2	2	no	—
30	C₂²×D₃₇	SD₁₆	1	1	yes	characteristic
31	C₂×#3866	—	4	2	no	—
32	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×Dic₃₇	Q₁₆	1	1	yes	characteristic
34	C₄×D₃₇	C₂²⋊C₄	1	1	yes	characteristic
35	C₄×D₃₇	C₄⋊C₄	1	1	yes	characteristic
36	C₄×D₃₇	D₈	1	1	yes	characteristic
37	C₄×D₃₇	SD₁₆	1	1	yes	characteristic
38	C₈×C₃₇	—	4	1	no	—
39	C₈×C₃₇	C₄²	1	1	yes	characteristic
40	C₈×C₃₇	C₄⋊C₄	1	1	yes	characteristic
41	#64436	—	4	1	no	—
42	#64436	C₂²⋊C₄	1	1	yes	characteristic
43	#64436	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₃₇	#103	1	1	yes	characteristic
45	C₂×D₃₇	—	2	2	no	—
46	C₂×D₃₇	#102	1	1	yes	characteristic
47	C₂×D₃₇	#106	1	1	yes	characteristic
48	#3866	—	8	2	no	—
49	C₄×C₃₇	#95	1	1	yes	characteristic
50	C₄×C₃₇	#102	1	1	yes	characteristic
51	Dic₃₇	#103	1	1	yes	characteristic
52	Dic₃₇	#107	1	1	yes	characteristic
53	#1382	—	37	1	no	maximal, Sylow
54	C₂×C₃₇	—	2	1	no	—
55	C₂×C₃₇	#340	1	1	yes	characteristic
56	D₃₇	—	2	1	no	—
57	D₃₇	#359	1	2	yes	—
58	C₂²×C₁₆	—	37	1	no	—
59	C₂×#108	—	37	1	no	—
60	#427	—	37	1	no	—
61	C₃₇	#1382	1	1	yes	characteristic, complemented, Sylow
62	C₂×M₁₆	—	74	1	no	—
63	C₂²×C₈	—	37	1	no	—
64	#117	—	74	1	no	—
65	C₂×C₁₆	—	37	2	no	—
66	C₂×C₁₆	—	74	1	no	—
67	C₂×C₁₆	—	74	1	no	—
68	#108	—	37	1	no	—
69	#108	—	74	1	no	—
70	#108	#3866	1	1	yes	characteristic, complemented
71	#107	—	37	2	no	—
72	#106	—	74	1	no	—
73	C₂²×C₄	—	37	1	no	—
74	M₁₆	—	2	1	no	—
75	M₁₆	—	74	1	no	—
76	M₁₆	—	148	1	no	—
77	C₂×C₈	—	37	1	no	—
78	C₂×C₈	—	37	1	no	—
79	C₂×C₈	—	37	1	no	—
80	C₂×C₈	—	37	1	no	—
81	C₂×C₈	—	37	1	no	—
82	C₂×C₈	—	148	1	no	—
83	C₂×C₈	C₂×#3866	1	1	yes	characteristic
84	C₄⋊C₄	—	74	2	no	—
85	C₂²⋊C₄	—	148	1	no	—
86	C₄²	—	148	1	no	—
87	C₁₆	—	74	2	no	—
88	C₂³	—	37	1	no	—
89	C₂×C₄	—	37	1	no	—
90	C₂×C₄	—	37	1	no	—
91	C₂×C₄	—	37	1	no	—
92	C₂×C₄	—	37	1	no	—
93	C₂×C₄	—	37	1	no	—
94	C₂×C₄	—	148	2	no	—
95	C₂×C₄	C₂²⋊#3866	1	1	yes	characteristic
96	C₈	—	4	1	no	—
97	C₈	—	37	1	no	—
98	C₈	—	37	1	no	—
99	C₈	—	148	1	no	—
100	C₈	C₄×#3866	1	1	yes	characteristic
101	C₈	C₄⋊#3866	1	1	yes	characteristic
102	C₂²	—	37	1	no	—
103	C₂²	—	37	1	no	—
104	C₂²	—	74	2	no	—
105	C₂²	#11924599	1	1	yes	characteristic
106	C₄	—	37	1	no	—
107	C₄	—	37	1	no	—
108	C₄	—	296	2	no	—
109	C₄	(C₂×#3866)⋊C₄	1	1	yes	characteristic
110	C₄	#11924598	1	1	yes	characteristic
111	C₂	—	2	1	no	—
112	C₂	—	37	2	no	—
113	C₂	—	74	1	no	—
114	C₂	#14558406	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₃₇⋊C₁₆

C₂

yes

maximal, characteristic

#14558543

C₂

yes

maximal, characteristic

#108×D₃₇

C₂

yes

maximal, characteristic

#11924615

—

D₃₇⋊C₁₆

—

D₃₇⋊C₁₆

C₄

yes

complemented

C₂×#10596927

—

#11924605

C₄

yes

—

#11924604

—

M₁₆×D₃₇

—

C₂×C₈×D₃₇

C₂²

yes

characteristic

#11924472

C₄

yes

characteristic, complemented

#11924471

—

#108×C₃₇

C₄

yes

characteristic, complemented

C₂×C₄×D₃₇

D₄

yes

characteristic

C₂²⋊#3866

—

C₄⋊#3866

—

C₄×#3866

—

M₁₆×C₃₇

—

C₂×C₈×C₃₇

C₂×C₄

yes

characteristic

#10596934

—

C₂×#64436

D₄

yes

characteristic

#10596929

—

C₈×D₃₇

—

C₈×D₃₇

C₂×C₄

yes

characteristic

C₈×D₃₇

C₂×C₄

yes

characteristic

C₈×D₃₇

D₄

yes

characteristic

C₈×D₃₇

Q₈

yes

characteristic

#10596927

—

C₂²×D₃₇

SD₁₆

yes

characteristic

C₂×#3866

—

C₂×C₄×C₃₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₃₇

Q₁₆

yes

characteristic

C₄×D₃₇

C₂²⋊C₄

yes

characteristic

C₄×D₃₇

C₄⋊C₄

yes

characteristic

C₄×D₃₇

D₈

yes

characteristic

C₄×D₃₇

SD₁₆

yes

characteristic

C₈×C₃₇

—

C₈×C₃₇

C₄²

yes

characteristic

C₈×C₃₇

C₄⋊C₄

yes

characteristic

#64436

—

#64436

C₂²⋊C₄

yes

characteristic

#64436

C₄⋊C₄

yes

characteristic

C₂²×C₃₇

#103

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

yes

C₂×D₃₇

yes

—

C₄×C₃₇

yes

C₄×C₃₇

yes

Dic₃₇

yes

Dic₃₇

yes

—

C₂×C₃₇

—

C₂×C₃₇

#340

yes

characteristic

D₃₇

—

D₃₇

#359

yes

—

C₂²×C₁₆

—

C₂×#108

—

#427

—

C₃₇

#1382

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

#117

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

yes

characteristic, complemented

#107

—

#106

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

M₁₆

—

148

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

148

—

C₂×C₈

C₂×#3866

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

148

—

C₄²

—

148

—

C₁₆

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

148

—

C₂×C₄

C₂²⋊#3866

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

148

—

100

C₈

C₄×#3866

yes

characteristic

101

C₈

C₄⋊#3866

yes

characteristic

102

C₂²

—

103

C₂²

—

104

C₂²

—

105

C₂²

#11924599

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

296

—

109

C₄

(C₂×#3866)⋊C₄

yes

characteristic

110

C₄

#11924598

yes

characteristic

111

C₂

—

112

C₂

—

113

C₂

—

114

C₂

#14558406

yes

characteristic

	subgroup	G/H
center	C₄	#11924598
commutator	C₈×C₃₇	C₄²
Frattini	C₂×C₈	C₂×#3866
Fitting	#108×C₃₇	C₄
socle	C₂×C₃₇	#340
prime radical	C₂²×D₃₇	SD₁₆
cosocle	C₂×C₈×D₃₇	C₂²
hypercenter	#108	#3866
hypocenter	C₃₇	#1382
LCS	G ▷ C₈×C₃₇ ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #108

subgroup

G/H

center

C₄

#11924598

commutator

C₈×C₃₇

C₄²

Frattini

C₂×C₈

C₂×#3866

Fitting

#108×C₃₇

C₄

socle

C₂×C₃₇

#340

prime radical

C₂²×D₃₇

SD₁₆

cosocle

C₂×C₈×D₃₇

C₂²

C₃₇

G ▷ C₈×C₃₇ ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #108

N	H ≅ G/N	type
6. D₃₇⋊C₁₆	C₄	semidirect
12. #11924472	C₄	semidirect
14. #108×C₃₇	C₄	semidirect
61. C₃₇	#1382	semidirect
70. #108	#3866	semidirect

H ≅ G/N

type

6. D₃₇⋊C₁₆

C₄

semidirect

12. #11924472

C₄

semidirect

14. #108×C₃₇

C₄

semidirect

61. C₃₇

#1382

semidirect

70. #108

#3866

semidirect

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

subgroup	index
C₂×D₃₇⋊C₁₆	2
#14558543	2
#108×D₃₇	2
#1382	37

subgroup

index

C₂×D₃₇⋊C₁₆

#14558543

#108×D₃₇

#1382

Subgroups — #18009659

Conjugacy classes of subgroups

Complements

Maximal subgroups