Group #18009478 — subgroups

Subgroups — #18009478

Order 4736.

190 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 48 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	D₃₇⋊(C₂²⋊C₈)	C₂	1	2	yes	maximal
2	#14557496	C₂	1	1	yes	maximal, characteristic
3	C₂×D₃₇⋊C₈	—	2	2	no	—
4	#11924593	C₄	1	4	yes	—
5	C₂²⋊#64438	—	2	2	no	—
6	C₂²×C₄×D₃₇	C₂²	1	1	yes	characteristic
7	C₂×#10596934	C₄	1	1	yes	characteristic
8	#11924448	—	2	2	no	—
9	C₂×M₁₆×C₃₇	C₄	1	1	yes	characteristic
10	C₂³×D₃₇	Q₈	1	1	yes	characteristic
11	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
12	C₂²×Dic₃₇	D₄	1	1	yes	characteristic
13	C₂×C₄×D₃₇	—	4	1	no	—
14	C₂×C₄×D₃₇	—	4	1	no	—
15	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	—
16	C₂×C₄×D₃₇	D₄	1	2	yes	—
17	C₂×#64438	—	2	4	no	—
18	D₃₇⋊C₈	—	4	2	no	—
19	D₃₇⋊C₈	—	4	2	no	—
20	M₁₆×C₃₇	—	4	1	no	—
21	C₂×C₈×C₃₇	—	2	1	no	—
22	#10596934	—	4	1	no	—
23	C₂×#64436	—	2	1	no	—
24	C₂³×C₃₇	C₄⋊C₄	1	1	yes	characteristic
25	C₂²×D₃₇	—	2	4	no	—
26	C₂²×D₃₇	—	4	1	no	—
27	C₂²×D₃₇	C₄⋊C₄	1	2	yes	—
28	C₂×C₄×C₃₇	—	2	1	no	—
29	C₂×C₄×C₃₇	—	2	1	no	—
30	C₂×C₄×C₃₇	C₄²	1	1	yes	characteristic
31	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×Dic₃₇	—	4	1	no	—
33	C₂×Dic₃₇	C₂²⋊C₄	1	2	yes	—
34	C₄×D₃₇	—	2	2	no	—
35	C₄×D₃₇	—	2	2	no	—
36	C₄×D₃₇	—	4	2	no	—
37	#64438	—	4	4	no	—
38	C₈×C₃₇	—	4	1	no	—
39	#64436	—	4	1	no	—
40	C₂²×C₃₇	—	2	2	no	—
41	C₂²×C₃₇	—	2	1	no	—
42	C₂²×C₃₇	#95	1	1	yes	characteristic
43	C₂×D₃₇	—	2	2	no	—
44	C₂×D₃₇	—	2	2	no	—
45	C₂×D₃₇	—	4	4	no	—
46	C₂×D₃₇	#108	1	4	yes	—
47	C₄×C₃₇	—	2	1	no	—
48	C₄×C₃₇	#99	1	1	yes	characteristic
49	C₄×C₃₇	#99	1	1	yes	characteristic
50	Dic₃₇	—	2	2	no	—
51	#1298	—	37	1	no	maximal, Sylow
52	C₂×C₃₇	—	4	1	no	—
53	C₂×C₃₇	#341	1	2	yes	—
54	C₂×C₃₇	#342	1	1	yes	characteristic
55	D₃₇	—	2	4	no	—
56	#407	—	37	1	no	—
57	C₂×C₂²⋊C₈	—	37	2	no	—
58	C₃₇	#1298	1	1	yes	characteristic, complemented, Sylow
59	C₂³×C₄	—	37	1	no	—
60	C₂×M₁₆	—	37	1	no	—
61	C₂×M₁₆	#3866	1	1	yes	characteristic
62	C₂²×C₈	—	74	2	no	—
63	C₂²⋊C₈	—	37	4	no	—
64	C₂²⋊C₈	—	74	2	no	—
65	C₂²⋊C₈	—	74	2	no	—
66	C₂⁴	—	37	1	no	—
67	C₂²×C₄	—	37	1	no	—
68	C₂²×C₄	—	37	2	no	—
69	C₂²×C₄	—	37	2	no	—
70	C₂²×C₄	—	148	1	no	—
71	C₂²×C₄	—	148	1	no	—
72	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
73	M₁₆	—	4	1	no	—
74	M₁₆	—	148	1	no	—
75	C₂×C₈	—	2	1	no	—
76	C₂×C₈	—	74	4	no	—
77	C₂×C₈	—	74	1	no	—
78	C₂×C₈	—	148	2	no	—
79	C₂×C₈	—	148	2	no	—
80	C₂³	—	37	2	no	—
81	C₂³	—	74	4	no	—
82	C₂³	—	148	1	no	—
83	C₂³	C₄⋊#3866	1	1	yes	characteristic
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	37	2	no	—
87	C₂×C₄	—	74	2	no	—
88	C₂×C₄	—	74	2	no	—
89	C₂×C₄	—	148	2	no	—
90	C₂×C₄	—	148	1	no	—
91	C₂×C₄	C₄×#3866	1	1	yes	characteristic
92	C₂×C₄	C₂²⋊#3866	1	1	yes	characteristic
93	C₈	—	4	1	no	—
94	C₈	—	148	4	no	—
95	C₈	—	148	1	no	—
96	C₂²	—	2	2	no	—
97	C₂²	—	2	1	no	—
98	C₂²	—	37	4	no	—
99	C₂²	—	74	2	no	—
100	C₂²	—	74	2	no	—
101	C₂²	—	148	4	no	—
102	C₂²	(C₂×#3866)⋊C₄	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	74	2	no	—
105	C₄	#11924594	1	1	yes	characteristic
106	C₄	#11924595	1	1	yes	characteristic
107	C₂	—	4	1	no	—
108	C₂	—	74	4	no	—
109	C₂	#14558408	1	2	yes	—
110	C₂	#14558411	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₃₇⋊(C₂²⋊C₈)

C₂

yes

maximal

#14557496

C₂

yes

maximal, characteristic

C₂×D₃₇⋊C₈

—

#11924593

C₄

yes

—

C₂²⋊#64438

—

C₂²×C₄×D₃₇

C₂²

yes

characteristic

C₂×#10596934

C₄

yes

characteristic

#11924448

—

C₂×M₁₆×C₃₇

C₄

yes

characteristic

C₂³×D₃₇

Q₈

yes

characteristic

C₂²×C₄×C₃₇

C₂×C₄

yes

characteristic

C₂²×Dic₃₇

D₄

yes

characteristic

C₂×C₄×D₃₇

—

C₂×C₄×D₃₇

—

C₂×C₄×D₃₇

C₂×C₄

yes

—

C₂×C₄×D₃₇

D₄

yes

—

C₂×#64438

—

D₃₇⋊C₈

—

D₃₇⋊C₈

—

M₁₆×C₃₇

—

C₂×C₈×C₃₇

—

#10596934

—

C₂×#64436

—

C₂³×C₃₇

C₄⋊C₄

yes

characteristic

C₂²×D₃₇

—

C₂²×D₃₇

—

C₂²×D₃₇

C₄⋊C₄

yes

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₄²

yes

characteristic

C₂×C₄×C₃₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂²⋊C₄

yes

—

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

—

#64438

—

C₈×C₃₇

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

#95

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

#108

yes

—

C₄×C₃₇

—

C₄×C₃₇

yes

C₄×C₃₇

yes

Dic₃₇

—

#1298

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

#341

yes

—

C₂×C₃₇

#342

yes

characteristic

D₃₇

—

#407

—

C₂×C₂²⋊C₈

—

C₃₇

#1298

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂×M₁₆

—

C₂×M₁₆

#3866

yes

characteristic

C₂²×C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

148

—

C₂²×C₄

—

148

—

C₂²×C₄

C₂×#3866

yes

characteristic

M₁₆

—

M₁₆

—

148

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

148

—

C₂×C₈

—

148

—

C₂³

—

C₂³

—

C₂³

—

148

—

C₂³

C₄⋊#3866

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

148

—

C₂×C₄

—

148

—

C₂×C₄

C₄×#3866

yes

characteristic

C₂×C₄

C₂²⋊#3866

yes

characteristic

C₈

—

C₈

—

148

—

C₈

—

148

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

—

148

—

102

C₂²

(C₂×#3866)⋊C₄

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

#11924594

yes

characteristic

106

C₄

#11924595

yes

characteristic

107

C₂

—

108

C₂

—

109

C₂

#14558408

yes

—

110

C₂

#14558411

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊#3866
commutator	C₂×C₄×C₃₇	C₄²
Frattini	C₂²×C₄	C₂×#3866
Fitting	C₂×M₁₆×C₃₇	C₄
socle	C₂²×C₃₇	#95
prime radical	C₂³×D₃₇	Q₈
cosocle	C₂²×C₄×D₃₇	C₂²
hypercenter	C₂×M₁₆	#3866
hypocenter	C₃₇	#1298
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

subgroup

G/H

center

C₂×C₄

C₂²⋊#3866

commutator

C₂×C₄×C₃₇

C₄²

Frattini

C₂²×C₄

C₂×#3866

Fitting

C₂×M₁₆×C₃₇

C₄

socle

C₂²×C₃₇

#95

prime radical

C₂³×D₃₇

Q₈

cosocle

C₂²×C₄×D₃₇

C₂²

C₂×M₁₆

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
58. C₃₇	#1298	semidirect

H ≅ G/N

type

58. C₃₇

#1298

semidirect

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

subgroup	index
D₃₇⋊(C₂²⋊C₈)	2
#14557496	2
#1298	37

subgroup

index

D₃₇⋊(C₂²⋊C₈)

#14557496

#1298

Subgroups — #18009478

Conjugacy classes of subgroups

Complements

Maximal subgroups