Group #18009800 — subgroups

Subgroups — #18009800

Order 4736.

212 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 106 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	#14558551	C₂	1	2	yes	maximal
2	#14558374	C₂	1	2	yes	maximal, complemented
3	C₈×#64438	C₂	1	2	yes	maximal, complemented
4	C₄×C₈×D₃₇	C₂	1	1	yes	maximal, characteristic
5	#11924603	C₄	1	4	yes	—
6	#11924593	C₄	1	4	yes	—
7	C₄×#64438	C₂²	1	4	yes	—
8	C₂×C₈×D₃₇	C₄	1	1	yes	characteristic
9	C₂×C₈×D₃₇	C₄	1	1	yes	characteristic
10	C₄²×D₃₇	C₂²	1	1	yes	characteristic
11	C₄×#64436	C₄	1	1	yes	characteristic
12	C₈×Dic₃₇	C₂²	1	1	yes	characteristic
13	C₈×Dic₃₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₃₇	C₄	1	1	yes	characteristic
15	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	—
16	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
17	C₂×#64438	C₂×C₄	1	8	yes	—
18	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
20	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
21	C₄×Dic₃₇	C₂×C₄	1	2	yes	—
22	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
23	C₂×#64436	C₂×C₄	1	1	yes	characteristic
24	C₂×#64436	C₂×C₄	1	1	yes	characteristic
25	C₈×D₃₇	C₈	1	4	yes	complemented
26	C₈×D₃₇	C₈	1	4	yes	complemented
27	C₂²×D₃₇	C₄²	1	1	yes	characteristic
28	C₂×C₄×C₃₇	C₄²	1	2	yes	—
29	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
30	C₂×Dic₃₇	C₄²	1	1	yes	characteristic
31	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
32	C₄×D₃₇	—	2	4	no	—
33	C₄×D₃₇	C₂×C₈	1	2	yes	—
34	C₄×D₃₇	C₂×C₈	1	2	yes	—
35	#64438	—	2	8	no	—
36	C₈×C₃₇	C₂×C₈	1	2	yes	—
37	C₈×C₃₇	C₂×C₈	1	2	yes	—
38	#64436	C₂×C₈	1	2	yes	—
39	#64436	C₂×C₈	1	2	yes	—
40	C₂²×C₃₇	C₂×C₄²	1	1	yes	characteristic
41	C₂×D₃₇	—	2	2	no	—
42	C₂×D₃₇	C₄×C₈	1	2	yes	—
43	C₄×C₃₇	—	2	2	no	—
44	C₄×C₃₇	C₂²×C₈	1	1	yes	characteristic
45	C₄×C₃₇	C₂²×C₈	1	1	yes	characteristic
46	Dic₃₇	C₄×C₈	1	2	yes	—
47	Dic₃₇	#131	1	4	yes	—
48	#1450	—	37	1	no	maximal, Sylow
49	C₂×C₃₇	C₂×C₄×C₈	1	1	yes	characteristic
50	C₂×C₃₇	#405	1	2	yes	—
51	D₃₇	—	2	2	no	—
52	#431	—	37	2	no	—
53	C₂×C₄×C₈	—	37	1	no	—
54	#322	—	37	2	no	—
55	C₈²	—	37	2	no	—
56	C₃₇	#1450	1	1	yes	characteristic, complemented, Sylow
57	C₂²×C₈	—	37	1	no	—
58	C₂²×C₈	—	37	1	no	—
59	C₂×C₄²	—	37	1	no	—
60	C₄⋊C₈	—	37	4	no	—
61	C₂²⋊C₈	—	37	4	no	—
62	C₄×C₈	—	37	4	no	—
63	C₄×C₈	—	37	1	no	—
64	C₄×C₈	—	37	1	no	—
65	C₄×C₈	—	37	1	no	—
66	C₄×C₈	#3866	1	1	yes	characteristic
67	C₂²×C₄	—	37	1	no	—
68	C₂²×C₄	—	37	2	no	—
69	C₂×C₈	—	37	1	no	—
70	C₂×C₈	—	37	1	no	—
71	C₂×C₈	—	37	4	no	—
72	C₂×C₈	—	37	4	no	—
73	C₂×C₈	—	37	8	no	—
74	C₂×C₈	C₂×#3866	1	1	yes	characteristic
75	C₂×C₈	C₂×#3866	1	1	yes	characteristic
76	C₄²	—	37	2	no	—
77	C₄²	—	37	1	no	—
78	C₄²	C₂×#3866	1	1	yes	characteristic
79	C₂³	—	37	1	no	—
80	C₂×C₄	—	37	2	no	—
81	C₂×C₄	—	37	1	no	—
82	C₂×C₄	—	37	2	no	—
83	C₂×C₄	—	37	2	no	—
84	C₂×C₄	—	74	4	no	—
85	C₂×C₄	C₄×#3866	1	2	yes	—
86	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
87	C₈	—	37	2	no	—
88	C₈	—	37	2	no	—
89	C₈	—	74	8	no	—
90	C₈	D₃₇⋊C₈	1	2	yes	—
91	C₈	D₃₇⋊C₈	1	2	yes	—
92	C₂²	—	37	2	no	—
93	C₂²	—	74	2	no	—
94	C₂²	C₂×C₄×#3866	1	1	yes	characteristic
95	C₄	—	2	2	no	—
96	C₄	—	37	2	no	—
97	C₄	—	37	4	no	—
98	C₄	C₂×D₃₇⋊C₈	1	1	yes	characteristic
99	C₄	C₂×D₃₇⋊C₈	1	1	yes	characteristic
100	C₂	—	74	2	no	—
101	C₂	C₄×D₃₇⋊C₈	1	1	yes	characteristic
102	C₂	#14558496	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14558551

C₂

yes

maximal

#14558374

C₂

yes

maximal, complemented

C₈×#64438

C₂

yes

maximal, complemented

C₄×C₈×D₃₇

C₂

yes

maximal, characteristic

#11924603

C₄

yes

—

#11924593

C₄

yes

—

C₄×#64438

C₂²

yes

—

C₂×C₈×D₃₇

C₄

yes

characteristic

C₂×C₈×D₃₇

C₄

yes

characteristic

C₄²×D₃₇

C₂²

yes

characteristic

C₄×#64436

C₄

yes

characteristic

C₈×Dic₃₇

C₂²

yes

characteristic

C₈×Dic₃₇

C₂²

yes

characteristic

C₄×C₈×C₃₇

C₄

yes

characteristic

C₂×C₄×D₃₇

C₂×C₄

yes

—

C₂×C₄×D₃₇

C₂×C₄

yes

characteristic

C₂×#64438

C₂×C₄

yes

—

C₂×C₈×C₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₃₇

C₂×C₄

yes

characteristic

C₄²×C₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

C₂×C₄

yes

—

C₄×Dic₃₇

C₂³

yes

characteristic

C₂×#64436

C₂×C₄

yes

characteristic

C₂×#64436

C₂×C₄

yes

characteristic

C₈×D₃₇

C₈

yes

complemented

C₈×D₃₇

C₈

yes

complemented

C₂²×D₃₇

C₄²

yes

characteristic

C₂×C₄×C₃₇

C₄²

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

C₄²

yes

characteristic

C₂×Dic₃₇

C₂²×C₄

yes

—

C₄×D₃₇

—

C₄×D₃₇

C₂×C₈

yes

—

C₄×D₃₇

C₂×C₈

yes

—

#64438

—

C₈×C₃₇

C₂×C₈

yes

—

C₈×C₃₇

C₂×C₈

yes

—

#64436

C₂×C₈

yes

—

#64436

C₂×C₈

yes

—

C₂²×C₃₇

C₂×C₄²

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

C₄×C₈

yes

—

C₄×C₃₇

—

C₄×C₃₇

C₂²×C₈

yes

characteristic

C₄×C₃₇

C₂²×C₈

yes

characteristic

Dic₃₇

C₄×C₈

yes

—

Dic₃₇

#131

yes

—

#1450

—

maximal, Sylow

C₂×C₃₇

C₂×C₄×C₈

yes

characteristic

C₂×C₃₇

#405

yes

—

D₃₇

—

#431

—

C₂×C₄×C₈

—

#322

—

C₈²

—

C₃₇

#1450

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

#3866

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₄²

—

C₄²

—

C₄²

C₂×#3866

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×#3866

yes

—

C₂×C₄

C₂²×#3866

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

D₃₇⋊C₈

yes

—

C₈

D₃₇⋊C₈

yes

—

C₂²

—

C₂²

—

C₂²

C₂×C₄×#3866

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

C₂×D₃₇⋊C₈

yes

characteristic

C₄

C₂×D₃₇⋊C₈

yes

characteristic

100

C₂

—

101

C₂

C₄×D₃₇⋊C₈

yes

characteristic

102

C₂

#14558496

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×#3866
commutator	C₂×C₃₇	C₂×C₄×C₈
Frattini	C₂×C₄	C₂²×#3866
Fitting	C₄×C₈×C₃₇	C₄
socle	C₂²×C₃₇	C₂×C₄²
prime radical	C₂²×D₃₇	C₄²
cosocle	C₄×Dic₃₇	C₂³
hypercenter	C₄×C₈	#3866
hypocenter	C₃₇	#1450
LCS	G ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈

subgroup

G/H

center

C₂×C₈

C₂×#3866

commutator

C₂×C₃₇

C₂×C₄×C₈

Frattini

C₂×C₄

C₂²×#3866

Fitting

C₄×C₈×C₃₇

C₄

socle

C₂²×C₃₇

C₂×C₄²

prime radical

C₂²×D₃₇

C₄²

cosocle

C₄×Dic₃₇

C₂³

C₄×C₈

C₃₇

G ▷ C₂×C₃₇ ▷ C₃₇

C₁ ◁ C₂×C₈ ◁ C₄×C₈

N	H ≅ G/N	type
2. #14558374	C₂	semidirect
3. C₈×#64438	C₂	semidirect
25. C₈×D₃₇	C₈	semidirect
26. C₈×D₃₇	C₈	semidirect
56. C₃₇	#1450	semidirect

H ≅ G/N

type

2. #14558374

C₂

semidirect

3. C₈×#64438

C₂

semidirect

25. C₈×D₃₇

C₈

semidirect

26. C₈×D₃₇

C₈

semidirect

56. C₃₇

#1450

semidirect

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

subgroup	index
#14558551	2
#14558374	2
C₈×#64438	2
C₄×C₈×D₃₇	2
#1450	37

subgroup

index

C₄×C₈×D₃₇

Subgroups — #18009800

Conjugacy classes of subgroups

Complements

Maximal subgroups