Group #17996192 — subgroups

Subgroups — #17996192

Order 4736.

184 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 125 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×#11924476	C₂	1	1	yes	maximal, characteristic
2	#14557488	C₂	1	1	yes	maximal, characteristic
3	#14557308	C₂	1	2	yes	maximal, complemented
4	C₁₆×Dic₃₇	C₂	1	2	yes	maximal, complemented
5	C₂²×C₁₆×C₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#10596934	C₄	1	1	yes	characteristic
7	#11924495	C₄	1	1	yes	characteristic
8	#11924476	C₄	1	4	yes	complemented
9	C₂×#10596925	C₂²	1	2	yes	—
10	#11924446	C₄	1	2	yes	—
11	C₈×Dic₃₇	C₂²	1	2	yes	—
12	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×C₃₇	C₄	1	4	yes	complemented
14	C₂×C₁₆×C₃₇	C₂²	1	1	yes	characteristic
15	C₂×C₁₆×C₃₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₃₇	C₂×C₄	1	2	yes	—
19	C₂×C₈×C₃₇	C₂×C₄	1	2	yes	—
20	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
21	C₂²⋊Dic₃₇	C₈	1	2	yes	—
22	C₄⋊Dic₃₇	C₈	1	2	yes	—
23	C₄×Dic₃₇	C₂×C₄	1	2	yes	—
24	#10596934	C₈	1	4	yes	—
25	C₂×#64436	C₂×C₄	1	2	yes	—
26	C₁₆×C₃₇	C₂×C₄	1	2	yes	—
27	C₁₆×C₃₇	C₂×C₄	1	2	yes	—
28	#10596925	C₂×C₄	1	4	yes	—
29	C₂³×C₃₇	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₃₇	C₄²	1	2	yes	—
31	C₂×C₄×C₃₇	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₃₇	C₂×C₈	1	2	yes	—
33	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
34	C₂×Dic₃₇	C₂×C₈	1	4	yes	—
35	C₈×C₃₇	C₄²	1	2	yes	—
36	C₈×C₃₇	C₂²×C₄	1	1	yes	characteristic
37	C₈×C₃₇	C₂²×C₄	1	1	yes	characteristic
38	#64436	C₂×C₈	1	4	yes	—
39	C₂²×C₃₇	—	2	2	no	—
40	C₂²×C₃₇	C₄×C₈	1	2	yes	—
41	C₂²×C₃₇	C₂²×C₈	1	1	yes	characteristic
42	C₄×C₃₇	C₄×C₈	1	2	yes	—
43	C₄×C₃₇	C₂×C₄²	1	1	yes	characteristic
44	C₄×C₃₇	C₂²×C₈	1	1	yes	characteristic
45	Dic₃₇	—	2	4	no	—
46	#2108	—	37	1	no	maximal, Sylow
47	C₂×C₃₇	—	2	2	no	—
48	C₂×C₃₇	C₂×C₄×C₈	1	1	yes	characteristic
49	C₂×C₃₇	#504	1	2	yes	—
50	C₂×M₃₂	—	37	1	no	—
51	C₂²×C₁₆	D₃₇	1	1	yes	characteristic
52	#405	—	37	1	no	—
53	#346	—	37	2	no	—
54	C₄×C₁₆	—	37	2	no	—
55	C₃₇	#2108	1	1	yes	characteristic, complemented, Sylow
56	C₂×M₁₆	—	37	1	no	—
57	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
58	#117	—	37	1	no	—
59	M₃₂	—	37	4	no	—
60	C₂×C₁₆	—	37	2	no	—
61	C₂×C₁₆	Dic₃₇	1	4	yes	complemented
62	C₂×C₁₆	C₂×D₃₇	1	1	yes	characteristic
63	C₂×C₁₆	C₂×D₃₇	1	1	yes	characteristic
64	#97	—	37	2	no	—
65	C₄×C₈	—	37	2	no	—
66	C₂²×C₄	C₄×D₃₇	1	1	yes	characteristic
67	M₁₆	—	37	4	no	—
68	C₂×C₈	—	37	2	no	—
69	C₂×C₈	C₄×D₃₇	1	1	yes	characteristic
70	C₂×C₈	C₄×D₃₇	1	2	yes	—
71	C₂×C₈	C₂×Dic₃₇	1	2	yes	—
72	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
73	C₄⋊C₄	—	37	2	no	—
74	C₂²⋊C₄	—	37	2	no	—
75	C₄²	—	37	2	no	—
76	C₁₆	—	37	4	no	—
77	C₁₆	C₂×Dic₃₇	1	2	yes	—
78	C₁₆	C₂×Dic₃₇	1	2	yes	—
79	C₂³	C₈×D₃₇	1	1	yes	characteristic
80	C₂×C₄	—	37	4	no	—
81	C₂×C₄	C₈×D₃₇	1	2	yes	—
82	C₂×C₄	C₈×D₃₇	1	1	yes	characteristic
83	C₂×C₄	C₄×Dic₃₇	1	2	yes	—
84	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
85	C₈	—	37	4	no	—
86	C₈	C₄×Dic₃₇	1	2	yes	—
87	C₈	C₂×C₄×D₃₇	1	1	yes	characteristic
88	C₈	C₂²×Dic₃₇	1	1	yes	characteristic
89	C₂²	—	2	2	no	—
90	C₂²	C₈×Dic₃₇	1	2	yes	—
91	C₂²	C₂×C₈×D₃₇	1	1	yes	characteristic
92	C₄	—	74	4	no	—
93	C₄	C₈×Dic₃₇	1	2	yes	—
94	C₄	C₂×C₄×Dic₃₇	1	1	yes	characteristic
95	C₄	C₂×C₈×D₃₇	1	1	yes	characteristic
96	C₂	—	2	2	no	—
97	C₂	C₂×C₈×Dic₃₇	1	1	yes	characteristic
98	C₂	#14557949	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924476

C₂

yes

maximal, characteristic

#14557488

C₂

yes

maximal, characteristic

#14557308

C₂

yes

maximal, complemented

C₁₆×Dic₃₇

C₂

yes

maximal, complemented

C₂²×C₁₆×C₃₇

C₂

yes

maximal, characteristic

C₂×#10596934

C₄

yes

characteristic

#11924495

C₄

yes

characteristic

#11924476

C₄

yes

complemented

C₂×#10596925

C₂²

yes

—

#11924446

C₄

yes

—

C₈×Dic₃₇

C₂²

yes

—

C₂²×C₈×C₃₇

C₂²

yes

characteristic

C₂×C₁₆×C₃₇

C₄

yes

complemented

C₂×C₁₆×C₃₇

C₂²

yes

characteristic

C₂×C₁₆×C₃₇

C₂²

yes

characteristic

C₂²×C₄×C₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₃₇

C₂×C₄

yes

—

C₂×C₈×C₃₇

C₂×C₄

yes

—

C₂×C₈×C₃₇

C₂³

yes

characteristic

C₂²⋊Dic₃₇

C₈

yes

—

C₄⋊Dic₃₇

C₈

yes

—

C₄×Dic₃₇

C₂×C₄

yes

—

#10596934

C₈

yes

—

C₂×#64436

C₂×C₄

yes

—

C₁₆×C₃₇

C₂×C₄

yes

—

C₁₆×C₃₇

C₂×C₄

yes

—

#10596925

C₂×C₄

yes

—

C₂³×C₃₇

C₂×C₈

yes

characteristic

C₂×C₄×C₃₇

C₄²

yes

—

C₂×C₄×C₃₇

C₂×C₈

yes

characteristic

C₂×C₄×C₃₇

C₂×C₈

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

C₂×C₈

yes

—

C₈×C₃₇

C₄²

yes

—

C₈×C₃₇

C₂²×C₄

yes

characteristic

C₈×C₃₇

C₂²×C₄

yes

characteristic

#64436

C₂×C₈

yes

—

C₂²×C₃₇

—

C₂²×C₃₇

C₄×C₈

yes

—

C₂²×C₃₇

C₂²×C₈

yes

characteristic

C₄×C₃₇

C₄×C₈

yes

—

C₄×C₃₇

C₂×C₄²

yes

characteristic

C₄×C₃₇

C₂²×C₈

yes

characteristic

Dic₃₇

—

#2108

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

C₂×C₄×C₈

yes

characteristic

C₂×C₃₇

#504

yes

—

C₂×M₃₂

—

C₂²×C₁₆

D₃₇

yes

characteristic

#405

—

#346

—

C₄×C₁₆

—

C₃₇

#2108

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

C₂×D₃₇

yes

characteristic

#117

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

Dic₃₇

yes

complemented

C₂×C₁₆

C₂×D₃₇

yes

characteristic

C₂×C₁₆

C₂×D₃₇

yes

characteristic

#97

—

C₄×C₈

—

C₂²×C₄

C₄×D₃₇

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

C₄×D₃₇

yes

characteristic

C₂×C₈

C₄×D₃₇

yes

—

C₂×C₈

C₂×Dic₃₇

yes

—

C₂×C₈

C₂²×D₃₇

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

C₁₆

C₂×Dic₃₇

yes

—

C₁₆

C₂×Dic₃₇

yes

—

C₂³

C₈×D₃₇

yes

characteristic

C₂×C₄

—

C₂×C₄

C₈×D₃₇

yes

—

C₂×C₄

C₈×D₃₇

yes

characteristic

C₂×C₄

C₄×Dic₃₇

yes

—

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

C₈

—

C₈

C₄×Dic₃₇

yes

—

C₈

C₂×C₄×D₃₇

yes

characteristic

C₈

C₂²×Dic₃₇

yes

characteristic

C₂²

—

C₂²

C₈×Dic₃₇

yes

—

C₂²

C₂×C₈×D₃₇

yes

characteristic

C₄

—

C₄

C₈×Dic₃₇

yes

—

C₄

C₂×C₄×Dic₃₇

yes

characteristic

C₄

C₂×C₈×D₃₇

yes

characteristic

C₂

—

C₂

C₂×C₈×Dic₃₇

yes

characteristic

C₂

#14557949

yes

—

	subgroup	G/H
center	C₂×C₁₆	C₂×D₃₇
commutator	C₂×C₃₇	C₂×C₄×C₈
Frattini	C₂×C₈	C₂²×D₃₇
Fitting	C₂²×C₁₆×C₃₇	C₂
socle	C₂²×C₃₇	C₂²×C₈
prime radical	C₂³×C₃₇	C₂×C₈
cosocle	C₂×C₈×C₃₇	C₂³
hypercenter	C₂²×C₁₆	D₃₇
hypocenter	C₃₇	#2108
LCS	G ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂×C₁₆

C₂×D₃₇

commutator

C₂×C₃₇

C₂×C₄×C₈

Frattini

C₂×C₈

C₂²×D₃₇

Fitting

C₂²×C₁₆×C₃₇

C₂

socle

C₂²×C₃₇

C₂²×C₈

prime radical

C₂³×C₃₇

C₂×C₈

cosocle

C₂×C₈×C₃₇

C₂³

hypercenter

C₂²×C₁₆

D₃₇

C₃₇

G ▷ C₂×C₃₇ ▷ C₃₇

C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆

N	H ≅ G/N	type
3. #14557308	C₂	semidirect
4. C₁₆×Dic₃₇	C₂	semidirect
8. #11924476	C₄	semidirect
13. C₂×C₁₆×C₃₇	C₄	semidirect
55. C₃₇	#2108	semidirect
61. C₂×C₁₆	Dic₃₇	semidirect

H ≅ G/N

type

3. #14557308

C₂

semidirect

4. C₁₆×Dic₃₇

C₂

semidirect

8. #11924476

C₄

semidirect

13. C₂×C₁₆×C₃₇

C₄

semidirect

55. C₃₇

#2108

semidirect

61. C₂×C₁₆

Dic₃₇

semidirect

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

subgroup	index
C₂×#11924476	2
#14557488	2
#14557308	2
C₁₆×Dic₃₇	2
C₂²×C₁₆×C₃₇	2
#2108	37

subgroup

index

C₂×#11924476

#14557488

#14557308

C₁₆×Dic₃₇

C₂²×C₁₆×C₃₇

#2108

Subgroups — #17996192

Conjugacy classes of subgroups

Complements

Maximal subgroups