Group #17996401 — subgroups

Subgroups — #17996401

Order 4736.

138 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 69 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	#14557487	C₂	1	1	yes	maximal, characteristic
2	#14557351	C₂	1	2	yes	maximal, complemented
3	#14557312	C₂	1	2	yes	maximal
4	#14557309	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₁₆×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10596929	C₄	1	1	yes	characteristic
7	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
8	#11924493	C₄	1	1	yes	characteristic
9	C₂×#10596925	C₂²	1	2	yes	—
10	#11924447	C₄	1	1	yes	characteristic
11	#11924446	C₄	1	1	yes	characteristic
12	C₈×Dic₃₇	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×C₃₇	C₂²	1	2	yes	—
14	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
18	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
19	#10596938	C₈	1	2	yes	—
20	#10596936	C₈	1	2	yes	—
21	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
22	C₂×#64436	C₂×C₄	1	1	yes	characteristic
23	C₂×#64436	C₂×C₄	1	1	yes	characteristic
24	#10596929	—	2	2	no	—
25	C₈×D₃₇	—	2	2	no	—
26	C₁₆×C₃₇	—	2	2	no	—
27	#10596925	—	2	2	no	—
28	C₂²×D₃₇	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₃₇	C₂×C₈	1	2	yes	—
30	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
31	C₂×Dic₃₇	C₂×C₈	1	2	yes	—
32	C₂×Dic₃₇	C₂×C₈	1	1	yes	characteristic
33	C₄×D₃₇	—	2	1	no	—
34	C₄×D₃₇	—	2	1	no	—
35	C₈×C₃₇	M₁₆	1	2	yes	—
36	C₈×C₃₇	#41	1	2	yes	—
37	#64436	—	2	1	no	—
38	#64436	—	2	1	no	—
39	C₂²×C₃₇	C₂²×C₈	1	1	yes	characteristic
40	C₂×D₃₇	—	2	1	no	—
41	C₂×D₃₇	—	2	2	no	—
42	C₄×C₃₇	—	2	2	no	—
43	C₄×C₃₇	#117	1	1	yes	characteristic
44	C₄×C₃₇	C₂×M₁₆	1	1	yes	characteristic
45	Dic₃₇	—	2	1	no	—
46	Dic₃₇	—	2	2	no	—
47	#2164	—	37	1	no	maximal, Sylow
48	C₂×C₃₇	#431	1	1	yes	characteristic
49	C₂×C₃₇	#504	1	2	yes	—
50	D₃₇	—	4	1	no	—
51	#405	—	37	1	no	—
52	C₄⋊C₁₆	—	37	2	no	—
53	C₂²⋊C₁₆	—	37	2	no	—
54	#346	—	37	1	no	—
55	C₄×C₁₆	D₃₇	1	1	yes	characteristic, complemented
56	C₃₇	#2164	1	1	yes	characteristic, complemented, Sylow
57	C₂×M₁₆	—	37	1	no	—
58	C₂²×C₈	—	37	1	no	—
59	#117	—	37	1	no	—
60	C₂×C₁₆	—	37	2	no	—
61	C₂×C₁₆	C₂×D₃₇	1	2	yes	—
62	#97	—	37	1	no	—
63	#97	—	37	1	no	—
64	C₄×C₈	—	37	1	no	—
65	C₄×C₈	C₂×D₃₇	1	1	yes	characteristic
66	C₂²×C₄	—	37	1	no	—
67	M₁₆	—	74	2	no	—
68	C₂×C₈	—	37	1	no	—
69	C₂×C₈	—	37	1	no	—
70	C₂×C₈	—	74	2	no	—
71	C₂×C₈	C₄×D₃₇	1	1	yes	characteristic
72	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
73	C₄⋊C₄	—	37	2	no	—
74	C₂²⋊C₄	—	37	2	no	—
75	C₄²	—	37	1	no	—
76	C₄²	C₄×D₃₇	1	1	yes	characteristic
77	C₁₆	—	2	2	no	—
78	C₁₆	—	74	2	no	—
79	C₂³	—	37	1	no	—
80	C₂×C₄	—	37	2	no	—
81	C₂×C₄	—	37	1	no	—
82	C₂×C₄	—	74	1	no	—
83	C₂×C₄	—	74	1	no	—
84	C₂×C₄	C₈×D₃₇	1	2	yes	—
85	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
86	C₈	—	74	1	no	—
87	C₈	—	74	1	no	—
88	C₈	#10596929	1	2	yes	—
89	C₈	#10596962	1	2	yes	—
90	C₂²	—	74	1	no	—
91	C₂²	—	74	2	no	—
92	C₂²	C₂×C₈×D₃₇	1	1	yes	characteristic
93	C₄	—	2	2	no	—
94	C₄	—	74	1	no	—
95	C₄	—	74	2	no	—
96	C₄	#11924493	1	1	yes	characteristic
97	C₄	C₂×#10596929	1	1	yes	characteristic
98	C₂	—	148	1	no	—
99	C₂	#14557584	1	1	yes	characteristic
100	C₂	#14557949	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14557487

C₂

yes

maximal, characteristic

#14557351

C₂

yes

maximal, complemented

#14557312

C₂

yes

maximal

#14557309

C₂

yes

maximal, characteristic, complemented

C₄×C₁₆×C₃₇

C₂

yes

maximal, characteristic, complemented

C₂×#10596929

C₄

yes

characteristic

C₂×C₈×D₃₇

C₂²

yes

characteristic

#11924493

C₄

yes

characteristic

C₂×#10596925

C₂²

yes

—

#11924447

C₄

yes

characteristic

#11924446

C₄

yes

characteristic

C₈×Dic₃₇

C₂²

yes

characteristic

C₂×C₁₆×C₃₇

C₂²

yes

—

C₄×C₈×C₃₇

C₂²

yes

characteristic

C₂×C₄×D₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₃₇

C₂³

yes

characteristic

C₄²×C₃₇

C₂×C₄

yes

characteristic

#10596938

C₈

yes

—

#10596936

C₈

yes

—

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₂×#64436

C₂×C₄

yes

characteristic

C₂×#64436

C₂×C₄

yes

characteristic

#10596929

—

C₈×D₃₇

—

C₁₆×C₃₇

—

#10596925

—

C₂²×D₃₇

C₂×C₈

yes

characteristic

C₂×C₄×C₃₇

C₂×C₈

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

C₂×C₈

yes

—

C₂×Dic₃₇

C₂×C₈

yes

characteristic

C₄×D₃₇

—

C₄×D₃₇

—

C₈×C₃₇

M₁₆

yes

—

C₈×C₃₇

#41

yes

—

#64436

—

#64436

—

C₂²×C₃₇

C₂²×C₈

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#117

yes

characteristic

C₄×C₃₇

C₂×M₁₆

yes

characteristic

Dic₃₇

—

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

—

D₃₇

—

#405

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

C₄×C₁₆

D₃₇

yes

characteristic, complemented

C₃₇

#2164

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

#117

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₃₇

yes

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×D₃₇

yes

characteristic

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₃₇

yes

characteristic

C₂×C₈

C₂²×D₃₇

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₄×D₃₇

yes

characteristic

C₁₆

—

C₁₆

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₈×D₃₇

yes

—

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

C₈

—

C₈

—

C₈

#10596929

yes

—

C₈

#10596962

yes

—

C₂²

—

C₂²

—

C₂²

C₂×C₈×D₃₇

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

yes

C₄

yes

C₂

—

148

—

C₂

#14557584

yes

characteristic

100

C₂

#14557949

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₃₇
commutator, socle	C₂²×C₃₇	C₂²×C₈
Fitting	C₄×C₁₆×C₃₇	C₂
prime radical	C₂²×D₃₇	C₂×C₈
cosocle	C₂×C₈×C₃₇	C₂³
hypercenter	C₄×C₁₆	D₃₇
hypocenter	C₃₇	#2164
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₃₇

commutator, socle

C₂²×C₃₇

C₂²×C₈

Fitting

C₄×C₁₆×C₃₇

C₂

prime radical

C₂²×D₃₇

C₂×C₈

cosocle

C₂×C₈×C₃₇

C₂³

hypercenter

C₄×C₁₆

D₃₇

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂×C₈ ◁ C₄×C₁₆

N	H ≅ G/N	type
2. #14557351	C₂	semidirect
4. #14557309	C₂	semidirect
5. C₄×C₁₆×C₃₇	C₂	semidirect
55. C₄×C₁₆	D₃₇	semidirect
56. C₃₇	#2164	semidirect

H ≅ G/N

type

2. #14557351

C₂

semidirect

4. #14557309

C₂

semidirect

5. C₄×C₁₆×C₃₇

C₂

semidirect

55. C₄×C₁₆

D₃₇

semidirect

56. C₃₇

#2164

semidirect

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

subgroup	index
#14557487	2
#14557351	2
#14557312	2
#14557309	2
C₄×C₁₆×C₃₇	2
#2164	37

subgroup

index

C₄×C₁₆×C₃₇

Subgroups — #17996401

Conjugacy classes of subgroups

Complements

Maximal subgroups