Group #17996351 — subgroups

Subgroups — #17996351

Order 4736.

172 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 97 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	C₂×C₈×Dic₃₇	C₂	1	1	yes	maximal, characteristic
3	#14557351	C₂	1	4	yes	maximal, complemented
4	C₂×M₃₂×C₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#64436	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₃₇	C₄	1	1	yes	characteristic
7	#11924476	—	2	2	no	—
8	C₂×#10596925	C₂²	1	2	yes	—
9	C₈×Dic₃₇	C₄	1	2	yes	—
10	C₈×Dic₃₇	C₂²	1	2	yes	—
11	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
12	M₃₂×C₃₇	—	2	2	no	—
13	C₂×C₁₆×C₃₇	C₂²	1	2	yes	—
14	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×Dic₃₇	C₈	1	2	yes	—
16	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₃₇	D₄	1	2	yes	—
18	C₂×C₈×C₃₇	Q₈	1	2	yes	—
19	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
20	C₄×Dic₃₇	C₈	1	1	yes	characteristic
21	C₄×Dic₃₇	C₈	1	1	yes	characteristic
22	C₄×Dic₃₇	C₂×C₄	1	2	yes	—
23	C₂×#64436	—	2	2	no	—
24	C₂×#64436	C₂×C₄	1	2	yes	—
25	C₁₆×C₃₇	—	2	2	no	—
26	#10596925	—	2	2	no	—
27	C₂³×C₃₇	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄×C₃₇	C₄⋊C₄	1	2	yes	—
29	C₂×C₄×C₃₇	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₃₇	M₁₆	1	2	yes	—
31	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
32	C₂×Dic₃₇	—	2	4	no	—
33	C₂×Dic₃₇	C₂×C₈	1	2	yes	—
34	C₂×Dic₃₇	C₂×C₈	1	2	yes	—
35	C₈×C₃₇	C₄⋊C₄	1	2	yes	—
36	C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
37	C₈×C₃₇	C₂×Q₈	1	1	yes	characteristic
38	#64436	—	2	2	no	—
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×C₄⋊C₄	1	1	yes	characteristic
44	C₄×C₃₇	C₂×M₁₆	1	1	yes	characteristic
45	Dic₃₇	—	2	2	no	—
46	Dic₃₇	M₃₂	1	4	yes	complemented
47	#2150	—	37	1	no	maximal, Sylow
48	C₂×C₃₇	—	2	2	no	—
49	C₂×C₃₇	C₂×C₄⋊C₈	1	1	yes	characteristic
50	C₂×C₃₇	C₂×M₃₂	1	2	yes	—
51	C₂×M₃₂	—	37	1	no	—
52	C₂×M₃₂	D₃₇	1	1	yes	characteristic
53	C₂×C₄×C₈	—	37	1	no	—
54	C₄⋊C₁₆	—	37	4	no	—
55	C₃₇	#2150	1	1	yes	characteristic, complemented, Sylow
56	C₂²×C₈	—	37	1	no	—
57	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
58	C₂×C₄²	—	37	1	no	—
59	M₃₂	—	2	2	no	—
60	M₃₂	—	74	2	no	—
61	C₂×C₁₆	—	37	2	no	—
62	C₂×C₁₆	C₂×D₃₇	1	2	yes	—
63	C₄×C₈	—	37	2	no	—
64	C₄×C₈	—	37	2	no	—
65	C₂²×C₄	—	37	2	no	—
66	C₂²×C₄	C₄×D₃₇	1	1	yes	characteristic
67	C₂×C₈	—	37	2	no	—
68	C₂×C₈	—	74	2	no	—
69	C₂×C₈	Dic₇₄	1	2	yes	—
70	C₂×C₈	C₄×D₃₇	1	1	yes	characteristic
71	C₂×C₈	C₂²⋊D₃₇	1	2	yes	—
72	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
73	C₄²	—	37	1	no	—
74	C₄²	—	37	1	no	—
75	C₄²	—	37	2	no	—
76	C₁₆	—	2	2	no	—
77	C₁₆	—	74	2	no	—
78	C₂³	C₈×D₃₇	1	1	yes	characteristic
79	C₂×C₄	—	37	2	no	—
80	C₂×C₄	—	37	2	no	—
81	C₂×C₄	—	74	4	no	—
82	C₂×C₄	C₈×D₃₇	1	1	yes	characteristic
83	C₂×C₄	#10596929	1	2	yes	—
84	C₂×C₄	#10596936	1	2	yes	—
85	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
86	C₈	—	74	2	no	—
87	C₈	#10596936	1	2	yes	—
88	C₈	C₂×Dic₇₄	1	1	yes	characteristic
89	C₈	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
90	C₂²	—	2	2	no	—
91	C₂²	#11924465	1	2	yes	—
92	C₂²	C₂×C₈×D₃₇	1	1	yes	characteristic
93	C₄	—	37	4	no	—
94	C₄	—	74	2	no	—
95	C₄	#11924465	1	2	yes	—
96	C₄	C₂×#10596936	1	1	yes	characteristic
97	C₄	C₂×#10596929	1	1	yes	characteristic
98	C₂	—	2	2	no	—
99	C₂	C₂×#11924465	1	1	yes	characteristic
100	C₂	M₃₂×D₃₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924476

C₂

yes

maximal, characteristic

C₂×C₈×Dic₃₇

C₂

yes

maximal, characteristic

#14557351

C₂

yes

maximal, complemented

C₂×M₃₂×C₃₇

C₂

yes

maximal, characteristic

C₂²×#64436

C₄

yes

characteristic

C₂×C₄×Dic₃₇

C₄

yes

characteristic

#11924476

—

C₂×#10596925

C₂²

yes

—

C₈×Dic₃₇

C₄

yes

—

C₈×Dic₃₇

C₂²

yes

—

C₂²×C₈×C₃₇

C₂²

yes

characteristic

M₃₂×C₃₇

—

C₂×C₁₆×C₃₇

C₂²

yes

—

C₂²×C₄×C₃₇

C₂×C₄

yes

characteristic

C₂²×Dic₃₇

C₈

yes

—

C₂×C₈×C₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₃₇

D₄

yes

—

C₂×C₈×C₃₇

Q₈

yes

—

C₂×C₈×C₃₇

C₂³

yes

characteristic

C₄×Dic₃₇

C₈

yes

characteristic

C₄×Dic₃₇

C₈

yes

characteristic

C₄×Dic₃₇

C₂×C₄

yes

—

C₂×#64436

—

C₂×#64436

C₂×C₄

yes

—

C₁₆×C₃₇

—

#10596925

—

C₂³×C₃₇

C₂×C₈

yes

characteristic

C₂×C₄×C₃₇

C₄⋊C₄

yes

—

C₂×C₄×C₃₇

C₂×C₈

yes

characteristic

C₂×C₄×C₃₇

M₁₆

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂×C₈

yes

—

C₂×Dic₃₇

C₂×C₈

yes

—

C₈×C₃₇

C₄⋊C₄

yes

—

C₈×C₃₇

C₂×D₄

yes

characteristic

C₈×C₃₇

C₂×Q₈

yes

characteristic

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

C₄⋊C₈

yes

—

C₂²×C₃₇

C₂²×C₈

yes

characteristic

C₄×C₃₇

C₄⋊C₈

yes

—

C₄×C₃₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₃₇

C₂×M₁₆

yes

characteristic

Dic₃₇

—

Dic₃₇

M₃₂

yes

complemented

#2150

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

C₂×C₄⋊C₈

yes

characteristic

C₂×C₃₇

C₂×M₃₂

yes

—

C₂×M₃₂

—

C₂×M₃₂

D₃₇

yes

characteristic

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₃₇

#2150

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×D₃₇

yes

characteristic

C₂×C₄²

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₃₇

yes

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

C₄×D₃₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₇₄

yes

—

C₂×C₈

C₄×D₃₇

yes

characteristic

C₂×C₈

C₂²⋊D₃₇

yes

—

C₂×C₈

C₂²×D₃₇

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

C₁₆

—

C₁₆

—

C₂³

C₈×D₃₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₈×D₃₇

yes

characteristic

C₂×C₄

#10596929

yes

—

C₂×C₄

#10596936

yes

—

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

C₈

—

C₈

#10596936

yes

—

C₈

C₂×Dic₇₄

yes

characteristic

C₈

C₂×C₂²⋊D₃₇

yes

characteristic

C₂²

—

C₂²

#11924465

yes

—

C₂²

C₂×C₈×D₃₇

yes

characteristic

C₄

—

C₄

—

C₄

#11924465

yes

—

C₄

yes

C₄

yes

C₂

—

C₂

C₂×#11924465

yes

characteristic

100

C₂

M₃₂×D₃₇

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₃₇
commutator, socle	C₂²×C₃₇	C₂²×C₈
Fitting	C₂×M₃₂×C₃₇	C₂
prime radical	C₂³×C₃₇	C₂×C₈
cosocle	C₂×C₈×C₃₇	C₂³
hypercenter	C₂×M₃₂	D₃₇
hypocenter	C₃₇	#2150
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₃₇

commutator, socle

C₂²×C₃₇

C₂²×C₈

Fitting

C₂×M₃₂×C₃₇

C₂

prime radical

C₂³×C₃₇

C₂×C₈

cosocle

C₂×C₈×C₃₇

C₂³

hypercenter

C₂×M₃₂

D₃₇

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
3. #14557351	C₂	semidirect
46. Dic₃₇	M₃₂	semidirect
55. C₃₇	#2150	semidirect

H ≅ G/N

type

3. #14557351

C₂

semidirect

46. Dic₃₇

M₃₂

semidirect

55. C₃₇

#2150

semidirect

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

subgroup	index
C₂×#11924476	2
C₂×C₈×Dic₃₇	2
#14557351	2
C₂×M₃₂×C₃₇	2
#2150	37

subgroup

index

C₂×#11924476

C₂×C₈×Dic₃₇

#14557351

C₂×M₃₂×C₃₇

#2150

Subgroups — #17996351

Conjugacy classes of subgroups

Complements

Maximal subgroups