Group #17992473 — subgroups

Subgroups — #17992473

Order 4736.

184 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 61 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	#14557497	C₂	1	2	yes	maximal
2	C₂×C₂²⋊C₈×C₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#10596934	—	2	2	no	—
4	C₂²⋊#64436	—	2	2	no	—
5	C₂²⋊#64436	C₄	1	4	yes	—
6	C₂³×C₄×C₃₇	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₃₇	C₄	1	2	yes	—
8	C₂²⋊C₈×C₃₇	—	2	2	no	—
9	C₂⁴×C₃₇	Q₈	1	1	yes	characteristic
10	C₂²×C₄×C₃₇	—	4	2	no	—
11	C₂²×C₄×C₃₇	C₂×C₄	1	2	yes	—
12	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₄×C₃₇	D₄	1	1	yes	characteristic
14	C₂²×C₄×C₃₇	D₄	1	2	yes	—
15	C₂×C₈×C₃₇	—	2	2	no	—
16	C₂×C₈×C₃₇	—	4	2	no	—
17	#10596934	—	4	4	no	—
18	C₂×#64436	—	2	4	no	—
19	C₂³×C₃₇	—	2	2	no	—
20	C₂³×C₃₇	—	4	1	no	—
21	C₂³×C₃₇	—	4	1	no	—
22	C₂³×C₃₇	C₄⋊C₄	1	2	yes	—
23	C₂³×C₃₇	C₄⋊C₄	1	1	yes	characteristic
24	C₂×C₄×C₃₇	—	2	4	no	—
25	C₂×C₄×C₃₇	—	2	2	no	—
26	C₂×C₄×C₃₇	—	4	2	no	—
27	C₂×C₄×C₃₇	—	4	1	no	—
28	C₂×C₄×C₃₇	C₄²	1	1	yes	characteristic
29	C₂×C₄×C₃₇	C₂²⋊C₄	1	2	yes	—
30	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
31	C₈×C₃₇	—	4	2	no	—
32	#64436	—	4	4	no	—
33	C₂²×C₃₇	—	2	1	no	—
34	C₂²×C₃₇	—	2	1	no	—
35	C₂²×C₃₇	—	2	2	no	—
36	C₂²×C₃₇	—	2	2	no	—
37	C₂²×C₃₇	—	2	2	no	—
38	C₂²×C₃₇	—	4	4	no	—
39	C₂²×C₃₇	#95	1	1	yes	characteristic
40	C₂²×C₃₇	#108	1	2	yes	—
41	C₄×C₃₇	—	2	2	no	—
42	C₄×C₃₇	—	2	1	no	—
43	C₄×C₃₇	#99	1	2	yes	—
44	#1293	—	37	1	no	maximal, Sylow
45	C₂×C₃₇	—	2	2	no	—
46	C₂×C₃₇	—	4	2	no	—
47	C₂×C₃₇	#338	1	1	yes	characteristic
48	C₂×C₃₇	#341	1	1	yes	characteristic
49	C₂×C₃₇	#342	1	1	yes	characteristic
50	#407	—	37	2	no	—
51	C₂×C₂²⋊C₈	D₃₇	1	1	yes	characteristic
52	C₃₇	#1293	1	1	yes	characteristic, complemented, Sylow
53	C₂³×C₄	C₂×D₃₇	1	1	yes	characteristic
54	C₂×M₁₆	—	74	2	no	—
55	C₂²×C₈	Dic₃₇	1	2	yes	—
56	C₂²⋊C₈	—	2	2	no	—
57	C₂²⋊C₈	—	37	4	no	—
58	C₂²⋊C₈	—	74	2	no	—
59	C₂⁴	Dic₇₄	1	1	yes	characteristic
60	C₂²×C₄	—	4	2	no	—
61	C₂²×C₄	C₄×D₃₇	1	2	yes	—
62	C₂²×C₄	D₁₄₈	1	1	yes	characteristic
63	C₂²×C₄	C₂×Dic₃₇	1	1	yes	characteristic
64	C₂²×C₄	C₂²⋊D₃₇	1	2	yes	—
65	M₁₆	—	148	4	no	—
66	C₂×C₈	—	2	2	no	—
67	C₂×C₈	—	4	2	no	—
68	C₂×C₈	—	74	4	no	—
69	C₂³	—	2	2	no	—
70	C₂³	—	4	1	no	—
71	C₂³	—	4	1	no	—
72	C₂³	#10596936	1	2	yes	—
73	C₂³	C₄⋊Dic₃₇	1	1	yes	characteristic
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	2	4	no	—
76	C₂×C₄	—	4	2	no	—
77	C₂×C₄	—	4	1	no	—
78	C₂×C₄	C₄×Dic₃₇	1	1	yes	characteristic
79	C₂×C₄	#10596938	1	2	yes	—
80	C₂×C₄	C₂²⋊Dic₃₇	1	1	yes	characteristic
81	C₈	—	4	2	no	—
82	C₈	—	148	4	no	—
83	C₂²	—	2	1	no	—
84	C₂²	—	2	1	no	—
85	C₂²	—	2	2	no	—
86	C₂²	—	2	2	no	—
87	C₂²	—	2	2	no	—
88	C₂²	—	4	4	no	—
89	C₂²	#11924443	1	1	yes	characteristic
90	C₂²	#11924472	1	2	yes	—
91	C₄	—	2	2	no	—
92	C₄	—	2	1	no	—
93	C₄	#11924451	1	2	yes	—
94	C₂	—	2	2	no	—
95	C₂	—	4	2	no	—
96	C₂	#14557291	1	1	yes	characteristic
97	C₂	#14557298	1	1	yes	characteristic
98	C₂	#14557300	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14557497

C₂

yes

maximal

C₂×C₂²⋊C₈×C₃₇

C₂

yes

maximal, characteristic

C₂×#10596934

—

C₂²⋊#64436

—

C₂²⋊#64436

C₄

yes

—

C₂³×C₄×C₃₇

C₂²

yes

characteristic

C₂²×C₈×C₃₇

C₄

yes

—

C₂²⋊C₈×C₃₇

—

C₂⁴×C₃₇

Q₈

yes

characteristic

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

C₂×C₄

yes

—

C₂²×C₄×C₃₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₇

D₄

yes

characteristic

C₂²×C₄×C₃₇

D₄

yes

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

—

#10596934

—

C₂×#64436

—

C₂³×C₃₇

—

C₂³×C₃₇

—

C₂³×C₃₇

—

C₂³×C₃₇

C₄⋊C₄

yes

—

C₂³×C₃₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₄²

yes

characteristic

C₂×C₄×C₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₃₇

C₂²⋊C₄

yes

characteristic

C₈×C₃₇

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

#95

yes

characteristic

C₂²×C₃₇

#108

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#99

yes

—

#1293

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

—

C₂×C₂²⋊C₈

D₃₇

yes

characteristic

C₃₇

#1293

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₃₇

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

Dic₃₇

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂⁴

Dic₇₄

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₄×D₃₇

yes

—

C₂²×C₄

D₁₄₈

yes

characteristic

C₂²×C₄

C₂×Dic₃₇

yes

characteristic

C₂²×C₄

C₂²⋊D₃₇

yes

—

M₁₆

—

148

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂³

—

C₂³

—

C₂³

—

C₂³

#10596936

yes

—

C₂³

C₄⋊Dic₃₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₃₇

yes

characteristic

C₂×C₄

#10596938

yes

—

C₂×C₄

C₂²⋊Dic₃₇

yes

characteristic

C₈

—

C₈

—

148

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#11924443

yes

characteristic

C₂²

#11924472

yes

—

C₄

—

C₄

—

C₄

#11924451

yes

—

C₂

—

C₂

—

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂²	#11924443
commutator	C₂×C₄×C₃₇	C₄²
Frattini	C₂³×C₄	C₂×D₃₇
Fitting	C₂×C₂²⋊C₈×C₃₇	C₂
socle	C₂²×C₃₇	#95
prime radical	C₂⁴×C₃₇	Q₈
cosocle	C₂³×C₄×C₃₇	C₂²
hypercenter	C₂×C₂²⋊C₈	D₃₇
hypocenter	C₃₇	#1293
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center

C₂²

#11924443

commutator

C₂×C₄×C₃₇

C₄²

Frattini

C₂³×C₄

C₂×D₃₇

Fitting

C₂×C₂²⋊C₈×C₃₇

C₂

socle

C₂²×C₃₇

#95

prime radical

C₂⁴×C₃₇

Q₈

cosocle

C₂³×C₄×C₃₇

C₂²

hypercenter

C₂×C₂²⋊C₈

D₃₇

hypocenter

C₃₇

#1293

LCS

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
52. C₃₇	#1293	semidirect

H ≅ G/N

type

52. C₃₇

#1293

semidirect

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

subgroup	index
#14557497	2
C₂×C₂²⋊C₈×C₃₇	2
#1293	37

subgroup

index

#14557497

C₂×C₂²⋊C₈×C₃₇

#1293

Subgroups — #17992473

Conjugacy classes of subgroups

Complements

Maximal subgroups