Group #17994501 — subgroups

Subgroups — #17994501

Order 4736.

220 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 105 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×#11924471	C₂	1	2	yes	maximal
2	M₁₆×Dic₃₇	C₂	1	2	yes	maximal
3	C₂×#11924446	C₂	1	1	yes	maximal, characteristic
4	#14557298	C₂	1	1	yes	maximal, characteristic
5	C₃₇×#341	C₂	1	1	yes	maximal, characteristic
6	C₂×#10596934	C₂²	1	2	yes	—
7	C₂²×#64436	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
9	#11924471	C₄	1	8	yes	complemented
10	#11924446	—	2	2	no	—
11	#11924446	C₄	1	4	yes	—
12	C₈×Dic₃₇	—	2	2	no	—
13	C₂×M₁₆×C₃₇	C₂²	1	2	yes	—
14	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
16	C₂²×Dic₃₇	Q₈	1	2	yes	—
17	M₁₆×C₃₇	—	2	2	no	—
18	M₁₆×C₃₇	C₂×C₄	1	4	yes	—
19	C₂×C₈×C₃₇	—	2	2	no	—
20	C₂×C₈×C₃₇	—	2	2	no	—
21	C₂×C₈×C₃₇	C₂×C₄	1	2	yes	—
22	C₄×Dic₃₇	C₂×C₄	1	2	yes	—
23	C₄×Dic₃₇	D₄	1	2	yes	—
24	#10596934	—	2	2	no	—
25	#10596934	C₂×C₄	1	4	yes	—
26	C₂×#64436	—	2	2	no	—
27	C₂×#64436	C₂×C₄	1	2	yes	—
28	C₂×#64436	C₂×C₄	1	4	yes	—
29	C₂³×C₃₇	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
32	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₃₇	#41	1	2	yes	—
34	C₂×Dic₃₇	—	4	2	no	—
35	C₂×Dic₃₇	C₄⋊C₄	1	4	yes	—
36	C₈×C₃₇	—	2	2	no	—
37	C₈×C₃₇	—	2	4	no	—
38	#64436	—	2	4	no	—
39	#64436	C₄²	1	4	yes	—
40	C₂²×C₃₇	—	2	2	no	—
41	C₂²×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₃₇	C₄×Q₈	1	2	yes	—
43	C₄×C₃₇	C₂×C₄²	1	1	yes	characteristic
44	C₄×C₃₇	#117	1	1	yes	characteristic
45	C₄×C₃₇	C₄×D₄	1	2	yes	—
46	Dic₃₇	—	2	4	no	—
47	#1778	—	37	1	no	maximal, Sylow
48	C₂×C₃₇	—	2	2	no	—
49	C₂×C₃₇	C₄×C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₃₇	#430	1	2	yes	—
51	C₂×#108	—	37	2	no	—
52	C₄×M₁₆	—	37	2	no	—
53	C₂×#97	—	37	1	no	—
54	#341	—	37	1	no	—
55	#341	D₃₇	1	1	yes	characteristic
56	C₃₇	#1778	1	1	yes	characteristic, complemented, Sylow
57	C₂×M₁₆	—	37	2	no	—
58	C₂×M₁₆	C₂×D₃₇	1	2	yes	—
59	C₂²×C₈	—	37	1	no	—
60	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
61	C₂×C₄²	—	37	1	no	—
62	#108	—	37	8	no	—
63	#97	—	37	4	no	—
64	#97	—	74	2	no	—
65	C₄×C₈	—	74	2	no	—
66	C₂²×C₄	—	37	2	no	—
67	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
68	M₁₆	—	2	2	no	—
69	M₁₆	—	37	4	no	—
70	M₁₆	—	74	2	no	—
71	M₁₆	C₄×D₃₇	1	4	yes	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	37	2	no	—
75	C₂×C₈	—	37	4	no	—
76	C₂×C₈	—	74	2	no	—
77	C₂×C₈	C₄×D₃₇	1	2	yes	—
78	C₄²	—	37	2	no	—
79	C₄²	—	37	2	no	—
80	C₂³	Q₈×D₃₇	1	1	yes	characteristic
81	C₂×C₄	—	37	4	no	—
82	C₂×C₄	—	148	2	no	—
83	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
84	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
85	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
86	C₂×C₄	#10596964	1	2	yes	—
87	C₈	—	2	2	no	—
88	C₈	—	2	4	no	—
89	C₈	—	37	4	no	—
90	C₈	—	74	4	no	—
91	C₂²	—	2	2	no	—
92	C₂²	C₄⋊C₄×D₃₇	1	1	yes	characteristic
93	C₂²	#11924501	1	2	yes	—
94	C₄	—	74	4	no	—
95	C₄	C₄²×D₃₇	1	1	yes	characteristic
96	C₄	#11924492	1	1	yes	characteristic
97	C₄	#11924496	1	2	yes	—
98	C₂	—	2	2	no	—
99	C₂	#14557384	1	1	yes	characteristic
100	C₂	#14557580	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924471

C₂

yes

maximal

M₁₆×Dic₃₇

C₂

yes

maximal

C₂×#11924446

C₂

yes

maximal, characteristic

#14557298

C₂

yes

maximal, characteristic

C₃₇×#341

C₂

yes

maximal, characteristic

C₂×#10596934

C₂²

yes

—

C₂²×#64436

C₂²

yes

characteristic

C₂×C₄×Dic₃₇

C₂²

yes

characteristic

#11924471

C₄

yes

complemented

#11924446

—

#11924446

C₄

yes

—

C₈×Dic₃₇

—

C₂×M₁₆×C₃₇

C₂²

yes

—

C₂²×C₈×C₃₇

C₂²

yes

characteristic

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

Q₈

yes

—

M₁₆×C₃₇

—

M₁₆×C₃₇

C₂×C₄

yes

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

C₂×C₄

yes

—

C₄×Dic₃₇

C₂×C₄

yes

—

C₄×Dic₃₇

D₄

yes

—

#10596934

—

#10596934

C₂×C₄

yes

—

C₂×#64436

—

C₂×#64436

C₂×C₄

yes

—

C₂×#64436

C₂×C₄

yes

—

C₂³×C₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

#41

yes

—

C₂×Dic₃₇

—

C₂×Dic₃₇

C₄⋊C₄

yes

—

C₈×C₃₇

—

C₈×C₃₇

—

#64436

—

#64436

C₄²

yes

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₇

C₄×Q₈

yes

—

C₄×C₃₇

C₂×C₄²

yes

characteristic

C₄×C₃₇

#117

yes

characteristic

C₄×C₃₇

C₄×D₄

yes

—

Dic₃₇

—

#1778

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

C₄×C₄⋊C₄

yes

characteristic

C₂×C₃₇

#430

yes

—

C₂×#108

—

C₄×M₁₆

—

C₂×#97

—

#341

—

#341

D₃₇

yes

characteristic

C₃₇

#1778

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃₇

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₃₇

yes

characteristic

C₂×C₄²

—

#108

—

#97

—

#97

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

C₄×D₃₇

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₃₇

yes

—

C₄²

—

C₄²

—

C₂³

Q₈×D₃₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

148

—

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

C₂×C₄

C₂×C₄×D₃₇

yes

—

C₂×C₄

D₄×D₃₇

yes

characteristic

C₂×C₄

#10596964

yes

—

C₈

—

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

C₄⋊C₄×D₃₇

yes

characteristic

C₂²

#11924501

yes

—

C₄

—

C₄

C₄²×D₃₇

yes

C₄

yes

C₄

yes

—

C₂

—

C₂

#14557384

yes

characteristic

100

C₂

#14557580

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₃₇
commutator	C₄×C₃₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₃₇×#341	C₂
socle	C₂²×C₃₇	C₂×C₄⋊C₄
prime radical	C₂³×C₃₇	C₂×Q₈
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	#341	D₃₇
hypocenter	C₃₇	#1778
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₃₇

commutator

C₄×C₃₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₃₇×#341

C₂

socle

C₂²×C₃₇

C₂×C₄⋊C₄

prime radical

C₂³×C₃₇

C₂×Q₈

cosocle

C₂²×C₄×C₃₇

C₂³

D₃₇

C₃₇

G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
9. #11924471	C₄	semidirect
56. C₃₇	#1778	semidirect

H ≅ G/N

type

9. #11924471

C₄

semidirect

56. C₃₇

#1778

semidirect

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

subgroup	index
C₂×#11924471	2
M₁₆×Dic₃₇	2
C₂×#11924446	2
#14557298	2
C₃₇×#341	2
#1778	37

subgroup

index

M₁₆×Dic₃₇

C₃₇×#341

Subgroups — #17994501

Conjugacy classes of subgroups

Complements

Maximal subgroups