Group #17994497 — subgroups

Subgroups — #17994497

Order 4736.

228 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 109 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×C₈×Dic₃₇	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	C₈×Dic₃₇	—	2	2	no	—
12	C₈×Dic₃₇	C₄	1	4	yes	—
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	1	yes	characteristic
24	C₄×Dic₃₇	D₄	1	1	yes	characteristic
25	#10596934	—	2	2	no	—
26	#10596934	C₂×C₄	1	4	yes	—
27	C₂×#64436	—	2	2	no	—
28	C₂×#64436	C₂×C₄	1	2	yes	—
29	C₂×#64436	C₂×C₄	1	4	yes	—
30	C₂³×C₃₇	C₂×Q₈	1	1	yes	characteristic
31	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
33	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₃₇	#41	1	2	yes	—
35	C₂×Dic₃₇	—	2	4	no	—
36	C₂×Dic₃₇	C₄⋊C₄	1	2	yes	—
37	C₂×Dic₃₇	C₄⋊C₄	1	2	yes	—
38	C₈×C₃₇	—	2	2	no	—
39	C₈×C₃₇	—	2	4	no	—
40	#64436	—	2	4	no	—
41	#64436	C₄²	1	4	yes	—
42	C₂²×C₃₇	—	2	2	no	—
43	C₂²×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₃₇	C₄×Q₈	1	2	yes	—
45	C₄×C₃₇	C₂×C₄²	1	1	yes	characteristic
46	C₄×C₃₇	#117	1	1	yes	characteristic
47	C₄×C₃₇	C₄×D₄	1	2	yes	—
48	Dic₃₇	—	2	2	no	—
49	Dic₃₇	#108	1	4	yes	complemented
50	C₄×#108	—	37	1	no	maximal, Sylow
51	C₂×C₃₇	—	2	2	no	—
52	C₂×C₃₇	C₄×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₃₇	C₂×#108	1	2	yes	—
54	C₂×#108	—	37	2	no	—
55	C₄×M₁₆	—	37	2	no	—
56	C₂×C₄×C₈	—	37	1	no	—
57	#341	—	37	1	no	—
58	#341	D₃₇	1	1	yes	characteristic
59	C₃₇	C₄×#108	1	1	yes	characteristic, complemented, Sylow
60	C₂×M₁₆	—	37	2	no	—
61	C₂×M₁₆	C₂×D₃₇	1	2	yes	—
62	C₂²×C₈	—	37	1	no	—
63	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
64	C₂×C₄²	—	37	1	no	—
65	#108	—	37	8	no	—
66	#97	—	74	2	no	—
67	C₄×C₈	—	37	4	no	—
68	C₄×C₈	—	74	2	no	—
69	C₂²×C₄	—	37	2	no	—
70	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
71	M₁₆	—	2	2	no	—
72	M₁₆	—	37	4	no	—
73	M₁₆	—	74	2	no	—
74	M₁₆	C₄×D₃₇	1	4	yes	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	37	2	no	—
78	C₂×C₈	—	37	4	no	—
79	C₂×C₈	—	74	2	no	—
80	C₂×C₈	C₄×D₃₇	1	2	yes	—
81	C₄²	—	37	2	no	—
82	C₄²	—	37	1	no	—
83	C₄²	—	37	1	no	—
84	C₂³	Q₈×D₃₇	1	1	yes	characteristic
85	C₂×C₄	—	37	2	no	—
86	C₂×C₄	—	37	2	no	—
87	C₂×C₄	—	74	4	no	—
88	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
89	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
90	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
91	C₂×C₄	#10596964	1	2	yes	—
92	C₈	—	2	2	no	—
93	C₈	—	2	4	no	—
94	C₈	—	37	4	no	—
95	C₈	—	74	4	no	—
96	C₂²	—	2	2	no	—
97	C₂²	C₄⋊C₄×D₃₇	1	1	yes	characteristic
98	C₂²	#11924501	1	2	yes	—
99	C₄	—	37	4	no	—
100	C₄	—	74	2	no	—
101	C₄	C₄²×D₃₇	1	1	yes	characteristic
102	C₄	#11924492	1	1	yes	characteristic
103	C₄	#11924496	1	2	yes	—
104	C₂	—	2	2	no	—
105	C₂	#14557384	1	1	yes	characteristic
106	C₂	#108×D₃₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924471

C₂

yes

maximal

M₁₆×Dic₃₇

C₂

yes

maximal

C₂×C₈×Dic₃₇

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

—

C₈×Dic₃₇

—

C₈×Dic₃₇

C₄

yes

—

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

characteristic

C₄×Dic₃₇

D₄

yes

characteristic

#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

—

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

—

Dic₃₇

yes

—

C₂×C₃₇

—

C₂×C₃₇

C₄×C₄⋊C₄

yes

characteristic

C₂×C₃₇

C₂×#108

yes

—

C₂×#108

—

C₄×M₁₆

—

C₂×C₄×C₈

—

#341

—

#341

D₃₇

yes

characteristic

C₃₇

C₄×#108

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃₇

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₃₇

yes

characteristic

C₂×C₄²

—

#108

—

#97

—

C₄×C₈

—

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

—

C₄²

—

C₂³

Q₈×D₃₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

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

—

100

C₄

—

101

C₄

C₄²×D₃₇

yes

characteristic

102

C₄

#11924492

yes

characteristic

103

C₄

#11924496

yes

—

104

C₂

—

105

C₂

#14557384

yes

characteristic

106

C₂

#108×D₃₇

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	C₄×#108
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
49. Dic₃₇	#108	semidirect
59. C₃₇	C₄×#108	semidirect

H ≅ G/N

type

9. #11924471

C₄

semidirect

49. Dic₃₇

#108

semidirect

59. C₃₇

C₄×#108

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×C₈×Dic₃₇	2
#14557298	2
C₃₇×#341	2
C₄×#108	37

subgroup

index

C₂×#11924471

M₁₆×Dic₃₇

C₂×C₈×Dic₃₇

#14557298

C₃₇×#341

C₄×#108

Subgroups — #17994497

Conjugacy classes of subgroups

Complements

Maximal subgroups