Group #17997702 — subgroups

Subgroups — #17997702

Order 4736.

448 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 221 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	#14557419	C₂	1	4	yes	maximal
2	#14557419	C₂	1	4	yes	maximal
3	C₄⋊C₄×Dic₃₇	C₂	1	4	yes	maximal
4	#14557379	C₂	1	2	yes	maximal
5	C₂×#125×C₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₃₇	C₂²	1	4	yes	—
7	C₂×C₄×Dic₃₇	C₂²	1	4	yes	—
8	C₂×C₄×Dic₃₇	C₂²	1	4	yes	—
9	#11924443	C₂²	1	2	yes	—
10	#11924443	C₂²	1	2	yes	—
11	#11924443	C₂²	1	4	yes	—
12	#11924443	C₂²	1	4	yes	—
13	#11924443	C₂²	1	4	yes	—
14	#125×C₃₇	C₄	1	8	yes	complemented
15	C₂×C₄⋊C₄×C₃₇	C₂²	1	4	yes	—
16	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
17	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
19	C₂²×C₄×C₃₇	C₂³	1	4	yes	—
20	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
21	C₂²×Dic₃₇	C₂³	1	4	yes	—
22	C₂²×Dic₃₇	C₂³	1	4	yes	—
23	C₄⋊C₄×C₃₇	C₂×C₄	1	16	yes	—
24	C₄⋊C₄×C₃₇	C₂×C₄	1	8	yes	—
25	C₄²×C₃₇	C₂×C₄	1	4	yes	—
26	C₄⋊Dic₃₇	—	2	8	no	—
27	C₄×Dic₃₇	—	2	8	no	—
28	C₄×Dic₃₇	—	2	8	no	—
29	C₂³×C₃₇	C₂⁴	1	1	yes	characteristic
30	C₂×C₄×C₃₇	—	2	4	no	—
31	C₂×C₄×C₃₇	—	2	4	no	—
32	C₂×C₄×C₃₇	—	2	4	no	—
33	C₂×C₄×C₃₇	—	2	2	no	—
34	C₂×C₄×C₃₇	C₂²×C₄	1	4	yes	—
35	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
36	C₂×C₄×C₃₇	C₂²×C₄	1	8	yes	—
37	C₂×Dic₃₇	—	2	4	no	—
38	C₂×Dic₃₇	—	2	4	no	—
39	C₂×Dic₃₇	—	2	4	no	—
40	C₂×Dic₃₇	—	2	4	no	—
41	C₂×Dic₃₇	—	2	4	no	—
42	C₂×Dic₃₇	#41	1	8	yes	—
43	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
44	C₂²×C₃₇	C₂×#41	1	4	yes	—
45	C₂²×C₃₇	2₋¹⁺⁴	1	2	yes	—
46	C₄×C₃₇	—	2	4	no	—
47	C₄×C₃₇	—	2	8	no	—
48	C₄×C₃₇	—	2	2	no	—
49	Dic₃₇	—	2	8	no	—
50	Dic₃₇	—	4	4	no	—
51	#2371	—	37	1	no	maximal, Sylow
52	C₂×C₃₇	C₄×#41	1	2	yes	—
53	C₂×C₃₇	#519	1	1	yes	characteristic
54	C₂×C₃₇	#552	1	4	yes	—
55	C₂×#125	D₃₇	1	1	yes	characteristic
56	#387	—	37	4	no	—
57	#387	—	37	4	no	—
58	C₄×C₄⋊C₄	—	37	4	no	—
59	#376	—	37	2	no	—
60	C₃₇	#2371	1	1	yes	characteristic, complemented, Sylow
61	#125	Dic₃₇	1	8	yes	complemented
62	C₂×C₄⋊C₄	—	37	4	no	—
63	C₂×C₄⋊C₄	C₂×D₃₇	1	4	yes	—
64	C₂×C₄⋊C₄	C₂×D₃₇	1	2	yes	—
65	C₂×C₄²	—	37	4	no	—
66	C₂×C₄²	—	37	4	no	—
67	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
68	#95	—	37	2	no	—
69	#95	—	37	4	no	—
70	#95	—	37	2	no	—
71	#95	—	37	4	no	—
72	#95	—	37	4	no	—
73	C₂²×C₄	—	37	4	no	—
74	C₂²×C₄	—	37	4	no	—
75	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
76	C₂²×C₄	C₂²×D₃₇	1	2	yes	—
77	C₂²×C₄	C₂²×D₃₇	1	4	yes	—
78	C₄⋊C₄	—	74	8	no	—
79	C₄⋊C₄	C₂×Dic₃₇	1	16	yes	—
80	C₄⋊C₄	C₂×Dic₃₇	1	8	yes	—
81	C₄²	—	74	8	no	—
82	C₄²	—	74	8	no	—
83	C₄²	C₂×Dic₃₇	1	4	yes	—
84	C₂³	C₂³×D₃₇	1	1	yes	characteristic
85	C₂×C₄	—	2	4	no	—
86	C₂×C₄	—	2	4	no	—
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	37	8	no	—
90	C₂×C₄	—	74	4	no	—
91	C₂×C₄	—	74	4	no	—
92	C₂×C₄	—	74	4	no	—
93	C₂×C₄	—	74	4	no	—
94	C₂×C₄	—	74	4	no	—
95	C₂×C₄	C₂²×Dic₃₇	1	4	yes	—
96	C₂×C₄	C₂²×Dic₃₇	1	2	yes	—
97	C₂×C₄	C₂²×Dic₃₇	1	8	yes	—
98	C₂²	C₂³×Dic₃₇	1	1	yes	characteristic
99	C₂²	#41×D₃₇	1	4	yes	—
100	C₂²	#11924584	1	2	yes	—
101	C₄	—	2	4	no	—
102	C₄	—	2	8	no	—
103	C₄	—	2	2	no	—
104	C₄	—	74	8	no	—
105	C₄	—	148	4	no	—
106	C₂	#41×Dic₃₇	1	2	yes	—
107	C₂	#14558011	1	1	yes	characteristic
108	C₂	#14558215	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14557419

C₂

yes

maximal

#14557419

C₂

yes

maximal

C₄⋊C₄×Dic₃₇

C₂

yes

maximal

#14557379

C₂

yes

maximal

C₂×#125×C₃₇

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₃₇

C₂²

yes

—

C₂×C₄×Dic₃₇

C₂²

yes

—

C₂×C₄×Dic₃₇

C₂²

yes

—

#11924443

C₂²

yes

—

#11924443

C₂²

yes

—

#11924443

C₂²

yes

—

#11924443

C₂²

yes

—

#11924443

C₂²

yes

—

#125×C₃₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₃₇

C₂²

yes

—

C₂×C₄⋊C₄×C₃₇

C₂²

yes

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

C₂²×C₄×C₃₇

C₂³

yes

—

C₂²×C₄×C₃₇

C₂³

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

C₂³

yes

—

C₂²×Dic₃₇

C₂³

yes

—

C₄⋊C₄×C₃₇

C₂×C₄

yes

—

C₄⋊C₄×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

C₂×C₄

yes

—

C₄⋊Dic₃₇

—

C₄×Dic₃₇

—

C₄×Dic₃₇

—

C₂³×C₃₇

C₂⁴

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

#41

yes

—

C₂²×C₃₇

C₂³×C₄

yes

characteristic

C₂²×C₃₇

C₂×#41

yes

—

C₂²×C₃₇

2₋¹⁺⁴

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

Dic₃₇

—

Dic₃₇

—

#2371

—

maximal, Sylow

C₂×C₃₇

C₄×#41

yes

—

C₂×C₃₇

#519

yes

characteristic

C₂×C₃₇

#552

yes

—

C₂×#125

D₃₇

yes

characteristic

#387

—

#387

—

C₄×C₄⋊C₄

—

#376

—

C₃₇

#2371

yes

characteristic, complemented, Sylow

#125

Dic₃₇

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₇

yes

—

C₂×C₄⋊C₄

C₂×D₃₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃₇

yes

—

—

—

—

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

C₂²×C₄

C₂²×D₃₇

yes

—

C₂²×C₄

C₂²×D₃₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₃₇

yes

—

C₄⋊C₄

C₂×Dic₃₇

yes

—

C₄²

—

C₄²

—

C₄²

C₂×Dic₃₇

yes

—

C₂³

C₂³×D₃₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×Dic₃₇

yes

—

C₂×C₄

C₂²×Dic₃₇

yes

—

C₂×C₄

C₂²×Dic₃₇

yes

—

C₂²

C₂³×Dic₃₇

yes

characteristic

C₂²

#41×D₃₇

yes

—

100

C₂²

#11924584

yes

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

148

—

106

C₂

#41×Dic₃₇

yes

—

107

C₂

#14558011

yes

characteristic

108

C₂

#14558215

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃₇
commutator	C₂²×C₃₇	C₂³×C₄
Fitting	C₂×#125×C₃₇	C₂
socle, prime radical, cosocle	C₂³×C₃₇	C₂⁴
hypercenter	C₂×#125	D₃₇
hypocenter	C₃₇	#2371
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center, Frattini

C₂³

C₂³×D₃₇

commutator

C₂²×C₃₇

C₂³×C₄

Fitting

C₂×#125×C₃₇

C₂

socle, prime radical, cosocle

C₂³×C₃₇

C₂⁴

D₃₇

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
14. #125×C₃₇	C₄	semidirect
60. C₃₇	#2371	semidirect
61. #125	Dic₃₇	semidirect

H ≅ G/N

type

14. #125×C₃₇

C₄

semidirect

60. C₃₇

#2371

semidirect

61. #125

Dic₃₇

semidirect

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

subgroup	index
#14557419	2
C₄⋊C₄×Dic₃₇	2
#14557379	2
C₂×#125×C₃₇	2
#2371	37

subgroup

index

#14557419

C₄⋊C₄×Dic₃₇

#14557379

C₂×#125×C₃₇

#2371

Subgroups — #17997702

Conjugacy classes of subgroups

Complements

Maximal subgroups