Group #17994547 — subgroups

Subgroups — #17994547

Order 4736.

548 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 121 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	#14557517	C₂	1	3	yes	maximal, complemented
2	#14557500	C₂	1	3	yes	maximal, complemented
3	C₃₇×#585	C₂	1	1	yes	maximal, characteristic
4	C₂×#10596934	C₂²	1	3	yes	complemented
5	C₂²×#64436	C₂²	1	1	yes	characteristic, complemented
6	#11924455	—	2	3	no	—
7	#11924453	—	2	3	no	—
8	C₂²⋊#64436	—	2	6	no	—
9	2₋¹⁺⁴×C₃₇	—	2	3	no	—
10	H(5,𝔽₂)×C₃₇	—	2	3	no	—
11	H(5,𝔽₂)×C₃₇	C₄	1	4	yes	—
12	C₂×#41×C₃₇	—	2	6	no	—
13	C₂×#41×C₃₇	C₂²	1	3	yes	—
14	#41×C₃₇	—	2	4	no	—
15	#41×C₃₇	—	2	12	no	—
16	#41×C₃₇	—	2	12	no	—
17	#41×C₃₇	—	2	6	no	—
18	#41×C₃₇	—	2	6	no	—
19	C₂×Q₈×C₃₇	—	2	6	no	—
20	C₂×Q₈×C₃₇	D₄	1	3	yes	—
21	C₂×D₄×C₃₇	—	2	6	no	—
22	C₂×D₄×C₃₇	—	2	12	no	—
23	C₂×D₄×C₃₇	C₂×C₄	1	6	yes	—
24	C₂×D₄×C₃₇	D₄	1	3	yes	—
25	C₂²×C₄×C₃₇	—	2	4	no	—
26	C₂²×C₄×C₃₇	D₄	1	6	yes	—
27	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
28	#10596934	—	2	6	no	—
29	C₂×#64436	—	2	3	no	—
30	C₂×#64436	—	2	3	no	—
31	C₂³×C₃₇	—	2	4	no	—
32	C₂³×C₃₇	C₂²⋊C₄	1	6	yes	—
33	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
34	Q₈×C₃₇	—	2	4	no	—
35	Q₈×C₃₇	—	2	6	no	—
36	D₄×C₃₇	—	2	12	no	—
37	D₄×C₃₇	—	2	12	no	—
38	D₄×C₃₇	—	2	6	no	—
39	C₂×C₄×C₃₇	—	2	12	no	—
40	C₂×C₄×C₃₇	—	2	6	no	—
41	C₂×C₄×C₃₇	—	2	6	no	—
42	C₂×C₄×C₃₇	C₂²⋊C₄	1	6	yes	—
43	C₂×C₄×C₃₇	C₂×D₄	1	3	yes	—
44	C₂×C₄×C₃₇	C₂×D₄	1	3	yes	—
45	#64436	—	4	1	no	—
46	#64436	—	4	3	no	—
47	C₂²×C₃₇	—	2	6	no	—
48	C₂²×C₃₇	—	4	1	no	—
49	C₂²×C₃₇	—	4	6	no	—
50	C₂²×C₃₇	—	8	4	no	—
51	C₂²×C₃₇	C₂×C₂²⋊C₄	1	3	yes	—
52	C₄×C₃₇	—	2	6	no	—
53	C₄×C₃₇	#120	1	3	yes	—
54	C₄×C₃₇	#120	1	1	yes	characteristic
55	#1791	—	37	1	no	maximal, Sylow
56	C₂×C₃₇	—	2	3	no	—
57	C₂×C₃₇	—	4	6	no	—
58	C₂×C₃₇	#379	1	1	yes	characteristic
59	#585	D₃₇	1	1	yes	characteristic
60	#413	—	37	3	no	—
61	#408	—	37	3	no	—
62	C₃₇	#1791	1	1	yes	characteristic, complemented, Sylow
63	2₋¹⁺⁴	—	2	3	no	—
64	H(5,𝔽₂)	—	2	3	no	—
65	H(5,𝔽₂)	Dic₃₇	1	4	yes	—
66	C₂×#41	—	2	6	no	—
67	C₂×#41	C₂×D₃₇	1	3	yes	—
68	C₂×M₁₆	—	37	3	no	—
69	C₂²×C₈	—	37	1	no	—
70	#101	—	74	3	no	—
71	#100	—	74	3	no	—
72	C₂²⋊C₈	—	74	6	no	—
73	#41	—	2	4	no	—
74	#41	—	2	12	no	—
75	#41	—	2	12	no	—
76	#41	—	2	6	no	—
77	#41	—	2	6	no	—
78	C₂×Q₈	—	2	6	no	—
79	C₂×Q₈	C₂²⋊D₃₇	1	3	yes	—
80	C₂×D₄	—	2	6	no	—
81	C₂×D₄	—	2	12	no	—
82	C₂×D₄	C₂×Dic₃₇	1	6	yes	—
83	C₂×D₄	C₂²⋊D₃₇	1	3	yes	—
84	C₂²×C₄	—	2	4	no	—
85	C₂²×C₄	C₂²⋊D₃₇	1	6	yes	—
86	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
87	M₁₆	—	74	6	no	—
88	C₂×C₈	—	74	3	no	—
89	C₂×C₈	—	74	3	no	—
90	C₂³	—	2	4	no	—
91	C₂³	C₂²⋊Dic₃₇	1	6	yes	—
92	C₂³	C₂²×Dic₃₇	1	1	yes	characteristic
93	Q₈	—	2	4	no	—
94	Q₈	—	2	6	no	—
95	D₄	—	2	12	no	—
96	D₄	—	2	12	no	—
97	D₄	—	2	6	no	—
98	C₂×C₄	—	2	12	no	—
99	C₂×C₄	—	2	6	no	—
100	C₂×C₄	—	2	6	no	—
101	C₂×C₄	C₂²⋊Dic₃₇	1	6	yes	—
102	C₂×C₄	C₂×C₂²⋊D₃₇	1	3	yes	—
103	C₂×C₄	C₂×C₂²⋊D₃₇	1	3	yes	—
104	C₈	—	148	3	no	—
105	C₈	—	148	1	no	—
106	C₂²	—	2	6	no	—
107	C₂²	—	4	1	no	—
108	C₂²	—	4	6	no	—
109	C₂²	—	8	4	no	—
110	C₂²	C₂×C₂²⋊Dic₃₇	1	3	yes	—
111	C₄	—	2	6	no	—
112	C₄	#11924505	1	3	yes	—
113	C₄	#11924505	1	1	yes	characteristic
114	C₂	—	2	3	no	—
115	C₂	—	4	6	no	—
116	C₂	#14557389	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14557517

C₂

yes

maximal, complemented

#14557500

C₂

yes

maximal, complemented

C₃₇×#585

C₂

yes

maximal, characteristic

C₂×#10596934

C₂²

yes

complemented

C₂²×#64436

C₂²

yes

characteristic, complemented

#11924455

—

#11924453

—

C₂²⋊#64436

—

2₋¹⁺⁴×C₃₇

—

H(5,𝔽₂)×C₃₇

—

H(5,𝔽₂)×C₃₇

C₄

yes

—

C₂×#41×C₃₇

—

C₂×#41×C₃₇

C₂²

yes

—

#41×C₃₇

—

#41×C₃₇

—

#41×C₃₇

—

#41×C₃₇

—

#41×C₃₇

—

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

D₄

yes

—

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

C₂×C₄

yes

—

C₂×D₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

—

—

—

C₂³×C₃₇

—

C₂³×C₃₇

C₂²⋊C₄

yes

—

C₂³×C₃₇

C₂²×C₄

yes

characteristic

Q₈×C₃₇

—

Q₈×C₃₇

—

D₄×C₃₇

—

D₄×C₃₇

—

D₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

#64436

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

—

C₄×C₃₇

—

C₄×C₃₇

#120

yes

—

C₄×C₃₇

yes

—

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

#379

yes

characteristic

#585

D₃₇

yes

characteristic

#413

—

#408

—

C₃₇

#1791

yes

characteristic, complemented, Sylow

2₋¹⁺⁴

—

H(5,𝔽₂)

—

H(5,𝔽₂)

Dic₃₇

yes

—

C₂×#41

—

C₂×#41

C₂×D₃₇

yes

—

C₂×M₁₆

—

C₂²×C₈

—

#101

—

#100

—

C₂²⋊C₈

—

—

—

—

—

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₃₇

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₃₇

yes

—

C₂×D₄

C₂²⋊D₃₇

yes

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₃₇

yes

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂³

—

C₂³

C₂²⋊Dic₃₇

yes

—

C₂³

C₂²×Dic₃₇

yes

characteristic

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₂²⋊Dic₃₇

yes

—

102

C₂×C₄

C₂×C₂²⋊D₃₇

yes

—

103

C₂×C₄

C₂×C₂²⋊D₃₇

yes

—

104

C₈

—

148

—

105

C₈

—

148

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

C₂×C₂²⋊Dic₃₇

yes

—

111

C₄

—

112

C₄

#11924505

yes

—

113

C₄

#11924505

yes

characteristic

114

C₂

—

115

C₂

—

116

C₂

#14557389

yes

characteristic

	subgroup	G/H
center	C₄	#11924505
commutator	C₂³×C₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting, prime radical	C₃₇×#585	C₂
socle	C₂×C₃₇	#379
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	#585	D₃₇
hypocenter	C₃₇	#1791
LCS	G ▷ C₂³×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #585

subgroup

G/H

center

C₄

#11924505

commutator

C₂³×C₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting, prime radical

C₃₇×#585

C₂

socle

C₂×C₃₇

#379

cosocle

C₂²×C₄×C₃₇

C₂³

D₃₇

C₃₇

G ▷ C₂³×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #585

N	H ≅ G/N	type
1. #14557517	C₂	semidirect
2. #14557500	C₂	semidirect
4. C₂×#10596934	C₂²	semidirect
5. C₂²×#64436	C₂²	semidirect
62. C₃₇	#1791	semidirect

H ≅ G/N

type

1. #14557517

C₂

semidirect

2. #14557500

C₂

semidirect

4. C₂×#10596934

C₂²

semidirect

5. C₂²×#64436

C₂²

semidirect

62. C₃₇

#1791

semidirect

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

subgroup	index
#14557517	2
#14557500	2
C₃₇×#585	2
#1791	37

subgroup

index

#14557517

#14557500

C₃₇×#585

#1791

Subgroups — #17994547

Conjugacy classes of subgroups

Complements

Maximal subgroups