Group #17996172 — subgroups

Subgroups — #17996172

Order 4736.

158 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 59 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	#14557582	C₂	1	1	yes	maximal, characteristic
2	#14557556	C₂	1	1	yes	maximal, characteristic
3	#14557303	C₂	1	2	yes	maximal
4	#14557292	C₂	1	2	yes	maximal
5	C₃₇×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
6	C₂×#10596934	C₂²	1	2	yes	—
7	C₂×#10596934	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
9	#11924472	—	2	2	no	—
10	#11924465	—	2	2	no	—
11	C₂×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
12	#117×C₃₇	C₂²	1	2	yes	—
13	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
14	C₂²×Dic₃₇	—	2	1	no	—
15	M₁₆×C₃₇	—	2	2	no	—
16	C₂×C₈×C₃₇	D₄	1	2	yes	—
17	C₄⋊C₄×C₃₇	—	2	2	no	—
18	C₂²⋊C₄×C₃₇	—	2	2	no	—
19	C₄²×C₃₇	—	2	2	no	—
20	C₄×Dic₃₇	—	2	1	no	—
21	C₄×Dic₃₇	Q₈	1	1	yes	characteristic
22	C₄×Dic₃₇	Q₈	1	1	yes	characteristic
23	#10596934	—	2	2	no	—
24	#10596934	—	2	2	no	—
25	#10596934	—	2	2	no	—
26	C₂×#64436	—	2	1	no	—
27	C₂×#64436	—	2	2	no	—
28	C₂³×C₃₇	#41	1	1	yes	characteristic
29	C₂×C₄×C₃₇	—	4	2	no	—
30	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₄×C₃₇	#41	1	2	yes	—
33	C₂×C₄×C₃₇	#41	1	2	yes	—
34	C₂×Dic₃₇	—	2	1	no	—
35	C₂×Dic₃₇	—	2	1	no	—
36	C₂×Dic₃₇	—	4	2	no	—
37	C₈×C₃₇	—	2	2	no	—
38	#64436	—	4	1	no	—
39	#64436	—	4	2	no	—
40	C₂²×C₃₇	—	4	1	no	—
41	C₂²×C₃₇	C₂²⋊Q₈	1	1	yes	characteristic
42	C₂²×C₃₇	#126	1	2	yes	—
43	C₄×C₃₇	—	8	2	no	—
44	C₄×C₃₇	#123	1	1	yes	characteristic
45	C₄×C₃₇	#124	1	1	yes	characteristic
46	C₄×C₃₇	#125	1	2	yes	—
47	Dic₃₇	—	2	2	no	—
48	Dic₃₇	—	4	1	no	—
49	#2102	—	37	1	no	maximal, Sylow
50	C₂×C₃₇	—	2	1	no	—
51	C₂×C₃₇	—	2	2	no	—
52	C₂×C₃₇	#400	1	1	yes	characteristic
53	#430	—	37	1	no	—
54	#423	—	37	1	no	—
55	#344	—	37	2	no	—
56	#339	—	37	2	no	—
57	H(3,ℤ/4)	D₃₇	1	1	yes	characteristic
58	C₃₇	#2102	1	1	yes	characteristic, complemented, Sylow
59	C₂×M₁₆	—	37	2	no	—
60	C₂×M₁₆	—	37	1	no	—
61	C₂×M₁₆	C₂×D₃₇	1	1	yes	characteristic
62	#117	C₂×D₃₇	1	2	yes	—
63	C₂×C₄²	—	37	1	no	—
64	#108	—	74	2	no	—
65	C₄⋊C₈	—	74	2	no	—
66	C₂²×C₄	—	74	1	no	—
67	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
68	M₁₆	—	2	2	no	—
69	M₁₆	—	74	2	no	—
70	M₁₆	—	74	2	no	—
71	M₁₆	—	74	2	no	—
72	C₂×C₈	—	74	1	no	—
73	C₂×C₈	—	74	2	no	—
74	C₂×C₈	D₁₄₈	1	2	yes	—
75	C₄⋊C₄	—	2	2	no	—
76	C₂²⋊C₄	—	2	2	no	—
77	C₄²	—	2	2	no	—
78	C₄²	—	37	1	no	—
79	C₄²	—	37	1	no	—
80	C₄²	—	74	1	no	—
81	C₂³	#10596964	1	1	yes	characteristic
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	74	1	no	—
84	C₂×C₄	—	74	1	no	—
85	C₂×C₄	—	148	2	no	—
86	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
87	C₂×C₄	#10596962	1	2	yes	—
88	C₂×C₄	#10596964	1	2	yes	—
89	C₂×C₄	Q₈×D₃₇	1	1	yes	characteristic
90	C₈	—	2	2	no	—
91	C₈	—	148	1	no	—
92	C₈	—	148	2	no	—
93	C₂²	—	4	1	no	—
94	C₂²	#11924516	1	1	yes	characteristic
95	C₂²	#11924529	1	2	yes	—
96	C₄	—	8	2	no	—
97	C₄	—	74	2	no	—
98	C₄	—	148	1	no	—
99	C₄	#11924520	1	1	yes	characteristic
100	C₄	#11924524	1	1	yes	characteristic
101	C₄	#11924526	1	2	yes	—
102	C₂	—	2	2	no	—
103	C₂	—	2	1	no	—
104	C₂	#14557473	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14557582

C₂

yes

maximal, characteristic

#14557556

C₂

yes

maximal, characteristic

#14557303

C₂

yes

maximal

#14557292

C₂

yes

maximal

C₃₇×H(3,ℤ/4)

C₂

yes

maximal, characteristic

C₂×#10596934

C₂²

yes

—

C₂×#10596934

C₂²

yes

characteristic

C₂×C₄×Dic₃₇

C₂²

yes

characteristic

#11924472

—

#11924465

—

C₂×M₁₆×C₃₇

C₂²

yes

characteristic

#117×C₃₇

C₂²

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

M₁₆×C₃₇

—

C₂×C₈×C₃₇

D₄

yes

—

C₄⋊C₄×C₃₇

—

C₂²⋊C₄×C₃₇

—

C₄²×C₃₇

—

C₄×Dic₃₇

—

C₄×Dic₃₇

Q₈

yes

characteristic

C₄×Dic₃₇

Q₈

yes

—

—

—

—

—

C₂³×C₃₇

#41

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₇

#41

yes

—

C₂×C₄×C₃₇

#41

yes

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₈×C₃₇

—

#64436

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₃₇

#126

yes

—

C₄×C₃₇

—

C₄×C₃₇

yes

C₄×C₃₇

yes

C₄×C₃₇

yes

—

Dic₃₇

—

Dic₃₇

—

#2102

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

yes

—

—

—

—

D₃₇

yes

characteristic

C₃₇

#2102

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃₇

yes

characteristic

#117

C₂×D₃₇

yes

—

C₂×C₄²

—

#108

—

C₄⋊C₈

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₁₄₈

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₂³

#10596964

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

148

—

C₂×C₄

C₂×D₁₄₈

yes

characteristic

C₂×C₄

#10596962

yes

—

C₂×C₄

#10596964

yes

—

C₂×C₄

Q₈×D₃₇

yes

characteristic

C₈

—

C₈

—

148

—

C₈

—

148

—

C₂²

—

C₂²

#11924516

yes

characteristic

C₂²

#11924529

yes

—

C₄

—

C₄

—

C₄

—

148

—

C₄

#11924520

yes

characteristic

100

C₄

#11924524

yes

characteristic

101

C₄

#11924526

yes

—

102

C₂

—

103

C₂

—

104

C₂

#14557473

yes

characteristic

	subgroup	G/H
center	C₄	#11924524
commutator, cosocle	C₂²×C₄×C₃₇	C₂³
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₃₇×H(3,ℤ/4)	C₂
socle	C₂×C₃₇	#400
prime radical	C₂³×C₃₇	#41
hypercenter	H(3,ℤ/4)	D₃₇
hypocenter	C₃₇	#2102
LCS	G ▷ C₂²×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#11924524

commutator, cosocle

C₂²×C₄×C₃₇

C₂³

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₃₇×H(3,ℤ/4)

C₂

C₂×C₃₇

C₂³×C₃₇

D₃₇

C₃₇

G ▷ C₂²×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
58. C₃₇	#2102	semidirect

H ≅ G/N

type

58. C₃₇

#2102

semidirect

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

subgroup	index
#14557582	2
#14557556	2
#14557303	2
#14557292	2
C₃₇×H(3,ℤ/4)	2
#2102	37

subgroup

index

C₃₇×H(3,ℤ/4)

Subgroups — #17996172

Conjugacy classes of subgroups

Complements

Maximal subgroups