Group #17996106 — subgroups

Subgroups — #17996106

Order 4736.

176 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 69 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	#14557576	C₂	1	2	yes	maximal
2	C₂×#11924468	C₂	1	1	yes	maximal, characteristic
3	#14557301	C₂	1	2	yes	maximal
4	#14557298	C₂	1	1	yes	maximal, characteristic
5	C₃₇×#424	C₂	1	1	yes	maximal, characteristic
6	C₂×#10596934	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
8	#11924469	—	2	2	no	—
9	#11924468	—	2	2	no	—
10	#11924467	—	2	2	no	—
11	C₂×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
13	#117×C₃₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₃₇	—	2	2	no	—
15	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
16	C₂²×Dic₃₇	—	2	2	no	—
17	M₁₆×C₃₇	—	2	2	no	—
18	C₂×C₈×C₃₇	—	2	2	no	—
19	C₂×C₈×C₃₇	—	2	1	no	—
20	C₂×C₈×C₃₇	D₄	1	2	yes	—
21	C₄⋊C₄×C₃₇	D₄	1	2	yes	—
22	C₂²⋊C₄×C₃₇	—	2	1	no	—
23	C₄²×C₃₇	—	2	1	no	—
24	C₄⋊Dic₃₇	—	2	2	no	—
25	C₄⋊Dic₃₇	—	2	2	no	—
26	#10596934	—	2	2	no	—
27	#10596934	Q₈	1	4	yes	—
28	C₂×#64436	—	2	2	no	—
29	C₂³×C₃₇	#41	1	1	yes	characteristic
30	C₂×C₄×C₃₇	—	2	2	no	—
31	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₃₇	C₂×Q₈	1	2	yes	—
34	C₂×C₄×C₃₇	#41	1	2	yes	—
35	C₂×Dic₃₇	—	4	2	no	—
36	C₂×Dic₃₇	—	8	2	no	—
37	C₈×C₃₇	—	2	2	no	—
38	C₈×C₃₇	—	4	1	no	—
39	#64436	—	2	4	no	—
40	C₂²×C₃₇	—	2	2	no	—
41	C₂²×C₃₇	#121	1	1	yes	characteristic
42	C₂²×C₃₇	#125	1	2	yes	—
43	C₄×C₃₇	—	4	2	no	—
44	C₄×C₃₇	C₂²⋊Q₈	1	2	yes	—
45	C₄×C₃₇	#123	1	1	yes	characteristic
46	C₄×C₃₇	C₄⋊Q₈	1	1	yes	characteristic
47	Dic₃₇	—	8	2	no	—
48	#2090	—	37	1	no	maximal, Sylow
49	C₂×C₃₇	—	2	2	no	—
50	C₂×C₃₇	#398	1	1	yes	characteristic
51	C₂×C₃₇	#471	1	2	yes	—
52	#428	—	37	2	no	—
53	C₂×#106	—	37	1	no	—
54	#424	D₃₇	1	1	yes	characteristic
55	#343	—	37	2	no	—
56	#341	—	37	1	no	—
57	C₃₇	#2090	1	1	yes	characteristic, complemented, Sylow
58	C₂×M₁₆	—	37	2	no	—
59	C₂×M₁₆	C₂×D₃₇	1	1	yes	characteristic
60	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
61	#117	C₂×D₃₇	1	1	yes	characteristic
62	C₂×C₄⋊C₄	—	37	2	no	—
63	#107	—	74	2	no	—
64	#106	—	74	2	no	—
65	#106	—	74	2	no	—
66	C₄⋊C₈	—	2	2	no	—
67	C₂²×C₄	—	74	2	no	—
68	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
69	M₁₆	—	2	2	no	—
70	M₁₆	—	37	4	no	—
71	M₁₆	—	74	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	1	no	—
74	C₂×C₈	—	74	2	no	—
75	C₂×C₈	C₂²⋊D₃₇	1	2	yes	—
76	C₄⋊C₄	—	74	2	no	—
77	C₄⋊C₄	—	74	2	no	—
78	C₄⋊C₄	D₁₄₈	1	2	yes	—
79	C₂²⋊C₄	—	2	1	no	—
80	C₄²	—	2	1	no	—
81	C₂³	#10596962	1	1	yes	characteristic
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	296	2	no	—
84	C₂×C₄	—	148	2	no	—
85	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
86	C₂×C₄	#10596964	1	2	yes	—
87	C₂×C₄	Q₈×D₃₇	1	2	yes	—
88	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
89	C₈	—	2	2	no	—
90	C₈	—	4	1	no	—
91	C₈	—	74	4	no	—
92	C₂²	—	2	2	no	—
93	C₂²	#11924510	1	1	yes	characteristic
94	C₂²	#11924526	1	2	yes	—
95	C₄	—	4	2	no	—
96	C₄	—	296	2	no	—
97	C₄	#11924516	1	2	yes	—
98	C₄	#11924517	1	1	yes	characteristic
99	C₄	#11924534	1	1	yes	characteristic
100	C₂	—	2	2	no	—
101	C₂	#14557463	1	1	yes	characteristic
102	C₂	#14557796	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14557576

C₂

yes

maximal

C₂×#11924468

C₂

yes

maximal, characteristic

#14557301

C₂

yes

maximal

#14557298

C₂

yes

maximal, characteristic

C₃₇×#424

C₂

yes

maximal, characteristic

C₂×#10596934

C₂²

yes

—

C₂×C₄⋊Dic₃₇

C₂²

yes

—

#11924469

—

#11924468

—

#11924467

—

C₂×M₁₆×C₃₇

C₂²

yes

characteristic

C₂²×C₈×C₃₇

C₂²

yes

characteristic

#117×C₃₇

C₂²

yes

characteristic

C₄⋊C₈×C₃₇

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

M₁₆×C₃₇

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

D₄

yes

—

C₄⋊C₄×C₃₇

D₄

yes

—

C₂²⋊C₄×C₃₇

—

C₄²×C₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

#10596934

—

#10596934

Q₈

yes

—

C₂×#64436

—

C₂³×C₃₇

#41

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×Q₈

yes

—

C₂×C₄×C₃₇

#41

yes

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₈×C₃₇

—

C₈×C₃₇

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

#121

yes

characteristic

C₂²×C₃₇

#125

yes

—

C₄×C₃₇

—

C₄×C₃₇

C₂²⋊Q₈

yes

—

C₄×C₃₇

#123

yes

characteristic

C₄×C₃₇

C₄⋊Q₈

yes

characteristic

Dic₃₇

—

#2090

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

#398

yes

characteristic

C₂×C₃₇

#471

yes

—

#428

—

C₂×#106

—

#424

D₃₇

yes

characteristic

#343

—

#341

—

C₃₇

#2090

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃₇

yes

characteristic

C₂²×C₈

C₂×D₃₇

yes

characteristic

#117

C₂×D₃₇

yes

characteristic

C₂×C₄⋊C₄

—

#107

—

#106

—

#106

—

C₄⋊C₈

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₃₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₁₄₈

yes

—

C₂²⋊C₄

—

C₄²

—

C₂³

#10596962

yes

characteristic

C₂×C₄

—

C₂×C₄

—

296

—

C₂×C₄

—

148

—

C₂×C₄

C₂×D₁₄₈

yes

characteristic

C₂×C₄

#10596964

yes

—

C₂×C₄

Q₈×D₃₇

yes

—

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

#11924510

yes

characteristic

C₂²

#11924526

yes

—

C₄

—

C₄

—

296

—

C₄

#11924516

yes

—

C₄

yes

C₄

yes

100

C₂

—

101

C₂

#14557463

yes

characteristic

102

C₂

#14557796

yes

—

	subgroup	G/H
center	C₂²	#11924510
commutator, cosocle	C₂²×C₄×C₃₇	C₂³
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₃₇×#424	C₂
socle	C₂²×C₃₇	#121
prime radical	C₂³×C₃₇	#41
hypercenter	#424	D₃₇
hypocenter	C₃₇	#2090
LCS	G ▷ C₂²×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

subgroup

G/H

center

C₂²

#11924510

commutator, cosocle

C₂²×C₄×C₃₇

C₂³

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₃₇×#424

C₂

C₂²×C₃₇

C₂³×C₃₇

D₃₇

C₃₇

G ▷ C₂²×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

N	H ≅ G/N	type
57. C₃₇	#2090	semidirect

H ≅ G/N

type

57. C₃₇

#2090

semidirect

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

subgroup	index
#14557576	2
C₂×#11924468	2
#14557301	2
#14557298	2
C₃₇×#424	2
#2090	37

subgroup

index

C₃₇×#424

Subgroups — #17996106

Conjugacy classes of subgroups

Complements

Maximal subgroups