Group #17996101 — subgroups

Subgroups — #17996101

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×#11924470	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	#11924470	—	2	2	no	—
9	#11924469	—	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	1	yes	characteristic
22	C₄⋊C₄×C₃₇	D₄	1	1	yes	characteristic
23	C₂²⋊C₄×C₃₇	—	2	1	no	—
24	C₄²×C₃₇	—	2	1	no	—
25	C₄⋊Dic₃₇	—	2	2	no	—
26	C₄⋊Dic₃₇	—	2	2	no	—
27	#10596934	—	2	2	no	—
28	#10596934	Q₈	1	4	yes	—
29	C₂×#64436	—	2	2	no	—
30	C₂³×C₃₇	#41	1	1	yes	characteristic
31	C₂×C₄×C₃₇	—	2	1	no	—
32	C₂×C₄×C₃₇	—	2	1	no	—
33	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₃₇	C₂×Q₈	1	2	yes	—
36	C₂×C₄×C₃₇	#41	1	2	yes	—
37	C₂×Dic₃₇	—	4	2	no	—
38	C₂×Dic₃₇	—	8	2	no	—
39	C₈×C₃₇	—	2	2	no	—
40	C₈×C₃₇	—	4	1	no	—
41	#64436	—	2	4	no	—
42	C₂²×C₃₇	—	2	1	no	—
43	C₂²×C₃₇	—	2	1	no	—
44	C₂²×C₃₇	#121	1	1	yes	characteristic
45	C₂²×C₃₇	#125	1	2	yes	—
46	C₄×C₃₇	—	4	1	no	—
47	C₄×C₃₇	—	4	1	no	—
48	C₄×C₃₇	C₂²⋊Q₈	1	2	yes	—
49	C₄×C₃₇	#123	1	1	yes	characteristic
50	C₄×C₃₇	C₄⋊Q₈	1	1	yes	characteristic
51	Dic₃₇	—	8	2	no	—
52	#2089	—	37	1	no	maximal, Sylow
53	C₂×C₃₇	—	2	2	no	—
54	C₂×C₃₇	#398	1	1	yes	characteristic
55	C₂×C₃₇	#472	1	1	yes	characteristic
56	C₂×C₃₇	#473	1	1	yes	characteristic
57	#428	—	37	2	no	—
58	C₂×#107	—	37	1	no	—
59	#424	D₃₇	1	1	yes	characteristic
60	#343	—	37	2	no	—
61	#341	—	37	1	no	—
62	C₃₇	#2089	1	1	yes	characteristic, complemented, Sylow
63	C₂×M₁₆	—	37	2	no	—
64	C₂×M₁₆	C₂×D₃₇	1	1	yes	characteristic
65	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
66	#117	C₂×D₃₇	1	1	yes	characteristic
67	C₂×C₄⋊C₄	—	37	2	no	—
68	#107	—	74	2	no	—
69	#107	—	74	2	no	—
70	#106	—	74	2	no	—
71	C₄⋊C₈	—	2	2	no	—
72	C₂²×C₄	—	74	2	no	—
73	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
74	M₁₆	—	2	2	no	—
75	M₁₆	—	37	4	no	—
76	M₁₆	—	74	2	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	74	2	no	—
80	C₂×C₈	C₂²⋊D₃₇	1	2	yes	—
81	C₄⋊C₄	—	74	2	no	—
82	C₄⋊C₄	—	74	2	no	—
83	C₄⋊C₄	D₁₄₈	1	1	yes	characteristic
84	C₄⋊C₄	D₁₄₈	1	1	yes	characteristic
85	C₂²⋊C₄	—	2	1	no	—
86	C₄²	—	2	1	no	—
87	C₂³	#10596962	1	1	yes	characteristic
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	296	2	no	—
91	C₂×C₄	—	148	2	no	—
92	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
93	C₂×C₄	#10596964	1	2	yes	—
94	C₂×C₄	Q₈×D₃₇	1	2	yes	—
95	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
96	C₈	—	2	2	no	—
97	C₈	—	4	1	no	—
98	C₈	—	74	4	no	—
99	C₂²	—	2	1	no	—
100	C₂²	—	2	1	no	—
101	C₂²	#11924510	1	1	yes	characteristic
102	C₂²	#11924526	1	2	yes	—
103	C₄	—	4	1	no	—
104	C₄	—	4	1	no	—
105	C₄	—	296	2	no	—
106	C₄	#11924516	1	2	yes	—
107	C₄	#11924517	1	1	yes	characteristic
108	C₄	#11924534	1	1	yes	characteristic
109	C₂	—	2	2	no	—
110	C₂	#14557463	1	1	yes	characteristic
111	C₂	#14557803	1	1	yes	characteristic
112	C₂	#14557808	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14557576

C₂

yes

maximal

C₂×#11924470

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

—

#11924470

—

#11924469

—

#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

characteristic

C₄⋊C₄×C₃₇

D₄

yes

characteristic

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×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

—

C₂²×C₃₇

#121

yes

characteristic

C₂²×C₃₇

#125

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂²⋊Q₈

yes

—

C₄×C₃₇

#123

yes

characteristic

C₄×C₃₇

C₄⋊Q₈

yes

characteristic

Dic₃₇

—

#2089

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

—

—

D₃₇

yes

characteristic

#343

—

#341

—

C₃₇

#2089

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

—

#107

—

#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

characteristic

C₄⋊C₄

D₁₄₈

yes

characteristic

C₂²⋊C₄

—

C₄²

—

C₂³

#10596962

yes

characteristic

C₂×C₄

—

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

—

100

C₂²

—

101

C₂²

#11924510

yes

characteristic

102

C₂²

#11924526

yes

—

103

C₄

—

104

C₄

—

105

C₄

—

296

—

106

C₄

#11924516

yes

—

107

C₄

#11924517

yes

characteristic

108

C₄

#11924534

yes

characteristic

109

C₂

—

110

C₂

#14557463

yes

characteristic

111

C₂

#14557803

yes

characteristic

112

C₂

#14557808

yes

characteristic

	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	#2089
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
62. C₃₇	#2089	semidirect

H ≅ G/N

type

62. C₃₇

#2089

semidirect

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

subgroup	index
#14557576	2
C₂×#11924470	2
#14557301	2
#14557298	2
C₃₇×#424	2
#2089	37

subgroup

index

C₃₇×#424

Subgroups — #17996101

Conjugacy classes of subgroups

Complements

Maximal subgroups