Group #16950717 — subgroups

Subgroups — #16950717

Order 4352.

274 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 147 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	#14528956	C₂	1	2	yes	maximal
2	#14526878	C₂	1	4	yes	maximal
3	C₂×C₁₇×#346	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587490	—	2	2	no	—
5	C₂×#10587490	C₂²	1	4	yes	—
6	#11762833	—	2	2	no	—
7	C₂²×C₁₆×C₁₇	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
9	C₁₇×#346	C₄	1	4	yes	—
10	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
11	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
12	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
13	C₂×#63223	—	2	4	no	—
14	C₂×C₁₆×C₁₇	—	2	4	no	—
15	C₂×C₁₆×C₁₇	C₂×C₄	1	4	yes	—
16	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
17	C₄×C₈×C₁₇	Q₈	1	2	yes	—
18	#10587490	—	2	8	no	—
19	#10587490	—	4	4	no	—
20	C₄⋊#3622	—	4	4	no	—
21	C₂²×C₄×C₁₇	Q₁₆	1	2	yes	—
22	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
23	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
24	C₂×C₈×C₁₇	SD₁₆	1	4	yes	—
25	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
26	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
27	C₂×C₈×C₁₇	#41	1	4	yes	—
28	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
29	C₄²×C₁₇	D₈	1	2	yes	—
30	#63223	—	4	4	no	—
31	#63223	—	4	4	no	—
32	C₂×#3622	—	4	4	no	—
33	C₁₆×C₁₇	—	2	4	no	—
34	#12892	—	17	1	no	maximal, Sylow
35	C₂³×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
36	C₂×C₄×C₁₇	—	4	2	no	—
37	C₂×C₄×C₁₇	#107	1	4	yes	—
38	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₇	#124	1	2	yes	—
40	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
41	C₂×C₄×C₁₇	C₂×SD₁₆	1	2	yes	—
42	C₈×C₁₇	#106	1	4	yes	—
43	C₈×C₁₇	#117	1	2	yes	—
44	C₈×C₁₇	#125	1	2	yes	—
45	#3622	—	8	4	no	—
46	C₂×#346	D₁₇	1	1	yes	characteristic
47	#1848	—	17	2	no	—
48	#1381	—	17	4	no	—
49	C₂²×C₁₇	—	2	2	no	—
50	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
51	C₂²×C₁₇	#487	1	2	yes	—
52	C₄×C₁₇	—	2	4	no	—
53	C₄×C₁₇	#382	1	1	yes	characteristic
54	C₄×C₁₇	C₂×#106	1	1	yes	characteristic
55	C₄×C₁₇	#486	1	2	yes	—
56	C₂²×C₁₆	C₂×D₁₇	1	2	yes	—
57	C₂×#108	—	17	4	no	—
58	C₂×#108	—	34	2	no	—
59	#423	—	34	2	no	—
60	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
61	#346	Dic₁₇	1	4	yes	—
62	C₂×C₁₇	—	2	2	no	—
63	C₂×C₁₇	#1834	1	1	yes	characteristic
64	C₂×C₁₇	#2161	1	2	yes	—
65	C₂×M₁₆	—	34	4	no	—
66	C₂²×C₈	D₆₈	1	1	yes	characteristic
67	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
68	C₂×C₄²	D₆₈	1	1	yes	characteristic
69	C₂×C₁₆	—	2	4	no	—
70	C₂×C₁₆	C₂×Dic₁₇	1	4	yes	—
71	#108	—	34	8	no	—
72	#108	—	68	4	no	—
73	C₄⋊C₈	—	68	4	no	—
74	C₄×C₈	Dic₃₄	1	2	yes	—
75	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
76	C₁₇	#12892	1	1	yes	characteristic, complemented, Sylow
77	C₂²×C₄	Dic₆₈	1	2	yes	—
78	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
79	M₁₆	—	68	4	no	—
80	M₁₆	—	68	4	no	—
81	C₂×C₈	—	68	4	no	—
82	C₂×C₈	#63219	1	4	yes	—
83	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
84	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
85	C₂×C₈	#63252	1	4	yes	—
86	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
87	C₄²	D₁₃₆	1	2	yes	—
88	C₄²	C₄⋊Dic₁₇	1	2	yes	—
89	C₁₆	—	2	4	no	—
90	C₂³	C₂×Dic₆₈	1	1	yes	characteristic
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	#10587489	1	4	yes	—
93	C₂×C₄	#10587561	1	2	yes	—
94	C₂×C₄	C₂×#63219	1	2	yes	—
95	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
96	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
97	C₈	—	136	4	no	—
98	C₈	#10587488	1	4	yes	—
99	C₈	#10587556	1	2	yes	—
100	C₈	#10587612	1	2	yes	—
101	C₂²	—	2	2	no	—
102	C₂²	C₂×#10587489	1	1	yes	characteristic
103	C₂²	#11763160	1	2	yes	—
104	C₄	—	2	4	no	—
105	C₄	#11762675	1	1	yes	characteristic
106	C₄	C₂×#10587488	1	1	yes	characteristic
107	C₄	#11763156	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	#14528912	1	1	yes	characteristic
110	C₂	#14530579	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14528956

C₂

yes

maximal

#14526878

C₂

yes

maximal

C₂×C₁₇×#346

C₂

yes

maximal, characteristic

C₂×#10587490

—

C₂×#10587490

C₂²

yes

—

#11762833

—

C₂²×C₁₆×C₁₇

C₂²

yes

—

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₁₇×#346

C₄

yes

—

C₂²×C₈×C₁₇

D₄

yes

characteristic

C₂²×C₈×C₁₇

C₂³

yes

characteristic

C₂×C₄²×C₁₇

D₄

yes

characteristic

C₂×#63223

—

C₂×C₁₆×C₁₇

—

C₂×C₁₆×C₁₇

C₂×C₄

yes

—

C₄×C₈×C₁₇

C₂×C₄

yes

—

C₄×C₈×C₁₇

Q₈

yes

—

#10587490

—

#10587490

—

C₄⋊#3622

—

C₂²×C₄×C₁₇

Q₁₆

yes

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₇

SD₁₆

yes

—

C₂×C₈×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₇

#41

yes

—

C₄²×C₁₇

C₄⋊C₄

yes

—

C₄²×C₁₇

D₈

yes

—

#63223

—

#63223

—

C₂×#3622

—

C₁₆×C₁₇

—

#12892

—

maximal, Sylow

C₂³×C₁₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#107

yes

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

#124

yes

—

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₇

C₂×SD₁₆

yes

—

C₈×C₁₇

#106

yes

—

C₈×C₁₇

#117

yes

—

C₈×C₁₇

#125

yes

—

#3622

—

C₂×#346

D₁₇

yes

characteristic

#1848

—

#1381

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂×#107

yes

characteristic

C₂²×C₁₇

#487

yes

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

—

C₂²×C₁₆

C₂×D₁₇

yes

—

C₂×#108

—

C₂×#108

—

#423

—

C₂×C₄×C₈

C₂×D₁₇

yes

characteristic

#346

Dic₁₇

yes

—

C₂×C₁₇

—

C₂×C₁₇

#1834

yes

characteristic

C₂×C₁₇

#2161

yes

—

C₂×M₁₆

—

C₂²×C₈

D₆₈

yes

characteristic

C₂²×C₈

C₂²×D₁₇

yes

characteristic

C₂×C₄²

D₆₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₁₇

yes

—

#108

—

#108

—

C₄⋊C₈

—

C₄×C₈

Dic₃₄

yes

—

C₄×C₈

C₂×Dic₁₇

yes

—

C₁₇

#12892

yes

characteristic, complemented, Sylow

C₂²×C₄

Dic₆₈

yes

—

C₂²×C₄

C₂×D₆₈

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

#63219

yes

—

C₂×C₈

C₄⋊Dic₁₇

yes

—

C₂×C₈

C₂×Dic₃₄

yes

characteristic

C₂×C₈

#63252

yes

—

C₂×C₈

C₂²×Dic₁₇

yes

characteristic

C₄²

D₁₃₆

yes

—

C₄²

C₄⋊Dic₁₇

yes

—

C₁₆

—

C₂³

C₂×Dic₆₈

yes

characteristic

C₂×C₄

—

C₂×C₄

#10587489

yes

—

C₂×C₄

#10587561

yes

—

C₂×C₄

C₂×#63219

yes

—

C₂×C₄

C₂×D₁₃₆

yes

characteristic

C₂×C₄

C₂×C₄⋊Dic₁₇

yes

characteristic

C₈

—

136

—

C₈

#10587488

yes

—

C₈

#10587556

yes

—

100

C₈

#10587612

yes

—

101

C₂²

—

102

C₂²

C₂×#10587489

yes

characteristic

103

C₂²

#11763160

yes

—

104

C₄

—

105

C₄

#11762675

yes

characteristic

106

C₄

C₂×#10587488

yes

characteristic

107

C₄

#11763156

yes

—

108

C₂

—

109

C₂

#14528912

yes

characteristic

110

C₂

#14530579

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₁₃₆
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₂×C₁₇×#346	C₂
socle	C₂²×C₁₇	C₂×#107
prime radical	C₂³×C₁₇	C₂×Q₁₆
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂×#346	D₁₇
hypocenter	C₁₇	#12892
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₄

C₂×D₁₃₆

commutator

C₂×C₈×C₁₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₁₇

Fitting

C₂×C₁₇×#346

C₂

socle

C₂²×C₁₇

C₂×#107

prime radical

C₂³×C₁₇

C₂×Q₁₆

cosocle

C₂²×C₈×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
76. C₁₇	#12892	semidirect

H ≅ G/N

type

76. C₁₇

#12892

semidirect

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

subgroup	index
#14528956	2
#14526878	2
C₂×C₁₇×#346	2
#12892	17

subgroup

index

#14528956

#14526878

C₂×C₁₇×#346

#12892

Subgroups — #16950717

Conjugacy classes of subgroups

Complements

Maximal subgroups