Group #16921200 — subgroups

Subgroups — #16921200

Order 4352.

214 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 119 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	#14527532	C₂	1	2	yes	maximal
2	C₂×C₁₇×#346	C₂	1	1	yes	maximal, characteristic
3	#11762862	—	2	2	no	—
4	#11762562	—	2	2	no	—
5	#11762560	C₄	1	4	yes	—
6	C₂²×C₁₆×C₁₇	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
8	C₁₇×#346	—	2	2	no	—
9	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
10	C₂²×C₈×C₁₇	Q₈	1	1	yes	characteristic
11	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
12	C₂×C₁₆×C₁₇	—	2	2	no	—
13	C₂×C₁₆×C₁₇	—	2	2	no	—
14	C₂×C₁₆×C₁₇	C₈	1	4	yes	complemented
15	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
16	C₄×C₈×C₁₇	D₄	1	2	yes	—
17	C₂²⋊#3622	—	4	2	no	—
18	C₄⋊#3622	—	2	4	no	—
19	C₄×#3622	—	4	2	no	—
20	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
22	C₂²×C₄×C₁₇	Q₁₆	1	1	yes	characteristic
23	C₂×C₈×C₁₇	—	2	2	no	—
24	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
25	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
28	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
29	C₄²×C₁₇	C₂²⋊C₄	1	2	yes	—
30	C₄²×C₁₇	SD₁₆	1	2	yes	—
31	C₂×#3622	—	4	4	no	—
32	C₁₆×C₁₇	—	2	4	no	—
33	#7157	—	17	1	no	maximal, Sylow
34	C₂³×C₁₇	#107	1	1	yes	characteristic
35	C₂×C₄×C₁₇	—	4	1	no	—
36	C₂×C₄×C₁₇	—	4	1	no	—
37	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
38	C₂×C₄×C₁₇	#102	1	2	yes	—
39	C₂×C₄×C₁₇	#103	1	2	yes	—
40	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
41	C₂×C₄×C₁₇	#106	1	1	yes	characteristic
42	C₂×C₄×C₁₇	#108	1	2	yes	—
43	C₈×C₁₇	—	2	2	no	—
44	C₈×C₁₇	C₄×C₈	1	1	yes	characteristic
45	C₈×C₁₇	#97	1	1	yes	characteristic
46	C₈×C₁₇	C₂²⋊C₈	1	2	yes	—
47	#3622	—	8	4	no	—
48	C₂×#346	D₁₇	1	1	yes	characteristic
49	#1564	—	17	2	no	—
50	C₂²×C₁₇	—	2	1	no	—
51	C₂²×C₁₇	—	2	1	no	—
52	C₂²×C₁₇	#335	1	2	yes	—
53	C₂²×C₁₇	#340	1	1	yes	characteristic
54	C₄×C₁₇	—	2	2	no	—
55	C₄×C₁₇	—	2	2	no	—
56	C₄×C₁₇	#334	1	2	yes	—
57	C₄×C₁₇	#336	1	1	yes	characteristic
58	C₄×C₁₇	#341	1	1	yes	characteristic
59	C₂²×C₁₆	Dic₁₇	1	2	yes	—
60	#431	—	34	2	no	—
61	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
62	#346	—	2	2	no	—
63	#335	—	34	2	no	—
64	#334	—	17	4	no	—
65	C₂×C₁₇	—	2	2	no	—
66	C₂×C₁₇	#1277	1	1	yes	characteristic
67	C₂×C₁₇	#1384	1	1	yes	characteristic
68	C₂×C₁₇	#1385	1	1	yes	characteristic
69	C₂²×C₈	Dic₃₄	1	1	yes	characteristic
70	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
71	C₂×C₄²	D₆₈	1	1	yes	characteristic
72	C₂×C₁₆	—	2	2	no	—
73	C₂×C₁₆	—	2	2	no	—
74	C₂×C₁₆	#3622	1	4	yes	complemented
75	C₄⋊C₈	—	34	4	no	—
76	C₂²⋊C₈	—	68	2	no	—
77	C₄×C₈	—	68	2	no	—
78	C₄×C₈	C₄×D₁₇	1	2	yes	—
79	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
80	C₁₇	#7157	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₄	D₁₃₆	1	1	yes	characteristic
82	C₂²×C₄	Dic₆₈	1	1	yes	characteristic
83	C₂²×C₄	C₄⋊Dic₁₇	1	1	yes	characteristic
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	68	4	no	—
86	C₂×C₈	C₂×#3622	1	2	yes	—
87	C₂×C₈	#63223	1	2	yes	—
88	C₂×C₈	C₄×Dic₁₇	1	1	yes	characteristic
89	C₂×C₈	#63225	1	2	yes	—
90	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
91	C₄²	#63219	1	2	yes	—
92	C₄²	#63227	1	2	yes	—
93	C₁₆	—	2	4	no	—
94	C₂³	#10587489	1	1	yes	characteristic
95	C₂×C₄	—	4	1	no	—
96	C₂×C₄	—	4	1	no	—
97	C₂×C₄	C₄⋊#3622	1	2	yes	—
98	C₂×C₄	#10587487	1	2	yes	—
99	C₂×C₄	#10587488	1	1	yes	characteristic
100	C₂×C₄	#10587490	1	2	yes	—
101	C₂×C₄	#10587492	1	2	yes	—
102	C₂×C₄	#10587502	1	1	yes	characteristic
103	C₈	—	2	2	no	—
104	C₈	—	136	4	no	—
105	C₈	C₄×#3622	1	1	yes	characteristic
106	C₈	#10587474	1	1	yes	characteristic
107	C₈	C₂²⋊#3622	1	2	yes	—
108	C₂²	—	2	1	no	—
109	C₂²	—	2	1	no	—
110	C₂²	#11762562	1	2	yes	—
111	C₂²	#11762572	1	1	yes	characteristic
112	C₄	—	2	2	no	—
113	C₄	—	2	2	no	—
114	C₄	#11762560	1	2	yes	—
115	C₄	#11762564	1	1	yes	characteristic
116	C₄	#11762574	1	1	yes	characteristic
117	C₂	—	2	2	no	—
118	C₂	#14526619	1	1	yes	characteristic
119	C₂	#14526886	1	1	yes	characteristic
120	C₂	#14526888	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14527532

C₂

yes

maximal

C₂×C₁₇×#346

C₂

yes

maximal, characteristic

#11762862

—

#11762562

—

#11762560

C₄

yes

—

C₂²×C₁₆×C₁₇

C₄

yes

—

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₁₇×#346

—

C₂²×C₈×C₁₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₇

Q₈

yes

characteristic

C₂×C₄²×C₁₇

D₄

yes

characteristic

C₂×C₁₆×C₁₇

—

C₂×C₁₆×C₁₇

—

C₂×C₁₆×C₁₇

C₈

yes

complemented

C₄×C₈×C₁₇

C₂×C₄

yes

—

C₄×C₈×C₁₇

D₄

yes

—

C₂²⋊#3622

—

C₄⋊#3622

—

C₄×#3622

—

C₂²×C₄×C₁₇

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₁₇

D₈

yes

characteristic

C₂²×C₄×C₁₇

Q₁₆

yes

characteristic

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄²

yes

characteristic

C₂×C₈×C₁₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₇

C₂×C₈

yes

—

C₂×C₈×C₁₇

M₁₆

yes

—

C₄²×C₁₇

C₂²⋊C₄

yes

—

C₄²×C₁₇

SD₁₆

yes

—

C₂×#3622

—

C₁₆×C₁₇

—

—

C₂³×C₁₇

yes

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#95

yes

characteristic

C₂×C₄×C₁₇

#102

yes

—

C₂×C₄×C₁₇

#103

yes

—

C₂×C₄×C₁₇

C₄⋊C₈

yes

—

C₂×C₄×C₁₇

#106

yes

characteristic

C₂×C₄×C₁₇

#108

yes

—

C₈×C₁₇

—

C₈×C₁₇

C₄×C₈

yes

characteristic

C₈×C₁₇

#97

yes

characteristic

C₈×C₁₇

C₂²⋊C₈

yes

—

#3622

—

C₂×#346

D₁₇

yes

characteristic

#1564

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#335

yes

—

C₂²×C₁₇

#340

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#334

yes

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₂²×C₁₆

Dic₁₇

yes

—

#431

—

C₂×C₄×C₈

C₂×D₁₇

yes

—

—

—

C₂×C₁₇

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂²×C₈

Dic₃₄

yes

characteristic

C₂²×C₈

C₂×Dic₁₇

yes

characteristic

C₂×C₄²

D₆₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

#3622

yes

complemented

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₁₇

yes

—

C₄×C₈

C₂²⋊D₁₇

yes

—

C₁₇

#7157

yes

characteristic, complemented, Sylow

C₂²×C₄

D₁₃₆

yes

characteristic

C₂²×C₄

Dic₆₈

yes

characteristic

C₂²×C₄

C₄⋊Dic₁₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×#3622

yes

—

C₂×C₈

#63223

yes

—

C₂×C₈

C₄×Dic₁₇

yes

characteristic

C₂×C₈

#63225

yes

—

C₂×C₈

C₂²⋊Dic₁₇

yes

characteristic

C₄²

#63219

yes

—

C₄²

#63227

yes

—

C₁₆

—

C₂³

#10587489

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊#3622

yes

—

C₂×C₄

#10587487

yes

—

C₂×C₄

#10587488

yes

characteristic

100

C₂×C₄

#10587490

yes

—

101

C₂×C₄

#10587492

yes

—

102

C₂×C₄

#10587502

yes

characteristic

103

C₈

—

104

C₈

—

136

—

105

C₈

C₄×#3622

yes

characteristic

106

C₈

#10587474

yes

characteristic

107

C₈

C₂²⋊#3622

yes

—

108

C₂²

—

109

C₂²

—

110

C₂²

#11762562

yes

—

111

C₂²

#11762572

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

#11762560

yes

—

115

C₄

#11762564

yes

characteristic

116

C₄

#11762574

yes

characteristic

117

C₂

—

118

C₂

#14526619

yes

characteristic

119

C₂

#14526886

yes

characteristic

120

C₂

#14526888

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#10587488
commutator	C₈×C₁₇	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₁₇
Fitting	C₂×C₁₇×#346	C₂
socle	C₂²×C₁₇	#340
prime radical	C₂³×C₁₇	#107
cosocle	C₂×C₄×C₈×C₁₇	C₂²
hypercenter	C₂×#346	D₁₇
hypocenter	C₁₇	#7157
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₄

#10587488

commutator

C₈×C₁₇

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₁₇

Fitting

C₂×C₁₇×#346

C₂

C₂²×C₁₇

C₂³×C₁₇

C₂×C₄×C₈×C₁₇

C₂²

D₁₇

C₁₇

G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

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

N	H ≅ G/N	type
14. C₂×C₁₆×C₁₇	C₈	semidirect
74. C₂×C₁₆	#3622	semidirect
80. C₁₇	#7157	semidirect

H ≅ G/N

type

14. C₂×C₁₆×C₁₇

C₈

semidirect

74. C₂×C₁₆

#3622

semidirect

80. C₁₇

#7157

semidirect

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

subgroup	index
#14527532	2
C₂×C₁₇×#346	2
#7157	17

subgroup

index

#14527532

C₂×C₁₇×#346

#7157

Subgroups — #16921200

Conjugacy classes of subgroups

Complements

Maximal subgroups