Group #16954827 — subgroups

Subgroups — #16954827

Order 4352.

190 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 115 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	#14530568	C₂	1	2	yes	maximal
2	#14526986	C₂	1	2	yes	maximal, complemented
3	#14526984	C₂	1	2	yes	maximal, complemented
4	C₂²×C₁₇×C₃₂	C₂	1	1	yes	maximal, characteristic
5	#11762847	—	2	2	no	—
6	#11762637	—	2	2	no	—
7	#11762635	C₂²	1	4	yes	—
8	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
9	C₂×C₁₇×C₃₂	C₄	1	4	yes	complemented
10	C₂×C₁₇×C₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₁₇×C₃₂	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
13	#10587612	—	4	2	no	—
14	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
15	C₂×C₁₆×C₁₇	D₄	1	1	yes	characteristic
16	C₂×C₁₆×C₁₇	Q₈	1	2	yes	—
17	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
18	#10587489	—	2	4	no	—
19	#10587488	—	4	2	no	—
20	C₁₇×C₃₂	C₂×C₄	1	2	yes	—
21	C₁₇×C₃₂	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
23	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
24	C₂×C₈×C₁₇	D₈	1	1	yes	characteristic
25	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
26	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
27	C₂²⋊Dic₁₇	—	8	2	no	—
28	C₄⋊Dic₁₇	—	4	4	no	—
29	C₄×Dic₁₇	—	8	2	no	—
30	C₁₆×C₁₇	C₄⋊C₄	1	2	yes	—
31	C₁₆×C₁₇	C₂²×C₄	1	1	yes	characteristic
32	C₁₆×C₁₇	C₂×Q₈	1	1	yes	characteristic
33	#13633	—	17	1	no	maximal, Sylow
34	C₂³×C₁₇	D₁₆	1	1	yes	characteristic
35	C₂×C₄×C₁₇	#107	1	2	yes	—
36	C₂×C₄×C₁₇	D₁₆	1	1	yes	characteristic
37	C₂×C₄×C₁₇	Q₃₂	1	2	yes	—
38	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
39	C₂×Dic₁₇	—	8	4	no	—
40	C₈×C₁₇	#107	1	2	yes	—
41	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₈×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
43	C₂²×C₃₂	D₁₇	1	1	yes	characteristic
44	#2158	—	17	2	no	—
45	#1424	—	17	2	no	—
46	#1423	—	17	2	no	—
47	C₂²×C₁₇	—	2	2	no	—
48	C₂²×C₁₇	#366	1	2	yes	—
49	C₂²×C₁₇	C₂×D₁₆	1	1	yes	characteristic
50	C₄×C₁₇	#366	1	2	yes	—
51	C₄×C₁₇	C₂×#107	1	1	yes	characteristic
52	C₄×C₁₇	C₂×Q₃₂	1	1	yes	characteristic
53	Dic₁₇	—	16	4	no	—
54	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
55	#427	—	34	2	no	—
56	C₂×C₃₂	Dic₁₇	1	4	yes	complemented
57	C₂×C₃₂	C₂×D₁₇	1	1	yes	characteristic
58	C₂×C₃₂	C₂×D₁₇	1	1	yes	characteristic
59	#367	—	34	2	no	—
60	#366	—	17	4	no	—
61	C₂×C₁₇	—	2	2	no	—
62	C₂×C₁₇	C₂×#366	1	1	yes	characteristic
63	C₂×C₁₇	#2262	1	2	yes	—
64	C₂²×C₈	D₆₈	1	1	yes	characteristic
65	#117	—	68	2	no	—
66	C₂×C₁₆	Dic₃₄	1	2	yes	—
67	C₂×C₁₆	D₆₈	1	1	yes	characteristic
68	C₂×C₁₆	C₂×Dic₁₇	1	2	yes	—
69	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
70	#107	—	34	4	no	—
71	#106	—	68	2	no	—
72	C₃₂	C₂×Dic₁₇	1	2	yes	—
73	C₃₂	C₂×Dic₁₇	1	2	yes	—
74	C₁₇	#13633	1	1	yes	characteristic, complemented, Sylow
75	C₂²×C₄	D₁₃₆	1	1	yes	characteristic
76	C₂×C₈	D₁₃₆	1	1	yes	characteristic
77	C₂×C₈	Dic₆₈	1	2	yes	—
78	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
79	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
80	C₄⋊C₄	—	68	4	no	—
81	C₂²⋊C₄	—	136	2	no	—
82	C₄²	—	136	2	no	—
83	C₁₆	C₄⋊Dic₁₇	1	2	yes	—
84	C₁₆	C₂×Dic₃₄	1	1	yes	characteristic
85	C₁₆	C₂²×Dic₁₇	1	1	yes	characteristic
86	C₂³	D₂₇₂	1	1	yes	characteristic
87	C₂×C₄	—	136	4	no	—
88	C₂×C₄	D₂₇₂	1	1	yes	characteristic
89	C₂×C₄	Dic₁₃₆	1	2	yes	—
90	C₂×C₄	#10587489	1	2	yes	—
91	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
92	C₈	#10587489	1	2	yes	—
93	C₈	C₂×Dic₆₈	1	1	yes	characteristic
94	C₈	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
95	C₂²	—	2	2	no	—
96	C₂²	#11762635	1	2	yes	—
97	C₂²	C₂×D₂₇₂	1	1	yes	characteristic
98	C₄	—	272	4	no	—
99	C₄	#11762635	1	2	yes	—
100	C₄	C₂×#10587489	1	1	yes	characteristic
101	C₄	C₂×Dic₁₃₆	1	1	yes	characteristic
102	C₂	—	2	2	no	—
103	C₂	C₂×#11762635	1	1	yes	characteristic
104	C₂	#14531093	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14530568

C₂

yes

maximal

#14526986

C₂

yes

maximal, complemented

#14526984

C₂

yes

maximal, complemented

C₂²×C₁₇×C₃₂

C₂

yes

maximal, characteristic

#11762847

—

#11762637

—

#11762635

C₂²

yes

—

C₂²×C₁₆×C₁₇

C₂²

yes

characteristic

C₂×C₁₇×C₃₂

C₄

yes

complemented

C₂×C₁₇×C₃₂

C₂²

yes

characteristic

C₂×C₁₇×C₃₂

C₂²

yes

characteristic

C₂²×C₈×C₁₇

D₄

yes

characteristic

#10587612

—

C₂×C₁₆×C₁₇

C₂×C₄

yes

—

C₂×C₁₆×C₁₇

D₄

yes

characteristic

C₂×C₁₆×C₁₇

Q₈

yes

—

C₂×C₁₆×C₁₇

C₂³

yes

characteristic

#10587489

—

#10587488

—

C₁₇×C₃₂

C₂×C₄

yes

—

C₁₇×C₃₂

C₂×C₄

yes

—

C₂²×C₄×C₁₇

D₈

yes

characteristic

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₇

D₈

yes

characteristic

C₂×C₈×C₁₇

Q₁₆

yes

—

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄×Dic₁₇

—

C₁₆×C₁₇

C₄⋊C₄

yes

—

C₁₆×C₁₇

C₂²×C₄

yes

characteristic

C₁₆×C₁₇

C₂×Q₈

yes

characteristic

#13633

—

maximal, Sylow

C₂³×C₁₇

D₁₆

yes

characteristic

C₂×C₄×C₁₇

#107

yes

—

C₂×C₄×C₁₇

D₁₆

yes

characteristic

C₂×C₄×C₁₇

Q₃₂

yes

—

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×Dic₁₇

—

C₈×C₁₇

#107

yes

—

C₈×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₈×C₁₇

C₂×Q₁₆

yes

characteristic

C₂²×C₃₂

D₁₇

yes

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

#366

yes

—

C₂²×C₁₇

C₂×D₁₆

yes

characteristic

C₄×C₁₇

#366

yes

—

C₄×C₁₇

C₂×#107

yes

characteristic

C₄×C₁₇

C₂×Q₃₂

yes

characteristic

Dic₁₇

—

C₂²×C₁₆

C₂×D₁₇

yes

characteristic

#427

—

C₂×C₃₂

Dic₁₇

yes

complemented

C₂×C₃₂

C₂×D₁₇

yes

characteristic

C₂×C₃₂

C₂×D₁₇

yes

characteristic

#367

—

#366

—

C₂×C₁₇

—

C₂×C₁₇

C₂×#366

yes

characteristic

C₂×C₁₇

#2262

yes

—

C₂²×C₈

D₆₈

yes

characteristic

#117

—

C₂×C₁₆

Dic₃₄

yes

—

C₂×C₁₆

D₆₈

yes

characteristic

C₂×C₁₆

C₂×Dic₁₇

yes

—

C₂×C₁₆

C₂²×D₁₇

yes

characteristic

#107

—

#106

—

C₃₂

C₂×Dic₁₇

yes

—

C₃₂

C₂×Dic₁₇

yes

—

C₁₇

#13633

yes

characteristic, complemented, Sylow

C₂²×C₄

D₁₃₆

yes

characteristic

C₂×C₈

D₁₃₆

yes

characteristic

C₂×C₈

Dic₆₈

yes

—

C₂×C₈

C₄⋊Dic₁₇

yes

—

C₂×C₈

C₂×D₆₈

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

136

—

C₄²

—

136

—

C₁₆

C₄⋊Dic₁₇

yes

—

C₁₆

C₂×Dic₃₄

yes

characteristic

C₁₆

C₂²×Dic₁₇

yes

characteristic

C₂³

D₂₇₂

yes

characteristic

C₂×C₄

—

136

—

C₂×C₄

D₂₇₂

yes

characteristic

C₂×C₄

Dic₁₃₆

yes

—

C₂×C₄

#10587489

yes

—

C₂×C₄

C₂×D₁₃₆

yes

characteristic

C₈

#10587489

yes

—

C₈

C₂×Dic₆₈

yes

characteristic

C₈

C₂×C₄⋊Dic₁₇

yes

characteristic

C₂²

—

C₂²

#11762635

yes

—

C₂²

C₂×D₂₇₂

yes

characteristic

C₄

—

272

—

C₄

#11762635

yes

—

100

C₄

C₂×#10587489

yes

characteristic

101

C₄

C₂×Dic₁₃₆

yes

characteristic

102

C₂

—

103

C₂

C₂×#11762635

yes

characteristic

104

C₂

#14531093

yes

—

	subgroup	G/H
center	C₂×C₄	D₂₇₂
commutator	C₁₆×C₁₇	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₁₇
Fitting	C₂²×C₁₇×C₃₂	C₂
socle	C₂²×C₁₇	C₂×D₁₆
prime radical	C₂³×C₁₇	D₁₆
cosocle	C₂×C₁₆×C₁₇	C₂³
hypercenter	C₂²×C₃₂	D₁₇
hypocenter	C₁₇	#13633
LCS	G ▷ C₁₆×C₁₇ ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²×C₃₂

subgroup

G/H

center

C₂×C₄

D₂₇₂

commutator

C₁₆×C₁₇

C₂²×C₄

Frattini

C₂×C₁₆

C₂²×D₁₇

Fitting

C₂²×C₁₇×C₃₂

C₂

socle

C₂²×C₁₇

C₂×D₁₆

prime radical

C₂³×C₁₇

D₁₆

cosocle

C₂×C₁₆×C₁₇

C₂³

hypercenter

C₂²×C₃₂

D₁₇

hypocenter

C₁₇

#13633

LCS

G ▷ C₁₆×C₁₇ ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²×C₃₂

N	H ≅ G/N	type
2. #14526986	C₂	semidirect
3. #14526984	C₂	semidirect
9. C₂×C₁₇×C₃₂	C₄	semidirect
56. C₂×C₃₂	Dic₁₇	semidirect
74. C₁₇	#13633	semidirect

H ≅ G/N

type

2. #14526986

C₂

semidirect

3. #14526984

C₂

semidirect

9. C₂×C₁₇×C₃₂

C₄

semidirect

56. C₂×C₃₂

Dic₁₇

semidirect

74. C₁₇

#13633

semidirect

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

subgroup	index
#14530568	2
#14526986	2
#14526984	2
C₂²×C₁₇×C₃₂	2
#13633	17

subgroup

index

C₂²×C₁₇×C₃₂

Subgroups — #16954827

Conjugacy classes of subgroups

Complements

Maximal subgroups