Group #14529749 — subgroups

Subgroups — #14529749

Order 2176.

252 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 109 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	C₂×#10587556	C₂	1	1	yes	maximal, characteristic
2	#11762675	C₂	1	1	yes	maximal, characteristic
3	#11762572	C₂	1	4	yes	maximal
4	C₂×C₄×C₈×C₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₈×C₁₇	C₂²	1	2	yes	—
6	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₁₇	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₁₇	C₂²	1	2	yes	—
9	C₂×#63225	—	2	2	no	—
10	#10587556	—	2	2	no	—
11	#10587556	C₄	1	4	yes	complemented
12	C₄×C₈×C₁₇	—	2	2	no	—
13	#10587502	—	2	2	no	—
14	C₂²×C₄×C₁₇	D₄	1	2	yes	—
15	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
16	C₂²×Dic₁₇	—	2	2	no	—
17	C₂²×Dic₁₇	—	2	2	no	—
18	C₂×C₈×C₁₇	—	2	2	no	—
19	C₂×C₈×C₁₇	—	2	2	no	—
20	C₂×C₈×C₁₇	Q₈	1	4	yes	—
21	C₄²×C₁₇	C₂×C₄	1	2	yes	—
22	C₄²×C₁₇	D₄	1	2	yes	—
23	C₄⋊Dic₁₇	—	2	2	no	—
24	C₄⋊Dic₁₇	—	2	4	no	—
25	C₄⋊Dic₁₇	C₂×C₄	1	4	yes	—
26	#63225	—	2	8	no	—
27	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
28	C₂×C₄×C₁₇	—	2	4	no	—
29	C₂×C₄×C₁₇	C₂²⋊C₄	1	4	yes	—
30	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₇	C₂×Q₈	1	2	yes	—
33	C₂×C₄×C₁₇	#41	1	2	yes	—
34	C₂×Dic₁₇	—	2	4	no	—
35	C₂×Dic₁₇	—	4	4	no	—
36	C₂×Dic₁₇	—	4	2	no	—
37	C₂×Dic₁₇	—	4	4	no	—
38	C₈×C₁₇	—	2	4	no	—
39	#1991	—	17	1	no	maximal, Sylow
40	C₂²×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
41	C₂²×C₁₇	#124	1	1	yes	characteristic
42	C₂²×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
43	C₂²×C₁₇	#135	1	4	yes	—
44	C₄×C₁₇	—	2	4	no	—
45	C₄×C₁₇	C₄×Q₈	1	2	yes	—
46	C₄×C₁₇	C₂²⋊Q₈	1	2	yes	—
47	Dic₁₇	—	4	4	no	—
48	Dic₁₇	—	8	2	no	—
49	C₂×#125	—	17	1	no	—
50	C₂×C₄×C₈	D₁₇	1	1	yes	characteristic
51	#382	—	17	1	no	—
52	#340	—	17	4	no	—
53	C₂×C₁₇	#391	1	1	yes	characteristic
54	C₂×C₁₇	#416	1	2	yes	—
55	C₂×C₁₇	#488	1	2	yes	—
56	C₂×C₁₇	#499	1	2	yes	—
57	C₂²×C₈	C₂×D₁₇	1	2	yes	—
58	#125	—	17	4	no	—
59	#125	—	34	2	no	—
60	C₂×C₄⋊C₄	—	17	2	no	—
61	C₂×C₄⋊C₄	—	17	2	no	—
62	C₂×C₄⋊C₄	—	34	2	no	—
63	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
64	C₄×C₈	—	2	2	no	—
65	#95	—	34	2	no	—
66	C₁₇	#1991	1	1	yes	characteristic, complemented, Sylow
67	C₂²×C₄	—	34	2	no	—
68	C₂²×C₄	—	34	2	no	—
69	C₂²×C₄	D₆₈	1	2	yes	—
70	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	Dic₃₄	1	4	yes	—
74	C₄⋊C₄	—	17	4	no	—
75	C₄⋊C₄	—	34	2	no	—
76	C₄⋊C₄	—	34	8	no	—
77	C₄⋊C₄	—	34	4	no	—
78	C₄²	C₄×D₁₇	1	2	yes	—
79	C₄²	C₂²⋊D₁₇	1	2	yes	—
80	C₂³	C₂×D₆₈	1	1	yes	characteristic
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	34	4	no	—
83	C₂×C₄	—	68	4	no	—
84	C₂×C₄	—	68	2	no	—
85	C₂×C₄	—	68	4	no	—
86	C₂×C₄	#63227	1	4	yes	—
87	C₂×C₄	C₂×Dic₃₄	1	2	yes	—
88	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
89	C₂×C₄	#63252	1	2	yes	—
90	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
91	C₈	—	2	4	no	—
92	C₂²	#10587555	1	1	yes	characteristic
93	C₂²	#10587561	1	1	yes	characteristic
94	C₂²	#10587590	1	4	yes	—
95	C₂²	C₂×#63227	1	1	yes	characteristic
96	C₄	—	2	4	no	—
97	C₄	—	68	4	no	—
98	C₄	—	136	2	no	—
99	C₄	C₄×Dic₃₄	1	2	yes	—
100	C₄	#10587610	1	2	yes	—
101	C₂	#11762715	1	1	yes	characteristic
102	C₂	#11762806	1	2	yes	—
103	C₂	#11763164	1	2	yes	—
104	C₂	#11763211	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10587556

C₂

yes

maximal, characteristic

#11762675

C₂

yes

maximal, characteristic

#11762572

C₂

yes

maximal

C₂×C₄×C₈×C₁₇

C₂

yes

maximal, characteristic

C₂²×C₈×C₁₇

C₂²

yes

—

C₂×C₄²×C₁₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₇

C₂²

yes

—

C₂×C₄⋊Dic₁₇

C₂²

yes

—

C₂×#63225

—

#10587556

—

#10587556

C₄

yes

complemented

C₄×C₈×C₁₇

—

#10587502

—

C₂²×C₄×C₁₇

D₄

yes

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

—

C₂²×Dic₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

Q₈

yes

—

C₄²×C₁₇

C₂×C₄

yes

—

C₄²×C₁₇

D₄

yes

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

C₂×C₄

yes

—

#63225

—

C₂³×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×Q₈

yes

—

C₂×C₄×C₁₇

#41

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₈×C₁₇

—

#1991

—

maximal, Sylow

C₂²×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₇

#124

yes

characteristic

C₂²×C₁₇

C₄⋊Q₈

yes

characteristic

C₂²×C₁₇

#135

yes

—

C₄×C₁₇

—

C₄×C₁₇

C₄×Q₈

yes

—

C₄×C₁₇

C₂²⋊Q₈

yes

—

Dic₁₇

—

Dic₁₇

—

C₂×#125

—

C₂×C₄×C₈

D₁₇

yes

characteristic

#382

—

#340

—

C₂×C₁₇

#391

yes

characteristic

C₂×C₁₇

#416

yes

—

C₂×C₁₇

#488

yes

—

C₂×C₁₇

#499

yes

—

C₂²×C₈

C₂×D₁₇

yes

—

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₁₇

yes

characteristic

C₄×C₈

—

#95

—

C₁₇

#1991

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₆₈

yes

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₃₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₄×D₁₇

yes

—

C₄²

C₂²⋊D₁₇

yes

—

C₂³

C₂×D₆₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#63227

yes

—

C₂×C₄

C₂×Dic₃₄

yes

—

C₂×C₄

C₂×C₄×D₁₇

yes

characteristic

C₂×C₄

#63252

yes

—

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

C₈

—

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

C₂×#63227

yes

characteristic

C₄

—

C₄

—

C₄

—

136

—

C₄

C₄×Dic₃₄

yes

—

100

C₄

#10587610

yes

—

101

C₂

#11762715

yes

characteristic

102

C₂

#11762806

yes

—

103

C₂

#11763164

yes

—

104

C₂

#11763211

yes

—

	subgroup	G/H
center	C₂³	C₂×D₆₈
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₂×C₄×C₈×C₁₇	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₄
cosocle	C₂²×C₄×C₁₇	C₂³
hypercenter	C₂×C₄×C₈	D₁₇
hypocenter	C₁₇	#1991
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₆₈

commutator

C₂×C₄×C₁₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₂×C₄×C₈×C₁₇

C₂

socle, prime radical

C₂³×C₁₇

C₂×D₄

cosocle

C₂²×C₄×C₁₇

C₂³

hypercenter

C₂×C₄×C₈

D₁₇

hypocenter

C₁₇

#1991

LCS

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
11. #10587556	C₄	semidirect
66. C₁₇	#1991	semidirect

H ≅ G/N

type

11. #10587556

C₄

semidirect

66. C₁₇

#1991

semidirect

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

subgroup	index
C₂×#10587556	2
#11762675	2
#11762572	2
C₂×C₄×C₈×C₁₇	2
#1991	17

subgroup

index

C₂×C₄×C₈×C₁₇

Subgroups — #14529749

Conjugacy classes of subgroups

Complements

Maximal subgroups