Group #14527680 — subgroups

Subgroups — #14527680

Order 2176.

152 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 65 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	#11762936	C₂	1	2	yes	maximal
2	C₄×Dic₆₈	C₂	1	1	yes	maximal, characteristic
3	#11762539	C₂	1	2	yes	maximal
4	C₈×#3622	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#335	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₆₈	C₄	1	1	yes	characteristic
7	C₄×Dic₃₄	C₂²	1	2	yes	—
8	C₁₇×C₄⋊C₈	C₂²	1	2	yes	—
9	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
10	#10587489	C₄	1	1	yes	characteristic
11	#10587487	C₄	1	2	yes	—
12	#10587486	—	2	2	no	—
13	#10587485	—	2	2	no	—
14	C₄×#3622	C₂²	1	1	yes	characteristic
15	C₄×#3622	C₂²	1	1	yes	characteristic
16	C₂×Dic₃₄	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₁₇	—	2	2	no	—
18	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
20	C₄²×C₁₇	C₂³	1	1	yes	characteristic
21	C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
22	#63225	—	2	2	no	—
23	C₄×Dic₁₇	—	2	2	no	—
24	C₂×#3622	—	2	1	no	—
25	C₂×#3622	D₄	1	1	yes	characteristic
26	C₂×#3622	D₄	1	1	yes	characteristic
27	Dic₆₈	—	2	2	no	—
28	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
31	C₂×Dic₁₇	—	2	2	no	—
32	C₂×Dic₁₇	—	2	2	no	—
33	Dic₃₄	—	2	2	no	—
34	Dic₃₄	—	2	2	no	—
35	C₈×C₁₇	—	2	1	no	—
36	C₈×C₁₇	—	4	2	no	—
37	C₈×C₁₇	M₁₆	1	2	yes	—
38	#3622	—	2	2	no	—
39	#3622	—	2	1	no	—
40	#3622	Q₁₆	1	2	yes	complemented
41	#1591	—	17	1	no	maximal, Sylow
42	C₂²×C₁₇	C₄×D₄	1	1	yes	characteristic
43	C₄×C₁₇	—	2	1	no	—
44	C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
45	C₄×C₁₇	#131	1	1	yes	characteristic
46	C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
47	C₄×C₁₇	#135	1	1	yes	characteristic
48	Dic₁₇	—	4	2	no	—
49	Dic₁₇	—	4	2	no	—
50	#446	—	17	2	no	—
51	C₄×Q₁₆	—	17	1	no	—
52	#335	D₁₇	1	1	yes	characteristic
53	Q₈⋊C₈	—	17	2	no	—
54	C₈²	—	17	1	no	—
55	C₂×C₁₇	#436	1	1	yes	characteristic
56	C₂×C₁₇	C₄×Q₁₆	1	1	yes	characteristic
57	C₂×C₁₇	#443	1	1	yes	characteristic
58	C₂×Q₁₆	—	17	1	no	—
59	C₄×Q₈	—	17	2	no	—
60	#107	—	17	1	no	—
61	C₄⋊C₈	—	34	2	no	—
62	C₄⋊C₈	C₂×D₁₇	1	2	yes	—
63	#103	—	17	2	no	—
64	#97	—	34	2	no	—
65	C₄×C₈	—	17	1	no	—
66	C₄×C₈	—	17	1	no	—
67	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
68	C₁₇	#1591	1	1	yes	characteristic, complemented, Sylow
69	C₂×Q₈	—	17	2	no	—
70	Q₁₆	—	34	2	no	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	17	1	no	—
73	C₂×C₈	—	17	1	no	—
74	C₂×C₈	—	34	1	no	—
75	C₂×C₈	C₄×D₁₇	1	1	yes	characteristic
76	C₂×C₈	C₄×D₁₇	1	1	yes	characteristic
77	C₄⋊C₄	—	17	2	no	—
78	C₄⋊C₄	—	34	2	no	—
79	C₄²	—	34	2	no	—
80	C₄²	C₂²×D₁₇	1	1	yes	characteristic
81	Q₈	—	34	2	no	—
82	Q₈	—	34	2	no	—
83	C₂×C₄	—	34	2	no	—
84	C₂×C₄	—	34	2	no	—
85	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
86	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
87	C₂×C₄	#63256	1	1	yes	characteristic
88	C₈	—	2	1	no	—
89	C₈	—	4	2	no	—
90	C₈	—	17	2	no	—
91	C₈	—	34	2	no	—
92	C₈	—	34	1	no	—
93	C₈	#63218	1	2	yes	—
94	C₂²	#10587579	1	1	yes	characteristic
95	C₄	—	2	1	no	—
96	C₄	—	68	2	no	—
97	C₄	—	68	2	no	—
98	C₄	C₂×#63218	1	1	yes	characteristic
99	C₄	#10587593	1	1	yes	characteristic
100	C₄	#10587598	1	1	yes	characteristic
101	C₄	Q₁₆×D₁₇	1	1	yes	characteristic
102	C₂	#11762886	1	1	yes	characteristic
103	C₂	#11762903	1	1	yes	characteristic
104	C₂	#11762925	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11762936

C₂

yes

maximal

C₄×Dic₆₈

C₂

yes

maximal, characteristic

#11762539

C₂

yes

maximal

C₈×#3622

C₂

yes

maximal, characteristic

C₁₇×#335

C₂

yes

maximal, characteristic

C₂×Dic₆₈

C₄

yes

characteristic

C₄×Dic₃₄

C₂²

yes

—

C₁₇×C₄⋊C₈

C₂²

yes

—

C₄×C₈×C₁₇

C₂²

yes

characteristic

#10587489

C₄

yes

characteristic

#10587487

C₄

yes

—

#10587486

—

#10587485

—

C₄×#3622

C₂²

yes

characteristic

C₄×#3622

C₂²

yes

characteristic

C₂×Dic₃₄

C₂×C₄

yes

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄²×C₁₇

C₂³

yes

characteristic

C₄⋊Dic₁₇

C₂×C₄

yes

—

#63225

—

C₄×Dic₁₇

—

C₂×#3622

—

C₂×#3622

D₄

yes

characteristic

C₂×#3622

D₄

yes

characteristic

Dic₆₈

—

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

#41

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

M₁₆

yes

—

#3622

—

#3622

—

#3622

Q₁₆

yes

complemented

#1591

—

maximal, Sylow

C₂²×C₁₇

C₄×D₄

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

C₂×M₁₆

yes

characteristic

C₄×C₁₇

#131

yes

characteristic

C₄×C₁₇

C₂×Q₁₆

yes

characteristic

C₄×C₁₇

#135

yes

characteristic

Dic₁₇

—

Dic₁₇

—

#446

—

C₄×Q₁₆

—

#335

D₁₇

yes

characteristic

Q₈⋊C₈

—

C₈²

—

C₂×C₁₇

#436

yes

characteristic

C₂×C₁₇

C₄×Q₁₆

yes

characteristic

C₂×C₁₇

#443

yes

characteristic

C₂×Q₁₆

—

C₄×Q₈

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₁₇

yes

—

#103

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₁₇

yes

characteristic

C₁₇

#1591

yes

characteristic, complemented, Sylow

C₂×Q₈

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₁₇

yes

characteristic

C₂×C₈

C₄×D₁₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₂²×D₁₇

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄×D₁₇

yes

characteristic

C₂×C₄

D₄×D₁₇

yes

characteristic

C₂×C₄

#63256

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

#63218

yes

—

C₂²

#10587579

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

yes

C₄

yes

100

C₄

#10587598

yes

characteristic

101

C₄

Q₁₆×D₁₇

yes

characteristic

102

C₂

#11762886

yes

characteristic

103

C₂

#11762903

yes

characteristic

104

C₂

#11762925

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₇
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₄²	C₂²×D₁₇
Fitting	C₁₇×#335	C₂
socle, prime radical	C₂²×C₁₇	C₄×D₄
cosocle	C₄²×C₁₇	C₂³
hypercenter	#335	D₁₇
hypocenter	C₁₇	#1591
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

subgroup

G/H

center

C₂×C₄

D₄×D₁₇

commutator

C₂×C₄×C₁₇

C₂²×C₄

Frattini

C₄²

C₂²×D₁₇

Fitting

C₁₇×#335

C₂

socle, prime radical

C₂²×C₁₇

C₄×D₄

cosocle

C₄²×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

N	H ≅ G/N	type
40. #3622	Q₁₆	semidirect
68. C₁₇	#1591	semidirect

H ≅ G/N

type

40. #3622

Q₁₆

semidirect

68. C₁₇

#1591

semidirect

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

subgroup	index
#11762936	2
C₄×Dic₆₈	2
#11762539	2
C₈×#3622	2
C₁₇×#335	2
#1591	17

subgroup

index

C₄×Dic₆₈

C₁₇×#335

Subgroups — #14527680

Conjugacy classes of subgroups

Complements

Maximal subgroups