Group #14528112 — subgroups

Subgroups — #14528112

Order 2176.

172 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 49 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	#11763219	C₂	1	2	yes	maximal
2	#11763179	C₂	1	1	yes	maximal, characteristic
3	#11762551	C₂	1	2	yes	maximal
4	#11762551	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#564	C₂	1	1	yes	maximal, characteristic
6	C₁₇×#125	—	2	2	no	—
7	C₁₇×#125	—	2	2	no	—
8	C₁₇×#125	—	2	2	no	—
9	C₁₇×#125	C₂²	1	2	yes	—
10	C₁₇×#125	C₂²	1	1	yes	characteristic
11	#10587555	C₂²	1	1	yes	characteristic
12	#10587481	—	2	2	no	—
13	#10587479	—	2	2	no	—
14	#10587478	—	2	2	no	—
15	#10587474	C₂²	1	2	yes	—
16	#10587474	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₄	—	2	1	no	—
18	C₄⋊C₄×C₁₇	—	2	2	no	—
19	C₄⋊C₄×C₁₇	—	2	2	no	—
20	C₄⋊C₄×C₁₇	—	2	2	no	—
21	C₄⋊C₄×C₁₇	—	2	2	no	—
22	C₄⋊C₄×C₁₇	—	2	2	no	—
23	C₄⋊C₄×C₁₇	—	2	2	no	—
24	C₄⋊C₄×C₁₇	D₄	1	2	yes	—
25	C₄⋊C₄×C₁₇	D₄	1	2	yes	—
26	C₄⋊C₄×C₁₇	D₄	1	2	yes	—
27	C₄²×C₁₇	—	2	2	no	—
28	C₄²×C₁₇	C₂³	1	1	yes	characteristic
29	C₄⋊Dic₁₇	—	2	2	no	—
30	C₂×#3622	—	2	2	no	—
31	C₂×#3622	—	2	1	no	—
32	C₂×C₄×C₁₇	—	2	2	no	—
33	C₂×C₄×C₁₇	—	2	2	no	—
34	C₂×C₄×C₁₇	—	2	2	no	—
35	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
37	C₂×Dic₁₇	—	4	1	no	—
38	Dic₃₄	—	4	2	no	—
39	#3622	—	2	4	no	—
40	#3622	—	4	1	no	—
41	#1663	—	17	1	no	maximal, Sylow
42	C₂²×C₁₇	#120	1	1	yes	characteristic
43	C₄×C₁₇	—	2	1	no	—
44	C₄×C₁₇	—	2	2	no	—
45	C₄×C₁₇	—	4	2	no	—
46	C₄×C₁₇	—	4	2	no	—
47	C₄×C₁₇	—	4	2	no	—
48	Dic₁₇	—	8	1	no	—
49	#564	D₁₇	1	1	yes	characteristic
50	#501	—	17	2	no	—
51	#491	—	17	1	no	—
52	#330	—	17	2	no	—
53	#330	—	17	1	no	—
54	C₂×C₁₇	#454	1	1	yes	characteristic
55	C₂×C₁₇	#456	1	2	yes	—
56	C₄⋊Q₈	—	17	1	no	—
57	#125	—	2	2	no	—
58	#125	—	2	2	no	—
59	#125	—	2	2	no	—
60	#125	C₂×D₁₇	1	1	yes	characteristic
61	#125	C₂×D₁₇	1	2	yes	—
62	#107	—	34	2	no	—
63	#106	—	34	2	no	—
64	#103	—	34	2	no	—
65	#97	—	17	1	no	—
66	#97	—	17	2	no	—
67	C₁₇	#1663	1	1	yes	characteristic, complemented, Sylow
68	C₂×Q₈	—	34	1	no	—
69	C₂×C₈	—	34	2	no	—
70	C₂×C₈	—	34	1	no	—
71	C₄⋊C₄	—	2	2	no	—
72	C₄⋊C₄	—	2	2	no	—
73	C₄⋊C₄	—	2	2	no	—
74	C₄⋊C₄	—	2	2	no	—
75	C₄⋊C₄	—	2	2	no	—
76	C₄⋊C₄	—	2	2	no	—
77	C₄⋊C₄	—	34	2	no	—
78	C₄⋊C₄	C₂²⋊D₁₇	1	2	yes	—
79	C₄⋊C₄	C₂²⋊D₁₇	1	2	yes	—
80	C₄⋊C₄	C₂²⋊D₁₇	1	2	yes	—
81	C₄²	—	2	2	no	—
82	C₄²	C₂²×D₁₇	1	1	yes	characteristic
83	Q₈	—	68	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	68	1	no	—
88	C₂×C₄	C₂×C₂²⋊D₁₇	1	2	yes	—
89	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
90	C₈	—	34	4	no	—
91	C₈	—	68	1	no	—
92	C₂²	#10587639	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	2	2	no	—
95	C₄	—	4	2	no	—
96	C₄	—	4	2	no	—
97	C₄	—	4	2	no	—
98	C₄	—	136	1	no	—
99	C₂	#11762982	1	1	yes	characteristic
100	C₂	#11762992	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11763219

C₂

yes

maximal

#11763179

C₂

yes

maximal, characteristic

#11762551

C₂

yes

maximal

#11762551

C₂

yes

maximal, characteristic

C₁₇×#564

C₂

yes

maximal, characteristic

C₁₇×#125

—

C₁₇×#125

—

C₁₇×#125

—

C₁₇×#125

C₂²

yes

—

C₁₇×#125

C₂²

yes

characteristic

#10587555

C₂²

yes

—

—

—

C₂²

yes

—

#10587474

C₂²

yes

characteristic

C₂×Dic₃₄

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

D₄

yes

—

C₄⋊C₄×C₁₇

D₄

yes

—

C₄⋊C₄×C₁₇

D₄

yes

—

C₄²×C₁₇

—

C₄²×C₁₇

C₂³

yes

characteristic

C₄⋊Dic₁₇

—

C₂×#3622

—

C₂×#3622

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×Dic₁₇

—

Dic₃₄

—

—

—

—

C₂²×C₁₇

yes

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

Dic₁₇

—

#564

D₁₇

yes

—

—

—

—

C₂×C₁₇

#454

yes

characteristic

C₂×C₁₇

#456

yes

—

C₄⋊Q₈

—

—

—

—

C₂×D₁₇

yes

characteristic

#125

C₂×D₁₇

yes

—

—

—

—

—

—

C₁₇

#1663

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²⋊D₁₇

yes

—

C₄⋊C₄

C₂²⋊D₁₇

yes

—

C₄⋊C₄

C₂²⋊D₁₇

yes

—

C₄²

—

C₄²

C₂²×D₁₇

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₂²⋊D₁₇

yes

—

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

C₈

—

C₈

—

C₂²

#10587639

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

136

—

C₂

#11762982

yes

characteristic

100

C₂

#11762992

yes

—

	subgroup	G/H
center	C₂²	#10587639
commutator, cosocle	C₄²×C₁₇	C₂³
Frattini	C₄²	C₂²×D₁₇
Fitting	C₁₇×#564	C₂
socle, prime radical	C₂²×C₁₇	#120
hypercenter	#564	D₁₇
hypocenter	C₁₇	#1663
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #564

subgroup

G/H

center

C₂²

#10587639

commutator, cosocle

C₄²×C₁₇

C₂³

Frattini

C₄²

C₂²×D₁₇

Fitting

C₁₇×#564

C₂

C₂²×C₁₇

D₁₇

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #564

N	H ≅ G/N	type
67. C₁₇	#1663	semidirect

H ≅ G/N

type

67. C₁₇

#1663

semidirect

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

subgroup	index
#11763219	2
#11763179	2
#11762551	2
C₁₇×#564	2
#1663	17

subgroup

index

C₁₇×#564

Subgroups — #14528112

Conjugacy classes of subgroups

Complements

Maximal subgroups