Group #14524312 — subgroups

Subgroups — C₁₇×#1306 #14524312

Order 2176.

192 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 64 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	C₁₇×#407	C₂	1	2	yes	maximal
2	C₁₇×#381	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₄×C₁₇	C₂²	1	1	yes	characteristic
4	C₂×M₁₆×C₁₇	—	2	2	no	—
5	C₂×C₂²⋊C₄×C₁₇	C₄	1	2	yes	—
6	C₁₇×C₂²⋊C₈	—	2	2	no	—
7	C₁₇×C₂²⋊C₈	C₄	1	4	yes	complemented
8	C₁₇×#95	—	2	2	no	—
9	C₂⁴×C₁₇	C₂×C₄	1	1	yes	characteristic
10	C₂²×C₄×C₁₇	—	2	2	no	—
11	C₂²×C₄×C₁₇	—	4	2	no	—
12	C₂²×C₄×C₁₇	C₂×C₄	1	2	yes	—
13	C₂²×C₄×C₁₇	D₄	1	2	yes	—
14	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
15	C₂²×C₄×C₁₇	Q₈	1	1	yes	characteristic
16	M₁₆×C₁₇	—	4	4	no	—
17	C₂×C₈×C₁₇	—	2	4	no	—
18	C₂²⋊C₄×C₁₇	—	4	2	no	—
19	C₂³×C₁₇	—	2	2	no	—
20	C₂³×C₁₇	—	4	2	no	—
21	C₂³×C₁₇	C₄²	1	1	yes	characteristic
22	C₂³×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₂³×C₁₇	C₄⋊C₄	1	1	yes	characteristic
24	C₂×C₄×C₁₇	—	2	4	no	—
25	C₂×C₄×C₁₇	—	4	2	no	—
26	C₂×C₄×C₁₇	—	4	1	no	—
27	C₂×C₄×C₁₇	—	4	1	no	—
28	C₂×C₄×C₁₇	—	4	4	no	—
29	C₂×C₄×C₁₇	—	4	2	no	—
30	C₂×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
31	C₂×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
32	C₈×C₁₇	—	4	4	no	—
33	#1306	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂²×C₁₇	—	2	2	no	—
35	C₂²×C₁₇	—	2	1	no	—
36	C₂²×C₁₇	—	2	1	no	—
37	C₂²×C₁₇	—	2	2	no	—
38	C₂²×C₁₇	—	2	2	no	—
39	C₂²×C₁₇	—	4	4	no	—
40	C₂²×C₁₇	#95	1	1	yes	characteristic
41	C₂²×C₁₇	#99	1	2	yes	—
42	C₄×C₁₇	—	2	2	no	—
43	C₄×C₁₇	—	2	2	no	—
44	C₄×C₁₇	—	8	2	no	—
45	#407	C₂×C₁₇	1	2	yes	—
46	#381	C₂×C₁₇	1	1	yes	characteristic
47	C₂×C₁₇	—	2	2	no	—
48	C₂×C₁₇	—	4	1	no	—
49	C₂×C₁₇	—	4	1	no	—
50	C₂×C₁₇	#342	1	1	yes	characteristic
51	C₂×C₁₇	#344	1	2	yes	—
52	C₂³×C₄	C₂²×C₁₇	1	1	yes	characteristic
53	C₂×M₁₆	—	2	2	no	—
54	C₂×C₂²⋊C₄	C₄×C₁₇	1	2	yes	—
55	C₂²⋊C₈	—	2	2	no	—
56	C₂²⋊C₈	C₄×C₁₇	1	4	yes	complemented
57	#95	—	2	2	no	—
58	C₁₇	#1306	1	1	yes	characteristic, direct product factor, Sylow
59	C₂⁴	C₂×C₄×C₁₇	1	1	yes	characteristic
60	C₂²×C₄	—	2	2	no	—
61	C₂²×C₄	—	4	2	no	—
62	C₂²×C₄	C₂×C₄×C₁₇	1	2	yes	—
63	C₂²×C₄	D₄×C₁₇	1	2	yes	—
64	C₂²×C₄	D₄×C₁₇	1	1	yes	characteristic
65	C₂²×C₄	Q₈×C₁₇	1	1	yes	characteristic
66	M₁₆	—	4	4	no	—
67	C₂×C₈	—	2	4	no	—
68	C₂²⋊C₄	—	4	2	no	—
69	C₂³	—	2	2	no	—
70	C₂³	—	4	2	no	—
71	C₂³	C₄²×C₁₇	1	1	yes	characteristic
72	C₂³	C₂²⋊C₄×C₁₇	1	1	yes	characteristic
73	C₂³	C₄⋊C₄×C₁₇	1	1	yes	characteristic
74	C₂×C₄	—	2	4	no	—
75	C₂×C₄	—	4	2	no	—
76	C₂×C₄	—	4	1	no	—
77	C₂×C₄	—	4	1	no	—
78	C₂×C₄	—	4	4	no	—
79	C₂×C₄	—	4	2	no	—
80	C₂×C₄	C₂²⋊C₄×C₁₇	1	2	yes	—
81	C₂×C₄	C₄⋊C₄×C₁₇	1	2	yes	—
82	C₈	—	4	4	no	—
83	C₂²	—	2	2	no	—
84	C₂²	—	2	1	no	—
85	C₂²	—	2	1	no	—
86	C₂²	—	2	2	no	—
87	C₂²	—	2	2	no	—
88	C₂²	—	4	4	no	—
89	C₂²	C₁₇×#95	1	1	yes	characteristic
90	C₂²	C₁₇×#99	1	2	yes	—
91	C₄	—	2	2	no	—
92	C₄	—	2	2	no	—
93	C₄	—	8	2	no	—
94	C₂	—	2	2	no	—
95	C₂	—	4	1	no	—
96	C₂	—	4	1	no	—
97	C₂	C₁₇×#342	1	1	yes	characteristic
98	C₂	C₁₇×#344	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₇×#407

C₂

yes

maximal

C₁₇×#381

C₂

yes

maximal, characteristic

C₂³×C₄×C₁₇

C₂²

yes

characteristic

C₂×M₁₆×C₁₇

—

C₂×C₂²⋊C₄×C₁₇

C₄

yes

—

C₁₇×C₂²⋊C₈

—

C₁₇×C₂²⋊C₈

C₄

yes

complemented

C₁₇×#95

—

C₂⁴×C₁₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂×C₄

yes

—

C₂²×C₄×C₁₇

D₄

yes

—

C₂²×C₄×C₁₇

D₄

yes

characteristic

C₂²×C₄×C₁₇

Q₈

yes

characteristic

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂²⋊C₄×C₁₇

—

C₂³×C₁₇

—

C₂³×C₁₇

—

C₂³×C₁₇

C₄²

yes

characteristic

C₂³×C₁₇

C₂²⋊C₄

yes

characteristic

C₂³×C₁₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₇

C₄⋊C₄

yes

—

C₈×C₁₇

—

#1306

C₁₇

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#95

yes

characteristic

C₂²×C₁₇

#99

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

#407

C₂×C₁₇

yes

—

#381

C₂×C₁₇

yes

characteristic

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

#342

yes

characteristic

C₂×C₁₇

#344

yes

—

C₂³×C₄

C₂²×C₁₇

yes

characteristic

C₂×M₁₆

—

C₂×C₂²⋊C₄

C₄×C₁₇

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

C₄×C₁₇

yes

complemented

#95

—

C₁₇

#1306

yes

characteristic, direct product factor, Sylow

C₂⁴

C₂×C₄×C₁₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄×C₁₇

yes

—

C₂²×C₄

D₄×C₁₇

yes

—

C₂²×C₄

D₄×C₁₇

yes

characteristic

C₂²×C₄

Q₈×C₁₇

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂²⋊C₄

—

C₂³

—

C₂³

—

C₂³

C₄²×C₁₇

yes

characteristic

C₂³

C₂²⋊C₄×C₁₇

yes

characteristic

C₂³

C₄⋊C₄×C₁₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊C₄×C₁₇

yes

—

C₂×C₄

C₄⋊C₄×C₁₇

yes

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₁₇×#95

yes

characteristic

C₂²

C₁₇×#99

yes

—

C₄

—

C₄

—

C₄

—

C₂

—

C₂

—

C₂

—

C₂

C₁₇×#342

yes

characteristic

C₂

C₁₇×#344

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₇	#95
commutator	C₂³	C₄²×C₁₇
Frattini, cosocle	C₂³×C₄	C₂²×C₁₇
prime radical	C₂⁴×C₁₇	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₇ ◁ C₂³×C₄×C₁₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₇

#95

commutator

C₂³

C₄²×C₁₇

Frattini, cosocle

C₂³×C₄

C₂²×C₁₇

prime radical

C₂⁴×C₁₇

C₂×C₄

LCS

G ▷ C₂³ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₁₇ ◁ C₂³×C₄×C₁₇ ◁ G

N	H ≅ G/N	type
7. C₁₇×C₂²⋊C₈	C₄	semidirect
33. #1306	C₁₇	direct (N × H)
56. C₂²⋊C₈	C₄×C₁₇	semidirect
58. C₁₇	#1306	direct (N × H)

H ≅ G/N

type

7. C₁₇×C₂²⋊C₈

C₄

semidirect

33. #1306

C₁₇

direct (N × H)

56. C₂²⋊C₈

C₄×C₁₇

semidirect

58. C₁₇

#1306

direct (N × H)

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

subgroup	index
C₁₇×#407	2
C₁₇×#381	2
#1306	17

subgroup

index

C₁₇×#407

C₁₇×#381

#1306

Subgroups — C₁₇×#1306 #14524312

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C17×#1306 #14524312

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#1306 #14524312