Group #14525170 — subgroups

Subgroups — C₁₇×#2164 #14525170

Order 2176.

138 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 96 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×#405	C₂	1	1	yes	maximal, characteristic
2	C₁₇×C₄⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	C₁₇×C₂²⋊C₁₆	C₂	1	2	yes	maximal
4	C₁₇×#346	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×M₁₆×C₁₇	C₄	1	1	yes	characteristic
7	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
8	C₁₇×#117	C₄	1	1	yes	characteristic
9	C₂×C₁₆×C₁₇	C₂²	1	2	yes	—
10	C₂×C₁₆×C₁₇	C₂²	1	2	yes	—
11	C₁₇×#97	C₄	1	1	yes	characteristic
12	C₁₇×#97	C₄	1	1	yes	characteristic
13	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
16	M₁₆×C₁₇	—	2	2	no	—
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	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₁₇	C₈	1	2	yes	—
23	C₂²⋊C₄×C₁₇	C₈	1	2	yes	—
24	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
26	C₁₆×C₁₇	—	2	2	no	—
27	C₁₆×C₁₇	—	2	2	no	—
28	C₂³×C₁₇	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₁₇	—	2	1	no	—
30	C₂×C₄×C₁₇	—	2	1	no	—
31	C₂×C₄×C₁₇	C₂×C₈	1	2	yes	—
32	C₂×C₄×C₁₇	C₂×C₈	1	2	yes	—
33	C₂×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
35	C₈×C₁₇	—	2	1	no	—
36	C₈×C₁₇	—	2	1	no	—
37	C₈×C₁₇	M₁₆	1	2	yes	—
38	C₈×C₁₇	#41	1	2	yes	—
39	#2164	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂²×C₁₇	—	2	1	no	—
41	C₂²×C₁₇	—	2	2	no	—
42	C₂²×C₁₇	C₂²×C₈	1	1	yes	characteristic
43	C₄×C₁₇	—	2	1	no	—
44	C₄×C₁₇	—	2	2	no	—
45	C₄×C₁₇	—	2	2	no	—
46	C₄×C₁₇	#117	1	1	yes	characteristic
47	C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
48	#405	C₂×C₁₇	1	1	yes	characteristic
49	C₄⋊C₁₆	C₂×C₁₇	1	2	yes	complemented
50	C₂²⋊C₁₆	C₂×C₁₇	1	2	yes	—
51	#346	C₂×C₁₇	1	1	yes	characteristic, complemented
52	C₄×C₁₆	C₂×C₁₇	1	1	yes	characteristic, complemented
53	C₂×C₁₇	—	4	1	no	—
54	C₂×C₁₇	#431	1	1	yes	characteristic
55	C₂×C₁₇	#504	1	2	yes	—
56	C₂×M₁₆	C₄×C₁₇	1	1	yes	characteristic
57	C₂²×C₈	C₂²×C₁₇	1	1	yes	characteristic
58	#117	C₄×C₁₇	1	1	yes	characteristic
59	C₂×C₁₆	C₂²×C₁₇	1	2	yes	—
60	C₂×C₁₆	C₂²×C₁₇	1	2	yes	—
61	#97	C₄×C₁₇	1	1	yes	characteristic
62	#97	C₄×C₁₇	1	1	yes	characteristic
63	C₄×C₈	C₂²×C₁₇	1	1	yes	characteristic
64	C₄×C₈	C₂²×C₁₇	1	1	yes	characteristic
65	C₁₇	#2164	1	1	yes	characteristic, direct product factor, Sylow
66	C₂²×C₄	C₂×C₄×C₁₇	1	1	yes	characteristic
67	M₁₆	—	2	2	no	—
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	C₂×C₄×C₁₇	1	1	yes	characteristic
70	C₂×C₈	C₂×C₄×C₁₇	1	1	yes	characteristic
71	C₂×C₈	C₂×C₄×C₁₇	1	1	yes	characteristic
72	C₂×C₈	C₂³×C₁₇	1	1	yes	characteristic
73	C₄⋊C₄	C₈×C₁₇	1	2	yes	—
74	C₂²⋊C₄	C₈×C₁₇	1	2	yes	—
75	C₄²	C₂×C₄×C₁₇	1	1	yes	characteristic
76	C₄²	C₂×C₄×C₁₇	1	1	yes	characteristic
77	C₁₆	—	2	2	no	—
78	C₁₆	—	2	2	no	—
79	C₂³	C₂×C₈×C₁₇	1	1	yes	characteristic
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	C₂×C₈×C₁₇	1	2	yes	—
83	C₂×C₄	C₂×C₈×C₁₇	1	2	yes	—
84	C₂×C₄	C₂×C₈×C₁₇	1	1	yes	characteristic
85	C₂×C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
86	C₈	—	2	1	no	—
87	C₈	—	2	1	no	—
88	C₈	M₁₆×C₁₇	1	2	yes	—
89	C₈	#41×C₁₇	1	2	yes	—
90	C₂²	—	2	2	no	—
91	C₂²	—	2	1	no	—
92	C₂²	C₂²×C₈×C₁₇	1	1	yes	characteristic
93	C₄	—	2	2	no	—
94	C₄	—	2	2	no	—
95	C₄	—	2	1	no	—
96	C₄	C₁₇×#117	1	1	yes	characteristic
97	C₄	C₂×M₁₆×C₁₇	1	1	yes	characteristic
98	C₂	—	4	1	no	—
99	C₂	C₁₇×#431	1	1	yes	characteristic
100	C₂	C₁₇×#504	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₇×#405

C₂

yes

maximal, characteristic

C₁₇×C₄⋊C₁₆

C₂

yes

maximal, complemented

C₁₇×C₂²⋊C₁₆

C₂

yes

maximal

C₁₇×#346

C₂

yes

maximal, characteristic, complemented

C₄×C₁₆×C₁₇

C₂

yes

maximal, characteristic, complemented

C₂×M₁₆×C₁₇

C₄

yes

characteristic

C₂²×C₈×C₁₇

C₂²

yes

characteristic

C₁₇×#117

C₄

yes

characteristic

C₂×C₁₆×C₁₇

C₂²

yes

—

C₂×C₁₆×C₁₇

C₂²

yes

—

C₁₇×#97

C₄

yes

characteristic

C₁₇×#97

C₄

yes

characteristic

C₄×C₈×C₁₇

C₂²

yes

characteristic

C₄×C₈×C₁₇

C₂²

yes

characteristic

C₂²×C₄×C₁₇

C₂×C₄

yes

characteristic

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₇

C₈

yes

—

C₂²⋊C₄×C₁₇

C₈

yes

—

C₄²×C₁₇

C₂×C₄

yes

characteristic

C₄²×C₁₇

C₂×C₄

yes

characteristic

C₁₆×C₁₇

—

C₁₆×C₁₇

—

C₂³×C₁₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×C₈

yes

—

C₂×C₄×C₁₇

C₂×C₈

yes

—

C₂×C₄×C₁₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

M₁₆

yes

—

C₈×C₁₇

#41

yes

—

#2164

C₁₇

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂²×C₈

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#117

yes

characteristic

C₄×C₁₇

C₂×M₁₆

yes

characteristic

#405

C₂×C₁₇

yes

characteristic

C₄⋊C₁₆

C₂×C₁₇

yes

complemented

C₂²⋊C₁₆

C₂×C₁₇

yes

—

#346

C₂×C₁₇

yes

characteristic, complemented

C₄×C₁₆

C₂×C₁₇

yes

characteristic, complemented

C₂×C₁₇

—

C₂×C₁₇

#431

yes

characteristic

C₂×C₁₇

#504

yes

—

C₂×M₁₆

C₄×C₁₇

yes

characteristic

C₂²×C₈

C₂²×C₁₇

yes

characteristic

#117

C₄×C₁₇

yes

characteristic

C₂×C₁₆

C₂²×C₁₇

yes

—

C₂×C₁₆

C₂²×C₁₇

yes

—

#97

C₄×C₁₇

yes

characteristic

#97

C₄×C₁₇

yes

characteristic

C₄×C₈

C₂²×C₁₇

yes

characteristic

C₄×C₈

C₂²×C₁₇

yes

characteristic

C₁₇

#2164

yes

characteristic, direct product factor, Sylow

C₂²×C₄

C₂×C₄×C₁₇

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂×C₄×C₁₇

yes

characteristic

C₂×C₈

C₂×C₄×C₁₇

yes

characteristic

C₂×C₈

C₂×C₄×C₁₇

yes

characteristic

C₂×C₈

C₂³×C₁₇

yes

characteristic

C₄⋊C₄

C₈×C₁₇

yes

—

C₂²⋊C₄

C₈×C₁₇

yes

—

C₄²

C₂×C₄×C₁₇

yes

characteristic

C₄²

C₂×C₄×C₁₇

yes

characteristic

C₁₆

—

C₁₆

—

C₂³

C₂×C₈×C₁₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₈×C₁₇

yes

—

C₂×C₄

C₂×C₈×C₁₇

yes

—

C₂×C₄

C₂×C₈×C₁₇

yes

characteristic

C₂×C₄

C₂²×C₄×C₁₇

yes

characteristic

C₈

—

C₈

—

C₈

M₁₆×C₁₇

yes

—

C₈

#41×C₁₇

yes

—

C₂²

—

C₂²

—

C₂²

C₂²×C₈×C₁₇

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

C₁₇×#117

yes

characteristic

C₄

C₂×M₁₆×C₁₇

yes

characteristic

C₂

—

C₂

C₁₇×#431

yes

characteristic

100

C₂

C₁₇×#504

yes

—

	subgroup	G/H
center	C₂×C₈×C₁₇	C₂³
commutator	C₂²	C₂²×C₈×C₁₇
Frattini, cosocle	C₂×C₈	C₂³×C₁₇
socle	C₂²×C₁₇	C₂²×C₈
prime radical	C₂³×C₁₇	C₂×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₇ ◁ G

subgroup

G/H

center

C₂×C₈×C₁₇

C₂³

commutator

C₂²

C₂²×C₈×C₁₇

Frattini, cosocle

C₂×C₈

C₂³×C₁₇

socle

C₂²×C₁₇

C₂²×C₈

prime radical

C₂³×C₁₇

C₂×C₈

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₈×C₁₇ ◁ G

N	H ≅ G/N	type
2. C₁₇×C₄⋊C₁₆	C₂	semidirect
4. C₁₇×#346	C₂	semidirect
5. C₄×C₁₆×C₁₇	C₂	semidirect
39. #2164	C₁₇	direct (N × H)
49. C₄⋊C₁₆	C₂×C₁₇	semidirect
51. #346	C₂×C₁₇	semidirect
52. C₄×C₁₆	C₂×C₁₇	semidirect
65. C₁₇	#2164	direct (N × H)

H ≅ G/N

type

2. C₁₇×C₄⋊C₁₆

C₂

semidirect

4. C₁₇×#346

C₂

semidirect

5. C₄×C₁₆×C₁₇

C₂

semidirect

39. #2164

C₁₇

direct (N × H)

49. C₄⋊C₁₆

C₂×C₁₇

semidirect

51. #346

C₂×C₁₇

semidirect

52. C₄×C₁₆

C₂×C₁₇

semidirect

65. C₁₇

#2164

direct (N × H)

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

subgroup	index
C₁₇×#405	2
C₁₇×C₄⋊C₁₆	2
C₁₇×C₂²⋊C₁₆	2
C₁₇×#346	2
C₄×C₁₆×C₁₇	2
#2164	17

subgroup

index

C₁₇×#405

C₁₇×C₄⋊C₁₆

C₁₇×C₂²⋊C₁₆

C₁₇×#346

C₄×C₁₆×C₁₇

#2164

Subgroups — C₁₇×#2164 #14525170

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C17×#2164 #14525170

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#2164 #14525170