Group #16868330 — subgroups

Subgroups — C₁₇×#10586 #16868330

Order 4352.

350 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 114 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×#2845	C₂	1	1	yes	maximal, characteristic
2	C₁₇×#1836	C₂	1	3	yes	maximal
3	C₁₇×#1295	C₂	1	3	yes	maximal
4	C₂×C₁₇×#97	C₂²	1	3	yes	—
5	C₁₇×#398	—	2	6	no	—
6	C₁₇×#384	C₂²	1	1	yes	characteristic
7	C₁₇×#382	C₂²	1	3	yes	—
8	C₁₇×#340	—	2	6	no	—
9	C₂²×C₈×C₁₇	—	2	3	no	—
10	C₂×C₄⋊C₄×C₁₇	—	2	3	no	—
11	C₂×C₄⋊C₄×C₁₇	—	2	6	no	—
12	C₂×C₄⋊C₄×C₁₇	D₄	1	6	yes	—
13	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
14	C₁₇×#97	—	2	3	no	—
15	C₁₇×#97	Q₈	1	6	yes	—
16	C₁₇×#95	—	2	2	no	—
17	C₁₇×#95	—	2	6	no	—
18	C₂²×C₄×C₁₇	—	2	6	no	—
19	C₂²×C₄×C₁₇	—	4	1	no	—
20	C₂²×C₄×C₁₇	C₂×D₄	1	3	yes	—
21	C₂×C₈×C₁₇	—	2	6	no	—
22	C₂×C₈×C₁₇	—	4	3	no	—
23	C₂×C₈×C₁₇	—	4	3	no	—
24	C₄⋊C₄×C₁₇	—	4	6	no	—
25	C₄⋊C₄×C₁₇	—	4	6	no	—
26	C₄⋊C₄×C₁₇	—	8	6	no	—
27	C₄²×C₁₇	C₂×Q₈	1	3	yes	—
28	C₄²×C₁₇	#41	1	1	yes	characteristic
29	#10586	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂³×C₁₇	#120	1	1	yes	characteristic
31	C₂×C₄×C₁₇	—	2	3	no	—
32	C₂×C₄×C₁₇	—	2	3	no	—
33	C₂×C₄×C₁₇	—	4	6	no	—
34	C₂×C₄×C₁₇	—	4	6	no	—
35	C₂×C₄×C₁₇	—	4	6	no	—
36	C₂×C₄×C₁₇	—	8	3	no	—
37	C₂×C₄×C₁₇	C₂²⋊Q₈	1	3	yes	—
38	C₂×C₄×C₁₇	C₄⋊Q₈	1	3	yes	—
39	C₈×C₁₇	—	4	6	no	—
40	#2845	C₂×C₁₇	1	1	yes	characteristic
41	#1836	C₂×C₁₇	1	3	yes	—
42	#1295	C₂×C₁₇	1	3	yes	—
43	C₂²×C₁₇	#395	1	1	yes	characteristic
44	C₂²×C₁₇	#454	1	3	yes	—
45	C₂²×C₁₇	#455	1	3	yes	—
46	C₄×C₁₇	—	2	3	no	—
47	C₄×C₁₇	—	2	3	no	—
48	C₄×C₁₇	—	8	6	no	—
49	C₄×C₁₇	—	16	1	no	—
50	C₂×#97	C₂²×C₁₇	1	3	yes	—
51	#398	—	2	6	no	—
52	#384	C₂²×C₁₇	1	1	yes	characteristic
53	#382	C₂²×C₁₇	1	3	yes	—
54	#340	—	2	6	no	—
55	C₂×C₁₇	#1658	1	1	yes	characteristic
56	C₂×C₁₇	#1662	1	3	yes	—
57	C₂×C₁₇	#2061	1	3	yes	—
58	C₂²×C₈	—	2	3	no	—
59	C₂×C₄⋊C₄	—	2	3	no	—
60	C₂×C₄⋊C₄	—	2	6	no	—
61	C₂×C₄⋊C₄	D₄×C₁₇	1	6	yes	—
62	C₂×C₄²	C₂³×C₁₇	1	1	yes	characteristic
63	#97	—	2	3	no	—
64	#97	Q₈×C₁₇	1	6	yes	—
65	#95	—	2	2	no	—
66	#95	—	2	6	no	—
67	C₁₇	#10586	1	1	yes	characteristic, direct product factor, Sylow
68	C₂²×C₄	—	2	6	no	—
69	C₂²×C₄	—	4	1	no	—
70	C₂²×C₄	C₂×D₄×C₁₇	1	3	yes	—
71	C₂×C₈	—	2	6	no	—
72	C₂×C₈	—	4	3	no	—
73	C₂×C₈	—	4	3	no	—
74	C₄⋊C₄	—	4	6	no	—
75	C₄⋊C₄	—	4	6	no	—
76	C₄⋊C₄	—	8	6	no	—
77	C₄²	C₂×Q₈×C₁₇	1	3	yes	—
78	C₄²	#41×C₁₇	1	1	yes	characteristic
79	C₂³	C₁₇×#120	1	1	yes	characteristic
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	—	2	3	no	—
82	C₂×C₄	—	4	6	no	—
83	C₂×C₄	—	4	6	no	—
84	C₂×C₄	—	4	6	no	—
85	C₂×C₄	—	8	3	no	—
86	C₂×C₄	C₁₇×C₂²⋊Q₈	1	3	yes	—
87	C₂×C₄	C₁₇×C₄⋊Q₈	1	3	yes	—
88	C₈	—	4	6	no	—
89	C₂²	C₁₇×#395	1	1	yes	characteristic
90	C₂²	C₁₇×#454	1	3	yes	—
91	C₂²	C₁₇×#455	1	3	yes	—
92	C₄	—	2	3	no	—
93	C₄	—	2	3	no	—
94	C₄	—	8	6	no	—
95	C₄	—	16	1	no	—
96	C₂	C₁₇×#1658	1	1	yes	characteristic
97	C₂	C₁₇×#1662	1	3	yes	—
98	C₂	C₁₇×#2061	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₇×#2845

C₂

yes

maximal, characteristic

C₁₇×#1836

C₂

yes

maximal

C₁₇×#1295

C₂

yes

maximal

C₂×C₁₇×#97

C₂²

yes

—

C₁₇×#398

—

C₁₇×#384

C₂²

yes

characteristic

C₁₇×#382

C₂²

yes

—

C₁₇×#340

—

C₂²×C₈×C₁₇

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄⋊C₄×C₁₇

D₄

yes

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₁₇×#97

—

C₁₇×#97

Q₈

yes

—

C₁₇×#95

—

C₁₇×#95

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

C₂×Q₈

yes

—

C₄²×C₁₇

#41

yes

characteristic

#10586

C₁₇

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₇

#120

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₇

C₄⋊Q₈

yes

—

C₈×C₁₇

—

#2845

C₂×C₁₇

yes

characteristic

#1836

C₂×C₁₇

yes

—

#1295

C₂×C₁₇

yes

—

C₂²×C₁₇

#395

yes

characteristic

C₂²×C₁₇

#454

yes

—

C₂²×C₁₇

#455

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₂×#97

C₂²×C₁₇

yes

—

#398

—

#384

C₂²×C₁₇

yes

characteristic

#382

C₂²×C₁₇

yes

—

#340

—

C₂×C₁₇

#1658

yes

characteristic

C₂×C₁₇

#1662

yes

—

C₂×C₁₇

#2061

yes

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₄×C₁₇

yes

—

C₂×C₄²

C₂³×C₁₇

yes

characteristic

#97

—

#97

Q₈×C₁₇

yes

—

#95

—

#95

—

C₁₇

#10586

yes

characteristic, direct product factor, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄×C₁₇

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×Q₈×C₁₇

yes

—

C₄²

#41×C₁₇

yes

characteristic

C₂³

C₁₇×#120

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₁₇×C₂²⋊Q₈

yes

—

C₂×C₄

C₁₇×C₄⋊Q₈

yes

—

C₈

—

C₂²

C₁₇×#395

yes

characteristic

C₂²

C₁₇×#454

yes

—

C₂²

C₁₇×#455

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

C₂

C₁₇×#1658

yes

characteristic

C₂

C₁₇×#1662

yes

—

C₂

C₁₇×#2061

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₇	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₇
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₇ ◁ C₂×C₄²×C₁₇ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₁₇

#120

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₁₇

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₁₇ ◁ C₂×C₄²×C₁₇ ◁ G

N	H ≅ G/N	type
29. #10586	C₁₇	direct (N × H)
67. C₁₇	#10586	direct (N × H)

H ≅ G/N

type

29. #10586

C₁₇

direct (N × H)

67. C₁₇

#10586

direct (N × H)

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

subgroup	index
C₁₇×#2845	2
C₁₇×#1836	2
C₁₇×#1295	2
#10586	17

subgroup

index

C₁₇×#2845

C₁₇×#1836

C₁₇×#1295

#10586

Subgroups — C₁₇×#10586 #16868330

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C17×#10586 #16868330

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#10586 #16868330