Group #14526164 — subgroups

Subgroups — C₂×C₁₇×#499 #14526164

Order 2176.

360 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 232 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	C₂×C₁₇×#125	C₂	1	2	yes	maximal
2	C₁₇×#499	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₁₇×#107	C₂	1	2	yes	maximal
4	C₂×C₁₇×#106	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
8	C₁₇×#125	—	2	4	no	—
9	C₁₇×#125	C₂²	1	8	yes	—
10	C₂×C₄⋊C₄×C₁₇	—	2	4	no	—
11	C₂×C₄⋊C₄×C₁₇	C₂²	1	4	yes	—
12	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
13	C₁₇×#107	C₂²	1	8	yes	—
14	C₁₇×#106	C₂²	1	8	yes	—
15	C₄×C₈×C₁₇	C₂²	1	4	yes	—
16	C₂²×C₄×C₁₇	—	2	4	no	—
17	C₂²×C₄×C₁₇	D₄	1	2	yes	—
18	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
19	C₂×C₈×C₁₇	Q₈	1	4	yes	—
20	C₂×C₈×C₁₇	Q₈	1	4	yes	—
21	C₂×C₈×C₁₇	C₂³	1	2	yes	—
22	C₂×C₈×C₁₇	C₂³	1	2	yes	—
23	C₄⋊C₄×C₁₇	—	2	4	no	—
24	C₄⋊C₄×C₁₇	—	2	16	no	—
25	C₄⋊C₄×C₁₇	C₂³	1	8	yes	—
26	C₄²×C₁₇	D₄	1	2	yes	—
27	C₄²×C₁₇	C₂³	1	2	yes	—
28	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₁₇	—	2	4	no	—
30	C₂×C₄×C₁₇	—	2	8	no	—
31	C₂×C₄×C₁₇	—	4	8	no	—
32	C₂×C₄×C₁₇	C₂×D₄	1	4	yes	—
33	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₇	C₂×Q₈	1	2	yes	—
35	C₂×C₄×C₁₇	C₂×Q₈	1	2	yes	—
36	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
37	C₈×C₁₇	C₂×Q₈	1	4	yes	—
38	C₈×C₁₇	C₂×Q₈	1	4	yes	—
39	C₂×#499	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂²×C₁₇	C₄⋊Q₈	1	2	yes	—
41	C₂²×C₁₇	#135	1	4	yes	—
42	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
43	C₄×C₁₇	—	2	4	no	—
44	C₄×C₁₇	—	4	8	no	—
45	C₄×C₁₇	C₄⋊Q₈	1	2	yes	—
46	C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
47	C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
48	C₂×#125	C₂×C₁₇	1	2	yes	—
49	#499	C₂×C₁₇	1	8	yes	direct product factor
50	C₂×#107	C₂×C₁₇	1	2	yes	—
51	C₂×#106	C₂×C₁₇	1	2	yes	—
52	C₂×C₄×C₈	C₂×C₁₇	1	1	yes	characteristic
53	C₂×C₁₇	#499	1	4	yes	direct product factor
54	C₂×C₁₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
55	C₂×C₁₇	C₂×#135	1	2	yes	—
56	C₂²×C₈	C₂²×C₁₇	1	1	yes	characteristic
57	C₂²×C₈	C₂²×C₁₇	1	1	yes	characteristic
58	#125	—	2	4	no	—
59	#125	C₂²×C₁₇	1	8	yes	—
60	C₂×C₄⋊C₄	—	2	4	no	—
61	C₂×C₄⋊C₄	C₂²×C₁₇	1	4	yes	—
62	C₂×C₄²	C₂²×C₁₇	1	1	yes	characteristic
63	#107	C₂²×C₁₇	1	8	yes	—
64	#106	C₂²×C₁₇	1	8	yes	—
65	C₄×C₈	C₂²×C₁₇	1	4	yes	—
66	C₁₇	C₂×#499	1	1	yes	characteristic, direct product factor, Sylow
67	C₂²×C₄	—	2	4	no	—
68	C₂²×C₄	D₄×C₁₇	1	2	yes	—
69	C₂²×C₄	C₂³×C₁₇	1	1	yes	characteristic
70	C₂×C₈	Q₈×C₁₇	1	4	yes	—
71	C₂×C₈	Q₈×C₁₇	1	4	yes	—
72	C₂×C₈	C₂³×C₁₇	1	2	yes	—
73	C₂×C₈	C₂³×C₁₇	1	2	yes	—
74	C₄⋊C₄	—	2	16	no	—
75	C₄⋊C₄	—	2	4	no	—
76	C₄⋊C₄	C₂³×C₁₇	1	8	yes	—
77	C₄²	D₄×C₁₇	1	2	yes	—
78	C₄²	C₂³×C₁₇	1	2	yes	—
79	C₂³	C₂×D₄×C₁₇	1	1	yes	characteristic
80	C₂×C₄	—	2	4	no	—
81	C₂×C₄	—	2	8	no	—
82	C₂×C₄	—	4	8	no	—
83	C₂×C₄	C₂×D₄×C₁₇	1	4	yes	—
84	C₂×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
85	C₂×C₄	C₂×Q₈×C₁₇	1	2	yes	—
86	C₂×C₄	C₂×Q₈×C₁₇	1	2	yes	—
87	C₂×C₄	C₂⁴×C₁₇	1	1	yes	characteristic
88	C₈	C₂×Q₈×C₁₇	1	4	yes	—
89	C₈	C₂×Q₈×C₁₇	1	4	yes	—
90	C₂²	C₁₇×C₄⋊Q₈	1	2	yes	—
91	C₂²	C₁₇×#135	1	4	yes	—
92	C₂²	C₂²×D₄×C₁₇	1	1	yes	characteristic
93	C₄	—	2	4	no	—
94	C₄	—	4	8	no	—
95	C₄	C₁₇×C₄⋊Q₈	1	2	yes	—
96	C₄	C₂²×Q₈×C₁₇	1	1	yes	characteristic
97	C₄	C₂²×Q₈×C₁₇	1	1	yes	characteristic
98	C₂	C₁₇×#499	1	4	yes	direct product factor
99	C₂	C₂×C₁₇×C₄⋊Q₈	1	1	yes	characteristic
100	C₂	C₂×C₁₇×#135	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₇×#125

C₂

yes

maximal

C₁₇×#499

C₂

yes

maximal, direct product factor

C₂×C₁₇×#107

C₂

yes

maximal

C₂×C₁₇×#106

C₂

yes

maximal

C₂×C₄×C₈×C₁₇

C₂

yes

maximal, characteristic

C₂²×C₈×C₁₇

C₂²

yes

characteristic

C₂²×C₈×C₁₇

C₂²

yes

characteristic

C₁₇×#125

—

C₁₇×#125

C₂²

yes

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄⋊C₄×C₁₇

C₂²

yes

—

C₂×C₄²×C₁₇

C₂²

yes

characteristic

C₁₇×#107

C₂²

yes

—

C₁₇×#106

C₂²

yes

—

C₄×C₈×C₁₇

C₂²

yes

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

D₄

yes

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂×C₈×C₁₇

Q₈

yes

—

C₂×C₈×C₁₇

Q₈

yes

—

C₂×C₈×C₁₇

C₂³

yes

—

C₂×C₈×C₁₇

C₂³

yes

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

C₂³

yes

—

C₄²×C₁₇

D₄

yes

—

C₄²×C₁₇

C₂³

yes

—

C₂³×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×Q₈

yes

—

C₂×C₄×C₁₇

C₂×Q₈

yes

—

C₂×C₄×C₁₇

C₂⁴

yes

characteristic

C₈×C₁₇

C₂×Q₈

yes

—

C₈×C₁₇

C₂×Q₈

yes

—

C₂×#499

C₁₇

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₁₇

C₄⋊Q₈

yes

—

C₂²×C₁₇

#135

yes

—

C₂²×C₁₇

C₂²×D₄

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₄⋊Q₈

yes

—

C₄×C₁₇

C₂²×Q₈

yes

characteristic

C₄×C₁₇

C₂²×Q₈

yes

characteristic

C₂×#125

C₂×C₁₇

yes

—

#499

C₂×C₁₇

yes

direct product factor

C₂×#107

C₂×C₁₇

yes

—

C₂×#106

C₂×C₁₇

yes

—

C₂×C₄×C₈

C₂×C₁₇

yes

characteristic

C₂×C₁₇

#499

yes

direct product factor

C₂×C₁₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₇

C₂×#135

yes

—

C₂²×C₈

C₂²×C₁₇

yes

characteristic

C₂²×C₈

C₂²×C₁₇

yes

characteristic

#125

—

#125

C₂²×C₁₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₁₇

yes

—

C₂×C₄²

C₂²×C₁₇

yes

characteristic

#107

C₂²×C₁₇

yes

—

#106

C₂²×C₁₇

yes

—

C₄×C₈

C₂²×C₁₇

yes

—

C₁₇

C₂×#499

yes

characteristic, direct product factor, Sylow

C₂²×C₄

—

C₂²×C₄

D₄×C₁₇

yes

—

C₂²×C₄

C₂³×C₁₇

yes

characteristic

C₂×C₈

Q₈×C₁₇

yes

—

C₂×C₈

Q₈×C₁₇

yes

—

C₂×C₈

C₂³×C₁₇

yes

—

C₂×C₈

C₂³×C₁₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂³×C₁₇

yes

—

C₄²

D₄×C₁₇

yes

—

C₄²

C₂³×C₁₇

yes

—

C₂³

C₂×D₄×C₁₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄×C₁₇

yes

—

C₂×C₄

C₂×D₄×C₁₇

yes

characteristic

C₂×C₄

C₂×Q₈×C₁₇

yes

—

C₂×C₄

C₂×Q₈×C₁₇

yes

—

C₂×C₄

C₂⁴×C₁₇

yes

characteristic

C₈

C₂×Q₈×C₁₇

yes

—

C₈

C₂×Q₈×C₁₇

yes

—

C₂²

C₁₇×C₄⋊Q₈

yes

—

C₂²

C₁₇×#135

yes

—

C₂²

C₂²×D₄×C₁₇

yes

characteristic

C₄

—

C₄

—

C₄

C₁₇×C₄⋊Q₈

yes

—

C₄

C₂²×Q₈×C₁₇

yes

characteristic

C₄

C₂²×Q₈×C₁₇

yes

characteristic

C₂

C₁₇×#499

yes

direct product factor

C₂

C₂×C₁₇×C₄⋊Q₈

yes

characteristic

100

C₂

C₂×C₁₇×#135

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₇ ◁ C₂×C₄²×C₁₇ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₁₇

C₂×D₄

commutator, Frattini, cosocle

C₂×C₄

C₂⁴×C₁₇

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₇ ◁ C₂×C₄²×C₁₇ ◁ G

N	H ≅ G/N	type
2. C₁₇×#499	C₂	direct (N × H)
39. C₂×#499	C₁₇	direct (N × H)
49. #499	C₂×C₁₇	direct (N × H)
53. C₂×C₁₇	#499	direct (N × H)
66. C₁₇	C₂×#499	direct (N × H)
98. C₂	C₁₇×#499	direct (N × H)

H ≅ G/N

type

2. C₁₇×#499

C₂

direct (N × H)

39. C₂×#499

C₁₇

direct (N × H)

49. #499

C₂×C₁₇

direct (N × H)

53. C₂×C₁₇

#499

direct (N × H)

66. C₁₇

C₂×#499

direct (N × H)

98. C₂

C₁₇×#499

direct (N × H)

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

subgroup	index
C₂×C₁₇×#125	2
C₁₇×#499	2
C₂×C₁₇×#107	2
C₂×C₁₇×#106	2
C₂×C₄×C₈×C₁₇	2
C₂×#499	17

subgroup

index

C₂×C₁₇×#125

C₁₇×#499

C₂×C₁₇×#107

C₂×C₁₇×#106

C₂×C₄×C₈×C₁₇

C₂×#499

Subgroups — C₂×C₁₇×#499 #14526164

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×C17×#499 #14526164

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₇×#499 #14526164