Group #14524583 — subgroups

Subgroups — C₈×Q₁₆×C₁₇ #14524583

Order 2176.

164 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 104 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	C₈×Q₈×C₁₇	C₂	1	2	yes	maximal
2	C₄×Q₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
3	C₁₇×#335	C₂	1	1	yes	maximal, characteristic
4	C₁₇×Q₈⋊C₈	C₂	1	2	yes	maximal
5	C₈²×C₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₁₆×C₁₇	C₄	1	1	yes	characteristic
7	C₄×Q₈×C₁₇	C₂²	1	2	yes	—
8	C₁₇×#107	C₄	1	1	yes	characteristic
9	C₁₇×C₄⋊C₈	—	2	2	no	—
10	C₁₇×C₄⋊C₈	C₂²	1	2	yes	—
11	C₁₇×#103	C₄	1	2	yes	—
12	C₄×C₈×C₁₇	—	2	2	no	—
13	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₁₇	C₂×C₄	1	2	yes	—
17	Q₁₆×C₁₇	C₈	1	4	yes	direct product factor
18	C₂×C₈×C₁₇	—	2	1	no	—
19	C₂×C₈×C₁₇	—	2	2	no	—
20	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₁₇	D₄	1	1	yes	characteristic
23	C₂×C₈×C₁₇	D₄	1	1	yes	characteristic
24	C₄⋊C₄×C₁₇	—	2	2	no	—
25	C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
26	C₄²×C₁₇	—	2	2	no	—
27	C₄²×C₁₇	C₂³	1	1	yes	characteristic
28	Q₈×C₁₇	—	2	2	no	—
29	Q₈×C₁₇	C₂×C₈	1	4	yes	—
30	C₂×C₄×C₁₇	—	2	2	no	—
31	C₂×C₄×C₁₇	—	2	2	no	—
32	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
35	C₈×C₁₇	—	2	2	no	—
36	C₈×C₁₇	—	2	1	no	—
37	C₈×C₁₇	—	2	1	no	—
38	C₈×C₁₇	—	4	2	no	—
39	C₈×C₁₇	C₂×C₈	1	2	yes	—
40	C₈×C₁₇	Q₁₆	1	2	yes	direct product factor
41	C₈×Q₁₆	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂²×C₁₇	C₄×D₄	1	1	yes	characteristic
43	C₄×C₁₇	—	2	4	no	—
44	C₄×C₁₇	—	2	1	no	—
45	C₄×C₁₇	—	4	2	no	—
46	C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
47	C₄×C₁₇	#131	1	1	yes	characteristic
48	C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
49	C₄×C₁₇	#135	1	1	yes	characteristic
50	C₈×Q₈	C₂×C₁₇	1	2	yes	—
51	C₄×Q₁₆	C₂×C₁₇	1	1	yes	characteristic
52	#335	C₂×C₁₇	1	1	yes	characteristic
53	Q₈⋊C₈	C₂×C₁₇	1	2	yes	—
54	C₈²	C₂×C₁₇	1	1	yes	characteristic
55	C₂×C₁₇	C₈×D₄	1	1	yes	characteristic
56	C₂×C₁₇	C₄×Q₁₆	1	1	yes	characteristic
57	C₂×C₁₇	#443	1	1	yes	characteristic
58	C₂×Q₁₆	C₄×C₁₇	1	1	yes	characteristic
59	C₄×Q₈	C₂²×C₁₇	1	2	yes	—
60	#107	C₄×C₁₇	1	1	yes	characteristic
61	C₄⋊C₈	—	2	2	no	—
62	C₄⋊C₈	C₂²×C₁₇	1	2	yes	—
63	#103	C₄×C₁₇	1	2	yes	—
64	C₄×C₈	—	2	2	no	—
65	C₄×C₈	C₂²×C₁₇	1	1	yes	characteristic
66	C₄×C₈	C₂²×C₁₇	1	1	yes	characteristic
67	C₄×C₈	C₂²×C₁₇	1	1	yes	characteristic
68	C₁₇	C₈×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
69	C₂×Q₈	C₂×C₄×C₁₇	1	2	yes	—
70	Q₁₆	C₈×C₁₇	1	4	yes	direct product factor
71	C₂×C₈	—	2	1	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	C₂×C₄×C₁₇	1	1	yes	characteristic
74	C₂×C₈	C₂×C₄×C₁₇	1	1	yes	characteristic
75	C₂×C₈	D₄×C₁₇	1	1	yes	characteristic
76	C₂×C₈	D₄×C₁₇	1	1	yes	characteristic
77	C₄⋊C₄	—	2	2	no	—
78	C₄⋊C₄	C₂×C₄×C₁₇	1	2	yes	—
79	C₄²	—	2	2	no	—
80	C₄²	C₂³×C₁₇	1	1	yes	characteristic
81	Q₈	—	2	2	no	—
82	Q₈	C₂×C₈×C₁₇	1	4	yes	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
86	C₂×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
87	C₂×C₄	#41×C₁₇	1	1	yes	characteristic
88	C₈	—	2	2	no	—
89	C₈	—	2	1	no	—
90	C₈	—	2	1	no	—
91	C₈	—	4	2	no	—
92	C₈	C₂×C₈×C₁₇	1	2	yes	—
93	C₈	Q₁₆×C₁₇	1	2	yes	direct product factor
94	C₂²	C₄×D₄×C₁₇	1	1	yes	characteristic
95	C₄	—	2	4	no	—
96	C₄	—	2	1	no	—
97	C₄	—	4	2	no	—
98	C₄	C₂²×C₈×C₁₇	1	1	yes	characteristic
99	C₄	C₁₇×#131	1	1	yes	characteristic
100	C₄	C₂×Q₁₆×C₁₇	1	1	yes	characteristic
101	C₄	C₁₇×#135	1	1	yes	characteristic
102	C₂	C₈×D₄×C₁₇	1	1	yes	characteristic
103	C₂	C₄×Q₁₆×C₁₇	1	1	yes	characteristic
104	C₂	C₁₇×#443	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈×Q₈×C₁₇

C₂

yes

maximal

C₄×Q₁₆×C₁₇

C₂

yes

maximal, characteristic

C₁₇×#335

C₂

yes

maximal, characteristic

C₁₇×Q₈⋊C₈

C₂

yes

maximal

C₈²×C₁₇

C₂

yes

maximal, characteristic

C₂×Q₁₆×C₁₇

C₄

yes

characteristic

C₄×Q₈×C₁₇

C₂²

yes

—

C₁₇×#107

C₄

yes

characteristic

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

C₂²

yes

—

C₁₇×#103

C₄

yes

—

C₄×C₈×C₁₇

—

C₄×C₈×C₁₇

C₂²

yes

characteristic

C₄×C₈×C₁₇

C₂²

yes

characteristic

C₄×C₈×C₁₇

C₂²

yes

characteristic

C₂×Q₈×C₁₇

C₂×C₄

yes

—

Q₁₆×C₁₇

C₈

yes

direct product factor

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₇

D₄

yes

characteristic

C₂×C₈×C₁₇

D₄

yes

characteristic

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

C₂×C₄

yes

—

C₄²×C₁₇

—

C₄²×C₁₇

C₂³

yes

characteristic

Q₈×C₁₇

—

Q₈×C₁₇

C₂×C₈

yes

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

#41

yes

characteristic

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

C₂×C₈

yes

—

C₈×C₁₇

Q₁₆

yes

direct product factor

C₈×Q₁₆

C₁₇

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₁₇

C₄×D₄

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂²×C₈

yes

characteristic

C₄×C₁₇

#131

yes

characteristic

C₄×C₁₇

C₂×Q₁₆

yes

characteristic

C₄×C₁₇

#135

yes

characteristic

C₈×Q₈

C₂×C₁₇

yes

—

C₄×Q₁₆

C₂×C₁₇

yes

characteristic

#335

C₂×C₁₇

yes

characteristic

Q₈⋊C₈

C₂×C₁₇

yes

—

C₈²

C₂×C₁₇

yes

characteristic

C₂×C₁₇

C₈×D₄

yes

characteristic

C₂×C₁₇

C₄×Q₁₆

yes

characteristic

C₂×C₁₇

#443

yes

characteristic

C₂×Q₁₆

C₄×C₁₇

yes

characteristic

C₄×Q₈

C₂²×C₁₇

yes

—

#107

C₄×C₁₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂²×C₁₇

yes

—

#103

C₄×C₁₇

yes

—

C₄×C₈

—

C₄×C₈

C₂²×C₁₇

yes

characteristic

C₄×C₈

C₂²×C₁₇

yes

characteristic

C₄×C₈

C₂²×C₁₇

yes

characteristic

C₁₇

C₈×Q₁₆

yes

characteristic, direct product factor, Sylow

C₂×Q₈

C₂×C₄×C₁₇

yes

—

Q₁₆

C₈×C₁₇

yes

direct product factor

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄×C₁₇

yes

characteristic

C₂×C₈

C₂×C₄×C₁₇

yes

characteristic

C₂×C₈

D₄×C₁₇

yes

characteristic

C₂×C₈

D₄×C₁₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

C₂×C₄×C₁₇

yes

—

C₄²

—

C₄²

C₂³×C₁₇

yes

characteristic

Q₈

—

Q₈

C₂×C₈×C₁₇

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×C₄×C₁₇

yes

characteristic

C₂×C₄

C₂×D₄×C₁₇

yes

characteristic

C₂×C₄

#41×C₁₇

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

C₈

C₂×C₈×C₁₇

yes

—

C₈

Q₁₆×C₁₇

yes

direct product factor

C₂²

C₄×D₄×C₁₇

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

C₂²×C₈×C₁₇

yes

characteristic

C₄

C₁₇×#131

yes

characteristic

100

C₄

C₂×Q₁₆×C₁₇

yes

characteristic

101

C₄

C₁₇×#135

yes

characteristic

102

C₂

C₈×D₄×C₁₇

yes

characteristic

103

C₂

C₄×Q₁₆×C₁₇

yes

characteristic

104

C₂

C₁₇×#443

yes

characteristic

	subgroup	G/H
center	C₂×C₈×C₁₇	D₄
commutator	C₄	C₂²×C₈×C₁₇
Frattini, cosocle	C₄²	C₂³×C₁₇
socle, prime radical	C₂²×C₁₇	C₄×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₇ ◁ C₄×C₈×C₁₇ ◁ G

subgroup

G/H

center

C₂×C₈×C₁₇

D₄

commutator

C₄

C₂²×C₈×C₁₇

Frattini, cosocle

C₄²

C₂³×C₁₇

socle, prime radical

C₂²×C₁₇

C₄×D₄

LCS

G ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₈×C₁₇ ◁ C₄×C₈×C₁₇ ◁ G

N	H ≅ G/N	type
17. Q₁₆×C₁₇	C₈	direct (N × H)
40. C₈×C₁₇	Q₁₆	direct (N × H)
41. C₈×Q₁₆	C₁₇	direct (N × H)
68. C₁₇	C₈×Q₁₆	direct (N × H)
70. Q₁₆	C₈×C₁₇	direct (N × H)
93. C₈	Q₁₆×C₁₇	direct (N × H)

H ≅ G/N

type

17. Q₁₆×C₁₇

C₈

direct (N × H)

40. C₈×C₁₇

Q₁₆

direct (N × H)

41. C₈×Q₁₆

C₁₇

direct (N × H)

68. C₁₇

C₈×Q₁₆

direct (N × H)

70. Q₁₆

C₈×C₁₇

direct (N × H)

93. C₈

Q₁₆×C₁₇

direct (N × H)

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

subgroup	index
C₈×Q₈×C₁₇	2
C₄×Q₁₆×C₁₇	2
C₁₇×#335	2
C₁₇×Q₈⋊C₈	2
C₈²×C₁₇	2
C₈×Q₁₆	17

subgroup

index

C₈×Q₈×C₁₇

C₄×Q₁₆×C₁₇

C₁₇×#335

C₁₇×Q₈⋊C₈

C₈²×C₁₇

C₈×Q₁₆

Subgroups — C₈×Q₁₆×C₁₇ #14524583

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C8×Q16×C17 #14524583

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×Q₁₆×C₁₇ #14524583