Subgroups — C₁₇×#1307 #14524313

Order 2176.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×#432	C₂	1	2	yes	maximal
2	C₁₇×#423	C₂	1	1	yes	maximal, characteristic
3	C₂×M₁₆×C₁₇	C₄	1	2	yes	—
4	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
5	C₁₇×C₄⋊C₈	—	2	2	no	—
6	C₁₇×C₄⋊C₈	—	2	2	no	—
7	C₁₇×C₂²⋊C₈	C₄	1	4	yes	—
8	C₁₇×#97	—	2	2	no	—
9	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
10	C₂²×C₄×C₁₇	C₂×C₄	1	2	yes	—
11	M₁₆×C₁₇	—	4	2	no	—
12	C₂×C₈×C₁₇	—	2	2	no	—
13	C₂×C₈×C₁₇	—	2	4	no	—
14	C₄²×C₁₇	D₄	1	1	yes	characteristic
15	C₄²×C₁₇	D₄	1	2	yes	—
16	C₄²×C₁₇	Q₈	1	1	yes	characteristic
17	C₂³×C₁₇	C₄²	1	1	yes	characteristic
18	C₂×C₄×C₁₇	—	2	1	no	—
19	C₂×C₄×C₁₇	—	2	1	no	—
20	C₂×C₄×C₁₇	—	4	2	no	—
21	C₂×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
22	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
24	C₂×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
25	C₈×C₁₇	—	4	4	no	—
26	C₈×C₁₇	—	4	2	no	—
27	#1307	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂²×C₁₇	—	2	1	no	—
29	C₂²×C₁₇	—	2	2	no	—
30	C₂²×C₁₇	#95	1	1	yes	characteristic
31	C₄×C₁₇	—	2	1	no	—
32	C₄×C₁₇	—	2	2	no	—
33	C₄×C₁₇	—	2	2	no	—
34	C₄×C₁₇	#100	1	1	yes	characteristic
35	C₄×C₁₇	#101	1	1	yes	characteristic
36	#432	C₂×C₁₇	1	2	yes	—
37	#423	C₂×C₁₇	1	1	yes	characteristic
38	C₂×C₁₇	—	4	1	no	—
39	C₂×C₁₇	#343	1	1	yes	characteristic
40	C₂×C₁₇	#344	1	2	yes	—
41	C₂×M₁₆	C₄×C₁₇	1	2	yes	—
42	C₂×C₄²	C₂²×C₁₇	1	1	yes	characteristic
43	C₄⋊C₈	—	2	2	no	—
44	C₄⋊C₈	—	2	2	no	—
45	C₂²⋊C₈	C₄×C₁₇	1	4	yes	—
46	#97	—	2	2	no	—
47	C₁₇	#1307	1	1	yes	characteristic, direct product factor, Sylow
48	C₂²×C₄	C₂×C₄×C₁₇	1	1	yes	characteristic
49	C₂²×C₄	C₂×C₄×C₁₇	1	2	yes	—
50	M₁₆	—	4	2	no	—
51	C₂×C₈	—	2	4	no	—
52	C₂×C₈	—	2	2	no	—
53	C₄²	D₄×C₁₇	1	1	yes	characteristic
54	C₄²	D₄×C₁₇	1	2	yes	—
55	C₄²	Q₈×C₁₇	1	1	yes	characteristic
56	C₂³	C₄²×C₁₇	1	1	yes	characteristic
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	4	2	no	—
60	C₂×C₄	C₂²⋊C₄×C₁₇	1	2	yes	—
61	C₂×C₄	C₂²⋊C₄×C₁₇	1	1	yes	characteristic
62	C₂×C₄	C₄⋊C₄×C₁₇	1	1	yes	characteristic
63	C₂×C₄	C₄⋊C₄×C₁₇	1	2	yes	—
64	C₈	—	4	4	no	—
65	C₈	—	4	2	no	—
66	C₂²	—	2	1	no	—
67	C₂²	—	2	2	no	—
68	C₂²	C₁₇×#95	1	1	yes	characteristic
69	C₄	—	2	1	no	—
70	C₄	—	2	2	no	—
71	C₄	—	2	2	no	—
72	C₄	C₁₇×#100	1	1	yes	characteristic
73	C₄	C₁₇×#101	1	1	yes	characteristic
74	C₂	—	4	1	no	—
75	C₂	C₁₇×#343	1	1	yes	characteristic
76	C₂	C₁₇×#344	1	2	yes	—

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

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
27. #1307	C₁₇	direct (N × H)
47. C₁₇	#1307	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₇×#432	2
C₁₇×#423	2
#1307	17

Subgroups — C17×#1307 #14524313

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#1307 #14524313