Subgroups — #17949088

Order 4352.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14543889	C₂	1	1	yes	maximal, characteristic
2	C₁₆×#3624	C₂	1	1	yes	maximal, characteristic
3	C₃₂×Dic₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587467	C₄	1	2	yes	—
5	#11763695	C₄	1	1	yes	characteristic
6	C₈×#3624	C₄	1	1	yes	characteristic
7	C₂×#10587465	C₄	1	1	yes	characteristic
8	C₁₆×Dic₁₇	C₂²	1	1	yes	characteristic
9	C₂×C₁₇×C₃₂	C₄	1	1	yes	characteristic
10	C₄×#3624	C₈	1	2	yes	—
11	C₂×#63216	C₈	1	2	yes	—
12	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
13	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
14	C₂×#63214	C₂×C₄	1	1	yes	characteristic
15	#10587467	C₈	1	4	yes	complemented
16	C₁₇×C₃₂	C₈	1	2	yes	complemented
17	#10587465	C₈	1	2	yes	complemented
18	C₂×#3624	C₁₆	1	4	yes	—
19	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
20	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
21	C₂×#3622	C₂×C₈	1	1	yes	characteristic
22	#63216	C₁₆	1	4	yes	—
23	C₁₆×C₁₇	C₂×C₈	1	1	yes	characteristic
24	C₁₆×C₁₇	C₂×C₈	1	1	yes	characteristic
25	#63214	C₂×C₈	1	2	yes	—
26	#7329	—	17	1	no	maximal, Sylow
27	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
28	C₂×Dic₁₇	C₂×C₁₆	1	2	yes	—
29	#3624	—	2	4	no	—
30	C₈×C₁₇	C₄×C₈	1	1	yes	characteristic
31	C₈×C₁₇	C₄×C₈	1	1	yes	characteristic
32	#3622	C₂×C₁₆	1	2	yes	—
33	C₄×C₃₂	—	17	1	no	—
34	C₄×C₃₂	—	17	1	no	—
35	C₈×C₁₆	—	17	1	no	—
36	C₂²×C₁₇	C₄×C₁₆	1	1	yes	characteristic
37	C₄×C₁₇	C₈²	1	1	yes	characteristic
38	C₄×C₁₇	C₄×C₁₆	1	1	yes	characteristic
39	Dic₁₇	M₆₄	1	4	yes	—
40	C₂×C₃₂	—	17	2	no	—
41	C₂×C₃₂	—	17	1	no	—
42	C₂×C₃₂	#595	1	1	yes	characteristic
43	C₄×C₁₆	—	17	1	no	—
44	C₄×C₁₆	—	17	1	no	—
45	C₈²	—	17	1	no	—
46	C₂×C₁₇	C₈×C₁₆	1	1	yes	characteristic
47	C₂×C₁₇	#1397	1	2	yes	—
48	C₂×C₁₆	—	17	2	no	—
49	C₂×C₁₆	—	17	1	no	—
50	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
51	C₄×C₈	—	17	2	no	—
52	C₄×C₈	—	17	1	no	—
53	C₃₂	—	17	4	no	—
54	C₃₂	—	17	2	no	—
55	C₃₂	#3624	1	2	yes	complemented
56	C₁₇	#7329	1	1	yes	characteristic, complemented, Sylow
57	C₂×C₈	—	17	1	no	—
58	C₂×C₈	—	17	4	no	—
59	C₂×C₈	C₄×#595	1	1	yes	characteristic
60	C₄²	—	17	1	no	—
61	C₁₆	—	17	4	no	—
62	C₁₆	—	17	2	no	—
63	C₁₆	#63242	1	1	yes	characteristic
64	C₁₆	C₂×#3624	1	1	yes	characteristic
65	C₂×C₄	—	17	2	no	—
66	C₂×C₄	C₈×#595	1	1	yes	characteristic
67	C₈	—	17	2	no	—
68	C₈	—	34	4	no	—
69	C₈	C₈×#595	1	1	yes	characteristic
70	C₈	C₄×#3624	1	1	yes	characteristic
71	C₂²	C₁₆×#595	1	1	yes	characteristic
72	C₄	—	17	4	no	—
73	C₄	C₈×#3624	1	1	yes	characteristic
74	C₄	C₁₆×#595	1	1	yes	characteristic
75	C₂	C₁₆×#3624	1	1	yes	characteristic
76	C₂	#14543890	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂×#595
commutator	C₂×C₁₇	C₈×C₁₆
Fitting	C₂×C₁₇×C₃₂	C₄
socle, prime radical	C₂²×C₁₇	C₄×C₁₆
cosocle	C₁₆×Dic₁₇	C₂²
hypercenter	C₂×C₃₂	#595
hypocenter	C₁₇	#7329
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
15. #10587467	C₈	semidirect
16. C₁₇×C₃₂	C₈	semidirect
17. #10587465	C₈	semidirect
55. C₃₂	#3624	semidirect
56. C₁₇	#7329	semidirect

Maximal subgroups

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

subgroup	index
#14543889	2
C₁₆×#3624	2
C₃₂×Dic₁₇	2
#7329	17