Subgroups — #16921997

Order 4352.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#10587465	C₂	1	2	yes	maximal
2	C₈×C₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#10587465	—	2	2	no	—
4	C₄×C₁₆×C₁₇	C₄	1	1	yes	characteristic
5	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
6	C₈²×C₁₇	C₄	1	1	yes	characteristic
7	C₂×C₁₆×C₁₇	—	2	1	no	—
8	C₂×C₁₆×C₁₇	D₄	1	1	yes	characteristic
9	C₂×C₁₆×C₁₇	Q₈	1	1	yes	characteristic
10	C₄×C₈×C₁₇	C₈	1	1	yes	characteristic
11	C₄×C₈×C₁₇	C₈	1	1	yes	characteristic
12	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
13	#10587465	—	4	2	no	—
14	C₂×C₈×C₁₇	—	2	1	no	—
15	C₂×C₈×C₁₇	C₁₆	1	1	yes	characteristic
16	C₂×C₈×C₁₇	C₁₆	1	1	yes	characteristic
17	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
18	C₂×C₈×C₁₇	M₁₆	1	1	yes	characteristic
19	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
20	C₁₆×C₁₇	—	2	2	no	—
21	C₁₆×C₁₇	—	2	1	no	—
22	C₁₆×C₁₇	D₈	1	1	yes	characteristic
23	C₁₆×C₁₇	Q₁₆	1	1	yes	characteristic
24	#7458	—	17	1	no	maximal, Sylow
25	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
26	C₂×C₄×C₁₇	C₂×C₁₆	1	1	yes	characteristic
27	C₂×C₄×C₁₇	M₃₂	1	1	yes	characteristic
28	C₈×C₁₇	—	2	2	no	—
29	C₈×C₁₇	—	2	1	no	—
30	C₈×C₁₇	—	2	1	no	—
31	C₈×C₁₇	C₃₂	1	2	yes	complemented
32	C₈×C₁₇	#107	1	1	yes	characteristic
33	C₈×C₁₇	#108	1	1	yes	characteristic
34	C₄⋊C₃₂	—	17	2	no	—
35	C₈×C₁₆	D₁₇	1	1	yes	characteristic
36	C₂²×C₁₇	C₄⋊C₁₆	1	1	yes	characteristic
37	C₄×C₁₇	—	2	1	no	—
38	C₄×C₁₇	#335	1	1	yes	characteristic
39	C₄×C₁₇	#364	1	1	yes	characteristic
40	C₄×C₁₇	C₂×C₃₂	1	1	yes	characteristic
41	C₄×C₁₇	M₆₄	1	1	yes	characteristic
42	C₂×C₃₂	—	34	2	no	—
43	C₄×C₁₆	Dic₁₇	1	1	yes	characteristic
44	C₄×C₁₆	C₂×D₁₇	1	1	yes	characteristic
45	C₈²	Dic₁₇	1	1	yes	characteristic
46	C₂×C₁₇	#1370	1	1	yes	characteristic
47	C₂×C₁₇	C₄⋊C₃₂	1	1	yes	characteristic
48	C₂×C₁₇	#1422	1	1	yes	characteristic
49	C₂×C₁₆	—	2	1	no	—
50	C₂×C₁₆	Dic₃₄	1	1	yes	characteristic
51	C₂×C₁₆	D₆₈	1	1	yes	characteristic
52	C₄×C₈	#3622	1	1	yes	characteristic
53	C₄×C₈	#3622	1	1	yes	characteristic
54	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
55	C₃₂	—	68	2	no	—
56	C₁₇	#7458	1	1	yes	characteristic, complemented, Sylow
57	C₂×C₈	—	2	1	no	—
58	C₂×C₈	#63214	1	1	yes	characteristic
59	C₂×C₈	#63214	1	1	yes	characteristic
60	C₂×C₈	#63223	1	1	yes	characteristic
61	C₂×C₈	C₄⋊Dic₁₇	1	1	yes	characteristic
62	C₄²	C₂×#3622	1	1	yes	characteristic
63	C₁₆	—	2	2	no	—
64	C₁₆	—	2	1	no	—
65	C₁₆	D₁₃₆	1	1	yes	characteristic
66	C₁₆	Dic₆₈	1	1	yes	characteristic
67	C₂×C₄	C₄⋊#3622	1	1	yes	characteristic
68	C₂×C₄	C₂×#63214	1	1	yes	characteristic
69	C₂×C₄	#10587483	1	1	yes	characteristic
70	C₈	—	2	2	no	—
71	C₈	—	2	1	no	—
72	C₈	—	2	1	no	—
73	C₈	#10587465	1	2	yes	complemented
74	C₈	#10587489	1	1	yes	characteristic
75	C₈	#10587490	1	1	yes	characteristic
76	C₂²	C₄⋊#63214	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	#11762562	1	1	yes	characteristic
79	C₄	#11762630	1	1	yes	characteristic
80	C₄	C₂×#10587465	1	1	yes	characteristic
81	C₄	#11762643	1	1	yes	characteristic
82	C₂	#14526854	1	1	yes	characteristic
83	C₂	C₄⋊#10587465	1	1	yes	characteristic
84	C₂	#14526982	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	D₆₈
commutator	C₄×C₁₇	C₂×C₃₂
Frattini	C₄×C₁₆	C₂×D₁₇
Fitting	C₈×C₁₆×C₁₇	C₂
socle, prime radical	C₂²×C₁₇	C₄⋊C₁₆
cosocle	C₄×C₁₆×C₁₇	C₂²
hypercenter	C₈×C₁₆	D₁₇
hypocenter	C₁₇	#7458
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄×C₁₆ ◁ C₈×C₁₆

Complements

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

N	H ≅ G/N	type
31. C₈×C₁₇	C₃₂	semidirect
56. C₁₇	#7458	semidirect
73. C₈	#10587465	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#10587465	2
C₈×C₁₆×C₁₇	2
#7458	17