Subgroups — #14526973

Order 2176.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₇₂	C₂	1	1	yes	maximal, characteristic
2	#11762643	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₇×#368	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₃₆	—	2	1	no	—
5	C₂×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
6	C₁₇×#108	—	2	1	no	—
7	D₂₇₂	—	2	1	no	—
8	D₂₇₂	C₄	1	2	yes	—
9	#10587465	—	2	1	no	—
10	C₂×D₆₈	—	4	1	no	—
11	M₁₆×C₁₇	—	4	1	no	—
12	C₂×C₈×C₁₇	D₄	1	1	yes	characteristic
13	D₁₃₆	—	2	2	no	—
14	D₁₃₆	—	4	1	no	—
15	C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₁₇	D₄	1	1	yes	characteristic
17	C₂²×D₁₇	—	8	1	no	—
18	C₂×C₄×C₁₇	D₈	1	1	yes	characteristic
19	D₆₈	—	4	2	no	—
20	D₆₈	—	8	1	no	—
21	C₈×C₁₇	—	8	1	no	—
22	C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₈×C₁₇	SD₁₆	1	1	yes	characteristic
24	#1419	—	17	1	no	maximal, Sylow
25	C₂²×C₁₇	SD₃₂	1	1	yes	characteristic
26	C₂×D₁₇	—	8	2	no	—
27	C₂×D₁₇	—	16	2	no	—
28	C₄×C₁₇	#102	1	1	yes	characteristic
29	C₄×C₁₇	D₁₆	1	1	yes	characteristic
30	C₂×D₁₆	—	17	1	no	—
31	M₆₄	—	17	1	no	—
32	#368	D₁₇	1	1	yes	characteristic, complemented
33	C₂×C₁₇	—	2	1	no	—
34	C₂×C₁₇	#357	1	1	yes	characteristic
35	D₁₇	—	16	2	no	—
36	C₂×D₈	—	34	1	no	—
37	D₁₆	—	17	2	no	—
38	D₁₆	—	34	1	no	—
39	C₂×C₁₆	C₂×D₁₇	1	1	yes	characteristic
40	#108	—	2	1	no	—
41	C₃₂	—	34	1	no	—
42	C₁₇	#1419	1	1	yes	characteristic, complemented, Sylow
43	C₂×D₄	—	68	1	no	—
44	D₈	—	34	2	no	—
45	D₈	—	68	1	no	—
46	M₁₆	—	4	1	no	—
47	C₂×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
48	C₁₆	C₄×D₁₇	1	1	yes	characteristic
49	C₁₆	D₆₈	1	1	yes	characteristic
50	C₂³	—	136	1	no	—
51	D₄	—	68	2	no	—
52	D₄	—	136	1	no	—
53	C₂×C₄	#63228	1	1	yes	characteristic
54	C₈	—	8	1	no	—
55	C₈	#63227	1	1	yes	characteristic
56	C₈	#63230	1	1	yes	characteristic
57	C₂²	—	272	2	no	—
58	C₂²	—	136	2	no	—
59	C₂²	#10587499	1	1	yes	characteristic
60	C₄	#10587480	1	1	yes	characteristic
61	C₄	#10587497	1	1	yes	characteristic
62	C₂	—	2	1	no	—
63	C₂	—	272	2	no	—
64	C₂	#11762609	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#11762609
commutator	C₁₆×C₁₇	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₇
Fitting	C₁₇×#368	C₂
socle	C₂×C₁₇	#357
prime radical	C₂×D₂₇₂	C₂
cosocle	C₂×C₁₆×C₁₇	C₂²
hypercenter	#368	D₁₇
hypocenter	C₁₇	#1419
LCS	G ▷ C₁₆×C₁₇ ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #368

Complements

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

N	H ≅ G/N	type
2. #11762643	C₂	semidirect
3. C₁₇×#368	C₂	semidirect
32. #368	D₁₇	semidirect
42. C₁₇	#1419	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₂₇₂	2
#11762643	2
C₁₇×#368	2
#1419	17