Subgroups — #14545842

Order 2176.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#63241	C₂	1	2	yes	maximal
2	C₈×#3624	C₂	1	1	yes	maximal, characteristic
3	C₂×#63241	—	2	2	no	—
4	C₄×#3624	C₄	1	1	yes	characteristic
5	C₄×#3624	C₂²	1	1	yes	characteristic
6	C₈×Dic₁₇	C₄	1	1	yes	characteristic
7	C₂×#3624	—	2	1	no	—
8	C₂×#3624	D₄	1	1	yes	characteristic
9	C₂×#3624	Q₈	1	1	yes	characteristic
10	#63241	—	4	2	no	—
11	C₂×C₈×C₁₇	C₈	1	1	yes	characteristic
12	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
13	C₂×#3622	C₈	1	1	yes	characteristic
14	C₂×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
15	C₂×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
16	C₂×Dic₁₇	M₁₆	1	1	yes	characteristic
17	#3624	—	2	2	no	—
18	#3624	—	2	1	no	—
19	#3624	SD₁₆	1	2	yes	—
20	C₈×C₁₇	C₁₆	1	2	yes	complemented
21	#3622	—	2	1	no	—
22	#1367	—	17	1	no	maximal, Sylow
23	C₂²×C₁₇	C₄⋊C₈	1	1	yes	characteristic
24	C₄×C₁₇	C₂×C₁₆	1	1	yes	characteristic
25	C₄×C₁₇	M₃₂	1	1	yes	characteristic
26	Dic₁₇	—	2	1	no	—
27	Dic₁₇	#106	1	1	yes	characteristic
28	Dic₁₇	#108	1	1	yes	characteristic
29	C₄⋊C₁₆	—	17	2	no	—
30	C₈²	—	17	1	no	—
31	C₂×C₁₇	#334	1	1	yes	characteristic
32	C₂×C₁₇	C₄⋊C₁₆	1	1	yes	characteristic
33	C₂×C₁₇	#364	1	1	yes	characteristic
34	C₂×C₁₆	—	34	2	no	—
35	C₄×C₈	—	17	1	no	—
36	C₄×C₈	—	17	1	no	—
37	C₄×C₈	—	17	1	no	—
38	C₁₇	#1367	1	1	yes	characteristic, complemented, Sylow
39	C₂×C₈	—	17	1	no	—
40	C₂×C₈	—	17	1	no	—
41	C₂×C₈	—	17	1	no	—
42	C₂×C₈	—	34	1	no	—
43	C₂×C₈	#3633	1	1	yes	characteristic
44	C₄²	—	17	1	no	—
45	C₁₆	—	68	2	no	—
46	C₂×C₄	—	17	1	no	—
47	C₂×C₄	—	17	1	no	—
48	C₂×C₄	C₂×#3633	1	1	yes	characteristic
49	C₈	—	17	2	no	—
50	C₈	—	34	1	no	—
51	C₈	—	34	2	no	—
52	C₈	—	34	1	no	—
53	C₈	#63241	1	2	yes	complemented
54	C₂²	C₄⋊#3633	1	1	yes	characteristic
55	C₄	—	17	1	no	—
56	C₄	—	17	1	no	—
57	C₄	—	34	1	no	—
58	C₄	#10587666	1	1	yes	characteristic
59	C₄	C₂×#63241	1	1	yes	characteristic
60	C₂	#11763898	1	1	yes	characteristic
61	C₂	C₄⋊#63241	1	1	yes	characteristic
62	C₂	#11763913	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₄⋊#3633
commutator	C₄×C₁₇	C₂×C₁₆
Frattini	C₂×C₄	C₂×#3633
Fitting	C₂×C₈×C₁₇	C₈
socle, prime radical	C₂²×C₁₇	C₄⋊C₈
cosocle	C₄×#3624	C₂²
hypercenter	C₂×C₈	#3633
hypocenter	C₁₇	#1367
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
20. C₈×C₁₇	C₁₆	semidirect
38. C₁₇	#1367	semidirect
53. C₈	#63241	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#63241	2
C₈×#3624	2
#1367	17