Subgroups — #16949039

Order 4352.

310 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14531037	C₂	1	4	yes	maximal
2	#14528402	C₂	1	2	yes	maximal
3	C₈×C₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₁₃₆	—	2	8	no	—
5	#11763214	—	2	4	no	—
6	#11763189	C₂²	1	4	yes	—
7	C₄×C₁₆×C₁₇	C₂²	1	2	yes	—
8	C₈²×C₁₇	C₂²	1	1	yes	characteristic
9	C₂×Dic₆₈	—	2	8	no	—
10	#10587555	—	2	4	no	—
11	C₂×C₁₆×C₁₇	D₄	1	4	yes	—
12	C₄×C₈×C₁₇	D₄	1	2	yes	—
13	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
14	#10587489	—	4	8	no	—
15	Dic₁₃₆	—	4	16	no	—
16	C₂×Dic₃₄	—	4	4	no	—
17	C₂×C₈×C₁₇	D₈	1	4	yes	—
18	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
19	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
20	C₄⋊Dic₁₇	—	4	8	no	—
21	Dic₆₈	—	4	16	no	—
22	C₁₆×C₁₇	D₈	1	8	yes	—
23	#12516	—	17	1	no	maximal, Sylow
24	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
25	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
26	C₂×Dic₁₇	—	8	4	no	—
27	Dic₃₄	—	8	8	no	—
28	C₈×C₁₇	Q₃₂	1	8	yes	complemented
29	C₈×C₁₇	C₂×D₈	1	4	yes	—
30	#2247	—	17	4	no	—
31	#1713	—	17	2	no	—
32	C₈×C₁₆	D₁₇	1	1	yes	characteristic
33	C₂²×C₁₇	#493	1	1	yes	characteristic
34	C₄×C₁₇	#493	1	2	yes	—
35	C₄×C₁₇	C₂×Q₃₂	1	4	yes	—
36	Dic₁₇	—	16	4	no	—
37	C₂×Q₃₂	—	34	8	no	—
38	#500	—	34	4	no	—
39	#494	—	17	4	no	—
40	C₄×C₁₆	C₂×D₁₇	1	2	yes	—
41	C₈²	C₂×D₁₇	1	1	yes	characteristic
42	C₂×C₁₇	#1712	1	1	yes	characteristic
43	C₂×C₁₇	#2247	1	2	yes	—
44	C₂×Q₁₆	—	34	8	no	—
45	C₄⋊Q₈	—	34	4	no	—
46	Q₃₂	—	68	16	no	—
47	C₂×C₁₆	D₆₈	1	4	yes	—
48	#107	—	68	8	no	—
49	C₄×C₈	D₆₈	1	2	yes	—
50	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
51	C₁₇	#12516	1	1	yes	characteristic, complemented, Sylow
52	C₂×Q₈	—	68	4	no	—
53	Q₁₆	—	68	16	no	—
54	C₂×C₈	D₁₃₆	1	4	yes	—
55	C₂×C₈	C₂×D₆₈	1	2	yes	—
56	C₄⋊C₄	—	68	8	no	—
57	C₄²	C₂×D₆₈	1	1	yes	characteristic
58	C₁₆	D₁₃₆	1	8	yes	—
59	Q₈	—	136	8	no	—
60	C₂×C₄	—	136	4	no	—
61	C₂×C₄	#10587560	1	1	yes	characteristic
62	C₂×C₄	C₂×D₁₃₆	1	2	yes	—
63	C₈	Dic₁₃₆	1	8	yes	complemented
64	C₈	C₂×D₁₃₆	1	4	yes	—
65	C₂²	#11763186	1	1	yes	characteristic
66	C₄	—	272	4	no	—
67	C₄	#11763186	1	2	yes	—
68	C₄	C₂×Dic₁₃₆	1	4	yes	—
69	C₂	#14528399	1	1	yes	characteristic
70	C₂	#14531037	1	2	yes	—

	subgroup	G/H
center	C₂²	#11763186
commutator, cosocle	C₄×C₈×C₁₇	C₂³
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₈×C₁₆×C₁₇	C₂
socle, prime radical	C₂²×C₁₇	#493
hypercenter	C₈×C₁₆	D₁₇
hypocenter	C₁₇	#12516
LCS	G ▷ C₄×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

Complements

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

N	H ≅ G/N	type
28. C₈×C₁₇	Q₃₂	semidirect
51. C₁₇	#12516	semidirect
63. C₈	Dic₁₃₆	semidirect

Maximal subgroups

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

subgroup	index
#14531037	2
#14528402	2
C₈×C₁₆×C₁₇	2
#12516	17