Subgroups — #16949061

Order 4352.

262 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14531059	C₂	1	4	yes	maximal
2	#14528402	C₂	1	2	yes	maximal
3	C₈×C₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
4	#11763214	—	2	2	no	—
5	#11763214	C₂²	1	4	yes	—
6	#11763189	—	2	2	no	—
7	#11762635	—	2	8	no	—
8	C₄×C₁₆×C₁₇	C₂²	1	2	yes	—
9	C₈²×C₁₇	C₂²	1	1	yes	characteristic
10	C₂×Dic₆₈	—	4	4	no	—
11	#10587555	—	2	4	no	—
12	C₂×C₁₆×C₁₇	D₄	1	4	yes	—
13	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
14	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
15	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
16	#10587489	—	2	8	no	—
17	#10587489	—	4	4	no	—
18	C₂×Dic₃₄	—	4	4	no	—
19	C₂×C₈×C₁₇	D₈	1	2	yes	—
20	C₂×C₈×C₁₇	D₈	1	2	yes	—
21	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
22	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
23	C₄⋊Dic₁₇	—	4	4	no	—
24	C₄⋊Dic₁₇	—	4	4	no	—
25	Dic₆₈	—	8	8	no	—
26	C₁₆×C₁₇	Q₁₆	1	8	yes	complemented
27	#12521	—	17	1	no	maximal, Sylow
28	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
29	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
30	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
31	C₂×Dic₁₇	—	8	4	no	—
32	Dic₃₄	—	8	8	no	—
33	C₈×C₁₇	D₁₆	1	4	yes	—
34	C₈×C₁₇	Q₃₂	1	4	yes	—
35	C₈×C₁₇	C₂×Q₁₆	1	4	yes	—
36	#2252	—	17	4	no	—
37	#1713	—	17	2	no	—
38	C₈×C₁₆	D₁₇	1	1	yes	characteristic
39	C₂²×C₁₇	#493	1	1	yes	characteristic
40	C₄×C₁₇	#494	1	2	yes	—
41	C₄×C₁₇	C₂×D₁₆	1	2	yes	—
42	C₄×C₁₇	C₂×Q₃₂	1	2	yes	—
43	Dic₁₇	—	16	4	no	—
44	#500	—	17	4	no	—
45	#500	—	34	2	no	—
46	#494	—	34	2	no	—
47	#366	—	34	8	no	—
48	C₄×C₁₆	C₂×D₁₇	1	2	yes	—
49	C₈²	C₂×D₁₇	1	1	yes	characteristic
50	C₂×C₁₇	#1713	1	1	yes	characteristic
51	C₂×C₁₇	#2246	1	1	yes	characteristic
52	C₂×C₁₇	#2247	1	1	yes	characteristic
53	C₂×Q₁₆	—	68	4	no	—
54	C₄⋊Q₈	—	34	4	no	—
55	C₂×C₁₆	D₆₈	1	4	yes	—
56	#107	—	34	8	no	—
57	#107	—	68	4	no	—
58	C₄×C₈	D₆₈	1	1	yes	characteristic
59	C₄×C₈	D₆₈	1	1	yes	characteristic
60	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
61	C₁₇	#12521	1	1	yes	characteristic, complemented, Sylow
62	C₂×Q₈	—	68	4	no	—
63	Q₁₆	—	136	8	no	—
64	C₂×C₈	D₁₃₆	1	2	yes	—
65	C₂×C₈	D₁₃₆	1	2	yes	—
66	C₂×C₈	C₂×D₆₈	1	2	yes	—
67	C₄⋊C₄	—	68	4	no	—
68	C₄⋊C₄	—	68	4	no	—
69	C₄²	C₂×D₆₈	1	1	yes	characteristic
70	C₁₆	Dic₆₈	1	8	yes	complemented
71	Q₈	—	136	8	no	—
72	C₂×C₄	—	136	4	no	—
73	C₂×C₄	#10587560	1	1	yes	characteristic
74	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
75	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
76	C₈	D₂₇₂	1	4	yes	—
77	C₈	Dic₁₃₆	1	4	yes	—
78	C₈	C₂×Dic₆₈	1	4	yes	—
79	C₂²	#11763186	1	1	yes	characteristic
80	C₄	—	272	4	no	—
81	C₄	#11763189	1	2	yes	—
82	C₄	C₂×D₂₇₂	1	2	yes	—
83	C₄	C₂×Dic₁₃₆	1	2	yes	—
84	C₂	#14528402	1	1	yes	characteristic
85	C₂	#14531034	1	1	yes	characteristic
86	C₂	#14531037	1	1	yes	characteristic

	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	#12521
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
26. C₁₆×C₁₇	Q₁₆	semidirect
61. C₁₇	#12521	semidirect
70. C₁₆	Dic₆₈	semidirect

Maximal subgroups

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

subgroup	index
#14531059	2
#14528402	2
C₈×C₁₆×C₁₇	2
#12521	17