Subgroups — #16949032

Order 4352.

246 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14531012	C₂	1	4	yes	maximal
2	#14528402	C₂	1	1	yes	maximal, characteristic
3	#14528394	C₂	1	1	yes	maximal, characteristic
4	C₁₇×#1311	C₂	1	1	yes	maximal, characteristic
5	#11763214	—	2	2	no	—
6	#11763214	C₂²	1	2	yes	—
7	#11763189	C₂²	1	2	yes	—
8	#11763160	—	2	2	no	—
9	#11762614	—	2	8	no	—
10	C₄×C₁₆×C₁₇	C₂²	1	2	yes	—
11	C₈²×C₁₇	C₂²	1	1	yes	characteristic
12	C₂×Dic₆₈	—	2	4	no	—
13	#10587555	—	2	2	no	—
14	#10587555	—	2	2	no	—
15	C₂×C₁₆×C₁₇	D₄	1	4	yes	—
16	C₄×C₈×C₁₇	D₄	1	2	yes	—
17	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
18	#10587489	—	2	4	no	—
19	#10587489	—	4	4	no	—
20	#10587487	—	4	4	no	—
21	C₂×Dic₃₄	—	4	2	no	—
22	C₂×Dic₃₄	—	4	2	no	—
23	C₂×C₈×C₁₇	D₈	1	4	yes	—
24	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
25	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
26	C₄⋊Dic₁₇	—	4	2	no	—
27	C₄⋊Dic₁₇	—	4	2	no	—
28	C₄⋊Dic₁₇	—	4	4	no	—
29	Dic₆₈	—	4	8	no	—
30	C₁₆×C₁₇	—	2	4	no	—
31	#12514	—	17	1	no	maximal, Sylow
32	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
33	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
34	C₂×Dic₁₇	—	8	2	no	—
35	C₂×Dic₁₇	—	8	2	no	—
36	Dic₃₄	—	8	4	no	—
37	Dic₃₄	—	8	4	no	—
38	C₈×C₁₇	—	2	4	no	—
39	C₈×C₁₇	#135	1	4	yes	—
40	#2241	—	17	4	no	—
41	#1713	—	17	1	no	—
42	#1710	—	17	1	no	—
43	#1311	D₁₇	1	1	yes	characteristic
44	C₂²×C₁₇	#493	1	1	yes	characteristic
45	C₄×C₁₇	#495	1	2	yes	—
46	C₄×C₁₇	#510	1	4	yes	—
47	Dic₁₇	—	16	2	no	—
48	Dic₁₇	—	16	2	no	—
49	#500	—	17	2	no	—
50	#500	—	34	2	no	—
51	#494	—	17	2	no	—
52	#487	—	34	2	no	—
53	#358	—	34	8	no	—
54	C₄×C₁₆	C₂×D₁₇	1	2	yes	—
55	C₈²	C₂×D₁₇	1	1	yes	characteristic
56	C₂×C₁₇	#1709	1	1	yes	characteristic
57	C₂×C₁₇	#2251	1	2	yes	—
58	C₂×Q₁₆	—	34	4	no	—
59	C₄⋊Q₈	—	34	2	no	—
60	C₄⋊Q₈	—	34	2	no	—
61	C₂×C₁₆	D₆₈	1	4	yes	—
62	#107	—	34	4	no	—
63	#107	—	68	4	no	—
64	#103	—	68	4	no	—
65	C₄×C₈	D₆₈	1	2	yes	—
66	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
67	C₁₇	#12514	1	1	yes	characteristic, complemented, Sylow
68	C₂×Q₈	—	68	2	no	—
69	C₂×Q₈	—	68	2	no	—
70	Q₁₆	—	68	8	no	—
71	C₂×C₈	D₁₃₆	1	4	yes	—
72	C₂×C₈	C₂×D₆₈	1	2	yes	—
73	C₄⋊C₄	—	68	2	no	—
74	C₄⋊C₄	—	68	2	no	—
75	C₄⋊C₄	—	68	4	no	—
76	C₄²	C₂×D₆₈	1	1	yes	characteristic
77	C₁₆	—	2	4	no	—
78	Q₈	—	136	4	no	—
79	Q₈	—	136	4	no	—
80	C₂×C₄	—	136	2	no	—
81	C₂×C₄	—	136	2	no	—
82	C₂×C₄	#10587560	1	1	yes	characteristic
83	C₂×C₄	C₂×D₁₃₆	1	2	yes	—
84	C₈	—	2	4	no	—
85	C₈	#10587590	1	4	yes	—
86	C₂²	#11763186	1	1	yes	characteristic
87	C₄	—	272	2	no	—
88	C₄	—	272	2	no	—
89	C₄	#11763193	1	2	yes	—
90	C₄	#11763252	1	4	yes	—
91	C₂	#14528390	1	1	yes	characteristic
92	C₂	#14531055	1	2	yes	—

	subgroup	G/H
center	C₂²	#11763186
commutator, cosocle	C₄×C₈×C₁₇	C₂³
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₁₇×#1311	C₂
socle, prime radical	C₂²×C₁₇	#493
hypercenter	#1311	D₁₇
hypocenter	C₁₇	#12514
LCS	G ▷ C₄×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

Complements

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

N	H ≅ G/N	type
67. C₁₇	#12514	semidirect

Maximal subgroups

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

subgroup	index
#14531012	2
#14528402	2
#14528394	2
C₁₇×#1311	2
#12514	17