Subgroups — #16955322

Order 4352.

230 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₂₇₂	C₂	1	4	yes	maximal
2	#14531037	C₂	1	2	yes	maximal
3	C₄×C₁₇×C₃₂	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₁₃₆	—	2	2	no	—
5	C₂×Dic₁₃₆	C₂²	1	4	yes	—
6	#11763189	—	2	2	no	—
7	Dic₂₇₂	—	2	8	no	—
8	C₂×C₁₇×C₃₂	C₂²	1	2	yes	—
9	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
10	C₂×Dic₆₈	—	2	4	no	—
11	C₂×Dic₆₈	—	4	2	no	—
12	#10587555	—	4	2	no	—
13	C₂×C₁₆×C₁₇	D₄	1	1	yes	characteristic
14	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
15	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
16	Dic₁₃₆	—	2	8	no	—
17	Dic₁₃₆	—	4	4	no	—
18	C₁₇×C₃₂	D₄	1	4	yes	—
19	C₂×Dic₃₄	—	4	4	no	—
20	C₂×C₈×C₁₇	D₈	1	1	yes	characteristic
21	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
22	C₄²×C₁₇	D₈	1	1	yes	characteristic
23	C₄⋊Dic₁₇	—	8	4	no	—
24	Dic₆₈	—	4	4	no	—
25	Dic₆₈	—	4	4	no	—
26	Dic₆₈	—	8	4	no	—
27	C₁₆×C₁₇	D₈	1	2	yes	—
28	C₁₆×C₁₇	C₂×D₄	1	2	yes	—
29	#13726	—	17	1	no	maximal, Sylow
30	C₂×C₄×C₁₇	D₁₆	1	2	yes	—
31	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
32	C₂×Dic₁₇	—	8	4	no	—
33	Dic₃₄	—	8	4	no	—
34	Dic₃₄	—	8	4	no	—
35	C₈×C₁₇	D₁₆	1	2	yes	—
36	C₈×C₁₇	#127	1	1	yes	characteristic
37	C₈×C₁₇	C₂×D₈	1	1	yes	characteristic
38	C₂×Q₆₄	—	17	4	no	—
39	#2247	—	17	2	no	—
40	C₄×C₃₂	D₁₇	1	1	yes	characteristic
41	C₂²×C₁₇	C₂×D₁₆	1	1	yes	characteristic
42	C₄×C₁₇	Q₆₄	1	4	yes	complemented
43	C₄×C₁₇	#493	1	1	yes	characteristic
44	C₄×C₁₇	C₂×D₁₆	1	1	yes	characteristic
45	Dic₁₇	—	16	4	no	—
46	C₂×Q₃₂	—	17	4	no	—
47	C₂×Q₃₂	—	34	2	no	—
48	#494	—	34	2	no	—
49	Q₆₄	—	34	8	no	—
50	C₂×C₃₂	C₂×D₁₇	1	2	yes	—
51	C₄×C₁₆	C₂×D₁₇	1	1	yes	characteristic
52	C₂×C₁₇	#2246	1	1	yes	characteristic
53	C₂×C₁₇	C₂×Q₆₄	1	2	yes	—
54	C₂×Q₁₆	—	34	4	no	—
55	C₂×Q₁₆	—	68	2	no	—
56	C₄⋊Q₈	—	68	2	no	—
57	Q₃₂	—	34	8	no	—
58	Q₃₂	—	68	4	no	—
59	C₂×C₁₆	D₆₈	1	1	yes	characteristic
60	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
61	C₄×C₈	D₆₈	1	1	yes	characteristic
62	C₃₂	D₆₈	1	4	yes	—
63	C₁₇	#13726	1	1	yes	characteristic, complemented, Sylow
64	C₂×Q₈	—	68	4	no	—
65	Q₁₆	—	68	4	no	—
66	Q₁₆	—	68	4	no	—
67	Q₁₆	—	136	4	no	—
68	C₂×C₈	D₁₃₆	1	1	yes	characteristic
69	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
70	C₄⋊C₄	—	136	4	no	—
71	C₄²	D₁₃₆	1	1	yes	characteristic
72	C₁₆	D₁₃₆	1	2	yes	—
73	C₁₆	C₂×D₆₈	1	2	yes	—
74	Q₈	—	136	4	no	—
75	Q₈	—	136	4	no	—
76	C₂×C₄	—	136	4	no	—
77	C₂×C₄	D₂₇₂	1	2	yes	—
78	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
79	C₈	D₂₇₂	1	2	yes	—
80	C₈	#10587560	1	1	yes	characteristic
81	C₈	C₂×D₁₃₆	1	1	yes	characteristic
82	C₂²	C₂×D₂₇₂	1	1	yes	characteristic
83	C₄	—	272	4	no	—
84	C₄	Dic₂₇₂	1	4	yes	complemented
85	C₄	#11763186	1	1	yes	characteristic
86	C₄	C₂×D₂₇₂	1	1	yes	characteristic
87	C₂	#14531034	1	1	yes	characteristic
88	C₂	C₂×Dic₂₇₂	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₂₇₂
commutator, cosocle	C₂×C₁₆×C₁₇	C₂³
Frattini	C₂×C₁₆	C₂²×D₁₇
Fitting	C₄×C₁₇×C₃₂	C₂
socle, prime radical	C₂²×C₁₇	C₂×D₁₆
hypercenter	C₄×C₃₂	D₁₇
hypocenter	C₁₇	#13726
LCS	G ▷ C₂×C₁₆×C₁₇ ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
42. C₄×C₁₇	Q₆₄	semidirect
63. C₁₇	#13726	semidirect
84. C₄	Dic₂₇₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₂₇₂	2
#14531037	2
C₄×C₁₇×C₃₂	2
#13726	17