Subgroups — #17949369

Order 4352.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14543943	C₂	1	2	yes	maximal
2	#14526700	C₂	1	1	yes	maximal, characteristic
3	#11763741	—	2	2	no	—
4	C₄×#63214	C₂²	1	1	yes	characteristic
5	C₈×#3622	C₄	1	1	yes	characteristic
6	C₄×C₁₆×C₁₇	C₄	1	1	yes	characteristic
7	C₂×C₁₆×C₁₇	C₈	1	1	yes	characteristic
8	C₂×C₁₆×C₁₇	C₈	1	1	yes	characteristic
9	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
10	C₂×#63214	D₄	1	1	yes	characteristic
11	C₂×#63214	Q₈	1	1	yes	characteristic
12	C₄×#3622	—	2	1	no	—
13	#10587467	—	4	2	no	—
14	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
15	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
16	C₄²×C₁₇	M₁₆	1	1	yes	characteristic
17	C₂×#3622	—	2	2	no	—
18	C₁₆×C₁₇	—	2	1	no	—
19	C₁₆×C₁₇	C₁₆	1	2	yes	—
20	#63214	—	2	1	no	—
21	#63214	SD₁₆	1	2	yes	—
22	#7456	—	17	1	no	maximal, Sylow
23	C₂×C₄×C₁₇	—	2	1	no	—
24	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
25	C₈×C₁₇	#106	1	1	yes	characteristic
26	C₈×C₁₇	#108	1	1	yes	characteristic
27	C₈×C₁₇	C₂×C₁₆	1	1	yes	characteristic
28	C₈×C₁₇	M₃₂	1	1	yes	characteristic
29	#3622	—	4	2	no	—
30	#1422	—	17	2	no	—
31	#1311	—	17	1	no	—
32	C₂²×C₁₇	#364	1	1	yes	characteristic
33	C₄×C₁₇	—	2	2	no	—
34	C₄×C₁₇	#334	1	1	yes	characteristic
35	C₄×C₁₇	C₄⋊C₁₆	1	1	yes	characteristic
36	M₆₄	—	34	2	no	—
37	C₄×C₁₆	—	17	1	no	—
38	C₄×C₁₆	#595	1	1	yes	characteristic
39	C₈²	—	17	1	no	—
40	C₂×C₁₇	—	2	1	no	—
41	C₂×C₁₇	#1367	1	1	yes	characteristic
42	C₂×C₁₆	—	17	1	no	—
43	C₂×C₁₆	—	17	1	no	—
44	C₂×C₁₆	#3624	1	1	yes	characteristic
45	C₂×C₁₆	#3624	1	1	yes	characteristic
46	C₄×C₈	—	34	1	no	—
47	C₄×C₈	C₂×#595	1	1	yes	characteristic
48	C₃₂	—	68	2	no	—
49	C₁₇	#7456	1	1	yes	characteristic, complemented, Sylow
50	C₂×C₈	—	34	2	no	—
51	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
52	C₂×C₈	C₂×#3624	1	1	yes	characteristic
53	C₄²	#63243	1	1	yes	characteristic
54	C₁₆	—	2	1	no	—
55	C₁₆	—	17	2	no	—
56	C₁₆	—	34	1	no	—
57	C₁₆	#63216	1	2	yes	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	C₄⋊#3624	1	1	yes	characteristic
60	C₈	—	68	2	no	—
61	C₈	#10587533	1	1	yes	characteristic
62	C₈	#10587535	1	1	yes	characteristic
63	C₈	C₂×#63216	1	1	yes	characteristic
64	C₈	#10587538	1	1	yes	characteristic
65	C₂²	#11763731	1	1	yes	characteristic
66	C₄	—	2	2	no	—
67	C₄	#11763670	1	1	yes	characteristic
68	C₄	C₄⋊#63216	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	#14543806	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#10587533
commutator	C₈×C₁₇	C₂×C₁₆
Frattini	C₄×C₈	C₂×#595
Fitting	C₄×C₁₆×C₁₇	C₄
socle	C₂×C₁₇	#1367
prime radical	C₂²×C₁₇	#364
cosocle	C₄×#63214	C₂²
hypercenter	C₄×C₁₆	#595
hypocenter	C₁₇	#7456
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
49. C₁₇	#7456	semidirect

Maximal subgroups

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

subgroup	index
#14543943	2
#14526700	2
#7456	17