Subgroups — #17948416

Order 4352.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14543813	C₂	1	2	yes	maximal
2	C₁₆×#3622	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#63216	—	2	2	no	—
4	C₄×#63214	C₄	1	1	yes	characteristic
5	C₈×#3622	C₂²	1	1	yes	characteristic
6	C₄×C₁₆×C₁₇	C₄	1	1	yes	characteristic
7	C₂×#63216	—	4	2	no	—
8	C₂×C₁₆×C₁₇	—	2	1	no	—
9	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
10	C₂×#63214	C₈	1	1	yes	characteristic
11	C₂×#63214	C₈	1	1	yes	characteristic
12	C₄×#3622	D₄	1	1	yes	characteristic
13	C₄×#3622	Q₈	1	1	yes	characteristic
14	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
15	C₂×C₈×C₁₇	M₁₆	1	1	yes	characteristic
16	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
17	C₂×#3622	—	2	1	no	—
18	C₂×#3622	D₈	1	1	yes	characteristic
19	C₂×#3622	Q₁₆	1	1	yes	characteristic
20	#63216	—	8	2	no	—
21	C₁₆×C₁₇	—	2	2	no	—
22	#63214	—	2	1	no	—
23	#63214	C₁₆	1	2	yes	complemented
24	#7101	—	17	1	no	maximal, Sylow
25	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
26	C₂×C₄×C₁₇	#107	1	1	yes	characteristic
27	C₂×C₄×C₁₇	#108	1	1	yes	characteristic
28	C₈×C₁₇	—	2	1	no	—
29	C₈×C₁₇	C₂×C₁₆	1	1	yes	characteristic
30	C₈×C₁₇	M₃₂	1	1	yes	characteristic
31	#3622	—	2	2	no	—
32	#3622	—	2	1	no	—
33	#3622	SD₃₂	1	2	yes	—
34	#1370	—	17	2	no	—
35	C₈×C₁₆	—	17	1	no	—
36	C₂²×C₁₇	#335	1	1	yes	characteristic
37	C₄×C₁₇	—	2	1	no	—
38	C₄×C₁₇	C₄⋊C₁₆	1	1	yes	characteristic
39	C₄×C₁₇	#364	1	1	yes	characteristic
40	C₄×C₁₇	#367	1	1	yes	characteristic
41	C₄×C₁₇	#368	1	1	yes	characteristic
42	C₄⋊C₁₆	—	34	2	no	—
43	C₄×C₁₆	—	17	1	no	—
44	C₄×C₁₆	#595	1	1	yes	characteristic
45	C₈²	—	17	1	no	—
46	C₂×C₁₇	#1370	1	1	yes	characteristic
47	C₂×C₁₇	#1372	1	1	yes	characteristic
48	C₂×C₁₇	#1373	1	1	yes	characteristic
49	C₂×C₁₆	—	2	1	no	—
50	C₂×C₁₆	—	17	1	no	—
51	C₂×C₁₆	—	17	1	no	—
52	C₂×C₁₆	—	68	2	no	—
53	C₄×C₈	—	17	1	no	—
54	C₄×C₈	—	17	1	no	—
55	C₄×C₈	C₂×#595	1	1	yes	characteristic
56	C₁₇	#7101	1	1	yes	characteristic, complemented, Sylow
57	C₂×C₈	—	17	1	no	—
58	C₂×C₈	—	17	1	no	—
59	C₂×C₈	—	34	1	no	—
60	C₂×C₈	#63242	1	1	yes	characteristic
61	C₂×C₈	#63247	1	1	yes	characteristic
62	C₄²	C₄⋊#595	1	1	yes	characteristic
63	C₁₆	—	2	2	no	—
64	C₁₆	—	17	2	no	—
65	C₁₆	—	34	1	no	—
66	C₁₆	—	136	2	no	—
67	C₂×C₄	#10587534	1	1	yes	characteristic
68	C₂×C₄	#10587535	1	1	yes	characteristic
69	C₂×C₄	#10587544	1	1	yes	characteristic
70	C₈	—	2	1	no	—
71	C₈	—	17	2	no	—
72	C₈	—	34	2	no	—
73	C₈	—	34	1	no	—
74	C₈	#10587529	1	1	yes	characteristic
75	C₈	#10587530	1	1	yes	characteristic
76	C₂²	#11763673	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	#11763730	1	1	yes	characteristic
79	C₄	#11763732	1	1	yes	characteristic
80	C₄	#11763736	1	1	yes	characteristic
81	C₄	#11763737	1	1	yes	characteristic
82	C₂	#14543814	1	1	yes	characteristic
83	C₂	#14543818	1	1	yes	characteristic
84	C₂	#14543820	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10587534
commutator	C₈×C₁₇	C₂×C₁₆
Frattini	C₄×C₈	C₂×#595
Fitting	C₄×C₁₆×C₁₇	C₄
socle, prime radical	C₂²×C₁₇	#335
cosocle	C₈×#3622	C₂²
hypercenter	C₄×C₁₆	#595
hypocenter	C₁₇	#7101
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
23. #63214	C₁₆	semidirect
56. C₁₇	#7101	semidirect

Maximal subgroups

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

subgroup	index
#14543813	2
C₁₆×#3622	2
#7101	17