Subgroups — #17961644

Order 4352.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14543822	C₂	1	6	yes	maximal
2	#14528471	C₂	1	1	yes	maximal, characteristic
3	C₂²×#63216	C₂²	1	4	yes	—
4	C₂×#10587474	C₄	1	3	yes	—
5	C₂×C₄×#3622	C₂²	1	3	yes	—
6	C₁₇×#376	C₄	1	1	yes	characteristic
7	C₂×C₄²×C₁₇	C₂×C₄	1	3	yes	—
8	C₂²×#3622	C₂×C₄	1	3	yes	—
9	C₂²×#3622	C₂³	1	1	yes	characteristic
10	C₂×#63216	—	2	12	no	—
11	C₁₇×#95	C₈	1	4	yes	—
12	#10587474	—	2	6	no	—
13	C₄×#3622	—	2	6	no	—
14	C₂²×C₄×C₁₇	C₂×C₈	1	6	yes	—
15	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
16	C₄²×C₁₇	—	2	3	no	—
17	C₄²×C₁₇	—	2	3	no	—
18	C₂×#3622	—	2	3	no	—
19	C₂×#3622	—	2	3	no	—
20	C₂×#3622	M₁₆	1	6	yes	—
21	C₂×#3622	#41	1	6	yes	—
22	#63216	—	4	4	no	—
23	#12895	—	17	1	no	maximal, Sylow
24	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
25	C₂×C₄×C₁₇	—	2	6	no	—
26	C₂×C₄×C₁₇	—	2	12	no	—
27	C₂×C₄×C₁₇	#117	1	3	yes	—
28	C₂×C₄×C₁₇	C₂×M₁₆	1	3	yes	—
29	#3622	—	2	6	no	—
30	#3622	#126	1	4	yes	—
31	#1726	—	17	1	no	—
32	#1374	—	17	6	no	—
33	C₂²×C₁₇	#431	1	3	yes	—
34	C₂²×C₁₇	#504	1	4	yes	—
35	C₄×C₁₇	—	4	6	no	—
36	C₄×C₁₇	#383	1	1	yes	characteristic
37	C₄×C₁₇	#432	1	3	yes	—
38	C₂²×C₁₆	—	17	4	no	—
39	C₂×#97	—	17	3	no	—
40	C₂×C₄×C₈	—	17	3	no	—
41	#376	#595	1	1	yes	characteristic
42	C₂×C₁₇	#1839	1	1	yes	characteristic
43	C₂×C₁₇	#2164	1	6	yes	—
44	C₂²×C₈	—	17	1	no	—
45	C₂²×C₈	—	17	3	no	—
46	C₂×C₄²	C₂×#595	1	3	yes	—
47	C₂×C₁₆	—	34	12	no	—
48	#97	—	34	6	no	—
49	C₄×C₈	—	34	6	no	—
50	#95	#3624	1	4	yes	—
51	C₁₇	#12895	1	1	yes	characteristic, complemented, Sylow
52	C₂²×C₄	C₂×#3624	1	6	yes	—
53	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
54	C₂×C₈	—	17	6	no	—
55	C₂×C₈	—	17	6	no	—
56	C₂×C₈	—	34	3	no	—
57	C₂×C₈	—	34	3	no	—
58	C₄²	—	2	3	no	—
59	C₄²	—	2	3	no	—
60	C₁₆	—	68	4	no	—
61	C₂³	C₂²×#3624	1	1	yes	characteristic
62	C₂×C₄	—	2	6	no	—
63	C₂×C₄	—	2	12	no	—
64	C₂×C₄	D₁₇⋊M₁₆	1	3	yes	—
65	C₂×C₄	#10587677	1	3	yes	—
66	C₈	—	17	4	no	—
67	C₈	—	34	6	no	—
68	C₂²	#11763825	1	3	yes	—
69	C₂²	#11763867	1	4	yes	—
70	C₄	—	4	6	no	—
71	C₄	#11763754	1	1	yes	characteristic
72	C₄	#11763832	1	3	yes	—
73	C₂	#14544711	1	1	yes	characteristic
74	C₂	#14545149	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#595
commutator, socle, prime radical	C₂³×C₁₇	C₂²×C₈
Fitting	C₁₇×#376	C₄
cosocle	C₂²×#3622	C₂³
hypercenter	#376	#595
hypocenter	C₁₇	#12895
LCS	G ▷ C₂³×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ #376

Complements

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

N	H ≅ G/N	type
51. C₁₇	#12895	semidirect

Maximal subgroups

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

subgroup	index
#14543822	2
#14528471	2
#12895	17