Subgroups — #17949597

Order 4352.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14543922	C₂	1	2	yes	maximal, complemented
2	#14528390	C₂	1	1	yes	maximal, characteristic
3	#11763720	—	2	2	no	—
4	#11763189	C₂²	1	1	yes	characteristic
5	#11763186	C₄	1	1	yes	characteristic
6	#11763156	—	2	2	no	—
7	C₈²×C₁₇	C₄	1	1	yes	characteristic
8	C₂×Dic₆₈	—	4	1	no	—
9	C₂×D₁₃₆	—	4	1	no	—
10	#10587560	—	2	1	no	—
11	#10587555	D₄	1	2	yes	—
12	#10587545	—	4	2	no	—
13	C₄×C₈×C₁₇	—	2	1	no	—
14	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
15	#10587492	—	4	4	no	—
16	C₂×D₆₈	—	4	2	no	—
17	C₂×D₆₈	—	4	1	no	—
18	C₂×Dic₃₄	—	2	2	no	—
19	#63247	—	8	2	no	—
20	C₂×C₈×C₁₇	—	2	1	no	—
21	C₂×C₈×C₁₇	—	2	2	no	—
22	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₄⋊Dic₁₇	—	4	2	no	—
24	Dic₆₈	—	8	2	no	—
25	D₁₃₆	—	8	2	no	—
26	#7531	—	17	1	no	maximal, Sylow
27	C₂²×D₁₇	—	8	1	no	—
28	C₂×C₄×C₁₇	—	2	1	no	—
29	C₂×C₄×C₁₇	#99	1	1	yes	characteristic
30	C₂×Dic₁₇	—	4	2	no	—
31	D₆₈	—	8	4	no	—
32	D₆₈	—	8	2	no	—
33	Dic₃₄	—	8	2	no	—
34	#3624	—	16	2	no	—
35	C₈×C₁₇	—	2	2	no	—
36	C₈×C₁₇	—	4	2	no	—
37	#1709	—	17	1	no	—
38	#1411	—	17	2	no	—
39	C₂²×C₁₇	#353	1	1	yes	characteristic
40	C₂×D₁₇	—	16	2	no	—
41	C₂×D₁₇	—	16	1	no	—
42	C₄×C₁₇	—	2	2	no	—
43	C₄×C₁₇	—	2	1	no	—
44	Dic₁₇	—	8	2	no	—
45	#494	—	17	1	no	—
46	#493	—	17	1	no	—
47	#486	—	34	2	no	—
48	#356	—	34	2	no	—
49	C₈²	#595	1	1	yes	characteristic
50	C₂×C₁₇	—	2	1	no	—
51	C₂×C₁₇	#1408	1	1	yes	characteristic
52	D₁₇	—	32	1	no	—
53	C₂×Q₁₆	—	68	1	no	—
54	C₂×D₈	—	68	1	no	—
55	C₄⋊Q₈	—	17	2	no	—
56	#127	—	34	1	no	—
57	#102	—	68	4	no	—
58	#101	—	68	2	no	—
59	C₄×C₈	—	2	1	no	—
60	C₄×C₈	C₂×#595	1	1	yes	characteristic
61	C₁₇	#7531	1	1	yes	characteristic, complemented, Sylow
62	C₂×Q₈	—	34	2	no	—
63	C₂×D₄	—	68	2	no	—
64	C₂×D₄	—	68	1	no	—
65	Q₁₆	—	136	2	no	—
66	D₈	—	136	2	no	—
67	M₁₆	—	136	2	no	—
68	C₂×C₈	—	2	1	no	—
69	C₂×C₈	—	2	2	no	—
70	C₄⋊C₄	—	68	2	no	—
71	C₄²	C₂²⋊#595	1	1	yes	characteristic
72	C₂³	—	136	1	no	—
73	Q₈	—	136	2	no	—
74	D₄	—	136	2	no	—
75	D₄	—	136	4	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	68	2	no	—
78	C₂×C₄	#10587539	1	1	yes	characteristic
79	C₈	—	2	2	no	—
80	C₈	—	4	2	no	—
81	C₈	—	272	2	no	—
82	C₂²	—	272	2	no	—
83	C₂²	—	272	1	no	—
84	C₂²	#11763714	1	1	yes	characteristic
85	C₄	—	2	2	no	—
86	C₄	—	2	1	no	—
87	C₄	—	136	2	no	—
88	C₂	—	2	1	no	—
89	C₂	—	544	1	no	—
90	C₂	#14543916	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#14543916
commutator	C₄×C₈×C₁₇	C₂×C₄
Frattini	C₄×C₈	C₂×#595
Fitting	C₈²×C₁₇	C₄
socle	C₂×C₁₇	#1408
prime radical	#11763186	C₄
cosocle	#11763189	C₂²
hypercenter	C₈²	#595
hypocenter	C₁₇	#7531
LCS	G ▷ C₄×C₈×C₁₇ ▷ C₄²×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₈²

Complements

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

N	H ≅ G/N	type
1. #14543922	C₂	semidirect
61. C₁₇	#7531	semidirect

Maximal subgroups

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

subgroup	index
#14543922	2
#14528390	2
#7531	17