Subgroups — #14528401

Order 2176.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763214	C₂	1	1	yes	maximal, characteristic
2	#11763212	C₂	1	2	yes	maximal
3	#11763188	C₂	1	2	yes	maximal
4	C₈×#3622	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#500	C₂	1	1	yes	maximal, characteristic
6	C₁₇×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₂×#63231	—	2	4	no	—
8	#10587555	C₂²	1	2	yes	—
9	C₁₇×#107	—	2	2	no	—
10	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
11	#10587489	—	2	2	no	—
12	#10587478	—	2	4	no	—
13	C₄×#3622	C₂²	1	1	yes	characteristic
14	C₄×#3622	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₁₇	—	2	2	no	—
16	C₂×Dic₃₄	—	2	2	no	—
17	C₂×C₈×C₁₇	D₄	1	2	yes	—
18	C₄⋊C₄×C₁₇	—	2	2	no	—
19	C₄⋊C₄×C₁₇	—	2	2	no	—
20	C₄²×C₁₇	C₂³	1	1	yes	characteristic
21	#63231	—	4	8	no	—
22	C₄⋊Dic₁₇	—	2	2	no	—
23	C₄⋊Dic₁₇	—	2	2	no	—
24	C₂×#3622	D₄	1	2	yes	—
25	C₂×#3622	D₄	1	2	yes	—
26	Q₈×C₁₇	—	4	4	no	—
27	C₂×C₄×C₁₇	—	4	2	no	—
28	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
31	C₂×Dic₁₇	—	4	2	no	—
32	Dic₃₄	—	4	4	no	—
33	C₈×C₁₇	Q₁₆	1	4	yes	complemented
34	#3622	D₈	1	4	yes	—
35	#3622	Q₁₆	1	4	yes	complemented
36	#1713	—	17	1	no	maximal, Sylow
37	C₂²×C₁₇	#127	1	1	yes	characteristic
38	C₄×C₁₇	—	8	2	no	—
39	C₄×C₁₇	C₂×D₈	1	2	yes	—
40	C₄×C₁₇	C₂×Q₁₆	1	2	yes	—
41	C₄×C₁₇	C₂×Q₁₆	1	2	yes	—
42	Dic₁₇	—	8	2	no	—
43	#500	—	17	2	no	—
44	#500	—	17	1	no	—
45	#500	D₁₇	1	1	yes	characteristic
46	#494	—	17	2	no	—
47	C₈²	—	17	1	no	—
48	C₂×C₁₇	#493	1	1	yes	characteristic
49	C₂×C₁₇	#494	1	1	yes	characteristic
50	C₂×C₁₇	#494	1	1	yes	characteristic
51	C₂×Q₁₆	—	34	4	no	—
52	C₄⋊Q₈	—	17	2	no	—
53	C₄⋊Q₈	C₂×D₁₇	1	2	yes	—
54	#107	—	2	2	no	—
55	#107	—	34	4	no	—
56	#107	—	34	2	no	—
57	C₄×C₈	—	17	1	no	—
58	C₄×C₈	—	17	1	no	—
59	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
60	C₁₇	#1713	1	1	yes	characteristic, complemented, Sylow
61	C₂×Q₈	—	2	2	no	—
62	C₂×Q₈	—	34	2	no	—
63	Q₁₆	—	68	8	no	—
64	C₂×C₈	—	17	2	no	—
65	C₂×C₈	—	17	2	no	—
66	C₂×C₈	C₂²⋊D₁₇	1	2	yes	—
67	C₄⋊C₄	—	2	2	no	—
68	C₄⋊C₄	—	2	2	no	—
69	C₄⋊C₄	—	34	2	no	—
70	C₄⋊C₄	—	34	2	no	—
71	C₄²	C₂²×D₁₇	1	1	yes	characteristic
72	Q₈	—	4	4	no	—
73	Q₈	—	68	4	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	68	2	no	—
76	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
77	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
78	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
79	C₈	—	17	4	no	—
80	C₈	—	17	4	no	—
81	C₈	#63231	1	4	yes	complemented
82	C₂²	#10587626	1	1	yes	characteristic
83	C₄	—	8	2	no	—
84	C₄	—	136	2	no	—
85	C₄	D₈×D₁₇	1	2	yes	—
86	C₄	Q₁₆×D₁₇	1	2	yes	—
87	C₄	C₂×#63231	1	2	yes	—
88	C₂	#11763184	1	1	yes	characteristic
89	C₂	#11763187	1	1	yes	characteristic
90	C₂	#11763188	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#10587626
commutator, cosocle	C₄²×C₁₇	C₂³
Frattini	C₄²	C₂²×D₁₇
Fitting	C₁₇×#500	C₂
socle, prime radical	C₂²×C₁₇	#127
hypercenter	#500	D₁₇
hypocenter	C₁₇	#1713
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

Complements

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

N	H ≅ G/N	type
33. C₈×C₁₇	Q₁₆	semidirect
35. #3622	Q₁₆	semidirect
60. C₁₇	#1713	semidirect
81. C₈	#63231	semidirect

Maximal subgroups

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

subgroup	index
#11763214	2
#11763212	2
#11763188	2
C₈×#3622	2
C₁₇×#500	2
#1713	17