Subgroups — #14530670

Order 2176.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11762926	C₂	1	2	yes	maximal, complemented
2	#11762639	C₂	1	1	yes	maximal, characteristic, complemented
3	#11762616	C₂	1	2	yes	maximal, complemented
4	#11762584	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#508	C₂	1	1	yes	maximal, characteristic
6	C₁₇×#135	C₂²	1	2	yes	—
7	#10587634	—	2	2	no	—
8	C₁₇×Q₃₂	C₄	1	1	yes	characteristic, complemented
9	C₁₇×SD₃₂	C₄	1	2	yes	complemented
10	C₁₇×D₁₆	C₄	1	1	yes	characteristic, complemented
11	C₂×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
12	#10587508	—	2	2	no	—
13	#10587490	C₂²	1	2	yes	—
14	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
15	C₂×#63214	C₂²	1	1	yes	characteristic
16	#41×C₁₇	—	2	2	no	—
17	Q₁₆×C₁₇	C₂×C₄	1	2	yes	—
18	SD₁₆×C₁₇	—	2	2	no	—
19	D₈×C₁₇	C₂×C₄	1	2	yes	—
20	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
21	C₄×Dic₁₇	D₄	1	1	yes	characteristic
22	#63223	—	2	2	no	—
23	#63223	—	4	2	no	—
24	C₂×#3622	—	4	2	no	—
25	C₂×#3622	D₄	1	1	yes	characteristic
26	C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
27	C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
28	#63214	—	2	1	no	—
29	Q₈×C₁₇	—	2	2	no	—
30	D₄×C₁₇	—	2	2	no	—
31	D₄×C₁₇	—	4	2	no	—
32	C₂×C₄×C₁₇	—	4	2	no	—
33	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
34	C₂×Dic₁₇	—	2	1	no	—
35	C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
36	C₈×C₁₇	#41	1	1	yes	characteristic
37	#3622	—	4	2	no	—
38	#3622	D₈	1	2	yes	—
39	#2179	—	17	1	no	maximal, Sylow
40	C₂²×C₁₇	—	4	2	no	—
41	C₂²×C₁₇	#135	1	1	yes	characteristic
42	C₄×C₁₇	—	4	2	no	—
43	C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
44	C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
45	Dic₁₇	—	4	1	no	—
46	#508	D₁₇	1	1	yes	characteristic
47	#443	—	17	2	no	—
48	#368	—	17	1	no	—
49	#359	—	17	2	no	—
50	#346	—	17	1	no	—
51	C₂×C₁₇	—	2	1	no	—
52	C₂×C₁₇	—	8	2	no	—
53	C₂×C₁₇	C₄×D₈	1	1	yes	characteristic
54	#135	C₂×D₁₇	1	2	yes	—
55	#131	—	34	2	no	—
56	Q₃₂	Dic₁₇	1	1	yes	characteristic, complemented
57	SD₃₂	Dic₁₇	1	2	yes	complemented
58	D₁₆	Dic₁₇	1	1	yes	characteristic, complemented
59	C₂×C₁₆	—	17	1	no	—
60	C₂×C₁₆	C₂×D₁₇	1	1	yes	characteristic
61	#108	—	17	2	no	—
62	#104	—	34	2	no	—
63	C₄×C₈	—	17	1	no	—
64	C₁₇	#2179	1	1	yes	characteristic, complemented, Sylow
65	#41	—	2	2	no	—
66	Q₁₆	C₂×Dic₁₇	1	2	yes	—
67	SD₁₆	—	2	2	no	—
68	D₈	C₂×Dic₁₇	1	2	yes	—
69	M₁₆	—	34	2	no	—
70	M₁₆	—	68	2	no	—
71	C₂×C₈	—	17	1	no	—
72	C₂×C₈	—	68	2	no	—
73	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
74	C₄²	—	17	1	no	—
75	C₁₆	—	34	1	no	—
76	C₁₆	C₂×Dic₁₇	1	1	yes	characteristic
77	C₁₆	C₂×Dic₁₇	1	1	yes	characteristic
78	Q₈	—	2	2	no	—
79	D₄	—	2	2	no	—
80	D₄	—	4	2	no	—
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	—	34	1	no	—
83	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
84	C₈	—	17	2	no	—
85	C₈	—	68	2	no	—
86	C₈	#63254	1	1	yes	characteristic
87	C₈	C₂²×Dic₁₇	1	1	yes	characteristic
88	C₂²	—	4	2	no	—
89	C₂²	#10587598	1	1	yes	characteristic
90	C₄	—	4	2	no	—
91	C₄	—	68	1	no	—
92	C₄	D₈×D₁₇	1	1	yes	characteristic
93	C₄	D₄×Dic₁₇	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	8	2	no	—
96	C₂	D₈×Dic₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₈×D₁₇
commutator	C₈×C₁₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₇
Fitting, prime radical	C₁₇×#508	C₂
socle	C₂×C₁₇	C₄×D₈
cosocle	C₂×C₈×C₁₇	C₂³
hypercenter	#508	D₁₇
hypocenter	C₁₇	#2179
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #508

Complements

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

N	H ≅ G/N	type
1. #11762926	C₂	semidirect
2. #11762639	C₂	semidirect
3. #11762616	C₂	semidirect
4. #11762584	C₂	semidirect
8. C₁₇×Q₃₂	C₄	semidirect
9. C₁₇×SD₃₂	C₄	semidirect
10. C₁₇×D₁₆	C₄	semidirect
56. Q₃₂	Dic₁₇	semidirect
57. SD₃₂	Dic₁₇	semidirect
58. D₁₆	Dic₁₇	semidirect
64. C₁₇	#2179	semidirect

Maximal subgroups

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

subgroup	index
#11762926	2
#11762639	2
#11762616	2
#11762584	2
C₁₇×#508	2
#2179	17