Subgroups — #14530567

Order 2176.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₆×D₁₇	C₂	1	1	yes	maximal, characteristic
2	#11762846	C₂	1	2	yes	maximal
3	#11762637	C₂	1	1	yes	maximal, characteristic, complemented
4	#11762634	C₂	1	2	yes	maximal, complemented
5	C₁₇×#367	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
7	#10587578	—	2	2	no	—
8	C₂×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
9	C₁₇×#107	C₂²	1	2	yes	—
10	#10587489	C₂²	1	2	yes	—
11	C₂×#63214	C₂²	1	1	yes	characteristic
12	#10587479	—	2	2	no	—
13	C₁₆×D₁₇	C₄	1	4	yes	complemented
14	C₂×C₄×D₁₇	D₄	1	1	yes	characteristic
15	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
16	C₄⋊C₄×C₁₇	—	2	2	no	—
17	#63227	—	4	2	no	—
18	C₄⋊Dic₁₇	—	2	2	no	—
19	C₄×Dic₁₇	—	4	2	no	—
20	C₂×#3622	D₄	1	1	yes	characteristic
21	C₈×D₁₇	C₂×C₄	1	2	yes	—
22	C₈×D₁₇	Q₈	1	2	yes	—
23	C₁₆×C₁₇	C₂×C₄	1	2	yes	—
24	#63214	C₂×C₄	1	2	yes	—
25	C₂²×D₁₇	D₈	1	1	yes	characteristic
26	C₂×C₄×C₁₇	—	4	2	no	—
27	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
28	C₂×Dic₁₇	—	4	2	no	—
29	C₂×Dic₁₇	D₈	1	1	yes	characteristic
30	C₄×D₁₇	C₄⋊C₄	1	2	yes	—
31	C₄×D₁₇	Q₁₆	1	2	yes	—
32	C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
34	#3622	C₄⋊C₄	1	2	yes	—
35	#2158	—	17	1	no	maximal, Sylow
36	C₂²×C₁₇	C₂×D₈	1	1	yes	characteristic
37	C₂×D₁₇	—	2	2	no	—
38	C₂×D₁₇	#107	1	2	yes	—
39	C₄×C₁₇	—	8	2	no	—
40	C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
42	Dic₁₇	—	8	2	no	—
43	Dic₁₇	#107	1	2	yes	—
44	C₂²×C₁₆	—	17	1	no	—
45	#427	—	17	2	no	—
46	#367	—	17	1	no	—
47	#367	D₁₇	1	1	yes	characteristic, complemented
48	#366	—	17	2	no	—
49	C₂×C₁₇	C₂×#107	1	1	yes	characteristic
50	C₂×C₁₇	#508	1	2	yes	—
51	D₁₇	—	2	2	no	—
52	C₂²×C₈	—	17	1	no	—
53	#117	—	34	2	no	—
54	C₂×C₁₆	—	17	1	no	—
55	C₂×C₁₆	—	17	4	no	—
56	C₂×C₁₆	C₂×D₁₇	1	1	yes	characteristic
57	#107	—	17	2	no	—
58	#107	C₂×D₁₇	1	2	yes	—
59	#106	—	34	2	no	—
60	C₁₇	#2158	1	1	yes	characteristic, complemented, Sylow
61	C₂²×C₄	—	17	1	no	—
62	C₂×C₈	—	17	1	no	—
63	C₂×C₈	—	17	2	no	—
64	C₂×C₈	—	17	2	no	—
65	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
66	C₄⋊C₄	—	2	2	no	—
67	C₄⋊C₄	—	34	2	no	—
68	C₂²⋊C₄	—	68	2	no	—
69	C₄²	—	68	2	no	—
70	C₁₆	—	17	2	no	—
71	C₁₆	C₄×D₁₇	1	2	yes	—
72	C₂³	—	17	1	no	—
73	C₂×C₄	—	4	2	no	—
74	C₂×C₄	—	17	1	no	—
75	C₂×C₄	—	17	2	no	—
76	C₂×C₄	—	17	2	no	—
77	C₂×C₄	—	68	2	no	—
78	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
79	C₈	—	17	2	no	—
80	C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
81	C₈	Q₈×D₁₇	1	1	yes	characteristic
82	C₂²	—	17	2	no	—
83	C₂²	—	34	2	no	—
84	C₂²	D₈×D₁₇	1	1	yes	characteristic
85	C₄	—	8	2	no	—
86	C₄	—	17	2	no	—
87	C₄	—	136	2	no	—
88	C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
89	C₄	Q₁₆×D₁₇	1	1	yes	characteristic
90	C₂	—	34	2	no	—
91	C₂	#107×D₁₇	1	1	yes	characteristic
92	C₂	#11763240	1	2	yes	—

	subgroup	G/H
center	C₂²	D₈×D₁₇
commutator	C₈×C₁₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₇
Fitting	C₁₇×#367	C₂
socle	C₂²×C₁₇	C₂×D₈
prime radical	C₂²×D₁₇	D₈
cosocle	C₂×C₈×C₁₇	C₂³
hypercenter	#367	D₁₇
hypocenter	C₁₇	#2158
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

Complements

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

N	H ≅ G/N	type
3. #11762637	C₂	semidirect
4. #11762634	C₂	semidirect
5. C₁₇×#367	C₂	semidirect
13. C₁₆×D₁₇	C₄	semidirect
47. #367	D₁₇	semidirect
60. C₁₇	#2158	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₆×D₁₇	2
#11762846	2
#11762637	2
#11762634	2
C₁₇×#367	2
#2158	17