Subgroups — #14545181

Order 2176.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₁₆×#595	C₂	1	2	yes	maximal
2	#11763736	C₂	1	1	yes	maximal, characteristic
3	#11763722	C₂	1	2	yes	maximal
4	#11763696	C₂	1	1	yes	maximal, characteristic
5	Q₃₂×D₁₇	C₂	1	1	yes	maximal, characteristic
6	Q₈×#595	—	2	2	no	—
7	Q₁₆×D₁₇	C₂²	1	2	yes	—
8	#10587549	—	2	2	no	—
9	#10587534	C₂²	1	2	yes	—
10	C₈×#595	C₂²	1	1	yes	characteristic
11	#10587529	C₂²	1	1	yes	characteristic
12	C₁₇×Q₃₂	C₄	1	1	yes	characteristic, complemented
13	#10587500	C₄	1	2	yes	complemented
14	Dic₁₃₆	C₄	1	1	yes	characteristic, complemented
15	C₁₆×D₁₇	C₂²	1	1	yes	characteristic
16	Q₈×D₁₇	—	2	2	no	—
17	C₄⋊#595	—	2	2	no	—
18	C₄⋊#595	—	4	2	no	—
19	C₄×#595	—	4	2	no	—
20	C₄×#595	D₄	1	1	yes	characteristic
21	#63242	D₄	1	1	yes	characteristic
22	Q₁₆×C₁₇	C₂×C₄	1	2	yes	—
23	#63231	—	2	2	no	—
24	Dic₆₈	C₂×C₄	1	2	yes	—
25	C₈×D₁₇	C₂³	1	1	yes	characteristic
26	#63216	—	2	1	no	—
27	C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
28	#63214	C₂×C₄	1	1	yes	characteristic
29	C₂×#595	—	4	2	no	—
30	C₂×#595	D₈	1	2	yes	—
31	Q₈×C₁₇	—	2	2	no	—
32	C₄×D₁₇	—	4	2	no	—
33	C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
34	Dic₃₄	—	2	2	no	—
35	Dic₃₄	—	4	2	no	—
36	#3624	—	2	1	no	—
37	C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
38	#3622	#41	1	1	yes	characteristic
39	#2176	—	17	1	no	maximal, Sylow
40	C₂×D₁₇	C₂×D₈	1	1	yes	characteristic
41	#595	—	2	2	no	—
42	#595	—	8	2	no	—
43	C₄×C₁₇	—	4	2	no	—
44	C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
45	Dic₁₇	—	4	2	no	—
46	Dic₁₇	#135	1	1	yes	characteristic
47	C₂×Q₃₂	—	17	1	no	—
48	C₄×Q₁₆	—	17	2	no	—
49	#367	—	17	1	no	—
50	#358	—	17	2	no	—
51	#346	—	17	1	no	—
52	C₂×C₁₇	C₄×D₈	1	1	yes	characteristic
53	D₁₇	#510	1	2	yes	—
54	C₂×Q₁₆	—	17	2	no	—
55	C₄×Q₈	—	34	2	no	—
56	Q₃₂	—	17	2	no	—
57	Q₃₂	—	17	1	no	—
58	Q₃₂	#595	1	1	yes	characteristic, complemented
59	C₂×C₁₆	—	17	1	no	—
60	C₂×C₁₆	—	17	1	no	—
61	#107	—	17	2	no	—
62	#103	—	34	2	no	—
63	C₄×C₈	—	17	1	no	—
64	C₁₇	#2176	1	1	yes	characteristic, complemented, Sylow
65	C₂×Q₈	—	34	2	no	—
66	Q₁₆	—	17	2	no	—
67	Q₁₆	—	34	2	no	—
68	Q₁₆	C₂×#595	1	2	yes	—
69	C₂×C₈	—	17	1	no	—
70	C₂×C₈	—	17	1	no	—
71	C₄⋊C₄	—	34	2	no	—
72	C₄⋊C₄	—	68	2	no	—
73	C₄²	—	17	1	no	—
74	C₄²	—	68	2	no	—
75	C₁₆	—	17	1	no	—
76	C₁₆	—	34	1	no	—
77	C₁₆	C₂×#595	1	1	yes	characteristic
78	Q₈	—	2	2	no	—
79	Q₈	—	34	2	no	—
80	Q₈	—	68	2	no	—
81	C₂×C₄	—	17	1	no	—
82	C₂×C₄	—	17	2	no	—
83	C₂×C₄	—	68	2	no	—
84	C₂×C₄	—	68	2	no	—
85	C₈	—	17	1	no	—
86	C₈	—	34	1	no	—
87	C₈	C₂²×#595	1	1	yes	characteristic
88	C₂²	—	17	1	no	—
89	C₄	—	4	2	no	—
90	C₄	—	17	1	no	—
91	C₄	—	34	2	no	—
92	C₄	—	68	2	no	—
93	C₄	—	136	2	no	—
94	C₄	D₄×#595	1	1	yes	characteristic
95	C₂	—	17	2	no	—
96	C₂	D₈×#595	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×#595
commutator	C₈×C₁₇	C₂²×C₄
Frattini	C₈	C₂²×#595
Fitting	C₁₇×Q₃₂	C₄
socle	C₂×C₁₇	C₄×D₈
prime radical	C₂×D₁₇	C₂×D₈
cosocle	C₈×D₁₇	C₂³
hypercenter	Q₃₂	#595
hypocenter	C₁₇	#2176
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
12. C₁₇×Q₃₂	C₄	semidirect
13. #10587500	C₄	semidirect
14. Dic₁₃₆	C₄	semidirect
58. Q₃₂	#595	semidirect
64. C₁₇	#2176	semidirect

Maximal subgroups

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

subgroup	index
Q₁₆×#595	2
#11763736	2
#11763722	2
#11763696	2
Q₃₂×D₁₇	2
#2176	17