Subgroups — #14543744

Order 2176.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763698	C₂	1	1	yes	maximal, characteristic
2	#11763670	C₂	1	1	yes	maximal, characteristic
3	Q₁₆×Dic₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₁₆×C₁₇	C₄	1	1	yes	characteristic
5	Q₈×Dic₁₇	—	2	1	no	—
6	C₄⋊#3624	—	2	1	no	—
7	C₂×#63216	—	2	1	no	—
8	#10587506	—	2	1	no	—
9	#10587489	C₄	1	1	yes	characteristic
10	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
11	C₂×Q₈×C₁₇	—	2	1	no	—
12	C₂×#3624	—	4	1	no	—
13	Q₁₆×C₁₇	—	2	1	no	—
14	Q₁₆×C₁₇	C₈	1	2	yes	complemented
15	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
16	C₄⋊Dic₁₇	—	2	1	no	—
17	C₄⋊Dic₁₇	—	4	1	no	—
18	C₄×Dic₁₇	—	4	1	no	—
19	C₄×Dic₁₇	D₄	1	1	yes	characteristic
20	C₂×#3622	D₄	1	1	yes	characteristic
21	#63216	—	2	2	no	—
22	Q₈×C₁₇	—	2	2	no	—
23	Q₈×C₁₇	—	4	1	no	—
24	C₂×C₄×C₁₇	—	4	1	no	—
25	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×Dic₁₇	—	4	1	no	—
27	C₂×Dic₁₇	D₈	1	1	yes	characteristic
28	C₂×Dic₁₇	SD₁₆	1	1	yes	characteristic
29	#3624	—	8	1	no	—
30	C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
31	C₈×C₁₇	M₁₆	1	1	yes	characteristic
32	#3622	—	2	1	no	—
33	#1334	—	17	1	no	maximal, Sylow
34	C₂²×C₁₇	#102	1	1	yes	characteristic
35	C₄×C₁₇	—	4	2	no	—
36	C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
37	C₄×C₁₇	#104	1	1	yes	characteristic
38	Dic₁₇	—	2	1	no	—
39	Dic₁₇	—	2	1	no	—
40	Dic₁₇	—	8	1	no	—
41	C₄×Q₁₆	—	17	1	no	—
42	#346	—	17	1	no	—
43	#334	—	17	1	no	—
44	C₂×C₁₇	D₄⋊C₈	1	1	yes	characteristic
45	C₂×C₁₇	#360	1	1	yes	characteristic
46	C₂×C₁₇	#362	1	1	yes	characteristic
47	C₂×Q₁₆	#595	1	1	yes	characteristic
48	C₄×Q₈	—	34	1	no	—
49	C₂×C₁₆	—	34	1	no	—
50	#107	—	17	1	no	—
51	C₄⋊C₈	—	34	1	no	—
52	#103	—	34	1	no	—
53	C₄×C₈	—	17	1	no	—
54	C₁₇	#1334	1	1	yes	characteristic, complemented, Sylow
55	C₂×Q₈	—	2	1	no	—
56	Q₁₆	—	2	1	no	—
57	Q₁₆	#3624	1	2	yes	complemented
58	C₂×C₈	—	17	1	no	—
59	C₂×C₈	—	68	1	no	—
60	C₂×C₈	C₂×#595	1	1	yes	characteristic
61	C₄⋊C₄	—	34	1	no	—
62	C₄⋊C₄	—	68	1	no	—
63	C₄²	—	17	1	no	—
64	C₄²	—	68	1	no	—
65	C₁₆	—	34	2	no	—
66	Q₈	—	2	2	no	—
67	Q₈	—	4	1	no	—
68	C₂×C₄	—	4	1	no	—
69	C₂×C₄	—	17	1	no	—
70	C₂×C₄	—	17	1	no	—
71	C₂×C₄	—	68	1	no	—
72	C₂×C₄	C₂²⋊#595	1	1	yes	characteristic
73	C₈	—	34	1	no	—
74	C₈	—	136	1	no	—
75	C₈	C₂×#3624	1	1	yes	characteristic
76	C₈	#63247	1	1	yes	characteristic
77	C₂²	#10587547	1	1	yes	characteristic
78	C₄	—	4	2	no	—
79	C₄	—	34	1	no	—
80	C₄	—	34	1	no	—
81	C₄	—	136	1	no	—
82	C₄	#10587548	1	1	yes	characteristic
83	C₄	C₂²⋊#3624	1	1	yes	characteristic
84	C₂	D₄⋊#3624	1	1	yes	characteristic
85	C₂	#11763726	1	1	yes	characteristic
86	C₂	#11763728	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#10587547
commutator	C₈×C₁₇	C₂×C₈
Frattini	C₂×C₈	C₂×#595
Fitting	C₂×Q₁₆×C₁₇	C₄
socle, prime radical	C₂²×C₁₇	#102
cosocle	C₈×Dic₁₇	C₂²
hypercenter	C₂×Q₁₆	#595
hypocenter	C₁₇	#1334
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
14. Q₁₆×C₁₇	C₈	semidirect
54. C₁₇	#1334	semidirect
57. Q₁₆	#3624	semidirect

Maximal subgroups

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

subgroup	index
#11763698	2
#11763670	2
Q₁₆×Dic₁₇	2
#1334	17