Subgroups — #14531012

Order 2176.

140 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	#11763189	C₂	1	1	yes	maximal, characteristic
3	#11762614	C₂	1	4	yes	maximal
4	C₄×C₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₆₈	—	2	1	no	—
6	C₂×Dic₆₈	C₂²	1	2	yes	—
7	#10587555	—	2	1	no	—
8	#10587555	—	2	1	no	—
9	C₂×C₁₆×C₁₇	C₂²	1	2	yes	—
10	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
11	#10587489	—	2	1	no	—
12	#10587489	C₂²	1	2	yes	—
13	C₂×Dic₃₄	—	2	2	no	—
14	C₂×Dic₃₄	—	4	1	no	—
15	C₂×C₈×C₁₇	D₄	1	1	yes	characteristic
16	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
17	C₄²×C₁₇	D₄	1	1	yes	characteristic
18	C₄⋊Dic₁₇	—	2	2	no	—
19	C₄⋊Dic₁₇	—	4	1	no	—
20	C₄⋊Dic₁₇	—	4	1	no	—
21	C₄⋊Dic₁₇	—	4	1	no	—
22	Dic₆₈	—	2	4	no	—
23	Dic₆₈	—	4	2	no	—
24	C₁₆×C₁₇	—	2	2	no	—
25	C₂×C₄×C₁₇	D₈	1	1	yes	characteristic
26	C₂×C₄×C₁₇	D₈	1	1	yes	characteristic
27	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
28	C₂×Dic₁₇	—	4	2	no	—
29	C₂×Dic₁₇	—	4	2	no	—
30	Dic₃₄	—	4	2	no	—
31	Dic₃₄	—	4	2	no	—
32	Dic₃₄	—	8	2	no	—
33	C₈×C₁₇	SD₁₆	1	2	yes	—
34	C₈×C₁₇	#41	1	2	yes	—
35	#2241	—	17	1	no	maximal, Sylow
36	C₂²×C₁₇	C₂×D₈	1	1	yes	characteristic
37	C₄×C₁₇	SD₃₂	1	2	yes	—
38	C₄×C₁₇	Q₃₂	1	2	yes	—
39	C₄×C₁₇	#124	1	1	yes	characteristic
40	C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
41	Dic₁₇	—	8	2	no	—
42	Dic₁₇	—	8	2	no	—
43	#500	—	17	1	no	—
44	#494	—	17	1	no	—
45	#358	—	17	4	no	—
46	C₄×C₁₆	D₁₇	1	1	yes	characteristic
47	C₂×C₁₇	#486	1	1	yes	characteristic
48	C₂×C₁₇	C₂×SD₃₂	1	1	yes	characteristic
49	C₂×C₁₇	C₂×Q₃₂	1	1	yes	characteristic
50	C₂×Q₁₆	—	17	2	no	—
51	C₂×Q₁₆	—	34	1	no	—
52	C₄⋊Q₈	—	34	1	no	—
53	C₄⋊Q₈	—	34	1	no	—
54	C₂×C₁₆	C₂×D₁₇	1	2	yes	—
55	#107	—	17	2	no	—
56	#107	—	34	1	no	—
57	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
58	C₁₇	#2241	1	1	yes	characteristic, complemented, Sylow
59	C₂×Q₈	—	34	2	no	—
60	C₂×Q₈	—	68	1	no	—
61	Q₁₆	—	34	4	no	—
62	Q₁₆	—	68	2	no	—
63	C₂×C₈	D₆₈	1	1	yes	characteristic
64	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
65	C₄⋊C₄	—	34	2	no	—
66	C₄⋊C₄	—	68	1	no	—
67	C₄⋊C₄	—	68	1	no	—
68	C₄⋊C₄	—	68	1	no	—
69	C₄²	D₆₈	1	1	yes	characteristic
70	C₁₆	—	2	2	no	—
71	Q₈	—	68	2	no	—
72	Q₈	—	68	2	no	—
73	Q₈	—	136	2	no	—
74	C₂×C₄	—	68	2	no	—
75	C₂×C₄	—	68	2	no	—
76	C₂×C₄	D₁₃₆	1	1	yes	characteristic
77	C₂×C₄	D₁₃₆	1	1	yes	characteristic
78	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
79	C₈	#63219	1	2	yes	—
80	C₈	#63252	1	2	yes	—
81	C₂²	C₂×D₁₃₆	1	1	yes	characteristic
82	C₄	—	136	2	no	—
83	C₄	—	136	2	no	—
84	C₄	#10587471	1	2	yes	—
85	C₄	Dic₁₃₆	1	2	yes	—
86	C₄	#10587561	1	1	yes	characteristic
87	C₄	C₂×#63219	1	1	yes	characteristic
88	C₂	#11763156	1	1	yes	characteristic
89	C₂	C₂×#10587471	1	1	yes	characteristic
90	C₂	C₂×Dic₁₃₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₃₆
commutator, cosocle	C₂×C₈×C₁₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₇
Fitting	C₄×C₁₆×C₁₇	C₂
socle, prime radical	C₂²×C₁₇	C₂×D₈
hypercenter	C₄×C₁₆	D₁₇
hypocenter	C₁₇	#2241
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
58. C₁₇	#2241	semidirect

Maximal subgroups

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

subgroup	index
#11763214	2
#11763189	2
#11762614	2
C₄×C₁₆×C₁₇	2
#2241	17