Subgroups — #14530686

Order 2176.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Dic₃₄	C₂	1	1	yes	maximal, characteristic
2	#11762627	C₂	1	2	yes	maximal
3	C₄×#63214	C₂	1	1	yes	maximal, characteristic
4	C₁₆×Dic₁₇	C₂	1	2	yes	maximal
5	C₁₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₃₄	C₄	1	1	yes	characteristic
7	C₂×C₁₆×C₁₇	C₂²	1	2	yes	—
8	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
9	#10587485	C₄	1	2	yes	—
10	C₈×Dic₁₇	C₂²	1	2	yes	—
11	C₂×#63214	C₂²	1	1	yes	characteristic
12	C₂×#63214	C₂²	1	1	yes	characteristic
13	C₄⋊#3622	C₄	1	1	yes	characteristic
14	C₂×Dic₃₄	C₈	1	1	yes	characteristic
15	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
17	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
18	C₄⋊Dic₁₇	C₈	1	1	yes	characteristic
19	#63225	C₈	1	2	yes	—
20	C₄×Dic₁₇	C₂×C₄	1	2	yes	—
21	C₂×#3622	C₂×C₄	1	2	yes	—
22	C₁₆×C₁₇	—	2	2	no	—
23	#63214	—	2	1	no	—
24	#63214	Q₈	1	2	yes	complemented
25	C₂×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
26	C₂×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
27	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
28	C₂×Dic₁₇	C₂×C₈	1	2	yes	—
29	C₂×Dic₁₇	C₂×C₈	1	2	yes	—
30	Dic₃₄	C₁₆	1	4	yes	complemented
31	C₈×C₁₇	—	2	1	no	—
32	C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
33	C₈×C₁₇	#41	1	1	yes	characteristic
34	#3622	—	2	2	no	—
35	Q₈×C₁₆	—	17	1	no	maximal, Sylow
36	C₂²×C₁₇	C₂²×C₈	1	1	yes	characteristic
37	C₄×C₁₇	—	2	1	no	—
38	C₄×C₁₇	C₂×C₁₆	1	2	yes	—
39	C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
40	C₄×C₁₇	#131	1	1	yes	characteristic
41	Dic₁₇	—	2	2	no	—
42	Dic₁₇	C₂×C₁₆	1	4	yes	—
43	C₈×Q₈	—	17	1	no	—
44	C₄⋊C₁₆	—	17	2	no	—
45	C₄⋊C₁₆	D₁₇	1	1	yes	characteristic
46	C₄×C₁₆	—	17	2	no	—
47	C₄×C₁₆	—	17	1	no	—
48	C₂×C₁₇	C₈×Q₈	1	1	yes	characteristic
49	C₂×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
50	C₂×C₁₇	#504	1	1	yes	characteristic
51	C₄×Q₈	—	17	1	no	—
52	C₂×C₁₆	—	17	1	no	—
53	C₂×C₁₆	—	17	1	no	—
54	C₂×C₁₆	C₂×D₁₇	1	2	yes	—
55	C₄⋊C₈	—	17	2	no	—
56	C₄⋊C₈	—	17	1	no	—
57	C₄×C₈	—	17	2	no	—
58	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
59	C₁₇	Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
60	C₂×Q₈	—	17	1	no	—
61	C₂×C₈	—	17	2	no	—
62	C₂×C₈	C₄×D₁₇	1	1	yes	characteristic
63	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
64	C₄⋊C₄	—	17	2	no	—
65	C₄⋊C₄	—	17	1	no	—
66	C₄²	—	17	2	no	—
67	C₄²	C₄×D₁₇	1	1	yes	characteristic
68	C₁₆	—	2	2	no	—
69	C₁₆	—	17	2	no	—
70	C₁₆	—	34	1	no	—
71	Q₈	—	17	4	no	—
72	C₂×C₄	—	17	2	no	—
73	C₂×C₄	—	17	2	no	—
74	C₂×C₄	C₈×D₁₇	1	1	yes	characteristic
75	C₂×C₄	C₈×D₁₇	1	1	yes	characteristic
76	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	34	2	no	—
79	C₈	#63254	1	1	yes	characteristic
80	C₈	Q₈×D₁₇	1	1	yes	characteristic
81	C₂²	C₂×C₈×D₁₇	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	17	4	no	—
84	C₄	—	34	2	no	—
85	C₄	C₁₆×D₁₇	1	2	yes	—
86	C₄	#10587573	1	1	yes	characteristic
87	C₄	#10587593	1	1	yes	characteristic
88	C₂	#11762933	1	1	yes	characteristic
89	C₂	C₂×C₁₆×D₁₇	1	1	yes	characteristic
90	C₂	#11763227	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₇
commutator	C₂×C₁₇	C₂²×C₁₆
Fitting	C₁₇×C₄⋊C₁₆	C₂
socle, prime radical	C₂²×C₁₇	C₂²×C₈
cosocle	C₂×C₈×C₁₇	C₂³
hypercenter	C₄⋊C₁₆	D₁₇
hypocenter	C₁₇	Q₈×C₁₆
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
24. #63214	Q₈	semidirect
30. Dic₃₄	C₁₆	semidirect
59. C₁₇	Q₈×C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₈×Dic₃₄	2
#11762627	2
C₄×#63214	2
C₁₆×Dic₁₇	2
C₁₇×C₄⋊C₁₆	2
Q₈×C₁₆	17