Subgroups — #11763535

Order 1088.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Q₈×C₁₇	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic
3	Q₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic
4	#10587576	C₂	1	3	yes	maximal
5	#10587574	C₂	1	3	yes	maximal
6	#10587556	C₂	1	3	yes	maximal
7	C₄×Dic₃₄	C₂	1	3	yes	maximal
8	C₂×Q₈×C₁₇	C₂²	1	1	yes	characteristic
9	C₂×Dic₃₄	C₂²	1	3	yes	—
10	C₄⋊C₄×C₁₇	C₂²	1	3	yes	—
11	C₄²×C₁₇	C₂²	1	3	yes	—
12	C₄⋊Dic₁₇	C₂²	1	3	yes	—
13	C₄⋊Dic₁₇	C₂²	1	3	yes	—
14	C₄⋊Dic₁₇	C₂²	1	3	yes	—
15	#63225	C₂²	1	1	yes	characteristic
16	#63225	C₂²	1	3	yes	—
17	#63225	C₂²	1	6	yes	—
18	C₄×Dic₁₇	C₂²	1	3	yes	—
19	C₄×Dic₁₇	C₂²	1	3	yes	—
20	Q₈×C₁₇	Q₈	1	4	yes	complemented
21	C₂×C₄×C₁₇	C₂³	1	1	yes	characteristic
22	C₂×C₄×C₁₇	C₂³	1	3	yes	—
23	C₂×C₄×C₁₇	C₂³	1	3	yes	—
24	C₂×Dic₁₇	C₂³	1	3	yes	—
25	C₂×Dic₁₇	C₂³	1	1	yes	characteristic
26	C₂×Dic₁₇	C₂³	1	3	yes	—
27	C₂×Dic₁₇	C₂³	1	1	yes	characteristic
28	Dic₃₄	—	2	6	no	—
29	C₂²×C₁₇	C₂⁴	1	1	yes	characteristic
30	C₄×C₁₇	—	2	3	no	—
31	C₄×C₁₇	—	2	1	no	—
32	C₄×C₁₇	C₂×Q₈	1	6	yes	—
33	Dic₁₇	—	2	3	no	—
34	Dic₁₇	—	2	1	no	—
35	Dic₁₇	—	2	3	no	—
36	Dic₁₇	#41	1	2	yes	—
37	#557	—	17	1	no	maximal, Sylow
38	C₂×C₁₇	C₂²×Q₈	1	1	yes	characteristic
39	C₂×C₁₇	C₂×#41	1	1	yes	characteristic
40	C₂×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
41	C₄⋊Q₈	—	17	3	no	—
42	#125	—	17	3	no	—
43	#125	—	17	3	no	—
44	C₄×Q₈	—	17	1	no	—
45	C₄×Q₈	—	17	1	no	—
46	C₄×Q₈	—	17	3	no	—
47	C₄×Q₈	D₁₇	1	1	yes	characteristic
48	C₁₇	#557	1	1	yes	characteristic, complemented, Sylow
49	C₂×Q₈	—	17	3	no	—
50	C₂×Q₈	C₂×D₁₇	1	1	yes	characteristic
51	C₄⋊C₄	—	17	1	no	—
52	C₄⋊C₄	—	17	3	no	—
53	C₄⋊C₄	—	17	3	no	—
54	C₄⋊C₄	—	17	3	no	—
55	C₄⋊C₄	—	17	3	no	—
56	C₄⋊C₄	—	17	6	no	—
57	C₄⋊C₄	C₂×D₁₇	1	3	yes	—
58	C₄²	—	17	3	no	—
59	C₄²	—	17	3	no	—
60	C₄²	C₂×D₁₇	1	3	yes	—
61	Q₈	—	34	6	no	—
62	Q₈	Dic₃₄	1	4	yes	complemented
63	C₂×C₄	—	17	1	no	—
64	C₂×C₄	—	17	1	no	—
65	C₂×C₄	—	17	3	no	—
66	C₂×C₄	—	17	3	no	—
67	C₂×C₄	C₂²×D₁₇	1	1	yes	characteristic
68	C₂×C₄	C₂²×D₁₇	1	3	yes	—
69	C₂×C₄	C₂²×D₁₇	1	3	yes	—
70	C₂²	C₂³×D₁₇	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	2	3	no	—
73	C₄	—	17	2	no	—
74	C₄	—	34	1	no	—
75	C₄	—	34	3	no	—
76	C₄	—	34	3	no	—
77	C₄	C₂×Dic₃₄	1	6	yes	—
78	C₂	C₂²×Dic₃₄	1	1	yes	characteristic
79	C₂	#10587694	1	1	yes	characteristic
80	C₂	#41×D₁₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₇
commutator, socle, prime radical, cosocle	C₂²×C₁₇	C₂⁴
Fitting	C₄×Q₈×C₁₇	C₂
hypercenter	C₄×Q₈	D₁₇
hypocenter	C₁₇	#557
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
20. Q₈×C₁₇	Q₈	semidirect
48. C₁₇	#557	semidirect
62. Q₈	Dic₃₄	semidirect

Maximal subgroups

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

subgroup	index
C₄×Q₈×C₁₇	2
Q₈×Dic₁₇	2
#10587576	2
#10587574	2
#10587556	2
C₄×Dic₃₄	2
#557	17