Subgroups — #11763539

Order 1088.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#10587630	C₂	1	4	yes	maximal
3	#10587574	C₂	1	4	yes	maximal
4	#10587573	C₂	1	4	yes	maximal
5	C₄×Dic₃₄	C₂	1	2	yes	maximal
6	C₂×Q₈×C₁₇	C₂²	1	2	yes	—
7	C₂×Dic₃₄	C₂²	1	4	yes	—
8	C₂×Dic₃₄	C₂²	1	2	yes	—
9	C₄⋊C₄×C₁₇	C₂²	1	4	yes	—
10	C₄²×C₁₇	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₁₇	C₂²	1	2	yes	—
12	#63225	C₂²	1	4	yes	—
13	#63225	C₂²	1	8	yes	—
14	C₄×Dic₁₇	C₂²	1	4	yes	—
15	C₄×Dic₁₇	C₂²	1	4	yes	—
16	Q₈×C₁₇	—	4	2	no	—
17	C₂×C₄×C₁₇	C₂³	1	1	yes	characteristic
18	C₂×C₄×C₁₇	C₂³	1	4	yes	—
19	C₂×C₄×C₁₇	C₂³	1	2	yes	—
20	C₂×Dic₁₇	C₂³	1	4	yes	—
21	C₂×Dic₁₇	C₂³	1	4	yes	—
22	Dic₃₄	—	4	4	no	—
23	Dic₃₄	Q₈	1	8	yes	complemented
24	C₂²×C₁₇	C₂⁴	1	1	yes	characteristic
25	C₄×C₁₇	—	2	4	no	—
26	C₄×C₁₇	—	2	1	no	—
27	C₄×C₁₇	C₂×Q₈	1	4	yes	—
28	Dic₁₇	—	2	4	no	—
29	Dic₁₇	C₂×Q₈	1	8	yes	—
30	Q₈²	—	17	1	no	maximal, Sylow
31	C₂×C₁₇	C₂²×Q₈	1	2	yes	—
32	C₂×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
33	C₄⋊Q₈	—	17	4	no	—
34	C₄⋊Q₈	—	17	4	no	—
35	C₄⋊Q₈	D₁₇	1	1	yes	characteristic
36	C₄×Q₈	—	17	4	no	—
37	C₄×Q₈	—	17	2	no	—
38	C₁₇	Q₈²	1	1	yes	characteristic, complemented, Sylow
39	C₂×Q₈	—	17	2	no	—
40	C₂×Q₈	—	17	4	no	—
41	C₂×Q₈	C₂×D₁₇	1	2	yes	—
42	C₄⋊C₄	—	17	8	no	—
43	C₄⋊C₄	—	17	4	no	—
44	C₄⋊C₄	—	17	2	no	—
45	C₄⋊C₄	C₂×D₁₇	1	4	yes	—
46	C₄²	—	17	4	no	—
47	C₄²	—	17	4	no	—
48	C₄²	C₂×D₁₇	1	1	yes	characteristic
49	Q₈	—	4	2	no	—
50	Q₈	—	17	8	no	—
51	Q₈	—	68	4	no	—
52	C₂×C₄	—	17	4	no	—
53	C₂×C₄	—	17	4	no	—
54	C₂×C₄	C₂²×D₁₇	1	4	yes	—
55	C₂×C₄	C₂²×D₁₇	1	1	yes	characteristic
56	C₂×C₄	C₂²×D₁₇	1	2	yes	—
57	C₂²	C₂³×D₁₇	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	2	4	no	—
60	C₄	—	17	8	no	—
61	C₄	—	34	4	no	—
62	C₄	Q₈×D₁₇	1	4	yes	—
63	C₂	#10587691	1	1	yes	characteristic
64	C₂	C₂×Q₈×D₁₇	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₇
commutator, socle, prime radical, cosocle	C₂²×C₁₇	C₂⁴
Fitting	C₁₇×C₄⋊Q₈	C₂
hypercenter	C₄⋊Q₈	D₁₇
hypocenter	C₁₇	Q₈²
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
23. Dic₃₄	Q₈	semidirect
38. C₁₇	Q₈²	semidirect

Maximal subgroups

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

subgroup	index
C₁₇×C₄⋊Q₈	2
#10587630	2
#10587574	2
#10587573	2
C₄×Dic₃₄	2
Q₈²	17