Subgroups — C₂²×Dic₆₈ #11763586

Order 1088.

258 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₃₄	C₂	1	2	yes	maximal
2	C₂²×C₈×C₁₇	C₂	1	1	yes	maximal, characteristic
3	C₂×Dic₆₈	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₄×C₁₇	C₂²	1	1	yes	characteristic
5	C₂²×Dic₁₇	—	2	2	no	—
6	C₂×Dic₃₄	—	2	6	no	—
7	C₂×Dic₃₄	C₂²	1	12	yes	—
8	C₂×C₈×C₁₇	C₂²	1	6	yes	—
9	Dic₆₈	C₂²	1	16	yes	direct product factor
10	C₂³×C₁₇	D₄	1	1	yes	characteristic
11	C₂×C₄×C₁₇	D₄	1	3	yes	—
12	C₂×C₄×C₁₇	C₂³	1	3	yes	—
13	C₂×Dic₁₇	—	2	12	no	—
14	Dic₃₄	—	2	12	no	—
15	Dic₃₄	C₂³	1	8	yes	—
16	C₈×C₁₇	C₂³	1	4	yes	—
17	C₂²×C₁₇	Q₁₆	1	4	yes	complemented
18	C₂²×C₁₇	C₂×D₄	1	3	yes	—
19	C₄×C₁₇	C₂×D₄	1	3	yes	—
20	C₄×C₁₇	C₂⁴	1	1	yes	characteristic
21	Dic₁₇	—	2	8	no	—
22	C₂²×Q₁₆	—	17	1	no	maximal, Sylow
23	C₂×C₁₇	C₂×Q₁₆	1	6	yes	complemented
24	C₂×C₁₇	C₂²×D₄	1	1	yes	characteristic
25	C₂²×Q₈	—	17	2	no	—
26	C₂×Q₁₆	—	17	12	no	—
27	C₂²×C₈	D₁₇	1	1	yes	characteristic
28	C₁₇	C₂²×Q₁₆	1	1	yes	characteristic, complemented, Sylow
29	C₂×Q₈	—	17	12	no	—
30	C₂×Q₈	—	34	6	no	—
31	C₂²×C₄	—	34	2	no	—
32	C₂²×C₄	C₂×D₁₇	1	1	yes	characteristic
33	Q₁₆	—	17	16	no	—
34	C₂×C₈	C₂×D₁₇	1	6	yes	—
35	C₂³	D₆₈	1	1	yes	characteristic
36	Q₈	—	17	8	no	—
37	Q₈	—	34	12	no	—
38	C₂×C₄	—	34	12	no	—
39	C₂×C₄	D₆₈	1	3	yes	—
40	C₂×C₄	C₂²×D₁₇	1	3	yes	—
41	C₈	C₂²×D₁₇	1	4	yes	—
42	C₂²	Dic₆₈	1	4	yes	direct product factor
43	C₂²	C₂×D₆₈	1	3	yes	—
44	C₄	—	34	8	no	—
45	C₄	C₂×D₆₈	1	3	yes	—
46	C₄	C₂³×D₁₇	1	1	yes	characteristic
47	C₂	C₂×Dic₆₈	1	6	yes	direct product factor
48	C₂	C₂²×D₆₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₆₈
commutator, cosocle	C₄×C₁₇	C₂⁴
Frattini	C₄	C₂³×D₁₇
Fitting	C₂²×C₈×C₁₇	C₂
socle, prime radical	C₂³×C₁₇	D₄
hypercenter	C₂²×C₈	D₁₇
hypocenter	C₁₇	C₂²×Q₁₆
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
3. C₂×Dic₆₈	C₂	direct (N × H)
9. Dic₆₈	C₂²	direct (N × H)
17. C₂²×C₁₇	Q₁₆	semidirect
23. C₂×C₁₇	C₂×Q₁₆	semidirect
28. C₁₇	C₂²×Q₁₆	semidirect
42. C₂²	Dic₆₈	direct (N × H)
47. C₂	C₂×Dic₆₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Dic₃₄	2
C₂²×C₈×C₁₇	2
C₂×Dic₆₈	2
C₂²×Q₁₆	17

Subgroups — C22×Dic68 #11763586

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Dic₆₈ #11763586