Subgroups — D₇×D₆₈ #14274140

Order 1904.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₇×D₁₇	C₂	1	2	yes	maximal, complemented
2	D₄₇₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₇×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×D₆₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#11745753	C₂	1	2	yes	maximal, complemented
6	C₂×D₁₁₉	C₂²	1	2	yes	complemented
7	C₂×D₇×C₁₇	C₂²	1	1	yes	characteristic
8	C₂×C₇×D₁₇	C₂²	1	2	yes	complemented
9	D₇×D₁₇	—	2	4	no	—
10	C₄×C₇×C₁₇	C₂²	1	1	yes	characteristic, complemented
11	C₁₇×Dic₇	C₂²	1	1	yes	characteristic, complemented
12	C₂×D₆₈	—	7	1	no	maximal, Hall
13	C₂×C₇×C₁₇	C₂³	1	1	yes	characteristic
14	D₁₁₉	—	2	2	no	—
15	C₇×D₁₇	—	2	2	no	—
16	D₇×C₁₇	D₄	1	2	yes	complemented
17	C₂²×D₁₇	—	7	2	no	—
18	C₂×C₄×C₁₇	—	7	1	no	—
19	D₆₈	—	7	2	no	—
20	D₆₈	—	7	1	no	—
21	D₆₈	D₇	1	1	yes	characteristic, direct product factor
22	C₇×C₁₇	C₂×D₄	1	1	yes	characteristic, complemented, Hall
23	D₄×D₇	—	17	1	no	maximal, Hall
24	C₂²×C₁₇	—	7	1	no	—
25	C₂×D₁₇	—	7	2	no	—
26	C₂×D₁₇	—	14	4	no	—
27	C₂×D₁₇	C₂×D₇	1	2	yes	complemented
28	C₄×C₁₇	—	7	1	no	—
29	C₄×C₁₇	C₂×D₇	1	1	yes	characteristic, complemented
30	C₂²×D₇	—	17	2	no	—
31	C₇×D₄	—	17	1	no	—
32	C₂²⋊D₇	—	17	2	no	—
33	D₂₈	—	17	1	no	—
34	C₄×D₇	D₁₇	1	1	yes	characteristic, complemented
35	C₂×C₁₇	—	7	2	no	—
36	C₂×C₁₇	C₂²×D₇	1	1	yes	characteristic
37	D₁₇	—	2	2	no	—
38	D₁₇	—	14	2	no	—
39	C₂²×C₇	—	17	2	no	—
40	C₂×D₇	—	17	2	no	—
41	C₂×D₇	—	34	4	no	—
42	C₂×D₇	C₂×D₁₇	1	1	yes	characteristic
43	C₄×C₇	C₂×D₁₇	1	1	yes	characteristic, complemented
44	Dic₇	C₂×D₁₇	1	1	yes	characteristic, complemented
45	C₁₇	D₄×D₇	1	1	yes	characteristic, complemented, Sylow
46	C₂×D₄	—	119	1	no	Sylow
47	C₂×C₇	—	34	2	no	—
48	C₂×C₇	C₂²×D₁₇	1	1	yes	characteristic
49	D₇	—	34	2	no	—
50	D₇	D₆₈	1	2	yes	direct product factor
51	C₂³	—	119	2	no	—
52	D₄	—	17	1	no	—
53	D₄	—	119	2	no	—
54	D₄	—	119	1	no	—
55	C₂×C₄	—	7	1	no	—
56	C₇	C₂×D₆₈	1	1	yes	characteristic, complemented, Sylow
57	C₂²	—	7	1	no	—
58	C₂²	—	17	2	no	—
59	C₂²	—	119	2	no	—
60	C₂²	—	238	4	no	—
61	C₄	—	7	1	no	—
62	C₄	D₇×D₁₇	1	1	yes	characteristic, complemented
63	C₂	—	7	2	no	—
64	C₂	—	34	2	no	—
65	C₂	—	238	2	no	—
66	C₂	C₂×D₇×D₁₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×D₇×D₁₇
commutator, socle, cosocle	C₂×C₇×C₁₇	C₂³
Fitting	C₄×C₇×C₁₇	C₂²
hypercenter	C₄	D₇×D₁₇
hypocenter	C₇×C₁₇	C₂×D₄
LCS	G ▷ C₂×C₇×C₁₇ ▷ C₇×C₁₇
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
1. C₂×D₇×D₁₇	C₂	semidirect
2. D₄₇₆	C₂	semidirect
3. C₄×D₇×C₁₇	C₂	semidirect
4. C₇×D₆₈	C₂	semidirect
5. #11745753	C₂	semidirect
6. C₂×D₁₁₉	C₂²	semidirect
8. C₂×C₇×D₁₇	C₂²	semidirect
10. C₄×C₇×C₁₇	C₂²	semidirect
11. C₁₇×Dic₇	C₂²	semidirect
16. D₇×C₁₇	D₄	semidirect
21. D₆₈	D₇	direct (N × H)
22. C₇×C₁₇	C₂×D₄	semidirect
27. C₂×D₁₇	C₂×D₇	semidirect
29. C₄×C₁₇	C₂×D₇	semidirect
34. C₄×D₇	D₁₇	semidirect
43. C₄×C₇	C₂×D₁₇	semidirect
44. Dic₇	C₂×D₁₇	semidirect
45. C₁₇	D₄×D₇	semidirect
50. D₇	D₆₈	direct (N × H)
56. C₇	C₂×D₆₈	semidirect
62. C₄	D₇×D₁₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₇×D₁₇	2
D₄₇₆	2
C₄×D₇×C₁₇	2
C₇×D₆₈	2
#11745753	2
C₂×D₆₈	7
D₄×D₇	17

Subgroups — D7×D68 #14274140

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇×D₆₈ #14274140