Subgroups — C₂²×SD₁₆×C₂₃₉ #29504474

Order 15296.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₄×C₂₃₉	C₂	1	1	yes	maximal, characteristic
3	C₂×SD₁₆×C₂₃₉	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₈×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×C₂₃₉	—	2	1	no	—
6	C₂×Q₈×C₂₃₉	—	2	3	no	—
7	C₂×Q₈×C₂₃₉	C₂²	1	6	yes	complemented
8	C₂×D₄×C₂₃₉	—	2	3	no	—
9	C₂×D₄×C₂₃₉	C₂²	1	6	yes	—
10	C₂²×C₄×C₂₃₉	—	2	1	no	—
11	C₂²×C₄×C₂₃₉	C₂²	1	1	yes	characteristic
12	SD₁₆×C₂₃₉	C₂²	1	16	yes	direct product factor
13	C₂×C₈×C₂₃₉	C₂²	1	6	yes	complemented
14	C₂³×C₂₃₉	—	2	6	no	—
15	C₂³×C₂₃₉	—	4	4	no	—
16	C₂³×C₂₃₉	D₄	1	1	yes	characteristic
17	Q₈×C₂₃₉	—	2	6	no	—
18	Q₈×C₂₃₉	C₂³	1	4	yes	complemented
19	D₄×C₂₃₉	—	2	6	no	—
20	D₄×C₂₃₉	C₂³	1	4	yes	—
21	C₂×C₄×C₂₃₉	—	2	6	no	—
22	C₂×C₄×C₂₃₉	D₄	1	3	yes	—
23	C₂×C₄×C₂₃₉	C₂³	1	3	yes	—
24	C₈×C₂₃₉	C₂³	1	4	yes	complemented
25	C₂²×C₂₃₉	—	2	4	no	—
26	C₂²×C₂₃₉	—	4	12	no	—
27	C₂²×C₂₃₉	SD₁₆	1	4	yes	direct product factor
28	C₂²×C₂₃₉	C₂×D₄	1	3	yes	—
29	C₄×C₂₃₉	—	2	4	no	—
30	C₄×C₂₃₉	C₂×D₄	1	3	yes	—
31	C₄×C₂₃₉	C₂⁴	1	1	yes	characteristic
32	C₂×C₂₃₉	—	4	4	no	—
33	C₂×C₂₃₉	C₂×SD₁₆	1	6	yes	direct product factor
34	C₂×C₂₃₉	C₂²×D₄	1	1	yes	characteristic
35	C₂₃₉	C₂²×SD₁₆	1	1	yes	characteristic, direct product factor, Sylow
36	C₂²×SD₁₆	C₂₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂²×Q₈	C₂×C₂₃₉	1	1	yes	characteristic, complemented
38	C₂²×D₄	C₂×C₂₃₉	1	1	yes	characteristic
39	C₂×SD₁₆	C₂×C₂₃₉	1	12	yes	direct product factor
40	C₂²×C₈	C₂×C₂₃₉	1	1	yes	characteristic, complemented
41	C₂⁴	—	2	1	no	—
42	C₂×Q₈	—	2	3	no	—
43	C₂×Q₈	C₂²×C₂₃₉	1	6	yes	complemented
44	C₂×D₄	—	2	3	no	—
45	C₂×D₄	C₂²×C₂₃₉	1	6	yes	—
46	C₂²×C₄	—	2	1	no	—
47	C₂²×C₄	C₂²×C₂₃₉	1	1	yes	characteristic
48	SD₁₆	C₂²×C₂₃₉	1	16	yes	direct product factor
49	C₂×C₈	C₂²×C₂₃₉	1	6	yes	complemented
50	C₂³	—	2	6	no	—
51	C₂³	—	4	4	no	—
52	C₂³	D₄×C₂₃₉	1	1	yes	characteristic
53	Q₈	—	2	6	no	—
54	Q₈	C₂³×C₂₃₉	1	4	yes	complemented
55	D₄	—	2	6	no	—
56	D₄	C₂³×C₂₃₉	1	4	yes	—
57	C₂×C₄	—	2	6	no	—
58	C₂×C₄	D₄×C₂₃₉	1	3	yes	—
59	C₂×C₄	C₂³×C₂₃₉	1	3	yes	—
60	C₈	C₂³×C₂₃₉	1	4	yes	complemented
61	C₂²	—	2	4	no	—
62	C₂²	—	4	12	no	—
63	C₂²	SD₁₆×C₂₃₉	1	4	yes	direct product factor
64	C₂²	C₂×D₄×C₂₃₉	1	3	yes	—
65	C₄	—	2	4	no	—
66	C₄	C₂×D₄×C₂₃₉	1	3	yes	—
67	C₄	C₂⁴×C₂₃₉	1	1	yes	characteristic
68	C₂	—	4	4	no	—
69	C₂	C₂×SD₁₆×C₂₃₉	1	6	yes	direct product factor
70	C₂	C₂²×D₄×C₂₃₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃₉	D₄
commutator, Frattini, cosocle	C₄	C₂⁴×C₂₃₉
prime radical	C₂²×D₄×C₂₃₉	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃₉ ◁ C₂²×C₄×C₂₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×Q₈×C₂₃₉	C₂	semidirect
3. C₂×SD₁₆×C₂₃₉	C₂	direct (N × H)
4. C₂²×C₈×C₂₃₉	C₂	semidirect
7. C₂×Q₈×C₂₃₉	C₂²	semidirect
12. SD₁₆×C₂₃₉	C₂²	direct (N × H)
13. C₂×C₈×C₂₃₉	C₂²	semidirect
18. Q₈×C₂₃₉	C₂³	semidirect
24. C₈×C₂₃₉	C₂³	semidirect
27. C₂²×C₂₃₉	SD₁₆	direct (N × H)
33. C₂×C₂₃₉	C₂×SD₁₆	direct (N × H)
35. C₂₃₉	C₂²×SD₁₆	direct (N × H)
36. C₂²×SD₁₆	C₂₃₉	direct (N × H)
37. C₂²×Q₈	C₂×C₂₃₉	semidirect
39. C₂×SD₁₆	C₂×C₂₃₉	direct (N × H)
40. C₂²×C₈	C₂×C₂₃₉	semidirect
43. C₂×Q₈	C₂²×C₂₃₉	semidirect
48. SD₁₆	C₂²×C₂₃₉	direct (N × H)
49. C₂×C₈	C₂²×C₂₃₉	semidirect
54. Q₈	C₂³×C₂₃₉	semidirect
60. C₈	C₂³×C₂₃₉	semidirect
63. C₂²	SD₁₆×C₂₃₉	direct (N × H)
69. C₂	C₂×SD₁₆×C₂₃₉	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×Q₈×C₂₃₉	2
C₂²×D₄×C₂₃₉	2
C₂×SD₁₆×C₂₃₉	2
C₂²×C₈×C₂₃₉	2
C₂²×SD₁₆	239

Subgroups — C22×SD16×C239 #29504474

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×SD₁₆×C₂₃₉ #29504474