Subgroups — #578×C₂₃₉ #29504482

Order 15296.

248 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	2₋¹⁺⁴×C₂₃₉	C₂	1	2	yes	maximal, complemented
2	#137×C₂₃₉	C₂	1	6	yes	maximal, complemented
3	#135×C₂₃₉	C₂	1	3	yes	maximal, complemented
4	C₂×Q₁₆×C₂₃₉	C₂	1	3	yes	maximal, complemented
5	#131×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#41×C₂₃₉	—	2	6	no	—
7	#41×C₂₃₉	C₂²	1	6	yes	—
8	#41×C₂₃₉	C₂²	1	1	yes	characteristic
9	C₂×Q₈×C₂₃₉	—	2	2	no	—
10	C₂×Q₈×C₂₃₉	C₂²	1	6	yes	—
11	Q₁₆×C₂₃₉	C₂²	1	3	yes	—
12	Q₁₆×C₂₃₉	C₂²	1	6	yes	—
13	SD₁₆×C₂₃₉	C₂²	1	6	yes	—
14	D₈×C₂₃₉	C₂²	1	1	yes	characteristic
15	M₁₆×C₂₃₉	C₂²	1	3	yes	—
16	C₂×C₈×C₂₃₉	C₂²	1	3	yes	—
17	Q₈×C₂₃₉	—	2	6	no	—
18	Q₈×C₂₃₉	D₄	1	1	yes	characteristic
19	Q₈×C₂₃₉	C₂³	1	6	yes	—
20	D₄×C₂₃₉	—	2	6	no	—
21	D₄×C₂₃₉	D₄	1	3	yes	—
22	D₄×C₂₃₉	C₂³	1	2	yes	—
23	C₂×C₄×C₂₃₉	—	2	6	no	—
24	C₂×C₄×C₂₃₉	—	2	6	no	—
25	C₂×C₄×C₂₃₉	C₂³	1	3	yes	—
26	C₈×C₂₃₉	C₂³	1	1	yes	characteristic
27	C₈×C₂₃₉	C₂³	1	3	yes	—
28	C₂²×C₂₃₉	—	2	2	no	—
29	C₂²×C₂₃₉	C₂×D₄	1	3	yes	—
30	C₄×C₂₃₉	—	2	6	no	—
31	C₄×C₂₃₉	C₂×D₄	1	3	yes	—
32	C₄×C₂₃₉	C₂⁴	1	1	yes	characteristic
33	C₂×C₂₃₉	—	2	3	no	—
34	C₂×C₂₃₉	—	4	2	no	—
35	C₂×C₂₃₉	C₂²×D₄	1	1	yes	characteristic
36	C₂₃₉	#578	1	1	yes	characteristic, direct product factor, Sylow
37	#578	C₂₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	2₋¹⁺⁴	C₂×C₂₃₉	1	2	yes	complemented
39	#137	C₂×C₂₃₉	1	6	yes	complemented
40	#135	C₂×C₂₃₉	1	3	yes	complemented
41	C₂×Q₁₆	C₂×C₂₃₉	1	3	yes	complemented
42	#131	C₂×C₂₃₉	1	1	yes	characteristic, complemented
43	#41	—	2	6	no	—
44	#41	C₂²×C₂₃₉	1	6	yes	—
45	#41	C₂²×C₂₃₉	1	1	yes	characteristic
46	C₂×Q₈	—	2	2	no	—
47	C₂×Q₈	C₂²×C₂₃₉	1	6	yes	—
48	Q₁₆	C₂²×C₂₃₉	1	3	yes	—
49	Q₁₆	C₂²×C₂₃₉	1	6	yes	—
50	SD₁₆	C₂²×C₂₃₉	1	6	yes	—
51	D₈	C₂²×C₂₃₉	1	1	yes	characteristic
52	M₁₆	C₂²×C₂₃₉	1	3	yes	—
53	C₂×C₈	C₂²×C₂₃₉	1	3	yes	—
54	Q₈	—	2	6	no	—
55	Q₈	D₄×C₂₃₉	1	1	yes	characteristic
56	Q₈	C₂³×C₂₃₉	1	6	yes	—
57	D₄	—	2	6	no	—
58	D₄	D₄×C₂₃₉	1	3	yes	—
59	D₄	C₂³×C₂₃₉	1	2	yes	—
60	C₂×C₄	—	2	6	no	—
61	C₂×C₄	—	2	6	no	—
62	C₂×C₄	C₂³×C₂₃₉	1	3	yes	—
63	C₈	C₂³×C₂₃₉	1	1	yes	characteristic
64	C₈	C₂³×C₂₃₉	1	3	yes	—
65	C₂²	—	2	2	no	—
66	C₂²	C₂×D₄×C₂₃₉	1	3	yes	—
67	C₄	—	2	6	no	—
68	C₄	C₂×D₄×C₂₃₉	1	3	yes	—
69	C₄	C₂⁴×C₂₃₉	1	1	yes	characteristic
70	C₂	—	2	3	no	—
71	C₂	—	4	2	no	—
72	C₂	C₂²×D₄×C₂₃₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₃₉	C₂²×D₄
commutator, Frattini, cosocle	C₄	C₂⁴×C₂₃₉
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₃₉ ◁ #41×C₂₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. 2₋¹⁺⁴×C₂₃₉	C₂	semidirect
2. #137×C₂₃₉	C₂	semidirect
3. #135×C₂₃₉	C₂	semidirect
4. C₂×Q₁₆×C₂₃₉	C₂	semidirect
5. #131×C₂₃₉	C₂	semidirect
36. C₂₃₉	#578	direct (N × H)
37. #578	C₂₃₉	direct (N × H)
38. 2₋¹⁺⁴	C₂×C₂₃₉	semidirect
39. #137	C₂×C₂₃₉	semidirect
40. #135	C₂×C₂₃₉	semidirect
41. C₂×Q₁₆	C₂×C₂₃₉	semidirect
42. #131	C₂×C₂₃₉	semidirect

Maximal subgroups

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

subgroup	index
2₋¹⁺⁴×C₂₃₉	2
#137×C₂₃₉	2
#135×C₂₃₉	2
C₂×Q₁₆×C₂₃₉	2
#131×C₂₃₉	2
#578	239

Subgroups — #578×C239 #29504482

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #578×C₂₃₉ #29504482