Subgroups — #106×D₂₃₉ #29504801

Order 15296.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₂₃₉	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₄×D₂₃₉	C₂	1	2	yes	maximal
3	#21439892	C₂	1	1	yes	maximal, characteristic, complemented
4	#21439891	C₂	1	2	yes	maximal, complemented
5	#106×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₂₃₉	—	2	2	no	—
7	C₂×C₄×D₂₃₉	C₂²	1	1	yes	characteristic
8	C₂×C₂₃₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₈×D₂₃₉	C₄	1	4	yes	complemented
10	C₄⋊Dic₂₃₉	C₂²	1	2	yes	—
11	Dic₂₃₉⋊C₄	—	2	2	no	—
12	C₂×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₂₃₉	C₂²	1	2	yes	—
14	C₂²×D₂₃₉	D₄	1	1	yes	characteristic
15	C₂×Dic₂₃₉	—	2	2	no	—
16	C₂×Dic₂₃₉	D₄	1	1	yes	characteristic
17	C₄×D₂₃₉	—	4	4	no	—
18	C₄×D₂₃₉	C₂×C₄	1	2	yes	—
19	C₄×D₂₃₉	Q₈	1	2	yes	—
20	C₂₃₉⋊C₈	C₂×C₄	1	2	yes	complemented
21	C₂×C₄×C₂₃₉	—	2	2	no	—
22	C₂×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
23	C₈×C₂₃₉	C₂×C₄	1	2	yes	complemented
24	C₂²×C₂₃₉	C₂×D₄	1	1	yes	characteristic
25	C₂×D₂₃₉	C₄⋊C₄	1	2	yes	—
26	C₂×D₂₃₉	SD₁₆	1	4	yes	—
27	C₄×C₂₃₉	—	4	2	no	—
28	C₄×C₂₃₉	C₂²×C₄	1	1	yes	characteristic
29	C₄×C₂₃₉	C₂×Q₈	1	1	yes	characteristic
30	Dic₂₃₉	—	4	2	no	—
31	Dic₂₃₉	C₄⋊C₄	1	2	yes	—
32	C₂×C₂₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₂₃₉	C₂×SD₁₆	1	2	yes	—
34	D₂₃₉	#106	1	4	yes	direct product factor
35	C₂₃₉	C₂×#106	1	1	yes	characteristic, complemented, Sylow
36	C₂×#106	—	239	1	no	maximal, Sylow
37	C₂²×C₈	—	239	1	no	—
38	C₂×C₄⋊C₄	—	239	2	no	—
39	#106	—	239	1	no	—
40	#106	—	239	2	no	—
41	#106	D₂₃₉	1	1	yes	characteristic, direct product factor
42	C₂²×C₄	—	478	2	no	—
43	C₂²×C₄	—	239	1	no	—
44	C₂×C₈	—	239	1	no	—
45	C₂×C₈	—	239	4	no	—
46	C₂×C₈	C₂×D₂₃₉	1	1	yes	characteristic
47	C₄⋊C₄	—	478	2	no	—
48	C₄⋊C₄	—	239	2	no	—
49	C₄⋊C₄	C₂×D₂₃₉	1	2	yes	—
50	C₂³	—	239	1	no	—
51	C₂×C₄	—	2	2	no	—
52	C₂×C₄	—	956	4	no	—
53	C₂×C₄	—	478	2	no	—
54	C₂×C₄	—	239	1	no	—
55	C₂×C₄	—	239	2	no	—
56	C₂×C₄	—	239	2	no	—
57	C₂×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
58	C₈	—	239	2	no	—
59	C₈	C₄×D₂₃₉	1	2	yes	complemented
60	C₂²	—	239	2	no	—
61	C₂²	—	239	4	no	—
62	C₂²	D₄×D₂₃₉	1	1	yes	characteristic
63	C₄	—	4	2	no	—
64	C₄	—	956	2	no	—
65	C₄	—	239	2	no	—
66	C₄	C₂×C₄×D₂₃₉	1	1	yes	characteristic
67	C₄	Q₈×D₂₃₉	1	1	yes	characteristic
68	C₂	—	239	4	no	—
69	C₂	C₄⋊C₄×D₂₃₉	1	1	yes	characteristic
70	C₂	SD₁₆×D₂₃₉	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₂₃₉
commutator	C₄×C₂₃₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₃₉
Fitting	#106×C₂₃₉	C₂
socle	C₂²×C₂₃₉	C₂×D₄
prime radical	C₂²×D₂₃₉	D₄
cosocle	C₂×C₄×C₂₃₉	C₂³
hypercenter	#106	D₂₃₉
hypocenter	C₂₃₉	C₂×#106
LCS	G ▷ C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
3. #21439892	C₂	semidirect
4. #21439891	C₂	semidirect
5. #106×C₂₃₉	C₂	semidirect
9. C₈×D₂₃₉	C₄	semidirect
20. C₂₃₉⋊C₈	C₂×C₄	semidirect
23. C₈×C₂₃₉	C₂×C₄	semidirect
34. D₂₃₉	#106	direct (N × H)
35. C₂₃₉	C₂×#106	semidirect
41. #106	D₂₃₉	direct (N × H)
59. C₈	C₄×D₂₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₂₃₉	2
C₄⋊C₄×D₂₃₉	2
#21439892	2
#21439891	2
#106×C₂₃₉	2
C₂×#106	239

Subgroups — #106×D239 #29504801

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #106×D₂₃₉ #29504801