Subgroups — #29505172

Order 15296.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21439951	C₂	1	2	yes	maximal
2	#21439893	C₂	1	2	yes	maximal
3	#21439892	C₂	1	1	yes	maximal, characteristic
4	C₈×Dic₂₃₉	C₂	1	1	yes	maximal, characteristic
5	#106×C₂₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂₃₉⋊C₈	C₂²	1	1	yes	characteristic
7	C₄⋊Dic₂₃₉	—	2	2	no	—
8	C₄⋊Dic₂₃₉	C₂²	1	2	yes	—
9	Dic₂₃₉⋊C₄	—	2	2	no	—
10	C₄×Dic₂₃₉	C₂²	1	1	yes	characteristic
11	C₂×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₂₃₉	C₂²	1	2	yes	—
13	C₂×Dic₂₃₉	—	2	2	no	—
14	C₂×Dic₂₃₉	D₄	1	2	yes	—
15	C₂₃₉⋊C₈	Q₈	1	2	yes	—
16	C₂×C₄×C₂₃₉	—	2	2	no	—
17	C₂×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
18	C₈×C₂₃₉	Q₈	1	2	yes	—
19	C₂²×C₂₃₉	C₂×D₄	1	1	yes	characteristic
20	C₄×C₂₃₉	—	4	2	no	—
21	C₄×C₂₃₉	C₂×Q₈	1	1	yes	characteristic
22	C₄×C₂₃₉	C₂×Q₈	1	1	yes	characteristic
23	Dic₂₃₉	—	2	2	no	—
24	Dic₂₃₉	—	4	2	no	—
25	C₂×C₂₃₉	C₄⋊Q₈	1	1	yes	characteristic
26	C₂×C₂₃₉	#135	1	2	yes	—
27	C₂₃₉	#499	1	1	yes	characteristic, complemented, Sylow
28	#499	—	239	1	no	maximal, Sylow
29	#125	—	239	2	no	—
30	#107	—	239	2	no	—
31	#106	—	239	1	no	—
32	#106	D₂₃₉	1	1	yes	characteristic
33	C₄×C₈	—	239	1	no	—
34	C₂×C₈	—	239	1	no	—
35	C₂×C₈	C₂×D₂₃₉	1	1	yes	characteristic
36	C₄⋊C₄	—	478	2	no	—
37	C₄⋊C₄	—	478	2	no	—
38	C₄⋊C₄	—	239	2	no	—
39	C₄⋊C₄	C₂×D₂₃₉	1	2	yes	—
40	C₄²	—	239	1	no	—
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	—	478	2	no	—
43	C₂×C₄	—	239	2	no	—
44	C₂×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
45	C₈	—	239	2	no	—
46	C₈	Dic₄₇₈	1	2	yes	—
47	C₂²	D₄×D₂₃₉	1	1	yes	characteristic
48	C₄	—	4	2	no	—
49	C₄	—	956	2	no	—
50	C₄	—	478	2	no	—
51	C₄	Q₈×D₂₃₉	1	1	yes	characteristic
52	C₄	C₂×Dic₄₇₈	1	1	yes	characteristic
53	C₂	#21439959	1	1	yes	characteristic
54	C₂	#21439981	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₂₃₉
commutator, cosocle	C₂×C₄×C₂₃₉	C₂³
Frattini	C₂×C₄	C₂²×D₂₃₉
Fitting	#106×C₂₃₉	C₂
socle, prime radical	C₂²×C₂₃₉	C₂×D₄
hypercenter	#106	D₂₃₉
hypocenter	C₂₃₉	#499
LCS	G ▷ C₂×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
27. C₂₃₉	#499	semidirect

Maximal subgroups

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

subgroup	index
#21439951	2
#21439893	2
#21439892	2
C₈×Dic₂₃₉	2
#106×C₂₃₉	2
#499	239