Subgroups — #29505169

Order 15296.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21439959	C₂	1	2	yes	maximal
2	#21439892	C₂	1	1	yes	maximal, characteristic
3	#21439891	C₂	1	2	yes	maximal
4	C₈×Dic₂₃₉	C₂	1	1	yes	maximal, characteristic
5	#106×C₂₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₇₈	—	2	2	no	—
7	C₂×C₂₃₉⋊C₈	C₂²	1	1	yes	characteristic
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	Dic₄₇₈	—	4	4	no	—
14	C₂×Dic₂₃₉	—	2	2	no	—
15	C₂×Dic₂₃₉	D₄	1	2	yes	—
16	C₂₃₉⋊C₈	Q₈	1	2	yes	complemented
17	C₂×C₄×C₂₃₉	—	2	2	no	—
18	C₂×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
19	C₈×C₂₃₉	Q₈	1	2	yes	complemented
20	C₂²×C₂₃₉	C₂×D₄	1	1	yes	characteristic
21	C₄×C₂₃₉	—	4	2	no	—
22	C₄×C₂₃₉	C₂×Q₈	1	1	yes	characteristic
23	C₄×C₂₃₉	C₂×Q₈	1	1	yes	characteristic
24	Dic₂₃₉	—	4	2	no	—
25	Dic₂₃₉	SD₁₆	1	4	yes	—
26	C₂×C₂₃₉	C₄⋊Q₈	1	1	yes	characteristic
27	C₂×C₂₃₉	C₂×SD₁₆	1	2	yes	—
28	C₂₃₉	#498	1	1	yes	characteristic, complemented, Sylow
29	#498	—	239	1	no	maximal, Sylow
30	C₄⋊Q₈	—	239	2	no	—
31	#106	—	239	1	no	—
32	#106	—	239	2	no	—
33	#106	D₂₃₉	1	1	yes	characteristic
34	C₄×C₈	—	239	1	no	—
35	C₂×Q₈	—	478	2	no	—
36	C₂×C₈	—	239	1	no	—
37	C₂×C₈	C₂×D₂₃₉	1	1	yes	characteristic
38	C₄⋊C₄	—	478	2	no	—
39	C₄⋊C₄	—	239	2	no	—
40	C₄⋊C₄	C₂×D₂₃₉	1	2	yes	—
41	C₄²	—	239	1	no	—
42	Q₈	—	956	4	no	—
43	C₂×C₄	—	2	2	no	—
44	C₂×C₄	—	478	2	no	—
45	C₂×C₄	—	239	2	no	—
46	C₂×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
47	C₈	—	239	2	no	—
48	C₈	Dic₄₇₈	1	2	yes	complemented
49	C₂²	D₄×D₂₃₉	1	1	yes	characteristic
50	C₄	—	4	2	no	—
51	C₄	—	956	2	no	—
52	C₄	—	239	4	no	—
53	C₄	Q₈×D₂₃₉	1	1	yes	characteristic
54	C₄	C₂×Dic₄₇₈	1	1	yes	characteristic
55	C₂	#21439959	1	1	yes	characteristic
56	C₂	SD₁₆×D₂₃₉	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	#498
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
16. C₂₃₉⋊C₈	Q₈	semidirect
19. C₈×C₂₃₉	Q₈	semidirect
28. C₂₃₉	#498	semidirect
48. C₈	Dic₄₇₈	semidirect

Maximal subgroups

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

subgroup	index
#21439959	2
#21439892	2
#21439891	2
C₈×Dic₂₃₉	2
#106×C₂₃₉	2
#498	239