Subgroups — #44482754

Order 30592.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#29504602	C₂	1	1	yes	maximal, characteristic
2	#29504600	C₂	1	1	yes	maximal, characteristic
3	M₆₄×C₂₃₉	C₂	1	1	yes	maximal, characteristic
4	#21439896	—	2	1	no	—
5	#21439894	—	2	1	no	—
6	C₂×C₁₆×C₂₃₉	C₂²	1	1	yes	characteristic
7	C₃₂×C₂₃₉	C₄	1	2	yes	complemented
8	#16838532	—	4	1	no	—
9	C₄⋊Dic₂₃₉	—	4	1	no	—
10	C₂×C₈×C₂₃₉	D₄	1	1	yes	characteristic
11	C₁₆×C₂₃₉	C₂×C₄	1	1	yes	characteristic
12	C₁₆×C₂₃₉	Q₈	1	1	yes	characteristic
13	C₂×Dic₂₃₉	—	8	1	no	—
14	C₂₃₉⋊C₈	—	8	1	no	—
15	C₂×C₄×C₂₃₉	D₈	1	1	yes	characteristic
16	C₈×C₂₃₉	C₄⋊C₄	1	1	yes	characteristic
17	C₈×C₂₃₉	Q₁₆	1	1	yes	characteristic
18	C₂²×C₂₃₉	SD₃₂	1	1	yes	characteristic
19	C₄×C₂₃₉	#107	1	1	yes	characteristic
20	C₄×C₂₃₉	SD₃₂	1	1	yes	characteristic
21	Dic₂₃₉	—	16	1	no	—
22	C₂×C₂₃₉	—	2	1	no	—
23	C₂×C₂₃₉	#367	1	1	yes	characteristic
24	C₂₃₉	#1426	1	1	yes	characteristic, complemented, Sylow
25	#1426	—	239	1	no	maximal, Sylow
26	M₆₄	D₂₃₉	1	1	yes	characteristic
27	#368	—	239	1	no	—
28	#367	—	239	1	no	—
29	C₂×C₁₆	C₂×D₂₃₉	1	1	yes	characteristic
30	#108	—	478	1	no	—
31	#107	—	478	1	no	—
32	C₃₂	Dic₂₃₉	1	2	yes	complemented
33	M₁₆	—	956	1	no	—
34	C₂×C₈	D₉₅₆	1	1	yes	characteristic
35	C₄⋊C₄	—	956	1	no	—
36	C₁₆	C₂×Dic₂₃₉	1	1	yes	characteristic
37	C₁₆	Dic₄₇₈	1	1	yes	characteristic
38	C₂×C₄	—	1912	1	no	—
39	C₂×C₄	D₁₉₁₂	1	1	yes	characteristic
40	C₈	—	1912	1	no	—
41	C₈	C₄⋊Dic₂₃₉	1	1	yes	characteristic
42	C₈	Dic₉₅₆	1	1	yes	characteristic
43	C₂²	#21439905	1	1	yes	characteristic
44	C₄	—	3824	1	no	—
45	C₄	#21439894	1	1	yes	characteristic
46	C₄	#21439905	1	1	yes	characteristic
47	C₂	—	2	1	no	—
48	C₂	#29504600	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#29504600
commutator	C₁₆×C₂₃₉	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₂₃₉
Fitting	M₆₄×C₂₃₉	C₂
socle	C₂×C₂₃₉	#367
prime radical	C₂²×C₂₃₉	SD₃₂
cosocle	C₂×C₁₆×C₂₃₉	C₂²
hypercenter	M₆₄	D₂₃₉
hypocenter	C₂₃₉	#1426
LCS	G ▷ C₁₆×C₂₃₉ ▷ C₈×C₂₃₉ ▷ C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
7. C₃₂×C₂₃₉	C₄	semidirect
24. C₂₃₉	#1426	semidirect
32. C₃₂	Dic₂₃₉	semidirect

Maximal subgroups

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

subgroup	index
#29504602	2
#29504600	2
M₆₄×C₂₃₉	2
#1426	239