Subgroups — #44482622

Order 30592.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#29504525	C₂	1	2	yes	maximal
2	C₄×C₁₆×C₂₃₉	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₂₃₉⋊C₈)	—	2	2	no	—
4	C₂×C₁₆×C₂₃₉	C₄	1	1	yes	characteristic
5	C₂×C₁₆×C₂₃₉	C₄	1	1	yes	characteristic
6	C₄×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
7	C₂×C₂₃₉⋊C₈	—	4	2	no	—
8	C₂×C₈×C₂₃₉	C₂×C₄	1	1	yes	characteristic
9	C₂×C₈×C₂₃₉	Q₈	1	1	yes	characteristic
10	C₄²×C₂₃₉	D₄	1	1	yes	characteristic
11	C₁₆×C₂₃₉	—	2	1	no	—
12	C₁₆×C₂₃₉	C₈	1	2	yes	complemented
13	C₂₃₉⋊C₈	—	8	2	no	—
14	C₂×C₄×C₂₃₉	C₄⋊C₄	1	1	yes	characteristic
15	C₂×C₄×C₂₃₉	D₈	1	1	yes	characteristic
16	C₂×C₄×C₂₃₉	Q₁₆	1	1	yes	characteristic
17	C₈×C₂₃₉	—	2	1	no	—
18	C₈×C₂₃₉	C₂×C₈	1	1	yes	characteristic
19	C₈×C₂₃₉	M₁₆	1	1	yes	characteristic
20	C₂²×C₂₃₉	#107	1	1	yes	characteristic
21	C₄×C₂₃₉	—	2	1	no	—
22	C₄×C₂₃₉	C₄⋊C₈	1	1	yes	characteristic
23	C₄×C₂₃₉	#108	1	1	yes	characteristic
24	C₄×C₂₃₉	SD₃₂	1	2	yes	—
25	C₂×C₂₃₉	#335	1	1	yes	characteristic
26	C₂×C₂₃₉	#367	1	1	yes	characteristic
27	C₂×C₂₃₉	#368	1	1	yes	characteristic
28	C₂₃₉	#1372	1	1	yes	characteristic, complemented, Sylow
29	#1372	—	239	1	no	maximal, Sylow
30	C₄×C₁₆	D₂₃₉	1	1	yes	characteristic
31	#335	—	239	2	no	—
32	C₂×C₁₆	Dic₂₃₉	1	1	yes	characteristic
33	C₂×C₁₆	Dic₂₃₉	1	1	yes	characteristic
34	C₄⋊C₈	—	478	2	no	—
35	C₄×C₈	C₂×D₂₃₉	1	1	yes	characteristic
36	C₂×C₈	—	956	2	no	—
37	C₂×C₈	C₂×Dic₂₃₉	1	1	yes	characteristic
38	C₂×C₈	Dic₄₇₈	1	1	yes	characteristic
39	C₄²	D₉₅₆	1	1	yes	characteristic
40	C₁₆	—	2	1	no	—
41	C₁₆	C₂₃₉⋊C₈	1	2	yes	complemented
42	C₂×C₄	C₄⋊Dic₂₃₉	1	1	yes	characteristic
43	C₂×C₄	D₁₉₁₂	1	1	yes	characteristic
44	C₂×C₄	Dic₉₅₆	1	1	yes	characteristic
45	C₈	—	2	1	no	—
46	C₈	—	1912	2	no	—
47	C₈	C₂×C₂₃₉⋊C₈	1	1	yes	characteristic
48	C₈	#16838532	1	1	yes	characteristic
49	C₂²	#21439894	1	1	yes	characteristic
50	C₄	—	2	1	no	—
51	C₄	C₄⋊(C₂₃₉⋊C₈)	1	1	yes	characteristic
52	C₄	#21439896	1	1	yes	characteristic
53	C₄	#21439905	1	2	yes	—
54	C₂	#29504525	1	1	yes	characteristic
55	C₂	#29504600	1	1	yes	characteristic
56	C₂	#29504602	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₁₉₁₂
commutator	C₈×C₂₃₉	C₂×C₈
Frattini	C₄×C₈	C₂×D₂₃₉
Fitting	C₄×C₁₆×C₂₃₉	C₂
socle, prime radical	C₂²×C₂₃₉	#107
cosocle	C₄×C₈×C₂₃₉	C₂²
hypercenter	C₄×C₁₆	D₂₃₉
hypocenter	C₂₃₉	#1372
LCS	G ▷ C₈×C₂₃₉ ▷ C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
12. C₁₆×C₂₃₉	C₈	semidirect
28. C₂₃₉	#1372	semidirect
41. C₁₆	C₂₃₉⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#29504525	2
C₄×C₁₆×C₂₃₉	2
#1372	239