Subgroups — #44482732

Order 30592.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃₉⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#29504602	C₂	1	1	yes	maximal, characteristic, complemented
3	#508×C₂₃₉	C₂	1	1	yes	maximal, characteristic
4	#21439896	—	2	1	no	—
5	C₂₃₉⋊C₃₂	—	2	1	no	—
6	#135×C₂₃₉	—	2	1	no	—
7	Q₃₂×C₂₃₉	C₄	1	1	yes	characteristic
8	SD₃₂×C₂₃₉	—	2	1	no	—
9	D₁₆×C₂₃₉	C₄	1	1	yes	characteristic
10	C₂×C₁₆×C₂₃₉	C₂²	1	1	yes	characteristic
11	#16838532	—	4	1	no	—
12	#41×C₂₃₉	—	4	1	no	—
13	Q₁₆×C₂₃₉	—	2	1	no	—
14	SD₁₆×C₂₃₉	—	4	1	no	—
15	D₈×C₂₃₉	—	2	1	no	—
16	C₂×C₈×C₂₃₉	D₄	1	1	yes	characteristic
17	C₁₆×C₂₃₉	C₂×C₄	1	1	yes	characteristic
18	C₁₆×C₂₃₉	D₄	1	1	yes	characteristic
19	C₂₃₉⋊C₈	—	8	1	no	—
20	Q₈×C₂₃₉	—	4	1	no	—
21	D₄×C₂₃₉	—	4	1	no	—
22	D₄×C₂₃₉	—	8	1	no	—
23	C₂×C₄×C₂₃₉	—	8	1	no	—
24	C₂×C₄×C₂₃₉	D₈	1	1	yes	characteristic
25	C₈×C₂₃₉	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₂₃₉	SD₁₆	1	1	yes	characteristic
27	C₂²×C₂₃₉	—	8	1	no	—
28	C₂²×C₂₃₉	SD₃₂	1	1	yes	characteristic
29	C₄×C₂₃₉	—	8	1	no	—
30	C₄×C₂₃₉	#102	1	1	yes	characteristic
31	C₄×C₂₃₉	D₁₆	1	1	yes	characteristic
32	C₂×C₂₃₉	—	2	1	no	—
33	C₂×C₂₃₉	—	16	1	no	—
34	C₂×C₂₃₉	#357	1	1	yes	characteristic
35	C₂₃₉	#1417	1	1	yes	characteristic, complemented, Sylow
36	#1417	—	239	1	no	maximal, Sylow
37	#508	D₂₃₉	1	1	yes	characteristic
38	C₂×C₃₂	—	239	1	no	—
39	#368	—	239	1	no	—
40	#135	—	2	1	no	—
41	Q₃₂	Dic₂₃₉	1	1	yes	characteristic
42	SD₃₂	—	2	1	no	—
43	D₁₆	Dic₂₃₉	1	1	yes	characteristic
44	C₂×C₁₆	C₂×D₂₃₉	1	1	yes	characteristic
45	#108	—	478	1	no	—
46	C₃₂	—	478	1	no	—
47	#41	—	4	1	no	—
48	Q₁₆	—	2	1	no	—
49	SD₁₆	—	4	1	no	—
50	D₈	—	2	1	no	—
51	M₁₆	—	956	1	no	—
52	C₂×C₈	C₂²⋊D₂₃₉	1	1	yes	characteristic
53	C₁₆	C₂×Dic₂₃₉	1	1	yes	characteristic
54	C₁₆	C₂²⋊D₂₃₉	1	1	yes	characteristic
55	Q₈	—	4	1	no	—
56	D₄	—	4	1	no	—
57	D₄	—	8	1	no	—
58	C₂×C₄	—	8	1	no	—
59	C₂×C₄	#16838533	1	1	yes	characteristic
60	C₈	—	1912	1	no	—
61	C₈	C₂²⋊Dic₂₃₉	1	1	yes	characteristic
62	C₈	#16838536	1	1	yes	characteristic
63	C₂²	—	8	1	no	—
64	C₂²	#21439904	1	1	yes	characteristic
65	C₄	—	8	1	no	—
66	C₄	#21439881	1	1	yes	characteristic
67	C₄	#21439901	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	16	1	no	—
70	C₂	#29504573	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#21439901
commutator	C₁₆×C₂₃₉	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₂₃₉
Fitting, prime radical	#508×C₂₃₉	C₂
socle	C₂×C₂₃₉	#357
cosocle	C₂×C₁₆×C₂₃₉	C₂²
hypercenter	#508	D₂₃₉
hypocenter	C₂₃₉	#1417
LCS	G ▷ C₁₆×C₂₃₉ ▷ C₈×C₂₃₉ ▷ C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #508

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃₉⋊C₃₂	C₂	semidirect
2. #29504602	C₂	semidirect
35. C₂₃₉	#1417	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃₉⋊C₃₂	2
#29504602	2
#508×C₂₃₉	2
#1417	239