Subgroups — #359×C₂₃₉ #29504263

Order 15296.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#135×C₂₃₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#108×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#41×C₂₃₉	—	2	1	no	—
5	Q₁₆×C₂₃₉	C₄	1	1	yes	characteristic
6	SD₁₆×C₂₃₉	—	2	1	no	—
7	D₈×C₂₃₉	C₄	1	1	yes	characteristic
8	M₁₆×C₂₃₉	—	2	1	no	—
9	C₂×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
10	C₁₆×C₂₃₉	—	2	1	no	—
11	Q₈×C₂₃₉	—	2	1	no	—
12	D₄×C₂₃₉	—	2	1	no	—
13	D₄×C₂₃₉	—	4	1	no	—
14	C₂×C₄×C₂₃₉	—	4	1	no	—
15	C₂×C₄×C₂₃₉	D₄	1	1	yes	characteristic
16	C₈×C₂₃₉	—	4	1	no	—
17	C₈×C₂₃₉	C₂×C₄	1	1	yes	characteristic
18	C₈×C₂₃₉	D₄	1	1	yes	characteristic
19	C₂²×C₂₃₉	—	4	1	no	—
20	C₂²×C₂₃₉	SD₁₆	1	1	yes	characteristic
21	C₄×C₂₃₉	—	4	1	no	—
22	C₄×C₂₃₉	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×C₂₃₉	D₈	1	1	yes	characteristic
24	C₂×C₂₃₉	—	2	1	no	—
25	C₂×C₂₃₉	—	8	1	no	—
26	C₂×C₂₃₉	#102	1	1	yes	characteristic
27	C₂₃₉	#359	1	1	yes	characteristic, direct product factor, Sylow
28	#359	C₂₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	#135	C₂×C₂₃₉	1	1	yes	characteristic
30	C₂×C₁₆	C₂×C₂₃₉	1	1	yes	characteristic, complemented
31	#108	C₂×C₂₃₉	1	1	yes	characteristic, complemented
32	#41	—	2	1	no	—
33	Q₁₆	C₄×C₂₃₉	1	1	yes	characteristic
34	SD₁₆	—	2	1	no	—
35	D₈	C₄×C₂₃₉	1	1	yes	characteristic
36	M₁₆	—	2	1	no	—
37	C₂×C₈	C₂²×C₂₃₉	1	1	yes	characteristic
38	C₁₆	—	2	1	no	—
39	Q₈	—	2	1	no	—
40	D₄	—	2	1	no	—
41	D₄	—	4	1	no	—
42	C₂×C₄	—	4	1	no	—
43	C₂×C₄	D₄×C₂₃₉	1	1	yes	characteristic
44	C₈	—	4	1	no	—
45	C₈	C₂×C₄×C₂₃₉	1	1	yes	characteristic
46	C₈	D₄×C₂₃₉	1	1	yes	characteristic
47	C₂²	—	4	1	no	—
48	C₂²	SD₁₆×C₂₃₉	1	1	yes	characteristic
49	C₄	—	4	1	no	—
50	C₄	C₂²⋊C₄×C₂₃₉	1	1	yes	characteristic
51	C₄	D₈×C₂₃₉	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	8	1	no	—
54	C₂	#102×C₂₃₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₃₉	D₈
commutator	C₈	C₂×C₄×C₂₃₉
Frattini, cosocle	C₂×C₈	C₂²×C₂₃₉
socle	C₂×C₂₃₉	#102
prime radical	#135×C₂₃₉	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₃₉ ◁ C₂×C₄×C₂₃₉ ◁ C₂×C₈×C₂₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₆×C₂₃₉	C₂	semidirect
3. #108×C₂₃₉	C₂	semidirect
27. C₂₃₉	#359	direct (N × H)
28. #359	C₂₃₉	direct (N × H)
30. C₂×C₁₆	C₂×C₂₃₉	semidirect
31. #108	C₂×C₂₃₉	semidirect

Maximal subgroups

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

subgroup	index
#135×C₂₃₉	2
C₂×C₁₆×C₂₃₉	2
#108×C₂₃₉	2
#359	239

Subgroups — #359×C239 #29504263

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #359×C₂₃₉ #29504263