Subgroups — #357×C₂₃₉ #29504261

Order 15296.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₈×C₂₃₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#107×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₄×C₂₃₉	—	2	1	no	—
5	D₈×C₂₃₉	—	2	1	no	—
6	D₈×C₂₃₉	C₄	1	2	yes	complemented
7	C₂×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
8	C₄⋊C₄×C₂₃₉	—	2	1	no	—
9	C₁₆×C₂₃₉	—	2	1	no	—
10	C₂³×C₂₃₉	—	4	1	no	—
11	D₄×C₂₃₉	—	2	2	no	—
12	D₄×C₂₃₉	—	4	1	no	—
13	C₂×C₄×C₂₃₉	—	4	1	no	—
14	C₂×C₄×C₂₃₉	D₄	1	1	yes	characteristic
15	C₈×C₂₃₉	C₂×C₄	1	1	yes	characteristic
16	C₈×C₂₃₉	D₄	1	1	yes	characteristic
17	C₂²×C₂₃₉	—	4	2	no	—
18	C₂²×C₂₃₉	—	8	1	no	—
19	C₂²×C₂₃₉	—	8	1	no	—
20	C₂²×C₂₃₉	D₈	1	1	yes	characteristic
21	C₄×C₂₃₉	—	8	1	no	—
22	C₄×C₂₃₉	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×C₂₃₉	SD₁₆	1	1	yes	characteristic
24	C₂×C₂₃₉	—	8	2	no	—
25	C₂×C₂₃₉	#102	1	1	yes	characteristic
26	C₂×C₂₃₉	D₁₆	1	1	yes	characteristic
27	C₂×C₂₃₉	SD₃₂	1	1	yes	characteristic
28	C₂₃₉	#357	1	1	yes	characteristic, direct product factor, Sylow
29	#357	C₂₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂×D₈	C₂×C₂₃₉	1	1	yes	characteristic
31	C₂×C₁₆	C₂×C₂₃₉	1	1	yes	characteristic, complemented
32	#107	C₂×C₂₃₉	1	1	yes	characteristic, complemented
33	C₂×D₄	—	2	1	no	—
34	D₈	—	2	1	no	—
35	D₈	C₄×C₂₃₉	1	2	yes	complemented
36	C₂×C₈	C₂²×C₂₃₉	1	1	yes	characteristic
37	C₄⋊C₄	—	2	1	no	—
38	C₁₆	—	2	1	no	—
39	C₂³	—	4	1	no	—
40	D₄	—	2	2	no	—
41	D₄	—	4	1	no	—
42	C₂×C₄	—	4	1	no	—
43	C₂×C₄	D₄×C₂₃₉	1	1	yes	characteristic
44	C₈	C₂×C₄×C₂₃₉	1	1	yes	characteristic
45	C₈	D₄×C₂₃₉	1	1	yes	characteristic
46	C₂²	—	4	2	no	—
47	C₂²	—	8	1	no	—
48	C₂²	—	8	1	no	—
49	C₂²	D₈×C₂₃₉	1	1	yes	characteristic
50	C₄	—	8	1	no	—
51	C₄	C₂²⋊C₄×C₂₃₉	1	1	yes	characteristic
52	C₄	SD₁₆×C₂₃₉	1	1	yes	characteristic
53	C₂	—	8	2	no	—
54	C₂	#102×C₂₃₉	1	1	yes	characteristic
55	C₂	D₁₆×C₂₃₉	1	1	yes	characteristic
56	C₂	SD₃₂×C₂₃₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃₉	D₈
commutator	C₈	C₂×C₄×C₂₃₉
Frattini, cosocle	C₂×C₈	C₂²×C₂₃₉
prime radical	C₂×D₈×C₂₃₉	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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. #107×C₂₃₉	C₂	semidirect
6. D₈×C₂₃₉	C₄	semidirect
28. C₂₃₉	#357	direct (N × H)
29. #357	C₂₃₉	direct (N × H)
31. C₂×C₁₆	C₂×C₂₃₉	semidirect
32. #107	C₂×C₂₃₉	semidirect
35. D₈	C₄×C₂₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₈×C₂₃₉	2
C₂×C₁₆×C₂₃₉	2
#107×C₂₃₉	2
#357	239

Subgroups — #357×C239 #29504261

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #357×C₂₃₉ #29504261