Subgroups — #427×C₂₃₉ #29504331

Order 15296.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×C₂₃₉	C₂	1	1	yes	maximal, characteristic
2	#117×C₂₃₉	C₂	1	2	yes	maximal
3	#107×C₂₃₉	C₂	1	2	yes	maximal, complemented
4	#106×C₂₃₉	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄×C₂₃₉	C₂²	1	1	yes	characteristic
6	C₂×C₈×C₂₃₉	C₄	1	4	yes	complemented
7	C₂×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
8	C₂×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
9	C₄⋊C₄×C₂₃₉	C₂²	1	4	yes	—
10	C₂²⋊C₄×C₂₃₉	—	2	2	no	—
11	C₄²×C₂₃₉	—	2	2	no	—
12	C₂³×C₂₃₉	D₄	1	1	yes	characteristic
13	C₂×C₄×C₂₃₉	—	2	4	no	—
14	C₂×C₄×C₂₃₉	C₂×C₄	1	2	yes	—
15	C₂×C₄×C₂₃₉	D₄	1	1	yes	characteristic
16	C₂×C₄×C₂₃₉	Q₈	1	2	yes	—
17	C₂×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
18	C₈×C₂₃₉	C₂×C₄	1	2	yes	—
19	C₈×C₂₃₉	C₂×C₄	1	2	yes	—
20	C₂²×C₂₃₉	—	2	2	no	—
21	C₂²×C₂₃₉	C₄⋊C₄	1	2	yes	—
22	C₂²×C₂₃₉	C₂×D₄	1	1	yes	characteristic
23	C₄×C₂₃₉	—	4	4	no	—
24	C₄×C₂₃₉	C₄⋊C₄	1	2	yes	—
25	C₄×C₂₃₉	C₂²×C₄	1	1	yes	characteristic
26	C₄×C₂₃₉	C₂×Q₈	1	1	yes	characteristic
27	C₂×C₂₃₉	—	2	2	no	—
28	C₂×C₂₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₂₃₉	#135	1	2	yes	—
30	C₂₃₉	#427	1	1	yes	characteristic, direct product factor, Sylow
31	#427	C₂₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂²×C₈	C₂×C₂₃₉	1	1	yes	characteristic
33	#117	C₂×C₂₃₉	1	2	yes	—
34	#107	C₂×C₂₃₉	1	2	yes	complemented
35	#106	C₂×C₂₃₉	1	2	yes	complemented
36	C₂²×C₄	C₂²×C₂₃₉	1	1	yes	characteristic
37	C₂×C₈	C₄×C₂₃₉	1	4	yes	complemented
38	C₂×C₈	C₂²×C₂₃₉	1	1	yes	characteristic
39	C₂×C₈	C₂²×C₂₃₉	1	1	yes	characteristic
40	C₄⋊C₄	C₂²×C₂₃₉	1	4	yes	—
41	C₂²⋊C₄	—	2	2	no	—
42	C₄²	—	2	2	no	—
43	C₂³	D₄×C₂₃₉	1	1	yes	characteristic
44	C₂×C₄	—	2	4	no	—
45	C₂×C₄	C₂×C₄×C₂₃₉	1	2	yes	—
46	C₂×C₄	D₄×C₂₃₉	1	1	yes	characteristic
47	C₂×C₄	Q₈×C₂₃₉	1	2	yes	—
48	C₂×C₄	C₂³×C₂₃₉	1	1	yes	characteristic
49	C₈	C₂×C₄×C₂₃₉	1	2	yes	—
50	C₈	C₂×C₄×C₂₃₉	1	2	yes	—
51	C₂²	—	2	2	no	—
52	C₂²	C₄⋊C₄×C₂₃₉	1	2	yes	—
53	C₂²	C₂×D₄×C₂₃₉	1	1	yes	characteristic
54	C₄	—	4	4	no	—
55	C₄	C₄⋊C₄×C₂₃₉	1	2	yes	—
56	C₄	C₂²×C₄×C₂₃₉	1	1	yes	characteristic
57	C₄	C₂×Q₈×C₂₃₉	1	1	yes	characteristic
58	C₂	—	2	2	no	—
59	C₂	C₂×C₄⋊C₄×C₂₃₉	1	1	yes	characteristic
60	C₂	#135×C₂₃₉	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃₉	D₄
commutator	C₄	C₂²×C₄×C₂₃₉
Frattini, cosocle	C₂×C₄	C₂³×C₂₃₉
socle	C₂²×C₂₃₉	C₂×D₄
prime radical	C₂³×C₂₃₉	D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃₉ ◁ C₂²×C₄×C₂₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. #107×C₂₃₉	C₂	semidirect
4. #106×C₂₃₉	C₂	semidirect
6. C₂×C₈×C₂₃₉	C₄	semidirect
30. C₂₃₉	#427	direct (N × H)
31. #427	C₂₃₉	direct (N × H)
34. #107	C₂×C₂₃₉	semidirect
35. #106	C₂×C₂₃₉	semidirect
37. C₂×C₈	C₄×C₂₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₈×C₂₃₉	2
#117×C₂₃₉	2
#107×C₂₃₉	2
#106×C₂₃₉	2
#427	239

Subgroups — #427×C239 #29504331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #427×C₂₃₉ #29504331