Subgroups — C₄×D₈×C₂₃₉ #29504341

Order 15296.

134 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	C₄×D₄×C₂₃₉	C₂	1	2	yes	maximal, complemented
3	#107×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#102×C₂₃₉	C₂	1	2	yes	maximal, complemented
5	C₄×C₈×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄×C₂₃₉	C₂²	1	2	yes	—
7	C₂²×C₄×C₂₃₉	—	2	2	no	—
8	D₈×C₂₃₉	C₄	1	4	yes	direct product factor
9	C₂×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
10	C₂×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
11	C₄⋊C₄×C₂₃₉	C₂²	1	2	yes	—
12	C₂²⋊C₄×C₂₃₉	—	2	2	no	—
13	C₄²×C₂₃₉	C₂²	1	1	yes	characteristic
14	C₂³×C₂₃₉	—	2	2	no	—
15	D₄×C₂₃₉	—	2	2	no	—
16	D₄×C₂₃₉	C₂×C₄	1	4	yes	complemented
17	C₂×C₄×C₂₃₉	—	2	2	no	—
18	C₂×C₄×C₂₃₉	—	4	4	no	—
19	C₂×C₄×C₂₃₉	D₄	1	1	yes	characteristic
20	C₂×C₄×C₂₃₉	D₄	1	1	yes	characteristic
21	C₂×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
22	C₈×C₂₃₉	—	2	1	no	—
23	C₈×C₂₃₉	C₂×C₄	1	2	yes	complemented
24	C₂²×C₂₃₉	—	2	4	no	—
25	C₂²×C₂₃₉	—	4	2	no	—
26	C₂²×C₂₃₉	—	4	2	no	—
27	C₂²×C₂₃₉	C₂×D₄	1	1	yes	characteristic
28	C₄×C₂₃₉	—	2	1	no	—
29	C₄×C₂₃₉	—	4	2	no	—
30	C₄×C₂₃₉	D₈	1	2	yes	direct product factor
31	C₄×C₂₃₉	C₂²×C₄	1	1	yes	characteristic
32	C₄×C₂₃₉	#41	1	1	yes	characteristic
33	C₂×C₂₃₉	—	4	4	no	—
34	C₂×C₂₃₉	C₄×D₄	1	1	yes	characteristic
35	C₂×C₂₃₉	C₂×D₈	1	1	yes	characteristic
36	C₂×C₂₃₉	#135	1	1	yes	characteristic
37	C₂₃₉	C₄×D₈	1	1	yes	characteristic, direct product factor, Sylow
38	C₄×D₈	C₂₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	C₂×D₈	C₂×C₂₃₉	1	1	yes	characteristic
40	C₄×D₄	C₂×C₂₃₉	1	2	yes	complemented
41	#107	C₂×C₂₃₉	1	1	yes	characteristic, complemented
42	#102	C₂×C₂₃₉	1	2	yes	complemented
43	C₄×C₈	C₂×C₂₃₉	1	1	yes	characteristic, complemented
44	C₂×D₄	C₂²×C₂₃₉	1	2	yes	—
45	C₂²×C₄	—	2	2	no	—
46	D₈	C₄×C₂₃₉	1	4	yes	direct product factor
47	C₂×C₈	C₂²×C₂₃₉	1	1	yes	characteristic
48	C₂×C₈	C₂²×C₂₃₉	1	1	yes	characteristic
49	C₄⋊C₄	C₂²×C₂₃₉	1	2	yes	—
50	C₂²⋊C₄	—	2	2	no	—
51	C₄²	C₂²×C₂₃₉	1	1	yes	characteristic
52	C₂³	—	2	2	no	—
53	D₄	—	2	2	no	—
54	D₄	C₂×C₄×C₂₃₉	1	4	yes	complemented
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	4	4	no	—
57	C₂×C₄	D₄×C₂₃₉	1	1	yes	characteristic
58	C₂×C₄	D₄×C₂₃₉	1	1	yes	characteristic
59	C₂×C₄	C₂³×C₂₃₉	1	1	yes	characteristic
60	C₈	—	2	1	no	—
61	C₈	C₂×C₄×C₂₃₉	1	2	yes	complemented
62	C₂²	—	2	4	no	—
63	C₂²	—	4	2	no	—
64	C₂²	—	4	2	no	—
65	C₂²	C₂×D₄×C₂₃₉	1	1	yes	characteristic
66	C₄	—	2	1	no	—
67	C₄	—	4	2	no	—
68	C₄	D₈×C₂₃₉	1	2	yes	direct product factor
69	C₄	C₂²×C₄×C₂₃₉	1	1	yes	characteristic
70	C₄	#41×C₂₃₉	1	1	yes	characteristic
71	C₂	—	4	4	no	—
72	C₂	C₄×D₄×C₂₃₉	1	1	yes	characteristic
73	C₂	C₂×D₈×C₂₃₉	1	1	yes	characteristic
74	C₂	#135×C₂₃₉	1	1	yes	characteristic

	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	C₂×D₈×C₂₃₉	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃₉ ◁ C₄²×C₂₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×D₄×C₂₃₉	C₂	semidirect
3. #107×C₂₃₉	C₂	semidirect
4. #102×C₂₃₉	C₂	semidirect
5. C₄×C₈×C₂₃₉	C₂	semidirect
8. D₈×C₂₃₉	C₄	direct (N × H)
16. D₄×C₂₃₉	C₂×C₄	semidirect
23. C₈×C₂₃₉	C₂×C₄	semidirect
30. C₄×C₂₃₉	D₈	direct (N × H)
37. C₂₃₉	C₄×D₈	direct (N × H)
38. C₄×D₈	C₂₃₉	direct (N × H)
40. C₄×D₄	C₂×C₂₃₉	semidirect
41. #107	C₂×C₂₃₉	semidirect
42. #102	C₂×C₂₃₉	semidirect
43. C₄×C₈	C₂×C₂₃₉	semidirect
46. D₈	C₄×C₂₃₉	direct (N × H)
54. D₄	C₂×C₄×C₂₃₉	semidirect
61. C₈	C₂×C₄×C₂₃₉	semidirect
68. C₄	D₈×C₂₃₉	direct (N × H)

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
C₄×D₄×C₂₃₉	2
#107×C₂₃₉	2
#102×C₂₃₉	2
C₄×C₈×C₂₃₉	2
C₄×D₈	239

Subgroups — C4×D8×C239 #29504341

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×D₈×C₂₃₉ #29504341