Subgroups — C₂×C₄×#1042 #11717423

Order 800.

224 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#1042	C₂	1	2	yes	maximal
2	C₂×C₄×C₅×D₅	C₂	1	1	yes	maximal, characteristic
3	C₄×#1042	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₅×D₅	C₂²	1	1	yes	characteristic
5	C₂×#1042	C₄	1	8	yes	direct product factor
6	C₂×#1042	C₂²	1	4	yes	—
7	C₂×C₄×C₅²	C₄	1	1	yes	characteristic, complemented
8	C₂×C₅×Dic₅	C₄	1	1	yes	characteristic, complemented
9	C₄×C₅×D₅	C₄	1	2	yes	complemented
10	C₄×C₅×D₅	C₂²	1	2	yes	—
11	C₂×C₄×AGL(1,𝔽₅)	—	5	1	no	maximal
12	C₂×C₄×Dic₅	—	5	1	no	maximal
13	C₂²×C₅²	C₂×C₄	1	1	yes	characteristic
14	C₂×C₅×D₅	C₂×C₄	1	4	yes	—
15	C₂×C₅×D₅	C₂×C₄	1	1	yes	characteristic
16	C₂×C₅×D₅	C₂³	1	1	yes	characteristic
17	#1042	C₂×C₄	1	8	yes	direct product factor
18	C₄×C₅²	C₂×C₄	1	2	yes	complemented
19	C₅×Dic₅	C₂×C₄	1	2	yes	complemented
20	C₂²×AGL(1,𝔽₅)	—	5	2	no	—
21	C₂²×C₄×C₅	—	5	1	no	—
22	C₂²×Dic₅	—	5	2	no	—
23	C₂×C₄×D₅	D₅	1	1	yes	characteristic
24	C₄×AGL(1,𝔽₅)	—	5	4	no	—
25	C₄×Dic₅	—	5	4	no	—
26	C₂×C₅²	C₄²	1	2	yes	complemented
27	C₂×C₅²	C₂²×C₄	1	1	yes	characteristic
28	C₅×D₅	C₄²	1	2	yes	complemented
29	C₅×D₅	C₂²×C₄	1	2	yes	—
30	C₂³×C₅	—	5	1	no	—
31	C₂²×D₅	C₂×D₅	1	1	yes	characteristic
32	C₂×AGL(1,𝔽₅)	—	5	8	no	—
33	C₂×AGL(1,𝔽₅)	—	5	4	no	—
34	C₂×C₄×C₅	—	4	1	no	—
35	C₂×C₄×C₅	—	5	1	no	—
36	C₂×C₄×C₅	—	5	2	no	—
37	C₂×C₄×C₅	—	5	2	no	—
38	C₂×C₄×C₅	Dic₅	1	1	yes	characteristic, complemented
39	C₂×C₄×C₅	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
40	C₂×Dic₅	—	5	4	no	—
41	C₂×Dic₅	—	5	8	no	—
42	C₂×Dic₅	Dic₅	1	1	yes	characteristic, complemented
43	C₄×D₅	Dic₅	1	2	yes	complemented
44	C₄×D₅	C₂×D₅	1	2	yes	—
45	C₂×C₄²	—	25	1	no	Sylow
46	C₅²	C₂×C₄²	1	1	yes	characteristic, complemented, Sylow
47	C₂²×C₅	—	4	1	no	—
48	C₂²×C₅	—	5	4	no	—
49	C₂²×C₅	—	5	1	no	—
50	C₂²×C₅	—	5	1	no	—
51	C₂²×C₅	C₂×Dic₅	1	1	yes	characteristic
52	C₂²×C₅	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
53	C₂×D₅	C₄×D₅	1	4	yes	—
54	C₂×D₅	C₂×Dic₅	1	1	yes	characteristic
55	C₂×D₅	C₂²×D₅	1	1	yes	characteristic
56	AGL(1,𝔽₅)	—	5	8	no	—
57	C₄×C₅	—	4	2	no	—
58	C₄×C₅	—	5	2	no	—
59	C₄×C₅	C₂×Dic₅	1	2	yes	complemented
60	C₄×C₅	C₂×AGL(1,𝔽₅)	1	2	yes	complemented
61	Dic₅	—	5	8	no	—
62	Dic₅	C₂×Dic₅	1	2	yes	complemented
63	C₂²×C₄	—	5	1	no	—
64	C₂²×C₄	—	25	2	no	—
65	C₄²	—	25	4	no	—
66	C₂×C₅	—	4	2	no	—
67	C₂×C₅	—	4	1	no	—
68	C₂×C₅	—	5	2	no	—
69	C₂×C₅	—	5	2	no	—
70	C₂×C₅	C₄×Dic₅	1	2	yes	complemented
71	C₂×C₅	C₄×AGL(1,𝔽₅)	1	2	yes	complemented
72	C₂×C₅	C₂²×Dic₅	1	1	yes	characteristic
73	C₂×C₅	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
74	D₅	C₄×Dic₅	1	2	yes	complemented
75	D₅	C₂×C₄×D₅	1	2	yes	—
76	C₂³	—	5	1	no	—
77	C₂×C₄	—	5	1	no	—
78	C₂×C₄	—	5	2	no	—
79	C₂×C₄	—	5	2	no	—
80	C₂×C₄	—	25	4	no	—
81	C₂×C₄	—	25	8	no	—
82	C₂×C₄	#1042	1	1	yes	characteristic, direct product factor
83	C₅	—	4	1	no	—
84	C₅	C₂×C₄×Dic₅	1	1	yes	characteristic, complemented
85	C₅	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
86	C₂²	—	5	4	no	—
87	C₂²	—	5	1	no	—
88	C₂²	—	5	1	no	—
89	C₂²	C₂×#1042	1	1	yes	characteristic
90	C₄	—	5	2	no	—
91	C₄	—	25	8	no	—
92	C₄	C₂×#1042	1	2	yes	direct product factor
93	C₂	—	5	2	no	—
94	C₂	—	5	2	no	—
95	C₂	C₄×#1042	1	2	yes	direct product factor
96	C₂	C₂²×#1042	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₄	#1042
commutator, hypocenter	C₅²	C₂×C₄²
Frattini	C₂	C₂²×#1042
Fitting	C₂×C₄×C₅²	C₄
socle	C₂²×C₅²	C₂×C₄
prime radical	C₂²×C₅×D₅	C₂²
cosocle	C₂×C₅×D₅	C₂³
LCS	G ▷ C₅²
UCS	C₁ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
3. C₄×#1042	C₂	direct (N × H)
5. C₂×#1042	C₄	direct (N × H)
7. C₂×C₄×C₅²	C₄	semidirect
8. C₂×C₅×Dic₅	C₄	semidirect
9. C₄×C₅×D₅	C₄	semidirect
17. #1042	C₂×C₄	direct (N × H)
18. C₄×C₅²	C₂×C₄	semidirect
19. C₅×Dic₅	C₂×C₄	semidirect
26. C₂×C₅²	C₄²	semidirect
28. C₅×D₅	C₄²	semidirect
38. C₂×C₄×C₅	Dic₅	semidirect
39. C₂×C₄×C₅	AGL(1,𝔽₅)	semidirect
42. C₂×Dic₅	Dic₅	semidirect
43. C₄×D₅	Dic₅	semidirect
46. C₅²	C₂×C₄²	semidirect
59. C₄×C₅	C₂×Dic₅	semidirect
60. C₄×C₅	C₂×AGL(1,𝔽₅)	semidirect
62. Dic₅	C₂×Dic₅	semidirect
70. C₂×C₅	C₄×Dic₅	semidirect
71. C₂×C₅	C₄×AGL(1,𝔽₅)	semidirect
74. D₅	C₄×Dic₅	semidirect
82. C₂×C₄	#1042	direct (N × H)
84. C₅	C₂×C₄×Dic₅	semidirect
85. C₅	C₂×C₄×AGL(1,𝔽₅)	semidirect
92. C₄	C₂×#1042	direct (N × H)
95. C₂	C₄×#1042	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#1042	2
C₂×C₄×C₅×D₅	2
C₄×#1042	2
C₂×C₄×AGL(1,𝔽₅)	5
C₂×C₄×Dic₅	5

Subgroups — C2×C4×#1042 #11717423

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#1042 #11717423