Subgroups — C₂×C₃×C₈×AGL(1,𝔽₅) #11752249

Order 960.

324 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×D₅⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₈×D₅	C₂	1	1	yes	maximal, characteristic
4	C₃×C₈×AGL(1,𝔽₅)	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₈×AGL(1,𝔽₅)	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×AGL(1,𝔽₅)	C₄	1	2	yes	—
7	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
8	C₂×C₃×#170	C₄	1	2	yes	complemented
9	C₃×C₄×AGL(1,𝔽₅)	C₄	1	2	yes	—
10	C₃×C₄×AGL(1,𝔽₅)	C₂²	1	2	yes	—
11	C₃×D₅⋊C₈	C₄	1	2	yes	complemented
12	C₃×D₅⋊C₈	C₂²	1	2	yes	—
13	C₂×C₃×C₅×C₈	C₄	1	1	yes	characteristic, complemented
14	C₂×C₃×#168	C₄	1	1	yes	characteristic, complemented
15	C₃×C₈×D₅	C₄	1	2	yes	complemented
16	C₃×C₈×D₅	C₂²	1	2	yes	—
17	C₂×C₃×C₄×C₈	—	5	1	no	maximal, Hall
18	C₂×C₄×AGL(1,𝔽₅)	C₂×C₃	1	1	yes	characteristic
19	C₂×D₅⋊C₈	C₂×C₃	1	1	yes	characteristic
20	C₂×C₈×D₅	C₂×C₃	1	1	yes	characteristic
21	C₈×AGL(1,𝔽₅)	C₂×C₃	1	4	yes	direct product factor
22	C₂²×C₃×D₅	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃×AGL(1,𝔽₅)	C₈	1	8	yes	direct product factor
24	C₂×C₃×AGL(1,𝔽₅)	C₂×C₄	1	4	yes	—
25	C₂×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
27	C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
28	C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
29	C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
30	C₃×C₄×D₅	C₂³	1	1	yes	characteristic
31	C₃×#170	C₂×C₄	1	4	yes	complemented
32	C₃×C₅×C₈	C₂×C₄	1	2	yes	complemented
33	C₃×#168	C₂×C₄	1	2	yes	complemented
34	C₂²×C₃×C₈	—	5	1	no	—
35	C₂²×C₃×C₈	—	5	1	no	—
36	C₂×C₃×C₄²	—	5	1	no	—
37	C₃×C₄×C₈	—	5	4	no	—
38	C₂²×AGL(1,𝔽₅)	C₃×C₄	1	2	yes	—
39	C₂×C₄×D₅	C₂²×C₃	1	1	yes	characteristic
40	C₂×#170	C₃×C₄	1	2	yes	complemented
41	C₄×AGL(1,𝔽₅)	C₃×C₄	1	2	yes	—
42	C₄×AGL(1,𝔽₅)	C₂²×C₃	1	2	yes	—
43	D₅⋊C₈	C₃×C₄	1	2	yes	complemented
44	D₅⋊C₈	C₂²×C₃	1	2	yes	—
45	C₂×C₅×C₈	C₃×C₄	1	1	yes	characteristic, complemented
46	C₂×#168	C₃×C₄	1	1	yes	characteristic, complemented
47	C₈×D₅	C₃×C₄	1	2	yes	complemented
48	C₈×D₅	C₂²×C₃	1	2	yes	—
49	C₂×C₄×C₈	—	5	1	no	Sylow
50	C₂²×C₃×C₅	C₄²	1	1	yes	characteristic
51	C₂×C₃×D₅	C₄²	1	1	yes	characteristic
52	C₂×C₃×D₅	C₂×C₈	1	4	yes	—
53	C₂×C₃×D₅	C₂²×C₄	1	1	yes	characteristic
54	C₃×AGL(1,𝔽₅)	C₂×C₈	1	8	yes	direct product factor
55	C₃×C₄×C₅	C₄²	1	1	yes	characteristic
56	C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₃×Dic₅	C₄²	1	1	yes	characteristic
58	C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₃×C₄	—	5	2	no	—
60	C₂²×C₃×C₄	—	5	1	no	—
61	C₂×C₃×C₈	—	5	2	no	—
62	C₂×C₃×C₈	—	5	2	no	—
63	C₂×C₃×C₈	—	5	2	no	—
64	C₂×C₃×C₈	—	5	1	no	—
65	C₂×C₃×C₈	—	5	2	no	—
66	C₂×C₃×C₈	—	5	2	no	—
67	C₂×C₃×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
68	C₃×C₄²	—	5	2	no	—
69	C₃×C₄²	—	5	2	no	—
70	C₂²×D₅	C₂×C₃×C₄	1	1	yes	characteristic
71	C₂×AGL(1,𝔽₅)	C₃×C₈	1	8	yes	direct product factor
72	C₂×AGL(1,𝔽₅)	C₂×C₃×C₄	1	4	yes	—
73	C₂×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
74	C₂×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
75	C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
76	C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
77	C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
78	C₄×D₅	C₂³×C₃	1	1	yes	characteristic
79	#170	C₂×C₃×C₄	1	4	yes	complemented
80	C₅×C₈	C₂×C₃×C₄	1	2	yes	complemented
81	#168	C₂×C₃×C₄	1	2	yes	complemented
82	C₂²×C₈	—	5	1	no	—
83	C₂²×C₈	—	5	1	no	—
84	C₂×C₄²	—	5	1	no	—
85	C₄×C₈	—	5	4	no	—
86	C₂×C₃×C₅	C₄×C₈	1	2	yes	complemented
87	C₂×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
88	C₃×D₅	C₄×C₈	1	2	yes	complemented
89	C₃×D₅	C₂²×C₈	1	2	yes	—
90	C₂³×C₃	—	5	1	no	—
91	C₂×C₃×C₄	—	5	8	no	—
92	C₂×C₃×C₄	—	5	4	no	—
93	C₂×C₃×C₄	—	5	1	no	—
94	C₂×C₃×C₄	—	5	1	no	—
95	C₂×C₃×C₄	—	5	1	no	—
96	C₂×C₃×C₄	—	5	1	no	—
97	C₂×C₃×C₄	—	5	1	no	—
98	C₂×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
99	C₃×C₈	—	5	4	no	—
100	C₃×C₈	—	5	2	no	—
101	C₃×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	direct product factor
102	C₂²×C₅	C₃×C₄²	1	1	yes	characteristic
103	C₂×D₅	C₃×C₄²	1	1	yes	characteristic
104	C₂×D₅	C₂×C₃×C₈	1	4	yes	—
105	C₂×D₅	C₂²×C₃×C₄	1	1	yes	characteristic
106	AGL(1,𝔽₅)	C₂×C₃×C₈	1	8	yes	direct product factor
107	C₄×C₅	C₃×C₄²	1	1	yes	characteristic
108	C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
109	Dic₅	C₃×C₄²	1	1	yes	characteristic
110	Dic₅	C₂²×C₃×C₄	1	1	yes	characteristic
111	C₂²×C₄	—	5	2	no	—
112	C₂²×C₄	—	5	1	no	—
113	C₂×C₈	—	5	2	no	—
114	C₂×C₈	—	5	2	no	—
115	C₂×C₈	—	5	2	no	—
116	C₂×C₈	—	5	1	no	—
117	C₂×C₈	—	5	2	no	—
118	C₂×C₈	—	5	2	no	—
119	C₂×C₈	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
120	C₄²	—	5	2	no	—
121	C₄²	—	5	2	no	—
122	C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic, complemented, Hall
123	C₂²×C₃	—	5	1	no	—
124	C₂²×C₃	—	5	1	no	—
125	C₂²×C₃	—	5	4	no	—
126	C₂²×C₃	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
127	C₃×C₄	—	5	8	no	—
128	C₃×C₄	—	5	1	no	—
129	C₃×C₄	—	5	1	no	—
130	C₃×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
131	C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
132	C₂×C₅	C₃×C₄×C₈	1	2	yes	complemented
133	C₂×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
134	D₅	C₃×C₄×C₈	1	2	yes	complemented
135	D₅	C₂²×C₃×C₈	1	2	yes	—
136	C₂³	—	5	1	no	—
137	C₂×C₄	—	5	4	no	—
138	C₂×C₄	—	5	8	no	—
139	C₂×C₄	—	5	1	no	—
140	C₂×C₄	—	5	1	no	—
141	C₂×C₄	—	5	1	no	—
142	C₂×C₄	—	5	1	no	—
143	C₂×C₄	—	5	1	no	—
144	C₂×C₄	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
145	C₈	—	5	4	no	—
146	C₈	—	5	2	no	—
147	C₈	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	direct product factor
148	C₂×C₃	—	5	2	no	—
149	C₂×C₃	—	5	2	no	—
150	C₂×C₃	C₈×AGL(1,𝔽₅)	1	2	yes	direct product factor
151	C₂×C₃	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
152	C₅	C₂×C₃×C₄×C₈	1	1	yes	characteristic, complemented, Sylow
153	C₂²	—	5	4	no	—
154	C₂²	—	5	1	no	—
155	C₂²	—	5	1	no	—
156	C₂²	C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
157	C₄	—	5	8	no	—
158	C₄	—	5	1	no	—
159	C₄	—	5	1	no	—
160	C₄	C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
161	C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
162	C₃	C₂×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor, Sylow
163	C₂	—	5	2	no	—
164	C₂	—	5	2	no	—
165	C₂	C₃×C₈×AGL(1,𝔽₅)	1	2	yes	direct product factor
166	C₂	C₂×C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃×C₈	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₂×C₃×C₄×C₈
Frattini	C₄	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₂×C₃×C₅×C₈	C₄
socle	C₂²×C₃×C₅	C₄²
prime radical	C₂²×C₃×D₅	C₂×C₄
cosocle	C₄×D₅	C₂³×C₃
LCS	G ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₈

Complements

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

N	H ≅ G/N	type
4. C₃×C₈×AGL(1,𝔽₅)	C₂	direct (N × H)
5. C₂×C₈×AGL(1,𝔽₅)	C₃	direct (N × H)
8. C₂×C₃×#170	C₄	semidirect
11. C₃×D₅⋊C₈	C₄	semidirect
13. C₂×C₃×C₅×C₈	C₄	semidirect
14. C₂×C₃×#168	C₄	semidirect
15. C₃×C₈×D₅	C₄	semidirect
21. C₈×AGL(1,𝔽₅)	C₂×C₃	direct (N × H)
23. C₂×C₃×AGL(1,𝔽₅)	C₈	direct (N × H)
31. C₃×#170	C₂×C₄	semidirect
32. C₃×C₅×C₈	C₂×C₄	semidirect
33. C₃×#168	C₂×C₄	semidirect
40. C₂×#170	C₃×C₄	semidirect
43. D₅⋊C₈	C₃×C₄	semidirect
45. C₂×C₅×C₈	C₃×C₄	semidirect
46. C₂×#168	C₃×C₄	semidirect
47. C₈×D₅	C₃×C₄	semidirect
54. C₃×AGL(1,𝔽₅)	C₂×C₈	direct (N × H)
67. C₂×C₃×C₈	AGL(1,𝔽₅)	direct (N × H)
71. C₂×AGL(1,𝔽₅)	C₃×C₈	direct (N × H)
79. #170	C₂×C₃×C₄	semidirect
80. C₅×C₈	C₂×C₃×C₄	semidirect
81. #168	C₂×C₃×C₄	semidirect
86. C₂×C₃×C₅	C₄×C₈	semidirect
88. C₃×D₅	C₄×C₈	semidirect
101. C₃×C₈	C₂×AGL(1,𝔽₅)	direct (N × H)
106. AGL(1,𝔽₅)	C₂×C₃×C₈	direct (N × H)
119. C₂×C₈	C₃×AGL(1,𝔽₅)	direct (N × H)
122. C₃×C₅	C₂×C₄×C₈	semidirect
132. C₂×C₅	C₃×C₄×C₈	semidirect
134. D₅	C₃×C₄×C₈	semidirect
147. C₈	C₂×C₃×AGL(1,𝔽₅)	direct (N × H)
150. C₂×C₃	C₈×AGL(1,𝔽₅)	direct (N × H)
152. C₅	C₂×C₃×C₄×C₈	semidirect
162. C₃	C₂×C₈×AGL(1,𝔽₅)	direct (N × H)
165. C₂	C₃×C₈×AGL(1,𝔽₅)	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×C₃×C₄×AGL(1,𝔽₅)	2
C₂×C₃×D₅⋊C₈	2
C₂×C₃×C₈×D₅	2
C₃×C₈×AGL(1,𝔽₅)	2
C₂×C₈×AGL(1,𝔽₅)	3
C₂×C₃×C₄×C₈	5

Subgroups — C2×C3×C8×AGL(1,𝔽5) #11752249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₈×AGL(1,𝔽₅) #11752249