Subgroups — C₂³×C₈×AGL(1,𝔽₅) #13044374

Order 1280.

4638 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic
2	C₂³×D₅⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₈×D₅	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₈×AGL(1,𝔽₅)	C₂	1	28	yes	maximal, direct product factor
5	C₂⁴×AGL(1,𝔽₅)	C₄	1	2	yes	—
6	C₂³×C₄×D₅	C₂²	1	1	yes	characteristic
7	C₂³×#170	C₄	1	2	yes	complemented
8	C₂²×C₄×AGL(1,𝔽₅)	C₄	1	14	yes	—
9	C₂²×C₄×AGL(1,𝔽₅)	C₂²	1	14	yes	—
10	C₂²×D₅⋊C₈	C₄	1	14	yes	complemented
11	C₂²×D₅⋊C₈	C₂²	1	14	yes	—
12	C₂³×C₅×C₈	C₄	1	1	yes	characteristic, complemented
13	C₂³×#168	C₄	1	1	yes	characteristic, complemented
14	C₂²×C₈×D₅	C₄	1	14	yes	complemented
15	C₂²×C₈×D₅	C₂²	1	14	yes	—
16	C₂×C₈×AGL(1,𝔽₅)	C₂²	1	112	yes	direct product factor
17	C₂³×C₄×C₈	—	5	1	no	maximal, Sylow
18	C₂⁴×D₅	C₂×C₄	1	1	yes	characteristic
19	C₂³×AGL(1,𝔽₅)	C₈	1	32	yes	direct product factor
20	C₂³×AGL(1,𝔽₅)	C₂×C₄	1	28	yes	—
21	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂³×Dic₅	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×D₅	C₂×C₄	1	7	yes	—
24	C₂²×C₄×D₅	C₂×C₄	1	7	yes	—
25	C₂²×C₄×D₅	C₂×C₄	1	7	yes	—
26	C₂²×C₄×D₅	C₂³	1	7	yes	—
27	C₂²×#170	C₂×C₄	1	28	yes	complemented
28	C₂×C₄×AGL(1,𝔽₅)	C₂×C₄	1	84	yes	—
29	C₂×C₄×AGL(1,𝔽₅)	C₂³	1	28	yes	—
30	C₂×D₅⋊C₈	C₂×C₄	1	84	yes	complemented
31	C₂×D₅⋊C₈	C₂³	1	28	yes	—
32	C₂²×C₅×C₈	C₂×C₄	1	14	yes	complemented
33	C₂²×#168	C₂×C₄	1	14	yes	complemented
34	C₂×C₈×D₅	C₂×C₄	1	84	yes	complemented
35	C₂×C₈×D₅	C₂³	1	28	yes	—
36	C₈×AGL(1,𝔽₅)	C₂³	1	64	yes	direct product factor
37	C₂⁴×C₈	—	5	1	no	—
38	C₂⁴×C₈	—	5	1	no	—
39	C₂³×C₄²	—	5	1	no	—
40	C₂²×C₄×C₈	—	5	28	no	—
41	C₂⁴×C₅	C₄²	1	1	yes	characteristic
42	C₂³×D₅	C₄²	1	7	yes	—
43	C₂³×D₅	C₂×C₈	1	16	yes	—
44	C₂³×D₅	C₂²×C₄	1	7	yes	—
45	C₂²×AGL(1,𝔽₅)	C₂×C₈	1	224	yes	direct product factor
46	C₂²×AGL(1,𝔽₅)	C₂²×C₄	1	56	yes	—
47	C₂²×C₄×C₅	C₄²	1	7	yes	—
48	C₂²×C₄×C₅	C₂²×C₄	1	7	yes	—
49	C₂²×Dic₅	C₄²	1	7	yes	—
50	C₂²×Dic₅	C₂²×C₄	1	7	yes	—
51	C₂×C₄×D₅	C₄²	1	42	yes	—
52	C₂×C₄×D₅	C₂²×C₄	1	21	yes	—
53	C₂×C₄×D₅	C₂²×C₄	1	21	yes	—
54	C₂×C₄×D₅	C₂²×C₄	1	21	yes	—
55	C₂×C₄×D₅	C₂⁴	1	7	yes	—
56	C₂×#170	C₂²×C₄	1	56	yes	complemented
57	C₄×AGL(1,𝔽₅)	C₂²×C₄	1	56	yes	—
58	C₄×AGL(1,𝔽₅)	C₂⁴	1	8	yes	—
59	D₅⋊C₈	C₂²×C₄	1	56	yes	complemented
60	D₅⋊C₈	C₂⁴	1	8	yes	—
61	C₂×C₅×C₈	C₂²×C₄	1	28	yes	complemented
62	C₂×#168	C₂²×C₄	1	28	yes	complemented
63	C₈×D₅	C₂²×C₄	1	56	yes	complemented
64	C₈×D₅	C₂⁴	1	8	yes	—
65	C₂⁴×C₄	—	5	2	no	—
66	C₂⁴×C₄	—	5	1	no	—
67	C₂³×C₈	—	5	14	no	—
68	C₂³×C₈	—	5	14	no	—
69	C₂³×C₈	—	5	2	no	—
70	C₂³×C₈	—	5	1	no	—
71	C₂³×C₈	—	5	14	no	—
72	C₂³×C₈	—	5	14	no	—
73	C₂³×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
74	C₂²×C₄²	—	5	14	no	—
75	C₂²×C₄²	—	5	14	no	—
76	C₂×C₄×C₈	—	5	112	no	—
77	C₂³×C₅	C₄×C₈	1	8	yes	complemented
78	C₂³×C₅	C₂×C₄²	1	7	yes	—
79	C₂²×D₅	C₄×C₈	1	56	yes	complemented
80	C₂²×D₅	C₂×C₄²	1	21	yes	—
81	C₂²×D₅	C₂²×C₈	1	56	yes	—
82	C₂²×D₅	C₂³×C₄	1	7	yes	—
83	C₂×AGL(1,𝔽₅)	C₂²×C₈	1	224	yes	direct product factor
84	C₂×AGL(1,𝔽₅)	C₂³×C₄	1	16	yes	—
85	C₂×C₄×C₅	C₂×C₄²	1	21	yes	—
86	C₂×C₄×C₅	C₂³×C₄	1	7	yes	—
87	C₂×Dic₅	C₂×C₄²	1	21	yes	—
88	C₂×Dic₅	C₂³×C₄	1	7	yes	—
89	C₄×D₅	C₂×C₄²	1	42	yes	—
90	C₄×D₅	C₂³×C₄	1	7	yes	—
91	C₄×D₅	C₂³×C₄	1	7	yes	—
92	C₄×D₅	C₂³×C₄	1	7	yes	—
93	C₄×D₅	C₂⁵	1	1	yes	characteristic
94	#170	C₂³×C₄	1	16	yes	complemented
95	C₅×C₈	C₂³×C₄	1	8	yes	complemented
96	#168	C₂³×C₄	1	8	yes	complemented
97	C₂⁵	—	5	1	no	—
98	C₂³×C₄	—	5	32	no	—
99	C₂³×C₄	—	5	28	no	—
100	C₂³×C₄	—	5	1	no	—
101	C₂³×C₄	—	5	7	no	—
102	C₂³×C₄	—	5	7	no	—
103	C₂³×C₄	—	5	7	no	—
104	C₂³×C₄	—	5	7	no	—
105	C₂³×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
106	C₂²×C₈	—	5	84	no	—
107	C₂²×C₈	—	5	28	no	—
108	C₂²×C₈	—	5	28	no	—
109	C₂²×C₈	—	5	14	no	—
110	C₂²×C₈	—	5	28	no	—
111	C₂²×C₈	—	5	84	no	—
112	C₂²×C₈	C₂×AGL(1,𝔽₅)	1	14	yes	direct product factor
113	C₂×C₄²	—	5	84	no	—
114	C₂×C₄²	—	5	28	no	—
115	C₄×C₈	—	5	64	no	—
116	C₂²×C₅	C₂×C₄×C₈	1	28	yes	complemented
117	C₂²×C₅	C₂²×C₄²	1	7	yes	—
118	C₂×D₅	C₂×C₄×C₈	1	84	yes	complemented
119	C₂×D₅	C₂²×C₄²	1	7	yes	—
120	C₂×D₅	C₂³×C₈	1	28	yes	—
121	C₂×D₅	C₂⁴×C₄	1	1	yes	characteristic
122	AGL(1,𝔽₅)	C₂³×C₈	1	32	yes	direct product factor
123	C₄×C₅	C₂²×C₄²	1	7	yes	—
124	C₄×C₅	C₂⁴×C₄	1	1	yes	characteristic
125	Dic₅	C₂²×C₄²	1	7	yes	—
126	Dic₅	C₂⁴×C₄	1	1	yes	characteristic
127	C₂⁴	—	5	7	no	—
128	C₂⁴	—	5	7	no	—
129	C₂⁴	—	5	16	no	—
130	C₂⁴	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
131	C₂²×C₄	—	5	56	no	—
132	C₂²×C₄	—	5	224	no	—
133	C₂²×C₄	—	5	7	no	—
134	C₂²×C₄	—	5	7	no	—
135	C₂²×C₄	—	5	21	no	—
136	C₂²×C₄	—	5	21	no	—
137	C₂²×C₄	—	5	42	no	—
138	C₂²×C₄	—	5	7	no	—
139	C₂²×C₄	—	5	21	no	—
140	C₂²×C₄	C₄×AGL(1,𝔽₅)	1	7	yes	—
141	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	7	yes	—
142	C₂×C₈	—	5	8	no	—
143	C₂×C₈	—	5	56	no	—
144	C₂×C₈	—	5	56	no	—
145	C₂×C₈	—	5	28	no	—
146	C₂×C₈	—	5	8	no	—
147	C₂×C₈	—	5	56	no	—
148	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	28	yes	direct product factor
149	C₄²	—	5	8	no	—
150	C₄²	—	5	56	no	—
151	C₂×C₅	C₂²×C₄×C₈	1	14	yes	complemented
152	C₂×C₅	C₂³×C₄²	1	1	yes	characteristic
153	D₅	C₂²×C₄×C₈	1	14	yes	complemented
154	D₅	C₂⁴×C₈	1	2	yes	—
155	C₂³	—	5	56	no	—
156	C₂³	—	5	56	no	—
157	C₂³	—	5	7	no	—
158	C₂³	—	5	21	no	—
159	C₂³	C₈×AGL(1,𝔽₅)	1	8	yes	direct product factor
160	C₂³	C₂×C₄×AGL(1,𝔽₅)	1	7	yes	—
161	C₂×C₄	—	5	16	no	—
162	C₂×C₄	—	5	224	no	—
163	C₂×C₄	—	5	7	no	—
164	C₂×C₄	—	5	21	no	—
165	C₂×C₄	—	5	7	no	—
166	C₂×C₄	—	5	7	no	—
167	C₂×C₄	—	5	42	no	—
168	C₂×C₄	—	5	1	no	—
169	C₂×C₄	—	5	7	no	—
170	C₂×C₄	C₂×C₄×AGL(1,𝔽₅)	1	21	yes	—
171	C₂×C₄	C₂³×AGL(1,𝔽₅)	1	7	yes	—
172	C₈	—	5	16	no	—
173	C₈	—	5	8	no	—
174	C₈	C₂³×AGL(1,𝔽₅)	1	8	yes	direct product factor
175	C₅	C₂³×C₄×C₈	1	1	yes	characteristic, complemented, Sylow
176	C₂²	—	5	1	no	—
177	C₂²	—	5	7	no	—
178	C₂²	—	5	28	no	—
179	C₂²	—	5	84	no	—
180	C₂²	C₂×C₈×AGL(1,𝔽₅)	1	28	yes	direct product factor
181	C₂²	C₂²×C₄×AGL(1,𝔽₅)	1	7	yes	—
182	C₄	—	5	32	no	—
183	C₄	—	5	1	no	—
184	C₄	—	5	7	no	—
185	C₄	C₂²×C₄×AGL(1,𝔽₅)	1	7	yes	—
186	C₄	C₂⁴×AGL(1,𝔽₅)	1	1	yes	characteristic
187	C₂	—	5	2	no	—
188	C₂	—	5	14	no	—
189	C₂	C₂²×C₈×AGL(1,𝔽₅)	1	14	yes	direct product factor
190	C₂	C₂³×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₈	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₂³×C₄×C₈
Frattini	C₄	C₂⁴×AGL(1,𝔽₅)
Fitting	C₂³×C₅×C₈	C₄
socle	C₂⁴×C₅	C₄²
prime radical	C₂⁴×D₅	C₂×C₄
cosocle	C₄×D₅	C₂⁵
LCS	G ▷ C₅
UCS	C₁ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
4. C₂²×C₈×AGL(1,𝔽₅)	C₂	direct (N × H)
7. C₂³×#170	C₄	semidirect
10. C₂²×D₅⋊C₈	C₄	semidirect
12. C₂³×C₅×C₈	C₄	semidirect
13. C₂³×#168	C₄	semidirect
14. C₂²×C₈×D₅	C₄	semidirect
16. C₂×C₈×AGL(1,𝔽₅)	C₂²	direct (N × H)
19. C₂³×AGL(1,𝔽₅)	C₈	direct (N × H)
27. C₂²×#170	C₂×C₄	semidirect
30. C₂×D₅⋊C₈	C₂×C₄	semidirect
32. C₂²×C₅×C₈	C₂×C₄	semidirect
33. C₂²×#168	C₂×C₄	semidirect
34. C₂×C₈×D₅	C₂×C₄	semidirect
36. C₈×AGL(1,𝔽₅)	C₂³	direct (N × H)
45. C₂²×AGL(1,𝔽₅)	C₂×C₈	direct (N × H)
56. C₂×#170	C₂²×C₄	semidirect
59. D₅⋊C₈	C₂²×C₄	semidirect
61. C₂×C₅×C₈	C₂²×C₄	semidirect
62. C₂×#168	C₂²×C₄	semidirect
63. C₈×D₅	C₂²×C₄	semidirect
73. C₂³×C₈	AGL(1,𝔽₅)	direct (N × H)
77. C₂³×C₅	C₄×C₈	semidirect
79. C₂²×D₅	C₄×C₈	semidirect
83. C₂×AGL(1,𝔽₅)	C₂²×C₈	direct (N × H)
94. #170	C₂³×C₄	semidirect
95. C₅×C₈	C₂³×C₄	semidirect
96. #168	C₂³×C₄	semidirect
112. C₂²×C₈	C₂×AGL(1,𝔽₅)	direct (N × H)
116. C₂²×C₅	C₂×C₄×C₈	semidirect
118. C₂×D₅	C₂×C₄×C₈	semidirect
122. AGL(1,𝔽₅)	C₂³×C₈	direct (N × H)
148. C₂×C₈	C₂²×AGL(1,𝔽₅)	direct (N × H)
151. C₂×C₅	C₂²×C₄×C₈	semidirect
153. D₅	C₂²×C₄×C₈	semidirect
159. C₂³	C₈×AGL(1,𝔽₅)	direct (N × H)
174. C₈	C₂³×AGL(1,𝔽₅)	direct (N × H)
175. C₅	C₂³×C₄×C₈	semidirect
180. C₂²	C₂×C₈×AGL(1,𝔽₅)	direct (N × H)
189. C₂	C₂²×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
C₂³×C₄×AGL(1,𝔽₅)	2
C₂³×D₅⋊C₈	2
C₂³×C₈×D₅	2
C₂²×C₈×AGL(1,𝔽₅)	2
C₂³×C₄×C₈	5

Subgroups — C23×C8×AGL(1,𝔽5) #13044374

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₈×AGL(1,𝔽₅) #13044374