Subgroups — C₃×C₄×C₈×AGL(1,𝔽₅) #14294188

Order 1920.

816 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₈×AGL(1,𝔽₅)	C₂	1	4	yes	maximal
2	C₃×C₄²×AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×D₅⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₃×C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic
5	C₄×C₈×AGL(1,𝔽₅)	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₃×C₄×AGL(1,𝔽₅)	C₄	1	4	yes	—
7	C₂×C₃×C₄×AGL(1,𝔽₅)	C₂²	1	2	yes	—
8	C₂×C₃×D₅⋊C₈	C₂²	1	2	yes	—
9	C₂×C₃×C₈×D₅	C₂²	1	2	yes	—
10	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
11	C₃×C₄×#170	C₄	1	4	yes	complemented
12	C₃×C₈×AGL(1,𝔽₅)	C₄	1	16	yes	direct product factor
13	C₃×C₄×C₅×C₈	C₄	1	1	yes	characteristic, complemented
14	C₃×C₈×Dic₅	C₄	1	2	yes	complemented
15	C₃×C₄×#168	C₄	1	1	yes	characteristic, complemented
16	C₃×C₄²×C₈	—	5	1	no	maximal, Hall
17	C₂×C₈×AGL(1,𝔽₅)	C₂×C₃	1	4	yes	—
18	C₄²×AGL(1,𝔽₅)	C₂×C₃	1	1	yes	characteristic
19	C₄×D₅⋊C₈	C₂×C₃	1	1	yes	characteristic
20	C₄×C₈×D₅	C₂×C₃	1	1	yes	characteristic
21	C₂²×C₃×AGL(1,𝔽₅)	C₂×C₄	1	4	yes	—
22	C₂×C₃×C₄×D₅	C₂×C₄	1	2	yes	—
23	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
24	C₂×C₃×#170	C₂×C₄	1	4	yes	—
25	C₃×C₄×AGL(1,𝔽₅)	C₈	1	16	yes	direct product factor
26	C₃×C₄×AGL(1,𝔽₅)	C₂×C₄	1	8	yes	—
27	C₃×D₅⋊C₈	C₂×C₄	1	8	yes	—
28	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
29	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
30	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
31	C₃×C₄×Dic₅	C₂×C₄	1	2	yes	—
32	C₂×C₃×#168	C₂×C₄	1	2	yes	—
33	C₃×C₈×D₅	C₂×C₄	1	8	yes	—
34	C₂×C₃×C₄×C₈	—	5	1	no	—
35	C₂×C₃×C₄×C₈	—	5	4	no	—
36	C₂×C₃×C₄×C₈	—	5	1	no	—
37	C₃×C₄³	—	5	1	no	—
38	C₂×C₄×AGL(1,𝔽₅)	C₃×C₄	1	4	yes	—
39	C₂×C₄×AGL(1,𝔽₅)	C₂²×C₃	1	2	yes	—
40	C₂×D₅⋊C₈	C₂²×C₃	1	2	yes	—
41	C₂×C₈×D₅	C₂²×C₃	1	2	yes	—
42	C₄²×D₅	C₂²×C₃	1	1	yes	characteristic
43	C₄×#170	C₃×C₄	1	4	yes	complemented
44	C₈×AGL(1,𝔽₅)	C₃×C₄	1	16	yes	direct product factor
45	C₄×C₅×C₈	C₃×C₄	1	1	yes	characteristic, complemented
46	C₈×Dic₅	C₃×C₄	1	2	yes	complemented
47	C₄×#168	C₃×C₄	1	1	yes	characteristic, complemented
48	C₄²×C₈	—	5	1	no	Sylow
49	C₂²×C₃×D₅	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₃×AGL(1,𝔽₅)	C₄²	1	8	yes	—
51	C₂×C₃×AGL(1,𝔽₅)	C₂×C₈	1	16	yes	—
52	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
53	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₃×Dic₅	C₄²	1	2	yes	—
55	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
56	C₃×C₄×D₅	C₂×C₈	1	8	yes	—
57	C₃×C₄×D₅	C₂²×C₄	1	4	yes	—
58	C₃×#170	C₄²	1	8	yes	complemented
59	C₃×C₅×C₈	C₄²	1	4	yes	complemented
60	C₃×#168	C₄²	1	4	yes	complemented
61	C₂²×C₃×C₈	—	5	2	no	—
62	C₂²×C₃×C₈	—	5	2	no	—
63	C₂×C₃×C₄²	—	5	4	no	—
64	C₂×C₃×C₄²	—	5	2	no	—
65	C₂×C₃×C₄²	—	5	1	no	—
66	C₃×C₄×C₈	—	5	16	no	—
67	C₃×C₄×C₈	—	5	4	no	—
68	C₃×C₄×C₈	—	5	2	no	—
69	C₃×C₄×C₈	—	5	1	no	—
70	C₃×C₄×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
71	C₂²×AGL(1,𝔽₅)	C₂×C₃×C₄	1	4	yes	—
72	C₂×C₄×D₅	C₂×C₃×C₄	1	2	yes	—
73	C₂×C₄×D₅	C₂³×C₃	1	1	yes	characteristic
74	C₂×#170	C₂×C₃×C₄	1	4	yes	—
75	C₄×AGL(1,𝔽₅)	C₃×C₈	1	16	yes	direct product factor
76	C₄×AGL(1,𝔽₅)	C₂×C₃×C₄	1	8	yes	—
77	D₅⋊C₈	C₂×C₃×C₄	1	8	yes	—
78	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
79	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
80	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
81	C₄×Dic₅	C₂×C₃×C₄	1	2	yes	—
82	C₂×#168	C₂×C₃×C₄	1	2	yes	—
83	C₈×D₅	C₂×C₃×C₄	1	8	yes	—
84	C₂×C₄×C₈	—	5	1	no	—
85	C₂×C₄×C₈	—	5	4	no	—
86	C₂×C₄×C₈	—	5	1	no	—
87	C₄³	—	5	1	no	—
88	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
89	C₂×C₃×D₅	C₂×C₄²	1	2	yes	—
90	C₂×C₃×D₅	C₂²×C₈	1	4	yes	—
91	C₃×AGL(1,𝔽₅)	C₄×C₈	1	16	yes	direct product factor
92	C₃×C₄×C₅	C₄×C₈	1	4	yes	complemented
93	C₃×C₄×C₅	C₂×C₄²	1	2	yes	—
94	C₃×Dic₅	C₄×C₈	1	4	yes	complemented
95	C₃×Dic₅	C₂×C₄²	1	2	yes	—
96	C₂²×C₃×C₄	—	5	4	no	—
97	C₂²×C₃×C₄	—	5	2	no	—
98	C₂²×C₃×C₄	—	5	1	no	—
99	C₂×C₃×C₈	—	5	8	no	—
100	C₂×C₃×C₈	—	5	4	no	—
101	C₂×C₃×C₈	—	5	2	no	—
102	C₂×C₃×C₈	—	5	8	no	—
103	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
104	C₃×C₄²	—	5	8	no	—
105	C₃×C₄²	—	5	16	no	—
106	C₃×C₄²	—	5	2	no	—
107	C₃×C₄²	—	5	1	no	—
108	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
109	C₂²×D₅	C₂²×C₃×C₄	1	1	yes	characteristic
110	C₂×AGL(1,𝔽₅)	C₃×C₄²	1	8	yes	—
111	C₂×AGL(1,𝔽₅)	C₂×C₃×C₈	1	16	yes	—
112	C₂×C₄×C₅	C₃×C₄²	1	2	yes	—
113	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
114	C₂×Dic₅	C₃×C₄²	1	2	yes	—
115	C₂×Dic₅	C₂²×C₃×C₄	1	1	yes	characteristic
116	C₄×D₅	C₂×C₃×C₈	1	8	yes	—
117	C₄×D₅	C₂²×C₃×C₄	1	4	yes	—
118	#170	C₃×C₄²	1	8	yes	complemented
119	C₅×C₈	C₃×C₄²	1	4	yes	complemented
120	#168	C₃×C₄²	1	4	yes	complemented
121	C₂²×C₈	—	5	2	no	—
122	C₂²×C₈	—	5	2	no	—
123	C₂×C₄²	—	5	4	no	—
124	C₂×C₄²	—	5	2	no	—
125	C₂×C₄²	—	5	1	no	—
126	C₄×C₈	—	5	16	no	—
127	C₄×C₈	—	5	4	no	—
128	C₄×C₈	—	5	2	no	—
129	C₄×C₈	—	5	1	no	—
130	C₄×C₈	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
131	C₂×C₃×C₅	C₄³	1	1	yes	characteristic
132	C₂×C₃×C₅	C₂×C₄×C₈	1	2	yes	—
133	C₃×D₅	C₂×C₄×C₈	1	4	yes	—
134	C₂³×C₃	—	5	1	no	—
135	C₂×C₃×C₄	—	5	8	no	—
136	C₂×C₃×C₄	—	5	16	no	—
137	C₂×C₃×C₄	—	5	2	no	—
138	C₂×C₃×C₄	—	5	1	no	—
139	C₂×C₃×C₄	—	5	4	no	—
140	C₂×C₃×C₄	—	5	8	no	—
141	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	2	yes	—
142	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
143	C₃×C₈	—	5	8	no	—
144	C₃×C₈	—	5	4	no	—
145	C₃×C₈	C₄×AGL(1,𝔽₅)	1	4	yes	direct product factor
146	C₂²×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
147	C₂×D₅	C₂×C₃×C₄²	1	2	yes	—
148	C₂×D₅	C₂²×C₃×C₈	1	4	yes	—
149	AGL(1,𝔽₅)	C₃×C₄×C₈	1	16	yes	direct product factor
150	C₄×C₅	C₃×C₄×C₈	1	4	yes	complemented
151	C₄×C₅	C₂×C₃×C₄²	1	2	yes	—
152	Dic₅	C₃×C₄×C₈	1	4	yes	complemented
153	Dic₅	C₂×C₃×C₄²	1	2	yes	—
154	C₂²×C₄	—	5	4	no	—
155	C₂²×C₄	—	5	2	no	—
156	C₂²×C₄	—	5	1	no	—
157	C₂×C₈	—	5	8	no	—
158	C₂×C₈	—	5	4	no	—
159	C₂×C₈	—	5	2	no	—
160	C₂×C₈	—	5	8	no	—
161	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	—
162	C₄²	—	5	8	no	—
163	C₄²	—	5	16	no	—
164	C₄²	—	5	2	no	—
165	C₄²	—	5	1	no	—
166	C₄²	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
167	C₃×C₅	C₄²×C₈	1	1	yes	characteristic, complemented, Hall
168	C₂²×C₃	—	5	2	no	—
169	C₂²×C₃	—	5	4	no	—
170	C₂²×C₃	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
171	C₃×C₄	—	5	16	no	—
172	C₃×C₄	—	5	2	no	—
173	C₃×C₄	—	5	4	no	—
174	C₃×C₄	C₈×AGL(1,𝔽₅)	1	4	yes	direct product factor
175	C₃×C₄	C₂×C₄×AGL(1,𝔽₅)	1	2	yes	—
176	C₂×C₅	C₃×C₄³	1	1	yes	characteristic
177	C₂×C₅	C₂×C₃×C₄×C₈	1	2	yes	—
178	D₅	C₂×C₃×C₄×C₈	1	4	yes	—
179	C₂³	—	5	1	no	—
180	C₂×C₄	—	5	8	no	—
181	C₂×C₄	—	5	16	no	—
182	C₂×C₄	—	5	1	no	—
183	C₂×C₄	—	5	2	no	—
184	C₂×C₄	—	5	8	no	—
185	C₂×C₄	—	5	4	no	—
186	C₂×C₄	C₃×C₄×AGL(1,𝔽₅)	1	2	yes	—
187	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
188	C₈	—	5	8	no	—
189	C₈	—	5	4	no	—
190	C₈	C₃×C₄×AGL(1,𝔽₅)	1	4	yes	direct product factor
191	C₂×C₃	—	5	4	no	—
192	C₂×C₃	C₄²×AGL(1,𝔽₅)	1	1	yes	characteristic
193	C₂×C₃	C₂×C₈×AGL(1,𝔽₅)	1	2	yes	—
194	C₅	C₃×C₄²×C₈	1	1	yes	characteristic, complemented, Sylow
195	C₂²	—	5	2	no	—
196	C₂²	—	5	4	no	—
197	C₂²	C₂×C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
198	C₄	—	5	16	no	—
199	C₄	—	5	2	no	—
200	C₄	—	5	4	no	—
201	C₄	C₃×C₈×AGL(1,𝔽₅)	1	4	yes	direct product factor
202	C₄	C₂×C₃×C₄×AGL(1,𝔽₅)	1	2	yes	—
203	C₃	C₄×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor, Sylow
204	C₂	—	5	4	no	—
205	C₂	C₃×C₄²×AGL(1,𝔽₅)	1	1	yes	characteristic
206	C₂	C₂×C₃×C₈×AGL(1,𝔽₅)	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
5. C₄×C₈×AGL(1,𝔽₅)	C₃	direct (N × H)
11. C₃×C₄×#170	C₄	semidirect
12. C₃×C₈×AGL(1,𝔽₅)	C₄	direct (N × H)
13. C₃×C₄×C₅×C₈	C₄	semidirect
14. C₃×C₈×Dic₅	C₄	semidirect
15. C₃×C₄×#168	C₄	semidirect
25. C₃×C₄×AGL(1,𝔽₅)	C₈	direct (N × H)
43. C₄×#170	C₃×C₄	semidirect
44. C₈×AGL(1,𝔽₅)	C₃×C₄	direct (N × H)
45. C₄×C₅×C₈	C₃×C₄	semidirect
46. C₈×Dic₅	C₃×C₄	semidirect
47. C₄×#168	C₃×C₄	semidirect
58. C₃×#170	C₄²	semidirect
59. C₃×C₅×C₈	C₄²	semidirect
60. C₃×#168	C₄²	semidirect
70. C₃×C₄×C₈	AGL(1,𝔽₅)	direct (N × H)
75. C₄×AGL(1,𝔽₅)	C₃×C₈	direct (N × H)
91. C₃×AGL(1,𝔽₅)	C₄×C₈	direct (N × H)
92. C₃×C₄×C₅	C₄×C₈	semidirect
94. C₃×Dic₅	C₄×C₈	semidirect
118. #170	C₃×C₄²	semidirect
119. C₅×C₈	C₃×C₄²	semidirect
120. #168	C₃×C₄²	semidirect
130. C₄×C₈	C₃×AGL(1,𝔽₅)	direct (N × H)
145. C₃×C₈	C₄×AGL(1,𝔽₅)	direct (N × H)
149. AGL(1,𝔽₅)	C₃×C₄×C₈	direct (N × H)
150. C₄×C₅	C₃×C₄×C₈	semidirect
152. Dic₅	C₃×C₄×C₈	semidirect
167. C₃×C₅	C₄²×C₈	semidirect
174. C₃×C₄	C₈×AGL(1,𝔽₅)	direct (N × H)
190. C₈	C₃×C₄×AGL(1,𝔽₅)	direct (N × H)
194. C₅	C₃×C₄²×C₈	semidirect
201. C₄	C₃×C₈×AGL(1,𝔽₅)	direct (N × H)
203. C₃	C₄×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×C₈×AGL(1,𝔽₅)	2
C₃×C₄²×AGL(1,𝔽₅)	2
C₃×C₄×D₅⋊C₈	2
C₃×C₄×C₈×D₅	2
C₄×C₈×AGL(1,𝔽₅)	3
C₃×C₄²×C₈	5

Subgroups — C3×C4×C8×AGL(1,𝔽5) #14294188

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×C₈×AGL(1,𝔽₅) #14294188