Subgroups — C₂×C₃×Dic₃×AGL(1,𝔽₅) #13088645

Order 1440.

486 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×ΓL(1,𝔽₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃²×AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×D₅×Dic₃	C₂	1	1	yes	maximal, characteristic
4	C₃×Dic₃×AGL(1,𝔽₅)	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₃×C₄×AGL(1,𝔽₅)	—	3	1	no	maximal
6	C₂×Dic₃×AGL(1,𝔽₅)	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂²×C₃²×D₅	C₂²	1	1	yes	characteristic
8	C₂×C₃×ΓL(1,𝔽₁₆)	C₄	1	4	yes	complemented
9	C₂×C₃×ΓL(1,𝔽₁₆)	C₂²	1	2	yes	—
10	C₂×C₃²×AGL(1,𝔽₅)	C₄	1	4	yes	complemented
11	C₂×C₃²×AGL(1,𝔽₅)	C₂²	1	2	yes	—
12	C₂×C₃×Dic₁₅	C₄	1	1	yes	characteristic, complemented
13	C₂×C₃×C₅×Dic₃	C₄	1	1	yes	characteristic, complemented
14	C₃×D₅×Dic₃	C₄	1	2	yes	complemented
15	C₃×D₅×Dic₃	C₂²	1	2	yes	—
16	C₂×C₃×C₄×Dic₃	—	5	1	no	maximal, Hall
17	C₂²×ΓL(1,𝔽₁₆)	C₂×C₃	1	1	yes	characteristic
18	C₂²×C₃×AGL(1,𝔽₅)	—	2	1	no	—
19	C₂²×C₃×AGL(1,𝔽₅)	—	3	1	no	—
20	C₂²×C₃×AGL(1,𝔽₅)	D₃	1	1	yes	characteristic
21	C₂²×C₃×AGL(1,𝔽₅)	C₂×C₃	1	1	yes	characteristic
22	C₂×C₃×C₄×D₅	—	3	1	no	—
23	C₂×D₅×Dic₃	C₂×C₃	1	1	yes	characteristic
24	C₃×C₄×AGL(1,𝔽₅)	—	3	4	no	—
25	Dic₃×AGL(1,𝔽₅)	C₂×C₃	1	4	yes	direct product factor
26	C₂²×C₃²×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃²×D₅	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃²×D₅	C₂×C₄	1	2	yes	—
29	C₂×C₃²×D₅	C₂×C₄	1	2	yes	—
30	C₂×C₃²×D₅	C₂³	1	1	yes	characteristic
31	C₃×ΓL(1,𝔽₁₆)	C₂×C₄	1	4	yes	complemented
32	C₃²×AGL(1,𝔽₅)	C₂×C₄	1	4	yes	complemented
33	C₃×Dic₁₅	C₂×C₄	1	2	yes	complemented
34	C₃×C₅×Dic₃	C₂×C₄	1	2	yes	complemented
35	C₂×C₄×AGL(1,𝔽₅)	—	3	1	no	Hall
36	C₂²×C₃²×C₄	—	5	1	no	—
37	C₂²×C₃×Dic₃	—	5	1	no	—
38	C₂²×C₃×Dic₃	—	5	1	no	—
39	C₃×C₄×Dic₃	—	5	4	no	—
40	C₂²×C₃×D₅	—	2	1	no	—
41	C₂²×C₃×D₅	C₂×D₃	1	1	yes	characteristic
42	C₂²×C₃×D₅	C₂²×C₃	1	1	yes	characteristic
43	C₂×ΓL(1,𝔽₁₆)	C₃×C₄	1	4	yes	complemented
44	C₂×ΓL(1,𝔽₁₆)	C₂²×C₃	1	2	yes	—
45	C₂×C₃×AGL(1,𝔽₅)	—	2	4	no	—
46	C₂×C₃×AGL(1,𝔽₅)	—	2	2	no	—
47	C₂×C₃×AGL(1,𝔽₅)	—	3	4	no	—
48	C₂×C₃×AGL(1,𝔽₅)	—	3	2	no	—
49	C₂×C₃×AGL(1,𝔽₅)	Dic₃	1	4	yes	direct product factor
50	C₂×C₃×AGL(1,𝔽₅)	C₃×C₄	1	4	yes	complemented
51	C₂×C₃×AGL(1,𝔽₅)	C₂×D₃	1	2	yes	—
52	C₂×C₃×AGL(1,𝔽₅)	C₂²×C₃	1	2	yes	—
53	C₂×C₃×C₄×C₅	—	3	1	no	—
54	C₂×Dic₁₅	C₃×C₄	1	1	yes	characteristic, complemented
55	C₂×C₅×Dic₃	C₃×C₄	1	1	yes	characteristic, complemented
56	C₂×C₃×Dic₅	—	3	1	no	—
57	C₃×C₄×D₅	—	3	2	no	—
58	C₃×C₄×D₅	—	3	2	no	—
59	D₅×Dic₃	C₃×C₄	1	2	yes	complemented
60	D₅×Dic₃	C₂²×C₃	1	2	yes	—
61	C₂×C₃×C₄²	—	15	1	no	—
62	C₂×C₄×Dic₃	—	5	1	no	—
63	C₂×C₃²×C₅	C₄²	1	2	yes	complemented
64	C₂×C₃²×C₅	C₂²×C₄	1	1	yes	characteristic
65	C₃²×D₅	C₄²	1	2	yes	complemented
66	C₃²×D₅	C₂²×C₄	1	1	yes	characteristic
67	C₃²×D₅	C₂²×C₄	1	1	yes	characteristic
68	C₂²×AGL(1,𝔽₅)	—	3	1	no	—
69	C₂²×AGL(1,𝔽₅)	C₃×D₃	1	1	yes	characteristic
70	C₂×C₄×D₅	—	3	1	no	—
71	C₄×AGL(1,𝔽₅)	—	3	4	no	—
72	C₂³×C₃²	—	5	1	no	—
73	C₂×C₃²×C₄	—	5	4	no	—
74	C₂×C₃²×C₄	—	5	2	no	—
75	C₂×C₃×Dic₃	—	5	4	no	—
76	C₂×C₃×Dic₃	—	5	2	no	—
77	C₂×C₃×Dic₃	—	5	1	no	—
78	C₂×C₃×Dic₃	—	5	2	no	—
79	C₂×C₃×Dic₃	—	5	2	no	—
80	C₂×C₃×Dic₃	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
81	C₂²×C₃×C₅	—	2	1	no	—
82	C₂²×C₃×C₅	C₄×D₃	1	1	yes	characteristic
83	C₂²×C₃×C₅	C₂×C₃×C₄	1	1	yes	characteristic
84	C₂×C₃×D₅	—	2	1	no	—
85	C₂×C₃×D₅	—	2	2	no	—
86	C₂×C₃×D₅	—	2	2	no	—
87	C₂×C₃×D₅	—	2	1	no	—
88	C₂×C₃×D₅	C₄×D₃	1	1	yes	characteristic
89	C₂×C₃×D₅	C₄×D₃	1	2	yes	—
90	C₂×C₃×D₅	C₂×Dic₃	1	2	yes	—
91	C₂×C₃×D₅	C₂×C₃×C₄	1	1	yes	characteristic
92	C₂×C₃×D₅	C₂×C₃×C₄	1	2	yes	—
93	C₂×C₃×D₅	C₂×C₃×C₄	1	2	yes	—
94	C₂×C₃×D₅	C₂²×D₃	1	1	yes	characteristic
95	C₂×C₃×D₅	C₂³×C₃	1	1	yes	characteristic
96	ΓL(1,𝔽₁₆)	C₂×C₃×C₄	1	4	yes	complemented
97	C₃×AGL(1,𝔽₅)	—	2	4	no	—
98	C₃×AGL(1,𝔽₅)	—	3	4	no	—
99	C₃×AGL(1,𝔽₅)	C₂×Dic₃	1	4	yes	direct product factor
100	C₃×AGL(1,𝔽₅)	C₂×C₃×C₄	1	4	yes	complemented
101	C₃×C₄×C₅	—	3	2	no	—
102	Dic₁₅	C₂×C₃×C₄	1	2	yes	complemented
103	C₃×Dic₅	—	3	2	no	—
104	C₅×Dic₃	C₂×C₃×C₄	1	2	yes	complemented
105	C₂²×C₃×C₄	—	5	1	no	—
106	C₂²×C₃×C₄	—	5	1	no	—
107	C₂²×C₃×C₄	—	10	1	no	—
108	C₂²×C₃×C₄	—	15	1	no	—
109	C₂²×C₃×C₄	—	15	1	no	—
110	C₂²×Dic₃	—	5	1	no	—
111	C₂²×Dic₃	—	5	1	no	—
112	C₃×C₄²	—	15	4	no	—
113	C₄×Dic₃	—	5	4	no	—
114	C₃²×C₅	C₂×C₄²	1	1	yes	characteristic, complemented, Hall
115	C₂²×D₅	C₂×C₃×D₃	1	1	yes	characteristic
116	C₂×AGL(1,𝔽₅)	—	3	4	no	—
117	C₂×AGL(1,𝔽₅)	—	3	2	no	—
118	C₂×AGL(1,𝔽₅)	C₃×Dic₃	1	4	yes	direct product factor
119	C₂×AGL(1,𝔽₅)	C₂×C₃×D₃	1	2	yes	—
120	C₂×C₄×C₅	—	3	1	no	—
121	C₂×Dic₅	—	3	1	no	—
122	C₄×D₅	—	3	2	no	—
123	C₄×D₅	—	3	2	no	—
124	C₂²×C₃²	—	5	2	no	—
125	C₂²×C₃²	—	5	2	no	—
126	C₂²×C₃²	—	5	1	no	—
127	C₂²×C₃²	—	5	1	no	—
128	C₂²×C₃²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
129	C₃²×C₄	—	5	4	no	—
130	C₃×Dic₃	—	5	4	no	—
131	C₃×Dic₃	—	5	2	no	—
132	C₃×Dic₃	C₂×AGL(1,𝔽₅)	1	2	yes	direct product factor
133	C₂×C₄²	—	15	1	no	Sylow
134	C₂×C₃×C₅	—	2	2	no	—
135	C₂×C₃×C₅	—	2	1	no	—
136	C₂×C₃×C₅	C₄×Dic₃	1	2	yes	complemented
137	C₂×C₃×C₅	C₃×C₄²	1	2	yes	complemented
138	C₂×C₃×C₅	C₂×C₄×D₃	1	1	yes	characteristic
139	C₂×C₃×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
140	C₃×D₅	—	2	2	no	—
141	C₃×D₅	—	2	1	no	—
142	C₃×D₅	—	2	1	no	—
143	C₃×D₅	C₄×Dic₃	1	2	yes	complemented
144	C₃×D₅	C₃×C₄²	1	2	yes	complemented
145	C₃×D₅	C₂×C₄×D₃	1	1	yes	characteristic
146	C₃×D₅	C₂²×Dic₃	1	1	yes	characteristic
147	C₃×D₅	C₂²×C₃×C₄	1	1	yes	characteristic
148	C₃×D₅	C₂²×C₃×C₄	1	1	yes	characteristic
149	C₂³×C₃	—	5	1	no	—
150	C₂³×C₃	—	5	1	no	—
151	C₂³×C₃	—	10	1	no	—
152	C₂×C₃×C₄	—	3	1	no	—
153	C₂×C₃×C₄	—	5	4	no	—
154	C₂×C₃×C₄	—	5	4	no	—
155	C₂×C₃×C₄	—	5	2	no	—
156	C₂×C₃×C₄	—	5	2	no	—
157	C₂×C₃×C₄	—	10	2	no	—
158	C₂×C₃×C₄	—	10	4	no	—
159	C₂×C₃×C₄	—	15	2	no	—
160	C₂×C₃×C₄	—	15	2	no	—
161	C₂×C₃×C₄	—	15	2	no	—
162	C₂×C₃×C₄	—	15	4	no	—
163	C₂×C₃×C₄	—	15	1	no	—
164	C₂×Dic₃	—	5	4	no	—
165	C₂×Dic₃	—	5	2	no	—
166	C₂×Dic₃	—	5	1	no	—
167	C₂×Dic₃	—	5	2	no	—
168	C₂×Dic₃	—	5	2	no	—
169	C₂×Dic₃	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
170	C₂²×C₅	C₃×C₄×D₃	1	1	yes	characteristic
171	C₂×D₅	C₃×C₄×D₃	1	1	yes	characteristic
172	C₂×D₅	C₃×C₄×D₃	1	2	yes	—
173	C₂×D₅	C₂×C₃×Dic₃	1	2	yes	—
174	C₂×D₅	C₂²×C₃×D₃	1	1	yes	characteristic
175	AGL(1,𝔽₅)	—	3	4	no	—
176	AGL(1,𝔽₅)	C₂×C₃×Dic₃	1	4	yes	direct product factor
177	C₄×C₅	—	3	2	no	—
178	Dic₅	—	3	2	no	—
179	C₂×C₃²	—	5	2	no	—
180	C₂×C₃²	—	5	1	no	—
181	C₂×C₃²	—	5	1	no	—
182	C₂×C₃²	C₄×AGL(1,𝔽₅)	1	2	yes	complemented
183	C₂×C₃²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
184	C₂²×C₄	—	5	1	no	—
185	C₂²×C₄	—	15	1	no	—
186	C₂²×C₄	—	15	1	no	—
187	C₄²	—	15	4	no	—
188	C₃×C₅	—	2	1	no	—
189	C₃×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic, complemented
190	C₃×C₅	C₂×C₃×C₄²	1	1	yes	characteristic, complemented
191	C₂²×C₃	—	2	1	no	—
192	C₂²×C₃	—	5	2	no	—
193	C₂²×C₃	—	5	2	no	—
194	C₂²×C₃	—	5	2	no	—
195	C₂²×C₃	—	5	2	no	—
196	C₂²×C₃	—	5	1	no	—
197	C₂²×C₃	—	5	1	no	—
198	C₂²×C₃	—	5	1	no	—
199	C₂²×C₃	—	5	1	no	—
200	C₂²×C₃	—	10	1	no	—
201	C₂²×C₃	—	10	2	no	—
202	C₂²×C₃	—	10	1	no	—
203	C₂²×C₃	—	10	2	no	—
204	C₂²×C₃	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
205	C₂²×C₃	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
206	C₃×C₄	—	3	2	no	—
207	C₃×C₄	—	5	4	no	—
208	C₃×C₄	—	5	4	no	—
209	C₃×C₄	—	10	4	no	—
210	C₃×C₄	—	15	4	no	—
211	C₃×C₄	—	15	2	no	—
212	Dic₃	—	5	4	no	—
213	Dic₃	—	5	2	no	—
214	Dic₃	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	direct product factor
215	C₂×C₅	C₃×C₄×Dic₃	1	2	yes	complemented
216	C₂×C₅	C₂×C₃×C₄×D₃	1	1	yes	characteristic
217	D₅	C₃×C₄×Dic₃	1	2	yes	complemented
218	D₅	C₂×C₃×C₄×D₃	1	1	yes	characteristic
219	D₅	C₂²×C₃×Dic₃	1	1	yes	characteristic
220	C₃²	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
221	C₂³	—	5	1	no	—
222	C₂×C₄	—	3	1	no	—
223	C₂×C₄	—	5	2	no	—
224	C₂×C₄	—	5	4	no	—
225	C₂×C₄	—	15	2	no	—
226	C₂×C₄	—	15	2	no	—
227	C₂×C₄	—	15	2	no	—
228	C₂×C₄	—	15	4	no	—
229	C₂×C₄	—	15	1	no	—
230	C₂×C₃	—	2	1	no	—
231	C₂×C₃	—	2	2	no	—
232	C₂×C₃	—	5	1	no	—
233	C₂×C₃	—	5	1	no	—
234	C₂×C₃	—	5	2	no	—
235	C₂×C₃	—	5	1	no	—
236	C₂×C₃	—	5	1	no	—
237	C₂×C₃	—	5	2	no	—
238	C₂×C₃	—	10	1	no	—
239	C₂×C₃	—	10	1	no	—
240	C₂×C₃	—	10	2	no	—
241	C₂×C₃	Dic₃×AGL(1,𝔽₅)	1	2	yes	direct product factor
242	C₂×C₃	C₃×C₄×AGL(1,𝔽₅)	1	2	yes	complemented
243	C₂×C₃	C₂×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
244	C₂×C₃	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
245	C₅	C₂×C₃×C₄×Dic₃	1	1	yes	characteristic, complemented, Sylow
246	C₂²	—	5	1	no	—
247	C₂²	—	5	1	no	—
248	C₂²	—	5	2	no	—
249	C₂²	—	5	2	no	—
250	C₂²	C₃×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
251	C₄	—	3	2	no	—
252	C₄	—	5	4	no	—
253	C₄	—	15	4	no	—
254	C₄	—	15	2	no	—
255	C₃	—	2	1	no	—
256	C₃	C₂×Dic₃×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
257	C₃	C₂×C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
258	C₂	—	5	1	no	—
259	C₂	—	5	1	no	—
260	C₂	—	5	2	no	—
261	C₂	C₃×Dic₃×AGL(1,𝔽₅)	1	2	yes	direct product factor
262	C₂	C₂×C₃×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₃	D₃×AGL(1,𝔽₅)
commutator, hypocenter	C₃×C₅	C₂×C₃×C₄²
Frattini	C₂	C₂×C₃×D₃×AGL(1,𝔽₅)
Fitting, socle	C₂²×C₃²×C₅	C₂×C₄
prime radical	C₂²×C₃²×D₅	C₂²
cosocle	C₂×C₃×D₅	C₂³×C₃
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₂²×C₃

Complements

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

N	H ≅ G/N	type
4. C₃×Dic₃×AGL(1,𝔽₅)	C₂	direct (N × H)
6. C₂×Dic₃×AGL(1,𝔽₅)	C₃	direct (N × H)
8. C₂×C₃×ΓL(1,𝔽₁₆)	C₄	semidirect
10. C₂×C₃²×AGL(1,𝔽₅)	C₄	semidirect
12. C₂×C₃×Dic₁₅	C₄	semidirect
13. C₂×C₃×C₅×Dic₃	C₄	semidirect
14. C₃×D₅×Dic₃	C₄	semidirect
25. Dic₃×AGL(1,𝔽₅)	C₂×C₃	direct (N × H)
31. C₃×ΓL(1,𝔽₁₆)	C₂×C₄	semidirect
32. C₃²×AGL(1,𝔽₅)	C₂×C₄	semidirect
33. C₃×Dic₁₅	C₂×C₄	semidirect
34. C₃×C₅×Dic₃	C₂×C₄	semidirect
43. C₂×ΓL(1,𝔽₁₆)	C₃×C₄	semidirect
49. C₂×C₃×AGL(1,𝔽₅)	Dic₃	direct (N × H)
50. C₂×C₃×AGL(1,𝔽₅)	C₃×C₄	semidirect
54. C₂×Dic₁₅	C₃×C₄	semidirect
55. C₂×C₅×Dic₃	C₃×C₄	semidirect
59. D₅×Dic₃	C₃×C₄	semidirect
63. C₂×C₃²×C₅	C₄²	semidirect
65. C₃²×D₅	C₄²	semidirect
80. C₂×C₃×Dic₃	AGL(1,𝔽₅)	direct (N × H)
96. ΓL(1,𝔽₁₆)	C₂×C₃×C₄	semidirect
99. C₃×AGL(1,𝔽₅)	C₂×Dic₃	direct (N × H)
100. C₃×AGL(1,𝔽₅)	C₂×C₃×C₄	semidirect
102. Dic₁₅	C₂×C₃×C₄	semidirect
104. C₅×Dic₃	C₂×C₃×C₄	semidirect
114. C₃²×C₅	C₂×C₄²	semidirect
118. C₂×AGL(1,𝔽₅)	C₃×Dic₃	direct (N × H)
132. C₃×Dic₃	C₂×AGL(1,𝔽₅)	direct (N × H)
136. C₂×C₃×C₅	C₄×Dic₃	semidirect
137. C₂×C₃×C₅	C₃×C₄²	semidirect
143. C₃×D₅	C₄×Dic₃	semidirect
144. C₃×D₅	C₃×C₄²	semidirect
169. C₂×Dic₃	C₃×AGL(1,𝔽₅)	direct (N × H)
176. AGL(1,𝔽₅)	C₂×C₃×Dic₃	direct (N × H)
182. C₂×C₃²	C₄×AGL(1,𝔽₅)	semidirect
189. C₃×C₅	C₂×C₄×Dic₃	semidirect
190. C₃×C₅	C₂×C₃×C₄²	semidirect
214. Dic₃	C₂×C₃×AGL(1,𝔽₅)	direct (N × H)
215. C₂×C₅	C₃×C₄×Dic₃	semidirect
217. D₅	C₃×C₄×Dic₃	semidirect
220. C₃²	C₂×C₄×AGL(1,𝔽₅)	semidirect
241. C₂×C₃	Dic₃×AGL(1,𝔽₅)	direct (N × H)
242. C₂×C₃	C₃×C₄×AGL(1,𝔽₅)	semidirect
245. C₅	C₂×C₃×C₄×Dic₃	semidirect
256. C₃	C₂×Dic₃×AGL(1,𝔽₅)	direct (N × H)
257. C₃	C₂×C₃×C₄×AGL(1,𝔽₅)	semidirect
261. C₂	C₃×Dic₃×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
C₂²×C₃×ΓL(1,𝔽₁₆)	2
C₂²×C₃²×AGL(1,𝔽₅)	2
C₂×C₃×D₅×Dic₃	2
C₃×Dic₃×AGL(1,𝔽₅)	2
C₂×C₃×C₄×AGL(1,𝔽₅)	3
C₂×Dic₃×AGL(1,𝔽₅)	3
C₂×C₃×C₄×Dic₃	5

Subgroups — C2×C3×Dic3×AGL(1,𝔽5) #13088645

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×Dic₃×AGL(1,𝔽₅) #13088645