Subgroups — C₃×#10604048 #14296671

Order 1920.

544 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#3994	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3992	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄⋊C₄×AGL(1,𝔽₅)	C₂	1	2	yes	maximal
4	C₃×D₅×#106	C₂	1	1	yes	maximal, characteristic
5	C₃×#64874	C₂	1	2	yes	maximal
6	#10604048	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₄⋊AGL(1,𝔽₅)	C₂²	1	2	yes	—
8	C₂×C₃×C₄×AGL(1,𝔽₅)	—	2	2	no	—
9	C₂×C₃×C₄×AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
10	C₂×C₃×D₅⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
12	C₃×D₅×C₄⋊C₄	C₂²	1	2	yes	—
13	C₃×(C₂×AGL(1,𝔽₅))⋊C₄	—	2	2	no	—
14	C₃×#3994	C₄	1	4	yes	complemented
15	C₃×#3992	C₄	1	4	yes	complemented
16	C₃×C₅×#106	C₄	1	1	yes	characteristic, complemented
17	C₃×#3949	C₄	1	1	yes	characteristic, complemented
18	C₃×#3940	C₄	1	2	yes	complemented
19	C₃×#1776	—	5	1	no	maximal, Hall
20	C₂×#3994	C₂×C₃	1	1	yes	characteristic
21	C₂×#3992	C₂×C₃	1	1	yes	characteristic
22	C₄⋊C₄×AGL(1,𝔽₅)	C₂×C₃	1	2	yes	—
23	D₅×#106	C₂×C₃	1	1	yes	characteristic
24	#64874	C₂×C₃	1	2	yes	—
25	C₂²×C₃×AGL(1,𝔽₅)	—	2	2	no	—
26	C₂²×C₃×AGL(1,𝔽₅)	D₄	1	2	yes	—
27	C₂×C₃×C₄×D₅	—	2	2	no	—
28	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
29	C₂×C₃×#170	—	2	1	no	—
30	C₃×C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
31	C₃×C₄⋊AGL(1,𝔽₅)	C₂×C₄	1	4	yes	—
32	C₃×C₄×AGL(1,𝔽₅)	—	4	4	no	—
33	C₃×C₄×AGL(1,𝔽₅)	C₂×C₄	1	2	yes	—
34	C₃×C₄×AGL(1,𝔽₅)	Q₈	1	2	yes	—
35	C₃×D₅⋊C₈	—	2	1	no	—
36	C₃×D₅⋊C₈	C₂×C₄	1	2	yes	—
37	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
38	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
39	C₃×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
40	C₃×#683	—	2	2	no	—
41	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
42	C₃×C₈×D₅	C₂×C₄	1	2	yes	—
43	C₃×C₈×D₅	C₂×C₄	1	2	yes	—
44	C₂×C₃×#107	—	5	1	no	—
45	C₂×C₃×#106	—	5	1	no	—
46	C₂×C₃×#97	—	5	1	no	—
47	C₃×C₄×C₄⋊C₄	—	5	2	no	—
48	C₃×#340	—	5	2	no	—
49	C₂×C₄⋊AGL(1,𝔽₅)	C₂²×C₃	1	2	yes	—
50	C₂×C₄×AGL(1,𝔽₅)	—	2	2	no	—
51	C₂×C₄×AGL(1,𝔽₅)	C₂²×C₃	1	1	yes	characteristic
52	C₂×D₅⋊C₈	C₂²×C₃	1	1	yes	characteristic
53	C₂×C₈×D₅	C₂²×C₃	1	1	yes	characteristic
54	D₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
55	(C₂×AGL(1,𝔽₅))⋊C₄	—	2	2	no	—
56	#3994	C₃×C₄	1	4	yes	complemented
57	#3992	C₃×C₄	1	4	yes	complemented
58	C₅×#106	C₃×C₄	1	1	yes	characteristic, complemented
59	#3949	C₃×C₄	1	1	yes	characteristic, complemented
60	#3940	C₃×C₄	1	2	yes	complemented
61	#1776	—	5	1	no	Sylow
62	C₂²×C₃×D₅	C₂×D₄	1	1	yes	characteristic
63	C₂×C₃×AGL(1,𝔽₅)	—	2	2	no	—
64	C₂×C₃×AGL(1,𝔽₅)	—	2	2	no	—
65	C₂×C₃×AGL(1,𝔽₅)	—	2	4	no	—
66	C₂×C₃×AGL(1,𝔽₅)	—	4	2	no	—
67	C₂×C₃×AGL(1,𝔽₅)	—	4	2	no	—
68	C₂×C₃×AGL(1,𝔽₅)	C₄⋊C₄	1	4	yes	—
69	C₂×C₃×C₄×C₅	—	2	2	no	—
70	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₃×Dic₅	—	2	2	no	—
72	C₂×C₃×Dic₅	#41	1	1	yes	characteristic
73	C₃×C₄×D₅	—	4	2	no	—
74	C₃×C₄×D₅	—	4	2	no	—
75	C₃×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
76	C₃×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
77	C₃×C₄×D₅	C₂×Q₈	1	1	yes	characteristic
78	C₃×C₄×D₅	#41	1	1	yes	characteristic
79	C₃×#170	—	2	2	no	—
80	C₃×C₅×C₈	C₄²	1	2	yes	complemented
81	C₃×#168	C₄²	1	2	yes	complemented
82	C₂²×C₃×C₈	—	5	1	no	—
83	C₂²×C₃×C₈	—	5	1	no	—
84	C₂×C₃×C₄⋊C₄	—	5	2	no	—
85	C₂×C₃×C₄⋊C₄	—	5	2	no	—
86	C₂×C₃×C₄²	—	5	1	no	—
87	C₂×C₃×C₄²	—	10	2	no	—
88	C₃×#107	—	5	4	no	—
89	C₃×#106	—	5	1	no	—
90	C₃×#106	—	5	2	no	—
91	C₃×#106	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
92	C₃×#97	—	5	4	no	—
93	C₃×#95	—	10	2	no	—
94	C₂²×AGL(1,𝔽₅)	—	2	2	no	—
95	C₂²×AGL(1,𝔽₅)	C₃×D₄	1	2	yes	—
96	C₂×C₄×D₅	—	2	2	no	—
97	C₂×C₄×D₅	C₂³×C₃	1	1	yes	characteristic
98	C₂×#170	—	2	1	no	—
99	C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
100	C₄⋊AGL(1,𝔽₅)	C₂×C₃×C₄	1	4	yes	—
101	C₄×AGL(1,𝔽₅)	—	4	4	no	—
102	C₄×AGL(1,𝔽₅)	C₂×C₃×C₄	1	2	yes	—
103	C₄×AGL(1,𝔽₅)	C₃×Q₈	1	2	yes	—
104	D₅⋊C₈	—	2	1	no	—
105	D₅⋊C₈	C₂×C₃×C₄	1	2	yes	—
106	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
107	C₅×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
108	C₄⋊Dic₅	C₂×C₃×C₄	1	2	yes	—
109	#683	—	2	2	no	—
110	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
111	C₈×D₅	C₂×C₃×C₄	1	2	yes	—
112	C₈×D₅	C₂×C₃×C₄	1	2	yes	—
113	C₂×#107	—	5	1	no	—
114	C₂×#106	—	5	1	no	—
115	C₂×#97	—	5	1	no	—
116	C₄×C₄⋊C₄	—	5	2	no	—
117	#340	—	5	2	no	—
118	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
119	C₂×C₃×D₅	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₃×D₅	C₄×D₄	1	1	yes	characteristic
121	C₂×C₃×D₅	AGL(1,ℤ/8)	1	2	yes	—
122	C₂×C₃×D₅	#137	1	2	yes	—
123	C₃×AGL(1,𝔽₅)	—	2	4	no	—
124	C₃×AGL(1,𝔽₅)	—	4	4	no	—
125	C₃×C₄×C₅	—	4	2	no	—
126	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
127	C₃×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
128	C₃×Dic₅	—	4	2	no	—
129	C₃×Dic₅	#117	1	1	yes	characteristic
130	C₃×Dic₅	C₄×Q₈	1	1	yes	characteristic
131	C₂²×C₃×C₄	—	5	1	no	—
132	C₂²×C₃×C₄	—	5	2	no	—
133	C₂²×C₃×C₄	—	10	2	no	—
134	C₂²×C₃×C₄	—	10	2	no	—
135	C₂×C₃×C₈	—	5	1	no	—
136	C₂×C₃×C₈	—	5	2	no	—
137	C₂×C₃×C₈	—	5	2	no	—
138	C₂×C₃×C₈	—	5	2	no	—
139	C₂×C₃×C₈	—	10	1	no	—
140	C₂×C₃×C₈	—	10	1	no	—
141	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
142	C₃×C₄⋊C₄	—	5	4	no	—
143	C₃×C₄⋊C₄	—	5	2	no	—
144	C₃×C₄⋊C₄	—	10	2	no	—
145	C₃×C₄⋊C₄	—	10	2	no	—
146	C₃×C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
147	C₃×C₄²	—	5	2	no	—
148	C₃×C₄²	—	5	2	no	—
149	C₃×C₄²	—	20	4	no	—
150	C₂²×D₅	C₂×C₃×D₄	1	1	yes	characteristic
151	C₂×AGL(1,𝔽₅)	—	2	2	no	—
152	C₂×AGL(1,𝔽₅)	—	2	2	no	—
153	C₂×AGL(1,𝔽₅)	—	2	4	no	—
154	C₂×AGL(1,𝔽₅)	—	4	2	no	—
155	C₂×AGL(1,𝔽₅)	—	4	2	no	—
156	C₂×AGL(1,𝔽₅)	C₃×C₄⋊C₄	1	4	yes	—
157	C₂×C₄×C₅	—	2	2	no	—
158	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
159	C₂×Dic₅	—	2	2	no	—
160	C₂×Dic₅	C₃×#41	1	1	yes	characteristic
161	C₄×D₅	—	4	2	no	—
162	C₄×D₅	—	4	2	no	—
163	C₄×D₅	C₂²×C₃×C₄	1	1	yes	characteristic
164	C₄×D₅	C₂²×C₃×C₄	1	1	yes	characteristic
165	C₄×D₅	C₂×C₃×Q₈	1	1	yes	characteristic
166	C₄×D₅	C₃×#41	1	1	yes	characteristic
167	#170	—	2	2	no	—
168	C₅×C₈	C₃×C₄²	1	2	yes	complemented
169	#168	C₃×C₄²	1	2	yes	complemented
170	C₂²×C₈	—	5	1	no	—
171	C₂²×C₈	—	5	1	no	—
172	C₂×C₄⋊C₄	—	5	2	no	—
173	C₂×C₄⋊C₄	—	5	2	no	—
174	C₂×C₄²	—	5	1	no	—
175	C₂×C₄²	—	10	2	no	—
176	#107	—	5	4	no	—
177	#106	—	5	1	no	—
178	#106	—	5	2	no	—
179	#106	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
180	#97	—	5	4	no	—
181	#95	—	10	2	no	—
182	C₂×C₃×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
183	C₂×C₃×C₅	#440	1	2	yes	—
184	C₃×D₅	#428	1	2	yes	—
185	C₃×D₅	#441	1	1	yes	characteristic
186	C₃×D₅	#442	1	1	yes	characteristic
187	C₂³×C₃	—	5	1	no	—
188	C₂×C₃×C₄	—	2	2	no	—
189	C₂×C₃×C₄	—	5	1	no	—
190	C₂×C₃×C₄	—	5	1	no	—
191	C₂×C₃×C₄	—	5	1	no	—
192	C₂×C₃×C₄	—	5	1	no	—
193	C₂×C₃×C₄	—	5	1	no	—
194	C₂×C₃×C₄	—	5	4	no	—
195	C₂×C₃×C₄	—	10	2	no	—
196	C₂×C₃×C₄	—	10	2	no	—
197	C₂×C₃×C₄	—	10	4	no	—
198	C₂×C₃×C₄	—	10	2	no	—
199	C₂×C₃×C₄	—	20	2	no	—
200	C₂×C₃×C₄	—	20	2	no	—
201	C₂×C₃×C₄	—	20	2	no	—
202	C₂×C₃×C₄	—	20	2	no	—
203	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
204	C₃×C₈	—	5	2	no	—
205	C₃×C₈	—	10	2	no	—
206	C₃×C₈	C₄×AGL(1,𝔽₅)	1	2	yes	complemented
207	C₂²×C₅	C₃×C₄×D₄	1	1	yes	characteristic
208	C₂×D₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
209	C₂×D₅	C₃×C₄×D₄	1	1	yes	characteristic
210	C₂×D₅	C₃×AGL(1,ℤ/8)	1	2	yes	—
211	C₂×D₅	C₃×#137	1	2	yes	—
212	AGL(1,𝔽₅)	—	2	4	no	—
213	AGL(1,𝔽₅)	—	4	4	no	—
214	C₄×C₅	—	4	2	no	—
215	C₄×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
216	C₄×C₅	C₃×C₄×Q₈	1	1	yes	characteristic
217	Dic₅	—	4	2	no	—
218	Dic₅	C₃×#117	1	1	yes	characteristic
219	Dic₅	C₃×C₄×Q₈	1	1	yes	characteristic
220	C₂²×C₄	—	5	1	no	—
221	C₂²×C₄	—	5	2	no	—
222	C₂²×C₄	—	10	2	no	—
223	C₂²×C₄	—	10	2	no	—
224	C₂×C₈	—	5	1	no	—
225	C₂×C₈	—	5	2	no	—
226	C₂×C₈	—	5	2	no	—
227	C₂×C₈	—	5	2	no	—
228	C₂×C₈	—	10	1	no	—
229	C₂×C₈	—	10	1	no	—
230	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
231	C₄⋊C₄	—	5	4	no	—
232	C₄⋊C₄	—	5	2	no	—
233	C₄⋊C₄	—	10	2	no	—
234	C₄⋊C₄	—	10	2	no	—
235	C₄⋊C₄	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	—
236	C₄²	—	5	2	no	—
237	C₄²	—	5	2	no	—
238	C₄²	—	20	4	no	—
239	C₃×C₅	#1776	1	1	yes	characteristic, complemented, Hall
240	C₂²×C₃	—	5	1	no	—
241	C₂²×C₃	—	5	1	no	—
242	C₂²×C₃	—	5	2	no	—
243	C₂²×C₃	—	5	2	no	—
244	C₂²×C₃	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
245	C₃×C₄	—	4	2	no	—
246	C₃×C₄	—	5	1	no	—
247	C₃×C₄	—	5	1	no	—
248	C₃×C₄	—	10	4	no	—
249	C₃×C₄	—	20	4	no	—
250	C₃×C₄	—	20	2	no	—
251	C₃×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
252	C₃×C₄	Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
253	C₂×C₅	C₃×C₄×C₄⋊C₄	1	1	yes	characteristic
254	C₂×C₅	C₃×#440	1	2	yes	—
255	D₅	C₃×#428	1	2	yes	—
256	D₅	C₃×#441	1	1	yes	characteristic
257	D₅	C₃×#442	1	1	yes	characteristic
258	C₂³	—	5	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	5	1	no	—
261	C₂×C₄	—	5	1	no	—
262	C₂×C₄	—	5	1	no	—
263	C₂×C₄	—	5	1	no	—
264	C₂×C₄	—	5	1	no	—
265	C₂×C₄	—	5	4	no	—
266	C₂×C₄	—	10	2	no	—
267	C₂×C₄	—	10	2	no	—
268	C₂×C₄	—	10	4	no	—
269	C₂×C₄	—	10	2	no	—
270	C₂×C₄	—	20	2	no	—
271	C₂×C₄	—	20	2	no	—
272	C₂×C₄	—	20	2	no	—
273	C₂×C₄	—	20	2	no	—
274	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
275	C₈	—	5	2	no	—
276	C₈	—	10	2	no	—
277	C₈	C₃×C₄×AGL(1,𝔽₅)	1	2	yes	complemented
278	C₂×C₃	—	5	2	no	—
279	C₂×C₃	—	5	1	no	—
280	C₂×C₃	—	5	1	no	—
281	C₂×C₃	C₄⋊C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
282	C₂×C₃	#65735	1	2	yes	—
283	C₅	C₃×#1776	1	1	yes	characteristic, complemented, Sylow
284	C₂²	—	5	1	no	—
285	C₂²	—	5	1	no	—
286	C₂²	—	5	2	no	—
287	C₂²	—	5	2	no	—
288	C₂²	C₃×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
289	C₄	—	4	2	no	—
290	C₄	—	5	1	no	—
291	C₄	—	5	1	no	—
292	C₄	—	10	4	no	—
293	C₄	—	20	4	no	—
294	C₄	—	20	2	no	—
295	C₄	C₂×C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
296	C₄	C₃×Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
297	C₃	#10604048	1	1	yes	characteristic, direct product factor, Sylow
298	C₂	—	5	2	no	—
299	C₂	—	5	1	no	—
300	C₂	—	5	1	no	—
301	C₂	C₃×C₄⋊C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
302	C₂	C₃×#65735	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	D₄×AGL(1,𝔽₅)
commutator	C₄×C₅	C₂×C₃×C₄²
Frattini	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₃×C₅×#106	C₄
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂²×C₃×D₅	C₂×D₄
cosocle	C₂×C₄×D₅	C₂³×C₃
hypercenter	C₃×#106	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#1776
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₃×#106

Complements

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

N	H ≅ G/N	type
6. #10604048	C₃	direct (N × H)
14. C₃×#3994	C₄	semidirect
15. C₃×#3992	C₄	semidirect
16. C₃×C₅×#106	C₄	semidirect
17. C₃×#3949	C₄	semidirect
18. C₃×#3940	C₄	semidirect
56. #3994	C₃×C₄	semidirect
57. #3992	C₃×C₄	semidirect
58. C₅×#106	C₃×C₄	semidirect
59. #3949	C₃×C₄	semidirect
60. #3940	C₃×C₄	semidirect
80. C₃×C₅×C₈	C₄²	semidirect
81. C₃×#168	C₄²	semidirect
91. C₃×#106	AGL(1,𝔽₅)	semidirect
168. C₅×C₈	C₃×C₄²	semidirect
169. #168	C₃×C₄²	semidirect
179. #106	C₃×AGL(1,𝔽₅)	semidirect
206. C₃×C₈	C₄×AGL(1,𝔽₅)	semidirect
239. C₃×C₅	#1776	semidirect
277. C₈	C₃×C₄×AGL(1,𝔽₅)	semidirect
283. C₅	C₃×#1776	semidirect
297. C₃	#10604048	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×#3994	2
C₂×C₃×#3992	2
C₃×C₄⋊C₄×AGL(1,𝔽₅)	2
C₃×D₅×#106	2
C₃×#64874	2
#10604048	3
C₃×#1776	5

Subgroups — C3×#10604048 #14296671

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10604048 #14296671