Subgroups — C₃×#10604730 #14294634

Order 1920.

784 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₅⋊M₁₆	C₂	1	2	yes	maximal
2	C₂²×C₃×C₈×D₅	C₂	1	1	yes	maximal, characteristic
3	C₃×#64896	C₂	1	4	yes	maximal, complemented
4	#10604730	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂²×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
6	C₂×C₃×#704	—	2	2	no	—
7	C₃×D₅⋊M₁₆	—	2	4	no	—
8	C₃×D₅⋊M₁₆	C₄	1	8	yes	—
9	C₂×C₃×#700	C₂²	1	4	yes	—
10	C₂×C₃×D₅⋊C₈	—	2	2	no	—
11	C₂²×C₃×C₅×C₈	C₄	1	1	yes	characteristic
12	C₂²×C₃×#168	C₄	1	1	yes	characteristic
13	C₂×C₃×C₈×D₅	—	2	4	no	—
14	C₂×C₃×C₈×D₅	C₄	1	2	yes	—
15	C₂×C₃×C₈×D₅	C₂²	1	2	yes	—
16	C₂×C₃×#341	—	5	1	no	maximal, Hall
17	C₂×D₅⋊M₁₆	C₂×C₃	1	2	yes	—
18	C₂²×C₈×D₅	C₂×C₃	1	1	yes	characteristic
19	#64896	C₂×C₃	1	4	yes	complemented
20	C₂³×C₃×D₅	Q₈	1	1	yes	characteristic
21	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₃×Dic₅	D₄	1	1	yes	characteristic
23	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×C₄×D₅	C₂×C₄	1	4	yes	—
25	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
26	C₂×C₃×C₄×D₅	D₄	1	4	yes	—
27	C₂×C₃×C₄×D₅	Q₈	1	1	yes	characteristic
28	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
29	C₃×#704	—	2	8	no	—
30	C₂×C₃×#170	—	2	4	no	—
31	C₃×#700	—	2	4	no	—
32	C₃×#700	C₂×C₄	1	8	yes	—
33	C₃×D₅⋊C₈	—	2	8	no	—
34	C₂×C₃×C₅×C₈	—	2	2	no	—
35	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
36	C₂×C₃×#168	—	2	2	no	—
37	C₂×C₃×#168	C₂×C₄	1	2	yes	—
38	C₃×C₈×D₅	—	2	4	no	—
39	C₃×C₈×D₅	—	2	4	no	—
40	C₂²×C₃×M₁₆	—	5	2	no	—
41	C₂³×C₃×C₈	—	5	1	no	—
42	C₃×#341	—	5	4	no	—
43	C₂²×C₄×D₅	C₂²×C₃	1	1	yes	characteristic
44	C₂×#704	—	2	2	no	—
45	D₅⋊M₁₆	—	2	4	no	—
46	D₅⋊M₁₆	C₃×C₄	1	8	yes	—
47	C₂×#700	C₂²×C₃	1	4	yes	—
48	C₂×D₅⋊C₈	—	2	2	no	—
49	C₂²×C₅×C₈	C₃×C₄	1	1	yes	characteristic
50	C₂²×#168	C₃×C₄	1	1	yes	characteristic
51	C₂×C₈×D₅	—	2	4	no	—
52	C₂×C₈×D₅	C₃×C₄	1	2	yes	—
53	C₂×C₈×D₅	C₂²×C₃	1	2	yes	—
54	C₂×#341	—	5	1	no	Sylow
55	C₂³×C₃×C₅	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₃×D₅	—	2	4	no	—
57	C₂²×C₃×D₅	C₄⋊C₄	1	4	yes	—
58	C₂²×C₃×D₅	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₃×D₅	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
61	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
62	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₃×Dic₅	C₂²⋊C₄	1	4	yes	—
65	C₂×C₃×Dic₅	C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
67	C₃×C₄×D₅	C₄²	1	2	yes	—
68	C₃×C₄×D₅	C₂²⋊C₄	1	2	yes	—
69	C₃×C₄×D₅	C₂²⋊C₄	1	4	yes	—
70	C₃×C₄×D₅	C₄⋊C₄	1	4	yes	—
71	C₃×C₄×D₅	C₂²×C₄	1	2	yes	—
72	C₃×C₄×D₅	C₂×D₄	1	2	yes	—
73	C₃×#170	—	2	8	no	—
74	C₃×C₅×C₈	—	2	2	no	—
75	C₃×#168	—	2	2	no	—
76	C₂³×C₃×C₄	—	5	1	no	—
77	C₂×C₃×M₁₆	—	5	4	no	—
78	C₂×C₃×M₁₆	—	5	8	no	—
79	C₂×C₃×M₁₆	—	10	4	no	—
80	C₂×C₃×M₁₆	—	10	2	no	—
81	C₂²×C₃×C₈	—	5	1	no	—
82	C₂²×C₃×C₈	—	5	2	no	—
83	C₂²×C₃×C₈	—	5	2	no	—
84	C₂²×C₃×C₈	—	10	4	no	—
85	C₂²×C₃×C₈	—	10	2	no	—
86	C₂²×C₃×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
87	C₂³×D₅	C₃×Q₈	1	1	yes	characteristic
88	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
89	C₂²×Dic₅	C₃×D₄	1	1	yes	characteristic
90	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
91	C₂×C₄×D₅	C₂×C₃×C₄	1	4	yes	—
92	C₂×C₄×D₅	C₃×D₄	1	1	yes	characteristic
93	C₂×C₄×D₅	C₃×D₄	1	4	yes	—
94	C₂×C₄×D₅	C₃×Q₈	1	1	yes	characteristic
95	C₂×C₄×D₅	C₂³×C₃	1	1	yes	characteristic
96	#704	—	2	8	no	—
97	C₂×#170	—	2	4	no	—
98	#700	—	2	4	no	—
99	#700	C₂×C₃×C₄	1	8	yes	—
100	D₅⋊C₈	—	2	8	no	—
101	C₂×C₅×C₈	—	2	2	no	—
102	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
103	C₂×#168	—	2	2	no	—
104	C₂×#168	C₂×C₃×C₄	1	2	yes	—
105	C₈×D₅	—	2	4	no	—
106	C₈×D₅	—	2	4	no	—
107	C₂²×M₁₆	—	5	2	no	—
108	C₂³×C₈	—	5	1	no	—
109	#341	—	5	4	no	—
110	C₂²×C₃×C₅	—	2	2	no	—
111	C₂²×C₃×C₅	#95	1	2	yes	—
112	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₃×D₅	—	2	2	no	—
114	C₂×C₃×D₅	—	2	8	no	—
115	C₂×C₃×D₅	#95	1	2	yes	—
116	C₂×C₃×D₅	#108	1	4	yes	—
117	C₂×C₃×D₅	C₂×C₄⋊C₄	1	2	yes	—
118	C₃×C₄×C₅	#95	1	2	yes	—
119	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
120	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₃×Dic₅	#95	1	2	yes	—
122	C₃×Dic₅	C₂×C₂²⋊C₄	1	2	yes	—
123	C₂⁴×C₃	—	5	1	no	—
124	C₂²×C₃×C₄	—	5	1	no	—
125	C₂²×C₃×C₄	—	5	4	no	—
126	C₂²×C₃×C₄	—	5	1	no	—
127	C₂²×C₃×C₄	—	5	1	no	—
128	C₂²×C₃×C₄	—	5	4	no	—
129	C₂²×C₃×C₄	—	5	1	no	—
130	C₂²×C₃×C₄	—	5	1	no	—
131	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
132	C₃×M₁₆	—	5	8	no	—
133	C₃×M₁₆	—	10	4	no	—
134	C₃×M₁₆	—	10	8	no	—
135	C₂×C₃×C₈	—	2	2	no	—
136	C₂×C₃×C₈	—	5	2	no	—
137	C₂×C₃×C₈	—	10	4	no	—
138	C₂×C₃×C₈	—	10	4	no	—
139	C₂×C₃×C₈	—	10	8	no	—
140	C₂×C₃×C₈	—	10	4	no	—
141	C₂×C₃×C₈	—	10	2	no	—
142	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
143	C₂³×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
144	C₂²×D₅	—	2	4	no	—
145	C₂²×D₅	C₃×C₄⋊C₄	1	4	yes	—
146	C₂²×D₅	C₃×C₄⋊C₄	1	1	yes	characteristic
147	C₂²×D₅	C₂×C₃×Q₈	1	1	yes	characteristic
148	C₂×C₄×C₅	C₃×C₄²	1	2	yes	—
149	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
150	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
152	C₂×Dic₅	C₃×C₂²⋊C₄	1	4	yes	—
153	C₂×Dic₅	C₃×C₄⋊C₄	1	1	yes	characteristic
154	C₂×Dic₅	C₂×C₃×D₄	1	1	yes	characteristic
155	C₄×D₅	C₃×C₄²	1	2	yes	—
156	C₄×D₅	C₃×C₂²⋊C₄	1	2	yes	—
157	C₄×D₅	C₃×C₂²⋊C₄	1	4	yes	—
158	C₄×D₅	C₃×C₄⋊C₄	1	4	yes	—
159	C₄×D₅	C₂²×C₃×C₄	1	2	yes	—
160	C₄×D₅	C₂×C₃×D₄	1	2	yes	—
161	#170	—	2	8	no	—
162	C₅×C₈	—	2	2	no	—
163	#168	—	2	2	no	—
164	C₂³×C₄	—	5	1	no	—
165	C₂×M₁₆	—	5	4	no	—
166	C₂×M₁₆	—	5	8	no	—
167	C₂×M₁₆	—	10	4	no	—
168	C₂×M₁₆	—	10	2	no	—
169	C₂²×C₈	—	5	1	no	—
170	C₂²×C₈	—	5	2	no	—
171	C₂²×C₈	—	5	2	no	—
172	C₂²×C₈	—	10	4	no	—
173	C₂²×C₈	—	10	2	no	—
174	C₂²×C₈	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
175	C₂×C₃×C₅	—	2	2	no	—
176	C₂×C₃×C₅	C₂×#95	1	1	yes	characteristic
177	C₂×C₃×C₅	C₂×#108	1	2	yes	—
178	C₃×D₅	—	2	2	no	—
179	C₃×D₅	#341	1	4	yes	complemented
180	C₂³×C₃	—	5	1	no	—
181	C₂³×C₃	—	5	1	no	—
182	C₂³×C₃	—	5	4	no	—
183	C₂³×C₃	—	10	4	no	—
184	C₂³×C₃	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
185	C₂×C₃×C₄	—	5	1	no	—
186	C₂×C₃×C₄	—	5	1	no	—
187	C₂×C₃×C₄	—	5	4	no	—
188	C₂×C₃×C₄	—	5	2	no	—
189	C₂×C₃×C₄	—	5	2	no	—
190	C₂×C₃×C₄	—	5	2	no	—
191	C₂×C₃×C₄	—	5	2	no	—
192	C₂×C₃×C₄	—	5	4	no	—
193	C₂×C₃×C₄	—	5	4	no	—
194	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	2	yes	—
195	C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
196	C₂×C₃×C₄	C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
197	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
198	C₃×C₈	—	2	2	no	—
199	C₃×C₈	—	10	8	no	—
200	C₃×C₈	—	10	2	no	—
201	C₂²×C₅	—	2	2	no	—
202	C₂²×C₅	C₃×#95	1	2	yes	—
203	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
204	C₂×D₅	—	2	2	no	—
205	C₂×D₅	—	2	8	no	—
206	C₂×D₅	C₃×#95	1	2	yes	—
207	C₂×D₅	C₃×#108	1	4	yes	—
208	C₂×D₅	C₂×C₃×C₄⋊C₄	1	2	yes	—
209	C₄×C₅	C₃×#95	1	2	yes	—
210	C₄×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
211	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
212	Dic₅	C₃×#95	1	2	yes	—
213	Dic₅	C₂×C₃×C₂²⋊C₄	1	2	yes	—
214	C₂⁴	—	5	1	no	—
215	C₂²×C₄	—	5	1	no	—
216	C₂²×C₄	—	5	4	no	—
217	C₂²×C₄	—	5	1	no	—
218	C₂²×C₄	—	5	1	no	—
219	C₂²×C₄	—	5	4	no	—
220	C₂²×C₄	—	5	1	no	—
221	C₂²×C₄	—	5	1	no	—
222	C₂²×C₄	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
223	M₁₆	—	5	8	no	—
224	M₁₆	—	10	4	no	—
225	M₁₆	—	10	8	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	5	2	no	—
228	C₂×C₈	—	10	4	no	—
229	C₂×C₈	—	10	4	no	—
230	C₂×C₈	—	10	8	no	—
231	C₂×C₈	—	10	4	no	—
232	C₂×C₈	—	10	2	no	—
233	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	—
234	C₃×C₅	C₂×#341	1	1	yes	characteristic, complemented, Hall
235	C₂²×C₃	—	2	2	no	—
236	C₂²×C₃	—	5	4	no	—
237	C₂²×C₃	—	5	2	no	—
238	C₂²×C₃	—	5	2	no	—
239	C₂²×C₃	—	10	8	no	—
240	C₂²×C₃	—	10	2	no	—
241	C₂²×C₃	(C₂×AGL(1,𝔽₅))⋊C₄	1	2	yes	—
242	C₂²×C₃	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
243	C₃×C₄	—	5	2	no	—
244	C₃×C₄	—	5	2	no	—
245	C₃×C₄	(C₂×AGL(1,𝔽₅))⋊C₄	1	2	yes	—
246	C₃×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
247	C₃×C₄	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
248	C₂×C₅	—	2	2	no	—
249	C₂×C₅	C₂×C₃×#95	1	1	yes	characteristic
250	C₂×C₅	C₂×C₃×#108	1	2	yes	—
251	D₅	—	2	2	no	—
252	D₅	C₃×#341	1	4	yes	complemented
253	C₂³	—	5	1	no	—
254	C₂³	—	5	1	no	—
255	C₂³	—	5	4	no	—
256	C₂³	—	10	4	no	—
257	C₂³	C₃×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
258	C₂×C₄	—	5	1	no	—
259	C₂×C₄	—	5	1	no	—
260	C₂×C₄	—	5	4	no	—
261	C₂×C₄	—	5	2	no	—
262	C₂×C₄	—	5	2	no	—
263	C₂×C₄	—	5	4	no	—
264	C₂×C₄	—	5	4	no	—
265	C₂×C₄	—	5	2	no	—
266	C₂×C₄	—	5	2	no	—
267	C₂×C₄	C₃×C₄×AGL(1,𝔽₅)	1	2	yes	—
268	C₂×C₄	C₃×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
269	C₂×C₄	C₃×C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
270	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
271	C₈	—	2	2	no	—
272	C₈	—	10	8	no	—
273	C₈	—	10	2	no	—
274	C₂×C₃	—	2	2	no	—
275	C₂×C₃	—	5	4	no	—
276	C₂×C₃	—	10	2	no	—
277	C₂×C₃	D₅⋊#108	1	2	yes	—
278	C₂×C₃	C₂×(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic
279	C₅	C₂×C₃×#341	1	1	yes	characteristic, complemented, Sylow
280	C₂²	—	2	2	no	—
281	C₂²	—	5	4	no	—
282	C₂²	—	5	2	no	—
283	C₂²	—	5	2	no	—
284	C₂²	—	10	8	no	—
285	C₂²	—	10	2	no	—
286	C₂²	C₃×(C₂×AGL(1,𝔽₅))⋊C₄	1	2	yes	—
287	C₂²	C₂×C₃×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
288	C₄	—	5	2	no	—
289	C₄	—	5	2	no	—
290	C₄	C₃×(C₂×AGL(1,𝔽₅))⋊C₄	1	2	yes	—
291	C₄	C₂×C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
292	C₄	C₂×C₃×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
293	C₃	#10604730	1	1	yes	characteristic, direct product factor, Sylow
294	C₂	—	2	2	no	—
295	C₂	—	5	4	no	—
296	C₂	—	10	2	no	—
297	C₂	C₃×D₅⋊#108	1	2	yes	—
298	C₂	C₂×C₃×(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)
commutator	C₄×C₅	C₂×C₃×C₄²
Frattini	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₂²×C₃×C₅×C₈	C₄
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂³×C₃×D₅	Q₈
cosocle	C₂×C₄×D₅	C₂³×C₃
hypercenter	C₂²×C₃×C₈	AGL(1,𝔽₅)
hypocenter	C₅	C₂×C₃×#341
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×C₈

Complements

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

N	H ≅ G/N	type
3. C₃×#64896	C₂	semidirect
4. #10604730	C₃	direct (N × H)
19. #64896	C₂×C₃	semidirect
179. C₃×D₅	#341	semidirect
234. C₃×C₅	C₂×#341	semidirect
252. D₅	C₃×#341	semidirect
279. C₅	C₂×C₃×#341	semidirect
293. C₃	#10604730	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×D₅⋊M₁₆	2
C₂²×C₃×C₈×D₅	2
C₃×#64896	2
#10604730	3
C₂×C₃×#341	5

Subgroups — C3×#10604730 #14294634

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10604730 #14294634