Subgroups — C₃×#10604895 #14294310

Order 1920.

960 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×#704	C₂	1	2	yes	maximal
2	C₂²×C₃×C₄×Dic₅	C₂	1	1	yes	maximal, characteristic
3	C₃×#64918	C₂	1	4	yes	maximal, complemented
4	#10604895	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂³×C₃×C₄×C₅	C₄	1	1	yes	characteristic
6	C₂³×C₃×Dic₅	C₄	1	1	yes	characteristic
7	C₂³×C₃×Dic₅	C₂²	1	1	yes	characteristic
8	C₂×C₃×#704	—	2	4	no	—
9	C₂×C₃×#704	—	2	4	no	—
10	C₂×C₃×#704	C₄	1	8	yes	complemented
11	C₂²×C₃×#170	C₂²	1	4	yes	—
12	C₂×C₃×C₄×Dic₅	—	2	4	no	—
13	C₂×C₃×C₄×Dic₅	C₄	1	2	yes	—
14	C₂×C₃×C₄×Dic₅	C₂²	1	2	yes	—
15	C₃×#1728	—	5	1	no	maximal, Hall
16	C₂²×#704	C₂×C₃	1	2	yes	—
17	C₂²×C₄×Dic₅	C₂×C₃	1	1	yes	characteristic
18	#64918	C₂×C₃	1	4	yes	complemented
19	C₂⁴×C₃×C₅	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₃×C₄×C₅	—	2	2	no	—
21	C₂²×C₃×C₄×C₅	—	2	4	no	—
22	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
23	C₂²×C₃×Dic₅	—	2	2	no	—
24	C₂²×C₃×Dic₅	—	2	4	no	—
25	C₂²×C₃×Dic₅	C₂×C₄	1	2	yes	—
26	C₂²×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃×Dic₅	C₂×C₄	1	4	yes	—
28	C₂²×C₃×Dic₅	D₄	1	2	yes	—
29	C₂²×C₃×Dic₅	D₄	1	4	yes	—
30	C₂²×C₃×Dic₅	Q₈	1	2	yes	—
31	C₂²×C₃×Dic₅	C₂³	1	1	yes	characteristic
32	C₃×#704	—	2	16	no	—
33	C₂×C₃×#170	—	2	4	no	—
34	C₂×C₃×#170	—	2	4	no	—
35	C₂×C₃×#170	C₂×C₄	1	8	yes	complemented
36	C₃×C₄×Dic₅	—	2	4	no	—
37	C₃×C₄×Dic₅	—	2	4	no	—
38	C₂²×C₃×M₁₆	—	5	2	no	—
39	C₂²×C₃×C₄²	—	5	1	no	—
40	C₃×#336	—	5	4	no	—
41	C₂³×C₄×C₅	C₃×C₄	1	1	yes	characteristic
42	C₂³×Dic₅	C₃×C₄	1	1	yes	characteristic
43	C₂³×Dic₅	C₂²×C₃	1	1	yes	characteristic
44	C₂×#704	—	2	4	no	—
45	C₂×#704	—	2	4	no	—
46	C₂×#704	C₃×C₄	1	8	yes	complemented
47	C₂²×#170	C₂²×C₃	1	4	yes	—
48	C₂×C₄×Dic₅	—	2	4	no	—
49	C₂×C₄×Dic₅	C₃×C₄	1	2	yes	—
50	C₂×C₄×Dic₅	C₂²×C₃	1	2	yes	—
51	#1728	—	5	1	no	Sylow
52	C₂³×C₃×C₅	—	2	2	no	—
53	C₂³×C₃×C₅	—	2	2	no	—
54	C₂³×C₃×C₅	C₄²	1	2	yes	—
55	C₂³×C₃×C₅	C₂²⋊C₄	1	2	yes	—
56	C₂³×C₃×C₅	C₄⋊C₄	1	2	yes	—
57	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	—	2	8	no	—
59	C₂×C₃×C₄×C₅	—	2	2	no	—
60	C₂×C₃×C₄×C₅	—	2	2	no	—
61	C₂×C₃×C₄×C₅	M₁₆	1	4	yes	complemented
62	C₂×C₃×Dic₅	—	2	8	no	—
63	C₂×C₃×Dic₅	—	2	2	no	—
64	C₂×C₃×Dic₅	—	2	2	no	—
65	C₂×C₃×Dic₅	C₄²	1	2	yes	—
66	C₂×C₃×Dic₅	C₂²⋊C₄	1	2	yes	—
67	C₂×C₃×Dic₅	C₂²⋊C₄	1	8	yes	—
68	C₂×C₃×Dic₅	C₄⋊C₄	1	2	yes	—
69	C₂×C₃×Dic₅	C₄⋊C₄	1	8	yes	—
70	C₂×C₃×Dic₅	M₁₆	1	4	yes	complemented
71	C₂×C₃×Dic₅	C₂²×C₄	1	2	yes	—
72	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
73	C₂×C₃×Dic₅	C₂×D₄	1	2	yes	—
74	C₂×C₃×Dic₅	C₂×Q₈	1	1	yes	characteristic
75	C₃×#170	—	2	8	no	—
76	C₂³×C₃×C₄	—	5	1	no	—
77	C₂³×C₃×C₄	—	5	1	no	—
78	C₂³×C₃×C₄	AGL(1,𝔽₅)	1	1	yes	characteristic
79	C₂×C₃×M₁₆	—	5	8	no	—
80	C₂×C₃×M₁₆	—	10	4	no	—
81	C₂×C₃×M₁₆	—	10	4	no	—
82	C₂²×C₃×C₈	—	5	4	no	—
83	C₂×C₃×C₄²	—	5	2	no	—
84	C₂×C₃×C₄²	—	5	2	no	—
85	C₂×C₃×C₄²	—	10	4	no	—
86	C₂⁴×C₅	C₂×C₃×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₅	—	2	2	no	—
88	C₂²×C₄×C₅	—	2	4	no	—
89	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
90	C₂²×Dic₅	—	2	2	no	—
91	C₂²×Dic₅	—	2	4	no	—
92	C₂²×Dic₅	C₂×C₃×C₄	1	2	yes	—
93	C₂²×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
94	C₂²×Dic₅	C₂×C₃×C₄	1	4	yes	—
95	C₂²×Dic₅	C₃×D₄	1	2	yes	—
96	C₂²×Dic₅	C₃×D₄	1	4	yes	—
97	C₂²×Dic₅	C₃×Q₈	1	2	yes	—
98	C₂²×Dic₅	C₂³×C₃	1	1	yes	characteristic
99	#704	—	2	16	no	—
100	C₂×#170	—	2	4	no	—
101	C₂×#170	—	2	4	no	—
102	C₂×#170	C₂×C₃×C₄	1	8	yes	complemented
103	C₄×Dic₅	—	2	4	no	—
104	C₄×Dic₅	—	2	4	no	—
105	C₂²×M₁₆	—	5	2	no	—
106	C₂²×C₄²	—	5	1	no	—
107	#336	—	5	4	no	—
108	C₂²×C₃×C₅	—	2	4	no	—
109	C₂²×C₃×C₅	—	2	2	no	—
110	C₂²×C₃×C₅	—	2	2	no	—
111	C₂²×C₃×C₅	—	2	4	no	—
112	C₂²×C₃×C₅	#95	1	4	yes	—
113	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
114	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
117	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
118	C₃×C₄×C₅	—	2	4	no	—
119	C₃×Dic₅	—	2	4	no	—
120	C₃×Dic₅	#95	1	4	yes	—
121	C₃×Dic₅	C₂×C₂²⋊C₄	1	2	yes	—
122	C₃×Dic₅	C₂×C₄⋊C₄	1	2	yes	—
123	C₂⁴×C₃	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
124	C₂²×C₃×C₄	—	2	2	no	—
125	C₂²×C₃×C₄	—	2	4	no	—
126	C₂²×C₃×C₄	—	5	4	no	—
127	C₂²×C₃×C₄	—	5	2	no	—
128	C₂²×C₃×C₄	—	5	2	no	—
129	C₂²×C₃×C₄	—	5	4	no	—
130	C₂²×C₃×C₄	—	5	1	no	—
131	C₂²×C₃×C₄	—	5	1	no	—
132	C₂²×C₃×C₄	—	5	2	no	—
133	C₂²×C₃×C₄	—	10	4	no	—
134	C₂²×C₃×C₄	—	10	2	no	—
135	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
136	C₃×M₁₆	—	10	16	no	—
137	C₂×C₃×C₈	—	5	8	no	—
138	C₂×C₃×C₈	—	10	4	no	—
139	C₂×C₃×C₈	—	10	4	no	—
140	C₃×C₄²	—	10	4	no	—
141	C₃×C₄²	—	10	4	no	—
142	C₂³×C₅	—	2	2	no	—
143	C₂³×C₅	—	2	2	no	—
144	C₂³×C₅	C₃×C₄²	1	2	yes	—
145	C₂³×C₅	C₃×C₂²⋊C₄	1	2	yes	—
146	C₂³×C₅	C₃×C₄⋊C₄	1	2	yes	—
147	C₂³×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
148	C₂×C₄×C₅	—	2	8	no	—
149	C₂×C₄×C₅	—	2	2	no	—
150	C₂×C₄×C₅	—	2	2	no	—
151	C₂×C₄×C₅	C₃×M₁₆	1	4	yes	complemented
152	C₂×Dic₅	—	2	8	no	—
153	C₂×Dic₅	—	2	2	no	—
154	C₂×Dic₅	—	2	2	no	—
155	C₂×Dic₅	C₃×C₄²	1	2	yes	—
156	C₂×Dic₅	C₃×C₂²⋊C₄	1	2	yes	—
157	C₂×Dic₅	C₃×C₂²⋊C₄	1	8	yes	—
158	C₂×Dic₅	C₃×C₄⋊C₄	1	2	yes	—
159	C₂×Dic₅	C₃×C₄⋊C₄	1	8	yes	—
160	C₂×Dic₅	C₃×M₁₆	1	4	yes	complemented
161	C₂×Dic₅	C₂²×C₃×C₄	1	2	yes	—
162	C₂×Dic₅	C₂×C₃×D₄	1	1	yes	characteristic
163	C₂×Dic₅	C₂×C₃×D₄	1	2	yes	—
164	C₂×Dic₅	C₂×C₃×Q₈	1	1	yes	characteristic
165	#170	—	2	8	no	—
166	C₂³×C₄	—	5	1	no	—
167	C₂³×C₄	—	5	1	no	—
168	C₂³×C₄	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
169	C₂×M₁₆	—	5	8	no	—
170	C₂×M₁₆	—	10	4	no	—
171	C₂×M₁₆	—	10	4	no	—
172	C₂²×C₈	—	5	4	no	—
173	C₂×C₄²	—	5	2	no	—
174	C₂×C₄²	—	5	2	no	—
175	C₂×C₄²	—	10	4	no	—
176	C₂×C₃×C₅	—	2	4	no	—
177	C₂×C₃×C₅	C₂×#95	1	1	yes	characteristic
178	C₂×C₃×C₅	C₄×M₁₆	1	2	yes	—
179	C₂×C₃×C₅	#407	1	1	yes	characteristic
180	C₂×C₃×C₅	#407	1	1	yes	characteristic
181	C₂×C₃×C₅	#423	1	2	yes	—
182	C₂³×C₃	—	2	2	no	—
183	C₂³×C₃	—	2	2	no	—
184	C₂³×C₃	C₄×AGL(1,𝔽₅)	1	2	yes	—
185	C₂³×C₃	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
186	C₂³×C₃	C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
187	C₂³×C₃	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
188	C₂×C₃×C₄	—	2	8	no	—
189	C₂×C₃×C₄	—	2	2	no	—
190	C₂×C₃×C₄	—	2	2	no	—
191	C₂×C₃×C₄	—	5	2	no	—
192	C₂×C₃×C₄	—	5	2	no	—
193	C₂×C₃×C₄	—	5	2	no	—
194	C₂×C₃×C₄	—	5	1	no	—
195	C₂×C₃×C₄	—	5	1	no	—
196	C₂×C₃×C₄	—	5	2	no	—
197	C₂×C₃×C₄	—	5	2	no	—
198	C₂×C₃×C₄	—	5	8	no	—
199	C₂×C₃×C₄	—	5	8	no	—
200	C₂×C₃×C₄	—	5	4	no	—
201	C₂×C₃×C₄	—	10	8	no	—
202	C₂×C₃×C₄	—	10	2	no	—
203	C₂×C₃×C₄	—	10	2	no	—
204	C₂×C₃×C₄	#704	1	4	yes	complemented
205	C₃×C₈	—	10	8	no	—
206	C₂²×C₅	—	2	4	no	—
207	C₂²×C₅	—	2	2	no	—
208	C₂²×C₅	—	2	2	no	—
209	C₂²×C₅	—	2	4	no	—
210	C₂²×C₅	C₃×#95	1	4	yes	—
211	C₂²×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
212	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
213	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
214	C₂²×C₅	C₂×C₃×M₁₆	1	2	yes	—
215	C₂²×C₅	C₂×C₃×M₁₆	1	2	yes	—
216	C₄×C₅	—	2	4	no	—
217	Dic₅	—	2	4	no	—
218	Dic₅	C₃×#95	1	4	yes	—
219	Dic₅	C₂×C₃×C₂²⋊C₄	1	2	yes	—
220	Dic₅	C₂×C₃×C₄⋊C₄	1	2	yes	—
221	C₂⁴	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	5	4	no	—
225	C₂²×C₄	—	5	2	no	—
226	C₂²×C₄	—	5	2	no	—
227	C₂²×C₄	—	5	4	no	—
228	C₂²×C₄	—	5	1	no	—
229	C₂²×C₄	—	5	1	no	—
230	C₂²×C₄	—	5	2	no	—
231	C₂²×C₄	—	10	4	no	—
232	C₂²×C₄	—	10	2	no	—
233	C₂²×C₄	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	—
234	M₁₆	—	10	16	no	—
235	C₂×C₈	—	5	8	no	—
236	C₂×C₈	—	10	4	no	—
237	C₂×C₈	—	10	4	no	—
238	C₄²	—	10	4	no	—
239	C₄²	—	10	4	no	—
240	C₃×C₅	#1728	1	1	yes	characteristic, complemented, Hall
241	C₂²×C₃	—	2	2	no	—
242	C₂²×C₃	—	2	4	no	—
243	C₂²×C₃	—	2	2	no	—
244	C₂²×C₃	—	2	4	no	—
245	C₂²×C₃	(C₂×AGL(1,𝔽₅))⋊C₄	1	4	yes	—
246	C₂²×C₃	D₅⋊M₁₆	1	2	yes	—
247	C₂²×C₃	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
248	C₂²×C₃	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
249	C₂²×C₃	C₂×#704	1	2	yes	—
250	C₂²×C₃	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
251	C₃×C₄	—	2	4	no	—
252	C₃×C₄	—	5	4	no	—
253	C₃×C₄	—	5	2	no	—
254	C₃×C₄	—	5	2	no	—
255	C₃×C₄	—	10	4	no	—
256	C₂×C₅	—	2	4	no	—
257	C₂×C₅	C₂×C₃×#95	1	1	yes	characteristic
258	C₂×C₅	C₃×C₄×M₁₆	1	2	yes	—
259	C₂×C₅	C₃×#407	1	1	yes	characteristic
260	C₂×C₅	C₃×#407	1	1	yes	characteristic
261	C₂×C₅	C₃×#423	1	2	yes	—
262	C₂³	—	2	2	no	—
263	C₂³	—	2	2	no	—
264	C₂³	C₃×C₄×AGL(1,𝔽₅)	1	2	yes	—
265	C₂³	C₃×C₄⋊AGL(1,𝔽₅)	1	2	yes	—
266	C₂³	C₃×C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
267	C₂³	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	5	2	no	—
272	C₂×C₄	—	5	2	no	—
273	C₂×C₄	—	5	1	no	—
274	C₂×C₄	—	5	2	no	—
275	C₂×C₄	—	5	1	no	—
276	C₂×C₄	—	5	8	no	—
277	C₂×C₄	—	5	8	no	—
278	C₂×C₄	—	5	2	no	—
279	C₂×C₄	—	5	2	no	—
280	C₂×C₄	—	5	4	no	—
281	C₂×C₄	—	10	8	no	—
282	C₂×C₄	—	10	2	no	—
283	C₂×C₄	—	10	2	no	—
284	C₂×C₄	C₃×#704	1	4	yes	complemented
285	C₈	—	10	8	no	—
286	C₂×C₃	—	2	4	no	—
287	C₂×C₃	C₄×#704	1	2	yes	—
288	C₂×C₃	#65755	1	1	yes	characteristic
289	C₂×C₃	#65758	1	2	yes	—
290	C₂×C₃	C₂×(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic
291	C₂×C₃	#65798	1	1	yes	characteristic
292	C₅	C₃×#1728	1	1	yes	characteristic, complemented, Sylow
293	C₂²	—	2	4	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	4	no	—
296	C₂²	—	2	2	no	—
297	C₂²	C₃×(C₂×AGL(1,𝔽₅))⋊C₄	1	4	yes	—
298	C₂²	C₃×D₅⋊M₁₆	1	2	yes	—
299	C₂²	C₂×C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
300	C₂²	C₂×C₃×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
301	C₂²	C₂×C₃×#704	1	2	yes	—
302	C₂²	C₂×C₃×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
303	C₄	—	2	4	no	—
304	C₄	—	5	2	no	—
305	C₄	—	5	4	no	—
306	C₄	—	5	2	no	—
307	C₄	—	10	4	no	—
308	C₃	#10604895	1	1	yes	characteristic, direct product factor, Sylow
309	C₂	—	2	4	no	—
310	C₂	C₃×C₄×#704	1	2	yes	—
311	C₂	C₃×#65755	1	1	yes	characteristic
312	C₂	C₃×#65758	1	2	yes	—
313	C₂	C₂×C₃×(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic
314	C₂	C₃×#65798	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	C₂²×AGL(1,𝔽₅)
commutator	C₂²×C₅	C₂×C₃×C₄²
Frattini	C₂³	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₂³×C₃×C₄×C₅	C₄
socle	C₂³×C₃×C₅	C₂²×C₄
prime radical	C₂⁴×C₃×C₅	C₂×C₄
cosocle	C₂²×Dic₅	C₂³×C₃
hypercenter	C₂³×C₃×C₄	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#1728
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂³×C₃×C₄

Complements

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

N	H ≅ G/N	type
3. C₃×#64918	C₂	semidirect
4. #10604895	C₃	direct (N × H)
10. C₂×C₃×#704	C₄	semidirect
18. #64918	C₂×C₃	semidirect
35. C₂×C₃×#170	C₂×C₄	semidirect
46. C₂×#704	C₃×C₄	semidirect
61. C₂×C₃×C₄×C₅	M₁₆	semidirect
70. C₂×C₃×Dic₅	M₁₆	semidirect
102. C₂×#170	C₂×C₃×C₄	semidirect
151. C₂×C₄×C₅	C₃×M₁₆	semidirect
160. C₂×Dic₅	C₃×M₁₆	semidirect
204. C₂×C₃×C₄	#704	semidirect
240. C₃×C₅	#1728	semidirect
284. C₂×C₄	C₃×#704	semidirect
292. C₅	C₃×#1728	semidirect
308. C₃	#10604895	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×#704	2
C₂²×C₃×C₄×Dic₅	2
C₃×#64918	2
#10604895	3
C₃×#1728	5

Subgroups — C3×#10604895 #14294310

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10604895 #14294310