Subgroups — C₃×#10604172 #14294312

Order 1920.

960 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×#700	C₂	1	2	yes	maximal
2	C₂×C₃×C₄²×D₅	C₂	1	1	yes	maximal, characteristic
3	C₃×#64918	C₂	1	4	yes	maximal, complemented
4	#10604172	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂²×C₃×C₄×D₅	C₄	1	2	yes	—
6	C₂²×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
7	C₂²×C₃×#170	C₂²	1	4	yes	—
8	C₂×C₃×#700	—	2	4	no	—
9	C₂×C₃×#700	—	2	4	no	—
10	C₂×C₃×#700	C₄	1	8	yes	complemented
11	C₂×C₃×C₄²×C₅	C₄	1	1	yes	characteristic
12	C₂×C₃×C₄×Dic₅	C₄	1	1	yes	characteristic
13	C₂×C₃×C₄×Dic₅	C₂²	1	2	yes	—
14	C₃×C₄²×D₅	—	2	4	no	—
15	C₃×#1728	—	5	1	no	maximal, Hall
16	C₂²×#700	C₂×C₃	1	2	yes	—
17	C₂×C₄²×D₅	C₂×C₃	1	1	yes	characteristic
18	#64918	C₂×C₃	1	4	yes	complemented
19	C₂³×C₃×D₅	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
21	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₃×Dic₅	C₂×C₄	1	2	yes	—
23	C₂²×C₃×Dic₅	C₂³	1	1	yes	characteristic
24	C₂×C₃×C₄×D₅	—	2	4	no	—
25	C₂×C₃×C₄×D₅	—	2	4	no	—
26	C₂×C₃×C₄×D₅	—	2	4	no	—
27	C₂×C₃×C₄×D₅	C₂×C₄	1	4	yes	—
28	C₂×C₃×C₄×D₅	D₄	1	2	yes	—
29	C₂×C₃×C₄×D₅	D₄	1	4	yes	—
30	C₂×C₃×C₄×D₅	Q₈	1	2	yes	—
31	C₂×C₃×#170	—	2	4	no	—
32	C₂×C₃×#170	—	2	4	no	—
33	C₂×C₃×#170	C₂×C₄	1	8	yes	complemented
34	C₃×#700	—	2	16	no	—
35	C₃×C₄²×C₅	—	2	2	no	—
36	C₃×C₄×Dic₅	—	2	2	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	C₂²×C₄×D₅	C₃×C₄	1	2	yes	—
42	C₂²×C₄×D₅	C₂²×C₃	1	1	yes	characteristic
43	C₂²×#170	C₂²×C₃	1	4	yes	—
44	C₂×#700	—	2	4	no	—
45	C₂×#700	—	2	4	no	—
46	C₂×#700	C₃×C₄	1	8	yes	complemented
47	C₂×C₄²×C₅	C₃×C₄	1	1	yes	characteristic
48	C₂×C₄×Dic₅	C₃×C₄	1	1	yes	characteristic
49	C₂×C₄×Dic₅	C₂²×C₃	1	2	yes	—
50	C₄²×D₅	—	2	4	no	—
51	#1728	—	5	1	no	Sylow
52	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₃×D₅	—	2	2	no	—
54	C₂²×C₃×D₅	—	2	2	no	—
55	C₂²×C₃×D₅	C₄²	1	2	yes	—
56	C₂²×C₃×D₅	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₃×D₅	C₄⋊C₄	1	2	yes	—
58	C₂×C₃×C₄×C₅	—	2	2	no	—
59	C₂×C₃×C₄×C₅	—	2	2	no	—
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	2	yes	—
63	C₂×C₃×C₄×C₅	M₁₆	1	4	yes	complemented
64	C₂×C₃×Dic₅	—	2	2	no	—
65	C₂×C₃×Dic₅	—	2	2	no	—
66	C₂×C₃×Dic₅	M₁₆	1	4	yes	complemented
67	C₂×C₃×Dic₅	C₂²×C₄	1	2	yes	—
68	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
69	C₂×C₃×Dic₅	C₂×D₄	1	2	yes	—
70	C₂×C₃×Dic₅	C₂×Q₈	1	1	yes	characteristic
71	C₃×C₄×D₅	—	2	16	no	—
72	C₃×C₄×D₅	C₂²⋊C₄	1	8	yes	—
73	C₃×C₄×D₅	C₄⋊C₄	1	8	yes	—
74	C₃×#170	—	2	8	no	—
75	C₂³×C₃×C₄	—	5	1	no	—
76	C₂³×C₃×C₄	—	5	2	no	—
77	C₂×C₃×M₁₆	—	5	8	no	—
78	C₂×C₃×M₁₆	—	10	4	no	—
79	C₂×C₃×M₁₆	—	10	4	no	—
80	C₂²×C₃×C₈	—	5	4	no	—
81	C₂×C₃×C₄²	—	5	2	no	—
82	C₂×C₃×C₄²	—	5	1	no	—
83	C₂×C₃×C₄²	—	10	4	no	—
84	C₂×C₃×C₄²	AGL(1,𝔽₅)	1	1	yes	characteristic
85	C₂³×D₅	C₂×C₃×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
87	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
88	C₂²×Dic₅	C₂×C₃×C₄	1	2	yes	—
89	C₂²×Dic₅	C₂³×C₃	1	1	yes	characteristic
90	C₂×C₄×D₅	—	2	4	no	—
91	C₂×C₄×D₅	—	2	4	no	—
92	C₂×C₄×D₅	—	2	4	no	—
93	C₂×C₄×D₅	C₂×C₃×C₄	1	4	yes	—
94	C₂×C₄×D₅	C₃×D₄	1	2	yes	—
95	C₂×C₄×D₅	C₃×D₄	1	4	yes	—
96	C₂×C₄×D₅	C₃×Q₈	1	2	yes	—
97	C₂×#170	—	2	4	no	—
98	C₂×#170	—	2	4	no	—
99	C₂×#170	C₂×C₃×C₄	1	8	yes	complemented
100	#700	—	2	16	no	—
101	C₄²×C₅	—	2	2	no	—
102	C₄×Dic₅	—	2	2	no	—
103	C₄×Dic₅	—	2	4	no	—
104	C₂²×M₁₆	—	5	2	no	—
105	C₂²×C₄²	—	5	1	no	—
106	#336	—	5	4	no	—
107	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
108	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
111	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
112	C₂×C₃×D₅	—	2	4	no	—
113	C₂×C₃×D₅	—	2	4	no	—
114	C₂×C₃×D₅	—	2	2	no	—
115	C₂×C₃×D₅	—	2	2	no	—
116	C₂×C₃×D₅	#95	1	4	yes	—
117	C₃×C₄×C₅	—	2	4	no	—
118	C₃×C₄×C₅	#95	1	4	yes	—
119	C₃×Dic₅	—	2	4	no	—
120	C₃×Dic₅	C₂×C₂²⋊C₄	1	2	yes	—
121	C₃×Dic₅	C₂×C₄⋊C₄	1	2	yes	—
122	C₂⁴×C₃	—	5	1	no	—
123	C₂²×C₃×C₄	—	5	1	no	—
124	C₂²×C₃×C₄	—	5	4	no	—
125	C₂²×C₃×C₄	—	5	2	no	—
126	C₂²×C₃×C₄	—	5	2	no	—
127	C₂²×C₃×C₄	—	5	4	no	—
128	C₂²×C₃×C₄	—	5	2	no	—
129	C₂²×C₃×C₄	—	10	4	no	—
130	C₂²×C₃×C₄	—	10	4	no	—
131	C₂²×C₃×C₄	—	10	4	no	—
132	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
133	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
134	C₃×M₁₆	—	10	16	no	—
135	C₂×C₃×C₈	—	5	8	no	—
136	C₂×C₃×C₈	—	10	4	no	—
137	C₂×C₃×C₈	—	10	4	no	—
138	C₃×C₄²	—	2	2	no	—
139	C₃×C₄²	—	10	4	no	—
140	C₃×C₄²	—	10	2	no	—
141	C₂³×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
142	C₂²×D₅	—	2	2	no	—
143	C₂²×D₅	—	2	2	no	—
144	C₂²×D₅	C₃×C₄²	1	2	yes	—
145	C₂²×D₅	C₃×C₂²⋊C₄	1	2	yes	—
146	C₂²×D₅	C₃×C₄⋊C₄	1	2	yes	—
147	C₂×C₄×C₅	—	2	2	no	—
148	C₂×C₄×C₅	—	2	2	no	—
149	C₂×C₄×C₅	C₃×C₄²	1	2	yes	—
150	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
151	C₂×C₄×C₅	C₃×C₄⋊C₄	1	2	yes	—
152	C₂×C₄×C₅	C₃×M₁₆	1	4	yes	complemented
153	C₂×Dic₅	—	2	2	no	—
154	C₂×Dic₅	—	2	2	no	—
155	C₂×Dic₅	C₃×M₁₆	1	4	yes	complemented
156	C₂×Dic₅	C₂²×C₃×C₄	1	2	yes	—
157	C₂×Dic₅	C₂×C₃×D₄	1	1	yes	characteristic
158	C₂×Dic₅	C₂×C₃×D₄	1	2	yes	—
159	C₂×Dic₅	C₂×C₃×Q₈	1	1	yes	characteristic
160	C₄×D₅	—	2	16	no	—
161	C₄×D₅	C₃×C₂²⋊C₄	1	8	yes	—
162	C₄×D₅	C₃×C₄⋊C₄	1	8	yes	—
163	#170	—	2	8	no	—
164	C₂³×C₄	—	5	1	no	—
165	C₂³×C₄	—	5	2	no	—
166	C₂×M₁₆	—	5	8	no	—
167	C₂×M₁₆	—	10	4	no	—
168	C₂×M₁₆	—	10	4	no	—
169	C₂²×C₈	—	5	4	no	—
170	C₂×C₄²	—	5	2	no	—
171	C₂×C₄²	—	5	1	no	—
172	C₂×C₄²	—	10	4	no	—
173	C₂×C₄²	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
174	C₂×C₃×C₅	C₂×#95	1	1	yes	characteristic
175	C₂×C₃×C₅	C₄×M₁₆	1	2	yes	—
176	C₂×C₃×C₅	#407	1	2	yes	—
177	C₂×C₃×C₅	#423	1	1	yes	characteristic
178	C₂×C₃×C₅	#423	1	1	yes	characteristic
179	C₃×D₅	—	2	4	no	—
180	C₂³×C₃	—	5	2	no	—
181	C₂³×C₃	—	5	2	no	—
182	C₂³×C₃	—	5	2	no	—
183	C₂³×C₃	—	10	2	no	—
184	C₂³×C₃	—	10	2	no	—
185	C₂³×C₃	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
186	C₂×C₃×C₄	—	2	2	no	—
187	C₂×C₃×C₄	—	2	2	no	—
188	C₂×C₃×C₄	—	5	2	no	—
189	C₂×C₃×C₄	—	5	1	no	—
190	C₂×C₃×C₄	—	5	1	no	—
191	C₂×C₃×C₄	—	5	2	no	—
192	C₂×C₃×C₄	—	5	8	no	—
193	C₂×C₃×C₄	—	5	8	no	—
194	C₂×C₃×C₄	—	5	4	no	—
195	C₂×C₃×C₄	—	10	16	no	—
196	C₂×C₃×C₄	—	10	2	no	—
197	C₂×C₃×C₄	—	10	2	no	—
198	C₂×C₃×C₄	#700	1	4	yes	complemented
199	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	2	yes	—
200	C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
201	C₂×C₃×C₄	C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
202	C₃×C₈	—	10	8	no	—
203	C₂²×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
204	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
205	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
206	C₂²×C₅	C₂×C₃×M₁₆	1	2	yes	—
207	C₂²×C₅	C₂×C₃×M₁₆	1	2	yes	—
208	C₂×D₅	—	2	4	no	—
209	C₂×D₅	—	2	4	no	—
210	C₂×D₅	—	2	2	no	—
211	C₂×D₅	—	2	2	no	—
212	C₂×D₅	C₃×#95	1	4	yes	—
213	C₄×C₅	—	2	4	no	—
214	C₄×C₅	C₃×#95	1	4	yes	—
215	Dic₅	—	2	4	no	—
216	Dic₅	C₂×C₃×C₂²⋊C₄	1	2	yes	—
217	Dic₅	C₂×C₃×C₄⋊C₄	1	2	yes	—
218	C₂⁴	—	5	1	no	—
219	C₂²×C₄	—	5	1	no	—
220	C₂²×C₄	—	5	4	no	—
221	C₂²×C₄	—	5	2	no	—
222	C₂²×C₄	—	5	2	no	—
223	C₂²×C₄	—	5	4	no	—
224	C₂²×C₄	—	5	2	no	—
225	C₂²×C₄	—	10	4	no	—
226	C₂²×C₄	—	10	4	no	—
227	C₂²×C₄	—	10	4	no	—
228	C₂²×C₄	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	—
229	C₂²×C₄	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
230	M₁₆	—	10	16	no	—
231	C₂×C₈	—	5	8	no	—
232	C₂×C₈	—	10	4	no	—
233	C₂×C₈	—	10	4	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	10	4	no	—
236	C₄²	—	10	2	no	—
237	C₃×C₅	#1728	1	1	yes	characteristic, complemented, Hall
238	C₂²×C₃	—	5	4	no	—
239	C₂²×C₃	—	10	4	no	—
240	C₂²×C₃	—	10	2	no	—
241	C₂²×C₃	—	10	2	no	—
242	C₂²×C₃	—	10	4	no	—
243	C₂²×C₃	C₂×#700	1	2	yes	—
244	C₂²×C₃	D₅⋊M₁₆	1	2	yes	—
245	C₂²×C₃	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
246	C₂²×C₃	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
247	C₂²×C₃	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
248	C₃×C₄	—	2	4	no	—
249	C₃×C₄	—	5	2	no	—
250	C₃×C₄	—	5	2	no	—
251	C₃×C₄	—	10	4	no	—
252	C₃×C₄	(C₂×AGL(1,𝔽₅))⋊C₄	1	4	yes	—
253	C₂×C₅	C₂×C₃×#95	1	1	yes	characteristic
254	C₂×C₅	C₃×C₄×M₁₆	1	2	yes	—
255	C₂×C₅	C₃×#407	1	2	yes	—
256	C₂×C₅	C₃×#423	1	1	yes	characteristic
257	C₂×C₅	C₃×#423	1	1	yes	characteristic
258	D₅	—	2	4	no	—
259	C₂³	—	5	2	no	—
260	C₂³	—	5	2	no	—
261	C₂³	—	5	2	no	—
262	C₂³	—	10	2	no	—
263	C₂³	—	10	2	no	—
264	C₂³	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	5	2	no	—
268	C₂×C₄	—	5	1	no	—
269	C₂×C₄	—	5	1	no	—
270	C₂×C₄	—	5	2	no	—
271	C₂×C₄	—	5	8	no	—
272	C₂×C₄	—	5	8	no	—
273	C₂×C₄	—	5	4	no	—
274	C₂×C₄	—	10	16	no	—
275	C₂×C₄	—	10	2	no	—
276	C₂×C₄	—	10	2	no	—
277	C₂×C₄	C₃×#700	1	4	yes	complemented
278	C₂×C₄	C₃×C₄×AGL(1,𝔽₅)	1	2	yes	—
279	C₂×C₄	C₃×C₄⋊AGL(1,𝔽₅)	1	2	yes	—
280	C₂×C₄	C₃×C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
281	C₈	—	10	8	no	—
282	C₂×C₃	—	10	4	no	—
283	C₂×C₃	C₄×#700	1	2	yes	—
284	C₂×C₃	#65681	1	1	yes	characteristic
285	C₂×C₃	#65682	1	1	yes	characteristic
286	C₂×C₃	#65756	1	2	yes	—
287	C₂×C₃	C₂×(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic
288	C₅	C₃×#1728	1	1	yes	characteristic, complemented, Sylow
289	C₂²	—	5	4	no	—
290	C₂²	—	10	4	no	—
291	C₂²	—	10	2	no	—
292	C₂²	—	10	2	no	—
293	C₂²	—	10	4	no	—
294	C₂²	C₂×C₃×#700	1	2	yes	—
295	C₂²	C₃×D₅⋊M₁₆	1	2	yes	—
296	C₂²	C₂×C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
297	C₂²	C₂×C₃×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
298	C₂²	C₂×C₃×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
299	C₄	—	2	4	no	—
300	C₄	—	5	2	no	—
301	C₄	—	5	2	no	—
302	C₄	—	10	4	no	—
303	C₄	C₃×(C₂×AGL(1,𝔽₅))⋊C₄	1	4	yes	—
304	C₃	#10604172	1	1	yes	characteristic, direct product factor, Sylow
305	C₂	—	10	4	no	—
306	C₂	C₃×C₄×#700	1	2	yes	—
307	C₂	C₃×#65681	1	1	yes	characteristic
308	C₂	C₃×#65682	1	1	yes	characteristic
309	C₂	C₃×#65756	1	2	yes	—
310	C₂	C₂×C₃×(C₂×AGL(1,𝔽₅))⋊C₄	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	C₂×C₃×C₄²×C₅	C₄
socle	C₂³×C₃×C₅	C₂²×C₄
prime radical	C₂³×C₃×D₅	C₂×C₄
cosocle	C₂²×Dic₅	C₂³×C₃
hypercenter	C₂×C₃×C₄²	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#1728
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄²

Complements

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

N	H ≅ G/N	type
3. C₃×#64918	C₂	semidirect
4. #10604172	C₃	direct (N × H)
10. C₂×C₃×#700	C₄	semidirect
18. #64918	C₂×C₃	semidirect
33. C₂×C₃×#170	C₂×C₄	semidirect
46. C₂×#700	C₃×C₄	semidirect
63. C₂×C₃×C₄×C₅	M₁₆	semidirect
66. C₂×C₃×Dic₅	M₁₆	semidirect
99. C₂×#170	C₂×C₃×C₄	semidirect
152. C₂×C₄×C₅	C₃×M₁₆	semidirect
155. C₂×Dic₅	C₃×M₁₆	semidirect
198. C₂×C₃×C₄	#700	semidirect
237. C₃×C₅	#1728	semidirect
277. C₂×C₄	C₃×#700	semidirect
288. C₅	C₃×#1728	semidirect
304. C₃	#10604172	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×#700	2
C₂×C₃×C₄²×D₅	2
C₃×#64918	2
#10604172	3
C₃×#1728	5

Subgroups — C3×#10604172 #14294312

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10604172 #14294312