Subgroups — C₃×#10599393 #14295038

Order 1920.

824 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄²×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3952	C₂	1	2	yes	maximal
3	C₂×C₃×#3947	C₂	1	2	yes	maximal, complemented
4	C₃×C₅×#336	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64744	C₂	1	1	yes	maximal, characteristic, complemented
6	#10599393	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₄×D₅	C₄	1	2	yes	—
8	C₂²×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄×Dic₅	C₄	1	2	yes	—
12	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
13	C₂²×C₃×#168	C₂²	1	2	yes	—
14	C₃×C₄²×D₅	—	2	4	no	—
15	C₃×#3952	C₄	1	8	yes	complemented
16	C₃×#3947	—	2	4	no	—
17	C₃×#1749	—	5	1	no	maximal, Hall
18	C₂×C₄²×D₅	C₂×C₃	1	1	yes	characteristic
19	C₂×#3952	C₂×C₃	1	2	yes	—
20	C₂×#3947	C₂×C₃	1	2	yes	complemented
21	C₅×#336	C₂×C₃	1	1	yes	characteristic, complemented
22	#64744	C₂×C₃	1	1	yes	characteristic, complemented
23	C₂³×C₃×D₅	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
25	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂²×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃×Dic₅	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₄×D₅	—	2	4	no	—
29	C₂×C₃×C₄×D₅	—	2	4	no	—
30	C₂×C₃×C₄×D₅	—	2	2	no	—
31	C₂×C₃×C₄×D₅	—	2	1	no	—
32	C₂×C₃×C₄×D₅	—	2	1	no	—
33	C₂×C₃×C₄×D₅	—	2	4	no	—
34	C₂×C₃×C₄×D₅	C₂×C₄	1	4	yes	—
35	C₂×C₃×C₅×C₈	—	2	2	no	—
36	C₂×C₃×C₅×C₈	—	2	2	no	—
37	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	complemented
38	C₃×C₄²×C₅	—	2	2	no	—
39	C₃×C₄×Dic₅	—	2	4	no	—
40	C₃×C₄×Dic₅	D₄	1	4	yes	—
41	C₂×C₃×#168	—	2	2	no	—
42	C₂×C₃×#168	—	2	2	no	—
43	C₂×C₃×#168	C₂×C₄	1	4	yes	complemented
44	C₂²×C₃×C₄²	—	5	1	no	—
45	C₂×C₃×C₂²⋊C₈	—	5	2	no	—
46	C₂×C₃×#97	—	5	2	no	—
47	C₃×#336	—	5	1	no	—
48	C₃×#336	D₅	1	1	yes	characteristic, complemented
49	C₂²×C₄×D₅	C₃×C₄	1	2	yes	—
50	C₂²×C₄×D₅	C₂²×C₃	1	1	yes	characteristic
51	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
52	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
53	C₂×C₄×Dic₅	C₃×C₄	1	2	yes	—
54	C₂×C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
55	C₂²×#168	C₂²×C₃	1	2	yes	—
56	C₄²×D₅	—	2	4	no	—
57	#3952	C₃×C₄	1	8	yes	complemented
58	#3947	—	2	4	no	—
59	#1749	—	5	1	no	Sylow
60	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₃×D₅	—	2	1	no	—
62	C₂²×C₃×D₅	—	2	1	no	—
63	C₂²×C₃×D₅	—	2	2	no	—
64	C₂²×C₃×D₅	C₄²	1	2	yes	—
65	C₂²×C₃×D₅	M₁₆	1	4	yes	—
66	C₂×C₃×C₄×C₅	—	2	2	no	—
67	C₂×C₃×C₄×C₅	—	2	2	no	—
68	C₂×C₃×C₄×C₅	M₁₆	1	4	yes	—
69	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
70	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
71	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
73	C₂×C₃×Dic₅	—	2	1	no	—
74	C₂×C₃×Dic₅	—	2	1	no	—
75	C₂×C₃×Dic₅	—	2	2	no	—
76	C₂×C₃×Dic₅	—	2	2	no	—
77	C₂×C₃×Dic₅	C₄²	1	2	yes	—
78	C₂×C₃×Dic₅	C₂²⋊C₄	1	4	yes	—
79	C₃×C₄×D₅	—	2	16	no	—
80	C₃×C₄×D₅	—	2	4	no	—
81	C₃×C₄×D₅	—	2	4	no	—
82	C₃×C₅×C₈	—	2	4	no	—
83	C₃×#168	—	2	4	no	—
84	C₂³×C₃×C₄	—	5	1	no	—
85	C₂³×C₃×C₄	—	5	2	no	—
86	C₂²×C₃×C₈	—	5	2	no	—
87	C₂²×C₃×C₈	C₂×D₅	1	2	yes	—
88	C₂×C₃×C₄²	—	5	1	no	—
89	C₂×C₃×C₄²	—	5	2	no	—
90	C₂×C₃×C₄²	—	10	4	no	—
91	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
92	C₃×C₂²⋊C₈	—	5	8	no	—
93	C₃×#97	—	10	4	no	—
94	C₂³×D₅	C₂×C₃×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
96	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
97	C₂²×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
98	C₂²×Dic₅	C₂×C₃×C₄	1	2	yes	—
99	C₂×C₄×D₅	—	2	4	no	—
100	C₂×C₄×D₅	—	2	4	no	—
101	C₂×C₄×D₅	—	2	2	no	—
102	C₂×C₄×D₅	—	2	1	no	—
103	C₂×C₄×D₅	—	2	1	no	—
104	C₂×C₄×D₅	—	2	4	no	—
105	C₂×C₄×D₅	C₂×C₃×C₄	1	4	yes	—
106	C₂×C₅×C₈	—	2	2	no	—
107	C₂×C₅×C₈	—	2	2	no	—
108	C₂×C₅×C₈	C₂×C₃×C₄	1	4	yes	complemented
109	C₄²×C₅	—	2	2	no	—
110	C₄×Dic₅	—	2	4	no	—
111	C₄×Dic₅	C₃×D₄	1	4	yes	—
112	C₂×#168	—	2	2	no	—
113	C₂×#168	—	2	2	no	—
114	C₂×#168	C₂×C₃×C₄	1	4	yes	complemented
115	C₂²×C₄²	—	5	1	no	—
116	C₂×C₂²⋊C₈	—	5	2	no	—
117	C₂×#97	—	5	2	no	—
118	#336	—	5	1	no	—
119	#336	C₃×D₅	1	1	yes	characteristic, complemented
120	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
121	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂²×C₃×C₅	#117	1	1	yes	characteristic
123	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
124	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
125	C₂×C₃×D₅	—	2	2	no	—
126	C₂×C₃×D₅	—	2	2	no	—
127	C₂×C₃×D₅	—	2	4	no	—
128	C₂×C₃×D₅	—	2	4	no	—
129	C₂×C₃×D₅	#97	1	4	yes	complemented
130	C₃×C₄×C₅	—	2	4	no	—
131	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
132	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
133	C₃×Dic₅	—	2	2	no	—
134	C₃×Dic₅	—	2	4	no	—
135	C₂⁴×C₃	—	5	1	no	—
136	C₂²×C₃×C₄	—	5	1	no	—
137	C₂²×C₃×C₄	—	5	2	no	—
138	C₂²×C₃×C₄	—	5	4	no	—
139	C₂²×C₃×C₄	—	10	4	no	—
140	C₂²×C₃×C₄	—	10	4	no	—
141	C₂²×C₃×C₄	—	10	4	no	—
142	C₂²×C₃×C₄	—	10	2	no	—
143	C₂²×C₃×C₄	—	10	1	no	—
144	C₂²×C₃×C₄	—	10	1	no	—
145	C₂²×C₃×C₄	C₄×D₅	1	2	yes	—
146	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
147	C₂×C₃×C₈	—	2	2	no	—
148	C₂×C₃×C₈	—	2	2	no	—
149	C₂×C₃×C₈	—	5	4	no	—
150	C₂×C₃×C₈	—	10	2	no	—
151	C₂×C₃×C₈	—	10	2	no	—
152	C₂×C₃×C₈	C₄×D₅	1	4	yes	complemented
153	C₃×C₄²	—	2	2	no	—
154	C₃×C₄²	—	5	4	no	—
155	C₃×C₄²	—	10	4	no	—
156	C₂³×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
157	C₂²×D₅	—	2	1	no	—
158	C₂²×D₅	—	2	1	no	—
159	C₂²×D₅	—	2	2	no	—
160	C₂²×D₅	C₃×C₄²	1	2	yes	—
161	C₂²×D₅	C₃×M₁₆	1	4	yes	—
162	C₂×C₄×C₅	—	2	2	no	—
163	C₂×C₄×C₅	—	2	2	no	—
164	C₂×C₄×C₅	C₃×M₁₆	1	4	yes	—
165	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
166	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
167	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
168	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
169	C₂×Dic₅	—	2	1	no	—
170	C₂×Dic₅	—	2	1	no	—
171	C₂×Dic₅	—	2	2	no	—
172	C₂×Dic₅	—	2	2	no	—
173	C₂×Dic₅	C₃×C₄²	1	2	yes	—
174	C₂×Dic₅	C₃×C₂²⋊C₄	1	4	yes	—
175	C₄×D₅	—	2	16	no	—
176	C₄×D₅	—	2	4	no	—
177	C₄×D₅	—	2	4	no	—
178	C₅×C₈	—	2	4	no	—
179	#168	—	2	4	no	—
180	C₂³×C₄	—	5	1	no	—
181	C₂³×C₄	—	5	2	no	—
182	C₂²×C₈	—	5	2	no	—
183	C₂²×C₈	C₂×C₃×D₅	1	2	yes	—
184	C₂×C₄²	—	5	1	no	—
185	C₂×C₄²	—	5	2	no	—
186	C₂×C₄²	—	10	4	no	—
187	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
188	C₂²⋊C₈	—	5	8	no	—
189	#97	—	10	4	no	—
190	C₂×C₃×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
191	C₂×C₃×C₅	C₂×#97	1	1	yes	characteristic
192	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
193	C₂×C₃×C₅	#407	1	2	yes	—
194	C₂×C₃×C₅	#432	1	2	yes	—
195	C₃×D₅	—	2	4	no	—
196	C₂³×C₃	—	5	2	no	—
197	C₂³×C₃	—	5	4	no	—
198	C₂³×C₃	—	10	1	no	—
199	C₂³×C₃	—	10	1	no	—
200	C₂³×C₃	—	10	2	no	—
201	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
202	C₂×C₃×C₄	—	2	2	no	—
203	C₂×C₃×C₄	—	2	2	no	—
204	C₂×C₃×C₄	—	5	2	no	—
205	C₂×C₃×C₄	—	5	4	no	—
206	C₂×C₃×C₄	—	10	16	no	—
207	C₂×C₃×C₄	—	10	1	no	—
208	C₂×C₃×C₄	—	10	1	no	—
209	C₂×C₃×C₄	—	10	2	no	—
210	C₂×C₃×C₄	—	10	2	no	—
211	C₂×C₃×C₄	—	10	4	no	—
212	C₂×C₃×C₄	—	10	4	no	—
213	C₂×C₃×C₄	#676	1	4	yes	—
214	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
215	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
216	C₂×C₃×C₄	#711	1	1	yes	characteristic
217	C₂×C₃×C₄	#713	1	1	yes	characteristic
218	C₃×C₈	—	2	4	no	—
219	C₃×C₈	—	10	4	no	—
220	C₂²×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
221	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
222	C₂²×C₅	C₃×#117	1	1	yes	characteristic
223	C₂²×C₅	C₂×C₃×M₁₆	1	2	yes	—
224	C₂²×C₅	C₂×C₃×M₁₆	1	2	yes	—
225	C₂×D₅	—	2	2	no	—
226	C₂×D₅	—	2	2	no	—
227	C₂×D₅	—	2	4	no	—
228	C₂×D₅	—	2	4	no	—
229	C₂×D₅	C₃×#97	1	4	yes	complemented
230	C₄×C₅	—	2	4	no	—
231	C₄×C₅	C₃×C₄×D₄	1	2	yes	—
232	C₄×C₅	C₃×C₄×D₄	1	2	yes	—
233	Dic₅	—	2	2	no	—
234	Dic₅	—	2	4	no	—
235	C₂⁴	—	5	1	no	—
236	C₂²×C₄	—	5	1	no	—
237	C₂²×C₄	—	5	2	no	—
238	C₂²×C₄	—	5	4	no	—
239	C₂²×C₄	—	10	4	no	—
240	C₂²×C₄	—	10	4	no	—
241	C₂²×C₄	—	10	4	no	—
242	C₂²×C₄	—	10	2	no	—
243	C₂²×C₄	—	10	1	no	—
244	C₂²×C₄	—	10	1	no	—
245	C₂²×C₄	C₃×C₄×D₅	1	2	yes	—
246	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	5	4	no	—
250	C₂×C₈	—	10	2	no	—
251	C₂×C₈	—	10	2	no	—
252	C₂×C₈	C₃×C₄×D₅	1	4	yes	complemented
253	C₄²	—	2	2	no	—
254	C₄²	—	5	4	no	—
255	C₄²	—	10	4	no	—
256	C₃×C₅	#1749	1	1	yes	characteristic, complemented, Hall
257	C₂²×C₃	—	5	4	no	—
258	C₂²×C₃	—	10	4	no	—
259	C₂²×C₃	—	10	2	no	—
260	C₂²×C₃	—	10	2	no	—
261	C₂²×C₃	—	10	4	no	—
262	C₂²×C₃	C₄²×D₅	1	1	yes	characteristic
263	C₂²×C₃	D₅×C₂²⋊C₄	1	1	yes	characteristic
264	C₂²×C₃	#4038	1	1	yes	characteristic
265	C₂²×C₃	C₂×#676	1	2	yes	—
266	C₂²×C₃	D₅×M₁₆	1	2	yes	—
267	C₃×C₄	—	2	4	no	—
268	C₃×C₄	—	10	2	no	—
269	C₃×C₄	—	10	4	no	—
270	C₃×C₄	#4027	1	2	yes	—
271	C₃×C₄	#4039	1	2	yes	—
272	C₂×C₅	C₃×C₄×C₂²⋊C₄	1	1	yes	characteristic
273	C₂×C₅	C₂×C₃×#97	1	1	yes	characteristic
274	C₂×C₅	C₃×C₄×M₁₆	1	1	yes	characteristic
275	C₂×C₅	C₃×#407	1	2	yes	—
276	C₂×C₅	C₃×#432	1	2	yes	—
277	D₅	—	2	4	no	—
278	C₂³	—	5	2	no	—
279	C₂³	—	5	4	no	—
280	C₂³	—	10	1	no	—
281	C₂³	—	10	1	no	—
282	C₂³	—	10	2	no	—
283	C₂³	C₂×C₃×C₄×D₅	1	1	yes	characteristic
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	5	2	no	—
287	C₂×C₄	—	5	4	no	—
288	C₂×C₄	—	10	16	no	—
289	C₂×C₄	—	10	1	no	—
290	C₂×C₄	—	10	1	no	—
291	C₂×C₄	—	10	2	no	—
292	C₂×C₄	—	10	2	no	—
293	C₂×C₄	—	10	4	no	—
294	C₂×C₄	—	10	4	no	—
295	C₂×C₄	C₃×#676	1	4	yes	—
296	C₂×C₄	C₂×C₃×C₄×D₅	1	2	yes	—
297	C₂×C₄	C₃×D₄×D₅	1	2	yes	—
298	C₂×C₄	C₃×#711	1	1	yes	characteristic
299	C₂×C₄	C₃×#713	1	1	yes	characteristic
300	C₈	—	2	4	no	—
301	C₈	—	10	4	no	—
302	C₂×C₃	—	10	4	no	—
303	C₂×C₃	#64955	1	1	yes	characteristic
304	C₂×C₃	C₄×#676	1	1	yes	characteristic
305	C₂×C₃	D₅×#97	1	1	yes	characteristic
306	C₂×C₃	#65015	1	2	yes	—
307	C₂×C₃	#65124	1	2	yes	—
308	C₅	C₃×#1749	1	1	yes	characteristic, complemented, Sylow
309	C₂²	—	5	4	no	—
310	C₂²	—	10	4	no	—
311	C₂²	—	10	2	no	—
312	C₂²	—	10	2	no	—
313	C₂²	—	10	4	no	—
314	C₂²	C₃×C₄²×D₅	1	1	yes	characteristic
315	C₂²	C₃×D₅×C₂²⋊C₄	1	1	yes	characteristic
316	C₂²	C₃×#4038	1	1	yes	characteristic
317	C₂²	C₂×C₃×#676	1	2	yes	—
318	C₂²	C₃×D₅×M₁₆	1	2	yes	—
319	C₄	—	2	4	no	—
320	C₄	—	10	2	no	—
321	C₄	—	10	4	no	—
322	C₄	C₃×#4027	1	2	yes	—
323	C₄	C₃×#4039	1	2	yes	—
324	C₃	#10599393	1	1	yes	characteristic, direct product factor, Sylow
325	C₂	—	10	4	no	—
326	C₂	C₃×#64955	1	1	yes	characteristic
327	C₂	C₃×C₄×#676	1	1	yes	characteristic
328	C₂	C₃×D₅×#97	1	1	yes	characteristic
329	C₂	C₃×#65015	1	2	yes	—
330	C₂	C₃×#65124	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃×C₄	C₂²×D₅
commutator	C₂²×C₅	C₂×C₃×C₄²
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₃×C₅×#336	C₂
socle	C₂³×C₃×C₅	C₂²×C₄
prime radical	C₂³×C₃×D₅	C₂×C₄
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₃×#336	D₅
hypocenter	C₅	C₃×#1749
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₃×#336

Complements

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

N	H ≅ G/N	type
3. C₂×C₃×#3947	C₂	semidirect
4. C₃×C₅×#336	C₂	semidirect
5. C₃×#64744	C₂	semidirect
6. #10599393	C₃	direct (N × H)
15. C₃×#3952	C₄	semidirect
20. C₂×#3947	C₂×C₃	semidirect
21. C₅×#336	C₂×C₃	semidirect
22. #64744	C₂×C₃	semidirect
37. C₂×C₃×C₅×C₈	C₂×C₄	semidirect
43. C₂×C₃×#168	C₂×C₄	semidirect
48. C₃×#336	D₅	semidirect
57. #3952	C₃×C₄	semidirect
108. C₂×C₅×C₈	C₂×C₃×C₄	semidirect
114. C₂×#168	C₂×C₃×C₄	semidirect
119. #336	C₃×D₅	semidirect
129. C₂×C₃×D₅	#97	semidirect
152. C₂×C₃×C₈	C₄×D₅	semidirect
229. C₂×D₅	C₃×#97	semidirect
252. C₂×C₈	C₃×C₄×D₅	semidirect
256. C₃×C₅	#1749	semidirect
308. C₅	C₃×#1749	semidirect
324. C₃	#10599393	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×C₄²×D₅	2
C₂×C₃×#3952	2
C₂×C₃×#3947	2
C₃×C₅×#336	2
C₃×#64744	2
#10599393	3
C₃×#1749	5

Subgroups — C3×#10599393 #14295038

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10599393 #14295038