Subgroups — C₃×#10600856 #14297554

Order 1920.

672 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₄×Dic₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×#100	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#3965	C₂	1	1	yes	maximal, characteristic
4	C₃×#65407	C₂	1	2	yes	maximal, complemented
5	C₃×#64774	C₂	1	2	yes	maximal, complemented
6	#10600856	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
8	C₂³×C₃×Dic₅	C₄	1	2	yes	—
9	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
10	C₂×C₃×C₂²⋊Dic₅	C₄	1	2	yes	—
11	C₃×D₄×Dic₅	—	2	2	no	—
12	C₃×D₄×Dic₅	—	2	2	no	—
13	C₂×C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
15	C₂×C₃×#681	C₂²	1	2	yes	—
16	C₃×C₅×#100	—	2	2	no	—
17	C₃×#3965	—	2	2	no	—
18	C₃×#3946	—	2	2	no	—
19	C₃×#3946	—	2	2	no	—
20	C₃×#1800	—	5	1	no	maximal, Hall
21	C₂×D₄×Dic₅	C₂×C₃	1	1	yes	characteristic
22	C₂×C₅×#100	C₂×C₃	1	1	yes	characteristic
23	C₂×#3965	C₂×C₃	1	1	yes	characteristic
24	#65407	C₂×C₃	1	2	yes	complemented
25	#64774	C₂×C₃	1	2	yes	complemented
26	C₂⁴×C₃×C₅	C₂×C₄	1	2	yes	—
27	C₂×C₃×C₅×D₄	—	2	2	no	—
28	C₂×C₃×C₅×D₄	—	2	2	no	—
29	C₂×C₃×C₅×D₄	D₄	1	2	yes	—
30	C₂×C₃×C₅×D₄	Q₈	1	2	yes	—
31	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂²×C₃×Dic₅	—	2	2	no	—
33	C₂²×C₃×Dic₅	—	2	2	no	—
34	C₂²×C₃×Dic₅	—	4	4	no	—
35	C₂²×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₃×Dic₅	C₂×C₄	1	2	yes	—
37	C₂²×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
38	C₃×C₅×M₁₆	—	2	2	no	—
39	C₃×C₅×M₁₆	—	2	2	no	—
40	C₂×C₃×C₅×C₈	—	2	2	no	—
41	C₃×C₂²⋊Dic₅	—	2	4	no	—
42	C₃×C₄⋊Dic₅	—	2	2	no	—
43	C₃×C₄×Dic₅	D₄	1	2	yes	—
44	C₃×C₄×Dic₅	D₄	1	2	yes	—
45	C₃×#681	—	2	2	no	—
46	C₃×#681	—	2	2	no	—
47	C₂×C₃×#168	—	2	2	no	—
48	C₂×C₃×C₄×D₄	—	5	1	no	—
49	C₃×#423	—	5	2	no	—
50	C₂×C₃×#100	—	5	1	no	—
51	C₂×C₃×#100	D₅	1	1	yes	characteristic
52	C₃×#343	—	5	2	no	—
53	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
54	C₂³×Dic₅	C₃×C₄	1	2	yes	—
55	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
56	C₂×C₂²⋊Dic₅	C₃×C₄	1	2	yes	—
57	D₄×Dic₅	—	2	2	no	—
58	D₄×Dic₅	—	2	2	no	—
59	C₂×C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
60	C₂×C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
61	C₂×#681	C₂²×C₃	1	2	yes	—
62	C₅×#100	—	2	2	no	—
63	#3965	—	2	2	no	—
64	#3946	—	2	2	no	—
65	#3946	—	2	2	no	—
66	#1800	—	5	1	no	Sylow
67	C₂³×C₃×C₅	—	2	2	no	—
68	C₂³×C₃×C₅	—	2	2	no	—
69	C₂³×C₃×C₅	—	4	2	no	—
70	C₂³×C₃×C₅	—	4	2	no	—
71	C₂³×C₃×C₅	C₄⋊C₄	1	4	yes	—
72	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
73	C₃×C₅×D₄	—	2	4	no	—
74	C₃×C₅×D₄	—	2	4	no	—
75	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
77	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	#41	1	2	yes	—
79	C₂×C₃×Dic₅	—	2	2	no	—
80	C₂×C₃×Dic₅	—	2	2	no	—
81	C₂×C₃×Dic₅	—	4	1	no	—
82	C₂×C₃×Dic₅	—	4	4	no	—
83	C₂×C₃×Dic₅	—	4	4	no	—
84	C₂×C₃×Dic₅	—	4	2	no	—
85	C₂×C₃×Dic₅	C₂²⋊C₄	1	2	yes	—
86	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
88	C₃×C₅×C₈	—	4	2	no	—
89	C₃×#168	—	4	2	no	—
90	C₂²×C₃×D₄	C₂×D₅	1	1	yes	characteristic
91	C₂³×C₃×C₄	—	5	2	no	—
92	C₂×C₃×M₁₆	—	5	2	no	—
93	C₂×C₃×M₁₆	C₂×D₅	1	2	yes	—
94	C₃×C₄×D₄	—	10	2	no	—
95	C₃×C₄×D₄	—	10	2	no	—
96	C₂×C₃×C₄⋊C₄	—	5	1	no	—
97	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
98	C₂×C₃×C₄²	—	5	1	no	—
99	C₃×C₄⋊C₈	—	10	2	no	—
100	C₃×C₄⋊C₈	—	10	2	no	—
101	C₃×#100	—	2	2	no	—
102	C₃×#100	—	10	2	no	—
103	C₂⁴×C₅	C₂×C₃×C₄	1	2	yes	—
104	C₂×C₅×D₄	—	2	2	no	—
105	C₂×C₅×D₄	—	2	2	no	—
106	C₂×C₅×D₄	C₃×D₄	1	2	yes	—
107	C₂×C₅×D₄	C₃×Q₈	1	2	yes	—
108	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
109	C₂²×Dic₅	—	2	2	no	—
110	C₂²×Dic₅	—	2	2	no	—
111	C₂²×Dic₅	—	4	4	no	—
112	C₂²×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
113	C₂²×Dic₅	C₂×C₃×C₄	1	2	yes	—
114	C₂²×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
115	C₅×M₁₆	—	2	2	no	—
116	C₅×M₁₆	—	2	2	no	—
117	C₂×C₅×C₈	—	2	2	no	—
118	C₂²⋊Dic₅	—	2	4	no	—
119	C₄⋊Dic₅	—	2	2	no	—
120	C₄×Dic₅	C₃×D₄	1	2	yes	—
121	C₄×Dic₅	C₃×D₄	1	2	yes	—
122	#681	—	2	2	no	—
123	#681	—	2	2	no	—
124	C₂×#168	—	2	2	no	—
125	C₂×C₄×D₄	—	5	1	no	—
126	#423	—	5	2	no	—
127	C₂×#100	—	5	1	no	—
128	C₂×#100	C₃×D₅	1	1	yes	characteristic
129	#343	—	5	2	no	—
130	C₂²×C₃×C₅	—	2	1	no	—
131	C₂²×C₃×C₅	—	2	1	no	—
132	C₂²×C₃×C₅	—	2	4	no	—
133	C₂²×C₃×C₅	—	4	4	no	—
134	C₂²×C₃×C₅	—	4	4	no	—
135	C₂²×C₃×C₅	—	4	2	no	—
136	C₂²×C₃×C₅	—	4	2	no	—
137	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂²×C₃×C₅	#117	1	1	yes	characteristic
140	C₃×C₄×C₅	#121	1	2	yes	—
141	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
142	C₃×Dic₅	—	2	1	no	—
143	C₃×Dic₅	—	2	2	no	—
144	C₃×Dic₅	—	4	2	no	—
145	C₃×Dic₅	#100	1	2	yes	complemented
146	C₂⁴×C₃	C₄×D₅	1	2	yes	—
147	C₂×C₃×D₄	—	2	2	no	—
148	C₂×C₃×D₄	—	2	2	no	—
149	C₂×C₃×D₄	Dic₁₀	1	2	yes	—
150	C₂×C₃×D₄	C₂²⋊D₅	1	2	yes	—
151	C₂²×C₃×C₄	—	5	1	no	—
152	C₂²×C₃×C₄	—	5	1	no	—
153	C₂²×C₃×C₄	—	5	2	no	—
154	C₂²×C₃×C₄	—	10	2	no	—
155	C₂²×C₃×C₄	—	10	2	no	—
156	C₂²×C₃×C₄	—	20	4	no	—
157	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
158	C₃×M₁₆	—	2	2	no	—
159	C₃×M₁₆	—	2	2	no	—
160	C₃×M₁₆	—	10	2	no	—
161	C₃×M₁₆	—	10	2	no	—
162	C₂×C₃×C₈	—	2	2	no	—
163	C₂×C₃×C₈	—	10	2	no	—
164	C₃×C₄⋊C₄	—	10	2	no	—
165	C₃×C₂²⋊C₄	—	10	4	no	—
166	C₃×C₄²	—	5	2	no	—
167	C₃×C₄²	—	5	2	no	—
168	C₂³×C₅	—	2	2	no	—
169	C₂³×C₅	—	2	2	no	—
170	C₂³×C₅	—	4	2	no	—
171	C₂³×C₅	—	4	2	no	—
172	C₂³×C₅	C₃×C₄⋊C₄	1	4	yes	—
173	C₂³×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
174	C₅×D₄	—	2	4	no	—
175	C₅×D₄	—	2	4	no	—
176	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
177	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
178	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
179	C₂×C₄×C₅	C₃×#41	1	2	yes	—
180	C₂×Dic₅	—	2	2	no	—
181	C₂×Dic₅	—	2	2	no	—
182	C₂×Dic₅	—	4	1	no	—
183	C₂×Dic₅	—	4	4	no	—
184	C₂×Dic₅	—	4	4	no	—
185	C₂×Dic₅	—	4	2	no	—
186	C₂×Dic₅	C₃×C₂²⋊C₄	1	2	yes	—
187	C₂×Dic₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
188	C₂×Dic₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
189	C₅×C₈	—	4	2	no	—
190	#168	—	4	2	no	—
191	C₂²×D₄	C₂×C₃×D₅	1	1	yes	characteristic
192	C₂³×C₄	—	5	2	no	—
193	C₂×M₁₆	—	5	2	no	—
194	C₂×M₁₆	C₂×C₃×D₅	1	2	yes	—
195	C₄×D₄	—	10	2	no	—
196	C₄×D₄	—	10	2	no	—
197	C₂×C₄⋊C₄	—	5	1	no	—
198	C₂×C₂²⋊C₄	—	5	2	no	—
199	C₂×C₄²	—	5	1	no	—
200	C₄⋊C₈	—	10	2	no	—
201	C₄⋊C₈	—	10	2	no	—
202	#100	—	2	2	no	—
203	#100	—	10	2	no	—
204	C₂×C₃×C₅	—	2	1	no	—
205	C₂×C₃×C₅	—	2	1	no	—
206	C₂×C₃×C₅	—	4	4	no	—
207	C₂×C₃×C₅	#380	1	1	yes	characteristic
208	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
209	C₂×C₃×C₅	#413	1	1	yes	characteristic
210	C₂³×C₃	—	2	2	no	—
211	C₂³×C₃	—	2	2	no	—
212	C₂³×C₃	—	4	2	no	—
213	C₂³×C₃	—	4	2	no	—
214	C₂³×C₃	#683	1	4	yes	—
215	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
216	C₃×D₄	—	2	4	no	—
217	C₃×D₄	—	2	4	no	—
218	C₂×C₃×C₄	—	5	1	no	—
219	C₂×C₃×C₄	—	5	2	no	—
220	C₂×C₃×C₄	—	5	1	no	—
221	C₂×C₃×C₄	—	10	2	no	—
222	C₂×C₃×C₄	—	10	2	no	—
223	C₂×C₃×C₄	—	20	4	no	—
224	C₂×C₃×C₄	—	20	1	no	—
225	C₂×C₃×C₄	—	20	4	no	—
226	C₂×C₃×C₄	—	20	2	no	—
227	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
228	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
229	C₂×C₃×C₄	#711	1	2	yes	—
230	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
231	C₃×C₈	—	4	2	no	—
232	C₃×C₈	—	20	2	no	—
233	C₂²×C₅	—	2	1	no	—
234	C₂²×C₅	—	2	1	no	—
235	C₂²×C₅	—	2	4	no	—
236	C₂²×C₅	—	4	4	no	—
237	C₂²×C₅	—	4	4	no	—
238	C₂²×C₅	—	4	2	no	—
239	C₂²×C₅	—	4	2	no	—
240	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
241	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
242	C₂²×C₅	C₃×#117	1	1	yes	characteristic
243	C₄×C₅	C₃×#121	1	2	yes	—
244	C₄×C₅	C₃×C₂²⋊Q₈	1	2	yes	—
245	Dic₅	—	2	1	no	—
246	Dic₅	—	2	2	no	—
247	Dic₅	—	4	2	no	—
248	Dic₅	C₃×#100	1	2	yes	complemented
249	C₂⁴	C₃×C₄×D₅	1	2	yes	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	C₃×Dic₁₀	1	2	yes	—
253	C₂×D₄	C₃×C₂²⋊D₅	1	2	yes	—
254	C₂²×C₄	—	5	1	no	—
255	C₂²×C₄	—	5	1	no	—
256	C₂²×C₄	—	5	2	no	—
257	C₂²×C₄	—	10	2	no	—
258	C₂²×C₄	—	10	2	no	—
259	C₂²×C₄	—	20	4	no	—
260	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
261	M₁₆	—	2	2	no	—
262	M₁₆	—	2	2	no	—
263	M₁₆	—	10	2	no	—
264	M₁₆	—	10	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	10	2	no	—
267	C₄⋊C₄	—	10	2	no	—
268	C₂²⋊C₄	—	10	4	no	—
269	C₄²	—	5	2	no	—
270	C₄²	—	5	2	no	—
271	C₃×C₅	#1800	1	1	yes	characteristic, complemented, Hall
272	C₂²×C₃	—	2	1	no	—
273	C₂²×C₃	—	2	1	no	—
274	C₂²×C₃	—	2	4	no	—
275	C₂²×C₃	—	4	4	no	—
276	C₂²×C₃	—	4	4	no	—
277	C₂²×C₃	—	4	2	no	—
278	C₂²×C₃	—	4	2	no	—
279	C₂²×C₃	#4023	1	1	yes	characteristic
280	C₂²×C₃	D₅×C₂²⋊C₄	1	1	yes	characteristic
281	C₂²×C₃	C₂×#683	1	1	yes	characteristic
282	C₃×C₄	—	5	2	no	—
283	C₃×C₄	—	10	1	no	—
284	C₃×C₄	—	10	2	no	—
285	C₃×C₄	—	20	2	no	—
286	C₃×C₄	#4024	1	2	yes	—
287	C₃×C₄	#4085	1	2	yes	—
288	C₂×C₅	—	2	1	no	—
289	C₂×C₅	—	2	1	no	—
290	C₂×C₅	—	4	4	no	—
291	C₂×C₅	C₃×#380	1	1	yes	characteristic
292	C₂×C₅	C₂×C₃×#100	1	1	yes	characteristic
293	C₂×C₅	C₃×#413	1	1	yes	characteristic
294	C₂³	—	2	2	no	—
295	C₂³	—	2	2	no	—
296	C₂³	—	4	2	no	—
297	C₂³	—	4	2	no	—
298	C₂³	C₃×#683	1	4	yes	—
299	C₂³	C₂×C₃×C₄×D₅	1	1	yes	characteristic
300	D₄	—	2	4	no	—
301	D₄	—	2	4	no	—
302	C₂×C₄	—	5	1	no	—
303	C₂×C₄	—	5	2	no	—
304	C₂×C₄	—	5	1	no	—
305	C₂×C₄	—	10	2	no	—
306	C₂×C₄	—	10	2	no	—
307	C₂×C₄	—	20	4	no	—
308	C₂×C₄	—	20	1	no	—
309	C₂×C₄	—	20	4	no	—
310	C₂×C₄	—	20	2	no	—
311	C₂×C₄	C₂×C₃×Dic₁₀	1	1	yes	characteristic
312	C₂×C₄	C₃×D₄×D₅	1	2	yes	—
313	C₂×C₄	C₃×#711	1	2	yes	—
314	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
315	C₈	—	4	2	no	—
316	C₈	—	20	2	no	—
317	C₂×C₃	—	2	1	no	—
318	C₂×C₃	—	2	1	no	—
319	C₂×C₃	—	4	4	no	—
320	C₂×C₃	D₅×#100	1	1	yes	characteristic
321	C₂×C₃	#65034	1	1	yes	characteristic
322	C₂×C₃	#65231	1	1	yes	characteristic
323	C₅	C₃×#1800	1	1	yes	characteristic, complemented, Sylow
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	4	no	—
327	C₂²	—	4	4	no	—
328	C₂²	—	4	4	no	—
329	C₂²	—	4	2	no	—
330	C₂²	—	4	2	no	—
331	C₂²	C₃×#4023	1	1	yes	characteristic
332	C₂²	C₃×D₅×C₂²⋊C₄	1	1	yes	characteristic
333	C₂²	C₂×C₃×#683	1	1	yes	characteristic
334	C₄	—	5	2	no	—
335	C₄	—	10	1	no	—
336	C₄	—	10	2	no	—
337	C₄	—	20	2	no	—
338	C₄	C₃×#4024	1	2	yes	—
339	C₄	C₃×#4085	1	2	yes	—
340	C₃	#10600856	1	1	yes	characteristic, direct product factor, Sylow
341	C₂	—	2	1	no	—
342	C₂	—	2	1	no	—
343	C₂	—	4	4	no	—
344	C₂	C₃×D₅×#100	1	1	yes	characteristic
345	C₂	C₃×#65034	1	1	yes	characteristic
346	C₂	C₃×#65231	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	D₅×C₂²⋊C₄
commutator	C₂³×C₅	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₂×C₃×C₅×#100	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×C₃×C₅×D₄	C₂²
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₂×C₃×#100	D₅
hypocenter	C₅	C₃×#1800
LCS	G ▷ C₂³×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#100

Complements

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

N	H ≅ G/N	type
4. C₃×#65407	C₂	semidirect
5. C₃×#64774	C₂	semidirect
6. #10600856	C₃	direct (N × H)
24. #65407	C₂×C₃	semidirect
25. #64774	C₂×C₃	semidirect
145. C₃×Dic₅	#100	semidirect
248. Dic₅	C₃×#100	semidirect
271. C₃×C₅	#1800	semidirect
323. C₅	C₃×#1800	semidirect
340. C₃	#10600856	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×D₄×Dic₅	2
C₂×C₃×C₅×#100	2
C₂×C₃×#3965	2
C₃×#65407	2
C₃×#64774	2
#10600856	3
C₃×#1800	5

Subgroups — C3×#10600856 #14297554

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10600856 #14297554