Subgroups — #14299860

Order 1920.

524 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#568	C₂	1	1	yes	maximal, characteristic
2	#11751337	C₂	1	1	yes	maximal, characteristic
3	C₂×#91121	C₂	1	1	yes	maximal, characteristic
4	#11751225	C₂	1	1	yes	maximal, characteristic, complemented
5	#11751219	C₂	1	1	yes	maximal, characteristic, complemented
6	#11746377	C₂	1	1	yes	maximal, characteristic, complemented
7	#11746376	C₂	1	1	yes	maximal, characteristic, complemented
8	#10602821	—	3	1	no	maximal, Hall
9	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
10	C₃×C₅×#131	—	2	2	no	—
11	C₃×C₅×#131	—	2	2	no	—
12	C₂×C₃×C₅×M₁₆	—	2	2	no	—
13	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
15	#91815	C₂²	1	1	yes	characteristic
16	C₂×C₄×Dic₁₅	C₂²	1	1	yes	characteristic
17	C₂×#6710	C₂²	1	1	yes	characteristic
18	C₂×#6710	C₂²	1	1	yes	characteristic
19	#91121	—	2	2	no	—
20	#91121	C₄	1	2	yes	complemented
21	#91121	C₄	1	2	yes	complemented
22	#91106	—	2	2	no	—
23	#91098	—	2	2	no	—
24	#72028	—	5	1	no	maximal, Hall
25	C₅×#568	D₃	1	1	yes	characteristic
26	#65525	—	3	1	no	—
27	C₂×#3969	—	3	1	no	—
28	#65413	—	3	1	no	—
29	#65407	—	3	1	no	—
30	#64776	—	3	1	no	—
31	#64775	—	3	1	no	—
32	C₃×C₅×#41	—	2	1	no	—
33	C₃×C₅×#41	—	2	1	no	—
34	C₃×C₅×#41	C₂×C₄	1	2	yes	—
35	C₃×C₅×#41	D₄	1	2	yes	—
36	C₂×C₃×C₅×Q₈	Q₈	1	1	yes	characteristic
37	C₂×C₃×C₅×D₄	—	2	1	no	—
38	C₂×C₃×C₅×D₄	Q₈	1	1	yes	characteristic
39	C₂²×C₃×C₄×C₅	—	2	1	no	—
40	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
41	C₂²×Dic₁₅	—	2	1	no	—
42	C₃×C₅×M₁₆	—	2	4	no	—
43	C₃×C₅×M₁₆	—	2	1	no	—
44	C₃×C₅×M₁₆	—	2	1	no	—
45	C₃×C₅×M₁₆	—	2	1	no	—
46	C₃×C₅×M₁₆	D₄	1	2	yes	—
47	C₂×C₃×C₅×C₈	—	2	4	no	—
48	C₂×C₃×C₅×C₈	—	2	1	no	—
49	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
50	#6719	—	2	1	no	—
51	C₄⋊Dic₁₅	—	2	1	no	—
52	C₄×Dic₁₅	—	2	1	no	—
53	C₄×Dic₁₅	—	2	1	no	—
54	C₄×Dic₁₅	C₂×C₄	1	1	yes	characteristic
55	C₄×Dic₁₅	C₂×C₄	1	1	yes	characteristic
56	#6710	—	2	1	no	—
57	#6710	—	2	1	no	—
58	#6710	—	2	1	no	—
59	#6710	C₂×C₄	1	2	yes	—
60	C₂×C₃⋊#168	—	2	1	no	—
61	C₂×C₃⋊#168	—	2	1	no	—
62	C₃×#568	D₅	1	1	yes	characteristic
63	#5236	—	5	1	no	—
64	C₂×#837	—	5	1	no	—
65	#5124	—	5	1	no	—
66	#5118	—	5	1	no	—
67	#4556	—	5	1	no	—
68	#4555	—	5	1	no	—
69	C₂×C₅×#41	C₂×D₃	1	1	yes	characteristic
70	C₅×#131	—	2	2	no	—
71	C₅×#131	—	2	2	no	—
72	C₂×C₅×M₁₆	—	2	2	no	—
73	C₂×C₅×M₁₆	C₂×D₃	1	1	yes	characteristic
74	C₂×C₅×M₁₆	C₂×D₃	1	1	yes	characteristic
75	#4072	—	3	1	no	—
76	C₂×C₄×Dic₅	—	3	1	no	—
77	C₂×#681	—	3	1	no	—
78	C₂×#681	—	3	1	no	—
79	#3969	—	3	2	no	—
80	#3969	—	3	2	no	—
81	#3969	—	6	2	no	—
82	#3954	—	6	2	no	—
83	#3946	—	6	2	no	—
84	#1861	—	15	1	no	Sylow
85	C₂³×C₃×C₅	—	2	1	no	—
86	C₂³×C₃×C₅	#41	1	1	yes	characteristic
87	C₃×C₅×Q₈	—	2	1	no	—
88	C₃×C₅×Q₈	C₄⋊C₄	1	2	yes	—
89	C₃×C₅×D₄	—	2	2	no	—
90	C₃×C₅×D₄	—	2	2	no	—
91	C₃×C₅×D₄	—	2	1	no	—
92	C₃×C₅×D₄	C₄⋊C₄	1	2	yes	—
93	C₂×C₃×C₄×C₅	—	2	1	no	—
94	C₂×C₃×C₄×C₅	—	2	2	no	—
95	C₂×C₃×C₄×C₅	—	2	2	no	—
96	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
98	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
99	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
100	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
101	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
102	C₂×Dic₁₅	—	2	1	no	—
103	C₂×Dic₁₅	—	2	1	no	—
104	C₂×Dic₁₅	—	4	2	no	—
105	C₂×Dic₁₅	—	4	1	no	—
106	C₃×C₅×C₈	—	2	2	no	—
107	C₃×C₅×C₈	—	2	2	no	—
108	C₃⋊#168	—	2	2	no	—
109	C₃⋊#168	—	4	1	no	—
110	C₂×C₃×#41	C₂×D₅	1	1	yes	characteristic
111	C₃×#131	—	2	2	no	—
112	C₃×#131	—	2	2	no	—
113	C₂×C₃×M₁₆	—	2	2	no	—
114	C₂×C₃×M₁₆	C₂×D₅	1	1	yes	characteristic
115	C₂×C₃×M₁₆	C₂×D₅	1	1	yes	characteristic
116	#926	—	5	1	no	—
117	C₂×C₄×Dic₃	—	5	1	no	—
118	C₂×#208	—	5	1	no	—
119	C₂×#208	—	5	1	no	—
120	#837	—	5	2	no	—
121	#837	—	5	2	no	—
122	#837	—	10	2	no	—
123	#822	—	10	2	no	—
124	Dic₃⋊C₈	—	10	2	no	—
125	C₅×#41	—	2	1	no	—
126	C₅×#41	—	2	1	no	—
127	C₅×#41	C₄×D₃	1	2	yes	—
128	C₅×#41	#67	1	2	yes	—
129	C₂×C₅×Q₈	Dic₆	1	1	yes	characteristic
130	C₂×C₅×D₄	—	2	1	no	—
131	C₂×C₅×D₄	Dic₆	1	1	yes	characteristic
132	C₂²×C₄×C₅	—	2	1	no	—
133	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
134	C₂²×Dic₅	—	6	1	no	—
135	C₅×M₁₆	—	2	4	no	—
136	C₅×M₁₆	—	2	1	no	—
137	C₅×M₁₆	—	2	1	no	—
138	C₅×M₁₆	—	2	1	no	—
139	C₅×M₁₆	#67	1	2	yes	—
140	C₂×C₅×C₈	—	2	4	no	—
141	C₂×C₅×C₈	—	2	1	no	—
142	C₂×C₅×C₈	#67	1	2	yes	—
143	C₂²⋊Dic₅	—	6	1	no	—
144	C₄⋊Dic₅	—	6	1	no	—
145	C₄×Dic₅	—	3	1	no	—
146	C₄×Dic₅	—	3	1	no	—
147	C₄×Dic₅	—	6	1	no	—
148	C₄×Dic₅	—	6	1	no	—
149	#681	—	3	2	no	—
150	#681	—	6	1	no	—
151	#681	—	6	1	no	—
152	#681	—	6	1	no	—
153	C₂×#168	—	6	1	no	—
154	C₂×#168	—	6	1	no	—
155	#568	D₁₅	1	1	yes	characteristic
156	#430	—	15	1	no	—
157	#423	—	15	1	no	—
158	#421	—	15	1	no	—
159	C₂×#104	—	15	1	no	—
160	#344	—	15	1	no	—
161	#339	—	15	1	no	—
162	C₂²×C₃×C₅	—	2	2	no	—
163	C₂²×C₃×C₅	—	4	1	no	—
164	C₂²×C₃×C₅	—	8	1	no	—
165	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
166	C₂²×C₃×C₅	C₂²⋊Q₈	1	1	yes	characteristic
167	C₂²×C₃×C₅	#123	1	1	yes	characteristic
168	C₃×C₄×C₅	—	2	2	no	—
169	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
170	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
171	C₃×C₄×C₅	#120	1	1	yes	characteristic
172	C₃×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
173	Dic₁₅	—	2	2	no	—
174	Dic₁₅	—	4	1	no	—
175	Dic₁₅	—	8	1	no	—
176	C₃×#41	—	2	1	no	—
177	C₃×#41	—	2	1	no	—
178	C₃×#41	C₄×D₅	1	2	yes	—
179	C₃×#41	C₂²⋊D₅	1	2	yes	—
180	C₂×C₃×Q₈	Dic₁₀	1	1	yes	characteristic
181	C₂×C₃×D₄	—	2	1	no	—
182	C₂×C₃×D₄	Dic₁₀	1	1	yes	characteristic
183	C₂²×C₃×C₄	—	2	1	no	—
184	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
185	C₂²×Dic₃	—	10	1	no	—
186	C₃×M₁₆	—	2	4	no	—
187	C₃×M₁₆	—	2	1	no	—
188	C₃×M₁₆	—	2	1	no	—
189	C₃×M₁₆	—	2	1	no	—
190	C₃×M₁₆	C₂²⋊D₅	1	2	yes	—
191	C₂×C₃×C₈	—	2	4	no	—
192	C₂×C₃×C₈	—	2	1	no	—
193	C₂×C₃×C₈	C₂²⋊D₅	1	2	yes	—
194	#217	—	10	1	no	—
195	C₄⋊Dic₃	—	10	1	no	—
196	C₄×Dic₃	—	5	1	no	—
197	C₄×Dic₃	—	5	1	no	—
198	C₄×Dic₃	—	10	1	no	—
199	C₄×Dic₃	—	10	1	no	—
200	#208	—	5	2	no	—
201	#208	—	10	1	no	—
202	#208	—	10	1	no	—
203	#208	—	10	1	no	—
204	C₂×C₃⋊C₈	—	10	1	no	—
205	C₂×C₃⋊C₈	—	10	1	no	—
206	C₂³×C₅	—	2	1	no	—
207	C₂³×C₅	#235	1	1	yes	characteristic
208	C₅×Q₈	—	2	1	no	—
209	C₅×Q₈	Dic₃⋊C₄	1	2	yes	—
210	C₅×D₄	—	2	2	no	—
211	C₅×D₄	—	2	2	no	—
212	C₅×D₄	—	2	1	no	—
213	C₅×D₄	Dic₃⋊C₄	1	2	yes	—
214	C₂×C₄×C₅	—	2	1	no	—
215	C₂×C₄×C₅	—	2	2	no	—
216	C₂×C₄×C₅	—	2	2	no	—
217	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
218	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
219	C₂×C₄×C₅	#235	1	1	yes	characteristic
220	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
221	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
222	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
223	C₂×Dic₅	—	6	1	no	—
224	C₂×Dic₅	—	6	1	no	—
225	C₂×Dic₅	—	12	2	no	—
226	C₂×Dic₅	—	12	1	no	—
227	C₅×C₈	—	2	2	no	—
228	C₅×C₈	—	2	2	no	—
229	#168	—	6	2	no	—
230	#168	—	12	1	no	—
231	C₂×#41	C₂×D₁₅	1	1	yes	characteristic
232	#131	—	2	2	no	—
233	#131	—	2	2	no	—
234	C₂×M₁₆	—	2	2	no	—
235	C₂×M₁₆	—	15	1	no	—
236	C₂×M₁₆	—	15	1	no	—
237	C₂×M₁₆	C₂×D₁₅	1	1	yes	characteristic
238	C₂×M₁₆	C₂×D₁₅	1	1	yes	characteristic
239	#117	—	15	1	no	—
240	C₂×C₄²	—	15	1	no	—
241	#108	—	30	2	no	—
242	C₄⋊C₈	—	30	2	no	—
243	#104	—	15	2	no	—
244	#104	—	15	2	no	—
245	#104	—	30	2	no	—
246	C₂×C₃×C₅	—	2	1	no	—
247	C₂×C₃×C₅	—	2	1	no	—
248	C₂×C₃×C₅	—	2	1	no	—
249	C₂×C₃×C₅	—	4	2	no	—
250	C₂×C₃×C₅	#385	1	1	yes	characteristic
251	C₂³×C₃	—	2	1	no	—
252	C₂³×C₃	#709	1	1	yes	characteristic
253	C₃×Q₈	—	2	1	no	—
254	C₃×Q₈	#683	1	2	yes	—
255	C₃×D₄	—	2	2	no	—
256	C₃×D₄	—	2	2	no	—
257	C₃×D₄	—	2	1	no	—
258	C₃×D₄	#683	1	2	yes	—
259	C₂×C₃×C₄	—	2	1	no	—
260	C₂×C₃×C₄	—	2	2	no	—
261	C₂×C₃×C₄	—	2	2	no	—
262	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
263	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
264	C₂×C₃×C₄	#709	1	1	yes	characteristic
265	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
266	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
267	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
268	C₂×Dic₃	—	10	1	no	—
269	C₂×Dic₃	—	10	1	no	—
270	C₂×Dic₃	—	20	2	no	—
271	C₂×Dic₃	—	20	1	no	—
272	C₃×C₈	—	2	2	no	—
273	C₃×C₈	—	2	2	no	—
274	C₃⋊C₈	—	10	2	no	—
275	C₃⋊C₈	—	20	1	no	—
276	C₂²×C₅	—	2	2	no	—
277	C₂²×C₅	—	4	1	no	—
278	C₂²×C₅	—	8	1	no	—
279	C₂²×C₅	#924	1	1	yes	characteristic
280	C₂²×C₅	C₄×#67	1	1	yes	characteristic
281	C₂²×C₅	#929	1	1	yes	characteristic
282	C₄×C₅	—	2	2	no	—
283	C₄×C₅	C₂×Dic₃⋊C₄	1	1	yes	characteristic
284	C₄×C₅	#924	1	1	yes	characteristic
285	C₄×C₅	C₄×#67	1	1	yes	characteristic
286	C₄×C₅	#953	1	1	yes	characteristic
287	Dic₅	—	6	2	no	—
288	Dic₅	—	12	1	no	—
289	Dic₅	—	24	1	no	—
290	#41	—	2	1	no	—
291	#41	—	2	1	no	—
292	#41	C₄×D₁₅	1	2	yes	—
293	#41	#1220	1	2	yes	—
294	C₂×Q₈	Dic₃₀	1	1	yes	characteristic
295	C₂×D₄	—	2	1	no	—
296	C₂×D₄	Dic₃₀	1	1	yes	characteristic
297	C₂²×C₄	—	2	1	no	—
298	C₂²×C₄	—	30	1	no	—
299	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
300	M₁₆	—	2	4	no	—
301	M₁₆	—	2	1	no	—
302	M₁₆	—	2	1	no	—
303	M₁₆	—	2	1	no	—
304	M₁₆	—	15	2	no	—
305	M₁₆	—	30	1	no	—
306	M₁₆	—	30	1	no	—
307	M₁₆	—	30	1	no	—
308	M₁₆	#1220	1	2	yes	—
309	C₂×C₈	—	2	4	no	—
310	C₂×C₈	—	2	1	no	—
311	C₂×C₈	—	30	1	no	—
312	C₂×C₈	—	30	1	no	—
313	C₂×C₈	#1220	1	2	yes	—
314	C₄⋊C₄	—	30	1	no	—
315	C₂²⋊C₄	—	30	1	no	—
316	C₄²	—	15	1	no	—
317	C₄²	—	15	1	no	—
318	C₄²	—	30	1	no	—
319	C₄²	—	30	1	no	—
320	C₃×C₅	#1861	1	1	yes	characteristic, complemented, Hall
321	C₂²×C₃	—	2	2	no	—
322	C₂²×C₃	—	4	1	no	—
323	C₂²×C₃	—	8	1	no	—
324	C₂²×C₃	#4070	1	1	yes	characteristic
325	C₂²×C₃	C₄×C₂²⋊D₅	1	1	yes	characteristic
326	C₂²×C₃	#4075	1	1	yes	characteristic
327	C₃×C₄	—	2	2	no	—
328	C₃×C₄	C₂×#683	1	1	yes	characteristic
329	C₃×C₄	#4070	1	1	yes	characteristic
330	C₃×C₄	C₄×C₂²⋊D₅	1	1	yes	characteristic
331	C₃×C₄	#4099	1	1	yes	characteristic
332	Dic₃	—	10	2	no	—
333	Dic₃	—	20	1	no	—
334	Dic₃	—	40	1	no	—
335	C₂×C₅	—	2	1	no	—
336	C₂×C₅	—	2	1	no	—
337	C₂×C₅	—	2	1	no	—
338	C₂×C₅	—	4	2	no	—
339	C₂×C₅	#5210	1	1	yes	characteristic
340	C₂³	—	2	1	no	—
341	C₂³	#6817	1	1	yes	characteristic
342	Q₈	—	2	1	no	—
343	Q₈	#6712	1	2	yes	—
344	D₄	—	2	2	no	—
345	D₄	—	2	2	no	—
346	D₄	—	2	1	no	—
347	D₄	#6712	1	2	yes	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	30	1	no	—
352	C₂×C₄	—	30	1	no	—
353	C₂×C₄	—	60	2	no	—
354	C₂×C₄	—	60	1	no	—
355	C₂×C₄	C₂×Dic₃₀	1	1	yes	characteristic
356	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
357	C₂×C₄	#6817	1	1	yes	characteristic
358	C₂×C₄	C₂×#1220	1	1	yes	characteristic
359	C₂×C₄	C₂×#1220	1	1	yes	characteristic
360	C₂×C₄	C₂×#1220	1	1	yes	characteristic
361	C₈	—	2	2	no	—
362	C₈	—	2	2	no	—
363	C₈	—	30	2	no	—
364	C₈	—	60	1	no	—
365	C₂×C₃	—	2	1	no	—
366	C₂×C₃	—	2	1	no	—
367	C₂×C₃	—	2	1	no	—
368	C₂×C₃	—	4	2	no	—
369	C₂×C₃	#65499	1	1	yes	characteristic
370	C₅	#72028	1	1	yes	characteristic, complemented, Sylow
371	C₂²	—	2	2	no	—
372	C₂²	—	4	1	no	—
373	C₂²	—	8	1	no	—
374	C₂²	#91813	1	1	yes	characteristic
375	C₂²	C₄×#1220	1	1	yes	characteristic
376	C₂²	#91818	1	1	yes	characteristic
377	C₄	—	2	2	no	—
378	C₄	—	30	2	no	—
379	C₄	—	60	1	no	—
380	C₄	—	120	1	no	—
381	C₄	C₂×#6712	1	1	yes	characteristic
382	C₄	#91813	1	1	yes	characteristic
383	C₄	C₄×#1220	1	1	yes	characteristic
384	C₄	#91842	1	1	yes	characteristic
385	C₃	#10602821	1	1	yes	characteristic, complemented, Sylow
386	C₂	—	2	1	no	—
387	C₂	—	2	1	no	—
388	C₂	—	2	1	no	—
389	C₂	—	4	2	no	—
390	C₂	#11751311	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#91842
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅
Fitting	C₃×C₅×#568	C₂
socle	C₂×C₃×C₅	#385
prime radical	C₂×C₃×C₅×#41	C₂²
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	#568	D₁₅
hypocenter	C₃×C₅	#1861
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #568

Complements

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

N	H ≅ G/N	type
4. #11751225	C₂	semidirect
5. #11751219	C₂	semidirect
6. #11746377	C₂	semidirect
7. #11746376	C₂	semidirect
20. #91121	C₄	semidirect
21. #91121	C₄	semidirect
320. C₃×C₅	#1861	semidirect
370. C₅	#72028	semidirect
385. C₃	#10602821	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×C₅×#568	2
#11751337	2
C₂×#91121	2
#11751225	2
#11751219	2
#11746377	2
#11746376	2
#10602821	3
#72028	5