Subgroups — #14303594

Order 1920.

372 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#424	C₂	1	1	yes	maximal, characteristic
2	#11751202	C₂	1	1	yes	maximal, characteristic
3	#11751099	C₂	1	1	yes	maximal, characteristic
4	#11746379	C₂	1	1	yes	maximal, characteristic
5	#11746376	C₂	1	1	yes	maximal, characteristic
6	#11746371	C₂	1	1	yes	maximal, characteristic
7	#11746353	C₂	1	1	yes	maximal, characteristic
8	#10601327	—	3	1	no	maximal, Hall
9	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₃×C₅×#117	C₂²	1	1	yes	characteristic
12	#91815	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₁₅	C₂²	1	1	yes	characteristic
14	C₂×#6710	C₂²	1	1	yes	characteristic
15	C₂²×C₃⋊#168	C₂²	1	1	yes	characteristic
16	C₃×C₅×C₄⋊C₈	—	2	1	no	—
17	C₃×C₅×C₄⋊C₈	—	2	1	no	—
18	#91099	—	2	1	no	—
19	C₈×Dic₁₅	—	2	1	no	—
20	#91092	C₄	1	1	yes	characteristic, complemented
21	#91092	C₄	1	1	yes	characteristic, complemented
22	#91091	C₄	1	1	yes	characteristic, complemented
23	#91091	C₄	1	1	yes	characteristic, complemented
24	#70534	—	5	1	no	maximal, Hall
25	C₅×#424	D₃	1	1	yes	characteristic
26	#65390	—	3	1	no	—
27	#65287	—	3	1	no	—
28	#64778	—	3	1	no	—
29	#64775	—	3	1	no	—
30	#64770	—	3	1	no	—
31	#64752	—	3	1	no	—
32	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂²×Dic₁₅	—	2	1	no	—
34	C₃×C₅×M₁₆	—	2	1	no	—
35	C₃×C₅×M₁₆	—	2	1	no	—
36	C₂×C₃×C₅×C₈	—	2	1	no	—
37	C₂×C₃×C₅×C₈	—	2	1	no	—
38	C₂×C₃×C₅×C₈	—	2	1	no	—
39	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
40	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
41	C₃×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
42	C₃×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
43	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
44	C₃×C₄²×C₅	—	2	1	no	—
45	#6719	—	2	1	no	—
46	C₄⋊Dic₁₅	C₂×C₄	1	1	yes	characteristic
47	C₄⋊Dic₁₅	C₂×C₄	1	1	yes	characteristic
48	C₄×Dic₁₅	—	2	1	no	—
49	C₄×Dic₁₅	—	2	1	no	—
50	C₄×Dic₁₅	—	2	1	no	—
51	#6710	—	2	1	no	—
52	#6710	Q₈	1	2	yes	—
53	C₂×C₃⋊#168	—	2	1	no	—
54	C₂×C₃⋊#168	—	2	1	no	—
55	C₂×C₃⋊#168	—	2	1	no	—
56	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
57	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
58	C₃×#424	D₅	1	1	yes	characteristic
59	#5101	—	5	1	no	—
60	#4998	—	5	1	no	—
61	#4558	—	5	1	no	—
62	#4555	—	5	1	no	—
63	#4550	—	5	1	no	—
64	#4532	—	5	1	no	—
65	C₂×C₅×M₁₆	C₂×D₃	1	1	yes	characteristic
66	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
67	C₅×#117	C₂×D₃	1	1	yes	characteristic
68	#4072	—	3	1	no	—
69	C₂×C₄×Dic₅	—	3	1	no	—
70	C₂×#681	—	3	1	no	—
71	C₂²×#168	—	3	1	no	—
72	C₅×C₄⋊C₈	—	2	1	no	—
73	C₅×C₄⋊C₈	—	2	1	no	—
74	#3947	—	6	1	no	—
75	C₈×Dic₅	—	6	1	no	—
76	#3940	—	3	1	no	—
77	#3940	—	3	1	no	—
78	#3939	—	3	1	no	—
79	#3939	—	3	1	no	—
80	#1922	—	15	1	no	Sylow
81	C₂³×C₃×C₅	#41	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	—	2	1	no	—
83	C₂×C₃×C₄×C₅	—	2	1	no	—
84	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
90	C₂×Dic₁₅	—	2	2	no	—
91	C₂×Dic₁₅	—	2	1	no	—
92	C₂×Dic₁₅	—	2	1	no	—
93	C₂×Dic₁₅	—	4	1	no	—
94	C₂×Dic₁₅	—	4	1	no	—
95	C₃×C₅×C₈	—	2	1	no	—
96	C₃×C₅×C₈	—	2	1	no	—
97	C₃×C₅×C₈	—	4	1	no	—
98	C₃⋊#168	—	2	1	no	—
99	C₃⋊#168	—	2	1	no	—
100	C₃⋊#168	—	2	2	no	—
101	C₂×C₃×M₁₆	C₂×D₅	1	1	yes	characteristic
102	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
103	C₃×#117	C₂×D₅	1	1	yes	characteristic
104	#926	—	5	1	no	—
105	C₂×C₄×Dic₃	—	5	1	no	—
106	C₂×#208	—	5	1	no	—
107	C₂²×C₃⋊C₈	—	5	1	no	—
108	C₃×C₄⋊C₈	—	2	1	no	—
109	C₃×C₄⋊C₈	—	2	1	no	—
110	#815	—	10	1	no	—
111	C₈×Dic₃	—	10	1	no	—
112	#808	—	5	1	no	—
113	#808	—	5	1	no	—
114	#807	—	5	1	no	—
115	#807	—	5	1	no	—
116	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
117	C₂²×Dic₅	—	6	1	no	—
118	C₅×M₁₆	—	2	1	no	—
119	C₅×M₁₆	—	2	1	no	—
120	C₂×C₅×C₈	—	2	1	no	—
121	C₂×C₅×C₈	—	2	1	no	—
122	C₂×C₅×C₈	—	2	1	no	—
123	C₂×C₅×C₈	#67	1	1	yes	characteristic
124	C₂×C₅×C₈	#67	1	1	yes	characteristic
125	C₅×C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
126	C₅×C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
127	C₅×C₂²⋊C₄	—	2	1	no	—
128	C₄²×C₅	—	2	1	no	—
129	C₂²⋊Dic₅	—	6	1	no	—
130	C₄⋊Dic₅	—	3	1	no	—
131	C₄⋊Dic₅	—	3	1	no	—
132	C₄×Dic₅	—	6	1	no	—
133	C₄×Dic₅	—	6	1	no	—
134	C₄×Dic₅	—	6	1	no	—
135	#681	—	3	2	no	—
136	#681	—	6	1	no	—
137	C₂×#168	—	3	1	no	—
138	C₂×#168	—	3	1	no	—
139	C₂×#168	—	6	1	no	—
140	C₂×#168	—	6	1	no	—
141	C₂×#168	—	6	1	no	—
142	#427	—	15	1	no	—
143	#424	D₁₅	1	1	yes	characteristic
144	#405	—	15	1	no	—
145	#341	—	15	1	no	—
146	#339	—	15	1	no	—
147	#339	—	15	1	no	—
148	#336	—	15	1	no	—
149	C₂²×C₃×C₅	—	2	1	no	—
150	C₂²×C₃×C₅	—	2	1	no	—
151	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
152	C₂²×C₃×C₅	#125	1	1	yes	characteristic
153	C₂²×C₃×C₅	#126	1	1	yes	characteristic
154	C₃×C₄×C₅	—	4	1	no	—
155	C₃×C₄×C₅	—	4	1	no	—
156	C₃×C₄×C₅	#117	1	1	yes	characteristic
157	C₃×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
158	C₃×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
159	C₃×C₄×C₅	#123	1	1	yes	characteristic
160	Dic₁₅	—	4	2	no	—
161	Dic₁₅	—	4	1	no	—
162	Dic₁₅	—	4	1	no	—
163	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
164	C₂²×Dic₃	—	10	1	no	—
165	C₃×M₁₆	—	2	1	no	—
166	C₃×M₁₆	—	2	1	no	—
167	C₂×C₃×C₈	—	2	1	no	—
168	C₂×C₃×C₈	—	2	1	no	—
169	C₂×C₃×C₈	—	2	1	no	—
170	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
171	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
172	C₃×C₄⋊C₄	C₄×D₅	1	1	yes	characteristic
173	C₃×C₄⋊C₄	C₄×D₅	1	1	yes	characteristic
174	C₃×C₂²⋊C₄	—	2	1	no	—
175	C₃×C₄²	—	2	1	no	—
176	#217	—	10	1	no	—
177	C₄⋊Dic₃	—	5	1	no	—
178	C₄⋊Dic₃	—	5	1	no	—
179	C₄×Dic₃	—	10	1	no	—
180	C₄×Dic₃	—	10	1	no	—
181	C₄×Dic₃	—	10	1	no	—
182	#208	—	5	2	no	—
183	#208	—	10	1	no	—
184	C₂×C₃⋊C₈	—	5	1	no	—
185	C₂×C₃⋊C₈	—	5	1	no	—
186	C₂×C₃⋊C₈	—	10	1	no	—
187	C₂×C₃⋊C₈	—	10	1	no	—
188	C₂×C₃⋊C₈	—	10	1	no	—
189	C₂³×C₅	#235	1	1	yes	characteristic
190	C₂×C₄×C₅	—	2	1	no	—
191	C₂×C₄×C₅	—	2	1	no	—
192	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
193	C₂×C₄×C₅	#237	1	1	yes	characteristic
194	C₂×C₄×C₅	#237	1	1	yes	characteristic
195	C₂×C₄×C₅	D₃×Q₈	1	1	yes	characteristic
196	C₂×C₄×C₅	#239	1	1	yes	characteristic
197	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
198	C₂×Dic₅	—	6	2	no	—
199	C₂×Dic₅	—	6	1	no	—
200	C₂×Dic₅	—	6	1	no	—
201	C₂×Dic₅	—	12	1	no	—
202	C₂×Dic₅	—	12	1	no	—
203	C₅×C₈	—	2	1	no	—
204	C₅×C₈	—	2	1	no	—
205	C₅×C₈	—	4	1	no	—
206	#168	—	6	1	no	—
207	#168	—	6	1	no	—
208	#168	—	6	2	no	—
209	C₂×M₁₆	—	15	1	no	—
210	C₂×M₁₆	C₂×D₁₅	1	1	yes	characteristic
211	C₂²×C₈	—	15	1	no	—
212	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
213	#117	—	15	1	no	—
214	#117	C₂×D₁₅	1	1	yes	characteristic
215	C₂×C₄²	—	15	1	no	—
216	#107	—	15	1	no	—
217	#107	—	15	1	no	—
218	#106	—	15	1	no	—
219	#106	—	15	1	no	—
220	C₄⋊C₈	—	2	1	no	—
221	C₄⋊C₈	—	2	1	no	—
222	#97	—	30	1	no	—
223	C₄×C₈	—	30	1	no	—
224	C₂×C₃×C₅	—	2	1	no	—
225	C₂×C₃×C₅	—	2	1	no	—
226	C₂×C₃×C₅	#387	1	1	yes	characteristic
227	C₂×C₃×C₅	#443	1	1	yes	characteristic
228	C₂×C₃×C₅	#444	1	1	yes	characteristic
229	C₂³×C₃	#709	1	1	yes	characteristic
230	C₂×C₃×C₄	—	2	1	no	—
231	C₂×C₃×C₄	—	2	1	no	—
232	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
233	C₂×C₃×C₄	#711	1	1	yes	characteristic
234	C₂×C₃×C₄	#711	1	1	yes	characteristic
235	C₂×C₃×C₄	Q₈×D₅	1	1	yes	characteristic
236	C₂×C₃×C₄	#713	1	1	yes	characteristic
237	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
238	C₂×Dic₃	—	10	2	no	—
239	C₂×Dic₃	—	10	1	no	—
240	C₂×Dic₃	—	10	1	no	—
241	C₂×Dic₃	—	20	1	no	—
242	C₂×Dic₃	—	20	1	no	—
243	C₃×C₈	—	2	1	no	—
244	C₃×C₈	—	2	1	no	—
245	C₃×C₈	—	4	1	no	—
246	C₃⋊C₈	—	10	1	no	—
247	C₃⋊C₈	—	10	1	no	—
248	C₃⋊C₈	—	10	2	no	—
249	C₂²×C₅	—	2	1	no	—
250	C₂²×C₅	—	2	1	no	—
251	C₂²×C₅	#889	1	1	yes	characteristic
252	C₂²×C₅	#898	1	1	yes	characteristic
253	C₂²×C₅	C₄×#67	1	1	yes	characteristic
254	C₄×C₅	—	4	1	no	—
255	C₄×C₅	—	4	1	no	—
256	C₄×C₅	#887	1	1	yes	characteristic
257	C₄×C₅	#892	1	1	yes	characteristic
258	C₄×C₅	#935	1	1	yes	characteristic
259	C₄×C₅	#946	1	1	yes	characteristic
260	Dic₅	—	12	2	no	—
261	Dic₅	—	12	1	no	—
262	Dic₅	—	12	1	no	—
263	C₂²×C₄	—	30	1	no	—
264	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
265	M₁₆	—	2	1	no	—
266	M₁₆	—	2	1	no	—
267	M₁₆	—	15	2	no	—
268	M₁₆	—	30	1	no	—
269	C₂×C₈	—	2	1	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	2	1	no	—
272	C₂×C₈	—	15	1	no	—
273	C₂×C₈	—	15	1	no	—
274	C₂×C₈	—	30	1	no	—
275	C₂×C₈	—	30	1	no	—
276	C₂×C₈	—	30	1	no	—
277	C₂×C₈	#1220	1	1	yes	characteristic
278	C₂×C₈	#1220	1	1	yes	characteristic
279	C₄⋊C₄	—	15	1	no	—
280	C₄⋊C₄	—	15	1	no	—
281	C₄⋊C₄	C₄×D₁₅	1	1	yes	characteristic
282	C₄⋊C₄	C₄×D₁₅	1	1	yes	characteristic
283	C₂²⋊C₄	—	2	1	no	—
284	C₂²⋊C₄	—	30	1	no	—
285	C₄²	—	2	1	no	—
286	C₄²	—	30	1	no	—
287	C₄²	—	30	1	no	—
288	C₄²	—	30	1	no	—
289	C₃×C₅	#1922	1	1	yes	characteristic, complemented, Hall
290	C₂²×C₃	—	2	1	no	—
291	C₂²×C₃	—	2	1	no	—
292	C₂²×C₃	#4035	1	1	yes	characteristic
293	C₂²×C₃	#4044	1	1	yes	characteristic
294	C₂²×C₃	C₄×C₂²⋊D₅	1	1	yes	characteristic
295	C₃×C₄	—	4	1	no	—
296	C₃×C₄	—	4	1	no	—
297	C₃×C₄	#4033	1	1	yes	characteristic
298	C₃×C₄	#4038	1	1	yes	characteristic
299	C₃×C₄	#4081	1	1	yes	characteristic
300	C₃×C₄	#4092	1	1	yes	characteristic
301	Dic₃	—	20	2	no	—
302	Dic₃	—	20	1	no	—
303	Dic₃	—	20	1	no	—
304	C₂×C₅	—	2	1	no	—
305	C₂×C₅	—	2	1	no	—
306	C₂×C₅	#4978	1	1	yes	characteristic
307	C₂×C₅	#5144	1	1	yes	characteristic
308	C₂×C₅	#5145	1	1	yes	characteristic
309	C₂³	#6817	1	1	yes	characteristic
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	30	2	no	—
313	C₂×C₄	—	30	1	no	—
314	C₂×C₄	—	30	1	no	—
315	C₂×C₄	—	60	1	no	—
316	C₂×C₄	—	60	1	no	—
317	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
318	C₂×C₄	#6819	1	1	yes	characteristic
319	C₂×C₄	#6819	1	1	yes	characteristic
320	C₂×C₄	Q₈×D₁₅	1	1	yes	characteristic
321	C₂×C₄	#6821	1	1	yes	characteristic
322	C₂×C₄	C₂×#1220	1	1	yes	characteristic
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	—	4	1	no	—
326	C₈	—	30	1	no	—
327	C₈	—	30	1	no	—
328	C₈	—	30	2	no	—
329	C₂×C₃	—	2	1	no	—
330	C₂×C₃	—	2	1	no	—
331	C₂×C₃	#65267	1	1	yes	characteristic
332	C₂×C₃	#65433	1	1	yes	characteristic
333	C₂×C₃	#65434	1	1	yes	characteristic
334	C₅	#70534	1	1	yes	characteristic, complemented, Sylow
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	#91778	1	1	yes	characteristic
338	C₂²	#91787	1	1	yes	characteristic
339	C₂²	C₄×#1220	1	1	yes	characteristic
340	C₄	—	4	1	no	—
341	C₄	—	4	1	no	—
342	C₄	—	60	2	no	—
343	C₄	—	60	1	no	—
344	C₄	—	60	1	no	—
345	C₄	#91776	1	1	yes	characteristic
346	C₄	#91781	1	1	yes	characteristic
347	C₄	#91824	1	1	yes	characteristic
348	C₄	#91835	1	1	yes	characteristic
349	C₃	#10601327	1	1	yes	characteristic, complemented, Sylow
350	C₂	—	2	1	no	—
351	C₂	—	2	1	no	—
352	C₂	#11751079	1	1	yes	characteristic
353	C₂	#11751245	1	1	yes	characteristic
354	C₂	#11751246	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#1220
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅
Fitting	C₃×C₅×#424	C₂
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂³×C₃×C₅	#41
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	#424	D₁₅
hypocenter	C₃×C₅	#1922
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #424

Complements

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

N	H ≅ G/N	type
20. #91092	C₄	semidirect
21. #91092	C₄	semidirect
22. #91091	C₄	semidirect
23. #91091	C₄	semidirect
289. C₃×C₅	#1922	semidirect
334. C₅	#70534	semidirect
349. C₃	#10601327	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#424	2
#11751202	2
#11751099	2
#11746379	2
#11746376	2
#11746371	2
#11746353	2
#10601327	3
#70534	5