Subgroups — #14300384

Order 1920.

520 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11754120	C₂	1	1	yes	maximal, characteristic
2	C₅×#4999	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#65295	C₂	1	1	yes	maximal, characteristic
4	#11748432	C₂	1	1	yes	maximal, characteristic, complemented
5	#11748342	C₂	1	1	yes	maximal, characteristic
6	#11746354	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745992	C₂	1	1	yes	maximal, characteristic, complemented
8	#10601339	—	3	1	no	maximal, Hall
9	C₂×C₃×#709	C₂²	1	1	yes	characteristic
10	C₃×C₅×#117	C₂²	1	1	yes	characteristic
11	C₂×#6710	C₂²	1	1	yes	characteristic
12	C₂×#6710	C₂²	1	1	yes	characteristic
13	C₂×C₅×#208	C₂²	1	1	yes	characteristic
14	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₃×#4072	C₂²	1	1	yes	characteristic
16	#91284	—	2	2	no	—
17	#91283	—	2	2	no	—
18	D₅×#208	—	2	2	no	—
19	C₅×C₄⋊(C₃⋊C₈)	—	2	2	no	—
20	C₃×#3938	C₄	1	4	yes	—
21	#90982	—	2	2	no	—
22	#90955	—	2	2	no	—
23	#70044	—	5	1	no	maximal, Hall
24	#66079	—	3	1	no	—
25	C₅×#424	—	3	1	no	—
26	#65399	—	3	1	no	—
27	#65391	—	3	1	no	—
28	#65295	D₃	1	1	yes	characteristic
29	#64776	—	3	1	no	—
30	#64753	—	3	1	no	—
31	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂×C₃×C₂²⋊D₅	C₂×C₄	1	2	yes	—
33	C₃×#709	—	2	1	no	—
34	C₃×#709	—	2	1	no	—
35	C₃×#709	—	2	1	no	—
36	C₃×#709	—	2	1	no	—
37	C₂×C₃×D₂₀	Q₈	1	1	yes	characteristic
38	C₂×C₃×C₄×D₅	D₄	1	2	yes	—
39	C₂×C₃×Dic₁₀	Q₈	1	1	yes	characteristic
40	C₃×C₅×C₄⋊C₄	—	2	1	no	—
41	C₃×C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
42	C₃×C₄²×C₅	—	2	1	no	—
43	#6710	—	2	1	no	—
44	#6710	—	2	1	no	—
45	#6710	—	2	1	no	—
46	#6710	—	2	1	no	—
47	C₂×C₃⋊#168	—	2	1	no	—
48	C₂×C₃⋊#168	D₄	1	2	yes	—
49	C₅×#208	—	2	1	no	—
50	C₅×#208	—	2	1	no	—
51	C₂×C₅×C₃⋊C₈	—	2	1	no	—
52	C₂×C₅×C₃⋊C₈	—	2	1	no	—
53	C₂×C₅×C₃⋊C₈	—	2	1	no	—
54	C₂×C₅×C₃⋊C₈	D₄	1	2	yes	—
55	C₃×C₂²⋊Dic₅	C₂×C₄	1	2	yes	—
56	C₃×C₄⋊Dic₅	—	2	1	no	—
57	C₃×C₄×Dic₅	—	2	1	no	—
58	#6649	—	2	4	no	—
59	D₅×C₃⋊C₈	—	2	4	no	—
60	#5819	—	5	1	no	—
61	C₃×#410	—	5	1	no	—
62	#5234	—	5	1	no	—
63	#4999	—	5	1	no	—
64	#4999	D₅	1	1	yes	characteristic, complemented
65	#4533	—	5	1	no	—
66	#4533	—	5	1	no	—
67	C₂×#709	C₂×D₃	1	1	yes	characteristic
68	C₂×C₅×M₁₆	—	3	1	no	—
69	C₂²×C₅×C₈	—	3	1	no	—
70	C₅×#117	C₂×D₃	1	1	yes	characteristic
71	#4072	C₂×D₃	1	1	yes	characteristic
72	C₂×#681	—	3	1	no	—
73	C₂×#681	—	3	1	no	—
74	#4053	—	6	2	no	—
75	D₅×M₁₆	—	6	2	no	—
76	#4047	—	6	2	no	—
77	C₅×C₄⋊C₈	—	6	2	no	—
78	#3952	—	6	2	no	—
79	#3946	—	6	2	no	—
80	#3938	Dic₃	1	4	yes	—
81	#1868	—	15	1	no	Sylow
82	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₃×D₅	C₄⋊C₄	1	2	yes	—
84	C₂×C₃×C₄×C₅	—	2	2	no	—
85	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
90	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
91	C₃×C₂²⋊D₅	—	2	2	no	—
92	C₃×C₂²⋊D₅	—	2	2	no	—
93	C₂×C₃×Dic₅	—	2	2	no	—
94	C₂×C₃×Dic₅	C₄⋊C₄	1	2	yes	—
95	C₃×D₂₀	—	2	1	no	—
96	C₃×D₂₀	—	2	1	no	—
97	C₃×C₄×D₅	—	2	2	no	—
98	C₃×C₄×D₅	—	2	2	no	—
99	C₃×Dic₁₀	—	2	1	no	—
100	C₃×Dic₁₀	—	2	1	no	—
101	C₃⋊#168	—	2	2	no	—
102	C₃⋊#168	—	4	1	no	—
103	C₅×C₃⋊C₈	—	2	2	no	—
104	C₅×C₃⋊C₈	—	4	1	no	—
105	C₂×C₃×#41	—	5	1	no	—
106	C₃×#117	—	5	1	no	—
107	C₃×#117	C₂×D₅	1	1	yes	characteristic
108	#948	—	10	2	no	—
109	#948	—	10	2	no	—
110	C₂×#208	—	5	1	no	—
111	C₂×#208	—	5	1	no	—
112	C₂×#208	—	10	2	no	—
113	C₂×#208	C₂×D₅	1	1	yes	characteristic
114	C₂²×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
115	C₃×#99	—	5	4	no	—
116	#830	—	10	2	no	—
117	C₄⋊(C₃⋊C₈)	—	2	2	no	—
118	C₄⋊(C₃⋊C₈)	—	10	2	no	—
119	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
120	C₂×C₂²⋊D₅	C₂×Dic₃	1	2	yes	—
121	#709	—	2	1	no	—
122	#709	—	2	1	no	—
123	#709	—	2	1	no	—
124	#709	—	2	1	no	—
125	C₂×D₂₀	Dic₆	1	1	yes	characteristic
126	C₂×C₄×D₅	D₁₂	1	2	yes	—
127	C₂×Dic₁₀	Dic₆	1	1	yes	characteristic
128	C₅×M₁₆	—	6	1	no	—
129	C₅×M₁₆	—	6	1	no	—
130	C₂×C₅×C₈	—	3	2	no	—
131	C₂×C₅×C₈	—	6	1	no	—
132	C₂×C₅×C₈	—	6	1	no	—
133	C₂×C₅×C₈	—	6	1	no	—
134	C₅×C₄⋊C₄	—	2	1	no	—
135	C₅×C₂²⋊C₄	C₂×Dic₃	1	2	yes	—
136	C₄²×C₅	—	2	1	no	—
137	C₂²⋊Dic₅	C₂×Dic₃	1	2	yes	—
138	C₄⋊Dic₅	—	2	1	no	—
139	C₄×Dic₅	—	2	1	no	—
140	#681	—	6	1	no	—
141	#681	—	6	1	no	—
142	#681	—	6	1	no	—
143	#681	—	6	1	no	—
144	C₂×#168	—	3	2	no	—
145	C₂×#168	—	6	1	no	—
146	#676	—	6	4	no	—
147	C₈×D₅	—	6	4	no	—
148	#568	—	15	1	no	—
149	#424	—	3	1	no	—
150	#424	—	15	1	no	—
151	#410	—	5	1	no	—
152	#408	—	15	1	no	—
153	#344	—	15	1	no	—
154	#344	—	15	1	no	—
155	C₂²×C₃×C₅	—	4	1	no	—
156	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
157	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
158	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
159	C₂×C₃×D₅	—	2	2	no	—
160	C₂×C₃×D₅	—	4	2	no	—
161	C₃×C₄×C₅	—	4	2	no	—
162	C₃×C₄×C₅	#120	1	1	yes	characteristic
163	C₃×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
164	C₃×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
165	C₃×C₄×C₅	#123	1	1	yes	characteristic
166	C₃×Dic₅	—	2	2	no	—
167	C₃×Dic₅	—	4	2	no	—
168	C₃×#41	—	10	1	no	—
169	C₃×#41	—	10	1	no	—
170	C₃×#41	—	10	1	no	—
171	C₃×#41	—	10	1	no	—
172	C₂×C₃×Q₈	—	5	1	no	—
173	C₂×C₃×D₄	—	5	2	no	—
174	C₂×C₃×D₄	—	5	1	no	—
175	C₂²×C₃×C₄	—	5	2	no	—
176	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
177	C₃×C₄⋊C₄	—	2	1	no	—
178	C₃×C₄⋊C₄	—	10	1	no	—
179	C₃×C₂²⋊C₄	—	5	2	no	—
180	C₃×C₂²⋊C₄	C₄×D₅	1	2	yes	—
181	C₃×C₄²	—	2	1	no	—
182	C₃×C₄²	—	10	1	no	—
183	#208	—	2	1	no	—
184	#208	—	2	1	no	—
185	#208	—	10	4	no	—
186	#208	—	10	1	no	—
187	#208	—	10	1	no	—
188	#208	—	10	1	no	—
189	#208	—	10	1	no	—
190	C₂×C₃⋊C₈	—	2	1	no	—
191	C₂×C₃⋊C₈	—	2	1	no	—
192	C₂×C₃⋊C₈	—	2	1	no	—
193	C₂×C₃⋊C₈	—	5	2	no	—
194	C₂×C₃⋊C₈	—	10	4	no	—
195	C₂×C₃⋊C₈	—	10	1	no	—
196	C₂×C₃⋊C₈	C₂²⋊D₅	1	2	yes	—
197	C₂³×C₅	C₂²×Dic₃	1	1	yes	characteristic
198	C₂²×D₅	C₄⋊Dic₃	1	2	yes	—
199	C₂×C₄×C₅	—	2	2	no	—
200	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
201	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
202	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
203	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
204	C₂×C₄×C₅	#237	1	1	yes	characteristic
205	C₂×C₄×C₅	#237	1	1	yes	characteristic
206	C₂²⋊D₅	—	2	2	no	—
207	C₂²⋊D₅	—	2	2	no	—
208	C₂×Dic₅	—	2	2	no	—
209	C₂×Dic₅	C₄⋊Dic₃	1	2	yes	—
210	D₂₀	—	2	1	no	—
211	D₂₀	—	2	1	no	—
212	C₄×D₅	—	2	2	no	—
213	C₄×D₅	—	2	2	no	—
214	Dic₁₀	—	2	1	no	—
215	Dic₁₀	—	2	1	no	—
216	C₅×C₈	—	6	2	no	—
217	C₅×C₈	—	12	1	no	—
218	#168	—	6	2	no	—
219	#168	—	12	1	no	—
220	C₂×#41	—	5	1	no	—
221	#131	—	30	2	no	—
222	#131	—	30	2	no	—
223	C₂×M₁₆	—	3	1	no	—
224	C₂×M₁₆	—	15	1	no	—
225	C₂×M₁₆	—	15	1	no	—
226	C₂×M₁₆	—	30	2	no	—
227	C₂²×C₈	—	3	1	no	—
228	#117	—	5	1	no	—
229	#117	D₃×D₅	1	1	yes	characteristic
230	C₄⋊C₈	—	6	2	no	—
231	C₄⋊C₈	—	30	2	no	—
232	#99	—	5	4	no	—
233	C₂²⋊C₈	—	30	2	no	—
234	C₂×C₃×C₅	—	2	1	no	—
235	C₂×C₃×C₅	—	2	1	no	—
236	C₂×C₃×C₅	—	2	1	no	—
237	C₂×C₃×C₅	#385	1	1	yes	characteristic
238	C₃×D₅	—	4	2	no	—
239	C₂³×C₃	—	5	2	no	—
240	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
241	C₃×Q₈	—	10	1	no	—
242	C₃×Q₈	—	10	1	no	—
243	C₃×D₄	—	10	2	no	—
244	C₃×D₄	—	10	2	no	—
245	C₃×D₄	—	10	1	no	—
246	C₃×D₄	—	10	1	no	—
247	C₂×C₃×C₄	—	2	2	no	—
248	C₂×C₃×C₄	—	5	2	no	—
249	C₂×C₃×C₄	—	10	2	no	—
250	C₂×C₃×C₄	—	10	2	no	—
251	C₂×C₃×C₄	—	10	2	no	—
252	C₂×C₃×C₄	#709	1	1	yes	characteristic
253	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
254	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
255	C₂×C₃×C₄	Q₈×D₅	1	1	yes	characteristic
256	C₂×C₃×C₄	#713	1	1	yes	characteristic
257	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
258	C₃⋊C₈	—	2	2	no	—
259	C₃⋊C₈	—	4	1	no	—
260	C₃⋊C₈	—	10	2	no	—
261	C₃⋊C₈	—	20	1	no	—
262	C₂²×C₅	—	4	1	no	—
263	C₂²×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
264	C₂²×C₅	D₄×Dic₃	1	1	yes	characteristic
265	C₂²×C₅	D₄×Dic₃	1	1	yes	characteristic
266	C₂×D₅	—	2	2	no	—
267	C₂×D₅	—	4	2	no	—
268	C₄×C₅	—	4	2	no	—
269	C₄×C₅	#878	1	1	yes	characteristic
270	C₄×C₅	#878	1	1	yes	characteristic
271	C₄×C₅	#882	1	1	yes	characteristic
272	C₄×C₅	#886	1	1	yes	characteristic
273	Dic₅	—	2	2	no	—
274	Dic₅	—	4	2	no	—
275	#41	—	10	1	no	—
276	#41	—	10	1	no	—
277	#41	—	10	1	no	—
278	#41	—	10	1	no	—
279	C₂×Q₈	—	5	1	no	—
280	C₂×D₄	—	5	2	no	—
281	C₂×D₄	—	5	1	no	—
282	C₂²×C₄	—	5	2	no	—
283	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
284	M₁₆	—	6	1	no	—
285	M₁₆	—	6	1	no	—
286	M₁₆	—	30	4	no	—
287	M₁₆	—	30	1	no	—
288	M₁₆	—	30	1	no	—
289	M₁₆	—	30	1	no	—
290	M₁₆	—	30	1	no	—
291	C₂×C₈	—	3	2	no	—
292	C₂×C₈	—	6	1	no	—
293	C₂×C₈	—	6	1	no	—
294	C₂×C₈	—	6	1	no	—
295	C₂×C₈	—	15	2	no	—
296	C₂×C₈	—	30	4	no	—
297	C₂×C₈	—	30	1	no	—
298	C₄⋊C₄	—	2	1	no	—
299	C₄⋊C₄	—	10	1	no	—
300	C₂²⋊C₄	—	5	2	no	—
301	C₂²⋊C₄	D₅×Dic₃	1	2	yes	—
302	C₄²	—	2	1	no	—
303	C₄²	—	10	1	no	—
304	C₃×C₅	#1868	1	1	yes	characteristic, complemented, Hall
305	C₂²×C₃	—	4	1	no	—
306	C₂²×C₃	—	10	2	no	—
307	C₂²×C₃	—	20	2	no	—
308	C₂²×C₃	D₅×C₄⋊C₄	1	1	yes	characteristic
309	C₂²×C₃	#4039	1	1	yes	characteristic
310	C₂²×C₃	C₄×C₂²⋊D₅	1	1	yes	characteristic
311	C₃×C₄	—	4	2	no	—
312	C₃×C₄	—	10	2	no	—
313	C₃×C₄	—	20	2	no	—
314	C₃×C₄	#4040	1	1	yes	characteristic
315	C₃×C₄	#4042	1	1	yes	characteristic
316	C₃×C₄	#4083	1	1	yes	characteristic
317	C₃×C₄	#4092	1	1	yes	characteristic
318	C₂×C₅	—	2	1	no	—
319	C₂×C₅	—	2	1	no	—
320	C₂×C₅	—	2	1	no	—
321	C₂×C₅	#4950	1	1	yes	characteristic
322	D₅	—	4	2	no	—
323	C₂³	—	5	2	no	—
324	C₂³	C₂×D₅×Dic₃	1	1	yes	characteristic
325	Q₈	—	10	1	no	—
326	Q₈	—	10	1	no	—
327	D₄	—	10	2	no	—
328	D₄	—	10	2	no	—
329	D₄	—	10	1	no	—
330	D₄	—	10	1	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	5	2	no	—
333	C₂×C₄	—	10	2	no	—
334	C₂×C₄	—	10	2	no	—
335	C₂×C₄	—	10	2	no	—
336	C₂×C₄	D₅×Dic₆	1	1	yes	characteristic
337	C₂×C₄	#6766	1	1	yes	characteristic
338	C₂×C₄	D₅×D₁₂	1	1	yes	characteristic
339	C₂×C₄	#6777	1	1	yes	characteristic
340	C₂×C₄	#6779	1	1	yes	characteristic
341	C₂×C₄	D₃×C₂²⋊D₅	1	1	yes	characteristic
342	C₈	—	6	2	no	—
343	C₈	—	12	1	no	—
344	C₈	—	30	2	no	—
345	C₈	—	60	1	no	—
346	C₂×C₃	—	2	1	no	—
347	C₂×C₃	—	2	1	no	—
348	C₂×C₃	—	2	1	no	—
349	C₂×C₃	—	20	2	no	—
350	C₂×C₃	#65278	1	1	yes	characteristic
351	C₅	#70044	1	1	yes	characteristic, complemented, Sylow
352	C₂²	—	4	1	no	—
353	C₂²	—	10	2	no	—
354	C₂²	—	20	2	no	—
355	C₂²	D₅×C₄⋊Dic₃	1	1	yes	characteristic
356	C₂²	#91434	1	1	yes	characteristic
357	C₂²	Dic₃×C₂²⋊D₅	1	1	yes	characteristic
358	C₄	—	4	2	no	—
359	C₄	—	10	2	no	—
360	C₄	—	20	2	no	—
361	C₄	#91413	1	1	yes	characteristic
362	C₄	#91421	1	1	yes	characteristic
363	C₄	#91430	1	1	yes	characteristic
364	C₄	#91475	1	1	yes	characteristic
365	C₃	#10601339	1	1	yes	characteristic, complemented, Sylow
366	C₂	—	2	1	no	—
367	C₂	—	2	1	no	—
368	C₂	—	2	1	no	—
369	C₂	—	20	2	no	—
370	C₂	#11748899	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#91475
commutator	C₂³×C₃×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃×D₅
Fitting	C₃×C₅×#117	C₂²
socle	C₂×C₃×C₅	#385
prime radical	C₂×C₃×#709	C₂²
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	#117	D₃×D₅
hypocenter	C₃×C₅	#1868
LCS	G ▷ C₂³×C₃×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
2. C₅×#4999	C₂	semidirect
4. #11748432	C₂	semidirect
6. #11746354	C₂	semidirect
7. #11745992	C₂	semidirect
64. #4999	D₅	semidirect
304. C₃×C₅	#1868	semidirect
351. C₅	#70044	semidirect
365. C₃	#10601339	semidirect

Maximal subgroups

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

subgroup	index
#11754120	2
C₅×#4999	2
C₃×#65295	2
#11748432	2
#11748342	2
#11746354	2
#11745992	2
#10601339	3
#70044	5