Subgroups — #14282516

Order 1920.

432 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11756238	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#423	C₂	1	1	yes	maximal, characteristic, complemented
3	#11751026	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746305	C₂	1	2	yes	maximal, complemented
5	#11746304	C₂	1	2	yes	maximal, complemented
6	#10601292	—	3	1	no	maximal, Hall
7	C₂×C₃×C₅×M₁₆	—	2	1	no	—
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	#91819	C₄	1	1	yes	characteristic
10	#91818	—	2	1	no	—
11	C₄×#1220	—	2	1	no	—
12	#91813	C₄	1	1	yes	characteristic
13	#91765	—	2	1	no	—
14	#91764	C₄	1	1	yes	characteristic
15	C₄×D₆₀	C₂²	1	1	yes	characteristic
16	#91761	—	2	1	no	—
17	#91759	C₄	1	1	yes	characteristic
18	C₄×Dic₃₀	C₂²	1	1	yes	characteristic
19	C₃×C₅×C₄⋊C₈	—	2	1	no	—
20	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
21	#91114	—	2	1	no	—
22	C₄⋊(C₃⋊#168)	—	2	1	no	—
23	C₄×C₃⋊#168	C₂²	1	2	yes	—
24	#70499	—	5	1	no	maximal, Hall
25	#65813	—	3	1	no	—
26	C₅×#423	D₃	1	1	yes	characteristic, complemented
27	#65214	—	3	1	no	—
28	#64704	—	3	2	no	—
29	#64703	—	3	2	no	—
30	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
32	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
33	C₂×#1220	—	2	1	no	—
34	C₂×D₆₀	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×D₁₅	—	2	1	no	—
36	C₂×Dic₃₀	C₂×C₄	1	1	yes	characteristic
37	C₃×C₅×M₁₆	—	4	2	no	—
38	C₂×C₃×C₅×C₈	—	2	2	no	—
39	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
40	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
41	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
43	#6719	—	2	1	no	—
44	#6714	—	2	1	no	—
45	#6714	—	2	1	no	—
46	#6714	—	2	1	no	—
47	#6714	—	2	1	no	—
48	C₄⋊Dic₁₅	C₂×C₄	1	1	yes	characteristic
49	C₄⋊Dic₁₅	C₂×C₄	1	1	yes	characteristic
50	#6712	—	2	1	no	—
51	#6712	—	2	1	no	—
52	#6712	—	2	1	no	—
53	#6712	—	2	1	no	—
54	C₄×Dic₁₅	—	2	1	no	—
55	C₂×C₃⋊#168	—	2	2	no	—
56	#5479	—	5	1	no	—
57	C₃×#423	D₅	1	1	yes	characteristic, complemented
58	#4925	—	5	1	no	—
59	#4484	—	5	2	no	—
60	#4483	—	5	2	no	—
61	C₂×C₅×M₁₆	—	2	1	no	—
62	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
63	#4076	—	3	1	no	—
64	#4075	—	6	1	no	—
65	C₄×C₂²⋊D₅	—	6	1	no	—
66	#4070	—	3	1	no	—
67	#4022	—	6	1	no	—
68	#4021	—	3	1	no	—
69	C₄×D₂₀	—	3	1	no	—
70	#4018	—	6	1	no	—
71	#4016	—	3	1	no	—
72	C₄×Dic₁₀	—	3	1	no	—
73	C₅×C₄⋊C₈	—	2	1	no	—
74	C₅×C₄⋊C₈	C₂×D₃	1	2	yes	—
75	C₂²⋊#168	—	6	1	no	—
76	C₄⋊#168	—	6	1	no	—
77	C₄×#168	—	3	2	no	—
78	#1495	—	15	1	no	Sylow
79	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×D₁₅	—	2	1	no	—
81	C₂×C₃×C₄×C₅	—	2	2	no	—
82	C₂×C₃×C₄×C₅	—	2	1	no	—
83	C₂×C₃×C₄×C₅	—	2	1	no	—
84	C₂×C₃×C₄×C₅	—	4	1	no	—
85	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
90	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
91	#1220	—	4	2	no	—
92	C₂×Dic₁₅	—	2	1	no	—
93	C₂×Dic₁₅	—	2	1	no	—
94	C₂×Dic₁₅	—	2	1	no	—
95	D₆₀	—	2	2	no	—
96	C₄×D₁₅	—	4	1	no	—
97	C₄×D₁₅	—	4	1	no	—
98	Dic₃₀	—	2	2	no	—
99	C₃×C₅×C₈	—	4	2	no	—
100	C₃⋊#168	—	4	2	no	—
101	C₂×C₃×M₁₆	—	2	1	no	—
102	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
103	#930	—	5	1	no	—
104	#929	—	10	1	no	—
105	C₄×#67	—	10	1	no	—
106	#924	—	5	1	no	—
107	#876	—	10	1	no	—
108	#875	—	5	1	no	—
109	C₄×D₁₂	—	5	1	no	—
110	#872	—	10	1	no	—
111	#870	—	5	1	no	—
112	C₄×Dic₆	—	5	1	no	—
113	C₃×C₄⋊C₈	—	2	1	no	—
114	C₃×C₄⋊C₈	C₂×D₅	1	2	yes	—
115	#830	—	10	1	no	—
116	C₄⋊(C₃⋊C₈)	—	10	1	no	—
117	C₄×C₃⋊C₈	—	5	2	no	—
118	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
119	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
120	C₂²×C₄×C₅	#67	1	1	yes	characteristic
121	C₂×C₂²⋊D₅	—	6	1	no	—
122	C₂×D₂₀	—	3	1	no	—
123	C₂×C₄×D₅	—	6	1	no	—
124	C₂×Dic₁₀	—	3	1	no	—
125	C₅×M₁₆	—	4	2	no	—
126	C₂×C₅×C₈	—	2	2	no	—
127	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
128	C₄²×C₅	D₁₂	1	1	yes	characteristic
129	C₄²×C₅	#67	1	1	yes	characteristic
130	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
131	C₂²⋊Dic₅	—	6	1	no	—
132	#685	—	6	1	no	—
133	#685	—	6	1	no	—
134	#685	—	6	1	no	—
135	#685	—	6	1	no	—
136	C₄⋊Dic₅	—	3	1	no	—
137	C₄⋊Dic₅	—	3	1	no	—
138	#683	—	6	1	no	—
139	#683	—	6	1	no	—
140	#683	—	6	1	no	—
141	#683	—	6	1	no	—
142	C₄×Dic₅	—	6	1	no	—
143	C₂×#168	—	6	2	no	—
144	#529	—	15	1	no	—
145	#431	—	15	1	no	—
146	#423	D₁₅	1	1	yes	characteristic, complemented
147	Q₈⋊C₈	—	15	2	no	—
148	D₄⋊C₈	—	15	2	no	—
149	C₂²×C₃×C₅	—	2	1	no	—
150	C₂²×C₃×C₅	—	2	1	no	—
151	C₂²×C₃×C₅	—	2	1	no	—
152	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
153	C₂×D₁₅	—	4	1	no	—
154	C₂×D₁₅	—	4	1	no	—
155	C₂×D₁₅	—	4	1	no	—
156	C₃×C₄×C₅	—	2	1	no	—
157	C₃×C₄×C₅	—	2	1	no	—
158	C₃×C₄×C₅	—	2	1	no	—
159	C₃×C₄×C₅	—	2	1	no	—
160	C₃×C₄×C₅	#131	1	2	yes	—
161	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
162	C₃×C₄×C₅	#137	1	1	yes	characteristic
163	Dic₁₅	—	4	1	no	—
164	Dic₁₅	—	4	1	no	—
165	Dic₁₅	—	4	1	no	—
166	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
167	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
168	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
169	C₂×#67	—	10	1	no	—
170	C₂×D₁₂	—	5	1	no	—
171	C₂×C₄×D₃	—	10	1	no	—
172	C₂×Dic₆	—	5	1	no	—
173	C₃×M₁₆	—	4	2	no	—
174	C₂×C₃×C₈	—	2	2	no	—
175	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
176	C₃×C₄²	D₂₀	1	1	yes	characteristic
177	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
178	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
179	#217	—	10	1	no	—
180	(C₂×D₃)⋊C₄	—	10	1	no	—
181	(C₂×D₃)⋊C₄	—	10	1	no	—
182	(C₂×D₃)⋊C₄	—	10	1	no	—
183	(C₂×D₃)⋊C₄	—	10	1	no	—
184	C₄⋊Dic₃	—	5	1	no	—
185	C₄⋊Dic₃	—	5	1	no	—
186	Dic₃⋊C₄	—	10	1	no	—
187	Dic₃⋊C₄	—	10	1	no	—
188	Dic₃⋊C₄	—	10	1	no	—
189	Dic₃⋊C₄	—	10	1	no	—
190	C₄×Dic₃	—	10	1	no	—
191	C₂×C₃⋊C₈	—	10	2	no	—
192	C₂³×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
193	C₂²×D₅	—	6	1	no	—
194	C₂×C₄×C₅	—	2	2	no	—
195	C₂×C₄×C₅	—	2	1	no	—
196	C₂×C₄×C₅	—	2	1	no	—
197	C₂×C₄×C₅	—	4	1	no	—
198	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
199	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
200	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
201	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
202	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
203	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
204	C₂²⋊D₅	—	12	2	no	—
205	C₂×Dic₅	—	6	1	no	—
206	C₂×Dic₅	—	6	1	no	—
207	C₂×Dic₅	—	6	1	no	—
208	D₂₀	—	6	2	no	—
209	C₄×D₅	—	12	1	no	—
210	C₄×D₅	—	12	1	no	—
211	Dic₁₀	—	6	2	no	—
212	C₅×C₈	—	4	2	no	—
213	#168	—	12	2	no	—
214	C₂×M₁₆	—	2	1	no	—
215	#126	—	30	1	no	—
216	#125	—	15	1	no	—
217	#124	—	15	1	no	—
218	#123	—	30	1	no	—
219	C₂²⋊Q₈	—	15	1	no	—
220	#121	—	15	1	no	—
221	C₄×Q₈	—	15	1	no	—
222	C₄×D₄	—	15	1	no	—
223	C₄×D₄	—	30	1	no	—
224	#117	—	30	1	no	—
225	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
226	C₄⋊C₈	—	2	1	no	—
227	C₄⋊C₈	—	30	1	no	—
228	C₄⋊C₈	C₂×D₁₅	1	2	yes	—
229	C₂²⋊C₈	—	30	1	no	—
230	C₄×C₈	—	15	2	no	—
231	C₂×C₃×C₅	—	4	1	no	—
232	C₂×C₃×C₅	#408	1	1	yes	characteristic
233	C₂×C₃×C₅	#417	1	1	yes	characteristic
234	C₂×C₃×C₅	#421	1	1	yes	characteristic
235	D₁₅	—	8	1	no	—
236	C₂³×C₃	#685	1	1	yes	characteristic
237	C₂²×D₃	—	10	1	no	—
238	C₂×C₃×C₄	—	2	2	no	—
239	C₂×C₃×C₄	—	2	1	no	—
240	C₂×C₃×C₄	—	2	1	no	—
241	C₂×C₃×C₄	—	4	1	no	—
242	C₂×C₃×C₄	#685	1	1	yes	characteristic
243	C₂×C₃×C₄	#685	1	1	yes	characteristic
244	C₂×C₃×C₄	#685	1	1	yes	characteristic
245	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
246	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
247	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
248	#67	—	20	2	no	—
249	C₂×Dic₃	—	10	1	no	—
250	C₂×Dic₃	—	10	1	no	—
251	C₂×Dic₃	—	10	1	no	—
252	D₁₂	—	10	2	no	—
253	C₄×D₃	—	20	1	no	—
254	C₄×D₃	—	20	1	no	—
255	Dic₆	—	10	2	no	—
256	C₃×C₈	—	4	2	no	—
257	C₃⋊C₈	—	20	2	no	—
258	C₂²×C₅	—	2	1	no	—
259	C₂²×C₅	—	2	1	no	—
260	C₂²×C₅	—	2	1	no	—
261	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
262	C₂×D₅	—	12	1	no	—
263	C₂×D₅	—	12	1	no	—
264	C₂×D₅	—	12	1	no	—
265	C₄×C₅	—	2	1	no	—
266	C₄×C₅	—	2	1	no	—
267	C₄×C₅	—	2	1	no	—
268	C₄×C₅	—	2	1	no	—
269	C₄×C₅	#907	1	2	yes	—
270	C₄×C₅	#932	1	1	yes	characteristic
271	C₄×C₅	#942	1	1	yes	characteristic
272	Dic₅	—	12	1	no	—
273	Dic₅	—	12	1	no	—
274	Dic₅	—	12	1	no	—
275	C₂×Q₈	—	15	1	no	—
276	C₂×D₄	—	15	1	no	—
277	C₂×D₄	—	30	1	no	—
278	C₂²×C₄	—	30	1	no	—
279	C₂²×C₄	C₄×D₁₅	1	1	yes	characteristic
280	C₂²×C₄	D₆₀	1	1	yes	characteristic
281	C₂²×C₄	#1220	1	1	yes	characteristic
282	M₁₆	—	4	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	30	2	no	—
285	C₄⋊C₄	—	15	1	no	—
286	C₄⋊C₄	—	15	1	no	—
287	C₄⋊C₄	—	30	1	no	—
288	C₄⋊C₄	—	30	1	no	—
289	C₄⋊C₄	—	30	1	no	—
290	C₄⋊C₄	—	30	1	no	—
291	C₂²⋊C₄	—	30	1	no	—
292	C₂²⋊C₄	—	30	1	no	—
293	C₂²⋊C₄	—	30	1	no	—
294	C₂²⋊C₄	—	30	1	no	—
295	C₂²⋊C₄	—	30	1	no	—
296	C₄²	—	30	1	no	—
297	C₄²	C₄×D₁₅	1	1	yes	characteristic
298	C₄²	D₆₀	1	1	yes	characteristic
299	C₄²	#1220	1	1	yes	characteristic
300	C₄²	C₂²×D₁₅	1	1	yes	characteristic
301	C₃×C₅	#1495	1	1	yes	characteristic, complemented, Hall
302	C₂²×C₃	—	2	1	no	—
303	C₂²×C₃	—	2	1	no	—
304	C₂²×C₃	—	2	1	no	—
305	C₂²×C₃	C₂×#685	1	1	yes	characteristic
306	C₂×D₃	—	20	1	no	—
307	C₂×D₃	—	20	1	no	—
308	C₂×D₃	—	20	1	no	—
309	C₃×C₄	—	2	1	no	—
310	C₃×C₄	—	2	1	no	—
311	C₃×C₄	—	2	1	no	—
312	C₃×C₄	—	2	1	no	—
313	C₃×C₄	#4053	1	2	yes	—
314	C₃×C₄	#4078	1	1	yes	characteristic
315	C₃×C₄	#4088	1	1	yes	characteristic
316	Dic₃	—	20	1	no	—
317	Dic₃	—	20	1	no	—
318	Dic₃	—	20	1	no	—
319	C₂×C₅	—	4	1	no	—
320	C₂×C₅	#4966	1	1	yes	characteristic
321	C₂×C₅	#5129	1	1	yes	characteristic
322	C₂×C₅	#5139	1	1	yes	characteristic
323	D₅	—	24	1	no	—
324	C₂³	—	30	1	no	—
325	C₂³	#6714	1	1	yes	characteristic
326	Q₈	—	30	2	no	—
327	D₄	—	30	2	no	—
328	D₄	—	60	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	30	1	no	—
334	C₂×C₄	—	30	1	no	—
335	C₂×C₄	—	30	1	no	—
336	C₂×C₄	—	60	1	no	—
337	C₂×C₄	—	60	1	no	—
338	C₂×C₄	#6714	1	1	yes	characteristic
339	C₂×C₄	#6714	1	1	yes	characteristic
340	C₂×C₄	#6714	1	1	yes	characteristic
341	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
342	C₂×C₄	C₂×D₆₀	1	1	yes	characteristic
343	C₂×C₄	C₂×#1220	1	1	yes	characteristic
344	C₈	—	4	2	no	—
345	C₈	—	60	2	no	—
346	C₂×C₃	—	4	1	no	—
347	C₂×C₃	#65255	1	1	yes	characteristic
348	C₂×C₃	#65418	1	1	yes	characteristic
349	C₂×C₃	#65428	1	1	yes	characteristic
350	D₃	—	40	1	no	—
351	C₅	#70499	1	1	yes	characteristic, complemented, Sylow
352	C₂²	—	2	1	no	—
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	60	1	no	—
356	C₂²	—	60	1	no	—
357	C₂²	—	60	1	no	—
358	C₂²	C₂×#6714	1	1	yes	characteristic
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	2	1	no	—
363	C₄	—	60	1	no	—
364	C₄	—	60	1	no	—
365	C₄	—	60	1	no	—
366	C₄	#91796	1	2	yes	—
367	C₄	#91821	1	1	yes	characteristic
368	C₄	#91831	1	1	yes	characteristic
369	C₃	#10601292	1	1	yes	characteristic, complemented, Sylow
370	C₂	—	4	1	no	—
371	C₂	—	120	1	no	—
372	C₂	#11751067	1	1	yes	characteristic
373	C₂	#11751230	1	1	yes	characteristic
374	C₂	#11751240	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#1220
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂²×D₁₅
Fitting	C₃×C₅×#423	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	#91819	C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	#423	D₁₅
hypocenter	C₃×C₅	#1495
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#423	C₂	semidirect
3. #11751026	C₂	semidirect
4. #11746305	C₂	semidirect
5. #11746304	C₂	semidirect
26. C₅×#423	D₃	semidirect
57. C₃×#423	D₅	semidirect
146. #423	D₁₅	semidirect
301. C₃×C₅	#1495	semidirect
351. C₅	#70499	semidirect
369. C₃	#10601292	semidirect

Maximal subgroups

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

subgroup	index
#11756238	2
C₃×C₅×#423	2
#11751026	2
#11746305	2
#11746304	2
#10601292	3
#70499	5