Subgroups — #14284878

Order 1920.

520 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11752674	C₂	1	1	yes	maximal, characteristic
2	C₃×#65126	C₂	1	1	yes	maximal, characteristic, complemented
3	D₅×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746306	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4482	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745858	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745854	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599214	—	3	1	no	maximal, Hall
9	C₂×#6773	—	2	1	no	—
10	#91763	C₂²	1	1	yes	characteristic
11	C₅×#869	C₂²	1	1	yes	characteristic
12	C₂×C₃×#676	—	2	1	no	—
13	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
14	#91450	—	2	1	no	—
15	#91450	C₄	1	2	yes	—
16	#91418	—	2	1	no	—
17	#91418	C₄	1	2	yes	—
18	#91375	—	2	1	no	—
19	C₂×D₅×C₃⋊C₈	—	2	1	no	—
20	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
22	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
23	C₃×#3946	—	2	1	no	—
24	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
25	#90982	—	2	1	no	—
26	#70503	—	5	1	no	maximal, Hall
27	#66059	—	3	1	no	—
28	#65126	D₃	1	1	yes	characteristic, complemented
29	D₅×C₄⋊C₈	—	3	1	no	—
30	C₅×#332	—	3	1	no	—
31	#64767	—	3	1	no	—
32	#64705	—	3	1	no	—
33	#64705	—	3	1	no	—
34	C₂×D₆₀	—	2	1	no	—
35	C₂×D₆₀	—	2	1	no	—
36	C₂×D₆₀	C₂×C₄	1	2	yes	—
37	C₂×C₅×Dic₆	—	2	1	no	—
38	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
40	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
41	C₂×#1202	—	2	2	no	—
42	C₂×#1200	—	2	2	no	—
43	#6773	—	4	2	no	—
44	#6773	—	4	2	no	—
45	C₂×C₃×C₅×C₈	—	2	1	no	—
46	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
47	C₂×C₃⋊#168	—	2	1	no	—
48	C₅×C₄⋊Dic₃	—	2	1	no	—
49	C₅×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
50	C₂×C₅×C₃⋊C₈	—	2	1	no	—
51	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
52	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
53	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
54	C₂×C₃×#168	—	2	1	no	—
55	C₃×#676	—	4	2	no	—
56	#6667	—	2	2	no	—
57	#6667	—	2	2	no	—
58	D₅×C₃⋊C₈	—	4	2	no	—
59	#5477	—	5	1	no	—
60	C₃×#423	—	5	1	no	—
61	C₂×C₄⋊(C₃⋊C₈)	—	5	1	no	—
62	#4485	—	5	1	no	—
63	#4485	—	5	1	no	—
64	#4482	—	5	1	no	—
65	#4482	D₅	1	1	yes	characteristic, complemented
66	C₂×#713	—	6	1	no	—
67	C₅×C₄⋊Q₈	—	3	1	no	—
68	#4093	—	6	1	no	—
69	C₂×#676	—	2	1	no	—
70	C₂×C₈×D₅	—	6	1	no	—
71	#4041	—	3	2	no	—
72	#4041	—	6	1	no	—
73	#4039	—	3	2	no	—
74	#4039	—	6	1	no	—
75	#4020	—	3	1	no	—
76	C₄²×D₅	C₂×D₃	1	1	yes	characteristic
77	C₅×C₄⋊C₈	—	3	1	no	—
78	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
79	#3946	—	2	1	no	—
80	#3946	—	6	1	no	—
81	C₄⋊#168	—	3	1	no	—
82	C₄⋊#168	C₂×D₃	1	1	yes	characteristic
83	#1547	—	15	1	no	Sylow
84	C₂²×D₁₅	—	2	2	no	—
85	C₂²×C₃×D₅	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
89	D₆₀	—	4	2	no	—
90	D₆₀	—	4	1	no	—
91	D₆₀	—	4	1	no	—
92	D₆₀	—	4	1	no	—
93	D₆₀	—	4	1	no	—
94	C₂×C₅×Dic₃	—	2	2	no	—
95	C₅×Dic₆	—	4	1	no	—
96	C₅×Dic₆	—	4	1	no	—
97	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
100	C₃×C₄×D₅	—	2	1	no	—
101	C₃×C₄×D₅	—	2	1	no	—
102	C₃×C₄×D₅	—	4	1	no	—
103	C₃×C₄×D₅	D₈	1	2	yes	—
104	C₃×C₄×D₅	SD₁₆	1	2	yes	—
105	#1202	—	4	4	no	—
106	#1200	—	4	4	no	—
107	C₃×C₅×C₈	—	4	1	no	—
108	C₃⋊#168	—	4	1	no	—
109	C₃×#168	—	4	1	no	—
110	C₅×C₃⋊C₈	—	4	1	no	—
111	C₂×#235	—	10	1	no	—
112	C₂×C₃×M₁₆	—	10	1	no	—
113	C₂×C₃×C₄²	—	5	1	no	—
114	#930	—	5	2	no	—
115	#930	—	10	1	no	—
116	C₂²×C₃⋊C₈	—	10	1	no	—
117	#875	—	10	1	no	—
118	#874	—	5	1	no	—
119	C₄×D₁₂	—	5	2	no	—
120	C₄×D₁₂	—	10	1	no	—
121	#869	C₂×D₅	1	1	yes	characteristic
122	C₃×C₄⋊C₈	—	5	1	no	—
123	C₃×C₄⋊C₈	—	10	1	no	—
124	C₃×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
125	C₄⋊(C₃⋊C₈)	—	5	1	no	—
126	C₄⋊(C₃⋊C₈)	—	10	1	no	—
127	C₄⋊(C₃⋊C₈)	C₂×D₅	1	1	yes	characteristic
128	C₂×C₅×Q₈	—	6	1	no	—
129	#713	—	12	2	no	—
130	#713	—	12	2	no	—
131	C₂×D₂₀	—	3	2	no	—
132	C₂×D₂₀	—	6	2	no	—
133	C₂×D₂₀	—	6	1	no	—
134	C₂×D₂₀	—	6	1	no	—
135	C₂×C₄×D₅	—	6	2	no	—
136	C₂×C₄×D₅	C₄×D₃	1	1	yes	characteristic
137	C₂×C₄×D₅	D₁₂	1	1	yes	characteristic
138	C₂×C₄×D₅	#67	1	1	yes	characteristic
139	C₂×C₅×C₈	—	2	1	no	—
140	C₂×C₅×C₈	—	6	1	no	—
141	C₅×C₄⋊C₄	—	3	2	no	—
142	C₅×C₄⋊C₄	—	6	1	no	—
143	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
144	#685	—	6	2	no	—
145	#685	—	6	2	no	—
146	C₄×Dic₅	C₄×D₃	1	1	yes	characteristic
147	C₄×Dic₅	D₁₂	1	1	yes	characteristic
148	C₄×Dic₅	#67	1	1	yes	characteristic
149	C₂×#168	—	2	1	no	—
150	C₂×#168	—	6	1	no	—
151	#676	—	4	2	no	—
152	C₈×D₅	—	12	2	no	—
153	#532	—	15	1	no	—
154	#423	—	5	1	no	—
155	C₂×C₄⋊C₈	—	15	1	no	—
156	#332	—	3	1	no	—
157	#332	—	15	1	no	—
158	#331	—	15	1	no	—
159	#331	—	15	1	no	—
160	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
161	C₂×D₁₅	—	4	2	no	—
162	C₂×D₁₅	—	4	2	no	—
163	C₂×D₁₅	—	4	2	no	—
164	C₂×C₃×D₅	—	2	1	no	—
165	C₂×C₃×D₅	—	2	1	no	—
166	C₂×C₃×D₅	#102	1	2	yes	—
167	C₃×C₄×C₅	—	2	1	no	—
168	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
169	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
170	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
171	C₃×C₄×C₅	#137	1	1	yes	characteristic
172	C₃×Dic₅	—	2	1	no	—
173	C₃×Dic₅	—	2	1	no	—
174	C₃×Dic₅	#102	1	2	yes	—
175	C₅×Dic₃	—	4	2	no	—
176	C₂²×C₃×C₄	—	5	1	no	—
177	C₂²×C₃×C₄	—	5	1	no	—
178	C₂²×C₃×C₄	—	5	1	no	—
179	C₂×#67	—	10	2	no	—
180	#235	—	20	2	no	—
181	#235	—	20	2	no	—
182	C₂×D₁₂	—	5	2	no	—
183	C₂×D₁₂	—	10	1	no	—
184	C₂×D₁₂	—	10	1	no	—
185	C₂×C₄×D₃	—	10	2	no	—
186	C₂×Dic₆	—	2	1	no	—
187	C₃×M₁₆	—	20	2	no	—
188	C₂×C₃×C₈	—	2	1	no	—
189	C₂×C₃×C₈	—	10	1	no	—
190	C₃×C₄²	—	5	1	no	—
191	C₃×C₄²	—	5	1	no	—
192	C₃×C₄²	—	5	1	no	—
193	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
194	(C₂×D₃)⋊C₄	—	10	2	no	—
195	(C₂×D₃)⋊C₄	—	10	2	no	—
196	C₄⋊Dic₃	—	2	1	no	—
197	C₄⋊Dic₃	C₄×D₅	1	2	yes	—
198	C₂×C₃⋊C₈	—	2	1	no	—
199	C₂×C₃⋊C₈	—	10	1	no	—
200	C₂×C₃⋊C₈	—	20	2	no	—
201	C₂²×D₅	—	6	2	no	—
202	C₂²×D₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
203	C₅×Q₈	—	12	1	no	—
204	C₅×Q₈	—	12	1	no	—
205	C₂×C₄×C₅	—	6	2	no	—
206	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
207	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
208	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
209	C₂×Dic₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
210	C₂×Dic₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
211	C₂×Dic₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
212	D₂₀	—	12	2	no	—
213	D₂₀	—	12	4	no	—
214	D₂₀	—	12	1	no	—
215	D₂₀	—	12	1	no	—
216	D₂₀	—	12	1	no	—
217	D₂₀	—	12	1	no	—
218	C₄×D₅	—	2	1	no	—
219	C₄×D₅	—	2	1	no	—
220	C₄×D₅	—	4	1	no	—
221	C₄×D₅	—	12	4	no	—
222	C₄×D₅	#213	1	2	yes	—
223	C₄×D₅	#215	1	2	yes	—
224	C₅×C₈	—	4	1	no	—
225	C₅×C₈	—	12	1	no	—
226	#168	—	4	1	no	—
227	#168	—	12	1	no	—
228	C₂×#41	—	30	1	no	—
229	C₂×M₁₆	—	10	1	no	—
230	C₂²×C₈	—	30	1	no	—
231	C₄⋊Q₈	—	3	1	no	—
232	#127	—	15	1	no	—
233	#124	—	30	1	no	—
234	#121	—	15	2	no	—
235	#121	—	30	1	no	—
236	C₄×D₄	—	15	2	no	—
237	C₄×D₄	—	30	1	no	—
238	C₂×C₄²	—	5	1	no	—
239	C₄⋊C₈	—	3	1	no	—
240	C₄⋊C₈	—	5	1	no	—
241	C₄⋊C₈	—	10	1	no	—
242	C₄⋊C₈	—	15	1	no	—
243	C₄⋊C₈	—	30	1	no	—
244	C₄⋊C₈	D₃×D₅	1	1	yes	characteristic
245	C₂×C₃×C₅	#413	1	1	yes	characteristic
246	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
247	C₂×C₃×C₅	#417	1	1	yes	characteristic
248	D₁₅	—	8	2	no	—
249	C₃×D₅	—	2	2	no	—
250	C₂³×C₃	—	5	1	no	—
251	C₂²×D₃	—	10	2	no	—
252	C₂×C₃×C₄	—	5	1	no	—
253	C₂×C₃×C₄	—	5	2	no	—
254	C₂×C₃×C₄	—	5	2	no	—
255	C₂×C₃×C₄	—	5	1	no	—
256	C₂×C₃×C₄	—	5	1	no	—
257	C₂×C₃×C₄	—	10	1	no	—
258	C₂×C₃×C₄	—	10	1	no	—
259	C₂×C₃×C₄	—	20	1	no	—
260	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
261	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
262	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
263	#67	—	20	4	no	—
264	C₂×Dic₃	—	2	2	no	—
265	D₁₂	—	20	2	no	—
266	D₁₂	—	20	1	no	—
267	D₁₂	—	20	1	no	—
268	D₁₂	—	20	1	no	—
269	D₁₂	—	20	1	no	—
270	C₄×D₃	—	20	4	no	—
271	Dic₆	—	4	1	no	—
272	Dic₆	—	4	1	no	—
273	C₃×C₈	—	4	1	no	—
274	C₃×C₈	—	20	1	no	—
275	C₃⋊C₈	—	4	1	no	—
276	C₃⋊C₈	—	20	1	no	—
277	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
278	C₂×D₅	—	2	1	no	—
279	C₂×D₅	—	2	1	no	—
280	C₂×D₅	—	12	2	no	—
281	C₂×D₅	—	12	2	no	—
282	C₂×D₅	—	12	2	no	—
283	C₂×D₅	#809	1	2	yes	—
284	C₄×C₅	—	2	1	no	—
285	C₄×C₅	—	12	2	no	—
286	C₄×C₅	#908	1	1	yes	characteristic
287	C₄×C₅	#909	1	1	yes	characteristic
288	C₄×C₅	C₂×#213	1	1	yes	characteristic
289	C₄×C₅	C₂×#215	1	1	yes	characteristic
290	Dic₅	—	2	1	no	—
291	Dic₅	—	2	1	no	—
292	Dic₅	#809	1	2	yes	—
293	#41	—	60	2	no	—
294	#41	—	60	2	no	—
295	C₂×Q₈	—	6	1	no	—
296	C₂×D₄	—	15	2	no	—
297	C₂×D₄	—	30	2	no	—
298	C₂×D₄	—	30	1	no	—
299	C₂×D₄	—	30	1	no	—
300	C₂²×C₄	—	5	1	no	—
301	C₂²×C₄	—	5	1	no	—
302	C₂²×C₄	—	5	1	no	—
303	C₂²×C₄	—	30	2	no	—
304	M₁₆	—	20	2	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	6	1	no	—
307	C₂×C₈	—	10	1	no	—
308	C₂×C₈	—	30	1	no	—
309	C₂×C₈	—	60	2	no	—
310	C₄⋊C₄	—	3	2	no	—
311	C₄⋊C₄	—	6	1	no	—
312	C₂²⋊C₄	—	30	2	no	—
313	C₂²⋊C₄	—	30	2	no	—
314	C₄²	—	5	1	no	—
315	C₄²	—	5	1	no	—
316	C₄²	—	5	1	no	—
317	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
318	C₃×C₅	#1547	1	1	yes	characteristic, complemented, Hall
319	C₂²×C₃	—	5	2	no	—
320	C₂²×C₃	—	10	1	no	—
321	C₂²×C₃	—	10	1	no	—
322	C₂²×C₃	D₅×C₂²⋊C₄	1	1	yes	characteristic
323	C₂×D₃	—	20	2	no	—
324	C₂×D₃	—	20	2	no	—
325	C₂×D₃	—	20	2	no	—
326	C₃×C₄	—	2	1	no	—
327	C₃×C₄	—	5	2	no	—
328	C₃×C₄	—	10	1	no	—
329	C₃×C₄	—	10	1	no	—
330	C₃×C₄	D₅×D₈	1	1	yes	characteristic
331	C₃×C₄	D₅×SD₁₆	1	1	yes	characteristic
332	C₃×C₄	#4060	1	1	yes	characteristic
333	C₃×C₄	#4064	1	1	yes	characteristic
334	Dic₃	—	4	2	no	—
335	C₂×C₅	C₂×#809	1	1	yes	characteristic
336	C₂×C₅	#5132	1	1	yes	characteristic
337	C₂×C₅	#5133	1	1	yes	characteristic
338	D₅	—	2	2	no	—
339	D₅	—	24	2	no	—
340	C₂³	—	5	1	no	—
341	C₂³	—	30	2	no	—
342	Q₈	—	12	1	no	—
343	Q₈	—	12	1	no	—
344	D₄	—	60	2	no	—
345	D₄	—	60	4	no	—
346	D₄	—	60	1	no	—
347	D₄	—	60	1	no	—
348	D₄	—	60	1	no	—
349	D₄	—	60	1	no	—
350	C₂×C₄	—	5	1	no	—
351	C₂×C₄	—	5	2	no	—
352	C₂×C₄	—	5	2	no	—
353	C₂×C₄	—	5	1	no	—
354	C₂×C₄	—	5	1	no	—
355	C₂×C₄	—	6	2	no	—
356	C₂×C₄	—	10	1	no	—
357	C₂×C₄	—	10	1	no	—
358	C₂×C₄	—	20	1	no	—
359	C₂×C₄	—	60	4	no	—
360	C₂×C₄	C₄×D₃×D₅	1	1	yes	characteristic
361	C₂×C₄	D₅×D₁₂	1	1	yes	characteristic
362	C₂×C₄	D₅×#67	1	1	yes	characteristic
363	C₈	—	4	1	no	—
364	C₈	—	12	1	no	—
365	C₈	—	20	1	no	—
366	C₈	—	60	1	no	—
367	C₂×C₃	—	10	2	no	—
368	C₂×C₃	#65040	1	1	yes	characteristic
369	C₂×C₃	D₅×#102	1	1	yes	characteristic
370	C₂×C₃	#65092	1	1	yes	characteristic
371	D₃	—	40	2	no	—
372	C₅	#70503	1	1	yes	characteristic, complemented, Sylow
373	C₂²	—	5	2	no	—
374	C₂²	—	10	1	no	—
375	C₂²	—	10	1	no	—
376	C₂²	—	60	2	no	—
377	C₂²	—	60	2	no	—
378	C₂²	—	60	2	no	—
379	C₂²	D₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
380	C₄	—	2	1	no	—
381	C₄	—	5	2	no	—
382	C₄	—	10	1	no	—
383	C₄	—	10	1	no	—
384	C₄	—	12	2	no	—
385	C₄	#91249	1	1	yes	characteristic
386	C₄	#91261	1	1	yes	characteristic
387	C₄	D₅×#213	1	1	yes	characteristic
388	C₄	D₅×#215	1	1	yes	characteristic
389	C₃	#10599214	1	1	yes	characteristic, complemented, Sylow
390	C₂	—	10	2	no	—
391	C₂	—	120	2	no	—
392	C₂	D₅×#809	1	1	yes	characteristic
393	C₂	#11747606	1	1	yes	characteristic
394	C₂	#11748231	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₅×(C₂×D₃)⋊C₄
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₃×C₅×C₄⋊C₈	C₂²
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	#11752674	C₂
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₄⋊C₈	D₃×D₅
hypocenter	C₃×C₅	#1547
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₃×#65126	C₂	semidirect
3. D₅×C₄⋊(C₃⋊C₈)	C₂	semidirect
4. #11746306	C₂	semidirect
5. C₅×#4482	C₂	semidirect
6. #11745858	C₂	semidirect
7. #11745854	C₂	semidirect
28. #65126	D₃	semidirect
65. #4482	D₅	semidirect
318. C₃×C₅	#1547	semidirect
372. C₅	#70503	semidirect
389. C₃	#10599214	semidirect

Maximal subgroups

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

subgroup	index
#11752674	2
C₃×#65126	2
D₅×C₄⋊(C₃⋊C₈)	2
#11746306	2
C₅×#4482	2
#11745858	2
#11745854	2
#10599214	3
#70503	5