Subgroups — #14284830

Order 1920.

480 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅×#869	C₂	1	1	yes	maximal, characteristic
2	C₃×#65126	C₂	1	1	yes	maximal, characteristic
3	#11746869	C₂	1	1	yes	maximal, characteristic
4	#11746303	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4482	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745860	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745858	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599213	—	3	1	no	maximal, Hall
9	C₂×D₅×Dic₆	—	2	1	no	—
10	#91758	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	#91468	—	2	1	no	—
15	#91468	C₄	1	2	yes	—
16	D₅×C₄⋊Dic₃	—	2	1	no	—
17	D₅×C₄⋊Dic₃	C₄	1	2	yes	—
18	#91376	—	2	1	no	—
19	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
21	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
22	C₃×#3946	—	2	1	no	—
23	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
24	#90955	—	2	1	no	—
25	Dic₅×C₃⋊C₈	—	2	1	no	—
26	#70506	—	5	1	no	maximal, Hall
27	D₅×C₄⋊Q₈	—	3	1	no	—
28	#65126	D₃	1	1	yes	characteristic
29	#65122	—	3	1	no	—
30	C₅×#332	—	3	1	no	—
31	#64767	—	3	1	no	—
32	#64706	—	3	1	no	—
33	#64702	—	3	1	no	—
34	C₂×Dic₃₀	—	2	1	no	—
35	C₂×C₅×Dic₆	—	2	1	no	—
36	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
38	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
39	C₂×#1204	—	2	2	no	—
40	C₂×D₅×Dic₃	—	2	2	no	—
41	D₅×Dic₆	—	4	2	no	—
42	D₅×Dic₆	—	4	2	no	—
43	C₂×C₃×C₅×C₈	—	2	1	no	—
44	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
45	C₄⋊Dic₁₅	—	2	1	no	—
46	C₄⋊Dic₁₅	C₂×C₄	1	2	yes	—
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	#6668	—	2	2	no	—
57	#6668	—	2	2	no	—
58	C₂×#869	—	5	1	no	—
59	C₃×#423	—	5	1	no	—
60	#4925	—	5	1	no	—
61	#4482	—	5	1	no	—
62	#4482	—	5	1	no	—
63	#4482	—	5	1	no	—
64	#4482	D₅	1	1	yes	characteristic, complemented
65	C₂×Q₈×D₅	—	6	1	no	—
66	C₅×C₄⋊Q₈	—	3	1	no	—
67	#4090	—	6	1	no	—
68	C₂×#676	—	2	1	no	—
69	D₅×C₄⋊C₄	—	3	2	no	—
70	D₅×C₄⋊C₄	—	6	1	no	—
71	#4034	—	3	2	no	—
72	#4034	—	6	1	no	—
73	C₄²×D₅	C₂×D₃	1	1	yes	characteristic
74	#4015	—	3	1	no	—
75	C₅×C₄⋊C₈	—	3	1	no	—
76	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
77	#3952	—	6	1	no	—
78	#3946	—	2	1	no	—
79	C₈×Dic₅	—	6	1	no	—
80	C₄⋊#168	—	3	1	no	—
81	C₄⋊#168	C₂×D₃	1	1	yes	characteristic
82	#1546	—	15	1	no	Sylow
83	C₂²×C₃×D₅	C₂²⋊C₄	1	1	yes	characteristic
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×D₄	1	1	yes	characteristic
87	C₂×Dic₁₅	—	2	2	no	—
88	Dic₃₀	—	4	1	no	—
89	Dic₃₀	—	4	1	no	—
90	C₂×C₅×Dic₃	—	2	2	no	—
91	C₅×Dic₆	—	4	1	no	—
92	C₅×Dic₆	—	4	1	no	—
93	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
96	C₃×C₄×D₅	—	2	1	no	—
97	C₃×C₄×D₅	—	2	1	no	—
98	C₃×C₄×D₅	—	2	2	no	—
99	C₃×C₄×D₅	D₈	1	2	yes	—
100	C₃×C₄×D₅	SD₁₆	1	2	yes	—
101	#1204	—	4	4	no	—
102	D₅×Dic₃	—	4	4	no	—
103	C₃×C₅×C₈	—	4	1	no	—
104	C₃⋊#168	—	4	1	no	—
105	C₃×#168	—	4	1	no	—
106	C₅×C₃⋊C₈	—	4	1	no	—
107	C₂²×Dic₆	—	10	1	no	—
108	C₂×C₃×M₁₆	—	10	1	no	—
109	C₂×C₃×C₄²	—	5	1	no	—
110	C₂×C₄⋊Dic₃	—	5	2	no	—
111	C₂×C₄⋊Dic₃	—	10	1	no	—
112	#869	—	5	1	no	—
113	#869	—	5	2	no	—
114	#869	—	10	1	no	—
115	#869	—	10	1	no	—
116	#869	C₂×D₅	1	1	yes	characteristic
117	C₃×C₄⋊C₈	—	5	1	no	—
118	C₃×C₄⋊C₈	—	10	1	no	—
119	C₃×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
120	#830	—	10	1	no	—
121	C₄⋊(C₃⋊C₈)	—	5	1	no	—
122	C₄⋊(C₃⋊C₈)	C₂×D₅	1	1	yes	characteristic
123	C₄×C₃⋊C₈	—	10	1	no	—
124	C₂×C₅×Q₈	—	6	1	no	—
125	Q₈×D₅	—	12	2	no	—
126	Q₈×D₅	—	12	2	no	—
127	C₂×C₄×D₅	—	6	2	no	—
128	C₂×C₄×D₅	C₄×D₃	1	1	yes	characteristic
129	C₂×C₄×D₅	D₁₂	1	1	yes	characteristic
130	C₂×C₄×D₅	#67	1	1	yes	characteristic
131	C₂×Dic₁₀	—	6	1	no	—
132	C₂×Dic₁₀	—	6	2	no	—
133	C₂×C₅×C₈	—	2	1	no	—
134	C₂×C₅×C₈	—	6	1	no	—
135	C₅×C₄⋊C₄	—	3	2	no	—
136	C₅×C₄⋊C₄	—	6	1	no	—
137	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
138	C₄⋊Dic₅	—	3	2	no	—
139	C₄⋊Dic₅	—	6	1	no	—
140	#683	—	6	2	no	—
141	#683	—	6	2	no	—
142	C₄×Dic₅	C₄×D₃	1	1	yes	characteristic
143	C₄×Dic₅	D₁₂	1	1	yes	characteristic
144	C₄×Dic₅	#67	1	1	yes	characteristic
145	C₂×#168	—	2	1	no	—
146	C₂×#168	—	6	1	no	—
147	#676	—	4	2	no	—
148	C₂×C₄⋊Q₈	—	15	1	no	—
149	#431	—	15	1	no	—
150	#423	—	5	1	no	—
151	#332	—	3	1	no	—
152	#332	—	15	1	no	—
153	#332	—	15	1	no	—
154	#332	—	15	1	no	—
155	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
156	C₂×C₃×D₅	—	2	1	no	—
157	C₂×C₃×D₅	—	2	1	no	—
158	C₂×C₃×D₅	#102	1	2	yes	—
159	C₃×C₄×C₅	—	2	1	no	—
160	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
161	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
162	C₃×C₄×C₅	#137	1	1	yes	characteristic
163	C₃×C₄×C₅	#137	1	1	yes	characteristic
164	Dic₁₅	—	4	2	no	—
165	C₃×Dic₅	—	2	1	no	—
166	C₃×Dic₅	#101	1	2	yes	—
167	C₃×Dic₅	#102	1	2	yes	—
168	C₅×Dic₃	—	4	2	no	—
169	C₂²×C₃×C₄	—	5	1	no	—
170	C₂²×C₃×C₄	—	5	1	no	—
171	C₂²×C₃×C₄	—	5	1	no	—
172	C₂²×Dic₃	—	10	2	no	—
173	C₂×Dic₆	—	2	1	no	—
174	C₂×Dic₆	—	10	2	no	—
175	C₂×Dic₆	—	10	1	no	—
176	C₂×Dic₆	—	20	2	no	—
177	C₂×Dic₆	—	20	2	no	—
178	C₃×M₁₆	—	20	2	no	—
179	C₂×C₃×C₈	—	2	1	no	—
180	C₂×C₃×C₈	—	10	1	no	—
181	C₃×C₄²	—	5	1	no	—
182	C₃×C₄²	—	5	1	no	—
183	C₃×C₄²	—	5	1	no	—
184	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
185	C₄⋊Dic₃	—	2	1	no	—
186	C₄⋊Dic₃	—	5	2	no	—
187	C₄⋊Dic₃	—	10	2	no	—
188	C₄⋊Dic₃	—	10	2	no	—
189	C₄⋊Dic₃	—	10	1	no	—
190	C₄⋊Dic₃	C₄×D₅	1	2	yes	—
191	C₂×C₃⋊C₈	—	2	1	no	—
192	C₂×C₃⋊C₈	—	10	1	no	—
193	C₂²×D₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
194	C₅×Q₈	—	12	1	no	—
195	C₅×Q₈	—	12	1	no	—
196	C₂×C₄×C₅	—	6	2	no	—
197	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
198	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
199	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
200	C₂×Dic₅	—	6	2	no	—
201	C₂×Dic₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
202	C₂×Dic₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
203	C₂×Dic₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
204	C₄×D₅	—	2	1	no	—
205	C₄×D₅	—	2	1	no	—
206	C₄×D₅	—	2	2	no	—
207	C₄×D₅	—	12	4	no	—
208	C₄×D₅	#213	1	2	yes	—
209	C₄×D₅	#215	1	2	yes	—
210	Dic₁₀	—	12	1	no	—
211	Dic₁₀	—	12	1	no	—
212	Dic₁₀	—	12	4	no	—
213	C₅×C₈	—	4	1	no	—
214	C₅×C₈	—	12	1	no	—
215	#168	—	4	1	no	—
216	#168	—	12	1	no	—
217	C₂²×Q₈	—	30	1	no	—
218	C₂×M₁₆	—	10	1	no	—
219	C₄⋊Q₈	—	3	1	no	—
220	C₄⋊Q₈	—	15	2	no	—
221	C₄⋊Q₈	—	15	1	no	—
222	C₄⋊Q₈	—	30	1	no	—
223	C₄⋊Q₈	—	30	1	no	—
224	C₂×C₄⋊C₄	—	15	2	no	—
225	C₂×C₄⋊C₄	—	30	1	no	—
226	C₂×C₄²	—	5	1	no	—
227	C₄⋊C₈	—	3	1	no	—
228	C₄⋊C₈	—	5	1	no	—
229	C₄⋊C₈	—	10	1	no	—
230	C₄⋊C₈	—	15	1	no	—
231	C₄⋊C₈	D₃×D₅	1	1	yes	characteristic
232	C₂²⋊C₈	—	30	1	no	—
233	C₄×C₈	—	30	1	no	—
234	C₂×C₃×C₅	C₂×#101	1	1	yes	characteristic
235	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
236	C₂×C₃×C₅	#419	1	1	yes	characteristic
237	C₃×D₅	—	2	2	no	—
238	C₂³×C₃	—	5	1	no	—
239	C₂×C₃×C₄	—	5	1	no	—
240	C₂×C₃×C₄	—	5	1	no	—
241	C₂×C₃×C₄	—	5	2	no	—
242	C₂×C₃×C₄	—	5	2	no	—
243	C₂×C₃×C₄	—	5	1	no	—
244	C₂×C₃×C₄	—	10	2	no	—
245	C₂×C₃×C₄	—	10	1	no	—
246	C₂×C₃×C₄	—	10	1	no	—
247	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
248	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
249	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
250	C₂×Dic₃	—	2	2	no	—
251	C₂×Dic₃	—	10	2	no	—
252	C₂×Dic₃	—	20	4	no	—
253	Dic₆	—	4	1	no	—
254	Dic₆	—	4	1	no	—
255	Dic₆	—	20	4	no	—
256	Dic₆	—	20	1	no	—
257	Dic₆	—	20	1	no	—
258	C₃×C₈	—	4	1	no	—
259	C₃×C₈	—	20	1	no	—
260	C₃⋊C₈	—	4	1	no	—
261	C₃⋊C₈	—	20	1	no	—
262	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
263	C₂×D₅	—	2	1	no	—
264	C₂×D₅	—	2	1	no	—
265	C₂×D₅	#809	1	2	yes	—
266	C₄×C₅	—	2	1	no	—
267	C₄×C₅	—	12	2	no	—
268	C₄×C₅	#909	1	1	yes	characteristic
269	C₄×C₅	#909	1	1	yes	characteristic
270	C₄×C₅	C₂×#213	1	1	yes	characteristic
271	C₄×C₅	C₂×#215	1	1	yes	characteristic
272	Dic₅	—	2	1	no	—
273	Dic₅	—	12	2	no	—
274	Dic₅	#809	1	2	yes	—
275	Dic₅	#824	1	2	yes	—
276	C₂×Q₈	—	6	1	no	—
277	C₂×Q₈	—	30	2	no	—
278	C₂×Q₈	—	30	1	no	—
279	C₂×Q₈	—	60	2	no	—
280	C₂×Q₈	—	60	2	no	—
281	C₂²×C₄	—	5	1	no	—
282	C₂²×C₄	—	5	1	no	—
283	C₂²×C₄	—	5	1	no	—
284	C₂²×C₄	—	30	2	no	—
285	M₁₆	—	20	2	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	6	1	no	—
288	C₂×C₈	—	10	1	no	—
289	C₂×C₈	—	30	1	no	—
290	C₄⋊C₄	—	3	2	no	—
291	C₄⋊C₄	—	6	1	no	—
292	C₄⋊C₄	—	15	2	no	—
293	C₄⋊C₄	—	30	2	no	—
294	C₄⋊C₄	—	30	2	no	—
295	C₄⋊C₄	—	30	1	no	—
296	C₄²	—	5	1	no	—
297	C₄²	—	5	1	no	—
298	C₄²	—	5	1	no	—
299	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
300	C₃×C₅	#1546	1	1	yes	characteristic, complemented, Hall
301	C₂²×C₃	—	5	2	no	—
302	C₂²×C₃	—	10	1	no	—
303	C₂²×C₃	—	10	1	no	—
304	C₂²×C₃	D₅×C₂²⋊C₄	1	1	yes	characteristic
305	C₃×C₄	—	2	1	no	—
306	C₃×C₄	—	5	2	no	—
307	C₃×C₄	—	5	2	no	—
308	C₃×C₄	—	10	1	no	—
309	C₃×C₄	D₅×D₈	1	1	yes	characteristic
310	C₃×C₄	D₅×SD₁₆	1	1	yes	characteristic
311	C₃×C₄	#4061	1	1	yes	characteristic
312	C₃×C₄	#4064	1	1	yes	characteristic
313	Dic₃	—	4	2	no	—
314	Dic₃	—	20	2	no	—
315	C₂×C₅	C₂×#809	1	1	yes	characteristic
316	C₂×C₅	#5120	1	1	yes	characteristic
317	C₂×C₅	C₂×#824	1	1	yes	characteristic
318	D₅	—	2	2	no	—
319	C₂³	—	5	1	no	—
320	Q₈	—	12	1	no	—
321	Q₈	—	12	1	no	—
322	Q₈	—	60	4	no	—
323	Q₈	—	60	1	no	—
324	Q₈	—	60	1	no	—
325	C₂×C₄	—	5	1	no	—
326	C₂×C₄	—	5	1	no	—
327	C₂×C₄	—	5	2	no	—
328	C₂×C₄	—	5	2	no	—
329	C₂×C₄	—	5	1	no	—
330	C₂×C₄	—	6	2	no	—
331	C₂×C₄	—	10	2	no	—
332	C₂×C₄	—	10	1	no	—
333	C₂×C₄	—	10	1	no	—
334	C₂×C₄	—	30	2	no	—
335	C₂×C₄	—	60	4	no	—
336	C₂×C₄	C₄×D₃×D₅	1	1	yes	characteristic
337	C₂×C₄	D₅×D₁₂	1	1	yes	characteristic
338	C₂×C₄	D₅×#67	1	1	yes	characteristic
339	C₈	—	4	1	no	—
340	C₈	—	12	1	no	—
341	C₈	—	20	1	no	—
342	C₈	—	60	1	no	—
343	C₂×C₃	—	10	2	no	—
344	C₂×C₃	D₅×#101	1	1	yes	characteristic
345	C₂×C₃	D₅×#102	1	1	yes	characteristic
346	C₂×C₃	#65091	1	1	yes	characteristic
347	C₅	#70506	1	1	yes	characteristic, complemented, Sylow
348	C₂²	—	5	2	no	—
349	C₂²	—	10	1	no	—
350	C₂²	—	10	1	no	—
351	C₂²	D₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
352	C₄	—	2	1	no	—
353	C₄	—	5	2	no	—
354	C₄	—	5	2	no	—
355	C₄	—	10	1	no	—
356	C₄	—	12	2	no	—
357	C₄	—	60	2	no	—
358	C₄	#91260	1	1	yes	characteristic
359	C₄	#91261	1	1	yes	characteristic
360	C₄	D₅×#213	1	1	yes	characteristic
361	C₄	D₅×#215	1	1	yes	characteristic
362	C₃	#10599213	1	1	yes	characteristic, complemented, Sylow
363	C₂	—	10	2	no	—
364	C₂	D₅×#809	1	1	yes	characteristic
365	C₂	#11748122	1	1	yes	characteristic
366	C₂	D₅×#824	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	C₂²×C₃×D₅	C₂²⋊C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₄⋊C₈	D₃×D₅
hypocenter	C₃×C₅	#1546
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
4. #11746303	C₂	semidirect
5. C₅×#4482	C₂	semidirect
6. #11745860	C₂	semidirect
7. #11745858	C₂	semidirect
64. #4482	D₅	semidirect
300. C₃×C₅	#1546	semidirect
347. C₅	#70506	semidirect
362. C₃	#10599213	semidirect

Maximal subgroups

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

subgroup	index
D₅×#869	2
C₃×#65126	2
#11746869	2
#11746303	2
C₅×#4482	2
#11745860	2
#11745858	2
#10599213	3
#70506	5