Subgroups — #14284717

Order 1920.

492 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11752542	C₂	1	1	yes	maximal, characteristic
2	C₅×#4833	C₂	1	1	yes	maximal, characteristic, complemented
3	#11746853	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746302	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64705	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745863	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745852	C₂	1	1	yes	maximal, characteristic, complemented
8	#10601566	—	3	1	no	maximal, Hall
9	C₂×#6765	C₄	1	2	yes	—
10	#91758	C₂²	1	1	yes	characteristic
11	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
12	C₃×#4020	C₂²	1	1	yes	characteristic
13	#91456	—	2	1	no	—
14	#91456	—	2	1	no	—
15	Dic₃×D₂₀	—	2	1	no	—
16	Dic₃×D₂₀	—	2	1	no	—
17	#91364	C₄	1	2	yes	—
18	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
20	C₅×(C₂×D₃)⋊C₈	—	2	1	no	—
21	C₅×C₈×Dic₃	—	2	1	no	—
22	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
23	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
24	(C₂×D₃)⋊#168	—	2	1	no	—
25	Dic₃×#168	—	2	1	no	—
26	#68382	—	5	1	no	maximal, Hall
27	#65811	—	3	1	no	—
28	C₅×#431	—	3	1	no	—
29	#65214	—	3	1	no	—
30	#64705	—	3	1	no	—
31	#64705	D₃	1	1	yes	characteristic, complemented
32	#64701	—	3	1	no	—
33	#64701	—	3	1	no	—
34	C₂×Dic₃₀	C₂×C₄	1	2	yes	—
35	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
37	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
38	C₂×C₃×D₂₀	—	2	1	no	—
39	C₂×C₃×D₂₀	—	2	1	no	—
40	C₂×C₃×D₂₀	C₂×C₄	1	2	yes	—
41	C₂×#1201	—	2	2	no	—
42	C₂×D₅×Dic₃	—	2	2	no	—
43	#6765	—	4	4	no	—
44	C₂×C₃×C₅×C₈	—	2	1	no	—
45	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
46	C₄⋊Dic₁₅	—	2	1	no	—
47	C₄⋊Dic₁₅	—	2	1	no	—
48	C₂×C₃⋊#168	—	2	1	no	—
49	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
50	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
51	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
52	C₂×C₅×C₃⋊C₈	—	2	1	no	—
53	C₂×C₃×#168	—	2	1	no	—
54	#6665	—	2	2	no	—
55	#6665	—	2	2	no	—
56	#5716	—	5	1	no	—
57	#4833	—	5	1	no	—
58	#4833	D₅	1	1	yes	characteristic, complemented
59	C₃×#331	—	5	1	no	—
60	#4544	—	5	1	no	—
61	#4481	—	5	1	no	—
62	#4481	—	5	1	no	—
63	C₂×#709	—	3	2	no	—
64	C₂×C₄²×C₅	—	3	1	no	—
65	#4076	—	6	1	no	—
66	#4076	—	6	1	no	—
67	#4021	—	3	2	no	—
68	#4020	C₂×D₃	1	1	yes	characteristic
69	C₄×D₂₀	—	6	1	no	—
70	C₄×D₂₀	—	6	1	no	—
71	#4015	—	3	1	no	—
72	C₅×C₄⋊C₈	—	3	1	no	—
73	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
74	C₅×C₂²⋊C₈	—	6	1	no	—
75	C₄×C₅×C₈	—	6	1	no	—
76	C₂²⋊#168	—	6	1	no	—
77	C₄⋊#168	—	3	1	no	—
78	C₄⋊#168	C₂×D₃	1	1	yes	characteristic
79	C₄×#168	—	6	1	no	—
80	#1543	—	15	1	no	Sylow
81	C₂²×C₅×D₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₂²×C₃×D₅	—	2	2	no	—
83	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
86	C₂×Dic₁₅	—	2	2	no	—
87	Dic₃₀	—	4	2	no	—
88	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
91	C₄×C₅×D₃	—	2	1	no	—
92	C₄×C₅×D₃	—	2	1	no	—
93	C₄×C₅×D₃	—	2	1	no	—
94	C₄×C₅×D₃	—	2	1	no	—
95	C₄×C₅×D₃	—	2	2	no	—
96	C₃×D₂₀	—	4	2	no	—
97	C₃×D₂₀	—	4	1	no	—
98	C₃×D₂₀	—	4	1	no	—
99	C₃×D₂₀	—	4	1	no	—
100	C₃×D₂₀	—	4	1	no	—
101	#1201	—	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	C₂×#237	—	5	2	no	—
108	C₃×#127	—	5	1	no	—
109	#938	—	10	1	no	—
110	#938	—	10	1	no	—
111	#936	—	5	2	no	—
112	D₄×Dic₃	—	10	1	no	—
113	D₄×Dic₃	—	10	1	no	—
114	C₄²×D₃	C₂×D₅	1	1	yes	characteristic
115	#869	—	5	1	no	—
116	C₃×C₄⋊C₈	—	5	1	no	—
117	C₃×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
118	(C₂×D₃)⋊C₈	—	2	1	no	—
119	(C₂×D₃)⋊C₈	—	10	1	no	—
120	C₈×Dic₃	—	2	1	no	—
121	C₈×Dic₃	—	10	1	no	—
122	C₄⋊(C₃⋊C₈)	—	5	1	no	—
123	C₄⋊(C₃⋊C₈)	C₂×D₅	1	1	yes	characteristic
124	C₂²×C₄×C₅	—	3	1	no	—
125	C₂²×C₄×C₅	—	3	1	no	—
126	C₂²×C₄×C₅	—	3	1	no	—
127	C₂×C₂²⋊D₅	—	6	2	no	—
128	#709	—	12	4	no	—
129	C₂×D₂₀	—	2	1	no	—
130	C₂×D₂₀	—	2	1	no	—
131	C₂×D₂₀	C₄×D₃	1	2	yes	—
132	C₂×C₄×D₅	—	6	2	no	—
133	C₂×Dic₁₀	—	3	2	no	—
134	C₂×C₅×C₈	—	2	1	no	—
135	C₂×C₅×C₈	—	6	1	no	—
136	C₄²×C₅	—	3	1	no	—
137	C₄²×C₅	—	3	1	no	—
138	C₄²×C₅	—	3	1	no	—
139	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
140	#685	—	6	2	no	—
141	#685	—	6	2	no	—
142	C₄⋊Dic₅	—	6	1	no	—
143	C₄⋊Dic₅	—	6	1	no	—
144	C₂×#168	—	2	1	no	—
145	C₂×#168	—	6	1	no	—
146	#532	—	15	1	no	—
147	#431	—	3	1	no	—
148	#431	—	15	1	no	—
149	#333	—	15	1	no	—
150	#333	—	15	1	no	—
151	#331	—	5	1	no	—
152	#331	—	15	1	no	—
153	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
154	C₂×C₅×D₃	—	2	1	no	—
155	C₂×C₅×D₃	—	2	1	no	—
156	C₂×C₅×D₃	—	2	1	no	—
157	C₂×C₃×D₅	—	4	2	no	—
158	C₂×C₃×D₅	—	4	2	no	—
159	C₂×C₃×D₅	—	4	2	no	—
160	C₃×C₄×C₅	—	2	1	no	—
161	C₃×C₄×C₅	#135	1	1	yes	characteristic
162	C₃×C₄×C₅	#135	1	1	yes	characteristic
163	C₃×C₄×C₅	#135	1	1	yes	characteristic
164	C₃×C₄×C₅	#135	1	1	yes	characteristic
165	Dic₁₅	—	4	2	no	—
166	C₅×Dic₃	—	2	1	no	—
167	C₅×Dic₃	—	2	1	no	—
168	C₅×Dic₃	#100	1	2	yes	—
169	C₂×C₃×D₄	—	5	2	no	—
170	C₂×C₃×D₄	—	10	1	no	—
171	C₂×C₃×D₄	—	10	1	no	—
172	C₂×#67	—	10	2	no	—
173	C₂²×Dic₃	—	10	2	no	—
174	#237	—	20	4	no	—
175	C₂×C₄×D₃	C₄×D₅	1	1	yes	characteristic
176	C₂×C₄×D₃	D₂₀	1	1	yes	characteristic
177	C₂×C₄×D₃	C₂²⋊D₅	1	1	yes	characteristic
178	C₂×Dic₆	—	5	2	no	—
179	C₂×C₃×C₈	—	2	1	no	—
180	C₂×C₃×C₈	—	10	1	no	—
181	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
182	#217	—	10	2	no	—
183	#217	—	10	2	no	—
184	C₄⋊Dic₃	—	10	1	no	—
185	C₄⋊Dic₃	—	10	1	no	—
186	C₄×Dic₃	C₄×D₅	1	1	yes	characteristic
187	C₄×Dic₃	D₂₀	1	1	yes	characteristic
188	C₄×Dic₃	C₂²⋊D₅	1	1	yes	characteristic
189	C₂×C₃⋊C₈	—	2	1	no	—
190	C₂×C₃⋊C₈	—	10	1	no	—
191	C₂³×C₅	—	3	1	no	—
192	C₂²×D₅	—	2	2	no	—
193	C₂×C₄×C₅	—	3	1	no	—
194	C₂×C₄×C₅	—	3	1	no	—
195	C₂×C₄×C₅	—	3	1	no	—
196	C₂×C₄×C₅	—	6	2	no	—
197	C₂×C₄×C₅	—	6	1	no	—
198	C₂×C₄×C₅	—	6	1	no	—
199	C₂×C₄×C₅	—	6	1	no	—
200	C₂×C₄×C₅	—	6	1	no	—
201	C₂×C₄×C₅	C₂×C₄×D₃	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	C₂²⋊D₅	—	12	4	no	—
205	C₂×Dic₅	—	6	2	no	—
206	D₂₀	—	4	2	no	—
207	D₂₀	—	4	1	no	—
208	D₂₀	—	4	1	no	—
209	D₂₀	—	4	1	no	—
210	D₂₀	—	4	1	no	—
211	C₄×D₅	—	12	4	no	—
212	Dic₁₀	—	12	2	no	—
213	C₅×C₈	—	4	1	no	—
214	C₅×C₈	—	12	1	no	—
215	#168	—	4	1	no	—
216	#168	—	12	1	no	—
217	C₂×#41	—	15	2	no	—
218	C₄⋊Q₈	—	15	1	no	—
219	#127	—	5	1	no	—
220	#124	—	15	2	no	—
221	#121	—	30	1	no	—
222	#121	—	30	1	no	—
223	C₄×D₄	—	30	1	no	—
224	C₄×D₄	—	30	1	no	—
225	C₂×C₄²	—	3	1	no	—
226	C₄⋊C₈	—	3	1	no	—
227	C₄⋊C₈	—	5	1	no	—
228	C₄⋊C₈	—	15	1	no	—
229	C₄⋊C₈	D₃×D₅	1	1	yes	characteristic
230	C₂²⋊C₈	—	6	1	no	—
231	C₂²⋊C₈	—	30	1	no	—
232	C₄×C₈	—	6	1	no	—
233	C₄×C₈	—	30	1	no	—
234	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
235	C₂×C₃×C₅	#416	1	1	yes	characteristic
236	C₂×C₃×C₅	#416	1	1	yes	characteristic
237	C₃×D₅	—	8	2	no	—
238	C₅×D₃	—	4	1	no	—
239	C₂³×C₃	—	10	2	no	—
240	C₂²×D₃	#685	1	1	yes	characteristic
241	C₃×D₄	—	20	2	no	—
242	C₃×D₄	—	20	1	no	—
243	C₃×D₄	—	20	1	no	—
244	C₃×D₄	—	20	1	no	—
245	C₃×D₄	—	20	1	no	—
246	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
247	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
248	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
249	#67	—	20	4	no	—
250	C₂×Dic₃	—	10	2	no	—
251	C₂×Dic₃	—	20	4	no	—
252	C₂×Dic₃	#685	1	1	yes	characteristic
253	C₂×Dic₃	#685	1	1	yes	characteristic
254	C₂×Dic₃	#685	1	1	yes	characteristic
255	C₄×D₃	—	2	1	no	—
256	C₄×D₃	—	2	1	no	—
257	C₄×D₃	—	2	1	no	—
258	C₄×D₃	—	2	1	no	—
259	C₄×D₃	—	2	2	no	—
260	Dic₆	—	20	2	no	—
261	C₃×C₈	—	4	1	no	—
262	C₃×C₈	—	20	1	no	—
263	C₃⋊C₈	—	4	1	no	—
264	C₃⋊C₈	—	20	1	no	—
265	C₂²×C₅	—	6	1	no	—
266	C₂²×C₅	—	6	1	no	—
267	C₂²×C₅	—	6	1	no	—
268	C₂²×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
269	C₂×D₅	—	4	2	no	—
270	C₂×D₅	—	4	2	no	—
271	C₂×D₅	—	4	2	no	—
272	C₄×C₅	—	2	1	no	—
273	C₄×C₅	—	3	2	no	—
274	C₄×C₅	—	6	1	no	—
275	C₄×C₅	—	6	1	no	—
276	C₄×C₅	#912	1	1	yes	characteristic
277	C₄×C₅	#912	1	1	yes	characteristic
278	C₄×C₅	#916	1	1	yes	characteristic
279	C₄×C₅	#916	1	1	yes	characteristic
280	Dic₅	—	12	2	no	—
281	#41	—	60	4	no	—
282	C₂×Q₈	—	15	2	no	—
283	C₂×D₄	—	5	2	no	—
284	C₂×D₄	—	10	1	no	—
285	C₂×D₄	—	10	1	no	—
286	C₂×D₄	—	30	2	no	—
287	C₂²×C₄	—	3	1	no	—
288	C₂²×C₄	—	3	1	no	—
289	C₂²×C₄	—	3	1	no	—
290	C₂²×C₄	—	30	2	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	6	1	no	—
293	C₂×C₈	—	10	1	no	—
294	C₂×C₈	—	30	1	no	—
295	C₄⋊C₄	—	30	1	no	—
296	C₄⋊C₄	—	30	1	no	—
297	C₂²⋊C₄	—	30	2	no	—
298	C₂²⋊C₄	—	30	2	no	—
299	C₄²	—	3	1	no	—
300	C₄²	—	3	1	no	—
301	C₄²	—	3	1	no	—
302	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
303	C₃×C₅	#1543	1	1	yes	characteristic, complemented, Hall
304	C₂²×C₃	—	20	2	no	—
305	C₂²×C₃	—	20	2	no	—
306	C₂²×C₃	—	20	2	no	—
307	C₂²×C₃	C₂×#685	1	1	yes	characteristic
308	C₂×D₃	—	2	1	no	—
309	C₂×D₃	—	2	1	no	—
310	C₂×D₃	—	2	1	no	—
311	C₃×C₄	—	2	1	no	—
312	C₃×C₄	#4050	1	1	yes	characteristic
313	C₃×C₄	#4050	1	1	yes	characteristic
314	C₃×C₄	#4096	1	1	yes	characteristic
315	C₃×C₄	#4096	1	1	yes	characteristic
316	Dic₃	—	2	1	no	—
317	Dic₃	—	2	1	no	—
318	Dic₃	—	20	2	no	—
319	Dic₃	#3955	1	2	yes	—
320	C₂×C₅	—	12	1	no	—
321	C₂×C₅	D₃×#100	1	1	yes	characteristic
322	C₂×C₅	#4772	1	1	yes	characteristic
323	C₂×C₅	#4772	1	1	yes	characteristic
324	D₅	—	8	2	no	—
325	C₂³	—	3	1	no	—
326	C₂³	—	10	2	no	—
327	Q₈	—	60	2	no	—
328	D₄	—	20	2	no	—
329	D₄	—	20	1	no	—
330	D₄	—	20	1	no	—
331	D₄	—	20	1	no	—
332	D₄	—	20	1	no	—
333	D₄	—	60	4	no	—
334	C₂×C₄	—	3	1	no	—
335	C₂×C₄	—	3	1	no	—
336	C₂×C₄	—	3	1	no	—
337	C₂×C₄	—	6	2	no	—
338	C₂×C₄	—	6	1	no	—
339	C₂×C₄	—	6	1	no	—
340	C₂×C₄	—	6	1	no	—
341	C₂×C₄	—	6	1	no	—
342	C₂×C₄	—	30	2	no	—
343	C₂×C₄	—	60	4	no	—
344	C₂×C₄	C₄×D₃×D₅	1	1	yes	characteristic
345	C₂×C₄	D₃×D₂₀	1	1	yes	characteristic
346	C₂×C₄	D₃×C₂²⋊D₅	1	1	yes	characteristic
347	C₈	—	4	1	no	—
348	C₈	—	12	1	no	—
349	C₈	—	20	1	no	—
350	C₈	—	60	1	no	—
351	C₂×C₃	—	40	2	no	—
352	C₂×C₃	#65291	1	1	yes	characteristic
353	C₂×C₃	#65402	1	1	yes	characteristic
354	C₂×C₃	C₂×#3955	1	1	yes	characteristic
355	D₃	—	4	1	no	—
356	C₅	#68382	1	1	yes	characteristic, complemented, Sylow
357	C₂²	—	6	1	no	—
358	C₂²	—	6	1	no	—
359	C₂²	—	6	1	no	—
360	C₂²	—	20	2	no	—
361	C₂²	—	20	2	no	—
362	C₂²	—	20	2	no	—
363	C₂²	D₃×#685	1	1	yes	characteristic
364	C₄	—	2	1	no	—
365	C₄	—	3	2	no	—
366	C₄	—	6	1	no	—
367	C₄	—	6	1	no	—
368	C₄	—	60	2	no	—
369	C₄	#91272	1	1	yes	characteristic
370	C₄	#91273	1	1	yes	characteristic
371	C₄	#91493	1	1	yes	characteristic
372	C₄	#91517	1	1	yes	characteristic
373	C₃	#10601566	1	1	yes	characteristic, complemented, Sylow
374	C₂	—	12	1	no	—
375	C₂	—	40	2	no	—
376	C₂	#11747502	1	1	yes	characteristic
377	C₂	#11748137	1	1	yes	characteristic
378	C₂	D₃×#3955	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₃×#685
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	#11752542	C₂
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₄⋊C₈	D₃×D₅
hypocenter	C₃×C₅	#1543
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. C₅×#4833	C₂	semidirect
3. #11746853	C₂	semidirect
4. #11746302	C₂	semidirect
5. C₃×#64705	C₂	semidirect
6. #11745863	C₂	semidirect
7. #11745852	C₂	semidirect
31. #64705	D₃	semidirect
58. #4833	D₅	semidirect
303. C₃×C₅	#1543	semidirect
356. C₅	#68382	semidirect
373. C₃	#10601566	semidirect

Maximal subgroups

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

subgroup	index
#11752542	2
C₅×#4833	2
#11746853	2
#11746302	2
C₃×#64705	2
#11745863	2
#11745852	2
#10601566	3
#68382	5