Subgroups — #14284607

Order 1920.

512 conjugacy classes of subgroups.

The subgroup lattice has 392 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×D₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#11746853	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746303	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64705	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745859	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745852	C₂	1	1	yes	maximal, characteristic, complemented
8	#10601224	—	3	1	no	maximal, Hall
9	C₂×#6765	—	2	1	no	—
10	#91758	C₂²	1	1	yes	characteristic
11	C₂×C₅×D₃×C₈	—	2	1	no	—
12	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
13	C₃×#4020	C₂²	1	1	yes	characteristic
14	#91456	—	2	1	no	—
15	#91456	C₄	1	2	yes	—
16	Dic₃×D₂₀	—	2	1	no	—
17	Dic₃×D₂₀	C₄	1	2	yes	—
18	#91364	—	2	1	no	—
19	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
21	C₅×Dic₃⋊C₈	—	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	#68381	—	5	1	no	maximal, Hall
27	#65811	—	3	1	no	—
28	C₂×C₅×C₄⋊C₈	—	3	1	no	—
29	#65214	—	3	1	no	—
30	#64705	—	3	1	no	—
31	#64705	D₃	1	1	yes	characteristic, complemented
32	#64702	—	3	1	no	—
33	#64702	—	3	1	no	—
34	C₂×Dic₃₀	—	2	1	no	—
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	2	no	—
44	#6765	—	4	2	no	—
45	C₂×C₃×C₅×C₈	—	2	1	no	—
46	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
47	C₄⋊Dic₁₅	—	2	1	no	—
48	C₄⋊Dic₁₅	C₂×C₄	1	2	yes	—
49	C₂×C₃⋊#168	—	2	1	no	—
50	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
51	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
52	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
53	C₂×C₅×C₃⋊C₈	—	2	1	no	—
54	C₅×D₃×C₈	—	4	2	no	—
55	C₂×C₃×#168	—	2	1	no	—
56	#6665	—	2	2	no	—
57	#6665	—	2	2	no	—
58	#5716	—	5	1	no	—
59	#4833	—	5	1	no	—
60	D₃×C₄⋊C₈	D₅	1	1	yes	characteristic, complemented
61	C₃×#331	—	5	1	no	—
62	#4544	—	5	1	no	—
63	#4486	—	5	1	no	—
64	#4482	—	5	1	no	—
65	C₂×#709	—	6	1	no	—
66	C₂²×C₅×C₈	—	6	1	no	—
67	C₂×C₄²×C₅	—	3	1	no	—
68	#4076	—	3	2	no	—
69	#4076	—	6	1	no	—
70	#4021	—	6	1	no	—
71	#4020	C₂×D₃	1	1	yes	characteristic
72	C₄×D₂₀	—	3	2	no	—
73	C₄×D₂₀	—	6	1	no	—
74	#4015	—	3	1	no	—
75	C₅×C₄⋊C₈	—	3	1	no	—
76	C₅×C₄⋊C₈	—	6	1	no	—
77	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
78	C₂²⋊#168	—	6	1	no	—
79	C₄⋊#168	—	3	1	no	—
80	C₄⋊#168	C₂×D₃	1	1	yes	characteristic
81	C₄×#168	—	6	1	no	—
82	#1540	—	15	1	no	Sylow
83	C₂²×C₅×D₃	C₂²⋊C₄	1	1	yes	characteristic
84	C₂²×C₃×D₅	—	2	2	no	—
85	C₂×C₃×C₄×C₅	C₂²×C₄	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×Dic₁₅	—	2	2	no	—
89	Dic₃₀	—	4	1	no	—
90	Dic₃₀	—	4	1	no	—
91	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
94	C₄×C₅×D₃	—	2	1	no	—
95	C₄×C₅×D₃	—	2	1	no	—
96	C₄×C₅×D₃	—	4	1	no	—
97	C₄×C₅×D₃	D₈	1	2	yes	—
98	C₄×C₅×D₃	SD₁₆	1	2	yes	—
99	C₃×D₂₀	—	4	2	no	—
100	C₃×D₂₀	—	4	1	no	—
101	C₃×D₂₀	—	4	1	no	—
102	C₃×D₂₀	—	4	1	no	—
103	C₃×D₂₀	—	4	1	no	—
104	#1201	—	4	4	no	—
105	D₅×Dic₃	—	4	4	no	—
106	C₃×C₅×C₈	—	4	1	no	—
107	C₃⋊#168	—	4	1	no	—
108	C₃×#168	—	4	1	no	—
109	C₅×C₃⋊C₈	—	4	1	no	—
110	C₂×#237	—	10	1	no	—
111	C₃×#127	—	5	1	no	—
112	#938	—	5	2	no	—
113	#938	—	10	1	no	—
114	#936	—	10	1	no	—
115	D₄×Dic₃	—	5	2	no	—
116	D₄×Dic₃	—	10	1	no	—
117	C₂×D₃×C₈	—	2	1	no	—
118	C₄²×D₃	C₂×D₅	1	1	yes	characteristic
119	#869	—	5	1	no	—
120	C₃×C₄⋊C₈	—	5	1	no	—
121	C₃×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
122	(C₂×D₃)⋊C₈	—	10	1	no	—
123	Dic₃⋊C₈	—	2	1	no	—
124	C₈×Dic₃	—	10	1	no	—
125	C₄⋊(C₃⋊C₈)	—	5	1	no	—
126	C₄⋊(C₃⋊C₈)	C₂×D₅	1	1	yes	characteristic
127	C₂²×C₄×C₅	—	3	1	no	—
128	C₂²×C₄×C₅	—	3	1	no	—
129	C₂²×C₄×C₅	—	3	1	no	—
130	C₂×C₂²⋊D₅	—	6	2	no	—
131	#709	—	12	2	no	—
132	#709	—	12	2	no	—
133	C₂×D₂₀	—	2	1	no	—
134	C₂×D₂₀	—	2	1	no	—
135	C₂×D₂₀	C₄×D₃	1	2	yes	—
136	C₂×C₄×D₅	—	6	2	no	—
137	C₂×Dic₁₀	—	6	1	no	—
138	C₂×C₅×C₈	—	2	1	no	—
139	C₂×C₅×C₈	—	6	1	no	—
140	C₂×C₅×C₈	—	12	2	no	—
141	C₄²×C₅	—	3	1	no	—
142	C₄²×C₅	—	3	1	no	—
143	C₄²×C₅	—	3	1	no	—
144	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
145	#685	—	6	2	no	—
146	#685	—	6	2	no	—
147	C₄⋊Dic₅	—	3	2	no	—
148	C₄⋊Dic₅	—	6	1	no	—
149	C₂×#168	—	2	1	no	—
150	C₂×#168	—	6	1	no	—
151	#532	—	15	1	no	—
152	#431	—	15	1	no	—
153	C₂×C₄⋊C₈	—	3	1	no	—
154	#332	—	15	1	no	—
155	#332	—	15	1	no	—
156	#331	—	5	1	no	—
157	#331	—	15	1	no	—
158	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
159	C₂×C₅×D₃	—	2	1	no	—
160	C₂×C₅×D₃	—	2	1	no	—
161	C₂×C₅×D₃	#102	1	2	yes	—
162	C₂×C₃×D₅	—	4	2	no	—
163	C₂×C₃×D₅	—	4	2	no	—
164	C₂×C₃×D₅	—	4	2	no	—
165	C₃×C₄×C₅	—	2	1	no	—
166	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
167	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
168	C₃×C₄×C₅	#135	1	1	yes	characteristic
169	C₃×C₄×C₅	#135	1	1	yes	characteristic
170	Dic₁₅	—	4	2	no	—
171	C₅×Dic₃	—	2	1	no	—
172	C₅×Dic₃	—	2	1	no	—
173	C₅×Dic₃	#102	1	2	yes	—
174	C₂×C₃×D₄	—	5	2	no	—
175	C₂×C₃×D₄	—	10	1	no	—
176	C₂×C₃×D₄	—	10	1	no	—
177	C₂×#67	—	10	2	no	—
178	C₂²×Dic₃	—	10	2	no	—
179	#237	—	20	2	no	—
180	#237	—	20	2	no	—
181	C₂×C₄×D₃	C₄×D₅	1	1	yes	characteristic
182	C₂×C₄×D₃	D₂₀	1	1	yes	characteristic
183	C₂×C₄×D₃	C₂²⋊D₅	1	1	yes	characteristic
184	C₂×Dic₆	—	10	1	no	—
185	C₂×C₃×C₈	—	2	1	no	—
186	C₂×C₃×C₈	—	10	1	no	—
187	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
188	#217	—	10	2	no	—
189	#217	—	10	2	no	—
190	C₄⋊Dic₃	—	5	2	no	—
191	C₄⋊Dic₃	—	10	1	no	—
192	C₄×Dic₃	C₄×D₅	1	1	yes	characteristic
193	C₄×Dic₃	D₂₀	1	1	yes	characteristic
194	C₄×Dic₃	C₂²⋊D₅	1	1	yes	characteristic
195	C₂×C₃⋊C₈	—	2	1	no	—
196	C₂×C₃⋊C₈	—	10	1	no	—
197	D₃×C₈	—	4	2	no	—
198	C₂³×C₅	—	3	1	no	—
199	C₂²×D₅	—	2	2	no	—
200	C₂×C₄×C₅	—	3	1	no	—
201	C₂×C₄×C₅	—	3	2	no	—
202	C₂×C₄×C₅	—	3	2	no	—
203	C₂×C₄×C₅	—	3	1	no	—
204	C₂×C₄×C₅	—	3	1	no	—
205	C₂×C₄×C₅	—	6	1	no	—
206	C₂×C₄×C₅	—	6	1	no	—
207	C₂×C₄×C₅	—	12	1	no	—
208	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
209	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
210	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
211	C₂²⋊D₅	—	12	4	no	—
212	C₂×Dic₅	—	6	2	no	—
213	D₂₀	—	4	2	no	—
214	D₂₀	—	4	1	no	—
215	D₂₀	—	4	1	no	—
216	D₂₀	—	4	1	no	—
217	D₂₀	—	4	1	no	—
218	C₄×D₅	—	12	4	no	—
219	Dic₁₀	—	12	1	no	—
220	Dic₁₀	—	12	1	no	—
221	C₅×C₈	—	4	1	no	—
222	C₅×C₈	—	12	1	no	—
223	#168	—	4	1	no	—
224	#168	—	12	1	no	—
225	C₂×#41	—	30	1	no	—
226	C₂²×C₈	—	6	1	no	—
227	C₄⋊Q₈	—	15	1	no	—
228	#127	—	5	1	no	—
229	#124	—	30	1	no	—
230	#121	—	15	2	no	—
231	#121	—	30	1	no	—
232	C₄×D₄	—	15	2	no	—
233	C₄×D₄	—	30	1	no	—
234	C₂×C₄²	—	3	1	no	—
235	C₄⋊C₈	—	3	1	no	—
236	C₄⋊C₈	—	5	1	no	—
237	C₄⋊C₈	—	6	1	no	—
238	C₄⋊C₈	—	15	1	no	—
239	C₄⋊C₈	D₃×D₅	1	1	yes	characteristic
240	C₂²⋊C₈	—	30	1	no	—
241	C₄×C₈	—	30	1	no	—
242	C₂×C₃×C₅	#413	1	1	yes	characteristic
243	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
244	C₂×C₃×C₅	#416	1	1	yes	characteristic
245	C₃×D₅	—	8	2	no	—
246	C₅×D₃	—	2	2	no	—
247	C₂³×C₃	—	10	2	no	—
248	C₂²×D₃	#685	1	1	yes	characteristic
249	C₃×D₄	—	20	2	no	—
250	C₃×D₄	—	20	1	no	—
251	C₃×D₄	—	20	1	no	—
252	C₃×D₄	—	20	1	no	—
253	C₃×D₄	—	20	1	no	—
254	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
255	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
256	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
257	#67	—	20	4	no	—
258	C₂×Dic₃	—	10	2	no	—
259	C₂×Dic₃	—	20	4	no	—
260	C₂×Dic₃	#685	1	1	yes	characteristic
261	C₂×Dic₃	#685	1	1	yes	characteristic
262	C₂×Dic₃	#685	1	1	yes	characteristic
263	C₄×D₃	—	2	1	no	—
264	C₄×D₃	—	2	1	no	—
265	C₄×D₃	—	4	1	no	—
266	C₄×D₃	#686	1	2	yes	—
267	C₄×D₃	#688	1	2	yes	—
268	Dic₆	—	20	1	no	—
269	Dic₆	—	20	1	no	—
270	C₃×C₈	—	4	1	no	—
271	C₃×C₈	—	20	1	no	—
272	C₃⋊C₈	—	4	1	no	—
273	C₃⋊C₈	—	20	1	no	—
274	C₂²×C₅	—	3	2	no	—
275	C₂²×C₅	—	6	1	no	—
276	C₂²×C₅	—	6	1	no	—
277	C₂²×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
278	C₂×D₅	—	4	2	no	—
279	C₂×D₅	—	4	2	no	—
280	C₂×D₅	—	4	2	no	—
281	C₄×C₅	—	2	1	no	—
282	C₄×C₅	—	3	2	no	—
283	C₄×C₅	—	6	1	no	—
284	C₄×C₅	—	6	1	no	—
285	C₄×C₅	D₃×D₈	1	1	yes	characteristic
286	C₄×C₅	#912	1	1	yes	characteristic
287	C₄×C₅	D₃×SD₁₆	1	1	yes	characteristic
288	C₄×C₅	#916	1	1	yes	characteristic
289	Dic₅	—	12	2	no	—
290	#41	—	60	2	no	—
291	#41	—	60	2	no	—
292	C₂×Q₈	—	30	1	no	—
293	C₂×D₄	—	5	2	no	—
294	C₂×D₄	—	10	1	no	—
295	C₂×D₄	—	10	1	no	—
296	C₂×D₄	—	30	2	no	—
297	C₂²×C₄	—	3	1	no	—
298	C₂²×C₄	—	3	1	no	—
299	C₂²×C₄	—	3	1	no	—
300	C₂²×C₄	—	30	2	no	—
301	C₂×C₈	—	2	1	no	—
302	C₂×C₈	—	6	1	no	—
303	C₂×C₈	—	10	1	no	—
304	C₂×C₈	—	12	2	no	—
305	C₂×C₈	—	30	1	no	—
306	C₄⋊C₄	—	15	2	no	—
307	C₄⋊C₄	—	30	1	no	—
308	C₂²⋊C₄	—	30	2	no	—
309	C₂²⋊C₄	—	30	2	no	—
310	C₄²	—	3	1	no	—
311	C₄²	—	3	1	no	—
312	C₄²	—	3	1	no	—
313	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
314	C₃×C₅	#1540	1	1	yes	characteristic, complemented, Hall
315	C₂²×C₃	—	20	2	no	—
316	C₂²×C₃	—	20	2	no	—
317	C₂²×C₃	—	20	2	no	—
318	C₂²×C₃	C₂×#685	1	1	yes	characteristic
319	C₂×D₃	—	2	1	no	—
320	C₂×D₃	—	2	1	no	—
321	C₂×D₃	#3941	1	2	yes	—
322	C₃×C₄	—	2	1	no	—
323	C₃×C₄	#4050	1	1	yes	characteristic
324	C₃×C₄	#4050	1	1	yes	characteristic
325	C₃×C₄	C₂×#686	1	1	yes	characteristic
326	C₃×C₄	C₂×#688	1	1	yes	characteristic
327	Dic₃	—	2	1	no	—
328	Dic₃	—	2	1	no	—
329	Dic₃	—	20	2	no	—
330	Dic₃	#3941	1	2	yes	—
331	C₂×C₅	—	6	2	no	—
332	C₂×C₅	#4745	1	1	yes	characteristic
333	C₂×C₅	D₃×#102	1	1	yes	characteristic
334	C₂×C₅	#4772	1	1	yes	characteristic
335	D₅	—	8	2	no	—
336	C₂³	—	3	1	no	—
337	C₂³	—	10	2	no	—
338	Q₈	—	60	1	no	—
339	Q₈	—	60	1	no	—
340	D₄	—	20	2	no	—
341	D₄	—	20	1	no	—
342	D₄	—	20	1	no	—
343	D₄	—	20	1	no	—
344	D₄	—	20	1	no	—
345	D₄	—	60	4	no	—
346	C₂×C₄	—	3	1	no	—
347	C₂×C₄	—	3	2	no	—
348	C₂×C₄	—	3	2	no	—
349	C₂×C₄	—	3	1	no	—
350	C₂×C₄	—	3	1	no	—
351	C₂×C₄	—	6	1	no	—
352	C₂×C₄	—	6	1	no	—
353	C₂×C₄	—	12	1	no	—
354	C₂×C₄	—	30	2	no	—
355	C₂×C₄	—	60	4	no	—
356	C₂×C₄	C₄×D₃×D₅	1	1	yes	characteristic
357	C₂×C₄	D₃×D₂₀	1	1	yes	characteristic
358	C₂×C₄	D₃×C₂²⋊D₅	1	1	yes	characteristic
359	C₈	—	4	1	no	—
360	C₈	—	12	1	no	—
361	C₈	—	20	1	no	—
362	C₈	—	60	1	no	—
363	C₂×C₃	—	40	2	no	—
364	C₂×C₃	C₂×#3941	1	1	yes	characteristic
365	C₂×C₃	#65402	1	1	yes	characteristic
366	C₂×C₃	#65421	1	1	yes	characteristic
367	D₃	—	2	2	no	—
368	C₅	#68381	1	1	yes	characteristic, complemented, Sylow
369	C₂²	—	3	2	no	—
370	C₂²	—	6	1	no	—
371	C₂²	—	6	1	no	—
372	C₂²	—	20	2	no	—
373	C₂²	—	20	2	no	—
374	C₂²	—	20	2	no	—
375	C₂²	D₃×#685	1	1	yes	characteristic
376	C₄	—	2	1	no	—
377	C₄	—	3	2	no	—
378	C₄	—	6	1	no	—
379	C₄	—	6	1	no	—
380	C₄	—	60	2	no	—
381	C₄	#91272	1	1	yes	characteristic
382	C₄	#91273	1	1	yes	characteristic
383	C₄	D₃×#686	1	1	yes	characteristic
384	C₄	D₃×#688	1	1	yes	characteristic
385	C₃	#10601224	1	1	yes	characteristic, complemented, Sylow
386	C₂	—	6	2	no	—
387	C₂	—	40	2	no	—
388	C₂	D₃×#3941	1	1	yes	characteristic
389	C₂	#11748137	1	1	yes	characteristic
390	C₂	#11748234	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	#1540
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×D₃×C₄⋊C₈	C₂	semidirect
3. #11746853	C₂	semidirect
4. #11746303	C₂	semidirect
5. C₃×#64705	C₂	semidirect
6. #11745859	C₂	semidirect
7. #11745852	C₂	semidirect
31. #64705	D₃	semidirect
60. D₃×C₄⋊C₈	D₅	semidirect
314. C₃×C₅	#1540	semidirect
368. C₅	#68381	semidirect
385. C₃	#10601224	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×D₃×C₄⋊C₈	2
#11746853	2
#11746303	2
C₃×#64705	2
#11745859	2
#11745852	2
#10601224	3
#68381	5