Subgroups — #14416280

Order 1920.

1000 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#91793	C₂	1	1	yes	maximal, characteristic
2	#11754436	C₂	1	2	yes	maximal, complemented
3	C₂×C₅×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65395	C₂	1	1	yes	maximal, characteristic, complemented
5	#11748411	C₂	1	2	yes	maximal, complemented
6	#11747873	C₂	1	2	yes	maximal, complemented
7	#11747793	C₂	1	2	yes	maximal, complemented
8	#11747789	C₂	1	4	yes	maximal, complemented
9	C₄×#4050	—	3	1	no	maximal, Hall
10	C₂×#6817	C₂²	1	2	yes	—
11	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₂×Dic₆₀	C₂²	1	1	yes	characteristic
13	#91793	C₄	1	8	yes	complemented
14	C₂×D₁₂₀	C₂²	1	1	yes	characteristic
15	C₂×#6706	C₂²	1	2	yes	—
16	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
17	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₃×#4072	C₂²	1	2	yes	—
19	#91556	—	2	4	no	—
20	#91428	C₂²	1	4	yes	—
21	C₄×#1200	—	2	2	no	—
22	#91386	—	2	2	no	—
23	#91340	C₂²	1	4	yes	—
24	C₅×C₈×Dic₃	C₂²	1	2	yes	—
25	C₅×C₈×Dic₃	C₂²	1	2	yes	—
26	C₃×#3950	C₂²	1	2	yes	—
27	C₃×#3949	C₂²	1	2	yes	—
28	#90974	C₂²	1	4	yes	—
29	#90968	C₂²	1	4	yes	—
30	#79165	—	5	1	no	maximal, Hall
31	C₂×#4050	—	3	1	no	—
32	C₄×#709	—	3	2	no	—
33	C₂×C₄×C₅×C₈	—	3	1	no	—
34	#65402	—	3	2	no	—
35	#65395	D₃	1	1	yes	characteristic, complemented
36	C₄×Dic₂₀	—	3	2	no	—
37	C₄×D₄₀	—	3	2	no	—
38	C₄×#677	—	3	4	no	—
39	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
40	C₂×#1220	—	2	2	no	—
41	#6817	—	2	4	no	—
42	#6817	C₂×C₄	1	8	yes	complemented
43	C₂×D₆₀	C₂³	1	2	yes	—
44	C₂×C₄×D₁₅	—	2	2	no	—
45	C₂×Dic₃₀	C₂³	1	2	yes	—
46	C₂²×C₅×Dic₃	D₄	1	2	yes	—
47	C₂×#1200	—	2	4	no	—
48	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	complemented
49	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
50	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
51	Dic₆₀	C₂×C₄	1	4	yes	complemented
52	D₁₂₀	C₂×C₄	1	4	yes	complemented
53	#6706	C₂×C₄	1	8	yes	complemented
54	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
55	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
56	C₄×C₅×Dic₃	C₂³	1	2	yes	—
57	C₂×C₅×C₃⋊C₈	—	2	2	no	—
58	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
59	C₃×C₂²⋊Dic₅	—	2	2	no	—
60	C₃×C₄⋊Dic₅	C₂³	1	4	yes	—
61	C₃×C₄×Dic₅	—	2	2	no	—
62	#6669	—	2	4	no	—
63	#6667	—	2	4	no	—
64	Dic₃×Dic₅	—	2	4	no	—
65	C₂×#904	—	5	1	no	—
66	#5522	—	5	2	no	—
67	C₃×#427	—	5	1	no	—
68	C₂×C₈×Dic₃	D₅	1	1	yes	characteristic, complemented
69	#5002	—	5	2	no	—
70	#4873	—	5	2	no	—
71	#4872	—	5	2	no	—
72	#4852	—	5	4	no	—
73	C₂×#709	—	3	2	no	—
74	C₂²×C₅×C₈	—	3	1	no	—
75	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
76	C₂×C₄²×C₅	—	3	1	no	—
77	C₄×C₂²⋊D₅	—	6	4	no	—
78	#4072	C₂×D₃	1	2	yes	—
79	C₂×Dic₂₀	—	3	1	no	—
80	#4050	—	3	8	no	—
81	C₂×D₄₀	—	3	1	no	—
82	C₂×#677	—	3	2	no	—
83	C₄×D₂₀	—	3	4	no	—
84	#4018	—	6	2	no	—
85	C₄²×D₅	—	6	2	no	—
86	C₄×Dic₁₀	—	3	4	no	—
87	C₄×C₅×C₈	—	3	2	no	—
88	C₄×C₅×C₈	—	3	2	no	—
89	#3953	—	3	4	no	—
90	#3950	C₂×D₃	1	2	yes	—
91	#3949	C₂×D₃	1	2	yes	—
92	#3948	—	3	4	no	—
93	C₄×#135	—	15	1	no	Sylow
94	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
95	C₂²×D₁₅	—	2	2	no	—
96	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
97	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
98	C₂×C₃×C₄×C₅	#41	1	2	yes	—
99	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
100	#1220	—	2	8	no	—
101	C₂×Dic₁₅	—	2	2	no	—
102	D₆₀	—	2	2	no	—
103	D₆₀	C₂²×C₄	1	4	yes	—
104	C₄×D₁₅	—	2	8	no	—
105	Dic₃₀	—	2	2	no	—
106	Dic₃₀	C₂²×C₄	1	4	yes	—
107	C₂×C₅×Dic₃	—	2	4	no	—
108	C₂×C₅×Dic₃	C₂×D₄	1	2	yes	—
109	C₂×C₅×Dic₃	C₂×D₄	1	2	yes	—
110	C₂×C₃×Dic₅	—	2	4	no	—
111	#1200	—	4	8	no	—
112	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
113	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
114	C₅×C₃⋊C₈	—	2	2	no	—
115	C₂×#235	—	5	2	no	—
116	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
117	C₃×#117	—	5	2	no	—
118	C₂×C₄×Dic₃	C₂×D₅	1	1	yes	characteristic
119	C₂²×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
120	C₂×Dic₁₂	—	5	1	no	—
121	#904	—	5	8	no	—
122	C₂×D₂₄	—	5	1	no	—
123	C₂×#204	—	5	2	no	—
124	#893	—	5	4	no	—
125	#887	—	5	4	no	—
126	#67⋊C₄	—	10	4	no	—
127	#872	—	10	2	no	—
128	C₄²×D₃	—	10	2	no	—
129	C₃×#107	—	5	2	no	—
130	C₃×#106	—	5	2	no	—
131	C₈×Dic₃	C₂×D₅	1	2	yes	—
132	C₈×Dic₃	C₂×D₅	1	2	yes	—
133	#810	—	5	4	no	—
134	#809	—	5	4	no	—
135	C₂²×C₄×C₅	—	3	2	no	—
136	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
137	C₂×C₂²⋊D₅	—	6	2	no	—
138	#709	—	3	8	no	—
139	#709	—	6	4	no	—
140	C₂×D₂₀	—	3	2	no	—
141	C₂×C₄×D₅	—	6	2	no	—
142	C₂×C₄×D₅	—	6	4	no	—
143	C₂×Dic₁₀	—	3	2	no	—
144	C₂×C₅×C₈	—	3	2	no	—
145	C₂×C₅×C₈	—	6	2	no	—
146	C₂×C₅×C₈	C₄×D₃	1	4	yes	complemented
147	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
148	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
149	C₄²×C₅	—	3	1	no	—
150	C₄²×C₅	—	3	1	no	—
151	C₄²×C₅	—	3	2	no	—
152	C₂²⋊Dic₅	—	2	2	no	—
153	#685	—	6	4	no	—
154	C₄⋊Dic₅	C₂²×D₃	1	4	yes	—
155	#683	—	6	4	no	—
156	C₄×Dic₅	—	2	2	no	—
157	C₄×Dic₅	—	6	4	no	—
158	Dic₂₀	—	3	4	no	—
159	D₄₀	—	3	4	no	—
160	#677	—	3	8	no	—
161	C₂×#135	—	15	1	no	—
162	C₄×#41	—	15	2	no	—
163	C₄×Q₁₆	—	15	2	no	—
164	C₄×SD₁₆	—	15	4	no	—
165	C₄×D₈	—	15	2	no	—
166	#427	—	5	1	no	—
167	#416	—	15	2	no	—
168	C₂×C₄×C₈	—	3	1	no	—
169	C₂²×C₃×C₅	—	2	2	no	—
170	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
171	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
172	C₂×D₁₅	—	2	4	no	—
173	C₂×D₁₅	—	4	4	no	—
174	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
175	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
176	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
177	Dic₁₅	—	2	4	no	—
178	C₃×Dic₅	—	4	4	no	—
179	C₅×Dic₃	—	2	2	no	—
180	C₅×Dic₃	#135	1	4	yes	complemented
181	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
182	C₂×#67	—	10	2	no	—
183	C₂²×Dic₃	D₂₀	1	2	yes	—
184	#235	—	5	8	no	—
185	#235	—	10	4	no	—
186	C₂×D₁₂	—	5	2	no	—
187	C₂×C₄×D₃	—	10	2	no	—
188	C₂×C₄×D₃	—	10	4	no	—
189	C₂×Dic₆	—	5	2	no	—
190	C₂×C₃×C₈	C₄×D₅	1	4	yes	complemented
191	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
192	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
193	C₃×C₄⋊C₄	—	5	4	no	—
194	C₃×C₂²⋊C₄	—	10	2	no	—
195	C₃×C₄²	—	10	2	no	—
196	(C₂×D₃)⋊C₄	—	10	4	no	—
197	Dic₃⋊C₄	—	10	4	no	—
198	C₄×Dic₃	—	10	4	no	—
199	C₄×Dic₃	D₂₀	1	1	yes	characteristic
200	C₄×Dic₃	D₂₀	1	1	yes	characteristic
201	C₄×Dic₃	C₂²×D₅	1	2	yes	—
202	C₂×C₃⋊C₈	—	2	2	no	—
203	C₂×C₃⋊C₈	C₂²×D₅	1	2	yes	—
204	Dic₁₂	—	5	4	no	—
205	D₂₄	—	5	4	no	—
206	#204	—	5	8	no	—
207	C₂³×C₅	D₃×D₄	1	1	yes	characteristic
208	C₂²×D₅	—	6	2	no	—
209	C₂×C₄×C₅	—	3	2	no	—
210	C₂×C₄×C₅	—	3	2	no	—
211	C₂×C₄×C₅	—	6	4	no	—
212	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
213	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
214	C₂×C₄×C₅	#239	1	2	yes	—
215	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
216	C₂²⋊D₅	—	6	8	no	—
217	C₂×Dic₅	—	2	4	no	—
218	C₂×Dic₅	—	6	2	no	—
219	D₂₀	—	3	4	no	—
220	D₂₀	—	6	2	no	—
221	C₄×D₅	—	6	8	no	—
222	C₄×D₅	—	12	8	no	—
223	Dic₁₀	—	3	4	no	—
224	Dic₁₀	—	6	2	no	—
225	C₅×C₈	—	6	2	no	—
226	C₅×C₈	C₂×C₄×D₃	1	2	yes	—
227	C₅×C₈	C₂×C₄×D₃	1	2	yes	—
228	C₂×#41	—	15	2	no	—
229	#135	—	15	8	no	—
230	C₂×Q₁₆	—	15	1	no	—
231	C₂×SD₁₆	—	15	2	no	—
232	C₂×D₈	—	15	1	no	—
233	C₂²×C₈	—	3	1	no	—
234	C₂²×C₈	D₃×D₅	1	1	yes	characteristic
235	C₄×Q₈	—	15	4	no	—
236	C₄×D₄	—	15	4	no	—
237	C₄×D₄	—	30	4	no	—
238	#117	—	5	2	no	—
239	#117	—	30	2	no	—
240	C₂×C₄²	—	3	1	no	—
241	C₂×C₄²	—	30	2	no	—
242	#107	—	5	2	no	—
243	#106	—	5	2	no	—
244	#103	—	15	4	no	—
245	#102	—	15	4	no	—
246	C₄×C₈	—	3	2	no	—
247	C₄×C₈	—	3	2	no	—
248	C₂×C₃×C₅	—	2	2	no	—
249	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
250	C₂×C₃×C₅	C₂×#135	1	2	yes	—
251	D₁₅	—	4	4	no	—
252	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
253	C₂²×D₃	—	10	2	no	—
254	C₂×C₃×C₄	—	10	4	no	—
255	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
256	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
257	C₂×C₃×C₄	#709	1	2	yes	—
258	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
259	#67	—	10	8	no	—
260	C₂×Dic₃	—	2	4	no	—
261	C₂×Dic₃	—	10	2	no	—
262	C₂×Dic₃	C₂×D₂₀	1	2	yes	—
263	C₂×Dic₃	C₂×D₂₀	1	2	yes	—
264	D₁₂	—	5	4	no	—
265	D₁₂	—	10	2	no	—
266	C₄×D₃	—	10	8	no	—
267	C₄×D₃	—	20	8	no	—
268	Dic₆	—	5	4	no	—
269	Dic₆	—	10	2	no	—
270	C₃×C₈	C₂×C₄×D₅	1	2	yes	—
271	C₃×C₈	C₂×C₄×D₅	1	2	yes	—
272	C₃⋊C₈	—	2	2	no	—
273	C₂²×C₅	—	2	2	no	—
274	C₂²×C₅	#893	1	2	yes	—
275	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
276	C₂×D₅	—	6	4	no	—
277	C₂×D₅	—	12	4	no	—
278	C₄×C₅	—	3	4	no	—
279	C₄×C₅	—	6	2	no	—
280	C₄×C₅	#893	1	2	yes	—
281	C₄×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
282	C₄×C₅	C₂×#239	1	1	yes	characteristic
283	Dic₅	—	4	4	no	—
284	Dic₅	—	6	4	no	—
285	#41	—	15	8	no	—
286	#41	—	30	4	no	—
287	C₂×Q₈	—	15	2	no	—
288	C₂×D₄	—	15	2	no	—
289	C₂×D₄	—	30	2	no	—
290	C₂²×C₄	—	3	2	no	—
291	C₂²×C₄	—	30	4	no	—
292	C₂²×C₄	—	30	2	no	—
293	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
294	Q₁₆	—	15	4	no	—
295	SD₁₆	—	15	8	no	—
296	D₈	—	15	4	no	—
297	C₂×C₈	—	3	2	no	—
298	C₂×C₈	—	6	2	no	—
299	C₂×C₈	#1200	1	4	yes	complemented
300	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
301	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
302	C₄⋊C₄	—	5	4	no	—
303	C₄⋊C₄	—	30	4	no	—
304	C₂²⋊C₄	—	10	2	no	—
305	C₂²⋊C₄	—	30	4	no	—
306	C₄²	—	3	1	no	—
307	C₄²	—	3	1	no	—
308	C₄²	—	3	2	no	—
309	C₄²	—	10	2	no	—
310	C₄²	—	30	4	no	—
311	C₃×C₅	C₄×#135	1	1	yes	characteristic, complemented, Hall
312	C₂²×C₃	—	2	2	no	—
313	C₂²×C₃	C₄×D₂₀	1	2	yes	—
314	C₂²×C₃	C₂²×D₂₀	1	1	yes	characteristic
315	C₂×D₃	—	10	4	no	—
316	C₂×D₃	—	20	4	no	—
317	C₃×C₄	—	20	4	no	—
318	C₃×C₄	C₄×D₂₀	1	2	yes	—
319	C₃×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
320	C₃×C₄	C₂×#709	1	1	yes	characteristic
321	Dic₃	—	2	2	no	—
322	Dic₃	—	10	4	no	—
323	Dic₃	#4050	1	4	yes	complemented
324	C₂×C₅	—	2	2	no	—
325	C₂×C₅	C₂×#893	1	1	yes	characteristic
326	C₂×C₅	D₃×#135	1	2	yes	—
327	D₅	—	12	4	no	—
328	C₂³	—	30	2	no	—
329	C₂³	D₃×D₂₀	1	1	yes	characteristic
330	Q₈	—	15	4	no	—
331	Q₈	—	30	2	no	—
332	D₄	—	15	4	no	—
333	D₄	—	30	8	no	—
334	D₄	—	30	2	no	—
335	C₂×C₄	—	3	2	no	—
336	C₂×C₄	—	3	2	no	—
337	C₂×C₄	—	6	4	no	—
338	C₂×C₄	—	10	4	no	—
339	C₂×C₄	—	30	2	no	—
340	C₂×C₄	—	30	8	no	—
341	C₂×C₄	—	60	8	no	—
342	C₂×C₄	#6769	1	2	yes	—
343	C₂×C₄	D₃×D₂₀	1	1	yes	characteristic
344	C₂×C₄	C₂×#1200	1	2	yes	—
345	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
346	C₈	—	6	2	no	—
347	C₈	C₂×#1200	1	2	yes	—
348	C₈	C₂×#1200	1	2	yes	—
349	C₂×C₃	—	2	2	no	—
350	C₂×C₃	C₂×C₄×D₂₀	1	1	yes	characteristic
351	C₂×C₃	C₂×#4050	1	2	yes	—
352	D₃	—	20	4	no	—
353	C₅	#79165	1	1	yes	characteristic, complemented, Sylow
354	C₂²	—	2	2	no	—
355	C₂²	—	30	4	no	—
356	C₂²	—	60	4	no	—
357	C₂²	#91428	1	2	yes	—
358	C₂²	C₂×D₃×D₂₀	1	1	yes	characteristic
359	C₄	—	3	4	no	—
360	C₄	—	6	2	no	—
361	C₄	—	20	4	no	—
362	C₄	—	30	4	no	—
363	C₄	#91428	1	2	yes	—
364	C₄	C₂×#6769	1	1	yes	characteristic
365	C₄	C₂²×#1200	1	1	yes	characteristic
366	C₃	C₄×#4050	1	1	yes	characteristic, complemented, Sylow
367	C₂	—	2	2	no	—
368	C₂	—	60	4	no	—
369	C₂	D₃×#4050	1	2	yes	—
370	C₂	C₂×#91428	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₃×D₂₀
commutator	C₃×C₄×C₅	C₂³×C₄
Frattini	C₂×C₄	C₂²×D₃×D₅
Fitting	C₂²×C₃×C₅×C₈	C₂²
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂×#91793	C₂
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	C₂²×C₈	D₃×D₅
hypocenter	C₃×C₅	C₄×#135
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. #11754436	C₂	semidirect
3. C₂×C₅×C₈×Dic₃	C₂	semidirect
4. C₃×#65395	C₂	semidirect
5. #11748411	C₂	semidirect
6. #11747873	C₂	semidirect
7. #11747793	C₂	semidirect
8. #11747789	C₂	semidirect
13. #91793	C₄	semidirect
35. #65395	D₃	semidirect
42. #6817	C₂×C₄	semidirect
48. C₂×C₃×C₅×C₈	C₂×C₄	semidirect
51. Dic₆₀	C₂×C₄	semidirect
52. D₁₂₀	C₂×C₄	semidirect
53. #6706	C₂×C₄	semidirect
68. C₂×C₈×Dic₃	D₅	semidirect
146. C₂×C₅×C₈	C₄×D₃	semidirect
180. C₅×Dic₃	#135	semidirect
190. C₂×C₃×C₈	C₄×D₅	semidirect
299. C₂×C₈	#1200	semidirect
311. C₃×C₅	C₄×#135	semidirect
323. Dic₃	#4050	semidirect
353. C₅	#79165	semidirect
366. C₃	C₄×#4050	semidirect

Maximal subgroups

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

subgroup	index
C₂×#91793	2
#11754436	2
C₂×C₅×C₈×Dic₃	2
C₃×#65395	2
#11748411	2
#11747873	2
#11747793	2
#11747789	2
C₄×#4050	3
#79165	5