Subgroups — #14417689

Order 1920.

968 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#904	C₂	1	1	yes	maximal, characteristic
2	#11754492	C₂	1	2	yes	maximal, complemented
3	#11751202	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#3950	C₂	1	1	yes	maximal, characteristic, complemented
5	#11748384	C₂	1	2	yes	maximal, complemented
6	#11747886	C₂	1	2	yes	maximal, complemented
7	#11747816	C₂	1	4	yes	maximal, complemented
8	#11747810	C₂	1	2	yes	maximal, complemented
9	#10615629	—	3	1	no	maximal, Hall
10	C₂×C₅×#235	C₂²	1	2	yes	—
11	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
12	#91815	C₂²	1	1	yes	characteristic
13	C₂×#6710	C₂²	1	1	yes	characteristic
14	C₂×C₅×Dic₁₂	C₂²	1	1	yes	characteristic
15	C₅×#904	C₄	1	8	yes	complemented
16	C₂×C₅×D₂₄	C₂²	1	1	yes	characteristic
17	C₂×C₅×#204	C₂²	1	2	yes	—
18	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
19	Dic₅×#67	—	2	4	no	—
20	#91435	C₂²	1	4	yes	—
21	D₃×C₄⋊Dic₅	—	2	2	no	—
22	#91344	C₂²	1	4	yes	—
23	#91338	—	2	2	no	—
24	#91099	C₂²	1	2	yes	—
25	C₈×Dic₁₅	C₂²	1	2	yes	—
26	C₃×#3950	C₂²	1	2	yes	—
27	C₃×#3950	C₂²	1	2	yes	—
28	#90972	C₂²	1	4	yes	—
29	#90966	C₂²	1	4	yes	—
30	#79136	—	5	1	no	maximal, Hall
31	C₂×C₅×#135	—	3	1	no	—
32	#66161	—	3	2	no	—
33	#65521	—	3	2	no	—
34	Q₁₆×Dic₅	—	3	2	no	—
35	#65453	—	3	4	no	—
36	D₈×Dic₅	—	3	2	no	—
37	C₂×#3950	D₃	1	1	yes	characteristic, complemented
38	#65390	—	3	1	no	—
39	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
40	C₂×C₅×#67	—	2	2	no	—
41	C₅×#235	—	2	4	no	—
42	C₅×#235	C₂×C₄	1	8	yes	complemented
43	C₂×C₅×D₁₂	C₂³	1	2	yes	—
44	C₂×C₄×C₅×D₃	—	2	2	no	—
45	C₂×C₅×Dic₆	C₂³	1	2	yes	—
46	C₂²×C₃×Dic₅	—	2	2	no	—
47	C₂×D₃×Dic₅	—	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	#6719	D₄	1	2	yes	—
52	C₄⋊Dic₁₅	D₄	1	1	yes	characteristic
53	C₄⋊Dic₁₅	D₄	1	1	yes	characteristic
54	C₄×Dic₁₅	C₂³	1	2	yes	—
55	#6710	—	2	2	no	—
56	C₂×C₃⋊#168	C₂³	1	2	yes	—
57	C₅×Dic₁₂	C₂×C₄	1	4	yes	complemented
58	C₅×D₂₄	C₂×C₄	1	4	yes	complemented
59	C₅×#204	C₂×C₄	1	8	yes	complemented
60	C₃×C₄⋊Dic₅	—	2	2	no	—
61	C₃×C₄⋊Dic₅	C₂³	1	4	yes	—
62	Dic₃⋊Dic₅	—	2	4	no	—
63	(C₂×D₃)⋊Dic₅	—	2	4	no	—
64	Dic₃×Dic₅	—	2	4	no	—
65	C₂×#904	D₅	1	1	yes	characteristic
66	#5504	—	5	2	no	—
67	C₂×C₃×#107	—	5	1	no	—
68	#5101	—	5	1	no	—
69	#4966	—	5	2	no	—
70	#4889	—	5	4	no	—
71	#4873	—	5	2	no	—
72	#4872	—	5	2	no	—
73	C₂×C₅×#41	—	3	2	no	—
74	C₅×#135	—	3	8	no	—
75	C₂×C₅×Q₁₆	—	3	1	no	—
76	C₂×C₅×SD₁₆	—	3	2	no	—
77	C₂×C₅×D₈	—	3	1	no	—
78	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
79	Q₈×Dic₅	—	3	4	no	—
80	D₄×Dic₅	—	3	4	no	—
81	D₄×Dic₅	—	6	4	no	—
82	#4072	—	3	1	no	—
83	#4072	—	6	2	no	—
84	C₂×C₄⋊Dic₅	—	6	2	no	—
85	C₂×C₄⋊Dic₅	C₂×D₃	1	2	yes	—
86	C₂×#681	—	3	1	no	—
87	#3967	—	3	4	no	—
88	#3964	—	3	4	no	—
89	#3950	C₂×D₃	1	2	yes	—
90	#3950	C₂×D₃	1	2	yes	—
91	#3947	—	3	2	no	—
92	C₈×Dic₅	—	3	2	no	—
93	#2951	—	15	1	no	Sylow
94	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₅×D₃	—	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	C₂×Dic₁₅	C₂×D₄	1	2	yes	—
101	C₂×Dic₁₅	C₂×D₄	1	2	yes	—
102	C₅×#67	—	2	8	no	—
103	C₂×C₅×Dic₃	—	2	2	no	—
104	C₅×D₁₂	—	2	2	no	—
105	C₅×D₁₂	C₂²×C₄	1	4	yes	—
106	C₄×C₅×D₃	—	2	8	no	—
107	C₅×Dic₆	—	2	2	no	—
108	C₅×Dic₆	C₂²×C₄	1	4	yes	—
109	C₂×C₃×Dic₅	—	2	4	no	—
110	C₂×C₃×Dic₅	—	8	2	no	—
111	D₃×Dic₅	—	8	4	no	—
112	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
113	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
114	C₃⋊#168	—	2	2	no	—
115	C₂×#235	C₂×D₅	1	2	yes	—
116	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
117	C₂×C₃×C₄⋊C₄	—	5	2	no	—
118	C₄×#67	—	10	4	no	—
119	#926	—	5	1	no	—
120	C₂×#208	—	5	1	no	—
121	C₂×Dic₁₂	C₂×D₅	1	1	yes	characteristic
122	#904	Dic₅	1	8	yes	complemented
123	C₂×D₂₄	C₂×D₅	1	1	yes	characteristic
124	C₂×#204	C₂×D₅	1	2	yes	—
125	#893	—	5	4	no	—
126	#892	—	10	2	no	—
127	D₃×C₄⋊C₄	—	10	2	no	—
128	#887	—	5	4	no	—
129	C₃×#107	—	5	2	no	—
130	C₃×#107	—	5	2	no	—
131	#815	—	5	2	no	—
132	C₈×Dic₃	—	5	2	no	—
133	#810	—	5	4	no	—
134	#809	—	5	4	no	—
135	C₅×#41	—	3	8	no	—
136	C₅×#41	—	6	4	no	—
137	C₂×C₅×Q₈	—	3	2	no	—
138	C₂×C₅×D₄	—	3	2	no	—
139	C₂×C₅×D₄	—	6	2	no	—
140	C₂²×C₄×C₅	—	6	2	no	—
141	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
142	C₂²×Dic₅	—	2	2	no	—
143	C₂²×Dic₅	—	6	4	no	—
144	C₅×Q₁₆	—	3	4	no	—
145	C₅×SD₁₆	—	3	8	no	—
146	C₅×D₈	—	3	4	no	—
147	C₂×C₅×C₈	C₄×D₃	1	4	yes	complemented
148	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
149	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
150	C₂²⋊Dic₅	—	3	2	no	—
151	C₂²⋊Dic₅	—	6	4	no	—
152	C₄⋊Dic₅	—	2	2	no	—
153	C₄⋊Dic₅	—	3	1	no	—
154	C₄⋊Dic₅	—	3	1	no	—
155	C₄⋊Dic₅	—	6	4	no	—
156	C₄⋊Dic₅	C₂²×D₃	1	4	yes	—
157	C₄×Dic₅	—	3	2	no	—
158	C₄×Dic₅	—	6	4	no	—
159	#681	—	6	2	no	—
160	C₂×#168	—	3	2	no	—
161	C₂×#135	—	3	1	no	—
162	#520	—	15	2	no	—
163	#440	—	15	4	no	—
164	C₄×Q₁₆	—	15	2	no	—
165	C₄×D₈	—	15	2	no	—
166	C₂×#107	—	5	1	no	—
167	#417	—	15	2	no	—
168	#405	—	15	1	no	—
169	C₂²×C₃×C₅	—	2	1	no	—
170	C₂²×C₃×C₅	—	2	1	no	—
171	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
172	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
173	C₂×C₅×D₃	—	2	4	no	—
174	C₂×C₅×D₃	—	4	2	no	—
175	C₂×C₅×D₃	—	4	2	no	—
176	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
177	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
178	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
179	Dic₁₅	—	2	2	no	—
180	Dic₁₅	—	2	2	no	—
181	C₃×Dic₅	—	4	4	no	—
182	C₅×Dic₃	—	2	4	no	—
183	C₂²×C₃×C₄	—	10	2	no	—
184	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
185	C₂×#67	—	2	2	no	—
186	#235	—	2	4	no	—
187	#235	C₂×Dic₅	1	8	yes	complemented
188	C₂×D₁₂	C₂²×D₅	1	2	yes	—
189	C₂×C₄×D₃	—	2	2	no	—
190	C₂×C₄×D₃	—	10	4	no	—
191	C₂×Dic₆	C₂²×D₅	1	2	yes	—
192	C₂×C₃×C₈	C₂×Dic₅	1	4	yes	complemented
193	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
194	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
195	C₃×C₄⋊C₄	—	5	4	no	—
196	C₃×C₄⋊C₄	—	10	2	no	—
197	#217	—	5	2	no	—
198	(C₂×D₃)⋊C₄	—	10	4	no	—
199	C₄⋊Dic₃	—	5	1	no	—
200	C₄⋊Dic₃	—	5	1	no	—
201	Dic₃⋊C₄	—	10	4	no	—
202	C₄×Dic₃	—	5	2	no	—
203	C₄×Dic₃	—	10	4	no	—
204	#208	—	10	2	no	—
205	C₂×C₃⋊C₈	—	5	2	no	—
206	Dic₁₂	C₂×Dic₅	1	4	yes	complemented
207	D₂₄	C₂×Dic₅	1	4	yes	complemented
208	#204	C₂×Dic₅	1	8	yes	complemented
209	C₂³×C₅	—	6	2	no	—
210	C₂³×C₅	D₃×D₄	1	1	yes	characteristic
211	C₅×Q₈	—	3	4	no	—
212	C₅×Q₈	—	6	2	no	—
213	C₅×D₄	—	3	4	no	—
214	C₅×D₄	—	6	8	no	—
215	C₅×D₄	—	6	2	no	—
216	C₂×C₄×C₅	—	6	2	no	—
217	C₂×C₄×C₅	—	6	8	no	—
218	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
219	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
220	C₂×C₄×C₅	#239	1	2	yes	—
221	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
222	C₂×Dic₅	—	2	4	no	—
223	C₂×Dic₅	—	3	2	no	—
224	C₂×Dic₅	—	3	2	no	—
225	C₂×Dic₅	—	8	2	no	—
226	C₂×Dic₅	—	24	4	no	—
227	C₅×C₈	C₂×C₄×D₃	1	2	yes	—
228	C₅×C₈	C₂×C₄×D₃	1	2	yes	—
229	#168	—	6	2	no	—
230	C₂×#41	—	3	2	no	—
231	#135	—	3	8	no	—
232	C₂×Q₁₆	—	3	1	no	—
233	C₂×SD₁₆	—	3	2	no	—
234	C₂×D₈	—	3	1	no	—
235	C₂×M₁₆	—	15	1	no	—
236	C₂²×C₈	D₃×D₅	1	1	yes	characteristic
237	C₄×Q₈	—	15	4	no	—
238	C₄×D₄	—	15	4	no	—
239	C₄×D₄	—	30	4	no	—
240	#117	—	15	1	no	—
241	#117	—	30	2	no	—
242	C₂×C₄⋊C₄	—	5	2	no	—
243	C₂×C₄⋊C₄	—	30	2	no	—
244	#107	—	5	2	no	—
245	#107	—	5	2	no	—
246	#103	—	15	4	no	—
247	#102	—	15	4	no	—
248	#97	—	15	2	no	—
249	C₄×C₈	—	15	2	no	—
250	C₂×C₃×C₅	—	2	2	no	—
251	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
252	C₂×C₃×C₅	#576	1	1	yes	characteristic
253	C₂×C₃×C₅	#578	1	1	yes	characteristic
254	C₅×D₃	—	4	4	no	—
255	C₂³×C₃	D₄×D₅	1	1	yes	characteristic
256	C₂²×D₃	—	2	2	no	—
257	C₂×C₃×C₄	—	10	4	no	—
258	C₂×C₃×C₄	—	40	2	no	—
259	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
260	C₂×C₃×C₄	#711	1	2	yes	—
261	C₂×C₃×C₄	C₂²×Dic₅	1	2	yes	—
262	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
263	#67	—	2	8	no	—
264	C₂×Dic₃	—	2	2	no	—
265	C₂×Dic₃	—	5	2	no	—
266	C₂×Dic₃	—	5	2	no	—
267	D₁₂	—	2	2	no	—
268	D₁₂	C₂²×Dic₅	1	4	yes	—
269	C₄×D₃	—	2	8	no	—
270	C₄×D₃	—	40	4	no	—
271	Dic₆	—	2	2	no	—
272	Dic₆	C₂²×Dic₅	1	4	yes	—
273	C₃×C₈	C₂²×Dic₅	1	2	yes	—
274	C₃×C₈	C₂²×Dic₅	1	2	yes	—
275	C₃⋊C₈	—	10	2	no	—
276	C₂²×C₅	—	2	1	no	—
277	C₂²×C₅	—	2	1	no	—
278	C₂²×C₅	—	6	4	no	—
279	C₂²×C₅	—	12	2	no	—
280	C₂²×C₅	—	12	2	no	—
281	C₂²×C₅	#893	1	2	yes	—
282	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
283	C₄×C₅	—	6	4	no	—
284	C₄×C₅	#893	1	2	yes	—
285	C₄×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
286	C₄×C₅	C₂×#239	1	1	yes	characteristic
287	Dic₅	—	4	4	no	—
288	Dic₅	—	6	2	no	—
289	Dic₅	—	6	2	no	—
290	#41	—	3	8	no	—
291	#41	—	6	4	no	—
292	C₂×Q₈	—	3	2	no	—
293	C₂×D₄	—	3	2	no	—
294	C₂×D₄	—	6	2	no	—
295	C₂²×C₄	—	6	2	no	—
296	C₂²×C₄	—	10	2	no	—
297	C₂²×C₄	—	30	4	no	—
298	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
299	Q₁₆	—	3	4	no	—
300	SD₁₆	—	3	8	no	—
301	D₈	—	3	4	no	—
302	M₁₆	—	30	2	no	—
303	C₂×C₈	—	15	2	no	—
304	C₂×C₈	D₃×Dic₅	1	4	yes	complemented
305	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
306	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
307	C₄⋊C₄	—	5	4	no	—
308	C₄⋊C₄	—	10	2	no	—
309	C₄⋊C₄	—	15	1	no	—
310	C₄⋊C₄	—	15	1	no	—
311	C₄⋊C₄	—	30	4	no	—
312	C₂²⋊C₄	—	15	2	no	—
313	C₂²⋊C₄	—	30	4	no	—
314	C₄²	—	15	2	no	—
315	C₄²	—	30	4	no	—
316	C₃×C₅	#2951	1	1	yes	characteristic, complemented, Hall
317	C₂²×C₃	—	2	1	no	—
318	C₂²×C₃	—	2	1	no	—
319	C₂²×C₃	D₄×Dic₅	1	2	yes	—
320	C₂²×C₃	C₂×D₄×D₅	1	1	yes	characteristic
321	C₂×D₃	—	2	4	no	—
322	C₂×D₃	—	4	2	no	—
323	C₂×D₃	—	4	2	no	—
324	C₃×C₄	—	20	4	no	—
325	C₃×C₄	D₄×Dic₅	1	2	yes	—
326	C₃×C₄	C₂×#711	1	1	yes	characteristic
327	C₃×C₄	C₂³×Dic₅	1	1	yes	characteristic
328	Dic₃	—	2	4	no	—
329	Dic₃	—	10	2	no	—
330	Dic₃	—	10	2	no	—
331	C₂×C₅	—	2	2	no	—
332	C₂×C₅	—	12	4	no	—
333	C₂×C₅	C₂×#893	1	1	yes	characteristic
334	C₂×C₅	#5757	1	1	yes	characteristic
335	C₂×C₅	#5766	1	1	yes	characteristic
336	C₂³	—	6	2	no	—
337	C₂³	#6777	1	1	yes	characteristic
338	Q₈	—	3	4	no	—
339	Q₈	—	6	2	no	—
340	D₄	—	3	4	no	—
341	D₄	—	6	8	no	—
342	D₄	—	6	2	no	—
343	C₂×C₄	—	6	2	no	—
344	C₂×C₄	—	6	8	no	—
345	C₂×C₄	—	10	4	no	—
346	C₂×C₄	—	15	2	no	—
347	C₂×C₄	—	15	2	no	—
348	C₂×C₄	—	40	2	no	—
349	C₂×C₄	—	120	4	no	—
350	C₂×C₄	#6768	1	2	yes	—
351	C₂×C₄	#6777	1	1	yes	characteristic
352	C₂×C₄	C₂×D₃×Dic₅	1	2	yes	—
353	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
354	C₈	—	30	2	no	—
355	C₈	C₂×D₃×Dic₅	1	2	yes	—
356	C₈	C₂×D₃×Dic₅	1	2	yes	—
357	C₂×C₃	—	2	2	no	—
358	C₂×C₃	#66103	1	1	yes	characteristic
359	C₂×C₃	#66105	1	1	yes	characteristic
360	C₂×C₃	C₂×D₄×Dic₅	1	1	yes	characteristic
361	D₃	—	4	4	no	—
362	C₅	#79136	1	1	yes	characteristic, complemented, Sylow
363	C₂²	—	2	1	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	6	4	no	—
366	C₂²	—	12	2	no	—
367	C₂²	—	12	2	no	—
368	C₂²	#91435	1	2	yes	—
369	C₂²	C₂×#6777	1	1	yes	characteristic
370	C₄	—	6	4	no	—
371	C₄	—	20	4	no	—
372	C₄	—	30	2	no	—
373	C₄	—	30	2	no	—
374	C₄	#91435	1	2	yes	—
375	C₄	C₂×#6768	1	1	yes	characteristic
376	C₄	C₂²×D₃×Dic₅	1	1	yes	characteristic
377	C₃	#10615629	1	1	yes	characteristic, complemented, Sylow
378	C₂	—	2	2	no	—
379	C₂	—	12	4	no	—
380	C₂	#11753981	1	1	yes	characteristic
381	C₂	#11754011	1	1	yes	characteristic
382	C₂	C₂×#91435	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#6777
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×C₅×#904	C₂
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	C₂²×C₈	D₃×D₅
hypocenter	C₃×C₅	#2951
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. #11754492	C₂	semidirect
3. #11751202	C₂	semidirect
4. C₂×C₃×#3950	C₂	semidirect
5. #11748384	C₂	semidirect
6. #11747886	C₂	semidirect
7. #11747816	C₂	semidirect
8. #11747810	C₂	semidirect
15. C₅×#904	C₄	semidirect
37. C₂×#3950	D₃	semidirect
42. C₅×#235	C₂×C₄	semidirect
48. C₂×C₃×C₅×C₈	C₂×C₄	semidirect
57. C₅×Dic₁₂	C₂×C₄	semidirect
58. C₅×D₂₄	C₂×C₄	semidirect
59. C₅×#204	C₂×C₄	semidirect
122. #904	Dic₅	semidirect
147. C₂×C₅×C₈	C₄×D₃	semidirect
187. #235	C₂×Dic₅	semidirect
192. C₂×C₃×C₈	C₂×Dic₅	semidirect
206. Dic₁₂	C₂×Dic₅	semidirect
207. D₂₄	C₂×Dic₅	semidirect
208. #204	C₂×Dic₅	semidirect
304. C₂×C₈	D₃×Dic₅	semidirect
316. C₃×C₅	#2951	semidirect
362. C₅	#79136	semidirect
377. C₃	#10615629	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#904	2
#11754492	2
#11751202	2
C₂×C₃×#3950	2
#11748384	2
#11747886	2
#11747816	2
#11747810	2
#10615629	3
#79136	5