Subgroups — #14417797

Order 1920.

968 conjugacy classes of subgroups.

The subgroup lattice has 364 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×#3949	C₂	1	1	yes	maximal, characteristic, complemented
5	#11748384	C₂	1	2	yes	maximal, complemented
6	#11747888	C₂	1	2	yes	maximal, complemented
7	#11747813	C₂	1	2	yes	maximal, complemented
8	#11747807	C₂	1	4	yes	maximal, complemented
9	#10615630	—	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×#3949	C₂²	1	2	yes	—
27	C₃×#3949	C₂²	1	2	yes	—
28	#90972	C₂²	1	4	yes	—
29	#90966	C₂²	1	4	yes	—
30	#79106	—	5	1	no	maximal, Hall
31	C₂×C₅×#135	—	3	1	no	—
32	#66161	—	3	2	no	—
33	#65521	—	3	2	no	—
34	#65473	—	3	2	no	—
35	SD₁₆×Dic₅	—	3	4	no	—
36	#65441	—	3	2	no	—
37	C₂×#3949	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	2	yes	—
53	C₄×Dic₁₅	C₂³	1	2	yes	—
54	#6710	—	2	2	no	—
55	C₂×C₃⋊#168	C₂³	1	2	yes	—
56	C₅×Dic₁₂	C₂×C₄	1	4	yes	complemented
57	C₅×D₂₄	C₂×C₄	1	4	yes	complemented
58	C₅×#204	C₂×C₄	1	8	yes	complemented
59	C₃×C₄⋊Dic₅	—	2	2	no	—
60	C₃×C₄⋊Dic₅	C₂³	1	4	yes	—
61	Dic₃⋊Dic₅	—	2	4	no	—
62	(C₂×D₃)⋊Dic₅	—	2	4	no	—
63	Dic₃×Dic₅	—	2	4	no	—
64	C₂×#904	D₅	1	1	yes	characteristic
65	#5504	—	5	2	no	—
66	C₂×C₃×#106	—	5	1	no	—
67	#5101	—	5	1	no	—
68	#4966	—	5	2	no	—
69	#4869	—	5	2	no	—
70	#4853	—	5	2	no	—
71	#4852	—	5	4	no	—
72	C₂×C₅×#41	—	3	2	no	—
73	C₅×#135	—	3	8	no	—
74	C₂×C₅×Q₁₆	—	3	1	no	—
75	C₂×C₅×SD₁₆	—	3	2	no	—
76	C₂×C₅×D₈	—	3	1	no	—
77	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
78	Q₈×Dic₅	—	3	4	no	—
79	D₄×Dic₅	—	3	4	no	—
80	D₄×Dic₅	—	6	4	no	—
81	#4072	—	3	1	no	—
82	#4072	—	6	2	no	—
83	C₂×C₄⋊Dic₅	—	6	2	no	—
84	C₂×C₄⋊Dic₅	C₂×D₃	1	2	yes	—
85	C₂×#681	—	3	1	no	—
86	#3967	—	3	4	no	—
87	#3964	—	3	4	no	—
88	#3949	C₂×D₃	1	2	yes	—
89	#3949	C₂×D₃	1	2	yes	—
90	#3947	—	3	2	no	—
91	C₈×Dic₅	—	3	2	no	—
92	#2952	—	15	1	no	Sylow
93	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₅×D₃	—	2	2	no	—
95	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
96	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
97	C₂×C₃×C₄×C₅	#41	1	2	yes	—
98	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
99	C₂×Dic₁₅	C₂×D₄	1	4	yes	—
100	C₅×#67	—	2	8	no	—
101	C₂×C₅×Dic₃	—	2	2	no	—
102	C₅×D₁₂	—	2	2	no	—
103	C₅×D₁₂	C₂²×C₄	1	4	yes	—
104	C₄×C₅×D₃	—	2	8	no	—
105	C₅×Dic₆	—	2	2	no	—
106	C₅×Dic₆	C₂²×C₄	1	4	yes	—
107	C₂×C₃×Dic₅	—	2	4	no	—
108	C₂×C₃×Dic₅	—	8	2	no	—
109	D₃×Dic₅	—	8	4	no	—
110	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
111	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
112	C₃⋊#168	—	2	2	no	—
113	C₂×#235	C₂×D₅	1	2	yes	—
114	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
115	C₂×C₃×C₄⋊C₄	—	5	2	no	—
116	C₄×#67	—	10	4	no	—
117	#926	—	5	1	no	—
118	C₂×#208	—	5	1	no	—
119	C₂×Dic₁₂	C₂×D₅	1	1	yes	characteristic
120	#904	Dic₅	1	8	yes	complemented
121	C₂×D₂₄	C₂×D₅	1	1	yes	characteristic
122	C₂×#204	C₂×D₅	1	2	yes	—
123	#893	—	5	4	no	—
124	#892	—	10	2	no	—
125	D₃×C₄⋊C₄	—	10	2	no	—
126	#887	—	5	4	no	—
127	C₃×#106	—	5	2	no	—
128	C₃×#106	—	5	2	no	—
129	#815	—	5	2	no	—
130	C₈×Dic₃	—	5	2	no	—
131	#810	—	5	4	no	—
132	#809	—	5	4	no	—
133	C₅×#41	—	3	8	no	—
134	C₅×#41	—	6	4	no	—
135	C₂×C₅×Q₈	—	3	2	no	—
136	C₂×C₅×D₄	—	3	2	no	—
137	C₂×C₅×D₄	—	6	2	no	—
138	C₂²×C₄×C₅	—	6	2	no	—
139	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
140	C₂²×Dic₅	—	2	2	no	—
141	C₂²×Dic₅	—	6	4	no	—
142	C₅×Q₁₆	—	3	4	no	—
143	C₅×SD₁₆	—	3	8	no	—
144	C₅×D₈	—	3	4	no	—
145	C₂×C₅×C₈	C₄×D₃	1	4	yes	complemented
146	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
147	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
148	C₂²⋊Dic₅	—	3	2	no	—
149	C₂²⋊Dic₅	—	6	4	no	—
150	C₄⋊Dic₅	—	2	2	no	—
151	C₄⋊Dic₅	—	3	2	no	—
152	C₄⋊Dic₅	—	6	4	no	—
153	C₄⋊Dic₅	C₂²×D₃	1	4	yes	—
154	C₄×Dic₅	—	3	2	no	—
155	C₄×Dic₅	—	6	4	no	—
156	#681	—	6	2	no	—
157	C₂×#168	—	3	2	no	—
158	C₂×#135	—	3	1	no	—
159	#520	—	15	2	no	—
160	#442	—	15	2	no	—
161	#441	—	15	2	no	—
162	C₄×SD₁₆	—	15	4	no	—
163	C₂×#106	—	5	1	no	—
164	#417	—	15	2	no	—
165	#405	—	15	1	no	—
166	C₂²×C₃×C₅	—	2	2	no	—
167	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
168	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
169	C₂×C₅×D₃	—	2	4	no	—
170	C₂×C₅×D₃	—	4	2	no	—
171	C₂×C₅×D₃	—	4	2	no	—
172	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
173	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
174	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
175	Dic₁₅	—	2	4	no	—
176	C₃×Dic₅	—	4	4	no	—
177	C₅×Dic₃	—	2	4	no	—
178	C₂²×C₃×C₄	—	10	2	no	—
179	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
180	C₂×#67	—	2	2	no	—
181	#235	—	2	4	no	—
182	#235	C₂×Dic₅	1	8	yes	complemented
183	C₂×D₁₂	C₂²×D₅	1	2	yes	—
184	C₂×C₄×D₃	—	2	2	no	—
185	C₂×C₄×D₃	—	10	4	no	—
186	C₂×Dic₆	C₂²×D₅	1	2	yes	—
187	C₂×C₃×C₈	C₂×Dic₅	1	4	yes	complemented
188	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
189	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
190	C₃×C₄⋊C₄	—	5	4	no	—
191	C₃×C₄⋊C₄	—	10	2	no	—
192	#217	—	5	2	no	—
193	(C₂×D₃)⋊C₄	—	10	4	no	—
194	C₄⋊Dic₃	—	5	2	no	—
195	Dic₃⋊C₄	—	10	4	no	—
196	C₄×Dic₃	—	5	2	no	—
197	C₄×Dic₃	—	10	4	no	—
198	#208	—	10	2	no	—
199	C₂×C₃⋊C₈	—	5	2	no	—
200	Dic₁₂	C₂×Dic₅	1	4	yes	complemented
201	D₂₄	C₂×Dic₅	1	4	yes	complemented
202	#204	C₂×Dic₅	1	8	yes	complemented
203	C₂³×C₅	—	6	2	no	—
204	C₂³×C₅	D₃×D₄	1	1	yes	characteristic
205	C₅×Q₈	—	3	4	no	—
206	C₅×Q₈	—	6	2	no	—
207	C₅×D₄	—	3	4	no	—
208	C₅×D₄	—	6	8	no	—
209	C₅×D₄	—	6	2	no	—
210	C₂×C₄×C₅	—	6	2	no	—
211	C₂×C₄×C₅	—	6	8	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	C₂×Dic₅	—	2	4	no	—
217	C₂×Dic₅	—	3	4	no	—
218	C₂×Dic₅	—	8	2	no	—
219	C₂×Dic₅	—	24	4	no	—
220	C₅×C₈	C₂×C₄×D₃	1	2	yes	—
221	C₅×C₈	C₂×C₄×D₃	1	2	yes	—
222	#168	—	6	2	no	—
223	C₂×#41	—	3	2	no	—
224	#135	—	3	8	no	—
225	C₂×Q₁₆	—	3	1	no	—
226	C₂×SD₁₆	—	3	2	no	—
227	C₂×D₈	—	3	1	no	—
228	C₂×M₁₆	—	15	1	no	—
229	C₂²×C₈	D₃×D₅	1	1	yes	characteristic
230	C₄×Q₈	—	15	4	no	—
231	C₄×D₄	—	15	4	no	—
232	C₄×D₄	—	30	4	no	—
233	#117	—	15	1	no	—
234	#117	—	30	2	no	—
235	C₂×C₄⋊C₄	—	5	2	no	—
236	C₂×C₄⋊C₄	—	30	2	no	—
237	#106	—	5	2	no	—
238	#106	—	5	2	no	—
239	#103	—	15	4	no	—
240	#102	—	15	4	no	—
241	#97	—	15	2	no	—
242	C₄×C₈	—	15	2	no	—
243	C₂×C₃×C₅	—	2	2	no	—
244	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
245	C₂×C₃×C₅	#577	1	2	yes	—
246	C₅×D₃	—	4	4	no	—
247	C₂³×C₃	D₄×D₅	1	1	yes	characteristic
248	C₂²×D₃	—	2	2	no	—
249	C₂×C₃×C₄	—	10	4	no	—
250	C₂×C₃×C₄	—	40	2	no	—
251	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
252	C₂×C₃×C₄	#711	1	2	yes	—
253	C₂×C₃×C₄	C₂²×Dic₅	1	2	yes	—
254	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
255	#67	—	2	8	no	—
256	C₂×Dic₃	—	2	2	no	—
257	C₂×Dic₃	—	5	4	no	—
258	D₁₂	—	2	2	no	—
259	D₁₂	C₂²×Dic₅	1	4	yes	—
260	C₄×D₃	—	2	8	no	—
261	C₄×D₃	—	40	4	no	—
262	Dic₆	—	2	2	no	—
263	Dic₆	C₂²×Dic₅	1	4	yes	—
264	C₃×C₈	C₂²×Dic₅	1	2	yes	—
265	C₃×C₈	C₂²×Dic₅	1	2	yes	—
266	C₃⋊C₈	—	10	2	no	—
267	C₂²×C₅	—	2	2	no	—
268	C₂²×C₅	—	6	4	no	—
269	C₂²×C₅	—	12	2	no	—
270	C₂²×C₅	—	12	2	no	—
271	C₂²×C₅	#893	1	2	yes	—
272	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
273	C₄×C₅	—	6	4	no	—
274	C₄×C₅	#893	1	2	yes	—
275	C₄×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
276	C₄×C₅	C₂×#239	1	1	yes	characteristic
277	Dic₅	—	4	4	no	—
278	Dic₅	—	6	4	no	—
279	#41	—	3	8	no	—
280	#41	—	6	4	no	—
281	C₂×Q₈	—	3	2	no	—
282	C₂×D₄	—	3	2	no	—
283	C₂×D₄	—	6	2	no	—
284	C₂²×C₄	—	6	2	no	—
285	C₂²×C₄	—	10	2	no	—
286	C₂²×C₄	—	30	4	no	—
287	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
288	Q₁₆	—	3	4	no	—
289	SD₁₆	—	3	8	no	—
290	D₈	—	3	4	no	—
291	M₁₆	—	30	2	no	—
292	C₂×C₈	—	15	2	no	—
293	C₂×C₈	D₃×Dic₅	1	4	yes	complemented
294	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
295	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
296	C₄⋊C₄	—	5	4	no	—
297	C₄⋊C₄	—	10	2	no	—
298	C₄⋊C₄	—	15	2	no	—
299	C₄⋊C₄	—	30	4	no	—
300	C₂²⋊C₄	—	15	2	no	—
301	C₂²⋊C₄	—	30	4	no	—
302	C₄²	—	15	2	no	—
303	C₄²	—	30	4	no	—
304	C₃×C₅	#2952	1	1	yes	characteristic, complemented, Hall
305	C₂²×C₃	—	2	2	no	—
306	C₂²×C₃	D₄×Dic₅	1	2	yes	—
307	C₂²×C₃	C₂×D₄×D₅	1	1	yes	characteristic
308	C₂×D₃	—	2	4	no	—
309	C₂×D₃	—	4	2	no	—
310	C₂×D₃	—	4	2	no	—
311	C₃×C₄	—	20	4	no	—
312	C₃×C₄	D₄×Dic₅	1	2	yes	—
313	C₃×C₄	C₂×#711	1	1	yes	characteristic
314	C₃×C₄	C₂³×Dic₅	1	1	yes	characteristic
315	Dic₃	—	2	4	no	—
316	Dic₃	—	10	4	no	—
317	C₂×C₅	—	2	2	no	—
318	C₂×C₅	—	12	4	no	—
319	C₂×C₅	C₂×#893	1	1	yes	characteristic
320	C₂×C₅	#5762	1	2	yes	—
321	C₂³	—	6	2	no	—
322	C₂³	#6777	1	1	yes	characteristic
323	Q₈	—	3	4	no	—
324	Q₈	—	6	2	no	—
325	D₄	—	3	4	no	—
326	D₄	—	6	8	no	—
327	D₄	—	6	2	no	—
328	C₂×C₄	—	6	2	no	—
329	C₂×C₄	—	6	8	no	—
330	C₂×C₄	—	10	4	no	—
331	C₂×C₄	—	15	4	no	—
332	C₂×C₄	—	40	2	no	—
333	C₂×C₄	—	120	4	no	—
334	C₂×C₄	#6768	1	2	yes	—
335	C₂×C₄	#6777	1	1	yes	characteristic
336	C₂×C₄	C₂×D₃×Dic₅	1	2	yes	—
337	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
338	C₈	—	30	2	no	—
339	C₈	C₂×D₃×Dic₅	1	2	yes	—
340	C₈	C₂×D₃×Dic₅	1	2	yes	—
341	C₂×C₃	—	2	2	no	—
342	C₂×C₃	#66104	1	2	yes	—
343	C₂×C₃	C₂×D₄×Dic₅	1	1	yes	characteristic
344	D₃	—	4	4	no	—
345	C₅	#79106	1	1	yes	characteristic, complemented, Sylow
346	C₂²	—	2	2	no	—
347	C₂²	—	6	4	no	—
348	C₂²	—	12	2	no	—
349	C₂²	—	12	2	no	—
350	C₂²	#91435	1	2	yes	—
351	C₂²	C₂×#6777	1	1	yes	characteristic
352	C₄	—	6	4	no	—
353	C₄	—	20	4	no	—
354	C₄	—	30	4	no	—
355	C₄	#91435	1	2	yes	—
356	C₄	C₂×#6768	1	1	yes	characteristic
357	C₄	C₂²×D₃×Dic₅	1	1	yes	characteristic
358	C₃	#10615630	1	1	yes	characteristic, complemented, Sylow
359	C₂	—	2	2	no	—
360	C₂	—	12	4	no	—
361	C₂	#11753980	1	2	yes	—
362	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	#2952
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×#3949	C₂	semidirect
5. #11748384	C₂	semidirect
6. #11747888	C₂	semidirect
7. #11747813	C₂	semidirect
8. #11747807	C₂	semidirect
15. C₅×#904	C₄	semidirect
37. C₂×#3949	D₃	semidirect
42. C₅×#235	C₂×C₄	semidirect
48. C₂×C₃×C₅×C₈	C₂×C₄	semidirect
56. C₅×Dic₁₂	C₂×C₄	semidirect
57. C₅×D₂₄	C₂×C₄	semidirect
58. C₅×#204	C₂×C₄	semidirect
120. #904	Dic₅	semidirect
145. C₂×C₅×C₈	C₄×D₃	semidirect
182. #235	C₂×Dic₅	semidirect
187. C₂×C₃×C₈	C₂×Dic₅	semidirect
200. Dic₁₂	C₂×Dic₅	semidirect
201. D₂₄	C₂×Dic₅	semidirect
202. #204	C₂×Dic₅	semidirect
293. C₂×C₈	D₃×Dic₅	semidirect
304. C₃×C₅	#2952	semidirect
345. C₅	#79106	semidirect
358. C₃	#10615630	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×#3949	2
#11748384	2
#11747888	2
#11747813	2
#11747807	2
#10615630	3
#79106	5