Subgroups — #14417691

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×#91793	C₂	1	1	yes	maximal, characteristic
2	#11754496	C₂	1	2	yes	maximal, complemented
3	C₅×#5101	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#3950	C₂	1	1	yes	maximal, characteristic, complemented
5	#11748413	C₂	1	2	yes	maximal, complemented
6	#11747873	C₂	1	2	yes	maximal, complemented
7	#11747800	C₂	1	4	yes	maximal, complemented
8	#11747793	C₂	1	2	yes	maximal, complemented
9	#10610293	—	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	C₅×#926	C₂²	1	1	yes	characteristic
17	C₂×C₅×#208	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
19	#91552	—	2	4	no	—
20	#91437	—	2	2	no	—
21	#91428	C₂²	1	4	yes	—
22	#91347	—	2	2	no	—
23	#91340	C₂²	1	4	yes	—
24	C₅×#815	C₂²	1	2	yes	—
25	C₅×C₈×Dic₃	C₂²	1	2	yes	—
26	C₃×#3950	C₂²	1	2	yes	—
27	C₃×#3950	C₂²	1	2	yes	—
28	#90974	C₂²	1	4	yes	—
29	#90968	C₂²	1	4	yes	—
30	#79136	—	5	1	no	maximal, Hall
31	C₂×#4050	—	3	1	no	—
32	#65857	—	3	2	no	—
33	C₅×#405	—	3	1	no	—
34	#65422	—	3	2	no	—
35	C₂×#3950	D₃	1	1	yes	characteristic, complemented
36	#64999	—	3	4	no	—
37	C₄×Dic₂₀	—	3	2	no	—
38	C₄×D₄₀	—	3	2	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×C₃×Dic₅	—	2	2	no	—
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	C₅×#217	D₄	1	2	yes	—
55	C₅×C₄⋊Dic₃	D₄	1	1	yes	characteristic
56	C₅×C₄⋊Dic₃	D₄	1	1	yes	characteristic
57	C₄×C₅×Dic₃	C₂³	1	2	yes	—
58	C₅×#208	—	2	2	no	—
59	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
60	C₃×C₄⋊Dic₅	—	2	2	no	—
61	C₃×C₄⋊Dic₅	C₂³	1	4	yes	—
62	#6669	—	2	4	no	—
63	#6667	—	2	4	no	—
64	Dic₃×Dic₅	—	2	4	no	—
65	C₂×#904	—	5	1	no	—
66	#5504	—	5	2	no	—
67	C₂×C₃×#107	—	5	1	no	—
68	#5101	D₅	1	1	yes	characteristic, complemented
69	#4966	—	5	2	no	—
70	#4889	—	5	4	no	—
71	#4873	—	5	2	no	—
72	#4872	—	5	2	no	—
73	C₂×#709	—	3	2	no	—
74	C₂×C₅×M₁₆	—	3	1	no	—
75	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
76	C₅×#117	—	3	1	no	—
77	C₂×C₄⋊Dic₅	C₂×D₃	1	2	yes	—
78	C₂×Dic₂₀	—	3	1	no	—
79	#4050	—	3	8	no	—
80	C₂×D₄₀	—	3	1	no	—
81	C₂×#677	—	3	2	no	—
82	#4038	—	6	2	no	—
83	D₅×C₄⋊C₄	—	6	2	no	—
84	#4027	—	6	4	no	—
85	C₄×D₂₀	—	3	4	no	—
86	C₄×Dic₁₀	—	3	4	no	—
87	C₅×#97	—	3	2	no	—
88	C₄×C₅×C₈	—	3	2	no	—
89	#3953	—	3	4	no	—
90	#3950	C₂×D₃	1	2	yes	—
91	#3950	C₂×D₃	1	2	yes	—
92	#3948	—	3	4	no	—
93	#2951	—	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	C₂×D₄	1	2	yes	—
108	C₂×C₅×Dic₃	C₂×D₄	1	2	yes	—
109	C₂×C₃×Dic₅	—	2	4	no	—
110	C₂×C₃×Dic₅	—	8	2	no	—
111	#1200	—	8	4	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	C₂×C₃×C₄⋊C₄	—	5	2	no	—
118	C₄×#67	—	10	4	no	—
119	#926	C₂×D₅	1	1	yes	characteristic
120	C₂×#208	C₂×D₅	1	1	yes	characteristic
121	C₂×Dic₁₂	—	5	1	no	—
122	#904	—	5	8	no	—
123	C₂×D₂₄	—	5	1	no	—
124	C₂×#204	—	5	2	no	—
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	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	C₂²×D₃	1	1	yes	characteristic
136	C₂×C₂²⋊D₅	—	6	2	no	—
137	C₂²×Dic₅	—	2	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	C₅×M₁₆	—	6	2	no	—
145	C₂×C₅×C₈	—	3	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	C₅×C₄⋊C₄	—	3	1	no	—
150	C₅×C₄⋊C₄	—	3	1	no	—
151	C₅×C₂²⋊C₄	—	3	2	no	—
152	C₄²×C₅	—	3	2	no	—
153	#685	—	6	4	no	—
154	C₄⋊Dic₅	—	2	2	no	—
155	C₄⋊Dic₅	C₂²×D₃	1	4	yes	—
156	#683	—	6	4	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	#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	—	3	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×D₁₅	—	2	4	no	—
174	C₂×D₁₅	—	4	2	no	—
175	C₂×D₁₅	—	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	4	no	—
180	C₃×Dic₅	—	4	4	no	—
181	C₅×Dic₃	—	2	2	no	—
182	C₅×Dic₃	—	2	2	no	—
183	C₂²×C₃×C₄	—	10	2	no	—
184	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
185	C₂×#67	—	10	2	no	—
186	#235	—	5	8	no	—
187	#235	—	10	4	no	—
188	C₂×D₁₂	—	5	2	no	—
189	C₂×C₄×D₃	—	10	2	no	—
190	C₂×C₄×D₃	—	10	4	no	—
191	C₂×Dic₆	—	5	2	no	—
192	C₂×C₃×C₈	C₄×D₅	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	D₂₀	1	2	yes	—
198	(C₂×D₃)⋊C₄	—	10	4	no	—
199	C₄⋊Dic₃	D₂₀	1	1	yes	characteristic
200	C₄⋊Dic₃	D₂₀	1	1	yes	characteristic
201	Dic₃⋊C₄	—	10	4	no	—
202	C₄×Dic₃	—	10	4	no	—
203	C₄×Dic₃	C₂²×D₅	1	2	yes	—
204	#208	—	2	2	no	—
205	C₂×C₃⋊C₈	C₂²×D₅	1	2	yes	—
206	Dic₁₂	—	5	4	no	—
207	D₂₄	—	5	4	no	—
208	#204	—	5	8	no	—
209	C₂³×C₅	D₃×D₄	1	1	yes	characteristic
210	C₂²×D₅	—	6	2	no	—
211	C₂×C₄×C₅	—	3	2	no	—
212	C₂×C₄×C₅	—	3	2	no	—
213	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
214	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
215	C₂×C₄×C₅	#239	1	2	yes	—
216	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
217	C₂²⋊D₅	—	6	8	no	—
218	C₂×Dic₅	—	2	4	no	—
219	C₂×Dic₅	—	6	2	no	—
220	C₂×Dic₅	—	8	2	no	—
221	D₂₀	—	3	4	no	—
222	D₂₀	—	6	2	no	—
223	C₄×D₅	—	6	8	no	—
224	C₄×D₅	—	24	4	no	—
225	Dic₁₀	—	3	4	no	—
226	Dic₁₀	—	6	2	no	—
227	C₅×C₈	—	6	2	no	—
228	C₅×C₈	C₂×C₄×D₃	1	2	yes	—
229	C₅×C₈	C₂×C₄×D₃	1	2	yes	—
230	C₂×#41	—	15	2	no	—
231	#135	—	15	8	no	—
232	C₂×Q₁₆	—	15	1	no	—
233	C₂×SD₁₆	—	15	2	no	—
234	C₂×D₈	—	15	1	no	—
235	C₂×M₁₆	—	3	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	—	3	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	—	3	2	no	—
249	C₄×C₈	—	3	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	D₁₅	—	4	4	no	—
255	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
256	C₂²×D₃	—	10	2	no	—
257	C₂×C₃×C₄	—	10	4	no	—
258	C₂×C₃×C₄	—	40	2	no	—
259	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
260	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
261	C₂×C₃×C₄	#709	1	2	yes	—
262	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
263	#67	—	10	8	no	—
264	C₂×Dic₃	—	10	2	no	—
265	C₂×Dic₃	C₂×D₂₀	1	2	yes	—
266	C₂×Dic₃	C₂×D₂₀	1	2	yes	—
267	D₁₂	—	5	4	no	—
268	D₁₂	—	10	2	no	—
269	C₄×D₃	—	10	8	no	—
270	C₄×D₃	—	40	4	no	—
271	Dic₆	—	5	4	no	—
272	Dic₆	—	10	2	no	—
273	C₃×C₈	C₂×C₄×D₅	1	2	yes	—
274	C₃×C₈	C₂×C₄×D₅	1	2	yes	—
275	C₃⋊C₈	—	2	2	no	—
276	C₂²×C₅	—	2	1	no	—
277	C₂²×C₅	—	2	1	no	—
278	C₂²×C₅	#893	1	2	yes	—
279	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
280	C₂×D₅	—	6	4	no	—
281	C₂×D₅	—	12	2	no	—
282	C₂×D₅	—	12	2	no	—
283	C₄×C₅	—	6	2	no	—
284	C₄×C₅	—	6	2	no	—
285	C₄×C₅	#893	1	2	yes	—
286	C₄×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
287	C₄×C₅	C₂×#239	1	1	yes	characteristic
288	Dic₅	—	4	4	no	—
289	Dic₅	—	6	4	no	—
290	#41	—	15	8	no	—
291	#41	—	30	4	no	—
292	C₂×Q₈	—	15	2	no	—
293	C₂×D₄	—	15	2	no	—
294	C₂×D₄	—	30	2	no	—
295	C₂²×C₄	—	10	2	no	—
296	C₂²×C₄	—	30	4	no	—
297	C₂²×C₄	—	30	2	no	—
298	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
299	Q₁₆	—	15	4	no	—
300	SD₁₆	—	15	8	no	—
301	D₈	—	15	4	no	—
302	M₁₆	—	6	2	no	—
303	C₂×C₈	—	3	2	no	—
304	C₂×C₈	#1200	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	—	3	1	no	—
308	C₄⋊C₄	—	3	1	no	—
309	C₄⋊C₄	—	5	4	no	—
310	C₄⋊C₄	—	10	2	no	—
311	C₄⋊C₄	—	30	4	no	—
312	C₂²⋊C₄	—	3	2	no	—
313	C₂²⋊C₄	—	30	4	no	—
314	C₄²	—	3	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	C₄×D₂₀	1	2	yes	—
320	C₂²×C₃	C₂²×D₂₀	1	1	yes	characteristic
321	C₂×D₃	—	10	4	no	—
322	C₂×D₃	—	20	2	no	—
323	C₂×D₃	—	20	2	no	—
324	C₃×C₄	—	20	4	no	—
325	C₃×C₄	C₄×D₂₀	1	2	yes	—
326	C₃×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
327	C₃×C₄	C₂×#709	1	1	yes	characteristic
328	Dic₃	—	2	2	no	—
329	Dic₃	—	2	2	no	—
330	Dic₃	—	10	4	no	—
331	C₂×C₅	—	2	2	no	—
332	C₂×C₅	C₂×#893	1	1	yes	characteristic
333	C₂×C₅	#5757	1	1	yes	characteristic
334	C₂×C₅	#5766	1	1	yes	characteristic
335	D₅	—	12	4	no	—
336	C₂³	—	30	2	no	—
337	C₂³	D₃×D₂₀	1	1	yes	characteristic
338	Q₈	—	15	4	no	—
339	Q₈	—	30	2	no	—
340	D₄	—	15	4	no	—
341	D₄	—	30	8	no	—
342	D₄	—	30	2	no	—
343	C₂×C₄	—	3	2	no	—
344	C₂×C₄	—	3	2	no	—
345	C₂×C₄	—	10	4	no	—
346	C₂×C₄	—	30	2	no	—
347	C₂×C₄	—	30	8	no	—
348	C₂×C₄	—	40	2	no	—
349	C₂×C₄	—	120	4	no	—
350	C₂×C₄	#6769	1	2	yes	—
351	C₂×C₄	D₃×D₂₀	1	1	yes	characteristic
352	C₂×C₄	C₂×#1200	1	2	yes	—
353	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
354	C₈	—	6	2	no	—
355	C₈	C₂×#1200	1	2	yes	—
356	C₈	C₂×#1200	1	2	yes	—
357	C₂×C₃	—	2	2	no	—
358	C₂×C₃	C₂×C₄×D₂₀	1	1	yes	characteristic
359	C₂×C₃	#66086	1	1	yes	characteristic
360	C₂×C₃	#66087	1	1	yes	characteristic
361	D₃	—	20	4	no	—
362	C₅	#79136	1	1	yes	characteristic, complemented, Sylow
363	C₂²	—	2	1	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	30	4	no	—
366	C₂²	—	60	2	no	—
367	C₂²	—	60	2	no	—
368	C₂²	#91428	1	2	yes	—
369	C₂²	C₂×D₃×D₂₀	1	1	yes	characteristic
370	C₄	—	6	2	no	—
371	C₄	—	6	2	no	—
372	C₄	—	20	4	no	—
373	C₄	—	30	4	no	—
374	C₄	#91428	1	2	yes	—
375	C₄	C₂×#6769	1	1	yes	characteristic
376	C₄	C₂²×#1200	1	1	yes	characteristic
377	C₃	#10610293	1	1	yes	characteristic, complemented, Sylow
378	C₂	—	2	2	no	—
379	C₂	—	60	4	no	—
380	C₂	#11753971	1	1	yes	characteristic
381	C₂	#11753991	1	1	yes	characteristic
382	C₂	C₂×#91428	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×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	#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. #11754496	C₂	semidirect
3. C₅×#5101	C₂	semidirect
4. C₂×C₃×#3950	C₂	semidirect
5. #11748413	C₂	semidirect
6. #11747873	C₂	semidirect
7. #11747800	C₂	semidirect
8. #11747793	C₂	semidirect
13. #91793	C₄	semidirect
35. C₂×#3950	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. #5101	D₅	semidirect
146. C₂×C₅×C₈	C₄×D₃	semidirect
192. C₂×C₃×C₈	C₄×D₅	semidirect
304. C₂×C₈	#1200	semidirect
316. C₃×C₅	#2951	semidirect
362. C₅	#79136	semidirect
377. C₃	#10610293	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
#11754496	2
C₅×#5101	2
C₂×C₃×#3950	2
#11748413	2
#11747873	2
#11747800	2
#11747793	2
#10610293	3
#79136	5