Subgroups — #14417625

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	#11756263	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×C₅×#107	C₂	1	1	yes	maximal, characteristic, complemented
4	#11751202	C₂	1	1	yes	maximal, characteristic, complemented
5	#11751067	C₂	1	2	yes	maximal, complemented
6	#11750990	C₂	1	4	yes	maximal, complemented
7	#11750974	C₂	1	2	yes	maximal, complemented
8	#11750973	C₂	1	2	yes	maximal, complemented
9	#10610867	—	3	1	no	maximal, Hall
10	C₂×#6817	C₂²	1	2	yes	—
11	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
13	C₄×#1220	—	2	4	no	—
14	#91815	C₂²	1	1	yes	characteristic
15	C₂×#6710	C₂²	1	1	yes	characteristic
16	C₂×Dic₆₀	C₂²	1	1	yes	characteristic
17	#91793	C₄	1	8	yes	complemented
18	C₂×D₁₂₀	C₂²	1	1	yes	characteristic
19	C₂×#6706	C₂²	1	2	yes	—
20	#91782	C₂²	1	4	yes	—
21	#91781	—	2	2	no	—
22	C₄⋊C₄×D₁₅	—	2	2	no	—
23	#91776	C₂²	1	4	yes	—
24	C₃×C₅×#107	C₂²	1	2	yes	—
25	C₃×C₅×#107	C₂²	1	2	yes	—
26	#91099	C₂²	1	2	yes	—
27	C₈×Dic₁₅	C₂²	1	2	yes	—
28	#91094	C₂²	1	4	yes	—
29	#91093	C₂²	1	4	yes	—
30	#79136	—	5	1	no	maximal, Hall
31	C₂×#4050	—	3	1	no	—
32	#65838	—	3	2	no	—
33	C₂×C₅×#107	D₃	1	1	yes	characteristic, complemented
34	#65390	—	3	1	no	—
35	#65255	—	3	2	no	—
36	#65178	—	3	4	no	—
37	#65162	—	3	2	no	—
38	#65161	—	3	2	no	—
39	C₂²×C₃×C₄×C₅	—	2	2	no	—
40	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
41	C₂×#1220	—	2	2	no	—
42	#6817	—	2	4	no	—
43	#6817	C₂×C₄	1	8	yes	complemented
44	C₂×D₆₀	C₂³	1	2	yes	—
45	C₂×C₄×D₁₅	—	2	2	no	—
46	C₂×C₄×D₁₅	—	2	4	no	—
47	C₂×Dic₃₀	C₂³	1	2	yes	—
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	C₃×C₅×C₄⋊C₄	—	2	2	no	—
52	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
53	#6719	D₄	1	2	yes	—
54	#6714	—	2	4	no	—
55	C₄⋊Dic₁₅	D₄	1	1	yes	characteristic
56	C₄⋊Dic₁₅	D₄	1	1	yes	characteristic
57	#6712	—	2	4	no	—
58	C₄×Dic₁₅	—	2	4	no	—
59	C₄×Dic₁₅	C₂³	1	2	yes	—
60	#6710	—	2	2	no	—
61	C₂×C₃⋊#168	C₂³	1	2	yes	—
62	Dic₆₀	C₂×C₄	1	4	yes	complemented
63	D₁₂₀	C₂×C₄	1	4	yes	complemented
64	#6706	C₂×C₄	1	8	yes	complemented
65	C₂×#904	—	5	1	no	—
66	#5504	—	5	2	no	—
67	C₂×C₃×#107	D₅	1	1	yes	characteristic, complemented
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×#709	—	3	2	no	—
74	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
75	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
76	C₄×C₂²⋊D₅	—	6	4	no	—
77	#4072	—	3	1	no	—
78	C₂×#681	—	3	1	no	—
79	C₂×Dic₂₀	—	3	1	no	—
80	#4050	—	3	8	no	—
81	C₂×D₄₀	—	3	1	no	—
82	C₂×#677	—	3	2	no	—
83	#4039	—	3	4	no	—
84	#4038	—	6	2	no	—
85	D₅×C₄⋊C₄	—	6	2	no	—
86	#4033	—	3	4	no	—
87	C₅×#107	C₂×D₃	1	2	yes	—
88	C₅×#107	C₂×D₃	1	2	yes	—
89	#3947	—	3	2	no	—
90	C₈×Dic₅	—	3	2	no	—
91	#3942	—	3	4	no	—
92	#3941	—	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	—	2	4	no	—
97	C₂×C₃×C₄×C₅	—	8	2	no	—
98	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
99	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
100	C₂×C₃×C₄×C₅	#41	1	2	yes	—
101	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
102	#1220	—	2	8	no	—
103	C₂×Dic₁₅	—	2	2	no	—
104	C₂×Dic₁₅	C₂×D₄	1	2	yes	—
105	C₂×Dic₁₅	C₂×D₄	1	2	yes	—
106	D₆₀	—	2	2	no	—
107	D₆₀	C₂²×C₄	1	4	yes	—
108	C₄×D₁₅	—	2	8	no	—
109	C₄×D₁₅	—	8	4	no	—
110	Dic₃₀	—	2	2	no	—
111	Dic₃₀	C₂²×C₄	1	4	yes	—
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	—	5	2	no	—
116	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
117	C₂×C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
118	C₄×#67	—	10	4	no	—
119	#926	—	5	1	no	—
120	C₂×#208	—	5	1	no	—
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	C₂×D₅	1	2	yes	—
130	C₃×#107	C₂×D₅	1	2	yes	—
131	#815	—	5	2	no	—
132	C₈×Dic₃	—	5	2	no	—
133	#810	—	5	4	no	—
134	#809	—	5	4	no	—
135	C₂²×C₄×C₅	—	2	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	C₄×D₃	1	4	yes	complemented
145	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
146	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
147	C₅×C₄⋊C₄	—	2	2	no	—
148	C₅×C₄⋊C₄	C₂²×D₃	1	4	yes	—
149	C₂²⋊Dic₅	—	3	2	no	—
150	#685	—	6	4	no	—
151	C₄⋊Dic₅	—	3	1	no	—
152	C₄⋊Dic₅	—	3	1	no	—
153	#683	—	6	4	no	—
154	C₄×Dic₅	—	3	2	no	—
155	C₄×Dic₅	—	6	4	no	—
156	#681	—	6	2	no	—
157	C₂×#168	—	3	2	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	D₁₅	1	1	yes	characteristic, complemented
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×D₁₅	—	2	4	no	—
174	C₂×D₁₅	—	4	2	no	—
175	C₂×D₁₅	—	4	2	no	—
176	C₃×C₄×C₅	—	4	4	no	—
177	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
178	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
179	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
180	Dic₁₅	—	2	4	no	—
181	Dic₁₅	—	2	2	no	—
182	Dic₁₅	—	2	2	no	—
183	C₂²×C₃×C₄	—	2	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	—	2	2	no	—
196	C₃×C₄⋊C₄	C₂²×D₅	1	4	yes	—
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	—	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	—	2	4	no	—
212	C₂×C₄×C₅	—	8	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	—	3	2	no	—
219	C₂×Dic₅	—	3	2	no	—
220	C₂×Dic₅	—	6	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	C₂×C₄×D₃	1	2	yes	—
228	C₅×C₈	C₂×C₄×D₃	1	2	yes	—
229	#168	—	6	2	no	—
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	—	15	1	no	—
236	C₂²×C₈	C₂×D₁₅	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	—	30	2	no	—
243	C₂×C₄⋊C₄	C₂×D₁₅	1	2	yes	—
244	#107	C₂×D₁₅	1	2	yes	—
245	#107	C₂×D₁₅	1	2	yes	—
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	D₁₅	—	4	4	no	—
255	C₂³×C₃	D₄×D₅	1	1	yes	characteristic
256	C₂²×D₃	—	10	2	no	—
257	C₂×C₃×C₄	—	2	4	no	—
258	C₂×C₃×C₄	—	8	2	no	—
259	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
260	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
261	C₂×C₃×C₄	#713	1	2	yes	—
262	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
263	#67	—	10	8	no	—
264	C₂×Dic₃	—	5	2	no	—
265	C₂×Dic₃	—	5	2	no	—
266	C₂×Dic₃	—	10	2	no	—
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	—	10	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	—	4	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	—	6	2	no	—
288	Dic₅	—	6	4	no	—
289	Dic₅	—	6	2	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	—	2	2	no	—
296	C₂²×C₄	—	30	4	no	—
297	C₂²×C₄	—	30	2	no	—
298	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
299	Q₁₆	—	15	4	no	—
300	SD₁₆	—	15	8	no	—
301	D₈	—	15	4	no	—
302	M₁₆	—	30	2	no	—
303	C₂×C₈	—	15	2	no	—
304	C₂×C₈	C₄×D₁₅	1	4	yes	complemented
305	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
306	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	—	15	1	no	—
309	C₄⋊C₄	—	15	1	no	—
310	C₄⋊C₄	—	30	4	no	—
311	C₄⋊C₄	C₂²×D₁₅	1	4	yes	—
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	#4039	1	2	yes	—
320	C₂²×C₃	C₂×D₄×D₅	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	—	4	4	no	—
325	C₃×C₄	#4039	1	2	yes	—
326	C₃×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
327	C₃×C₄	C₂×#713	1	1	yes	characteristic
328	Dic₃	—	10	2	no	—
329	Dic₃	—	10	4	no	—
330	Dic₃	—	10	2	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	D₄×D₁₅	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	—	2	4	no	—
344	C₂×C₄	—	8	2	no	—
345	C₂×C₄	—	15	2	no	—
346	C₂×C₄	—	15	2	no	—
347	C₂×C₄	—	30	2	no	—
348	C₂×C₄	—	30	8	no	—
349	C₂×C₄	—	120	4	no	—
350	C₂×C₄	C₂×C₄×D₁₅	1	2	yes	—
351	C₂×C₄	D₄×D₁₅	1	1	yes	characteristic
352	C₂×C₄	#6821	1	2	yes	—
353	C₂×C₄	C₂³×D₁₅	1	1	yes	characteristic
354	C₈	—	30	2	no	—
355	C₈	C₂×C₄×D₁₅	1	2	yes	—
356	C₈	C₂×C₄×D₁₅	1	2	yes	—
357	C₂×C₃	—	2	2	no	—
358	C₂×C₃	C₂×#4039	1	1	yes	characteristic
359	C₂×C₃	#66091	1	1	yes	characteristic
360	C₂×C₃	#66100	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	#91782	1	2	yes	—
369	C₂²	C₂×D₄×D₁₅	1	1	yes	characteristic
370	C₄	—	4	4	no	—
371	C₄	—	30	2	no	—
372	C₄	—	30	4	no	—
373	C₄	—	30	2	no	—
374	C₄	#91782	1	2	yes	—
375	C₄	C₂²×C₄×D₁₅	1	1	yes	characteristic
376	C₄	C₂×#6821	1	1	yes	characteristic
377	C₃	#10610867	1	1	yes	characteristic, complemented, Sylow
378	C₂	—	2	2	no	—
379	C₂	—	60	4	no	—
380	C₂	C₂×#91782	1	1	yes	characteristic
381	C₂	#11756516	1	1	yes	characteristic
382	C₂	#11756525	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₅
commutator	C₃×C₄×C₅	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₅
Fitting	C₂×C₃×C₅×#107	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂×#91793	C₂
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	C₂×#107	D₁₅
hypocenter	C₃×C₅	#2951
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#107

Complements

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

N	H ≅ G/N	type
2. #11756263	C₂	semidirect
3. C₂×C₃×C₅×#107	C₂	semidirect
4. #11751202	C₂	semidirect
5. #11751067	C₂	semidirect
6. #11750990	C₂	semidirect
7. #11750974	C₂	semidirect
8. #11750973	C₂	semidirect
17. #91793	C₄	semidirect
33. C₂×C₅×#107	D₃	semidirect
43. #6817	C₂×C₄	semidirect
48. C₂×C₃×C₅×C₈	C₂×C₄	semidirect
62. Dic₆₀	C₂×C₄	semidirect
63. D₁₂₀	C₂×C₄	semidirect
64. #6706	C₂×C₄	semidirect
67. C₂×C₃×#107	D₅	semidirect
144. C₂×C₅×C₈	C₄×D₃	semidirect
166. C₂×#107	D₁₅	semidirect
192. C₂×C₃×C₈	C₄×D₅	semidirect
304. C₂×C₈	C₄×D₁₅	semidirect
316. C₃×C₅	#2951	semidirect
362. C₅	#79136	semidirect
377. C₃	#10610867	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
#11756263	2
C₂×C₃×C₅×#107	2
#11751202	2
#11751067	2
#11750990	2
#11750974	2
#11750973	2
#10610867	3
#79136	5