Subgroups — #14458232

Order 1920.

888 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11756530	C₂	1	2	yes	maximal, complemented
2	#11756512	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#443	C₂	1	1	yes	maximal, characteristic, complemented
4	#11751244	C₂	1	2	yes	maximal, complemented
5	#11750997	C₂	1	2	yes	maximal, complemented
6	#11750927	C₂	1	4	yes	maximal, complemented
7	#11750801	C₂	1	1	yes	maximal, characteristic, complemented
8	#11750800	C₂	1	1	yes	maximal, characteristic, complemented
9	#10611626	—	3	1	no	maximal, Hall
10	#92101	C₂²	1	4	yes	—
11	C₃×C₅×#135	C₂²	1	1	yes	characteristic
12	C₃×C₅×#131	C₂²	1	2	yes	—
13	#91840	C₂²	1	4	yes	—
14	Q₁₆×D₁₅	—	2	2	no	—
15	#91804	—	2	4	no	—
16	#91801	—	2	2	no	—
17	#91798	—	2	4	no	—
18	#91798	C₂²	1	4	yes	—
19	C₂×Dic₆₀	—	2	2	no	—
20	C₂×Dic₆₀	C₂²	1	2	yes	—
21	C₂×Dic₆₀	C₂²	1	2	yes	—
22	#91793	—	2	2	no	—
23	#91793	C₂²	1	1	yes	characteristic
24	#91790	C₂²	1	2	yes	—
25	#91758	C₂²	1	2	yes	—
26	C₃×C₅×#108	C₂²	1	1	yes	characteristic
27	C₃×C₅×#104	C₂²	1	2	yes	—
28	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
29	#91108	C₂²	1	4	yes	—
30	#91103	C₂²	1	1	yes	characteristic
31	#91089	C₂²	1	2	yes	—
32	#79895	—	5	1	no	maximal, Hall
33	#66105	—	3	2	no	—
34	#66087	—	3	2	no	—
35	C₅×#443	D₃	1	1	yes	characteristic, complemented
36	#65432	—	3	2	no	—
37	#65185	—	3	2	no	—
38	#65115	—	3	4	no	—
39	#64989	—	3	1	no	—
40	#64988	—	3	1	no	—
41	C₃×C₅×#41	C₂³	1	2	yes	—
42	Q₈×D₁₅	—	2	4	no	—
43	#6819	—	2	4	no	—
44	#6819	—	4	4	no	—
45	#6817	—	2	4	no	—
46	#6817	C₂³	1	2	yes	—
47	C₂×Dic₃₀	—	2	4	no	—
48	C₂×Dic₃₀	C₂³	1	4	yes	—
49	C₃×C₅×Q₁₆	D₄	1	1	yes	characteristic
50	C₃×C₅×SD₁₆	D₄	1	2	yes	—
51	C₃×C₅×D₈	D₄	1	1	yes	characteristic
52	C₃×C₅×M₁₆	—	2	2	no	—
53	C₃×C₅×M₁₆	C₂³	1	2	yes	—
54	C₂×C₃×C₅×C₈	—	2	2	no	—
55	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
56	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
57	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
58	#6718	—	2	4	no	—
59	#6716	—	2	4	no	—
60	C₄⋊Dic₁₅	—	4	2	no	—
61	#6710	C₂³	1	2	yes	—
62	Dic₆₀	—	2	4	no	—
63	Dic₆₀	—	2	2	no	—
64	Dic₆₀	—	2	2	no	—
65	Dic₆₀	—	2	2	no	—
66	Dic₆₀	—	2	2	no	—
67	Dic₆₀	D₄	1	1	yes	characteristic
68	D₁₂₀	D₄	1	1	yes	characteristic
69	#6706	—	2	4	no	—
70	#6706	D₄	1	2	yes	—
71	#6705	—	2	2	no	—
72	C₈×D₁₅	—	2	2	no	—
73	#5771	—	5	2	no	—
74	#5753	—	5	2	no	—
75	C₃×#443	D₅	1	1	yes	characteristic, complemented
76	#5143	—	5	2	no	—
77	#4896	—	5	2	no	—
78	#4826	—	5	4	no	—
79	#4700	—	5	1	no	—
80	#4699	—	5	1	no	—
81	#4150	—	3	4	no	—
82	C₅×#135	C₂×D₃	1	1	yes	characteristic
83	C₅×#131	C₂×D₃	1	2	yes	—
84	#4097	—	3	4	no	—
85	D₅×Q₁₆	—	6	2	no	—
86	#4061	—	6	4	no	—
87	#4058	—	6	2	no	—
88	#4055	—	3	4	no	—
89	#4055	—	6	4	no	—
90	C₂×Dic₂₀	—	3	2	no	—
91	C₂×Dic₂₀	—	3	2	no	—
92	C₂×Dic₂₀	—	6	2	no	—
93	#4050	—	3	1	no	—
94	#4050	—	6	2	no	—
95	#4047	—	3	2	no	—
96	#4015	—	3	2	no	—
97	C₅×#108	C₂×D₃	1	1	yes	characteristic
98	C₅×#104	C₂×D₃	1	2	yes	—
99	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
100	#3956	—	3	4	no	—
101	#3951	—	3	1	no	—
102	#3937	—	3	2	no	—
103	#3293	—	15	1	no	Sylow
104	C₃×C₅×Q₈	C₂×D₄	1	2	yes	—
105	C₃×C₅×D₄	—	2	2	no	—
106	C₃×C₅×D₄	C₂×D₄	1	2	yes	—
107	C₂×C₃×C₄×C₅	—	2	1	no	—
108	C₂×C₃×C₄×C₅	—	2	2	no	—
109	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
110	#1220	—	2	2	no	—
111	#1220	—	4	4	no	—
112	C₂×Dic₁₅	—	2	4	no	—
113	C₂×Dic₁₅	—	4	4	no	—
114	D₆₀	C₂×D₄	1	2	yes	—
115	C₄×D₁₅	—	2	2	no	—
116	C₄×D₁₅	—	4	4	no	—
117	Dic₃₀	—	2	4	no	—
118	Dic₃₀	—	2	4	no	—
119	Dic₃₀	—	4	4	no	—
120	Dic₃₀	C₂×D₄	1	2	yes	—
121	C₃×C₅×C₈	—	2	2	no	—
122	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
123	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
124	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
125	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
126	C₃⋊#168	—	2	2	no	—
127	#1010	—	5	4	no	—
128	C₃×#135	C₂×D₅	1	1	yes	characteristic
129	C₃×#131	C₂×D₅	1	2	yes	—
130	#951	—	5	4	no	—
131	D₃×Q₁₆	—	10	2	no	—
132	#915	—	10	4	no	—
133	#912	—	10	2	no	—
134	#909	—	5	4	no	—
135	#909	—	10	4	no	—
136	C₂×Dic₁₂	—	5	2	no	—
137	C₂×Dic₁₂	—	5	2	no	—
138	C₂×Dic₁₂	—	10	2	no	—
139	#904	—	5	1	no	—
140	#904	—	10	2	no	—
141	#901	—	5	2	no	—
142	#869	—	5	2	no	—
143	C₃×#108	C₂×D₅	1	1	yes	characteristic
144	C₃×#104	C₂×D₅	1	2	yes	—
145	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
146	#824	—	5	4	no	—
147	#819	—	5	1	no	—
148	#805	—	5	2	no	—
149	C₅×#41	C₂²×D₃	1	2	yes	—
150	Q₈×D₅	—	6	4	no	—
151	#711	—	6	4	no	—
152	#711	—	12	4	no	—
153	#709	—	3	2	no	—
154	#709	—	6	4	no	—
155	C₂×Dic₁₀	—	3	4	no	—
156	C₂×Dic₁₀	—	6	4	no	—
157	C₅×Q₁₆	D₁₂	1	1	yes	characteristic
158	C₅×SD₁₆	D₁₂	1	2	yes	—
159	C₅×D₈	D₁₂	1	1	yes	characteristic
160	C₅×M₁₆	—	2	2	no	—
161	C₅×M₁₆	C₂²×D₃	1	2	yes	—
162	C₂×C₅×C₈	—	2	2	no	—
163	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
164	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
165	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
166	#689	—	6	4	no	—
167	#687	—	6	4	no	—
168	C₄⋊Dic₅	—	12	2	no	—
169	#681	—	3	2	no	—
170	Dic₂₀	—	3	1	no	—
171	Dic₂₀	—	6	4	no	—
172	Dic₂₀	—	6	2	no	—
173	Dic₂₀	—	6	2	no	—
174	Dic₂₀	—	6	2	no	—
175	Dic₂₀	—	6	2	no	—
176	D₄₀	—	3	1	no	—
177	#677	—	3	2	no	—
178	#677	—	6	4	no	—
179	#676	—	6	2	no	—
180	C₈×D₅	—	6	2	no	—
181	#578	—	15	2	no	—
182	#578	—	15	2	no	—
183	#494	—	15	1	no	—
184	#473	—	15	2	no	—
185	#473	—	15	2	no	—
186	#456	—	15	4	no	—
187	#443	—	15	1	no	—
188	#443	D₁₅	1	1	yes	characteristic, complemented
189	C₂²×C₃×C₅	—	2	2	no	—
190	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
191	C₂×D₁₅	—	2	2	no	—
192	C₃×C₄×C₅	—	2	2	no	—
193	C₃×C₄×C₅	—	2	2	no	—
194	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
195	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
196	Dic₁₅	—	2	2	no	—
197	Dic₁₅	—	4	4	no	—
198	C₃×#41	C₂²×D₅	1	2	yes	—
199	D₃×Q₈	—	10	4	no	—
200	#237	—	10	4	no	—
201	#237	—	20	4	no	—
202	#235	—	5	2	no	—
203	#235	—	10	4	no	—
204	C₂×Dic₆	—	5	4	no	—
205	C₂×Dic₆	—	10	4	no	—
206	C₃×Q₁₆	D₂₀	1	1	yes	characteristic
207	C₃×SD₁₆	D₂₀	1	2	yes	—
208	C₃×D₈	D₂₀	1	1	yes	characteristic
209	C₃×M₁₆	—	2	2	no	—
210	C₃×M₁₆	C₂²×D₅	1	2	yes	—
211	C₂×C₃×C₈	—	2	2	no	—
212	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
213	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
214	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
215	#216	—	10	4	no	—
216	#214	—	10	4	no	—
217	C₄⋊Dic₃	—	20	2	no	—
218	#208	—	5	2	no	—
219	Dic₁₂	—	5	1	no	—
220	Dic₁₂	—	10	4	no	—
221	Dic₁₂	—	10	2	no	—
222	Dic₁₂	—	10	2	no	—
223	Dic₁₂	—	10	2	no	—
224	Dic₁₂	—	10	2	no	—
225	D₂₄	—	5	1	no	—
226	#204	—	5	2	no	—
227	#204	—	10	4	no	—
228	ΓL(1,𝔽₂₅)	—	10	2	no	—
229	D₃×C₈	—	10	2	no	—
230	C₅×Q₈	C₂×D₁₂	1	2	yes	—
231	C₅×D₄	—	2	2	no	—
232	C₅×D₄	C₂×D₁₂	1	2	yes	—
233	C₂×C₄×C₅	—	2	1	no	—
234	C₂×C₄×C₅	—	2	2	no	—
235	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
236	C₂²⋊D₅	—	6	2	no	—
237	C₂²⋊D₅	—	12	4	no	—
238	C₂×Dic₅	—	6	4	no	—
239	C₂×Dic₅	—	12	4	no	—
240	D₂₀	—	3	2	no	—
241	C₄×D₅	—	6	2	no	—
242	C₄×D₅	—	12	4	no	—
243	Dic₁₀	—	3	2	no	—
244	Dic₁₀	—	6	4	no	—
245	Dic₁₀	—	6	4	no	—
246	Dic₁₀	—	12	4	no	—
247	C₅×C₈	—	2	2	no	—
248	C₅×C₈	C₂×D₁₂	1	1	yes	characteristic
249	C₅×C₈	C₂×D₁₂	1	1	yes	characteristic
250	C₅×C₈	D₃×D₄	1	1	yes	characteristic
251	C₅×C₈	D₃×D₄	1	1	yes	characteristic
252	#168	—	6	2	no	—
253	2₋¹⁺⁴	—	15	4	no	—
254	#137	—	15	4	no	—
255	#137	—	15	4	no	—
256	#137	—	30	4	no	—
257	#137	—	30	4	no	—
258	#135	—	15	1	no	—
259	#135	—	30	2	no	—
260	#135	—	30	2	no	—
261	#135	C₂×D₁₅	1	1	yes	characteristic
262	C₂×Q₁₆	—	15	2	no	—
263	C₂×Q₁₆	—	15	2	no	—
264	C₂×Q₁₆	—	30	2	no	—
265	C₂×Q₁₆	—	30	2	no	—
266	#131	—	15	2	no	—
267	#131	C₂×D₁₅	1	2	yes	—
268	C₄⋊Q₈	—	15	2	no	—
269	#108	—	15	1	no	—
270	#108	C₂×D₁₅	1	1	yes	characteristic
271	#104	—	15	2	no	—
272	#104	C₂×D₁₅	1	2	yes	—
273	#101	—	15	4	no	—
274	C₄×C₈	C₂×D₁₅	1	1	yes	characteristic
275	C₂×C₃×C₅	—	2	1	no	—
276	C₂×C₃×C₅	—	4	2	no	—
277	C₂×C₃×C₅	D₄²	1	1	yes	characteristic
278	D₁₅	—	4	2	no	—
279	C₃×Q₈	C₂×D₂₀	1	2	yes	—
280	C₃×D₄	—	2	2	no	—
281	C₃×D₄	C₂×D₂₀	1	2	yes	—
282	C₂×C₃×C₄	—	2	1	no	—
283	C₂×C₃×C₄	—	2	2	no	—
284	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
285	#67	—	10	2	no	—
286	#67	—	20	4	no	—
287	C₂×Dic₃	—	10	4	no	—
288	C₂×Dic₃	—	20	4	no	—
289	D₁₂	—	5	2	no	—
290	C₄×D₃	—	10	2	no	—
291	C₄×D₃	—	20	4	no	—
292	Dic₆	—	5	2	no	—
293	Dic₆	—	10	4	no	—
294	Dic₆	—	10	4	no	—
295	Dic₆	—	20	4	no	—
296	C₃×C₈	—	2	2	no	—
297	C₃×C₈	C₂×D₂₀	1	1	yes	characteristic
298	C₃×C₈	C₂×D₂₀	1	1	yes	characteristic
299	C₃×C₈	D₄×D₅	1	1	yes	characteristic
300	C₃×C₈	D₄×D₅	1	1	yes	characteristic
301	C₃⋊C₈	—	10	2	no	—
302	C₂²×C₅	—	2	2	no	—
303	C₂²×C₅	#1009	1	1	yes	characteristic
304	C₂×D₅	—	6	2	no	—
305	C₄×C₅	—	2	2	no	—
306	C₄×C₅	—	2	2	no	—
307	C₄×C₅	C₂²×D₁₂	1	1	yes	characteristic
308	C₄×C₅	C₂×D₃×D₄	1	1	yes	characteristic
309	Dic₅	—	6	2	no	—
310	Dic₅	—	12	4	no	—
311	#41	—	15	2	no	—
312	#41	—	30	4	no	—
313	#41	—	30	4	no	—
314	#41	—	60	4	no	—
315	#41	C₂²×D₁₅	1	2	yes	—
316	C₂×Q₈	—	15	4	no	—
317	C₂×Q₈	—	30	4	no	—
318	C₂×Q₈	—	30	4	no	—
319	Q₁₆	—	15	1	no	—
320	Q₁₆	—	30	4	no	—
321	Q₁₆	—	30	4	no	—
322	Q₁₆	—	30	2	no	—
323	Q₁₆	—	30	2	no	—
324	Q₁₆	—	30	2	no	—
325	Q₁₆	—	30	2	no	—
326	Q₁₆	D₆₀	1	1	yes	characteristic
327	SD₁₆	—	15	2	no	—
328	SD₁₆	—	30	4	no	—
329	SD₁₆	—	30	4	no	—
330	SD₁₆	D₆₀	1	2	yes	—
331	D₈	—	15	1	no	—
332	D₈	D₆₀	1	1	yes	characteristic
333	M₁₆	—	2	2	no	—
334	M₁₆	—	15	2	no	—
335	M₁₆	—	30	2	no	—
336	M₁₆	C₂²×D₁₅	1	2	yes	—
337	C₂×C₈	—	2	2	no	—
338	C₂×C₈	—	30	2	no	—
339	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
340	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
341	C₄⋊C₄	—	60	2	no	—
342	C₄²	C₂²×D₁₅	1	1	yes	characteristic
343	C₃×C₅	#3293	1	1	yes	characteristic, complemented, Hall
344	C₂²×C₃	—	2	2	no	—
345	C₂²×C₃	#4149	1	1	yes	characteristic
346	C₂×D₃	—	10	2	no	—
347	C₃×C₄	—	2	2	no	—
348	C₃×C₄	—	2	2	no	—
349	C₃×C₄	C₂²×D₂₀	1	1	yes	characteristic
350	C₃×C₄	C₂×D₄×D₅	1	1	yes	characteristic
351	Dic₃	—	10	2	no	—
352	Dic₃	—	20	4	no	—
353	C₂×C₅	—	2	1	no	—
354	C₂×C₅	—	4	2	no	—
355	C₂×C₅	D₄×D₁₂	1	1	yes	characteristic
356	D₅	—	12	2	no	—
357	Q₈	—	15	2	no	—
358	Q₈	—	30	4	no	—
359	Q₈	—	30	4	no	—
360	Q₈	—	60	4	no	—
361	Q₈	C₂×D₆₀	1	2	yes	—
362	D₄	—	2	2	no	—
363	D₄	—	15	2	no	—
364	D₄	—	30	2	no	—
365	D₄	—	60	4	no	—
366	D₄	C₂×D₆₀	1	2	yes	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	30	2	no	—
370	C₂×C₄	—	30	4	no	—
371	C₂×C₄	—	60	4	no	—
372	C₂×C₄	—	60	4	no	—
373	C₂×C₄	C₂³×D₁₅	1	1	yes	characteristic
374	C₈	—	2	2	no	—
375	C₈	—	30	2	no	—
376	C₈	C₂×D₆₀	1	1	yes	characteristic
377	C₈	C₂×D₆₀	1	1	yes	characteristic
378	C₈	D₄×D₁₅	1	1	yes	characteristic
379	C₈	D₄×D₁₅	1	1	yes	characteristic
380	C₂×C₃	—	2	1	no	—
381	C₂×C₃	—	4	2	no	—
382	C₂×C₃	D₄×D₂₀	1	1	yes	characteristic
383	D₃	—	20	2	no	—
384	C₅	#79895	1	1	yes	characteristic, complemented, Sylow
385	C₂²	—	2	2	no	—
386	C₂²	—	30	2	no	—
387	C₂²	#92100	1	1	yes	characteristic
388	C₄	—	2	2	no	—
389	C₄	—	2	2	no	—
390	C₄	—	30	2	no	—
391	C₄	—	60	4	no	—
392	C₄	C₂²×D₆₀	1	1	yes	characteristic
393	C₄	C₂×D₄×D₁₅	1	1	yes	characteristic
394	C₃	#10611626	1	1	yes	characteristic, complemented, Sylow
395	C₂	—	2	1	no	—
396	C₂	—	4	2	no	—
397	C₂	—	60	2	no	—
398	C₂	D₄×D₆₀	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₆₀
commutator, cosocle	C₂×C₃×C₄×C₅	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅
Fitting	C₃×C₅×#443	C₂
socle	C₂×C₃×C₅	D₄²
hypercenter	#443	D₁₅
hypocenter	C₃×C₅	#3293
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #443

Complements

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

N	H ≅ G/N	type
1. #11756530	C₂	semidirect
2. #11756512	C₂	semidirect
3. C₃×C₅×#443	C₂	semidirect
4. #11751244	C₂	semidirect
5. #11750997	C₂	semidirect
6. #11750927	C₂	semidirect
7. #11750801	C₂	semidirect
8. #11750800	C₂	semidirect
35. C₅×#443	D₃	semidirect
75. C₃×#443	D₅	semidirect
188. #443	D₁₅	semidirect
343. C₃×C₅	#3293	semidirect
384. C₅	#79895	semidirect
394. C₃	#10611626	semidirect

Maximal subgroups

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

subgroup	index
#11756530	2
#11756512	2
C₃×C₅×#443	2
#11751244	2
#11750997	2
#11750927	2
#11750801	2
#11750800	2
#10611626	3
#79895	5