Subgroups — #14299897

Order 1920.

524 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11754120	C₂	1	1	yes	maximal, characteristic
2	#11751226	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#3957	C₂	1	1	yes	maximal, characteristic
4	#11748431	C₂	1	1	yes	maximal, characteristic, complemented
5	#11748421	C₂	1	1	yes	maximal, characteristic
6	C₅×#4559	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745977	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599826	—	3	1	no	maximal, Hall
9	C₂×C₃×#709	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×#6710	C₂²	1	1	yes	characteristic
12	C₂×#6710	C₂²	1	1	yes	characteristic
13	C₅×#926	C₂²	1	1	yes	characteristic
14	C₂×C₅×#208	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
16	#91284	—	2	2	no	—
17	#91283	—	2	2	no	—
18	D₅×#208	—	2	2	no	—
19	#91103	—	2	2	no	—
20	C₃×#3957	C₄	1	2	yes	complemented
21	C₃×#3957	C₄	1	2	yes	complemented
22	#90982	—	2	2	no	—
23	#90977	—	2	2	no	—
24	#70375	—	5	1	no	maximal, Hall
25	#66079	—	3	1	no	—
26	C₂×#3957	D₃	1	1	yes	characteristic
27	#65414	—	3	1	no	—
28	#65407	—	3	1	no	—
29	#65294	—	3	1	no	—
30	C₅×#344	—	3	1	no	—
31	#64752	—	3	1	no	—
32	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂×C₃×C₂²⋊D₅	—	2	1	no	—
34	C₂²×C₃×Dic₅	—	2	1	no	—
35	C₃×#709	—	2	1	no	—
36	C₃×#709	—	2	1	no	—
37	C₃×#709	C₂×C₄	1	2	yes	—
38	C₃×#709	D₄	1	2	yes	—
39	C₂×C₃×D₂₀	Q₈	1	1	yes	characteristic
40	C₂×C₃×C₄×D₅	—	2	1	no	—
41	C₂×C₃×Dic₁₀	Q₈	1	1	yes	characteristic
42	C₃×C₅×M₁₆	—	2	1	no	—
43	C₃×C₅×M₁₆	C₂×C₄	1	2	yes	—
44	C₂×C₃×C₅×C₈	—	2	1	no	—
45	#6710	—	2	1	no	—
46	#6710	—	2	1	no	—
47	#6710	—	2	1	no	—
48	#6710	D₄	1	2	yes	—
49	C₂×C₃⋊#168	—	2	1	no	—
50	C₂×C₃⋊#168	—	2	1	no	—
51	C₅×#217	—	2	1	no	—
52	C₅×C₄⋊Dic₃	—	2	1	no	—
53	C₄×C₅×Dic₃	—	2	1	no	—
54	C₅×#208	—	2	1	no	—
55	C₅×#208	—	2	1	no	—
56	C₂×C₅×C₃⋊C₈	D₄	1	2	yes	—
57	C₃×C₄×Dic₅	—	2	1	no	—
58	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
59	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
60	#6649	—	2	4	no	—
61	D₅×C₃⋊C₈	—	2	4	no	—
62	#5819	—	5	1	no	—
63	C₂×C₃×#104	—	5	1	no	—
64	#5236	—	5	1	no	—
65	#5125	—	5	1	no	—
66	#4924	—	5	1	no	—
67	#4559	D₅	1	1	yes	characteristic, complemented
68	#4523	—	5	1	no	—
69	C₂×#709	C₂×D₃	1	1	yes	characteristic
70	C₂×C₅×M₁₆	—	3	1	no	—
71	C₂×C₅×M₁₆	C₂×D₃	1	1	yes	characteristic
72	C₅×#117	—	3	1	no	—
73	C₂×C₄×Dic₅	C₂×D₃	1	1	yes	characteristic
74	C₂×#681	—	3	1	no	—
75	C₂×#681	—	3	1	no	—
76	#4053	—	6	2	no	—
77	D₅×M₁₆	—	6	2	no	—
78	#4047	—	6	2	no	—
79	#3957	Dic₃	1	2	yes	complemented
80	#3957	Dic₃	1	2	yes	complemented
81	#3951	—	6	2	no	—
82	#3946	—	6	2	no	—
83	#3937	—	6	2	no	—
84	#1861	—	15	1	no	Sylow
85	C₂³×C₃×C₅	#41	1	1	yes	characteristic
86	C₂²×C₃×D₅	—	2	1	no	—
87	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
90	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
91	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
92	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
93	C₂×C₅×Dic₃	—	4	1	no	—
94	C₃×C₂²⋊D₅	—	2	2	no	—
95	C₃×C₂²⋊D₅	—	2	2	no	—
96	C₂×C₃×Dic₅	—	2	1	no	—
97	C₂×C₃×Dic₅	—	2	1	no	—
98	C₂×C₃×Dic₅	—	2	1	no	—
99	C₂×C₃×Dic₅	—	4	2	no	—
100	C₃×D₂₀	—	2	1	no	—
101	C₃×D₂₀	C₄⋊C₄	1	2	yes	—
102	C₃×C₄×D₅	—	2	2	no	—
103	C₃×C₄×D₅	—	2	2	no	—
104	C₃×Dic₁₀	—	2	1	no	—
105	C₃×Dic₁₀	C₄⋊C₄	1	2	yes	—
106	C₃×C₅×C₈	—	2	2	no	—
107	C₃⋊#168	—	2	2	no	—
108	C₃⋊#168	—	4	1	no	—
109	C₅×C₃⋊C₈	—	2	2	no	—
110	C₂×C₃×#41	—	5	1	no	—
111	C₂×C₃×M₁₆	C₂×D₅	1	1	yes	characteristic
112	C₂×C₃×C₄²	—	5	1	no	—
113	#948	—	10	2	no	—
114	#948	—	10	2	no	—
115	#926	C₂×D₅	1	1	yes	characteristic
116	C₂×#208	—	5	1	no	—
117	C₂×#208	—	5	1	no	—
118	C₂×#208	—	10	2	no	—
119	C₂×#208	C₂×D₅	1	1	yes	characteristic
120	C₃×#104	—	5	2	no	—
121	C₃×#104	—	5	2	no	—
122	#837	—	10	2	no	—
123	#819	—	10	2	no	—
124	C₄⋊(C₃⋊C₈)	—	10	2	no	—
125	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
126	C₂×C₂²⋊D₅	—	2	1	no	—
127	C₂²×Dic₅	—	2	1	no	—
128	#709	—	2	1	no	—
129	#709	—	2	1	no	—
130	#709	D₁₂	1	2	yes	—
131	#709	C₂×Dic₃	1	2	yes	—
132	C₂×D₂₀	Dic₆	1	1	yes	characteristic
133	C₂×C₄×D₅	—	2	1	no	—
134	C₂×Dic₁₀	Dic₆	1	1	yes	characteristic
135	C₅×M₁₆	—	2	1	no	—
136	C₅×M₁₆	—	6	1	no	—
137	C₅×M₁₆	—	6	1	no	—
138	C₅×M₁₆	C₂×Dic₃	1	2	yes	—
139	C₂×C₅×C₈	—	2	1	no	—
140	C₂×C₅×C₈	—	3	2	no	—
141	C₅×C₄⋊C₄	—	6	1	no	—
142	C₅×C₂²⋊C₄	—	6	1	no	—
143	C₄²×C₅	—	6	1	no	—
144	C₄×Dic₅	—	2	1	no	—
145	C₄×Dic₅	C₂×Dic₃	1	1	yes	characteristic
146	C₄×Dic₅	C₂×Dic₃	1	1	yes	characteristic
147	#681	—	3	2	no	—
148	#681	—	6	1	no	—
149	#681	—	6	1	no	—
150	#681	—	6	1	no	—
151	C₂×#168	—	6	1	no	—
152	C₂×#168	—	6	1	no	—
153	#676	—	6	4	no	—
154	C₈×D₅	—	6	4	no	—
155	#568	—	15	1	no	—
156	#430	—	15	1	no	—
157	#423	—	15	1	no	—
158	#421	—	15	1	no	—
159	C₂×#104	—	5	1	no	—
160	#344	—	3	1	no	—
161	#339	—	15	1	no	—
162	C₂²×C₃×C₅	—	4	1	no	—
163	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
164	C₂²×C₃×C₅	C₂²⋊Q₈	1	1	yes	characteristic
165	C₂²×C₃×C₅	#123	1	1	yes	characteristic
166	C₂×C₃×D₅	—	2	2	no	—
167	C₂×C₃×D₅	—	8	1	no	—
168	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
169	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
170	C₃×C₄×C₅	#120	1	1	yes	characteristic
171	C₃×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
172	C₃×Dic₅	—	2	2	no	—
173	C₃×Dic₅	—	2	2	no	—
174	C₃×Dic₅	—	4	1	no	—
175	C₅×Dic₃	—	8	1	no	—
176	C₃×#41	—	5	2	no	—
177	C₃×#41	—	5	2	no	—
178	C₃×#41	—	10	1	no	—
179	C₃×#41	—	10	1	no	—
180	C₂×C₃×Q₈	—	5	1	no	—
181	C₂×C₃×D₄	—	5	1	no	—
182	C₂×C₃×D₄	—	10	1	no	—
183	C₂²×C₃×C₄	—	10	1	no	—
184	C₂²×C₃×C₄	—	10	1	no	—
185	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
186	C₃×M₁₆	—	2	1	no	—
187	C₃×M₁₆	C₄×D₅	1	2	yes	—
188	C₂×C₃×C₈	—	2	1	no	—
189	C₃×C₄²	—	5	1	no	—
190	C₃×C₄²	—	5	1	no	—
191	C₃×C₄²	—	10	1	no	—
192	#217	—	2	1	no	—
193	C₄⋊Dic₃	—	2	1	no	—
194	C₄×Dic₃	—	2	1	no	—
195	#208	—	2	1	no	—
196	#208	—	2	1	no	—
197	#208	—	5	2	no	—
198	#208	—	10	4	no	—
199	#208	—	10	1	no	—
200	#208	—	10	1	no	—
201	#208	—	10	1	no	—
202	C₂×C₃⋊C₈	—	10	4	no	—
203	C₂×C₃⋊C₈	—	10	1	no	—
204	C₂×C₃⋊C₈	—	10	1	no	—
205	C₂×C₃⋊C₈	D₂₀	1	2	yes	—
206	C₂³×C₅	#237	1	1	yes	characteristic
207	C₂²×D₅	—	2	1	no	—
208	C₂×C₄×C₅	—	12	1	no	—
209	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
210	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
211	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
212	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
213	C₂×C₄×C₅	#237	1	1	yes	characteristic
214	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
215	C₂²⋊D₅	—	2	2	no	—
216	C₂²⋊D₅	—	2	2	no	—
217	C₂×Dic₅	—	2	1	no	—
218	C₂×Dic₅	—	2	1	no	—
219	C₂×Dic₅	—	2	1	no	—
220	C₂×Dic₅	—	4	2	no	—
221	D₂₀	—	2	1	no	—
222	D₂₀	C₄⋊Dic₃	1	2	yes	—
223	C₄×D₅	—	2	2	no	—
224	C₄×D₅	—	2	2	no	—
225	Dic₁₀	—	2	1	no	—
226	Dic₁₀	C₄⋊Dic₃	1	2	yes	—
227	C₅×C₈	—	2	2	no	—
228	C₅×C₈	—	6	2	no	—
229	#168	—	6	2	no	—
230	#168	—	12	1	no	—
231	C₂×#41	—	5	1	no	—
232	#131	—	30	2	no	—
233	#131	—	30	2	no	—
234	C₂×M₁₆	—	3	1	no	—
235	C₂×M₁₆	—	15	1	no	—
236	C₂×M₁₆	—	15	1	no	—
237	C₂×M₁₆	—	30	2	no	—
238	C₂×M₁₆	D₃×D₅	1	1	yes	characteristic
239	#117	—	3	1	no	—
240	C₂×C₄²	—	5	1	no	—
241	#108	—	30	2	no	—
242	C₄⋊C₈	—	30	2	no	—
243	#104	—	5	2	no	—
244	#104	—	5	2	no	—
245	#104	—	30	2	no	—
246	C₂×C₃×C₅	—	2	1	no	—
247	C₂×C₃×C₅	—	2	1	no	—
248	C₂×C₃×C₅	—	2	1	no	—
249	C₂×C₃×C₅	#385	1	1	yes	characteristic
250	C₃×D₅	—	4	2	no	—
251	C₂³×C₃	—	10	1	no	—
252	C₂³×C₃	#709	1	1	yes	characteristic
253	C₃×Q₈	—	5	2	no	—
254	C₃×Q₈	—	10	1	no	—
255	C₃×D₄	—	5	2	no	—
256	C₃×D₄	—	10	2	no	—
257	C₃×D₄	—	10	2	no	—
258	C₃×D₄	—	10	1	no	—
259	C₂×C₃×C₄	—	10	1	no	—
260	C₂×C₃×C₄	—	10	1	no	—
261	C₂×C₃×C₄	—	10	2	no	—
262	C₂×C₃×C₄	—	10	2	no	—
263	C₂×C₃×C₄	—	10	1	no	—
264	C₂×C₃×C₄	—	20	2	no	—
265	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
266	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
267	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
268	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
269	C₂×C₃×C₄	#711	1	1	yes	characteristic
270	C₂×C₃×C₄	Q₈×D₅	1	1	yes	characteristic
271	C₂×Dic₃	—	4	1	no	—
272	C₃×C₈	—	2	2	no	—
273	C₃⋊C₈	—	2	2	no	—
274	C₃⋊C₈	—	10	2	no	—
275	C₃⋊C₈	—	20	1	no	—
276	C₂²×C₅	—	4	1	no	—
277	C₂²×C₅	#878	1	1	yes	characteristic
278	C₂²×C₅	#886	1	1	yes	characteristic
279	C₂²×C₅	D₄×Dic₃	1	1	yes	characteristic
280	C₂×D₅	—	2	2	no	—
281	C₂×D₅	—	8	1	no	—
282	C₄×C₅	—	24	1	no	—
283	C₄×C₅	#878	1	1	yes	characteristic
284	C₄×C₅	#882	1	1	yes	characteristic
285	C₄×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
286	C₄×C₅	D₄×Dic₃	1	1	yes	characteristic
287	Dic₅	—	2	2	no	—
288	Dic₅	—	2	2	no	—
289	Dic₅	—	4	1	no	—
290	#41	—	5	2	no	—
291	#41	—	5	2	no	—
292	#41	—	10	1	no	—
293	#41	—	10	1	no	—
294	C₂×Q₈	—	5	1	no	—
295	C₂×D₄	—	5	1	no	—
296	C₂×D₄	—	10	1	no	—
297	C₂²×C₄	—	10	1	no	—
298	C₂²×C₄	—	10	1	no	—
299	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
300	M₁₆	—	2	1	no	—
301	M₁₆	—	6	1	no	—
302	M₁₆	—	6	1	no	—
303	M₁₆	—	15	2	no	—
304	M₁₆	—	30	4	no	—
305	M₁₆	—	30	1	no	—
306	M₁₆	—	30	1	no	—
307	M₁₆	—	30	1	no	—
308	M₁₆	D₅×Dic₃	1	2	yes	—
309	C₂×C₈	—	2	1	no	—
310	C₂×C₈	—	3	2	no	—
311	C₂×C₈	—	30	4	no	—
312	C₂×C₈	—	30	1	no	—
313	C₂×C₈	—	30	1	no	—
314	C₄⋊C₄	—	6	1	no	—
315	C₂²⋊C₄	—	6	1	no	—
316	C₄²	—	5	1	no	—
317	C₄²	—	5	1	no	—
318	C₄²	—	6	1	no	—
319	C₄²	—	10	1	no	—
320	C₃×C₅	#1861	1	1	yes	characteristic, complemented, Hall
321	C₂²×C₃	—	4	1	no	—
322	C₂²×C₃	—	10	2	no	—
323	C₂²×C₃	—	40	1	no	—
324	C₂²×C₃	C₄×D₂₀	1	1	yes	characteristic
325	C₂²×C₃	#4029	1	1	yes	characteristic
326	C₂²×C₃	#4042	1	1	yes	characteristic
327	C₃×C₄	—	10	2	no	—
328	C₃×C₄	—	10	2	no	—
329	C₃×C₄	—	20	1	no	—
330	C₃×C₄	#4027	1	1	yes	characteristic
331	C₃×C₄	#4028	1	1	yes	characteristic
332	C₃×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
333	C₃×C₄	#4043	1	1	yes	characteristic
334	Dic₃	—	8	1	no	—
335	C₂×C₅	—	2	1	no	—
336	C₂×C₅	—	2	1	no	—
337	C₂×C₅	—	2	1	no	—
338	C₂×C₅	#4950	1	1	yes	characteristic
339	D₅	—	4	2	no	—
340	C₂³	—	10	1	no	—
341	C₂³	#6765	1	1	yes	characteristic
342	Q₈	—	5	2	no	—
343	Q₈	—	10	1	no	—
344	D₄	—	5	2	no	—
345	D₄	—	10	2	no	—
346	D₄	—	10	2	no	—
347	D₄	—	10	1	no	—
348	C₂×C₄	—	10	1	no	—
349	C₂×C₄	—	10	1	no	—
350	C₂×C₄	—	10	2	no	—
351	C₂×C₄	—	10	2	no	—
352	C₂×C₄	—	10	1	no	—
353	C₂×C₄	—	12	1	no	—
354	C₂×C₄	—	20	2	no	—
355	C₂×C₄	D₅×Dic₆	1	1	yes	characteristic
356	C₂×C₄	D₅×D₁₂	1	1	yes	characteristic
357	C₂×C₄	D₃×D₂₀	1	1	yes	characteristic
358	C₂×C₄	C₂×D₅×Dic₃	1	1	yes	characteristic
359	C₂×C₄	#6780	1	1	yes	characteristic
360	C₂×C₄	#6790	1	1	yes	characteristic
361	C₈	—	2	2	no	—
362	C₈	—	6	2	no	—
363	C₈	—	30	2	no	—
364	C₈	—	60	1	no	—
365	C₂×C₃	—	2	1	no	—
366	C₂×C₃	—	2	1	no	—
367	C₂×C₃	—	2	1	no	—
368	C₂×C₃	—	20	2	no	—
369	C₂×C₃	#64957	1	1	yes	characteristic
370	C₅	#70375	1	1	yes	characteristic, complemented, Sylow
371	C₂²	—	4	1	no	—
372	C₂²	—	10	2	no	—
373	C₂²	—	40	1	no	—
374	C₂²	#91414	1	1	yes	characteristic
375	C₂²	Dic₃×D₂₀	1	1	yes	characteristic
376	C₂²	#91431	1	1	yes	characteristic
377	C₄	—	10	2	no	—
378	C₄	—	10	2	no	—
379	C₄	—	20	1	no	—
380	C₄	—	24	1	no	—
381	C₄	D₅×C₄⋊Dic₃	1	1	yes	characteristic
382	C₄	#91422	1	1	yes	characteristic
383	C₄	#91476	1	1	yes	characteristic
384	C₄	#91559	1	1	yes	characteristic
385	C₃	#10599826	1	1	yes	characteristic, complemented, Sylow
386	C₂	—	2	1	no	—
387	C₂	—	2	1	no	—
388	C₂	—	2	1	no	—
389	C₂	—	20	2	no	—
390	C₂	#11748900	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#91476
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃×D₅
Fitting	C₂×C₃×C₅×M₁₆	C₂²
socle	C₂×C₃×C₅	#385
prime radical	C₂×C₃×#709	C₂²
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×M₁₆	D₃×D₅
hypocenter	C₃×C₅	#1861
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
2. #11751226	C₂	semidirect
4. #11748431	C₂	semidirect
6. C₅×#4559	C₂	semidirect
7. #11745977	C₂	semidirect
20. C₃×#3957	C₄	semidirect
21. C₃×#3957	C₄	semidirect
67. #4559	D₅	semidirect
79. #3957	Dic₃	semidirect
80. #3957	Dic₃	semidirect
320. C₃×C₅	#1861	semidirect
370. C₅	#70375	semidirect
385. C₃	#10599826	semidirect

Maximal subgroups

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

subgroup	index
#11754120	2
#11751226	2
C₂×C₃×#3957	2
#11748431	2
#11748421	2
C₅×#4559	2
#11745977	2
#10599826	3
#70375	5