Subgroups — #14303606

Order 1920.

372 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11751207	C₂	1	1	yes	maximal, characteristic
2	#11751203	C₂	1	1	yes	maximal, characteristic
3	#11751202	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#336	C₂	1	1	yes	maximal, characteristic
5	#11746373	C₂	1	1	yes	maximal, characteristic
6	#11746344	C₂	1	1	yes	maximal, characteristic
7	#11746344	C₂	1	1	yes	maximal, characteristic
8	#10599438	—	3	1	no	maximal, Hall
9	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	#91815	C₂²	1	1	yes	characteristic
13	#91815	C₂²	1	1	yes	characteristic
14	C₂×#6710	C₂²	1	1	yes	characteristic
15	C₂×#6710	C₂²	1	1	yes	characteristic
16	#91102	C₄	1	1	yes	characteristic, complemented
17	#91102	C₄	1	1	yes	characteristic, complemented
18	#91101	C₄	1	1	yes	characteristic, complemented
19	#91101	C₄	1	1	yes	characteristic, complemented
20	#91099	—	2	1	no	—
21	#91098	—	2	1	no	—
22	#91098	—	2	1	no	—
23	C₈×Dic₁₅	—	2	1	no	—
24	#68645	—	5	1	no	maximal, Hall
25	#65395	—	3	1	no	—
26	#65391	—	3	1	no	—
27	#65390	—	3	1	no	—
28	C₅×#336	D₃	1	1	yes	characteristic
29	#64772	—	3	1	no	—
30	#64743	—	3	1	no	—
31	#64743	—	3	1	no	—
32	C₂²×C₃×C₄×C₅	—	2	1	no	—
33	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
34	C₂×C₃×C₅×C₈	—	2	1	no	—
35	C₂×C₃×C₅×C₈	—	2	1	no	—
36	C₂×C₃×C₅×C₈	—	2	1	no	—
37	C₂×C₃×C₅×C₈	—	2	1	no	—
38	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
41	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
42	C₃×C₄²×C₅	—	2	1	no	—
43	C₃×C₄²×C₅	—	2	1	no	—
44	#6719	—	2	1	no	—
45	#6719	—	2	1	no	—
46	C₄⋊Dic₁₅	C₂×C₄	1	1	yes	characteristic
47	C₄⋊Dic₁₅	C₂×C₄	1	1	yes	characteristic
48	C₄⋊Dic₁₅	C₂×C₄	1	1	yes	characteristic
49	C₄⋊Dic₁₅	C₂×C₄	1	1	yes	characteristic
50	C₄×Dic₁₅	—	2	1	no	—
51	C₄×Dic₁₅	—	2	1	no	—
52	#6710	—	2	1	no	—
53	#6710	—	2	1	no	—
54	#6710	—	2	1	no	—
55	#6710	Q₈	1	2	yes	—
56	C₂×C₃⋊#168	—	2	1	no	—
57	C₂×C₃⋊#168	—	2	1	no	—
58	#5106	—	5	1	no	—
59	#5102	—	5	1	no	—
60	#5101	—	5	1	no	—
61	C₃×#336	D₅	1	1	yes	characteristic
62	#4552	—	5	1	no	—
63	#4523	—	5	1	no	—
64	#4523	—	5	1	no	—
65	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
66	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
67	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
68	#4072	—	3	1	no	—
69	#4072	—	3	1	no	—
70	C₂×#681	—	3	1	no	—
71	C₂×#681	—	3	1	no	—
72	#3950	—	3	1	no	—
73	#3950	—	3	1	no	—
74	#3949	—	3	1	no	—
75	#3949	—	3	1	no	—
76	#3947	—	6	1	no	—
77	#3946	—	6	1	no	—
78	#3946	—	6	1	no	—
79	C₈×Dic₅	—	6	1	no	—
80	#1922	—	15	1	no	Sylow
81	C₂³×C₃×C₅	#41	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	—	2	2	no	—
83	C₂×C₃×C₄×C₅	—	4	1	no	—
84	C₂×C₃×C₄×C₅	—	4	1	no	—
85	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
90	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
91	C₂×Dic₁₅	—	2	1	no	—
92	C₂×Dic₁₅	—	2	1	no	—
93	C₂×Dic₁₅	—	2	1	no	—
94	C₂×Dic₁₅	—	2	1	no	—
95	C₃×C₅×C₈	—	2	1	no	—
96	C₃×C₅×C₈	—	2	1	no	—
97	C₃×C₅×C₈	—	2	1	no	—
98	C₃×C₅×C₈	—	2	1	no	—
99	C₃⋊#168	—	2	2	no	—
100	C₃⋊#168	—	4	1	no	—
101	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
102	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
103	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
104	#926	—	5	1	no	—
105	#926	—	5	1	no	—
106	C₂×#208	—	5	1	no	—
107	C₂×#208	—	5	1	no	—
108	#818	—	5	1	no	—
109	#818	—	5	1	no	—
110	#817	—	5	1	no	—
111	#817	—	5	1	no	—
112	#815	—	10	1	no	—
113	Dic₃⋊C₈	—	10	1	no	—
114	Dic₃⋊C₈	—	10	1	no	—
115	C₈×Dic₃	—	10	1	no	—
116	C₂²×C₄×C₅	—	2	1	no	—
117	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
118	C₂×C₅×C₈	—	2	1	no	—
119	C₂×C₅×C₈	—	2	1	no	—
120	C₂×C₅×C₈	—	2	1	no	—
121	C₂×C₅×C₈	—	2	1	no	—
122	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
123	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
124	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
125	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
126	C₄²×C₅	—	2	1	no	—
127	C₄²×C₅	—	2	1	no	—
128	C₂²⋊Dic₅	—	6	1	no	—
129	C₂²⋊Dic₅	—	6	1	no	—
130	C₄⋊Dic₅	—	3	1	no	—
131	C₄⋊Dic₅	—	3	1	no	—
132	C₄⋊Dic₅	—	3	1	no	—
133	C₄⋊Dic₅	—	3	1	no	—
134	C₄×Dic₅	—	6	1	no	—
135	C₄×Dic₅	—	6	1	no	—
136	#681	—	3	2	no	—
137	#681	—	6	1	no	—
138	#681	—	6	1	no	—
139	#681	—	6	1	no	—
140	C₂×#168	—	6	1	no	—
141	C₂×#168	—	6	1	no	—
142	#427	—	15	1	no	—
143	#424	—	15	1	no	—
144	#405	—	15	1	no	—
145	#341	—	15	1	no	—
146	#339	—	15	1	no	—
147	#339	—	15	1	no	—
148	#336	D₁₅	1	1	yes	characteristic
149	C₂²×C₃×C₅	—	2	1	no	—
150	C₂²×C₃×C₅	—	2	1	no	—
151	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
152	C₂²×C₃×C₅	#125	1	1	yes	characteristic
153	C₂²×C₃×C₅	#126	1	1	yes	characteristic
154	C₃×C₄×C₅	—	4	2	no	—
155	C₃×C₄×C₅	#117	1	1	yes	characteristic
156	C₃×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
157	C₃×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
158	C₃×C₄×C₅	#123	1	1	yes	characteristic
159	Dic₁₅	—	4	1	no	—
160	Dic₁₅	—	4	1	no	—
161	Dic₁₅	—	4	1	no	—
162	Dic₁₅	—	4	1	no	—
163	C₂²×C₃×C₄	—	2	1	no	—
164	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
165	C₂×C₃×C₈	—	2	1	no	—
166	C₂×C₃×C₈	—	2	1	no	—
167	C₂×C₃×C₈	—	2	1	no	—
168	C₂×C₃×C₈	—	2	1	no	—
169	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
170	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
171	C₂×C₃×C₈	D₂₀	1	1	yes	characteristic
172	C₂×C₃×C₈	D₂₀	1	1	yes	characteristic
173	C₃×C₄²	—	2	1	no	—
174	C₃×C₄²	—	2	1	no	—
175	#217	—	10	1	no	—
176	#217	—	10	1	no	—
177	C₄⋊Dic₃	—	5	1	no	—
178	C₄⋊Dic₃	—	5	1	no	—
179	C₄⋊Dic₃	—	5	1	no	—
180	C₄⋊Dic₃	—	5	1	no	—
181	C₄×Dic₃	—	10	1	no	—
182	C₄×Dic₃	—	10	1	no	—
183	#208	—	5	2	no	—
184	#208	—	10	1	no	—
185	#208	—	10	1	no	—
186	#208	—	10	1	no	—
187	C₂×C₃⋊C₈	—	10	1	no	—
188	C₂×C₃⋊C₈	—	10	1	no	—
189	C₂³×C₅	#235	1	1	yes	characteristic
190	C₂×C₄×C₅	—	2	2	no	—
191	C₂×C₄×C₅	—	4	1	no	—
192	C₂×C₄×C₅	—	4	1	no	—
193	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
194	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
195	C₂×C₄×C₅	#237	1	1	yes	characteristic
196	C₂×C₄×C₅	#237	1	1	yes	characteristic
197	C₂×C₄×C₅	#237	1	1	yes	characteristic
198	C₂×C₄×C₅	D₃×Q₈	1	1	yes	characteristic
199	C₂×Dic₅	—	6	1	no	—
200	C₂×Dic₅	—	6	1	no	—
201	C₂×Dic₅	—	6	1	no	—
202	C₂×Dic₅	—	6	1	no	—
203	C₅×C₈	—	2	1	no	—
204	C₅×C₈	—	2	1	no	—
205	C₅×C₈	—	2	1	no	—
206	C₅×C₈	—	2	1	no	—
207	#168	—	6	2	no	—
208	#168	—	12	1	no	—
209	C₂×M₁₆	—	15	1	no	—
210	C₂×M₁₆	—	15	1	no	—
211	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
212	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
213	#117	—	15	1	no	—
214	#117	—	15	1	no	—
215	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
216	#107	—	15	1	no	—
217	#107	—	15	1	no	—
218	#106	—	15	1	no	—
219	#106	—	15	1	no	—
220	C₄⋊C₈	—	30	1	no	—
221	C₄⋊C₈	—	30	1	no	—
222	#97	—	30	1	no	—
223	C₄×C₈	—	30	1	no	—
224	C₂×C₃×C₅	—	2	1	no	—
225	C₂×C₃×C₅	—	2	1	no	—
226	C₂×C₃×C₅	#387	1	1	yes	characteristic
227	C₂×C₃×C₅	#443	1	1	yes	characteristic
228	C₂×C₃×C₅	#444	1	1	yes	characteristic
229	C₂³×C₃	#709	1	1	yes	characteristic
230	C₂×C₃×C₄	—	2	2	no	—
231	C₂×C₃×C₄	—	4	1	no	—
232	C₂×C₃×C₄	—	4	1	no	—
233	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
234	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
235	C₂×C₃×C₄	#711	1	1	yes	characteristic
236	C₂×C₃×C₄	#711	1	1	yes	characteristic
237	C₂×C₃×C₄	#711	1	1	yes	characteristic
238	C₂×C₃×C₄	Q₈×D₅	1	1	yes	characteristic
239	C₂×Dic₃	—	10	1	no	—
240	C₂×Dic₃	—	10	1	no	—
241	C₂×Dic₃	—	10	1	no	—
242	C₂×Dic₃	—	10	1	no	—
243	C₃×C₈	—	2	1	no	—
244	C₃×C₈	—	2	1	no	—
245	C₃×C₈	—	2	1	no	—
246	C₃×C₈	—	2	1	no	—
247	C₃⋊C₈	—	10	2	no	—
248	C₃⋊C₈	—	20	1	no	—
249	C₂²×C₅	—	2	1	no	—
250	C₂²×C₅	—	2	1	no	—
251	C₂²×C₅	C₄×D₁₂	1	1	yes	characteristic
252	C₂²×C₅	#879	1	1	yes	characteristic
253	C₂²×C₅	#889	1	1	yes	characteristic
254	C₄×C₅	—	4	2	no	—
255	C₄×C₅	#877	1	1	yes	characteristic
256	C₄×C₅	#886	1	1	yes	characteristic
257	C₄×C₅	#887	1	1	yes	characteristic
258	C₄×C₅	#897	1	1	yes	characteristic
259	Dic₅	—	12	1	no	—
260	Dic₅	—	12	1	no	—
261	Dic₅	—	12	1	no	—
262	Dic₅	—	12	1	no	—
263	C₂²×C₄	—	2	1	no	—
264	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
265	M₁₆	—	15	2	no	—
266	M₁₆	—	30	1	no	—
267	M₁₆	—	30	1	no	—
268	M₁₆	—	30	1	no	—
269	C₂×C₈	—	2	1	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	2	1	no	—
272	C₂×C₈	—	2	1	no	—
273	C₂×C₈	—	30	1	no	—
274	C₂×C₈	—	30	1	no	—
275	C₂×C₈	C₄×D₁₅	1	1	yes	characteristic
276	C₂×C₈	C₄×D₁₅	1	1	yes	characteristic
277	C₂×C₈	D₆₀	1	1	yes	characteristic
278	C₂×C₈	D₆₀	1	1	yes	characteristic
279	C₄⋊C₄	—	15	1	no	—
280	C₄⋊C₄	—	15	1	no	—
281	C₄⋊C₄	—	15	1	no	—
282	C₄⋊C₄	—	15	1	no	—
283	C₂²⋊C₄	—	30	1	no	—
284	C₂²⋊C₄	—	30	1	no	—
285	C₄²	—	2	1	no	—
286	C₄²	—	2	1	no	—
287	C₄²	—	30	1	no	—
288	C₄²	—	30	1	no	—
289	C₃×C₅	#1922	1	1	yes	characteristic, complemented, Hall
290	C₂²×C₃	—	2	1	no	—
291	C₂²×C₃	—	2	1	no	—
292	C₂²×C₃	C₄×D₂₀	1	1	yes	characteristic
293	C₂²×C₃	#4025	1	1	yes	characteristic
294	C₂²×C₃	#4035	1	1	yes	characteristic
295	C₃×C₄	—	4	2	no	—
296	C₃×C₄	#4023	1	1	yes	characteristic
297	C₃×C₄	#4032	1	1	yes	characteristic
298	C₃×C₄	#4033	1	1	yes	characteristic
299	C₃×C₄	#4043	1	1	yes	characteristic
300	Dic₃	—	20	1	no	—
301	Dic₃	—	20	1	no	—
302	Dic₃	—	20	1	no	—
303	Dic₃	—	20	1	no	—
304	C₂×C₅	—	2	1	no	—
305	C₂×C₅	—	2	1	no	—
306	C₂×C₅	#4653	1	1	yes	characteristic
307	C₂×C₅	#4700	1	1	yes	characteristic
308	C₂×C₅	#4717	1	1	yes	characteristic
309	C₂³	#6817	1	1	yes	characteristic
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	30	1	no	—
314	C₂×C₄	—	30	1	no	—
315	C₂×C₄	—	30	1	no	—
316	C₂×C₄	—	30	1	no	—
317	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
318	C₂×C₄	C₂×D₆₀	1	1	yes	characteristic
319	C₂×C₄	#6819	1	1	yes	characteristic
320	C₂×C₄	#6819	1	1	yes	characteristic
321	C₂×C₄	#6819	1	1	yes	characteristic
322	C₂×C₄	Q₈×D₁₅	1	1	yes	characteristic
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	—	2	1	no	—
326	C₈	—	2	1	no	—
327	C₈	—	30	2	no	—
328	C₈	—	60	1	no	—
329	C₂×C₃	—	2	1	no	—
330	C₂×C₃	—	2	1	no	—
331	C₂×C₃	#64942	1	1	yes	characteristic
332	C₂×C₃	#64989	1	1	yes	characteristic
333	C₂×C₃	#65006	1	1	yes	characteristic
334	C₅	#68645	1	1	yes	characteristic, complemented, Sylow
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	C₄×D₆₀	1	1	yes	characteristic
338	C₂²	#91768	1	1	yes	characteristic
339	C₂²	#91778	1	1	yes	characteristic
340	C₄	—	4	2	no	—
341	C₄	—	60	1	no	—
342	C₄	—	60	1	no	—
343	C₄	—	60	1	no	—
344	C₄	—	60	1	no	—
345	C₄	#91766	1	1	yes	characteristic
346	C₄	#91775	1	1	yes	characteristic
347	C₄	#91776	1	1	yes	characteristic
348	C₄	#91786	1	1	yes	characteristic
349	C₃	#10599438	1	1	yes	characteristic, complemented, Sylow
350	C₂	—	2	1	no	—
351	C₂	—	2	1	no	—
352	C₂	#11750754	1	1	yes	characteristic
353	C₂	#11750801	1	1	yes	characteristic
354	C₂	#11750818	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₆₀
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅
Fitting	C₃×C₅×#336	C₂
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂³×C₃×C₅	#41
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	#336	D₁₅
hypocenter	C₃×C₅	#1922
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #336

Complements

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

N	H ≅ G/N	type
16. #91102	C₄	semidirect
17. #91102	C₄	semidirect
18. #91101	C₄	semidirect
19. #91101	C₄	semidirect
289. C₃×C₅	#1922	semidirect
334. C₅	#68645	semidirect
349. C₃	#10599438	semidirect

Maximal subgroups

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

subgroup	index
#11751207	2
#11751203	2
#11751202	2
C₃×C₅×#336	2
#11746373	2
#11746344	2
#10599438	3
#68645	5