Subgroups — #14303621

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	C₅×#5119	C₂	1	1	yes	maximal, characteristic
2	#11748469	C₂	1	1	yes	maximal, characteristic
3	#11748318	C₂	1	1	yes	maximal, characteristic
4	#11746379	C₂	1	1	yes	maximal, characteristic
5	C₃×#64775	C₂	1	1	yes	maximal, characteristic
6	#11745977	C₂	1	1	yes	maximal, characteristic
7	#11745966	C₂	1	1	yes	maximal, characteristic
8	#10601327	—	3	1	no	maximal, Hall
9	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
10	C₂×#6710	C₂²	1	1	yes	characteristic
11	C₂²×C₃⋊#168	C₂²	1	1	yes	characteristic
12	C₅×#926	C₂²	1	1	yes	characteristic
13	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₃×#4072	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
16	C₅×Dic₃⋊C₈	—	2	1	no	—
17	C₅×Dic₃⋊C₈	—	2	1	no	—
18	#90989	C₄	1	1	yes	characteristic, complemented
19	#90989	C₄	1	1	yes	characteristic, complemented
20	#90986	C₄	1	1	yes	characteristic, complemented
21	#90986	C₄	1	1	yes	characteristic, complemented
22	#90952	—	2	1	no	—
23	Dic₅×C₃⋊C₈	—	2	1	no	—
24	#68743	—	5	1	no	maximal, Hall
25	C₅×#424	—	3	1	no	—
26	#65390	—	3	1	no	—
27	#65287	—	3	1	no	—
28	#64778	—	3	1	no	—
29	#64775	D₃	1	1	yes	characteristic
30	#64770	—	3	1	no	—
31	#64752	—	3	1	no	—
32	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂²×C₃×Dic₅	—	2	1	no	—
34	C₃×C₅×M₁₆	—	2	1	no	—
35	C₃×C₅×M₁₆	—	2	1	no	—
36	C₂×C₃×C₅×C₈	—	2	1	no	—
37	#6710	—	2	1	no	—
38	#6710	Q₈	1	2	yes	—
39	C₂×C₃⋊#168	—	2	1	no	—
40	C₂×C₃⋊#168	—	2	1	no	—
41	C₂×C₃⋊#168	—	2	1	no	—
42	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
43	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
44	C₅×#217	—	2	1	no	—
45	C₅×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
46	C₅×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
47	C₄×C₅×Dic₃	—	2	1	no	—
48	C₂×C₅×C₃⋊C₈	—	2	1	no	—
49	C₂×C₅×C₃⋊C₈	—	2	1	no	—
50	C₂×C₅×C₃⋊C₈	D₄	1	1	yes	characteristic
51	C₂×C₅×C₃⋊C₈	D₄	1	1	yes	characteristic
52	C₃×C₂²⋊Dic₅	—	2	1	no	—
53	C₃×C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
54	C₃×C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
55	C₃×C₄×Dic₅	—	2	1	no	—
56	C₃×C₄×Dic₅	—	2	1	no	—
57	C₃×C₄×Dic₅	—	2	1	no	—
58	#5119	D₅	1	1	yes	characteristic
59	#4998	—	5	1	no	—
60	#4997	—	5	1	no	—
61	C₃×#339	—	5	1	no	—
62	#4558	—	5	1	no	—
63	#4524	—	5	1	no	—
64	#4523	—	5	1	no	—
65	C₂×C₅×M₁₆	C₂×D₃	1	1	yes	characteristic
66	C₂²×C₅×C₈	—	3	1	no	—
67	C₅×#117	—	3	1	no	—
68	#4072	C₂×D₃	1	1	yes	characteristic
69	C₂×C₄×Dic₅	C₂×D₃	1	1	yes	characteristic
70	C₂×#681	—	3	1	no	—
71	C₂²×#168	—	3	1	no	—
72	C₅×C₄⋊C₈	—	6	1	no	—
73	C₅×C₄⋊C₈	—	6	1	no	—
74	#3947	—	6	1	no	—
75	C₈×Dic₅	—	6	1	no	—
76	#3940	—	3	1	no	—
77	#3940	—	3	1	no	—
78	#3939	—	3	1	no	—
79	#3939	—	3	1	no	—
80	#1922	—	15	1	no	Sylow
81	C₂³×C₃×C₅	#41	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
88	C₂×C₅×Dic₃	—	2	1	no	—
89	C₂×C₅×Dic₃	—	2	1	no	—
90	C₂×C₃×Dic₅	—	2	2	no	—
91	C₂×C₃×Dic₅	—	2	1	no	—
92	C₂×C₃×Dic₅	—	2	1	no	—
93	C₂×C₃×Dic₅	—	4	1	no	—
94	C₂×C₃×Dic₅	—	4	1	no	—
95	C₃×C₅×C₈	—	4	1	no	—
96	C₃⋊#168	—	2	1	no	—
97	C₃⋊#168	—	2	1	no	—
98	C₃⋊#168	—	2	2	no	—
99	C₅×C₃⋊C₈	—	2	1	no	—
100	C₅×C₃⋊C₈	—	2	1	no	—
101	C₂×C₃×M₁₆	C₂×D₅	1	1	yes	characteristic
102	C₃×#117	—	5	1	no	—
103	C₂×C₃×C₄²	—	5	1	no	—
104	#926	C₂×D₅	1	1	yes	characteristic
105	C₂×#208	—	5	1	no	—
106	C₂²×C₃⋊C₈	—	5	1	no	—
107	C₂²×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
108	Dic₃⋊C₈	—	2	1	no	—
109	Dic₃⋊C₈	—	2	1	no	—
110	#808	—	5	1	no	—
111	#808	—	5	1	no	—
112	#807	—	5	1	no	—
113	#807	—	5	1	no	—
114	#803	—	10	1	no	—
115	C₄×C₃⋊C₈	—	10	1	no	—
116	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
117	C₂²×Dic₅	—	2	1	no	—
118	C₅×M₁₆	—	2	1	no	—
119	C₅×M₁₆	—	2	1	no	—
120	C₂×C₅×C₈	—	2	1	no	—
121	C₂×C₅×C₈	—	3	1	no	—
122	C₂×C₅×C₈	—	3	1	no	—
123	C₂×C₅×C₈	—	6	1	no	—
124	C₂×C₅×C₈	—	6	1	no	—
125	C₅×C₄⋊C₄	—	3	1	no	—
126	C₅×C₄⋊C₄	—	3	1	no	—
127	C₅×C₂²⋊C₄	—	6	1	no	—
128	C₄²×C₅	—	6	1	no	—
129	C₂²⋊Dic₅	—	2	1	no	—
130	C₄⋊Dic₅	C₄×D₃	1	1	yes	characteristic
131	C₄⋊Dic₅	C₄×D₃	1	1	yes	characteristic
132	C₄×Dic₅	—	2	1	no	—
133	C₄×Dic₅	—	2	1	no	—
134	C₄×Dic₅	—	2	1	no	—
135	#681	—	3	2	no	—
136	#681	—	6	1	no	—
137	C₂×#168	—	3	1	no	—
138	C₂×#168	—	3	1	no	—
139	C₂×#168	—	6	1	no	—
140	C₂×#168	—	6	1	no	—
141	C₂×#168	—	6	1	no	—
142	#427	—	15	1	no	—
143	#424	—	3	1	no	—
144	#405	—	15	1	no	—
145	#341	—	15	1	no	—
146	#339	—	5	1	no	—
147	#339	—	15	1	no	—
148	#336	—	15	1	no	—
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	#117	1	1	yes	characteristic
155	C₃×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
156	C₃×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
157	C₃×C₄×C₅	#123	1	1	yes	characteristic
158	C₃×Dic₅	—	4	2	no	—
159	C₃×Dic₅	—	4	1	no	—
160	C₃×Dic₅	—	4	1	no	—
161	C₅×Dic₃	—	4	1	no	—
162	C₅×Dic₃	—	4	1	no	—
163	C₂²×C₃×C₄	—	10	1	no	—
164	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
165	C₃×M₁₆	—	2	1	no	—
166	C₃×M₁₆	—	2	1	no	—
167	C₂×C₃×C₈	—	2	1	no	—
168	C₃×C₄⋊C₄	—	5	1	no	—
169	C₃×C₄⋊C₄	—	5	1	no	—
170	C₃×C₂²⋊C₄	—	10	1	no	—
171	C₃×C₄²	—	10	1	no	—
172	C₃×C₄²	—	10	1	no	—
173	C₃×C₄²	—	10	1	no	—
174	#217	—	2	1	no	—
175	C₄⋊Dic₃	C₄×D₅	1	1	yes	characteristic
176	C₄⋊Dic₃	C₄×D₅	1	1	yes	characteristic
177	C₄×Dic₃	—	2	1	no	—
178	#208	—	5	2	no	—
179	#208	—	10	1	no	—
180	C₂×C₃⋊C₈	—	2	1	no	—
181	C₂×C₃⋊C₈	—	2	1	no	—
182	C₂×C₃⋊C₈	—	5	1	no	—
183	C₂×C₃⋊C₈	—	5	1	no	—
184	C₂×C₃⋊C₈	—	10	1	no	—
185	C₂×C₃⋊C₈	—	10	1	no	—
186	C₂×C₃⋊C₈	—	10	1	no	—
187	C₂×C₃⋊C₈	C₂²⋊D₅	1	1	yes	characteristic
188	C₂×C₃⋊C₈	C₂²⋊D₅	1	1	yes	characteristic
189	C₂³×C₅	#237	1	1	yes	characteristic
190	C₂×C₄×C₅	—	6	1	no	—
191	C₂×C₄×C₅	—	6	1	no	—
192	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
193	C₂×C₄×C₅	#235	1	1	yes	characteristic
194	C₂×C₄×C₅	#235	1	1	yes	characteristic
195	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
196	C₂×C₄×C₅	#237	1	1	yes	characteristic
197	C₂×C₄×C₅	D₃×Q₈	1	1	yes	characteristic
198	C₂×Dic₅	—	2	2	no	—
199	C₂×Dic₅	—	2	1	no	—
200	C₂×Dic₅	—	2	1	no	—
201	C₂×Dic₅	—	4	1	no	—
202	C₂×Dic₅	—	4	1	no	—
203	C₅×C₈	—	4	1	no	—
204	C₅×C₈	—	6	1	no	—
205	C₅×C₈	—	6	1	no	—
206	#168	—	6	1	no	—
207	#168	—	6	1	no	—
208	#168	—	6	2	no	—
209	C₂×M₁₆	—	15	1	no	—
210	C₂×M₁₆	D₃×D₅	1	1	yes	characteristic
211	C₂²×C₈	—	3	1	no	—
212	C₂²×C₈	—	15	1	no	—
213	#117	—	3	1	no	—
214	#117	—	5	1	no	—
215	C₂×C₄²	—	5	1	no	—
216	#107	—	15	1	no	—
217	#107	—	15	1	no	—
218	#106	—	15	1	no	—
219	#106	—	15	1	no	—
220	C₄⋊C₈	—	6	1	no	—
221	C₄⋊C₈	—	6	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	—	10	2	no	—
231	C₂×C₃×C₄	—	10	1	no	—
232	C₂×C₃×C₄	—	10	1	no	—
233	C₂×C₃×C₄	—	20	1	no	—
234	C₂×C₃×C₄	—	20	1	no	—
235	C₂×C₃×C₄	C₂×C₄×D₅	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×C₃×C₄	#713	1	1	yes	characteristic
240	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
241	C₂×Dic₃	—	2	1	no	—
242	C₂×Dic₃	—	2	1	no	—
243	C₃×C₈	—	4	1	no	—
244	C₃⋊C₈	—	2	1	no	—
245	C₃⋊C₈	—	2	1	no	—
246	C₃⋊C₈	—	10	1	no	—
247	C₃⋊C₈	—	10	1	no	—
248	C₃⋊C₈	—	10	2	no	—
249	C₂²×C₅	—	2	1	no	—
250	C₂²×C₅	—	2	1	no	—
251	C₂²×C₅	#879	1	1	yes	characteristic
252	C₂²×C₅	#67⋊C₄	1	1	yes	characteristic
253	C₂²×C₅	#889	1	1	yes	characteristic
254	C₄×C₅	—	12	1	no	—
255	C₄×C₅	—	12	1	no	—
256	C₄×C₅	#872	1	1	yes	characteristic
257	C₄×C₅	#883	1	1	yes	characteristic
258	C₄×C₅	#887	1	1	yes	characteristic
259	C₄×C₅	#896	1	1	yes	characteristic
260	Dic₅	—	4	2	no	—
261	Dic₅	—	4	1	no	—
262	Dic₅	—	4	1	no	—
263	C₂²×C₄	—	10	1	no	—
264	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
265	M₁₆	—	2	1	no	—
266	M₁₆	—	2	1	no	—
267	M₁₆	—	15	2	no	—
268	M₁₆	—	30	1	no	—
269	C₂×C₈	—	2	1	no	—
270	C₂×C₈	—	3	1	no	—
271	C₂×C₈	—	3	1	no	—
272	C₂×C₈	—	6	1	no	—
273	C₂×C₈	—	6	1	no	—
274	C₂×C₈	—	15	1	no	—
275	C₂×C₈	—	15	1	no	—
276	C₂×C₈	—	30	1	no	—
277	C₂×C₈	—	30	1	no	—
278	C₂×C₈	—	30	1	no	—
279	C₄⋊C₄	—	3	1	no	—
280	C₄⋊C₄	—	3	1	no	—
281	C₄⋊C₄	—	5	1	no	—
282	C₄⋊C₄	—	5	1	no	—
283	C₂²⋊C₄	—	6	1	no	—
284	C₂²⋊C₄	—	10	1	no	—
285	C₄²	—	6	1	no	—
286	C₄²	—	10	1	no	—
287	C₄²	—	10	1	no	—
288	C₄²	—	10	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	#4035	1	1	yes	characteristic
293	C₂²×C₃	#4044	1	1	yes	characteristic
294	C₂²×C₃	C₄×C₂²⋊D₅	1	1	yes	characteristic
295	C₃×C₄	—	20	2	no	—
296	C₃×C₄	—	20	1	no	—
297	C₃×C₄	—	20	1	no	—
298	C₃×C₄	#4033	1	1	yes	characteristic
299	C₃×C₄	#4038	1	1	yes	characteristic
300	C₃×C₄	#4081	1	1	yes	characteristic
301	C₃×C₄	#4092	1	1	yes	characteristic
302	Dic₃	—	4	1	no	—
303	Dic₃	—	4	1	no	—
304	C₂×C₅	—	2	1	no	—
305	C₂×C₅	—	2	1	no	—
306	C₂×C₅	#4652	1	1	yes	characteristic
307	C₂×C₅	#4823	1	1	yes	characteristic
308	C₂×C₅	#4824	1	1	yes	characteristic
309	C₂³	#6765	1	1	yes	characteristic
310	C₂×C₄	—	6	1	no	—
311	C₂×C₄	—	6	1	no	—
312	C₂×C₄	—	10	2	no	—
313	C₂×C₄	—	10	1	no	—
314	C₂×C₄	—	10	1	no	—
315	C₂×C₄	—	20	1	no	—
316	C₂×C₄	—	20	1	no	—
317	C₂×C₄	D₁₅⋊Q₈	1	1	yes	characteristic
318	C₂×C₄	#6773	1	1	yes	characteristic
319	C₂×C₄	C₄×D₃×D₅	1	1	yes	characteristic
320	C₂×C₄	#6780	1	1	yes	characteristic
321	C₂×C₄	#6782	1	1	yes	characteristic
322	C₂×C₄	D₃×C₂²⋊D₅	1	1	yes	characteristic
323	C₈	—	4	1	no	—
324	C₈	—	6	1	no	—
325	C₈	—	6	1	no	—
326	C₈	—	30	1	no	—
327	C₈	—	30	1	no	—
328	C₈	—	30	2	no	—
329	C₂×C₃	—	2	1	no	—
330	C₂×C₃	—	2	1	no	—
331	C₂×C₃	#65267	1	1	yes	characteristic
332	C₂×C₃	#65433	1	1	yes	characteristic
333	C₂×C₃	#65434	1	1	yes	characteristic
334	C₅	#68743	1	1	yes	characteristic, complemented, Sylow
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	#91339	1	1	yes	characteristic
338	C₂²	#91362	1	1	yes	characteristic
339	C₂²	#91441	1	1	yes	characteristic
340	C₄	—	12	1	no	—
341	C₄	—	12	1	no	—
342	C₄	—	20	2	no	—
343	C₄	—	20	1	no	—
344	C₄	—	20	1	no	—
345	C₄	#91325	1	1	yes	characteristic
346	C₄	#91402	1	1	yes	characteristic
347	C₄	#91424	1	1	yes	characteristic
348	C₄	#91555	1	1	yes	characteristic
349	C₃	#10601327	1	1	yes	characteristic, complemented, Sylow
350	C₂	—	2	1	no	—
351	C₂	—	2	1	no	—
352	C₂	#11748204	1	1	yes	characteristic
353	C₂	#11748205	1	1	yes	characteristic
354	C₂	#11748662	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₃×C₂²⋊D₅
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃×D₅
Fitting	C₂×C₃×C₅×M₁₆	C₂²
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂³×C₃×C₅	#41
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×M₁₆	D₃×D₅
hypocenter	C₃×C₅	#1922
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
18. #90989	C₄	semidirect
19. #90989	C₄	semidirect
20. #90986	C₄	semidirect
21. #90986	C₄	semidirect
289. C₃×C₅	#1922	semidirect
334. C₅	#68743	semidirect
349. C₃	#10601327	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5119	2
#11748469	2
#11748318	2
#11746379	2
C₃×#64775	2
#11745977	2
#11745966	2
#10601327	3
#68743	5