Subgroups — #14313460

Order 1920.

572 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#535	C₂	1	1	yes	maximal, characteristic
2	C₂×#91118	C₂	1	2	yes	maximal, complemented
3	#11746366	C₂	1	2	yes	maximal, complemented
4	#11746362	C₂	1	2	yes	maximal, complemented
5	#10602653	—	3	1	no	maximal, Hall
6	C₂×C₃×C₅×#41	C₄	1	1	yes	characteristic, complemented
7	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
8	C₃×C₅×#127	C₄	1	1	yes	characteristic, complemented
9	C₃×C₅×#127	C₂²	1	1	yes	characteristic
10	C₃×C₅×#124	C₄	1	1	yes	characteristic, complemented
11	C₃×C₅×#124	C₂²	1	1	yes	characteristic
12	C₃×C₅×#121	—	2	2	no	—
13	C₃×C₅×#120	—	2	2	no	—
14	C₃×C₅×#117	C₄	1	1	yes	characteristic, complemented
15	C₂×#6719	—	2	2	no	—
16	#91118	—	2	2	no	—
17	#91118	C₂²	1	4	yes	—
18	#71860	—	5	1	no	maximal, Hall
19	C₅×#535	D₃	1	1	yes	characteristic
20	C₂×#3966	—	3	2	no	—
21	#64765	—	3	2	no	—
22	#64761	—	3	2	no	—
23	C₂⁴×C₃×C₅	D₄	1	2	yes	—
24	C₃×C₅×#41	—	4	2	no	—
25	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic, complemented
26	C₂×C₃×C₅×D₄	—	2	1	no	—
27	C₂×C₃×C₅×D₄	—	2	2	no	—
28	C₂×C₃×C₅×D₄	—	2	2	no	—
29	C₂×C₃×C₅×D₄	—	4	2	no	—
30	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic, complemented
32	C₂×C₃×C₅×D₄	D₄	1	2	yes	—
33	C₂×C₃×C₅×D₄	C₂³	1	1	yes	characteristic
34	C₂²×C₃×C₄×C₅	—	2	1	no	—
35	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
36	C₂²×Dic₁₅	—	4	2	no	—
37	C₃×C₅×C₄⋊C₄	—	2	1	no	—
38	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
39	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
40	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
41	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic, complemented
42	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic, complemented
43	#6719	—	2	4	no	—
44	#6719	—	4	2	no	—
45	C₃×#535	D₅	1	1	yes	characteristic
46	C₂×#834	—	5	2	no	—
47	#4545	—	5	2	no	—
48	#4541	—	5	2	no	—
49	C₂×C₅×#41	Dic₃	1	1	yes	characteristic, complemented
50	C₂²×C₅×D₄	C₂×D₃	1	1	yes	characteristic
51	C₅×#127	Dic₃	1	1	yes	characteristic, complemented
52	C₅×#127	C₂×D₃	1	1	yes	characteristic
53	C₅×#124	Dic₃	1	1	yes	characteristic, complemented
54	C₅×#124	C₂×D₃	1	1	yes	characteristic
55	C₅×#121	—	2	2	no	—
56	C₅×#120	—	2	2	no	—
57	C₅×#117	Dic₃	1	1	yes	characteristic, complemented
58	C₂×C₂²⋊Dic₅	—	6	2	no	—
59	#3966	—	3	4	no	—
60	#3966	—	6	2	no	—
61	#2127	—	15	1	no	Sylow
62	C₂³×C₃×C₅	—	2	2	no	—
63	C₂³×C₃×C₅	—	2	2	no	—
64	C₂³×C₃×C₅	—	2	1	no	—
65	C₂³×C₃×C₅	—	8	2	no	—
66	C₂³×C₃×C₅	C₂²⋊C₄	1	2	yes	—
67	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
68	C₂³×C₃×C₅	C₂×D₄	1	2	yes	—
69	C₃×C₅×Q₈	—	4	1	no	—
70	C₃×C₅×D₄	—	4	1	no	—
71	C₃×C₅×D₄	—	4	2	no	—
72	C₃×C₅×D₄	—	4	1	no	—
73	C₃×C₅×D₄	—	4	1	no	—
74	C₃×C₅×D₄	—	4	2	no	—
75	C₃×C₅×D₄	—	8	2	no	—
76	C₃×C₅×D₄	—	8	2	no	—
77	C₂×C₃×C₄×C₅	—	2	2	no	—
78	C₂×C₃×C₄×C₅	—	2	1	no	—
79	C₂×C₃×C₄×C₅	—	2	1	no	—
80	C₂×C₃×C₄×C₅	—	2	1	no	—
81	C₂×C₃×C₄×C₅	—	4	2	no	—
82	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₁₅	—	4	4	no	—
85	C₂×Dic₁₅	—	8	4	no	—
86	C₂×C₃×#41	Dic₅	1	1	yes	characteristic, complemented
87	C₂²×C₃×D₄	C₂×D₅	1	1	yes	characteristic
88	C₃×#127	Dic₅	1	1	yes	characteristic, complemented
89	C₃×#127	C₂×D₅	1	1	yes	characteristic
90	C₃×#124	Dic₅	1	1	yes	characteristic, complemented
91	C₃×#124	C₂×D₅	1	1	yes	characteristic
92	C₃×#121	—	2	2	no	—
93	C₃×#120	—	2	2	no	—
94	C₃×#117	Dic₅	1	1	yes	characteristic, complemented
95	C₂×#217	—	10	2	no	—
96	#834	—	5	4	no	—
97	#834	—	10	2	no	—
98	C₂⁴×C₅	#67	1	2	yes	—
99	C₅×#41	—	4	2	no	—
100	C₂×C₅×Q₈	C₂×Dic₃	1	1	yes	characteristic, complemented
101	C₂×C₅×D₄	—	2	1	no	—
102	C₂×C₅×D₄	—	2	2	no	—
103	C₂×C₅×D₄	—	2	2	no	—
104	C₂×C₅×D₄	—	4	2	no	—
105	C₂×C₅×D₄	C₂×Dic₃	1	1	yes	characteristic
106	C₂×C₅×D₄	C₂×Dic₃	1	1	yes	characteristic, complemented
107	C₂×C₅×D₄	#67	1	2	yes	—
108	C₂×C₅×D₄	C₂²×D₃	1	1	yes	characteristic
109	C₂²×C₄×C₅	—	2	1	no	—
110	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
111	C₂²×Dic₅	—	12	2	no	—
112	C₅×C₄⋊C₄	—	2	1	no	—
113	C₅×C₂²⋊C₄	—	2	1	no	—
114	C₅×C₂²⋊C₄	—	2	2	no	—
115	C₅×C₂²⋊C₄	—	2	2	no	—
116	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic, complemented
117	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic, complemented
118	C₂²⋊Dic₅	—	6	4	no	—
119	C₂²⋊Dic₅	—	12	2	no	—
120	#535	D₁₅	1	1	yes	characteristic
121	C₂×#99	—	15	2	no	—
122	#354	—	15	2	no	—
123	#353	—	15	2	no	—
124	C₂²×C₃×C₅	—	2	2	no	—
125	C₂²×C₃×C₅	—	2	2	no	—
126	C₂²×C₃×C₅	—	4	1	no	—
127	C₂²×C₃×C₅	—	4	2	no	—
128	C₂²×C₃×C₅	—	4	2	no	—
129	C₂²×C₃×C₅	—	4	4	no	—
130	C₂²×C₃×C₅	—	4	2	no	—
131	C₂²×C₃×C₅	—	4	1	no	—
132	C₂²×C₃×C₅	—	8	2	no	—
133	C₂²×C₃×C₅	#99	1	2	yes	—
134	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₃×C₄×C₅	—	2	1	no	—
136	C₃×C₄×C₅	—	2	1	no	—
137	C₃×C₄×C₅	—	4	1	no	—
138	C₃×C₄×C₅	—	4	1	no	—
139	C₃×C₄×C₅	—	4	1	no	—
140	Dic₁₅	—	8	4	no	—
141	C₂⁴×C₃	C₂²⋊D₅	1	2	yes	—
142	C₃×#41	—	4	2	no	—
143	C₂×C₃×Q₈	C₂×Dic₅	1	1	yes	characteristic, complemented
144	C₂×C₃×D₄	—	2	1	no	—
145	C₂×C₃×D₄	—	2	2	no	—
146	C₂×C₃×D₄	—	2	2	no	—
147	C₂×C₃×D₄	—	4	2	no	—
148	C₂×C₃×D₄	C₂×Dic₅	1	1	yes	characteristic
149	C₂×C₃×D₄	C₂×Dic₅	1	1	yes	characteristic, complemented
150	C₂×C₃×D₄	C₂²⋊D₅	1	2	yes	—
151	C₂×C₃×D₄	C₂²×D₅	1	1	yes	characteristic
152	C₂²×C₃×C₄	—	2	1	no	—
153	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
154	C₂²×Dic₃	—	20	2	no	—
155	C₃×C₄⋊C₄	—	2	1	no	—
156	C₃×C₂²⋊C₄	—	2	1	no	—
157	C₃×C₂²⋊C₄	—	2	2	no	—
158	C₃×C₂²⋊C₄	—	2	2	no	—
159	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic, complemented
160	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic, complemented
161	#217	—	10	4	no	—
162	#217	—	20	2	no	—
163	C₂³×C₅	—	2	2	no	—
164	C₂³×C₅	—	2	2	no	—
165	C₂³×C₅	—	2	1	no	—
166	C₂³×C₅	—	8	2	no	—
167	C₂³×C₅	#217	1	2	yes	—
168	C₂³×C₅	#217	1	1	yes	characteristic
169	C₂³×C₅	C₂×#67	1	2	yes	—
170	C₅×Q₈	—	4	1	no	—
171	C₅×D₄	—	4	1	no	—
172	C₅×D₄	—	4	2	no	—
173	C₅×D₄	—	4	1	no	—
174	C₅×D₄	—	4	1	no	—
175	C₅×D₄	—	4	2	no	—
176	C₅×D₄	—	8	2	no	—
177	C₅×D₄	—	8	2	no	—
178	C₂×C₄×C₅	—	2	2	no	—
179	C₂×C₄×C₅	—	2	1	no	—
180	C₂×C₄×C₅	—	2	1	no	—
181	C₂×C₄×C₅	—	2	1	no	—
182	C₂×C₄×C₅	—	4	2	no	—
183	C₂×C₄×C₅	#217	1	1	yes	characteristic
184	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
185	C₂×Dic₅	—	12	4	no	—
186	C₂×Dic₅	—	24	4	no	—
187	C₂×#41	Dic₁₅	1	1	yes	characteristic, complemented
188	C₂²×D₄	C₂×D₁₅	1	1	yes	characteristic
189	#127	Dic₁₅	1	1	yes	characteristic, complemented
190	#127	C₂×D₁₅	1	1	yes	characteristic
191	#124	Dic₁₅	1	1	yes	characteristic, complemented
192	#124	C₂×D₁₅	1	1	yes	characteristic
193	#121	—	2	2	no	—
194	#120	—	2	2	no	—
195	#117	Dic₁₅	1	1	yes	characteristic, complemented
196	C₂×C₂²⋊C₄	—	30	2	no	—
197	#99	—	15	4	no	—
198	#99	—	30	2	no	—
199	C₂×C₃×C₅	—	2	1	no	—
200	C₂×C₃×C₅	—	2	2	no	—
201	C₂×C₃×C₅	—	4	2	no	—
202	C₂×C₃×C₅	—	4	2	no	—
203	C₂×C₃×C₅	—	8	1	no	—
204	C₂×C₃×C₅	C₂×#99	1	1	yes	characteristic
205	C₂³×C₃	—	2	2	no	—
206	C₂³×C₃	—	2	2	no	—
207	C₂³×C₃	—	2	1	no	—
208	C₂³×C₃	—	8	2	no	—
209	C₂³×C₃	C₂²⋊Dic₅	1	2	yes	—
210	C₂³×C₃	C₂²⋊Dic₅	1	1	yes	characteristic
211	C₂³×C₃	C₂×C₂²⋊D₅	1	2	yes	—
212	C₃×Q₈	—	4	1	no	—
213	C₃×D₄	—	4	1	no	—
214	C₃×D₄	—	4	2	no	—
215	C₃×D₄	—	4	1	no	—
216	C₃×D₄	—	4	1	no	—
217	C₃×D₄	—	4	2	no	—
218	C₃×D₄	—	8	2	no	—
219	C₃×D₄	—	8	2	no	—
220	C₂×C₃×C₄	—	2	2	no	—
221	C₂×C₃×C₄	—	2	1	no	—
222	C₂×C₃×C₄	—	2	1	no	—
223	C₂×C₃×C₄	—	2	1	no	—
224	C₂×C₃×C₄	—	4	2	no	—
225	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
226	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
227	C₂×Dic₃	—	20	4	no	—
228	C₂×Dic₃	—	40	4	no	—
229	C₂²×C₅	—	2	2	no	—
230	C₂²×C₅	—	2	2	no	—
231	C₂²×C₅	—	4	1	no	—
232	C₂²×C₅	—	4	2	no	—
233	C₂²×C₅	—	4	2	no	—
234	C₂²×C₅	—	4	4	no	—
235	C₂²×C₅	—	4	2	no	—
236	C₂²×C₅	—	4	1	no	—
237	C₂²×C₅	—	8	2	no	—
238	C₂²×C₅	#834	1	2	yes	—
239	C₂²×C₅	C₂×#217	1	1	yes	characteristic
240	C₄×C₅	—	2	1	no	—
241	C₄×C₅	—	2	1	no	—
242	C₄×C₅	—	4	1	no	—
243	C₄×C₅	—	4	1	no	—
244	C₄×C₅	—	4	1	no	—
245	Dic₅	—	24	4	no	—
246	C₂⁴	#1220	1	2	yes	—
247	#41	—	4	2	no	—
248	C₂×Q₈	C₂×Dic₁₅	1	1	yes	characteristic, complemented
249	C₂×D₄	—	2	1	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	—	4	2	no	—
253	C₂×D₄	C₂×Dic₁₅	1	1	yes	characteristic
254	C₂×D₄	C₂×Dic₁₅	1	1	yes	characteristic, complemented
255	C₂×D₄	#1220	1	2	yes	—
256	C₂×D₄	C₂²×D₁₅	1	1	yes	characteristic
257	C₂²×C₄	—	2	1	no	—
258	C₂²×C₄	—	60	2	no	—
259	C₂²×C₄	C₂×Dic₁₅	1	1	yes	characteristic
260	C₄⋊C₄	—	2	1	no	—
261	C₂²⋊C₄	—	2	2	no	—
262	C₂²⋊C₄	—	2	1	no	—
263	C₂²⋊C₄	—	2	2	no	—
264	C₂²⋊C₄	—	30	4	no	—
265	C₂²⋊C₄	—	60	2	no	—
266	C₄²	C₂×Dic₁₅	1	1	yes	characteristic, complemented
267	C₄²	C₂×Dic₁₅	1	1	yes	characteristic, complemented
268	C₃×C₅	#2127	1	1	yes	characteristic, complemented, Hall
269	C₂²×C₃	—	2	2	no	—
270	C₂²×C₃	—	2	2	no	—
271	C₂²×C₃	—	4	1	no	—
272	C₂²×C₃	—	4	2	no	—
273	C₂²×C₃	—	4	2	no	—
274	C₂²×C₃	—	4	4	no	—
275	C₂²×C₃	—	4	2	no	—
276	C₂²×C₃	—	4	1	no	—
277	C₂²×C₃	—	8	2	no	—
278	C₂²×C₃	#3966	1	2	yes	—
279	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
280	C₃×C₄	—	2	1	no	—
281	C₃×C₄	—	2	1	no	—
282	C₃×C₄	—	4	1	no	—
283	C₃×C₄	—	4	1	no	—
284	C₃×C₄	—	4	1	no	—
285	Dic₃	—	40	4	no	—
286	C₂×C₅	—	2	1	no	—
287	C₂×C₅	—	2	2	no	—
288	C₂×C₅	—	4	2	no	—
289	C₂×C₅	—	4	2	no	—
290	C₂×C₅	—	8	1	no	—
291	C₂×C₅	C₂×#834	1	1	yes	characteristic
292	C₂³	—	2	2	no	—
293	C₂³	—	2	2	no	—
294	C₂³	—	2	1	no	—
295	C₂³	—	8	2	no	—
296	C₂³	#6719	1	2	yes	—
297	C₂³	#6719	1	1	yes	characteristic
298	C₂³	C₂×#1220	1	2	yes	—
299	Q₈	—	4	1	no	—
300	D₄	—	4	1	no	—
301	D₄	—	4	2	no	—
302	D₄	—	4	1	no	—
303	D₄	—	4	1	no	—
304	D₄	—	4	2	no	—
305	D₄	—	8	2	no	—
306	D₄	—	8	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	60	4	no	—
313	C₂×C₄	—	120	4	no	—
314	C₂×C₄	#6719	1	1	yes	characteristic
315	C₂×C₄	C₂²×Dic₁₅	1	1	yes	characteristic
316	C₂×C₃	—	2	1	no	—
317	C₂×C₃	—	2	2	no	—
318	C₂×C₃	—	4	2	no	—
319	C₂×C₃	—	4	2	no	—
320	C₂×C₃	—	8	1	no	—
321	C₂×C₃	C₂×#3966	1	1	yes	characteristic
322	C₅	#71860	1	1	yes	characteristic, complemented, Sylow
323	C₂²	—	2	2	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	4	1	no	—
326	C₂²	—	4	2	no	—
327	C₂²	—	4	2	no	—
328	C₂²	—	4	4	no	—
329	C₂²	—	4	2	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	8	2	no	—
332	C₂²	#91118	1	2	yes	—
333	C₂²	C₂×#6719	1	1	yes	characteristic
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	4	1	no	—
338	C₄	—	4	1	no	—
339	C₄	—	120	4	no	—
340	C₃	#10602653	1	1	yes	characteristic, complemented, Sylow
341	C₂	—	2	1	no	—
342	C₂	—	2	2	no	—
343	C₂	—	4	2	no	—
344	C₂	—	4	2	no	—
345	C₂	—	8	1	no	—
346	C₂	C₂×#91118	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#91118
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂×D₄	C₂²×D₁₅
Fitting, prime radical	C₃×C₅×#535	C₂
socle	C₂×C₃×C₅	C₂×#99
cosocle	C₂×C₃×C₅×D₄	C₂³
hypercenter	#535	D₁₅
hypocenter	C₃×C₅	#2127
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂²×C₃×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×D₄ ◁ #535

Complements

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

N	H ≅ G/N	type
2. C₂×#91118	C₂	semidirect
3. #11746366	C₂	semidirect
4. #11746362	C₂	semidirect
6. C₂×C₃×C₅×#41	C₄	semidirect
8. C₃×C₅×#127	C₄	semidirect
10. C₃×C₅×#124	C₄	semidirect
14. C₃×C₅×#117	C₄	semidirect
25. C₂×C₃×C₅×Q₈	C₂×C₄	semidirect
31. C₂×C₃×C₅×D₄	C₂×C₄	semidirect
41. C₃×C₄²×C₅	C₂×C₄	semidirect
42. C₃×C₄²×C₅	C₂×C₄	semidirect
49. C₂×C₅×#41	Dic₃	semidirect
51. C₅×#127	Dic₃	semidirect
53. C₅×#124	Dic₃	semidirect
57. C₅×#117	Dic₃	semidirect
86. C₂×C₃×#41	Dic₅	semidirect
88. C₃×#127	Dic₅	semidirect
90. C₃×#124	Dic₅	semidirect
94. C₃×#117	Dic₅	semidirect
100. C₂×C₅×Q₈	C₂×Dic₃	semidirect
106. C₂×C₅×D₄	C₂×Dic₃	semidirect
116. C₄²×C₅	C₂×Dic₃	semidirect
117. C₄²×C₅	C₂×Dic₃	semidirect
143. C₂×C₃×Q₈	C₂×Dic₅	semidirect
149. C₂×C₃×D₄	C₂×Dic₅	semidirect
159. C₃×C₄²	C₂×Dic₅	semidirect
160. C₃×C₄²	C₂×Dic₅	semidirect
187. C₂×#41	Dic₁₅	semidirect
189. #127	Dic₁₅	semidirect
191. #124	Dic₁₅	semidirect
195. #117	Dic₁₅	semidirect
248. C₂×Q₈	C₂×Dic₁₅	semidirect
254. C₂×D₄	C₂×Dic₁₅	semidirect
266. C₄²	C₂×Dic₁₅	semidirect
267. C₄²	C₂×Dic₁₅	semidirect
268. C₃×C₅	#2127	semidirect
322. C₅	#71860	semidirect
340. C₃	#10602653	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#535	2
C₂×#91118	2
#11746366	2
#11746362	2
#10602653	3
#71860	5