Subgroups — #14282901

Order 1920.

568 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#65811	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4925	C₂	1	1	yes	maximal, characteristic, complemented
4	#11745955	C₂	1	4	yes	maximal, complemented
5	#10601582	—	3	1	no	maximal, Hall
6	C₂×C₃×#709	—	2	1	no	—
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₅	—	2	1	no	—
9	C₂×C₄×Dic₁₅	—	2	1	no	—
10	C₃×#4076	—	2	1	no	—
11	C₃×#4076	C₂²	1	2	yes	—
12	C₃×#4021	—	2	1	no	—
13	C₃×#4020	C₄	1	1	yes	characteristic, complemented
14	C₃×C₄×D₂₀	—	2	1	no	—
15	C₃×C₄×D₂₀	C₄	1	2	yes	complemented
16	C₃×#4015	C₄	1	1	yes	characteristic, complemented
17	#91115	C₂²	1	2	yes	—
18	C₅×#830	C₂²	1	2	yes	—
19	C₅×C₄⋊(C₃⋊C₈)	—	2	1	no	—
20	C₄×C₅×C₃⋊C₈	—	2	1	no	—
21	#71811	—	5	1	no	maximal, Hall
22	#65811	D₃	1	1	yes	characteristic
23	C₅×#431	—	3	1	no	—
24	C₄×C₄⋊Dic₅	—	3	1	no	—
25	#64690	—	3	4	no	—
26	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂²×Dic₁₅	—	2	2	no	—
30	C₂×C₃×C₂²⋊D₅	—	2	2	no	—
31	C₃×#709	—	4	2	no	—
32	C₃×#709	—	4	2	no	—
33	C₂×C₃×D₂₀	—	2	1	no	—
34	C₂×C₃×D₂₀	—	2	1	no	—
35	C₂×C₃×D₂₀	C₂×C₄	1	2	yes	—
36	C₂×C₃×C₄×D₅	—	2	2	no	—
37	C₂×C₃×Dic₁₀	—	2	1	no	—
38	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
39	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
40	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
41	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₄⋊Dic₁₅	—	2	4	no	—
43	C₄×Dic₁₅	—	4	2	no	—
44	C₂×C₅×C₃⋊C₈	—	2	2	no	—
45	C₃×#685	—	2	2	no	—
46	C₃×#685	—	2	2	no	—
47	C₃×C₄⋊Dic₅	—	2	1	no	—
48	C₃×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
49	C₃×#532	—	5	1	no	—
50	C₄×C₄⋊Dic₃	—	5	1	no	—
51	#4925	D₅	1	1	yes	characteristic, complemented
52	#4537	—	5	4	no	—
53	C₂×#709	—	2	1	no	—
54	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
55	#4076	—	2	1	no	—
56	#4076	C₂×D₃	1	2	yes	—
57	C₂×C₄⋊Dic₅	—	6	1	no	—
58	C₂×C₄×Dic₅	—	6	1	no	—
59	#4021	—	2	1	no	—
60	#4020	Dic₃	1	1	yes	characteristic, complemented
61	C₄×D₂₀	—	2	1	no	—
62	C₄×D₂₀	Dic₃	1	2	yes	complemented
63	#4015	Dic₃	1	1	yes	characteristic, complemented
64	C₅×C₄⋊C₈	—	6	1	no	—
65	C₅×C₂²⋊C₈	—	3	2	no	—
66	C₄×C₅×C₈	—	6	1	no	—
67	#3963	—	3	2	no	—
68	#1502	—	15	1	no	Sylow
69	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₃×D₅	—	2	2	no	—
71	C₂×C₃×C₄×C₅	—	2	2	no	—
72	C₂×C₃×C₄×C₅	—	4	1	no	—
73	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	D₈	1	2	yes	—
78	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
79	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂×Dic₁₅	—	2	4	no	—
82	C₂×Dic₁₅	—	4	2	no	—
83	C₂×Dic₁₅	—	4	2	no	—
84	C₃×C₂²⋊D₅	—	4	2	no	—
85	C₃×C₂²⋊D₅	—	4	2	no	—
86	C₂×C₃×Dic₅	—	2	2	no	—
87	C₃×D₂₀	—	4	2	no	—
88	C₃×D₂₀	—	4	2	no	—
89	C₃×D₂₀	—	4	2	no	—
90	C₃×C₄×D₅	—	4	4	no	—
91	C₃×Dic₁₀	—	4	2	no	—
92	C₅×C₃⋊C₈	—	4	2	no	—
93	C₂×C₃×#41	—	10	1	no	—
94	C₃×C₄⋊Q₈	—	5	1	no	—
95	C₃×#127	—	5	1	no	—
96	C₃×#124	—	10	1	no	—
97	C₃×#121	—	5	2	no	—
98	C₃×#121	—	10	1	no	—
99	C₃×C₄×D₄	—	5	2	no	—
100	C₃×C₄×D₄	—	10	1	no	—
101	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
102	C₂×C₄⋊Dic₃	—	10	1	no	—
103	C₂×C₄×Dic₃	—	10	1	no	—
104	C₃⋊#95	—	5	2	no	—
105	#830	C₂×D₅	1	2	yes	—
106	C₄⋊(C₃⋊C₈)	—	2	1	no	—
107	C₄×C₃⋊C₈	—	2	1	no	—
108	C₂²×C₄×C₅	#67	1	1	yes	characteristic
109	C₂²×C₄×C₅	#67	1	1	yes	characteristic
110	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
111	C₂×C₂²⋊D₅	—	2	2	no	—
112	C₂²×Dic₅	—	6	2	no	—
113	#709	—	4	2	no	—
114	#709	—	4	2	no	—
115	C₂×D₂₀	—	2	1	no	—
116	C₂×D₂₀	—	2	1	no	—
117	C₂×D₂₀	C₂×Dic₃	1	2	yes	—
118	C₂×C₄×D₅	—	2	2	no	—
119	C₂×Dic₁₀	—	2	1	no	—
120	C₂×C₅×C₈	—	6	2	no	—
121	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
122	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
123	C₄²×C₅	#67	1	1	yes	characteristic
124	C₄²×C₅	#67	1	1	yes	characteristic
125	#685	—	2	2	no	—
126	#685	—	2	2	no	—
127	C₄⋊Dic₅	—	2	1	no	—
128	C₄⋊Dic₅	—	6	4	no	—
129	C₄⋊Dic₅	C₂×Dic₃	1	2	yes	—
130	C₄×Dic₅	—	12	2	no	—
131	#532	—	5	1	no	—
132	#431	—	3	1	no	—
133	C₄×C₄⋊C₄	—	15	1	no	—
134	#327	—	15	4	no	—
135	C₂²×C₃×C₅	—	2	1	no	—
136	C₂²×C₃×C₅	—	2	1	no	—
137	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
139	C₂²×C₃×C₅	C₂×SD₁₆	1	1	yes	characteristic
140	C₂×C₃×D₅	—	4	2	no	—
141	C₂×C₃×D₅	—	4	2	no	—
142	C₂×C₃×D₅	—	4	2	no	—
143	C₃×C₄×C₅	—	2	1	no	—
144	C₃×C₄×C₅	—	2	1	no	—
145	C₃×C₄×C₅	—	2	1	no	—
146	C₃×C₄×C₅	#102	1	2	yes	—
147	C₃×C₄×C₅	#102	1	2	yes	—
148	C₃×C₄×C₅	#104	1	2	yes	—
149	Dic₁₅	—	4	4	no	—
150	C₃×Dic₅	—	4	2	no	—
151	C₃×#41	—	20	2	no	—
152	C₃×#41	—	20	2	no	—
153	C₂×C₃×Q₈	—	10	1	no	—
154	C₂×C₃×D₄	—	5	2	no	—
155	C₂×C₃×D₄	—	10	2	no	—
156	C₂×C₃×D₄	—	10	1	no	—
157	C₂×C₃×D₄	—	10	1	no	—
158	C₂²×C₃×C₄	—	10	2	no	—
159	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
160	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
161	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
162	C₂²×Dic₃	—	10	2	no	—
163	C₃×C₄⋊C₄	—	5	2	no	—
164	C₃×C₄⋊C₄	—	10	1	no	—
165	C₃×C₂²⋊C₄	—	10	2	no	—
166	C₃×C₂²⋊C₄	—	10	2	no	—
167	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
168	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
169	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
170	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
171	C₄⋊Dic₃	—	10	4	no	—
172	C₄×Dic₃	—	20	2	no	—
173	C₂×C₃⋊C₈	—	2	2	no	—
174	C₂³×C₅	C₂×#67	1	1	yes	characteristic
175	C₂²×D₅	—	2	2	no	—
176	C₂×C₄×C₅	—	2	2	no	—
177	C₂×C₄×C₅	—	4	1	no	—
178	C₂×C₄×C₅	#213	1	2	yes	—
179	C₂×C₄×C₅	#214	1	2	yes	—
180	C₂×C₄×C₅	#217	1	1	yes	characteristic
181	C₂×C₄×C₅	#217	1	1	yes	characteristic
182	C₂×C₄×C₅	#217	1	1	yes	characteristic
183	C₂×C₄×C₅	#217	1	1	yes	characteristic
184	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
185	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
186	C₂²⋊D₅	—	4	2	no	—
187	C₂²⋊D₅	—	4	2	no	—
188	C₂×Dic₅	—	2	2	no	—
189	C₂×Dic₅	—	6	4	no	—
190	C₂×Dic₅	—	12	2	no	—
191	C₂×Dic₅	—	12	2	no	—
192	D₂₀	—	4	2	no	—
193	D₂₀	—	4	2	no	—
194	D₂₀	—	4	2	no	—
195	C₄×D₅	—	4	4	no	—
196	Dic₁₀	—	4	2	no	—
197	C₅×C₈	—	12	2	no	—
198	C₂×#41	—	10	1	no	—
199	C₄⋊Q₈	—	5	1	no	—
200	#127	—	5	1	no	—
201	#124	—	10	1	no	—
202	#121	—	5	2	no	—
203	#121	—	10	1	no	—
204	C₄×D₄	—	5	2	no	—
205	C₄×D₄	—	10	1	no	—
206	C₂×C₄⋊C₄	—	30	1	no	—
207	C₂×C₄²	—	30	1	no	—
208	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
209	C₄⋊C₈	—	6	1	no	—
210	C₂²⋊C₈	—	3	2	no	—
211	C₄×C₈	—	6	1	no	—
212	#95	—	15	2	no	—
213	C₂×C₃×C₅	—	2	1	no	—
214	C₂×C₃×C₅	—	2	1	no	—
215	C₂×C₃×C₅	#410	1	1	yes	characteristic
216	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
217	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
218	C₃×D₅	—	8	2	no	—
219	C₂³×C₃	—	10	2	no	—
220	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
221	C₃×Q₈	—	20	2	no	—
222	C₃×D₄	—	20	2	no	—
223	C₃×D₄	—	20	2	no	—
224	C₃×D₄	—	20	2	no	—
225	C₃×D₄	—	20	2	no	—
226	C₃×D₄	—	20	2	no	—
227	C₂×C₃×C₄	—	2	2	no	—
228	C₂×C₃×C₄	—	4	1	no	—
229	C₂×C₃×C₄	—	10	2	no	—
230	C₂×C₃×C₄	—	20	4	no	—
231	C₂×C₃×C₄	#677	1	2	yes	—
232	C₂×C₃×C₄	D₄₀	1	2	yes	—
233	C₂×C₃×C₄	#685	1	1	yes	characteristic
234	C₂×C₃×C₄	#685	1	1	yes	characteristic
235	C₂×C₃×C₄	#685	1	1	yes	characteristic
236	C₂×C₃×C₄	#685	1	1	yes	characteristic
237	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
238	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
239	C₂×Dic₃	—	10	4	no	—
240	C₂×Dic₃	—	20	2	no	—
241	C₂×Dic₃	—	20	2	no	—
242	C₃⋊C₈	—	4	2	no	—
243	C₂²×C₅	—	2	1	no	—
244	C₂²×C₅	—	2	1	no	—
245	C₂²×C₅	C₂×#213	1	1	yes	characteristic
246	C₂²×C₅	C₂×#214	1	1	yes	characteristic
247	C₂²×C₅	C₂×#217	1	1	yes	characteristic
248	C₂×D₅	—	4	2	no	—
249	C₂×D₅	—	4	2	no	—
250	C₂×D₅	—	4	2	no	—
251	C₄×C₅	—	2	1	no	—
252	C₄×C₅	—	2	1	no	—
253	C₄×C₅	—	2	1	no	—
254	C₄×C₅	#832	1	2	yes	—
255	C₄×C₅	#832	1	2	yes	—
256	C₄×C₅	#837	1	2	yes	—
257	Dic₅	—	4	2	no	—
258	Dic₅	—	12	4	no	—
259	#41	—	20	2	no	—
260	#41	—	20	2	no	—
261	C₂×Q₈	—	10	1	no	—
262	C₂×D₄	—	5	2	no	—
263	C₂×D₄	—	10	2	no	—
264	C₂×D₄	—	10	1	no	—
265	C₂×D₄	—	10	1	no	—
266	C₂²×C₄	—	10	2	no	—
267	C₂²×C₄	—	30	2	no	—
268	C₂²×C₄	#1202	1	1	yes	characteristic
269	C₂²×C₄	#1202	1	1	yes	characteristic
270	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
271	C₂×C₈	—	6	2	no	—
272	C₄⋊C₄	—	5	2	no	—
273	C₄⋊C₄	—	10	1	no	—
274	C₄⋊C₄	—	30	4	no	—
275	C₂²⋊C₄	—	10	2	no	—
276	C₂²⋊C₄	—	10	2	no	—
277	C₄²	—	60	2	no	—
278	C₄²	D₅×Dic₃	1	1	yes	characteristic
279	C₄²	D₅×Dic₃	1	1	yes	characteristic
280	C₄²	#1201	1	1	yes	characteristic
281	C₄²	#1201	1	1	yes	characteristic
282	C₃×C₅	#1502	1	1	yes	characteristic, complemented, Hall
283	C₂²×C₃	—	2	1	no	—
284	C₂²×C₃	—	2	1	no	—
285	C₂²×C₃	—	20	2	no	—
286	C₂²×C₃	—	20	2	no	—
287	C₂²×C₃	—	20	2	no	—
288	C₂²×C₃	C₂×#677	1	1	yes	characteristic
289	C₂²×C₃	C₂×D₄₀	1	1	yes	characteristic
290	C₂²×C₃	C₂×#685	1	1	yes	characteristic
291	C₃×C₄	—	2	1	no	—
292	C₃×C₄	—	2	1	no	—
293	C₃×C₄	—	2	1	no	—
294	C₃×C₄	—	20	2	no	—
295	C₃×C₄	#3953	1	2	yes	—
296	C₃×C₄	#3953	1	2	yes	—
297	C₃×C₄	#3957	1	2	yes	—
298	Dic₃	—	20	4	no	—
299	C₂×C₅	—	2	1	no	—
300	C₂×C₅	—	2	1	no	—
301	C₂×C₅	C₂×#832	1	1	yes	characteristic
302	C₂×C₅	C₂×#837	1	1	yes	characteristic
303	C₂×C₅	#5240	1	1	yes	characteristic
304	D₅	—	8	2	no	—
305	C₂³	—	10	2	no	—
306	C₂³	C₂×#1202	1	1	yes	characteristic
307	Q₈	—	20	2	no	—
308	D₄	—	20	2	no	—
309	D₄	—	20	2	no	—
310	D₄	—	20	2	no	—
311	D₄	—	20	2	no	—
312	D₄	—	20	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	10	2	no	—
316	C₂×C₄	—	20	4	no	—
317	C₂×C₄	—	30	4	no	—
318	C₂×C₄	—	60	2	no	—
319	C₂×C₄	—	60	2	no	—
320	C₂×C₄	#6653	1	2	yes	—
321	C₂×C₄	#6657	1	2	yes	—
322	C₂×C₄	#6665	1	1	yes	characteristic
323	C₂×C₄	#6665	1	1	yes	characteristic
324	C₂×C₄	#6665	1	1	yes	characteristic
325	C₂×C₄	#6665	1	1	yes	characteristic
326	C₂×C₄	C₂×D₅×Dic₃	1	1	yes	characteristic
327	C₂×C₄	C₂×#1201	1	1	yes	characteristic
328	C₈	—	12	2	no	—
329	C₂×C₃	—	2	1	no	—
330	C₂×C₃	—	2	1	no	—
331	C₂×C₃	—	40	2	no	—
332	C₂×C₃	#65295	1	1	yes	characteristic
333	C₂×C₃	C₂×#3953	1	1	yes	characteristic
334	C₂×C₃	C₂×#3957	1	1	yes	characteristic
335	C₅	#71811	1	1	yes	characteristic, complemented, Sylow
336	C₂²	—	2	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	20	2	no	—
339	C₂²	—	20	2	no	—
340	C₂²	—	20	2	no	—
341	C₂²	C₂×#6653	1	1	yes	characteristic
342	C₂²	C₂×#6657	1	1	yes	characteristic
343	C₂²	C₂×#6665	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	2	1	no	—
347	C₄	—	20	2	no	—
348	C₄	—	60	4	no	—
349	C₄	#90965	1	2	yes	—
350	C₄	#90965	1	2	yes	—
351	C₄	#90976	1	2	yes	—
352	C₃	#10601582	1	1	yes	characteristic, complemented, Sylow
353	C₂	—	2	1	no	—
354	C₂	—	2	1	no	—
355	C₂	—	40	2	no	—
356	C₂	C₂×#90965	1	1	yes	characteristic
357	C₂	C₂×#90976	1	1	yes	characteristic
358	C₂	#11748624	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6665
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₃×#65811	C₂
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#1502
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. C₄×C₄⋊Dic₁₅	C₂	semidirect
3. C₅×#4925	C₂	semidirect
4. #11745955	C₂	semidirect
13. C₃×#4020	C₄	semidirect
15. C₃×C₄×D₂₀	C₄	semidirect
16. C₃×#4015	C₄	semidirect
51. #4925	D₅	semidirect
60. #4020	Dic₃	semidirect
62. C₄×D₂₀	Dic₃	semidirect
63. #4015	Dic₃	semidirect
282. C₃×C₅	#1502	semidirect
335. C₅	#71811	semidirect
352. C₃	#10601582	semidirect

Maximal subgroups

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

subgroup	index
C₃×#65811	2
C₄×C₄⋊Dic₁₅	2
C₅×#4925	2
#11745955	2
#10601582	3
#71811	5