Subgroups — #14284603

Order 1920.

512 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	#11756236	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#65214	C₂	1	1	yes	maximal, characteristic, complemented
4	#11745859	C₂	1	2	yes	maximal, complemented
5	#11745854	C₂	1	2	yes	maximal, complemented
6	#10601224	—	3	1	no	maximal, Hall
7	C₂×#6817	—	2	1	no	—
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	#91819	—	2	1	no	—
10	#91819	C₄	1	2	yes	—
11	#91764	—	2	1	no	—
12	#91763	C₂²	1	1	yes	characteristic
13	C₄×D₆₀	—	2	1	no	—
14	C₄×D₆₀	C₄	1	2	yes	—
15	#91758	C₂²	1	1	yes	characteristic
16	C₂²×C₅×C₃⋊C₈	—	2	1	no	—
17	C₅×C₄⋊(C₃⋊C₈)	—	2	1	no	—
18	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
19	C₃×C₂²⋊#168	—	2	1	no	—
20	C₃×C₄⋊#168	C₂²	1	2	yes	—
21	C₃×C₄×#168	—	2	1	no	—
22	#70772	—	5	1	no	maximal, Hall
23	#65811	—	3	1	no	—
24	C₂×C₅×C₄⋊C₈	—	3	1	no	—
25	#65214	D₃	1	1	yes	characteristic, complemented
26	#64705	—	3	2	no	—
27	#64702	—	3	2	no	—
28	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
30	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
31	C₂×#1220	—	2	2	no	—
32	#6817	—	4	2	no	—
33	#6817	—	4	2	no	—
34	C₂×D₆₀	—	2	1	no	—
35	C₂×D₆₀	—	2	1	no	—
36	C₂×D₆₀	C₂×C₄	1	2	yes	—
37	C₂×C₄×D₁₅	—	2	2	no	—
38	C₂×Dic₃₀	—	2	1	no	—
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	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
43	#6714	—	2	2	no	—
44	#6714	—	2	2	no	—
45	C₄⋊Dic₁₅	—	2	1	no	—
46	C₄⋊Dic₁₅	C₂×C₄	1	2	yes	—
47	C₂×C₅×C₃⋊C₈	—	2	2	no	—
48	C₂×C₅×C₃⋊C₈	—	4	2	no	—
49	C₂×C₃×#168	—	2	2	no	—
50	#5477	—	5	1	no	—
51	C₃×#431	—	5	1	no	—
52	C₂×C₄⋊(C₃⋊C₈)	D₅	1	1	yes	characteristic, complemented
53	#4486	—	5	2	no	—
54	#4485	—	5	2	no	—
55	C₂×#709	—	6	1	no	—
56	C₂²×C₅×C₈	—	6	1	no	—
57	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
58	#4076	—	3	2	no	—
59	#4076	—	6	1	no	—
60	#4021	—	6	1	no	—
61	#4020	—	3	1	no	—
62	C₄×D₂₀	—	3	2	no	—
63	C₄×D₂₀	—	6	1	no	—
64	#4015	—	3	1	no	—
65	C₅×C₄⋊C₈	—	3	2	no	—
66	C₅×C₄⋊C₈	—	6	1	no	—
67	C₂²⋊#168	—	2	1	no	—
68	C₄⋊#168	C₂×D₃	1	2	yes	—
69	C₄×#168	—	2	1	no	—
70	#1540	—	15	1	no	Sylow
71	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×D₁₅	—	2	2	no	—
73	C₂×C₃×C₄×C₅	—	2	2	no	—
74	C₂×C₃×C₄×C₅	—	4	1	no	—
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	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	D₈	1	2	yes	—
79	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
80	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	#1220	—	4	4	no	—
84	C₂×Dic₁₅	—	2	2	no	—
85	D₆₀	—	4	2	no	—
86	D₆₀	—	4	1	no	—
87	D₆₀	—	4	1	no	—
88	D₆₀	—	4	2	no	—
89	C₄×D₁₅	—	4	4	no	—
90	Dic₃₀	—	4	1	no	—
91	Dic₃₀	—	4	1	no	—
92	C₃×#168	—	4	2	no	—
93	C₅×C₃⋊C₈	—	4	2	no	—
94	C₂×#235	—	10	1	no	—
95	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
96	#930	—	5	2	no	—
97	#930	—	10	1	no	—
98	C₂²×C₃⋊C₈	—	2	1	no	—
99	#875	—	10	1	no	—
100	#874	—	5	1	no	—
101	C₄×D₁₂	—	5	2	no	—
102	C₄×D₁₂	—	10	1	no	—
103	#869	—	5	1	no	—
104	C₃×C₄⋊C₈	—	5	2	no	—
105	C₃×C₂²⋊C₈	—	10	1	no	—
106	C₃×C₄×C₈	—	10	1	no	—
107	C₄⋊(C₃⋊C₈)	—	2	1	no	—
108	C₄⋊(C₃⋊C₈)	C₂×D₅	1	2	yes	—
109	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
110	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
111	C₂²×C₄×C₅	#67	1	1	yes	characteristic
112	C₂×C₂²⋊D₅	—	6	2	no	—
113	#709	—	12	2	no	—
114	#709	—	12	2	no	—
115	C₂×D₂₀	—	3	2	no	—
116	C₂×D₂₀	—	6	1	no	—
117	C₂×D₂₀	—	6	1	no	—
118	C₂×C₄×D₅	—	6	2	no	—
119	C₂×Dic₁₀	—	6	1	no	—
120	C₂×C₅×C₈	—	6	2	no	—
121	C₂×C₅×C₈	—	12	2	no	—
122	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
123	C₄²×C₅	D₁₂	1	1	yes	characteristic
124	C₄²×C₅	#67	1	1	yes	characteristic
125	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
126	#685	—	6	2	no	—
127	#685	—	6	2	no	—
128	C₄⋊Dic₅	—	3	2	no	—
129	C₄⋊Dic₅	—	6	1	no	—
130	C₂×#168	—	2	2	no	—
131	#532	—	15	1	no	—
132	#431	—	5	1	no	—
133	C₂×C₄⋊C₈	—	3	1	no	—
134	#332	—	15	2	no	—
135	#331	—	15	2	no	—
136	C₂²×C₃×C₅	—	2	1	no	—
137	C₂²×C₃×C₅	—	2	1	no	—
138	C₂²×C₃×C₅	#102	1	2	yes	—
139	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₂×D₁₅	—	4	2	no	—
141	C₂×D₁₅	—	4	2	no	—
142	C₂×D₁₅	—	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	C₂×D₈	1	1	yes	characteristic
148	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
149	C₃×C₄×C₅	#135	1	2	yes	—
150	Dic₁₅	—	4	2	no	—
151	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
152	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
153	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
154	C₂×#67	—	10	2	no	—
155	#235	—	20	2	no	—
156	#235	—	20	2	no	—
157	C₂×D₁₂	—	5	2	no	—
158	C₂×D₁₂	—	10	1	no	—
159	C₂×D₁₂	—	10	1	no	—
160	C₂×C₄×D₃	—	10	2	no	—
161	C₂×Dic₆	—	10	1	no	—
162	C₂×C₃×C₈	—	10	2	no	—
163	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
164	C₃×C₄²	D₂₀	1	1	yes	characteristic
165	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
166	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
167	(C₂×D₃)⋊C₄	—	10	2	no	—
168	(C₂×D₃)⋊C₄	—	10	2	no	—
169	C₄⋊Dic₃	—	5	2	no	—
170	C₄⋊Dic₃	—	10	1	no	—
171	C₂×C₃⋊C₈	—	2	2	no	—
172	C₂×C₃⋊C₈	—	4	2	no	—
173	C₂³×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
174	C₂²×D₅	—	6	2	no	—
175	C₂×C₄×C₅	—	2	2	no	—
176	C₂×C₄×C₅	—	4	1	no	—
177	C₂×C₄×C₅	#204	1	2	yes	—
178	C₂×C₄×C₅	D₂₄	1	2	yes	—
179	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
180	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
181	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
182	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
183	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
184	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
185	C₂²⋊D₅	—	12	4	no	—
186	C₂×Dic₅	—	6	2	no	—
187	D₂₀	—	12	2	no	—
188	D₂₀	—	12	1	no	—
189	D₂₀	—	12	1	no	—
190	D₂₀	—	12	2	no	—
191	C₄×D₅	—	12	4	no	—
192	Dic₁₀	—	12	1	no	—
193	Dic₁₀	—	12	1	no	—
194	C₅×C₈	—	12	2	no	—
195	#168	—	4	2	no	—
196	C₂×#41	—	30	1	no	—
197	C₂²×C₈	—	6	1	no	—
198	C₄⋊Q₈	—	15	1	no	—
199	#127	—	15	1	no	—
200	#124	—	30	1	no	—
201	#121	—	15	2	no	—
202	#121	—	30	1	no	—
203	C₄×D₄	—	15	2	no	—
204	C₄×D₄	—	30	1	no	—
205	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
206	C₄⋊C₈	—	3	2	no	—
207	C₄⋊C₈	—	5	2	no	—
208	C₄⋊C₈	—	6	1	no	—
209	C₂²⋊C₈	—	10	1	no	—
210	C₄×C₈	—	10	1	no	—
211	C₂×C₃×C₅	—	2	2	no	—
212	C₂×C₃×C₅	#413	1	1	yes	characteristic
213	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
214	C₂×C₃×C₅	#416	1	1	yes	characteristic
215	D₁₅	—	8	2	no	—
216	C₂³×C₃	#685	1	1	yes	characteristic
217	C₂²×D₃	—	10	2	no	—
218	C₂×C₃×C₄	—	2	2	no	—
219	C₂×C₃×C₄	—	4	1	no	—
220	C₂×C₃×C₄	#685	1	1	yes	characteristic
221	C₂×C₃×C₄	#685	1	1	yes	characteristic
222	C₂×C₃×C₄	#685	1	1	yes	characteristic
223	C₂×C₃×C₄	#686	1	2	yes	—
224	C₂×C₃×C₄	#688	1	2	yes	—
225	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
226	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
227	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
228	#67	—	20	4	no	—
229	C₂×Dic₃	—	10	2	no	—
230	D₁₂	—	20	2	no	—
231	D₁₂	—	20	2	no	—
232	D₁₂	—	20	1	no	—
233	D₁₂	—	20	1	no	—
234	C₄×D₃	—	20	4	no	—
235	Dic₆	—	20	1	no	—
236	Dic₆	—	20	1	no	—
237	C₃×C₈	—	20	2	no	—
238	C₃⋊C₈	—	4	2	no	—
239	C₂²×C₅	—	2	1	no	—
240	C₂²×C₅	—	2	1	no	—
241	C₂²×C₅	#821	1	2	yes	—
242	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
243	C₂×D₅	—	12	2	no	—
244	C₂×D₅	—	12	2	no	—
245	C₂×D₅	—	12	2	no	—
246	C₄×C₅	—	2	1	no	—
247	C₄×C₅	—	2	1	no	—
248	C₄×C₅	—	2	1	no	—
249	C₄×C₅	#821	1	2	yes	—
250	C₄×C₅	C₂×#204	1	1	yes	characteristic
251	C₄×C₅	C₂×D₂₄	1	1	yes	characteristic
252	C₄×C₅	#950	1	2	yes	—
253	Dic₅	—	12	2	no	—
254	#41	—	60	2	no	—
255	#41	—	60	2	no	—
256	C₂×Q₈	—	30	1	no	—
257	C₂×D₄	—	15	2	no	—
258	C₂×D₄	—	30	2	no	—
259	C₂×D₄	—	30	1	no	—
260	C₂×D₄	—	30	1	no	—
261	C₂²×C₄	—	30	2	no	—
262	C₂²×C₄	#1200	1	1	yes	characteristic
263	C₂²×C₄	#1202	1	1	yes	characteristic
264	C₂²×C₄	#1203	1	1	yes	characteristic
265	C₂×C₈	—	6	2	no	—
266	C₂×C₈	—	10	2	no	—
267	C₂×C₈	—	12	2	no	—
268	C₄⋊C₄	—	15	2	no	—
269	C₄⋊C₄	—	30	1	no	—
270	C₂²⋊C₄	—	30	2	no	—
271	C₂²⋊C₄	—	30	2	no	—
272	C₄²	#1200	1	1	yes	characteristic
273	C₄²	#1202	1	1	yes	characteristic
274	C₄²	#1203	1	1	yes	characteristic
275	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
276	C₃×C₅	#1540	1	1	yes	characteristic, complemented, Hall
277	C₂²×C₃	—	2	1	no	—
278	C₂²×C₃	—	2	1	no	—
279	C₂²×C₃	#3941	1	2	yes	—
280	C₂²×C₃	C₂×#685	1	1	yes	characteristic
281	C₂×D₃	—	20	2	no	—
282	C₂×D₃	—	20	2	no	—
283	C₂×D₃	—	20	2	no	—
284	C₃×C₄	—	2	1	no	—
285	C₃×C₄	—	2	1	no	—
286	C₃×C₄	—	2	1	no	—
287	C₃×C₄	#3941	1	2	yes	—
288	C₃×C₄	#4050	1	2	yes	—
289	C₃×C₄	C₂×#686	1	1	yes	characteristic
290	C₃×C₄	C₂×#688	1	1	yes	characteristic
291	Dic₃	—	20	2	no	—
292	C₂×C₅	—	2	2	no	—
293	C₂×C₅	#5002	1	1	yes	characteristic
294	C₂×C₅	C₂×#821	1	1	yes	characteristic
295	C₂×C₅	#5132	1	1	yes	characteristic
296	D₅	—	24	2	no	—
297	C₂³	—	30	2	no	—
298	C₂³	#6667	1	1	yes	characteristic
299	Q₈	—	60	1	no	—
300	Q₈	—	60	1	no	—
301	D₄	—	60	2	no	—
302	D₄	—	60	4	no	—
303	D₄	—	60	1	no	—
304	D₄	—	60	2	no	—
305	D₄	—	60	1	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	30	2	no	—
309	C₂×C₄	—	60	4	no	—
310	C₂×C₄	#6654	1	2	yes	—
311	C₂×C₄	#6660	1	2	yes	—
312	C₂×C₄	#6667	1	1	yes	characteristic
313	C₂×C₄	#6667	1	1	yes	characteristic
314	C₂×C₄	#6667	1	1	yes	characteristic
315	C₂×C₄	C₂×#1200	1	1	yes	characteristic
316	C₂×C₄	C₂×#1202	1	1	yes	characteristic
317	C₂×C₄	C₂×#1203	1	1	yes	characteristic
318	C₈	—	12	2	no	—
319	C₈	—	20	2	no	—
320	C₂×C₃	—	2	2	no	—
321	C₂×C₃	C₂×#3941	1	1	yes	characteristic
322	C₂×C₃	#65402	1	1	yes	characteristic
323	C₂×C₃	#65421	1	1	yes	characteristic
324	D₃	—	40	2	no	—
325	C₅	#70772	1	1	yes	characteristic, complemented, Sylow
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	60	2	no	—
329	C₂²	—	60	2	no	—
330	C₂²	—	60	2	no	—
331	C₂²	#90969	1	2	yes	—
332	C₂²	C₂×#6667	1	1	yes	characteristic
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	60	2	no	—
337	C₄	#90969	1	2	yes	—
338	C₄	C₂×#6654	1	1	yes	characteristic
339	C₄	#91311	1	2	yes	—
340	C₄	C₂×#6660	1	1	yes	characteristic
341	C₃	#10601224	1	1	yes	characteristic, complemented, Sylow
342	C₂	—	2	2	no	—
343	C₂	—	120	2	no	—
344	C₂	#11748357	1	1	yes	characteristic
345	C₂	C₂×#90969	1	1	yes	characteristic
346	C₂	#11748411	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6667
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	#11756236	C₂
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#1540
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. C₂×C₅×C₄⋊(C₃⋊C₈)	C₂	semidirect
3. C₃×#65214	C₂	semidirect
4. #11745859	C₂	semidirect
5. #11745854	C₂	semidirect
25. #65214	D₃	semidirect
52. C₂×C₄⋊(C₃⋊C₈)	D₅	semidirect
276. C₃×C₅	#1540	semidirect
325. C₅	#70772	semidirect
341. C₃	#10601224	semidirect

Maximal subgroups

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

subgroup	index
#11756236	2
C₂×C₅×C₄⋊(C₃⋊C₈)	2
C₃×#65214	2
#11745859	2
#11745854	2
#10601224	3
#70772	5