Subgroups — C₅×#68081 #14286979

Order 1920.

364 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×D₂₄	C₂	1	1	yes	maximal, characteristic
2	C₅×#4688	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×C₈×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#322	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4459	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4459	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4457	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1586	—	3	1	no	maximal, Hall
9	C₂×C₅×D₂₄	C₄	1	1	yes	characteristic
10	C₂×C₅×ΓL(1,𝔽₂₅)	—	2	1	no	—
11	C₂×C₅×D₃×C₈	—	2	1	no	—
12	C₄×C₅×D₁₂	C₂²	1	1	yes	characteristic
13	C₄×C₅×D₁₂	C₂²	1	1	yes	characteristic
14	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₅×#821	C₄	1	1	yes	characteristic
18	C₅×#821	C₄	1	1	yes	characteristic
19	C₅×(C₂×D₃)⋊C₈	—	2	1	no	—
20	C₅×(C₂×D₃)⋊C₈	—	2	1	no	—
21	C₅×#818	C₄	1	1	yes	characteristic
22	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
23	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
24	#68081	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₄×C₅×D₈	—	3	1	no	—
26	C₅×#436	—	3	1	no	—
27	C₅×C₈×D₄	—	3	1	no	—
28	C₅×#334	—	3	1	no	—
29	C₅×D₄⋊C₈	—	3	1	no	—
30	C₅×D₄⋊C₈	—	3	1	no	—
31	C₅×#322	D₃	1	1	yes	characteristic, complemented
32	C₂×C₅×D₁₂	C₂×C₄	1	1	yes	characteristic
33	C₂×C₅×D₁₂	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×C₅×D₃	—	2	1	no	—
35	C₂×C₄×C₅×D₃	—	2	1	no	—
36	C₂×C₃×C₅×C₈	—	2	1	no	—
37	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
40	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
41	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
42	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
43	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
44	C₅×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
45	C₅×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₅×C₃⋊C₈	—	2	1	no	—
47	C₂×C₅×C₃⋊C₈	—	2	1	no	—
48	C₅×D₂₄	—	2	2	no	—
49	C₅×ΓL(1,𝔽₂₅)	—	4	2	no	—
50	C₅×D₃×C₈	—	4	2	no	—
51	C₄×D₂₄	C₂×C₅	1	1	yes	characteristic
52	#4688	C₂×C₅	1	1	yes	characteristic, complemented
53	C₈×D₁₂	C₂×C₅	1	1	yes	characteristic, complemented
54	C₃×#322	C₂×C₅	1	1	yes	characteristic, complemented
55	#4459	C₂×C₅	1	1	yes	characteristic, complemented
56	#4459	C₂×C₅	1	1	yes	characteristic, complemented
57	#4457	C₂×C₅	1	1	yes	characteristic, complemented
58	C₂×C₅×D₈	—	3	1	no	—
59	C₂×C₅×M₁₆	—	6	1	no	—
60	C₂²×C₅×C₈	—	6	1	no	—
61	C₄×C₅×D₄	—	3	1	no	—
62	C₄×C₅×D₄	—	3	1	no	—
63	C₅×#107	—	3	1	no	—
64	C₅×C₄⋊C₈	—	3	1	no	—
65	C₅×C₄⋊C₈	—	3	1	no	—
66	C₅×#102	—	3	1	no	—
67	C₅×#102	—	3	1	no	—
68	C₅×C₂²⋊C₈	—	6	1	no	—
69	C₅×C₂²⋊C₈	—	6	1	no	—
70	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
71	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
72	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
73	#1586	—	3	1	no	Sylow
74	C₂²×C₅×D₃	—	2	1	no	—
75	C₂²×C₅×D₃	—	2	1	no	—
76	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
79	C₂×C₅×Dic₃	—	2	1	no	—
80	C₂×C₅×Dic₃	—	2	1	no	—
81	C₅×D₁₂	—	2	1	no	—
82	C₅×D₁₂	—	2	1	no	—
83	C₅×D₁₂	—	2	1	no	—
84	C₅×D₁₂	M₁₆	1	2	yes	complemented
85	C₄×C₅×D₃	—	4	1	no	—
86	C₄×C₅×D₃	—	4	1	no	—
87	C₄×C₅×D₃	—	4	1	no	—
88	C₄×C₅×D₃	—	4	1	no	—
89	C₃×C₅×C₈	—	2	2	no	—
90	C₃×C₅×C₈	—	2	1	no	—
91	C₃×C₅×C₈	—	2	1	no	—
92	C₃×C₅×C₈	—	2	1	no	—
93	C₃×C₅×C₈	—	2	1	no	—
94	C₅×C₃⋊C₈	—	4	1	no	—
95	C₅×C₃⋊C₈	—	4	1	no	—
96	C₂×D₂₄	C₄×C₅	1	1	yes	characteristic
97	C₂×ΓL(1,𝔽₂₅)	—	2	1	no	—
98	C₂×D₃×C₈	—	2	1	no	—
99	C₄×D₁₂	C₂²×C₅	1	1	yes	characteristic
100	C₄×D₁₂	C₂²×C₅	1	1	yes	characteristic
101	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
102	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
103	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
104	#821	C₄×C₅	1	1	yes	characteristic
105	#821	C₄×C₅	1	1	yes	characteristic
106	(C₂×D₃)⋊C₈	—	2	1	no	—
107	(C₂×D₃)⋊C₈	—	2	1	no	—
108	#818	C₄×C₅	1	1	yes	characteristic
109	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
110	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
111	C₂×C₅×D₄	—	3	1	no	—
112	C₂×C₅×D₄	—	3	1	no	—
113	C₂²×C₄×C₅	—	6	1	no	—
114	C₂²×C₄×C₅	—	6	1	no	—
115	C₅×D₈	—	6	2	no	—
116	C₅×M₁₆	—	12	2	no	—
117	C₂×C₅×C₈	—	2	1	no	—
118	C₂×C₅×C₈	—	6	1	no	—
119	C₂×C₅×C₈	—	6	1	no	—
120	C₂×C₅×C₈	—	12	2	no	—
121	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
122	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
123	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
124	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
125	C₅×C₄⋊C₄	—	3	1	no	—
126	C₅×C₄⋊C₄	—	3	1	no	—
127	C₅×C₂²⋊C₄	—	6	1	no	—
128	C₅×C₂²⋊C₄	—	6	1	no	—
129	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
130	C₄×D₈	—	3	1	no	—
131	#436	—	3	1	no	—
132	C₈×D₄	—	3	1	no	—
133	#334	—	3	1	no	—
134	D₄⋊C₈	—	3	1	no	—
135	D₄⋊C₈	—	3	1	no	—
136	#322	C₅×D₃	1	1	yes	characteristic, complemented
137	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
138	C₂×C₅×D₃	—	2	2	no	—
139	C₂×C₅×D₃	—	4	1	no	—
140	C₂×C₅×D₃	—	4	1	no	—
141	C₂×C₅×D₃	—	4	1	no	—
142	C₂×C₅×D₃	—	4	1	no	—
143	C₂×C₅×D₃	—	4	1	no	—
144	C₃×C₄×C₅	—	2	1	no	—
145	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
146	C₃×C₄×C₅	#131	1	1	yes	characteristic
147	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₅×Dic₃	—	4	1	no	—
150	C₅×Dic₃	—	4	1	no	—
151	C₂×D₁₂	C₂×C₄×C₅	1	1	yes	characteristic
152	C₂×D₁₂	C₂×C₄×C₅	1	1	yes	characteristic
153	C₂×C₄×D₃	—	2	1	no	—
154	C₂×C₄×D₃	—	2	1	no	—
155	C₂×C₃×C₈	—	2	1	no	—
156	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
157	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
158	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
159	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
160	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
161	(C₂×D₃)⋊C₄	—	2	1	no	—
162	(C₂×D₃)⋊C₄	—	2	1	no	—
163	C₄⋊Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
164	C₄⋊Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
165	C₂×C₃⋊C₈	—	2	1	no	—
166	C₂×C₃⋊C₈	—	2	1	no	—
167	D₂₄	—	2	2	no	—
168	ΓL(1,𝔽₂₅)	—	4	2	no	—
169	D₃×C₈	—	4	2	no	—
170	C₂³×C₅	—	6	1	no	—
171	C₂³×C₅	—	6	1	no	—
172	C₅×D₄	—	3	2	no	—
173	C₅×D₄	—	6	1	no	—
174	C₅×D₄	—	6	1	no	—
175	C₅×D₄	—	6	1	no	—
176	C₂×C₄×C₅	—	6	1	no	—
177	C₂×C₄×C₅	—	6	1	no	—
178	C₂×C₄×C₅	—	12	1	no	—
179	C₂×C₄×C₅	—	12	1	no	—
180	C₂×C₄×C₅	—	12	1	no	—
181	C₂×C₄×C₅	—	12	1	no	—
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	#235	1	1	yes	characteristic
185	C₅×C₈	—	2	2	no	—
186	C₅×C₈	—	2	1	no	—
187	C₅×C₈	—	2	1	no	—
188	C₅×C₈	—	2	1	no	—
189	C₅×C₈	—	2	1	no	—
190	C₅×C₈	—	12	1	no	—
191	C₅×C₈	—	12	1	no	—
192	C₂×D₈	—	3	1	no	—
193	C₂×M₁₆	—	6	1	no	—
194	C₂²×C₈	—	6	1	no	—
195	C₄×D₄	—	3	1	no	—
196	C₄×D₄	—	3	1	no	—
197	#107	—	3	1	no	—
198	C₄⋊C₈	—	3	1	no	—
199	C₄⋊C₈	—	3	1	no	—
200	#102	—	3	1	no	—
201	#102	—	3	1	no	—
202	C₂²⋊C₈	—	6	1	no	—
203	C₂²⋊C₈	—	6	1	no	—
204	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
205	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
206	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
207	C₂×C₃×C₅	#435	1	1	yes	characteristic
208	C₂×C₃×C₅	#442	1	1	yes	characteristic
209	C₂×C₃×C₅	#443	1	1	yes	characteristic
210	C₅×D₃	—	4	2	no	—
211	C₅×D₃	—	8	1	no	—
212	C₂²×D₃	—	2	1	no	—
213	C₂²×D₃	—	2	1	no	—
214	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
215	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
216	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
217	C₂×Dic₃	—	2	1	no	—
218	C₂×Dic₃	—	2	1	no	—
219	D₁₂	—	2	1	no	—
220	D₁₂	—	2	1	no	—
221	D₁₂	—	2	1	no	—
222	D₁₂	C₅×M₁₆	1	2	yes	complemented
223	C₄×D₃	—	4	1	no	—
224	C₄×D₃	—	4	1	no	—
225	C₄×D₃	—	4	1	no	—
226	C₄×D₃	—	4	1	no	—
227	C₃×C₈	—	2	2	no	—
228	C₃×C₈	—	2	1	no	—
229	C₃×C₈	—	2	1	no	—
230	C₃×C₈	—	2	1	no	—
231	C₃×C₈	—	2	1	no	—
232	C₃⋊C₈	—	4	1	no	—
233	C₃⋊C₈	—	4	1	no	—
234	C₂²×C₅	—	6	2	no	—
235	C₂²×C₅	—	12	1	no	—
236	C₂²×C₅	—	12	1	no	—
237	C₂²×C₅	—	12	1	no	—
238	C₂²×C₅	—	12	1	no	—
239	C₂²×C₅	—	12	1	no	—
240	C₂²×C₅	C₄×D₁₂	1	1	yes	characteristic
241	C₄×C₅	—	2	1	no	—
242	C₄×C₅	—	12	1	no	—
243	C₄×C₅	—	12	1	no	—
244	C₄×C₅	D₃×M₁₆	1	1	yes	characteristic
245	C₄×C₅	#907	1	1	yes	characteristic
246	C₄×C₅	#908	1	1	yes	characteristic
247	C₄×C₅	#908	1	1	yes	characteristic
248	C₂×D₄	—	3	1	no	—
249	C₂×D₄	—	3	1	no	—
250	C₂²×C₄	—	6	1	no	—
251	C₂²×C₄	—	6	1	no	—
252	D₈	—	6	2	no	—
253	M₁₆	—	12	2	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	6	1	no	—
256	C₂×C₈	—	6	1	no	—
257	C₂×C₈	—	12	2	no	—
258	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
259	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
260	C₂×C₈	C₅×D₁₂	1	1	yes	characteristic
261	C₂×C₈	C₅×D₁₂	1	1	yes	characteristic
262	C₄⋊C₄	—	3	1	no	—
263	C₄⋊C₄	—	3	1	no	—
264	C₂²⋊C₄	—	6	1	no	—
265	C₂²⋊C₄	—	6	1	no	—
266	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
267	C₃×C₅	#1586	1	1	yes	characteristic, complemented, Hall
268	C₂²×C₃	C₄×C₅×D₄	1	1	yes	characteristic
269	C₂×D₃	—	2	2	no	—
270	C₂×D₃	—	4	1	no	—
271	C₂×D₃	—	4	1	no	—
272	C₂×D₃	—	4	1	no	—
273	C₂×D₃	—	4	1	no	—
274	C₂×D₃	—	4	1	no	—
275	C₃×C₄	—	2	1	no	—
276	C₃×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
277	C₃×C₄	C₅×#131	1	1	yes	characteristic
278	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
279	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
280	Dic₃	—	4	1	no	—
281	Dic₃	—	4	1	no	—
282	C₂×C₅	—	12	2	no	—
283	C₂×C₅	—	24	1	no	—
284	C₂×C₅	#4700	1	1	yes	characteristic
285	C₂×C₅	#4706	1	1	yes	characteristic
286	C₂×C₅	#4711	1	1	yes	characteristic
287	C₂³	—	6	1	no	—
288	C₂³	—	6	1	no	—
289	D₄	—	3	2	no	—
290	D₄	—	6	1	no	—
291	D₄	—	6	1	no	—
292	D₄	—	6	1	no	—
293	C₂×C₄	—	6	1	no	—
294	C₂×C₄	—	6	1	no	—
295	C₂×C₄	—	12	1	no	—
296	C₂×C₄	—	12	1	no	—
297	C₂×C₄	—	12	1	no	—
298	C₂×C₄	—	12	1	no	—
299	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
300	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
301	C₂×C₄	C₅×#235	1	1	yes	characteristic
302	C₈	—	2	2	no	—
303	C₈	—	2	1	no	—
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	12	1	no	—
308	C₈	—	12	1	no	—
309	C₂×C₃	C₅×#435	1	1	yes	characteristic
310	C₂×C₃	C₅×#442	1	1	yes	characteristic
311	C₂×C₃	C₅×#443	1	1	yes	characteristic
312	D₃	—	4	2	no	—
313	D₃	—	8	1	no	—
314	C₅	#68081	1	1	yes	characteristic, direct product factor, Sylow
315	C₂²	—	6	2	no	—
316	C₂²	—	12	1	no	—
317	C₂²	—	12	1	no	—
318	C₂²	—	12	1	no	—
319	C₂²	—	12	1	no	—
320	C₂²	—	12	1	no	—
321	C₂²	C₄×C₅×D₁₂	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	12	1	no	—
324	C₄	—	12	1	no	—
325	C₄	C₅×D₃×M₁₆	1	1	yes	characteristic
326	C₄	C₅×#907	1	1	yes	characteristic
327	C₄	C₅×#908	1	1	yes	characteristic
328	C₄	C₅×#908	1	1	yes	characteristic
329	C₃	C₅×#1586	1	1	yes	characteristic, complemented, Sylow
330	C₂	—	12	2	no	—
331	C₂	—	24	1	no	—
332	C₂	C₅×#4700	1	1	yes	characteristic
333	C₂	C₅×#4706	1	1	yes	characteristic
334	C₂	C₅×#4711	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#322	C₂
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂×C₅×D₂₄	C₄
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₅×#322	D₃
hypocenter	C₃	C₅×#1586
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₄×C₅×C₈ ◁ C₅×#322

Complements

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

N	H ≅ G/N	type
2. C₅×#4688	C₂	semidirect
3. C₅×C₈×D₁₂	C₂	semidirect
4. C₃×C₅×#322	C₂	semidirect
5. C₅×#4459	C₂	semidirect
6. C₅×#4459	C₂	semidirect
7. C₅×#4457	C₂	semidirect
24. #68081	C₅	direct (N × H)
31. C₅×#322	D₃	semidirect
52. #4688	C₂×C₅	semidirect
53. C₈×D₁₂	C₂×C₅	semidirect
54. C₃×#322	C₂×C₅	semidirect
55. #4459	C₂×C₅	semidirect
56. #4459	C₂×C₅	semidirect
57. #4457	C₂×C₅	semidirect
84. C₅×D₁₂	M₁₆	semidirect
136. #322	C₅×D₃	semidirect
222. D₁₂	C₅×M₁₆	semidirect
267. C₃×C₅	#1586	semidirect
314. C₅	#68081	direct (N × H)
329. C₃	C₅×#1586	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×D₂₄	2
C₅×#4688	2
C₅×C₈×D₁₂	2
C₃×C₅×#322	2
C₅×#4459	2
C₅×#4457	2
C₅×#1586	3
#68081	5

Subgroups — C5×#68081 #14286979

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68081 #14286979