Subgroups — C₅×#68580 #14292105

Order 1920.

364 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#4843	C₂	1	1	yes	maximal, characteristic
2	C₅×#4841	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4694	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×C₈×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#335	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4485	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4482	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1681	—	3	1	no	maximal, Hall
9	C₂×C₅×D₂₄	—	2	1	no	—
10	C₂×C₅×#204	—	2	1	no	—
11	C₂×C₅×D₃×C₈	—	2	1	no	—
12	C₅×#874	C₂²	1	1	yes	characteristic
13	C₄×C₅×D₁₂	C₂²	1	1	yes	characteristic
14	C₅×#869	C₂²	1	1	yes	characteristic
15	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₅×#821	—	2	1	no	—
19	C₅×#821	—	2	1	no	—
20	C₅×#821	—	2	1	no	—
21	C₅×(C₂×D₃)⋊C₈	—	2	1	no	—
22	C₅×#816	—	2	1	no	—
23	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
24	#68580	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₅×#486	—	3	1	no	—
26	C₅×#461	—	3	1	no	—
27	C₅×#459	—	3	1	no	—
28	C₅×C₈×D₄	—	3	1	no	—
29	C₅×#335	D₃	1	1	yes	characteristic, complemented
30	C₅×#332	—	3	1	no	—
31	C₅×#331	—	3	1	no	—
32	C₂×C₅×D₁₂	—	2	1	no	—
33	C₂×C₅×D₁₂	—	2	1	no	—
34	C₂×C₅×D₁₂	—	2	1	no	—
35	C₂×C₅×D₁₂	D₄	1	1	yes	characteristic
36	C₂×C₄×C₅×D₃	—	2	1	no	—
37	C₂×C₅×Dic₆	—	2	1	no	—
38	C₂×C₃×C₅×C₈	—	2	1	no	—
39	C₂×C₃×C₅×C₈	—	2	1	no	—
40	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
41	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
42	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
43	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
44	C₅×C₄⋊Dic₃	—	2	1	no	—
45	C₅×C₄⋊Dic₃	—	2	1	no	—
46	C₅×C₄⋊Dic₃	D₄	1	1	yes	characteristic
47	C₂×C₅×C₃⋊C₈	—	2	1	no	—
48	C₅×D₂₄	—	4	2	no	—
49	C₅×#204	—	4	2	no	—
50	C₅×D₃×C₈	—	4	2	no	—
51	#4843	C₂×C₅	1	1	yes	characteristic
52	#4841	C₂×C₅	1	1	yes	characteristic, complemented
53	#4694	C₂×C₅	1	1	yes	characteristic, complemented
54	C₈×D₁₂	C₂×C₅	1	1	yes	characteristic, complemented
55	C₃×#335	C₂×C₅	1	1	yes	characteristic, complemented
56	#4485	C₂×C₅	1	1	yes	characteristic, complemented
57	#4482	C₂×C₅	1	1	yes	characteristic, complemented
58	C₂×C₅×SD₁₆	—	6	1	no	—
59	C₂×C₅×D₈	—	6	1	no	—
60	C₂²×C₅×C₈	—	6	1	no	—
61	C₅×C₄⋊Q₈	—	3	1	no	—
62	C₅×#127	—	3	1	no	—
63	C₄×C₅×D₄	—	3	1	no	—
64	C₅×C₄⋊C₈	—	3	1	no	—
65	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
66	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
67	C₅×#103	—	6	1	no	—
68	C₅×#102	—	6	1	no	—
69	C₅×#102	—	6	1	no	—
70	C₅×#102	—	6	1	no	—
71	C₅×C₂²⋊C₈	—	6	1	no	—
72	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
73	#1681	—	3	1	no	Sylow
74	C₂²×C₅×D₃	—	2	1	no	—
75	C₂²×C₅×D₃	—	4	1	no	—
76	C₂×C₃×C₄×C₅	C₂×D₄	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	—	4	1	no	—
81	C₅×D₁₂	—	2	1	no	—
82	C₅×D₁₂	—	4	1	no	—
83	C₅×D₁₂	—	4	1	no	—
84	C₅×D₁₂	—	4	1	no	—
85	C₅×D₁₂	—	4	1	no	—
86	C₅×D₁₂	—	8	1	no	—
87	C₅×D₁₂	D₈	1	2	yes	—
88	C₄×C₅×D₃	—	4	1	no	—
89	C₄×C₅×D₃	—	4	1	no	—
90	C₅×Dic₆	—	4	1	no	—
91	C₅×Dic₆	—	4	1	no	—
92	C₃×C₅×C₈	—	2	1	no	—
93	C₃×C₅×C₈	—	2	1	no	—
94	C₃×C₅×C₈	—	4	1	no	—
95	C₃×C₅×C₈	—	4	1	no	—
96	C₅×C₃⋊C₈	—	4	1	no	—
97	C₂×D₂₄	—	2	1	no	—
98	C₂×#204	—	2	1	no	—
99	C₂×D₃×C₈	—	2	1	no	—
100	#874	C₂²×C₅	1	1	yes	characteristic
101	C₄×D₁₂	C₂²×C₅	1	1	yes	characteristic
102	#869	C₂²×C₅	1	1	yes	characteristic
103	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
104	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
105	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
106	#821	—	2	1	no	—
107	#821	—	2	1	no	—
108	#821	—	2	1	no	—
109	(C₂×D₃)⋊C₈	—	2	1	no	—
110	#816	—	2	1	no	—
111	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
112	C₂×C₅×Q₈	—	6	1	no	—
113	C₂×C₅×D₄	—	3	1	no	—
114	C₂×C₅×D₄	—	6	1	no	—
115	C₂×C₅×D₄	—	6	1	no	—
116	C₂×C₅×D₄	—	6	1	no	—
117	C₂²×C₄×C₅	—	6	1	no	—
118	C₅×SD₁₆	—	12	2	no	—
119	C₅×D₈	—	12	2	no	—
120	C₂×C₅×C₈	—	2	1	no	—
121	C₂×C₅×C₈	—	2	1	no	—
122	C₂×C₅×C₈	—	6	1	no	—
123	C₂×C₅×C₈	—	12	2	no	—
124	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
125	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
126	C₅×C₄⋊C₄	—	3	1	no	—
127	C₅×C₄⋊C₄	—	6	1	no	—
128	C₅×C₄⋊C₄	—	6	1	no	—
129	C₅×C₂²⋊C₄	—	6	1	no	—
130	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
131	#486	—	3	1	no	—
132	#461	—	3	1	no	—
133	#459	—	3	1	no	—
134	C₈×D₄	—	3	1	no	—
135	#335	C₅×D₃	1	1	yes	characteristic, complemented
136	#332	—	3	1	no	—
137	#331	—	3	1	no	—
138	C₂²×C₃×C₅	#121	1	1	yes	characteristic
139	C₂×C₅×D₃	—	2	2	no	—
140	C₂×C₅×D₃	—	4	1	no	—
141	C₂×C₅×D₃	—	4	1	no	—
142	C₂×C₅×D₃	—	8	1	no	—
143	C₂×C₅×D₃	—	8	1	no	—
144	C₂×C₅×D₃	—	8	1	no	—
145	C₃×C₄×C₅	—	2	1	no	—
146	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
147	C₃×C₄×C₅	#135	1	1	yes	characteristic
148	C₃×C₄×C₅	#135	1	1	yes	characteristic
149	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
150	C₅×Dic₃	—	4	1	no	—
151	C₅×Dic₃	—	8	1	no	—
152	C₂×D₁₂	—	2	1	no	—
153	C₂×D₁₂	—	2	1	no	—
154	C₂×D₁₂	—	2	1	no	—
155	C₂×D₁₂	C₅×D₄	1	1	yes	characteristic
156	C₂×C₄×D₃	—	2	1	no	—
157	C₂×Dic₆	—	2	1	no	—
158	C₂×C₃×C₈	—	2	1	no	—
159	C₂×C₃×C₈	—	2	1	no	—
160	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
161	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
162	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
163	(C₂×D₃)⋊C₄	—	2	1	no	—
164	C₄⋊Dic₃	—	2	1	no	—
165	C₄⋊Dic₃	—	2	1	no	—
166	C₄⋊Dic₃	C₅×D₄	1	1	yes	characteristic
167	C₂×C₃⋊C₈	—	2	1	no	—
168	D₂₄	—	4	2	no	—
169	#204	—	4	2	no	—
170	D₃×C₈	—	4	2	no	—
171	C₂³×C₅	—	6	1	no	—
172	C₂³×C₅	—	12	1	no	—
173	C₅×Q₈	—	12	1	no	—
174	C₅×Q₈	—	12	1	no	—
175	C₅×D₄	—	3	2	no	—
176	C₅×D₄	—	6	1	no	—
177	C₅×D₄	—	12	1	no	—
178	C₅×D₄	—	12	1	no	—
179	C₅×D₄	—	12	1	no	—
180	C₅×D₄	—	12	1	no	—
181	C₅×D₄	—	24	1	no	—
182	C₂×C₄×C₅	—	6	1	no	—
183	C₂×C₄×C₅	—	12	1	no	—
184	C₂×C₄×C₅	—	12	1	no	—
185	C₂×C₄×C₅	—	12	1	no	—
186	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
187	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
188	C₂×C₄×C₅	#239	1	1	yes	characteristic
189	C₅×C₈	—	2	1	no	—
190	C₅×C₈	—	2	1	no	—
191	C₅×C₈	—	4	1	no	—
192	C₅×C₈	—	4	1	no	—
193	C₅×C₈	—	12	1	no	—
194	C₂×SD₁₆	—	6	1	no	—
195	C₂×D₈	—	6	1	no	—
196	C₂²×C₈	—	6	1	no	—
197	C₄⋊Q₈	—	3	1	no	—
198	#127	—	3	1	no	—
199	C₄×D₄	—	3	1	no	—
200	C₄⋊C₈	—	3	1	no	—
201	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
202	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
203	#103	—	6	1	no	—
204	#102	—	6	1	no	—
205	#102	—	6	1	no	—
206	#102	—	6	1	no	—
207	C₂²⋊C₈	—	6	1	no	—
208	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
209	C₂×C₃×C₅	#463	1	1	yes	characteristic
210	C₂×C₃×C₅	#466	1	1	yes	characteristic
211	C₂×C₃×C₅	#471	1	1	yes	characteristic
212	C₅×D₃	—	4	2	no	—
213	C₅×D₃	—	16	1	no	—
214	C₂²×D₃	—	2	1	no	—
215	C₂²×D₃	—	4	1	no	—
216	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
217	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
218	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
219	C₂×Dic₃	—	2	1	no	—
220	C₂×Dic₃	—	4	1	no	—
221	D₁₂	—	2	1	no	—
222	D₁₂	—	4	1	no	—
223	D₁₂	—	4	1	no	—
224	D₁₂	—	4	1	no	—
225	D₁₂	—	4	1	no	—
226	D₁₂	—	8	1	no	—
227	D₁₂	C₅×D₈	1	2	yes	—
228	C₄×D₃	—	4	1	no	—
229	C₄×D₃	—	4	1	no	—
230	Dic₆	—	4	1	no	—
231	Dic₆	—	4	1	no	—
232	C₃×C₈	—	2	1	no	—
233	C₃×C₈	—	2	1	no	—
234	C₃×C₈	—	4	1	no	—
235	C₃×C₈	—	4	1	no	—
236	C₃⋊C₈	—	4	1	no	—
237	C₂²×C₅	—	6	2	no	—
238	C₂²×C₅	—	12	1	no	—
239	C₂²×C₅	—	12	1	no	—
240	C₂²×C₅	—	24	1	no	—
241	C₂²×C₅	—	24	1	no	—
242	C₂²×C₅	—	24	1	no	—
243	C₂²×C₅	#895	1	1	yes	characteristic
244	C₄×C₅	—	2	1	no	—
245	C₄×C₅	—	12	1	no	—
246	C₄×C₅	—	24	1	no	—
247	C₄×C₅	#904	1	1	yes	characteristic
248	C₄×C₅	#908	1	1	yes	characteristic
249	C₄×C₅	D₃×D₈	1	1	yes	characteristic
250	C₄×C₅	#919	1	1	yes	characteristic
251	C₂×Q₈	—	6	1	no	—
252	C₂×D₄	—	3	1	no	—
253	C₂×D₄	—	6	1	no	—
254	C₂×D₄	—	6	1	no	—
255	C₂×D₄	—	6	1	no	—
256	C₂²×C₄	—	6	1	no	—
257	SD₁₆	—	12	2	no	—
258	D₈	—	12	2	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	6	1	no	—
262	C₂×C₈	—	12	2	no	—
263	C₂×C₈	C₅×D₁₂	1	1	yes	characteristic
264	C₂×C₈	C₅×D₁₂	1	1	yes	characteristic
265	C₄⋊C₄	—	3	1	no	—
266	C₄⋊C₄	—	6	1	no	—
267	C₄⋊C₄	—	6	1	no	—
268	C₂²⋊C₄	—	6	1	no	—
269	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
270	C₃×C₅	#1681	1	1	yes	characteristic, complemented, Hall
271	C₂²×C₃	C₅×#121	1	1	yes	characteristic
272	C₂×D₃	—	2	2	no	—
273	C₂×D₃	—	4	1	no	—
274	C₂×D₃	—	4	1	no	—
275	C₂×D₃	—	8	1	no	—
276	C₂×D₃	—	8	1	no	—
277	C₂×D₃	—	8	1	no	—
278	C₃×C₄	—	2	1	no	—
279	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
280	C₃×C₄	C₅×#135	1	1	yes	characteristic
281	C₃×C₄	C₅×#135	1	1	yes	characteristic
282	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
283	Dic₃	—	4	1	no	—
284	Dic₃	—	8	1	no	—
285	C₂×C₅	—	12	2	no	—
286	C₂×C₅	—	48	1	no	—
287	C₂×C₅	#4844	1	1	yes	characteristic
288	C₂×C₅	#4883	1	1	yes	characteristic
289	C₂×C₅	#4898	1	1	yes	characteristic
290	C₂³	—	6	1	no	—
291	C₂³	—	12	1	no	—
292	Q₈	—	12	1	no	—
293	Q₈	—	12	1	no	—
294	D₄	—	3	2	no	—
295	D₄	—	6	1	no	—
296	D₄	—	12	1	no	—
297	D₄	—	12	1	no	—
298	D₄	—	12	1	no	—
299	D₄	—	12	1	no	—
300	D₄	—	24	1	no	—
301	C₂×C₄	—	6	1	no	—
302	C₂×C₄	—	12	1	no	—
303	C₂×C₄	—	12	1	no	—
304	C₂×C₄	—	12	1	no	—
305	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
306	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
307	C₂×C₄	C₅×#239	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	4	1	no	—
311	C₈	—	4	1	no	—
312	C₈	—	12	1	no	—
313	C₂×C₃	C₅×#463	1	1	yes	characteristic
314	C₂×C₃	C₅×#466	1	1	yes	characteristic
315	C₂×C₃	C₅×#471	1	1	yes	characteristic
316	D₃	—	4	2	no	—
317	D₃	—	16	1	no	—
318	C₅	#68580	1	1	yes	characteristic, direct product factor, Sylow
319	C₂²	—	6	2	no	—
320	C₂²	—	12	1	no	—
321	C₂²	—	12	1	no	—
322	C₂²	—	24	1	no	—
323	C₂²	—	24	1	no	—
324	C₂²	—	24	1	no	—
325	C₂²	C₅×#895	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	12	1	no	—
328	C₄	—	24	1	no	—
329	C₄	C₅×#904	1	1	yes	characteristic
330	C₄	C₅×#908	1	1	yes	characteristic
331	C₄	C₅×D₃×D₈	1	1	yes	characteristic
332	C₄	C₅×#919	1	1	yes	characteristic
333	C₃	C₅×#1681	1	1	yes	characteristic, complemented, Sylow
334	C₂	—	12	2	no	—
335	C₂	—	48	1	no	—
336	C₂	C₅×#4844	1	1	yes	characteristic
337	C₂	C₅×#4883	1	1	yes	characteristic
338	C₂	C₅×#4898	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	#895
commutator, cosocle	C₃×C₄²	C₂³×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#335	C₂
socle	C₂²×C₃×C₅	#121
prime radical	C₅×#4843	C₂
hypercenter	C₅×#335	D₃
hypocenter	C₃	C₅×#1681
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#335

Complements

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

N	H ≅ G/N	type
2. C₅×#4841	C₂	semidirect
3. C₅×#4694	C₂	semidirect
4. C₅×C₈×D₁₂	C₂	semidirect
5. C₃×C₅×#335	C₂	semidirect
6. C₅×#4485	C₂	semidirect
7. C₅×#4482	C₂	semidirect
24. #68580	C₅	direct (N × H)
29. C₅×#335	D₃	semidirect
52. #4841	C₂×C₅	semidirect
53. #4694	C₂×C₅	semidirect
54. C₈×D₁₂	C₂×C₅	semidirect
55. C₃×#335	C₂×C₅	semidirect
56. #4485	C₂×C₅	semidirect
57. #4482	C₂×C₅	semidirect
135. #335	C₅×D₃	semidirect
270. C₃×C₅	#1681	semidirect
318. C₅	#68580	direct (N × H)
333. C₃	C₅×#1681	semidirect

Maximal subgroups

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

subgroup	index
C₅×#4843	2
C₅×#4841	2
C₅×#4694	2
C₅×C₈×D₁₂	2
C₃×C₅×#335	2
C₅×#4485	2
C₅×#4482	2
C₅×#1681	3
#68580	5

Subgroups — C5×#68580 #14292105

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68580 #14292105