Subgroups — C₅×#78463 #14437785

Order 1920.

800 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5477	C₂	1	1	yes	maximal, characteristic
2	C₅×#5470	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5113	C₂	1	4	yes	maximal, complemented
5	C₅×#4697	C₂	1	4	yes	maximal, complemented
6	C₅×#4694	C₂	1	2	yes	maximal, complemented
7	C₅×#4681	C₂	1	2	yes	maximal, complemented
8	C₅×#3131	—	3	1	no	maximal, Hall
9	C₂×C₅×#235	C₂²	1	2	yes	—
10	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₅×#930	—	2	2	no	—
13	C₅×#926	C₂²	1	2	yes	—
14	C₅×#924	—	2	2	no	—
15	C₅×#875	C₂²	1	2	yes	—
16	C₅×#874	C₂²	1	1	yes	characteristic
17	C₄×C₅×D₁₂	—	2	2	no	—
18	C₅×#870	C₂²	1	2	yes	—
19	C₅×#869	C₂²	1	1	yes	characteristic
20	C₅×#869	C₂²	1	2	yes	—
21	C₄×C₅×Dic₆	—	2	2	no	—
22	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
23	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
24	C₅×#821	C₂²	1	8	yes	—
25	C₅×#816	C₂²	1	8	yes	—
26	#78463	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
27	C₅×#533	—	3	1	no	—
28	C₅×#532	—	3	1	no	—
29	C₅×#488	—	3	4	no	—
30	C₅×#487	—	3	2	no	—
31	C₅×#486	—	3	2	no	—
32	C₅×#416	—	3	4	no	—
33	C₂×C₄×C₅×C₈	D₃	1	1	yes	characteristic, complemented
34	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
35	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
36	C₂×C₅×#67	—	2	2	no	—
37	C₅×#235	—	2	8	no	—
38	C₂×C₅×D₁₂	—	2	2	no	—
39	C₂×C₅×D₁₂	C₂³	1	2	yes	—
40	C₂×C₄×C₅×D₃	—	2	2	no	—
41	C₂×C₅×Dic₆	—	2	2	no	—
42	C₂×C₅×Dic₆	C₂³	1	2	yes	—
43	C₂×C₃×C₅×C₈	—	2	4	no	—
44	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
45	C₃×C₄²×C₅	D₄	1	2	yes	—
46	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
47	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
48	C₅×#217	—	2	2	no	—
49	C₅×(C₂×D₃)⋊C₄	—	2	4	no	—
50	C₅×C₄⋊Dic₃	—	2	2	no	—
51	C₅×C₄⋊Dic₃	—	2	2	no	—
52	C₅×C₄⋊Dic₃	C₂³	1	4	yes	—
53	C₅×Dic₃⋊C₄	—	2	4	no	—
54	C₄×C₅×Dic₃	—	2	2	no	—
55	#5477	C₂×C₅	1	1	yes	characteristic
56	#5470	C₂×C₅	1	1	yes	characteristic, complemented
57	C₂×C₃×C₄×C₈	C₂×C₅	1	1	yes	characteristic, complemented
58	#5113	C₂×C₅	1	4	yes	complemented
59	#4697	C₂×C₅	1	4	yes	complemented
60	#4694	C₂×C₅	1	2	yes	complemented
61	#4681	C₂×C₅	1	2	yes	complemented
62	C₂×C₅×#41	—	3	2	no	—
63	C₂²×C₅×C₈	C₂×D₃	1	2	yes	—
64	C₅×C₄⋊Q₈	—	3	2	no	—
65	C₅×C₄⋊Q₈	—	3	1	no	—
66	C₅×#127	—	3	1	no	—
67	C₅×#125	—	3	2	no	—
68	C₅×#124	—	3	2	no	—
69	C₅×C₂²⋊Q₈	—	6	2	no	—
70	C₅×#121	—	6	2	no	—
71	C₄×C₅×Q₈	—	6	2	no	—
72	C₄×C₅×D₄	—	6	2	no	—
73	C₅×#117	—	3	2	no	—
74	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
75	C₅×#103	—	3	8	no	—
76	C₅×#102	—	3	8	no	—
77	C₄×C₅×C₈	C₂×D₃	1	2	yes	—
78	C₄×C₅×C₈	C₂×D₃	1	2	yes	—
79	#3131	—	3	1	no	Sylow
80	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₅×D₃	—	2	2	no	—
82	C₂×C₃×C₄×C₅	—	2	4	no	—
83	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
84	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
86	C₂×C₃×C₄×C₅	#41	1	4	yes	—
87	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
88	C₅×#67	—	4	4	no	—
89	C₂×C₅×Dic₃	—	2	4	no	—
90	C₂×C₅×Dic₃	—	2	2	no	—
91	C₅×D₁₂	—	2	4	no	—
92	C₅×D₁₂	—	4	4	no	—
93	C₄×C₅×D₃	—	4	4	no	—
94	C₅×Dic₆	—	2	4	no	—
95	C₅×Dic₆	—	4	4	no	—
96	C₃×C₅×C₈	—	2	4	no	—
97	C₂×#235	C₂²×C₅	1	2	yes	—
98	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
99	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
100	#930	—	2	2	no	—
101	#926	C₂²×C₅	1	2	yes	—
102	#924	—	2	2	no	—
103	#875	C₂²×C₅	1	2	yes	—
104	#874	C₂²×C₅	1	1	yes	characteristic
105	C₄×D₁₂	—	2	2	no	—
106	#870	C₂²×C₅	1	2	yes	—
107	#869	C₂²×C₅	1	1	yes	characteristic
108	#869	C₂²×C₅	1	2	yes	—
109	C₄×Dic₆	—	2	2	no	—
110	C₃×C₄×C₈	C₂²×C₅	1	2	yes	—
111	C₃×C₄×C₈	C₂²×C₅	1	2	yes	—
112	#821	C₂²×C₅	1	8	yes	—
113	#816	C₂²×C₅	1	8	yes	—
114	C₅×#41	—	6	8	no	—
115	C₂×C₅×Q₈	—	3	2	no	—
116	C₂×C₅×Q₈	—	6	2	no	—
117	C₂×C₅×D₄	—	3	2	no	—
118	C₂×C₅×D₄	—	6	2	no	—
119	C₂×C₅×D₄	—	6	2	no	—
120	C₂²×C₄×C₅	—	6	2	no	—
121	C₂²×C₄×C₅	D₁₂	1	2	yes	—
122	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
123	C₂×C₅×C₈	—	2	4	no	—
124	C₂×C₅×C₈	C₂²×D₃	1	4	yes	—
125	C₅×C₄⋊C₄	—	3	4	no	—
126	C₅×C₄⋊C₄	—	6	2	no	—
127	C₅×C₄⋊C₄	—	6	4	no	—
128	C₅×C₄⋊C₄	—	6	2	no	—
129	C₅×C₂²⋊C₄	—	6	2	no	—
130	C₅×C₂²⋊C₄	—	6	4	no	—
131	C₄²×C₅	—	6	2	no	—
132	C₄²×C₅	D₁₂	1	2	yes	—
133	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
134	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
135	#533	—	3	1	no	—
136	#532	—	3	1	no	—
137	#488	—	3	4	no	—
138	#487	—	3	2	no	—
139	#486	—	3	2	no	—
140	#416	—	3	4	no	—
141	C₂×C₄×C₈	C₅×D₃	1	1	yes	characteristic, complemented
142	C₂²×C₃×C₅	—	2	2	no	—
143	C₂²×C₃×C₅	#124	1	2	yes	—
144	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₅×D₃	—	4	2	no	—
146	C₂×C₅×D₃	—	4	4	no	—
147	C₃×C₄×C₅	—	2	2	no	—
148	C₃×C₄×C₅	#124	1	2	yes	—
149	C₃×C₄×C₅	#135	1	4	yes	—
150	C₃×C₄×C₅	C₂×#41	1	2	yes	—
151	C₅×Dic₃	—	4	2	no	—
152	C₅×Dic₃	—	4	4	no	—
153	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
154	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
155	C₂×#67	—	2	2	no	—
156	#235	—	2	8	no	—
157	C₂×D₁₂	—	2	2	no	—
158	C₂×D₁₂	C₂³×C₅	1	2	yes	—
159	C₂×C₄×D₃	—	2	2	no	—
160	C₂×Dic₆	—	2	2	no	—
161	C₂×Dic₆	C₂³×C₅	1	2	yes	—
162	C₂×C₃×C₈	—	2	4	no	—
163	C₂×C₃×C₈	C₂³×C₅	1	4	yes	—
164	C₃×C₄²	C₅×D₄	1	2	yes	—
165	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
166	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
167	#217	—	2	2	no	—
168	(C₂×D₃)⋊C₄	—	2	4	no	—
169	C₄⋊Dic₃	—	2	2	no	—
170	C₄⋊Dic₃	—	2	2	no	—
171	C₄⋊Dic₃	C₂³×C₅	1	4	yes	—
172	Dic₃⋊C₄	—	2	4	no	—
173	C₄×Dic₃	—	2	2	no	—
174	C₂³×C₅	—	6	2	no	—
175	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
176	C₅×Q₈	—	6	4	no	—
177	C₅×Q₈	—	12	4	no	—
178	C₅×D₄	—	6	4	no	—
179	C₅×D₄	—	12	4	no	—
180	C₅×D₄	—	12	4	no	—
181	C₂×C₄×C₅	—	2	4	no	—
182	C₂×C₄×C₅	—	6	4	no	—
183	C₂×C₄×C₅	—	6	2	no	—
184	C₂×C₄×C₅	—	12	4	no	—
185	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
186	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
187	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
188	C₂×C₄×C₅	#235	1	4	yes	—
189	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
190	C₅×C₈	—	2	4	no	—
191	C₂×#41	—	3	2	no	—
192	C₂²×C₈	C₂×C₅×D₃	1	2	yes	—
193	C₄⋊Q₈	—	3	2	no	—
194	C₄⋊Q₈	—	3	1	no	—
195	#127	—	3	1	no	—
196	#125	—	3	2	no	—
197	#124	—	3	2	no	—
198	C₂²⋊Q₈	—	6	2	no	—
199	#121	—	6	2	no	—
200	C₄×Q₈	—	6	2	no	—
201	C₄×D₄	—	6	2	no	—
202	#117	—	3	2	no	—
203	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
204	#103	—	3	8	no	—
205	#102	—	3	8	no	—
206	C₄×C₈	C₂×C₅×D₃	1	2	yes	—
207	C₄×C₈	C₂×C₅×D₃	1	2	yes	—
208	C₂×C₃×C₅	—	2	2	no	—
209	C₂×C₃×C₅	C₂×#124	1	1	yes	characteristic
210	C₂×C₃×C₅	C₂×#135	1	2	yes	—
211	C₅×D₃	—	8	2	no	—
212	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
213	C₂²×D₃	—	2	2	no	—
214	C₂×C₃×C₄	—	2	4	no	—
215	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
216	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
217	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
218	C₂×C₃×C₄	C₅×#41	1	4	yes	—
219	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
220	#67	—	4	4	no	—
221	C₂×Dic₃	—	2	4	no	—
222	C₂×Dic₃	—	2	2	no	—
223	D₁₂	—	2	4	no	—
224	D₁₂	—	4	4	no	—
225	C₄×D₃	—	4	4	no	—
226	Dic₆	—	2	4	no	—
227	Dic₆	—	4	4	no	—
228	C₃×C₈	—	2	4	no	—
229	C₂²×C₅	—	2	2	no	—
230	C₂²×C₅	—	12	2	no	—
231	C₂²×C₅	—	12	4	no	—
232	C₂²×C₅	#875	1	2	yes	—
233	C₂²×C₅	C₂²×D₁₂	1	1	yes	characteristic
234	C₄×C₅	—	2	2	no	—
235	C₄×C₅	—	12	2	no	—
236	C₄×C₅	—	12	4	no	—
237	C₄×C₅	#875	1	2	yes	—
238	C₄×C₅	#904	1	4	yes	—
239	C₄×C₅	C₂×#235	1	2	yes	—
240	#41	—	6	8	no	—
241	C₂×Q₈	—	3	2	no	—
242	C₂×Q₈	—	6	2	no	—
243	C₂×D₄	—	3	2	no	—
244	C₂×D₄	—	6	2	no	—
245	C₂×D₄	—	6	2	no	—
246	C₂²×C₄	—	6	2	no	—
247	C₂²×C₄	C₅×D₁₂	1	2	yes	—
248	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	C₂²×C₅×D₃	1	4	yes	—
251	C₄⋊C₄	—	3	4	no	—
252	C₄⋊C₄	—	6	2	no	—
253	C₄⋊C₄	—	6	4	no	—
254	C₄⋊C₄	—	6	2	no	—
255	C₂²⋊C₄	—	6	2	no	—
256	C₂²⋊C₄	—	6	4	no	—
257	C₄²	—	6	2	no	—
258	C₄²	C₅×D₁₂	1	2	yes	—
259	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
260	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
261	C₃×C₅	#3131	1	1	yes	characteristic, complemented, Hall
262	C₂²×C₃	—	2	2	no	—
263	C₂²×C₃	C₅×#124	1	2	yes	—
264	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
265	C₂×D₃	—	4	2	no	—
266	C₂×D₃	—	4	4	no	—
267	C₃×C₄	—	2	2	no	—
268	C₃×C₄	C₅×#124	1	2	yes	—
269	C₃×C₄	C₅×#135	1	4	yes	—
270	C₃×C₄	C₂×C₅×#41	1	2	yes	—
271	Dic₃	—	4	2	no	—
272	Dic₃	—	4	4	no	—
273	C₂×C₅	—	2	2	no	—
274	C₂×C₅	—	24	2	no	—
275	C₂×C₅	C₂×#875	1	1	yes	characteristic
276	C₂×C₅	C₂×#904	1	2	yes	—
277	C₂³	—	6	2	no	—
278	C₂³	C₂×C₅×D₁₂	1	1	yes	characteristic
279	Q₈	—	6	4	no	—
280	Q₈	—	12	4	no	—
281	D₄	—	6	4	no	—
282	D₄	—	12	4	no	—
283	D₄	—	12	4	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	6	4	no	—
286	C₂×C₄	—	6	2	no	—
287	C₂×C₄	—	12	4	no	—
288	C₂×C₄	C₂×C₅×D₁₂	1	2	yes	—
289	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
290	C₂×C₄	C₂×C₅×D₁₂	1	2	yes	—
291	C₂×C₄	C₅×#235	1	4	yes	—
292	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
293	C₈	—	2	4	no	—
294	C₂×C₃	—	2	2	no	—
295	C₂×C₃	C₂×C₅×#124	1	1	yes	characteristic
296	C₂×C₃	C₂×C₅×#135	1	2	yes	—
297	D₃	—	8	2	no	—
298	C₅	#78463	1	1	yes	characteristic, direct product factor, Sylow
299	C₂²	—	2	2	no	—
300	C₂²	—	12	2	no	—
301	C₂²	—	12	4	no	—
302	C₂²	C₅×#875	1	2	yes	—
303	C₂²	C₂²×C₅×D₁₂	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	12	2	no	—
306	C₄	—	12	4	no	—
307	C₄	C₅×#875	1	2	yes	—
308	C₄	C₅×#904	1	4	yes	—
309	C₄	C₂×C₅×#235	1	2	yes	—
310	C₃	C₅×#3131	1	1	yes	characteristic, complemented, Sylow
311	C₂	—	2	2	no	—
312	C₂	—	24	2	no	—
313	C₂	C₂×C₅×#875	1	1	yes	characteristic
314	C₂	C₂×C₅×#904	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₂×C₃×C₄×C₅×C₈	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₅×#5477	C₂
hypercenter	C₂×C₄×C₅×C₈	D₃
hypocenter	C₃	C₅×#3131
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈

Complements

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

N	H ≅ G/N	type
2. C₅×#5470	C₂	semidirect
3. C₂×C₃×C₄×C₅×C₈	C₂	semidirect
4. C₅×#5113	C₂	semidirect
5. C₅×#4697	C₂	semidirect
6. C₅×#4694	C₂	semidirect
7. C₅×#4681	C₂	semidirect
26. #78463	C₅	direct (N × H)
33. C₂×C₄×C₅×C₈	D₃	semidirect
56. #5470	C₂×C₅	semidirect
57. C₂×C₃×C₄×C₈	C₂×C₅	semidirect
58. #5113	C₂×C₅	semidirect
59. #4697	C₂×C₅	semidirect
60. #4694	C₂×C₅	semidirect
61. #4681	C₂×C₅	semidirect
141. C₂×C₄×C₈	C₅×D₃	semidirect
261. C₃×C₅	#3131	semidirect
298. C₅	#78463	direct (N × H)
310. C₃	C₅×#3131	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5477	2
C₅×#5470	2
C₂×C₃×C₄×C₅×C₈	2
C₅×#5113	2
C₅×#4697	2
C₅×#4694	2
C₅×#4681	2
C₅×#3131	3
#78463	5

Subgroups — C5×#78463 #14437785

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#78463 #14437785