Subgroups — C₅×#8193 #11930913

Order 1280.

482 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₈×#41	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅×#104	C₂	1	1	yes	maximal, characteristic
3	C₅×#1744	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1476	C₂	1	1	yes	maximal, characteristic
5	C₅×#1451	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1278	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1276	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₅×#41	C₂²	1	1	yes	characteristic
9	C₅×C₈×Q₈	C₄	1	2	yes	complemented
10	C₅×C₈×D₄	—	2	2	no	—
11	C₅×C₈×D₄	C₄	1	2	yes	complemented
12	C₅×#431	—	2	1	no	—
13	C₅×#431	C₂²	1	1	yes	characteristic
14	C₅×#431	C₂²	1	1	yes	characteristic
15	C₂×C₅×#104	C₄	1	2	yes	—
16	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₅×C₈	—	2	1	no	—
18	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	C₄³×C₅	C₂²	1	1	yes	characteristic
21	C₅×#339	C₄	1	2	yes	—
22	C₅×#336	—	2	2	no	—
23	C₅×Q₈⋊C₈	C₄	1	2	yes	complemented
24	C₅×D₄⋊C₈	C₄	1	2	yes	complemented
25	C₅×#322	—	2	1	no	—
26	C₅×#322	—	2	1	no	—
27	#8193	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
29	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
30	C₂²×C₅×C₈	—	2	1	no	—
31	C₂²×C₅×C₈	—	2	2	no	—
32	C₂²×C₅×C₈	D₄	1	2	yes	—
33	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
34	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
35	C₄×C₅×D₄	—	2	1	no	—
36	C₄×C₅×D₄	—	2	1	no	—
37	C₄×C₅×D₄	C₂×C₄	1	1	yes	characteristic
38	C₄×C₅×D₄	C₂×C₄	1	1	yes	characteristic
39	C₅×#117	—	2	1	no	—
40	C₅×#117	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅	—	2	1	no	—
42	C₂×C₄²×C₅	—	2	1	no	—
43	C₂×C₄²×C₅	—	2	1	no	—
44	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
46	C₅×C₄⋊C₈	—	2	2	no	—
47	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
48	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
49	C₅×#104	—	2	4	no	—
50	C₅×C₂²⋊C₈	—	2	1	no	—
51	C₅×C₂²⋊C₈	—	2	1	no	—
52	C₅×C₂²⋊C₈	—	2	2	no	—
53	C₅×#97	—	2	1	no	—
54	C₄×C₅×C₈	—	2	2	no	—
55	C₄×C₅×C₈	—	2	1	no	—
56	C₄×C₅×C₈	—	2	1	no	—
57	C₄×C₅×C₈	—	2	1	no	—
58	C₄×C₅×C₈	—	2	1	no	—
59	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
60	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
61	C₄×C₅×C₈	D₄	1	1	yes	characteristic
62	C₄×C₅×C₈	D₄	1	1	yes	characteristic
63	C₈×#41	C₂×C₅	1	1	yes	characteristic
64	C₄×#104	C₂×C₅	1	1	yes	characteristic
65	#1744	C₂×C₅	1	1	yes	characteristic, complemented
66	#1476	C₂×C₅	1	1	yes	characteristic
67	#1451	C₂×C₅	1	1	yes	characteristic, complemented
68	#1278	C₂×C₅	1	1	yes	characteristic, complemented
69	#1276	C₂×C₅	1	1	yes	characteristic, complemented
70	C₅×#41	—	2	2	no	—
71	C₅×#41	M₁₆	1	4	yes	—
72	C₂×C₅×Q₈	C₄²	1	1	yes	characteristic
73	C₂×C₅×D₄	—	2	1	no	—
74	C₂×C₅×D₄	C₄²	1	1	yes	characteristic
75	C₂²×C₄×C₅	—	2	1	no	—
76	C₂²×C₄×C₅	—	2	2	no	—
77	C₂²×C₄×C₅	—	2	1	no	—
78	C₂²×C₄×C₅	—	2	1	no	—
79	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₅	#41	1	1	yes	characteristic
82	C₅×M₁₆	—	2	2	no	—
83	C₅×M₁₆	—	2	2	no	—
84	C₂×C₅×C₈	—	2	2	no	—
85	C₂×C₅×C₈	—	2	2	no	—
86	C₂×C₅×C₈	—	2	2	no	—
87	C₂×C₅×C₈	—	2	2	no	—
88	C₂×C₅×C₈	—	4	2	no	—
89	C₂×C₅×C₈	—	4	4	no	—
90	C₂×C₅×C₈	—	4	2	no	—
91	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
92	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
93	C₅×C₄⋊C₄	—	2	1	no	—
94	C₅×C₄⋊C₄	—	2	1	no	—
95	C₅×C₄⋊C₄	C₄²	1	1	yes	characteristic
96	C₅×C₄⋊C₄	C₄²	1	1	yes	characteristic
97	C₅×C₂²⋊C₄	—	2	1	no	—
98	C₅×C₂²⋊C₄	—	2	1	no	—
99	C₅×C₂²⋊C₄	—	2	1	no	—
100	C₄²×C₅	—	2	4	no	—
101	C₄²×C₅	—	2	1	no	—
102	C₄²×C₅	—	2	1	no	—
103	C₄²×C₅	—	2	1	no	—
104	C₄²×C₅	—	2	2	no	—
105	C₄²×C₅	—	2	2	no	—
106	C₄²×C₅	—	4	2	no	—
107	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₅	#41	1	1	yes	characteristic
111	C₄×#41	C₂²×C₅	1	1	yes	characteristic
112	C₈×Q₈	C₄×C₅	1	2	yes	complemented
113	C₈×D₄	—	2	2	no	—
114	C₈×D₄	C₄×C₅	1	2	yes	complemented
115	#431	—	2	1	no	—
116	#431	C₂²×C₅	1	1	yes	characteristic
117	#431	C₂²×C₅	1	1	yes	characteristic
118	C₂×#104	C₄×C₅	1	2	yes	—
119	C₄×M₁₆	C₂²×C₅	1	1	yes	characteristic
120	C₂×C₄×C₈	—	2	1	no	—
121	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
122	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
123	C₄³	C₂²×C₅	1	1	yes	characteristic
124	#339	C₄×C₅	1	2	yes	—
125	#336	—	2	2	no	—
126	Q₈⋊C₈	C₄×C₅	1	2	yes	complemented
127	D₄⋊C₈	C₄×C₅	1	2	yes	complemented
128	#322	—	2	1	no	—
129	#322	—	2	1	no	—
130	C₂³×C₅	—	2	1	no	—
131	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
132	C₅×Q₈	—	2	1	no	—
133	C₅×Q₈	#97	1	2	yes	—
134	C₅×D₄	—	2	4	no	—
135	C₅×D₄	—	2	1	no	—
136	C₅×D₄	#97	1	2	yes	—
137	C₂×C₄×C₅	—	2	1	no	—
138	C₂×C₄×C₅	—	2	2	no	—
139	C₂×C₄×C₅	—	2	4	no	—
140	C₂×C₄×C₅	—	2	2	no	—
141	C₂×C₄×C₅	—	2	4	no	—
142	C₂×C₄×C₅	—	2	1	no	—
143	C₂×C₄×C₅	—	2	1	no	—
144	C₂×C₄×C₅	—	4	2	no	—
145	C₂×C₄×C₅	—	4	2	no	—
146	C₂×C₄×C₅	—	4	2	no	—
147	C₂×C₄×C₅	—	4	2	no	—
148	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₅	#117	1	1	yes	characteristic
151	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
152	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
153	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
155	C₂×C₄×C₅	#131	1	2	yes	—
156	C₅×C₈	—	2	2	no	—
157	C₅×C₈	—	2	2	no	—
158	C₅×C₈	—	4	2	no	—
159	C₅×C₈	—	4	2	no	—
160	C₅×C₈	—	4	2	no	—
161	C₅×C₈	—	4	2	no	—
162	C₂×#41	C₂×C₄×C₅	1	1	yes	characteristic
163	C₂×M₁₆	C₂×C₄×C₅	1	2	yes	—
164	C₂²×C₈	—	2	1	no	—
165	C₂²×C₈	—	2	2	no	—
166	C₂²×C₈	C₅×D₄	1	2	yes	—
167	C₄×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
168	C₄×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
169	C₄×D₄	—	2	1	no	—
170	C₄×D₄	—	2	1	no	—
171	C₄×D₄	C₂×C₄×C₅	1	1	yes	characteristic
172	C₄×D₄	C₂×C₄×C₅	1	1	yes	characteristic
173	#117	—	2	1	no	—
174	#117	C₂×C₄×C₅	1	1	yes	characteristic
175	C₂×C₄²	—	2	1	no	—
176	C₂×C₄²	—	2	1	no	—
177	C₂×C₄²	—	2	1	no	—
178	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
179	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
180	C₄⋊C₈	—	2	2	no	—
181	C₄⋊C₈	C₂×C₄×C₅	1	2	yes	—
182	C₄⋊C₈	C₂×C₄×C₅	1	2	yes	—
183	#104	—	2	4	no	—
184	C₂²⋊C₈	—	2	1	no	—
185	C₂²⋊C₈	—	2	1	no	—
186	C₂²⋊C₈	—	2	2	no	—
187	#97	—	2	1	no	—
188	C₄×C₈	—	2	2	no	—
189	C₄×C₈	—	2	1	no	—
190	C₄×C₈	—	2	1	no	—
191	C₄×C₈	—	2	1	no	—
192	C₄×C₈	—	2	1	no	—
193	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
194	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
195	C₄×C₈	C₅×D₄	1	1	yes	characteristic
196	C₄×C₈	C₅×D₄	1	1	yes	characteristic
197	C₂²×C₅	—	2	1	no	—
198	C₂²×C₅	—	2	1	no	—
199	C₂²×C₅	—	2	2	no	—
200	C₂²×C₅	—	4	1	no	—
201	C₂²×C₅	—	4	1	no	—
202	C₂²×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
203	C₂²×C₅	#435	1	2	yes	—
204	C₄×C₅	—	2	2	no	—
205	C₄×C₅	—	2	1	no	—
206	C₄×C₅	—	2	1	no	—
207	C₄×C₅	—	4	4	no	—
208	C₄×C₅	—	4	1	no	—
209	C₄×C₅	—	4	1	no	—
210	C₄×C₅	C₂×#97	1	1	yes	characteristic
211	C₄×C₅	#405	1	1	yes	characteristic
212	C₄×C₅	#421	1	2	yes	—
213	C₄×C₅	#435	1	2	yes	—
214	C₄×C₅	#443	1	2	yes	—
215	#41	—	2	2	no	—
216	#41	C₅×M₁₆	1	4	yes	—
217	C₂×Q₈	C₄²×C₅	1	1	yes	characteristic
218	C₂×D₄	—	2	1	no	—
219	C₂×D₄	C₄²×C₅	1	1	yes	characteristic
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
225	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
226	C₂²×C₄	C₅×#41	1	1	yes	characteristic
227	M₁₆	—	2	2	no	—
228	M₁₆	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	—	4	4	no	—
235	C₂×C₈	—	4	2	no	—
236	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
237	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
238	C₄⋊C₄	—	2	1	no	—
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	C₄²×C₅	1	1	yes	characteristic
241	C₄⋊C₄	C₄²×C₅	1	1	yes	characteristic
242	C₂²⋊C₄	—	2	1	no	—
243	C₂²⋊C₄	—	2	1	no	—
244	C₂²⋊C₄	—	2	1	no	—
245	C₄²	—	2	4	no	—
246	C₄²	—	2	1	no	—
247	C₄²	—	2	1	no	—
248	C₄²	—	2	1	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	4	2	no	—
252	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
253	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
254	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
255	C₄²	C₅×#41	1	1	yes	characteristic
256	C₂×C₅	—	2	2	no	—
257	C₂×C₅	—	4	2	no	—
258	C₂×C₅	#1752	1	1	yes	characteristic
259	C₂×C₅	#1759	1	1	yes	characteristic
260	C₂×C₅	#1764	1	1	yes	characteristic
261	C₂³	—	2	1	no	—
262	C₂³	C₄×C₅×D₄	1	1	yes	characteristic
263	Q₈	—	2	1	no	—
264	Q₈	C₅×#97	1	2	yes	—
265	D₄	—	2	4	no	—
266	D₄	—	2	1	no	—
267	D₄	C₅×#97	1	2	yes	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
280	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
281	C₂×C₄	C₅×#117	1	1	yes	characteristic
282	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
283	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
284	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
285	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
286	C₂×C₄	C₅×#131	1	2	yes	—
287	C₈	—	2	2	no	—
288	C₈	—	2	2	no	—
289	C₈	—	4	2	no	—
290	C₈	—	4	2	no	—
291	C₈	—	4	2	no	—
292	C₈	—	4	2	no	—
293	C₅	#8193	1	1	yes	characteristic, direct product factor, Sylow
294	C₂²	—	2	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	4	1	no	—
298	C₂²	—	4	1	no	—
299	C₂²	C₄×C₅×C₂²⋊C₄	1	1	yes	characteristic
300	C₂²	C₅×#435	1	2	yes	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	2	no	—
304	C₄	—	4	4	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	1	no	—
307	C₄	C₂×C₅×#97	1	1	yes	characteristic
308	C₄	C₅×#405	1	1	yes	characteristic
309	C₄	C₅×#421	1	2	yes	—
310	C₄	C₅×#435	1	2	yes	—
311	C₄	C₅×#443	1	2	yes	—
312	C₂	—	2	2	no	—
313	C₂	—	4	2	no	—
314	C₂	C₅×#1752	1	1	yes	characteristic
315	C₂	C₅×#1759	1	1	yes	characteristic
316	C₂	C₅×#1764	1	1	yes	characteristic

	subgroup	G/H
center	C₄²×C₅	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂²×C₅	C₄×C₂²⋊C₄
prime radical	C₂×C₅×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#1744	C₂	semidirect
5. C₅×#1451	C₂	semidirect
6. C₅×#1278	C₂	semidirect
7. C₅×#1276	C₂	semidirect
9. C₅×C₈×Q₈	C₄	semidirect
11. C₅×C₈×D₄	C₄	semidirect
23. C₅×Q₈⋊C₈	C₄	semidirect
24. C₅×D₄⋊C₈	C₄	semidirect
27. #8193	C₅	direct (N × H)
65. #1744	C₂×C₅	semidirect
67. #1451	C₂×C₅	semidirect
68. #1278	C₂×C₅	semidirect
69. #1276	C₂×C₅	semidirect
112. C₈×Q₈	C₄×C₅	semidirect
114. C₈×D₄	C₄×C₅	semidirect
126. Q₈⋊C₈	C₄×C₅	semidirect
127. D₄⋊C₈	C₄×C₅	semidirect
293. C₅	#8193	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×C₈×#41	2
C₄×C₅×#104	2
C₅×#1744	2
C₅×#1476	2
C₅×#1451	2
C₅×#1278	2
C₅×#1276	2
#8193	5

Subgroups — C5×#8193 #11930913

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8193 #11930913