Subgroups — C₅×#10843 #11933563

Order 1280.

998 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#97	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#384	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#382	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#340	C₂	1	4	yes	maximal
5	C₂³×C₅×C₈	C₂²	1	2	yes	—
6	C₂²×C₅×C₄⋊C₄	—	2	2	no	—
7	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	—
8	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₅×#97	—	2	2	no	—
11	C₂×C₅×#97	—	2	2	no	—
12	C₂×C₅×#97	C₄	1	4	yes	complemented
13	C₅×#384	—	2	2	no	—
14	C₅×#384	C₄	1	4	yes	complemented
15	C₅×#382	—	2	2	no	—
16	C₅×#382	C₄	1	4	yes	complemented
17	C₂×C₅×#95	—	2	2	no	—
18	C₅×#340	—	2	8	no	—
19	#10843	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂³×C₄×C₅	—	2	2	no	—
21	C₂³×C₄×C₅	—	2	2	no	—
22	C₂³×C₄×C₅	D₄	1	2	yes	—
23	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
24	C₂²×C₅×C₈	—	2	2	no	—
25	C₂²×C₅×C₈	—	2	2	no	—
26	C₂²×C₅×C₈	—	2	2	no	—
27	C₂²×C₅×C₈	—	2	2	no	—
28	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
29	C₂²×C₅×C₈	D₄	1	4	yes	—
30	C₂²×C₅×C₈	Q₈	1	4	yes	—
31	C₂×C₅×C₄⋊C₄	—	2	4	no	—
32	C₂×C₅×C₄⋊C₄	—	2	4	no	—
33	C₂×C₅×C₄⋊C₄	—	2	4	no	—
34	C₂×C₅×C₄⋊C₄	—	2	4	no	—
35	C₂×C₅×C₄⋊C₄	—	2	8	no	—
36	C₂×C₅×C₄⋊C₄	—	2	2	no	—
37	C₂×C₅×C₄⋊C₄	—	2	2	no	—
38	C₂×C₅×C₄⋊C₄	—	4	8	no	—
39	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
40	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₅	D₄	1	2	yes	—
45	C₂×C₄²×C₅	D₄	1	2	yes	—
46	C₂×C₄²×C₅	Q₈	1	2	yes	—
47	C₅×#97	—	2	8	no	—
48	C₅×#95	—	2	8	no	—
49	C₂²×#97	C₂×C₅	1	1	yes	characteristic
50	C₂×#384	C₂×C₅	1	1	yes	characteristic
51	C₂×#382	C₂×C₅	1	1	yes	characteristic
52	C₂×#340	C₂×C₅	1	4	yes	—
53	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅	—	2	4	no	—
55	C₂²×C₄×C₅	—	2	4	no	—
56	C₂²×C₄×C₅	—	2	4	no	—
57	C₂²×C₄×C₅	—	2	4	no	—
58	C₂²×C₄×C₅	—	4	2	no	—
59	C₂²×C₄×C₅	—	4	2	no	—
60	C₂²×C₄×C₅	—	4	2	no	—
61	C₂²×C₄×C₅	—	4	2	no	—
62	C₂²×C₄×C₅	—	4	8	no	—
63	C₂²×C₄×C₅	—	4	4	no	—
64	C₂²×C₄×C₅	—	4	4	no	—
65	C₂²×C₄×C₅	C₂²⋊C₄	1	4	yes	—
66	C₂²×C₄×C₅	C₂²⋊C₄	1	4	yes	—
67	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
68	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₅	#41	1	2	yes	—
77	C₂²×C₄×C₅	#41	1	2	yes	—
78	C₂×C₅×C₈	—	2	4	no	—
79	C₂×C₅×C₈	—	2	4	no	—
80	C₂×C₅×C₈	—	2	4	no	—
81	C₂×C₅×C₈	—	2	4	no	—
82	C₂×C₅×C₈	—	2	4	no	—
83	C₂×C₅×C₈	—	2	4	no	—
84	C₂×C₅×C₈	C₄⋊C₄	1	8	yes	complemented
85	C₅×C₄⋊C₄	—	2	8	no	—
86	C₅×C₄⋊C₄	—	2	8	no	—
87	C₅×C₄⋊C₄	—	4	16	no	—
88	C₅×C₄⋊C₄	—	4	4	no	—
89	C₅×C₄⋊C₄	—	4	4	no	—
90	C₄²×C₅	C₄²	1	4	yes	—
91	C₄²×C₅	C₂²⋊C₄	1	4	yes	—
92	C₄²×C₅	C₄⋊C₄	1	4	yes	—
93	C₄²×C₅	C₄⋊C₄	1	4	yes	—
94	C₂³×C₈	C₂²×C₅	1	2	yes	—
95	C₂²×C₄⋊C₄	—	2	2	no	—
96	C₂²×C₄⋊C₄	C₂²×C₅	1	2	yes	—
97	C₂²×C₄⋊C₄	C₂²×C₅	1	2	yes	—
98	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
99	C₂×#97	—	2	2	no	—
100	C₂×#97	—	2	2	no	—
101	C₂×#97	C₄×C₅	1	4	yes	complemented
102	#384	—	2	2	no	—
103	#384	C₄×C₅	1	4	yes	complemented
104	#382	—	2	2	no	—
105	#382	C₄×C₅	1	4	yes	complemented
106	C₂×#95	—	2	2	no	—
107	#340	—	2	8	no	—
108	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂³×C₅	#124	1	1	yes	characteristic
112	C₂³×C₅	#124	1	1	yes	characteristic
113	C₂³×C₅	#127	1	1	yes	characteristic
114	C₂³×C₅	C₄⋊Q₈	1	1	yes	characteristic
115	C₂³×C₅	AGL(1,ℤ/8)	1	4	yes	—
116	C₂³×C₅	#137	1	4	yes	—
117	C₂×C₄×C₅	—	2	8	no	—
118	C₂×C₄×C₅	—	2	4	no	—
119	C₂×C₄×C₅	—	2	4	no	—
120	C₂×C₄×C₅	—	2	8	no	—
121	C₂×C₄×C₅	—	4	4	no	—
122	C₂×C₄×C₅	—	4	4	no	—
123	C₂×C₄×C₅	—	4	8	no	—
124	C₂×C₄×C₅	—	4	8	no	—
125	C₂×C₄×C₅	—	8	4	no	—
126	C₂×C₄×C₅	—	8	4	no	—
127	C₂×C₄×C₅	—	8	2	no	—
128	C₂×C₄×C₅	—	8	4	no	—
129	C₂×C₄×C₅	—	8	2	no	—
130	C₂×C₄×C₅	#95	1	8	yes	—
131	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
134	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₅	#117	1	2	yes	—
137	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
138	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
139	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
140	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
141	C₂×C₄×C₅	#121	1	2	yes	—
142	C₂×C₄×C₅	#121	1	2	yes	—
143	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
144	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
145	C₂×C₄×C₅	#125	1	2	yes	—
146	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
147	C₅×C₈	—	2	8	no	—
148	C₂³×C₄	—	2	2	no	—
149	C₂³×C₄	—	2	2	no	—
150	C₂³×C₄	C₅×D₄	1	2	yes	—
151	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
152	C₂²×C₈	—	2	2	no	—
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
157	C₂²×C₈	C₅×D₄	1	4	yes	—
158	C₂²×C₈	C₅×Q₈	1	4	yes	—
159	C₂×C₄⋊C₄	—	2	4	no	—
160	C₂×C₄⋊C₄	—	2	4	no	—
161	C₂×C₄⋊C₄	—	2	4	no	—
162	C₂×C₄⋊C₄	—	2	4	no	—
163	C₂×C₄⋊C₄	—	2	8	no	—
164	C₂×C₄⋊C₄	—	2	2	no	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄⋊C₄	—	4	8	no	—
167	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
168	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
169	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
170	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
171	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
172	C₂×C₄²	C₅×D₄	1	2	yes	—
173	C₂×C₄²	C₅×D₄	1	2	yes	—
174	C₂×C₄²	C₅×Q₈	1	2	yes	—
175	#97	—	2	8	no	—
176	#95	—	2	8	no	—
177	C₂²×C₅	C₂×#95	1	1	yes	characteristic
178	C₂²×C₅	#382	1	1	yes	characteristic
179	C₂²×C₅	#384	1	1	yes	characteristic
180	C₂²×C₅	#390	1	1	yes	characteristic
181	C₂²×C₅	#390	1	1	yes	characteristic
182	C₂²×C₅	#391	1	1	yes	characteristic
183	C₂²×C₅	#391	1	1	yes	characteristic
184	C₂²×C₅	#417	1	4	yes	—
185	C₂²×C₅	#418	1	2	yes	—
186	C₂²×C₅	#419	1	2	yes	—
187	C₂²×C₅	#428	1	4	yes	—
188	C₂²×C₅	#489	1	4	yes	—
189	C₂²×C₅	#490	1	2	yes	—
190	C₂²×C₅	#491	1	2	yes	—
191	C₂²×C₅	#496	1	2	yes	—
192	C₂²×C₅	#497	1	2	yes	—
193	C₂²×C₅	#501	1	4	yes	—
194	C₄×C₅	—	2	8	no	—
195	C₄×C₅	—	8	4	no	—
196	C₄×C₅	—	8	4	no	—
197	C₄×C₅	C₄×C₄⋊C₄	1	2	yes	—
198	C₄×C₅	#380	1	2	yes	—
199	C₄×C₅	#389	1	2	yes	—
200	C₄×C₅	#389	1	2	yes	—
201	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	4	2	no	—
207	C₂²×C₄	—	4	2	no	—
208	C₂²×C₄	—	4	2	no	—
209	C₂²×C₄	—	4	2	no	—
210	C₂²×C₄	—	4	8	no	—
211	C₂²×C₄	—	4	4	no	—
212	C₂²×C₄	—	4	4	no	—
213	C₂²×C₄	C₅×C₂²⋊C₄	1	4	yes	—
214	C₂²×C₄	C₅×C₂²⋊C₄	1	4	yes	—
215	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
216	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
217	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
218	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
222	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
224	C₂²×C₄	C₅×#41	1	2	yes	—
225	C₂²×C₄	C₅×#41	1	2	yes	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	C₅×C₄⋊C₄	1	8	yes	complemented
233	C₄⋊C₄	—	2	8	no	—
234	C₄⋊C₄	—	2	8	no	—
235	C₄⋊C₄	—	4	16	no	—
236	C₄⋊C₄	—	4	4	no	—
237	C₄⋊C₄	—	4	4	no	—
238	C₄²	C₄²×C₅	1	4	yes	—
239	C₄²	C₅×C₂²⋊C₄	1	4	yes	—
240	C₄²	C₅×C₄⋊C₄	1	4	yes	—
241	C₄²	C₅×C₄⋊C₄	1	4	yes	—
242	C₂×C₅	C₄⋊#95	1	1	yes	characteristic
243	C₂×C₅	#1736	1	2	yes	—
244	C₂×C₅	#1836	1	2	yes	—
245	C₂×C₅	#1847	1	2	yes	—
246	C₂×C₅	#1959	1	2	yes	—
247	C₂×C₅	#1960	1	1	yes	characteristic
248	C₂×C₅	#1961	1	1	yes	characteristic
249	C₂×C₅	#1992	1	2	yes	—
250	C₂×C₅	#1993	1	2	yes	—
251	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
252	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
253	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
254	C₂³	C₅×#124	1	1	yes	characteristic
255	C₂³	C₅×#124	1	1	yes	characteristic
256	C₂³	C₅×#127	1	1	yes	characteristic
257	C₂³	C₅×C₄⋊Q₈	1	1	yes	characteristic
258	C₂³	C₅×AGL(1,ℤ/8)	1	4	yes	—
259	C₂³	C₅×#137	1	4	yes	—
260	C₂×C₄	—	2	8	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	8	no	—
267	C₂×C₄	—	4	8	no	—
268	C₂×C₄	—	8	4	no	—
269	C₂×C₄	—	8	4	no	—
270	C₂×C₄	—	8	2	no	—
271	C₂×C₄	—	8	4	no	—
272	C₂×C₄	—	8	2	no	—
273	C₂×C₄	C₅×#95	1	8	yes	—
274	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
275	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
276	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
277	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
278	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
279	C₂×C₄	C₅×#117	1	2	yes	—
280	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
281	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
282	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
283	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
284	C₂×C₄	C₅×#121	1	2	yes	—
285	C₂×C₄	C₅×#121	1	2	yes	—
286	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
287	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
288	C₂×C₄	C₅×#125	1	2	yes	—
289	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
290	C₈	—	2	8	no	—
291	C₅	#10843	1	1	yes	characteristic, direct product factor, Sylow
292	C₂²	C₂×C₅×#95	1	1	yes	characteristic
293	C₂²	C₅×#382	1	1	yes	characteristic
294	C₂²	C₅×#384	1	1	yes	characteristic
295	C₂²	C₅×#390	1	1	yes	characteristic
296	C₂²	C₅×#390	1	1	yes	characteristic
297	C₂²	C₅×#391	1	1	yes	characteristic
298	C₂²	C₅×#391	1	1	yes	characteristic
299	C₂²	C₅×#417	1	4	yes	—
300	C₂²	C₅×#418	1	2	yes	—
301	C₂²	C₅×#419	1	2	yes	—
302	C₂²	C₅×#428	1	4	yes	—
303	C₂²	C₅×#489	1	4	yes	—
304	C₂²	C₅×#490	1	2	yes	—
305	C₂²	C₅×#491	1	2	yes	—
306	C₂²	C₅×#496	1	2	yes	—
307	C₂²	C₅×#497	1	2	yes	—
308	C₂²	C₅×#501	1	4	yes	—
309	C₄	—	2	8	no	—
310	C₄	—	8	4	no	—
311	C₄	—	8	4	no	—
312	C₄	C₄×C₅×C₄⋊C₄	1	2	yes	—
313	C₄	C₅×#380	1	2	yes	—
314	C₄	C₅×#389	1	2	yes	—
315	C₄	C₅×#389	1	2	yes	—
316	C₂	C₅×C₄⋊#95	1	1	yes	characteristic
317	C₂	C₅×#1736	1	2	yes	—
318	C₂	C₅×#1836	1	2	yes	—
319	C₂	C₅×#1847	1	2	yes	—
320	C₂	C₅×#1959	1	2	yes	—
321	C₂	C₅×#1960	1	1	yes	characteristic
322	C₂	C₅×#1961	1	1	yes	characteristic
323	C₂	C₅×#1992	1	2	yes	—
324	C₂	C₅×#1993	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₅	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₂×C₅×#97	C₄	semidirect
14. C₅×#384	C₄	semidirect
16. C₅×#382	C₄	semidirect
19. #10843	C₅	direct (N × H)
84. C₂×C₅×C₈	C₄⋊C₄	semidirect
101. C₂×#97	C₄×C₅	semidirect
103. #384	C₄×C₅	semidirect
105. #382	C₄×C₅	semidirect
232. C₂×C₈	C₅×C₄⋊C₄	semidirect
291. C₅	#10843	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×#97	2
C₂×C₅×#384	2
C₂×C₅×#382	2
C₂×C₅×#340	2
#10843	5

Subgroups — C5×#10843 #11933563

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#10843 #11933563