Subgroups — C₅×#20667 #11943387

Order 1280.

1438 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#430	C₂	1	2	yes	maximal, complemented
3	C₂×C₅×#407	C₂	1	2	yes	maximal
4	C₅×#1811	C₂	1	4	yes	maximal, complemented
5	C₅×#1810	C₂	1	4	yes	maximal, complemented
6	C₅×#1738	C₂	1	2	yes	maximal, complemented
7	C₂⁴×C₄×C₅	C₂²	1	1	yes	characteristic
8	C₂²×C₅×M₁₆	—	2	8	no	—
9	C₂²×C₅×M₁₆	C₄	1	8	yes	—
10	C₂²×C₅×M₁₆	C₂²	1	4	yes	—
11	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
12	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
14	C₂³×C₅×C₈	C₂²	1	2	yes	—
15	C₅×#430	—	2	8	no	—
16	C₂×C₅×#108	C₂²	1	8	yes	complemented
17	C₅×#407	—	2	4	no	—
18	C₅×#407	C₂²	1	4	yes	—
19	C₅×#407	C₂²	1	4	yes	—
20	C₂×C₅×C₂²⋊C₈	—	2	4	no	—
21	C₅×#341	C₂²	1	8	yes	complemented
22	#20667	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂⁵×C₅	Q₈	1	1	yes	characteristic
24	C₂³×C₄×C₅	—	2	4	no	—
25	C₂³×C₄×C₅	—	2	4	no	—
26	C₂³×C₄×C₅	C₂×C₄	1	4	yes	—
27	C₂³×C₄×C₅	D₄	1	4	yes	—
28	C₂³×C₄×C₅	Q₈	1	1	yes	characteristic
29	C₂³×C₄×C₅	Q₈	1	1	yes	characteristic
30	C₂³×C₄×C₅	Q₈	1	1	yes	characteristic
31	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
34	C₂×C₅×M₁₆	—	2	4	no	—
35	C₂×C₅×M₁₆	—	2	32	no	—
36	C₂×C₅×M₁₆	—	2	4	no	—
37	C₂×C₅×M₁₆	—	2	8	no	—
38	C₂×C₅×M₁₆	—	2	8	no	—
39	C₂×C₅×M₁₆	—	2	8	no	—
40	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
41	C₂×C₅×M₁₆	C₂×C₄	1	8	yes	—
42	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
43	C₂×C₅×M₁₆	D₄	1	8	yes	—
44	C₂×C₅×M₁₆	C₂³	1	8	yes	—
45	C₂²×C₅×C₈	—	2	8	no	—
46	C₂²×C₅×C₈	—	2	4	no	—
47	C₂²×C₅×C₈	C₂×C₄	1	8	yes	—
48	C₂²×C₅×C₈	C₂³	1	2	yes	—
49	C₂²×C₅×C₈	C₂³	1	2	yes	—
50	C₅×#108	—	2	16	no	—
51	C₅×C₂²⋊C₈	—	2	8	no	—
52	C₅×C₂²⋊C₈	—	2	8	no	—
53	C₂³×M₁₆	C₂×C₅	1	1	yes	characteristic
54	C₂×#430	C₂×C₅	1	2	yes	complemented
55	C₂×#407	C₂×C₅	1	2	yes	—
56	#1811	C₂×C₅	1	4	yes	complemented
57	#1810	C₂×C₅	1	4	yes	complemented
58	#1738	C₂×C₅	1	2	yes	complemented
59	C₂⁴×C₅	—	2	4	no	—
60	C₂⁴×C₅	—	4	4	no	—
61	C₂⁴×C₅	C₄⋊C₄	1	4	yes	—
62	C₂⁴×C₅	C₂×Q₈	1	1	yes	characteristic
63	C₂⁴×C₅	C₂×Q₈	1	1	yes	characteristic
64	C₂⁴×C₅	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄×C₅	—	2	4	no	—
66	C₂²×C₄×C₅	—	2	8	no	—
67	C₂²×C₄×C₅	—	2	8	no	—
68	C₂²×C₄×C₅	—	2	16	no	—
69	C₂²×C₄×C₅	—	2	4	no	—
70	C₂²×C₄×C₅	—	2	4	no	—
71	C₂²×C₄×C₅	—	2	2	no	—
72	C₂²×C₄×C₅	—	2	2	no	—
73	C₂²×C₄×C₅	—	2	2	no	—
74	C₂²×C₄×C₅	—	2	2	no	—
75	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
76	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
77	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
78	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
84	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
86	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
87	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
88	C₂²×C₄×C₅	#41	1	4	yes	—
89	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
90	C₅×M₁₆	—	2	8	no	—
91	C₅×M₁₆	—	2	16	no	—
92	C₅×M₁₆	—	2	8	no	—
93	C₅×M₁₆	—	2	8	no	—
94	C₅×M₁₆	—	4	8	no	—
95	C₅×M₁₆	C₂²⋊C₄	1	16	yes	—
96	C₂×C₅×C₈	—	2	4	no	—
97	C₂×C₅×C₈	—	2	16	no	—
98	C₂×C₅×C₈	—	2	8	no	—
99	C₂×C₅×C₈	—	4	8	no	—
100	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
101	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
102	C₂×C₅×C₈	C₂×D₄	1	8	yes	—
103	C₂⁴×C₄	C₂²×C₅	1	1	yes	characteristic
104	C₂²×M₁₆	—	2	8	no	—
105	C₂²×M₁₆	C₄×C₅	1	8	yes	—
106	C₂²×M₁₆	C₂²×C₅	1	4	yes	—
107	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
108	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
109	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
110	C₂³×C₈	C₂²×C₅	1	2	yes	—
111	#430	—	2	8	no	—
112	C₂×#108	C₂²×C₅	1	8	yes	complemented
113	#407	—	2	4	no	—
114	#407	C₂²×C₅	1	4	yes	—
115	#407	C₂²×C₅	1	4	yes	—
116	C₂×C₂²⋊C₈	—	2	4	no	—
117	#341	C₂²×C₅	1	8	yes	complemented
118	C₂³×C₅	—	2	2	no	—
119	C₂³×C₅	—	2	4	no	—
120	C₂³×C₅	—	2	2	no	—
121	C₂³×C₅	—	2	8	no	—
122	C₂³×C₅	—	2	8	no	—
123	C₂³×C₅	—	4	4	no	—
124	C₂³×C₅	—	4	2	no	—
125	C₂³×C₅	—	4	2	no	—
126	C₂³×C₅	—	4	16	no	—
127	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
128	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
129	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
130	C₂³×C₅	C₂²⋊Q₈	1	4	yes	—
131	C₂³×C₅	C₂²×Q₈	1	1	yes	characteristic
132	C₂×C₄×C₅	—	2	2	no	—
133	C₂×C₄×C₅	—	2	2	no	—
134	C₂×C₄×C₅	—	2	8	no	—
135	C₂×C₄×C₅	—	2	2	no	—
136	C₂×C₄×C₅	—	2	2	no	—
137	C₂×C₄×C₅	—	2	2	no	—
138	C₂×C₄×C₅	—	2	2	no	—
139	C₂×C₄×C₅	—	2	4	no	—
140	C₂×C₄×C₅	—	2	4	no	—
141	C₂×C₄×C₅	—	2	8	no	—
142	C₂×C₄×C₅	—	2	8	no	—
143	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	4	yes	—
144	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
145	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
146	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
147	C₂×C₄×C₅	#117	1	4	yes	—
148	C₂×C₄×C₅	#121	1	8	yes	—
149	C₂×C₄×C₅	C₂²⋊Q₈	1	4	yes	—
150	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
152	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂×#41	1	2	yes	—
154	C₅×C₈	—	2	4	no	—
155	C₅×C₈	—	4	8	no	—
156	C₅×C₈	C₂×C₂²⋊C₄	1	8	yes	—
157	C₂⁵	C₅×Q₈	1	1	yes	characteristic
158	C₂³×C₄	—	2	4	no	—
159	C₂³×C₄	—	2	4	no	—
160	C₂³×C₄	C₂×C₄×C₅	1	4	yes	—
161	C₂³×C₄	C₅×D₄	1	4	yes	—
162	C₂³×C₄	C₅×Q₈	1	1	yes	characteristic
163	C₂³×C₄	C₅×Q₈	1	1	yes	characteristic
164	C₂³×C₄	C₅×Q₈	1	1	yes	characteristic
165	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
166	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
167	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
168	C₂×M₁₆	—	2	4	no	—
169	C₂×M₁₆	—	2	32	no	—
170	C₂×M₁₆	—	2	4	no	—
171	C₂×M₁₆	—	2	8	no	—
172	C₂×M₁₆	—	2	8	no	—
173	C₂×M₁₆	—	2	8	no	—
174	C₂×M₁₆	C₂×C₄×C₅	1	4	yes	—
175	C₂×M₁₆	C₂×C₄×C₅	1	8	yes	—
176	C₂×M₁₆	C₂×C₄×C₅	1	4	yes	—
177	C₂×M₁₆	C₅×D₄	1	8	yes	—
178	C₂×M₁₆	C₂³×C₅	1	8	yes	—
179	C₂²×C₈	—	2	8	no	—
180	C₂²×C₈	—	2	4	no	—
181	C₂²×C₈	C₂×C₄×C₅	1	8	yes	—
182	C₂²×C₈	C₂³×C₅	1	2	yes	—
183	C₂²×C₈	C₂³×C₅	1	2	yes	—
184	#108	—	2	16	no	—
185	C₂²⋊C₈	—	2	8	no	—
186	C₂²⋊C₈	—	2	8	no	—
187	C₂²×C₅	—	2	8	no	—
188	C₂²×C₅	—	2	2	no	—
189	C₂²×C₅	—	2	4	no	—
190	C₂²×C₅	—	2	2	no	—
191	C₂²×C₅	—	2	2	no	—
192	C₂²×C₅	—	2	4	no	—
193	C₂²×C₅	—	2	2	no	—
194	C₂²×C₅	—	4	4	no	—
195	C₂²×C₅	—	4	16	no	—
196	C₂²×C₅	—	4	4	no	—
197	C₂²×C₅	#380	1	4	yes	—
198	C₂²×C₅	#430	1	4	yes	complemented
199	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
200	C₂²×C₅	C₂×C₂²⋊Q₈	1	2	yes	—
201	C₄×C₅	—	2	2	no	—
202	C₄×C₅	—	2	2	no	—
203	C₄×C₅	#380	1	4	yes	—
204	C₄×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
205	C₄×C₅	C₂×#117	1	1	yes	characteristic
206	C₄×C₅	C₂×#121	1	2	yes	—
207	C₂⁴	—	2	4	no	—
208	C₂⁴	—	4	4	no	—
209	C₂⁴	C₅×C₄⋊C₄	1	4	yes	—
210	C₂⁴	C₂×C₅×Q₈	1	1	yes	characteristic
211	C₂⁴	C₂×C₅×Q₈	1	1	yes	characteristic
212	C₂⁴	C₂×C₅×Q₈	1	1	yes	characteristic
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	2	8	no	—
215	C₂²×C₄	—	2	8	no	—
216	C₂²×C₄	—	2	16	no	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
224	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
225	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
226	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
227	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
228	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
229	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
230	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
231	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
232	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
233	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
234	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
235	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
236	C₂²×C₄	C₅×#41	1	4	yes	—
237	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
238	M₁₆	—	2	8	no	—
239	M₁₆	—	2	16	no	—
240	M₁₆	—	2	8	no	—
241	M₁₆	—	2	8	no	—
242	M₁₆	—	4	8	no	—
243	M₁₆	C₅×C₂²⋊C₄	1	16	yes	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	2	16	no	—
246	C₂×C₈	—	2	8	no	—
247	C₂×C₈	—	4	8	no	—
248	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
249	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
250	C₂×C₈	C₂×C₅×D₄	1	8	yes	—
251	C₂×C₅	—	2	2	no	—
252	C₂×C₅	—	2	4	no	—
253	C₂×C₅	—	2	4	no	—
254	C₂×C₅	—	4	2	no	—
255	C₂×C₅	C₂×#380	1	1	yes	characteristic
256	C₂×C₅	C₂×#430	1	2	yes	—
257	C₂³	—	2	4	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	2	8	no	—
260	C₂³	—	2	8	no	—
261	C₂³	—	2	2	no	—
262	C₂³	—	4	4	no	—
263	C₂³	—	4	2	no	—
264	C₂³	—	4	2	no	—
265	C₂³	—	4	16	no	—
266	C₂³	C₂×C₅×C₄⋊C₄	1	2	yes	—
267	C₂³	C₂×C₅×C₄⋊C₄	1	2	yes	—
268	C₂³	C₂×C₅×C₄⋊C₄	1	2	yes	—
269	C₂³	C₅×C₂²⋊Q₈	1	4	yes	—
270	C₂³	C₂²×C₅×Q₈	1	1	yes	characteristic
271	C₂×C₄	—	2	8	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	8	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	4	yes	—
283	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
284	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
285	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
286	C₂×C₄	C₅×#117	1	4	yes	—
287	C₂×C₄	C₅×#121	1	8	yes	—
288	C₂×C₄	C₅×C₂²⋊Q₈	1	4	yes	—
289	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
290	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
291	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
292	C₂×C₄	C₂×C₅×#41	1	2	yes	—
293	C₈	—	2	4	no	—
294	C₈	—	4	8	no	—
295	C₈	C₂×C₅×C₂²⋊C₄	1	8	yes	—
296	C₅	#20667	1	1	yes	characteristic, direct product factor, Sylow
297	C₂²	—	2	8	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	4	no	—
301	C₂²	—	2	2	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	4	no	—
304	C₂²	—	4	4	no	—
305	C₂²	—	4	16	no	—
306	C₂²	—	4	4	no	—
307	C₂²	C₅×#380	1	4	yes	—
308	C₂²	C₅×#430	1	4	yes	complemented
309	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
310	C₂²	C₂×C₅×C₂²⋊Q₈	1	2	yes	—
311	C₄	—	2	2	no	—
312	C₄	—	2	2	no	—
313	C₄	C₅×#380	1	4	yes	—
314	C₄	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
315	C₄	C₂×C₅×#117	1	1	yes	characteristic
316	C₄	C₂×C₅×#121	1	2	yes	—
317	C₂	—	2	4	no	—
318	C₂	—	2	2	no	—
319	C₂	—	2	4	no	—
320	C₂	—	4	2	no	—
321	C₂	C₂×C₅×#380	1	1	yes	characteristic
322	C₂	C₂×C₅×#430	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂²×C₅	C₂²×C₄⋊C₄
prime radical	C₂⁵×C₅	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂⁴×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#430	C₂	semidirect
4. C₅×#1811	C₂	semidirect
5. C₅×#1810	C₂	semidirect
6. C₅×#1738	C₂	semidirect
16. C₂×C₅×#108	C₂²	semidirect
21. C₅×#341	C₂²	semidirect
22. #20667	C₅	direct (N × H)
54. C₂×#430	C₂×C₅	semidirect
56. #1811	C₂×C₅	semidirect
57. #1810	C₂×C₅	semidirect
58. #1738	C₂×C₅	semidirect
112. C₂×#108	C₂²×C₅	semidirect
117. #341	C₂²×C₅	semidirect
198. C₂²×C₅	#430	semidirect
296. C₅	#20667	direct (N × H)
308. C₂²	C₅×#430	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₅×M₁₆	2
C₂×C₅×#430	2
C₂×C₅×#407	2
C₅×#1811	2
C₅×#1810	2
C₅×#1738	2
#20667	5

Subgroups — C5×#20667 #11943387

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#20667 #11943387