Subgroups — C₅×#20819 #11943539

Order 1280.

886 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₅×#2285	C₂	1	1	yes	maximal, characteristic
3	C₅×#2283	C₂	1	1	yes	maximal, characteristic
4	C₅×#1847	C₂	1	2	yes	maximal, complemented
5	C₅×#1836	C₂	1	2	yes	maximal, complemented
6	C₅×#1809	C₂	1	4	yes	maximal
7	C₄×C₅×#107	C₂	1	2	yes	maximal, complemented
8	C₄×C₅×#106	C₂	1	2	yes	maximal, complemented
9	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
10	C₂×C₅×#117	—	2	1	no	—
11	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₂×C₅×#107	C₂²	1	4	yes	—
13	C₂×C₅×#106	C₂²	1	4	yes	—
14	C₄×C₅×M₁₆	C₄	1	8	yes	complemented
15	C₂×C₅×#97	C₂²	1	1	yes	characteristic
16	C₂×C₅×#97	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
18	C₅×#384	C₂²	1	2	yes	—
19	C₅×#383	—	2	1	no	—
20	C₅×#382	—	2	1	no	—
21	C₅×#382	C₂²	1	2	yes	—
22	C₅×#381	—	2	1	no	—
23	C₅×#380	—	2	1	no	—
24	C₅×#380	C₂²	1	2	yes	—
25	C₅×#380	C₂²	1	2	yes	—
26	C₄×C₅×C₄⋊C₄	—	2	1	no	—
27	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
28	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
29	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
30	C₄×C₅×C₂²⋊C₄	—	2	1	no	—
31	C₅×#376	—	2	1	no	—
32	C₅×#340	C₂²	1	4	yes	—
33	C₅×#340	C₂²	1	4	yes	—
34	#20819	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
38	C₂×C₅×M₁₆	—	2	8	no	—
39	C₂×C₅×M₁₆	C₂×C₄	1	8	yes	—
40	C₂²×C₅×C₈	C₂³	1	2	yes	—
41	C₂²×C₅×C₈	C₂³	1	2	yes	—
42	C₅×#117	—	4	4	no	—
43	C₂×C₅×C₄⋊C₄	—	2	2	no	—
44	C₂×C₅×C₄⋊C₄	—	2	2	no	—
45	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
46	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
47	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
48	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
49	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
50	C₂×C₄²×C₅	—	2	1	no	—
51	C₂×C₄²×C₅	—	2	1	no	—
52	C₂×C₄²×C₅	—	2	2	no	—
53	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
54	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₅	Q₈	1	4	yes	—
57	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
59	C₅×#107	—	2	8	no	—
60	C₅×#106	—	2	8	no	—
61	C₅×#97	C₂×C₄	1	4	yes	—
62	C₅×#97	C₂×C₄	1	4	yes	—
63	C₄×C₅×C₈	C₂×C₄	1	8	yes	—
64	C₅×#95	—	2	2	no	—
65	C₅×#95	—	2	2	no	—
66	C₅×#95	—	2	2	no	—
67	C₅×#95	—	2	2	no	—
68	C₅×#95	—	2	2	no	—
69	C₅×#95	—	2	1	no	—
70	C₅×#95	—	2	1	no	—
71	C₂×C₄×M₁₆	C₂×C₅	1	1	yes	characteristic
72	#2285	C₂×C₅	1	1	yes	characteristic
73	#2283	C₂×C₅	1	1	yes	characteristic
74	#1847	C₂×C₅	1	2	yes	complemented
75	#1836	C₂×C₅	1	2	yes	complemented
76	#1809	C₂×C₅	1	4	yes	—
77	C₄×#107	C₂×C₅	1	2	yes	complemented
78	C₄×#106	C₂×C₅	1	2	yes	complemented
79	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₅	—	2	2	no	—
81	C₂²×C₄×C₅	—	2	2	no	—
82	C₂²×C₄×C₅	—	2	4	no	—
83	C₂²×C₄×C₅	—	2	4	no	—
84	C₂²×C₄×C₅	—	2	2	no	—
85	C₂²×C₄×C₅	—	2	2	no	—
86	C₂²×C₄×C₅	—	4	2	no	—
87	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
88	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
89	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
96	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
97	C₅×M₁₆	—	2	16	no	—
98	C₂×C₅×C₈	—	2	4	no	—
99	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
100	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
101	C₂×C₅×C₈	#41	1	8	yes	—
102	C₅×C₄⋊C₄	—	2	4	no	—
103	C₅×C₄⋊C₄	—	2	4	no	—
104	C₅×C₄⋊C₄	—	2	4	no	—
105	C₅×C₄⋊C₄	—	2	4	no	—
106	C₅×C₄⋊C₄	—	4	4	no	—
107	C₅×C₄⋊C₄	—	4	4	no	—
108	C₅×C₂²⋊C₄	—	4	4	no	—
109	C₅×C₂²⋊C₄	—	4	4	no	—
110	C₄²×C₅	—	4	2	no	—
111	C₄²×C₅	—	4	4	no	—
112	C₄²×C₅	—	4	2	no	—
113	C₄²×C₅	C₄⋊C₄	1	4	yes	—
114	C₄²×C₅	C₄⋊C₄	1	4	yes	—
115	C₄²×C₅	C₂²×C₄	1	2	yes	—
116	C₄²×C₅	C₂²×C₄	1	2	yes	—
117	C₄²×C₅	C₂×Q₈	1	2	yes	—
118	C₄²×C₅	C₂×Q₈	1	2	yes	—
119	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
120	C₂×#117	—	2	1	no	—
121	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
122	C₂×#107	C₂²×C₅	1	4	yes	—
123	C₂×#106	C₂²×C₅	1	4	yes	—
124	C₄×M₁₆	C₄×C₅	1	8	yes	complemented
125	C₂×#97	C₂²×C₅	1	1	yes	characteristic
126	C₂×#97	C₂²×C₅	1	1	yes	characteristic
127	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
128	#384	C₂²×C₅	1	2	yes	—
129	#383	—	2	1	no	—
130	#382	—	2	1	no	—
131	#382	C₂²×C₅	1	2	yes	—
132	#381	—	2	1	no	—
133	#380	—	2	1	no	—
134	#380	C₂²×C₅	1	2	yes	—
135	#380	C₂²×C₅	1	2	yes	—
136	C₄×C₄⋊C₄	—	2	1	no	—
137	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
138	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
139	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
140	C₄×C₂²⋊C₄	—	2	1	no	—
141	#376	—	2	1	no	—
142	#340	C₂²×C₅	1	4	yes	—
143	#340	C₂²×C₅	1	4	yes	—
144	C₂³×C₅	—	2	2	no	—
145	C₂³×C₅	—	2	1	no	—
146	C₂³×C₅	—	2	1	no	—
147	C₂³×C₅	—	2	2	no	—
148	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
149	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₅	—	2	2	no	—
151	C₂×C₄×C₅	—	2	1	no	—
152	C₂×C₄×C₅	—	2	1	no	—
153	C₂×C₄×C₅	—	2	2	no	—
154	C₂×C₄×C₅	—	2	2	no	—
155	C₂×C₄×C₅	—	2	4	no	—
156	C₂×C₄×C₅	—	2	4	no	—
157	C₂×C₄×C₅	—	2	8	no	—
158	C₂×C₄×C₅	—	4	4	no	—
159	C₂×C₄×C₅	—	4	4	no	—
160	C₂×C₄×C₅	—	4	2	no	—
161	C₂×C₄×C₅	—	4	2	no	—
162	C₂×C₄×C₅	—	4	4	no	—
163	C₂×C₄×C₅	—	4	2	no	—
164	C₂×C₄×C₅	—	4	2	no	—
165	C₂×C₄×C₅	—	4	2	no	—
166	C₂×C₄×C₅	—	4	2	no	—
167	C₂×C₄×C₅	—	4	4	no	—
168	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
169	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
170	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
171	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
172	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
173	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
174	C₂×C₄×C₅	#137	1	2	yes	—
175	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
176	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂×#41	1	4	yes	—
178	C₅×C₈	—	2	4	no	—
179	C₅×C₈	#117	1	8	yes	complemented
180	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
181	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
182	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
183	C₂×M₁₆	—	2	8	no	—
184	C₂×M₁₆	C₂×C₄×C₅	1	8	yes	—
185	C₂²×C₈	C₂³×C₅	1	2	yes	—
186	C₂²×C₈	C₂³×C₅	1	2	yes	—
187	#117	—	4	4	no	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
191	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
192	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
193	C₂×C₂²⋊C₄	—	2	2	no	—
194	C₂×C₂²⋊C₄	—	2	2	no	—
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	—	2	1	no	—
197	C₂×C₄²	—	2	1	no	—
198	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
199	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
200	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
201	C₂×C₄²	C₅×Q₈	1	4	yes	—
202	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
203	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
204	#107	—	2	8	no	—
205	#106	—	2	8	no	—
206	#97	C₂×C₄×C₅	1	4	yes	—
207	#97	C₂×C₄×C₅	1	4	yes	—
208	C₄×C₈	C₂×C₄×C₅	1	8	yes	—
209	#95	—	2	2	no	—
210	#95	—	2	2	no	—
211	#95	—	2	2	no	—
212	#95	—	2	2	no	—
213	#95	—	2	2	no	—
214	#95	—	2	1	no	—
215	#95	—	2	1	no	—
216	C₂²×C₅	—	2	2	no	—
217	C₂²×C₅	—	2	2	no	—
218	C₂²×C₅	—	2	2	no	—
219	C₂²×C₅	—	4	2	no	—
220	C₂²×C₅	—	4	2	no	—
221	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
222	C₂²×C₅	#532	1	2	yes	—
223	C₂²×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
224	C₂²×C₅	C₂×#137	1	1	yes	characteristic
225	C₂²×C₅	#575	1	2	yes	—
226	C₄×C₅	—	2	2	no	—
227	C₄×C₅	—	2	2	no	—
228	C₄×C₅	—	2	2	no	—
229	C₄×C₅	—	2	2	no	—
230	C₄×C₅	—	8	4	no	—
231	C₄×C₅	—	8	2	no	—
232	C₄×C₅	—	8	2	no	—
233	C₄×C₅	#428	1	4	yes	—
234	C₄×C₅	C₂×#117	1	2	yes	—
235	C₄×C₅	#533	1	2	yes	—
236	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	4	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	4	no	—
243	C₂²×C₄	—	4	2	no	—
244	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
245	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
246	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
247	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
248	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
250	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
251	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
252	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
253	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
254	M₁₆	—	2	16	no	—
255	C₂×C₈	—	2	4	no	—
256	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
257	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
258	C₂×C₈	C₅×#41	1	8	yes	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	2	4	no	—
263	C₄⋊C₄	—	4	4	no	—
264	C₄⋊C₄	—	4	4	no	—
265	C₂²⋊C₄	—	4	4	no	—
266	C₂²⋊C₄	—	4	4	no	—
267	C₄²	—	4	4	no	—
268	C₄²	—	4	2	no	—
269	C₄²	—	4	2	no	—
270	C₄²	C₅×C₄⋊C₄	1	4	yes	—
271	C₄²	C₅×C₄⋊C₄	1	4	yes	—
272	C₄²	C₂²×C₄×C₅	1	2	yes	—
273	C₄²	C₂²×C₄×C₅	1	2	yes	—
274	C₄²	C₂×C₅×Q₈	1	2	yes	—
275	C₄²	C₂×C₅×Q₈	1	2	yes	—
276	C₂×C₅	—	4	2	no	—
277	C₂×C₅	#2285	1	1	yes	characteristic
278	C₂×C₅	C₂×#428	1	1	yes	characteristic
279	C₂×C₅	#2911	1	1	yes	characteristic
280	C₂×C₅	#3171	1	2	yes	—
281	C₂×C₅	#3172	1	2	yes	—
282	C₂³	—	2	2	no	—
283	C₂³	—	2	1	no	—
284	C₂³	—	2	1	no	—
285	C₂³	—	2	2	no	—
286	C₂³	C₂×C₅×C₄⋊C₄	1	2	yes	—
287	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	8	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	4	no	—
306	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
307	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
308	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
309	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
310	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
311	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
312	C₂×C₄	C₅×#137	1	2	yes	—
313	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
314	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
315	C₂×C₄	C₂×C₅×#41	1	4	yes	—
316	C₈	—	2	4	no	—
317	C₈	C₅×#117	1	8	yes	complemented
318	C₅	#20819	1	1	yes	characteristic, direct product factor, Sylow
319	C₂²	—	2	2	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	4	2	no	—
323	C₂²	—	4	2	no	—
324	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
325	C₂²	C₅×#532	1	2	yes	—
326	C₂²	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
327	C₂²	C₂×C₅×#137	1	1	yes	characteristic
328	C₂²	C₅×#575	1	2	yes	—
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	2	no	—
333	C₄	—	8	4	no	—
334	C₄	—	8	2	no	—
335	C₄	—	8	2	no	—
336	C₄	C₅×#428	1	4	yes	—
337	C₄	C₂×C₅×#117	1	2	yes	—
338	C₄	C₅×#533	1	2	yes	—
339	C₂	—	4	2	no	—
340	C₂	C₅×#2285	1	1	yes	characteristic
341	C₂	C₂×C₅×#428	1	1	yes	characteristic
342	C₂	C₅×#2911	1	1	yes	characteristic
343	C₂	C₅×#3171	1	2	yes	—
344	C₂	C₅×#3172	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₅×#1847	C₂	semidirect
5. C₅×#1836	C₂	semidirect
7. C₄×C₅×#107	C₂	semidirect
8. C₄×C₅×#106	C₂	semidirect
14. C₄×C₅×M₁₆	C₄	semidirect
34. #20819	C₅	direct (N × H)
74. #1847	C₂×C₅	semidirect
75. #1836	C₂×C₅	semidirect
77. C₄×#107	C₂×C₅	semidirect
78. C₄×#106	C₂×C₅	semidirect
124. C₄×M₁₆	C₄×C₅	semidirect
179. C₅×C₈	#117	semidirect
317. C₈	C₅×#117	semidirect
318. C₅	#20819	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×C₅×M₁₆	2
C₅×#2285	2
C₅×#2283	2
C₅×#1847	2
C₅×#1836	2
C₅×#1809	2
C₄×C₅×#107	2
C₄×C₅×#106	2
#20819	5

Subgroups — C5×#20819 #11943539

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#20819 #11943539