Subgroups — C₅×#20385 #11943105

Order 1280.

1774 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#407	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#405	C₂	1	2	yes	maximal, complemented
4	C₅×#2280	C₂	1	1	yes	maximal, characteristic
5	C₅×#1752	C₂	1	4	yes	maximal, complemented
6	C₅×C₈×C₂²⋊C₄	C₂	1	4	yes	maximal, complemented
7	C₅×#1729	C₂	1	2	yes	maximal, complemented
8	C₂⁴×C₄×C₅	C₂²	1	1	yes	characteristic
9	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
10	C₂³×C₅×C₈	—	2	8	no	—
11	C₂³×C₅×C₈	C₄	1	8	yes	complemented
12	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₂³×C₅×C₈	C₂²	1	2	yes	—
14	C₂³×C₅×C₈	C₂²	1	2	yes	—
15	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₂×C₅×#117	C₂²	1	2	yes	—
17	C₅×#407	C₄	1	8	yes	complemented
18	C₂×C₅×C₂²⋊C₈	C₂²	1	4	yes	—
19	C₅×#405	—	2	8	no	—
20	C₂×C₅×#97	C₂²	1	4	yes	complemented
21	C₂×C₄×C₅×C₈	C₂²	1	4	yes	complemented
22	C₅×#381	C₄	1	2	yes	—
23	C₅×#380	C₄	1	4	yes	—
24	C₅×#379	C₄	1	2	yes	—
25	C₄×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
26	C₅×#336	C₂²	1	8	yes	complemented
27	#20385	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂⁵×C₅	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₅	—	2	4	no	—
30	C₂³×C₄×C₅	—	2	4	no	—
31	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
32	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₅	C₂×C₄	1	4	yes	—
34	C₂³×C₄×C₅	C₂×C₄	1	4	yes	—
35	C₂³×C₄×C₅	C₂³	1	2	yes	—
36	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
37	C₂×C₅×M₁₆	—	2	4	no	—
38	C₂×C₅×M₁₆	—	2	4	no	—
39	C₂×C₅×M₁₆	C₂×C₄	1	8	yes	—
40	C₂²×C₅×C₈	—	2	4	no	—
41	C₂²×C₅×C₈	—	2	32	no	—
42	C₂²×C₅×C₈	—	2	8	no	—
43	C₂²×C₅×C₈	—	2	8	no	—
44	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
45	C₂²×C₅×C₈	C₂×C₄	1	8	yes	—
46	C₂²×C₅×C₈	C₂×C₄	1	8	yes	—
47	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
48	C₂²×C₅×C₈	D₄	1	8	yes	—
49	C₂²×C₅×C₈	C₂³	1	4	yes	—
50	C₂²×C₅×C₈	C₂³	1	2	yes	—
51	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
52	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
53	C₅×#117	—	2	4	no	—
54	C₅×#117	—	2	4	no	—
55	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
56	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	8	yes	—
57	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	4	yes	—
58	C₂×C₄²×C₅	C₂³	1	4	yes	—
59	C₅×C₂²⋊C₈	C₂×C₄	1	16	yes	—
60	C₅×#97	—	2	8	no	—
61	C₄×C₅×C₈	—	2	8	no	—
62	C₅×#95	C₂×C₄	1	8	yes	—
63	C₂⁴×C₈	C₂×C₅	1	1	yes	characteristic
64	C₂×#407	C₂×C₅	1	1	yes	characteristic
65	C₂×#405	C₂×C₅	1	2	yes	complemented
66	#2280	C₂×C₅	1	1	yes	characteristic
67	#1752	C₂×C₅	1	4	yes	complemented
68	C₈×C₂²⋊C₄	C₂×C₅	1	4	yes	complemented
69	#1729	C₂×C₅	1	2	yes	complemented
70	C₂⁴×C₅	—	2	4	no	—
71	C₂⁴×C₅	—	2	8	no	—
72	C₂⁴×C₅	C₄²	1	4	yes	—
73	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂⁴×C₅	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₅	—	2	4	no	—
76	C₂²×C₄×C₅	—	2	8	no	—
77	C₂²×C₄×C₅	—	2	8	no	—
78	C₂²×C₄×C₅	—	2	8	no	—
79	C₂²×C₄×C₅	—	2	16	no	—
80	C₂²×C₄×C₅	—	2	4	no	—
81	C₂²×C₄×C₅	—	2	4	no	—
82	C₂²×C₄×C₅	C₄²	1	4	yes	—
83	C₂²×C₄×C₅	C₄²	1	8	yes	—
84	C₂²×C₄×C₅	C₂²×C₄	1	8	yes	—
85	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
87	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
88	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
92	C₂²×C₄×C₅	#41	1	4	yes	—
93	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
94	C₅×M₁₆	—	2	16	no	—
95	C₂×C₅×C₈	—	2	8	no	—
96	C₂×C₅×C₈	—	2	4	no	—
97	C₂×C₅×C₈	—	2	4	no	—
98	C₂×C₅×C₈	—	2	4	no	—
99	C₂×C₅×C₈	—	2	16	no	—
100	C₂×C₅×C₈	—	2	16	no	—
101	C₂×C₅×C₈	C₂²⋊C₄	1	16	yes	complemented
102	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
103	C₂×C₅×C₈	C₂²×C₄	1	8	yes	—
104	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
105	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
106	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
107	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
108	C₅×C₄⋊C₄	—	2	8	no	—
109	C₅×C₂²⋊C₄	—	2	8	no	—
110	C₅×C₂²⋊C₄	—	2	16	no	—
111	C₄²×C₅	—	2	4	no	—
112	C₄²×C₅	—	2	4	no	—
113	C₂⁴×C₄	C₂²×C₅	1	1	yes	characteristic
114	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
115	C₂³×C₈	—	2	8	no	—
116	C₂³×C₈	C₄×C₅	1	8	yes	complemented
117	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
118	C₂³×C₈	C₂²×C₅	1	2	yes	—
119	C₂³×C₈	C₂²×C₅	1	2	yes	—
120	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
121	C₂×#117	C₂²×C₅	1	2	yes	—
122	#407	C₄×C₅	1	8	yes	complemented
123	C₂×C₂²⋊C₈	C₂²×C₅	1	4	yes	—
124	#405	—	2	8	no	—
125	C₂×#97	C₂²×C₅	1	4	yes	complemented
126	C₂×C₄×C₈	C₂²×C₅	1	4	yes	complemented
127	#381	C₄×C₅	1	2	yes	—
128	#380	C₄×C₅	1	4	yes	—
129	#379	C₄×C₅	1	2	yes	—
130	C₄×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
131	#336	C₂²×C₅	1	8	yes	complemented
132	C₂³×C₅	—	2	32	no	—
133	C₂³×C₅	—	2	4	no	—
134	C₂³×C₅	—	2	4	no	—
135	C₂³×C₅	—	2	4	no	—
136	C₂³×C₅	—	2	8	no	—
137	C₂³×C₅	—	2	16	no	—
138	C₂³×C₅	C₂×C₄²	1	2	yes	—
139	C₂³×C₅	C₂×C₄²	1	4	yes	—
140	C₂³×C₅	C₄×D₄	1	4	yes	—
141	C₂³×C₅	#131	1	8	yes	—
142	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₄×C₅	—	2	4	no	—
144	C₂×C₄×C₅	—	2	8	no	—
145	C₂×C₄×C₅	—	2	4	no	—
146	C₂×C₄×C₅	—	2	4	no	—
147	C₂×C₄×C₅	—	2	8	no	—
148	C₂×C₄×C₅	—	2	16	no	—
149	C₂×C₄×C₅	—	2	8	no	—
150	C₂×C₄×C₅	—	2	16	no	—
151	C₂×C₄×C₅	C₂×C₄²	1	4	yes	—
152	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
153	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	4	yes	—
154	C₂×C₄×C₅	#117	1	4	yes	—
155	C₂×C₄×C₅	C₄×D₄	1	8	yes	—
156	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
157	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
158	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
159	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
160	C₂×C₄×C₅	C₂×#41	1	2	yes	—
161	C₅×C₈	—	2	4	no	—
162	C₅×C₈	—	2	8	no	—
163	C₅×C₈	C₂×C₂²⋊C₄	1	4	yes	—
164	C₅×C₈	C₂×C₂²⋊C₄	1	4	yes	—
165	C₂⁵	C₂×C₄×C₅	1	1	yes	characteristic
166	C₂³×C₄	—	2	4	no	—
167	C₂³×C₄	—	2	4	no	—
168	C₂³×C₄	C₂×C₄×C₅	1	2	yes	—
169	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
170	C₂³×C₄	C₂×C₄×C₅	1	4	yes	—
171	C₂³×C₄	C₂×C₄×C₅	1	4	yes	—
172	C₂³×C₄	C₂³×C₅	1	2	yes	—
173	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
174	C₂×M₁₆	—	2	4	no	—
175	C₂×M₁₆	—	2	4	no	—
176	C₂×M₁₆	C₂×C₄×C₅	1	8	yes	—
177	C₂²×C₈	—	2	4	no	—
178	C₂²×C₈	—	2	32	no	—
179	C₂²×C₈	—	2	8	no	—
180	C₂²×C₈	—	2	8	no	—
181	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
182	C₂²×C₈	C₂×C₄×C₅	1	8	yes	—
183	C₂²×C₈	C₂×C₄×C₅	1	8	yes	—
184	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
185	C₂²×C₈	C₅×D₄	1	8	yes	—
186	C₂²×C₈	C₂³×C₅	1	2	yes	—
187	C₂²×C₈	C₂³×C₅	1	4	yes	—
188	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
189	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
190	#117	—	2	4	no	—
191	#117	—	2	4	no	—
192	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
193	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	4	yes	—
194	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	8	yes	—
195	C₂×C₄²	C₂³×C₅	1	4	yes	—
196	C₂²⋊C₈	C₂×C₄×C₅	1	16	yes	—
197	#97	—	2	8	no	—
198	C₄×C₈	—	2	8	no	—
199	#95	C₂×C₄×C₅	1	8	yes	—
200	C₂²×C₅	—	2	32	no	—
201	C₂²×C₅	—	2	4	no	—
202	C₂²×C₅	—	2	4	no	—
203	C₂²×C₅	—	2	8	no	—
204	C₂²×C₅	—	2	16	no	—
205	C₂²×C₅	—	2	4	no	—
206	C₂²×C₅	C₄×C₂²⋊C₄	1	4	yes	—
207	C₂²×C₅	#405	1	8	yes	complemented
208	C₂²×C₅	C₂²×C₄²	1	1	yes	characteristic
209	C₂²×C₅	C₂×C₄×D₄	1	2	yes	—
210	C₂²×C₅	C₂×#131	1	4	yes	—
211	C₄×C₅	—	2	4	no	—
212	C₄×C₅	—	4	8	no	—
213	C₄×C₅	C₄×C₂²⋊C₄	1	4	yes	—
214	C₄×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
215	C₄×C₅	C₂×#117	1	1	yes	characteristic
216	C₄×C₅	C₂×C₄×D₄	1	2	yes	—
217	C₂⁴	—	2	4	no	—
218	C₂⁴	—	2	8	no	—
219	C₂⁴	C₄²×C₅	1	4	yes	—
220	C₂⁴	C₂²×C₄×C₅	1	1	yes	characteristic
221	C₂⁴	C₂²×C₄×C₅	1	2	yes	—
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	—	2	8	no	—
224	C₂²×C₄	—	2	8	no	—
225	C₂²×C₄	—	2	16	no	—
226	C₂²×C₄	—	2	8	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	C₄²×C₅	1	4	yes	—
230	C₂²×C₄	C₄²×C₅	1	8	yes	—
231	C₂²×C₄	C₂²×C₄×C₅	1	8	yes	—
232	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
233	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
234	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
235	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
236	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
237	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
238	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
239	C₂²×C₄	C₅×#41	1	4	yes	—
240	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
241	M₁₆	—	2	16	no	—
242	C₂×C₈	—	2	8	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	2	16	no	—
247	C₂×C₈	—	2	16	no	—
248	C₂×C₈	C₅×C₂²⋊C₄	1	16	yes	complemented
249	C₂×C₈	C₂²×C₄×C₅	1	8	yes	—
250	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
251	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
252	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
253	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
254	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
255	C₄⋊C₄	—	2	8	no	—
256	C₂²⋊C₄	—	2	8	no	—
257	C₂²⋊C₄	—	2	16	no	—
258	C₄²	—	2	4	no	—
259	C₄²	—	2	4	no	—
260	C₂×C₅	—	2	4	no	—
261	C₂×C₅	—	2	8	no	—
262	C₂×C₅	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic
263	C₂×C₅	C₂×#405	1	2	yes	—
264	C₂×C₅	#2929	1	4	yes	—
265	C₂³	—	2	32	no	—
266	C₂³	—	2	8	no	—
267	C₂³	—	2	16	no	—
268	C₂³	—	2	4	no	—
269	C₂³	—	2	4	no	—
270	C₂³	—	2	4	no	—
271	C₂³	C₂×C₄²×C₅	1	2	yes	—
272	C₂³	C₂×C₄²×C₅	1	4	yes	—
273	C₂³	C₄×C₅×D₄	1	4	yes	—
274	C₂³	C₅×#131	1	8	yes	—
275	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
276	C₂×C₄	—	2	8	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	16	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	8	no	—
282	C₂×C₄	—	2	8	no	—
283	C₂×C₄	—	2	16	no	—
284	C₂×C₄	C₂×C₄²×C₅	1	4	yes	—
285	C₂×C₄	C₂×C₄²×C₅	1	2	yes	—
286	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	4	yes	—
287	C₂×C₄	C₅×#117	1	4	yes	—
288	C₂×C₄	C₄×C₅×D₄	1	4	yes	—
289	C₂×C₄	C₄×C₅×D₄	1	8	yes	—
290	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
291	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
292	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
293	C₂×C₄	C₂×C₅×#41	1	2	yes	—
294	C₈	—	2	4	no	—
295	C₈	—	2	8	no	—
296	C₈	C₂×C₅×C₂²⋊C₄	1	4	yes	—
297	C₈	C₂×C₅×C₂²⋊C₄	1	4	yes	—
298	C₅	#20385	1	1	yes	characteristic, direct product factor, Sylow
299	C₂²	—	2	32	no	—
300	C₂²	—	2	4	no	—
301	C₂²	—	2	4	no	—
302	C₂²	—	2	4	no	—
303	C₂²	—	2	8	no	—
304	C₂²	—	2	16	no	—
305	C₂²	C₄×C₅×C₂²⋊C₄	1	4	yes	—
306	C₂²	C₅×#405	1	8	yes	complemented
307	C₂²	C₂²×C₄²×C₅	1	1	yes	characteristic
308	C₂²	C₂×C₄×C₅×D₄	1	2	yes	—
309	C₂²	C₂×C₅×#131	1	4	yes	—
310	C₄	—	2	4	no	—
311	C₄	—	4	8	no	—
312	C₄	C₄×C₅×C₂²⋊C₄	1	4	yes	—
313	C₄	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
314	C₄	C₂×C₅×#117	1	1	yes	characteristic
315	C₄	C₂×C₄×C₅×D₄	1	2	yes	—
316	C₂	—	2	4	no	—
317	C₂	—	2	8	no	—
318	C₂	C₂×C₄×C₅×C₂²⋊C₄	1	1	yes	characteristic
319	C₂	C₂×C₅×#405	1	2	yes	—
320	C₂	C₅×#2929	1	4	yes	—

	subgroup	G/H
center	C₂²×C₅×C₈	C₂³
commutator	C₂²	C₂²×C₄²×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂³×C₅	C₂³×C₄
prime radical	C₂⁵×C₅	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₅×#405	C₂	semidirect
5. C₅×#1752	C₂	semidirect
6. C₅×C₈×C₂²⋊C₄	C₂	semidirect
7. C₅×#1729	C₂	semidirect
11. C₂³×C₅×C₈	C₄	semidirect
17. C₅×#407	C₄	semidirect
20. C₂×C₅×#97	C₂²	semidirect
21. C₂×C₄×C₅×C₈	C₂²	semidirect
26. C₅×#336	C₂²	semidirect
27. #20385	C₅	direct (N × H)
65. C₂×#405	C₂×C₅	semidirect
67. #1752	C₂×C₅	semidirect
68. C₈×C₂²⋊C₄	C₂×C₅	semidirect
69. #1729	C₂×C₅	semidirect
101. C₂×C₅×C₈	C₂²⋊C₄	semidirect
116. C₂³×C₈	C₄×C₅	semidirect
122. #407	C₄×C₅	semidirect
125. C₂×#97	C₂²×C₅	semidirect
126. C₂×C₄×C₈	C₂²×C₅	semidirect
131. #336	C₂²×C₅	semidirect
207. C₂²×C₅	#405	semidirect
248. C₂×C₈	C₅×C₂²⋊C₄	semidirect
298. C₅	#20385	direct (N × H)
306. C₂²	C₅×#405	semidirect

Maximal subgroups

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

subgroup	index
C₂⁴×C₅×C₈	2
C₂×C₅×#407	2
C₂×C₅×#405	2
C₅×#2280	2
C₅×#1752	2
C₅×C₈×C₂²⋊C₄	2
C₅×#1729	2
#20385	5

Subgroups — C5×#20385 #11943105

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#20385 #11943105