Subgroups — C₅×#21349 #11944069

Order 1280.

1030 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#431	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×#117	C₂	1	1	yes	maximal, characteristic
4	C₅×#1939	C₂	1	4	yes	maximal, complemented
5	C₅×#1850	C₂	1	4	yes	maximal
6	C₄×C₅×C₄⋊C₈	C₂	1	4	yes	maximal, complemented
7	C₂³×C₅×C₈	C₂²	1	2	yes	—
8	C₂×C₅×#117	C₄	1	1	yes	characteristic
9	C₂×C₅×#117	C₄	1	1	yes	characteristic
10	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₅×#431	—	2	4	no	—
12	C₂×C₅×C₄⋊C₈	C₂²	1	8	yes	—
13	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₅×C₂²⋊C₈	C₂²	1	2	yes	—
15	C₂×C₄×C₅×C₈	—	2	4	no	—
16	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
17	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
18	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
19	C₄×C₅×C₄⋊C₄	C₄	1	2	yes	—
20	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
21	C₄×C₅×C₂²⋊C₄	C₄	1	2	yes	—
22	C₄×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₅×#376	C₄	1	2	yes	—
24	C₄³×C₅	C₂²	1	2	yes	—
25	C₅×#336	C₂²	1	8	yes	—
26	#21349	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
30	C₂²×C₅×C₈	—	2	4	no	—
31	C₂²×C₅×C₈	—	2	8	no	—
32	C₂²×C₅×C₈	C₂³	1	4	yes	—
33	C₂²×C₅×C₈	C₂³	1	4	yes	—
34	C₅×#117	—	2	4	no	—
35	C₅×#117	C₈	1	8	yes	complemented
36	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
37	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
38	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
39	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₅	D₄	1	4	yes	—
46	C₂×C₄²×C₅	Q₈	1	4	yes	—
47	C₂×C₄²×C₅	C₂³	1	4	yes	—
48	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
50	C₅×C₄⋊C₈	—	2	16	no	—
51	C₅×C₄⋊C₈	—	2	4	no	—
52	C₅×C₂²⋊C₈	—	2	4	no	—
53	C₄×C₅×C₈	—	2	4	no	—
54	C₄×C₅×C₈	—	2	4	no	—
55	C₄×C₅×C₈	—	2	4	no	—
56	C₅×#95	C₂×C₄	1	4	yes	—
57	C₅×#95	C₂×C₄	1	4	yes	—
58	C₂×#431	C₂×C₅	1	1	yes	characteristic
59	C₂²×C₄×C₈	C₂×C₅	1	1	yes	characteristic
60	C₄×#117	C₂×C₅	1	1	yes	characteristic
61	#1939	C₂×C₅	1	4	yes	complemented
62	#1850	C₂×C₅	1	4	yes	—
63	C₄×C₄⋊C₈	C₂×C₅	1	4	yes	complemented
64	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅	—	2	4	no	—
66	C₂²×C₄×C₅	—	2	4	no	—
67	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
68	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
69	C₂²×C₄×C₅	C₂×C₈	1	4	yes	—
70	C₂²×C₄×C₅	M₁₆	1	4	yes	—
71	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
75	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
80	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
81	C₂×C₅×C₈	—	2	16	no	—
82	C₂×C₅×C₈	—	2	4	no	—
83	C₂×C₅×C₈	—	2	4	no	—
84	C₂×C₅×C₈	—	2	4	no	—
85	C₂×C₅×C₈	—	2	8	no	—
86	C₂×C₅×C₈	#41	1	8	yes	—
87	C₅×C₄⋊C₄	—	2	4	no	—
88	C₅×C₄⋊C₄	C₂×C₈	1	8	yes	complemented
89	C₅×C₂²⋊C₄	—	2	4	no	—
90	C₅×C₂²⋊C₄	C₂×C₈	1	8	yes	—
91	C₄²×C₅	—	2	4	no	—
92	C₄²×C₅	—	2	8	no	—
93	C₄²×C₅	—	2	8	no	—
94	C₄²×C₅	C₄⋊C₄	1	4	yes	—
95	C₄²×C₅	C₄⋊C₄	1	4	yes	—
96	C₄²×C₅	C₂×C₈	1	8	yes	complemented
97	C₄²×C₅	C₂×D₄	1	2	yes	—
98	C₄²×C₅	C₂×D₄	1	2	yes	—
99	C₄²×C₅	C₂×Q₈	1	2	yes	—
100	C₄²×C₅	C₂×Q₈	1	2	yes	—
101	C₂³×C₈	C₂²×C₅	1	2	yes	—
102	C₂×#117	C₄×C₅	1	1	yes	characteristic
103	C₂×#117	C₄×C₅	1	1	yes	characteristic
104	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
105	#431	—	2	4	no	—
106	C₂×C₄⋊C₈	C₂²×C₅	1	8	yes	—
107	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
108	C₂×C₂²⋊C₈	C₂²×C₅	1	2	yes	—
109	C₂×C₄×C₈	—	2	4	no	—
110	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
111	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
112	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
113	C₄×C₄⋊C₄	C₄×C₅	1	2	yes	—
114	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
115	C₄×C₂²⋊C₄	C₄×C₅	1	2	yes	—
116	C₄×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
117	#376	C₄×C₅	1	2	yes	—
118	C₄³	C₂²×C₅	1	2	yes	—
119	#336	C₂²×C₅	1	8	yes	—
120	C₂³×C₅	—	2	2	no	—
121	C₂³×C₅	—	2	2	no	—
122	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
123	C₂³×C₅	C₂²×C₈	1	2	yes	—
124	C₂³×C₅	C₂×M₁₆	1	2	yes	—
125	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₅	—	2	4	no	—
127	C₂×C₄×C₅	—	2	2	no	—
128	C₂×C₄×C₅	—	2	4	no	—
129	C₂×C₄×C₅	—	2	2	no	—
130	C₂×C₄×C₅	—	2	4	no	—
131	C₂×C₄×C₅	—	2	4	no	—
132	C₂×C₄×C₅	—	2	8	no	—
133	C₂×C₄×C₅	—	2	4	no	—
134	C₂×C₄×C₅	—	2	4	no	—
135	C₂×C₄×C₅	—	2	4	no	—
136	C₂×C₄×C₅	C₄⋊C₈	1	16	yes	complemented
137	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
138	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
139	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
140	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
141	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
142	C₂×C₄×C₅	C₂²×C₈	1	8	yes	—
143	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
144	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
145	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
146	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
147	C₂×C₄×C₅	#131	1	4	yes	—
148	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂×#41	1	4	yes	—
151	C₅×C₈	—	2	8	no	—
152	C₅×C₈	—	4	4	no	—
153	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
154	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
155	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
156	C₂²×C₈	—	2	4	no	—
157	C₂²×C₈	—	2	8	no	—
158	C₂²×C₈	C₂³×C₅	1	4	yes	—
159	C₂²×C₈	C₂³×C₅	1	4	yes	—
160	#117	—	2	4	no	—
161	#117	C₅×C₈	1	8	yes	complemented
162	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
163	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
164	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	2	yes	—
165	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	2	yes	—
166	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
167	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
168	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
169	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
170	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
171	C₂×C₄²	C₅×D₄	1	4	yes	—
172	C₂×C₄²	C₅×Q₈	1	4	yes	—
173	C₂×C₄²	C₂³×C₅	1	4	yes	—
174	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
175	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
176	C₄⋊C₈	—	2	16	no	—
177	C₄⋊C₈	—	2	4	no	—
178	C₂²⋊C₈	—	2	4	no	—
179	C₄×C₈	—	2	4	no	—
180	C₄×C₈	—	2	4	no	—
181	C₄×C₈	—	2	4	no	—
182	#95	C₂×C₄×C₅	1	4	yes	—
183	#95	C₂×C₄×C₅	1	4	yes	—
184	C₂²×C₅	—	2	2	no	—
185	C₂²×C₅	—	2	4	no	—
186	C₂²×C₅	—	2	4	no	—
187	C₂²×C₅	—	2	2	no	—
188	C₂²×C₅	C₂×C₄⋊C₈	1	4	yes	—
189	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
190	C₂²×C₅	C₄×#41	1	2	yes	—
191	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
192	C₂²×C₅	C₂²×M₁₆	1	1	yes	characteristic
193	C₂²×C₅	C₂×#131	1	2	yes	—
194	C₄×C₅	—	2	2	no	—
195	C₄×C₅	—	2	8	no	—
196	C₄×C₅	—	2	2	no	—
197	C₄×C₅	—	4	4	no	—
198	C₄×C₅	C₂×C₄⋊C₈	1	4	yes	—
199	C₄×C₅	C₂×C₄⋊C₈	1	4	yes	—
200	C₄×C₅	#424	1	4	yes	—
201	C₄×C₅	#525	1	2	yes	—
202	C₄×C₅	#533	1	2	yes	—
203	C₂⁴	C₂²×C₄×C₅	1	1	yes	characteristic
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
207	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
208	C₂²×C₄	C₂×C₅×C₈	1	4	yes	—
209	C₂²×C₄	C₅×M₁₆	1	4	yes	—
210	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
211	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
212	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
213	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
214	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
215	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
216	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
218	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
219	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
220	C₂×C₈	—	2	16	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	2	8	no	—
225	C₂×C₈	C₅×#41	1	8	yes	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	C₂×C₅×C₈	1	8	yes	complemented
228	C₂²⋊C₄	—	2	4	no	—
229	C₂²⋊C₄	C₂×C₅×C₈	1	8	yes	—
230	C₄²	—	2	4	no	—
231	C₄²	—	2	8	no	—
232	C₄²	—	2	8	no	—
233	C₄²	C₅×C₄⋊C₄	1	4	yes	—
234	C₄²	C₅×C₄⋊C₄	1	4	yes	—
235	C₄²	C₂×C₅×C₈	1	8	yes	complemented
236	C₄²	C₂×C₅×D₄	1	2	yes	—
237	C₄²	C₂×C₅×D₄	1	2	yes	—
238	C₄²	C₂×C₅×Q₈	1	2	yes	—
239	C₄²	C₂×C₅×Q₈	1	2	yes	—
240	C₂×C₅	—	2	4	no	—
241	C₂×C₅	#2296	1	1	yes	characteristic
242	C₂×C₅	C₂²×C₄⋊C₈	1	1	yes	characteristic
243	C₂×C₅	C₂×#424	1	1	yes	characteristic
244	C₂×C₅	C₈×#41	1	2	yes	—
245	C₂×C₅	#2965	1	2	yes	—
246	C₂³	—	2	2	no	—
247	C₂³	—	2	2	no	—
248	C₂³	C₂×C₅×C₄⋊C₄	1	2	yes	—
249	C₂³	C₂²×C₅×C₈	1	2	yes	—
250	C₂³	C₂×C₅×M₁₆	1	2	yes	—
251	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	8	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	C₅×C₄⋊C₈	1	16	yes	complemented
263	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
264	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
265	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
266	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
267	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
268	C₂×C₄	C₂²×C₅×C₈	1	8	yes	—
269	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
270	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
271	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
272	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
273	C₂×C₄	C₅×#131	1	4	yes	—
274	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
275	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
276	C₂×C₄	C₂×C₅×#41	1	4	yes	—
277	C₈	—	2	8	no	—
278	C₈	—	4	4	no	—
279	C₅	#21349	1	1	yes	characteristic, direct product factor, Sylow
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	4	no	—
283	C₂²	—	2	4	no	—
284	C₂²	C₂×C₅×C₄⋊C₈	1	4	yes	—
285	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
286	C₂²	C₄×C₅×#41	1	2	yes	—
287	C₂²	C₂³×C₅×C₈	1	1	yes	characteristic
288	C₂²	C₂²×C₅×M₁₆	1	1	yes	characteristic
289	C₂²	C₂×C₅×#131	1	2	yes	—
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	2	8	no	—
293	C₄	—	4	4	no	—
294	C₄	C₂×C₅×C₄⋊C₈	1	4	yes	—
295	C₄	C₂×C₅×C₄⋊C₈	1	4	yes	—
296	C₄	C₅×#424	1	4	yes	—
297	C₄	C₅×#525	1	2	yes	—
298	C₄	C₅×#533	1	2	yes	—
299	C₂	—	2	4	no	—
300	C₂	C₅×#2296	1	1	yes	characteristic
301	C₂	C₂²×C₅×C₄⋊C₈	1	1	yes	characteristic
302	C₂	C₂×C₅×#424	1	1	yes	characteristic
303	C₂	C₅×C₈×#41	1	2	yes	—
304	C₂	C₅×#2965	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₅×#1939	C₂	semidirect
6. C₄×C₅×C₄⋊C₈	C₂	semidirect
26. #21349	C₅	direct (N × H)
35. C₅×#117	C₈	semidirect
61. #1939	C₂×C₅	semidirect
63. C₄×C₄⋊C₈	C₂×C₅	semidirect
88. C₅×C₄⋊C₄	C₂×C₈	semidirect
96. C₄²×C₅	C₂×C₈	semidirect
136. C₂×C₄×C₅	C₄⋊C₈	semidirect
161. #117	C₅×C₈	semidirect
227. C₄⋊C₄	C₂×C₅×C₈	semidirect
235. C₄²	C₂×C₅×C₈	semidirect
262. C₂×C₄	C₅×C₄⋊C₈	semidirect
279. C₅	#21349	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×C₅×#431	2
C₂²×C₄×C₅×C₈	2
C₄×C₅×#117	2
C₅×#1939	2
C₅×#1850	2
C₄×C₅×C₄⋊C₈	2
#21349	5

Subgroups — C5×#21349 #11944069

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21349 #11944069