Subgroups — C₅×#10706 #11933426

Order 1280.

926 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#405	C₂	1	2	yes	maximal
2	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#341	C₂	1	1	yes	maximal, characteristic
4	C₅×#1735	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅×#339	C₂	1	2	yes	maximal
6	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
7	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₅×#117	C₂²	1	2	yes	—
10	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₅×#405	—	2	4	no	—
12	C₅×#405	C₄	1	8	yes	complemented
13	C₂×C₅×#97	—	2	2	no	—
14	C₂×C₄×C₅×C₈	—	2	4	no	—
15	C₂×C₄×C₅×C₈	—	2	2	no	—
16	C₂×C₄×C₅×C₈	C₄	1	4	yes	complemented
17	C₅×#341	C₄	1	4	yes	complemented
18	C₅×#340	C₄	1	4	yes	complemented
19	C₅×#339	C₄	1	8	yes	complemented
20	#10706	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂³×C₄×C₅	—	2	1	no	—
22	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
23	C₂×C₅×M₁₆	—	2	2	no	—
24	C₂×C₅×M₁₆	—	2	2	no	—
25	C₂×C₅×M₁₆	—	2	2	no	—
26	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
27	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
28	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
29	C₂²×C₅×C₈	—	2	2	no	—
30	C₂²×C₅×C₈	—	2	2	no	—
31	C₂²×C₅×C₈	—	2	2	no	—
32	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
33	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
34	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
35	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
36	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
37	C₂²×C₅×C₈	D₄	1	4	yes	—
38	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
39	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
40	C₂²×C₅×C₈	Q₈	1	2	yes	—
41	C₅×#117	—	2	2	no	—
42	C₅×#117	—	2	2	no	—
43	C₅×#117	C₂×C₄	1	4	yes	—
44	C₅×#117	C₂×C₄	1	4	yes	—
45	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
46	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
47	C₂×C₄²×C₅	—	2	2	no	—
48	C₂×C₄²×C₅	—	2	2	no	—
49	C₂×C₄²×C₅	—	2	1	no	—
50	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
52	C₅×#97	—	2	8	no	—
53	C₄×C₅×C₈	—	2	8	no	—
54	C₄×C₅×C₈	—	2	8	no	—
55	C₂×#405	C₂×C₅	1	2	yes	—
56	C₂²×C₄×C₈	C₂×C₅	1	1	yes	characteristic
57	C₂×#341	C₂×C₅	1	1	yes	characteristic
58	#1735	C₂×C₅	1	1	yes	characteristic
59	C₂×#339	C₂×C₅	1	2	yes	—
60	C₂⁴×C₅	#41	1	1	yes	characteristic
61	C₂²×C₄×C₅	—	2	8	no	—
62	C₂²×C₄×C₅	—	2	2	no	—
63	C₂²×C₄×C₅	—	2	4	no	—
64	C₂²×C₄×C₅	—	2	4	no	—
65	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
73	C₂²×C₄×C₅	#41	1	1	yes	characteristic
74	C₂²×C₄×C₅	#41	1	2	yes	—
75	C₅×M₁₆	—	2	4	no	—
76	C₅×M₁₆	—	2	4	no	—
77	C₅×M₁₆	—	2	4	no	—
78	C₅×M₁₆	C₄²	1	8	yes	—
79	C₂×C₅×C₈	—	2	2	no	—
80	C₂×C₅×C₈	—	2	4	no	—
81	C₂×C₅×C₈	—	2	2	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	4	no	—
86	C₂×C₅×C₈	C₄²	1	4	yes	—
87	C₂×C₅×C₈	C₄²	1	4	yes	—
88	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
89	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
90	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
91	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
92	C₂×C₅×C₈	C₄⋊C₄	1	8	yes	—
93	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
94	C₅×C₄⋊C₄	—	2	4	no	—
95	C₅×C₄⋊C₄	C₄²	1	8	yes	—
96	C₅×C₂²⋊C₄	—	2	4	no	—
97	C₅×C₂²⋊C₄	—	2	4	no	—
98	C₄²×C₅	—	2	2	no	—
99	C₄²×C₅	—	2	4	no	—
100	C₄²×C₅	—	2	4	no	—
101	C₄²×C₅	—	2	4	no	—
102	C₄²×C₅	C₄²	1	4	yes	—
103	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
104	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
105	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
106	C₂×#117	C₂²×C₅	1	2	yes	—
107	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
108	#405	—	2	4	no	—
109	#405	C₄×C₅	1	8	yes	complemented
110	C₂×#97	—	2	2	no	—
111	C₂×C₄×C₈	—	2	4	no	—
112	C₂×C₄×C₈	—	2	2	no	—
113	C₂×C₄×C₈	C₄×C₅	1	4	yes	complemented
114	#341	C₄×C₅	1	4	yes	complemented
115	#340	C₄×C₅	1	4	yes	complemented
116	#339	C₄×C₅	1	8	yes	complemented
117	C₂³×C₅	—	2	4	no	—
118	C₂³×C₅	#117	1	2	yes	—
119	C₂³×C₅	#117	1	1	yes	characteristic
120	C₂³×C₅	C₄×D₄	1	2	yes	—
121	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
122	C₂³×C₅	C₄×Q₈	1	1	yes	characteristic
123	C₂×C₄×C₅	—	2	16	no	—
124	C₂×C₄×C₅	—	2	8	no	—
125	C₂×C₄×C₅	—	2	4	no	—
126	C₂×C₄×C₅	—	2	8	no	—
127	C₂×C₄×C₅	—	4	8	no	—
128	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
129	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
131	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
133	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
135	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
136	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₅	#117	1	1	yes	characteristic
138	C₂×C₄×C₅	#117	1	2	yes	—
139	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
141	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
142	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
143	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
144	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
145	C₂×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
146	C₅×C₈	—	2	4	no	—
147	C₅×C₈	—	2	8	no	—
148	C₅×C₈	#95	1	4	yes	—
149	C₅×C₈	#95	1	4	yes	—
150	C₂³×C₄	—	2	1	no	—
151	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
152	C₂×M₁₆	—	2	2	no	—
153	C₂×M₁₆	—	2	2	no	—
154	C₂×M₁₆	—	2	2	no	—
155	C₂×M₁₆	C₂×C₄×C₅	1	4	yes	—
156	C₂×M₁₆	C₂×C₄×C₅	1	4	yes	—
157	C₂×M₁₆	C₂×C₄×C₅	1	4	yes	—
158	C₂²×C₈	—	2	2	no	—
159	C₂²×C₈	—	2	2	no	—
160	C₂²×C₈	—	2	2	no	—
161	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
162	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
163	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
164	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
165	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
166	C₂²×C₈	C₅×D₄	1	4	yes	—
167	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
168	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
169	C₂²×C₈	C₅×Q₈	1	2	yes	—
170	#117	—	2	2	no	—
171	#117	—	2	2	no	—
172	#117	C₂×C₄×C₅	1	4	yes	—
173	#117	C₂×C₄×C₅	1	4	yes	—
174	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
175	C₂×C₂²⋊C₄	—	2	2	no	—
176	C₂×C₄²	—	2	2	no	—
177	C₂×C₄²	—	2	2	no	—
178	C₂×C₄²	—	2	1	no	—
179	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
180	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
181	#97	—	2	8	no	—
182	C₄×C₈	—	2	8	no	—
183	C₄×C₈	—	2	8	no	—
184	C₂²×C₅	—	2	4	no	—
185	C₂²×C₅	—	2	4	no	—
186	C₂²×C₅	—	2	2	no	—
187	C₂²×C₅	—	2	2	no	—
188	C₂²×C₅	#376	1	1	yes	characteristic
189	C₂²×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
190	C₂²×C₅	C₄×C₂²⋊C₄	1	2	yes	—
191	C₂²×C₅	C₄×C₄⋊C₄	1	2	yes	—
192	C₂²×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
193	C₂²×C₅	#443	1	4	yes	—
194	C₄×C₅	—	2	8	no	—
195	C₄×C₅	—	4	8	no	—
196	C₄×C₅	C₄³	1	1	yes	characteristic
197	C₄×C₅	C₂×#95	1	1	yes	characteristic
198	C₄×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
199	C₄×C₅	C₄×C₂²⋊C₄	1	2	yes	—
200	C₄×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
201	C₄×C₅	C₄×C₄⋊C₄	1	2	yes	—
202	C₂⁴	C₅×#41	1	1	yes	characteristic
203	C₂²×C₄	—	2	8	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
208	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
209	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
210	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
211	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
214	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
215	C₂²×C₄	C₅×#41	1	1	yes	characteristic
216	C₂²×C₄	C₅×#41	1	2	yes	—
217	M₁₆	—	2	4	no	—
218	M₁₆	—	2	4	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	C₄²×C₅	1	8	yes	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	C₄²×C₅	1	4	yes	—
229	C₂×C₈	C₄²×C₅	1	4	yes	—
230	C₂×C₈	C₅×C₂²⋊C₄	1	4	yes	—
231	C₂×C₈	C₅×C₂²⋊C₄	1	4	yes	—
232	C₂×C₈	C₅×C₂²⋊C₄	1	4	yes	—
233	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
234	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
235	C₂×C₈	C₅×C₄⋊C₄	1	8	yes	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	C₄²×C₅	1	8	yes	—
238	C₂²⋊C₄	—	2	4	no	—
239	C₂²⋊C₄	—	2	4	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	4	no	—
242	C₄²	—	2	4	no	—
243	C₄²	—	2	4	no	—
244	C₄²	C₄²×C₅	1	4	yes	—
245	C₂×C₅	—	2	1	no	—
246	C₂×C₅	—	2	1	no	—
247	C₂×C₅	—	2	2	no	—
248	C₂×C₅	C₄×#95	1	1	yes	characteristic
249	C₂×C₅	#1764	1	4	yes	—
250	C₂×C₅	#1772	1	2	yes	—
251	C₂³	—	2	4	no	—
252	C₂³	C₅×#117	1	2	yes	—
253	C₂³	C₅×#117	1	1	yes	characteristic
254	C₂³	C₄×C₅×D₄	1	2	yes	—
255	C₂³	C₄×C₅×D₄	1	1	yes	characteristic
256	C₂³	C₄×C₅×Q₈	1	1	yes	characteristic
257	C₂×C₄	—	2	16	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	8	no	—
261	C₂×C₄	—	4	8	no	—
262	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
263	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
264	C₂×C₄	C₂×C₄²×C₅	1	2	yes	—
265	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
266	C₂×C₄	C₂×C₄²×C₅	1	2	yes	—
267	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
268	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
269	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
270	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
271	C₂×C₄	C₅×#117	1	1	yes	characteristic
272	C₂×C₄	C₅×#117	1	2	yes	—
273	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
274	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
275	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
276	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
277	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
278	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
279	C₂×C₄	C₄×C₅×Q₈	1	1	yes	characteristic
280	C₈	—	2	4	no	—
281	C₈	—	2	8	no	—
282	C₈	C₅×#95	1	4	yes	—
283	C₈	C₅×#95	1	4	yes	—
284	C₅	#10706	1	1	yes	characteristic, direct product factor, Sylow
285	C₂²	—	2	4	no	—
286	C₂²	—	2	4	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	C₅×#376	1	1	yes	characteristic
290	C₂²	C₄×C₅×C₂²⋊C₄	1	1	yes	characteristic
291	C₂²	C₄×C₅×C₂²⋊C₄	1	2	yes	—
292	C₂²	C₄×C₅×C₄⋊C₄	1	2	yes	—
293	C₂²	C₄×C₅×C₄⋊C₄	1	1	yes	characteristic
294	C₂²	C₅×#443	1	4	yes	—
295	C₄	—	2	8	no	—
296	C₄	—	4	8	no	—
297	C₄	C₄³×C₅	1	1	yes	characteristic
298	C₄	C₂×C₅×#95	1	1	yes	characteristic
299	C₄	C₄×C₅×C₂²⋊C₄	1	1	yes	characteristic
300	C₄	C₄×C₅×C₂²⋊C₄	1	2	yes	—
301	C₄	C₄×C₅×C₄⋊C₄	1	1	yes	characteristic
302	C₄	C₄×C₅×C₄⋊C₄	1	2	yes	—
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	2	2	no	—
306	C₂	C₄×C₅×#95	1	1	yes	characteristic
307	C₂	C₅×#1764	1	4	yes	—
308	C₂	C₅×#1772	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅×C₈	D₄
commutator	C₄	C₄³×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
socle	C₂³×C₅	C₄×D₄
prime radical	C₂⁴×C₅	#41
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅×C₈ ◁ C₂³×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₅×#405	C₄	semidirect
16. C₂×C₄×C₅×C₈	C₄	semidirect
17. C₅×#341	C₄	semidirect
18. C₅×#340	C₄	semidirect
19. C₅×#339	C₄	semidirect
20. #10706	C₅	direct (N × H)
109. #405	C₄×C₅	semidirect
113. C₂×C₄×C₈	C₄×C₅	semidirect
114. #341	C₄×C₅	semidirect
115. #340	C₄×C₅	semidirect
116. #339	C₄×C₅	semidirect
284. C₅	#10706	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×#405	2
C₂²×C₄×C₅×C₈	2
C₂×C₅×#341	2
C₅×#1735	2
C₂×C₅×#339	2
#10706	5

Subgroups — C5×#10706 #11933426

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#10706 #11933426