Subgroups — C₅×#10707 #11933427

Order 1280.

886 conjugacy classes of subgroups.

The subgroup lattice has 304 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×C₅×#97	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	4	no	—
14	C₂×C₅×#97	—	2	2	no	—
15	C₂×C₅×#97	C₄	1	4	yes	complemented
16	C₂×C₄×C₅×C₈	—	2	2	no	—
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	#10707	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	2	yes	—
39	C₂²×C₅×C₈	Q₈	1	2	yes	—
40	C₅×#117	—	2	2	no	—
41	C₅×#117	—	2	2	no	—
42	C₅×#117	C₂×C₄	1	4	yes	—
43	C₅×#117	C₂×C₄	1	4	yes	—
44	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
45	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
46	C₂×C₄²×C₅	—	2	2	no	—
47	C₂×C₄²×C₅	—	2	2	no	—
48	C₂×C₄²×C₅	—	2	1	no	—
49	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
51	C₅×#97	—	2	8	no	—
52	C₅×#97	—	2	8	no	—
53	C₄×C₅×C₈	—	2	8	no	—
54	C₂×#405	C₂×C₅	1	2	yes	—
55	C₂²×#97	C₂×C₅	1	1	yes	characteristic
56	C₂×#341	C₂×C₅	1	1	yes	characteristic
57	#1735	C₂×C₅	1	1	yes	characteristic
58	C₂×#339	C₂×C₅	1	2	yes	—
59	C₂⁴×C₅	#41	1	1	yes	characteristic
60	C₂²×C₄×C₅	—	2	2	no	—
61	C₂²×C₄×C₅	—	2	4	no	—
62	C₂²×C₄×C₅	—	2	4	no	—
63	C₂²×C₄×C₅	—	4	4	no	—
64	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
72	C₂²×C₄×C₅	#41	1	1	yes	characteristic
73	C₂²×C₄×C₅	#41	1	2	yes	—
74	C₅×M₁₆	—	2	4	no	—
75	C₅×M₁₆	—	2	4	no	—
76	C₅×M₁₆	—	2	4	no	—
77	C₅×M₁₆	C₄²	1	8	yes	—
78	C₂×C₅×C₈	—	2	2	no	—
79	C₂×C₅×C₈	—	2	4	no	—
80	C₂×C₅×C₈	—	2	2	no	—
81	C₂×C₅×C₈	—	2	4	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	C₄²	1	4	yes	—
86	C₂×C₅×C₈	C₄²	1	4	yes	—
87	C₂×C₅×C₈	C₂²⋊C₄	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	C₄⋊C₄	1	8	yes	—
91	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
92	C₅×C₄⋊C₄	—	2	4	no	—
93	C₅×C₄⋊C₄	C₄²	1	8	yes	—
94	C₅×C₂²⋊C₄	—	2	4	no	—
95	C₅×C₂²⋊C₄	—	2	4	no	—
96	C₄²×C₅	—	2	2	no	—
97	C₄²×C₅	—	2	4	no	—
98	C₄²×C₅	—	2	4	no	—
99	C₄²×C₅	—	2	4	no	—
100	C₄²×C₅	C₄²	1	4	yes	—
101	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
102	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
103	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
104	C₂×#117	C₂²×C₅	1	2	yes	—
105	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
106	#405	—	2	4	no	—
107	#405	C₄×C₅	1	8	yes	complemented
108	C₂×#97	—	2	4	no	—
109	C₂×#97	—	2	2	no	—
110	C₂×#97	C₄×C₅	1	4	yes	complemented
111	C₂×C₄×C₈	—	2	2	no	—
112	#341	C₄×C₅	1	4	yes	complemented
113	#340	C₄×C₅	1	4	yes	complemented
114	#339	C₄×C₅	1	8	yes	complemented
115	C₂³×C₅	—	2	4	no	—
116	C₂³×C₅	#117	1	2	yes	—
117	C₂³×C₅	#117	1	1	yes	characteristic
118	C₂³×C₅	C₄×D₄	1	2	yes	—
119	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
120	C₂³×C₅	C₄×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₅	—	2	4	no	—
122	C₂×C₄×C₅	—	2	8	no	—
123	C₂×C₄×C₅	—	4	8	no	—
124	C₂×C₄×C₅	—	4	4	no	—
125	C₂×C₄×C₅	—	4	8	no	—
126	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
127	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
128	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
129	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
131	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
133	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
134	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₅	#117	1	1	yes	characteristic
136	C₂×C₄×C₅	#117	1	2	yes	—
137	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
139	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
140	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
141	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
142	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
143	C₂×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
144	C₅×C₈	—	2	4	no	—
145	C₅×C₈	—	2	8	no	—
146	C₅×C₈	#95	1	8	yes	—
147	C₂³×C₄	—	2	1	no	—
148	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
149	C₂×M₁₆	—	2	2	no	—
150	C₂×M₁₆	—	2	2	no	—
151	C₂×M₁₆	—	2	2	no	—
152	C₂×M₁₆	C₂×C₄×C₅	1	4	yes	—
153	C₂×M₁₆	C₂×C₄×C₅	1	4	yes	—
154	C₂×M₁₆	C₂×C₄×C₅	1	4	yes	—
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
159	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
160	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
161	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
162	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
163	C₂²×C₈	C₅×D₄	1	2	yes	—
164	C₂²×C₈	C₅×D₄	1	4	yes	—
165	C₂²×C₈	C₅×Q₈	1	2	yes	—
166	#117	—	2	2	no	—
167	#117	—	2	2	no	—
168	#117	C₂×C₄×C₅	1	4	yes	—
169	#117	C₂×C₄×C₅	1	4	yes	—
170	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
171	C₂×C₂²⋊C₄	—	2	2	no	—
172	C₂×C₄²	—	2	2	no	—
173	C₂×C₄²	—	2	2	no	—
174	C₂×C₄²	—	2	1	no	—
175	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
176	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
177	#97	—	2	8	no	—
178	#97	—	2	8	no	—
179	C₄×C₈	—	2	8	no	—
180	C₂²×C₅	—	2	4	no	—
181	C₂²×C₅	—	2	4	no	—
182	C₂²×C₅	—	2	2	no	—
183	C₂²×C₅	—	2	2	no	—
184	C₂²×C₅	#376	1	1	yes	characteristic
185	C₂²×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
186	C₂²×C₅	C₄×C₂²⋊C₄	1	2	yes	—
187	C₂²×C₅	C₄×C₄⋊C₄	1	2	yes	—
188	C₂²×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
189	C₂²×C₅	#444	1	4	yes	—
190	C₄×C₅	—	4	4	no	—
191	C₄×C₅	—	4	8	no	—
192	C₄×C₅	C₄³	1	1	yes	characteristic
193	C₄×C₅	C₂×#95	1	1	yes	characteristic
194	C₄×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
195	C₄×C₅	C₄×C₂²⋊C₄	1	2	yes	—
196	C₄×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
197	C₄×C₅	C₄×C₄⋊C₄	1	2	yes	—
198	C₂⁴	C₅×#41	1	1	yes	characteristic
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	4	4	no	—
203	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
204	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
205	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
206	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
207	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
210	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
211	C₂²×C₄	C₅×#41	1	1	yes	characteristic
212	C₂²×C₄	C₅×#41	1	2	yes	—
213	M₁₆	—	2	4	no	—
214	M₁₆	—	2	4	no	—
215	M₁₆	—	2	4	no	—
216	M₁₆	C₄²×C₅	1	8	yes	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	C₄²×C₅	1	4	yes	—
225	C₂×C₈	C₄²×C₅	1	4	yes	—
226	C₂×C₈	C₅×C₂²⋊C₄	1	4	yes	—
227	C₂×C₈	C₅×C₂²⋊C₄	1	4	yes	—
228	C₂×C₈	C₅×C₂²⋊C₄	1	4	yes	—
229	C₂×C₈	C₅×C₄⋊C₄	1	4	yes	—
230	C₂×C₈	C₅×C₄⋊C₄	1	8	yes	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	C₄²×C₅	1	8	yes	—
233	C₂²⋊C₄	—	2	4	no	—
234	C₂²⋊C₄	—	2	4	no	—
235	C₄²	—	2	4	no	—
236	C₄²	—	2	2	no	—
237	C₄²	—	2	4	no	—
238	C₄²	—	2	4	no	—
239	C₄²	C₄²×C₅	1	4	yes	—
240	C₂×C₅	—	2	1	no	—
241	C₂×C₅	—	2	1	no	—
242	C₂×C₅	—	2	2	no	—
243	C₂×C₅	C₄×#95	1	1	yes	characteristic
244	C₂×C₅	#1765	1	4	yes	—
245	C₂×C₅	#1773	1	2	yes	—
246	C₂³	—	2	4	no	—
247	C₂³	C₅×#117	1	2	yes	—
248	C₂³	C₅×#117	1	1	yes	characteristic
249	C₂³	C₄×C₅×D₄	1	2	yes	—
250	C₂³	C₄×C₅×D₄	1	1	yes	characteristic
251	C₂³	C₄×C₅×Q₈	1	1	yes	characteristic
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	8	no	—
254	C₂×C₄	—	4	8	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	8	no	—
257	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
258	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
259	C₂×C₄	C₂×C₄²×C₅	1	2	yes	—
260	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
261	C₂×C₄	C₂×C₄²×C₅	1	2	yes	—
262	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
263	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
264	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
265	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
266	C₂×C₄	C₅×#117	1	1	yes	characteristic
267	C₂×C₄	C₅×#117	1	2	yes	—
268	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
269	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
270	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
271	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
272	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
273	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
274	C₂×C₄	C₄×C₅×Q₈	1	1	yes	characteristic
275	C₈	—	2	4	no	—
276	C₈	—	2	8	no	—
277	C₈	C₅×#95	1	8	yes	—
278	C₅	#10707	1	1	yes	characteristic, direct product factor, Sylow
279	C₂²	—	2	4	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	C₅×#376	1	1	yes	characteristic
284	C₂²	C₄×C₅×C₂²⋊C₄	1	1	yes	characteristic
285	C₂²	C₄×C₅×C₂²⋊C₄	1	2	yes	—
286	C₂²	C₄×C₅×C₄⋊C₄	1	2	yes	—
287	C₂²	C₄×C₅×C₄⋊C₄	1	1	yes	characteristic
288	C₂²	C₅×#444	1	4	yes	—
289	C₄	—	4	4	no	—
290	C₄	—	4	8	no	—
291	C₄	C₄³×C₅	1	1	yes	characteristic
292	C₄	C₂×C₅×#95	1	1	yes	characteristic
293	C₄	C₄×C₅×C₂²⋊C₄	1	1	yes	characteristic
294	C₄	C₄×C₅×C₂²⋊C₄	1	2	yes	—
295	C₄	C₄×C₅×C₄⋊C₄	1	1	yes	characteristic
296	C₄	C₄×C₅×C₄⋊C₄	1	2	yes	—
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	—	2	2	no	—
300	C₂	C₄×C₅×#95	1	1	yes	characteristic
301	C₂	C₅×#1765	1	4	yes	—
302	C₂	C₅×#1773	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×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×C₄×C₅ ◁ 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
15. C₂×C₅×#97	C₄	semidirect
17. C₅×#341	C₄	semidirect
18. C₅×#340	C₄	semidirect
19. C₅×#339	C₄	semidirect
20. #10707	C₅	direct (N × H)
107. #405	C₄×C₅	semidirect
110. C₂×#97	C₄×C₅	semidirect
112. #341	C₄×C₅	semidirect
113. #340	C₄×C₅	semidirect
114. #339	C₄×C₅	semidirect
278. C₅	#10707	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×C₅×#97	2
C₂×C₅×#341	2
C₅×#1735	2
C₂×C₅×#339	2
#10707	5

Subgroups — C5×#10707 #11933427

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#10707 #11933427