Subgroups — C₅×#10856 #11933576

Order 1280.

810 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#423	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#423	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×#117	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#344	C₂	1	4	yes	maximal
5	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
6	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
7	C₂×C₅×#117	C₂²	1	2	yes	—
8	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₅×#423	—	2	2	no	—
10	C₅×#423	—	2	1	no	—
11	C₅×#423	—	2	1	no	—
12	C₅×#423	—	2	2	no	—
13	C₅×#423	C₄	1	2	yes	complemented
14	C₅×#423	C₄	1	2	yes	complemented
15	C₂×C₅×C₄⋊C₈	—	2	2	no	—
16	C₂×C₅×C₄⋊C₈	C₄	1	4	yes	complemented
17	C₄×C₅×C₄⋊C₄	—	2	2	no	—
18	C₄×C₅×C₂²⋊C₄	C₄	1	4	yes	—
19	C₅×#376	—	2	1	no	—
20	C₄³×C₅	—	2	1	no	—
21	C₅×#344	—	2	8	no	—
22	#10856	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₂×C₅×M₁₆	—	2	4	no	—
26	C₂×C₅×M₁₆	—	2	2	no	—
27	C₂×C₅×M₁₆	—	2	2	no	—
28	C₂×C₅×M₁₆	—	2	2	no	—
29	C₂×C₅×M₁₆	—	2	4	no	—
30	C₂×C₅×M₁₆	—	2	4	no	—
31	C₂×C₅×M₁₆	—	2	4	no	—
32	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
33	C₂²×C₅×C₈	—	2	2	no	—
34	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
35	C₅×#117	—	2	2	no	—
36	C₅×#117	—	2	2	no	—
37	C₅×#117	D₄	1	4	yes	—
38	C₅×#117	Q₈	1	4	yes	—
39	C₂×C₅×C₄⋊C₄	—	2	2	no	—
40	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₅	—	2	2	no	—
42	C₂×C₄²×C₅	—	2	4	no	—
43	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₅	D₄	1	2	yes	—
47	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₅	D₄	1	2	yes	—
50	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
51	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
52	C₅×C₄⋊C₈	—	2	4	no	—
53	C₅×C₄⋊C₈	—	2	4	no	—
54	C₅×C₄⋊C₈	—	2	4	no	—
55	C₅×C₄⋊C₈	—	2	4	no	—
56	C₅×#95	—	2	4	no	—
57	C₂×#423	C₂×C₅	1	1	yes	characteristic
58	C₂×#423	C₂×C₅	1	1	yes	characteristic
59	C₄×#117	C₂×C₅	1	1	yes	characteristic
60	C₂×#344	C₂×C₅	1	4	yes	—
61	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	—	2	4	no	—
63	C₂²×C₄×C₅	—	4	4	no	—
64	C₂²×C₄×C₅	C₄²	1	2	yes	—
65	C₂²×C₄×C₅	C₄²	1	2	yes	—
66	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
78	C₂²×C₄×C₅	#41	1	2	yes	—
79	C₂²×C₄×C₅	#41	1	2	yes	—
80	C₅×M₁₆	—	2	8	no	—
81	C₅×M₁₆	—	4	4	no	—
82	C₅×M₁₆	—	4	4	no	—
83	C₅×M₁₆	—	4	2	no	—
84	C₅×M₁₆	—	4	2	no	—
85	C₂×C₅×C₈	—	2	4	no	—
86	C₂×C₅×C₈	—	2	4	no	—
87	C₂×C₅×C₈	—	2	8	no	—
88	C₂×C₅×C₈	—	4	2	no	—
89	C₂×C₅×C₈	—	4	2	no	—
90	C₅×C₄⋊C₄	—	2	4	no	—
91	C₅×C₄⋊C₄	—	2	4	no	—
92	C₅×C₂²⋊C₄	—	2	4	no	—
93	C₅×C₂²⋊C₄	C₄⋊C₄	1	8	yes	—
94	C₄²×C₅	—	2	4	no	—
95	C₄²×C₅	—	2	4	no	—
96	C₄²×C₅	—	4	8	no	—
97	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
98	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
99	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
100	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
101	C₄²×C₅	C₄⋊C₄	1	2	yes	—
102	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
103	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
104	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
105	C₄²×C₅	C₄⋊C₄	1	2	yes	—
106	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
107	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
108	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
109	C₂×#117	C₂²×C₅	1	2	yes	—
110	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
111	#423	—	2	2	no	—
112	#423	—	2	2	no	—
113	#423	—	2	1	no	—
114	#423	—	2	1	no	—
115	#423	C₄×C₅	1	2	yes	complemented
116	#423	C₄×C₅	1	2	yes	complemented
117	C₂×C₄⋊C₈	—	2	2	no	—
118	C₂×C₄⋊C₈	C₄×C₅	1	4	yes	complemented
119	C₄×C₄⋊C₄	—	2	2	no	—
120	C₄×C₂²⋊C₄	C₄×C₅	1	4	yes	—
121	#376	—	2	1	no	—
122	C₄³	—	2	1	no	—
123	#344	—	2	8	no	—
124	C₂³×C₅	—	4	2	no	—
125	C₂³×C₅	C₂×C₄²	1	1	yes	characteristic
126	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
129	C₂³×C₅	#117	1	2	yes	—
130	C₂×C₄×C₅	—	2	4	no	—
131	C₂×C₄×C₅	—	2	4	no	—
132	C₂×C₄×C₅	—	2	4	no	—
133	C₂×C₄×C₅	—	2	8	no	—
134	C₂×C₄×C₅	—	4	2	no	—
135	C₂×C₄×C₅	—	4	2	no	—
136	C₂×C₄×C₅	—	4	2	no	—
137	C₂×C₄×C₅	—	4	8	no	—
138	C₂×C₄×C₅	#95	1	2	yes	—
139	C₂×C₄×C₅	#95	1	2	yes	—
140	C₂×C₄×C₅	#95	1	2	yes	—
141	C₂×C₄×C₅	#95	1	2	yes	—
142	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
147	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
148	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
149	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
150	C₂×C₄×C₅	#121	1	2	yes	—
151	C₂×C₄×C₅	#121	1	2	yes	—
152	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
153	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
154	C₂×C₄×C₅	#124	1	1	yes	characteristic
155	C₂×C₄×C₅	#124	1	1	yes	characteristic
156	C₂×C₄×C₅	#125	1	2	yes	—
157	C₂×C₄×C₅	#127	1	1	yes	characteristic
158	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
159	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
160	C₅×C₈	—	4	4	no	—
161	C₅×C₈	—	4	4	no	—
162	C₂³×C₄	C₂×C₄×C₅	1	2	yes	—
163	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
164	C₂×M₁₆	—	2	4	no	—
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	4	no	—
169	C₂×M₁₆	—	2	4	no	—
170	C₂×M₁₆	—	2	4	no	—
171	C₂×M₁₆	C₂×C₄×C₅	1	4	yes	—
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
174	#117	—	2	2	no	—
175	#117	—	2	2	no	—
176	#117	C₅×D₄	1	4	yes	—
177	#117	C₅×Q₈	1	4	yes	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	4	yes	—
180	C₂×C₄²	—	2	2	no	—
181	C₂×C₄²	—	2	4	no	—
182	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
183	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
184	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
185	C₂×C₄²	C₅×D₄	1	2	yes	—
186	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
187	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
188	C₂×C₄²	C₅×D₄	1	2	yes	—
189	C₂×C₄²	C₅×Q₈	1	1	yes	characteristic
190	C₂×C₄²	C₅×Q₈	1	1	yes	characteristic
191	C₄⋊C₈	—	2	4	no	—
192	C₄⋊C₈	—	2	4	no	—
193	C₄⋊C₈	—	2	4	no	—
194	C₄⋊C₈	—	2	4	no	—
195	#95	—	2	4	no	—
196	C₂²×C₅	—	2	2	no	—
197	C₂²×C₅	—	2	2	no	—
198	C₂²×C₅	—	2	2	no	—
199	C₂²×C₅	—	4	2	no	—
200	C₂²×C₅	—	4	2	no	—
201	C₂²×C₅	C₂×#95	1	1	yes	characteristic
202	C₂²×C₅	C₄×C₄⋊C₄	1	2	yes	—
203	C₂²×C₅	#380	1	2	yes	—
204	C₂²×C₅	#382	1	1	yes	characteristic
205	C₂²×C₅	#384	1	1	yes	characteristic
206	C₄×C₅	—	2	2	no	—
207	C₄×C₅	—	2	2	no	—
208	C₄×C₅	—	2	2	no	—
209	C₄×C₅	—	4	8	no	—
210	C₄×C₅	#344	1	4	yes	complemented
211	C₄×C₅	#389	1	2	yes	—
212	C₄×C₅	#389	1	2	yes	—
213	C₄×C₅	#390	1	1	yes	characteristic
214	C₄×C₅	#390	1	1	yes	characteristic
215	C₄×C₅	#391	1	1	yes	characteristic
216	C₄×C₅	#391	1	1	yes	characteristic
217	C₂⁴	C₂²×C₄×C₅	1	1	yes	characteristic
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	4	4	no	—
220	C₂²×C₄	C₄²×C₅	1	2	yes	—
221	C₂²×C₄	C₄²×C₅	1	2	yes	—
222	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
223	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
224	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
225	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
226	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
227	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
232	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
234	C₂²×C₄	C₅×#41	1	2	yes	—
235	C₂²×C₄	C₅×#41	1	2	yes	—
236	M₁₆	—	2	8	no	—
237	M₁₆	—	4	4	no	—
238	M₁₆	—	4	4	no	—
239	M₁₆	—	4	2	no	—
240	M₁₆	—	4	2	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	2	8	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	4	2	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₂²⋊C₄	—	2	4	no	—
249	C₂²⋊C₄	C₅×C₄⋊C₄	1	8	yes	—
250	C₄²	—	2	4	no	—
251	C₄²	—	2	4	no	—
252	C₄²	—	4	8	no	—
253	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
254	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
255	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
256	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
257	C₄²	C₅×C₄⋊C₄	1	2	yes	—
258	C₄²	C₅×C₄⋊C₄	1	1	yes	characteristic
259	C₄²	C₅×C₄⋊C₄	1	1	yes	characteristic
260	C₄²	C₅×C₄⋊C₄	1	1	yes	characteristic
261	C₄²	C₅×C₄⋊C₄	1	2	yes	—
262	C₄²	C₅×C₄⋊C₄	1	1	yes	characteristic
263	C₂×C₅	—	2	2	no	—
264	C₂×C₅	—	2	2	no	—
265	C₂×C₅	—	2	1	no	—
266	C₂×C₅	—	2	1	no	—
267	C₂×C₅	C₄⋊#95	1	1	yes	characteristic
268	C₂×C₅	C₂×#344	1	2	yes	—
269	C₂³	—	4	2	no	—
270	C₂³	C₂×C₄²×C₅	1	1	yes	characteristic
271	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
272	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
273	C₂³	C₂×C₅×C₄⋊C₄	1	2	yes	—
274	C₂³	C₅×#117	1	2	yes	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	8	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	8	no	—
283	C₂×C₄	C₅×#95	1	2	yes	—
284	C₂×C₄	C₅×#95	1	2	yes	—
285	C₂×C₄	C₅×#95	1	2	yes	—
286	C₂×C₄	C₅×#95	1	2	yes	—
287	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
288	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
289	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
290	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
291	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
292	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
293	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
294	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
295	C₂×C₄	C₅×#121	1	2	yes	—
296	C₂×C₄	C₅×#121	1	2	yes	—
297	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
298	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
299	C₂×C₄	C₅×#124	1	1	yes	characteristic
300	C₂×C₄	C₅×#124	1	1	yes	characteristic
301	C₂×C₄	C₅×#125	1	2	yes	—
302	C₂×C₄	C₅×#127	1	1	yes	characteristic
303	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
304	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
305	C₈	—	4	4	no	—
306	C₈	—	4	4	no	—
307	C₅	#10856	1	1	yes	characteristic, direct product factor, Sylow
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	4	2	no	—
312	C₂²	—	4	2	no	—
313	C₂²	C₂×C₅×#95	1	1	yes	characteristic
314	C₂²	C₄×C₅×C₄⋊C₄	1	2	yes	—
315	C₂²	C₅×#380	1	2	yes	—
316	C₂²	C₅×#382	1	1	yes	characteristic
317	C₂²	C₅×#384	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	2	2	no	—
321	C₄	—	4	8	no	—
322	C₄	C₅×#344	1	4	yes	complemented
323	C₄	C₅×#389	1	2	yes	—
324	C₄	C₅×#389	1	2	yes	—
325	C₄	C₅×#390	1	1	yes	characteristic
326	C₄	C₅×#390	1	1	yes	characteristic
327	C₄	C₅×#391	1	1	yes	characteristic
328	C₄	C₅×#391	1	1	yes	characteristic
329	C₂	—	2	2	no	—
330	C₂	—	2	2	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	C₅×C₄⋊#95	1	1	yes	characteristic
334	C₂	C₂×C₅×#344	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	C₂×C₂²⋊C₄
commutator	C₂³	C₂×C₄²×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
socle	C₂²×C₅	C₂×#95
prime radical	C₂⁴×C₅	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₅×#423	C₄	semidirect
14. C₅×#423	C₄	semidirect
16. C₂×C₅×C₄⋊C₈	C₄	semidirect
22. #10856	C₅	direct (N × H)
115. #423	C₄×C₅	semidirect
116. #423	C₄×C₅	semidirect
118. C₂×C₄⋊C₈	C₄×C₅	semidirect
210. C₄×C₅	#344	semidirect
307. C₅	#10856	direct (N × H)
322. C₄	C₅×#344	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#423	2
C₄×C₅×#117	2
C₂×C₅×#344	2
#10856	5

Subgroups — C5×#10856 #11933576

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#10856 #11933576