Subgroups — C₅×#10855 #11933575

Order 1280.

802 conjugacy classes of subgroups.

The subgroup lattice has 344 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×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
3	C₅×#2285	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	1	no	—
10	C₅×#423	—	2	1	no	—
11	C₅×#423	—	2	2	no	—
12	C₂×C₅×C₄⋊C₈	C₄	1	4	yes	complemented
13	C₄×C₅×M₁₆	—	2	1	no	—
14	C₄×C₅×M₁₆	—	2	1	no	—
15	C₄×C₅×M₁₆	C₄	1	2	yes	complemented
16	C₄×C₅×M₁₆	C₄	1	2	yes	complemented
17	C₂×C₅×#97	—	2	1	no	—
18	C₂×C₄×C₅×C₈	—	2	1	no	—
19	C₅×#384	—	2	1	no	—
20	C₅×#382	—	2	1	no	—
21	C₅×#380	C₄	1	4	yes	—
22	C₄×C₅×C₄⋊C₄	—	2	2	no	—
23	C₅×#344	—	2	8	no	—
24	#10855	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
27	C₂×C₅×M₁₆	—	2	2	no	—
28	C₂×C₅×M₁₆	—	2	2	no	—
29	C₂×C₅×M₁₆	—	2	2	no	—
30	C₂×C₅×M₁₆	—	2	4	no	—
31	C₂×C₅×M₁₆	—	2	2	no	—
32	C₂×C₅×M₁₆	—	2	2	no	—
33	C₂×C₅×M₁₆	—	2	4	no	—
34	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
35	C₂×C₅×M₁₆	D₄	1	4	yes	—
36	C₂×C₅×M₁₆	Q₈	1	4	yes	—
37	C₂²×C₅×C₈	—	2	2	no	—
38	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
39	C₅×#117	—	2	4	no	—
40	C₅×#117	—	2	4	no	—
41	C₂×C₅×C₄⋊C₄	—	2	4	no	—
42	C₂×C₅×C₄⋊C₄	—	2	2	no	—
43	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₅	—	2	2	no	—
45	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₅	D₄	1	2	yes	—
51	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₅	Q₈	1	2	yes	—
54	C₅×C₄⋊C₈	—	2	4	no	—
55	C₅×C₄⋊C₈	—	2	4	no	—
56	C₅×#97	—	2	2	no	—
57	C₅×#97	—	2	2	no	—
58	C₄×C₅×C₈	—	2	2	no	—
59	C₄×C₅×C₈	—	2	2	no	—
60	C₅×#95	—	2	4	no	—
61	C₂×#423	C₂×C₅	1	1	yes	characteristic
62	C₂×C₄×M₁₆	C₂×C₅	1	1	yes	characteristic
63	#2285	C₂×C₅	1	1	yes	characteristic
64	C₂×#344	C₂×C₅	1	4	yes	—
65	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅	—	2	4	no	—
67	C₂²×C₄×C₅	—	4	4	no	—
68	C₂²×C₄×C₅	C₄²	1	2	yes	—
69	C₂²×C₄×C₅	C₄²	1	2	yes	—
70	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
71	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
72	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
74	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
82	C₂²×C₄×C₅	#41	1	2	yes	—
83	C₂²×C₄×C₅	#41	1	2	yes	—
84	C₅×M₁₆	—	2	4	no	—
85	C₅×M₁₆	—	2	4	no	—
86	C₅×M₁₆	—	2	4	no	—
87	C₅×M₁₆	—	4	4	no	—
88	C₅×M₁₆	—	4	2	no	—
89	C₅×M₁₆	—	4	2	no	—
90	C₅×M₁₆	C₄⋊C₄	1	8	yes	complemented
91	C₂×C₅×C₈	—	2	4	no	—
92	C₂×C₅×C₈	—	2	4	no	—
93	C₂×C₅×C₈	—	2	4	no	—
94	C₂×C₅×C₈	—	2	4	no	—
95	C₂×C₅×C₈	—	2	8	no	—
96	C₅×C₄⋊C₄	—	4	8	no	—
97	C₅×C₄⋊C₄	—	4	4	no	—
98	C₅×C₂²⋊C₄	—	2	8	no	—
99	C₄²×C₅	—	4	2	no	—
100	C₄²×C₅	—	4	2	no	—
101	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
102	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
103	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
104	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
105	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
106	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
107	C₄²×C₅	C₄⋊C₄	1	2	yes	—
108	C₄²×C₅	C₄⋊C₄	1	2	yes	—
109	C₄²×C₅	C₄⋊C₄	1	2	yes	—
110	C₄²×C₅	C₄⋊C₄	1	2	yes	—
111	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
112	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
113	C₂×#117	C₂²×C₅	1	2	yes	—
114	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
115	#423	—	2	1	no	—
116	#423	—	2	1	no	—
117	#423	—	2	2	no	—
118	C₂×C₄⋊C₈	C₄×C₅	1	4	yes	complemented
119	C₄×M₁₆	—	2	1	no	—
120	C₄×M₁₆	—	2	1	no	—
121	C₄×M₁₆	C₄×C₅	1	2	yes	complemented
122	C₄×M₁₆	C₄×C₅	1	2	yes	complemented
123	C₂×#97	—	2	1	no	—
124	C₂×C₄×C₈	—	2	1	no	—
125	#384	—	2	1	no	—
126	#382	—	2	1	no	—
127	#380	C₄×C₅	1	4	yes	—
128	C₄×C₄⋊C₄	—	2	2	no	—
129	#344	—	2	8	no	—
130	C₂³×C₅	—	4	2	no	—
131	C₂³×C₅	C₂×C₄²	1	1	yes	characteristic
132	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂³×C₅	C₄×D₄	1	2	yes	—
135	C₂³×C₅	C₄×Q₈	1	2	yes	—
136	C₂×C₄×C₅	—	2	4	no	—
137	C₂×C₄×C₅	—	2	4	no	—
138	C₂×C₄×C₅	—	2	4	no	—
139	C₂×C₄×C₅	—	4	2	no	—
140	C₂×C₄×C₅	—	4	2	no	—
141	C₂×C₄×C₅	—	4	2	no	—
142	C₂×C₄×C₅	—	4	8	no	—
143	C₂×C₄×C₅	—	4	4	no	—
144	C₂×C₄×C₅	#95	1	2	yes	—
145	C₂×C₄×C₅	#95	1	2	yes	—
146	C₂×C₄×C₅	#95	1	2	yes	—
147	C₂×C₄×C₅	#95	1	2	yes	—
148	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
151	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₂×C₄×C₅	#117	1	2	yes	—
154	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
155	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
156	C₂×C₄×C₅	#121	1	2	yes	—
157	C₂×C₄×C₅	#121	1	2	yes	—
158	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
159	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
160	C₂×C₄×C₅	#124	1	1	yes	characteristic
161	C₂×C₄×C₅	#124	1	1	yes	characteristic
162	C₂×C₄×C₅	#125	1	2	yes	—
163	C₂×C₄×C₅	#127	1	1	yes	characteristic
164	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
165	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
166	C₅×C₈	—	2	8	no	—
167	C₅×C₈	—	4	4	no	—
168	C₂³×C₄	C₂×C₄×C₅	1	2	yes	—
169	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	4	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	4	no	—
177	C₂×M₁₆	C₂×C₄×C₅	1	4	yes	—
178	C₂×M₁₆	C₅×D₄	1	4	yes	—
179	C₂×M₁₆	C₅×Q₈	1	4	yes	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
182	#117	—	2	4	no	—
183	#117	—	2	4	no	—
184	C₂×C₄⋊C₄	—	2	4	no	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	4	yes	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
189	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
190	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
191	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
192	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
193	C₂×C₄²	C₅×D₄	1	2	yes	—
194	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
195	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
196	C₂×C₄²	C₅×Q₈	1	2	yes	—
197	C₄⋊C₈	—	2	4	no	—
198	C₄⋊C₈	—	2	4	no	—
199	#97	—	2	2	no	—
200	#97	—	2	2	no	—
201	C₄×C₈	—	2	2	no	—
202	C₄×C₈	—	2	2	no	—
203	#95	—	2	4	no	—
204	C₂²×C₅	—	2	2	no	—
205	C₂²×C₅	—	2	2	no	—
206	C₂²×C₅	—	2	2	no	—
207	C₂²×C₅	—	4	2	no	—
208	C₂²×C₅	—	4	2	no	—
209	C₂²×C₅	C₂×#95	1	1	yes	characteristic
210	C₂²×C₅	C₄×C₄⋊C₄	1	2	yes	—
211	C₂²×C₅	#389	1	2	yes	—
212	C₂²×C₅	#390	1	1	yes	characteristic
213	C₂²×C₅	#391	1	1	yes	characteristic
214	C₄×C₅	—	2	2	no	—
215	C₄×C₅	—	2	2	no	—
216	C₄×C₅	—	2	2	no	—
217	C₄×C₅	—	8	4	no	—
218	C₄×C₅	#344	1	4	yes	complemented
219	C₄×C₅	#380	1	2	yes	—
220	C₄×C₅	#382	1	1	yes	characteristic
221	C₄×C₅	#384	1	1	yes	characteristic
222	C₄×C₅	#389	1	2	yes	—
223	C₄×C₅	#390	1	1	yes	characteristic
224	C₄×C₅	#391	1	1	yes	characteristic
225	C₂⁴	C₂²×C₄×C₅	1	1	yes	characteristic
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	4	4	no	—
228	C₂²×C₄	C₄²×C₅	1	2	yes	—
229	C₂²×C₄	C₄²×C₅	1	2	yes	—
230	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
231	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
232	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
233	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
234	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
235	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
240	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
242	C₂²×C₄	C₅×#41	1	2	yes	—
243	C₂²×C₄	C₅×#41	1	2	yes	—
244	M₁₆	—	2	4	no	—
245	M₁₆	—	2	4	no	—
246	M₁₆	—	2	4	no	—
247	M₁₆	—	4	4	no	—
248	M₁₆	—	4	2	no	—
249	M₁₆	—	4	2	no	—
250	M₁₆	C₅×C₄⋊C₄	1	8	yes	complemented
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	—	2	4	no	—
255	C₂×C₈	—	2	8	no	—
256	C₄⋊C₄	—	4	8	no	—
257	C₄⋊C₄	—	4	4	no	—
258	C₂²⋊C₄	—	2	8	no	—
259	C₄²	—	4	2	no	—
260	C₄²	—	4	2	no	—
261	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
262	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
263	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
264	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
265	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
266	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
267	C₄²	C₅×C₄⋊C₄	1	2	yes	—
268	C₄²	C₅×C₄⋊C₄	1	2	yes	—
269	C₄²	C₅×C₄⋊C₄	1	2	yes	—
270	C₄²	C₅×C₄⋊C₄	1	2	yes	—
271	C₂×C₅	—	2	1	no	—
272	C₂×C₅	—	2	2	no	—
273	C₂×C₅	—	2	2	no	—
274	C₂×C₅	—	2	1	no	—
275	C₂×C₅	C₄⋊#95	1	1	yes	characteristic
276	C₂×C₅	C₂×#344	1	2	yes	—
277	C₂³	—	4	2	no	—
278	C₂³	C₂×C₄²×C₅	1	1	yes	characteristic
279	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
280	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
281	C₂³	C₄×C₅×D₄	1	2	yes	—
282	C₂³	C₄×C₅×Q₈	1	2	yes	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	8	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	C₅×#95	1	2	yes	—
292	C₂×C₄	C₅×#95	1	2	yes	—
293	C₂×C₄	C₅×#95	1	2	yes	—
294	C₂×C₄	C₅×#95	1	2	yes	—
295	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
296	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
297	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
298	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
299	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
300	C₂×C₄	C₅×#117	1	2	yes	—
301	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
302	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
303	C₂×C₄	C₅×#121	1	2	yes	—
304	C₂×C₄	C₅×#121	1	2	yes	—
305	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
306	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
307	C₂×C₄	C₅×#124	1	1	yes	characteristic
308	C₂×C₄	C₅×#124	1	1	yes	characteristic
309	C₂×C₄	C₅×#125	1	2	yes	—
310	C₂×C₄	C₅×#127	1	1	yes	characteristic
311	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
312	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
313	C₈	—	2	8	no	—
314	C₈	—	4	4	no	—
315	C₅	#10855	1	1	yes	characteristic, direct product factor, Sylow
316	C₂²	—	2	2	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	4	2	no	—
321	C₂²	C₂×C₅×#95	1	1	yes	characteristic
322	C₂²	C₄×C₅×C₄⋊C₄	1	2	yes	—
323	C₂²	C₅×#389	1	2	yes	—
324	C₂²	C₅×#390	1	1	yes	characteristic
325	C₂²	C₅×#391	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	2	no	—
329	C₄	—	8	4	no	—
330	C₄	C₅×#344	1	4	yes	complemented
331	C₄	C₅×#380	1	2	yes	—
332	C₄	C₅×#382	1	1	yes	characteristic
333	C₄	C₅×#384	1	1	yes	characteristic
334	C₄	C₅×#389	1	2	yes	—
335	C₄	C₅×#390	1	1	yes	characteristic
336	C₄	C₅×#391	1	1	yes	characteristic
337	C₂	—	2	1	no	—
338	C₂	—	2	2	no	—
339	C₂	—	2	2	no	—
340	C₂	—	2	1	no	—
341	C₂	C₅×C₄⋊#95	1	1	yes	characteristic
342	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
12. C₂×C₅×C₄⋊C₈	C₄	semidirect
15. C₄×C₅×M₁₆	C₄	semidirect
16. C₄×C₅×M₁₆	C₄	semidirect
24. #10855	C₅	direct (N × H)
90. C₅×M₁₆	C₄⋊C₄	semidirect
118. C₂×C₄⋊C₈	C₄×C₅	semidirect
121. C₄×M₁₆	C₄×C₅	semidirect
122. C₄×M₁₆	C₄×C₅	semidirect
218. C₄×C₅	#344	semidirect
250. M₁₆	C₅×C₄⋊C₄	semidirect
315. C₅	#10855	direct (N × H)
330. 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
C₂×C₄×C₅×M₁₆	2
C₅×#2285	2
C₂×C₅×#344	2
#10855	5

Subgroups — C5×#10855 #11933575

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#10855 #11933575