Subgroups — C₅×#21026 #11943746

Order 1280.

906 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#424	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#405	C₂	1	1	yes	maximal, characteristic
3	C₅×#2285	C₂	1	1	yes	maximal, characteristic
4	C₅×#1917	C₂	1	4	yes	maximal, complemented
5	C₅×#1916	C₂	1	4	yes	maximal, complemented
6	C₅×#1729	C₂	1	1	yes	maximal, characteristic
7	C₅×#1729	C₂	1	1	yes	maximal, characteristic
8	C₅×#1728	C₂	1	1	yes	maximal, characteristic
9	C₂×C₅×#336	C₂	1	1	yes	maximal, characteristic
10	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
12	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₅×#117	C₂²	1	1	yes	characteristic
15	C₂×C₅×#117	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
17	C₅×#424	—	2	4	no	—
18	C₂×C₅×C₄⋊C₈	C₂²	1	4	yes	—
19	C₅×#405	—	2	4	no	—
20	C₂×C₅×#97	C₂²	1	2	yes	—
21	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
22	C₅×#384	C₄	1	2	yes	—
23	C₅×#382	C₄	1	2	yes	—
24	C₅×#380	C₄	1	4	yes	—
25	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
26	C₅×#336	C₂²	1	2	yes	—
27	C₅×#336	C₂²	1	2	yes	—
28	C₅×#336	C₂²	1	4	yes	—
29	C₅×#336	C₂²	1	4	yes	—
30	C₅×#336	C₂²	1	2	yes	—
31	C₅×#336	C₂²	1	2	yes	—
32	#21026	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
34	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
35	C₂×C₅×M₁₆	—	2	2	no	—
36	C₂×C₅×M₁₆	—	2	2	no	—
37	C₂×C₅×M₁₆	—	2	2	no	—
38	C₂×C₅×M₁₆	—	2	2	no	—
39	C₂×C₅×M₁₆	—	2	2	no	—
40	C₂×C₅×M₁₆	D₄	1	4	yes	—
41	C₂²×C₅×C₈	—	2	2	no	—
42	C₂²×C₅×C₈	—	2	2	no	—
43	C₂²×C₅×C₈	—	2	2	no	—
44	C₂²×C₅×C₈	—	2	2	no	—
45	C₂²×C₅×C₈	—	2	2	no	—
46	C₂²×C₅×C₈	Q₈	1	4	yes	—
47	C₂²×C₅×C₈	C₂³	1	2	yes	—
48	C₂²×C₅×C₈	C₂³	1	2	yes	—
49	C₂²×C₅×C₈	C₂³	1	2	yes	—
50	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
51	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
52	C₅×#117	—	2	2	no	—
53	C₅×#117	—	2	2	no	—
54	C₅×#117	—	2	2	no	—
55	C₅×#117	—	2	2	no	—
56	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
57	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
58	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
59	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
60	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
61	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
62	C₂×C₄²×C₅	—	2	2	no	—
63	C₂×C₄²×C₅	—	2	2	no	—
64	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₅	C₂³	1	2	yes	—
67	C₂×C₄²×C₅	C₂³	1	2	yes	—
68	C₂×C₄²×C₅	C₂³	1	2	yes	—
69	C₅×C₄⋊C₈	—	2	8	no	—
70	C₅×#97	—	2	4	no	—
71	C₄×C₅×C₈	—	2	4	no	—
72	C₅×#95	C₂×C₄	1	8	yes	—
73	C₂×#424	C₂×C₅	1	1	yes	characteristic
74	C₂×#405	C₂×C₅	1	1	yes	characteristic
75	#2285	C₂×C₅	1	1	yes	characteristic
76	#1917	C₂×C₅	1	4	yes	complemented
77	#1916	C₂×C₅	1	4	yes	complemented
78	#1729	C₂×C₅	1	1	yes	characteristic
79	#1729	C₂×C₅	1	1	yes	characteristic
80	#1728	C₂×C₅	1	1	yes	characteristic
81	C₂×#336	C₂×C₅	1	1	yes	characteristic
82	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅	—	2	4	no	—
84	C₂²×C₄×C₅	—	2	4	no	—
85	C₂²×C₄×C₅	—	4	2	no	—
86	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
88	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
89	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
93	C₂²×C₄×C₅	#41	1	2	yes	—
94	C₂²×C₄×C₅	#41	1	2	yes	—
95	C₂²×C₄×C₅	#41	1	2	yes	—
96	C₂²×C₄×C₅	#41	1	2	yes	—
97	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
98	C₅×M₁₆	—	2	8	no	—
99	C₅×M₁₆	—	4	4	no	—
100	C₂×C₅×C₈	—	2	8	no	—
101	C₂×C₅×C₈	—	2	1	no	—
102	C₂×C₅×C₈	—	2	1	no	—
103	C₂×C₅×C₈	—	2	1	no	—
104	C₂×C₅×C₈	—	2	1	no	—
105	C₂×C₅×C₈	—	2	2	no	—
106	C₂×C₅×C₈	—	2	2	no	—
107	C₂×C₅×C₈	—	2	2	no	—
108	C₂×C₅×C₈	—	2	2	no	—
109	C₂×C₅×C₈	—	2	2	no	—
110	C₂×C₅×C₈	—	2	2	no	—
111	C₂×C₅×C₈	—	2	2	no	—
112	C₂×C₅×C₈	—	2	2	no	—
113	C₂×C₅×C₈	—	4	4	no	—
114	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
115	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
116	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
117	C₅×C₄⋊C₄	—	2	4	no	—
118	C₅×C₄⋊C₄	—	2	4	no	—
119	C₅×C₄⋊C₄	—	2	8	no	—
120	C₅×C₄⋊C₄	—	4	4	no	—
121	C₅×C₂²⋊C₄	—	2	4	no	—
122	C₅×C₂²⋊C₄	—	2	4	no	—
123	C₄²×C₅	—	2	2	no	—
124	C₄²×C₅	—	2	2	no	—
125	C₄²×C₅	—	2	2	no	—
126	C₄²×C₅	—	2	2	no	—
127	C₄²×C₅	—	2	2	no	—
128	C₄²×C₅	—	2	2	no	—
129	C₄²×C₅	—	2	2	no	—
130	C₄²×C₅	—	2	2	no	—
131	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
132	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
133	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
134	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
135	C₂×#117	C₂²×C₅	1	1	yes	characteristic
136	C₂×#117	C₂²×C₅	1	1	yes	characteristic
137	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
138	#424	—	2	4	no	—
139	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
140	#405	—	2	4	no	—
141	C₂×#97	C₂²×C₅	1	2	yes	—
142	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
143	#384	C₄×C₅	1	2	yes	—
144	#382	C₄×C₅	1	2	yes	—
145	#380	C₄×C₅	1	4	yes	—
146	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	—
147	#336	C₂²×C₅	1	2	yes	—
148	#336	C₂²×C₅	1	2	yes	—
149	#336	C₂²×C₅	1	4	yes	—
150	#336	C₂²×C₅	1	4	yes	—
151	#336	C₂²×C₅	1	2	yes	—
152	#336	C₂²×C₅	1	2	yes	—
153	C₂³×C₅	—	2	2	no	—
154	C₂³×C₅	—	2	2	no	—
155	C₂³×C₅	#117	1	2	yes	—
156	C₂³×C₅	C₄×D₄	1	2	yes	—
157	C₂³×C₅	C₄×Q₈	1	2	yes	—
158	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
159	C₂×C₄×C₅	—	2	2	no	—
160	C₂×C₄×C₅	—	2	2	no	—
161	C₂×C₄×C₅	—	2	2	no	—
162	C₂×C₄×C₅	—	2	4	no	—
163	C₂×C₄×C₅	—	2	4	no	—
164	C₂×C₄×C₅	—	2	4	no	—
165	C₂×C₄×C₅	—	2	4	no	—
166	C₂×C₄×C₅	—	2	2	no	—
167	C₂×C₄×C₅	—	2	4	no	—
168	C₂×C₄×C₅	—	2	4	no	—
169	C₂×C₄×C₅	—	2	2	no	—
170	C₂×C₄×C₅	—	4	8	no	—
171	C₂×C₄×C₅	#117	1	2	yes	—
172	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
173	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
174	C₂×C₄×C₅	C₂²⋊Q₈	1	4	yes	—
175	C₂×C₄×C₅	#123	1	4	yes	—
176	C₂×C₄×C₅	#125	1	4	yes	—
177	C₂×C₄×C₅	#126	1	4	yes	—
178	C₂×C₄×C₅	#131	1	4	yes	—
179	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
180	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
181	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
182	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
183	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
184	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
185	C₅×C₈	—	2	2	no	—
186	C₅×C₈	—	2	2	no	—
187	C₅×C₈	—	2	4	no	—
188	C₅×C₈	—	4	2	no	—
189	C₅×C₈	—	4	2	no	—
190	C₂³×C₄	C₂×C₄×C₅	1	2	yes	—
191	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
192	C₂×M₁₆	—	2	2	no	—
193	C₂×M₁₆	—	2	2	no	—
194	C₂×M₁₆	—	2	2	no	—
195	C₂×M₁₆	—	2	2	no	—
196	C₂×M₁₆	—	2	2	no	—
197	C₂×M₁₆	C₅×D₄	1	4	yes	—
198	C₂²×C₈	—	2	2	no	—
199	C₂²×C₈	—	2	2	no	—
200	C₂²×C₈	—	2	2	no	—
201	C₂²×C₈	—	2	2	no	—
202	C₂²×C₈	—	2	2	no	—
203	C₂²×C₈	C₅×Q₈	1	4	yes	—
204	C₂²×C₈	C₂³×C₅	1	2	yes	—
205	C₂²×C₈	C₂³×C₅	1	2	yes	—
206	C₂²×C₈	C₂³×C₅	1	2	yes	—
207	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
208	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
209	#117	—	2	2	no	—
210	#117	—	2	2	no	—
211	#117	—	2	2	no	—
212	#117	—	2	2	no	—
213	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
214	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
215	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
216	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
217	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	2	yes	—
218	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	2	yes	—
219	C₂×C₄²	—	2	2	no	—
220	C₂×C₄²	—	2	2	no	—
221	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
222	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
223	C₂×C₄²	C₂³×C₅	1	2	yes	—
224	C₂×C₄²	C₂³×C₅	1	2	yes	—
225	C₂×C₄²	C₂³×C₅	1	2	yes	—
226	C₄⋊C₈	—	2	8	no	—
227	#97	—	2	4	no	—
228	C₄×C₈	—	2	4	no	—
229	#95	C₂×C₄×C₅	1	8	yes	—
230	C₂²×C₅	—	2	4	no	—
231	C₂²×C₅	—	2	4	no	—
232	C₂²×C₅	—	2	2	no	—
233	C₂²×C₅	—	2	2	no	—
234	C₂²×C₅	#387	1	4	yes	—
235	C₂²×C₅	C₂×#117	1	1	yes	characteristic
236	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
237	C₂²×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
238	C₂²×C₅	C₂×#131	1	2	yes	—
239	C₂²×C₅	#568	1	2	yes	—
240	C₄×C₅	—	2	4	no	—
241	C₄×C₅	—	4	2	no	—
242	C₄×C₅	—	4	4	no	—
243	C₄×C₅	—	4	4	no	—
244	C₄×C₅	#387	1	4	yes	—
245	C₄×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
246	C₄×C₅	C₂×#123	1	1	yes	characteristic
247	C₄×C₅	C₂×#125	1	1	yes	characteristic
248	C₄×C₅	C₂×#126	1	1	yes	characteristic
249	C₂⁴	C₂²×C₄×C₅	1	1	yes	characteristic
250	C₂²×C₄	—	2	4	no	—
251	C₂²×C₄	—	2	4	no	—
252	C₂²×C₄	—	4	2	no	—
253	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
254	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
255	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
256	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
257	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
258	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
259	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
260	C₂²×C₄	C₅×#41	1	2	yes	—
261	C₂²×C₄	C₅×#41	1	2	yes	—
262	C₂²×C₄	C₅×#41	1	2	yes	—
263	C₂²×C₄	C₅×#41	1	2	yes	—
264	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
265	M₁₆	—	2	8	no	—
266	M₁₆	—	4	4	no	—
267	C₂×C₈	—	2	8	no	—
268	C₂×C₈	—	2	1	no	—
269	C₂×C₈	—	2	1	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	2	1	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	4	4	no	—
281	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
282	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
283	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
284	C₄⋊C₄	—	2	4	no	—
285	C₄⋊C₄	—	2	4	no	—
286	C₄⋊C₄	—	2	8	no	—
287	C₄⋊C₄	—	4	4	no	—
288	C₂²⋊C₄	—	2	4	no	—
289	C₂²⋊C₄	—	2	4	no	—
290	C₄²	—	2	2	no	—
291	C₄²	—	2	2	no	—
292	C₄²	—	2	2	no	—
293	C₄²	—	2	2	no	—
294	C₄²	—	2	2	no	—
295	C₄²	—	2	2	no	—
296	C₄²	—	2	2	no	—
297	C₄²	—	2	2	no	—
298	C₂×C₅	—	2	4	no	—
299	C₂×C₅	C₂×#387	1	1	yes	characteristic
300	C₂×C₅	#2924	1	2	yes	—
301	C₂×C₅	#2935	1	2	yes	—
302	C₂×C₅	#2960	1	1	yes	characteristic
303	C₂×C₅	#2961	1	1	yes	characteristic
304	C₂³	—	2	2	no	—
305	C₂³	—	2	2	no	—
306	C₂³	C₅×#117	1	2	yes	—
307	C₂³	C₄×C₅×D₄	1	2	yes	—
308	C₂³	C₄×C₅×Q₈	1	2	yes	—
309	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	4	no	—
314	C₂×C₄	—	2	4	no	—
315	C₂×C₄	—	2	4	no	—
316	C₂×C₄	—	2	4	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	4	no	—
319	C₂×C₄	—	2	4	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	4	8	no	—
322	C₂×C₄	C₅×#117	1	2	yes	—
323	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
324	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
325	C₂×C₄	C₅×C₂²⋊Q₈	1	4	yes	—
326	C₂×C₄	C₅×#123	1	4	yes	—
327	C₂×C₄	C₅×#125	1	4	yes	—
328	C₂×C₄	C₅×#126	1	4	yes	—
329	C₂×C₄	C₅×#131	1	4	yes	—
330	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
331	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
332	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
333	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
334	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
335	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
336	C₈	—	2	2	no	—
337	C₈	—	2	2	no	—
338	C₈	—	2	4	no	—
339	C₈	—	4	2	no	—
340	C₈	—	4	2	no	—
341	C₅	#21026	1	1	yes	characteristic, direct product factor, Sylow
342	C₂²	—	2	4	no	—
343	C₂²	—	2	4	no	—
344	C₂²	—	2	2	no	—
345	C₂²	—	2	2	no	—
346	C₂²	C₅×#387	1	4	yes	—
347	C₂²	C₂×C₅×#117	1	1	yes	characteristic
348	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
349	C₂²	C₂×C₄×C₅×Q₈	1	1	yes	characteristic
350	C₂²	C₂×C₅×#131	1	2	yes	—
351	C₂²	C₅×#568	1	2	yes	—
352	C₄	—	2	4	no	—
353	C₄	—	4	2	no	—
354	C₄	—	4	4	no	—
355	C₄	—	4	4	no	—
356	C₄	C₅×#387	1	4	yes	—
357	C₄	C₂×C₅×C₂²⋊Q₈	1	1	yes	characteristic
358	C₄	C₂×C₅×#123	1	1	yes	characteristic
359	C₄	C₂×C₅×#125	1	1	yes	characteristic
360	C₄	C₂×C₅×#126	1	1	yes	characteristic
361	C₂	—	2	4	no	—
362	C₂	C₂×C₅×#387	1	1	yes	characteristic
363	C₂	C₅×#2924	1	2	yes	—
364	C₂	C₅×#2935	1	2	yes	—
365	C₂	C₅×#2960	1	1	yes	characteristic
366	C₂	C₅×#2961	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₅×#1917	C₂	semidirect
5. C₅×#1916	C₂	semidirect
32. #21026	C₅	direct (N × H)
76. #1917	C₂×C₅	semidirect
77. #1916	C₂×C₅	semidirect
341. C₅	#21026	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×#424	2
C₂×C₅×#405	2
C₅×#2285	2
C₅×#1917	2
C₅×#1916	2
C₅×#1729	2
C₅×#1728	2
C₂×C₅×#336	2
#21026	5

Subgroups — C5×#21026 #11943746

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21026 #11943746