Subgroups — C₅×#20671 #11943391

Order 1280.

1324 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#568	C₂	1	1	yes	maximal, characteristic
2	C₅×#2911	C₂	1	2	yes	maximal, complemented
3	C₂×C₅×#410	C₂	1	2	yes	maximal
4	C₅×#1812	C₂	1	8	yes	maximal, complemented
5	C₂×C₅×H(3,ℤ/4)	C₂	1	2	yes	maximal, complemented
6	C₂²×C₅×#41	C₂²	1	1	yes	characteristic
7	C₅×#568	—	2	8	no	—
8	C₂×C₅×#131	C₄	1	4	yes	complemented
9	C₂×C₅×#131	C₂²	1	4	yes	—
10	C₂²×C₅×M₁₆	C₄	1	4	yes	complemented
11	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₅×#117	C₂²	1	4	yes	—
14	C₅×#428	—	2	8	no	—
15	C₅×#427	C₂²	1	8	yes	complemented
16	C₅×#410	—	2	4	no	—
17	C₅×#410	C₂²	1	8	yes	—
18	C₂×C₅×#99	—	2	4	no	—
19	C₅×H(3,ℤ/4)	C₂²	1	8	yes	complemented
20	#20671	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂×C₅×#41	—	2	4	no	—
22	C₂×C₅×#41	—	2	4	no	—
23	C₂×C₅×#41	C₂×C₄	1	2	yes	—
24	C₂×C₅×#41	D₄	1	2	yes	—
25	C₂×C₅×#41	Q₈	1	2	yes	—
26	C₂×C₅×#41	C₂³	1	2	yes	—
27	C₂²×C₅×Q₈	Q₈	1	1	yes	characteristic
28	C₂²×C₅×D₄	D₄	1	2	yes	—
29	C₂²×C₅×D₄	Q₈	1	1	yes	characteristic
30	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
31	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₅×#131	—	2	16	no	—
33	C₅×#131	—	2	16	no	—
34	C₂×C₅×M₁₆	—	2	4	no	—
35	C₂×C₅×M₁₆	—	2	16	no	—
36	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
37	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
38	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
39	C₂×C₅×M₁₆	C₂×C₄	1	8	yes	complemented
40	C₂×C₅×M₁₆	D₄	1	8	yes	—
41	C₂×C₅×M₁₆	C₂³	1	2	yes	—
42	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
43	C₂²×C₅×C₈	C₂×C₄	1	8	yes	complemented
44	C₂²×C₅×C₈	C₂³	1	2	yes	—
45	C₅×#117	—	2	4	no	—
46	C₅×#117	—	2	4	no	—
47	C₅×#117	—	2	4	no	—
48	C₅×#117	C₂³	1	8	yes	—
49	C₂×C₅×C₄⋊C₄	—	2	4	no	—
50	C₂×C₅×C₂²⋊C₄	—	2	4	no	—
51	C₂×C₄²×C₅	—	2	4	no	—
52	C₅×#107	—	2	8	no	—
53	C₅×#106	—	2	8	no	—
54	C₅×#99	—	2	16	no	—
55	C₂×#568	C₂×C₅	1	1	yes	characteristic
56	#2911	C₂×C₅	1	2	yes	complemented
57	C₂×#410	C₂×C₅	1	2	yes	—
58	#1812	C₂×C₅	1	8	yes	complemented
59	C₂×H(3,ℤ/4)	C₂×C₅	1	2	yes	complemented
60	C₂⁴×C₅	C₄⋊C₄	1	2	yes	—
61	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
62	C₅×#41	—	2	4	no	—
63	C₅×#41	—	2	4	no	—
64	C₅×#41	—	2	8	no	—
65	C₅×#41	—	2	8	no	—
66	C₅×#41	—	2	4	no	—
67	C₅×#41	—	2	4	no	—
68	C₂×C₅×Q₈	—	2	4	no	—
69	C₂×C₅×Q₈	C₄⋊C₄	1	2	yes	—
70	C₂×C₅×Q₈	C₂×Q₈	1	2	yes	—
71	C₂×C₅×D₄	—	2	8	no	—
72	C₂×C₅×D₄	—	2	4	no	—
73	C₂×C₅×D₄	C₄⋊C₄	1	4	yes	—
74	C₂×C₅×D₄	C₄⋊C₄	1	2	yes	—
75	C₂×C₅×D₄	C₂×D₄	1	4	yes	—
76	C₂×C₅×D₄	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₅	—	2	4	no	—
78	C₂²×C₄×C₅	—	2	4	no	—
79	C₂²×C₄×C₅	—	4	4	no	—
80	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
81	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
82	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
83	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
86	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
88	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
89	C₂²×C₄×C₅	#41	1	4	yes	—
90	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
91	C₅×M₁₆	—	2	4	no	—
92	C₅×M₁₆	—	2	4	no	—
93	C₅×M₁₆	—	2	16	no	—
94	C₅×M₁₆	C₂²⋊C₄	1	8	yes	—
95	C₅×M₁₆	C₂²⋊C₄	1	8	yes	—
96	C₂×C₅×C₈	—	2	4	no	—
97	C₂×C₅×C₈	—	2	16	no	—
98	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
99	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
100	C₂×C₅×C₈	C₂×D₄	1	8	yes	—
101	C₅×C₄⋊C₄	—	2	8	no	—
102	C₅×C₄⋊C₄	—	4	4	no	—
103	C₅×C₂²⋊C₄	—	2	8	no	—
104	C₅×C₂²⋊C₄	—	4	4	no	—
105	C₄²×C₅	—	2	8	no	—
106	C₄²×C₅	—	4	4	no	—
107	C₂²×#41	C₂²×C₅	1	1	yes	characteristic
108	#568	—	2	8	no	—
109	C₂×#131	C₄×C₅	1	4	yes	complemented
110	C₂×#131	C₂²×C₅	1	4	yes	—
111	C₂²×M₁₆	C₄×C₅	1	4	yes	complemented
112	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
113	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
114	C₂×#117	C₂²×C₅	1	4	yes	—
115	#428	—	2	8	no	—
116	#427	C₂²×C₅	1	8	yes	complemented
117	#410	—	2	4	no	—
118	#410	C₂²×C₅	1	8	yes	—
119	C₂×#99	—	2	4	no	—
120	H(3,ℤ/4)	C₂²×C₅	1	8	yes	complemented
121	C₂³×C₅	—	2	4	no	—
122	C₂³×C₅	—	8	1	no	—
123	C₂³×C₅	—	8	2	no	—
124	C₂³×C₅	C₂×C₄⋊C₄	1	4	yes	—
125	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂³×C₅	#121	1	4	yes	—
128	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
129	C₅×Q₈	—	2	4	no	—
130	C₅×Q₈	—	2	4	no	—
131	C₅×D₄	—	2	8	no	—
132	C₅×D₄	—	2	8	no	—
133	C₅×D₄	—	2	4	no	—
134	C₅×D₄	—	2	4	no	—
135	C₂×C₄×C₅	—	2	4	no	—
136	C₂×C₄×C₅	—	2	4	no	—
137	C₂×C₄×C₅	—	2	4	no	—
138	C₂×C₄×C₅	—	2	8	no	—
139	C₂×C₄×C₅	—	4	8	no	—
140	C₂×C₄×C₅	—	8	8	no	—
141	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
142	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
143	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
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	#117	1	2	yes	—
147	C₂×C₄×C₅	#117	1	2	yes	—
148	C₂×C₄×C₅	#121	1	4	yes	—
149	C₂×C₄×C₅	C₂²⋊Q₈	1	4	yes	—
150	C₂×C₄×C₅	C₂²⋊Q₈	1	4	yes	—
151	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
153	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂×#41	1	2	yes	—
155	C₅×C₈	—	2	4	no	—
156	C₅×C₈	C₂×C₂²⋊C₄	1	8	yes	—
157	C₂×#41	—	2	4	no	—
158	C₂×#41	—	2	4	no	—
159	C₂×#41	C₂×C₄×C₅	1	2	yes	—
160	C₂×#41	C₅×D₄	1	2	yes	—
161	C₂×#41	C₅×Q₈	1	2	yes	—
162	C₂×#41	C₂³×C₅	1	2	yes	—
163	C₂²×Q₈	C₅×Q₈	1	1	yes	characteristic
164	C₂²×D₄	C₅×D₄	1	2	yes	—
165	C₂²×D₄	C₅×Q₈	1	1	yes	characteristic
166	C₂³×C₄	C₂×C₄×C₅	1	2	yes	—
167	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
168	#131	—	2	16	no	—
169	#131	—	2	16	no	—
170	C₂×M₁₆	—	2	4	no	—
171	C₂×M₁₆	—	2	16	no	—
172	C₂×M₁₆	C₂×C₄×C₅	1	2	yes	—
173	C₂×M₁₆	C₂×C₄×C₅	1	2	yes	—
174	C₂×M₁₆	C₂×C₄×C₅	1	2	yes	—
175	C₂×M₁₆	C₂×C₄×C₅	1	8	yes	complemented
176	C₂×M₁₆	C₅×D₄	1	8	yes	—
177	C₂×M₁₆	C₂³×C₅	1	2	yes	—
178	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
179	C₂²×C₈	C₂×C₄×C₅	1	8	yes	complemented
180	C₂²×C₈	C₂³×C₅	1	2	yes	—
181	#117	—	2	4	no	—
182	#117	—	2	4	no	—
183	#117	—	2	4	no	—
184	#117	C₂³×C₅	1	8	yes	—
185	C₂×C₄⋊C₄	—	2	4	no	—
186	C₂×C₂²⋊C₄	—	2	4	no	—
187	C₂×C₄²	—	2	4	no	—
188	#107	—	2	8	no	—
189	#106	—	2	8	no	—
190	#99	—	2	16	no	—
191	C₂²×C₅	—	2	4	no	—
192	C₂²×C₅	—	4	4	no	—
193	C₂²×C₅	—	4	1	no	—
194	C₂²×C₅	—	4	1	no	—
195	C₂²×C₅	—	4	1	no	—
196	C₂²×C₅	—	4	4	no	—
197	C₂²×C₅	—	8	4	no	—
198	C₂²×C₅	#380	1	2	yes	—
199	C₂²×C₅	#380	1	2	yes	—
200	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
201	C₂²×C₅	C₂×#121	1	2	yes	—
202	C₄×C₅	—	2	4	no	—
203	C₄×C₅	—	8	8	no	—
204	C₄×C₅	#380	1	2	yes	—
205	C₄×C₅	#380	1	2	yes	—
206	C₄×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
207	C₄×C₅	C₂×#117	1	1	yes	characteristic
208	C₄×C₅	C₂×C₂²⋊Q₈	1	2	yes	—
209	C₂⁴	C₅×C₄⋊C₄	1	2	yes	—
210	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
211	#41	—	2	4	no	—
212	#41	—	2	4	no	—
213	#41	—	2	8	no	—
214	#41	—	2	8	no	—
215	#41	—	2	4	no	—
216	#41	—	2	4	no	—
217	C₂×Q₈	—	2	4	no	—
218	C₂×Q₈	C₅×C₄⋊C₄	1	2	yes	—
219	C₂×Q₈	C₂×C₅×Q₈	1	2	yes	—
220	C₂×D₄	—	2	8	no	—
221	C₂×D₄	—	2	4	no	—
222	C₂×D₄	C₅×C₄⋊C₄	1	4	yes	—
223	C₂×D₄	C₅×C₄⋊C₄	1	2	yes	—
224	C₂×D₄	C₂×C₅×D₄	1	4	yes	—
225	C₂×D₄	C₂×C₅×Q₈	1	2	yes	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	4	4	no	—
229	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
230	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
231	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
232	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
233	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
234	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
235	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
238	C₂²×C₄	C₅×#41	1	4	yes	—
239	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
240	M₁₆	—	2	4	no	—
241	M₁₆	—	2	4	no	—
242	M₁₆	—	2	16	no	—
243	M₁₆	C₅×C₂²⋊C₄	1	8	yes	—
244	M₁₆	C₅×C₂²⋊C₄	1	8	yes	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	2	16	no	—
247	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
248	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
249	C₂×C₈	C₂×C₅×D₄	1	8	yes	—
250	C₄⋊C₄	—	2	8	no	—
251	C₄⋊C₄	—	4	4	no	—
252	C₂²⋊C₄	—	2	8	no	—
253	C₂²⋊C₄	—	4	4	no	—
254	C₄²	—	2	8	no	—
255	C₄²	—	4	4	no	—
256	C₂×C₅	—	2	2	no	—
257	C₂×C₅	—	2	1	no	—
258	C₂×C₅	—	2	2	no	—
259	C₂×C₅	—	2	2	no	—
260	C₂×C₅	—	4	4	no	—
261	C₂×C₅	C₂×#380	1	1	yes	characteristic
262	C₂³	—	2	4	no	—
263	C₂³	—	8	1	no	—
264	C₂³	—	8	2	no	—
265	C₂³	C₂×C₅×C₄⋊C₄	1	4	yes	—
266	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
267	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
268	C₂³	C₅×#121	1	4	yes	—
269	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
270	Q₈	—	2	4	no	—
271	Q₈	—	2	4	no	—
272	D₄	—	2	8	no	—
273	D₄	—	2	8	no	—
274	D₄	—	2	4	no	—
275	D₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	8	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	4	8	no	—
281	C₂×C₄	—	8	8	no	—
282	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
283	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
284	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
285	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
286	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
287	C₂×C₄	C₅×#117	1	2	yes	—
288	C₂×C₄	C₅×#117	1	2	yes	—
289	C₂×C₄	C₅×#121	1	4	yes	—
290	C₂×C₄	C₅×C₂²⋊Q₈	1	4	yes	—
291	C₂×C₄	C₅×C₂²⋊Q₈	1	4	yes	—
292	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
293	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
294	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
295	C₂×C₄	C₂×C₅×#41	1	2	yes	—
296	C₈	—	2	4	no	—
297	C₈	C₂×C₅×C₂²⋊C₄	1	8	yes	—
298	C₅	#20671	1	1	yes	characteristic, direct product factor, Sylow
299	C₂²	—	2	4	no	—
300	C₂²	—	4	4	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	4	1	no	—
303	C₂²	—	4	1	no	—
304	C₂²	—	4	4	no	—
305	C₂²	—	8	4	no	—
306	C₂²	C₅×#380	1	2	yes	—
307	C₂²	C₅×#380	1	2	yes	—
308	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
309	C₂²	C₂×C₅×#121	1	2	yes	—
310	C₄	—	2	4	no	—
311	C₄	—	8	8	no	—
312	C₄	C₅×#380	1	2	yes	—
313	C₄	C₅×#380	1	2	yes	—
314	C₄	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
315	C₄	C₂×C₅×#117	1	1	yes	characteristic
316	C₄	C₂×C₅×C₂²⋊Q₈	1	2	yes	—
317	C₂	—	2	2	no	—
318	C₂	—	2	1	no	—
319	C₂	—	2	2	no	—
320	C₂	—	2	2	no	—
321	C₂	—	4	4	no	—
322	C₂	C₂×C₅×#380	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	C₂×#117
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂×C₅	C₂×#380
prime radical	C₂²×C₅×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₅×M₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2911	C₂	semidirect
4. C₅×#1812	C₂	semidirect
5. C₂×C₅×H(3,ℤ/4)	C₂	semidirect
8. C₂×C₅×#131	C₄	semidirect
10. C₂²×C₅×M₁₆	C₄	semidirect
15. C₅×#427	C₂²	semidirect
19. C₅×H(3,ℤ/4)	C₂²	semidirect
20. #20671	C₅	direct (N × H)
39. C₂×C₅×M₁₆	C₂×C₄	semidirect
43. C₂²×C₅×C₈	C₂×C₄	semidirect
56. #2911	C₂×C₅	semidirect
58. #1812	C₂×C₅	semidirect
59. C₂×H(3,ℤ/4)	C₂×C₅	semidirect
109. C₂×#131	C₄×C₅	semidirect
111. C₂²×M₁₆	C₄×C₅	semidirect
116. #427	C₂²×C₅	semidirect
120. H(3,ℤ/4)	C₂²×C₅	semidirect
175. C₂×M₁₆	C₂×C₄×C₅	semidirect
179. C₂²×C₈	C₂×C₄×C₅	semidirect
298. C₅	#20671	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×#568	2
C₅×#2911	2
C₂×C₅×#410	2
C₅×#1812	2
C₂×C₅×H(3,ℤ/4)	2
#20671	5

Subgroups — C5×#20671 #11943391

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#20671 #11943391