Subgroups — C₅×#22689 #11945409

Order 1280.

862 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×#443	C₂	1	1	yes	maximal, characteristic
2	C₅×#2952	C₂	1	2	yes	maximal
3	C₅×#2912	C₂	1	2	yes	maximal, complemented
4	C₅×#1951	C₂	1	2	yes	maximal, complemented
5	C₅×#1859	C₂	1	4	yes	maximal, complemented
6	C₅×#1844	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1815	C₂	1	2	yes	maximal, complemented
8	C₅×#1772	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₅×#135	C₂²	1	1	yes	characteristic
10	C₂×C₅×#131	C₂²	1	2	yes	—
11	C₅×#520	C₂²	1	4	yes	—
12	C₅×#443	C₄	1	8	yes	complemented
13	C₅×#442	—	2	2	no	—
14	C₅×#441	—	2	2	no	—
15	C₄×C₅×SD₁₆	—	2	4	no	—
16	C₂×C₅×#108	C₂²	1	1	yes	characteristic
17	C₂×C₅×#108	C₂²	1	1	yes	characteristic
18	C₅×#428	—	2	4	no	—
19	C₅×#428	C₂²	1	4	yes	—
20	C₅×#427	—	2	2	no	—
21	C₅×#427	C₂²	1	1	yes	characteristic
22	C₂×C₅×#106	—	2	2	no	—
23	C₂×C₅×#106	C₂²	1	2	yes	—
24	C₂×C₅×#106	C₂²	1	2	yes	—
25	C₂×C₅×#104	C₂²	1	2	yes	—
26	C₅×#417	C₂²	1	4	yes	—
27	C₅×#405	C₂²	1	2	yes	—
28	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
29	C₅×#384	C₂²	1	2	yes	—
30	C₅×#343	C₂²	1	4	yes	—
31	C₅×#339	C₂²	1	2	yes	—
32	#22689	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂×C₅×#41	C₂³	1	2	yes	—
34	C₅×#135	C₂×C₄	1	4	yes	—
35	C₅×#135	Q₈	1	4	yes	—
36	C₂×C₅×Q₁₆	D₄	1	1	yes	characteristic
37	C₂×C₅×SD₁₆	D₄	1	2	yes	—
38	C₂×C₅×D₈	D₄	1	1	yes	characteristic
39	C₅×#131	—	2	4	no	—
40	C₅×#131	C₂×C₄	1	8	yes	complemented
41	C₂×C₅×M₁₆	—	2	2	no	—
42	C₂×C₅×M₁₆	C₂³	1	2	yes	—
43	C₂×C₅×M₁₆	C₂³	1	2	yes	—
44	C₂²×C₅×C₈	—	2	2	no	—
45	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
46	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
47	C₄×C₅×Q₈	—	2	4	no	—
48	C₄×C₅×D₄	—	2	4	no	—
49	C₄×C₅×D₄	—	4	4	no	—
50	C₅×#117	—	2	4	no	—
51	C₅×#117	C₂³	1	2	yes	—
52	C₂×C₅×C₄⋊C₄	—	2	4	no	—
53	C₂×C₅×C₄⋊C₄	—	4	2	no	—
54	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
55	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
56	C₅×#108	—	2	2	no	—
57	C₅×#108	C₂×C₄	1	4	yes	complemented
58	C₅×#107	—	2	4	no	—
59	C₅×#107	D₄	1	2	yes	—
60	C₅×#106	—	2	2	no	—
61	C₅×#106	—	2	2	no	—
62	C₅×#106	—	2	2	no	—
63	C₅×#106	—	2	2	no	—
64	C₅×#106	—	2	4	no	—
65	C₅×#106	D₄	1	2	yes	—
66	C₅×#104	C₂×C₄	1	8	yes	complemented
67	C₅×#103	—	2	4	no	—
68	C₅×#102	—	2	4	no	—
69	C₅×#97	—	2	2	no	—
70	C₄×C₅×C₈	—	2	2	no	—
71	C₄×C₅×C₈	C₂×C₄	1	4	yes	complemented
72	C₂×#443	C₂×C₅	1	1	yes	characteristic
73	#2952	C₂×C₅	1	2	yes	—
74	#2912	C₂×C₅	1	2	yes	complemented
75	#1951	C₂×C₅	1	2	yes	complemented
76	#1859	C₂×C₅	1	4	yes	complemented
77	#1844	C₂×C₅	1	1	yes	characteristic, complemented
78	#1815	C₂×C₅	1	2	yes	complemented
79	#1772	C₂×C₅	1	1	yes	characteristic, complemented
80	C₅×#41	—	2	2	no	—
81	C₅×#41	—	2	2	no	—
82	C₅×#41	C₂²×C₄	1	4	yes	—
83	C₅×#41	C₂×Q₈	1	4	yes	—
84	C₂×C₅×Q₈	C₂×D₄	1	2	yes	—
85	C₂×C₅×D₄	—	2	2	no	—
86	C₂×C₅×D₄	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₅	—	2	1	no	—
88	C₂²×C₄×C₅	—	2	2	no	—
89	C₂²×C₄×C₅	—	2	4	no	—
90	C₂²×C₄×C₅	—	4	4	no	—
91	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
92	C₅×Q₁₆	C₄⋊C₄	1	4	yes	complemented
93	C₅×SD₁₆	C₄⋊C₄	1	8	yes	complemented
94	C₅×D₈	C₄⋊C₄	1	4	yes	complemented
95	C₅×M₁₆	—	2	2	no	—
96	C₅×M₁₆	—	2	8	no	—
97	C₅×M₁₆	—	2	2	no	—
98	C₅×M₁₆	—	2	2	no	—
99	C₅×M₁₆	C₂²×C₄	1	4	yes	—
100	C₂×C₅×C₈	—	2	2	no	—
101	C₂×C₅×C₈	—	2	8	no	—
102	C₂×C₅×C₈	—	2	2	no	—
103	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
104	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
105	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
106	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
108	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
109	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
110	C₂×C₅×C₈	#41	1	2	yes	—
111	C₅×C₄⋊C₄	—	2	4	no	—
112	C₅×C₄⋊C₄	—	2	4	no	—
113	C₅×C₄⋊C₄	—	4	4	no	—
114	C₅×C₄⋊C₄	—	8	4	no	—
115	C₅×C₄⋊C₄	C₂×D₄	1	4	yes	—
116	C₅×C₂²⋊C₄	—	2	2	no	—
117	C₅×C₂²⋊C₄	—	4	4	no	—
118	C₄²×C₅	—	2	2	no	—
119	C₄²×C₅	—	2	1	no	—
120	C₄²×C₅	—	4	4	no	—
121	C₄²×C₅	C₂²×C₄	1	2	yes	—
122	C₂×#135	C₂²×C₅	1	1	yes	characteristic
123	C₂×#131	C₂²×C₅	1	2	yes	—
124	#520	C₂²×C₅	1	4	yes	—
125	#443	C₄×C₅	1	8	yes	complemented
126	#442	—	2	2	no	—
127	#441	—	2	2	no	—
128	C₄×SD₁₆	—	2	4	no	—
129	C₂×#108	C₂²×C₅	1	1	yes	characteristic
130	C₂×#108	C₂²×C₅	1	1	yes	characteristic
131	#428	—	2	4	no	—
132	#428	C₂²×C₅	1	4	yes	—
133	#427	—	2	2	no	—
134	#427	C₂²×C₅	1	1	yes	characteristic
135	C₂×#106	—	2	2	no	—
136	C₂×#106	C₂²×C₅	1	2	yes	—
137	C₂×#106	C₂²×C₅	1	2	yes	—
138	C₂×#104	C₂²×C₅	1	2	yes	—
139	#417	C₂²×C₅	1	4	yes	—
140	#405	C₂²×C₅	1	2	yes	—
141	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
142	#384	C₂²×C₅	1	2	yes	—
143	#343	C₂²×C₅	1	4	yes	—
144	#339	C₂²×C₅	1	2	yes	—
145	C₂³×C₅	—	2	2	no	—
146	C₂³×C₅	H(5,𝔽₂)	1	1	yes	characteristic
147	C₅×Q₈	—	2	2	no	—
148	C₅×Q₈	C₂×C₄⋊C₄	1	4	yes	—
149	C₅×D₄	—	2	4	no	—
150	C₅×D₄	—	2	4	no	—
151	C₅×D₄	—	2	2	no	—
152	C₅×D₄	C₂×C₄⋊C₄	1	4	yes	—
153	C₂×C₄×C₅	—	2	4	no	—
154	C₂×C₄×C₅	—	2	2	no	—
155	C₂×C₄×C₅	—	2	2	no	—
156	C₂×C₄×C₅	—	2	4	no	—
157	C₂×C₄×C₅	—	2	4	no	—
158	C₂×C₄×C₅	—	2	4	no	—
159	C₂×C₄×C₅	—	4	4	no	—
160	C₂×C₄×C₅	—	8	4	no	—
161	C₂×C₄×C₅	—	16	4	no	—
162	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
163	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
165	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
166	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
167	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
168	C₅×C₈	—	2	4	no	—
169	C₅×C₈	—	4	2	no	—
170	C₅×C₈	C₂×C₄⋊C₄	1	2	yes	—
171	C₅×C₈	C₂×C₄⋊C₄	1	2	yes	—
172	C₅×C₈	C₄×D₄	1	2	yes	—
173	C₅×C₈	C₄×D₄	1	2	yes	—
174	C₂×#41	C₂³×C₅	1	2	yes	—
175	#135	C₂×C₄×C₅	1	4	yes	—
176	#135	C₅×Q₈	1	4	yes	—
177	C₂×Q₁₆	C₅×D₄	1	1	yes	characteristic
178	C₂×SD₁₆	C₅×D₄	1	2	yes	—
179	C₂×D₈	C₅×D₄	1	1	yes	characteristic
180	#131	—	2	4	no	—
181	#131	C₂×C₄×C₅	1	8	yes	complemented
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	C₂³×C₅	1	2	yes	—
184	C₂×M₁₆	C₂³×C₅	1	2	yes	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
187	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
188	C₄×Q₈	—	2	4	no	—
189	C₄×D₄	—	2	4	no	—
190	C₄×D₄	—	4	4	no	—
191	#117	—	2	4	no	—
192	#117	C₂³×C₅	1	2	yes	—
193	C₂×C₄⋊C₄	—	2	4	no	—
194	C₂×C₄⋊C₄	—	4	2	no	—
195	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
196	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
197	#108	—	2	2	no	—
198	#108	C₂×C₄×C₅	1	4	yes	complemented
199	#107	—	2	4	no	—
200	#107	C₅×D₄	1	2	yes	—
201	#106	—	2	2	no	—
202	#106	—	2	2	no	—
203	#106	—	2	2	no	—
204	#106	—	2	2	no	—
205	#106	—	2	4	no	—
206	#106	C₅×D₄	1	2	yes	—
207	#104	C₂×C₄×C₅	1	8	yes	complemented
208	#103	—	2	4	no	—
209	#102	—	2	4	no	—
210	#97	—	2	2	no	—
211	C₄×C₈	—	2	2	no	—
212	C₄×C₈	C₂×C₄×C₅	1	4	yes	complemented
213	C₂²×C₅	—	2	2	no	—
214	C₂²×C₅	—	2	4	no	—
215	C₂²×C₅	—	4	4	no	—
216	C₂²×C₅	#520	1	1	yes	characteristic
217	C₂²×C₅	D₄²	1	1	yes	characteristic
218	C₂²×C₅	#554	1	1	yes	characteristic
219	C₄×C₅	—	2	4	no	—
220	C₄×C₅	—	2	4	no	—
221	C₄×C₅	—	4	4	no	—
222	C₄×C₅	—	8	4	no	—
223	C₄×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
224	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
225	C₄×C₅	#547	1	1	yes	characteristic
226	C₄×C₅	D₄×Q₈	1	1	yes	characteristic
227	#41	—	2	2	no	—
228	#41	—	2	2	no	—
229	#41	C₂²×C₄×C₅	1	4	yes	—
230	#41	C₂×C₅×Q₈	1	4	yes	—
231	C₂×Q₈	C₂×C₅×D₄	1	2	yes	—
232	C₂×D₄	—	2	2	no	—
233	C₂×D₄	C₂×C₅×D₄	1	2	yes	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	4	4	no	—
238	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
239	Q₁₆	C₅×C₄⋊C₄	1	4	yes	complemented
240	SD₁₆	C₅×C₄⋊C₄	1	8	yes	complemented
241	D₈	C₅×C₄⋊C₄	1	4	yes	complemented
242	M₁₆	—	2	2	no	—
243	M₁₆	—	2	8	no	—
244	M₁₆	—	2	2	no	—
245	M₁₆	—	2	2	no	—
246	M₁₆	C₂²×C₄×C₅	1	4	yes	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	8	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
251	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
252	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
253	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
254	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
255	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
256	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
257	C₂×C₈	C₅×#41	1	2	yes	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	4	4	no	—
261	C₄⋊C₄	—	8	4	no	—
262	C₄⋊C₄	C₂×C₅×D₄	1	4	yes	—
263	C₂²⋊C₄	—	2	2	no	—
264	C₂²⋊C₄	—	4	4	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	2	1	no	—
267	C₄²	—	4	4	no	—
268	C₄²	C₂²×C₄×C₅	1	2	yes	—
269	C₂×C₅	—	2	1	no	—
270	C₂×C₅	—	2	1	no	—
271	C₂×C₅	—	4	4	no	—
272	C₂×C₅	D₄×C₄⋊C₄	1	1	yes	characteristic
273	C₂×C₅	#3291	1	2	yes	—
274	C₂³	—	2	2	no	—
275	C₂³	C₅×H(5,𝔽₂)	1	1	yes	characteristic
276	Q₈	—	2	2	no	—
277	Q₈	C₂×C₅×C₄⋊C₄	1	4	yes	—
278	D₄	—	2	4	no	—
279	D₄	—	2	4	no	—
280	D₄	—	2	2	no	—
281	D₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	—	8	4	no	—
290	C₂×C₄	—	16	4	no	—
291	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
292	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
293	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
294	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
295	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
296	C₂×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
297	C₈	—	2	4	no	—
298	C₈	—	4	2	no	—
299	C₈	C₂×C₅×C₄⋊C₄	1	2	yes	—
300	C₈	C₂×C₅×C₄⋊C₄	1	2	yes	—
301	C₈	C₄×C₅×D₄	1	2	yes	—
302	C₈	C₄×C₅×D₄	1	2	yes	—
303	C₅	#22689	1	1	yes	characteristic, direct product factor, Sylow
304	C₂²	—	2	2	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	4	4	no	—
307	C₂²	C₅×#520	1	1	yes	characteristic
308	C₂²	C₅×D₄²	1	1	yes	characteristic
309	C₂²	C₅×#554	1	1	yes	characteristic
310	C₄	—	2	4	no	—
311	C₄	—	2	4	no	—
312	C₄	—	4	4	no	—
313	C₄	—	8	4	no	—
314	C₄	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
315	C₄	C₂×C₄×C₅×D₄	1	1	yes	characteristic
316	C₄	C₅×#547	1	1	yes	characteristic
317	C₄	C₅×D₄×Q₈	1	1	yes	characteristic
318	C₂	—	2	1	no	—
319	C₂	—	2	1	no	—
320	C₂	—	4	4	no	—
321	C₂	C₅×D₄×C₄⋊C₄	1	1	yes	characteristic
322	C₂	C₅×#3291	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	D₄²
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₂×C₅×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#2912	C₂	semidirect
4. C₅×#1951	C₂	semidirect
5. C₅×#1859	C₂	semidirect
6. C₅×#1844	C₂	semidirect
7. C₅×#1815	C₂	semidirect
8. C₅×#1772	C₂	semidirect
12. C₅×#443	C₄	semidirect
32. #22689	C₅	direct (N × H)
40. C₅×#131	C₂×C₄	semidirect
57. C₅×#108	C₂×C₄	semidirect
66. C₅×#104	C₂×C₄	semidirect
71. C₄×C₅×C₈	C₂×C₄	semidirect
74. #2912	C₂×C₅	semidirect
75. #1951	C₂×C₅	semidirect
76. #1859	C₂×C₅	semidirect
77. #1844	C₂×C₅	semidirect
78. #1815	C₂×C₅	semidirect
79. #1772	C₂×C₅	semidirect
92. C₅×Q₁₆	C₄⋊C₄	semidirect
93. C₅×SD₁₆	C₄⋊C₄	semidirect
94. C₅×D₈	C₄⋊C₄	semidirect
125. #443	C₄×C₅	semidirect
181. #131	C₂×C₄×C₅	semidirect
198. #108	C₂×C₄×C₅	semidirect
207. #104	C₂×C₄×C₅	semidirect
212. C₄×C₈	C₂×C₄×C₅	semidirect
239. Q₁₆	C₅×C₄⋊C₄	semidirect
240. SD₁₆	C₅×C₄⋊C₄	semidirect
241. D₈	C₅×C₄⋊C₄	semidirect
303. C₅	#22689	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×#443	2
C₅×#2952	2
C₅×#2912	2
C₅×#1951	2
C₅×#1859	2
C₅×#1844	2
C₅×#1815	2
C₅×#1772	2
#22689	5

Subgroups — C5×#22689 #11945409

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22689 #11945409