Subgroups — C₅×#22707 #11945427

Order 1280.

910 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₈×D₄	C₂	1	1	yes	maximal, characteristic
2	C₅×#2354	C₂	1	2	yes	maximal
3	C₅×#1947	C₂	1	2	yes	maximal, complemented
4	C₅×#1854	C₂	1	4	yes	maximal, complemented
5	C₅×#1834	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1808	C₂	1	2	yes	maximal, complemented
7	C₄×C₅×#106	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1735	C₂	1	2	yes	maximal, complemented
9	C₂³×C₅×C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
11	C₂×C₅×#117	C₂²	1	4	yes	—
12	C₅×C₈×D₄	C₄	1	8	yes	complemented
13	C₂×C₅×#107	C₂²	1	4	yes	—
14	C₂×C₅×#106	C₂²	1	2	yes	—
15	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₅×C₂²⋊C₈	C₂²	1	2	yes	—
17	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₅×#390	—	2	2	no	—
19	C₅×#388	—	2	4	no	—
20	C₅×#384	C₂²	1	2	yes	—
21	C₅×#380	C₂²	1	4	yes	—
22	C₄×C₅×C₄⋊C₄	—	2	2	no	—
23	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
24	C₅×#340	C₂²	1	1	yes	characteristic
25	C₅×#340	C₂²	1	1	yes	characteristic
26	C₅×#340	C₂²	1	2	yes	—
27	C₅×#340	C₂²	1	2	yes	—
28	C₅×#340	C₂²	1	4	yes	—
29	#22707	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂²×C₅×D₄	D₄	1	1	yes	characteristic
31	C₂³×C₄×C₅	C₂³	1	2	yes	—
32	C₂²×C₅×C₈	—	4	4	no	—
33	C₂²×C₅×C₈	C₂×C₄	1	8	yes	—
34	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
35	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
36	C₂²×C₅×C₈	C₂³	1	2	yes	—
37	C₄×C₅×D₄	C₂×C₄	1	4	yes	—
38	C₄×C₅×D₄	Q₈	1	4	yes	—
39	C₅×#117	—	2	8	no	—
40	C₅×#117	—	4	4	no	—
41	C₂×C₅×C₄⋊C₄	—	2	2	no	—
42	C₂×C₅×C₄⋊C₄	—	2	2	no	—
43	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
44	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
45	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
46	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
47	C₂×C₅×C₂²⋊C₄	—	2	4	no	—
48	C₂×C₅×C₂²⋊C₄	—	2	4	no	—
49	C₂×C₅×C₂²⋊C₄	D₄	1	2	yes	—
50	C₂×C₄²×C₅	—	2	2	no	—
51	C₂×C₄²×C₅	—	2	4	no	—
52	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
53	C₅×#107	—	4	4	no	—
54	C₅×#106	—	2	4	no	—
55	C₅×C₄⋊C₈	C₂×C₄	1	4	yes	—
56	C₅×C₂²⋊C₈	C₂×C₄	1	8	yes	—
57	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
58	C₅×#95	—	2	4	no	—
59	C₅×#95	—	2	2	no	—
60	C₂×C₈×D₄	C₂×C₅	1	1	yes	characteristic
61	#2354	C₂×C₅	1	2	yes	—
62	#1947	C₂×C₅	1	2	yes	complemented
63	#1854	C₂×C₅	1	4	yes	complemented
64	#1834	C₂×C₅	1	1	yes	characteristic, complemented
65	#1808	C₂×C₅	1	2	yes	complemented
66	C₄×#106	C₂×C₅	1	1	yes	characteristic, complemented
67	#1735	C₂×C₅	1	2	yes	complemented
68	C₂⁴×C₅	C₂×D₄	1	2	yes	—
69	C₂×C₅×D₄	—	2	4	no	—
70	C₂×C₅×D₄	C₄⋊C₄	1	4	yes	—
71	C₂²×C₄×C₅	—	2	4	no	—
72	C₂²×C₄×C₅	—	2	4	no	—
73	C₂²×C₄×C₅	—	2	2	no	—
74	C₂²×C₄×C₅	—	2	4	no	—
75	C₂²×C₄×C₅	—	2	2	no	—
76	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
77	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
81	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
82	C₂×C₅×C₈	—	2	2	no	—
83	C₂×C₅×C₈	—	2	2	no	—
84	C₂×C₅×C₈	—	2	4	no	—
85	C₂×C₅×C₈	—	4	8	no	—
86	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
87	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
88	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
89	C₅×C₄⋊C₄	—	2	2	no	—
90	C₅×C₄⋊C₄	—	2	4	no	—
91	C₅×C₄⋊C₄	—	2	2	no	—
92	C₅×C₄⋊C₄	—	2	4	no	—
93	C₅×C₄⋊C₄	—	2	2	no	—
94	C₅×C₄⋊C₄	—	4	4	no	—
95	C₅×C₄⋊C₄	—	8	2	no	—
96	C₅×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
97	C₅×C₄⋊C₄	#41	1	4	yes	—
98	C₅×C₂²⋊C₄	—	4	8	no	—
99	C₅×C₂²⋊C₄	—	8	4	no	—
100	C₅×C₂²⋊C₄	C₄⋊C₄	1	8	yes	—
101	C₄²×C₅	—	4	8	no	—
102	C₄²×C₅	—	4	4	no	—
103	C₄²×C₅	C₂²×C₄	1	2	yes	—
104	C₄²×C₅	C₂×Q₈	1	2	yes	—
105	C₂³×C₈	C₂²×C₅	1	2	yes	—
106	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
107	C₂×#117	C₂²×C₅	1	4	yes	—
108	C₈×D₄	C₄×C₅	1	8	yes	complemented
109	C₂×#107	C₂²×C₅	1	4	yes	—
110	C₂×#106	C₂²×C₅	1	2	yes	—
111	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
112	C₂×C₂²⋊C₈	C₂²×C₅	1	2	yes	—
113	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
114	#390	—	2	2	no	—
115	#388	—	2	4	no	—
116	#384	C₂²×C₅	1	2	yes	—
117	#380	C₂²×C₅	1	4	yes	—
118	C₄×C₄⋊C₄	—	2	2	no	—
119	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
120	#340	C₂²×C₅	1	1	yes	characteristic
121	#340	C₂²×C₅	1	2	yes	—
122	#340	C₂²×C₅	1	2	yes	—
123	#340	C₂²×C₅	1	4	yes	—
124	#340	C₂²×C₅	1	1	yes	characteristic
125	C₂³×C₅	—	2	4	no	—
126	C₂³×C₅	—	2	4	no	—
127	C₂³×C₅	—	2	2	no	—
128	C₂³×C₅	—	2	2	no	—
129	C₂³×C₅	C₂×C₄⋊C₄	1	4	yes	—
130	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
131	C₅×D₄	—	2	8	no	—
132	C₂×C₄×C₅	—	2	4	no	—
133	C₂×C₄×C₅	—	2	1	no	—
134	C₂×C₄×C₅	—	2	1	no	—
135	C₂×C₄×C₅	—	2	4	no	—
136	C₂×C₄×C₅	—	2	4	no	—
137	C₂×C₄×C₅	—	2	8	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	4	no	—
142	C₂×C₄×C₅	—	4	2	no	—
143	C₂×C₄×C₅	—	4	2	no	—
144	C₂×C₄×C₅	—	4	4	no	—
145	C₂×C₄×C₅	—	4	2	no	—
146	C₂×C₄×C₅	—	4	4	no	—
147	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
148	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
149	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
150	C₂×C₄×C₅	#135	1	4	yes	—
151	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂×#41	1	2	yes	—
154	C₂×C₄×C₅	H(5,𝔽₂)	1	2	yes	—
155	C₅×C₈	—	2	2	no	—
156	C₅×C₈	—	2	2	no	—
157	C₅×C₈	—	2	4	no	—
158	C₂²×D₄	C₅×D₄	1	1	yes	characteristic
159	C₂³×C₄	C₂³×C₅	1	2	yes	—
160	C₂²×C₈	—	4	4	no	—
161	C₂²×C₈	C₂×C₄×C₅	1	8	yes	—
162	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
163	C₂²×C₈	C₂³×C₅	1	2	yes	—
164	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
165	C₄×D₄	C₂×C₄×C₅	1	4	yes	—
166	C₄×D₄	C₅×Q₈	1	4	yes	—
167	#117	—	2	8	no	—
168	#117	—	4	4	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
172	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
173	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
174	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
175	C₂×C₂²⋊C₄	—	2	4	no	—
176	C₂×C₂²⋊C₄	—	2	4	no	—
177	C₂×C₂²⋊C₄	C₅×D₄	1	2	yes	—
178	C₂×C₄²	—	2	4	no	—
179	C₂×C₄²	—	2	2	no	—
180	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
181	#107	—	4	4	no	—
182	#106	—	2	4	no	—
183	C₄⋊C₈	C₂×C₄×C₅	1	4	yes	—
184	C₂²⋊C₈	C₂×C₄×C₅	1	8	yes	—
185	C₄×C₈	C₂×C₄×C₅	1	4	yes	—
186	#95	—	2	4	no	—
187	#95	—	2	2	no	—
188	C₂²×C₅	—	2	4	no	—
189	C₂²×C₅	—	2	4	no	—
190	C₂²×C₅	—	2	4	no	—
191	C₂²×C₅	—	4	4	no	—
192	C₂²×C₅	—	4	4	no	—
193	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
194	C₂²×C₅	#546	1	2	yes	—
195	C₂²×C₅	C₂×#135	1	2	yes	—
196	C₂²×C₅	#576	1	1	yes	characteristic
197	C₂²×C₅	#578	1	1	yes	characteristic
198	C₄×C₅	—	2	4	no	—
199	C₄×C₅	—	2	2	no	—
200	C₄×C₅	—	4	4	no	—
201	C₄×C₅	—	8	4	no	—
202	C₄×C₅	—	8	2	no	—
203	C₄×C₅	#427	1	4	yes	—
204	C₄×C₅	C₄×#41	1	1	yes	characteristic
205	C₄×C₅	#518	1	1	yes	characteristic
206	C₄×C₅	#554	1	2	yes	—
207	C₂⁴	C₂×C₅×D₄	1	2	yes	—
208	C₂×D₄	—	2	4	no	—
209	C₂×D₄	C₅×C₄⋊C₄	1	4	yes	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
216	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
217	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₅×Q₈	1	4	yes	—
220	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	4	8	no	—
225	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
226	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
227	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	4	4	no	—
234	C₄⋊C₄	—	8	2	no	—
235	C₄⋊C₄	C₅×C₄⋊C₄	1	4	yes	—
236	C₄⋊C₄	C₅×#41	1	4	yes	—
237	C₂²⋊C₄	—	4	8	no	—
238	C₂²⋊C₄	—	8	4	no	—
239	C₂²⋊C₄	C₅×C₄⋊C₄	1	8	yes	—
240	C₄²	—	4	8	no	—
241	C₄²	—	4	4	no	—
242	C₄²	C₂²×C₄×C₅	1	2	yes	—
243	C₄²	C₂×C₅×Q₈	1	2	yes	—
244	C₂×C₅	—	4	4	no	—
245	C₂×C₅	#2349	1	1	yes	characteristic
246	C₂×C₅	C₂×#427	1	1	yes	characteristic
247	C₂×C₅	#2913	1	1	yes	characteristic
248	C₂×C₅	#3302	1	2	yes	—
249	C₂×C₅	#3305	1	2	yes	—
250	C₂³	—	2	4	no	—
251	C₂³	—	2	4	no	—
252	C₂³	—	2	2	no	—
253	C₂³	—	2	2	no	—
254	C₂³	C₂×C₅×C₄⋊C₄	1	4	yes	—
255	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
256	D₄	—	2	8	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
273	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
274	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
275	C₂×C₄	C₅×#135	1	4	yes	—
276	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
277	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
278	C₂×C₄	C₂×C₅×#41	1	2	yes	—
279	C₂×C₄	C₅×H(5,𝔽₂)	1	2	yes	—
280	C₈	—	2	2	no	—
281	C₈	—	2	2	no	—
282	C₈	—	2	4	no	—
283	C₅	#22707	1	1	yes	characteristic, direct product factor, Sylow
284	C₂²	—	2	4	no	—
285	C₂²	—	2	4	no	—
286	C₂²	—	2	4	no	—
287	C₂²	—	4	4	no	—
288	C₂²	—	4	4	no	—
289	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
290	C₂²	C₅×#546	1	2	yes	—
291	C₂²	C₂×C₅×#135	1	2	yes	—
292	C₂²	C₅×#576	1	1	yes	characteristic
293	C₂²	C₅×#578	1	1	yes	characteristic
294	C₄	—	2	4	no	—
295	C₄	—	2	2	no	—
296	C₄	—	4	4	no	—
297	C₄	—	8	4	no	—
298	C₄	—	8	2	no	—
299	C₄	C₅×#427	1	4	yes	—
300	C₄	C₄×C₅×#41	1	1	yes	characteristic
301	C₄	C₅×#518	1	1	yes	characteristic
302	C₄	C₅×#554	1	2	yes	—
303	C₂	—	4	4	no	—
304	C₂	C₅×#2349	1	1	yes	characteristic
305	C₂	C₂×C₅×#427	1	1	yes	characteristic
306	C₂	C₅×#2913	1	1	yes	characteristic
307	C₂	C₅×#3302	1	2	yes	—
308	C₂	C₅×#3305	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₂²×C₅×D₄	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄×C₅×D₄ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#1947	C₂	semidirect
4. C₅×#1854	C₂	semidirect
5. C₅×#1834	C₂	semidirect
6. C₅×#1808	C₂	semidirect
7. C₄×C₅×#106	C₂	semidirect
8. C₅×#1735	C₂	semidirect
12. C₅×C₈×D₄	C₄	semidirect
29. #22707	C₅	direct (N × H)
62. #1947	C₂×C₅	semidirect
63. #1854	C₂×C₅	semidirect
64. #1834	C₂×C₅	semidirect
65. #1808	C₂×C₅	semidirect
66. C₄×#106	C₂×C₅	semidirect
67. #1735	C₂×C₅	semidirect
108. C₈×D₄	C₄×C₅	semidirect
283. C₅	#22707	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×C₈×D₄	2
C₅×#2354	2
C₅×#1947	2
C₅×#1854	2
C₅×#1834	2
C₅×#1808	2
C₄×C₅×#106	2
C₅×#1735	2
#22707	5

Subgroups — C5×#22707 #11945427

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22707 #11945427