Subgroups — C₅×#22241 #11944961

Order 1280.

694 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#446	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
3	C₅×#2399	C₂	1	1	yes	maximal, characteristic
4	C₅×#1985	C₂	1	1	yes	maximal, characteristic
5	C₅×#1985	C₂	1	1	yes	maximal, characteristic
6	C₅×#1917	C₂	1	4	yes	maximal
7	C₅×#1916	C₂	1	4	yes	maximal
8	C₅×#1832	C₂	1	1	yes	maximal, characteristic
9	C₅×#1746	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
12	C₅×#446	—	2	4	no	—
13	C₅×C₈×Q₈	—	2	4	no	—
14	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂×C₅×#97	C₂²	1	2	yes	—
19	C₂×C₅×#97	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
21	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
22	C₅×#398	C₄	1	2	yes	—
23	C₅×#398	C₄	1	2	yes	—
24	C₅×#395	C₄	1	2	yes	—
25	C₅×#395	C₄	1	2	yes	—
26	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
27	C₅×#376	C₂²	1	1	yes	characteristic
28	C₅×#336	C₂²	1	4	yes	—
29	C₅×#336	C₂²	1	2	yes	—
30	C₅×#336	C₂²	1	4	yes	—
31	C₅×#336	C₂²	1	4	yes	—
32	C₅×#336	C₂²	1	2	yes	—
33	#22241	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂²×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₅×C₈	C₂³	1	2	yes	—
37	C₂²×C₅×C₈	C₂³	1	2	yes	—
38	C₂²×C₅×C₈	C₂³	1	2	yes	—
39	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
40	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
41	C₄×C₅×Q₈	—	2	2	no	—
42	C₄×C₅×Q₈	—	2	2	no	—
43	C₄×C₅×Q₈	—	2	2	no	—
44	C₄×C₅×Q₈	—	2	2	no	—
45	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
46	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
47	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
48	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
49	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
50	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
51	C₂×C₄²×C₅	C₂³	1	2	yes	—
52	C₂×C₄²×C₅	C₂³	1	2	yes	—
53	C₂×C₄²×C₅	C₂³	1	2	yes	—
54	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
55	C₅×C₄⋊C₈	—	2	4	no	—
56	C₅×C₄⋊C₈	—	2	4	no	—
57	C₅×C₄⋊C₈	—	2	2	no	—
58	C₅×C₄⋊C₈	—	2	2	no	—
59	C₅×#97	—	2	4	no	—
60	C₅×#97	—	2	2	no	—
61	C₄×C₅×C₈	—	2	4	no	—
62	C₄×C₅×C₈	D₄	1	4	yes	—
63	C₅×#95	C₂×C₄	1	2	yes	—
64	C₅×#95	C₂×C₄	1	2	yes	—
65	C₅×#95	C₂×C₄	1	4	yes	—
66	C₅×#95	C₂×C₄	1	4	yes	—
67	C₂×#446	C₂×C₅	1	1	yes	characteristic
68	C₂×C₈×Q₈	C₂×C₅	1	1	yes	characteristic
69	#2399	C₂×C₅	1	1	yes	characteristic
70	#1985	C₂×C₅	1	1	yes	characteristic
71	#1985	C₂×C₅	1	1	yes	characteristic
72	#1917	C₂×C₅	1	4	yes	—
73	#1916	C₂×C₅	1	4	yes	—
74	#1832	C₂×C₅	1	1	yes	characteristic
75	#1746	C₂×C₅	1	1	yes	characteristic
76	C₂×C₅×Q₈	—	2	2	no	—
77	C₂×C₅×Q₈	—	2	2	no	—
78	C₂×C₅×Q₈	—	2	4	no	—
79	C₂×C₅×Q₈	—	4	2	no	—
80	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
84	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
88	C₂×C₅×C₈	—	2	2	no	—
89	C₂×C₅×C₈	—	2	2	no	—
90	C₂×C₅×C₈	—	2	2	no	—
91	C₂×C₅×C₈	—	2	2	no	—
92	C₂×C₅×C₈	—	2	2	no	—
93	C₂×C₅×C₈	—	2	4	no	—
94	C₂×C₅×C₈	—	2	4	no	—
95	C₂×C₅×C₈	—	2	2	no	—
96	C₂×C₅×C₈	—	2	2	no	—
97	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
98	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
99	C₅×C₄⋊C₄	—	2	4	no	—
100	C₅×C₄⋊C₄	—	2	2	no	—
101	C₅×C₄⋊C₄	—	2	2	no	—
102	C₅×C₄⋊C₄	—	2	4	no	—
103	C₅×C₄⋊C₄	—	2	8	no	—
104	C₅×C₄⋊C₄	—	4	4	no	—
105	C₄²×C₅	—	2	2	no	—
106	C₄²×C₅	—	2	2	no	—
107	C₄²×C₅	—	2	2	no	—
108	C₄²×C₅	—	2	2	no	—
109	C₄²×C₅	—	2	2	no	—
110	C₄²×C₅	—	2	2	no	—
111	C₄²×C₅	C₂×D₄	1	2	yes	—
112	C₄²×C₅	#41	1	2	yes	—
113	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
114	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
115	#446	—	2	4	no	—
116	C₈×Q₈	—	2	4	no	—
117	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
118	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
119	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
120	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
121	C₂×#97	C₂²×C₅	1	2	yes	—
122	C₂×#97	C₂²×C₅	1	1	yes	characteristic
123	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
124	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
125	#398	C₄×C₅	1	2	yes	—
126	#398	C₄×C₅	1	2	yes	—
127	#395	C₄×C₅	1	2	yes	—
128	#395	C₄×C₅	1	2	yes	—
129	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	—
130	#376	C₂²×C₅	1	1	yes	characteristic
131	#336	C₂²×C₅	1	4	yes	—
132	#336	C₂²×C₅	1	2	yes	—
133	#336	C₂²×C₅	1	4	yes	—
134	#336	C₂²×C₅	1	4	yes	—
135	#336	C₂²×C₅	1	2	yes	—
136	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
137	C₅×Q₈	—	2	8	no	—
138	C₅×Q₈	—	4	4	no	—
139	C₂×C₄×C₅	—	2	2	no	—
140	C₂×C₄×C₅	—	2	2	no	—
141	C₂×C₄×C₅	—	2	2	no	—
142	C₂×C₄×C₅	—	2	2	no	—
143	C₂×C₄×C₅	—	2	2	no	—
144	C₂×C₄×C₅	—	2	2	no	—
145	C₂×C₄×C₅	—	2	2	no	—
146	C₂×C₄×C₅	—	2	4	no	—
147	C₂×C₄×C₅	—	2	4	no	—
148	C₂×C₄×C₅	—	2	4	no	—
149	C₂×C₄×C₅	—	2	2	no	—
150	C₂×C₄×C₅	—	2	4	no	—
151	C₂×C₄×C₅	—	2	4	no	—
152	C₂×C₄×C₅	—	2	2	no	—
153	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
154	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
155	C₂×C₄×C₅	#131	1	4	yes	—
156	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
158	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
159	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
160	C₅×C₈	—	2	2	no	—
161	C₅×C₈	—	2	2	no	—
162	C₅×C₈	—	4	2	no	—
163	C₅×C₈	—	4	2	no	—
164	C₅×C₈	—	4	2	no	—
165	C₂²×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
166	C₂²×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
167	C₂²×C₈	C₂³×C₅	1	2	yes	—
168	C₂²×C₈	C₂³×C₅	1	2	yes	—
169	C₂²×C₈	C₂³×C₅	1	2	yes	—
170	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
171	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
172	C₄×Q₈	—	2	2	no	—
173	C₄×Q₈	—	2	2	no	—
174	C₄×Q₈	—	2	2	no	—
175	C₄×Q₈	—	2	2	no	—
176	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
177	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
178	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
179	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
180	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
181	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
182	C₂×C₄²	C₂³×C₅	1	2	yes	—
183	C₂×C₄²	C₂³×C₅	1	2	yes	—
184	C₂×C₄²	C₂³×C₅	1	2	yes	—
185	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
186	C₄⋊C₈	—	2	4	no	—
187	C₄⋊C₈	—	2	4	no	—
188	C₄⋊C₈	—	2	2	no	—
189	C₄⋊C₈	—	2	2	no	—
190	#97	—	2	4	no	—
191	#97	—	2	2	no	—
192	C₄×C₈	—	2	4	no	—
193	C₄×C₈	C₅×D₄	1	4	yes	—
194	#95	C₂×C₄×C₅	1	2	yes	—
195	#95	C₂×C₄×C₅	1	2	yes	—
196	#95	C₂×C₄×C₅	1	4	yes	—
197	#95	C₂×C₄×C₅	1	4	yes	—
198	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
199	C₂²×C₅	#519	1	2	yes	—
200	C₂²×C₅	C₂×#131	1	2	yes	—
201	C₂²×C₅	#568	1	2	yes	—
202	C₄×C₅	—	2	2	no	—
203	C₄×C₅	—	2	4	no	—
204	C₄×C₅	—	2	2	no	—
205	C₄×C₅	—	4	2	no	—
206	C₄×C₅	—	4	2	no	—
207	C₄×C₅	—	4	2	no	—
208	C₄×C₅	—	4	4	no	—
209	C₄×C₅	#536	1	1	yes	characteristic
210	C₄×C₅	#536	1	1	yes	characteristic
211	C₄×C₅	#541	1	2	yes	—
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	—	2	4	no	—
215	C₂×Q₈	—	4	2	no	—
216	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
217	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
218	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
219	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
220	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
221	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
222	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
223	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
234	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	2	8	no	—
240	C₄⋊C₄	—	4	4	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	2	no	—
247	C₄²	C₂×C₅×D₄	1	2	yes	—
248	C₄²	C₅×#41	1	2	yes	—
249	C₂×C₅	#2327	1	1	yes	characteristic
250	C₂×C₅	#2929	1	1	yes	characteristic
251	C₂×C₅	#2930	1	1	yes	characteristic
252	C₂×C₅	#2985	1	4	yes	—
253	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
254	Q₈	—	2	8	no	—
255	Q₈	—	4	4	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
271	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
272	C₂×C₄	C₅×#131	1	4	yes	—
273	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
274	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
275	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
276	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
277	C₈	—	2	2	no	—
278	C₈	—	2	2	no	—
279	C₈	—	4	2	no	—
280	C₈	—	4	2	no	—
281	C₈	—	4	2	no	—
282	C₅	#22241	1	1	yes	characteristic, direct product factor, Sylow
283	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
284	C₂²	C₅×#519	1	2	yes	—
285	C₂²	C₂×C₅×#131	1	2	yes	—
286	C₂²	C₅×#568	1	2	yes	—
287	C₄	—	2	2	no	—
288	C₄	—	2	4	no	—
289	C₄	—	2	2	no	—
290	C₄	—	4	4	no	—
291	C₄	—	4	2	no	—
292	C₄	—	4	2	no	—
293	C₄	—	4	2	no	—
294	C₄	C₅×#536	1	1	yes	characteristic
295	C₄	C₅×#536	1	1	yes	characteristic
296	C₄	C₅×#541	1	2	yes	—
297	C₂	C₅×#2327	1	1	yes	characteristic
298	C₂	C₅×#2929	1	1	yes	characteristic
299	C₂	C₅×#2930	1	1	yes	characteristic
300	C₂	C₅×#2985	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂⁴
commutator	C₂³	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle, prime radical	C₂³×C₅	C₂³×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
33. #22241	C₅	direct (N × H)
282. C₅	#22241	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×#446	2
C₂×C₅×C₈×Q₈	2
C₅×#2399	2
C₅×#1985	2
C₅×#1917	2
C₅×#1916	2
C₅×#1832	2
C₅×#1746	2
#22241	5

Subgroups — C5×#22241 #11944961

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22241 #11944961