Subgroups — C₅×#21282 #11944002

Order 1280.

862 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×D₈	C₂	1	1	yes	maximal, characteristic
2	C₅×#2319	C₂	1	2	yes	maximal, complemented
3	C₅×#1970	C₂	1	2	yes	maximal
4	C₅×#1942	C₂	1	2	yes	maximal, complemented
5	C₅×#1936	C₂	1	4	yes	maximal, complemented
6	C₄×C₅×#107	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1762	C₂	1	2	yes	maximal, complemented
8	C₅×#1726	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₅×D₄	C₂²	1	2	yes	—
11	C₄×C₅×D₈	C₄	1	8	yes	complemented
12	C₂×C₅×#107	C₂²	1	2	yes	—
13	C₂×C₅×#107	C₂²	1	1	yes	characteristic
14	C₂×C₅×#102	C₂²	1	4	yes	—
15	C₂×C₅×#102	C₂²	1	2	yes	—
16	C₂×C₅×#97	C₂²	1	1	yes	characteristic
17	C₂×C₅×#97	C₂²	1	2	yes	—
18	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
19	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	C₅×#390	C₂²	1	4	yes	—
21	C₅×#389	C₂²	1	4	yes	—
22	C₅×#388	—	2	4	no	—
23	C₅×#381	—	2	2	no	—
24	C₅×#380	—	2	2	no	—
25	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
26	C₅×#376	C₂²	1	1	yes	characteristic
27	C₅×#340	C₂²	1	4	yes	—
28	C₅×#340	C₂²	1	2	yes	—
29	#21282	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂²×C₅×D₄	C₂³	1	2	yes	—
31	C₂³×C₄×C₅	—	2	2	no	—
32	C₂×C₅×D₈	—	2	4	no	—
33	C₂×C₅×D₈	C₂×C₄	1	4	yes	—
34	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
35	C₂²×C₅×C₈	C₂³	1	2	yes	—
36	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
37	C₄×C₅×D₄	—	2	4	no	—
38	C₄×C₅×D₄	C₂×C₄	1	8	yes	—
39	C₂×C₅×C₄⋊C₄	—	2	4	no	—
40	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
41	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
42	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
43	C₂×C₅×C₂²⋊C₄	—	2	4	no	—
44	C₂×C₅×C₂²⋊C₄	—	2	4	no	—
45	C₂×C₄²×C₅	—	2	2	no	—
46	C₂×C₄²×C₅	C₂³	1	2	yes	—
47	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
48	C₅×#107	—	2	4	no	—
49	C₅×#107	C₂×C₄	1	4	yes	—
50	C₅×#102	—	2	8	no	—
51	C₅×#102	C₂×C₄	1	8	yes	—
52	C₅×#97	—	2	4	no	—
53	C₅×#97	—	2	2	no	—
54	C₄×C₅×C₈	—	2	4	no	—
55	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
56	C₅×#95	—	2	2	no	—
57	C₅×#95	—	2	4	no	—
58	C₅×#95	D₄	1	1	yes	characteristic
59	C₅×#95	D₄	1	1	yes	characteristic
60	C₅×#95	D₄	1	2	yes	—
61	C₂×C₄×D₈	C₂×C₅	1	1	yes	characteristic
62	#2319	C₂×C₅	1	2	yes	complemented
63	#1970	C₂×C₅	1	2	yes	—
64	#1942	C₂×C₅	1	2	yes	complemented
65	#1936	C₂×C₅	1	4	yes	complemented
66	C₄×#107	C₂×C₅	1	1	yes	characteristic, complemented
67	#1762	C₂×C₅	1	2	yes	complemented
68	#1726	C₂×C₅	1	1	yes	characteristic, complemented
69	C₂⁴×C₅	—	2	2	no	—
70	C₂×C₅×D₄	—	2	4	no	—
71	C₂×C₅×D₄	—	2	4	no	—
72	C₂×C₅×D₄	—	2	2	no	—
73	C₂×C₅×D₄	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₅	—	2	2	no	—
75	C₂²×C₄×C₅	—	2	8	no	—
76	C₂²×C₄×C₅	—	2	4	no	—
77	C₂²×C₄×C₅	—	8	4	no	—
78	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
83	C₅×D₈	—	2	8	no	—
84	C₂×C₅×C₈	—	2	2	no	—
85	C₂×C₅×C₈	—	2	2	no	—
86	C₂×C₅×C₈	—	2	2	no	—
87	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
88	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
89	C₂×C₅×C₈	#41	1	4	yes	—
90	C₂×C₅×C₈	#41	1	4	yes	—
91	C₅×C₄⋊C₄	—	2	2	no	—
92	C₅×C₄⋊C₄	—	2	4	no	—
93	C₅×C₄⋊C₄	—	2	4	no	—
94	C₅×C₄⋊C₄	—	8	4	no	—
95	C₅×C₄⋊C₄	C₂²×C₄	1	4	yes	—
96	C₅×C₂²⋊C₄	—	2	8	no	—
97	C₅×C₂²⋊C₄	—	8	4	no	—
98	C₅×C₂²⋊C₄	—	8	4	no	—
99	C₄²×C₅	—	2	2	no	—
100	C₄²×C₅	—	2	2	no	—
101	C₄²×C₅	—	8	2	no	—
102	C₄²×C₅	C₂²×C₄	1	2	yes	—
103	C₄²×C₅	#41	1	2	yes	—
104	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
105	C₂×C₄×D₄	C₂²×C₅	1	2	yes	—
106	C₄×D₈	C₄×C₅	1	8	yes	complemented
107	C₂×#107	C₂²×C₅	1	2	yes	—
108	C₂×#107	C₂²×C₅	1	1	yes	characteristic
109	C₂×#102	C₂²×C₅	1	4	yes	—
110	C₂×#102	C₂²×C₅	1	2	yes	—
111	C₂×#97	C₂²×C₅	1	2	yes	—
112	C₂×#97	C₂²×C₅	1	1	yes	characteristic
113	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
114	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
115	#390	C₂²×C₅	1	4	yes	—
116	#389	C₂²×C₅	1	4	yes	—
117	#388	—	2	4	no	—
118	#381	—	2	2	no	—
119	#380	—	2	2	no	—
120	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
121	#376	C₂²×C₅	1	1	yes	characteristic
122	#340	C₂²×C₅	1	2	yes	—
123	#340	C₂²×C₅	1	4	yes	—
124	C₂³×C₅	—	2	4	no	—
125	C₂³×C₅	—	4	4	no	—
126	C₂³×C₅	—	4	2	no	—
127	C₂³×C₅	—	4	4	no	—
128	C₂³×C₅	—	4	2	no	—
129	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
130	C₅×D₄	—	2	4	no	—
131	C₅×D₄	—	2	4	no	—
132	C₅×D₄	—	4	4	no	—
133	C₂×C₄×C₅	—	2	1	no	—
134	C₂×C₄×C₅	—	2	2	no	—
135	C₂×C₄×C₅	—	2	1	no	—
136	C₂×C₄×C₅	—	2	4	no	—
137	C₂×C₄×C₅	—	2	2	no	—
138	C₂×C₄×C₅	—	2	2	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	—	4	4	no	—
144	C₂×C₄×C₅	—	4	4	no	—
145	C₂×C₄×C₅	—	4	4	no	—
146	C₂×C₄×C₅	—	4	4	no	—
147	C₂×C₄×C₅	—	8	8	no	—
148	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
149	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
150	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂×#41	1	2	yes	—
152	C₂×C₄×C₅	C₂×#41	1	2	yes	—
153	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
154	C₅×C₈	—	2	2	no	—
155	C₅×C₈	—	2	4	no	—
156	C₅×C₈	#117	1	4	yes	—
157	C₂²×D₄	C₂³×C₅	1	2	yes	—
158	C₂³×C₄	—	2	2	no	—
159	C₂×D₈	—	2	4	no	—
160	C₂×D₈	C₂×C₄×C₅	1	4	yes	—
161	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
162	C₂²×C₈	C₂³×C₅	1	2	yes	—
163	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
164	C₄×D₄	—	2	4	no	—
165	C₄×D₄	C₂×C₄×C₅	1	8	yes	—
166	C₂×C₄⋊C₄	—	2	4	no	—
167	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
168	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
169	C₂×C₂²⋊C₄	—	2	2	no	—
170	C₂×C₂²⋊C₄	—	2	4	no	—
171	C₂×C₂²⋊C₄	—	2	4	no	—
172	C₂×C₄²	—	2	2	no	—
173	C₂×C₄²	C₂³×C₅	1	2	yes	—
174	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
175	#107	—	2	4	no	—
176	#107	C₂×C₄×C₅	1	4	yes	—
177	#102	—	2	8	no	—
178	#102	C₂×C₄×C₅	1	8	yes	—
179	#97	—	2	4	no	—
180	#97	—	2	2	no	—
181	C₄×C₈	—	2	4	no	—
182	C₄×C₈	C₂×C₄×C₅	1	4	yes	—
183	#95	—	2	2	no	—
184	#95	—	2	4	no	—
185	#95	C₅×D₄	1	1	yes	characteristic
186	#95	C₅×D₄	1	1	yes	characteristic
187	#95	C₅×D₄	1	2	yes	—
188	C₂²×C₅	—	4	4	no	—
189	C₂²×C₅	—	4	4	no	—
190	C₂²×C₅	—	4	4	no	—
191	C₂²×C₅	—	8	4	no	—
192	C₂²×C₅	—	8	4	no	—
193	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
194	C₂²×C₅	#532	1	2	yes	—
195	C₂²×C₅	#576	1	2	yes	—
196	C₂²×C₅	#576	1	1	yes	characteristic
197	C₂²×C₅	#578	1	1	yes	characteristic
198	C₄×C₅	—	2	2	no	—
199	C₄×C₅	—	2	2	no	—
200	C₄×C₅	—	4	2	no	—
201	C₄×C₅	—	4	2	no	—
202	C₄×C₅	—	4	4	no	—
203	C₄×C₅	—	8	4	no	—
204	C₄×C₅	C₂×#117	1	1	yes	characteristic
205	C₄×C₅	C₄×#41	1	1	yes	characteristic
206	C₄×C₅	#529	1	2	yes	—
207	C₂⁴	—	2	2	no	—
208	C₂×D₄	—	2	4	no	—
209	C₂×D₄	—	2	4	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	C₂²×C₄×C₅	1	4	yes	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	8	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	8	4	no	—
216	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
217	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
218	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
220	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
221	D₈	—	2	8	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
226	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
227	C₂×C₈	C₅×#41	1	4	yes	—
228	C₂×C₈	C₅×#41	1	4	yes	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	8	4	no	—
233	C₄⋊C₄	C₂²×C₄×C₅	1	4	yes	—
234	C₂²⋊C₄	—	2	8	no	—
235	C₂²⋊C₄	—	8	4	no	—
236	C₂²⋊C₄	—	8	4	no	—
237	C₄²	—	2	2	no	—
238	C₄²	—	2	2	no	—
239	C₄²	—	8	2	no	—
240	C₄²	C₂²×C₄×C₅	1	2	yes	—
241	C₄²	C₅×#41	1	2	yes	—
242	C₂×C₅	—	8	4	no	—
243	C₂×C₅	#2293	1	1	yes	characteristic
244	C₂×C₅	#2948	1	1	yes	characteristic
245	C₂×C₅	#2951	1	1	yes	characteristic
246	C₂×C₅	#3175	1	2	yes	—
247	C₂×C₅	#3176	1	2	yes	—
248	C₂³	—	2	4	no	—
249	C₂³	—	4	4	no	—
250	C₂³	—	4	2	no	—
251	C₂³	—	4	4	no	—
252	C₂³	—	4	2	no	—
253	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
254	D₄	—	2	4	no	—
255	D₄	—	2	4	no	—
256	D₄	—	4	4	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	8	8	no	—
272	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
273	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
274	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
275	C₂×C₄	C₂×C₅×#41	1	2	yes	—
276	C₂×C₄	C₂×C₅×#41	1	2	yes	—
277	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
278	C₈	—	2	2	no	—
279	C₈	—	2	4	no	—
280	C₈	C₅×#117	1	4	yes	—
281	C₅	#21282	1	1	yes	characteristic, direct product factor, Sylow
282	C₂²	—	4	4	no	—
283	C₂²	—	4	4	no	—
284	C₂²	—	4	4	no	—
285	C₂²	—	8	4	no	—
286	C₂²	—	8	4	no	—
287	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
288	C₂²	C₅×#532	1	2	yes	—
289	C₂²	C₅×#576	1	2	yes	—
290	C₂²	C₅×#576	1	1	yes	characteristic
291	C₂²	C₅×#578	1	1	yes	characteristic
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	4	2	no	—
295	C₄	—	4	2	no	—
296	C₄	—	4	4	no	—
297	C₄	—	8	4	no	—
298	C₄	C₂×C₅×#117	1	1	yes	characteristic
299	C₄	C₄×C₅×#41	1	1	yes	characteristic
300	C₄	C₅×#529	1	2	yes	—
301	C₂	—	8	4	no	—
302	C₂	C₅×#2293	1	1	yes	characteristic
303	C₂	C₅×#2948	1	1	yes	characteristic
304	C₂	C₅×#2951	1	1	yes	characteristic
305	C₂	C₅×#3175	1	2	yes	—
306	C₂	C₅×#3176	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×D₈	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₅×#376 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2319	C₂	semidirect
4. C₅×#1942	C₂	semidirect
5. C₅×#1936	C₂	semidirect
6. C₄×C₅×#107	C₂	semidirect
7. C₅×#1762	C₂	semidirect
8. C₅×#1726	C₂	semidirect
11. C₄×C₅×D₈	C₄	semidirect
29. #21282	C₅	direct (N × H)
62. #2319	C₂×C₅	semidirect
64. #1942	C₂×C₅	semidirect
65. #1936	C₂×C₅	semidirect
66. C₄×#107	C₂×C₅	semidirect
67. #1762	C₂×C₅	semidirect
68. #1726	C₂×C₅	semidirect
106. C₄×D₈	C₄×C₅	semidirect
281. C₅	#21282	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×C₄×C₅×D₈	2
C₅×#2319	2
C₅×#1970	2
C₅×#1942	2
C₅×#1936	2
C₄×C₅×#107	2
C₅×#1762	2
C₅×#1726	2
#21282	5

Subgroups — C5×#21282 #11944002

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21282 #11944002