Subgroups — C₅×#7900 #11930620

Order 1280.

638 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×#432	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#431	C₂	1	1	yes	maximal, characteristic
3	C₅×#2285	C₂	1	1	yes	maximal, characteristic
4	C₅×#1279	C₂	1	4	yes	maximal, complemented
5	C₂×C₅×#117	—	2	1	no	—
6	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₅×#432	—	2	2	no	—
8	C₅×#432	C₄	1	4	yes	complemented
9	C₅×#431	—	2	2	no	—
10	C₅×#431	C₄	1	4	yes	complemented
11	C₂×C₅×C₄⋊C₈	—	2	1	no	—
12	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₅×C₂²⋊C₈	—	2	1	no	—
14	C₂×C₅×C₂²⋊C₈	—	2	1	no	—
15	C₂×C₅×#97	—	2	1	no	—
16	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
17	C₅×#384	C₄	1	2	yes	—
18	C₅×#382	—	2	1	no	—
19	C₅×#380	—	2	1	no	—
20	C₅×#380	C₄	1	2	yes	—
21	C₄×C₅×C₄⋊C₄	—	2	1	no	—
22	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
23	#7900	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	—	2	2	no	—
28	C₂²×C₅×C₈	—	2	2	no	—
29	C₅×#117	—	4	4	no	—
30	C₂×C₅×C₄⋊C₄	—	2	2	no	—
31	C₂×C₅×C₄⋊C₄	—	2	2	no	—
32	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
33	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
34	C₂×C₄²×C₅	—	2	2	no	—
35	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅	D₄	1	2	yes	—
41	C₂×C₄²×C₅	Q₈	1	2	yes	—
42	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
43	C₅×C₄⋊C₈	—	2	4	no	—
44	C₅×C₄⋊C₈	—	2	2	no	—
45	C₅×C₄⋊C₈	C₂×C₄	1	4	yes	—
46	C₅×C₂²⋊C₈	—	2	4	no	—
47	C₅×C₂²⋊C₈	—	2	4	no	—
48	C₅×#97	—	2	4	no	—
49	C₄×C₅×C₈	—	2	2	no	—
50	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
51	C₅×#95	—	2	2	no	—
52	C₅×#95	—	2	2	no	—
53	C₂×#432	C₂×C₅	1	1	yes	characteristic
54	C₂×#431	C₂×C₅	1	1	yes	characteristic
55	#2285	C₂×C₅	1	1	yes	characteristic
56	#1279	C₂×C₅	1	4	yes	complemented
57	C₂⁴×C₅	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅	—	2	4	no	—
59	C₂²×C₄×C₅	—	2	2	no	—
60	C₂²×C₄×C₅	—	2	2	no	—
61	C₂²×C₄×C₅	—	2	4	no	—
62	C₂²×C₄×C₅	—	4	2	no	—
63	C₂²×C₄×C₅	C₄²	1	2	yes	—
64	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
75	C₂×C₅×C₈	—	2	4	no	—
76	C₂×C₅×C₈	—	2	4	no	—
77	C₂×C₅×C₈	—	4	4	no	—
78	C₂×C₅×C₈	—	4	2	no	—
79	C₂×C₅×C₈	—	4	2	no	—
80	C₅×C₄⋊C₄	—	2	8	no	—
81	C₅×C₄⋊C₄	—	4	4	no	—
82	C₅×C₄⋊C₄	—	4	4	no	—
83	C₅×C₂²⋊C₄	—	4	4	no	—
84	C₄²×C₅	—	4	4	no	—
85	C₄²×C₅	C₄²	1	2	yes	—
86	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
87	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
88	C₄²×C₅	C₄⋊C₄	1	2	yes	—
89	C₄²×C₅	C₄⋊C₄	1	2	yes	—
90	C₄²×C₅	C₄⋊C₄	1	2	yes	—
91	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
95	C₂×#117	—	2	1	no	—
96	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
97	#432	—	2	2	no	—
98	#432	C₄×C₅	1	4	yes	complemented
99	#431	—	2	2	no	—
100	#431	C₄×C₅	1	4	yes	complemented
101	C₂×C₄⋊C₈	—	2	1	no	—
102	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
103	C₂×C₂²⋊C₈	—	2	1	no	—
104	C₂×C₂²⋊C₈	—	2	1	no	—
105	C₂×#97	—	2	1	no	—
106	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
107	#384	C₄×C₅	1	2	yes	—
108	#382	—	2	1	no	—
109	#380	—	2	1	no	—
110	#380	C₄×C₅	1	2	yes	—
111	C₄×C₄⋊C₄	—	2	1	no	—
112	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
113	C₂³×C₅	—	2	2	no	—
114	C₂³×C₅	—	2	1	no	—
115	C₂³×C₅	—	2	1	no	—
116	C₂³×C₅	—	2	2	no	—
117	C₂³×C₅	#95	1	2	yes	—
118	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₅	—	2	2	no	—
120	C₂×C₄×C₅	—	2	1	no	—
121	C₂×C₄×C₅	—	2	1	no	—
122	C₂×C₄×C₅	—	2	2	no	—
123	C₂×C₄×C₅	—	2	2	no	—
124	C₂×C₄×C₅	—	2	4	no	—
125	C₂×C₄×C₅	—	2	4	no	—
126	C₂×C₄×C₅	—	2	4	no	—
127	C₂×C₄×C₅	—	2	4	no	—
128	C₂×C₄×C₅	—	4	2	no	—
129	C₂×C₄×C₅	—	4	2	no	—
130	C₂×C₄×C₅	—	4	4	no	—
131	C₂×C₄×C₅	—	4	4	no	—
132	C₂×C₄×C₅	—	4	4	no	—
133	C₂×C₄×C₅	#95	1	2	yes	—
134	C₂×C₄×C₅	#95	1	2	yes	—
135	C₂×C₄×C₅	#95	1	2	yes	—
136	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₅	#131	1	4	yes	—
143	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
144	C₂×C₄×C₅	#137	1	2	yes	—
145	C₅×C₈	—	4	4	no	—
146	C₅×C₈	—	4	4	no	—
147	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
148	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
149	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
150	C₂²×C₈	—	2	2	no	—
151	C₂²×C₈	—	2	2	no	—
152	#117	—	4	4	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	—	2	2	no	—
155	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
156	C₂×C₂²⋊C₄	—	2	2	no	—
157	C₂×C₄²	—	2	2	no	—
158	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
159	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
160	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
161	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
162	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
163	C₂×C₄²	C₅×D₄	1	2	yes	—
164	C₂×C₄²	C₅×Q₈	1	2	yes	—
165	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
166	C₄⋊C₈	—	2	4	no	—
167	C₄⋊C₈	—	2	2	no	—
168	C₄⋊C₈	C₂×C₄×C₅	1	4	yes	—
169	C₂²⋊C₈	—	2	4	no	—
170	C₂²⋊C₈	—	2	4	no	—
171	#97	—	2	4	no	—
172	C₄×C₈	—	2	2	no	—
173	C₄×C₈	C₂×C₄×C₅	1	4	yes	—
174	#95	—	2	2	no	—
175	#95	—	2	2	no	—
176	C₂²×C₅	—	2	2	no	—
177	C₂²×C₅	—	2	2	no	—
178	C₂²×C₅	—	2	2	no	—
179	C₂²×C₅	—	4	2	no	—
180	C₂²×C₅	—	4	2	no	—
181	C₂²×C₅	C₂×#95	1	1	yes	characteristic
182	C₂²×C₅	#408	1	2	yes	—
183	C₂²×C₅	#418	1	1	yes	characteristic
184	C₂²×C₅	#419	1	1	yes	characteristic
185	C₂²×C₅	#421	1	2	yes	—
186	C₄×C₅	—	2	2	no	—
187	C₄×C₅	—	2	1	no	—
188	C₄×C₅	—	2	2	no	—
189	C₄×C₅	—	2	1	no	—
190	C₄×C₅	—	2	2	no	—
191	C₄×C₅	—	8	4	no	—
192	C₄×C₅	#405	1	2	yes	—
193	C₄×C₅	#417	1	2	yes	—
194	C₄×C₅	#424	1	2	yes	—
195	C₄×C₅	#428	1	2	yes	—
196	C₂⁴	C₅×C₂²⋊C₄	1	1	yes	characteristic
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	C₄²×C₅	1	2	yes	—
203	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
204	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
205	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
206	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
207	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
208	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
209	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
210	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
211	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	4	4	no	—
217	C₂×C₈	—	4	2	no	—
218	C₂×C₈	—	4	2	no	—
219	C₄⋊C₄	—	2	8	no	—
220	C₄⋊C₄	—	4	4	no	—
221	C₄⋊C₄	—	4	4	no	—
222	C₂²⋊C₄	—	4	4	no	—
223	C₄²	—	4	4	no	—
224	C₄²	C₄²×C₅	1	2	yes	—
225	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
226	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
227	C₄²	C₅×C₄⋊C₄	1	2	yes	—
228	C₄²	C₅×C₄⋊C₄	1	2	yes	—
229	C₄²	C₅×C₄⋊C₄	1	2	yes	—
230	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
231	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
232	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
233	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
234	C₂×C₅	—	4	2	no	—
235	C₂×C₅	#1496	1	2	yes	—
236	C₂×C₅	#1497	1	2	yes	—
237	C₂×C₅	#1729	1	1	yes	characteristic
238	C₂×C₅	#1733	1	1	yes	characteristic
239	C₂×C₅	#1736	1	1	yes	characteristic
240	C₂³	—	2	2	no	—
241	C₂³	—	2	1	no	—
242	C₂³	—	2	1	no	—
243	C₂³	—	2	2	no	—
244	C₂³	C₅×#95	1	2	yes	—
245	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	C₅×#95	1	2	yes	—
261	C₂×C₄	C₅×#95	1	2	yes	—
262	C₂×C₄	C₅×#95	1	2	yes	—
263	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
264	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
265	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
266	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
267	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
268	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
269	C₂×C₄	C₅×#131	1	4	yes	—
270	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
271	C₂×C₄	C₅×#137	1	2	yes	—
272	C₈	—	4	4	no	—
273	C₈	—	4	4	no	—
274	C₅	#7900	1	1	yes	characteristic, direct product factor, Sylow
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	4	2	no	—
280	C₂²	C₂×C₅×#95	1	1	yes	characteristic
281	C₂²	C₅×#408	1	2	yes	—
282	C₂²	C₅×#418	1	1	yes	characteristic
283	C₂²	C₅×#419	1	1	yes	characteristic
284	C₂²	C₅×#421	1	2	yes	—
285	C₄	—	2	2	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	2	no	—
289	C₄	—	2	2	no	—
290	C₄	—	8	4	no	—
291	C₄	C₅×#405	1	2	yes	—
292	C₄	C₅×#417	1	2	yes	—
293	C₄	C₅×#424	1	2	yes	—
294	C₄	C₅×#428	1	2	yes	—
295	C₂	—	4	2	no	—
296	C₂	C₅×#1496	1	2	yes	—
297	C₂	C₅×#1497	1	2	yes	—
298	C₂	C₅×#1729	1	1	yes	characteristic
299	C₂	C₅×#1733	1	1	yes	characteristic
300	C₂	C₅×#1736	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂³×C₅	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₅	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅×#1279	C₂	semidirect
8. C₅×#432	C₄	semidirect
10. C₅×#431	C₄	semidirect
23. #7900	C₅	direct (N × H)
56. #1279	C₂×C₅	semidirect
98. #432	C₄×C₅	semidirect
100. #431	C₄×C₅	semidirect
274. C₅	#7900	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×#432	2
C₂×C₅×#431	2
C₅×#2285	2
C₅×#1279	2
#7900	5

Subgroups — C5×#7900 #11930620

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#7900 #11930620