Subgroups — C₅×#8724 #11931444

Order 1280.

630 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×C₅×#384	C₂	1	1	yes	maximal, characteristic
2	C₅×#1797	C₂	1	1	yes	maximal, characteristic
3	C₅×#1797	C₂	1	1	yes	maximal, characteristic
4	C₅×#1296	C₂	1	4	yes	maximal, complemented
5	C₂²×C₅×C₄⋊C₄	—	2	1	no	—
6	C₂²×C₅×C₄⋊C₄	—	2	1	no	—
7	C₂²×C₅×C₄⋊C₄	C₄	1	2	yes	—
8	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₅×C₂²⋊C₈	—	2	1	no	—
12	C₂×C₅×C₂²⋊C₈	—	2	1	no	—
13	C₅×#384	—	2	1	no	—
14	C₅×#384	—	2	1	no	—
15	C₅×#384	C₄	1	2	yes	—
16	C₅×#384	C₂²	1	2	yes	—
17	C₅×#336	—	2	1	no	—
18	C₅×#336	—	2	1	no	—
19	#8724	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂³×C₄×C₅	—	2	2	no	—
21	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
24	C₂²×C₅×C₈	—	2	2	no	—
25	C₂²×C₅×C₈	—	2	2	no	—
26	C₂×C₅×C₄⋊C₄	—	2	2	no	—
27	C₂×C₅×C₄⋊C₄	—	2	2	no	—
28	C₂×C₅×C₄⋊C₄	—	2	2	no	—
29	C₂×C₅×C₄⋊C₄	—	2	2	no	—
30	C₂×C₅×C₄⋊C₄	—	2	2	no	—
31	C₂×C₅×C₄⋊C₄	—	4	4	no	—
32	C₂×C₅×C₄⋊C₄	—	4	2	no	—
33	C₂×C₅×C₄⋊C₄	—	4	2	no	—
34	C₂×C₅×C₄⋊C₄	—	4	4	no	—
35	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₅	—	2	4	no	—
37	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
41	C₅×C₄⋊C₈	—	2	4	no	—
42	C₅×C₄⋊C₈	—	2	4	no	—
43	C₅×C₂²⋊C₈	—	4	2	no	—
44	C₅×C₂²⋊C₈	—	4	2	no	—
45	C₂×#384	C₂×C₅	1	1	yes	characteristic
46	#1797	C₂×C₅	1	1	yes	characteristic
47	#1797	C₂×C₅	1	1	yes	characteristic
48	#1296	C₂×C₅	1	4	yes	complemented
49	C₂⁴×C₅	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅	—	2	2	no	—
51	C₂²×C₄×C₅	—	2	2	no	—
52	C₂²×C₄×C₅	—	2	2	no	—
53	C₂²×C₄×C₅	—	2	2	no	—
54	C₂²×C₄×C₅	—	2	2	no	—
55	C₂²×C₄×C₅	—	2	2	no	—
56	C₂²×C₄×C₅	—	2	2	no	—
57	C₂²×C₄×C₅	—	2	4	no	—
58	C₂²×C₄×C₅	—	4	4	no	—
59	C₂²×C₄×C₅	—	4	4	no	—
60	C₂²×C₄×C₅	—	4	4	no	—
61	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅	D₈	1	2	yes	—
65	C₂²×C₄×C₅	SD₁₆	1	2	yes	—
66	C₂²×C₄×C₅	SD₁₆	1	2	yes	—
67	C₂²×C₄×C₅	Q₁₆	1	2	yes	—
68	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
71	C₂×C₅×C₈	—	4	2	no	—
72	C₂×C₅×C₈	—	4	2	no	—
73	C₂×C₅×C₈	—	4	2	no	—
74	C₂×C₅×C₈	—	4	2	no	—
75	C₂×C₅×C₈	—	4	2	no	—
76	C₂×C₅×C₈	—	4	2	no	—
77	C₅×C₄⋊C₄	—	4	4	no	—
78	C₅×C₄⋊C₄	—	4	4	no	—
79	C₅×C₄⋊C₄	—	4	4	no	—
80	C₅×C₄⋊C₄	—	4	4	no	—
81	C₅×C₄⋊C₄	—	4	2	no	—
82	C₅×C₄⋊C₄	—	4	2	no	—
83	C₅×C₄⋊C₄	—	4	4	no	—
84	C₄²×C₅	—	2	2	no	—
85	C₄²×C₅	—	2	2	no	—
86	C₄²×C₅	—	2	2	no	—
87	C₄²×C₅	#41	1	4	yes	—
88	C₂²×C₄⋊C₄	—	2	1	no	—
89	C₂²×C₄⋊C₄	—	2	1	no	—
90	C₂²×C₄⋊C₄	C₄×C₅	1	2	yes	—
91	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
92	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
93	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
94	C₂×C₂²⋊C₈	—	2	1	no	—
95	C₂×C₂²⋊C₈	—	2	1	no	—
96	#384	—	2	1	no	—
97	#384	—	2	1	no	—
98	#384	C₄×C₅	1	2	yes	—
99	#384	C₂²×C₅	1	2	yes	—
100	#336	—	2	1	no	—
101	#336	—	2	1	no	—
102	C₂³×C₅	—	2	1	no	—
103	C₂³×C₅	—	2	1	no	—
104	C₂³×C₅	—	2	1	no	—
105	C₂³×C₅	—	2	1	no	—
106	C₂³×C₅	#101	1	2	yes	—
107	C₂³×C₅	#102	1	2	yes	—
108	C₂³×C₅	#103	1	2	yes	—
109	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₅	—	2	8	no	—
111	C₂×C₄×C₅	—	2	1	no	—
112	C₂×C₄×C₅	—	2	1	no	—
113	C₂×C₄×C₅	—	2	1	no	—
114	C₂×C₄×C₅	—	2	1	no	—
115	C₂×C₄×C₅	—	2	1	no	—
116	C₂×C₄×C₅	—	2	1	no	—
117	C₂×C₄×C₅	—	2	1	no	—
118	C₂×C₄×C₅	—	2	1	no	—
119	C₂×C₄×C₅	—	2	4	no	—
120	C₂×C₄×C₅	—	2	4	no	—
121	C₂×C₄×C₅	—	2	4	no	—
122	C₂×C₄×C₅	—	2	4	no	—
123	C₂×C₄×C₅	—	4	4	no	—
124	C₂×C₄×C₅	—	4	4	no	—
125	C₂×C₄×C₅	—	4	4	no	—
126	C₂×C₄×C₅	—	8	8	no	—
127	C₂×C₄×C₅	#100	1	2	yes	—
128	C₂×C₄×C₅	#102	1	2	yes	—
129	C₂×C₄×C₅	#103	1	2	yes	—
130	C₂×C₄×C₅	#117	1	2	yes	—
131	C₂×C₄×C₅	#123	1	2	yes	—
132	C₂×C₄×C₅	#123	1	2	yes	—
133	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
137	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
138	C₂×C₄×C₅	#137	1	2	yes	—
139	C₅×C₈	—	8	2	no	—
140	C₅×C₈	—	8	2	no	—
141	C₂³×C₄	—	2	2	no	—
142	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
143	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
144	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
145	C₂²×C₈	—	2	2	no	—
146	C₂²×C₈	—	2	2	no	—
147	C₂×C₄⋊C₄	—	2	2	no	—
148	C₂×C₄⋊C₄	—	2	2	no	—
149	C₂×C₄⋊C₄	—	2	2	no	—
150	C₂×C₄⋊C₄	—	2	2	no	—
151	C₂×C₄⋊C₄	—	2	2	no	—
152	C₂×C₄⋊C₄	—	4	4	no	—
153	C₂×C₄⋊C₄	—	4	2	no	—
154	C₂×C₄⋊C₄	—	4	2	no	—
155	C₂×C₄⋊C₄	—	4	4	no	—
156	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
157	C₂×C₄²	—	2	4	no	—
158	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
159	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
160	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
161	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
162	C₄⋊C₈	—	2	4	no	—
163	C₄⋊C₈	—	2	4	no	—
164	C₂²⋊C₈	—	4	2	no	—
165	C₂²⋊C₈	—	4	2	no	—
166	C₂²×C₅	—	2	2	no	—
167	C₂²×C₅	—	2	2	no	—
168	C₂²×C₅	—	2	2	no	—
169	C₂²×C₅	—	2	2	no	—
170	C₂²×C₅	—	2	2	no	—
171	C₂²×C₅	—	2	2	no	—
172	C₂²×C₅	#332	1	4	yes	—
173	C₂²×C₅	#381	1	1	yes	characteristic
174	C₂²×C₅	C₂×#100	1	1	yes	characteristic
175	C₂²×C₅	C₂×#101	1	1	yes	characteristic
176	C₂²×C₅	C₂×#102	1	1	yes	characteristic
177	C₂²×C₅	C₂×#103	1	1	yes	characteristic
178	C₂²×C₅	#418	1	1	yes	characteristic
179	C₂²×C₅	#419	1	1	yes	characteristic
180	C₄×C₅	—	2	2	no	—
181	C₄×C₅	—	2	2	no	—
182	C₄×C₅	—	2	2	no	—
183	C₄×C₅	—	2	2	no	—
184	C₄×C₅	—	8	4	no	—
185	C₄×C₅	#480	1	2	yes	—
186	C₄×C₅	#481	1	2	yes	—
187	C₄×C₅	#483	1	2	yes	—
188	C₄×C₅	#484	1	2	yes	—
189	C₂⁴	C₅×C₂²⋊C₄	1	1	yes	characteristic
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	4	4	no	—
199	C₂²×C₄	—	4	4	no	—
200	C₂²×C₄	—	4	4	no	—
201	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
202	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
203	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
204	C₂²×C₄	C₅×D₈	1	2	yes	—
205	C₂²×C₄	C₅×SD₁₆	1	2	yes	—
206	C₂²×C₄	C₅×SD₁₆	1	2	yes	—
207	C₂²×C₄	C₅×Q₁₆	1	2	yes	—
208	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
211	C₂×C₈	—	4	2	no	—
212	C₂×C₈	—	4	2	no	—
213	C₂×C₈	—	4	2	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	4	2	no	—
217	C₄⋊C₄	—	4	4	no	—
218	C₄⋊C₄	—	4	4	no	—
219	C₄⋊C₄	—	4	2	no	—
220	C₄⋊C₄	—	4	2	no	—
221	C₄⋊C₄	—	4	4	no	—
222	C₄⋊C₄	—	4	4	no	—
223	C₄⋊C₄	—	4	4	no	—
224	C₄²	—	2	2	no	—
225	C₄²	—	2	2	no	—
226	C₄²	—	2	2	no	—
227	C₄²	C₅×#41	1	4	yes	—
228	C₂×C₅	—	2	4	no	—
229	C₂×C₅	C₂×#332	1	1	yes	characteristic
230	C₂×C₅	#1545	1	1	yes	characteristic
231	C₂×C₅	#1548	1	1	yes	characteristic
232	C₂×C₅	#1556	1	1	yes	characteristic
233	C₂×C₅	#1821	1	1	yes	characteristic
234	C₂×C₅	#1823	1	1	yes	characteristic
235	C₂×C₅	#1824	1	1	yes	characteristic
236	C₂³	—	2	1	no	—
237	C₂³	—	2	1	no	—
238	C₂³	—	2	1	no	—
239	C₂³	—	2	1	no	—
240	C₂³	C₅×#101	1	2	yes	—
241	C₂³	C₅×#102	1	2	yes	—
242	C₂³	C₅×#103	1	2	yes	—
243	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	8	8	no	—
261	C₂×C₄	C₅×#100	1	2	yes	—
262	C₂×C₄	C₅×#102	1	2	yes	—
263	C₂×C₄	C₅×#103	1	2	yes	—
264	C₂×C₄	C₅×#117	1	2	yes	—
265	C₂×C₄	C₅×#123	1	2	yes	—
266	C₂×C₄	C₅×#123	1	2	yes	—
267	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
268	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
269	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
270	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
271	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
272	C₂×C₄	C₅×#137	1	2	yes	—
273	C₈	—	8	2	no	—
274	C₈	—	8	2	no	—
275	C₅	#8724	1	1	yes	characteristic, direct product factor, Sylow
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	C₅×#332	1	4	yes	—
283	C₂²	C₅×#381	1	1	yes	characteristic
284	C₂²	C₂×C₅×#100	1	1	yes	characteristic
285	C₂²	C₂×C₅×#101	1	1	yes	characteristic
286	C₂²	C₂×C₅×#102	1	1	yes	characteristic
287	C₂²	C₂×C₅×#103	1	1	yes	characteristic
288	C₂²	C₅×#418	1	1	yes	characteristic
289	C₂²	C₅×#419	1	1	yes	characteristic
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	8	4	no	—
295	C₄	C₅×#480	1	2	yes	—
296	C₄	C₅×#481	1	2	yes	—
297	C₄	C₅×#483	1	2	yes	—
298	C₄	C₅×#484	1	2	yes	—
299	C₂	—	2	4	no	—
300	C₂	C₂×C₅×#332	1	1	yes	characteristic
301	C₂	C₅×#1545	1	1	yes	characteristic
302	C₂	C₅×#1548	1	1	yes	characteristic
303	C₂	C₅×#1556	1	1	yes	characteristic
304	C₂	C₅×#1821	1	1	yes	characteristic
305	C₂	C₅×#1823	1	1	yes	characteristic
306	C₂	C₅×#1824	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₅×#1296	C₂	semidirect
19. #8724	C₅	direct (N × H)
48. #1296	C₂×C₅	semidirect
275. C₅	#8724	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×#384	2
C₅×#1797	2
C₅×#1296	2
#8724	5

Subgroups — C5×#8724 #11931444

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8724 #11931444