Subgroups — C₅×#8590 #11931310

Order 1280.

496 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₅×#1842	C₂	1	1	yes	maximal, characteristic
3	C₅×#1831	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#333	C₂	1	2	yes	maximal
5	C₅×#1296	C₂	1	2	yes	maximal
6	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₅×Q₈	C₄	1	2	yes	—
8	C₂×C₅×C₄⋊C₈	—	2	1	no	—
9	C₂×C₅×C₄⋊C₈	—	2	1	no	—
10	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
12	C₅×#391	C₄	1	2	yes	—
13	C₅×#389	—	2	1	no	—
14	C₅×#389	—	2	1	no	—
15	C₅×#384	C₂²	1	1	yes	characteristic
16	C₄×C₅×C₄⋊C₄	—	2	1	no	—
17	C₄×C₅×C₄⋊C₄	—	2	1	no	—
18	C₄³×C₅	C₂²	1	1	yes	characteristic
19	C₅×#336	—	2	1	no	—
20	C₅×#336	—	2	1	no	—
21	C₅×#333	—	2	4	no	—
22	#8590	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂²×C₅×Q₈	C₂×C₄	1	2	yes	—
24	C₂²×C₅×C₈	—	2	2	no	—
25	C₂²×C₅×C₈	—	2	2	no	—
26	C₅×C₄⋊Q₈	—	2	1	no	—
27	C₅×C₄⋊Q₈	—	2	1	no	—
28	C₅×C₄⋊Q₈	D₄	1	2	yes	—
29	C₅×C₄⋊Q₈	Q₈	1	2	yes	—
30	C₄×C₅×Q₈	—	4	4	no	—
31	C₂×C₅×C₄⋊C₄	—	2	1	no	—
32	C₂×C₅×C₄⋊C₄	—	2	1	no	—
33	C₂×C₅×C₄⋊C₄	—	2	2	no	—
34	C₂×C₅×C₄⋊C₄	—	2	1	no	—
35	C₂×C₅×C₄⋊C₄	—	2	1	no	—
36	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₅	—	2	2	no	—
38	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
45	C₅×C₄⋊C₈	—	2	2	no	—
46	C₅×C₄⋊C₈	—	2	2	no	—
47	C₅×C₄⋊C₈	—	2	2	no	—
48	C₅×C₄⋊C₈	—	2	2	no	—
49	C₅×C₄⋊C₈	—	4	2	no	—
50	C₅×C₄⋊C₈	—	4	2	no	—
51	C₅×#95	—	2	2	no	—
52	C₅×#95	—	2	2	no	—
53	C₄×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
54	#1842	C₂×C₅	1	1	yes	characteristic
55	#1831	C₂×C₅	1	1	yes	characteristic
56	C₂×#333	C₂×C₅	1	2	yes	—
57	#1296	C₂×C₅	1	2	yes	—
58	C₂×C₅×Q₈	—	2	2	no	—
59	C₂×C₅×Q₈	—	4	2	no	—
60	C₂×C₅×Q₈	—	4	2	no	—
61	C₂×C₅×Q₈	C₄⋊C₄	1	4	yes	—
62	C₂²×C₄×C₅	—	2	2	no	—
63	C₂²×C₄×C₅	—	2	2	no	—
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	1	yes	characteristic
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	—	2	2	no	—
78	C₅×C₄⋊C₄	—	2	2	no	—
79	C₅×C₄⋊C₄	—	2	2	no	—
80	C₅×C₄⋊C₄	—	2	2	no	—
81	C₅×C₄⋊C₄	—	4	2	no	—
82	C₅×C₄⋊C₄	—	4	4	no	—
83	C₅×C₄⋊C₄	—	4	2	no	—
84	C₅×C₄⋊C₄	—	4	1	no	—
85	C₅×C₄⋊C₄	—	4	1	no	—
86	C₄²×C₅	—	2	2	no	—
87	C₄²×C₅	—	2	2	no	—
88	C₄²×C₅	—	2	1	no	—
89	C₄²×C₅	—	2	1	no	—
90	C₄²×C₅	—	2	2	no	—
91	C₄²×C₅	—	4	1	no	—
92	C₄²×C₅	—	4	4	no	—
93	C₄²×C₅	SD₁₆	1	2	yes	—
94	C₄²×C₅	Q₁₆	1	2	yes	—
95	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
97	C₄²×C₅	#41	1	1	yes	characteristic
98	C₄²×C₅	#41	1	1	yes	characteristic
99	C₂×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
100	C₂×C₄×Q₈	C₄×C₅	1	2	yes	—
101	C₂×C₄⋊C₈	—	2	1	no	—
102	C₂×C₄⋊C₈	—	2	1	no	—
103	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
104	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
105	#391	C₄×C₅	1	2	yes	—
106	#389	—	2	1	no	—
107	#389	—	2	1	no	—
108	#384	C₂²×C₅	1	1	yes	characteristic
109	C₄×C₄⋊C₄	—	2	1	no	—
110	C₄×C₄⋊C₄	—	2	1	no	—
111	C₄³	C₂²×C₅	1	1	yes	characteristic
112	#336	—	2	1	no	—
113	#336	—	2	1	no	—
114	#333	—	2	4	no	—
115	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₅×Q₈	—	4	4	no	—
117	C₅×Q₈	—	4	4	no	—
118	C₂×C₄×C₅	—	2	1	no	—
119	C₂×C₄×C₅	—	2	1	no	—
120	C₂×C₄×C₅	—	2	1	no	—
121	C₂×C₄×C₅	—	2	1	no	—
122	C₂×C₄×C₅	—	2	1	no	—
123	C₂×C₄×C₅	—	2	1	no	—
124	C₂×C₄×C₅	—	2	1	no	—
125	C₂×C₄×C₅	—	2	1	no	—
126	C₂×C₄×C₅	—	2	4	no	—
127	C₂×C₄×C₅	—	2	1	no	—
128	C₂×C₄×C₅	—	2	1	no	—
129	C₂×C₄×C₅	—	2	1	no	—
130	C₂×C₄×C₅	—	4	2	no	—
131	C₂×C₄×C₅	—	4	2	no	—
132	C₂×C₄×C₅	—	4	2	no	—
133	C₂×C₄×C₅	—	4	2	no	—
134	C₂×C₄×C₅	—	4	2	no	—
135	C₂×C₄×C₅	#100	1	2	yes	—
136	C₂×C₄×C₅	#103	1	2	yes	—
137	C₂×C₄×C₅	#103	1	2	yes	—
138	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₅	#117	1	1	yes	characteristic
140	C₂×C₄×C₅	#121	1	1	yes	characteristic
141	C₂×C₄×C₅	#121	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
146	C₂×C₄×C₅	#135	1	2	yes	—
147	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₂×C₄×C₅	#137	1	1	yes	characteristic
149	C₂×C₄×C₅	#137	1	2	yes	—
150	C₅×C₈	—	8	2	no	—
151	C₅×C₈	—	8	2	no	—
152	C₂²×Q₈	C₂×C₄×C₅	1	2	yes	—
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	—	2	2	no	—
155	C₄⋊Q₈	—	2	1	no	—
156	C₄⋊Q₈	—	2	1	no	—
157	C₄⋊Q₈	C₅×D₄	1	2	yes	—
158	C₄⋊Q₈	C₅×Q₈	1	2	yes	—
159	C₄×Q₈	—	4	4	no	—
160	C₂×C₄⋊C₄	—	2	1	no	—
161	C₂×C₄⋊C₄	—	2	1	no	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	—	2	1	no	—
164	C₂×C₄⋊C₄	—	2	1	no	—
165	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
166	C₂×C₄²	—	2	2	no	—
167	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
168	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
169	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
170	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
171	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
172	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
173	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
174	C₄⋊C₈	—	2	2	no	—
175	C₄⋊C₈	—	2	2	no	—
176	C₄⋊C₈	—	2	2	no	—
177	C₄⋊C₈	—	2	2	no	—
178	C₄⋊C₈	—	4	2	no	—
179	C₄⋊C₈	—	4	2	no	—
180	#95	—	2	2	no	—
181	#95	—	2	2	no	—
182	C₂²×C₅	#380	1	1	yes	characteristic
183	C₂²×C₅	C₂×#100	1	1	yes	characteristic
184	C₂²×C₅	#413	1	1	yes	characteristic
185	C₂²×C₅	C₂×#103	1	1	yes	characteristic
186	C₂²×C₅	#416	1	1	yes	characteristic
187	C₂²×C₅	#417	1	1	yes	characteristic
188	C₂²×C₅	#419	1	1	yes	characteristic
189	C₄×C₅	—	2	2	no	—
190	C₄×C₅	—	2	1	no	—
191	C₄×C₅	—	2	1	no	—
192	C₄×C₅	—	2	2	no	—
193	C₄×C₅	—	2	2	no	—
194	C₄×C₅	—	4	4	no	—
195	C₄×C₅	—	4	1	no	—
196	C₄×C₅	—	8	2	no	—
197	C₄×C₅	#460	1	1	yes	characteristic
198	C₄×C₅	#462	1	1	yes	characteristic
199	C₄×C₅	#464	1	2	yes	—
200	C₄×C₅	#475	1	1	yes	characteristic
201	C₄×C₅	#477	1	1	yes	characteristic
202	C₄×C₅	#479	1	2	yes	—
203	C₂×Q₈	—	2	2	no	—
204	C₂×Q₈	—	4	2	no	—
205	C₂×Q₈	—	4	2	no	—
206	C₂×Q₈	C₅×C₄⋊C₄	1	4	yes	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
210	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
211	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
212	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
213	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	—	4	2	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	4	2	no	—
221	C₂×C₈	—	4	2	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	4	2	no	—
227	C₄⋊C₄	—	4	2	no	—
228	C₄⋊C₄	—	4	1	no	—
229	C₄⋊C₄	—	4	1	no	—
230	C₄⋊C₄	—	4	4	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	2	1	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	4	1	no	—
237	C₄²	—	4	4	no	—
238	C₄²	C₅×SD₁₆	1	2	yes	—
239	C₄²	C₅×Q₁₆	1	2	yes	—
240	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
241	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
242	C₄²	C₅×#41	1	1	yes	characteristic
243	C₄²	C₅×#41	1	1	yes	characteristic
244	C₂×C₅	#1542	1	1	yes	characteristic
245	C₂×C₅	#1548	1	1	yes	characteristic
246	C₂×C₅	#1551	1	1	yes	characteristic
247	C₂×C₅	#1559	1	1	yes	characteristic
248	C₂×C₅	#1800	1	1	yes	characteristic
249	C₂×C₅	#1803	1	1	yes	characteristic
250	C₂×C₅	#1805	1	1	yes	characteristic
251	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
252	Q₈	—	4	4	no	—
253	Q₈	—	4	4	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	C₅×#100	1	2	yes	—
272	C₂×C₄	C₅×#103	1	2	yes	—
273	C₂×C₄	C₅×#103	1	2	yes	—
274	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
275	C₂×C₄	C₅×#117	1	1	yes	characteristic
276	C₂×C₄	C₅×#121	1	1	yes	characteristic
277	C₂×C₄	C₅×#121	1	1	yes	characteristic
278	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
279	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
280	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
281	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
282	C₂×C₄	C₅×#135	1	2	yes	—
283	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
284	C₂×C₄	C₅×#137	1	1	yes	characteristic
285	C₂×C₄	C₅×#137	1	2	yes	—
286	C₈	—	8	2	no	—
287	C₈	—	8	2	no	—
288	C₅	#8590	1	1	yes	characteristic, direct product factor, Sylow
289	C₂²	C₅×#380	1	1	yes	characteristic
290	C₂²	C₂×C₅×#100	1	1	yes	characteristic
291	C₂²	C₅×#413	1	1	yes	characteristic
292	C₂²	C₂×C₅×#103	1	1	yes	characteristic
293	C₂²	C₅×#416	1	1	yes	characteristic
294	C₂²	C₅×#417	1	1	yes	characteristic
295	C₂²	C₅×#419	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	2	no	—
301	C₄	—	4	4	no	—
302	C₄	—	4	1	no	—
303	C₄	—	8	2	no	—
304	C₄	C₅×#460	1	1	yes	characteristic
305	C₄	C₅×#462	1	1	yes	characteristic
306	C₄	C₅×#464	1	2	yes	—
307	C₄	C₅×#475	1	1	yes	characteristic
308	C₄	C₅×#477	1	1	yes	characteristic
309	C₄	C₅×#479	1	2	yes	—
310	C₂	C₅×#1542	1	1	yes	characteristic
311	C₂	C₅×#1548	1	1	yes	characteristic
312	C₂	C₅×#1551	1	1	yes	characteristic
313	C₂	C₅×#1559	1	1	yes	characteristic
314	C₂	C₅×#1800	1	1	yes	characteristic
315	C₂	C₅×#1803	1	1	yes	characteristic
316	C₂	C₅×#1805	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₄³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
22. #8590	C₅	direct (N × H)
288. C₅	#8590	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×C₅×C₄⋊Q₈	2
C₅×#1842	2
C₅×#1831	2
C₂×C₅×#333	2
C₅×#1296	2
#8590	5

Subgroups — C5×#8590 #11931310

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8590 #11931310