Subgroups — C₅×#7562 #11930282

Order 1280.

462 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2295	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₅×#1744	C₂	1	1	yes	maximal, characteristic
4	C₅×#1272	C₂	1	2	yes	maximal
5	C₅×#1271	C₂	1	2	yes	maximal
6	C₂×C₄×C₅×Q₈	C₄	1	1	yes	characteristic
7	C₂×C₅×C₄⋊C₈	—	2	1	no	—
8	C₂×C₄×C₅×C₈	—	2	1	no	—
9	C₂×C₄×C₅×C₈	—	2	1	no	—
10	C₅×#391	C₄	1	1	yes	characteristic
11	C₅×#391	C₂²	1	1	yes	characteristic
12	C₅×#389	C₄	1	1	yes	characteristic
13	C₅×#389	C₂²	1	1	yes	characteristic
14	C₅×#387	—	2	1	no	—
15	C₅×#387	—	2	1	no	—
16	C₄×C₅×C₄⋊C₄	—	2	1	no	—
17	C₄×C₅×C₄⋊C₄	C₄	1	1	yes	characteristic
18	C₅×#376	—	2	1	no	—
19	C₄³×C₅	C₂²	1	1	yes	characteristic
20	C₅×#336	—	2	1	no	—
21	C₅×#336	C₂²	1	2	yes	—
22	C₅×#336	C₂²	1	2	yes	—
23	#7562	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂²×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₅×C₈	—	2	2	no	—
26	C₂²×C₅×C₈	—	2	2	no	—
27	C₄×C₅×Q₈	—	2	2	no	—
28	C₄×C₅×Q₈	C₈	1	4	yes	complemented
29	C₂×C₅×C₄⋊C₄	—	2	1	no	—
30	C₂×C₅×C₄⋊C₄	—	2	1	no	—
31	C₂×C₅×C₄⋊C₄	—	2	1	no	—
32	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
33	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
34	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₅	—	2	1	no	—
36	C₂×C₄²×C₅	—	2	1	no	—
37	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₅	D₄	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	C₂³	1	1	yes	characteristic
44	C₅×C₄⋊C₈	—	2	4	no	—
45	C₄×C₅×C₈	—	2	4	no	—
46	C₄×C₅×C₈	—	4	2	no	—
47	C₅×#95	—	2	1	no	—
48	C₅×#95	—	2	1	no	—
49	C₅×#95	—	2	1	no	—
50	C₅×#95	—	2	1	no	—
51	C₅×#95	—	2	1	no	—
52	C₅×#95	—	2	1	no	—
53	C₅×#95	—	2	1	no	—
54	#2295	C₂×C₅	1	1	yes	characteristic
55	C₄×C₄⋊C₈	C₂×C₅	1	1	yes	characteristic
56	#1744	C₂×C₅	1	1	yes	characteristic
57	#1272	C₂×C₅	1	2	yes	—
58	#1271	C₂×C₅	1	2	yes	—
59	C₂×C₅×Q₈	—	2	1	no	—
60	C₂×C₅×Q₈	—	4	1	no	—
61	C₂×C₅×Q₈	—	4	1	no	—
62	C₂×C₅×Q₈	C₂×C₈	1	2	yes	—
63	C₂²×C₄×C₅	—	2	1	no	—
64	C₂²×C₄×C₅	—	2	1	no	—
65	C₂²×C₄×C₅	—	2	1	no	—
66	C₂²×C₄×C₅	—	2	1	no	—
67	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
74	C₂×C₅×C₈	—	2	4	no	—
75	C₂×C₅×C₈	—	2	4	no	—
76	C₂×C₅×C₈	—	4	2	no	—
77	C₂×C₅×C₈	—	4	2	no	—
78	C₂×C₅×C₈	—	4	2	no	—
79	C₂×C₅×C₈	—	4	2	no	—
80	C₅×C₄⋊C₄	—	2	4	no	—
81	C₅×C₄⋊C₄	—	2	1	no	—
82	C₅×C₄⋊C₄	—	4	1	no	—
83	C₅×C₄⋊C₄	—	4	1	no	—
84	C₅×C₄⋊C₄	—	4	2	no	—
85	C₅×C₄⋊C₄	—	4	2	no	—
86	C₅×C₄⋊C₄	C₂×C₈	1	2	yes	—
87	C₄²×C₅	—	2	4	no	—
88	C₄²×C₅	—	2	2	no	—
89	C₄²×C₅	—	2	2	no	—
90	C₄²×C₅	—	2	2	no	—
91	C₄²×C₅	—	4	1	no	—
92	C₄²×C₅	—	4	1	no	—
93	C₄²×C₅	—	4	2	no	—
94	C₄²×C₅	C₂×C₈	1	2	yes	—
95	C₄²×C₅	M₁₆	1	2	yes	—
96	C₄²×C₅	SD₁₆	1	2	yes	—
97	C₄²×C₅	Q₁₆	1	2	yes	—
98	C₂×C₄×Q₈	C₄×C₅	1	1	yes	characteristic
99	C₂×C₄⋊C₈	—	2	1	no	—
100	C₂×C₄×C₈	—	2	1	no	—
101	C₂×C₄×C₈	—	2	1	no	—
102	#391	C₄×C₅	1	1	yes	characteristic
103	#391	C₂²×C₅	1	1	yes	characteristic
104	#389	C₄×C₅	1	1	yes	characteristic
105	#389	C₂²×C₅	1	1	yes	characteristic
106	#387	—	2	1	no	—
107	#387	—	2	1	no	—
108	C₄×C₄⋊C₄	—	2	1	no	—
109	C₄×C₄⋊C₄	C₄×C₅	1	1	yes	characteristic
110	#376	—	2	1	no	—
111	C₄³	C₂²×C₅	1	1	yes	characteristic
112	#336	—	2	1	no	—
113	#336	C₂²×C₅	1	2	yes	—
114	#336	C₂²×C₅	1	2	yes	—
115	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₅×Q₈	—	4	2	no	—
117	C₅×Q₈	—	4	2	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	1	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	—	2	2	no	—
131	C₂×C₄×C₅	—	2	2	no	—
132	C₂×C₄×C₅	—	4	1	no	—
133	C₂×C₄×C₅	—	4	1	no	—
134	C₂×C₄×C₅	—	4	1	no	—
135	C₂×C₄×C₅	—	4	1	no	—
136	C₂×C₄×C₅	—	4	1	no	—
137	C₂×C₄×C₅	—	4	1	no	—
138	C₂×C₄×C₅	—	4	1	no	—
139	C₂×C₄×C₅	—	4	1	no	—
140	C₂×C₄×C₅	—	4	1	no	—
141	C₂×C₄×C₅	—	4	1	no	—
142	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
143	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
144	C₂×C₄×C₅	#103	1	2	yes	—
145	C₂×C₄×C₅	#103	1	2	yes	—
146	C₂×C₄×C₅	#104	1	2	yes	—
147	C₂×C₄×C₅	#104	1	2	yes	—
148	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
152	C₂×C₄×C₅	#135	1	2	yes	—
153	C₅×C₈	—	4	4	no	—
154	C₅×C₈	—	4	4	no	—
155	C₂²×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	—	2	2	no	—
158	C₄×Q₈	—	2	2	no	—
159	C₄×Q₈	C₅×C₈	1	4	yes	complemented
160	C₂×C₄⋊C₄	—	2	1	no	—
161	C₂×C₄⋊C₄	—	2	1	no	—
162	C₂×C₄⋊C₄	—	2	1	no	—
163	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
164	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
165	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
166	C₂×C₄²	—	2	1	no	—
167	C₂×C₄²	—	2	1	no	—
168	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
169	C₂×C₄²	C₂×C₄×C₅	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	C₅×D₄	1	1	yes	characteristic
174	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
175	C₄⋊C₈	—	2	4	no	—
176	C₄×C₈	—	2	4	no	—
177	C₄×C₈	—	4	2	no	—
178	#95	—	2	1	no	—
179	#95	—	2	1	no	—
180	#95	—	2	1	no	—
181	#95	—	2	1	no	—
182	#95	—	2	1	no	—
183	#95	—	2	1	no	—
184	#95	—	2	1	no	—
185	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
186	C₂²×C₅	#410	1	1	yes	characteristic
187	C₂²×C₅	#413	1	1	yes	characteristic
188	C₂²×C₅	C₂×#103	1	1	yes	characteristic
189	C₂²×C₅	#416	1	1	yes	characteristic
190	C₂²×C₅	C₂×#104	1	1	yes	characteristic
191	C₂²×C₅	C₂×#104	1	1	yes	characteristic
192	C₄×C₅	—	2	2	no	—
193	C₄×C₅	—	2	1	no	—
194	C₄×C₅	—	2	1	no	—
195	C₄×C₅	—	2	2	no	—
196	C₄×C₅	—	2	2	no	—
197	C₄×C₅	—	2	1	no	—
198	C₄×C₅	—	2	1	no	—
199	C₄×C₅	—	2	2	no	—
200	C₄×C₅	—	4	2	no	—
201	C₄×C₅	—	4	2	no	—
202	C₄×C₅	—	8	1	no	—
203	C₄×C₅	—	8	1	no	—
204	C₄×C₅	Q₈⋊C₈	1	4	yes	—
205	C₂×Q₈	—	2	1	no	—
206	C₂×Q₈	—	4	1	no	—
207	C₂×Q₈	—	4	1	no	—
208	C₂×Q₈	C₂×C₅×C₈	1	2	yes	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
214	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
215	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
216	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
217	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₄⋊C₄	—	2	1	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	4	1	no	—
229	C₄⋊C₄	—	4	1	no	—
230	C₄⋊C₄	—	4	2	no	—
231	C₄⋊C₄	—	4	2	no	—
232	C₄⋊C₄	C₂×C₅×C₈	1	2	yes	—
233	C₄²	—	2	2	no	—
234	C₄²	—	2	4	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	2	2	no	—
237	C₄²	—	4	1	no	—
238	C₄²	—	4	1	no	—
239	C₄²	—	4	2	no	—
240	C₄²	C₂×C₅×C₈	1	2	yes	—
241	C₄²	C₅×M₁₆	1	2	yes	—
242	C₄²	C₅×SD₁₆	1	2	yes	—
243	C₄²	C₅×Q₁₆	1	2	yes	—
244	C₂×C₅	#1458	1	1	yes	characteristic
245	C₂×C₅	C₂×Q₈⋊C₈	1	1	yes	characteristic
246	C₂×C₅	#1476	1	1	yes	characteristic
247	C₂×C₅	#1500	1	1	yes	characteristic
248	C₂×C₅	#1503	1	1	yes	characteristic
249	C₂×C₅	#1524	1	1	yes	characteristic
250	C₂×C₅	#1542	1	1	yes	characteristic
251	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
252	Q₈	—	4	2	no	—
253	Q₈	—	4	2	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	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	C₅×C₂²⋊C₈	1	2	yes	—
279	C₂×C₄	C₅×C₂²⋊C₈	1	2	yes	—
280	C₂×C₄	C₅×#103	1	2	yes	—
281	C₂×C₄	C₅×#103	1	2	yes	—
282	C₂×C₄	C₅×#104	1	2	yes	—
283	C₂×C₄	C₅×#104	1	2	yes	—
284	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
285	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
286	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
287	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
288	C₂×C₄	C₅×#135	1	2	yes	—
289	C₈	—	4	4	no	—
290	C₈	—	4	4	no	—
291	C₅	#7562	1	1	yes	characteristic, direct product factor, Sylow
292	C₂²	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
293	C₂²	C₅×#410	1	1	yes	characteristic
294	C₂²	C₅×#413	1	1	yes	characteristic
295	C₂²	C₂×C₅×#103	1	1	yes	characteristic
296	C₂²	C₅×#416	1	1	yes	characteristic
297	C₂²	C₂×C₅×#104	1	1	yes	characteristic
298	C₂²	C₂×C₅×#104	1	1	yes	characteristic
299	C₄	—	2	2	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	2	no	—
303	C₄	—	2	2	no	—
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	2	no	—
307	C₄	—	4	2	no	—
308	C₄	—	4	2	no	—
309	C₄	—	8	1	no	—
310	C₄	—	8	1	no	—
311	C₄	C₅×Q₈⋊C₈	1	4	yes	—
312	C₂	C₅×#1458	1	1	yes	characteristic
313	C₂	C₂×C₅×Q₈⋊C₈	1	1	yes	characteristic
314	C₂	C₅×#1476	1	1	yes	characteristic
315	C₂	C₅×#1500	1	1	yes	characteristic
316	C₂	C₅×#1503	1	1	yes	characteristic
317	C₂	C₅×#1524	1	1	yes	characteristic
318	C₂	C₅×#1542	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₅×C₈
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₄³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
23. #7562	C₅	direct (N × H)
28. C₄×C₅×Q₈	C₈	semidirect
159. C₄×Q₈	C₅×C₈	semidirect
291. C₅	#7562	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#2295	2
C₄×C₅×C₄⋊C₈	2
C₅×#1744	2
C₅×#1272	2
C₅×#1271	2
#7562	5

Subgroups — C5×#7562 #11930282

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#7562 #11930282