Subgroups — C₅×#11331 #11934051

Order 1280.

722 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#428	C₂	1	2	yes	maximal
2	C₂²×C₅×#107	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#343	C₂	1	2	yes	maximal
4	C₂×C₅×#341	C₂	1	1	yes	maximal, characteristic
5	C₅×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
8	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₅×#117	C₂²	1	1	yes	characteristic
10	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	—
11	C₅×#428	—	2	4	no	—
12	C₅×#428	—	2	4	no	—
13	C₂×C₅×#107	—	2	2	no	—
14	C₂×C₅×#107	—	2	4	no	—
15	C₂×C₅×#107	—	2	2	no	—
16	C₂×C₅×#106	—	2	2	no	—
17	C₅×#343	—	2	4	no	—
18	C₅×#341	C₄	1	4	yes	complemented
19	C₅×#336	—	2	2	no	—
20	#11331	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂³×C₄×C₅	—	2	2	no	—
22	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
23	C₂×C₅×M₁₆	—	2	2	no	—
24	C₂×C₅×M₁₆	—	2	2	no	—
25	C₂×C₅×M₁₆	—	2	2	no	—
26	C₂×C₅×M₁₆	—	2	1	no	—
27	C₂×C₅×M₁₆	—	2	2	no	—
28	C₂×C₅×M₁₆	—	2	2	no	—
29	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
30	C₂×C₅×M₁₆	D₄	1	4	yes	—
31	C₂×C₅×M₁₆	Q₈	1	2	yes	—
32	C₂×C₅×M₁₆	Q₈	1	4	yes	—
33	C₂²×C₅×C₈	—	2	2	no	—
34	C₂²×C₅×C₈	—	2	2	no	—
35	C₂²×C₅×C₈	—	2	2	no	—
36	C₂²×C₅×C₈	—	2	1	no	—
37	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
38	C₂²×C₅×C₈	D₄	1	2	yes	—
39	C₂²×C₅×C₈	Q₈	1	2	yes	—
40	C₅×#117	—	2	2	no	—
41	C₅×#117	—	2	2	no	—
42	C₂×C₅×C₄⋊C₄	—	2	2	no	—
43	C₂×C₅×C₄⋊C₄	—	2	2	no	—
44	C₂×C₅×C₄⋊C₄	—	2	2	no	—
45	C₂×C₅×C₄⋊C₄	—	2	2	no	—
46	C₂×C₅×C₄⋊C₄	—	2	2	no	—
47	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
48	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
49	C₂×C₅×C₄⋊C₄	D₄	1	4	yes	—
50	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
51	C₂×C₄²×C₅	—	2	1	no	—
52	C₅×#107	—	2	8	no	—
53	C₅×#107	—	2	8	no	—
54	C₅×#106	—	2	8	no	—
55	C₂×#428	C₂×C₅	1	2	yes	—
56	C₂²×#107	C₂×C₅	1	1	yes	characteristic
57	C₂×#343	C₂×C₅	1	2	yes	—
58	C₂×#341	C₂×C₅	1	1	yes	characteristic
59	#1729	C₂×C₅	1	1	yes	characteristic
60	C₂⁴×C₅	#41	1	1	yes	characteristic
61	C₂²×C₄×C₅	—	2	1	no	—
62	C₂²×C₄×C₅	—	2	1	no	—
63	C₂²×C₄×C₅	—	2	4	no	—
64	C₂²×C₄×C₅	—	4	2	no	—
65	C₂²×C₄×C₅	—	4	2	no	—
66	C₂²×C₄×C₅	—	8	4	no	—
67	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
74	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
75	C₂²×C₄×C₅	#41	1	2	yes	—
76	C₂²×C₄×C₅	#41	1	1	yes	characteristic
77	C₅×M₁₆	—	2	4	no	—
78	C₅×M₁₆	—	2	4	no	—
79	C₅×M₁₆	—	2	4	no	—
80	C₅×M₁₆	—	2	4	no	—
81	C₅×M₁₆	—	4	2	no	—
82	C₅×M₁₆	C₄⋊C₄	1	8	yes	—
83	C₂×C₅×C₈	—	2	4	no	—
84	C₂×C₅×C₈	—	2	4	no	—
85	C₂×C₅×C₈	—	2	2	no	—
86	C₂×C₅×C₈	—	2	4	no	—
87	C₂×C₅×C₈	—	2	4	no	—
88	C₂×C₅×C₈	—	2	4	no	—
89	C₂×C₅×C₈	—	2	2	no	—
90	C₂×C₅×C₈	—	2	2	no	—
91	C₂×C₅×C₈	—	4	1	no	—
92	C₂×C₅×C₈	—	4	1	no	—
93	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
94	C₅×C₄⋊C₄	—	2	4	no	—
95	C₅×C₄⋊C₄	—	2	4	no	—
96	C₅×C₄⋊C₄	—	2	2	no	—
97	C₅×C₄⋊C₄	—	2	2	no	—
98	C₅×C₄⋊C₄	—	4	2	no	—
99	C₅×C₄⋊C₄	—	4	2	no	—
100	C₅×C₂²⋊C₄	—	4	2	no	—
101	C₄²×C₅	—	4	1	no	—
102	C₄²×C₅	—	4	1	no	—
103	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
104	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
105	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
106	C₂×#117	C₂²×C₅	1	1	yes	characteristic
107	C₂²×C₄⋊C₄	C₂²×C₅	1	2	yes	—
108	#428	—	2	4	no	—
109	#428	—	2	4	no	—
110	C₂×#107	—	2	2	no	—
111	C₂×#107	—	2	4	no	—
112	C₂×#107	—	2	2	no	—
113	C₂×#106	—	2	2	no	—
114	#343	—	2	4	no	—
115	#341	C₄×C₅	1	4	yes	complemented
116	#336	—	2	2	no	—
117	C₂³×C₅	—	2	2	no	—
118	C₂³×C₅	—	2	2	no	—
119	C₂³×C₅	C₄×Q₈	1	1	yes	characteristic
120	C₂³×C₅	#121	1	2	yes	—
121	C₂³×C₅	#121	1	1	yes	characteristic
122	C₂³×C₅	#125	1	2	yes	—
123	C₂³×C₅	#125	1	1	yes	characteristic
124	C₂×C₄×C₅	—	4	1	no	—
125	C₂×C₄×C₅	—	4	1	no	—
126	C₂×C₄×C₅	—	4	4	no	—
127	C₂×C₄×C₅	—	4	2	no	—
128	C₂×C₄×C₅	—	4	2	no	—
129	C₂×C₄×C₅	—	8	4	no	—
130	C₂×C₄×C₅	—	8	4	no	—
131	C₂×C₄×C₅	—	8	4	no	—
132	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
134	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
136	C₂×C₄×C₅	#120	1	1	yes	characteristic
137	C₂×C₄×C₅	#120	1	2	yes	—
138	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
139	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
140	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
141	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
143	C₂×C₄×C₅	#123	1	1	yes	characteristic
144	C₂×C₄×C₅	#123	1	2	yes	—
145	C₂×C₄×C₅	#127	1	1	yes	characteristic
146	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
147	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
148	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
149	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
150	C₅×C₈	—	2	4	no	—
151	C₅×C₈	—	2	8	no	—
152	C₅×C₈	—	4	2	no	—
153	C₂³×C₄	—	2	2	no	—
154	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
155	C₂×M₁₆	—	2	2	no	—
156	C₂×M₁₆	—	2	2	no	—
157	C₂×M₁₆	—	2	2	no	—
158	C₂×M₁₆	—	2	1	no	—
159	C₂×M₁₆	—	2	2	no	—
160	C₂×M₁₆	—	2	2	no	—
161	C₂×M₁₆	C₂×C₄×C₅	1	4	yes	—
162	C₂×M₁₆	C₅×D₄	1	4	yes	—
163	C₂×M₁₆	C₅×Q₈	1	2	yes	—
164	C₂×M₁₆	C₅×Q₈	1	4	yes	—
165	C₂²×C₈	—	2	2	no	—
166	C₂²×C₈	—	2	2	no	—
167	C₂²×C₈	—	2	2	no	—
168	C₂²×C₈	—	2	1	no	—
169	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
170	C₂²×C₈	C₅×D₄	1	2	yes	—
171	C₂²×C₈	C₅×Q₈	1	2	yes	—
172	#117	—	2	2	no	—
173	#117	—	2	2	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₄⋊C₄	—	2	2	no	—
176	C₂×C₄⋊C₄	—	2	2	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₄⋊C₄	C₅×D₄	1	4	yes	—
180	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
181	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
182	C₂×C₂²⋊C₄	—	2	1	no	—
183	C₂×C₄²	—	2	1	no	—
184	#107	—	2	8	no	—
185	#107	—	2	8	no	—
186	#106	—	2	8	no	—
187	C₂²×C₅	—	2	4	no	—
188	C₂²×C₅	—	2	2	no	—
189	C₂²×C₅	—	2	2	no	—
190	C₂²×C₅	—	2	2	no	—
191	C₂²×C₅	—	2	2	no	—
192	C₂²×C₅	#389	1	2	yes	—
193	C₂²×C₅	#389	1	1	yes	characteristic
194	C₂²×C₅	#392	1	1	yes	characteristic
195	C₂²×C₅	#398	1	2	yes	—
196	C₂²×C₅	#398	1	1	yes	characteristic
197	C₂²×C₅	#472	1	2	yes	—
198	C₂²×C₅	#473	1	2	yes	—
199	C₄×C₅	—	8	1	no	—
200	C₄×C₅	—	8	4	no	—
201	C₄×C₅	—	8	1	no	—
202	C₄×C₅	#384	1	1	yes	characteristic
203	C₄×C₅	#385	1	1	yes	characteristic
204	C₄×C₅	#385	1	2	yes	—
205	C₄×C₅	#395	1	1	yes	characteristic
206	C₄×C₅	#395	1	2	yes	—
207	C₄×C₅	#399	1	1	yes	characteristic
208	C₂⁴	C₅×#41	1	1	yes	characteristic
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	4	2	no	—
213	C₂²×C₄	—	4	2	no	—
214	C₂²×C₄	—	8	4	no	—
215	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
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	2	yes	—
220	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
222	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
223	C₂²×C₄	C₅×#41	1	2	yes	—
224	C₂²×C₄	C₅×#41	1	1	yes	characteristic
225	M₁₆	—	2	4	no	—
226	M₁₆	—	2	4	no	—
227	M₁₆	—	2	4	no	—
228	M₁₆	—	2	4	no	—
229	M₁₆	—	4	2	no	—
230	M₁₆	C₅×C₄⋊C₄	1	8	yes	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	4	1	no	—
240	C₂×C₈	—	4	1	no	—
241	C₂×C₈	C₅×C₄⋊C₄	1	4	yes	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	—	4	2	no	—
248	C₂²⋊C₄	—	4	2	no	—
249	C₄²	—	4	1	no	—
250	C₄²	—	4	1	no	—
251	C₂×C₅	—	2	2	no	—
252	C₂×C₅	—	2	2	no	—
253	C₂×C₅	#1444	1	1	yes	characteristic
254	C₂×C₅	#1952	1	2	yes	—
255	C₂×C₅	#2018	1	1	yes	characteristic
256	C₂×C₅	#2020	1	1	yes	characteristic
257	C₂×C₅	#2089	1	2	yes	—
258	C₂³	—	2	2	no	—
259	C₂³	—	2	2	no	—
260	C₂³	C₄×C₅×Q₈	1	1	yes	characteristic
261	C₂³	C₅×#121	1	2	yes	—
262	C₂³	C₅×#121	1	1	yes	characteristic
263	C₂³	C₅×#125	1	2	yes	—
264	C₂³	C₅×#125	1	1	yes	characteristic
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	8	4	no	—
271	C₂×C₄	—	8	4	no	—
272	C₂×C₄	—	8	4	no	—
273	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
274	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
275	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
276	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
277	C₂×C₄	C₅×#120	1	1	yes	characteristic
278	C₂×C₄	C₅×#120	1	2	yes	—
279	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
280	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
281	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
282	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
283	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
284	C₂×C₄	C₅×#123	1	1	yes	characteristic
285	C₂×C₄	C₅×#123	1	2	yes	—
286	C₂×C₄	C₅×#127	1	1	yes	characteristic
287	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
288	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
289	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
290	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
291	C₈	—	2	4	no	—
292	C₈	—	2	8	no	—
293	C₈	—	4	2	no	—
294	C₅	#11331	1	1	yes	characteristic, direct product factor, Sylow
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	4	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	C₅×#389	1	2	yes	—
301	C₂²	C₅×#389	1	1	yes	characteristic
302	C₂²	C₅×#392	1	1	yes	characteristic
303	C₂²	C₅×#398	1	2	yes	—
304	C₂²	C₅×#398	1	1	yes	characteristic
305	C₂²	C₅×#472	1	2	yes	—
306	C₂²	C₅×#473	1	2	yes	—
307	C₄	—	8	1	no	—
308	C₄	—	8	4	no	—
309	C₄	—	8	1	no	—
310	C₄	C₅×#384	1	1	yes	characteristic
311	C₄	C₅×#385	1	1	yes	characteristic
312	C₄	C₅×#385	1	2	yes	—
313	C₄	C₅×#395	1	1	yes	characteristic
314	C₄	C₅×#395	1	2	yes	—
315	C₄	C₅×#399	1	1	yes	characteristic
316	C₂	—	2	2	no	—
317	C₂	—	2	2	no	—
318	C₂	C₅×#1444	1	1	yes	characteristic
319	C₂	C₅×#1952	1	2	yes	—
320	C₂	C₅×#2018	1	1	yes	characteristic
321	C₂	C₅×#2020	1	1	yes	characteristic
322	C₂	C₅×#2089	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅	#121
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
prime radical	C₂⁴×C₅	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂³×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
18. C₅×#341	C₄	semidirect
20. #11331	C₅	direct (N × H)
115. #341	C₄×C₅	semidirect
294. C₅	#11331	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×#428	2
C₂²×C₅×#107	2
C₂×C₅×#343	2
C₂×C₅×#341	2
C₅×#1729	2
#11331	5

Subgroups — C5×#11331 #11934051

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#11331 #11934051