Subgroups — C₅×#11329 #11934049

Order 1280.

750 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#108	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#424	C₂	1	2	yes	maximal
3	C₅×#1735	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#339	C₂	1	2	yes	maximal
5	C₂×C₅×#336	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
7	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₅×#117	C₂²	1	2	yes	—
10	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₅×#108	—	2	4	no	—
12	C₂×C₅×#108	—	2	2	no	—
13	C₅×#424	—	2	4	no	—
14	C₅×#424	—	2	4	no	—
15	C₂×C₅×C₄⋊C₈	—	2	4	no	—
16	C₅×#340	C₄	1	4	yes	complemented
17	C₅×#339	—	2	4	no	—
18	C₅×#336	—	2	2	no	—
19	#11329	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂³×C₄×C₅	—	2	1	no	—
21	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
22	C₂×C₅×M₁₆	—	2	2	no	—
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	2	no	—
27	C₂×C₅×M₁₆	D₄	1	4	yes	—
28	C₂²×C₅×C₈	—	2	2	no	—
29	C₂²×C₅×C₈	—	2	2	no	—
30	C₂²×C₅×C₈	—	2	1	no	—
31	C₂²×C₅×C₈	—	2	2	no	—
32	C₂²×C₅×C₈	—	2	2	no	—
33	C₂²×C₅×C₈	—	2	2	no	—
34	C₂²×C₅×C₈	—	2	1	no	—
35	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
36	C₂²×C₅×C₈	D₄	1	2	yes	—
37	C₂²×C₅×C₈	D₄	1	2	yes	—
38	C₂²×C₅×C₈	Q₈	1	2	yes	—
39	C₅×#117	—	2	2	no	—
40	C₅×#117	—	2	2	no	—
41	C₅×#117	D₄	1	4	yes	—
42	C₅×#117	Q₈	1	4	yes	—
43	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
44	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
45	C₂×C₄²×C₅	—	2	2	no	—
46	C₂×C₄²×C₅	—	2	2	no	—
47	C₂×C₄²×C₅	—	2	2	no	—
48	C₂×C₄²×C₅	—	2	1	no	—
49	C₂×C₄²×C₅	Q₈	1	2	yes	—
50	C₅×#108	—	2	8	no	—
51	C₅×C₄⋊C₈	—	2	16	no	—
52	C₂²×#108	C₂×C₅	1	1	yes	characteristic
53	C₂×#424	C₂×C₅	1	2	yes	—
54	#1735	C₂×C₅	1	1	yes	characteristic
55	C₂×#339	C₂×C₅	1	2	yes	—
56	C₂×#336	C₂×C₅	1	1	yes	characteristic
57	C₂⁴×C₅	#41	1	1	yes	characteristic
58	C₂²×C₄×C₅	—	2	4	no	—
59	C₂²×C₄×C₅	—	2	4	no	—
60	C₂²×C₄×C₅	—	2	2	no	—
61	C₂²×C₄×C₅	—	4	2	no	—
62	C₂²×C₄×C₅	—	4	2	no	—
63	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
71	C₂²×C₄×C₅	#41	1	2	yes	—
72	C₂²×C₄×C₅	#41	1	1	yes	characteristic
73	C₅×M₁₆	—	2	4	no	—
74	C₅×M₁₆	—	2	4	no	—
75	C₅×M₁₆	—	4	2	no	—
76	C₅×M₁₆	—	4	2	no	—
77	C₂×C₅×C₈	—	2	4	no	—
78	C₂×C₅×C₈	—	2	2	no	—
79	C₂×C₅×C₈	—	2	4	no	—
80	C₂×C₅×C₈	—	2	2	no	—
81	C₂×C₅×C₈	—	2	4	no	—
82	C₂×C₅×C₈	—	2	4	no	—
83	C₂×C₅×C₈	—	2	2	no	—
84	C₂×C₅×C₈	—	4	1	no	—
85	C₂×C₅×C₈	—	4	2	no	—
86	C₂×C₅×C₈	—	4	2	no	—
87	C₂×C₅×C₈	—	4	2	no	—
88	C₂×C₅×C₈	—	4	1	no	—
89	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
90	C₅×C₄⋊C₄	—	2	4	no	—
91	C₅×C₄⋊C₄	C₄⋊C₄	1	8	yes	—
92	C₅×C₂²⋊C₄	—	2	4	no	—
93	C₅×C₂²⋊C₄	—	2	4	no	—
94	C₄²×C₅	—	2	4	no	—
95	C₄²×C₅	—	2	4	no	—
96	C₄²×C₅	—	2	4	no	—
97	C₄²×C₅	—	4	1	no	—
98	C₄²×C₅	—	4	1	no	—
99	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
100	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
101	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
102	C₂×#117	C₂²×C₅	1	2	yes	—
103	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
104	C₂×#108	—	2	4	no	—
105	C₂×#108	—	2	2	no	—
106	#424	—	2	4	no	—
107	#424	—	2	4	no	—
108	C₂×C₄⋊C₈	—	2	4	no	—
109	#340	C₄×C₅	1	4	yes	complemented
110	#339	—	2	4	no	—
111	#336	—	2	2	no	—
112	C₂³×C₅	—	2	2	no	—
113	C₂³×C₅	—	2	2	no	—
114	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
115	C₂³×C₅	C₂²⋊Q₈	1	2	yes	—
116	C₂³×C₅	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂³×C₅	#123	1	2	yes	—
118	C₂³×C₅	#125	1	1	yes	characteristic
119	C₂×C₄×C₅	—	2	4	no	—
120	C₂×C₄×C₅	—	2	8	no	—
121	C₂×C₄×C₅	—	4	8	no	—
122	C₂×C₄×C₅	—	4	2	no	—
123	C₂×C₄×C₅	—	4	2	no	—
124	C₂×C₄×C₅	—	4	4	no	—
125	C₂×C₄×C₅	—	4	4	no	—
126	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
129	C₂×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
130	C₂×C₄×C₅	#120	1	1	yes	characteristic
131	C₂×C₄×C₅	#120	1	2	yes	—
132	C₂×C₄×C₅	#121	1	2	yes	—
133	C₂×C₄×C₅	#121	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
135	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
136	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
137	C₂×C₄×C₅	#123	1	1	yes	characteristic
138	C₂×C₄×C₅	#125	1	2	yes	—
139	C₂×C₄×C₅	#127	1	1	yes	characteristic
140	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
141	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
142	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
143	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
144	C₅×C₈	—	2	4	no	—
145	C₅×C₈	—	4	2	no	—
146	C₅×C₈	—	4	4	no	—
147	C₂³×C₄	—	2	1	no	—
148	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
149	C₂×M₁₆	—	2	2	no	—
150	C₂×M₁₆	—	2	2	no	—
151	C₂×M₁₆	—	2	2	no	—
152	C₂×M₁₆	—	2	2	no	—
153	C₂×M₁₆	—	2	2	no	—
154	C₂×M₁₆	C₅×D₄	1	4	yes	—
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	—	2	1	no	—
158	C₂²×C₈	—	2	2	no	—
159	C₂²×C₈	—	2	2	no	—
160	C₂²×C₈	—	2	2	no	—
161	C₂²×C₈	—	2	1	no	—
162	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
163	C₂²×C₈	C₅×D₄	1	2	yes	—
164	C₂²×C₈	C₅×D₄	1	2	yes	—
165	C₂²×C₈	C₅×Q₈	1	2	yes	—
166	#117	—	2	2	no	—
167	#117	—	2	2	no	—
168	#117	C₅×D₄	1	4	yes	—
169	#117	C₅×Q₈	1	4	yes	—
170	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
171	C₂×C₂²⋊C₄	—	2	2	no	—
172	C₂×C₄²	—	2	2	no	—
173	C₂×C₄²	—	2	2	no	—
174	C₂×C₄²	—	2	2	no	—
175	C₂×C₄²	—	2	1	no	—
176	C₂×C₄²	C₅×Q₈	1	2	yes	—
177	#108	—	2	8	no	—
178	C₄⋊C₈	—	2	16	no	—
179	C₂²×C₅	—	2	4	no	—
180	C₂²×C₅	—	2	4	no	—
181	C₂²×C₅	—	2	1	no	—
182	C₂²×C₅	—	2	1	no	—
183	C₂²×C₅	—	2	1	no	—
184	C₂²×C₅	—	2	1	no	—
185	C₂²×C₅	#385	1	2	yes	—
186	C₂²×C₅	#389	1	1	yes	characteristic
187	C₂²×C₅	#395	1	1	yes	characteristic
188	C₂²×C₅	#398	1	2	yes	—
189	C₂²×C₅	#399	1	1	yes	characteristic
190	C₂²×C₅	#443	1	2	yes	—
191	C₂²×C₅	#444	1	2	yes	—
192	C₄×C₅	—	4	4	no	—
193	C₄×C₅	—	4	8	no	—
194	C₄×C₅	#384	1	1	yes	characteristic
195	C₄×C₅	#385	1	1	yes	characteristic
196	C₄×C₅	#389	1	2	yes	—
197	C₄×C₅	#392	1	1	yes	characteristic
198	C₄×C₅	#395	1	2	yes	—
199	C₄×C₅	#398	1	1	yes	characteristic
200	C₂⁴	C₅×#41	1	1	yes	characteristic
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	4	2	no	—
205	C₂²×C₄	—	4	2	no	—
206	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
208	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
210	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
213	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
214	C₂²×C₄	C₅×#41	1	2	yes	—
215	C₂²×C₄	C₅×#41	1	1	yes	characteristic
216	M₁₆	—	2	4	no	—
217	M₁₆	—	2	4	no	—
218	M₁₆	—	4	2	no	—
219	M₁₆	—	4	2	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	C₅×C₄⋊C₄	1	4	yes	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	C₅×C₄⋊C₄	1	8	yes	—
235	C₂²⋊C₄	—	2	4	no	—
236	C₂²⋊C₄	—	2	4	no	—
237	C₄²	—	2	4	no	—
238	C₄²	—	2	4	no	—
239	C₄²	—	2	4	no	—
240	C₄²	—	4	1	no	—
241	C₄²	—	4	1	no	—
242	C₂×C₅	—	2	1	no	—
243	C₂×C₅	—	2	1	no	—
244	C₂×C₅	—	2	2	no	—
245	C₂×C₅	#1444	1	1	yes	characteristic
246	C₂×C₅	#1866	1	4	yes	—
247	C₂×C₅	#1945	1	1	yes	characteristic
248	C₂×C₅	#1946	1	1	yes	characteristic
249	C₂³	—	2	2	no	—
250	C₂³	—	2	2	no	—
251	C₂³	C₄×C₅×D₄	1	1	yes	characteristic
252	C₂³	C₅×C₂²⋊Q₈	1	2	yes	—
253	C₂³	C₅×C₂²⋊Q₈	1	1	yes	characteristic
254	C₂³	C₅×#123	1	2	yes	—
255	C₂³	C₅×#125	1	1	yes	characteristic
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	4	8	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
264	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
265	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
266	C₂×C₄	C₄×C₅×Q₈	1	1	yes	characteristic
267	C₂×C₄	C₅×#120	1	1	yes	characteristic
268	C₂×C₄	C₅×#120	1	2	yes	—
269	C₂×C₄	C₅×#121	1	2	yes	—
270	C₂×C₄	C₅×#121	1	1	yes	characteristic
271	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
272	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
273	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
274	C₂×C₄	C₅×#123	1	1	yes	characteristic
275	C₂×C₄	C₅×#125	1	2	yes	—
276	C₂×C₄	C₅×#127	1	1	yes	characteristic
277	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
278	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
279	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
280	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
281	C₈	—	2	4	no	—
282	C₈	—	4	2	no	—
283	C₈	—	4	4	no	—
284	C₅	#11329	1	1	yes	characteristic, direct product factor, Sylow
285	C₂²	—	2	4	no	—
286	C₂²	—	2	4	no	—
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	C₅×#385	1	2	yes	—
292	C₂²	C₅×#389	1	1	yes	characteristic
293	C₂²	C₅×#395	1	1	yes	characteristic
294	C₂²	C₅×#398	1	2	yes	—
295	C₂²	C₅×#399	1	1	yes	characteristic
296	C₂²	C₅×#443	1	2	yes	—
297	C₂²	C₅×#444	1	2	yes	—
298	C₄	—	4	4	no	—
299	C₄	—	4	8	no	—
300	C₄	C₅×#384	1	1	yes	characteristic
301	C₄	C₅×#385	1	1	yes	characteristic
302	C₄	C₅×#389	1	2	yes	—
303	C₄	C₅×#392	1	1	yes	characteristic
304	C₄	C₅×#395	1	2	yes	—
305	C₄	C₅×#398	1	1	yes	characteristic
306	C₂	—	2	1	no	—
307	C₂	—	2	1	no	—
308	C₂	—	2	2	no	—
309	C₂	C₅×#1444	1	1	yes	characteristic
310	C₂	C₅×#1866	1	4	yes	—
311	C₂	C₅×#1945	1	1	yes	characteristic
312	C₂	C₅×#1946	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
socle	C₂³×C₅	C₄×D₄
prime radical	C₂⁴×C₅	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
16. C₅×#340	C₄	semidirect
19. #11329	C₅	direct (N × H)
109. #340	C₄×C₅	semidirect
284. C₅	#11329	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×#108	2
C₂×C₅×#424	2
C₅×#1735	2
C₂×C₅×#339	2
C₂×C₅×#336	2
#11329	5

Subgroups — C5×#11329 #11934049

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#11329 #11934049