Subgroups — C₅×#11326 #11934046

Order 1280.

730 conjugacy classes of subgroups.

The subgroup lattice has 322 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	2	yes	maximal
2	C₂×C₅×#423	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#341	C₂	1	2	yes	maximal
4	C₅×#1728	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅×#336	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
7	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
9	C₂³×C₅×C₈	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	4	no	—
13	C₂×C₅×#108	—	2	4	no	—
14	C₅×#423	—	2	4	no	—
15	C₂×C₅×C₄⋊C₈	—	2	2	no	—
16	C₅×#341	—	2	4	no	—
17	C₅×#336	—	2	2	no	—
18	C₅×#336	C₄	1	2	yes	—
19	C₅×#336	C₄	1	2	yes	—
20	#11326	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂³×C₄×C₅	Q₈	1	2	yes	—
22	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
23	C₂×C₅×M₁₆	—	2	2	no	—
24	C₂×C₅×M₁₆	—	2	1	no	—
25	C₂×C₅×M₁₆	—	2	1	no	—
26	C₂×C₅×M₁₆	—	2	2	no	—
27	C₂×C₅×M₁₆	—	2	2	no	—
28	C₂×C₅×M₁₆	—	2	2	no	—
29	C₂×C₅×M₁₆	—	2	2	no	—
30	C₂×C₅×M₁₆	—	2	2	no	—
31	C₂×C₅×M₁₆	—	2	2	no	—
32	C₂×C₅×M₁₆	—	2	2	no	—
33	C₂×C₅×M₁₆	—	2	2	no	—
34	C₂×C₅×M₁₆	D₄	1	4	yes	—
35	C₂×C₅×M₁₆	D₄	1	2	yes	—
36	C₂×C₅×M₁₆	Q₈	1	2	yes	—
37	C₂²×C₅×C₈	—	2	2	no	—
38	C₂²×C₅×C₈	—	2	2	no	—
39	C₂²×C₅×C₈	—	2	2	no	—
40	C₂²×C₅×C₈	—	2	2	no	—
41	C₂²×C₅×C₈	—	2	1	no	—
42	C₂²×C₅×C₈	—	2	1	no	—
43	C₂²×C₅×C₈	—	2	2	no	—
44	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
45	C₂²×C₅×C₈	D₄	1	4	yes	—
46	C₂²×C₅×C₈	Q₈	1	4	yes	—
47	C₂×C₄²×C₅	—	2	2	no	—
48	C₂×C₄²×C₅	—	2	2	no	—
49	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₅	D₄	1	2	yes	—
52	C₅×#108	—	2	16	no	—
53	C₅×C₄⋊C₈	—	2	8	no	—
54	C₂²×#108	C₂×C₅	1	2	yes	—
55	C₂×#423	C₂×C₅	1	1	yes	characteristic
56	C₂×#341	C₂×C₅	1	2	yes	—
57	#1728	C₂×C₅	1	1	yes	characteristic
58	C₂×#336	C₂×C₅	1	1	yes	characteristic
59	C₂⁴×C₅	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₅	—	2	4	no	—
61	C₂²×C₄×C₅	—	2	4	no	—
62	C₂²×C₄×C₅	—	4	2	no	—
63	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₅	C₂²×C₄	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×D₄	1	2	yes	—
69	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
72	C₂²×C₄×C₅	#41	1	1	yes	characteristic
73	C₂²×C₄×C₅	#41	1	2	yes	—
74	C₂²×C₄×C₅	#41	1	1	yes	characteristic
75	C₅×M₁₆	—	2	4	no	—
76	C₅×M₁₆	—	2	4	no	—
77	C₅×M₁₆	—	2	4	no	—
78	C₅×M₁₆	—	2	4	no	—
79	C₅×M₁₆	—	4	2	no	—
80	C₅×M₁₆	—	4	2	no	—
81	C₅×M₁₆	—	4	2	no	—
82	C₅×M₁₆	—	4	2	no	—
83	C₂×C₅×C₈	—	2	4	no	—
84	C₂×C₅×C₈	—	2	4	no	—
85	C₂×C₅×C₈	—	2	4	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	4	no	—
90	C₂×C₅×C₈	—	2	2	no	—
91	C₂×C₅×C₈	—	2	2	no	—
92	C₂×C₅×C₈	—	4	1	no	—
93	C₂×C₅×C₈	—	4	2	no	—
94	C₂×C₅×C₈	—	4	1	no	—
95	C₂×C₅×C₈	—	4	1	no	—
96	C₂×C₅×C₈	—	4	2	no	—
97	C₂×C₅×C₈	—	4	1	no	—
98	C₂×C₅×C₈	C₄⋊C₄	1	8	yes	—
99	C₄²×C₅	—	2	2	no	—
100	C₄²×C₅	—	2	2	no	—
101	C₄²×C₅	—	2	2	no	—
102	C₄²×C₅	—	2	2	no	—
103	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
104	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
105	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
106	C₂³×C₈	C₂²×C₅	1	2	yes	—
107	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
108	C₂×#108	—	2	4	no	—
109	C₂×#108	—	2	4	no	—
110	C₂×#108	—	2	4	no	—
111	#423	—	2	4	no	—
112	C₂×C₄⋊C₈	—	2	2	no	—
113	#341	—	2	4	no	—
114	#336	—	2	2	no	—
115	#336	C₄×C₅	1	2	yes	—
116	#336	C₄×C₅	1	2	yes	—
117	C₂³×C₅	—	2	2	no	—
118	C₂³×C₅	—	2	2	no	—
119	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂³×C₅	C₂²⋊Q₈	1	2	yes	—
121	C₂³×C₅	#125	1	1	yes	characteristic
122	C₂³×C₅	C₄⋊Q₈	1	2	yes	—
123	C₂³×C₅	C₄⋊Q₈	1	1	yes	characteristic
124	C₂×C₄×C₅	—	2	2	no	—
125	C₂×C₄×C₅	—	2	2	no	—
126	C₂×C₄×C₅	—	2	2	no	—
127	C₂×C₄×C₅	—	2	2	no	—
128	C₂×C₄×C₅	—	2	2	no	—
129	C₂×C₄×C₅	—	4	8	no	—
130	C₂×C₄×C₅	#108	1	4	yes	complemented
131	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
132	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
135	C₂×C₄×C₅	#120	1	2	yes	—
136	C₂×C₄×C₅	#120	1	1	yes	characteristic
137	C₂×C₄×C₅	#121	1	2	yes	—
138	C₂×C₄×C₅	#121	1	1	yes	characteristic
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	2	yes	—
142	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
143	C₂×C₄×C₅	#123	1	2	yes	—
144	C₂×C₄×C₅	#123	1	1	yes	characteristic
145	C₂×C₄×C₅	#125	1	2	yes	—
146	C₂×C₄×C₅	#127	1	1	yes	characteristic
147	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
148	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
149	C₅×C₈	—	2	8	no	—
150	C₅×C₈	—	4	4	no	—
151	C₅×C₈	—	4	2	no	—
152	C₅×C₈	—	4	2	no	—
153	C₂³×C₄	C₅×Q₈	1	2	yes	—
154	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
155	C₂×M₁₆	—	2	2	no	—
156	C₂×M₁₆	—	2	1	no	—
157	C₂×M₁₆	—	2	1	no	—
158	C₂×M₁₆	—	2	2	no	—
159	C₂×M₁₆	—	2	2	no	—
160	C₂×M₁₆	—	2	2	no	—
161	C₂×M₁₆	—	2	2	no	—
162	C₂×M₁₆	—	2	2	no	—
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	C₅×D₄	1	4	yes	—
167	C₂×M₁₆	C₅×D₄	1	2	yes	—
168	C₂×M₁₆	C₅×Q₈	1	2	yes	—
169	C₂²×C₈	—	2	2	no	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	—	2	2	no	—
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	—	2	1	no	—
174	C₂²×C₈	—	2	1	no	—
175	C₂²×C₈	—	2	2	no	—
176	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
177	C₂²×C₈	C₅×D₄	1	4	yes	—
178	C₂²×C₈	C₅×Q₈	1	4	yes	—
179	C₂×C₄²	—	2	2	no	—
180	C₂×C₄²	—	2	2	no	—
181	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
182	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
183	C₂×C₄²	C₅×D₄	1	2	yes	—
184	#108	—	2	16	no	—
185	C₄⋊C₈	—	2	8	no	—
186	C₂²×C₅	—	2	2	no	—
187	C₂²×C₅	—	2	2	no	—
188	C₂²×C₅	—	2	4	no	—
189	C₂²×C₅	—	2	2	no	—
190	C₂²×C₅	—	2	2	no	—
191	C₂²×C₅	#384	1	1	yes	characteristic
192	C₂²×C₅	#385	1	1	yes	characteristic
193	C₂²×C₅	#389	1	1	yes	characteristic
194	C₂²×C₅	#395	1	2	yes	—
195	C₂²×C₅	#398	1	2	yes	—
196	C₂²×C₅	C₂×#108	1	2	yes	—
197	C₂²×C₅	#430	1	2	yes	—
198	C₄×C₅	—	2	4	no	—
199	C₄×C₅	—	4	2	no	—
200	C₄×C₅	#385	1	2	yes	—
201	C₄×C₅	#389	1	2	yes	—
202	C₄×C₅	#392	1	1	yes	characteristic
203	C₄×C₅	#395	1	1	yes	characteristic
204	C₄×C₅	#398	1	1	yes	characteristic
205	C₄×C₅	#399	1	1	yes	characteristic
206	C₂⁴	C₂×C₅×Q₈	1	1	yes	characteristic
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	4	2	no	—
210	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
211	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
212	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
214	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
216	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
218	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
219	C₂²×C₄	C₅×#41	1	1	yes	characteristic
220	C₂²×C₄	C₅×#41	1	2	yes	—
221	C₂²×C₄	C₅×#41	1	1	yes	characteristic
222	M₁₆	—	2	4	no	—
223	M₁₆	—	2	4	no	—
224	M₁₆	—	2	4	no	—
225	M₁₆	—	2	4	no	—
226	M₁₆	—	4	2	no	—
227	M₁₆	—	4	2	no	—
228	M₁₆	—	4	2	no	—
229	M₁₆	—	4	2	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	4	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	2	no	—
241	C₂×C₈	—	4	1	no	—
242	C₂×C₈	—	4	1	no	—
243	C₂×C₈	—	4	2	no	—
244	C₂×C₈	—	4	1	no	—
245	C₂×C₈	C₅×C₄⋊C₄	1	8	yes	—
246	C₄²	—	2	2	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₂×C₅	—	2	2	no	—
251	C₂×C₅	—	2	2	no	—
252	C₂×C₅	#1444	1	1	yes	characteristic
253	C₂×C₅	#1848	1	1	yes	characteristic
254	C₂×C₅	#1849	1	1	yes	characteristic
255	C₂×C₅	#1855	1	2	yes	—
256	C₂×C₅	#1950	1	2	yes	—
257	C₂³	—	2	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
260	C₂³	C₅×C₂²⋊Q₈	1	2	yes	—
261	C₂³	C₅×#125	1	1	yes	characteristic
262	C₂³	C₅×C₄⋊Q₈	1	2	yes	—
263	C₂³	C₅×C₄⋊Q₈	1	1	yes	characteristic
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	4	8	no	—
270	C₂×C₄	C₅×#108	1	4	yes	complemented
271	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
272	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
273	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
274	C₂×C₄	C₄×C₅×Q₈	1	1	yes	characteristic
275	C₂×C₄	C₅×#120	1	2	yes	—
276	C₂×C₄	C₅×#120	1	1	yes	characteristic
277	C₂×C₄	C₅×#121	1	2	yes	—
278	C₂×C₄	C₅×#121	1	1	yes	characteristic
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×#123	1	2	yes	—
284	C₂×C₄	C₅×#123	1	1	yes	characteristic
285	C₂×C₄	C₅×#125	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	C₈	—	2	8	no	—
290	C₈	—	4	4	no	—
291	C₈	—	4	2	no	—
292	C₈	—	4	2	no	—
293	C₅	#11326	1	1	yes	characteristic, direct product factor, Sylow
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	4	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	C₅×#384	1	1	yes	characteristic
300	C₂²	C₅×#385	1	1	yes	characteristic
301	C₂²	C₅×#389	1	1	yes	characteristic
302	C₂²	C₅×#395	1	2	yes	—
303	C₂²	C₅×#398	1	2	yes	—
304	C₂²	C₂×C₅×#108	1	2	yes	—
305	C₂²	C₅×#430	1	2	yes	—
306	C₄	—	2	4	no	—
307	C₄	—	4	2	no	—
308	C₄	C₅×#385	1	2	yes	—
309	C₄	C₅×#389	1	2	yes	—
310	C₄	C₅×#392	1	1	yes	characteristic
311	C₄	C₅×#395	1	1	yes	characteristic
312	C₄	C₅×#398	1	1	yes	characteristic
313	C₄	C₅×#399	1	1	yes	characteristic
314	C₂	—	2	2	no	—
315	C₂	—	2	2	no	—
316	C₂	C₅×#1444	1	1	yes	characteristic
317	C₂	C₅×#1848	1	1	yes	characteristic
318	C₂	C₅×#1849	1	1	yes	characteristic
319	C₂	C₅×#1855	1	2	yes	—
320	C₂	C₅×#1950	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
socle	C₂³×C₅	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
20. #11326	C₅	direct (N × H)
130. C₂×C₄×C₅	#108	semidirect
270. C₂×C₄	C₅×#108	semidirect
293. C₅	#11326	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×#423	2
C₂×C₅×#341	2
C₅×#1728	2
C₂×C₅×#336	2
#11326	5

Subgroups — C5×#11326 #11934046

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#11326 #11934046