Subgroups — C₅×#10162 #11932882

Order 1280.

510 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3206	C₂	1	1	yes	maximal, characteristic
2	C₅×#3141	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#487	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1997	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1548	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1492	C₂	1	1	yes	maximal, characteristic
7	C₅×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#532	C₂²	1	1	yes	characteristic
9	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₅×#492	—	2	1	no	—
12	C₅×#489	—	2	2	no	—
13	C₅×#487	—	2	2	no	—
14	C₅×#487	—	2	2	no	—
15	C₅×#487	—	2	1	no	—
16	C₅×#487	—	2	1	no	—
17	C₅×#486	—	2	1	no	—
18	C₅×#461	—	2	2	no	—
19	C₅×#448	—	2	2	no	—
20	C₅×#431	C₂²	1	1	yes	characteristic
21	C₅×#423	C₂²	1	1	yes	characteristic
22	C₅×#417	—	2	2	no	—
23	C₂×C₅×#103	—	2	2	no	—
24	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
26	C₅×#343	—	2	2	no	—
27	C₅×#333	—	2	2	no	—
28	C₅×D₄⋊C₈	—	2	2	no	—
29	#10162	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂×C₅×#41	—	2	1	no	—
31	C₂²×C₅×Q₈	—	2	1	no	—
32	C₂²×C₅×Q₈	D₄	1	2	yes	—
33	C₂×C₅×SD₁₆	—	2	4	no	—
34	C₂×C₅×M₁₆	—	2	1	no	—
35	C₂×C₅×M₁₆	D₄	1	2	yes	—
36	C₂²×C₅×C₈	—	2	1	no	—
37	C₅×C₄⋊Q₈	—	2	1	no	—
38	C₅×C₄⋊Q₈	—	2	1	no	—
39	C₅×C₄⋊Q₈	—	2	1	no	—
40	C₅×C₄⋊Q₈	—	2	1	no	—
41	C₅×C₄⋊Q₈	—	2	1	no	—
42	C₅×C₄⋊Q₈	—	2	1	no	—
43	C₅×C₄⋊Q₈	—	2	1	no	—
44	C₅×C₄⋊Q₈	D₄	1	1	yes	characteristic
45	C₅×C₄⋊Q₈	D₄	1	2	yes	—
46	C₅×#127	D₄	1	1	yes	characteristic
47	C₅×#124	—	2	1	no	—
48	C₅×#121	—	2	1	no	—
49	C₅×#121	—	2	1	no	—
50	C₄×C₅×D₄	—	2	1	no	—
51	C₄×C₅×D₄	—	2	1	no	—
52	C₂×C₅×C₄⋊C₄	—	2	1	no	—
53	C₂×C₅×C₄⋊C₄	—	2	1	no	—
54	C₂×C₅×C₄⋊C₄	—	2	1	no	—
55	C₂×C₅×C₄⋊C₄	—	2	1	no	—
56	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
57	C₅×C₄⋊C₈	—	2	1	no	—
58	C₅×C₄⋊C₈	—	2	1	no	—
59	C₅×C₄⋊C₈	D₄	1	2	yes	—
60	C₅×#103	—	2	8	no	—
61	C₅×#103	—	2	4	no	—
62	C₅×#103	—	2	4	no	—
63	C₅×#102	—	2	4	no	—
64	C₅×C₂²⋊C₈	—	2	1	no	—
65	C₅×#97	—	2	1	no	—
66	C₄×C₅×C₈	—	2	1	no	—
67	C₄×C₅×C₈	—	2	1	no	—
68	C₄×C₅×C₈	—	2	1	no	—
69	C₄×C₅×C₈	D₄	1	2	yes	—
70	#3206	C₂×C₅	1	1	yes	characteristic
71	#3141	C₂×C₅	1	1	yes	characteristic
72	C₂×#487	C₂×C₅	1	1	yes	characteristic, complemented
73	#1997	C₂×C₅	1	1	yes	characteristic, complemented
74	#1548	C₂×C₅	1	1	yes	characteristic, complemented
75	#1492	C₂×C₅	1	1	yes	characteristic
76	#1278	C₂×C₅	1	1	yes	characteristic, complemented
77	C₅×#41	—	4	2	no	—
78	C₅×#41	—	4	2	no	—
79	C₂×C₅×Q₈	—	2	1	no	—
80	C₂×C₅×Q₈	—	2	2	no	—
81	C₂×C₅×Q₈	—	2	2	no	—
82	C₂×C₅×Q₈	—	2	2	no	—
83	C₂×C₅×Q₈	—	4	4	no	—
84	C₂×C₅×Q₈	—	4	1	no	—
85	C₂×C₅×Q₈	—	8	2	no	—
86	C₂×C₅×D₄	—	2	1	no	—
87	C₂×C₅×D₄	—	2	1	no	—
88	C₂×C₅×D₄	—	2	1	no	—
89	C₂×C₅×D₄	—	4	1	no	—
90	C₂²×C₄×C₅	—	2	2	no	—
91	C₂²×C₄×C₅	—	4	1	no	—
92	C₂²×C₄×C₅	—	4	1	no	—
93	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₅	#41	1	1	yes	characteristic
96	C₅×SD₁₆	—	8	4	no	—
97	C₅×M₁₆	—	2	2	no	—
98	C₅×M₁₆	—	2	2	no	—
99	C₅×M₁₆	—	4	2	no	—
100	C₂×C₅×C₈	—	2	2	no	—
101	C₂×C₅×C₈	—	2	2	no	—
102	C₂×C₅×C₈	—	2	2	no	—
103	C₂×C₅×C₈	—	4	2	no	—
104	C₂×C₅×C₈	—	4	1	no	—
105	C₅×C₄⋊C₄	—	2	2	no	—
106	C₅×C₄⋊C₄	—	2	2	no	—
107	C₅×C₄⋊C₄	—	2	1	no	—
108	C₅×C₄⋊C₄	—	2	1	no	—
109	C₅×C₄⋊C₄	—	4	1	no	—
110	C₅×C₄⋊C₄	—	4	1	no	—
111	C₅×C₄⋊C₄	—	4	1	no	—
112	C₅×C₄⋊C₄	—	4	1	no	—
113	C₅×C₄⋊C₄	—	4	1	no	—
114	C₅×C₄⋊C₄	—	4	1	no	—
115	C₅×C₂²⋊C₄	—	4	1	no	—
116	C₅×C₂²⋊C₄	—	4	1	no	—
117	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
121	#532	C₂²×C₅	1	1	yes	characteristic
122	C₂×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
123	C₂×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
124	#492	—	2	1	no	—
125	#489	—	2	2	no	—
126	#487	—	2	2	no	—
127	#487	—	2	1	no	—
128	#487	—	2	1	no	—
129	#487	—	2	2	no	—
130	#486	—	2	1	no	—
131	#461	—	2	2	no	—
132	#448	—	2	2	no	—
133	#431	C₂²×C₅	1	1	yes	characteristic
134	#423	C₂²×C₅	1	1	yes	characteristic
135	#417	—	2	2	no	—
136	C₂×#103	—	2	2	no	—
137	C₄×M₁₆	C₂²×C₅	1	1	yes	characteristic
138	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
139	#343	—	2	2	no	—
140	#333	—	2	2	no	—
141	D₄⋊C₈	—	2	2	no	—
142	C₂³×C₅	—	4	1	no	—
143	C₂³×C₅	#121	1	1	yes	characteristic
144	C₅×Q₈	—	4	4	no	—
145	C₅×Q₈	—	4	2	no	—
146	C₅×Q₈	—	4	4	no	—
147	C₅×Q₈	—	8	2	no	—
148	C₅×Q₈	—	8	2	no	—
149	C₅×D₄	—	4	2	no	—
150	C₅×D₄	—	4	2	no	—
151	C₅×D₄	—	4	2	no	—
152	C₅×D₄	—	8	2	no	—
153	C₂×C₄×C₅	—	2	2	no	—
154	C₂×C₄×C₅	—	2	2	no	—
155	C₂×C₄×C₅	—	4	1	no	—
156	C₂×C₄×C₅	—	4	2	no	—
157	C₂×C₄×C₅	—	4	2	no	—
158	C₂×C₄×C₅	—	8	2	no	—
159	C₂×C₄×C₅	—	8	1	no	—
160	C₂×C₄×C₅	—	8	1	no	—
161	C₂×C₄×C₅	—	8	2	no	—
162	C₂×C₄×C₅	#120	1	1	yes	characteristic
163	C₂×C₄×C₅	#120	1	1	yes	characteristic
164	C₂×C₄×C₅	#120	1	1	yes	characteristic
165	C₂×C₄×C₅	#121	1	1	yes	characteristic
166	C₂×C₄×C₅	#121	1	1	yes	characteristic
167	C₂×C₄×C₅	#127	1	1	yes	characteristic
168	C₂×C₄×C₅	#135	1	2	yes	—
169	C₂×C₄×C₅	#137	1	2	yes	—
170	C₅×C₈	—	4	2	no	—
171	C₅×C₈	—	4	2	no	—
172	C₅×C₈	—	4	2	no	—
173	C₅×C₈	—	8	1	no	—
174	C₂×#41	—	2	1	no	—
175	C₂²×Q₈	—	2	1	no	—
176	C₂²×Q₈	C₅×D₄	1	2	yes	—
177	C₂×SD₁₆	—	2	4	no	—
178	C₂×M₁₆	—	2	1	no	—
179	C₂×M₁₆	C₅×D₄	1	2	yes	—
180	C₂²×C₈	—	2	1	no	—
181	C₄⋊Q₈	—	2	1	no	—
182	C₄⋊Q₈	—	2	1	no	—
183	C₄⋊Q₈	—	2	1	no	—
184	C₄⋊Q₈	—	2	1	no	—
185	C₄⋊Q₈	—	2	1	no	—
186	C₄⋊Q₈	—	2	1	no	—
187	C₄⋊Q₈	—	2	1	no	—
188	C₄⋊Q₈	C₅×D₄	1	2	yes	—
189	C₄⋊Q₈	C₅×D₄	1	1	yes	characteristic
190	#127	C₅×D₄	1	1	yes	characteristic
191	#124	—	2	1	no	—
192	#121	—	2	1	no	—
193	#121	—	2	1	no	—
194	C₄×D₄	—	2	1	no	—
195	C₄×D₄	—	2	1	no	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₄⋊C₄	—	2	1	no	—
198	C₂×C₄⋊C₄	—	2	1	no	—
199	C₂×C₄⋊C₄	—	2	1	no	—
200	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
201	C₄⋊C₈	—	2	1	no	—
202	C₄⋊C₈	—	2	1	no	—
203	C₄⋊C₈	C₅×D₄	1	2	yes	—
204	#103	—	2	8	no	—
205	#103	—	2	4	no	—
206	#103	—	2	4	no	—
207	#102	—	2	4	no	—
208	C₂²⋊C₈	—	2	1	no	—
209	#97	—	2	1	no	—
210	C₄×C₈	—	2	1	no	—
211	C₄×C₈	—	2	1	no	—
212	C₄×C₈	—	2	1	no	—
213	C₄×C₈	C₅×D₄	1	2	yes	—
214	C₂²×C₅	—	2	1	no	—
215	C₂²×C₅	—	2	1	no	—
216	C₂²×C₅	—	8	1	no	—
217	C₂²×C₅	—	8	1	no	—
218	C₂²×C₅	—	8	1	no	—
219	C₂²×C₅	#392	1	1	yes	characteristic
220	C₂²×C₅	#464	1	2	yes	—
221	C₄×C₅	—	2	1	no	—
222	C₄×C₅	—	2	1	no	—
223	C₄×C₅	—	8	2	no	—
224	C₄×C₅	—	8	1	no	—
225	C₄×C₅	—	8	2	no	—
226	C₄×C₅	#452	1	2	yes	—
227	C₄×C₅	#455	1	2	yes	—
228	C₄×C₅	#471	1	2	yes	—
229	C₄×C₅	#495	1	1	yes	characteristic
230	C₄×C₅	#497	1	1	yes	characteristic
231	#41	—	4	2	no	—
232	#41	—	4	2	no	—
233	C₂×Q₈	—	2	1	no	—
234	C₂×Q₈	—	2	2	no	—
235	C₂×Q₈	—	2	2	no	—
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	4	4	no	—
238	C₂×Q₈	—	4	1	no	—
239	C₂×Q₈	—	8	2	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	—	2	1	no	—
242	C₂×D₄	—	2	1	no	—
243	C₂×D₄	—	4	1	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	4	1	no	—
246	C₂²×C₄	—	4	1	no	—
247	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
248	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
249	C₂²×C₄	C₅×#41	1	1	yes	characteristic
250	SD₁₆	—	8	4	no	—
251	M₁₆	—	2	2	no	—
252	M₁₆	—	2	2	no	—
253	M₁₆	—	4	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	4	2	no	—
258	C₂×C₈	—	4	1	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	1	no	—
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₂²⋊C₄	—	4	1	no	—
270	C₂²⋊C₄	—	4	1	no	—
271	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
272	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
273	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
274	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
275	C₂×C₅	—	2	2	no	—
276	C₂×C₅	—	16	1	no	—
277	C₂×C₅	#2004	1	1	yes	characteristic
278	C₂×C₅	#2015	1	1	yes	characteristic
279	C₂×C₅	#2019	1	1	yes	characteristic
280	C₂³	—	4	1	no	—
281	C₂³	C₅×#121	1	1	yes	characteristic
282	Q₈	—	4	4	no	—
283	Q₈	—	4	2	no	—
284	Q₈	—	4	4	no	—
285	Q₈	—	8	2	no	—
286	Q₈	—	8	2	no	—
287	D₄	—	4	2	no	—
288	D₄	—	4	2	no	—
289	D₄	—	4	2	no	—
290	D₄	—	8	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	8	2	no	—
297	C₂×C₄	—	8	1	no	—
298	C₂×C₄	—	8	1	no	—
299	C₂×C₄	—	8	2	no	—
300	C₂×C₄	C₅×#120	1	1	yes	characteristic
301	C₂×C₄	C₅×#120	1	1	yes	characteristic
302	C₂×C₄	C₅×#120	1	1	yes	characteristic
303	C₂×C₄	C₅×#121	1	1	yes	characteristic
304	C₂×C₄	C₅×#121	1	1	yes	characteristic
305	C₂×C₄	C₅×#127	1	1	yes	characteristic
306	C₂×C₄	C₅×#135	1	2	yes	—
307	C₂×C₄	C₅×#137	1	2	yes	—
308	C₈	—	4	2	no	—
309	C₈	—	4	2	no	—
310	C₈	—	4	2	no	—
311	C₈	—	8	1	no	—
312	C₅	#10162	1	1	yes	characteristic, direct product factor, Sylow
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	8	1	no	—
316	C₂²	—	8	1	no	—
317	C₂²	—	8	1	no	—
318	C₂²	C₅×#392	1	1	yes	characteristic
319	C₂²	C₅×#464	1	2	yes	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	8	1	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	2	no	—
325	C₄	C₅×#452	1	2	yes	—
326	C₄	C₅×#455	1	2	yes	—
327	C₄	C₅×#471	1	2	yes	—
328	C₄	C₅×#495	1	1	yes	characteristic
329	C₄	C₅×#497	1	1	yes	characteristic
330	C₂	—	2	2	no	—
331	C₂	—	16	1	no	—
332	C₂	C₅×#2004	1	1	yes	characteristic
333	C₂	C₅×#2015	1	1	yes	characteristic
334	C₂	C₅×#2019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₅×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₅×#487	C₂	semidirect
4. C₅×#1997	C₂	semidirect
5. C₅×#1548	C₂	semidirect
7. C₅×#1278	C₂	semidirect
29. #10162	C₅	direct (N × H)
72. C₂×#487	C₂×C₅	semidirect
73. #1997	C₂×C₅	semidirect
74. #1548	C₂×C₅	semidirect
76. #1278	C₂×C₅	semidirect
312. C₅	#10162	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×#3206	2
C₅×#3141	2
C₂×C₅×#487	2
C₅×#1997	2
C₅×#1548	2
C₅×#1492	2
C₅×#1278	2
#10162	5

Subgroups — C5×#10162 #11932882

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#10162 #11932882