Subgroups — C₅×#8672 #11931392

Order 1280.

386 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2965	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅×#108	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#1566	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1489	C₂	1	2	yes	maximal
5	C₅×#1299	C₂	1	2	yes	maximal, complemented
6	C₄×C₅×#41	C₂²	1	1	yes	characteristic
7	C₅×#446	C₄	1	1	yes	characteristic
8	C₅×#446	C₄	1	1	yes	characteristic
9	C₅×#436	C₄	1	1	yes	characteristic
10	C₅×#436	C₄	1	1	yes	characteristic
11	C₅×#435	—	2	1	no	—
12	C₅×#435	—	2	1	no	—
13	C₅×#431	—	2	1	no	—
14	C₅×#431	C₂²	1	1	yes	characteristic
15	C₂×C₅×#108	—	2	1	no	—
16	C₅×#423	C₂²	1	2	yes	—
17	C₄×C₅×M₁₆	—	2	1	no	—
18	C₄×C₅×M₁₆	C₂²	1	2	yes	—
19	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	C₅×#341	—	2	1	no	—
21	C₅×#334	—	2	1	no	—
22	C₅×#334	—	2	1	no	—
23	C₅×Q₈⋊C₈	—	2	2	no	—
24	C₅×D₄⋊C₈	—	2	2	no	—
25	#8672	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂×C₅×#41	Q₈	1	1	yes	characteristic
27	C₂×C₅×M₁₆	—	2	1	no	—
28	C₂×C₅×M₁₆	—	2	1	no	—
29	C₂×C₅×M₁₆	—	2	2	no	—
30	C₂×C₅×M₁₆	—	2	2	no	—
31	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
32	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
33	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
34	C₄×C₅×Q₈	D₄	1	1	yes	characteristic
35	C₄×C₅×D₄	—	2	1	no	—
36	C₄×C₅×D₄	—	2	1	no	—
37	C₄×C₅×D₄	C₂×C₄	1	1	yes	characteristic
38	C₄×C₅×D₄	D₄	1	1	yes	characteristic
39	C₅×#117	—	2	1	no	—
40	C₅×#117	Q₈	1	1	yes	characteristic
41	C₂×C₄²×C₅	—	2	1	no	—
42	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
43	C₅×#108	—	2	4	no	—
44	C₅×C₄⋊C₈	—	2	1	no	—
45	C₅×C₄⋊C₈	—	2	1	no	—
46	C₅×C₄⋊C₈	—	2	2	no	—
47	C₅×C₄⋊C₈	—	2	2	no	—
48	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
49	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
50	C₅×C₂²⋊C₈	—	2	1	no	—
51	C₅×C₂²⋊C₈	—	2	1	no	—
52	C₅×C₂²⋊C₈	—	2	1	no	—
53	C₅×#97	—	2	1	no	—
54	C₅×#97	—	2	1	no	—
55	C₅×#97	—	2	2	no	—
56	C₄×C₅×C₈	—	2	1	no	—
57	C₄×C₅×C₈	—	2	2	no	—
58	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
59	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
60	C₄×C₅×C₈	D₄	1	1	yes	characteristic
61	C₄×C₅×C₈	D₄	1	1	yes	characteristic
62	#2965	C₂×C₅	1	1	yes	characteristic
63	C₄×#108	C₂×C₅	1	1	yes	characteristic, complemented
64	#1566	C₂×C₅	1	1	yes	characteristic, complemented
65	#1489	C₂×C₅	1	2	yes	—
66	#1299	C₂×C₅	1	2	yes	complemented
67	C₅×#41	—	4	2	no	—
68	C₂×C₅×Q₈	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₅×D₄	—	2	1	no	—
70	C₂×C₅×D₄	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	—	2	1	no	—
72	C₂²×C₄×C₅	—	2	1	no	—
73	C₂²×C₄×C₅	—	2	1	no	—
74	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
76	C₂²×C₄×C₅	#41	1	1	yes	characteristic
77	C₅×M₁₆	—	2	4	no	—
78	C₅×M₁₆	—	4	2	no	—
79	C₅×M₁₆	—	4	2	no	—
80	C₅×M₁₆	—	4	2	no	—
81	C₅×M₁₆	—	8	2	no	—
82	C₂×C₅×C₈	—	2	2	no	—
83	C₂×C₅×C₈	—	2	1	no	—
84	C₂×C₅×C₈	—	2	1	no	—
85	C₂×C₅×C₈	—	4	1	no	—
86	C₂×C₅×C₈	—	4	2	no	—
87	C₂×C₅×C₈	—	4	2	no	—
88	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
92	C₅×C₄⋊C₄	—	2	1	no	—
93	C₅×C₄⋊C₄	—	2	1	no	—
94	C₅×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
95	C₅×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
96	C₅×C₂²⋊C₄	—	2	1	no	—
97	C₅×C₂²⋊C₄	—	2	1	no	—
98	C₅×C₂²⋊C₄	—	2	1	no	—
99	C₄²×C₅	—	2	1	no	—
100	C₄²×C₅	—	2	1	no	—
101	C₄²×C₅	—	2	1	no	—
102	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₅	#41	1	1	yes	characteristic
106	C₄×#41	C₂²×C₅	1	1	yes	characteristic
107	#446	C₄×C₅	1	1	yes	characteristic
108	#446	C₄×C₅	1	1	yes	characteristic
109	#436	C₄×C₅	1	1	yes	characteristic
110	#436	C₄×C₅	1	1	yes	characteristic
111	#435	—	2	1	no	—
112	#435	—	2	1	no	—
113	#431	—	2	1	no	—
114	#431	C₂²×C₅	1	1	yes	characteristic
115	C₂×#108	—	2	1	no	—
116	#423	C₂²×C₅	1	2	yes	—
117	C₄×M₁₆	—	2	1	no	—
118	C₄×M₁₆	C₂²×C₅	1	2	yes	—
119	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
120	#341	—	2	1	no	—
121	#334	—	2	1	no	—
122	#334	—	2	1	no	—
123	Q₈⋊C₈	—	2	2	no	—
124	D₄⋊C₈	—	2	2	no	—
125	C₂³×C₅	—	2	1	no	—
126	C₂³×C₅	C₂²⋊Q₈	1	1	yes	characteristic
127	C₅×Q₈	—	2	2	no	—
128	C₅×D₄	—	2	2	no	—
129	C₅×D₄	—	4	2	no	—
130	C₂×C₄×C₅	—	2	1	no	—
131	C₂×C₄×C₅	—	2	1	no	—
132	C₂×C₄×C₅	—	2	2	no	—
133	C₂×C₄×C₅	—	2	2	no	—
134	C₂×C₄×C₅	—	2	1	no	—
135	C₂×C₄×C₅	—	2	1	no	—
136	C₂×C₄×C₅	—	2	1	no	—
137	C₂×C₄×C₅	—	4	2	no	—
138	C₂×C₄×C₅	—	4	1	no	—
139	C₂×C₄×C₅	—	4	1	no	—
140	C₂×C₄×C₅	—	4	2	no	—
141	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₅	#117	1	1	yes	characteristic
144	C₂×C₄×C₅	#121	1	1	yes	characteristic
145	C₂×C₄×C₅	#121	1	1	yes	characteristic
146	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
147	C₅×C₈	—	2	1	no	—
148	C₅×C₈	—	2	1	no	—
149	C₅×C₈	—	2	1	no	—
150	C₅×C₈	—	4	4	no	—
151	C₅×C₈	—	4	1	no	—
152	C₅×C₈	—	4	1	no	—
153	C₅×C₈	—	8	2	no	—
154	C₅×C₈	#104	1	2	yes	—
155	C₂×#41	C₅×Q₈	1	1	yes	characteristic
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	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
161	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
162	C₄×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
163	C₄×Q₈	C₅×D₄	1	1	yes	characteristic
164	C₄×D₄	—	2	1	no	—
165	C₄×D₄	—	2	1	no	—
166	C₄×D₄	C₂×C₄×C₅	1	1	yes	characteristic
167	C₄×D₄	C₅×D₄	1	1	yes	characteristic
168	#117	—	2	1	no	—
169	#117	C₅×Q₈	1	1	yes	characteristic
170	C₂×C₄²	—	2	1	no	—
171	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
172	#108	—	2	4	no	—
173	C₄⋊C₈	—	2	1	no	—
174	C₄⋊C₈	—	2	1	no	—
175	C₄⋊C₈	—	2	2	no	—
176	C₄⋊C₈	—	2	2	no	—
177	C₄⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
178	C₄⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
179	C₂²⋊C₈	—	2	1	no	—
180	C₂²⋊C₈	—	2	1	no	—
181	C₂²⋊C₈	—	2	1	no	—
182	#97	—	2	1	no	—
183	#97	—	2	1	no	—
184	#97	—	2	2	no	—
185	C₄×C₈	—	2	1	no	—
186	C₄×C₈	—	2	2	no	—
187	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
188	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
189	C₄×C₈	C₅×D₄	1	1	yes	characteristic
190	C₄×C₈	C₅×D₄	1	1	yes	characteristic
191	C₂²×C₅	—	2	1	no	—
192	C₂²×C₅	—	2	1	no	—
193	C₂²×C₅	—	2	1	no	—
194	C₂²×C₅	—	4	1	no	—
195	C₂²×C₅	—	4	1	no	—
196	C₂²×C₅	—	4	1	no	—
197	C₂²×C₅	#380	1	1	yes	characteristic
198	C₄×C₅	—	2	1	no	—
199	C₄×C₅	—	2	1	no	—
200	C₄×C₅	—	2	1	no	—
201	C₄×C₅	—	4	1	no	—
202	C₄×C₅	—	4	1	no	—
203	C₄×C₅	—	4	1	no	—
204	C₄×C₅	C₂×#104	1	1	yes	characteristic
205	C₄×C₅	#421	1	1	yes	characteristic
206	C₄×C₅	#430	1	2	yes	—
207	C₄×C₅	#471	1	2	yes	—
208	#41	—	4	2	no	—
209	C₂×Q₈	C₅×C₄⋊C₄	1	1	yes	characteristic
210	C₂×D₄	—	2	1	no	—
211	C₂×D₄	C₅×C₄⋊C₄	1	1	yes	characteristic
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
217	C₂²×C₄	C₅×#41	1	1	yes	characteristic
218	M₁₆	—	2	4	no	—
219	M₁₆	—	4	2	no	—
220	M₁₆	—	4	2	no	—
221	M₁₆	—	4	2	no	—
222	M₁₆	—	8	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	4	1	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
230	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
231	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
232	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
236	C₄⋊C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
237	C₂²⋊C₄	—	2	1	no	—
238	C₂²⋊C₄	—	2	1	no	—
239	C₂²⋊C₄	—	2	1	no	—
240	C₄²	—	2	1	no	—
241	C₄²	—	2	1	no	—
242	C₄²	—	2	1	no	—
243	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
244	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
245	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
246	C₄²	C₅×#41	1	1	yes	characteristic
247	C₂×C₅	—	4	1	no	—
248	C₂×C₅	—	8	1	no	—
249	C₂×C₅	#1806	1	1	yes	characteristic
250	C₂×C₅	#1811	1	1	yes	characteristic
251	C₂×C₅	#1815	1	1	yes	characteristic
252	C₂³	—	2	1	no	—
253	C₂³	C₅×C₂²⋊Q₈	1	1	yes	characteristic
254	Q₈	—	2	2	no	—
255	D₄	—	2	2	no	—
256	D₄	—	4	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
269	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
270	C₂×C₄	C₅×#117	1	1	yes	characteristic
271	C₂×C₄	C₅×#121	1	1	yes	characteristic
272	C₂×C₄	C₅×#121	1	1	yes	characteristic
273	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
274	C₈	—	2	1	no	—
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	4	4	no	—
278	C₈	—	4	1	no	—
279	C₈	—	4	1	no	—
280	C₈	—	8	2	no	—
281	C₈	C₅×#104	1	2	yes	—
282	C₅	#8672	1	1	yes	characteristic, direct product factor, Sylow
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	4	1	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	1	no	—
289	C₂²	C₅×#380	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	4	1	no	—
294	C₄	—	4	1	no	—
295	C₄	—	4	1	no	—
296	C₄	C₂×C₅×#104	1	1	yes	characteristic
297	C₄	C₅×#421	1	1	yes	characteristic
298	C₄	C₅×#430	1	2	yes	—
299	C₄	C₅×#471	1	2	yes	—
300	C₂	—	4	1	no	—
301	C₂	—	8	1	no	—
302	C₂	C₅×#1806	1	1	yes	characteristic
303	C₂	C₅×#1811	1	1	yes	characteristic
304	C₂	C₅×#1815	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	#121
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂²×C₅	#380
prime radical	C₂×C₅×#41	Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₅×#108	C₂	semidirect
3. C₅×#1566	C₂	semidirect
5. C₅×#1299	C₂	semidirect
25. #8672	C₅	direct (N × H)
63. C₄×#108	C₂×C₅	semidirect
64. #1566	C₂×C₅	semidirect
66. #1299	C₂×C₅	semidirect
282. C₅	#8672	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×#2965	2
C₄×C₅×#108	2
C₅×#1566	2
C₅×#1489	2
C₅×#1299	2
#8672	5

Subgroups — C5×#8672 #11931392

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8672 #11931392