Subgroups — C₅×#9639 #11932359

Order 1280.

618 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	C₄×C₅×#124	C₂	1	1	yes	maximal, characteristic
2	C₅×#1831	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄²×C₅×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×Q₈⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₂×C₅×D₄⋊C₈	C₂	1	2	yes	maximal
6	C₂×C₅×#124	C₄	1	1	yes	characteristic
7	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
9	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₅×C₈	—	2	2	no	—
11	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
12	C₅×#391	C₄	1	1	yes	characteristic
13	C₅×#390	C₄	1	1	yes	characteristic
14	C₅×#388	—	2	2	no	—
15	C₅×#382	C₄	1	1	yes	characteristic
16	C₄×C₅×C₂²⋊C₄	—	2	2	no	—
17	C₄³×C₅	C₂²	1	1	yes	characteristic
18	C₅×#336	—	2	2	no	—
19	C₅×Q₈⋊C₈	—	2	4	no	—
20	C₅×D₄⋊C₈	—	2	4	no	—
21	#9639	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂²×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₅×D₄	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₅	—	2	1	no	—
25	C₂²×C₅×C₈	—	2	2	no	—
26	C₂²×C₅×C₈	—	2	2	no	—
27	C₅×#124	—	2	2	no	—
28	C₅×#124	C₈	1	4	yes	complemented
29	C₄×C₅×Q₈	—	2	2	no	—
30	C₄×C₅×Q₈	—	2	2	no	—
31	C₄×C₅×D₄	—	2	2	no	—
32	C₄×C₅×D₄	—	2	2	no	—
33	C₂×C₅×C₄⋊C₄	—	2	1	no	—
34	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
35	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
36	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
37	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
38	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
39	C₂×C₄²×C₅	—	2	1	no	—
40	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₅	D₄	1	2	yes	—
43	C₂×C₄²×C₅	D₄	1	2	yes	—
44	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
45	C₅×C₄⋊C₈	—	2	2	no	—
46	C₅×C₄⋊C₈	—	2	2	no	—
47	C₄×C₅×C₈	—	2	8	no	—
48	C₄×C₅×C₈	—	2	2	no	—
49	C₄×C₅×C₈	D₄	1	4	yes	—
50	C₅×#95	—	2	2	no	—
51	C₅×#95	—	2	2	no	—
52	C₄×#124	C₂×C₅	1	1	yes	characteristic
53	#1831	C₂×C₅	1	1	yes	characteristic, complemented
54	C₄²×C₈	C₂×C₅	1	1	yes	characteristic, complemented
55	C₂×Q₈⋊C₈	C₂×C₅	1	2	yes	complemented
56	C₂×D₄⋊C₈	C₂×C₅	1	2	yes	—
57	C₂⁴×C₅	—	2	1	no	—
58	C₂×C₅×Q₈	—	2	1	no	—
59	C₂×C₅×Q₈	—	2	2	no	—
60	C₂×C₅×Q₈	—	2	2	no	—
61	C₂×C₅×Q₈	C₂×C₈	1	2	yes	—
62	C₂×C₅×D₄	—	2	2	no	—
63	C₂×C₅×D₄	—	2	1	no	—
64	C₂×C₅×D₄	—	2	2	no	—
65	C₂×C₅×D₄	C₂×C₈	1	2	yes	—
66	C₂²×C₄×C₅	—	2	1	no	—
67	C₂²×C₄×C₅	—	2	1	no	—
68	C₂²×C₄×C₅	—	2	1	no	—
69	C₂²×C₄×C₅	—	4	1	no	—
70	C₂²×C₄×C₅	—	4	1	no	—
71	C₂²×C₄×C₅	—	4	4	no	—
72	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
73	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
74	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
77	C₂×C₅×C₈	—	2	8	no	—
78	C₂×C₅×C₈	—	2	4	no	—
79	C₂×C₅×C₈	—	4	2	no	—
80	C₂×C₅×C₈	—	4	4	no	—
81	C₅×C₄⋊C₄	—	2	2	no	—
82	C₅×C₄⋊C₄	—	2	2	no	—
83	C₅×C₄⋊C₄	—	4	2	no	—
84	C₅×C₂²⋊C₄	—	2	8	no	—
85	C₅×C₂²⋊C₄	—	4	4	no	—
86	C₅×C₂²⋊C₄	—	4	2	no	—
87	C₄²×C₅	—	2	4	no	—
88	C₄²×C₅	—	2	2	no	—
89	C₄²×C₅	—	2	4	no	—
90	C₄²×C₅	—	4	1	no	—
91	C₄²×C₅	—	4	1	no	—
92	C₄²×C₅	C₂×C₈	1	2	yes	—
93	C₄²×C₅	M₁₆	1	2	yes	—
94	C₄²×C₅	C₂×D₄	1	2	yes	—
95	C₄²×C₅	#41	1	2	yes	—
96	C₂×#124	C₄×C₅	1	1	yes	characteristic
97	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
98	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
99	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
100	C₂×C₄×C₈	—	2	2	no	—
101	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
102	#391	C₄×C₅	1	1	yes	characteristic
103	#390	C₄×C₅	1	1	yes	characteristic
104	#388	—	2	2	no	—
105	#382	C₄×C₅	1	1	yes	characteristic
106	C₄×C₂²⋊C₄	—	2	2	no	—
107	C₄³	C₂²×C₅	1	1	yes	characteristic
108	#336	—	2	2	no	—
109	Q₈⋊C₈	—	2	4	no	—
110	D₄⋊C₈	—	2	4	no	—
111	C₂³×C₅	—	2	2	no	—
112	C₂³×C₅	—	4	1	no	—
113	C₂³×C₅	—	4	1	no	—
114	C₂³×C₅	—	4	2	no	—
115	C₂³×C₅	—	4	2	no	—
116	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₅×Q₈	—	2	4	no	—
118	C₅×Q₈	—	4	2	no	—
119	C₅×D₄	—	2	4	no	—
120	C₅×D₄	—	4	2	no	—
121	C₂×C₄×C₅	—	2	2	no	—
122	C₂×C₄×C₅	—	2	2	no	—
123	C₂×C₄×C₅	—	2	2	no	—
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	—	4	1	no	—
128	C₂×C₄×C₅	—	4	1	no	—
129	C₂×C₄×C₅	—	4	1	no	—
130	C₂×C₄×C₅	—	4	1	no	—
131	C₂×C₄×C₅	—	4	2	no	—
132	C₂×C₄×C₅	—	4	2	no	—
133	C₂×C₄×C₅	—	8	2	no	—
134	C₂×C₄×C₅	—	8	2	no	—
135	C₂×C₄×C₅	C₂²⋊C₈	1	4	yes	—
136	C₂×C₄×C₅	#104	1	4	yes	—
137	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
138	C₂×C₄×C₅	#121	1	2	yes	—
139	C₂×C₄×C₅	#124	1	1	yes	characteristic
140	C₂×C₄×C₅	#127	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
143	C₂×C₄×C₅	#131	1	2	yes	—
144	C₂×C₄×C₅	#135	1	4	yes	—
145	C₅×C₈	—	2	8	no	—
146	C₅×C₈	—	8	2	no	—
147	C₂²×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
148	C₂²×D₄	C₂×C₄×C₅	1	1	yes	characteristic
149	C₂³×C₄	—	2	1	no	—
150	C₂²×C₈	—	2	2	no	—
151	C₂²×C₈	—	2	2	no	—
152	#124	—	2	2	no	—
153	#124	C₅×C₈	1	4	yes	complemented
154	C₄×Q₈	—	2	2	no	—
155	C₄×Q₈	—	2	2	no	—
156	C₄×D₄	—	2	2	no	—
157	C₄×D₄	—	2	2	no	—
158	C₂×C₄⋊C₄	—	2	1	no	—
159	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
160	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
161	C₂×C₂²⋊C₄	—	2	2	no	—
162	C₂×C₂²⋊C₄	—	2	2	no	—
163	C₂×C₂²⋊C₄	—	2	1	no	—
164	C₂×C₄²	—	2	1	no	—
165	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
166	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
167	C₂×C₄²	C₅×D₄	1	2	yes	—
168	C₂×C₄²	C₅×D₄	1	2	yes	—
169	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	—	2	2	no	—
172	C₄×C₈	—	2	8	no	—
173	C₄×C₈	—	2	2	no	—
174	C₄×C₈	C₅×D₄	1	4	yes	—
175	#95	—	2	2	no	—
176	#95	—	2	2	no	—
177	C₂²×C₅	—	4	2	no	—
178	C₂²×C₅	—	4	4	no	—
179	C₂²×C₅	—	8	1	no	—
180	C₂²×C₅	—	8	1	no	—
181	C₂²×C₅	—	8	2	no	—
182	C₂²×C₅	#390	1	1	yes	characteristic
183	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
184	C₂²×C₅	#408	1	1	yes	characteristic
185	C₂²×C₅	#416	1	2	yes	—
186	C₂²×C₅	C₂×#104	1	2	yes	—
187	C₄×C₅	—	2	2	no	—
188	C₄×C₅	—	2	4	no	—
189	C₄×C₅	—	2	4	no	—
190	C₄×C₅	—	4	2	no	—
191	C₄×C₅	—	8	1	no	—
192	C₄×C₅	—	8	1	no	—
193	C₄×C₅	C₈×D₄	1	2	yes	—
194	C₄×C₅	#436	1	2	yes	—
195	C₄×C₅	#488	1	2	yes	—
196	C₄×C₅	#495	1	2	yes	—
197	C₂⁴	—	2	1	no	—
198	C₂×Q₈	—	2	1	no	—
199	C₂×Q₈	—	2	2	no	—
200	C₂×Q₈	—	2	2	no	—
201	C₂×Q₈	C₂×C₅×C₈	1	2	yes	—
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	—	2	1	no	—
204	C₂×D₄	—	2	2	no	—
205	C₂×D₄	C₂×C₅×C₈	1	2	yes	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	4	1	no	—
210	C₂²×C₄	—	4	1	no	—
211	C₂²×C₄	—	4	4	no	—
212	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
213	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
214	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
217	C₂×C₈	—	2	8	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	4	4	no	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	4	2	no	—
224	C₂²⋊C₄	—	2	8	no	—
225	C₂²⋊C₄	—	4	2	no	—
226	C₂²⋊C₄	—	4	4	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	2	4	no	—
229	C₄²	—	2	4	no	—
230	C₄²	—	4	1	no	—
231	C₄²	—	4	1	no	—
232	C₄²	C₂×C₅×C₈	1	2	yes	—
233	C₄²	C₅×M₁₆	1	2	yes	—
234	C₄²	C₂×C₅×D₄	1	2	yes	—
235	C₄²	C₅×#41	1	2	yes	—
236	C₂×C₅	—	8	2	no	—
237	C₂×C₅	#1476	1	2	yes	—
238	C₂×C₅	#1494	1	2	yes	—
239	C₂×C₅	#1954	1	1	yes	characteristic
240	C₂×C₅	#1958	1	1	yes	characteristic
241	C₂×C₅	#1962	1	1	yes	characteristic
242	C₂³	—	2	2	no	—
243	C₂³	—	4	1	no	—
244	C₂³	—	4	1	no	—
245	C₂³	—	4	2	no	—
246	C₂³	—	4	2	no	—
247	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
248	Q₈	—	2	4	no	—
249	Q₈	—	4	2	no	—
250	D₄	—	2	4	no	—
251	D₄	—	4	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	8	2	no	—
265	C₂×C₄	—	8	2	no	—
266	C₂×C₄	C₅×C₂²⋊C₈	1	4	yes	—
267	C₂×C₄	C₅×#104	1	4	yes	—
268	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
269	C₂×C₄	C₅×#121	1	2	yes	—
270	C₂×C₄	C₅×#124	1	1	yes	characteristic
271	C₂×C₄	C₅×#127	1	1	yes	characteristic
272	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
273	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
274	C₂×C₄	C₅×#131	1	2	yes	—
275	C₂×C₄	C₅×#135	1	4	yes	—
276	C₈	—	2	8	no	—
277	C₈	—	8	2	no	—
278	C₅	#9639	1	1	yes	characteristic, direct product factor, Sylow
279	C₂²	—	4	2	no	—
280	C₂²	—	4	4	no	—
281	C₂²	—	8	1	no	—
282	C₂²	—	8	1	no	—
283	C₂²	—	8	2	no	—
284	C₂²	C₅×#390	1	1	yes	characteristic
285	C₂²	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
286	C₂²	C₅×#408	1	1	yes	characteristic
287	C₂²	C₅×#416	1	2	yes	—
288	C₂²	C₂×C₅×#104	1	2	yes	—
289	C₄	—	2	2	no	—
290	C₄	—	2	4	no	—
291	C₄	—	2	4	no	—
292	C₄	—	4	2	no	—
293	C₄	—	8	1	no	—
294	C₄	—	8	1	no	—
295	C₄	C₅×C₈×D₄	1	2	yes	—
296	C₄	C₅×#436	1	2	yes	—
297	C₄	C₅×#488	1	2	yes	—
298	C₄	C₅×#495	1	2	yes	—
299	C₂	—	8	2	no	—
300	C₂	C₅×#1476	1	2	yes	—
301	C₂	C₅×#1494	1	2	yes	—
302	C₂	C₅×#1954	1	1	yes	characteristic
303	C₂	C₅×#1958	1	1	yes	characteristic
304	C₂	C₅×#1962	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator	C₂×C₄	C₂²×C₅×C₈
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂³×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×C₅×D₄	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₄³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#1831	C₂	semidirect
3. C₄²×C₅×C₈	C₂	semidirect
4. C₂×C₅×Q₈⋊C₈	C₂	semidirect
21. #9639	C₅	direct (N × H)
28. C₅×#124	C₈	semidirect
53. #1831	C₂×C₅	semidirect
54. C₄²×C₈	C₂×C₅	semidirect
55. C₂×Q₈⋊C₈	C₂×C₅	semidirect
153. #124	C₅×C₈	semidirect
278. C₅	#9639	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₅×#124	2
C₅×#1831	2
C₄²×C₅×C₈	2
C₂×C₅×Q₈⋊C₈	2
C₂×C₅×D₄⋊C₈	2
#9639	5

Subgroups — C5×#9639 #11932359

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#9639 #11932359