Subgroups — C₅×#8684 #11931404

Order 1280.

430 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×#2931	C₂	1	1	yes	maximal, characteristic
2	C₅×#1848	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#335	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1547	C₂	1	2	yes	maximal
5	C₅×#1292	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#131	—	2	1	no	—
7	C₅×#532	C₂²	1	1	yes	characteristic
8	C₅×#436	—	2	1	no	—
9	C₅×#436	C₄	1	2	yes	—
10	C₅×C₈×D₄	—	2	1	no	—
11	C₅×C₈×D₄	C₄	1	2	yes	—
12	C₂×C₅×#108	—	2	1	no	—
13	C₂×C₅×#108	—	2	1	no	—
14	C₅×#423	C₂²	1	2	yes	—
15	C₅×#423	C₂²	1	1	yes	characteristic
16	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
17	C₅×#408	—	2	1	no	—
18	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
19	C₅×#335	—	2	1	no	—
20	C₅×#335	—	2	1	no	—
21	C₅×#332	—	2	2	no	—
22	C₅×#331	—	2	2	no	—
23	#8684	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂×C₅×#41	—	2	1	no	—
25	C₅×#131	—	4	2	no	—
26	C₅×#131	—	4	2	no	—
27	C₂×C₅×M₁₆	—	2	1	no	—
28	C₂×C₅×M₁₆	—	2	2	no	—
29	C₂×C₅×M₁₆	—	2	2	no	—
30	C₂²×C₅×C₈	—	2	2	no	—
31	C₂²×C₅×C₈	—	2	2	no	—
32	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
33	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
34	C₅×C₄⋊Q₈	D₄	1	1	yes	characteristic
35	C₅×#127	D₄	1	1	yes	characteristic
36	C₅×#124	—	2	1	no	—
37	C₅×#121	—	2	1	no	—
38	C₅×#121	Q₈	1	2	yes	—
39	C₄×C₅×D₄	—	2	1	no	—
40	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
42	C₅×#108	—	4	2	no	—
43	C₅×#108	—	4	2	no	—
44	C₅×C₄⋊C₈	—	2	1	no	—
45	C₅×C₄⋊C₈	—	2	2	no	—
46	C₅×C₄⋊C₈	—	2	2	no	—
47	C₅×C₄⋊C₈	—	2	2	no	—
48	C₅×C₄⋊C₈	—	2	2	no	—
49	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
50	C₅×C₂²⋊C₈	—	2	2	no	—
51	C₅×C₂²⋊C₈	—	2	2	no	—
52	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
53	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
54	C₄×C₅×C₈	D₄	1	1	yes	characteristic
55	C₄×C₅×C₈	D₄	1	1	yes	characteristic
56	#2931	C₂×C₅	1	1	yes	characteristic
57	#1848	C₂×C₅	1	1	yes	characteristic, complemented
58	C₂×#335	C₂×C₅	1	1	yes	characteristic, complemented
59	#1547	C₂×C₅	1	2	yes	—
60	#1292	C₂×C₅	1	2	yes	complemented
61	C₅×#41	—	4	1	no	—
62	C₅×#41	—	4	1	no	—
63	C₅×#41	—	4	1	no	—
64	C₅×#41	—	4	1	no	—
65	C₂×C₅×Q₈	—	2	1	no	—
66	C₂×C₅×D₄	—	2	2	no	—
67	C₂×C₅×D₄	—	2	1	no	—
68	C₂×C₅×D₄	—	2	1	no	—
69	C₂×C₅×D₄	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₅	—	2	2	no	—
71	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
73	C₂²×C₄×C₅	#41	1	1	yes	characteristic
74	C₅×M₁₆	—	4	1	no	—
75	C₅×M₁₆	—	4	1	no	—
76	C₅×M₁₆	—	4	4	no	—
77	C₅×M₁₆	—	4	2	no	—
78	C₅×M₁₆	—	4	2	no	—
79	C₂×C₅×C₈	—	2	1	no	—
80	C₂×C₅×C₈	—	2	1	no	—
81	C₂×C₅×C₈	—	2	2	no	—
82	C₂×C₅×C₈	—	4	4	no	—
83	C₂×C₅×C₈	—	4	2	no	—
84	C₂×C₅×C₈	—	4	2	no	—
85	C₂×C₅×C₈	—	8	2	no	—
86	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₅×C₈	D₈	1	2	yes	—
91	C₂×C₅×C₈	SD₁₆	1	2	yes	—
92	C₅×C₄⋊C₄	—	2	1	no	—
93	C₅×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
94	C₅×C₂²⋊C₄	—	2	2	no	—
95	C₅×C₂²⋊C₄	—	2	2	no	—
96	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₅	#41	1	1	yes	characteristic
100	C₂×#131	—	2	1	no	—
101	#532	C₂²×C₅	1	1	yes	characteristic
102	#436	—	2	1	no	—
103	#436	C₄×C₅	1	2	yes	—
104	C₈×D₄	—	2	1	no	—
105	C₈×D₄	C₄×C₅	1	2	yes	—
106	C₂×#108	—	2	1	no	—
107	C₂×#108	—	2	1	no	—
108	#423	C₂²×C₅	1	2	yes	—
109	#423	C₂²×C₅	1	1	yes	characteristic
110	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
111	#408	—	2	1	no	—
112	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
113	#335	—	2	1	no	—
114	#335	—	2	1	no	—
115	#332	—	2	2	no	—
116	#331	—	2	2	no	—
117	C₂³×C₅	—	2	2	no	—
118	C₂³×C₅	C₂²⋊Q₈	1	1	yes	characteristic
119	C₅×Q₈	—	4	1	no	—
120	C₅×Q₈	—	4	1	no	—
121	C₅×D₄	—	4	2	no	—
122	C₅×D₄	—	4	2	no	—
123	C₅×D₄	—	4	2	no	—
124	C₅×D₄	—	4	1	no	—
125	C₅×D₄	—	4	1	no	—
126	C₅×D₄	—	4	1	no	—
127	C₅×D₄	—	4	1	no	—
128	C₂×C₄×C₅	—	2	1	no	—
129	C₂×C₄×C₅	—	2	1	no	—
130	C₂×C₄×C₅	—	2	2	no	—
131	C₂×C₄×C₅	—	4	1	no	—
132	C₂×C₄×C₅	—	4	2	no	—
133	C₂×C₄×C₅	—	4	2	no	—
134	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₅	#117	1	1	yes	characteristic
137	C₂×C₄×C₅	#121	1	1	yes	characteristic
138	C₂×C₄×C₅	#121	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
142	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂×C₄×C₅	#137	1	1	yes	characteristic
144	C₅×C₈	—	2	1	no	—
145	C₅×C₈	—	2	1	no	—
146	C₅×C₈	—	4	2	no	—
147	C₅×C₈	—	8	2	no	—
148	C₅×C₈	—	8	2	no	—
149	C₅×C₈	#102	1	2	yes	—
150	C₅×C₈	#102	1	2	yes	—
151	C₂×#41	—	2	1	no	—
152	#131	—	4	2	no	—
153	#131	—	4	2	no	—
154	C₂×M₁₆	—	2	1	no	—
155	C₂×M₁₆	—	2	2	no	—
156	C₂×M₁₆	—	2	2	no	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	—	2	2	no	—
159	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
160	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
161	C₄⋊Q₈	C₅×D₄	1	1	yes	characteristic
162	#127	C₅×D₄	1	1	yes	characteristic
163	#124	—	2	1	no	—
164	#121	—	2	1	no	—
165	#121	C₅×Q₈	1	2	yes	—
166	C₄×D₄	—	2	1	no	—
167	C₄×D₄	C₂×C₄×C₅	1	2	yes	—
168	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
169	#108	—	4	2	no	—
170	#108	—	4	2	no	—
171	C₄⋊C₈	—	2	1	no	—
172	C₄⋊C₈	—	2	2	no	—
173	C₄⋊C₈	—	2	2	no	—
174	C₄⋊C₈	—	2	2	no	—
175	C₄⋊C₈	—	2	2	no	—
176	C₄⋊C₈	C₂×C₄×C₅	1	2	yes	—
177	C₂²⋊C₈	—	2	2	no	—
178	C₂²⋊C₈	—	2	2	no	—
179	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
180	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
181	C₄×C₈	C₅×D₄	1	1	yes	characteristic
182	C₄×C₈	C₅×D₄	1	1	yes	characteristic
183	C₂²×C₅	—	2	1	no	—
184	C₂²×C₅	—	2	1	no	—
185	C₂²×C₅	—	4	2	no	—
186	C₂²×C₅	—	4	2	no	—
187	C₂²×C₅	—	4	2	no	—
188	C₂²×C₅	#380	1	1	yes	characteristic
189	C₂²×C₅	#474	1	1	yes	characteristic
190	C₂²×C₅	#476	1	1	yes	characteristic
191	C₄×C₅	—	2	1	no	—
192	C₄×C₅	—	2	1	no	—
193	C₄×C₅	—	2	1	no	—
194	C₄×C₅	—	4	2	no	—
195	C₄×C₅	C₂×#102	1	1	yes	characteristic
196	C₄×C₅	#417	1	1	yes	characteristic
197	C₄×C₅	#459	1	1	yes	characteristic
198	C₄×C₅	#461	1	1	yes	characteristic
199	C₄×C₅	#472	1	1	yes	characteristic
200	C₄×C₅	#473	1	1	yes	characteristic
201	#41	—	4	1	no	—
202	#41	—	4	1	no	—
203	#41	—	4	1	no	—
204	#41	—	4	1	no	—
205	C₂×Q₈	—	2	1	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	2	1	no	—
208	C₂×D₄	—	2	1	no	—
209	C₂×D₄	C₅×C₄⋊C₄	1	2	yes	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
213	C₂²×C₄	C₅×#41	1	1	yes	characteristic
214	M₁₆	—	4	1	no	—
215	M₁₆	—	4	1	no	—
216	M₁₆	—	4	4	no	—
217	M₁₆	—	4	2	no	—
218	M₁₆	—	4	2	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	4	4	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	8	2	no	—
226	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
227	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
228	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
229	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
230	C₂×C₈	C₅×D₈	1	2	yes	—
231	C₂×C₈	C₅×SD₁₆	1	2	yes	—
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	C₅×C₄⋊C₄	1	2	yes	—
234	C₂²⋊C₄	—	2	2	no	—
235	C₂²⋊C₄	—	2	2	no	—
236	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
237	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
238	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
239	C₄²	C₅×#41	1	1	yes	characteristic
240	C₂×C₅	—	2	1	no	—
241	C₂×C₅	—	2	1	no	—
242	C₂×C₅	—	8	2	no	—
243	C₂×C₅	#1802	1	1	yes	characteristic
244	C₂×C₅	#1813	1	1	yes	characteristic
245	C₂×C₅	#1816	1	1	yes	characteristic
246	C₂³	—	2	2	no	—
247	C₂³	C₅×C₂²⋊Q₈	1	1	yes	characteristic
248	Q₈	—	4	1	no	—
249	Q₈	—	4	1	no	—
250	D₄	—	4	2	no	—
251	D₄	—	4	2	no	—
252	D₄	—	4	2	no	—
253	D₄	—	4	1	no	—
254	D₄	—	4	1	no	—
255	D₄	—	4	1	no	—
256	D₄	—	4	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
264	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
265	C₂×C₄	C₅×#117	1	1	yes	characteristic
266	C₂×C₄	C₅×#121	1	1	yes	characteristic
267	C₂×C₄	C₅×#121	1	1	yes	characteristic
268	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
269	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
270	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
271	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
272	C₂×C₄	C₅×#137	1	1	yes	characteristic
273	C₈	—	2	1	no	—
274	C₈	—	2	1	no	—
275	C₈	—	4	2	no	—
276	C₈	—	8	2	no	—
277	C₈	—	8	2	no	—
278	C₈	C₅×#102	1	2	yes	—
279	C₈	C₅×#102	1	2	yes	—
280	C₅	#8684	1	1	yes	characteristic, direct product factor, Sylow
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	4	2	no	—
286	C₂²	C₅×#380	1	1	yes	characteristic
287	C₂²	C₅×#474	1	1	yes	characteristic
288	C₂²	C₅×#476	1	1	yes	characteristic
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	4	2	no	—
293	C₄	C₂×C₅×#102	1	1	yes	characteristic
294	C₄	C₅×#417	1	1	yes	characteristic
295	C₄	C₅×#459	1	1	yes	characteristic
296	C₄	C₅×#461	1	1	yes	characteristic
297	C₄	C₅×#472	1	1	yes	characteristic
298	C₄	C₅×#473	1	1	yes	characteristic
299	C₂	—	2	1	no	—
300	C₂	—	2	1	no	—
301	C₂	—	8	2	no	—
302	C₂	C₅×#1802	1	1	yes	characteristic
303	C₂	C₅×#1813	1	1	yes	characteristic
304	C₂	C₅×#1816	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	#380
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₅×#532	C₂²
LCS	G ▷ 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
2. C₅×#1848	C₂	semidirect
3. C₂×C₅×#335	C₂	semidirect
5. C₅×#1292	C₂	semidirect
23. #8684	C₅	direct (N × H)
57. #1848	C₂×C₅	semidirect
58. C₂×#335	C₂×C₅	semidirect
60. #1292	C₂×C₅	semidirect
280. C₅	#8684	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×#2931	2
C₅×#1848	2
C₂×C₅×#335	2
C₅×#1547	2
C₅×#1292	2
#8684	5

Subgroups — C5×#8684 #11931404

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8684 #11931404