Subgroups — C₅×#3045 #10616640

Order 640.

504 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#135	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#535	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#520	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#464	C₂	1	2	yes	maximal, complemented
6	C₅×#463	C₂	1	2	yes	maximal, complemented
7	C₅×#460	C₂	1	2	yes	maximal, complemented
8	C₅×#459	C₂	1	2	yes	maximal, complemented
9	C₅×#424	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#417	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₅×#102	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₅×#41	C₂²	1	1	yes	characteristic
13	C₂×C₅×#41	C₂²	1	1	yes	characteristic
14	C₂²×C₅×D₄	C₂²	1	1	yes	characteristic
15	C₅×AGL(1,ℤ/8)	—	2	4	no	—
16	C₅×#135	—	2	4	no	—
17	C₂×C₅×Q₁₆	C₂²	1	1	yes	characteristic, complemented
18	C₂×C₅×SD₁₆	C₂²	1	2	yes	—
19	C₂×C₅×SD₁₆	C₂²	1	1	yes	characteristic, complemented
20	C₂×C₅×SD₁₆	C₂²	1	1	yes	characteristic, complemented
21	C₂×C₅×D₈	C₂²	1	2	yes	—
22	C₂×C₅×D₈	C₂²	1	1	yes	characteristic, complemented
23	C₂×C₅×M₁₆	C₂²	1	1	yes	characteristic
24	C₂²×C₅×C₈	C₂²	1	1	yes	characteristic
25	C₅×#127	C₂²	1	2	yes	—
26	C₅×#124	C₂²	1	2	yes	—
27	C₅×#121	—	2	1	no	—
28	C₅×#121	—	2	1	no	—
29	C₅×#120	—	2	2	no	—
30	C₄×C₅×Q₈	C₂²	1	2	yes	complemented
31	C₄×C₅×D₄	—	2	2	no	—
32	C₄×C₅×D₄	C₂²	1	2	yes	complemented
33	C₅×#117	—	2	1	no	—
34	C₅×#117	C₂²	1	1	yes	characteristic
35	C₂×C₅×C₄⋊C₄	—	2	1	no	—
36	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
37	C₅×C₄⋊C₈	C₂²	1	2	yes	complemented
38	C₅×C₄⋊C₈	C₂²	1	2	yes	complemented
39	C₅×#103	C₂²	1	2	yes	complemented
40	C₅×#102	C₂²	1	2	yes	—
41	C₅×#102	C₂²	1	2	yes	—
42	C₅×#102	C₂²	1	2	yes	complemented
43	#3045	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂⁴×C₅	—	2	1	no	—
45	C₅×#41	—	2	2	no	—
46	C₅×#41	—	2	2	no	—
47	C₅×#41	—	2	2	no	—
48	C₅×#41	D₄	1	4	yes	complemented
49	C₂×C₅×Q₈	C₂³	1	1	yes	characteristic
50	C₂×C₅×Q₈	C₂³	1	1	yes	characteristic
51	C₂×C₅×D₄	—	2	1	no	—
52	C₂×C₅×D₄	—	2	1	no	—
53	C₂×C₅×D₄	—	2	2	no	—
54	C₂×C₅×D₄	—	2	1	no	—
55	C₂×C₅×D₄	—	2	2	no	—
56	C₂×C₅×D₄	—	2	2	no	—
57	C₂×C₅×D₄	—	2	2	no	—
58	C₂×C₅×D₄	C₂³	1	2	yes	—
59	C₂×C₅×D₄	C₂³	1	1	yes	characteristic
60	C₂×C₅×D₄	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₅	—	2	1	no	—
62	C₂²×C₄×C₅	—	2	1	no	—
63	C₂²×C₄×C₅	—	2	1	no	—
64	C₂²×C₄×C₅	—	2	2	no	—
65	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
66	C₅×Q₁₆	—	2	2	no	—
67	C₅×SD₁₆	—	2	4	no	—
68	C₅×SD₁₆	—	2	2	no	—
69	C₅×SD₁₆	—	2	2	no	—
70	C₅×D₈	—	2	4	no	—
71	C₅×D₈	—	2	2	no	—
72	C₅×M₁₆	—	2	2	no	—
73	C₂×C₅×C₈	—	2	2	no	—
74	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
75	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
76	C₂×C₅×C₈	C₂³	1	2	yes	—
77	C₅×C₄⋊C₄	—	2	1	no	—
78	C₅×C₄⋊C₄	—	2	2	no	—
79	C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
80	C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
81	C₅×C₄⋊C₄	C₂³	1	2	yes	—
82	C₅×C₂²⋊C₄	—	2	2	no	—
83	C₅×C₂²⋊C₄	—	2	2	no	—
84	C₅×C₂²⋊C₄	—	2	2	no	—
85	C₅×C₂²⋊C₄	D₄	1	2	yes	—
86	C₄²×C₅	—	2	2	no	—
87	C₄²×C₅	C₂³	1	2	yes	—
88	C₂×AGL(1,ℤ/8)	C₂×C₅	1	1	yes	characteristic, complemented
89	C₂×#135	C₂×C₅	1	1	yes	characteristic, complemented
90	#535	C₂×C₅	1	1	yes	characteristic, complemented
91	#520	C₂×C₅	1	1	yes	characteristic, complemented
92	#464	C₂×C₅	1	2	yes	complemented
93	#463	C₂×C₅	1	2	yes	complemented
94	#460	C₂×C₅	1	2	yes	complemented
95	#459	C₂×C₅	1	2	yes	complemented
96	#424	C₂×C₅	1	1	yes	characteristic, complemented
97	#417	C₂×C₅	1	1	yes	characteristic, complemented
98	C₂×#102	C₂×C₅	1	1	yes	characteristic, complemented
99	C₂³×C₅	—	2	1	no	—
100	C₂³×C₅	—	2	2	no	—
101	C₂³×C₅	—	2	1	no	—
102	C₂³×C₅	—	4	2	no	—
103	C₂³×C₅	—	8	1	no	—
104	C₂³×C₅	—	8	1	no	—
105	C₂³×C₅	C₂×D₄	1	1	yes	characteristic
106	C₅×Q₈	—	2	1	no	—
107	C₅×Q₈	—	2	1	no	—
108	C₅×Q₈	—	2	1	no	—
109	C₅×Q₈	C₂×D₄	1	2	yes	—
110	C₅×D₄	—	2	4	no	—
111	C₅×D₄	—	2	2	no	—
112	C₅×D₄	—	2	1	no	—
113	C₅×D₄	—	2	1	no	—
114	C₅×D₄	—	2	2	no	—
115	C₅×D₄	—	2	1	no	—
116	C₅×D₄	—	4	2	no	—
117	C₅×D₄	—	4	2	no	—
118	C₅×D₄	—	8	2	no	—
119	C₅×D₄	—	8	2	no	—
120	C₅×D₄	C₂×D₄	1	2	yes	—
121	C₂×C₄×C₅	—	2	1	no	—
122	C₂×C₄×C₅	—	2	4	no	—
123	C₂×C₄×C₅	—	2	2	no	—
124	C₂×C₄×C₅	—	2	1	no	—
125	C₂×C₄×C₅	—	4	2	no	—
126	C₂×C₄×C₅	—	8	2	no	—
127	C₂×C₄×C₅	—	8	1	no	—
128	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
129	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
130	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
132	C₂×C₄×C₅	#41	1	2	yes	—
133	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
134	C₅×C₈	—	2	1	no	—
135	C₅×C₈	—	2	1	no	—
136	C₅×C₈	—	2	2	no	—
137	C₂×#41	C₂²×C₅	1	1	yes	characteristic
138	C₂×#41	C₂²×C₅	1	1	yes	characteristic
139	C₂²×D₄	C₂²×C₅	1	1	yes	characteristic
140	AGL(1,ℤ/8)	—	2	4	no	—
141	#135	—	2	4	no	—
142	C₂×Q₁₆	C₂²×C₅	1	1	yes	characteristic, complemented
143	C₂×SD₁₆	C₂²×C₅	1	2	yes	—
144	C₂×SD₁₆	C₂²×C₅	1	1	yes	characteristic, complemented
145	C₂×SD₁₆	C₂²×C₅	1	1	yes	characteristic, complemented
146	C₂×D₈	C₂²×C₅	1	2	yes	—
147	C₂×D₈	C₂²×C₅	1	1	yes	characteristic, complemented
148	C₂×M₁₆	C₂²×C₅	1	1	yes	characteristic
149	C₂²×C₈	C₂²×C₅	1	1	yes	characteristic
150	#127	C₂²×C₅	1	2	yes	—
151	#124	C₂²×C₅	1	2	yes	—
152	#121	—	2	1	no	—
153	#121	—	2	1	no	—
154	#120	—	2	2	no	—
155	C₄×Q₈	C₂²×C₅	1	2	yes	complemented
156	C₄×D₄	—	2	2	no	—
157	C₄×D₄	C₂²×C₅	1	2	yes	complemented
158	#117	—	2	1	no	—
159	#117	C₂²×C₅	1	1	yes	characteristic
160	C₂×C₄⋊C₄	—	2	1	no	—
161	C₂×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
162	C₄⋊C₈	C₂²×C₅	1	2	yes	complemented
163	C₄⋊C₈	C₂²×C₅	1	2	yes	complemented
164	#103	C₂²×C₅	1	2	yes	complemented
165	#102	C₂²×C₅	1	2	yes	—
166	#102	C₂²×C₅	1	2	yes	—
167	#102	C₂²×C₅	1	2	yes	complemented
168	C₂²×C₅	—	2	1	no	—
169	C₂²×C₅	—	2	1	no	—
170	C₂²×C₅	—	2	2	no	—
171	C₂²×C₅	—	4	1	no	—
172	C₂²×C₅	—	4	1	no	—
173	C₂²×C₅	—	4	1	no	—
174	C₂²×C₅	—	4	2	no	—
175	C₂²×C₅	—	4	1	no	—
176	C₂²×C₅	—	4	2	no	—
177	C₂²×C₅	—	4	1	no	—
178	C₂²×C₅	—	4	2	no	—
179	C₂²×C₅	—	8	4	no	—
180	C₂²×C₅	#121	1	2	yes	—
181	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
182	C₄×C₅	—	2	2	no	—
183	C₄×C₅	—	2	2	no	—
184	C₄×C₅	—	2	2	no	—
185	C₄×C₅	—	4	1	no	—
186	C₄×C₅	—	4	2	no	—
187	C₄×C₅	#121	1	2	yes	—
188	C₄×C₅	C₂²×D₄	1	1	yes	characteristic
189	C₄×C₅	C₂×#41	1	1	yes	characteristic
190	C₂⁴	—	2	1	no	—
191	#41	—	2	2	no	—
192	#41	—	2	2	no	—
193	#41	—	2	2	no	—
194	#41	C₅×D₄	1	4	yes	complemented
195	C₂×Q₈	C₂³×C₅	1	1	yes	characteristic
196	C₂×Q₈	C₂³×C₅	1	1	yes	characteristic
197	C₂×D₄	—	2	1	no	—
198	C₂×D₄	—	2	1	no	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	—	2	1	no	—
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	C₂³×C₅	1	2	yes	—
205	C₂×D₄	C₂³×C₅	1	1	yes	characteristic
206	C₂×D₄	C₂³×C₅	1	1	yes	characteristic
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
212	Q₁₆	—	2	2	no	—
213	SD₁₆	—	2	4	no	—
214	SD₁₆	—	2	2	no	—
215	SD₁₆	—	2	2	no	—
216	D₈	—	2	4	no	—
217	D₈	—	2	2	no	—
218	M₁₆	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	C₂³×C₅	1	1	yes	characteristic
221	C₂×C₈	C₂³×C₅	1	1	yes	characteristic
222	C₂×C₈	C₂³×C₅	1	2	yes	—
223	C₄⋊C₄	—	2	1	no	—
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
226	C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
227	C₄⋊C₄	C₂³×C₅	1	2	yes	—
228	C₂²⋊C₄	—	2	2	no	—
229	C₂²⋊C₄	—	2	2	no	—
230	C₂²⋊C₄	—	2	2	no	—
231	C₂²⋊C₄	C₅×D₄	1	2	yes	—
232	C₄²	—	2	2	no	—
233	C₄²	C₂³×C₅	1	2	yes	—
234	C₂×C₅	—	2	2	no	—
235	C₂×C₅	—	4	2	no	—
236	C₂×C₅	—	8	1	no	—
237	C₂×C₅	—	8	2	no	—
238	C₂×C₅	C₂×#121	1	1	yes	characteristic
239	C₂×C₅	#576	1	1	yes	characteristic
240	C₂×C₅	#577	1	1	yes	characteristic
241	C₂³	—	2	1	no	—
242	C₂³	—	2	2	no	—
243	C₂³	—	2	1	no	—
244	C₂³	—	4	2	no	—
245	C₂³	—	8	1	no	—
246	C₂³	—	8	1	no	—
247	C₂³	C₂×C₅×D₄	1	1	yes	characteristic
248	Q₈	—	2	1	no	—
249	Q₈	—	2	1	no	—
250	Q₈	—	2	1	no	—
251	Q₈	C₂×C₅×D₄	1	2	yes	—
252	D₄	—	2	4	no	—
253	D₄	—	2	2	no	—
254	D₄	—	2	1	no	—
255	D₄	—	2	1	no	—
256	D₄	—	2	2	no	—
257	D₄	—	2	1	no	—
258	D₄	—	4	2	no	—
259	D₄	—	4	2	no	—
260	D₄	—	8	2	no	—
261	D₄	—	8	2	no	—
262	D₄	C₂×C₅×D₄	1	2	yes	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	8	2	no	—
269	C₂×C₄	—	8	1	no	—
270	C₂×C₄	C₂×C₅×D₄	1	2	yes	—
271	C₂×C₄	C₂×C₅×D₄	1	2	yes	—
272	C₂×C₄	C₂×C₅×D₄	1	1	yes	characteristic
273	C₂×C₄	C₂×C₅×D₄	1	2	yes	—
274	C₂×C₄	C₅×#41	1	2	yes	—
275	C₂×C₄	C₂⁴×C₅	1	1	yes	characteristic
276	C₈	—	2	1	no	—
277	C₈	—	2	1	no	—
278	C₈	—	2	2	no	—
279	C₅	#3045	1	1	yes	characteristic, direct product factor, Sylow
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	4	1	no	—
284	C₂²	—	4	1	no	—
285	C₂²	—	4	1	no	—
286	C₂²	—	4	2	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	2	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	2	no	—
291	C₂²	—	8	4	no	—
292	C₂²	C₅×#121	1	2	yes	—
293	C₂²	C₂²×C₅×D₄	1	1	yes	characteristic
294	C₄	—	2	2	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	4	1	no	—
298	C₄	—	4	2	no	—
299	C₄	C₅×#121	1	2	yes	—
300	C₄	C₂²×C₅×D₄	1	1	yes	characteristic
301	C₄	C₂×C₅×#41	1	1	yes	characteristic
302	C₂	—	2	2	no	—
303	C₂	—	4	2	no	—
304	C₂	—	8	2	no	—
305	C₂	—	8	1	no	—
306	C₂	C₂×C₅×#121	1	1	yes	characteristic
307	C₂	C₅×#576	1	1	yes	characteristic
308	C₂	C₅×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×AGL(1,ℤ/8)	C₂	semidirect
2. C₂×C₅×#135	C₂	semidirect
3. C₅×#535	C₂	semidirect
4. C₅×#520	C₂	semidirect
5. C₅×#464	C₂	semidirect
6. C₅×#463	C₂	semidirect
7. C₅×#460	C₂	semidirect
8. C₅×#459	C₂	semidirect
9. C₅×#424	C₂	semidirect
10. C₅×#417	C₂	semidirect
11. C₂×C₅×#102	C₂	semidirect
17. C₂×C₅×Q₁₆	C₂²	semidirect
19. C₂×C₅×SD₁₆	C₂²	semidirect
20. C₂×C₅×SD₁₆	C₂²	semidirect
22. C₂×C₅×D₈	C₂²	semidirect
30. C₄×C₅×Q₈	C₂²	semidirect
32. C₄×C₅×D₄	C₂²	semidirect
37. C₅×C₄⋊C₈	C₂²	semidirect
38. C₅×C₄⋊C₈	C₂²	semidirect
39. C₅×#103	C₂²	semidirect
42. C₅×#102	C₂²	semidirect
43. #3045	C₅	direct (N × H)
48. C₅×#41	D₄	semidirect
88. C₂×AGL(1,ℤ/8)	C₂×C₅	semidirect
89. C₂×#135	C₂×C₅	semidirect
90. #535	C₂×C₅	semidirect
91. #520	C₂×C₅	semidirect
92. #464	C₂×C₅	semidirect
93. #463	C₂×C₅	semidirect
94. #460	C₂×C₅	semidirect
95. #459	C₂×C₅	semidirect
96. #424	C₂×C₅	semidirect
97. #417	C₂×C₅	semidirect
98. C₂×#102	C₂×C₅	semidirect
142. C₂×Q₁₆	C₂²×C₅	semidirect
144. C₂×SD₁₆	C₂²×C₅	semidirect
145. C₂×SD₁₆	C₂²×C₅	semidirect
147. C₂×D₈	C₂²×C₅	semidirect
155. C₄×Q₈	C₂²×C₅	semidirect
157. C₄×D₄	C₂²×C₅	semidirect
162. C₄⋊C₈	C₂²×C₅	semidirect
163. C₄⋊C₈	C₂²×C₅	semidirect
164. #103	C₂²×C₅	semidirect
167. #102	C₂²×C₅	semidirect
194. #41	C₅×D₄	semidirect
279. C₅	#3045	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×AGL(1,ℤ/8)	2
C₂×C₅×#135	2
C₅×#535	2
C₅×#520	2
C₅×#464	2
C₅×#463	2
C₅×#460	2
C₅×#459	2
C₅×#424	2
C₅×#417	2
C₂×C₅×#102	2
#3045	5

Subgroups — C5×#3045 #10616640

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#3045 #10616640