Subgroups — C₅×#3018 #10616613

Order 640.

724 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#585	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#575	C₂	1	2	yes	maximal, complemented
3	C₅×#532	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#456	C₂	1	2	yes	maximal, complemented
5	C₅×#455	C₂	1	2	yes	maximal, complemented
6	C₅×#454	C₂	1	2	yes	maximal, complemented
7	C₅×#453	C₂	1	2	yes	maximal, complemented
8	C₂×C₅×#104	C₂	1	2	yes	maximal, complemented
9	C₅×#413	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×2₋¹⁺⁴	—	2	2	no	—
11	C₅×2₋¹⁺⁴	C₂²	1	2	yes	—
12	C₅×H(5,𝔽₂)	—	2	2	no	—
13	C₅×H(5,𝔽₂)	—	2	2	no	—
14	C₅×H(5,𝔽₂)	C₂²	1	2	yes	—
15	C₂×C₅×#41	—	2	4	no	—
16	C₂×C₅×#41	—	2	2	no	—
17	C₂×C₅×#41	C₂²	1	2	yes	—
18	C₂×C₅×#41	C₂²	1	1	yes	characteristic
19	C₂×C₅×#41	C₂²	1	1	yes	characteristic, complemented
20	C₅×#137	—	2	2	no	—
21	C₅×#137	C₂²	1	4	yes	complemented
22	C₅×AGL(1,ℤ/8)	—	2	2	no	—
23	C₅×AGL(1,ℤ/8)	C₂²	1	4	yes	complemented
24	C₅×#135	—	2	4	no	—
25	C₂×C₅×M₁₆	C₂²	1	2	yes	complemented
26	C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
27	C₅×#127	C₂²	1	1	yes	characteristic, complemented
28	C₅×#124	C₂²	1	1	yes	characteristic, complemented
29	C₅×#124	C₂²	1	1	yes	characteristic, complemented
30	C₅×#121	—	2	2	no	—
31	C₄×C₅×D₄	—	2	2	no	—
32	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic, complemented
33	C₅×#104	C₂²	1	4	yes	complemented
34	C₅×#104	C₂²	1	4	yes	complemented
35	C₅×#101	C₂²	1	2	yes	complemented
36	C₅×#100	C₂²	1	2	yes	complemented
37	#3018	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₅×#41	—	2	4	no	—
39	C₅×#41	—	2	8	no	—
40	C₅×#41	—	2	4	no	—
41	C₅×#41	—	2	4	no	—
42	C₅×#41	—	2	4	no	—
43	C₅×#41	—	2	2	no	—
44	C₅×#41	—	2	2	no	—
45	C₅×#41	—	2	4	no	—
46	C₅×#41	—	2	2	no	—
47	C₅×#41	—	2	2	no	—
48	C₅×#41	—	2	2	no	—
49	C₅×#41	—	2	2	no	—
50	C₅×#41	D₄	1	4	yes	complemented
51	C₅×#41	C₂³	1	4	yes	—
52	C₂×C₅×Q₈	—	2	2	no	—
53	C₂×C₅×Q₈	—	2	4	no	—
54	C₂×C₅×Q₈	D₄	1	2	yes	complemented
55	C₂×C₅×Q₈	C₂³	1	1	yes	characteristic
56	C₂×C₅×Q₈	C₂³	1	1	yes	characteristic
57	C₂×C₅×D₄	—	2	8	no	—
58	C₂×C₅×D₄	—	2	2	no	—
59	C₂×C₅×D₄	—	2	4	no	—
60	C₂×C₅×D₄	—	2	2	no	—
61	C₂×C₅×D₄	—	2	4	no	—
62	C₂×C₅×D₄	—	2	2	no	—
63	C₂×C₅×D₄	—	2	1	no	—
64	C₂×C₅×D₄	D₄	1	2	yes	—
65	C₂×C₅×D₄	D₄	1	2	yes	complemented
66	C₂×C₅×D₄	C₂³	1	1	yes	characteristic
67	C₂×C₅×D₄	C₂³	1	1	yes	characteristic
68	C₂²×C₄×C₅	—	2	4	no	—
69	C₂²×C₄×C₅	—	2	2	no	—
70	C₂²×C₄×C₅	—	2	1	no	—
71	C₂²×C₄×C₅	—	2	1	no	—
72	C₂²×C₄×C₅	D₄	1	2	yes	—
73	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
74	C₅×Q₁₆	—	2	4	no	—
75	C₅×SD₁₆	—	2	4	no	—
76	C₅×SD₁₆	—	2	4	no	—
77	C₅×D₈	—	2	4	no	—
78	C₅×M₁₆	—	2	2	no	—
79	C₅×M₁₆	C₂³	1	4	yes	—
80	C₂×C₅×C₈	—	2	2	no	—
81	C₅×C₄⋊C₄	—	2	2	no	—
82	C₅×C₂²⋊C₄	—	2	2	no	—
83	C₅×C₂²⋊C₄	—	2	2	no	—
84	C₄²×C₅	—	2	1	no	—
85	C₄²×C₅	C₂³	1	1	yes	characteristic
86	C₄²×C₅	C₂³	1	1	yes	characteristic
87	#585	C₂×C₅	1	1	yes	characteristic, complemented
88	#575	C₂×C₅	1	2	yes	complemented
89	#532	C₂×C₅	1	1	yes	characteristic, complemented
90	#456	C₂×C₅	1	2	yes	complemented
91	#455	C₂×C₅	1	2	yes	complemented
92	#454	C₂×C₅	1	2	yes	complemented
93	#453	C₂×C₅	1	2	yes	complemented
94	C₂×#104	C₂×C₅	1	2	yes	complemented
95	#413	C₂×C₅	1	1	yes	characteristic, complemented
96	C₂³×C₅	—	2	4	no	—
97	C₂³×C₅	—	2	2	no	—
98	C₂³×C₅	—	2	1	no	—
99	C₂³×C₅	C₂×D₄	1	2	yes	—
100	C₂³×C₅	C₂×D₄	1	1	yes	characteristic
101	C₅×Q₈	—	2	4	no	—
102	C₅×Q₈	—	2	2	no	—
103	C₅×Q₈	—	2	2	no	—
104	C₅×Q₈	—	2	2	no	—
105	C₅×Q₈	C₂×D₄	1	4	yes	—
106	C₅×D₄	—	2	8	no	—
107	C₅×D₄	—	2	4	no	—
108	C₅×D₄	—	2	4	no	—
109	C₅×D₄	—	2	4	no	—
110	C₅×D₄	—	2	4	no	—
111	C₅×D₄	—	2	2	no	—
112	C₅×D₄	—	2	2	no	—
113	C₅×D₄	—	2	2	no	—
114	C₅×D₄	—	4	4	no	—
115	C₅×D₄	—	4	2	no	—
116	C₅×D₄	C₂×D₄	1	4	yes	—
117	C₂×C₄×C₅	—	2	4	no	—
118	C₂×C₄×C₅	—	2	8	no	—
119	C₂×C₄×C₅	—	2	2	no	—
120	C₂×C₄×C₅	—	2	1	no	—
121	C₂×C₄×C₅	—	2	2	no	—
122	C₂×C₄×C₅	—	2	4	no	—
123	C₂×C₄×C₅	—	2	2	no	—
124	C₂×C₄×C₅	—	2	4	no	—
125	C₂×C₄×C₅	—	2	1	no	—
126	C₂×C₄×C₅	—	2	1	no	—
127	C₂×C₄×C₅	—	4	2	no	—
128	C₂×C₄×C₅	—	4	1	no	—
129	C₂×C₄×C₅	—	4	1	no	—
130	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
131	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
133	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
134	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
135	C₅×C₈	—	2	4	no	—
136	2₋¹⁺⁴	—	2	2	no	—
137	2₋¹⁺⁴	C₂²×C₅	1	2	yes	—
138	H(5,𝔽₂)	—	2	2	no	—
139	H(5,𝔽₂)	—	2	2	no	—
140	H(5,𝔽₂)	C₂²×C₅	1	2	yes	—
141	C₂×#41	—	2	4	no	—
142	C₂×#41	—	2	2	no	—
143	C₂×#41	C₂²×C₅	1	2	yes	—
144	C₂×#41	C₂²×C₅	1	1	yes	characteristic
145	C₂×#41	C₂²×C₅	1	1	yes	characteristic, complemented
146	#137	—	2	2	no	—
147	#137	C₂²×C₅	1	4	yes	complemented
148	AGL(1,ℤ/8)	—	2	2	no	—
149	AGL(1,ℤ/8)	C₂²×C₅	1	4	yes	complemented
150	#135	—	2	4	no	—
151	C₂×M₁₆	C₂²×C₅	1	2	yes	complemented
152	C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic, complemented
153	#127	C₂²×C₅	1	1	yes	characteristic, complemented
154	#124	C₂²×C₅	1	1	yes	characteristic, complemented
155	#124	C₂²×C₅	1	1	yes	characteristic, complemented
156	#121	—	2	2	no	—
157	C₄×D₄	—	2	2	no	—
158	C₂×C₄²	C₂²×C₅	1	1	yes	characteristic, complemented
159	#104	C₂²×C₅	1	4	yes	complemented
160	#104	C₂²×C₅	1	4	yes	complemented
161	#101	C₂²×C₅	1	2	yes	complemented
162	#100	C₂²×C₅	1	2	yes	complemented
163	C₂²×C₅	—	2	2	no	—
164	C₂²×C₅	—	2	4	no	—
165	C₂²×C₅	—	4	1	no	—
166	C₂²×C₅	—	4	2	no	—
167	C₂²×C₅	—	4	2	no	—
168	C₂²×C₅	—	4	1	no	—
169	C₂²×C₅	—	8	2	no	—
170	C₂²×C₅	—	8	4	no	—
171	C₂²×C₅	#120	1	2	yes	—
172	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
173	C₄×C₅	—	2	2	no	—
174	C₄×C₅	—	2	4	no	—
175	C₄×C₅	—	2	2	no	—
176	C₄×C₅	—	4	1	no	—
177	C₄×C₅	—	4	1	no	—
178	C₄×C₅	#120	1	1	yes	characteristic
179	C₄×C₅	#120	1	1	yes	characteristic
180	C₄×C₅	C₂²×D₄	1	2	yes	—
181	#41	—	2	4	no	—
182	#41	—	2	8	no	—
183	#41	—	2	4	no	—
184	#41	—	2	4	no	—
185	#41	—	2	4	no	—
186	#41	—	2	2	no	—
187	#41	—	2	2	no	—
188	#41	—	2	4	no	—
189	#41	—	2	2	no	—
190	#41	—	2	2	no	—
191	#41	—	2	2	no	—
192	#41	—	2	2	no	—
193	#41	C₅×D₄	1	4	yes	complemented
194	#41	C₂³×C₅	1	4	yes	—
195	C₂×Q₈	—	2	2	no	—
196	C₂×Q₈	—	2	4	no	—
197	C₂×Q₈	C₅×D₄	1	2	yes	complemented
198	C₂×Q₈	C₂³×C₅	1	1	yes	characteristic
199	C₂×Q₈	C₂³×C₅	1	1	yes	characteristic
200	C₂×D₄	—	2	8	no	—
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	—	2	4	no	—
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	—	2	4	no	—
205	C₂×D₄	—	2	2	no	—
206	C₂×D₄	—	2	1	no	—
207	C₂×D₄	C₅×D₄	1	2	yes	—
208	C₂×D₄	C₅×D₄	1	2	yes	complemented
209	C₂×D₄	C₂³×C₅	1	1	yes	characteristic
210	C₂×D₄	C₂³×C₅	1	1	yes	characteristic
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	C₅×D₄	1	2	yes	—
216	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
217	Q₁₆	—	2	4	no	—
218	SD₁₆	—	2	4	no	—
219	SD₁₆	—	2	4	no	—
220	D₈	—	2	4	no	—
221	M₁₆	—	2	2	no	—
222	M₁₆	C₂³×C₅	1	4	yes	—
223	C₂×C₈	—	2	2	no	—
224	C₄⋊C₄	—	2	2	no	—
225	C₂²⋊C₄	—	2	2	no	—
226	C₂²⋊C₄	—	2	2	no	—
227	C₄²	—	2	1	no	—
228	C₄²	C₂³×C₅	1	1	yes	characteristic
229	C₄²	C₂³×C₅	1	1	yes	characteristic
230	C₂×C₅	—	2	1	no	—
231	C₂×C₅	—	2	2	no	—
232	C₂×C₅	—	4	2	no	—
233	C₂×C₅	—	4	4	no	—
234	C₂×C₅	—	8	1	no	—
235	C₂×C₅	C₂×#120	1	1	yes	characteristic
236	C₂³	—	2	4	no	—
237	C₂³	—	2	2	no	—
238	C₂³	—	2	1	no	—
239	C₂³	C₂×C₅×D₄	1	2	yes	—
240	C₂³	C₂×C₅×D₄	1	1	yes	characteristic
241	Q₈	—	2	4	no	—
242	Q₈	—	2	2	no	—
243	Q₈	—	2	2	no	—
244	Q₈	—	2	2	no	—
245	Q₈	C₂×C₅×D₄	1	4	yes	—
246	D₄	—	2	8	no	—
247	D₄	—	2	4	no	—
248	D₄	—	2	4	no	—
249	D₄	—	2	4	no	—
250	D₄	—	2	4	no	—
251	D₄	—	2	2	no	—
252	D₄	—	2	2	no	—
253	D₄	—	2	2	no	—
254	D₄	—	4	4	no	—
255	D₄	—	4	2	no	—
256	D₄	C₂×C₅×D₄	1	4	yes	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	C₂×C₅×D₄	1	2	yes	—
271	C₂×C₄	C₂×C₅×D₄	1	1	yes	characteristic
272	C₂×C₄	C₂×C₅×D₄	1	2	yes	—
273	C₂×C₄	C₂×C₅×D₄	1	2	yes	—
274	C₂×C₄	C₂⁴×C₅	1	1	yes	characteristic
275	C₈	—	2	4	no	—
276	C₅	#3018	1	1	yes	characteristic, direct product factor, Sylow
277	C₂²	—	2	2	no	—
278	C₂²	—	2	4	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	—	4	1	no	—
283	C₂²	—	8	2	no	—
284	C₂²	—	8	4	no	—
285	C₂²	C₅×#120	1	2	yes	—
286	C₂²	C₂²×C₅×D₄	1	1	yes	characteristic
287	C₄	—	2	2	no	—
288	C₄	—	2	4	no	—
289	C₄	—	2	2	no	—
290	C₄	—	4	1	no	—
291	C₄	—	4	1	no	—
292	C₄	C₅×#120	1	1	yes	characteristic
293	C₄	C₅×#120	1	1	yes	characteristic
294	C₄	C₂²×C₅×D₄	1	2	yes	—
295	C₂	—	2	1	no	—
296	C₂	—	2	2	no	—
297	C₂	—	4	2	no	—
298	C₂	—	4	4	no	—
299	C₂	—	8	1	no	—
300	C₂	C₂×C₅×#120	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	#120
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₅
socle	C₂×C₅	C₂×#120
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#585	C₂	semidirect
2. C₅×#575	C₂	semidirect
3. C₅×#532	C₂	semidirect
4. C₅×#456	C₂	semidirect
5. C₅×#455	C₂	semidirect
6. C₅×#454	C₂	semidirect
7. C₅×#453	C₂	semidirect
8. C₂×C₅×#104	C₂	semidirect
9. C₅×#413	C₂	semidirect
19. C₂×C₅×#41	C₂²	semidirect
21. C₅×#137	C₂²	semidirect
23. C₅×AGL(1,ℤ/8)	C₂²	semidirect
25. C₂×C₅×M₁₆	C₂²	semidirect
26. C₅×C₄⋊Q₈	C₂²	semidirect
27. C₅×#127	C₂²	semidirect
28. C₅×#124	C₂²	semidirect
29. C₅×#124	C₂²	semidirect
32. C₂×C₄²×C₅	C₂²	semidirect
33. C₅×#104	C₂²	semidirect
34. C₅×#104	C₂²	semidirect
35. C₅×#101	C₂²	semidirect
36. C₅×#100	C₂²	semidirect
37. #3018	C₅	direct (N × H)
50. C₅×#41	D₄	semidirect
54. C₂×C₅×Q₈	D₄	semidirect
65. C₂×C₅×D₄	D₄	semidirect
87. #585	C₂×C₅	semidirect
88. #575	C₂×C₅	semidirect
89. #532	C₂×C₅	semidirect
90. #456	C₂×C₅	semidirect
91. #455	C₂×C₅	semidirect
92. #454	C₂×C₅	semidirect
93. #453	C₂×C₅	semidirect
94. C₂×#104	C₂×C₅	semidirect
95. #413	C₂×C₅	semidirect
145. C₂×#41	C₂²×C₅	semidirect
147. #137	C₂²×C₅	semidirect
149. AGL(1,ℤ/8)	C₂²×C₅	semidirect
151. C₂×M₁₆	C₂²×C₅	semidirect
152. C₄⋊Q₈	C₂²×C₅	semidirect
153. #127	C₂²×C₅	semidirect
154. #124	C₂²×C₅	semidirect
155. #124	C₂²×C₅	semidirect
158. C₂×C₄²	C₂²×C₅	semidirect
159. #104	C₂²×C₅	semidirect
160. #104	C₂²×C₅	semidirect
161. #101	C₂²×C₅	semidirect
162. #100	C₂²×C₅	semidirect
193. #41	C₅×D₄	semidirect
197. C₂×Q₈	C₅×D₄	semidirect
208. C₂×D₄	C₅×D₄	semidirect
276. C₅	#3018	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×#585	2
C₅×#575	2
C₅×#532	2
C₅×#456	2
C₅×#455	2
C₅×#454	2
C₅×#453	2
C₂×C₅×#104	2
C₅×#413	2
#3018	5

Subgroups — C5×#3018 #10616613

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#3018 #10616613