Subgroups — C₅×#3508 #10619293

Order 640.

1232 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#559	C₂	1	4	yes	maximal, complemented
2	C₅×#550	C₂	1	4	yes	maximal, complemented
3	C₅×#546	C₂	1	8	yes	maximal, complemented
4	C₅×#540	C₂	1	4	yes	maximal, complemented
5	C₅×#535	C₂	1	2	yes	maximal, complemented
6	C₅×#533	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#532	C₂	1	4	yes	maximal, complemented
8	C₂×C₅×#127	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#518	C₂	1	2	yes	maximal, complemented
10	C₄×C₅×#41	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₅×#41	C₂²	1	1	yes	characteristic
12	C₂×C₅×#41	C₂²	1	8	yes	complemented
13	C₂²×C₅×D₄	C₂²	1	2	yes	—
14	C₂²×C₅×D₄	C₂²	1	4	yes	—
15	C₅×C₄⋊Q₈	C₂²	1	2	yes	complemented
16	C₅×#127	C₂²	1	2	yes	—
17	C₅×#127	C₂²	1	2	yes	—
18	C₅×#127	C₂²	1	4	yes	—
19	C₅×#127	C₂²	1	4	yes	complemented
20	C₅×#125	C₂²	1	2	yes	complemented
21	C₅×#124	C₂²	1	8	yes	complemented
22	C₅×#124	C₂²	1	4	yes	complemented
23	C₅×#123	C₂²	1	16	yes	complemented
24	C₅×C₂²⋊Q₈	C₂²	1	4	yes	—
25	C₅×#121	C₂²	1	8	yes	complemented
26	C₅×#121	C₂²	1	8	yes	complemented
27	C₅×#121	C₂²	1	16	yes	complemented
28	C₅×#120	C₂²	1	8	yes	—
29	C₄×C₅×Q₈	C₂²	1	2	yes	complemented
30	C₄×C₅×Q₈	C₂²	1	4	yes	complemented
31	C₄×C₅×D₄	C₂²	1	4	yes	—
32	C₄×C₅×D₄	C₂²	1	2	yes	complemented
33	C₄×C₅×D₄	C₂²	1	8	yes	complemented
34	C₄×C₅×D₄	C₂²	1	8	yes	complemented
35	C₄×C₅×D₄	C₂²	1	8	yes	complemented
36	C₅×#117	C₂²	1	1	yes	characteristic
37	C₅×#117	C₂²	1	2	yes	—
38	C₅×#117	C₂²	1	2	yes	complemented
39	C₂×C₅×C₂²⋊C₄	C₂²	1	8	yes	—
40	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
41	C₂×C₄²×C₅	C₂²	1	2	yes	complemented
42	#3508	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂⁴×C₅	C₂³	1	4	yes	—
44	C₅×#41	—	2	4	no	—
45	C₅×#41	—	4	16	no	—
46	C₂×C₅×Q₈	C₂³	1	1	yes	characteristic
47	C₂×C₅×Q₈	C₂³	1	4	yes	—
48	C₂×C₅×D₄	—	2	8	no	—
49	C₂×C₅×D₄	—	4	8	no	—
50	C₂×C₅×D₄	C₂³	1	2	yes	—
51	C₂×C₅×D₄	C₂³	1	4	yes	—
52	C₂×C₅×D₄	C₂³	1	4	yes	—
53	C₂×C₅×D₄	C₂³	1	1	yes	characteristic
54	C₂×C₅×D₄	C₂³	1	8	yes	—
55	C₂×C₅×D₄	C₂³	1	16	yes	—
56	C₂×C₅×D₄	C₂³	1	8	yes	—
57	C₂²×C₄×C₅	C₂³	1	2	yes	—
58	C₂²×C₄×C₅	C₂³	1	2	yes	—
59	C₂²×C₄×C₅	C₂³	1	4	yes	—
60	C₂²×C₄×C₅	C₂³	1	16	yes	—
61	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
62	C₅×C₄⋊C₄	C₂³	1	2	yes	—
63	C₅×C₄⋊C₄	C₂³	1	4	yes	—
64	C₅×C₄⋊C₄	C₂³	1	4	yes	—
65	C₅×C₄⋊C₄	C₂³	1	4	yes	—
66	C₅×C₄⋊C₄	C₂³	1	8	yes	—
67	C₅×C₂²⋊C₄	C₂³	1	2	yes	—
68	C₅×C₂²⋊C₄	C₂³	1	4	yes	—
69	C₅×C₂²⋊C₄	C₂³	1	16	yes	—
70	C₅×C₂²⋊C₄	C₂³	1	4	yes	—
71	C₅×C₂²⋊C₄	C₂³	1	16	yes	—
72	C₄²×C₅	C₂³	1	1	yes	characteristic
73	C₄²×C₅	C₂³	1	1	yes	characteristic
74	C₄²×C₅	C₂³	1	2	yes	—
75	C₄²×C₅	C₂³	1	2	yes	—
76	C₄²×C₅	C₂³	1	4	yes	—
77	C₄²×C₅	C₂³	1	4	yes	—
78	#559	C₂×C₅	1	4	yes	complemented
79	#550	C₂×C₅	1	4	yes	complemented
80	#546	C₂×C₅	1	8	yes	complemented
81	#540	C₂×C₅	1	4	yes	complemented
82	#535	C₂×C₅	1	2	yes	complemented
83	#533	C₂×C₅	1	1	yes	characteristic, complemented
84	#532	C₂×C₅	1	4	yes	complemented
85	C₂×#127	C₂×C₅	1	1	yes	characteristic, complemented
86	#518	C₂×C₅	1	2	yes	complemented
87	C₄×#41	C₂×C₅	1	1	yes	characteristic, complemented
88	C₂³×C₅	—	2	8	no	—
89	C₂³×C₅	—	4	8	no	—
90	C₂³×C₅	C₂⁴	1	8	yes	—
91	C₂³×C₅	C₂⁴	1	2	yes	—
92	C₂³×C₅	C₂⁴	1	1	yes	characteristic
93	C₅×Q₈	—	2	2	no	—
94	C₅×Q₈	—	2	8	no	—
95	C₅×D₄	—	2	4	no	—
96	C₅×D₄	—	2	8	no	—
97	C₅×D₄	—	2	8	no	—
98	C₅×D₄	—	2	2	no	—
99	C₅×D₄	—	2	16	no	—
100	C₅×D₄	—	4	8	no	—
101	C₅×D₄	—	4	16	no	—
102	C₂×C₄×C₅	—	2	4	no	—
103	C₂×C₄×C₅	—	2	4	no	—
104	C₂×C₄×C₅	—	2	8	no	—
105	C₂×C₄×C₅	—	4	16	no	—
106	C₂×C₄×C₅	#41	1	4	yes	complemented
107	C₂×C₄×C₅	C₂⁴	1	2	yes	—
108	C₂×C₄×C₅	C₂⁴	1	4	yes	—
109	C₂×C₄×C₅	C₂⁴	1	2	yes	—
110	C₂×C₄×C₅	C₂⁴	1	8	yes	—
111	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
112	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
113	C₂×C₄×C₅	C₂⁴	1	2	yes	—
114	C₂×#41	C₂²×C₅	1	1	yes	characteristic
115	C₂×#41	C₂²×C₅	1	8	yes	complemented
116	C₂²×D₄	C₂²×C₅	1	2	yes	—
117	C₂²×D₄	C₂²×C₅	1	4	yes	—
118	C₄⋊Q₈	C₂²×C₅	1	2	yes	complemented
119	#127	C₂²×C₅	1	2	yes	—
120	#127	C₂²×C₅	1	2	yes	—
121	#127	C₂²×C₅	1	4	yes	—
122	#127	C₂²×C₅	1	4	yes	complemented
123	#125	C₂²×C₅	1	2	yes	complemented
124	#124	C₂²×C₅	1	8	yes	complemented
125	#124	C₂²×C₅	1	4	yes	complemented
126	#123	C₂²×C₅	1	16	yes	complemented
127	C₂²⋊Q₈	C₂²×C₅	1	4	yes	—
128	#121	C₂²×C₅	1	8	yes	complemented
129	#121	C₂²×C₅	1	8	yes	complemented
130	#121	C₂²×C₅	1	16	yes	complemented
131	#120	C₂²×C₅	1	8	yes	—
132	C₄×Q₈	C₂²×C₅	1	2	yes	complemented
133	C₄×Q₈	C₂²×C₅	1	4	yes	complemented
134	C₄×D₄	C₂²×C₅	1	4	yes	—
135	C₄×D₄	C₂²×C₅	1	2	yes	complemented
136	C₄×D₄	C₂²×C₅	1	8	yes	complemented
137	C₄×D₄	C₂²×C₅	1	8	yes	complemented
138	C₄×D₄	C₂²×C₅	1	8	yes	complemented
139	#117	C₂²×C₅	1	1	yes	characteristic
140	#117	C₂²×C₅	1	2	yes	—
141	#117	C₂²×C₅	1	2	yes	complemented
142	C₂×C₂²⋊C₄	C₂²×C₅	1	8	yes	—
143	C₂×C₄²	C₂²×C₅	1	1	yes	characteristic
144	C₂×C₄²	C₂²×C₅	1	2	yes	complemented
145	C₂²×C₅	—	2	2	no	—
146	C₂²×C₅	—	2	2	no	—
147	C₂²×C₅	—	2	8	no	—
148	C₂²×C₅	—	2	4	no	—
149	C₂²×C₅	—	2	16	no	—
150	C₂²×C₅	—	8	8	no	—
151	C₂²×C₅	C₂×#41	1	2	yes	—
152	C₂²×C₅	C₂⁵	1	1	yes	characteristic
153	C₄×C₅	—	2	2	no	—
154	C₄×C₅	—	2	4	no	—
155	C₄×C₅	—	2	2	no	—
156	C₄×C₅	—	2	8	no	—
157	C₄×C₅	C₂×#41	1	2	yes	—
158	C₄×C₅	C₂×#41	1	2	yes	—
159	C₄×C₅	H(5,𝔽₂)	1	4	yes	—
160	C₂⁴	C₂³×C₅	1	4	yes	—
161	#41	—	2	4	no	—
162	#41	—	4	16	no	—
163	C₂×Q₈	C₂³×C₅	1	1	yes	characteristic
164	C₂×Q₈	C₂³×C₅	1	4	yes	—
165	C₂×D₄	—	2	8	no	—
166	C₂×D₄	—	4	8	no	—
167	C₂×D₄	C₂³×C₅	1	2	yes	—
168	C₂×D₄	C₂³×C₅	1	4	yes	—
169	C₂×D₄	C₂³×C₅	1	4	yes	—
170	C₂×D₄	C₂³×C₅	1	1	yes	characteristic
171	C₂×D₄	C₂³×C₅	1	8	yes	—
172	C₂×D₄	C₂³×C₅	1	16	yes	—
173	C₂×D₄	C₂³×C₅	1	8	yes	—
174	C₂²×C₄	C₂³×C₅	1	2	yes	—
175	C₂²×C₄	C₂³×C₅	1	2	yes	—
176	C₂²×C₄	C₂³×C₅	1	4	yes	—
177	C₂²×C₄	C₂³×C₅	1	16	yes	—
178	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
179	C₄⋊C₄	C₂³×C₅	1	2	yes	—
180	C₄⋊C₄	C₂³×C₅	1	4	yes	—
181	C₄⋊C₄	C₂³×C₅	1	4	yes	—
182	C₄⋊C₄	C₂³×C₅	1	4	yes	—
183	C₄⋊C₄	C₂³×C₅	1	8	yes	—
184	C₂²⋊C₄	C₂³×C₅	1	2	yes	—
185	C₂²⋊C₄	C₂³×C₅	1	4	yes	—
186	C₂²⋊C₄	C₂³×C₅	1	16	yes	—
187	C₂²⋊C₄	C₂³×C₅	1	4	yes	—
188	C₂²⋊C₄	C₂³×C₅	1	16	yes	—
189	C₄²	C₂³×C₅	1	1	yes	characteristic
190	C₄²	C₂³×C₅	1	1	yes	characteristic
191	C₄²	C₂³×C₅	1	2	yes	—
192	C₄²	C₂³×C₅	1	2	yes	—
193	C₄²	C₂³×C₅	1	4	yes	—
194	C₄²	C₂³×C₅	1	4	yes	—
195	C₂×C₅	—	2	2	no	—
196	C₂×C₅	—	4	2	no	—
197	C₂×C₅	—	4	8	no	—
198	C₂×C₅	C₂²×#41	1	1	yes	characteristic
199	C₂×C₅	C₂×H(5,𝔽₂)	1	2	yes	—
200	C₂³	—	2	8	no	—
201	C₂³	—	4	8	no	—
202	C₂³	C₂⁴×C₅	1	8	yes	—
203	C₂³	C₂⁴×C₅	1	2	yes	—
204	C₂³	C₂⁴×C₅	1	1	yes	characteristic
205	Q₈	—	2	2	no	—
206	Q₈	—	2	8	no	—
207	D₄	—	2	4	no	—
208	D₄	—	2	8	no	—
209	D₄	—	2	8	no	—
210	D₄	—	2	2	no	—
211	D₄	—	2	16	no	—
212	D₄	—	4	8	no	—
213	D₄	—	4	16	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	8	no	—
217	C₂×C₄	—	4	16	no	—
218	C₂×C₄	C₅×#41	1	4	yes	complemented
219	C₂×C₄	C₂⁴×C₅	1	2	yes	—
220	C₂×C₄	C₂⁴×C₅	1	4	yes	—
221	C₂×C₄	C₂⁴×C₅	1	2	yes	—
222	C₂×C₄	C₂⁴×C₅	1	8	yes	—
223	C₂×C₄	C₂⁴×C₅	1	1	yes	characteristic
224	C₂×C₄	C₂⁴×C₅	1	1	yes	characteristic
225	C₂×C₄	C₂⁴×C₅	1	2	yes	—
226	C₅	#3508	1	1	yes	characteristic, direct product factor, Sylow
227	C₂²	—	2	2	no	—
228	C₂²	—	2	2	no	—
229	C₂²	—	2	8	no	—
230	C₂²	—	2	4	no	—
231	C₂²	—	2	16	no	—
232	C₂²	—	8	8	no	—
233	C₂²	C₂×C₅×#41	1	2	yes	—
234	C₂²	C₂⁵×C₅	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	2	4	no	—
237	C₄	—	2	2	no	—
238	C₄	—	2	8	no	—
239	C₄	C₂×C₅×#41	1	2	yes	—
240	C₄	C₂×C₅×#41	1	2	yes	—
241	C₄	C₅×H(5,𝔽₂)	1	4	yes	—
242	C₂	—	2	2	no	—
243	C₂	—	4	8	no	—
244	C₂	—	4	2	no	—
245	C₂	C₂²×C₅×#41	1	1	yes	characteristic
246	C₂	C₂×C₅×H(5,𝔽₂)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₅
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#559	C₂	semidirect
2. C₅×#550	C₂	semidirect
3. C₅×#546	C₂	semidirect
4. C₅×#540	C₂	semidirect
5. C₅×#535	C₂	semidirect
6. C₅×#533	C₂	semidirect
7. C₅×#532	C₂	semidirect
8. C₂×C₅×#127	C₂	semidirect
9. C₅×#518	C₂	semidirect
10. C₄×C₅×#41	C₂	semidirect
12. C₂×C₅×#41	C₂²	semidirect
15. C₅×C₄⋊Q₈	C₂²	semidirect
19. C₅×#127	C₂²	semidirect
20. C₅×#125	C₂²	semidirect
21. C₅×#124	C₂²	semidirect
22. C₅×#124	C₂²	semidirect
23. C₅×#123	C₂²	semidirect
25. C₅×#121	C₂²	semidirect
26. C₅×#121	C₂²	semidirect
27. C₅×#121	C₂²	semidirect
29. C₄×C₅×Q₈	C₂²	semidirect
30. C₄×C₅×Q₈	C₂²	semidirect
32. C₄×C₅×D₄	C₂²	semidirect
33. C₄×C₅×D₄	C₂²	semidirect
34. C₄×C₅×D₄	C₂²	semidirect
35. C₄×C₅×D₄	C₂²	semidirect
38. C₅×#117	C₂²	semidirect
41. C₂×C₄²×C₅	C₂²	semidirect
42. #3508	C₅	direct (N × H)
78. #559	C₂×C₅	semidirect
79. #550	C₂×C₅	semidirect
80. #546	C₂×C₅	semidirect
81. #540	C₂×C₅	semidirect
82. #535	C₂×C₅	semidirect
83. #533	C₂×C₅	semidirect
84. #532	C₂×C₅	semidirect
85. C₂×#127	C₂×C₅	semidirect
86. #518	C₂×C₅	semidirect
87. C₄×#41	C₂×C₅	semidirect
106. C₂×C₄×C₅	#41	semidirect
115. C₂×#41	C₂²×C₅	semidirect
118. C₄⋊Q₈	C₂²×C₅	semidirect
122. #127	C₂²×C₅	semidirect
123. #125	C₂²×C₅	semidirect
124. #124	C₂²×C₅	semidirect
125. #124	C₂²×C₅	semidirect
126. #123	C₂²×C₅	semidirect
128. #121	C₂²×C₅	semidirect
129. #121	C₂²×C₅	semidirect
130. #121	C₂²×C₅	semidirect
132. C₄×Q₈	C₂²×C₅	semidirect
133. C₄×Q₈	C₂²×C₅	semidirect
135. C₄×D₄	C₂²×C₅	semidirect
136. C₄×D₄	C₂²×C₅	semidirect
137. C₄×D₄	C₂²×C₅	semidirect
138. C₄×D₄	C₂²×C₅	semidirect
141. #117	C₂²×C₅	semidirect
144. C₂×C₄²	C₂²×C₅	semidirect
218. C₂×C₄	C₅×#41	semidirect
226. C₅	#3508	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×#559	2
C₅×#550	2
C₅×#546	2
C₅×#540	2
C₅×#535	2
C₅×#533	2
C₅×#532	2
C₂×C₅×#127	2
C₅×#518	2
C₄×C₅×#41	2
#3508	5

Subgroups — C5×#3508 #10619293

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#3508 #10619293