Subgroups — C₂²×C₅×#509 #11985349

Order 1280.

2334 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#135	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₅×D₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#509	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₅×SD₃₂	C₂	1	2	yes	maximal, complemented
5	C₂²×C₅×D₁₆	C₂	1	2	yes	maximal, complemented
6	C₂²×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₅×#41	—	2	1	no	—
8	C₂³×C₅×D₄	—	2	1	no	—
9	C₂×C₅×#135	—	2	6	no	—
10	C₂×C₅×#135	C₂²	1	12	yes	complemented
11	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic, complemented
12	C₂²×C₅×SD₁₆	—	2	1	no	—
13	C₂²×C₅×D₈	—	2	1	no	—
14	C₂²×C₅×D₈	—	2	6	no	—
15	C₂²×C₅×D₈	C₂²	1	2	yes	—
16	C₂²×C₅×D₈	C₂²	1	12	yes	complemented
17	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic, complemented
18	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
19	C₅×#509	C₂²	1	64	yes	direct product factor
20	C₂×C₅×SD₃₂	C₂²	1	24	yes	complemented
21	C₂×C₅×D₁₆	C₂²	1	24	yes	complemented
22	C₂×C₅×M₃₂	C₂²	1	12	yes	complemented
23	C₂²×C₅×C₁₆	C₂²	1	2	yes	complemented
24	C₂²×#509	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂⁵×C₅	—	4	1	no	—
26	C₂×C₅×#41	—	2	12	no	—
27	C₂×C₅×#41	—	4	6	no	—
28	C₂²×C₅×Q₈	—	2	1	no	—
29	C₂²×C₅×D₄	—	2	2	no	—
30	C₂²×C₅×D₄	—	2	12	no	—
31	C₂²×C₅×D₄	—	2	1	no	—
32	C₂²×C₅×D₄	—	4	1	no	—
33	C₂²×C₅×D₄	—	4	6	no	—
34	C₂²×C₅×D₄	—	4	1	no	—
35	C₂³×C₄×C₅	—	4	1	no	—
36	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
37	C₅×#135	—	2	24	no	—
38	C₅×#135	C₂³	1	16	yes	complemented
39	C₂×C₅×Q₁₆	—	2	3	no	—
40	C₂×C₅×Q₁₆	C₂³	1	6	yes	complemented
41	C₂×C₅×SD₁₆	—	2	12	no	—
42	C₂×C₅×D₈	—	2	6	no	—
43	C₂×C₅×D₈	—	2	12	no	—
44	C₂×C₅×D₈	—	2	24	no	—
45	C₂×C₅×D₈	—	2	3	no	—
46	C₂×C₅×D₈	C₂³	1	12	yes	—
47	C₂×C₅×D₈	C₂³	1	16	yes	complemented
48	C₂×C₅×D₈	C₂³	1	6	yes	complemented
49	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
50	C₂²×C₅×C₈	D₄	1	6	yes	—
51	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
52	C₂²×C₅×C₈	C₂³	1	6	yes	—
53	C₅×SD₃₂	C₂³	1	32	yes	complemented
54	C₅×D₁₆	C₂³	1	32	yes	complemented
55	C₅×M₃₂	C₂³	1	16	yes	complemented
56	C₂×C₅×C₁₆	C₂³	1	12	yes	complemented
57	C₂²×#135	C₂×C₅	1	1	yes	characteristic, complemented
58	C₂³×D₈	C₂×C₅	1	1	yes	characteristic, complemented
59	C₂×#509	C₂×C₅	1	24	yes	direct product factor
60	C₂²×SD₃₂	C₂×C₅	1	2	yes	complemented
61	C₂²×D₁₆	C₂×C₅	1	2	yes	complemented
62	C₂²×M₃₂	C₂×C₅	1	1	yes	characteristic, complemented
63	C₂⁴×C₅	—	4	12	no	—
64	C₂⁴×C₅	—	4	1	no	—
65	C₂⁴×C₅	—	4	2	no	—
66	C₂⁴×C₅	—	8	8	no	—
67	C₂⁴×C₅	D₈	1	1	yes	characteristic
68	C₅×#41	—	2	16	no	—
69	C₅×#41	—	4	24	no	—
70	C₂×C₅×Q₈	—	2	6	no	—
71	C₂×C₅×Q₈	—	4	3	no	—
72	C₂×C₅×D₄	—	2	12	no	—
73	C₂×C₅×D₄	—	2	16	no	—
74	C₂×C₅×D₄	—	2	6	no	—
75	C₂×C₅×D₄	—	4	12	no	—
76	C₂×C₅×D₄	—	4	12	no	—
77	C₂×C₅×D₄	—	4	6	no	—
78	C₂×C₅×D₄	—	4	24	no	—
79	C₂×C₅×D₄	—	4	3	no	—
80	C₂²×C₄×C₅	—	4	1	no	—
81	C₂²×C₄×C₅	—	4	12	no	—
82	C₂²×C₄×C₅	D₈	1	6	yes	—
83	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₂×D₄	1	6	yes	—
85	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
86	C₅×Q₁₆	—	2	6	no	—
87	C₅×Q₁₆	C₂⁴	1	4	yes	complemented
88	C₅×SD₁₆	—	2	16	no	—
89	C₅×D₈	—	2	12	no	—
90	C₅×D₈	—	2	16	no	—
91	C₅×D₈	—	2	6	no	—
92	C₅×D₈	C₂⁴	1	8	yes	—
93	C₅×D₈	C₂⁴	1	4	yes	complemented
94	C₂×C₅×C₈	C₂×D₄	1	3	yes	—
95	C₂×C₅×C₈	C₂×D₄	1	6	yes	—
96	C₂×C₅×C₈	C₂×D₄	1	12	yes	—
97	C₂×C₅×C₈	C₂⁴	1	3	yes	—
98	C₂×C₅×C₈	C₂⁴	1	4	yes	—
99	C₅×C₁₆	C₂⁴	1	8	yes	complemented
100	C₂²×#41	—	2	1	no	—
101	C₂³×D₄	—	2	1	no	—
102	C₂×#135	—	2	6	no	—
103	C₂×#135	C₂²×C₅	1	12	yes	complemented
104	C₂²×Q₁₆	C₂²×C₅	1	1	yes	characteristic, complemented
105	C₂²×SD₁₆	—	2	1	no	—
106	C₂²×D₈	—	2	1	no	—
107	C₂²×D₈	—	2	6	no	—
108	C₂²×D₈	C₂²×C₅	1	2	yes	—
109	C₂²×D₈	C₂²×C₅	1	12	yes	complemented
110	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic, complemented
111	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
112	#509	C₂²×C₅	1	64	yes	direct product factor
113	C₂×SD₃₂	C₂²×C₅	1	24	yes	complemented
114	C₂×D₁₆	C₂²×C₅	1	24	yes	complemented
115	C₂×M₃₂	C₂²×C₅	1	12	yes	complemented
116	C₂²×C₁₆	C₂²×C₅	1	2	yes	complemented
117	C₂³×C₅	—	2	4	no	—
118	C₂³×C₅	—	4	6	no	—
119	C₂³×C₅	—	4	16	no	—
120	C₂³×C₅	—	4	12	no	—
121	C₂³×C₅	—	8	48	no	—
122	C₂³×C₅	—	8	4	no	—
123	C₂³×C₅	—	8	8	no	—
124	C₂³×C₅	C₂×D₈	1	6	yes	—
125	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
126	C₅×Q₈	—	2	4	no	—
127	C₅×Q₈	—	4	6	no	—
128	C₅×D₄	—	2	8	no	—
129	C₅×D₄	—	2	4	no	—
130	C₅×D₄	—	4	16	no	—
131	C₅×D₄	—	4	16	no	—
132	C₅×D₄	—	4	12	no	—
133	C₅×D₄	—	4	6	no	—
134	C₂×C₄×C₅	—	4	6	no	—
135	C₂×C₄×C₅	—	4	16	no	—
136	C₂×C₄×C₅	C₂×D₈	1	6	yes	—
137	C₂×C₄×C₅	C₂×D₈	1	3	yes	—
138	C₂×C₄×C₅	C₂×D₈	1	12	yes	—
139	C₂×C₄×C₅	C₂²×D₄	1	3	yes	—
140	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
141	C₅×C₈	C₂²×D₄	1	3	yes	—
142	C₅×C₈	C₂²×D₄	1	4	yes	—
143	C₅×C₈	C₂⁵	1	1	yes	characteristic
144	C₂⁵	—	4	1	no	—
145	C₂×#41	—	2	12	no	—
146	C₂×#41	—	4	6	no	—
147	C₂²×Q₈	—	2	1	no	—
148	C₂²×D₄	—	2	2	no	—
149	C₂²×D₄	—	2	12	no	—
150	C₂²×D₄	—	2	1	no	—
151	C₂²×D₄	—	4	1	no	—
152	C₂²×D₄	—	4	1	no	—
153	C₂²×D₄	—	4	6	no	—
154	C₂³×C₄	—	4	1	no	—
155	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
156	#135	—	2	24	no	—
157	#135	C₂³×C₅	1	16	yes	complemented
158	C₂×Q₁₆	—	2	3	no	—
159	C₂×Q₁₆	C₂³×C₅	1	6	yes	complemented
160	C₂×SD₁₆	—	2	12	no	—
161	C₂×D₈	—	2	6	no	—
162	C₂×D₈	—	2	12	no	—
163	C₂×D₈	—	2	24	no	—
164	C₂×D₈	—	2	3	no	—
165	C₂×D₈	C₂³×C₅	1	12	yes	—
166	C₂×D₈	C₂³×C₅	1	16	yes	complemented
167	C₂×D₈	C₂³×C₅	1	6	yes	complemented
168	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
169	C₂²×C₈	C₅×D₄	1	6	yes	—
170	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
171	C₂²×C₈	C₂³×C₅	1	6	yes	—
172	SD₃₂	C₂³×C₅	1	32	yes	complemented
173	D₁₆	C₂³×C₅	1	32	yes	complemented
174	M₃₂	C₂³×C₅	1	16	yes	complemented
175	C₂×C₁₆	C₂³×C₅	1	12	yes	complemented
176	C₂²×C₅	—	2	12	no	—
177	C₂²×C₅	—	4	4	no	—
178	C₂²×C₅	—	4	8	no	—
179	C₂²×C₅	—	8	32	no	—
180	C₂²×C₅	—	8	12	no	—
181	C₂²×C₅	—	8	24	no	—
182	C₂²×C₅	#509	1	4	yes	direct product factor
183	C₂²×C₅	C₂²×D₈	1	4	yes	—
184	C₂²×C₅	C₂²×D₈	1	3	yes	—
185	C₄×C₅	—	4	4	no	—
186	C₄×C₅	C₂²×D₈	1	4	yes	—
187	C₄×C₅	C₂²×D₈	1	3	yes	—
188	C₄×C₅	C₂³×D₄	1	1	yes	characteristic
189	C₂⁴	—	4	1	no	—
190	C₂⁴	—	4	12	no	—
191	C₂⁴	—	4	2	no	—
192	C₂⁴	—	8	8	no	—
193	C₂⁴	C₅×D₈	1	1	yes	characteristic
194	#41	—	2	16	no	—
195	#41	—	4	24	no	—
196	C₂×Q₈	—	2	6	no	—
197	C₂×Q₈	—	4	3	no	—
198	C₂×D₄	—	2	12	no	—
199	C₂×D₄	—	2	16	no	—
200	C₂×D₄	—	2	6	no	—
201	C₂×D₄	—	4	12	no	—
202	C₂×D₄	—	4	12	no	—
203	C₂×D₄	—	4	6	no	—
204	C₂×D₄	—	4	24	no	—
205	C₂×D₄	—	4	3	no	—
206	C₂²×C₄	—	4	1	no	—
207	C₂²×C₄	—	4	12	no	—
208	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
209	C₂²×C₄	C₅×D₈	1	6	yes	—
210	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₅×D₄	1	6	yes	—
212	Q₁₆	—	2	6	no	—
213	Q₁₆	C₂⁴×C₅	1	4	yes	complemented
214	SD₁₆	—	2	16	no	—
215	D₈	—	2	12	no	—
216	D₈	—	2	16	no	—
217	D₈	—	2	6	no	—
218	D₈	C₂⁴×C₅	1	8	yes	—
219	D₈	C₂⁴×C₅	1	4	yes	complemented
220	C₂×C₈	C₂×C₅×D₄	1	3	yes	—
221	C₂×C₈	C₂×C₅×D₄	1	6	yes	—
222	C₂×C₈	C₂×C₅×D₄	1	12	yes	—
223	C₂×C₈	C₂⁴×C₅	1	3	yes	—
224	C₂×C₈	C₂⁴×C₅	1	4	yes	—
225	C₁₆	C₂⁴×C₅	1	8	yes	complemented
226	C₂×C₅	—	2	4	no	—
227	C₂×C₅	—	8	4	no	—
228	C₂×C₅	—	8	8	no	—
229	C₂×C₅	C₂×#509	1	6	yes	direct product factor
230	C₂×C₅	C₂³×D₈	1	1	yes	characteristic
231	C₂³	—	2	4	no	—
232	C₂³	—	4	6	no	—
233	C₂³	—	4	16	no	—
234	C₂³	—	4	12	no	—
235	C₂³	—	8	48	no	—
236	C₂³	—	8	8	no	—
237	C₂³	—	8	4	no	—
238	C₂³	C₂×C₅×D₈	1	6	yes	—
239	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
240	Q₈	—	2	4	no	—
241	Q₈	—	4	6	no	—
242	D₄	—	2	8	no	—
243	D₄	—	2	4	no	—
244	D₄	—	4	16	no	—
245	D₄	—	4	16	no	—
246	D₄	—	4	12	no	—
247	D₄	—	4	6	no	—
248	C₂×C₄	—	4	6	no	—
249	C₂×C₄	—	4	16	no	—
250	C₂×C₄	C₂×C₅×D₈	1	6	yes	—
251	C₂×C₄	C₂×C₅×D₈	1	3	yes	—
252	C₂×C₄	C₂×C₅×D₈	1	12	yes	—
253	C₂×C₄	C₂²×C₅×D₄	1	3	yes	—
254	C₂×C₄	C₂²×C₅×D₄	1	4	yes	—
255	C₈	C₂²×C₅×D₄	1	3	yes	—
256	C₈	C₂²×C₅×D₄	1	4	yes	—
257	C₈	C₂⁵×C₅	1	1	yes	characteristic
258	C₅	C₂²×#509	1	1	yes	characteristic, direct product factor, Sylow
259	C₂²	—	2	12	no	—
260	C₂²	—	4	4	no	—
261	C₂²	—	4	8	no	—
262	C₂²	—	8	32	no	—
263	C₂²	—	8	24	no	—
264	C₂²	—	8	12	no	—
265	C₂²	C₅×#509	1	4	yes	direct product factor
266	C₂²	C₂²×C₅×D₈	1	4	yes	—
267	C₂²	C₂²×C₅×D₈	1	3	yes	—
268	C₄	—	4	4	no	—
269	C₄	C₂²×C₅×D₈	1	4	yes	—
270	C₄	C₂²×C₅×D₈	1	3	yes	—
271	C₄	C₂³×C₅×D₄	1	1	yes	characteristic
272	C₂	—	2	4	no	—
273	C₂	—	8	8	no	—
274	C₂	—	8	4	no	—
275	C₂	C₂×C₅×#509	1	6	yes	direct product factor
276	C₂	C₂³×C₅×D₈	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	C₂×D₈
commutator, Frattini, cosocle	C₈	C₂⁵×C₅
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂³×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×#135	C₂	semidirect
2. C₂³×C₅×D₈	C₂	semidirect
3. C₂×C₅×#509	C₂	direct (N × H)
4. C₂²×C₅×SD₃₂	C₂	semidirect
5. C₂²×C₅×D₁₆	C₂	semidirect
6. C₂²×C₅×M₃₂	C₂	semidirect
10. C₂×C₅×#135	C₂²	semidirect
11. C₂²×C₅×Q₁₆	C₂²	semidirect
16. C₂²×C₅×D₈	C₂²	semidirect
17. C₂²×C₅×D₈	C₂²	semidirect
19. C₅×#509	C₂²	direct (N × H)
20. C₂×C₅×SD₃₂	C₂²	semidirect
21. C₂×C₅×D₁₆	C₂²	semidirect
22. C₂×C₅×M₃₂	C₂²	semidirect
23. C₂²×C₅×C₁₆	C₂²	semidirect
24. C₂²×#509	C₅	direct (N × H)
38. C₅×#135	C₂³	semidirect
40. C₂×C₅×Q₁₆	C₂³	semidirect
47. C₂×C₅×D₈	C₂³	semidirect
48. C₂×C₅×D₈	C₂³	semidirect
53. C₅×SD₃₂	C₂³	semidirect
54. C₅×D₁₆	C₂³	semidirect
55. C₅×M₃₂	C₂³	semidirect
56. C₂×C₅×C₁₆	C₂³	semidirect
57. C₂²×#135	C₂×C₅	semidirect
58. C₂³×D₈	C₂×C₅	semidirect
59. C₂×#509	C₂×C₅	direct (N × H)
60. C₂²×SD₃₂	C₂×C₅	semidirect
61. C₂²×D₁₆	C₂×C₅	semidirect
62. C₂²×M₃₂	C₂×C₅	semidirect
87. C₅×Q₁₆	C₂⁴	semidirect
93. C₅×D₈	C₂⁴	semidirect
99. C₅×C₁₆	C₂⁴	semidirect
103. C₂×#135	C₂²×C₅	semidirect
104. C₂²×Q₁₆	C₂²×C₅	semidirect
109. C₂²×D₈	C₂²×C₅	semidirect
110. C₂²×D₈	C₂²×C₅	semidirect
112. #509	C₂²×C₅	direct (N × H)
113. C₂×SD₃₂	C₂²×C₅	semidirect
114. C₂×D₁₆	C₂²×C₅	semidirect
115. C₂×M₃₂	C₂²×C₅	semidirect
116. C₂²×C₁₆	C₂²×C₅	semidirect
157. #135	C₂³×C₅	semidirect
159. C₂×Q₁₆	C₂³×C₅	semidirect
166. C₂×D₈	C₂³×C₅	semidirect
167. C₂×D₈	C₂³×C₅	semidirect
172. SD₃₂	C₂³×C₅	semidirect
173. D₁₆	C₂³×C₅	semidirect
174. M₃₂	C₂³×C₅	semidirect
175. C₂×C₁₆	C₂³×C₅	semidirect
182. C₂²×C₅	#509	direct (N × H)
213. Q₁₆	C₂⁴×C₅	semidirect
219. D₈	C₂⁴×C₅	semidirect
225. C₁₆	C₂⁴×C₅	semidirect
229. C₂×C₅	C₂×#509	direct (N × H)
258. C₅	C₂²×#509	direct (N × H)
265. C₂²	C₅×#509	direct (N × H)
275. C₂	C₂×C₅×#509	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×#135	2
C₂³×C₅×D₈	2
C₂×C₅×#509	2
C₂²×C₅×SD₃₂	2
C₂²×C₅×D₁₆	2
C₂²×C₅×M₃₂	2
C₂²×#509	5

Subgroups — C22×C5×#509 #11985349

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×#509 #11985349