Subgroups — C₂²×C₅×#510 #11985350

Order 1280.

2078 conjugacy classes of subgroups.

The subgroup lattice has 266 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
2	C₂³×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#510	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₅×Q₃₂	C₂	1	2	yes	maximal, complemented
5	C₂²×C₅×SD₃₂	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×Q₈	—	2	1	no	—
9	C₂×C₅×#135	—	2	6	no	—
10	C₂×C₅×#135	C₂²	1	12	yes	—
11	C₂²×C₅×Q₁₆	—	2	1	no	—
12	C₂²×C₅×Q₁₆	—	2	6	no	—
13	C₂²×C₅×Q₁₆	C₂²	1	2	yes	—
14	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₅×Q₁₆	C₂²	1	12	yes	complemented
16	C₂²×C₅×SD₁₆	—	2	1	no	—
17	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
18	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
19	C₅×#510	C₂²	1	64	yes	direct product factor
20	C₂×C₅×Q₃₂	C₂²	1	24	yes	complemented
21	C₂×C₅×SD₃₂	C₂²	1	24	yes	complemented
22	C₂×C₅×M₃₂	C₂²	1	12	yes	complemented
23	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
24	C₂²×#510	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂×C₅×#41	—	2	12	no	—
26	C₂×C₅×#41	—	4	6	no	—
27	C₂²×C₅×Q₈	—	2	1	no	—
28	C₂²×C₅×Q₈	—	2	2	no	—
29	C₂²×C₅×Q₈	—	2	12	no	—
30	C₂²×C₅×Q₈	—	4	6	no	—
31	C₂²×C₅×Q₈	—	4	1	no	—
32	C₂²×C₅×D₄	—	2	1	no	—
33	C₂²×C₅×D₄	—	4	1	no	—
34	C₂³×C₄×C₅	—	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	—
39	C₂×C₅×Q₁₆	—	2	6	no	—
40	C₂×C₅×Q₁₆	—	2	12	no	—
41	C₂×C₅×Q₁₆	—	2	24	no	—
42	C₂×C₅×Q₁₆	—	2	3	no	—
43	C₂×C₅×Q₁₆	C₂³	1	12	yes	—
44	C₂×C₅×Q₁₆	C₂³	1	6	yes	—
45	C₂×C₅×Q₁₆	C₂³	1	16	yes	complemented
46	C₂×C₅×SD₁₆	—	2	12	no	—
47	C₂×C₅×D₈	—	2	3	no	—
48	C₂×C₅×D₈	C₂³	1	6	yes	—
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×Q₃₂	C₂³	1	32	yes	complemented
54	C₅×SD₃₂	C₂³	1	32	yes	complemented
55	C₅×M₃₂	C₂³	1	16	yes	complemented
56	C₂×C₅×C₁₆	C₂³	1	12	yes	—
57	C₂²×#135	C₂×C₅	1	1	yes	characteristic
58	C₂³×Q₁₆	C₂×C₅	1	1	yes	characteristic, complemented
59	C₂×#510	C₂×C₅	1	24	yes	direct product factor
60	C₂²×Q₃₂	C₂×C₅	1	2	yes	complemented
61	C₂²×SD₃₂	C₂×C₅	1	2	yes	complemented
62	C₂²×M₃₂	C₂×C₅	1	1	yes	characteristic, complemented
63	C₂⁴×C₅	—	4	1	no	—
64	C₂⁴×C₅	D₈	1	1	yes	characteristic
65	C₅×#41	—	2	16	no	—
66	C₅×#41	—	4	24	no	—
67	C₂×C₅×Q₈	—	2	6	no	—
68	C₂×C₅×Q₈	—	2	12	no	—
69	C₂×C₅×Q₈	—	2	16	no	—
70	C₂×C₅×Q₈	—	4	12	no	—
71	C₂×C₅×Q₈	—	4	6	no	—
72	C₂×C₅×Q₈	—	4	24	no	—
73	C₂×C₅×Q₈	—	4	3	no	—
74	C₂×C₅×D₄	—	2	6	no	—
75	C₂×C₅×D₄	—	4	12	no	—
76	C₂×C₅×D₄	—	4	3	no	—
77	C₂²×C₄×C₅	—	4	12	no	—
78	C₂²×C₄×C₅	—	4	1	no	—
79	C₂²×C₄×C₅	—	4	2	no	—
80	C₂²×C₄×C₅	—	4	12	no	—
81	C₂²×C₄×C₅	D₈	1	6	yes	—
82	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
83	C₂²×C₄×C₅	C₂×D₄	1	6	yes	—
84	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₅×Q₁₆	—	2	12	no	—
86	C₅×Q₁₆	—	2	16	no	—
87	C₅×Q₁₆	—	2	6	no	—
88	C₅×Q₁₆	C₂⁴	1	8	yes	—
89	C₅×Q₁₆	C₂⁴	1	4	yes	—
90	C₅×SD₁₆	—	2	16	no	—
91	C₅×D₈	—	2	6	no	—
92	C₅×D₈	C₂⁴	1	4	yes	—
93	C₂×C₅×C₈	C₂×D₄	1	3	yes	—
94	C₂×C₅×C₈	C₂×D₄	1	6	yes	—
95	C₂×C₅×C₈	C₂×D₄	1	12	yes	—
96	C₂×C₅×C₈	C₂⁴	1	3	yes	—
97	C₂×C₅×C₈	C₂⁴	1	4	yes	—
98	C₅×C₁₆	C₂⁴	1	8	yes	—
99	C₂²×#41	—	2	1	no	—
100	C₂³×Q₈	—	2	1	no	—
101	C₂×#135	—	2	6	no	—
102	C₂×#135	C₂²×C₅	1	12	yes	—
103	C₂²×Q₁₆	—	2	1	no	—
104	C₂²×Q₁₆	—	2	6	no	—
105	C₂²×Q₁₆	C₂²×C₅	1	2	yes	—
106	C₂²×Q₁₆	C₂²×C₅	1	1	yes	characteristic
107	C₂²×Q₁₆	C₂²×C₅	1	12	yes	complemented
108	C₂²×SD₁₆	—	2	1	no	—
109	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
110	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
111	#510	C₂²×C₅	1	64	yes	direct product factor
112	C₂×Q₃₂	C₂²×C₅	1	24	yes	complemented
113	C₂×SD₃₂	C₂²×C₅	1	24	yes	complemented
114	C₂×M₃₂	C₂²×C₅	1	12	yes	complemented
115	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
116	C₂³×C₅	—	2	4	no	—
117	C₂³×C₅	—	4	6	no	—
118	C₂³×C₅	—	8	4	no	—
119	C₂³×C₅	C₂×D₈	1	6	yes	—
120	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
121	C₅×Q₈	—	2	4	no	—
122	C₅×Q₈	—	2	8	no	—
123	C₅×Q₈	—	4	16	no	—
124	C₅×Q₈	—	4	12	no	—
125	C₅×Q₈	—	4	6	no	—
126	C₅×D₄	—	2	4	no	—
127	C₅×D₄	—	4	16	no	—
128	C₅×D₄	—	4	6	no	—
129	C₂×C₄×C₅	—	4	6	no	—
130	C₂×C₄×C₅	—	4	16	no	—
131	C₂×C₄×C₅	—	4	12	no	—
132	C₂×C₄×C₅	—	4	16	no	—
133	C₂×C₄×C₅	C₂×D₈	1	6	yes	—
134	C₂×C₄×C₅	C₂×D₈	1	3	yes	—
135	C₂×C₄×C₅	C₂×D₈	1	12	yes	—
136	C₂×C₄×C₅	C₂²×D₄	1	3	yes	—
137	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
138	C₅×C₈	C₂²×D₄	1	3	yes	—
139	C₅×C₈	C₂²×D₄	1	4	yes	—
140	C₅×C₈	C₂⁵	1	1	yes	characteristic
141	C₂×#41	—	2	12	no	—
142	C₂×#41	—	4	6	no	—
143	C₂²×Q₈	—	2	1	no	—
144	C₂²×Q₈	—	2	2	no	—
145	C₂²×Q₈	—	2	12	no	—
146	C₂²×Q₈	—	4	1	no	—
147	C₂²×Q₈	—	4	6	no	—
148	C₂²×D₄	—	2	1	no	—
149	C₂²×D₄	—	4	1	no	—
150	C₂³×C₄	—	4	1	no	—
151	C₂³×C₄	—	4	1	no	—
152	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
153	#135	—	2	24	no	—
154	#135	C₂³×C₅	1	16	yes	—
155	C₂×Q₁₆	—	2	6	no	—
156	C₂×Q₁₆	—	2	12	no	—
157	C₂×Q₁₆	—	2	24	no	—
158	C₂×Q₁₆	—	2	3	no	—
159	C₂×Q₁₆	C₂³×C₅	1	12	yes	—
160	C₂×Q₁₆	C₂³×C₅	1	6	yes	—
161	C₂×Q₁₆	C₂³×C₅	1	16	yes	complemented
162	C₂×SD₁₆	—	2	12	no	—
163	C₂×D₈	—	2	3	no	—
164	C₂×D₈	C₂³×C₅	1	6	yes	—
165	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
166	C₂²×C₈	C₅×D₄	1	6	yes	—
167	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
168	C₂²×C₈	C₂³×C₅	1	6	yes	—
169	Q₃₂	C₂³×C₅	1	32	yes	complemented
170	SD₃₂	C₂³×C₅	1	32	yes	complemented
171	M₃₂	C₂³×C₅	1	16	yes	complemented
172	C₂×C₁₆	C₂³×C₅	1	12	yes	—
173	C₂²×C₅	—	2	12	no	—
174	C₂²×C₅	—	4	4	no	—
175	C₂²×C₅	—	8	12	no	—
176	C₂²×C₅	#510	1	4	yes	direct product factor
177	C₂²×C₅	C₂²×D₈	1	4	yes	—
178	C₂²×C₅	C₂²×D₈	1	3	yes	—
179	C₄×C₅	—	4	4	no	—
180	C₄×C₅	—	4	8	no	—
181	C₄×C₅	C₂²×D₈	1	4	yes	—
182	C₄×C₅	C₂²×D₈	1	3	yes	—
183	C₄×C₅	C₂³×D₄	1	1	yes	characteristic
184	C₂⁴	—	4	1	no	—
185	C₂⁴	C₅×D₈	1	1	yes	characteristic
186	#41	—	2	16	no	—
187	#41	—	4	24	no	—
188	C₂×Q₈	—	2	6	no	—
189	C₂×Q₈	—	2	12	no	—
190	C₂×Q₈	—	2	16	no	—
191	C₂×Q₈	—	4	12	no	—
192	C₂×Q₈	—	4	6	no	—
193	C₂×Q₈	—	4	24	no	—
194	C₂×Q₈	—	4	3	no	—
195	C₂×D₄	—	2	6	no	—
196	C₂×D₄	—	4	12	no	—
197	C₂×D₄	—	4	3	no	—
198	C₂²×C₄	—	4	1	no	—
199	C₂²×C₄	—	4	12	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	—	4	12	no	—
202	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
203	C₂²×C₄	C₅×D₈	1	6	yes	—
204	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₅×D₄	1	6	yes	—
206	Q₁₆	—	2	12	no	—
207	Q₁₆	—	2	16	no	—
208	Q₁₆	—	2	6	no	—
209	Q₁₆	C₂⁴×C₅	1	8	yes	—
210	Q₁₆	C₂⁴×C₅	1	4	yes	—
211	SD₁₆	—	2	16	no	—
212	D₈	—	2	6	no	—
213	D₈	C₂⁴×C₅	1	4	yes	—
214	C₂×C₈	C₂×C₅×D₄	1	3	yes	—
215	C₂×C₈	C₂×C₅×D₄	1	6	yes	—
216	C₂×C₈	C₂×C₅×D₄	1	12	yes	—
217	C₂×C₈	C₂⁴×C₅	1	3	yes	—
218	C₂×C₈	C₂⁴×C₅	1	4	yes	—
219	C₁₆	C₂⁴×C₅	1	8	yes	—
220	C₂×C₅	—	2	4	no	—
221	C₂×C₅	—	8	4	no	—
222	C₂×C₅	C₂×#510	1	6	yes	direct product factor
223	C₂×C₅	C₂³×D₈	1	1	yes	characteristic
224	C₂³	—	2	4	no	—
225	C₂³	—	4	6	no	—
226	C₂³	—	8	4	no	—
227	C₂³	C₂×C₅×D₈	1	6	yes	—
228	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
229	Q₈	—	2	4	no	—
230	Q₈	—	2	8	no	—
231	Q₈	—	4	16	no	—
232	Q₈	—	4	12	no	—
233	Q₈	—	4	6	no	—
234	D₄	—	2	4	no	—
235	D₄	—	4	16	no	—
236	D₄	—	4	6	no	—
237	C₂×C₄	—	4	6	no	—
238	C₂×C₄	—	4	16	no	—
239	C₂×C₄	—	4	12	no	—
240	C₂×C₄	—	4	16	no	—
241	C₂×C₄	C₂×C₅×D₈	1	6	yes	—
242	C₂×C₄	C₂×C₅×D₈	1	3	yes	—
243	C₂×C₄	C₂×C₅×D₈	1	12	yes	—
244	C₂×C₄	C₂²×C₅×D₄	1	3	yes	—
245	C₂×C₄	C₂²×C₅×D₄	1	4	yes	—
246	C₈	C₂²×C₅×D₄	1	3	yes	—
247	C₈	C₂²×C₅×D₄	1	4	yes	—
248	C₈	C₂⁵×C₅	1	1	yes	characteristic
249	C₅	C₂²×#510	1	1	yes	characteristic, direct product factor, Sylow
250	C₂²	—	2	12	no	—
251	C₂²	—	4	4	no	—
252	C₂²	—	8	12	no	—
253	C₂²	C₅×#510	1	4	yes	direct product factor
254	C₂²	C₂²×C₅×D₈	1	4	yes	—
255	C₂²	C₂²×C₅×D₈	1	3	yes	—
256	C₄	—	4	4	no	—
257	C₄	—	4	8	no	—
258	C₄	C₂²×C₅×D₈	1	4	yes	—
259	C₄	C₂²×C₅×D₈	1	3	yes	—
260	C₄	C₂³×C₅×D₄	1	1	yes	characteristic
261	C₂	—	2	4	no	—
262	C₂	—	8	4	no	—
263	C₂	C₂×C₅×#510	1	6	yes	direct product factor
264	C₂	C₂³×C₅×D₈	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	C₂×D₈
commutator, Frattini, cosocle	C₈	C₂⁵×C₅
prime radical	C₂²×C₅×#135	C₂
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
2. C₂³×C₅×Q₁₆	C₂	semidirect
3. C₂×C₅×#510	C₂	direct (N × H)
4. C₂²×C₅×Q₃₂	C₂	semidirect
5. C₂²×C₅×SD₃₂	C₂	semidirect
6. C₂²×C₅×M₃₂	C₂	semidirect
15. C₂²×C₅×Q₁₆	C₂²	semidirect
19. C₅×#510	C₂²	direct (N × H)
20. C₂×C₅×Q₃₂	C₂²	semidirect
21. C₂×C₅×SD₃₂	C₂²	semidirect
22. C₂×C₅×M₃₂	C₂²	semidirect
24. C₂²×#510	C₅	direct (N × H)
45. C₂×C₅×Q₁₆	C₂³	semidirect
53. C₅×Q₃₂	C₂³	semidirect
54. C₅×SD₃₂	C₂³	semidirect
55. C₅×M₃₂	C₂³	semidirect
58. C₂³×Q₁₆	C₂×C₅	semidirect
59. C₂×#510	C₂×C₅	direct (N × H)
60. C₂²×Q₃₂	C₂×C₅	semidirect
61. C₂²×SD₃₂	C₂×C₅	semidirect
62. C₂²×M₃₂	C₂×C₅	semidirect
107. C₂²×Q₁₆	C₂²×C₅	semidirect
111. #510	C₂²×C₅	direct (N × H)
112. C₂×Q₃₂	C₂²×C₅	semidirect
113. C₂×SD₃₂	C₂²×C₅	semidirect
114. C₂×M₃₂	C₂²×C₅	semidirect
161. C₂×Q₁₆	C₂³×C₅	semidirect
169. Q₃₂	C₂³×C₅	semidirect
170. SD₃₂	C₂³×C₅	semidirect
171. M₃₂	C₂³×C₅	semidirect
176. C₂²×C₅	#510	direct (N × H)
222. C₂×C₅	C₂×#510	direct (N × H)
249. C₅	C₂²×#510	direct (N × H)
253. C₂²	C₅×#510	direct (N × H)
263. C₂	C₂×C₅×#510	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×Q₁₆	2
C₂×C₅×#510	2
C₂²×C₅×Q₃₂	2
C₂²×C₅×SD₃₂	2
C₂²×C₅×M₃₂	2
C₂²×#510	5

Subgroups — C22×C5×#510 #11985350

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×#510 #11985350