Subgroups — C₂×C₅×#2812 #11959366

Order 1280.

4438 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#120	C₂	1	2	yes	maximal, complemented
2	C₅×#2812	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₅×#397	C₂	1	3	yes	maximal, complemented
4	C₂×C₅×#394	C₂	1	6	yes	maximal, complemented
5	C₂×C₅×#392	C₂	1	3	yes	maximal, complemented
6	C₂×C₅×#379	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁶×C₅	C₂²	1	1	yes	characteristic
8	C₂³×C₅×D₄	C₂²	1	6	yes	—
9	C₂³×C₅×D₄	C₂²	1	1	yes	characteristic, complemented
10	C₂×C₅×#120	—	2	16	no	—
11	C₂×C₅×#120	C₂²	1	16	yes	complemented
12	C₂²×C₅×C₄⋊C₄	C₂²	1	3	yes	complemented
13	C₂²×C₅×C₂²⋊C₄	C₂²	1	6	yes	—
14	C₂²×C₅×C₂²⋊C₄	C₂²	1	3	yes	—
15	C₂²×C₅×C₂²⋊C₄	C₂²	1	3	yes	—
16	C₂²×C₅×C₂²⋊C₄	C₂²	1	6	yes	complemented
17	C₅×#397	C₂²	1	24	yes	complemented
18	C₅×#394	C₂²	1	48	yes	complemented
19	C₅×#392	C₂²	1	24	yes	complemented
20	C₅×#379	C₂²	1	8	yes	complemented
21	C₂×C₅×#95	C₂²	1	3	yes	complemented
22	C₂×C₅×#95	C₂²	1	3	yes	complemented
23	C₂×#2812	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂⁵×C₅	—	4	6	no	—
25	C₂⁵×C₅	—	4	6	no	—
26	C₂⁵×C₅	D₄	1	8	yes	complemented
27	C₂⁵×C₅	C₂³	1	2	yes	—
28	C₂⁵×C₅	C₂³	1	3	yes	—
29	C₂⁵×C₅	C₂³	1	4	yes	—
30	C₂²×C₅×D₄	—	2	24	no	—
31	C₂²×C₅×D₄	—	4	12	no	—
32	C₂²×C₅×D₄	—	4	12	no	—
33	C₂²×C₅×D₄	—	4	6	no	—
34	C₂²×C₅×D₄	C₂³	1	24	yes	—
35	C₂²×C₅×D₄	C₂³	1	4	yes	complemented
36	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
37	C₂³×C₄×C₅	C₂³	1	6	yes	—
38	C₂³×C₄×C₅	C₂³	1	3	yes	—
39	C₅×#120	—	2	64	no	—
40	C₂×C₅×C₄⋊C₄	—	4	6	no	—
41	C₂×C₅×C₄⋊C₄	C₂³	1	12	yes	complemented
42	C₂×C₅×C₂²⋊C₄	—	2	24	no	—
43	C₂×C₅×C₂²⋊C₄	—	4	6	no	—
44	C₂×C₅×C₂²⋊C₄	—	4	6	no	—
45	C₂×C₅×C₂²⋊C₄	—	4	12	no	—
46	C₂×C₅×C₂²⋊C₄	C₂³	1	24	yes	—
47	C₂×C₅×C₂²⋊C₄	C₂³	1	12	yes	—
48	C₂×C₅×C₂²⋊C₄	C₂³	1	12	yes	—
49	C₂×C₅×C₂²⋊C₄	C₂³	1	24	yes	complemented
50	C₅×#95	C₂³	1	12	yes	complemented
51	C₅×#95	C₂³	1	12	yes	complemented
52	C₂²×#120	C₂×C₅	1	2	yes	complemented
53	#2812	C₂×C₅	1	16	yes	direct product factor
54	C₂×#397	C₂×C₅	1	3	yes	complemented
55	C₂×#394	C₂×C₅	1	6	yes	complemented
56	C₂×#392	C₂×C₅	1	3	yes	complemented
57	C₂×#379	C₂×C₅	1	1	yes	characteristic, complemented
58	C₂⁴×C₅	—	2	6	no	—
59	C₂⁴×C₅	—	2	6	no	—
60	C₂⁴×C₅	—	2	12	no	—
61	C₂⁴×C₅	—	2	12	no	—
62	C₂⁴×C₅	—	2	4	no	—
63	C₂⁴×C₅	—	2	12	no	—
64	C₂⁴×C₅	—	2	12	no	—
65	C₂⁴×C₅	—	4	48	no	—
66	C₂⁴×C₅	—	4	12	no	—
67	C₂⁴×C₅	—	4	12	no	—
68	C₂⁴×C₅	—	4	48	no	—
69	C₂⁴×C₅	—	4	16	no	—
70	C₂⁴×C₅	C₂×D₄	1	12	yes	—
71	C₂⁴×C₅	C₂×D₄	1	16	yes	complemented
72	C₂⁴×C₅	C₂⁴	1	4	yes	—
73	C₂⁴×C₅	C₂⁴	1	6	yes	—
74	C₂⁴×C₅	C₂⁴	1	1	yes	characteristic
75	C₂×C₅×D₄	—	2	48	no	—
76	C₂×C₅×D₄	—	4	24	no	—
77	C₂×C₅×D₄	—	4	24	no	—
78	C₂×C₅×D₄	—	4	12	no	—
79	C₂×C₅×D₄	—	4	96	no	—
80	C₂×C₅×D₄	—	4	16	no	—
81	C₂²×C₄×C₅	—	2	12	no	—
82	C₂²×C₄×C₅	—	2	12	no	—
83	C₂²×C₄×C₅	—	2	6	no	—
84	C₂²×C₄×C₅	—	2	12	no	—
85	C₂²×C₄×C₅	—	2	6	no	—
86	C₂²×C₄×C₅	—	2	12	no	—
87	C₂²×C₄×C₅	C₂⁴	1	6	yes	—
88	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
89	C₂²×C₄×C₅	C₂⁴	1	12	yes	—
90	C₅×C₄⋊C₄	—	4	12	no	—
91	C₅×C₂²⋊C₄	—	2	48	no	—
92	C₅×C₂²⋊C₄	—	4	12	no	—
93	C₅×C₂²⋊C₄	—	4	12	no	—
94	C₅×C₂²⋊C₄	—	4	24	no	—
95	C₂⁶	C₂²×C₅	1	1	yes	characteristic
96	C₂³×D₄	C₂²×C₅	1	6	yes	—
97	C₂³×D₄	C₂²×C₅	1	1	yes	characteristic, complemented
98	C₂×#120	—	2	16	no	—
99	C₂×#120	C₂²×C₅	1	16	yes	complemented
100	C₂²×C₄⋊C₄	C₂²×C₅	1	3	yes	complemented
101	C₂²×C₂²⋊C₄	C₂²×C₅	1	6	yes	—
102	C₂²×C₂²⋊C₄	C₂²×C₅	1	3	yes	—
103	C₂²×C₂²⋊C₄	C₂²×C₅	1	3	yes	—
104	C₂²×C₂²⋊C₄	C₂²×C₅	1	6	yes	complemented
105	#397	C₂²×C₅	1	24	yes	complemented
106	#394	C₂²×C₅	1	48	yes	complemented
107	#392	C₂²×C₅	1	24	yes	complemented
108	#379	C₂²×C₅	1	8	yes	complemented
109	C₂×#95	C₂²×C₅	1	3	yes	complemented
110	C₂×#95	C₂²×C₅	1	3	yes	complemented
111	C₂³×C₅	—	2	48	no	—
112	C₂³×C₅	—	2	24	no	—
113	C₂³×C₅	—	2	6	no	—
114	C₂³×C₅	—	2	6	no	—
115	C₂³×C₅	—	2	12	no	—
116	C₂³×C₅	—	2	12	no	—
117	C₂³×C₅	—	2	4	no	—
118	C₂³×C₅	—	2	12	no	—
119	C₂³×C₅	—	2	12	no	—
120	C₂³×C₅	—	4	48	no	—
121	C₂³×C₅	—	4	48	no	—
122	C₂³×C₅	—	4	16	no	—
123	C₂³×C₅	—	4	128	no	—
124	C₂³×C₅	—	4	24	no	—
125	C₂³×C₅	—	4	12	no	—
126	C₂³×C₅	—	4	12	no	—
127	C₂³×C₅	—	4	24	no	—
128	C₂³×C₅	—	4	24	no	—
129	C₂³×C₅	—	4	8	no	—
130	C₂³×C₅	C₂²×D₄	1	12	yes	—
131	C₂³×C₅	C₂²×D₄	1	2	yes	—
132	C₂³×C₅	H(5,𝔽₂)	1	6	yes	—
133	C₂³×C₅	H(5,𝔽₂)	1	6	yes	—
134	C₂³×C₅	C₂⁵	1	1	yes	characteristic
135	C₅×D₄	—	4	96	no	—
136	C₅×D₄	—	4	16	no	—
137	C₂×C₄×C₅	—	2	12	no	—
138	C₂×C₄×C₅	—	2	12	no	—
139	C₂×C₄×C₅	—	2	6	no	—
140	C₂×C₄×C₅	—	2	12	no	—
141	C₂×C₄×C₅	—	2	6	no	—
142	C₂×C₄×C₅	—	2	12	no	—
143	C₂×C₄×C₅	—	4	24	no	—
144	C₂×C₄×C₅	—	4	12	no	—
145	C₂×C₄×C₅	—	4	4	no	—
146	C₂⁵	—	4	6	no	—
147	C₂⁵	—	4	6	no	—
148	C₂⁵	C₅×D₄	1	8	yes	complemented
149	C₂⁵	C₂³×C₅	1	2	yes	—
150	C₂⁵	C₂³×C₅	1	3	yes	—
151	C₂⁵	C₂³×C₅	1	4	yes	—
152	C₂²×D₄	—	2	24	no	—
153	C₂²×D₄	—	4	12	no	—
154	C₂²×D₄	—	4	12	no	—
155	C₂²×D₄	—	4	6	no	—
156	C₂²×D₄	C₂³×C₅	1	24	yes	—
157	C₂²×D₄	C₂³×C₅	1	4	yes	complemented
158	C₂³×C₄	C₂³×C₅	1	3	yes	—
159	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
160	C₂³×C₄	C₂³×C₅	1	6	yes	—
161	#120	—	2	64	no	—
162	C₂×C₄⋊C₄	—	4	6	no	—
163	C₂×C₄⋊C₄	C₂³×C₅	1	12	yes	complemented
164	C₂×C₂²⋊C₄	—	2	24	no	—
165	C₂×C₂²⋊C₄	—	4	6	no	—
166	C₂×C₂²⋊C₄	—	4	6	no	—
167	C₂×C₂²⋊C₄	—	4	12	no	—
168	C₂×C₂²⋊C₄	C₂³×C₅	1	24	yes	—
169	C₂×C₂²⋊C₄	C₂³×C₅	1	12	yes	—
170	C₂×C₂²⋊C₄	C₂³×C₅	1	12	yes	—
171	C₂×C₂²⋊C₄	C₂³×C₅	1	24	yes	complemented
172	#95	C₂³×C₅	1	12	yes	complemented
173	#95	C₂³×C₅	1	12	yes	complemented
174	C₂²×C₅	—	2	24	no	—
175	C₂²×C₅	—	2	12	no	—
176	C₂²×C₅	—	4	64	no	—
177	C₂²×C₅	—	4	12	no	—
178	C₂²×C₅	—	4	6	no	—
179	C₂²×C₅	—	4	6	no	—
180	C₂²×C₅	—	4	48	no	—
181	C₂²×C₅	—	4	12	no	—
182	C₂²×C₅	—	4	12	no	—
183	C₂²×C₅	—	4	4	no	—
184	C₂²×C₅	—	8	16	no	—
185	C₂²×C₅	#534	1	12	yes	—
186	C₂²×C₅	#560	1	12	yes	—
187	C₂²×C₅	H(3,𝔽₄)	1	4	yes	—
188	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
189	C₂²×C₅	C₂×H(5,𝔽₂)	1	3	yes	—
190	C₂²×C₅	C₂×H(5,𝔽₂)	1	3	yes	—
191	C₄×C₅	—	4	12	no	—
192	C₄×C₅	—	4	6	no	—
193	C₄×C₅	—	4	2	no	—
194	C₂⁴	—	2	6	no	—
195	C₂⁴	—	2	6	no	—
196	C₂⁴	—	2	12	no	—
197	C₂⁴	—	2	12	no	—
198	C₂⁴	—	2	4	no	—
199	C₂⁴	—	2	12	no	—
200	C₂⁴	—	2	12	no	—
201	C₂⁴	—	4	48	no	—
202	C₂⁴	—	4	12	no	—
203	C₂⁴	—	4	12	no	—
204	C₂⁴	—	4	48	no	—
205	C₂⁴	—	4	16	no	—
206	C₂⁴	C₂×C₅×D₄	1	12	yes	—
207	C₂⁴	C₂×C₅×D₄	1	16	yes	complemented
208	C₂⁴	C₂⁴×C₅	1	4	yes	—
209	C₂⁴	C₂⁴×C₅	1	6	yes	—
210	C₂⁴	C₂⁴×C₅	1	1	yes	characteristic
211	C₂×D₄	—	2	48	no	—
212	C₂×D₄	—	4	24	no	—
213	C₂×D₄	—	4	24	no	—
214	C₂×D₄	—	4	12	no	—
215	C₂×D₄	—	4	96	no	—
216	C₂×D₄	—	4	16	no	—
217	C₂²×C₄	—	2	12	no	—
218	C₂²×C₄	—	2	12	no	—
219	C₂²×C₄	—	2	6	no	—
220	C₂²×C₄	—	2	12	no	—
221	C₂²×C₄	—	2	6	no	—
222	C₂²×C₄	—	2	12	no	—
223	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
224	C₂²×C₄	C₂⁴×C₅	1	12	yes	—
225	C₂²×C₄	C₂⁴×C₅	1	6	yes	—
226	C₄⋊C₄	—	4	12	no	—
227	C₂²⋊C₄	—	2	48	no	—
228	C₂²⋊C₄	—	4	12	no	—
229	C₂²⋊C₄	—	4	12	no	—
230	C₂²⋊C₄	—	4	24	no	—
231	C₂×C₅	—	4	12	no	—
232	C₂×C₅	—	8	4	no	—
233	C₂×C₅	#2812	1	8	yes	direct product factor
234	C₂×C₅	C₂×#534	1	3	yes	—
235	C₂×C₅	C₂×#560	1	3	yes	—
236	C₂×C₅	C₂×H(3,𝔽₄)	1	1	yes	characteristic
237	C₂³	—	2	48	no	—
238	C₂³	—	2	24	no	—
239	C₂³	—	2	6	no	—
240	C₂³	—	2	6	no	—
241	C₂³	—	2	12	no	—
242	C₂³	—	2	12	no	—
243	C₂³	—	2	4	no	—
244	C₂³	—	2	12	no	—
245	C₂³	—	2	12	no	—
246	C₂³	—	4	48	no	—
247	C₂³	—	4	128	no	—
248	C₂³	—	4	48	no	—
249	C₂³	—	4	16	no	—
250	C₂³	—	4	24	no	—
251	C₂³	—	4	12	no	—
252	C₂³	—	4	12	no	—
253	C₂³	—	4	24	no	—
254	C₂³	—	4	24	no	—
255	C₂³	—	4	8	no	—
256	C₂³	C₂²×C₅×D₄	1	12	yes	—
257	C₂³	C₂²×C₅×D₄	1	2	yes	—
258	C₂³	C₅×H(5,𝔽₂)	1	6	yes	—
259	C₂³	C₅×H(5,𝔽₂)	1	6	yes	—
260	C₂³	C₂⁵×C₅	1	1	yes	characteristic
261	D₄	—	4	96	no	—
262	D₄	—	4	16	no	—
263	C₂×C₄	—	2	12	no	—
264	C₂×C₄	—	2	12	no	—
265	C₂×C₄	—	2	6	no	—
266	C₂×C₄	—	2	12	no	—
267	C₂×C₄	—	2	6	no	—
268	C₂×C₄	—	2	12	no	—
269	C₂×C₄	—	4	24	no	—
270	C₂×C₄	—	4	12	no	—
271	C₂×C₄	—	4	4	no	—
272	C₅	C₂×#2812	1	1	yes	characteristic, direct product factor, Sylow
273	C₂²	—	2	24	no	—
274	C₂²	—	2	12	no	—
275	C₂²	—	4	64	no	—
276	C₂²	—	4	48	no	—
277	C₂²	—	4	12	no	—
278	C₂²	—	4	6	no	—
279	C₂²	—	4	6	no	—
280	C₂²	—	4	12	no	—
281	C₂²	—	4	12	no	—
282	C₂²	—	4	4	no	—
283	C₂²	—	8	16	no	—
284	C₂²	C₅×#534	1	12	yes	—
285	C₂²	C₅×#560	1	12	yes	—
286	C₂²	C₅×H(3,𝔽₄)	1	4	yes	—
287	C₂²	C₂³×C₅×D₄	1	1	yes	characteristic
288	C₂²	C₂×C₅×H(5,𝔽₂)	1	3	yes	—
289	C₂²	C₂×C₅×H(5,𝔽₂)	1	3	yes	—
290	C₄	—	4	12	no	—
291	C₄	—	4	6	no	—
292	C₄	—	4	2	no	—
293	C₂	—	4	12	no	—
294	C₂	—	8	4	no	—
295	C₂	C₅×#2812	1	8	yes	direct product factor
296	C₂	C₂×C₅×#534	1	3	yes	—
297	C₂	C₂×C₅×#560	1	3	yes	—
298	C₂	C₂×C₅×H(3,𝔽₄)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×#120	C₂	semidirect
2. C₅×#2812	C₂	direct (N × H)
3. C₂×C₅×#397	C₂	semidirect
4. C₂×C₅×#394	C₂	semidirect
5. C₂×C₅×#392	C₂	semidirect
6. C₂×C₅×#379	C₂	semidirect
9. C₂³×C₅×D₄	C₂²	semidirect
11. C₂×C₅×#120	C₂²	semidirect
12. C₂²×C₅×C₄⋊C₄	C₂²	semidirect
16. C₂²×C₅×C₂²⋊C₄	C₂²	semidirect
17. C₅×#397	C₂²	semidirect
18. C₅×#394	C₂²	semidirect
19. C₅×#392	C₂²	semidirect
20. C₅×#379	C₂²	semidirect
21. C₂×C₅×#95	C₂²	semidirect
22. C₂×C₅×#95	C₂²	semidirect
23. C₂×#2812	C₅	direct (N × H)
26. C₂⁵×C₅	D₄	semidirect
35. C₂²×C₅×D₄	C₂³	semidirect
41. C₂×C₅×C₄⋊C₄	C₂³	semidirect
49. C₂×C₅×C₂²⋊C₄	C₂³	semidirect
50. C₅×#95	C₂³	semidirect
51. C₅×#95	C₂³	semidirect
52. C₂²×#120	C₂×C₅	semidirect
53. #2812	C₂×C₅	direct (N × H)
54. C₂×#397	C₂×C₅	semidirect
55. C₂×#394	C₂×C₅	semidirect
56. C₂×#392	C₂×C₅	semidirect
57. C₂×#379	C₂×C₅	semidirect
71. C₂⁴×C₅	C₂×D₄	semidirect
97. C₂³×D₄	C₂²×C₅	semidirect
99. C₂×#120	C₂²×C₅	semidirect
100. C₂²×C₄⋊C₄	C₂²×C₅	semidirect
104. C₂²×C₂²⋊C₄	C₂²×C₅	semidirect
105. #397	C₂²×C₅	semidirect
106. #394	C₂²×C₅	semidirect
107. #392	C₂²×C₅	semidirect
108. #379	C₂²×C₅	semidirect
109. C₂×#95	C₂²×C₅	semidirect
110. C₂×#95	C₂²×C₅	semidirect
148. C₂⁵	C₅×D₄	semidirect
157. C₂²×D₄	C₂³×C₅	semidirect
163. C₂×C₄⋊C₄	C₂³×C₅	semidirect
171. C₂×C₂²⋊C₄	C₂³×C₅	semidirect
172. #95	C₂³×C₅	semidirect
173. #95	C₂³×C₅	semidirect
207. C₂⁴	C₂×C₅×D₄	semidirect
233. C₂×C₅	#2812	direct (N × H)
272. C₅	C₂×#2812	direct (N × H)
295. C₂	C₅×#2812	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×#120	2
C₅×#2812	2
C₂×C₅×#397	2
C₂×C₅×#394	2
C₂×C₅×#392	2
C₂×C₅×#379	2
C₂×#2812	5

Subgroups — C2×C5×#2812 #11959366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2812 #11959366