Subgroups — C₂×C₅×#2234 #11956361

Order 1280.

638 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#466	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#427	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2234	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₅×#367	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅×#366	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×#357	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅×#357	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
10	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₅×#121	—	2	1	no	—
12	C₂×C₅×#117	—	2	1	no	—
13	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
15	C₅×#466	—	2	2	no	—
16	C₅×#466	C₂²	1	4	yes	—
17	C₅×#427	—	2	2	no	—
18	C₅×#427	C₂²	1	4	yes	complemented
19	C₂×C₅×#107	C₂²	1	1	yes	characteristic
20	C₂×C₅×#107	C₂²	1	1	yes	characteristic
21	C₂×C₅×#107	C₂²	1	1	yes	characteristic
22	C₂×C₅×#106	—	2	1	no	—
23	C₂×C₅×#102	—	2	1	no	—
24	C₅×#367	C₂²	1	4	yes	complemented
25	C₅×#366	C₂²	1	4	yes	complemented
26	C₅×#357	C₂²	1	4	yes	complemented
27	C₅×#357	C₂²	1	4	yes	complemented
28	C₅×C₂²⋊C₁₆	C₂²	1	4	yes	complemented
29	C₂×#2234	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂²×C₅×D₄	—	2	1	no	—
31	C₂²×C₅×D₄	—	4	1	no	—
32	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
33	C₂×C₅×D₈	—	2	1	no	—
34	C₂×C₅×D₈	—	2	2	no	—
35	C₂×C₅×D₈	—	2	2	no	—
36	C₂×C₅×D₈	C₂³	1	2	yes	—
37	C₂²×C₅×C₈	—	2	2	no	—
38	C₂²×C₅×C₈	—	2	2	no	—
39	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
40	C₂²×C₅×C₈	D₄	1	2	yes	—
41	C₂²×C₅×C₈	C₂³	1	2	yes	—
42	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
43	C₅×#121	—	2	4	no	—
44	C₅×#121	—	4	2	no	—
45	C₅×#117	—	2	4	no	—
46	C₅×#117	—	4	2	no	—
47	C₂×C₅×C₄⋊C₄	—	2	1	no	—
48	C₂×C₅×C₄⋊C₄	—	2	1	no	—
49	C₂×C₅×C₄⋊C₄	—	2	1	no	—
50	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
51	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
52	C₂×C₄²×C₅	—	4	1	no	—
53	C₂×C₅×C₁₆	—	2	2	no	—
54	C₂×C₅×C₁₆	—	2	2	no	—
55	C₂×C₅×C₁₆	C₂³	1	2	yes	—
56	C₂×C₅×C₁₆	C₂³	1	2	yes	—
57	C₅×#107	—	2	1	no	—
58	C₅×#107	—	2	1	no	—
59	C₅×#107	—	2	1	no	—
60	C₅×#107	C₂³	1	2	yes	—
61	C₅×#107	C₂³	1	2	yes	—
62	C₅×#107	C₂³	1	2	yes	—
63	C₅×#106	—	2	4	no	—
64	C₅×#102	—	2	4	no	—
65	C₂×#466	C₂×C₅	1	1	yes	characteristic
66	C₂×#427	C₂×C₅	1	1	yes	characteristic, complemented
67	#2234	C₂×C₅	1	8	yes	direct product factor
68	C₂×#367	C₂×C₅	1	1	yes	characteristic, complemented
69	C₂×#366	C₂×C₅	1	1	yes	characteristic, complemented
70	C₂×#357	C₂×C₅	1	1	yes	characteristic, complemented
71	C₂×#357	C₂×C₅	1	1	yes	characteristic, complemented
72	C₂×C₂²⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
73	C₂⁴×C₅	—	4	1	no	—
74	C₂⁴×C₅	D₈	1	1	yes	characteristic
75	C₂×C₅×D₄	—	2	2	no	—
76	C₂×C₅×D₄	—	4	4	no	—
77	C₂×C₅×D₄	—	4	2	no	—
78	C₂×C₅×D₄	—	4	1	no	—
79	C₂×C₅×D₄	—	4	2	no	—
80	C₂×C₅×D₄	—	8	4	no	—
81	C₂²×C₄×C₅	—	2	2	no	—
82	C₂²×C₄×C₅	—	2	2	no	—
83	C₂²×C₄×C₅	—	4	1	no	—
84	C₂²×C₄×C₅	—	4	1	no	—
85	C₂²×C₄×C₅	—	4	1	no	—
86	C₂²×C₄×C₅	D₈	1	2	yes	—
87	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
88	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
90	C₅×D₈	—	2	2	no	—
91	C₅×D₈	—	2	2	no	—
92	C₅×D₈	—	4	2	no	—
93	C₂×C₅×C₈	—	2	2	no	—
94	C₂×C₅×C₈	—	2	4	no	—
95	C₂×C₅×C₈	—	2	2	no	—
96	C₂×C₅×C₈	—	2	2	no	—
97	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
98	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
99	C₂×C₅×C₈	#41	1	2	yes	—
100	C₂×C₅×C₈	#41	1	2	yes	—
101	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
102	C₅×C₄⋊C₄	—	2	2	no	—
103	C₅×C₄⋊C₄	—	2	2	no	—
104	C₅×C₄⋊C₄	—	2	2	no	—
105	C₅×C₄⋊C₄	—	4	1	no	—
106	C₅×C₄⋊C₄	—	4	1	no	—
107	C₅×C₄⋊C₄	—	4	1	no	—
108	C₅×C₂²⋊C₄	—	4	4	no	—
109	C₅×C₂²⋊C₄	—	4	4	no	—
110	C₄²×C₅	—	4	4	no	—
111	C₅×C₁₆	—	2	2	no	—
112	C₅×C₁₆	—	2	2	no	—
113	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
114	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
115	C₂×#121	—	2	1	no	—
116	C₂×#117	—	2	1	no	—
117	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
118	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
119	#466	—	2	2	no	—
120	#466	C₂²×C₅	1	4	yes	—
121	#427	—	2	2	no	—
122	#427	C₂²×C₅	1	4	yes	complemented
123	C₂×#107	C₂²×C₅	1	1	yes	characteristic
124	C₂×#107	C₂²×C₅	1	1	yes	characteristic
125	C₂×#107	C₂²×C₅	1	1	yes	characteristic
126	C₂×#106	—	2	1	no	—
127	C₂×#102	—	2	1	no	—
128	#367	C₂²×C₅	1	4	yes	complemented
129	#366	C₂²×C₅	1	4	yes	complemented
130	#357	C₂²×C₅	1	4	yes	complemented
131	#357	C₂²×C₅	1	4	yes	complemented
132	C₂²⋊C₁₆	C₂²×C₅	1	4	yes	complemented
133	C₂³×C₅	—	2	2	no	—
134	C₂³×C₅	—	2	2	no	—
135	C₂³×C₅	—	2	2	no	—
136	C₂³×C₅	—	4	2	no	—
137	C₂³×C₅	—	8	2	no	—
138	C₂³×C₅	—	8	2	no	—
139	C₂³×C₅	—	8	2	no	—
140	C₂³×C₅	C₂×D₈	1	2	yes	—
141	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
142	C₅×D₄	—	4	2	no	—
143	C₅×D₄	—	4	2	no	—
144	C₅×D₄	—	8	8	no	—
145	C₅×D₄	—	8	2	no	—
146	C₂×C₄×C₅	—	2	2	no	—
147	C₂×C₄×C₅	—	2	2	no	—
148	C₂×C₄×C₅	—	2	2	no	—
149	C₂×C₄×C₅	—	2	4	no	—
150	C₂×C₄×C₅	—	4	2	no	—
151	C₂×C₄×C₅	—	4	2	no	—
152	C₂×C₄×C₅	—	4	2	no	—
153	C₂×C₄×C₅	—	8	2	no	—
154	C₂×C₄×C₅	—	8	2	no	—
155	C₂×C₄×C₅	—	8	2	no	—
156	C₂×C₄×C₅	#123	1	2	yes	—
157	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
158	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
159	C₂×C₄×C₅	#137	1	2	yes	—
160	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
161	C₅×C₈	—	2	2	no	—
162	C₅×C₈	#123	1	2	yes	—
163	C₅×C₈	C₂×#41	1	1	yes	characteristic
164	C₅×C₈	C₂×#41	1	1	yes	characteristic
165	C₂²×D₄	—	2	1	no	—
166	C₂²×D₄	—	4	1	no	—
167	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
168	C₂×D₈	—	2	1	no	—
169	C₂×D₈	—	2	2	no	—
170	C₂×D₈	—	2	2	no	—
171	C₂×D₈	C₂³×C₅	1	2	yes	—
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
175	C₂²×C₈	C₅×D₄	1	2	yes	—
176	C₂²×C₈	C₂³×C₅	1	2	yes	—
177	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
178	#121	—	2	4	no	—
179	#121	—	4	2	no	—
180	#117	—	2	4	no	—
181	#117	—	4	2	no	—
182	C₂×C₄⋊C₄	—	2	1	no	—
183	C₂×C₄⋊C₄	—	2	1	no	—
184	C₂×C₄⋊C₄	—	2	1	no	—
185	C₂×C₂²⋊C₄	—	4	1	no	—
186	C₂×C₂²⋊C₄	—	4	1	no	—
187	C₂×C₄²	—	4	1	no	—
188	C₂×C₁₆	—	2	2	no	—
189	C₂×C₁₆	—	2	2	no	—
190	C₂×C₁₆	C₂³×C₅	1	2	yes	—
191	C₂×C₁₆	C₂³×C₅	1	2	yes	—
192	#107	—	2	1	no	—
193	#107	—	2	1	no	—
194	#107	—	2	1	no	—
195	#107	C₂³×C₅	1	2	yes	—
196	#107	C₂³×C₅	1	2	yes	—
197	#107	C₂³×C₅	1	2	yes	—
198	#106	—	2	4	no	—
199	#102	—	2	4	no	—
200	C₂²×C₅	—	2	2	no	—
201	C₂²×C₅	—	2	2	no	—
202	C₂²×C₅	—	2	2	no	—
203	C₂²×C₅	—	4	4	no	—
204	C₂²×C₅	—	8	2	no	—
205	C₂²×C₅	—	8	2	no	—
206	C₂²×C₅	—	8	2	no	—
207	C₂²×C₅	—	16	4	no	—
208	C₂²×C₅	#480	1	2	yes	—
209	C₂²×C₅	#508	1	2	yes	—
210	C₂²×C₅	#509	1	2	yes	—
211	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
212	C₄×C₅	—	2	2	no	—
213	C₄×C₅	—	8	2	no	—
214	C₄×C₅	—	8	2	no	—
215	C₄×C₅	—	8	2	no	—
216	C₄×C₅	#480	1	2	yes	—
217	C₄×C₅	C₂×#123	1	1	yes	characteristic
218	C₄×C₅	C₂×#137	1	1	yes	characteristic
219	C₂⁴	—	4	1	no	—
220	C₂⁴	C₅×D₈	1	1	yes	characteristic
221	C₂×D₄	—	2	2	no	—
222	C₂×D₄	—	4	4	no	—
223	C₂×D₄	—	4	2	no	—
224	C₂×D₄	—	4	1	no	—
225	C₂×D₄	—	4	2	no	—
226	C₂×D₄	—	8	4	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	4	1	no	—
230	C₂²×C₄	—	4	1	no	—
231	C₂²×C₄	—	4	1	no	—
232	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
233	C₂²×C₄	C₅×D₈	1	2	yes	—
234	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
235	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
236	D₈	—	2	2	no	—
237	D₈	—	2	2	no	—
238	D₈	—	4	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
244	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
245	C₂×C₈	C₅×#41	1	2	yes	—
246	C₂×C₈	C₅×#41	1	2	yes	—
247	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	4	1	no	—
252	C₄⋊C₄	—	4	1	no	—
253	C₄⋊C₄	—	4	1	no	—
254	C₂²⋊C₄	—	4	4	no	—
255	C₂²⋊C₄	—	4	4	no	—
256	C₄²	—	4	4	no	—
257	C₁₆	—	2	2	no	—
258	C₁₆	—	2	2	no	—
259	C₂×C₅	—	4	2	no	—
260	C₂×C₅	—	16	2	no	—
261	C₂×C₅	#2234	1	4	yes	direct product factor
262	C₂×C₅	C₂×#480	1	1	yes	characteristic
263	C₂×C₅	C₂×#508	1	1	yes	characteristic
264	C₂×C₅	C₂×#509	1	1	yes	characteristic
265	C₂³	—	2	2	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	—	4	2	no	—
269	C₂³	—	8	2	no	—
270	C₂³	—	8	2	no	—
271	C₂³	—	8	2	no	—
272	C₂³	C₂×C₅×D₈	1	2	yes	—
273	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
274	D₄	—	4	2	no	—
275	D₄	—	4	2	no	—
276	D₄	—	8	8	no	—
277	D₄	—	8	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	8	2	no	—
286	C₂×C₄	—	8	2	no	—
287	C₂×C₄	—	8	2	no	—
288	C₂×C₄	C₅×#123	1	2	yes	—
289	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
290	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
291	C₂×C₄	C₅×#137	1	2	yes	—
292	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
293	C₈	—	2	2	no	—
294	C₈	C₅×#123	1	2	yes	—
295	C₈	C₂×C₅×#41	1	1	yes	characteristic
296	C₈	C₂×C₅×#41	1	1	yes	characteristic
297	C₅	C₂×#2234	1	1	yes	characteristic, direct product factor, Sylow
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	4	4	no	—
302	C₂²	—	8	2	no	—
303	C₂²	—	8	2	no	—
304	C₂²	—	8	2	no	—
305	C₂²	—	16	4	no	—
306	C₂²	C₅×#480	1	2	yes	—
307	C₂²	C₅×#508	1	2	yes	—
308	C₂²	C₅×#509	1	2	yes	—
309	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	8	2	no	—
312	C₄	—	8	2	no	—
313	C₄	—	8	2	no	—
314	C₄	C₅×#480	1	2	yes	—
315	C₄	C₂×C₅×#123	1	1	yes	characteristic
316	C₄	C₂×C₅×#137	1	1	yes	characteristic
317	C₂	—	4	2	no	—
318	C₂	—	16	2	no	—
319	C₂	C₅×#2234	1	4	yes	direct product factor
320	C₂	C₂×C₅×#480	1	1	yes	characteristic
321	C₂	C₂×C₅×#508	1	1	yes	characteristic
322	C₂	C₂×C₅×#509	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₂×C₅×#466	C₂
LCS	G ▷ C₂×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. C₂×C₅×#427	C₂	semidirect
3. C₅×#2234	C₂	direct (N × H)
4. C₂×C₅×#367	C₂	semidirect
5. C₂×C₅×#366	C₂	semidirect
6. C₂×C₅×#357	C₂	semidirect
7. C₂×C₅×#357	C₂	semidirect
8. C₂×C₅×C₂²⋊C₁₆	C₂	semidirect
18. C₅×#427	C₂²	semidirect
24. C₅×#367	C₂²	semidirect
25. C₅×#366	C₂²	semidirect
26. C₅×#357	C₂²	semidirect
27. C₅×#357	C₂²	semidirect
28. C₅×C₂²⋊C₁₆	C₂²	semidirect
29. C₂×#2234	C₅	direct (N × H)
66. C₂×#427	C₂×C₅	semidirect
67. #2234	C₂×C₅	direct (N × H)
68. C₂×#367	C₂×C₅	semidirect
69. C₂×#366	C₂×C₅	semidirect
70. C₂×#357	C₂×C₅	semidirect
71. C₂×#357	C₂×C₅	semidirect
72. C₂×C₂²⋊C₁₆	C₂×C₅	semidirect
122. #427	C₂²×C₅	semidirect
128. #367	C₂²×C₅	semidirect
129. #366	C₂²×C₅	semidirect
130. #357	C₂²×C₅	semidirect
131. #357	C₂²×C₅	semidirect
132. C₂²⋊C₁₆	C₂²×C₅	semidirect
261. C₂×C₅	#2234	direct (N × H)
297. C₅	C₂×#2234	direct (N × H)
319. C₂	C₅×#2234	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#466	2
C₂×C₅×#427	2
C₅×#2234	2
C₂×C₅×#367	2
C₂×C₅×#366	2
C₂×C₅×#357	2
C₂×C₅×C₂²⋊C₁₆	2
C₂×#2234	5

Subgroups — C2×C5×#2234 #11956361

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2234 #11956361