Subgroups — C₂×C₅×#2213 #11956312

Order 1280.

790 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#509	C₂	1	2	yes	maximal, complemented
2	C₂×C₅×#493	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#443	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2213	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅×#364	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×#361	C₂	1	2	yes	maximal, complemented
7	C₂×C₅×#135	C₂²	1	1	yes	characteristic
8	C₂²×C₅×D₈	—	2	1	no	—
9	C₂²×C₅×D₈	C₂²	1	2	yes	—
10	C₂×C₅×#131	—	2	1	no	—
11	C₂×C₅×#127	—	2	1	no	—
12	C₅×#509	—	2	4	no	—
13	C₅×#509	C₂²	1	8	yes	complemented
14	C₂×C₅×SD₃₂	—	2	2	no	—
15	C₂×C₅×D₁₆	—	2	2	no	—
16	C₂×C₅×M₃₂	C₂²	1	2	yes	—
17	C₅×#493	—	2	2	no	—
18	C₅×#493	C₂²	1	4	yes	complemented
19	C₅×#443	—	2	2	no	—
20	C₅×#443	C₂²	1	4	yes	complemented
21	C₂×C₅×#108	C₂²	1	1	yes	characteristic
22	C₂×C₅×#104	—	2	1	no	—
23	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
24	C₅×#364	C₂²	1	4	yes	complemented
25	C₅×#361	C₂²	1	8	yes	complemented
26	C₂×#2213	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂×C₅×#41	—	2	1	no	—
28	C₂²×C₅×D₄	—	2	2	no	—
29	C₂²×C₅×D₄	—	4	2	no	—
30	C₅×#135	—	2	1	no	—
31	C₅×#135	—	2	1	no	—
32	C₅×#135	D₄	1	2	yes	complemented
33	C₅×#135	C₂³	1	2	yes	—
34	C₂×C₅×Q₁₆	D₄	1	1	yes	characteristic, complemented
35	C₂×C₅×SD₁₆	—	2	1	no	—
36	C₂×C₅×D₈	—	2	2	no	—
37	C₂×C₅×D₈	—	2	2	no	—
38	C₂×C₅×D₈	—	2	2	no	—
39	C₂×C₅×D₈	—	2	4	no	—
40	C₂×C₅×D₈	—	2	2	no	—
41	C₂×C₅×D₈	—	2	2	no	—
42	C₂×C₅×D₈	—	4	1	no	—
43	C₂×C₅×D₈	—	4	1	no	—
44	C₂×C₅×D₈	—	4	2	no	—
45	C₂×C₅×D₈	D₄	1	1	yes	characteristic, complemented
46	C₂×C₅×D₈	C₂³	1	4	yes	—
47	C₅×#131	—	2	4	no	—
48	C₅×#131	—	4	2	no	—
49	C₂×C₅×M₁₆	—	2	1	no	—
50	C₂×C₅×M₁₆	—	4	1	no	—
51	C₂²×C₅×C₈	—	4	1	no	—
52	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
53	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
54	C₅×#127	—	2	4	no	—
55	C₅×#127	—	4	2	no	—
56	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
57	C₅×SD₃₂	—	2	8	no	—
58	C₅×D₁₆	—	2	8	no	—
59	C₅×M₃₂	—	2	2	no	—
60	C₅×M₃₂	C₂³	1	4	yes	—
61	C₂×C₅×C₁₆	—	2	2	no	—
62	C₅×#108	—	2	1	no	—
63	C₅×#108	C₂³	1	2	yes	—
64	C₅×#104	—	2	4	no	—
65	C₄×C₅×C₈	D₄	1	2	yes	—
66	C₄×C₅×C₈	C₂³	1	2	yes	—
67	C₂×#509	C₂×C₅	1	2	yes	complemented
68	C₂×#493	C₂×C₅	1	1	yes	characteristic, complemented
69	C₂×#443	C₂×C₅	1	1	yes	characteristic, complemented
70	#2213	C₂×C₅	1	8	yes	direct product factor
71	C₂×#364	C₂×C₅	1	1	yes	characteristic, complemented
72	C₂×#361	C₂×C₅	1	2	yes	complemented
73	C₂⁴×C₅	—	4	2	no	—
74	C₅×#41	—	2	2	no	—
75	C₅×#41	—	2	2	no	—
76	C₅×#41	—	4	1	no	—
77	C₅×#41	—	4	1	no	—
78	C₂×C₅×Q₈	—	2	1	no	—
79	C₂×C₅×D₄	—	2	1	no	—
80	C₂×C₅×D₄	—	2	4	no	—
81	C₂×C₅×D₄	—	4	8	no	—
82	C₂×C₅×D₄	—	4	1	no	—
83	C₂×C₅×D₄	—	4	2	no	—
84	C₂×C₅×D₄	—	4	2	no	—
85	C₂×C₅×D₄	—	4	2	no	—
86	C₂×C₅×D₄	—	4	2	no	—
87	C₂×C₅×D₄	—	4	2	no	—
88	C₂×C₅×D₄	—	8	8	no	—
89	C₂²×C₄×C₅	—	2	1	no	—
90	C₂²×C₄×C₅	—	4	1	no	—
91	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
92	C₅×Q₁₆	—	2	1	no	—
93	C₅×Q₁₆	C₂×D₄	1	2	yes	complemented
94	C₅×SD₁₆	—	2	2	no	—
95	C₅×SD₁₆	—	2	2	no	—
96	C₅×D₈	—	2	1	no	—
97	C₅×D₈	—	2	4	no	—
98	C₅×D₈	—	2	4	no	—
99	C₅×D₈	—	4	2	no	—
100	C₅×D₈	—	4	2	no	—
101	C₅×D₈	—	4	4	no	—
102	C₅×D₈	—	4	4	no	—
103	C₅×D₈	C₂×D₄	1	2	yes	complemented
104	C₅×M₁₆	—	2	2	no	—
105	C₅×M₁₆	—	4	1	no	—
106	C₅×M₁₆	—	4	4	no	—
107	C₂×C₅×C₈	—	4	1	no	—
108	C₂×C₅×C₈	—	4	4	no	—
109	C₂×C₅×C₈	D₈	1	1	yes	characteristic
110	C₂×C₅×C₈	D₈	1	1	yes	characteristic
111	C₂×C₅×C₈	D₈	1	2	yes	—
112	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
113	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
114	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
115	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
116	C₂×C₅×C₈	#41	1	1	yes	characteristic
117	C₂×C₅×C₈	#41	1	1	yes	characteristic
118	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
119	C₄²×C₅	—	2	1	no	—
120	C₄²×C₅	C₂×D₄	1	2	yes	—
121	C₅×C₁₆	—	2	4	no	—
122	C₂×#135	C₂²×C₅	1	1	yes	characteristic
123	C₂²×D₈	—	2	1	no	—
124	C₂²×D₈	C₂²×C₅	1	2	yes	—
125	C₂×#131	—	2	1	no	—
126	C₂×#127	—	2	1	no	—
127	#509	—	2	4	no	—
128	#509	C₂²×C₅	1	8	yes	complemented
129	C₂×SD₃₂	—	2	2	no	—
130	C₂×D₁₆	—	2	2	no	—
131	C₂×M₃₂	C₂²×C₅	1	2	yes	—
132	#493	—	2	2	no	—
133	#493	C₂²×C₅	1	4	yes	complemented
134	#443	—	2	2	no	—
135	#443	C₂²×C₅	1	4	yes	complemented
136	C₂×#108	C₂²×C₅	1	1	yes	characteristic
137	C₂×#104	—	2	1	no	—
138	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
139	#364	C₂²×C₅	1	4	yes	complemented
140	#361	C₂²×C₅	1	8	yes	complemented
141	C₂³×C₅	—	4	1	no	—
142	C₂³×C₅	—	4	4	no	—
143	C₂³×C₅	—	8	8	no	—
144	C₂³×C₅	—	8	2	no	—
145	C₂³×C₅	—	8	2	no	—
146	C₂³×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
147	C₅×Q₈	—	2	2	no	—
148	C₅×Q₈	—	4	1	no	—
149	C₅×D₄	—	2	2	no	—
150	C₅×D₄	—	4	2	no	—
151	C₅×D₄	—	4	2	no	—
152	C₅×D₄	—	4	4	no	—
153	C₅×D₄	—	4	4	no	—
154	C₅×D₄	—	4	1	no	—
155	C₅×D₄	—	8	16	no	—
156	C₅×D₄	—	8	2	no	—
157	C₅×D₄	—	8	2	no	—
158	C₂×C₄×C₅	—	2	4	no	—
159	C₂×C₄×C₅	—	2	2	no	—
160	C₂×C₄×C₅	—	4	1	no	—
161	C₂×C₄×C₅	—	4	2	no	—
162	C₂×C₄×C₅	—	4	2	no	—
163	C₂×C₄×C₅	#121	1	1	yes	characteristic
164	C₂×C₄×C₅	#121	1	1	yes	characteristic
165	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
166	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
167	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
168	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
169	C₅×C₈	—	4	2	no	—
170	C₅×C₈	#121	1	1	yes	characteristic
171	C₅×C₈	#121	1	1	yes	characteristic
172	C₅×C₈	C₂×D₈	1	2	yes	—
173	C₅×C₈	C₂×D₈	1	2	yes	—
174	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
175	C₅×C₈	C₂×#41	1	1	yes	characteristic
176	C₂×#41	—	2	1	no	—
177	C₂²×D₄	—	2	2	no	—
178	C₂²×D₄	—	4	2	no	—
179	#135	—	2	1	no	—
180	#135	—	2	1	no	—
181	#135	C₅×D₄	1	2	yes	complemented
182	#135	C₂³×C₅	1	2	yes	—
183	C₂×Q₁₆	C₅×D₄	1	1	yes	characteristic, complemented
184	C₂×SD₁₆	—	2	1	no	—
185	C₂×D₈	—	2	2	no	—
186	C₂×D₈	—	2	2	no	—
187	C₂×D₈	—	2	4	no	—
188	C₂×D₈	—	2	2	no	—
189	C₂×D₈	—	2	2	no	—
190	C₂×D₈	—	2	2	no	—
191	C₂×D₈	—	4	1	no	—
192	C₂×D₈	—	4	1	no	—
193	C₂×D₈	—	4	2	no	—
194	C₂×D₈	C₅×D₄	1	1	yes	characteristic, complemented
195	C₂×D₈	C₂³×C₅	1	4	yes	—
196	#131	—	2	4	no	—
197	#131	—	4	2	no	—
198	C₂×M₁₆	—	2	1	no	—
199	C₂×M₁₆	—	4	1	no	—
200	C₂²×C₈	—	4	1	no	—
201	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
202	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
203	#127	—	2	4	no	—
204	#127	—	4	2	no	—
205	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
206	SD₃₂	—	2	8	no	—
207	D₁₆	—	2	8	no	—
208	M₃₂	—	2	2	no	—
209	M₃₂	C₂³×C₅	1	4	yes	—
210	C₂×C₁₆	—	2	2	no	—
211	#108	—	2	1	no	—
212	#108	C₂³×C₅	1	2	yes	—
213	#104	—	2	4	no	—
214	C₄×C₈	C₅×D₄	1	2	yes	—
215	C₄×C₈	C₂³×C₅	1	2	yes	—
216	C₂²×C₅	—	2	2	no	—
217	C₂²×C₅	—	4	2	no	—
218	C₂²×C₅	—	8	8	no	—
219	C₂²×C₅	—	8	2	no	—
220	C₂²×C₅	—	8	4	no	—
221	C₂²×C₅	—	16	4	no	—
222	C₂²×C₅	—	16	4	no	—
223	C₂²×C₅	#459	1	1	yes	characteristic
224	C₂²×C₅	#459	1	1	yes	characteristic
225	C₂²×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
226	C₄×C₅	—	2	4	no	—
227	C₄×C₅	—	4	2	no	—
228	C₄×C₅	#459	1	1	yes	characteristic
229	C₄×C₅	#459	1	1	yes	characteristic
230	C₄×C₅	C₂×#121	1	1	yes	characteristic
231	C₄×C₅	C₂²×D₈	1	1	yes	characteristic
232	C₂⁴	—	4	2	no	—
233	#41	—	2	2	no	—
234	#41	—	2	2	no	—
235	#41	—	4	1	no	—
236	#41	—	4	1	no	—
237	C₂×Q₈	—	2	1	no	—
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	2	4	no	—
240	C₂×D₄	—	4	8	no	—
241	C₂×D₄	—	4	1	no	—
242	C₂×D₄	—	4	2	no	—
243	C₂×D₄	—	4	2	no	—
244	C₂×D₄	—	4	2	no	—
245	C₂×D₄	—	4	2	no	—
246	C₂×D₄	—	4	2	no	—
247	C₂×D₄	—	8	8	no	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
251	Q₁₆	—	2	1	no	—
252	Q₁₆	C₂×C₅×D₄	1	2	yes	complemented
253	SD₁₆	—	2	2	no	—
254	SD₁₆	—	2	2	no	—
255	D₈	—	2	1	no	—
256	D₈	—	2	4	no	—
257	D₈	—	2	4	no	—
258	D₈	—	4	2	no	—
259	D₈	—	4	2	no	—
260	D₈	—	4	4	no	—
261	D₈	—	4	4	no	—
262	D₈	C₂×C₅×D₄	1	2	yes	complemented
263	M₁₆	—	2	2	no	—
264	M₁₆	—	4	1	no	—
265	M₁₆	—	4	4	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	—	4	4	no	—
268	C₂×C₈	C₅×D₈	1	1	yes	characteristic
269	C₂×C₈	C₅×D₈	1	1	yes	characteristic
270	C₂×C₈	C₅×D₈	1	2	yes	—
271	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
272	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
273	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
274	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
275	C₂×C₈	C₅×#41	1	1	yes	characteristic
276	C₂×C₈	C₅×#41	1	1	yes	characteristic
277	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
278	C₄²	—	2	1	no	—
279	C₄²	C₂×C₅×D₄	1	2	yes	—
280	C₁₆	—	2	4	no	—
281	C₂×C₅	—	2	1	no	—
282	C₂×C₅	—	2	1	no	—
283	C₂×C₅	—	8	2	no	—
284	C₂×C₅	—	16	4	no	—
285	C₂×C₅	#2213	1	2	yes	direct product factor
286	C₂×C₅	C₂×#459	1	1	yes	characteristic
287	C₂³	—	4	1	no	—
288	C₂³	—	4	4	no	—
289	C₂³	—	8	8	no	—
290	C₂³	—	8	2	no	—
291	C₂³	—	8	2	no	—
292	C₂³	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
293	Q₈	—	2	2	no	—
294	Q₈	—	4	1	no	—
295	D₄	—	2	2	no	—
296	D₄	—	4	2	no	—
297	D₄	—	4	2	no	—
298	D₄	—	4	1	no	—
299	D₄	—	4	4	no	—
300	D₄	—	4	4	no	—
301	D₄	—	8	16	no	—
302	D₄	—	8	2	no	—
303	D₄	—	8	2	no	—
304	C₂×C₄	—	2	4	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	C₅×#121	1	1	yes	characteristic
310	C₂×C₄	C₅×#121	1	1	yes	characteristic
311	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
312	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
313	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
314	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
315	C₈	—	4	2	no	—
316	C₈	C₅×#121	1	1	yes	characteristic
317	C₈	C₅×#121	1	1	yes	characteristic
318	C₈	C₂×C₅×D₈	1	2	yes	—
319	C₈	C₂×C₅×D₈	1	2	yes	—
320	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
321	C₈	C₂×C₅×#41	1	1	yes	characteristic
322	C₅	C₂×#2213	1	1	yes	characteristic, direct product factor, Sylow
323	C₂²	—	2	2	no	—
324	C₂²	—	4	2	no	—
325	C₂²	—	8	8	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	8	4	no	—
328	C₂²	—	16	4	no	—
329	C₂²	—	16	4	no	—
330	C₂²	C₅×#459	1	1	yes	characteristic
331	C₂²	C₅×#459	1	1	yes	characteristic
332	C₂²	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
333	C₄	—	2	4	no	—
334	C₄	—	4	2	no	—
335	C₄	C₅×#459	1	1	yes	characteristic
336	C₄	C₅×#459	1	1	yes	characteristic
337	C₄	C₂×C₅×#121	1	1	yes	characteristic
338	C₄	C₂²×C₅×D₈	1	1	yes	characteristic
339	C₂	—	2	1	no	—
340	C₂	—	2	1	no	—
341	C₂	—	8	2	no	—
342	C₂	—	16	4	no	—
343	C₂	C₅×#2213	1	2	yes	direct product factor
344	C₂	C₂×C₅×#459	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	#459
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#509	C₂	semidirect
2. C₂×C₅×#493	C₂	semidirect
3. C₂×C₅×#443	C₂	semidirect
4. C₅×#2213	C₂	direct (N × H)
5. C₂×C₅×#364	C₂	semidirect
6. C₂×C₅×#361	C₂	semidirect
13. C₅×#509	C₂²	semidirect
18. C₅×#493	C₂²	semidirect
20. C₅×#443	C₂²	semidirect
24. C₅×#364	C₂²	semidirect
25. C₅×#361	C₂²	semidirect
26. C₂×#2213	C₅	direct (N × H)
32. C₅×#135	D₄	semidirect
34. C₂×C₅×Q₁₆	D₄	semidirect
45. C₂×C₅×D₈	D₄	semidirect
67. C₂×#509	C₂×C₅	semidirect
68. C₂×#493	C₂×C₅	semidirect
69. C₂×#443	C₂×C₅	semidirect
70. #2213	C₂×C₅	direct (N × H)
71. C₂×#364	C₂×C₅	semidirect
72. C₂×#361	C₂×C₅	semidirect
93. C₅×Q₁₆	C₂×D₄	semidirect
103. C₅×D₈	C₂×D₄	semidirect
128. #509	C₂²×C₅	semidirect
133. #493	C₂²×C₅	semidirect
135. #443	C₂²×C₅	semidirect
139. #364	C₂²×C₅	semidirect
140. #361	C₂²×C₅	semidirect
181. #135	C₅×D₄	semidirect
183. C₂×Q₁₆	C₅×D₄	semidirect
194. C₂×D₈	C₅×D₄	semidirect
252. Q₁₆	C₂×C₅×D₄	semidirect
262. D₈	C₂×C₅×D₄	semidirect
285. C₂×C₅	#2213	direct (N × H)
322. C₅	C₂×#2213	direct (N × H)
343. C₂	C₅×#2213	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×#509	2
C₂×C₅×#493	2
C₂×C₅×#443	2
C₅×#2213	2
C₂×C₅×#364	2
C₂×C₅×#361	2
C₂×#2213	5

Subgroups — C2×C5×#2213 #11956312

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2213 #11956312