Subgroups — C₂×C₅×#2206 #11956293

Order 1280.

870 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×D₁₆	C₂	1	2	yes	maximal, complemented
2	C₂×C₅×#493	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×C₅×D₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2206	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×#357	C₂	1	2	yes	maximal, complemented
7	C₂²×C₅×D₈	—	2	1	no	—
8	C₂²×C₅×D₈	C₂²	1	2	yes	—
9	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
10	C₂×C₅×#127	—	2	1	no	—
11	C₂×C₄×C₅×D₄	—	2	1	no	—
12	C₂×C₅×D₁₆	—	2	8	no	—
13	C₂×C₅×D₁₆	C₂²	1	8	yes	complemented
14	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
15	C₅×#493	—	2	2	no	—
16	C₅×#493	C₂²	1	4	yes	complemented
17	C₄×C₅×D₈	—	2	2	no	—
18	C₄×C₅×D₈	C₂²	1	4	yes	complemented
19	C₂×C₅×#107	C₂²	1	1	yes	characteristic
20	C₂×C₅×#102	—	2	1	no	—
21	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
22	C₅×C₄⋊C₁₆	C₂²	1	4	yes	complemented
23	C₅×#357	C₂²	1	8	yes	complemented
24	C₂×#2206	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂²×C₅×D₄	—	2	1	no	—
26	C₂²×C₅×D₄	—	2	2	no	—
27	C₂²×C₅×D₄	—	4	1	no	—
28	C₂²×C₅×D₄	—	4	1	no	—
29	C₂³×C₄×C₅	—	4	1	no	—
30	C₂×C₅×D₈	—	2	1	no	—
31	C₂×C₅×D₈	—	2	4	no	—
32	C₂×C₅×D₈	—	2	2	no	—
33	C₂×C₅×D₈	—	2	4	no	—
34	C₂×C₅×D₈	—	2	2	no	—
35	C₂×C₅×D₈	—	2	4	no	—
36	C₂×C₅×D₈	—	4	4	no	—
37	C₂×C₅×D₈	D₄	1	4	yes	complemented
38	C₂×C₅×D₈	C₂³	1	4	yes	—
39	C₂×C₅×D₈	C₂³	1	2	yes	—
40	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
41	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
42	C₅×#127	—	2	4	no	—
43	C₅×#127	—	4	2	no	—
44	C₄×C₅×D₄	—	2	4	no	—
45	C₄×C₅×D₄	—	4	2	no	—
46	C₂×C₅×C₄⋊C₄	—	2	1	no	—
47	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
48	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
49	C₅×D₁₆	—	2	16	no	—
50	C₂×C₅×C₁₆	—	2	4	no	—
51	C₂×C₅×C₁₆	C₂³	1	4	yes	—
52	C₅×#107	—	2	1	no	—
53	C₅×#107	C₂³	1	2	yes	—
54	C₅×#102	—	2	4	no	—
55	C₄×C₅×C₈	D₄	1	2	yes	—
56	C₄×C₅×C₈	C₂³	1	2	yes	—
57	C₂²×D₁₆	C₂×C₅	1	2	yes	complemented
58	C₂×#493	C₂×C₅	1	1	yes	characteristic, complemented
59	C₂×C₄×D₈	C₂×C₅	1	1	yes	characteristic, complemented
60	#2206	C₂×C₅	1	8	yes	direct product factor
61	C₂×C₄⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
62	C₂×#357	C₂×C₅	1	2	yes	complemented
63	C₂⁴×C₅	—	4	1	no	—
64	C₂⁴×C₅	—	4	2	no	—
65	C₂×C₅×D₄	—	2	4	no	—
66	C₂×C₅×D₄	—	2	2	no	—
67	C₂×C₅×D₄	—	2	4	no	—
68	C₂×C₅×D₄	—	4	4	no	—
69	C₂×C₅×D₄	—	4	1	no	—
70	C₂×C₅×D₄	—	4	4	no	—
71	C₂×C₅×D₄	—	4	4	no	—
72	C₂×C₅×D₄	—	4	2	no	—
73	C₂×C₅×D₄	—	4	2	no	—
74	C₂×C₅×D₄	—	4	4	no	—
75	C₂×C₅×D₄	—	8	4	no	—
76	C₂×C₅×D₄	—	8	4	no	—
77	C₂²×C₄×C₅	—	4	1	no	—
78	C₂²×C₄×C₅	—	4	4	no	—
79	C₂²×C₄×C₅	—	8	4	no	—
80	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
81	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₅×D₈	—	2	4	no	—
84	C₅×D₈	—	2	4	no	—
85	C₅×D₈	—	2	2	no	—
86	C₅×D₈	—	2	4	no	—
87	C₅×D₈	—	4	8	no	—
88	C₅×D₈	—	4	4	no	—
89	C₅×D₈	C₂×D₄	1	4	yes	complemented
90	C₂×C₅×C₈	—	2	2	no	—
91	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
92	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
93	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
94	C₂×C₅×C₈	#41	1	2	yes	—
95	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
96	C₅×C₄⋊C₄	—	2	2	no	—
97	C₅×C₄⋊C₄	—	4	1	no	—
98	C₅×C₂²⋊C₄	—	4	4	no	—
99	C₄²×C₅	D₈	1	2	yes	—
100	C₄²×C₅	C₂×D₄	1	2	yes	—
101	C₅×C₁₆	—	2	4	no	—
102	C₂²×D₈	—	2	1	no	—
103	C₂²×D₈	C₂²×C₅	1	2	yes	—
104	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
105	C₂×#127	—	2	1	no	—
106	C₂×C₄×D₄	—	2	1	no	—
107	C₂×D₁₆	—	2	8	no	—
108	C₂×D₁₆	C₂²×C₅	1	8	yes	complemented
109	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
110	#493	—	2	2	no	—
111	#493	C₂²×C₅	1	4	yes	complemented
112	C₄×D₈	—	2	2	no	—
113	C₄×D₈	C₂²×C₅	1	4	yes	complemented
114	C₂×#107	C₂²×C₅	1	1	yes	characteristic
115	C₂×#102	—	2	1	no	—
116	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
117	C₄⋊C₁₆	C₂²×C₅	1	4	yes	complemented
118	#357	C₂²×C₅	1	8	yes	complemented
119	C₂³×C₅	—	4	2	no	—
120	C₂³×C₅	—	4	4	no	—
121	C₂³×C₅	—	4	4	no	—
122	C₂³×C₅	—	8	2	no	—
123	C₂³×C₅	—	8	2	no	—
124	C₂³×C₅	—	8	4	no	—
125	C₂³×C₅	—	8	4	no	—
126	C₂³×C₅	—	8	4	no	—
127	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
128	C₅×D₄	—	2	4	no	—
129	C₅×D₄	—	4	4	no	—
130	C₅×D₄	—	4	2	no	—
131	C₅×D₄	—	4	4	no	—
132	C₅×D₄	—	4	4	no	—
133	C₅×D₄	—	8	8	no	—
134	C₅×D₄	—	8	4	no	—
135	C₅×D₄	—	8	8	no	—
136	C₂×C₄×C₅	—	2	2	no	—
137	C₂×C₄×C₅	—	4	2	no	—
138	C₂×C₄×C₅	—	8	2	no	—
139	C₂×C₄×C₅	—	8	8	no	—
140	C₂×C₄×C₅	D₁₆	1	4	yes	complemented
141	C₂×C₄×C₅	#121	1	2	yes	—
142	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
143	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
145	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
146	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
147	C₅×C₈	—	2	2	no	—
148	C₅×C₈	#121	1	2	yes	—
149	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
150	C₅×C₈	C₂×#41	1	1	yes	characteristic
151	C₂²×D₄	—	2	1	no	—
152	C₂²×D₄	—	2	2	no	—
153	C₂²×D₄	—	4	1	no	—
154	C₂²×D₄	—	4	1	no	—
155	C₂³×C₄	—	4	1	no	—
156	C₂×D₈	—	2	1	no	—
157	C₂×D₈	—	2	4	no	—
158	C₂×D₈	—	2	4	no	—
159	C₂×D₈	—	2	2	no	—
160	C₂×D₈	—	2	2	no	—
161	C₂×D₈	—	2	4	no	—
162	C₂×D₈	—	4	4	no	—
163	C₂×D₈	C₅×D₄	1	4	yes	complemented
164	C₂×D₈	C₂³×C₅	1	4	yes	—
165	C₂×D₈	C₂³×C₅	1	2	yes	—
166	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
167	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
168	#127	—	2	4	no	—
169	#127	—	4	2	no	—
170	C₄×D₄	—	2	4	no	—
171	C₄×D₄	—	4	2	no	—
172	C₂×C₄⋊C₄	—	2	1	no	—
173	C₂×C₂²⋊C₄	—	4	1	no	—
174	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
175	D₁₆	—	2	16	no	—
176	C₂×C₁₆	—	2	4	no	—
177	C₂×C₁₆	C₂³×C₅	1	4	yes	—
178	#107	—	2	1	no	—
179	#107	C₂³×C₅	1	2	yes	—
180	#102	—	2	4	no	—
181	C₄×C₈	C₅×D₄	1	2	yes	—
182	C₄×C₈	C₂³×C₅	1	2	yes	—
183	C₂²×C₅	—	4	4	no	—
184	C₂²×C₅	—	8	2	no	—
185	C₂²×C₅	—	8	2	no	—
186	C₂²×C₅	—	8	8	no	—
187	C₂²×C₅	—	8	4	no	—
188	C₂²×C₅	—	8	4	no	—
189	C₂²×C₅	—	8	4	no	—
190	C₂²×C₅	—	16	8	no	—
191	C₂²×C₅	#459	1	2	yes	—
192	C₂²×C₅	C₂×D₁₆	1	2	yes	—
193	C₂²×C₅	#509	1	2	yes	—
194	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
195	C₄×C₅	—	2	2	no	—
196	C₄×C₅	—	8	2	no	—
197	C₄×C₅	#459	1	2	yes	—
198	C₄×C₅	C₂×D₁₆	1	4	yes	complemented
199	C₄×C₅	C₂×#121	1	1	yes	characteristic
200	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
201	C₂⁴	—	4	1	no	—
202	C₂⁴	—	4	2	no	—
203	C₂×D₄	—	2	4	no	—
204	C₂×D₄	—	2	2	no	—
205	C₂×D₄	—	2	4	no	—
206	C₂×D₄	—	4	4	no	—
207	C₂×D₄	—	4	1	no	—
208	C₂×D₄	—	4	4	no	—
209	C₂×D₄	—	4	4	no	—
210	C₂×D₄	—	4	2	no	—
211	C₂×D₄	—	4	2	no	—
212	C₂×D₄	—	4	4	no	—
213	C₂×D₄	—	8	4	no	—
214	C₂×D₄	—	8	4	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	—	4	4	no	—
217	C₂²×C₄	—	8	4	no	—
218	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
219	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
221	D₈	—	2	4	no	—
222	D₈	—	2	4	no	—
223	D₈	—	2	2	no	—
224	D₈	—	2	4	no	—
225	D₈	—	4	8	no	—
226	D₈	—	4	4	no	—
227	D₈	C₂×C₅×D₄	1	4	yes	complemented
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
230	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
231	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
232	C₂×C₈	C₅×#41	1	2	yes	—
233	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	4	1	no	—
236	C₂²⋊C₄	—	4	4	no	—
237	C₄²	C₅×D₈	1	2	yes	—
238	C₄²	C₂×C₅×D₄	1	2	yes	—
239	C₁₆	—	2	4	no	—
240	C₂×C₅	—	8	4	no	—
241	C₂×C₅	—	16	4	no	—
242	C₂×C₅	#2206	1	4	yes	direct product factor
243	C₂×C₅	C₂×#459	1	1	yes	characteristic
244	C₂×C₅	C₂²×D₁₆	1	1	yes	characteristic
245	C₂×C₅	C₂×#509	1	1	yes	characteristic
246	C₂³	—	4	4	no	—
247	C₂³	—	4	2	no	—
248	C₂³	—	4	4	no	—
249	C₂³	—	8	2	no	—
250	C₂³	—	8	2	no	—
251	C₂³	—	8	4	no	—
252	C₂³	—	8	4	no	—
253	C₂³	—	8	4	no	—
254	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
255	D₄	—	2	4	no	—
256	D₄	—	4	4	no	—
257	D₄	—	4	4	no	—
258	D₄	—	4	2	no	—
259	D₄	—	4	4	no	—
260	D₄	—	8	8	no	—
261	D₄	—	8	4	no	—
262	D₄	—	8	8	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	8	2	no	—
266	C₂×C₄	—	8	8	no	—
267	C₂×C₄	C₅×D₁₆	1	4	yes	complemented
268	C₂×C₄	C₅×#121	1	2	yes	—
269	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
270	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
271	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
272	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
273	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
274	C₈	—	2	2	no	—
275	C₈	C₅×#121	1	2	yes	—
276	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
277	C₈	C₂×C₅×#41	1	1	yes	characteristic
278	C₅	C₂×#2206	1	1	yes	characteristic, direct product factor, Sylow
279	C₂²	—	4	4	no	—
280	C₂²	—	8	2	no	—
281	C₂²	—	8	2	no	—
282	C₂²	—	8	8	no	—
283	C₂²	—	8	4	no	—
284	C₂²	—	8	4	no	—
285	C₂²	—	8	4	no	—
286	C₂²	—	16	8	no	—
287	C₂²	C₅×#459	1	2	yes	—
288	C₂²	C₂×C₅×D₁₆	1	2	yes	—
289	C₂²	C₅×#509	1	2	yes	—
290	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
291	C₄	—	2	2	no	—
292	C₄	—	8	2	no	—
293	C₄	C₅×#459	1	2	yes	—
294	C₄	C₂×C₅×D₁₆	1	4	yes	complemented
295	C₄	C₂×C₅×#121	1	1	yes	characteristic
296	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
297	C₂	—	8	4	no	—
298	C₂	—	16	4	no	—
299	C₂	C₅×#2206	1	4	yes	direct product factor
300	C₂	C₂×C₅×#459	1	1	yes	characteristic
301	C₂	C₂²×C₅×D₁₆	1	1	yes	characteristic
302	C₂	C₂×C₅×#509	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×D₁₆	C₂	semidirect
2. C₂×C₅×#493	C₂	semidirect
3. C₂×C₄×C₅×D₈	C₂	semidirect
4. C₅×#2206	C₂	direct (N × H)
5. C₂×C₅×C₄⋊C₁₆	C₂	semidirect
6. C₂×C₅×#357	C₂	semidirect
13. C₂×C₅×D₁₆	C₂²	semidirect
16. C₅×#493	C₂²	semidirect
18. C₄×C₅×D₈	C₂²	semidirect
22. C₅×C₄⋊C₁₆	C₂²	semidirect
23. C₅×#357	C₂²	semidirect
24. C₂×#2206	C₅	direct (N × H)
37. C₂×C₅×D₈	D₄	semidirect
57. C₂²×D₁₆	C₂×C₅	semidirect
58. C₂×#493	C₂×C₅	semidirect
59. C₂×C₄×D₈	C₂×C₅	semidirect
60. #2206	C₂×C₅	direct (N × H)
61. C₂×C₄⋊C₁₆	C₂×C₅	semidirect
62. C₂×#357	C₂×C₅	semidirect
89. C₅×D₈	C₂×D₄	semidirect
108. C₂×D₁₆	C₂²×C₅	semidirect
111. #493	C₂²×C₅	semidirect
113. C₄×D₈	C₂²×C₅	semidirect
117. C₄⋊C₁₆	C₂²×C₅	semidirect
118. #357	C₂²×C₅	semidirect
140. C₂×C₄×C₅	D₁₆	semidirect
163. C₂×D₈	C₅×D₄	semidirect
198. C₄×C₅	C₂×D₁₆	semidirect
227. D₈	C₂×C₅×D₄	semidirect
242. C₂×C₅	#2206	direct (N × H)
267. C₂×C₄	C₅×D₁₆	semidirect
278. C₅	C₂×#2206	direct (N × H)
294. C₄	C₂×C₅×D₁₆	semidirect
299. C₂	C₅×#2206	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×D₁₆	2
C₂×C₅×#493	2
C₂×C₄×C₅×D₈	2
C₅×#2206	2
C₂×C₅×C₄⋊C₁₆	2
C₂×C₅×#357	2
C₂×#2206	5

Subgroups — C2×C5×#2206 #11956293

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2206 #11956293