Subgroups — C₂×C₅×#2221 #11956334

Order 1280.

678 conjugacy classes of subgroups.

The subgroup lattice has 310 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	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#504	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#472	C₂	1	2	yes	maximal, complemented
4	C₅×#2221	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅×#368	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×#361	C₂	1	2	yes	maximal, complemented
7	C₂×C₅×AGL(1,ℤ/8)	—	2	2	no	—
8	C₂²×C₅×D₈	C₂²	1	2	yes	—
9	C₂×C₅×#131	C₂²	1	1	yes	characteristic
10	C₂×C₅×D₁₆	—	2	1	no	—
11	C₂×C₅×D₁₆	—	2	1	no	—
12	C₂×C₅×D₁₆	—	2	2	no	—
13	C₂×C₅×D₁₆	C₂²	1	4	yes	complemented
14	C₅×#504	—	2	2	no	—
15	C₅×#504	C₂²	1	4	yes	complemented
16	C₂×C₅×M₃₂	—	2	1	no	—
17	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
18	C₂²×C₅×C₁₆	—	2	1	no	—
19	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
20	C₅×#472	—	2	4	no	—
21	C₅×#472	C₂²	1	8	yes	complemented
22	C₂×C₅×#108	C₂²	1	2	yes	—
23	C₂×C₅×#100	—	2	2	no	—
24	C₅×#368	C₂²	1	4	yes	complemented
25	C₅×#361	C₂²	1	8	yes	complemented
26	C₂×#2221	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂×C₅×#41	D₄	1	1	yes	characteristic
28	C₂²×C₅×D₄	—	2	2	no	—
29	C₅×AGL(1,ℤ/8)	—	2	8	no	—
30	C₅×AGL(1,ℤ/8)	—	4	4	no	—
31	C₂×C₅×SD₁₆	—	4	2	no	—
32	C₂×C₅×D₈	—	2	2	no	—
33	C₂×C₅×D₈	—	2	2	no	—
34	C₂×C₅×D₈	—	2	2	no	—
35	C₂×C₅×D₈	—	2	2	no	—
36	C₂×C₅×D₈	—	2	2	no	—
37	C₂×C₅×D₈	—	4	2	no	—
38	C₂×C₅×D₈	C₂³	1	4	yes	—
39	C₅×#131	—	2	1	no	—
40	C₅×#131	—	2	1	no	—
41	C₅×#131	D₄	1	2	yes	—
42	C₅×#131	C₂³	1	2	yes	—
43	C₂×C₅×M₁₆	—	2	1	no	—
44	C₂×C₅×M₁₆	—	2	2	no	—
45	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
46	C₂²×C₅×C₈	—	2	1	no	—
47	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
48	C₅×D₁₆	—	2	4	no	—
49	C₅×D₁₆	—	2	4	no	—
50	C₅×M₃₂	—	2	1	no	—
51	C₅×M₃₂	—	2	4	no	—
52	C₅×M₃₂	C₂³	1	2	yes	—
53	C₂×C₅×C₁₆	—	2	1	no	—
54	C₂×C₅×C₁₆	—	2	4	no	—
55	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
56	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
57	C₂×C₅×C₁₆	D₄	1	2	yes	—
58	C₂×C₅×C₁₆	C₂³	1	2	yes	—
59	C₅×#108	—	2	2	no	—
60	C₅×#108	C₂³	1	4	yes	—
61	C₅×#100	—	2	8	no	—
62	C₂²×D₁₆	C₂×C₅	1	1	yes	characteristic, complemented
63	C₂×#504	C₂×C₅	1	1	yes	characteristic, complemented
64	C₂×#472	C₂×C₅	1	2	yes	complemented
65	#2221	C₂×C₅	1	8	yes	direct product factor
66	C₂×#368	C₂×C₅	1	1	yes	characteristic, complemented
67	C₂×#361	C₂×C₅	1	2	yes	complemented
68	C₂⁴×C₅	—	4	2	no	—
69	C₅×#41	—	2	1	no	—
70	C₅×#41	—	2	1	no	—
71	C₅×#41	D₈	1	2	yes	—
72	C₅×#41	C₂×D₄	1	2	yes	—
73	C₂×C₅×Q₈	D₈	1	1	yes	characteristic
74	C₂×C₅×D₄	—	2	1	no	—
75	C₂×C₅×D₄	—	2	4	no	—
76	C₂×C₅×D₄	—	4	2	no	—
77	C₂×C₅×D₄	—	4	2	no	—
78	C₂×C₅×D₄	—	4	2	no	—
79	C₂×C₅×D₄	—	4	2	no	—
80	C₂×C₅×D₄	—	4	2	no	—
81	C₂×C₅×D₄	D₈	1	1	yes	characteristic
82	C₂²×C₄×C₅	—	2	1	no	—
83	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₅×SD₁₆	—	4	8	no	—
85	C₅×D₈	—	2	4	no	—
86	C₅×D₈	—	2	4	no	—
87	C₅×D₈	—	4	8	no	—
88	C₅×D₈	—	4	2	no	—
89	C₅×D₈	—	4	2	no	—
90	C₅×M₁₆	—	2	1	no	—
91	C₅×M₁₆	—	2	2	no	—
92	C₅×M₁₆	—	2	2	no	—
93	C₅×M₁₆	—	2	4	no	—
94	C₅×M₁₆	—	4	2	no	—
95	C₅×M₁₆	C₂×D₄	1	2	yes	—
96	C₂×C₅×C₈	—	2	1	no	—
97	C₂×C₅×C₈	—	2	2	no	—
98	C₂×C₅×C₈	—	2	2	no	—
99	C₂×C₅×C₈	—	4	2	no	—
100	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
101	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
102	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
103	C₂×C₅×C₈	#41	1	1	yes	characteristic
104	C₂×C₅×C₈	#41	1	1	yes	characteristic
105	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
106	C₅×C₁₆	—	2	2	no	—
107	C₅×C₁₆	C₂×D₄	1	2	yes	—
108	C₅×C₁₆	C₂×D₄	1	2	yes	—
109	C₂×AGL(1,ℤ/8)	—	2	2	no	—
110	C₂²×D₈	C₂²×C₅	1	2	yes	—
111	C₂×#131	C₂²×C₅	1	1	yes	characteristic
112	C₂×D₁₆	—	2	1	no	—
113	C₂×D₁₆	—	2	1	no	—
114	C₂×D₁₆	—	2	2	no	—
115	C₂×D₁₆	C₂²×C₅	1	4	yes	complemented
116	#504	—	2	2	no	—
117	#504	C₂²×C₅	1	4	yes	complemented
118	C₂×M₃₂	—	2	1	no	—
119	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
120	C₂²×C₁₆	—	2	1	no	—
121	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
122	#472	—	2	4	no	—
123	#472	C₂²×C₅	1	8	yes	complemented
124	C₂×#108	C₂²×C₅	1	2	yes	—
125	C₂×#100	—	2	2	no	—
126	#368	C₂²×C₅	1	4	yes	complemented
127	#361	C₂²×C₅	1	8	yes	complemented
128	C₂³×C₅	—	2	1	no	—
129	C₂³×C₅	—	4	4	no	—
130	C₂³×C₅	—	8	2	no	—
131	C₂³×C₅	—	8	2	no	—
132	C₂³×C₅	—	16	4	no	—
133	C₂³×C₅	#135	1	1	yes	characteristic
134	C₅×Q₈	—	2	1	no	—
135	C₅×Q₈	C₂×D₈	1	2	yes	—
136	C₅×D₄	—	2	2	no	—
137	C₅×D₄	—	2	2	no	—
138	C₅×D₄	—	2	1	no	—
139	C₅×D₄	—	4	4	no	—
140	C₅×D₄	—	4	4	no	—
141	C₅×D₄	—	8	2	no	—
142	C₅×D₄	—	8	2	no	—
143	C₅×D₄	C₂×D₈	1	2	yes	—
144	C₂×C₄×C₅	—	2	1	no	—
145	C₂×C₄×C₅	—	2	2	no	—
146	C₂×C₄×C₅	—	2	2	no	—
147	C₂×C₄×C₅	#121	1	1	yes	characteristic
148	C₂×C₄×C₅	#121	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₅	#135	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
153	C₅×C₈	—	2	2	no	—
154	C₅×C₈	—	4	4	no	—
155	C₅×C₈	#121	1	1	yes	characteristic
156	C₅×C₈	#121	1	1	yes	characteristic
157	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
158	C₅×C₈	C₂×#41	1	1	yes	characteristic
159	C₂×#41	C₅×D₄	1	1	yes	characteristic
160	C₂²×D₄	—	2	2	no	—
161	AGL(1,ℤ/8)	—	2	8	no	—
162	AGL(1,ℤ/8)	—	4	4	no	—
163	C₂×SD₁₆	—	4	2	no	—
164	C₂×D₈	—	2	2	no	—
165	C₂×D₈	—	2	2	no	—
166	C₂×D₈	—	2	2	no	—
167	C₂×D₈	—	2	2	no	—
168	C₂×D₈	—	2	2	no	—
169	C₂×D₈	—	4	2	no	—
170	C₂×D₈	C₂³×C₅	1	4	yes	—
171	#131	—	2	1	no	—
172	#131	—	2	1	no	—
173	#131	C₅×D₄	1	2	yes	—
174	#131	C₂³×C₅	1	2	yes	—
175	C₂×M₁₆	—	2	1	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
178	C₂²×C₈	—	2	1	no	—
179	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
180	D₁₆	—	2	4	no	—
181	D₁₆	—	2	4	no	—
182	M₃₂	—	2	1	no	—
183	M₃₂	—	2	4	no	—
184	M₃₂	C₂³×C₅	1	2	yes	—
185	C₂×C₁₆	—	2	1	no	—
186	C₂×C₁₆	—	2	4	no	—
187	C₂×C₁₆	C₅×D₄	1	1	yes	characteristic
188	C₂×C₁₆	C₅×D₄	1	1	yes	characteristic
189	C₂×C₁₆	C₅×D₄	1	2	yes	—
190	C₂×C₁₆	C₂³×C₅	1	2	yes	—
191	#108	—	2	2	no	—
192	#108	C₂³×C₅	1	4	yes	—
193	#100	—	2	8	no	—
194	C₂²×C₅	—	2	2	no	—
195	C₂²×C₅	—	2	2	no	—
196	C₂²×C₅	—	4	2	no	—
197	C₂²×C₅	—	8	4	no	—
198	C₂²×C₅	—	16	4	no	—
199	C₂²×C₅	—	16	4	no	—
200	C₂²×C₅	—	16	4	no	—
201	C₂²×C₅	#466	1	1	yes	characteristic
202	C₂²×C₅	#466	1	1	yes	characteristic
203	C₂²×C₅	C₂×#135	1	1	yes	characteristic
204	C₄×C₅	—	2	2	no	—
205	C₄×C₅	#466	1	1	yes	characteristic
206	C₄×C₅	#466	1	1	yes	characteristic
207	C₄×C₅	C₂×#121	1	1	yes	characteristic
208	C₄×C₅	C₂²×D₈	1	1	yes	characteristic
209	C₂⁴	—	4	2	no	—
210	#41	—	2	1	no	—
211	#41	—	2	1	no	—
212	#41	C₅×D₈	1	2	yes	—
213	#41	C₂×C₅×D₄	1	2	yes	—
214	C₂×Q₈	C₅×D₈	1	1	yes	characteristic
215	C₂×D₄	—	2	1	no	—
216	C₂×D₄	—	2	4	no	—
217	C₂×D₄	—	4	2	no	—
218	C₂×D₄	—	4	2	no	—
219	C₂×D₄	—	4	2	no	—
220	C₂×D₄	—	4	2	no	—
221	C₂×D₄	—	4	2	no	—
222	C₂×D₄	C₅×D₈	1	1	yes	characteristic
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
225	SD₁₆	—	4	8	no	—
226	D₈	—	2	4	no	—
227	D₈	—	2	4	no	—
228	D₈	—	4	8	no	—
229	D₈	—	4	2	no	—
230	D₈	—	4	2	no	—
231	M₁₆	—	2	1	no	—
232	M₁₆	—	2	2	no	—
233	M₁₆	—	2	2	no	—
234	M₁₆	—	2	4	no	—
235	M₁₆	—	4	2	no	—
236	M₁₆	C₂×C₅×D₄	1	2	yes	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
242	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
243	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
244	C₂×C₈	C₅×#41	1	1	yes	characteristic
245	C₂×C₈	C₅×#41	1	1	yes	characteristic
246	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
247	C₁₆	—	2	2	no	—
248	C₁₆	C₂×C₅×D₄	1	2	yes	—
249	C₁₆	C₂×C₅×D₄	1	2	yes	—
250	C₂×C₅	—	2	1	no	—
251	C₂×C₅	—	2	1	no	—
252	C₂×C₅	—	4	2	no	—
253	C₂×C₅	—	16	4	no	—
254	C₂×C₅	#2221	1	2	yes	direct product factor
255	C₂×C₅	C₂×#466	1	1	yes	characteristic
256	C₂³	—	2	1	no	—
257	C₂³	—	4	4	no	—
258	C₂³	—	8	2	no	—
259	C₂³	—	8	2	no	—
260	C₂³	—	16	4	no	—
261	C₂³	C₅×#135	1	1	yes	characteristic
262	Q₈	—	2	1	no	—
263	Q₈	C₂×C₅×D₈	1	2	yes	—
264	D₄	—	2	2	no	—
265	D₄	—	2	2	no	—
266	D₄	—	2	1	no	—
267	D₄	—	4	4	no	—
268	D₄	—	4	4	no	—
269	D₄	—	8	2	no	—
270	D₄	—	8	2	no	—
271	D₄	C₂×C₅×D₈	1	2	yes	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	C₅×#121	1	1	yes	characteristic
276	C₂×C₄	C₅×#121	1	1	yes	characteristic
277	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
278	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
279	C₂×C₄	C₅×#135	1	1	yes	characteristic
280	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
281	C₈	—	2	2	no	—
282	C₈	—	4	4	no	—
283	C₈	C₅×#121	1	1	yes	characteristic
284	C₈	C₅×#121	1	1	yes	characteristic
285	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
286	C₈	C₂×C₅×#41	1	1	yes	characteristic
287	C₅	C₂×#2221	1	1	yes	characteristic, direct product factor, Sylow
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	4	2	no	—
291	C₂²	—	8	4	no	—
292	C₂²	—	16	4	no	—
293	C₂²	—	16	4	no	—
294	C₂²	—	16	4	no	—
295	C₂²	C₅×#466	1	1	yes	characteristic
296	C₂²	C₅×#466	1	1	yes	characteristic
297	C₂²	C₂×C₅×#135	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	C₅×#466	1	1	yes	characteristic
300	C₄	C₅×#466	1	1	yes	characteristic
301	C₄	C₂×C₅×#121	1	1	yes	characteristic
302	C₄	C₂²×C₅×D₈	1	1	yes	characteristic
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	4	2	no	—
306	C₂	—	16	4	no	—
307	C₂	C₅×#2221	1	2	yes	direct product factor
308	C₂	C₂×C₅×#466	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	#466
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#131 ◁ 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×#504	C₂	semidirect
3. C₂×C₅×#472	C₂	semidirect
4. C₅×#2221	C₂	direct (N × H)
5. C₂×C₅×#368	C₂	semidirect
6. C₂×C₅×#361	C₂	semidirect
13. C₂×C₅×D₁₆	C₂²	semidirect
15. C₅×#504	C₂²	semidirect
21. C₅×#472	C₂²	semidirect
24. C₅×#368	C₂²	semidirect
25. C₅×#361	C₂²	semidirect
26. C₂×#2221	C₅	direct (N × H)
62. C₂²×D₁₆	C₂×C₅	semidirect
63. C₂×#504	C₂×C₅	semidirect
64. C₂×#472	C₂×C₅	semidirect
65. #2221	C₂×C₅	direct (N × H)
66. C₂×#368	C₂×C₅	semidirect
67. C₂×#361	C₂×C₅	semidirect
115. C₂×D₁₆	C₂²×C₅	semidirect
117. #504	C₂²×C₅	semidirect
123. #472	C₂²×C₅	semidirect
126. #368	C₂²×C₅	semidirect
127. #361	C₂²×C₅	semidirect
254. C₂×C₅	#2221	direct (N × H)
287. C₅	C₂×#2221	direct (N × H)
307. C₂	C₅×#2221	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×#504	2
C₂×C₅×#472	2
C₅×#2221	2
C₂×C₅×#368	2
C₂×C₅×#361	2
C₂×#2221	5

Subgroups — C2×C5×#2221 #11956334

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2221 #11956334