Subgroups — C₂×C₅×#2226 #11956341

Order 1280.

638 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×C₅×D₈	C₂	1	1	yes	maximal, characteristic
3	C₅×#2226	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₅×#367	C₂	1	2	yes	maximal, complemented
5	C₂×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×#357	C₂	1	2	yes	maximal
7	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
8	C₂×C₅×C₄⋊Q₈	—	2	1	no	—
9	C₂×C₄×C₅×D₄	—	2	1	no	—
10	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
11	C₅×#500	—	2	2	no	—
12	C₅×#500	C₂²	1	4	yes	complemented
13	C₄×C₅×D₈	—	2	2	no	—
14	C₄×C₅×D₈	C₂²	1	4	yes	—
15	C₂×C₅×#107	—	2	1	no	—
16	C₂×C₅×#107	C₂²	1	2	yes	—
17	C₂×C₅×#107	C₂²	1	1	yes	characteristic
18	C₂×C₅×#102	—	2	1	no	—
19	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	C₅×#367	C₂²	1	8	yes	complemented
21	C₅×C₄⋊C₁₆	C₂²	1	4	yes	complemented
22	C₅×#357	C₂²	1	8	yes	—
23	C₂×#2226	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂²×C₅×Q₈	—	4	1	no	—
25	C₂²×C₅×D₄	—	2	1	no	—
26	C₂³×C₄×C₅	—	4	1	no	—
27	C₂×C₅×D₈	—	2	1	no	—
28	C₂×C₅×D₈	—	2	2	no	—
29	C₂×C₅×D₈	Q₈	1	4	yes	complemented
30	C₂×C₅×D₈	C₂³	1	2	yes	—
31	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
32	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
33	C₅×C₄⋊Q₈	—	2	4	no	—
34	C₅×C₄⋊Q₈	—	4	2	no	—
35	C₄×C₅×D₄	—	2	4	no	—
36	C₄×C₅×D₄	—	4	2	no	—
37	C₂×C₅×C₄⋊C₄	—	2	1	no	—
38	C₂×C₅×C₄⋊C₄	—	2	2	no	—
39	C₂×C₅×C₄⋊C₄	—	4	1	no	—
40	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
41	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₂×C₅×C₁₆	—	2	4	no	—
43	C₂×C₅×C₁₆	C₂³	1	4	yes	—
44	C₅×#107	—	2	4	no	—
45	C₅×#107	—	2	1	no	—
46	C₅×#107	—	2	2	no	—
47	C₅×#107	C₂³	1	4	yes	—
48	C₅×#107	C₂³	1	2	yes	—
49	C₅×#102	—	2	4	no	—
50	C₄×C₅×C₈	D₄	1	2	yes	—
51	C₄×C₅×C₈	C₂³	1	2	yes	—
52	C₂×#500	C₂×C₅	1	1	yes	characteristic, complemented
53	C₂×C₄×D₈	C₂×C₅	1	1	yes	characteristic
54	#2226	C₂×C₅	1	8	yes	direct product factor
55	C₂×#367	C₂×C₅	1	2	yes	complemented
56	C₂×C₄⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
57	C₂×#357	C₂×C₅	1	2	yes	—
58	C₂⁴×C₅	—	4	1	no	—
59	C₂×C₅×Q₈	—	4	4	no	—
60	C₂×C₅×Q₈	—	8	4	no	—
61	C₂×C₅×D₄	—	2	4	no	—
62	C₂×C₅×D₄	—	2	2	no	—
63	C₂×C₅×D₄	—	4	1	no	—
64	C₂×C₅×D₄	—	4	2	no	—
65	C₂²×C₄×C₅	—	4	1	no	—
66	C₂²×C₄×C₅	—	4	2	no	—
67	C₂²×C₄×C₅	—	4	4	no	—
68	C₂²×C₄×C₅	—	8	4	no	—
69	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
70	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₅×D₈	—	2	4	no	—
73	C₅×D₈	—	2	2	no	—
74	C₅×D₈	C₂×Q₈	1	4	yes	complemented
75	C₂×C₅×C₈	—	2	2	no	—
76	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
77	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
78	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
79	C₂×C₅×C₈	#41	1	2	yes	—
80	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
81	C₅×C₄⋊C₄	—	2	2	no	—
82	C₅×C₄⋊C₄	—	2	4	no	—
83	C₅×C₄⋊C₄	—	4	1	no	—
84	C₅×C₄⋊C₄	—	4	2	no	—
85	C₅×C₄⋊C₄	—	4	4	no	—
86	C₅×C₂²⋊C₄	—	4	4	no	—
87	C₄²×C₅	D₈	1	2	yes	—
88	C₄²×C₅	C₂×D₄	1	2	yes	—
89	C₅×C₁₆	—	2	4	no	—
90	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
91	C₂×C₄⋊Q₈	—	2	1	no	—
92	C₂×C₄×D₄	—	2	1	no	—
93	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
94	#500	—	2	2	no	—
95	#500	C₂²×C₅	1	4	yes	complemented
96	C₄×D₈	—	2	2	no	—
97	C₄×D₈	C₂²×C₅	1	4	yes	—
98	C₂×#107	—	2	1	no	—
99	C₂×#107	C₂²×C₅	1	2	yes	—
100	C₂×#107	C₂²×C₅	1	1	yes	characteristic
101	C₂×#102	—	2	1	no	—
102	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
103	#367	C₂²×C₅	1	8	yes	complemented
104	C₄⋊C₁₆	C₂²×C₅	1	4	yes	complemented
105	#357	C₂²×C₅	1	8	yes	—
106	C₂³×C₅	—	4	2	no	—
107	C₂³×C₅	—	4	4	no	—
108	C₂³×C₅	—	8	2	no	—
109	C₂³×C₅	—	8	2	no	—
110	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
111	C₅×Q₈	—	8	8	no	—
112	C₅×D₄	—	2	4	no	—
113	C₅×D₄	—	4	2	no	—
114	C₅×D₄	—	4	4	no	—
115	C₂×C₄×C₅	—	2	2	no	—
116	C₂×C₄×C₅	—	4	2	no	—
117	C₂×C₄×C₅	—	4	4	no	—
118	C₂×C₄×C₅	—	8	2	no	—
119	C₂×C₄×C₅	—	8	4	no	—
120	C₂×C₄×C₅	—	8	8	no	—
121	C₂×C₄×C₅	SD₃₂	1	4	yes	—
122	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
123	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
124	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
126	C₂×C₄×C₅	#137	1	2	yes	—
127	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
128	C₅×C₈	—	2	2	no	—
129	C₅×C₈	C₂²⋊Q₈	1	2	yes	—
130	C₅×C₈	C₂²×Q₈	1	1	yes	characteristic
131	C₅×C₈	C₂×#41	1	1	yes	characteristic
132	C₂²×Q₈	—	4	1	no	—
133	C₂²×D₄	—	2	1	no	—
134	C₂³×C₄	—	4	1	no	—
135	C₂×D₈	—	2	1	no	—
136	C₂×D₈	—	2	2	no	—
137	C₂×D₈	C₅×Q₈	1	4	yes	complemented
138	C₂×D₈	C₂³×C₅	1	2	yes	—
139	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
140	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
141	C₄⋊Q₈	—	2	4	no	—
142	C₄⋊Q₈	—	4	2	no	—
143	C₄×D₄	—	2	4	no	—
144	C₄×D₄	—	4	2	no	—
145	C₂×C₄⋊C₄	—	2	1	no	—
146	C₂×C₄⋊C₄	—	2	2	no	—
147	C₂×C₄⋊C₄	—	4	1	no	—
148	C₂×C₂²⋊C₄	—	4	1	no	—
149	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
150	C₂×C₁₆	—	2	4	no	—
151	C₂×C₁₆	C₂³×C₅	1	4	yes	—
152	#107	—	2	4	no	—
153	#107	—	2	1	no	—
154	#107	—	2	2	no	—
155	#107	C₂³×C₅	1	4	yes	—
156	#107	C₂³×C₅	1	2	yes	—
157	#102	—	2	4	no	—
158	C₄×C₈	C₅×D₄	1	2	yes	—
159	C₄×C₈	C₂³×C₅	1	2	yes	—
160	C₂²×C₅	—	4	4	no	—
161	C₂²×C₅	—	8	2	no	—
162	C₂²×C₅	—	8	2	no	—
163	C₂²×C₅	—	8	8	no	—
164	C₂²×C₅	#474	1	2	yes	—
165	C₂²×C₅	C₂×SD₃₂	1	2	yes	—
166	C₂²×C₅	#509	1	2	yes	—
167	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
168	C₄×C₅	—	2	2	no	—
169	C₄×C₅	—	8	2	no	—
170	C₄×C₅	—	8	4	no	—
171	C₄×C₅	#474	1	2	yes	—
172	C₄×C₅	C₂×SD₃₂	1	4	yes	—
173	C₄×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
174	C₄×C₅	C₂×#137	1	1	yes	characteristic
175	C₂⁴	—	4	1	no	—
176	C₂×Q₈	—	4	4	no	—
177	C₂×Q₈	—	8	4	no	—
178	C₂×D₄	—	2	4	no	—
179	C₂×D₄	—	2	2	no	—
180	C₂×D₄	—	4	1	no	—
181	C₂×D₄	—	4	2	no	—
182	C₂²×C₄	—	4	1	no	—
183	C₂²×C₄	—	4	2	no	—
184	C₂²×C₄	—	4	4	no	—
185	C₂²×C₄	—	8	4	no	—
186	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
187	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
188	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
189	D₈	—	2	4	no	—
190	D₈	—	2	2	no	—
191	D₈	C₂×C₅×Q₈	1	4	yes	complemented
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
194	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
195	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
196	C₂×C₈	C₅×#41	1	2	yes	—
197	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	2	4	no	—
200	C₄⋊C₄	—	4	1	no	—
201	C₄⋊C₄	—	4	2	no	—
202	C₄⋊C₄	—	4	4	no	—
203	C₂²⋊C₄	—	4	4	no	—
204	C₄²	C₅×D₈	1	2	yes	—
205	C₄²	C₂×C₅×D₄	1	2	yes	—
206	C₁₆	—	2	4	no	—
207	C₂×C₅	—	8	4	no	—
208	C₂×C₅	#2226	1	4	yes	direct product factor
209	C₂×C₅	C₂×#474	1	1	yes	characteristic
210	C₂×C₅	C₂²×SD₃₂	1	1	yes	characteristic
211	C₂×C₅	C₂×#509	1	1	yes	characteristic
212	C₂³	—	4	4	no	—
213	C₂³	—	4	2	no	—
214	C₂³	—	8	2	no	—
215	C₂³	—	8	2	no	—
216	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
217	Q₈	—	8	8	no	—
218	D₄	—	2	4	no	—
219	D₄	—	4	4	no	—
220	D₄	—	4	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	—	8	2	no	—
225	C₂×C₄	—	8	4	no	—
226	C₂×C₄	—	8	8	no	—
227	C₂×C₄	C₅×SD₃₂	1	4	yes	—
228	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
229	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
230	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
231	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
232	C₂×C₄	C₅×#137	1	2	yes	—
233	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
234	C₈	—	2	2	no	—
235	C₈	C₅×C₂²⋊Q₈	1	2	yes	—
236	C₈	C₂²×C₅×Q₈	1	1	yes	characteristic
237	C₈	C₂×C₅×#41	1	1	yes	characteristic
238	C₅	C₂×#2226	1	1	yes	characteristic, direct product factor, Sylow
239	C₂²	—	4	4	no	—
240	C₂²	—	8	2	no	—
241	C₂²	—	8	2	no	—
242	C₂²	—	8	8	no	—
243	C₂²	C₅×#474	1	2	yes	—
244	C₂²	C₂×C₅×SD₃₂	1	2	yes	—
245	C₂²	C₅×#509	1	2	yes	—
246	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
247	C₄	—	2	2	no	—
248	C₄	—	8	2	no	—
249	C₄	—	8	4	no	—
250	C₄	C₅×#474	1	2	yes	—
251	C₄	C₂×C₅×SD₃₂	1	4	yes	—
252	C₄	C₂×C₅×C₂²⋊Q₈	1	1	yes	characteristic
253	C₄	C₂×C₅×#137	1	1	yes	characteristic
254	C₂	—	8	4	no	—
255	C₂	C₅×#2226	1	4	yes	direct product factor
256	C₂	C₂×C₅×#474	1	1	yes	characteristic
257	C₂	C₂²×C₅×SD₃₂	1	1	yes	characteristic
258	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	C₂²×C₅×D₈	C₂²
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. C₂×C₅×#500	C₂	semidirect
3. C₅×#2226	C₂	direct (N × H)
4. C₂×C₅×#367	C₂	semidirect
5. C₂×C₅×C₄⋊C₁₆	C₂	semidirect
12. C₅×#500	C₂²	semidirect
20. C₅×#367	C₂²	semidirect
21. C₅×C₄⋊C₁₆	C₂²	semidirect
23. C₂×#2226	C₅	direct (N × H)
29. C₂×C₅×D₈	Q₈	semidirect
52. C₂×#500	C₂×C₅	semidirect
54. #2226	C₂×C₅	direct (N × H)
55. C₂×#367	C₂×C₅	semidirect
56. C₂×C₄⋊C₁₆	C₂×C₅	semidirect
74. C₅×D₈	C₂×Q₈	semidirect
95. #500	C₂²×C₅	semidirect
103. #367	C₂²×C₅	semidirect
104. C₄⋊C₁₆	C₂²×C₅	semidirect
137. C₂×D₈	C₅×Q₈	semidirect
191. D₈	C₂×C₅×Q₈	semidirect
208. C₂×C₅	#2226	direct (N × H)
238. C₅	C₂×#2226	direct (N × H)
255. C₂	C₅×#2226	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×#500	2
C₂×C₄×C₅×D₈	2
C₅×#2226	2
C₂×C₅×#367	2
C₂×C₅×C₄⋊C₁₆	2
C₂×C₅×#357	2
C₂×#2226	5

Subgroups — C2×C5×#2226 #11956341

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2226 #11956341