Subgroups — C₂×C₅×#2218 #11956321

Order 1280.

786 conjugacy classes of subgroups.

The subgroup lattice has 256 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	C₂²×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#466	C₂	1	2	yes	maximal, complemented
4	C₅×#2218	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅×#367	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×#357	C₂	1	2	yes	maximal, complemented
7	C₂²×C₅×D₈	C₂²	1	2	yes	—
8	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₅×#121	—	2	2	no	—
10	C₂×C₅×D₁₆	—	2	4	no	—
11	C₂×C₅×D₁₆	C₂²	1	4	yes	complemented
12	C₂×C₅×M₃₂	—	2	4	no	—
13	C₂×C₅×M₃₂	C₂²	1	4	yes	complemented
14	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
16	C₅×#466	—	2	4	no	—
17	C₅×#466	C₂²	1	8	yes	complemented
18	C₂×C₅×#107	C₂²	1	2	yes	—
19	C₂×C₅×#102	—	2	2	no	—
20	C₅×#367	C₂²	1	4	yes	complemented
21	C₅×#357	C₂²	1	8	yes	complemented
22	C₂×#2218	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂²×C₅×D₄	—	2	2	no	—
24	C₂²×C₅×D₄	—	4	2	no	—
25	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
26	C₂×C₅×D₈	—	2	4	no	—
27	C₂×C₅×D₈	—	2	2	no	—
28	C₂×C₅×D₈	—	2	4	no	—
29	C₂×C₅×D₈	C₂³	1	4	yes	—
30	C₂²×C₅×C₈	—	2	2	no	—
31	C₂²×C₅×C₈	—	2	2	no	—
32	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
33	C₂²×C₅×C₈	D₄	1	2	yes	—
34	C₂²×C₅×C₈	C₂³	1	2	yes	—
35	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
36	C₅×#121	—	2	8	no	—
37	C₅×#121	—	4	4	no	—
38	C₂×C₅×C₄⋊C₄	—	2	2	no	—
39	C₂×C₅×C₂²⋊C₄	—	4	2	no	—
40	C₅×D₁₆	—	2	8	no	—
41	C₅×M₃₂	—	2	8	no	—
42	C₂×C₅×C₁₆	—	2	2	no	—
43	C₂×C₅×C₁₆	D₄	1	4	yes	complemented
44	C₂×C₅×C₁₆	C₂³	1	2	yes	—
45	C₂×C₅×C₁₆	C₂³	1	2	yes	—
46	C₅×#107	—	2	2	no	—
47	C₅×#107	C₂³	1	4	yes	—
48	C₅×#102	—	2	8	no	—
49	C₂²×D₁₆	C₂×C₅	1	1	yes	characteristic, complemented
50	C₂²×M₃₂	C₂×C₅	1	1	yes	characteristic, complemented
51	C₂×#466	C₂×C₅	1	2	yes	complemented
52	#2218	C₂×C₅	1	8	yes	direct product factor
53	C₂×#367	C₂×C₅	1	1	yes	characteristic, complemented
54	C₂×#357	C₂×C₅	1	2	yes	complemented
55	C₂⁴×C₅	—	4	2	no	—
56	C₂⁴×C₅	D₈	1	1	yes	characteristic
57	C₂×C₅×D₄	—	2	4	no	—
58	C₂×C₅×D₄	—	4	8	no	—
59	C₂×C₅×D₄	—	4	4	no	—
60	C₂×C₅×D₄	—	4	2	no	—
61	C₂×C₅×D₄	—	4	4	no	—
62	C₂×C₅×D₄	—	8	8	no	—
63	C₂²×C₄×C₅	—	2	2	no	—
64	C₂²×C₄×C₅	—	2	2	no	—
65	C₂²×C₄×C₅	—	4	2	no	—
66	C₂²×C₄×C₅	D₈	1	2	yes	—
67	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
70	C₅×D₈	—	2	4	no	—
71	C₅×D₈	—	2	4	no	—
72	C₅×D₈	—	4	4	no	—
73	C₂×C₅×C₈	—	2	2	no	—
74	C₂×C₅×C₈	—	2	4	no	—
75	C₂×C₅×C₈	—	2	2	no	—
76	C₂×C₅×C₈	—	2	2	no	—
77	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
78	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
79	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
80	C₂×C₅×C₈	#41	1	2	yes	—
81	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
82	C₅×C₄⋊C₄	—	2	4	no	—
83	C₅×C₄⋊C₄	—	4	2	no	—
84	C₅×C₂²⋊C₄	—	4	8	no	—
85	C₅×C₁₆	—	2	2	no	—
86	C₅×C₁₆	C₂×D₄	1	4	yes	complemented
87	C₂²×D₈	C₂²×C₅	1	2	yes	—
88	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
89	C₂×#121	—	2	2	no	—
90	C₂×D₁₆	—	2	4	no	—
91	C₂×D₁₆	C₂²×C₅	1	4	yes	complemented
92	C₂×M₃₂	—	2	4	no	—
93	C₂×M₃₂	C₂²×C₅	1	4	yes	complemented
94	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
95	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
96	#466	—	2	4	no	—
97	#466	C₂²×C₅	1	8	yes	complemented
98	C₂×#107	C₂²×C₅	1	2	yes	—
99	C₂×#102	—	2	2	no	—
100	#367	C₂²×C₅	1	4	yes	complemented
101	#357	C₂²×C₅	1	8	yes	complemented
102	C₂³×C₅	—	2	2	no	—
103	C₂³×C₅	—	2	4	no	—
104	C₂³×C₅	—	4	4	no	—
105	C₂³×C₅	—	8	4	no	—
106	C₂³×C₅	—	8	4	no	—
107	C₂³×C₅	—	8	4	no	—
108	C₂³×C₅	C₂×D₈	1	2	yes	—
109	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
110	C₅×D₄	—	4	4	no	—
111	C₅×D₄	—	4	4	no	—
112	C₅×D₄	—	8	16	no	—
113	C₅×D₄	—	8	4	no	—
114	C₂×C₄×C₅	—	2	2	no	—
115	C₂×C₄×C₅	—	2	2	no	—
116	C₂×C₄×C₅	—	2	2	no	—
117	C₂×C₄×C₅	—	2	4	no	—
118	C₂×C₄×C₅	—	4	4	no	—
119	C₂×C₄×C₅	—	8	4	no	—
120	C₂×C₄×C₅	#121	1	2	yes	—
121	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
122	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₅	#135	1	2	yes	—
124	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
125	C₅×C₈	—	2	2	no	—
126	C₅×C₈	#121	1	2	yes	—
127	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
128	C₅×C₈	C₂×#41	1	1	yes	characteristic
129	C₂²×D₄	—	2	2	no	—
130	C₂²×D₄	—	4	2	no	—
131	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
132	C₂×D₈	—	2	4	no	—
133	C₂×D₈	—	2	2	no	—
134	C₂×D₈	—	2	4	no	—
135	C₂×D₈	C₂³×C₅	1	4	yes	—
136	C₂²×C₈	—	2	2	no	—
137	C₂²×C₈	—	2	2	no	—
138	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
139	C₂²×C₈	C₅×D₄	1	2	yes	—
140	C₂²×C₈	C₂³×C₅	1	2	yes	—
141	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
142	#121	—	2	8	no	—
143	#121	—	4	4	no	—
144	C₂×C₄⋊C₄	—	2	2	no	—
145	C₂×C₂²⋊C₄	—	4	2	no	—
146	D₁₆	—	2	8	no	—
147	M₃₂	—	2	8	no	—
148	C₂×C₁₆	—	2	2	no	—
149	C₂×C₁₆	C₅×D₄	1	4	yes	complemented
150	C₂×C₁₆	C₂³×C₅	1	2	yes	—
151	C₂×C₁₆	C₂³×C₅	1	2	yes	—
152	#107	—	2	2	no	—
153	#107	C₂³×C₅	1	4	yes	—
154	#102	—	2	8	no	—
155	C₂²×C₅	—	2	4	no	—
156	C₂²×C₅	—	2	2	no	—
157	C₂²×C₅	—	4	4	no	—
158	C₂²×C₅	—	8	4	no	—
159	C₂²×C₅	—	8	4	no	—
160	C₂²×C₅	—	8	4	no	—
161	C₂²×C₅	—	16	8	no	—
162	C₂²×C₅	#466	1	2	yes	—
163	C₂²×C₅	#509	1	4	yes	—
164	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
165	C₄×C₅	—	2	2	no	—
166	C₄×C₅	—	8	4	no	—
167	C₄×C₅	#466	1	2	yes	—
168	C₄×C₅	C₂×#121	1	1	yes	characteristic
169	C₄×C₅	C₂×#135	1	1	yes	characteristic
170	C₂⁴	—	4	2	no	—
171	C₂⁴	C₅×D₈	1	1	yes	characteristic
172	C₂×D₄	—	2	4	no	—
173	C₂×D₄	—	4	8	no	—
174	C₂×D₄	—	4	4	no	—
175	C₂×D₄	—	4	2	no	—
176	C₂×D₄	—	4	4	no	—
177	C₂×D₄	—	8	8	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	4	2	no	—
181	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
182	C₂²×C₄	C₅×D₈	1	2	yes	—
183	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
184	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
185	D₈	—	2	4	no	—
186	D₈	—	2	4	no	—
187	D₈	—	4	4	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
193	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
194	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
195	C₂×C₈	C₅×#41	1	2	yes	—
196	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	—	4	2	no	—
199	C₂²⋊C₄	—	4	8	no	—
200	C₁₆	—	2	2	no	—
201	C₁₆	C₂×C₅×D₄	1	4	yes	complemented
202	C₂×C₅	—	4	2	no	—
203	C₂×C₅	—	16	4	no	—
204	C₂×C₅	#2218	1	4	yes	direct product factor
205	C₂×C₅	C₂×#466	1	1	yes	characteristic
206	C₂×C₅	C₂×#509	1	2	yes	—
207	C₂³	—	2	2	no	—
208	C₂³	—	2	4	no	—
209	C₂³	—	4	4	no	—
210	C₂³	—	8	4	no	—
211	C₂³	—	8	4	no	—
212	C₂³	—	8	4	no	—
213	C₂³	C₂×C₅×D₈	1	2	yes	—
214	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
215	D₄	—	4	4	no	—
216	D₄	—	4	4	no	—
217	D₄	—	8	16	no	—
218	D₄	—	8	4	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	—	8	4	no	—
225	C₂×C₄	C₅×#121	1	2	yes	—
226	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
227	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
228	C₂×C₄	C₅×#135	1	2	yes	—
229	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
230	C₈	—	2	2	no	—
231	C₈	C₅×#121	1	2	yes	—
232	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
233	C₈	C₂×C₅×#41	1	1	yes	characteristic
234	C₅	C₂×#2218	1	1	yes	characteristic, direct product factor, Sylow
235	C₂²	—	2	2	no	—
236	C₂²	—	2	4	no	—
237	C₂²	—	4	4	no	—
238	C₂²	—	8	4	no	—
239	C₂²	—	8	4	no	—
240	C₂²	—	8	4	no	—
241	C₂²	—	16	8	no	—
242	C₂²	C₅×#466	1	2	yes	—
243	C₂²	C₅×#509	1	4	yes	—
244	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	8	4	no	—
247	C₄	C₅×#466	1	2	yes	—
248	C₄	C₂×C₅×#121	1	1	yes	characteristic
249	C₄	C₂×C₅×#135	1	1	yes	characteristic
250	C₂	—	4	2	no	—
251	C₂	—	16	4	no	—
252	C₂	C₅×#2218	1	4	yes	direct product factor
253	C₂	C₂×C₅×#466	1	1	yes	characteristic
254	C₂	C₂×C₅×#509	1	2	yes	—

	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 ◁ C₂³×C₄×C₅ ◁ C₂³×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. C₂²×C₅×M₃₂	C₂	semidirect
3. C₂×C₅×#466	C₂	semidirect
4. C₅×#2218	C₂	direct (N × H)
5. C₂×C₅×#367	C₂	semidirect
6. C₂×C₅×#357	C₂	semidirect
11. C₂×C₅×D₁₆	C₂²	semidirect
13. C₂×C₅×M₃₂	C₂²	semidirect
17. C₅×#466	C₂²	semidirect
20. C₅×#367	C₂²	semidirect
21. C₅×#357	C₂²	semidirect
22. C₂×#2218	C₅	direct (N × H)
43. C₂×C₅×C₁₆	D₄	semidirect
49. C₂²×D₁₆	C₂×C₅	semidirect
50. C₂²×M₃₂	C₂×C₅	semidirect
51. C₂×#466	C₂×C₅	semidirect
52. #2218	C₂×C₅	direct (N × H)
53. C₂×#367	C₂×C₅	semidirect
54. C₂×#357	C₂×C₅	semidirect
86. C₅×C₁₆	C₂×D₄	semidirect
91. C₂×D₁₆	C₂²×C₅	semidirect
93. C₂×M₃₂	C₂²×C₅	semidirect
97. #466	C₂²×C₅	semidirect
100. #367	C₂²×C₅	semidirect
101. #357	C₂²×C₅	semidirect
149. C₂×C₁₆	C₅×D₄	semidirect
201. C₁₆	C₂×C₅×D₄	semidirect
204. C₂×C₅	#2218	direct (N × H)
234. C₅	C₂×#2218	direct (N × H)
252. C₂	C₅×#2218	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
C₂²×C₅×M₃₂	2
C₂×C₅×#466	2
C₅×#2218	2
C₂×C₅×#367	2
C₂×C₅×#357	2
C₂×#2218	5

Subgroups — C2×C5×#2218 #11956321

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2218 #11956321