Subgroups — C₂×C₅×#2216 #11956318

Order 1280.

702 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#467	C₂	1	2	yes	maximal
4	C₅×#2216	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅×#366	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×#358	C₂	1	2	yes	maximal, complemented
7	C₂²×C₅×Q₁₆	C₂²	1	2	yes	—
8	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₅×C₂²⋊Q₈	—	2	2	no	—
10	C₂×C₅×Q₃₂	—	2	4	no	—
11	C₂×C₅×Q₃₂	C₂²	1	4	yes	complemented
12	C₂²×C₅×C₁₆	—	2	4	no	—
13	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₅×C₁₆	C₂²	1	4	yes	—
15	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
16	C₅×#467	—	2	4	no	—
17	C₅×#467	C₂²	1	8	yes	—
18	C₂×C₅×#107	C₂²	1	2	yes	—
19	C₂×C₅×#103	—	2	2	no	—
20	C₅×#366	C₂²	1	4	yes	complemented
21	C₅×#358	C₂²	1	8	yes	complemented
22	C₂×#2216	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂²×C₅×Q₈	—	2	2	no	—
24	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
25	C₂×C₅×Q₁₆	—	2	4	no	—
26	C₂×C₅×Q₁₆	—	2	2	no	—
27	C₂×C₅×Q₁₆	—	2	4	no	—
28	C₂×C₅×Q₁₆	C₂³	1	4	yes	—
29	C₂²×C₅×C₈	—	2	2	no	—
30	C₂²×C₅×C₈	—	2	2	no	—
31	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
32	C₂²×C₅×C₈	D₄	1	2	yes	—
33	C₂²×C₅×C₈	C₂³	1	2	yes	—
34	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
35	C₅×C₂²⋊Q₈	—	2	8	no	—
36	C₅×C₂²⋊Q₈	—	4	4	no	—
37	C₂×C₅×C₄⋊C₄	—	2	2	no	—
38	C₂×C₅×C₄⋊C₄	—	4	2	no	—
39	C₂×C₅×C₂²⋊C₄	—	4	2	no	—
40	C₅×Q₃₂	—	2	8	no	—
41	C₂×C₅×C₁₆	—	2	2	no	—
42	C₂×C₅×C₁₆	—	2	8	no	—
43	C₂×C₅×C₁₆	D₄	1	4	yes	—
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×#103	—	2	8	no	—
49	C₂²×Q₃₂	C₂×C₅	1	1	yes	characteristic, complemented
50	C₂³×C₁₆	C₂×C₅	1	1	yes	characteristic
51	C₂×#467	C₂×C₅	1	2	yes	—
52	#2216	C₂×C₅	1	8	yes	direct product factor
53	C₂×#366	C₂×C₅	1	1	yes	characteristic, complemented
54	C₂×#358	C₂×C₅	1	2	yes	complemented
55	C₂⁴×C₅	D₈	1	1	yes	characteristic
56	C₂×C₅×Q₈	—	2	4	no	—
57	C₂×C₅×Q₈	—	4	2	no	—
58	C₂×C₅×Q₈	—	4	4	no	—
59	C₂×C₅×Q₈	—	4	4	no	—
60	C₂²×C₄×C₅	—	2	2	no	—
61	C₂²×C₄×C₅	—	2	2	no	—
62	C₂²×C₄×C₅	—	4	2	no	—
63	C₂²×C₄×C₅	—	4	2	no	—
64	C₂²×C₄×C₅	D₈	1	2	yes	—
65	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
68	C₅×Q₁₆	—	2	4	no	—
69	C₅×Q₁₆	—	2	4	no	—
70	C₅×Q₁₆	—	4	4	no	—
71	C₂×C₅×C₈	—	2	2	no	—
72	C₂×C₅×C₈	—	2	4	no	—
73	C₂×C₅×C₈	—	2	2	no	—
74	C₂×C₅×C₈	—	2	2	no	—
75	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
76	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
77	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
78	C₂×C₅×C₈	#41	1	2	yes	—
79	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
80	C₅×C₄⋊C₄	—	2	4	no	—
81	C₅×C₄⋊C₄	—	4	2	no	—
82	C₅×C₄⋊C₄	—	4	8	no	—
83	C₅×C₂²⋊C₄	—	4	8	no	—
84	C₅×C₁₆	—	2	2	no	—
85	C₅×C₁₆	C₂×D₄	1	4	yes	—
86	C₂²×Q₁₆	C₂²×C₅	1	2	yes	—
87	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
88	C₂×C₂²⋊Q₈	—	2	2	no	—
89	C₂×Q₃₂	—	2	4	no	—
90	C₂×Q₃₂	C₂²×C₅	1	4	yes	complemented
91	C₂²×C₁₆	—	2	4	no	—
92	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
93	C₂²×C₁₆	C₂²×C₅	1	4	yes	—
94	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
95	#467	—	2	4	no	—
96	#467	C₂²×C₅	1	8	yes	—
97	C₂×#107	C₂²×C₅	1	2	yes	—
98	C₂×#103	—	2	2	no	—
99	#366	C₂²×C₅	1	4	yes	complemented
100	#358	C₂²×C₅	1	8	yes	complemented
101	C₂³×C₅	—	2	2	no	—
102	C₂³×C₅	—	2	2	no	—
103	C₂³×C₅	Q₃₂	1	4	yes	complemented
104	C₂³×C₅	C₂×D₈	1	2	yes	—
105	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
106	C₅×Q₈	—	4	4	no	—
107	C₅×Q₈	—	4	4	no	—
108	C₅×Q₈	—	8	4	no	—
109	C₂×C₄×C₅	—	2	2	no	—
110	C₂×C₄×C₅	—	2	2	no	—
111	C₂×C₄×C₅	—	2	2	no	—
112	C₂×C₄×C₅	—	2	4	no	—
113	C₂×C₄×C₅	—	4	4	no	—
114	C₂×C₄×C₅	—	4	4	no	—
115	C₂×C₄×C₅	—	8	4	no	—
116	C₂×C₄×C₅	—	8	4	no	—
117	C₂×C₄×C₅	#121	1	2	yes	—
118	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
119	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₅	#135	1	2	yes	—
121	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
122	C₅×C₈	—	2	2	no	—
123	C₅×C₈	#121	1	2	yes	—
124	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
125	C₅×C₈	C₂×#41	1	1	yes	characteristic
126	C₂²×Q₈	—	2	2	no	—
127	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
128	C₂×Q₁₆	—	2	4	no	—
129	C₂×Q₁₆	—	2	2	no	—
130	C₂×Q₁₆	—	2	4	no	—
131	C₂×Q₁₆	C₂³×C₅	1	4	yes	—
132	C₂²×C₈	—	2	2	no	—
133	C₂²×C₈	—	2	2	no	—
134	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
135	C₂²×C₈	C₅×D₄	1	2	yes	—
136	C₂²×C₈	C₂³×C₅	1	2	yes	—
137	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
138	C₂²⋊Q₈	—	2	8	no	—
139	C₂²⋊Q₈	—	4	4	no	—
140	C₂×C₄⋊C₄	—	2	2	no	—
141	C₂×C₄⋊C₄	—	4	2	no	—
142	C₂×C₂²⋊C₄	—	4	2	no	—
143	Q₃₂	—	2	8	no	—
144	C₂×C₁₆	—	2	2	no	—
145	C₂×C₁₆	—	2	8	no	—
146	C₂×C₁₆	C₅×D₄	1	4	yes	—
147	C₂×C₁₆	C₂³×C₅	1	2	yes	—
148	C₂×C₁₆	C₂³×C₅	1	2	yes	—
149	#107	—	2	2	no	—
150	#107	C₂³×C₅	1	4	yes	—
151	#103	—	2	8	no	—
152	C₂²×C₅	—	2	2	no	—
153	C₂²×C₅	—	2	8	no	—
154	C₂²×C₅	—	2	2	no	—
155	C₂²×C₅	#466	1	2	yes	—
156	C₂²×C₅	C₂×Q₃₂	1	2	yes	—
157	C₂²×C₅	C₂×Q₃₂	1	4	yes	complemented
158	C₂²×C₅	#508	1	2	yes	—
159	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
160	C₄×C₅	—	2	2	no	—
161	C₄×C₅	—	8	4	no	—
162	C₄×C₅	—	8	4	no	—
163	C₄×C₅	#466	1	2	yes	—
164	C₄×C₅	C₂×#121	1	1	yes	characteristic
165	C₄×C₅	C₂×#135	1	1	yes	characteristic
166	C₂⁴	C₅×D₈	1	1	yes	characteristic
167	C₂×Q₈	—	2	4	no	—
168	C₂×Q₈	—	4	2	no	—
169	C₂×Q₈	—	4	4	no	—
170	C₂×Q₈	—	4	4	no	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	4	2	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
176	C₂²×C₄	C₅×D₈	1	2	yes	—
177	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
178	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
179	Q₁₆	—	2	4	no	—
180	Q₁₆	—	2	4	no	—
181	Q₁₆	—	4	4	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
187	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
188	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
189	C₂×C₈	C₅×#41	1	2	yes	—
190	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
191	C₄⋊C₄	—	2	4	no	—
192	C₄⋊C₄	—	4	2	no	—
193	C₄⋊C₄	—	4	8	no	—
194	C₂²⋊C₄	—	4	8	no	—
195	C₁₆	—	2	2	no	—
196	C₁₆	C₂×C₅×D₄	1	4	yes	—
197	C₂×C₅	—	2	4	no	—
198	C₂×C₅	#2216	1	4	yes	direct product factor
199	C₂×C₅	C₂×#466	1	1	yes	characteristic
200	C₂×C₅	C₂²×Q₃₂	1	1	yes	characteristic
201	C₂×C₅	C₂×#508	1	1	yes	characteristic
202	C₂³	—	2	2	no	—
203	C₂³	—	2	2	no	—
204	C₂³	C₅×Q₃₂	1	4	yes	complemented
205	C₂³	C₂×C₅×D₈	1	2	yes	—
206	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
207	Q₈	—	4	4	no	—
208	Q₈	—	4	4	no	—
209	Q₈	—	8	4	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	8	4	no	—
217	C₂×C₄	—	8	4	no	—
218	C₂×C₄	C₅×#121	1	2	yes	—
219	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
220	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
221	C₂×C₄	C₅×#135	1	2	yes	—
222	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	C₅×#121	1	2	yes	—
225	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
226	C₈	C₂×C₅×#41	1	1	yes	characteristic
227	C₅	C₂×#2216	1	1	yes	characteristic, direct product factor, Sylow
228	C₂²	—	2	2	no	—
229	C₂²	—	2	2	no	—
230	C₂²	—	2	8	no	—
231	C₂²	C₅×#466	1	2	yes	—
232	C₂²	C₂×C₅×Q₃₂	1	2	yes	—
233	C₂²	C₂×C₅×Q₃₂	1	4	yes	complemented
234	C₂²	C₅×#508	1	2	yes	—
235	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	8	4	no	—
238	C₄	—	8	4	no	—
239	C₄	C₅×#466	1	2	yes	—
240	C₄	C₂×C₅×#121	1	1	yes	characteristic
241	C₄	C₂×C₅×#135	1	1	yes	characteristic
242	C₂	—	2	4	no	—
243	C₂	C₅×#2216	1	4	yes	direct product factor
244	C₂	C₂×C₅×#466	1	1	yes	characteristic
245	C₂	C₂²×C₅×Q₃₂	1	1	yes	characteristic
246	C₂	C₂×C₅×#508	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₂⁴×C₅	D₈
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×Q₃₂	C₂	semidirect
4. C₅×#2216	C₂	direct (N × H)
5. C₂×C₅×#366	C₂	semidirect
6. C₂×C₅×#358	C₂	semidirect
11. C₂×C₅×Q₃₂	C₂²	semidirect
20. C₅×#366	C₂²	semidirect
21. C₅×#358	C₂²	semidirect
22. C₂×#2216	C₅	direct (N × H)
49. C₂²×Q₃₂	C₂×C₅	semidirect
52. #2216	C₂×C₅	direct (N × H)
53. C₂×#366	C₂×C₅	semidirect
54. C₂×#358	C₂×C₅	semidirect
90. C₂×Q₃₂	C₂²×C₅	semidirect
99. #366	C₂²×C₅	semidirect
100. #358	C₂²×C₅	semidirect
103. C₂³×C₅	Q₃₂	semidirect
157. C₂²×C₅	C₂×Q₃₂	semidirect
198. C₂×C₅	#2216	direct (N × H)
204. C₂³	C₅×Q₃₂	semidirect
227. C₅	C₂×#2216	direct (N × H)
233. C₂²	C₂×C₅×Q₃₂	semidirect
243. C₂	C₅×#2216	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×Q₃₂	2
C₂³×C₅×C₁₆	2
C₂×C₅×#467	2
C₅×#2216	2
C₂×C₅×#366	2
C₂×C₅×#358	2
C₂×#2216	5

Subgroups — C2×C5×#2216 #11956318

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2216 #11956318