Subgroups — C₂×C₅×#2112 #11956051

Order 1280.

1406 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×M₃₂	C₂	1	2	yes	maximal, complemented
3	C₅×#2112	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₅×C₂²⋊C₁₆	C₂	1	4	yes	maximal, complemented
5	C₂⁴×C₄×C₅	C₄	1	1	yes	characteristic
6	C₂³×C₅×C₈	—	2	8	no	—
7	C₂³×C₅×C₈	C₄	1	1	yes	characteristic
8	C₂³×C₅×C₈	C₄	1	4	yes	—
9	C₂³×C₅×C₈	C₄	1	2	yes	—
10	C₂³×C₅×C₈	C₂²	1	4	yes	—
11	C₂³×C₅×C₈	C₂²	1	2	yes	—
12	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₅×M₃₂	—	2	8	no	—
14	C₂×C₅×M₃₂	C₂²	1	8	yes	complemented
15	C₂²×C₅×C₁₆	C₂²	1	4	yes	complemented
16	C₅×C₂²⋊C₁₆	C₂²	1	16	yes	complemented
17	C₂×#2112	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂⁵×C₅	C₈	1	1	yes	characteristic
19	C₂³×C₄×C₅	—	2	4	no	—
20	C₂³×C₄×C₅	—	2	4	no	—
21	C₂³×C₄×C₅	C₈	1	1	yes	characteristic
22	C₂³×C₄×C₅	C₈	1	4	yes	—
23	C₂³×C₄×C₅	C₈	1	2	yes	—
24	C₂³×C₄×C₅	C₂×C₄	1	4	yes	—
25	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	—	2	4	no	—
28	C₂²×C₅×C₈	—	2	16	no	—
29	C₂²×C₅×C₈	—	2	8	no	—
30	C₂²×C₅×C₈	—	2	16	no	—
31	C₂²×C₅×C₈	—	2	8	no	—
32	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
33	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
34	C₂²×C₅×C₈	C₂×C₄	1	8	yes	—
35	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
36	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
38	C₂²×C₅×C₈	D₄	1	8	yes	—
39	C₂²×C₅×C₈	C₂³	1	4	yes	—
40	C₂²×C₅×C₈	C₂³	1	2	yes	—
41	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
42	C₅×M₃₂	—	2	16	no	—
43	C₂×C₅×C₁₆	—	2	8	no	—
44	C₂×C₅×C₁₆	C₂³	1	8	yes	complemented
45	C₂⁴×C₈	C₂×C₅	1	1	yes	characteristic
46	C₂²×M₃₂	C₂×C₅	1	2	yes	complemented
47	#2112	C₂×C₅	1	8	yes	direct product factor
48	C₂×C₂²⋊C₁₆	C₂×C₅	1	4	yes	complemented
49	C₂⁴×C₅	—	2	4	no	—
50	C₂⁴×C₅	—	2	8	no	—
51	C₂⁴×C₅	C₂×C₈	1	4	yes	—
52	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
53	C₂⁴×C₅	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₅	—	2	4	no	—
55	C₂²×C₄×C₅	—	2	16	no	—
56	C₂²×C₄×C₅	—	2	8	no	—
57	C₂²×C₄×C₅	—	2	8	no	—
58	C₂²×C₄×C₅	—	2	4	no	—
59	C₂²×C₄×C₅	—	2	8	no	—
60	C₂²×C₄×C₅	—	2	4	no	—
61	C₂²×C₄×C₅	C₂²⋊C₄	1	4	yes	—
62	C₂²×C₄×C₅	C₂×C₈	1	4	yes	—
63	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
64	C₂²×C₄×C₅	C₂×C₈	1	8	yes	—
65	C₂²×C₄×C₅	C₂×C₈	1	4	yes	—
66	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
67	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
68	C₂²×C₄×C₅	M₁₆	1	4	yes	—
69	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
70	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₅×C₈	—	2	8	no	—
73	C₂×C₅×C₈	—	2	4	no	—
74	C₂×C₅×C₈	—	2	16	no	—
75	C₂×C₅×C₈	—	2	8	no	—
76	C₂×C₅×C₈	—	2	8	no	—
77	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
78	C₂×C₅×C₈	C₂²⋊C₄	1	8	yes	—
79	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
80	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
81	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₅×C₈	C₂×D₄	1	8	yes	—
83	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
84	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
85	C₅×C₁₆	—	2	8	no	—
86	C₂⁴×C₄	C₄×C₅	1	1	yes	characteristic
87	C₂³×C₈	—	2	8	no	—
88	C₂³×C₈	C₄×C₅	1	1	yes	characteristic
89	C₂³×C₈	C₄×C₅	1	2	yes	—
90	C₂³×C₈	C₄×C₅	1	4	yes	—
91	C₂³×C₈	C₂²×C₅	1	2	yes	—
92	C₂³×C₈	C₂²×C₅	1	4	yes	—
93	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
94	C₂×M₃₂	—	2	8	no	—
95	C₂×M₃₂	C₂²×C₅	1	8	yes	complemented
96	C₂²×C₁₆	C₂²×C₅	1	4	yes	complemented
97	C₂²⋊C₁₆	C₂²×C₅	1	16	yes	complemented
98	C₂³×C₅	—	2	32	no	—
99	C₂³×C₅	—	2	8	no	—
100	C₂³×C₅	—	2	4	no	—
101	C₂³×C₅	—	2	4	no	—
102	C₂³×C₅	—	2	4	no	—
103	C₂³×C₅	—	2	16	no	—
104	C₂³×C₅	C₂²⋊C₈	1	4	yes	—
105	C₂³×C₅	M₃₂	1	8	yes	complemented
106	C₂³×C₅	C₂²×C₈	1	2	yes	—
107	C₂³×C₅	C₂²×C₈	1	4	yes	—
108	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₅	—	2	8	no	—
110	C₂×C₄×C₅	—	2	4	no	—
111	C₂×C₄×C₅	—	2	4	no	—
112	C₂×C₄×C₅	—	2	4	no	—
113	C₂×C₄×C₅	—	2	8	no	—
114	C₂×C₄×C₅	—	2	16	no	—
115	C₂×C₄×C₅	C₂²⋊C₈	1	8	yes	—
116	C₂×C₄×C₅	C₂²⋊C₈	1	4	yes	—
117	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	4	yes	—
118	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
119	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
120	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
121	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
123	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
124	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
125	C₅×C₈	—	2	4	no	—
126	C₅×C₈	C₂×C₂²⋊C₄	1	4	yes	—
127	C₅×C₈	C₂×C₂²⋊C₄	1	2	yes	—
128	C₅×C₈	C₂²×D₄	1	2	yes	—
129	C₂⁵	C₅×C₈	1	1	yes	characteristic
130	C₂³×C₄	—	2	4	no	—
131	C₂³×C₄	—	2	4	no	—
132	C₂³×C₄	C₅×C₈	1	1	yes	characteristic
133	C₂³×C₄	C₅×C₈	1	4	yes	—
134	C₂³×C₄	C₅×C₈	1	2	yes	—
135	C₂³×C₄	C₂×C₄×C₅	1	4	yes	—
136	C₂³×C₄	C₂×C₄×C₅	1	2	yes	—
137	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
138	C₂²×C₈	—	2	4	no	—
139	C₂²×C₈	—	2	16	no	—
140	C₂²×C₈	—	2	8	no	—
141	C₂²×C₈	—	2	8	no	—
142	C₂²×C₈	—	2	16	no	—
143	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
144	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
145	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
146	C₂²×C₈	C₂×C₄×C₅	1	8	yes	—
147	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
148	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
149	C₂²×C₈	C₅×D₄	1	8	yes	—
150	C₂²×C₈	C₂³×C₅	1	4	yes	—
151	C₂²×C₈	C₂³×C₅	1	2	yes	—
152	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
153	M₃₂	—	2	16	no	—
154	C₂×C₁₆	—	2	8	no	—
155	C₂×C₁₆	C₂³×C₅	1	8	yes	complemented
156	C₂²×C₅	—	2	32	no	—
157	C₂²×C₅	—	2	4	no	—
158	C₂²×C₅	—	2	8	no	—
159	C₂²×C₅	—	2	16	no	—
160	C₂²×C₅	—	2	4	no	—
161	C₂²×C₅	—	2	4	no	—
162	C₂²×C₅	C₂×C₂²⋊C₈	1	4	yes	—
163	C₂²×C₅	C₂×C₂²⋊C₈	1	2	yes	—
164	C₂²×C₅	C₂×M₃₂	1	4	yes	—
165	C₂²×C₅	C₂×M₃₂	1	8	yes	complemented
166	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
167	C₄×C₅	—	2	4	no	—
168	C₄×C₅	C₂×C₂²⋊C₈	1	2	yes	—
169	C₄×C₅	C₂×C₂²⋊C₈	1	4	yes	—
170	C₄×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
171	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
172	C₂⁴	—	2	4	no	—
173	C₂⁴	—	2	8	no	—
174	C₂⁴	C₂×C₅×C₈	1	4	yes	—
175	C₂⁴	C₂×C₅×C₈	1	1	yes	characteristic
176	C₂⁴	C₂×C₅×C₈	1	2	yes	—
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	—	2	8	no	—
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	2	8	no	—
181	C₂²×C₄	—	2	4	no	—
182	C₂²×C₄	—	2	16	no	—
183	C₂²×C₄	—	2	8	no	—
184	C₂²×C₄	C₅×C₂²⋊C₄	1	4	yes	—
185	C₂²×C₄	C₂×C₅×C₈	1	4	yes	—
186	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
187	C₂²×C₄	C₂×C₅×C₈	1	4	yes	—
188	C₂²×C₄	C₂×C₅×C₈	1	8	yes	—
189	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
190	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
191	C₂²×C₄	C₅×M₁₆	1	4	yes	—
192	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
193	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
194	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
195	C₂×C₈	—	2	8	no	—
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	—	2	8	no	—
198	C₂×C₈	—	2	16	no	—
199	C₂×C₈	—	2	8	no	—
200	C₂×C₈	C₅×C₂²⋊C₄	1	8	yes	—
201	C₂×C₈	C₅×C₂²⋊C₄	1	4	yes	—
202	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
203	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
204	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
205	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
206	C₂×C₈	C₂×C₅×D₄	1	8	yes	—
207	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
208	C₁₆	—	2	8	no	—
209	C₂×C₅	—	2	4	no	—
210	C₂×C₅	—	2	8	no	—
211	C₂×C₅	#2112	1	4	yes	direct product factor
212	C₂×C₅	C₂²×C₂²⋊C₈	1	1	yes	characteristic
213	C₂×C₅	C₂²×M₃₂	1	2	yes	—
214	C₂³	—	2	32	no	—
215	C₂³	—	2	8	no	—
216	C₂³	—	2	4	no	—
217	C₂³	—	2	4	no	—
218	C₂³	—	2	4	no	—
219	C₂³	—	2	16	no	—
220	C₂³	C₅×C₂²⋊C₈	1	4	yes	—
221	C₂³	C₅×M₃₂	1	8	yes	complemented
222	C₂³	C₂²×C₅×C₈	1	2	yes	—
223	C₂³	C₂²×C₅×C₈	1	4	yes	—
224	C₂³	C₂²×C₅×C₈	1	1	yes	characteristic
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	16	no	—
228	C₂×C₄	—	2	8	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	C₅×C₂²⋊C₈	1	8	yes	—
232	C₂×C₄	C₅×C₂²⋊C₈	1	4	yes	—
233	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	4	yes	—
234	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
235	C₂×C₄	C₂²×C₅×C₈	1	4	yes	—
236	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
237	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
238	C₂×C₄	C₂×C₅×M₁₆	1	4	yes	—
239	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
240	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
241	C₈	—	2	4	no	—
242	C₈	C₂×C₅×C₂²⋊C₄	1	2	yes	—
243	C₈	C₂×C₅×C₂²⋊C₄	1	4	yes	—
244	C₈	C₂²×C₅×D₄	1	2	yes	—
245	C₅	C₂×#2112	1	1	yes	characteristic, direct product factor, Sylow
246	C₂²	—	2	32	no	—
247	C₂²	—	2	8	no	—
248	C₂²	—	2	16	no	—
249	C₂²	—	2	4	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	4	no	—
252	C₂²	C₂×C₅×C₂²⋊C₈	1	4	yes	—
253	C₂²	C₂×C₅×C₂²⋊C₈	1	2	yes	—
254	C₂²	C₂×C₅×M₃₂	1	4	yes	—
255	C₂²	C₂×C₅×M₃₂	1	8	yes	complemented
256	C₂²	C₂³×C₅×C₈	1	1	yes	characteristic
257	C₄	—	2	4	no	—
258	C₄	C₂×C₅×C₂²⋊C₈	1	4	yes	—
259	C₄	C₂×C₅×C₂²⋊C₈	1	2	yes	—
260	C₄	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
261	C₄	C₂²×C₅×M₁₆	1	1	yes	characteristic
262	C₂	—	2	4	no	—
263	C₂	—	2	8	no	—
264	C₂	C₅×#2112	1	4	yes	direct product factor
265	C₂	C₂²×C₅×C₂²⋊C₈	1	1	yes	characteristic
266	C₂	C₂²×C₅×M₃₂	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅×C₈	C₂³
commutator	C₂²	C₂³×C₅×C₈
Frattini, cosocle	C₂×C₈	C₂⁴×C₅
socle	C₂³×C₅	C₂²×C₈
prime radical	C₂⁵×C₅	C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₅×M₃₂	C₂	semidirect
3. C₅×#2112	C₂	direct (N × H)
4. C₂×C₅×C₂²⋊C₁₆	C₂	semidirect
14. C₂×C₅×M₃₂	C₂²	semidirect
15. C₂²×C₅×C₁₆	C₂²	semidirect
16. C₅×C₂²⋊C₁₆	C₂²	semidirect
17. C₂×#2112	C₅	direct (N × H)
44. C₂×C₅×C₁₆	C₂³	semidirect
46. C₂²×M₃₂	C₂×C₅	semidirect
47. #2112	C₂×C₅	direct (N × H)
48. C₂×C₂²⋊C₁₆	C₂×C₅	semidirect
95. C₂×M₃₂	C₂²×C₅	semidirect
96. C₂²×C₁₆	C₂²×C₅	semidirect
97. C₂²⋊C₁₆	C₂²×C₅	semidirect
105. C₂³×C₅	M₃₂	semidirect
155. C₂×C₁₆	C₂³×C₅	semidirect
165. C₂²×C₅	C₂×M₃₂	semidirect
211. C₂×C₅	#2112	direct (N × H)
221. C₂³	C₅×M₃₂	semidirect
245. C₅	C₂×#2112	direct (N × H)
255. C₂²	C₂×C₅×M₃₂	semidirect
264. C₂	C₅×#2112	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂⁴×C₅×C₈	2
C₂²×C₅×M₃₂	2
C₅×#2112	2
C₂×C₅×C₂²⋊C₁₆	2
C₂×#2112	5

Subgroups — C2×C5×#2112 #11956051

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2112 #11956051