Subgroups — C₂²×C₅×#350 #11956058

Order 1280.

1174 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#131	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×#350	C₂	1	12	yes	maximal, direct product factor
5	C₂²×C₅×#41	C₄	1	1	yes	characteristic
6	C₂×C₅×#131	—	2	3	no	—
7	C₂×C₅×#131	—	2	3	no	—
8	C₂×C₅×#131	C₄	1	6	yes	—
9	C₂×C₅×#131	C₂²	1	6	yes	—
10	C₂²×C₅×M₁₆	—	2	1	no	—
11	C₂²×C₅×M₁₆	C₄	1	1	yes	characteristic
12	C₂³×C₅×C₈	—	2	1	no	—
13	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₅×M₃₂	—	2	3	no	—
15	C₂×C₅×M₃₂	C₂²	1	6	yes	complemented
16	C₂²×C₅×C₁₆	—	2	1	no	—
17	C₂²×C₅×C₁₆	—	2	3	no	—
18	C₂²×C₅×C₁₆	—	2	1	no	—
19	C₂²×C₅×C₁₆	C₂²	1	6	yes	complemented
20	C₅×#350	C₂²	1	16	yes	direct product factor
21	C₂²×#350	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂×C₅×#41	—	2	3	no	—
23	C₂×C₅×#41	—	2	3	no	—
24	C₂×C₅×#41	C₈	1	6	yes	—
25	C₂×C₅×#41	C₂×C₄	1	6	yes	—
26	C₂²×C₅×Q₈	C₈	1	1	yes	characteristic
27	C₂²×C₅×D₄	—	2	1	no	—
28	C₂²×C₅×D₄	C₈	1	1	yes	characteristic
29	C₂³×C₄×C₅	—	2	1	no	—
30	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
31	C₅×#131	—	2	12	no	—
32	C₅×#131	—	2	6	no	—
33	C₅×#131	—	2	6	no	—
34	C₅×#131	C₂×C₄	1	12	yes	—
35	C₅×#131	C₂³	1	4	yes	—
36	C₂×C₅×M₁₆	—	2	3	no	—
37	C₂×C₅×M₁₆	—	2	6	no	—
38	C₂×C₅×M₁₆	—	2	6	no	—
39	C₂×C₅×M₁₆	C₂×C₄	1	6	yes	—
40	C₂²×C₅×C₈	—	2	1	no	—
41	C₂²×C₅×C₈	—	2	6	no	—
42	C₂²×C₅×C₈	—	2	6	no	—
43	C₂²×C₅×C₈	C₂×C₄	1	3	yes	—
44	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
45	C₂²×C₅×C₈	D₄	1	3	yes	—
46	C₂²×C₅×C₈	D₄	1	3	yes	—
47	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
48	C₂²×C₅×C₈	C₂³	1	3	yes	—
49	C₅×M₃₂	—	2	6	no	—
50	C₅×M₃₂	C₂³	1	4	yes	complemented
51	C₂×C₅×C₁₆	—	2	6	no	—
52	C₂×C₅×C₁₆	—	2	6	no	—
53	C₂×C₅×C₁₆	—	2	6	no	—
54	C₂×C₅×C₁₆	C₂³	1	4	yes	complemented
55	C₂²×#131	C₂×C₅	1	1	yes	characteristic
56	C₂²×M₃₂	C₂×C₅	1	1	yes	characteristic, complemented
57	C₂³×C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
58	C₂×#350	C₂×C₅	1	12	yes	direct product factor
59	C₂⁴×C₅	—	2	1	no	—
60	C₂⁴×C₅	M₁₆	1	1	yes	characteristic
61	C₅×#41	—	2	12	no	—
62	C₅×#41	—	2	6	no	—
63	C₅×#41	—	2	6	no	—
64	C₅×#41	C₂×C₈	1	12	yes	—
65	C₅×#41	C₂²×C₄	1	4	yes	—
66	C₂×C₅×Q₈	—	2	3	no	—
67	C₂×C₅×Q₈	C₂×C₈	1	6	yes	—
68	C₂×C₅×D₄	—	2	6	no	—
69	C₂×C₅×D₄	—	2	6	no	—
70	C₂×C₅×D₄	—	2	3	no	—
71	C₂×C₅×D₄	C₂×C₈	1	6	yes	—
72	C₂²×C₄×C₅	—	2	1	no	—
73	C₂²×C₄×C₅	—	2	6	no	—
74	C₂²×C₄×C₅	—	2	6	no	—
75	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂²⋊C₄	1	3	yes	—
77	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂×C₈	1	3	yes	—
79	C₂²×C₄×C₅	M₁₆	1	3	yes	—
80	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
81	C₅×M₁₆	—	2	6	no	—
82	C₅×M₁₆	—	2	12	no	—
83	C₅×M₁₆	—	2	4	no	—
84	C₅×M₁₆	C₂²×C₄	1	4	yes	—
85	C₂×C₅×C₈	—	2	6	no	—
86	C₂×C₅×C₈	—	2	12	no	—
87	C₂×C₅×C₈	—	2	4	no	—
88	C₂×C₅×C₈	C₂²⋊C₄	1	3	yes	—
89	C₂×C₅×C₈	C₂²⋊C₄	1	3	yes	—
90	C₂×C₅×C₈	C₂²⋊C₄	1	6	yes	—
91	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
92	C₂×C₅×C₈	C₂×D₄	1	3	yes	—
93	C₂×C₅×C₈	C₂×D₄	1	3	yes	—
94	C₂×C₅×C₈	C₂×D₄	1	3	yes	—
95	C₂×C₅×C₈	C₂×D₄	1	3	yes	—
96	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
97	C₅×C₁₆	—	2	4	no	—
98	C₅×C₁₆	—	2	4	no	—
99	C₂²×#41	C₄×C₅	1	1	yes	characteristic
100	C₂×#131	—	2	3	no	—
101	C₂×#131	—	2	3	no	—
102	C₂×#131	C₄×C₅	1	6	yes	—
103	C₂×#131	C₂²×C₅	1	6	yes	—
104	C₂²×M₁₆	—	2	1	no	—
105	C₂²×M₁₆	C₄×C₅	1	1	yes	characteristic
106	C₂³×C₈	—	2	1	no	—
107	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
108	C₂×M₃₂	—	2	3	no	—
109	C₂×M₃₂	C₂²×C₅	1	6	yes	complemented
110	C₂²×C₁₆	—	2	1	no	—
111	C₂²×C₁₆	—	2	3	no	—
112	C₂²×C₁₆	—	2	1	no	—
113	C₂²×C₁₆	C₂²×C₅	1	6	yes	complemented
114	#350	C₂²×C₅	1	16	yes	direct product factor
115	C₂³×C₅	—	2	4	no	—
116	C₂³×C₅	—	2	6	no	—
117	C₂³×C₅	—	4	4	no	—
118	C₂³×C₅	C₂²⋊C₈	1	3	yes	—
119	C₂³×C₅	C₂²⋊C₈	1	1	yes	characteristic
120	C₂³×C₅	C₂×M₁₆	1	3	yes	—
121	C₅×Q₈	—	2	6	no	—
122	C₅×Q₈	C₂²×C₈	1	4	yes	—
123	C₅×D₄	—	2	4	no	—
124	C₅×D₄	—	2	12	no	—
125	C₅×D₄	—	2	6	no	—
126	C₅×D₄	C₂²×C₈	1	4	yes	—
127	C₂×C₄×C₅	—	2	6	no	—
128	C₂×C₄×C₅	—	2	4	no	—
129	C₂×C₄×C₅	—	2	12	no	—
130	C₂×C₄×C₅	C₂²⋊C₈	1	3	yes	—
131	C₂×C₄×C₅	C₂²⋊C₈	1	3	yes	—
132	C₂×C₄×C₅	C₂²⋊C₈	1	6	yes	—
133	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	3	yes	—
134	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	3	yes	—
135	C₂×C₄×C₅	C₂²×C₈	1	3	yes	—
136	C₂×C₄×C₅	C₂²×C₈	1	3	yes	—
137	C₂×C₄×C₅	C₂×M₁₆	1	3	yes	—
138	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
139	C₅×C₈	—	2	4	no	—
140	C₅×C₈	C₂×C₂²⋊C₄	1	3	yes	—
141	C₅×C₈	C₂×C₂²⋊C₄	1	3	yes	—
142	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
143	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
144	C₂×#41	—	2	3	no	—
145	C₂×#41	—	2	3	no	—
146	C₂×#41	C₅×C₈	1	6	yes	—
147	C₂×#41	C₂×C₄×C₅	1	6	yes	—
148	C₂²×Q₈	C₅×C₈	1	1	yes	characteristic
149	C₂²×D₄	—	2	1	no	—
150	C₂²×D₄	C₅×C₈	1	1	yes	characteristic
151	C₂³×C₄	—	2	1	no	—
152	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
153	#131	—	2	12	no	—
154	#131	—	2	6	no	—
155	#131	—	2	6	no	—
156	#131	C₂×C₄×C₅	1	12	yes	—
157	#131	C₂³×C₅	1	4	yes	—
158	C₂×M₁₆	—	2	3	no	—
159	C₂×M₁₆	—	2	6	no	—
160	C₂×M₁₆	—	2	6	no	—
161	C₂×M₁₆	C₂×C₄×C₅	1	6	yes	—
162	C₂²×C₈	—	2	1	no	—
163	C₂²×C₈	—	2	6	no	—
164	C₂²×C₈	—	2	6	no	—
165	C₂²×C₈	C₂×C₄×C₅	1	3	yes	—
166	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
167	C₂²×C₈	C₅×D₄	1	3	yes	—
168	C₂²×C₈	C₅×D₄	1	3	yes	—
169	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
170	C₂²×C₈	C₂³×C₅	1	3	yes	—
171	M₃₂	—	2	6	no	—
172	M₃₂	C₂³×C₅	1	4	yes	complemented
173	C₂×C₁₆	—	2	6	no	—
174	C₂×C₁₆	—	2	6	no	—
175	C₂×C₁₆	—	2	6	no	—
176	C₂×C₁₆	C₂³×C₅	1	4	yes	complemented
177	C₂²×C₅	—	2	6	no	—
178	C₂²×C₅	—	2	6	no	—
179	C₂²×C₅	—	2	4	no	—
180	C₂²×C₅	—	4	12	no	—
181	C₂²×C₅	#350	1	4	yes	direct product factor
182	C₂²×C₅	C₂×C₂²⋊C₈	1	3	yes	—
183	C₂²×C₅	C₂×C₂²⋊C₈	1	3	yes	—
184	C₂²×C₅	C₂²×M₁₆	1	1	yes	characteristic
185	C₄×C₅	—	2	4	no	—
186	C₄×C₅	C₂×C₂²⋊C₈	1	3	yes	—
187	C₄×C₅	C₂×C₂²⋊C₈	1	3	yes	—
188	C₄×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
189	C₄×C₅	C₂³×C₈	1	1	yes	characteristic
190	C₂⁴	—	2	1	no	—
191	C₂⁴	C₅×M₁₆	1	1	yes	characteristic
192	#41	—	2	6	no	—
193	#41	—	2	6	no	—
194	#41	—	2	12	no	—
195	#41	C₂×C₅×C₈	1	12	yes	—
196	#41	C₂²×C₄×C₅	1	4	yes	—
197	C₂×Q₈	—	2	3	no	—
198	C₂×Q₈	C₂×C₅×C₈	1	6	yes	—
199	C₂×D₄	—	2	6	no	—
200	C₂×D₄	—	2	6	no	—
201	C₂×D₄	—	2	3	no	—
202	C₂×D₄	C₂×C₅×C₈	1	6	yes	—
203	C₂²×C₄	—	2	1	no	—
204	C₂²×C₄	—	2	6	no	—
205	C₂²×C₄	—	2	6	no	—
206	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
207	C₂²×C₄	C₅×C₂²⋊C₄	1	3	yes	—
208	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₅×C₈	1	3	yes	—
210	C₂²×C₄	C₅×M₁₆	1	3	yes	—
211	C₂²×C₄	C₂²×C₄×C₅	1	3	yes	—
212	M₁₆	—	2	6	no	—
213	M₁₆	—	2	12	no	—
214	M₁₆	—	2	4	no	—
215	M₁₆	C₂²×C₄×C₅	1	4	yes	—
216	C₂×C₈	—	2	6	no	—
217	C₂×C₈	—	2	12	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	C₅×C₂²⋊C₄	1	3	yes	—
220	C₂×C₈	C₅×C₂²⋊C₄	1	3	yes	—
221	C₂×C₈	C₅×C₂²⋊C₄	1	6	yes	—
222	C₂×C₈	C₂²×C₄×C₅	1	3	yes	—
223	C₂×C₈	C₂×C₅×D₄	1	3	yes	—
224	C₂×C₈	C₂×C₅×D₄	1	3	yes	—
225	C₂×C₈	C₂×C₅×D₄	1	3	yes	—
226	C₂×C₈	C₂×C₅×D₄	1	3	yes	—
227	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
228	C₁₆	—	2	4	no	—
229	C₁₆	—	2	4	no	—
230	C₂×C₅	—	2	1	no	—
231	C₂×C₅	—	2	3	no	—
232	C₂×C₅	—	4	4	no	—
233	C₂×C₅	C₂×#350	1	6	yes	direct product factor
234	C₂×C₅	C₂²×C₂²⋊C₈	1	1	yes	characteristic
235	C₂³	—	2	4	no	—
236	C₂³	—	2	6	no	—
237	C₂³	—	4	4	no	—
238	C₂³	C₅×C₂²⋊C₈	1	3	yes	—
239	C₂³	C₅×C₂²⋊C₈	1	1	yes	characteristic
240	C₂³	C₂×C₅×M₁₆	1	3	yes	—
241	Q₈	—	2	6	no	—
242	Q₈	C₂²×C₅×C₈	1	4	yes	—
243	D₄	—	2	12	no	—
244	D₄	—	2	4	no	—
245	D₄	—	2	6	no	—
246	D₄	C₂²×C₅×C₈	1	4	yes	—
247	C₂×C₄	—	2	6	no	—
248	C₂×C₄	—	2	12	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	C₅×C₂²⋊C₈	1	3	yes	—
251	C₂×C₄	C₅×C₂²⋊C₈	1	3	yes	—
252	C₂×C₄	C₅×C₂²⋊C₈	1	6	yes	—
253	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	3	yes	—
254	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	3	yes	—
255	C₂×C₄	C₂²×C₅×C₈	1	3	yes	—
256	C₂×C₄	C₂²×C₅×C₈	1	3	yes	—
257	C₂×C₄	C₂×C₅×M₁₆	1	3	yes	—
258	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
259	C₈	—	2	4	no	—
260	C₈	C₂×C₅×C₂²⋊C₄	1	3	yes	—
261	C₈	C₂×C₅×C₂²⋊C₄	1	3	yes	—
262	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
263	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
264	C₅	C₂²×#350	1	1	yes	characteristic, direct product factor, Sylow
265	C₂²	—	2	6	no	—
266	C₂²	—	2	6	no	—
267	C₂²	—	2	4	no	—
268	C₂²	—	4	12	no	—
269	C₂²	C₅×#350	1	4	yes	direct product factor
270	C₂²	C₂×C₅×C₂²⋊C₈	1	3	yes	—
271	C₂²	C₂×C₅×C₂²⋊C₈	1	3	yes	—
272	C₂²	C₂²×C₅×M₁₆	1	1	yes	characteristic
273	C₄	—	2	4	no	—
274	C₄	C₂×C₅×C₂²⋊C₈	1	3	yes	—
275	C₄	C₂×C₅×C₂²⋊C₈	1	3	yes	—
276	C₄	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
277	C₄	C₂³×C₅×C₈	1	1	yes	characteristic
278	C₂	—	2	3	no	—
279	C₂	—	2	1	no	—
280	C₂	—	4	4	no	—
281	C₂	C₂×C₅×#350	1	6	yes	direct product factor
282	C₂	C₂²×C₅×C₂²⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅×C₈	D₄
commutator	C₄	C₂³×C₅×C₈
Frattini, cosocle	C₂×C₈	C₂⁴×C₅
socle	C₂³×C₅	C₂²⋊C₈
prime radical	C₂²×C₅×#41	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅×C₈ ◁ C₂³×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. C₂³×C₅×C₁₆	C₂	semidirect
4. C₂×C₅×#350	C₂	direct (N × H)
15. C₂×C₅×M₃₂	C₂²	semidirect
19. C₂²×C₅×C₁₆	C₂²	semidirect
20. C₅×#350	C₂²	direct (N × H)
21. C₂²×#350	C₅	direct (N × H)
50. C₅×M₃₂	C₂³	semidirect
54. C₂×C₅×C₁₆	C₂³	semidirect
56. C₂²×M₃₂	C₂×C₅	semidirect
57. C₂³×C₁₆	C₂×C₅	semidirect
58. C₂×#350	C₂×C₅	direct (N × H)
109. C₂×M₃₂	C₂²×C₅	semidirect
113. C₂²×C₁₆	C₂²×C₅	semidirect
114. #350	C₂²×C₅	direct (N × H)
172. M₃₂	C₂³×C₅	semidirect
176. C₂×C₁₆	C₂³×C₅	semidirect
181. C₂²×C₅	#350	direct (N × H)
233. C₂×C₅	C₂×#350	direct (N × H)
264. C₅	C₂²×#350	direct (N × H)
269. C₂²	C₅×#350	direct (N × H)
281. C₂	C₂×C₅×#350	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×#131	2
C₂²×C₅×M₃₂	2
C₂³×C₅×C₁₆	2
C₂×C₅×#350	2
C₂²×#350	5

Subgroups — C22×C5×#350 #11956058

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×#350 #11956058