Subgroups — C₂×C₅×#1292 #11930719

Order 1280.

690 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#423	C₂	1	2	yes	maximal
2	C₂²×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₅×#1292	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₅×M₁₆	—	2	2	no	—
5	C₂³×C₅×C₈	C₄	1	2	yes	—
6	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₅×#423	—	2	2	no	—
8	C₅×#423	—	2	2	no	—
9	C₅×#423	C₄	1	4	yes	—
10	C₅×#423	C₂²	1	4	yes	—
11	C₂×C₅×C₄⋊C₈	—	2	2	no	—
12	C₂×C₅×C₄⋊C₈	—	2	2	no	—
13	C₂×C₅×C₄⋊C₈	—	2	2	no	—
14	C₂×C₅×C₄⋊C₈	C₄	1	4	yes	—
15	C₂×C₅×C₄⋊C₈	C₄	1	2	yes	—
16	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×#1292	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
20	C₂³×C₄×C₅	Q₈	1	1	yes	characteristic
21	C₂×C₅×M₁₆	—	2	4	no	—
22	C₂×C₅×M₁₆	—	2	4	no	—
23	C₂×C₅×M₁₆	—	4	4	no	—
24	C₂×C₅×M₁₆	—	4	4	no	—
25	C₂²×C₅×C₈	—	2	2	no	—
26	C₂²×C₅×C₈	—	2	2	no	—
27	C₂²×C₅×C₈	—	2	4	no	—
28	C₂²×C₅×C₈	—	4	4	no	—
29	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₅	D₄	1	2	yes	—
35	C₂×C₄²×C₅	D₄	1	2	yes	—
36	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
37	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
38	C₅×C₄⋊C₈	—	2	4	no	—
39	C₅×C₄⋊C₈	—	2	4	no	—
40	C₅×C₄⋊C₈	—	2	4	no	—
41	C₅×C₄⋊C₈	—	2	4	no	—
42	C₅×C₄⋊C₈	—	2	4	no	—
43	C₅×C₄⋊C₈	C₂×C₄	1	8	yes	—
44	C₂×#423	C₂×C₅	1	2	yes	—
45	C₂²×C₄⋊C₈	C₂×C₅	1	1	yes	characteristic
46	#1292	C₂×C₅	1	4	yes	direct product factor
47	C₂⁴×C₅	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅	—	2	2	no	—
49	C₂²×C₄×C₅	—	2	2	no	—
50	C₂²×C₄×C₅	—	4	2	no	—
51	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
52	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
53	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	D₈	1	2	yes	—
63	C₂²×C₄×C₅	SD₁₆	1	4	yes	—
64	C₂²×C₄×C₅	Q₁₆	1	2	yes	—
65	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
68	C₅×M₁₆	—	4	8	no	—
69	C₅×M₁₆	—	4	4	no	—
70	C₅×M₁₆	—	4	4	no	—
71	C₂×C₅×C₈	—	2	4	no	—
72	C₂×C₅×C₈	—	2	8	no	—
73	C₂×C₅×C₈	—	4	4	no	—
74	C₂×C₅×C₈	—	4	4	no	—
75	C₂×C₅×C₈	—	4	8	no	—
76	C₂×C₅×C₈	—	4	4	no	—
77	C₄²×C₅	C₄²	1	2	yes	—
78	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
79	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
80	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
81	C₄²×C₅	C₄⋊C₄	1	2	yes	—
82	C₄²×C₅	C₄⋊C₄	1	2	yes	—
83	C₄²×C₅	C₂²×C₄	1	2	yes	—
84	C₄²×C₅	C₂×D₄	1	2	yes	—
85	C₂²×M₁₆	—	2	2	no	—
86	C₂³×C₈	C₄×C₅	1	2	yes	—
87	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
88	#423	—	2	2	no	—
89	#423	—	2	2	no	—
90	#423	C₄×C₅	1	4	yes	—
91	#423	C₂²×C₅	1	4	yes	—
92	C₂×C₄⋊C₈	—	2	2	no	—
93	C₂×C₄⋊C₈	—	2	2	no	—
94	C₂×C₄⋊C₈	—	2	2	no	—
95	C₂×C₄⋊C₈	C₄×C₅	1	2	yes	—
96	C₂×C₄⋊C₈	C₄×C₅	1	4	yes	—
97	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
98	C₂³×C₅	—	2	2	no	—
99	C₂³×C₅	—	2	2	no	—
100	C₂³×C₅	#95	1	1	yes	characteristic
101	C₂³×C₅	#95	1	1	yes	characteristic
102	C₂³×C₅	#106	1	2	yes	—
103	C₂³×C₅	#107	1	2	yes	—
104	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₅	—	2	4	no	—
106	C₂×C₄×C₅	—	2	2	no	—
107	C₂×C₄×C₅	—	2	2	no	—
108	C₂×C₄×C₅	—	2	2	no	—
109	C₂×C₄×C₅	—	2	1	no	—
110	C₂×C₄×C₅	—	2	1	no	—
111	C₂×C₄×C₅	—	2	1	no	—
112	C₂×C₄×C₅	—	2	1	no	—
113	C₂×C₄×C₅	—	4	1	no	—
114	C₂×C₄×C₅	—	4	2	no	—
115	C₂×C₄×C₅	—	4	1	no	—
116	C₂×C₄×C₅	#95	1	2	yes	—
117	C₂×C₄×C₅	#95	1	1	yes	characteristic
118	C₂×C₄×C₅	#95	1	2	yes	—
119	C₂×C₄×C₅	#95	1	1	yes	characteristic
120	C₂×C₄×C₅	#100	1	2	yes	—
121	C₂×C₄×C₅	#101	1	2	yes	—
122	C₂×C₄×C₅	#102	1	4	yes	—
123	C₂×C₄×C₅	#102	1	4	yes	—
124	C₂×C₄×C₅	#103	1	4	yes	—
125	C₂×C₄×C₅	#103	1	4	yes	—
126	C₂×C₄×C₅	#106	1	2	yes	—
127	C₂×C₄×C₅	#107	1	2	yes	—
128	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
129	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
131	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
132	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
134	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
135	C₅×C₈	—	4	4	no	—
136	C₅×C₈	—	4	8	no	—
137	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
138	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
139	C₂³×C₄	C₅×Q₈	1	1	yes	characteristic
140	C₂×M₁₆	—	2	4	no	—
141	C₂×M₁₆	—	2	4	no	—
142	C₂×M₁₆	—	4	4	no	—
143	C₂×M₁₆	—	4	4	no	—
144	C₂²×C₈	—	2	2	no	—
145	C₂²×C₈	—	2	2	no	—
146	C₂²×C₈	—	2	4	no	—
147	C₂²×C₈	—	4	4	no	—
148	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
149	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
150	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
151	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
152	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
153	C₂×C₄²	C₅×D₄	1	2	yes	—
154	C₂×C₄²	C₅×D₄	1	2	yes	—
155	C₂×C₄²	C₅×Q₈	1	1	yes	characteristic
156	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
157	C₄⋊C₈	—	2	4	no	—
158	C₄⋊C₈	—	2	4	no	—
159	C₄⋊C₈	—	2	4	no	—
160	C₄⋊C₈	—	2	4	no	—
161	C₄⋊C₈	—	2	4	no	—
162	C₄⋊C₈	C₂×C₄×C₅	1	8	yes	—
163	C₂²×C₅	—	2	4	no	—
164	C₂²×C₅	—	2	4	no	—
165	C₂²×C₅	—	2	1	no	—
166	C₂²×C₅	—	2	1	no	—
167	C₂²×C₅	—	2	1	no	—
168	C₂²×C₅	—	2	1	no	—
169	C₂²×C₅	#338	1	2	yes	—
170	C₂²×C₅	#340	1	2	yes	—
171	C₂²×C₅	#340	1	2	yes	—
172	C₂²×C₅	#343	1	2	yes	—
173	C₂²×C₅	C₂×#95	1	1	yes	characteristic
174	C₂²×C₅	C₂×#106	1	1	yes	characteristic
175	C₂²×C₅	C₂×#107	1	1	yes	characteristic
176	C₄×C₅	—	2	2	no	—
177	C₄×C₅	—	2	2	no	—
178	C₄×C₅	—	2	1	no	—
179	C₄×C₅	—	2	1	no	—
180	C₄×C₅	#340	1	4	yes	—
181	C₄×C₅	#343	1	2	yes	—
182	C₄×C₅	C₂×#100	1	1	yes	characteristic
183	C₄×C₅	C₂×#101	1	1	yes	characteristic
184	C₄×C₅	C₂×#102	1	2	yes	—
185	C₄×C₅	C₂×#103	1	2	yes	—
186	C₂⁴	C₅×C₄⋊C₄	1	1	yes	characteristic
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	C₄²×C₅	1	1	yes	characteristic
191	C₂²×C₄	C₄²×C₅	1	1	yes	characteristic
192	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
193	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
194	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
195	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
196	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
197	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
198	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
199	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
200	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
201	C₂²×C₄	C₅×D₈	1	2	yes	—
202	C₂²×C₄	C₅×SD₁₆	1	4	yes	—
203	C₂²×C₄	C₅×Q₁₆	1	2	yes	—
204	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
207	M₁₆	—	4	4	no	—
208	M₁₆	—	4	4	no	—
209	M₁₆	—	4	8	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	2	8	no	—
212	C₂×C₈	—	4	4	no	—
213	C₂×C₈	—	4	4	no	—
214	C₂×C₈	—	4	8	no	—
215	C₂×C₈	—	4	4	no	—
216	C₄²	C₄²×C₅	1	2	yes	—
217	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
218	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
219	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
220	C₄²	C₅×C₄⋊C₄	1	2	yes	—
221	C₄²	C₅×C₄⋊C₄	1	2	yes	—
222	C₄²	C₂²×C₄×C₅	1	2	yes	—
223	C₄²	C₂×C₅×D₄	1	2	yes	—
224	C₂×C₅	—	2	2	no	—
225	C₂×C₅	—	2	1	no	—
226	C₂×C₅	—	2	1	no	—
227	C₂×C₅	#1292	1	4	yes	direct product factor
228	C₂×C₅	C₂×#338	1	1	yes	characteristic
229	C₂×C₅	C₂×#340	1	1	yes	characteristic
230	C₂×C₅	C₂×#343	1	1	yes	characteristic
231	C₂³	—	2	2	no	—
232	C₂³	—	2	2	no	—
233	C₂³	C₅×#95	1	1	yes	characteristic
234	C₂³	C₅×#95	1	1	yes	characteristic
235	C₂³	C₅×#106	1	2	yes	—
236	C₂³	C₅×#107	1	2	yes	—
237	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	C₅×#95	1	2	yes	—
250	C₂×C₄	C₅×#95	1	1	yes	characteristic
251	C₂×C₄	C₅×#95	1	2	yes	—
252	C₂×C₄	C₅×#95	1	1	yes	characteristic
253	C₂×C₄	C₅×#100	1	2	yes	—
254	C₂×C₄	C₅×#101	1	2	yes	—
255	C₂×C₄	C₅×#102	1	4	yes	—
256	C₂×C₄	C₅×#102	1	4	yes	—
257	C₂×C₄	C₅×#103	1	4	yes	—
258	C₂×C₄	C₅×#103	1	4	yes	—
259	C₂×C₄	C₅×#106	1	2	yes	—
260	C₂×C₄	C₅×#107	1	2	yes	—
261	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
262	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
263	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
264	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
265	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
266	C₂×C₄	C₂×C₅×SD₁₆	1	2	yes	—
267	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
268	C₈	—	4	4	no	—
269	C₈	—	4	8	no	—
270	C₅	C₂×#1292	1	1	yes	characteristic, direct product factor, Sylow
271	C₂²	—	2	4	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	C₅×#338	1	2	yes	—
278	C₂²	C₅×#340	1	2	yes	—
279	C₂²	C₅×#340	1	2	yes	—
280	C₂²	C₅×#343	1	2	yes	—
281	C₂²	C₂×C₅×#95	1	1	yes	characteristic
282	C₂²	C₂×C₅×#106	1	1	yes	characteristic
283	C₂²	C₂×C₅×#107	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	C₅×#340	1	4	yes	—
289	C₄	C₅×#343	1	2	yes	—
290	C₄	C₂×C₅×#100	1	1	yes	characteristic
291	C₄	C₂×C₅×#101	1	1	yes	characteristic
292	C₄	C₂×C₅×#102	1	2	yes	—
293	C₄	C₂×C₅×#103	1	2	yes	—
294	C₂	—	2	2	no	—
295	C₂	—	2	1	no	—
296	C₂	—	2	1	no	—
297	C₂	C₅×#1292	1	4	yes	direct product factor
298	C₂	C₂×C₅×#338	1	1	yes	characteristic
299	C₂	C₂×C₅×#340	1	1	yes	characteristic
300	C₂	C₂×C₅×#343	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	#95
commutator	C₂×C₄	C₂×C₄²×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂⁴×C₅	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#1292	C₂	direct (N × H)
17. C₂×#1292	C₅	direct (N × H)
46. #1292	C₂×C₅	direct (N × H)
227. C₂×C₅	#1292	direct (N × H)
270. C₅	C₂×#1292	direct (N × H)
297. C₂	C₅×#1292	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×#423	2
C₂²×C₅×C₄⋊C₈	2
C₅×#1292	2
C₂×#1292	5

Subgroups — C2×C5×#1292 #11930719

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#1292 #11930719