Subgroups — C₂×C₅×#1382 #11935280

Order 1280.

590 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×#108	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#427	C₂	1	1	yes	maximal, characteristic
4	C₅×#1382	C₂	1	4	yes	maximal, direct product factor
5	C₂²×C₅×M₁₆	—	2	1	no	—
6	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₅×#117	—	2	1	no	—
8	C₂²×C₅×C₁₆	—	2	1	no	—
9	C₂²×C₅×C₁₆	—	2	1	no	—
10	C₂²×C₅×C₁₆	—	2	2	no	—
11	C₂²×C₅×C₁₆	C₄	1	4	yes	complemented
12	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
13	C₂×C₅×#108	—	2	1	no	—
14	C₂×C₅×#108	—	2	2	no	—
15	C₂×C₅×#108	C₄	1	4	yes	complemented
16	C₂×C₅×#108	C₂²	1	2	yes	—
17	C₅×#427	—	2	1	no	—
18	C₅×#427	—	2	1	no	—
19	C₅×#427	C₄	1	2	yes	—
20	C₅×#427	C₂²	1	2	yes	—
21	C₂×C₅×#107	C₄	1	2	yes	—
22	C₂×C₅×#106	—	2	1	no	—
23	C₂×#1382	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
25	C₂×C₅×M₁₆	—	2	4	no	—
26	C₂×C₅×M₁₆	—	2	2	no	—
27	C₂×C₅×M₁₆	—	4	1	no	—
28	C₂×C₅×M₁₆	—	4	2	no	—
29	C₂²×C₅×C₈	—	4	1	no	—
30	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
31	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
34	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
35	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
36	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
37	C₂²×C₅×C₈	D₄	1	2	yes	—
38	C₂²×C₅×C₈	Q₈	1	2	yes	—
39	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
40	C₅×#117	—	2	2	no	—
41	C₅×#117	—	2	2	no	—
42	C₅×#117	—	4	1	no	—
43	C₅×#117	—	4	1	no	—
44	C₂×C₅×C₄⋊C₄	—	2	2	no	—
45	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
46	C₂×C₄²×C₅	—	4	1	no	—
47	C₂×C₅×C₁₆	—	2	4	no	—
48	C₂×C₅×C₁₆	—	2	4	no	—
49	C₂×C₅×C₁₆	—	2	2	no	—
50	C₂×C₅×C₁₆	—	2	2	no	—
51	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	complemented
52	C₅×#108	—	2	4	no	—
53	C₅×#108	—	2	2	no	—
54	C₅×#108	C₂×C₄	1	4	yes	complemented
55	C₅×#107	—	2	2	no	—
56	C₅×#107	C₂×C₄	1	4	yes	—
57	C₅×#106	—	2	2	no	—
58	C₅×#106	—	2	2	no	—
59	C₂³×C₁₆	C₂×C₅	1	1	yes	characteristic
60	C₂²×#108	C₂×C₅	1	1	yes	characteristic
61	C₂×#427	C₂×C₅	1	1	yes	characteristic
62	#1382	C₂×C₅	1	4	yes	direct product factor
63	C₂⁴×C₅	SD₁₆	1	1	yes	characteristic
64	C₂²×C₄×C₅	—	4	2	no	—
65	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
70	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
71	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
72	C₂²×C₄×C₅	SD₁₆	1	2	yes	—
73	C₂²×C₄×C₅	Q₁₆	1	2	yes	—
74	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
75	C₅×M₁₆	—	2	4	no	—
76	C₅×M₁₆	—	4	4	no	—
77	C₅×M₁₆	—	4	2	no	—
78	C₂×C₅×C₈	—	4	4	no	—
79	C₂×C₅×C₈	—	4	2	no	—
80	C₂×C₅×C₈	C₄²	1	2	yes	—
81	C₂×C₅×C₈	C₄²	1	2	yes	—
82	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
85	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
86	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
87	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
88	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
89	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
90	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
91	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
92	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
96	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
98	C₅×C₄⋊C₄	—	2	4	no	—
99	C₅×C₄⋊C₄	—	4	2	no	—
100	C₅×C₂²⋊C₄	—	4	2	no	—
101	C₅×C₂²⋊C₄	—	4	2	no	—
102	C₄²×C₅	—	4	2	no	—
103	C₄²×C₅	—	4	2	no	—
104	C₅×C₁₆	—	2	4	no	—
105	C₂²×M₁₆	—	2	1	no	—
106	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
107	C₂×#117	—	2	1	no	—
108	C₂²×C₁₆	—	2	1	no	—
109	C₂²×C₁₆	—	2	1	no	—
110	C₂²×C₁₆	—	2	2	no	—
111	C₂²×C₁₆	C₄×C₅	1	4	yes	complemented
112	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
113	C₂×#108	—	2	1	no	—
114	C₂×#108	—	2	2	no	—
115	C₂×#108	C₄×C₅	1	4	yes	complemented
116	C₂×#108	C₂²×C₅	1	2	yes	—
117	#427	—	2	1	no	—
118	#427	—	2	1	no	—
119	#427	C₄×C₅	1	2	yes	—
120	#427	C₂²×C₅	1	2	yes	—
121	C₂×#107	C₄×C₅	1	2	yes	—
122	C₂×#106	—	2	1	no	—
123	C₂³×C₅	—	2	4	no	—
124	C₂³×C₅	#102	1	1	yes	characteristic
125	C₂³×C₅	#102	1	1	yes	characteristic
126	C₂³×C₅	#103	1	2	yes	—
127	C₂³×C₅	#106	1	2	yes	—
128	C₂³×C₅	C₂×SD₁₆	1	1	yes	characteristic
129	C₂×C₄×C₅	—	4	4	no	—
130	C₂×C₄×C₅	—	8	4	no	—
131	C₂×C₄×C₅	#95	1	2	yes	—
132	C₂×C₄×C₅	#95	1	2	yes	—
133	C₂×C₄×C₅	#102	1	1	yes	characteristic
134	C₂×C₄×C₅	#102	1	1	yes	characteristic
135	C₂×C₄×C₅	#102	1	2	yes	—
136	C₂×C₄×C₅	#102	1	2	yes	—
137	C₂×C₄×C₅	#103	1	2	yes	—
138	C₂×C₄×C₅	#103	1	2	yes	—
139	C₂×C₄×C₅	#103	1	2	yes	—
140	C₂×C₄×C₅	#106	1	2	yes	—
141	C₂×C₄×C₅	#107	1	2	yes	—
142	C₂×C₄×C₅	#107	1	2	yes	—
143	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
148	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
149	C₅×C₈	—	4	4	no	—
150	C₅×C₈	#95	1	2	yes	—
151	C₅×C₈	#95	1	2	yes	—
152	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
153	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
154	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
156	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
157	C₂×M₁₆	—	2	4	no	—
158	C₂×M₁₆	—	2	2	no	—
159	C₂×M₁₆	—	4	1	no	—
160	C₂×M₁₆	—	4	2	no	—
161	C₂²×C₈	—	4	1	no	—
162	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
163	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
164	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
165	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
166	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
167	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
168	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
169	C₂²×C₈	C₅×D₄	1	2	yes	—
170	C₂²×C₈	C₅×Q₈	1	2	yes	—
171	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
172	#117	—	2	2	no	—
173	#117	—	2	2	no	—
174	#117	—	4	1	no	—
175	#117	—	4	1	no	—
176	C₂×C₄⋊C₄	—	2	2	no	—
177	C₂×C₂²⋊C₄	—	4	1	no	—
178	C₂×C₄²	—	4	1	no	—
179	C₂×C₁₆	—	2	4	no	—
180	C₂×C₁₆	—	2	4	no	—
181	C₂×C₁₆	—	2	2	no	—
182	C₂×C₁₆	—	2	2	no	—
183	C₂×C₁₆	C₂×C₄×C₅	1	4	yes	complemented
184	#108	—	2	4	no	—
185	#108	—	2	2	no	—
186	#108	C₂×C₄×C₅	1	4	yes	complemented
187	#107	—	2	2	no	—
188	#107	C₂×C₄×C₅	1	4	yes	—
189	#106	—	2	2	no	—
190	#106	—	2	2	no	—
191	C₂²×C₅	—	2	2	no	—
192	C₂²×C₅	—	2	2	no	—
193	C₂²×C₅	—	2	4	no	—
194	C₂²×C₅	—	2	4	no	—
195	C₂²×C₅	#340	1	2	yes	—
196	C₂²×C₅	#340	1	2	yes	—
197	C₂²×C₅	#359	1	4	yes	—
198	C₂²×C₅	C₂×#102	1	1	yes	characteristic
199	C₂²×C₅	C₂×#103	1	1	yes	characteristic
200	C₂²×C₅	C₂×#106	1	1	yes	characteristic
201	C₄×C₅	—	8	4	no	—
202	C₄×C₅	#340	1	2	yes	—
203	C₄×C₅	#340	1	2	yes	—
204	C₄×C₅	C₂×#95	1	1	yes	characteristic
205	C₄×C₅	C₂×#102	1	1	yes	characteristic
206	C₄×C₅	C₂×#103	1	1	yes	characteristic
207	C₄×C₅	C₂×#107	1	1	yes	characteristic
208	C₂⁴	C₅×SD₁₆	1	1	yes	characteristic
209	C₂²×C₄	—	4	2	no	—
210	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
211	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
212	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
213	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
214	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
215	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
216	C₂²×C₄	C₅×SD₁₆	1	2	yes	—
217	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
218	C₂²×C₄	C₅×Q₁₆	1	2	yes	—
219	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
220	M₁₆	—	2	4	no	—
221	M₁₆	—	4	2	no	—
222	M₁₆	—	4	4	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	4	4	no	—
225	C₂×C₈	C₄²×C₅	1	2	yes	—
226	C₂×C₈	C₄²×C₅	1	2	yes	—
227	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
228	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
229	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
230	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
231	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
232	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
233	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
234	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
235	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
236	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
237	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
238	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
239	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
240	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
241	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
242	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₂²⋊C₄	—	4	2	no	—
246	C₂²⋊C₄	—	4	2	no	—
247	C₄²	—	4	2	no	—
248	C₄²	—	4	2	no	—
249	C₁₆	—	2	4	no	—
250	C₂×C₅	—	2	2	no	—
251	C₂×C₅	—	2	1	no	—
252	C₂×C₅	—	2	1	no	—
253	C₂×C₅	#1382	1	4	yes	direct product factor
254	C₂×C₅	C₂×#340	1	1	yes	characteristic
255	C₂×C₅	C₂×#359	1	2	yes	—
256	C₂³	—	2	4	no	—
257	C₂³	C₅×#102	1	1	yes	characteristic
258	C₂³	C₅×#102	1	1	yes	characteristic
259	C₂³	C₅×#103	1	2	yes	—
260	C₂³	C₅×#106	1	2	yes	—
261	C₂³	C₂×C₅×SD₁₆	1	1	yes	characteristic
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	8	4	no	—
264	C₂×C₄	C₅×#95	1	2	yes	—
265	C₂×C₄	C₅×#95	1	2	yes	—
266	C₂×C₄	C₅×#102	1	1	yes	characteristic
267	C₂×C₄	C₅×#102	1	1	yes	characteristic
268	C₂×C₄	C₅×#102	1	2	yes	—
269	C₂×C₄	C₅×#102	1	2	yes	—
270	C₂×C₄	C₅×#103	1	2	yes	—
271	C₂×C₄	C₅×#103	1	2	yes	—
272	C₂×C₄	C₅×#103	1	2	yes	—
273	C₂×C₄	C₅×#106	1	2	yes	—
274	C₂×C₄	C₅×#107	1	2	yes	—
275	C₂×C₄	C₅×#107	1	2	yes	—
276	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
277	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
278	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
279	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
280	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
281	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
282	C₈	—	4	4	no	—
283	C₈	C₅×#95	1	2	yes	—
284	C₈	C₅×#95	1	2	yes	—
285	C₈	C₂×C₄²×C₅	1	1	yes	characteristic
286	C₈	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
287	C₈	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
288	C₈	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
289	C₅	C₂×#1382	1	1	yes	characteristic, direct product factor, Sylow
290	C₂²	—	2	4	no	—
291	C₂²	—	2	4	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	C₅×#340	1	2	yes	—
295	C₂²	C₅×#340	1	2	yes	—
296	C₂²	C₅×#359	1	4	yes	—
297	C₂²	C₂×C₅×#102	1	1	yes	characteristic
298	C₂²	C₂×C₅×#103	1	1	yes	characteristic
299	C₂²	C₂×C₅×#106	1	1	yes	characteristic
300	C₄	—	8	4	no	—
301	C₄	C₅×#340	1	2	yes	—
302	C₄	C₅×#340	1	2	yes	—
303	C₄	C₂×C₅×#95	1	1	yes	characteristic
304	C₄	C₂×C₅×#102	1	1	yes	characteristic
305	C₄	C₂×C₅×#103	1	1	yes	characteristic
306	C₄	C₂×C₅×#107	1	1	yes	characteristic
307	C₂	—	2	2	no	—
308	C₂	—	2	1	no	—
309	C₂	—	2	1	no	—
310	C₂	C₅×#1382	1	4	yes	direct product factor
311	C₂	C₂×C₅×#340	1	1	yes	characteristic
312	C₂	C₂×C₅×#359	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	D₈
commutator	C₈	C₂×C₄²×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂³×C₅	#102
prime radical	C₂⁴×C₅	SD₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂³×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅×#1382	C₂	direct (N × H)
11. C₂²×C₅×C₁₆	C₄	semidirect
15. C₂×C₅×#108	C₄	semidirect
23. C₂×#1382	C₅	direct (N × H)
51. C₂×C₅×C₁₆	C₂×C₄	semidirect
54. C₅×#108	C₂×C₄	semidirect
62. #1382	C₂×C₅	direct (N × H)
111. C₂²×C₁₆	C₄×C₅	semidirect
115. C₂×#108	C₄×C₅	semidirect
183. C₂×C₁₆	C₂×C₄×C₅	semidirect
186. #108	C₂×C₄×C₅	semidirect
253. C₂×C₅	#1382	direct (N × H)
289. C₅	C₂×#1382	direct (N × H)
310. C₂	C₅×#1382	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₅×C₁₆	2
C₂²×C₅×#108	2
C₂×C₅×#427	2
C₅×#1382	2
C₂×#1382	5

Subgroups — C2×C5×#1382 #11935280

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#1382 #11935280