Subgroups — C₂×C₅×#1941 #11943869

Order 1280.

1182 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#106	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×#106	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#389	C₂	1	2	yes	maximal
5	C₅×#1941	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₅×#340	C₂	1	2	yes	maximal
7	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₅×C₄⋊C₄	—	2	2	no	—
10	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₂×C₅×#106	—	2	4	no	—
14	C₂×C₅×#106	C₄	1	8	yes	complemented
15	C₂×C₅×#106	C₂²	1	4	yes	—
16	C₂×C₅×#106	C₂²	1	4	yes	—
17	C₂×C₄×C₅×C₈	—	2	4	no	—
18	C₂×C₄×C₅×C₈	C₂²	1	4	yes	—
19	C₅×#389	—	2	4	no	—
20	C₅×#389	C₂²	1	8	yes	—
21	C₂×C₅×#95	—	2	2	no	—
22	C₅×#340	C₂²	1	8	yes	—
23	C₂×#1941	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂³×C₄×C₅	—	2	2	no	—
25	C₂³×C₄×C₅	—	2	2	no	—
26	C₂³×C₄×C₅	D₄	1	2	yes	—
27	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂²×C₅×C₈	—	2	2	no	—
29	C₂²×C₅×C₈	—	2	2	no	—
30	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
31	C₂²×C₅×C₈	D₄	1	4	yes	—
32	C₂²×C₅×C₈	Q₈	1	4	yes	—
33	C₂²×C₅×C₈	Q₈	1	4	yes	—
34	C₂²×C₅×C₈	C₂³	1	2	yes	—
35	C₂²×C₅×C₈	C₂³	1	2	yes	—
36	C₂×C₅×C₄⋊C₄	—	2	2	no	—
37	C₂×C₅×C₄⋊C₄	—	2	4	no	—
38	C₂×C₅×C₄⋊C₄	—	2	4	no	—
39	C₂×C₅×C₄⋊C₄	—	2	4	no	—
40	C₂×C₅×C₄⋊C₄	—	2	2	no	—
41	C₂×C₅×C₄⋊C₄	—	2	8	no	—
42	C₂×C₅×C₄⋊C₄	—	4	8	no	—
43	C₂×C₅×C₄⋊C₄	C₂×C₄	1	8	yes	—
44	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
45	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
46	C₂×C₄²×C₅	—	2	2	no	—
47	C₂×C₄²×C₅	—	2	2	no	—
48	C₂×C₄²×C₅	D₄	1	2	yes	—
49	C₂×C₄²×C₅	C₂³	1	2	yes	—
50	C₅×#106	—	2	8	no	—
51	C₅×#106	C₂×C₄	1	16	yes	complemented
52	C₄×C₅×C₈	—	2	8	no	—
53	C₅×#95	—	2	8	no	—
54	C₂²×#106	C₂×C₅	1	1	yes	characteristic
55	C₂²×#106	C₂×C₅	1	1	yes	characteristic
56	C₂²×C₄×C₈	C₂×C₅	1	1	yes	characteristic
57	C₂×#389	C₂×C₅	1	2	yes	—
58	#1941	C₂×C₅	1	8	yes	direct product factor
59	C₂×#340	C₂×C₅	1	2	yes	—
60	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₅	—	2	4	no	—
62	C₂²×C₄×C₅	—	2	4	no	—
63	C₂²×C₄×C₅	—	2	8	no	—
64	C₂²×C₄×C₅	—	2	4	no	—
65	C₂²×C₄×C₅	—	2	4	no	—
66	C₂²×C₄×C₅	—	4	4	no	—
67	C₂²×C₄×C₅	—	4	4	no	—
68	C₂²×C₄×C₅	—	4	4	no	—
69	C₂²×C₄×C₅	—	4	4	no	—
70	C₂²×C₄×C₅	—	4	4	no	—
71	C₂²×C₄×C₅	SD₁₆	1	8	yes	—
72	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
74	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
78	C₂²×C₄×C₅	#41	1	2	yes	—
79	C₂²×C₄×C₅	#41	1	2	yes	—
80	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
81	C₂×C₅×C₈	—	2	8	no	—
82	C₂×C₅×C₈	—	2	2	no	—
83	C₂×C₅×C₈	—	2	2	no	—
84	C₂×C₅×C₈	C₄⋊C₄	1	16	yes	complemented
85	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
86	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
87	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
88	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
89	C₅×C₄⋊C₄	—	2	8	no	—
90	C₅×C₄⋊C₄	—	2	4	no	—
91	C₅×C₄⋊C₄	—	2	4	no	—
92	C₅×C₄⋊C₄	—	2	4	no	—
93	C₅×C₄⋊C₄	—	4	4	no	—
94	C₅×C₄⋊C₄	—	4	16	no	—
95	C₅×C₄⋊C₄	C₂²×C₄	1	8	yes	—
96	C₄²×C₅	—	2	4	no	—
97	C₄²×C₅	—	2	2	no	—
98	C₄²×C₅	—	2	2	no	—
99	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
100	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
101	C₂²×C₄⋊C₄	—	2	2	no	—
102	C₂²×C₄⋊C₄	C₂²×C₅	1	2	yes	—
103	C₂²×C₄⋊C₄	C₂²×C₅	1	2	yes	—
104	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
105	C₂×#106	—	2	4	no	—
106	C₂×#106	C₄×C₅	1	8	yes	complemented
107	C₂×#106	C₂²×C₅	1	4	yes	—
108	C₂×#106	C₂²×C₅	1	4	yes	—
109	C₂×C₄×C₈	—	2	4	no	—
110	C₂×C₄×C₈	C₂²×C₅	1	4	yes	—
111	#389	—	2	4	no	—
112	#389	C₂²×C₅	1	8	yes	—
113	C₂×#95	—	2	2	no	—
114	#340	C₂²×C₅	1	8	yes	—
115	C₂³×C₅	C₄×D₄	1	2	yes	—
116	C₂³×C₅	#121	1	2	yes	—
117	C₂³×C₅	C₄⋊Q₈	1	2	yes	—
118	C₂³×C₅	C₂×SD₁₆	1	4	yes	—
119	C₂³×C₅	#135	1	4	yes	—
120	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₅	—	2	4	no	—
122	C₂×C₄×C₅	—	2	4	no	—
123	C₂×C₄×C₅	—	2	16	no	—
124	C₂×C₄×C₅	—	2	8	no	—
125	C₂×C₄×C₅	—	4	4	no	—
126	C₂×C₄×C₅	—	4	4	no	—
127	C₂×C₄×C₅	—	4	16	no	—
128	C₂×C₄×C₅	—	4	4	no	—
129	C₂×C₄×C₅	—	4	4	no	—
130	C₂×C₄×C₅	—	4	4	no	—
131	C₂×C₄×C₅	—	8	8	no	—
132	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
133	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₅	C₄×Q₈	1	4	yes	—
135	C₂×C₄×C₅	#121	1	2	yes	—
136	C₂×C₄×C₅	C₂²⋊Q₈	1	4	yes	—
137	C₂×C₄×C₅	#125	1	4	yes	—
138	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
139	C₂×C₄×C₅	C₂×SD₁₆	1	8	yes	—
140	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
146	C₅×C₈	—	2	4	no	—
147	C₅×C₈	C₂×C₄⋊C₄	1	8	yes	complemented
148	C₂³×C₄	—	2	2	no	—
149	C₂³×C₄	—	2	2	no	—
150	C₂³×C₄	C₅×D₄	1	2	yes	—
151	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
152	C₂²×C₈	—	2	2	no	—
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
155	C₂²×C₈	C₅×D₄	1	4	yes	—
156	C₂²×C₈	C₅×Q₈	1	4	yes	—
157	C₂²×C₈	C₅×Q₈	1	4	yes	—
158	C₂²×C₈	C₂³×C₅	1	2	yes	—
159	C₂²×C₈	C₂³×C₅	1	2	yes	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	—	2	4	no	—
162	C₂×C₄⋊C₄	—	2	4	no	—
163	C₂×C₄⋊C₄	—	2	4	no	—
164	C₂×C₄⋊C₄	—	2	8	no	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄⋊C₄	—	4	8	no	—
167	C₂×C₄⋊C₄	C₂×C₄×C₅	1	8	yes	—
168	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
169	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
170	C₂×C₄²	—	2	2	no	—
171	C₂×C₄²	—	2	2	no	—
172	C₂×C₄²	C₅×D₄	1	2	yes	—
173	C₂×C₄²	C₂³×C₅	1	2	yes	—
174	#106	—	2	8	no	—
175	#106	C₂×C₄×C₅	1	16	yes	complemented
176	C₄×C₈	—	2	8	no	—
177	#95	—	2	8	no	—
178	C₂²×C₅	#389	1	4	yes	—
179	C₂²×C₅	C₄×SD₁₆	1	8	yes	—
180	C₂²×C₅	#465	1	8	yes	—
181	C₂²×C₅	#498	1	4	yes	—
182	C₂²×C₅	#499	1	4	yes	—
183	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
184	C₂²×C₅	C₂×#121	1	1	yes	characteristic
185	C₂²×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
186	C₂²×C₅	C₂²×SD₁₆	1	2	yes	—
187	C₂²×C₅	C₂×#135	1	2	yes	—
188	C₄×C₅	—	2	8	no	—
189	C₄×C₅	—	4	8	no	—
190	C₄×C₅	—	8	4	no	—
191	C₄×C₅	#389	1	4	yes	—
192	C₄×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
193	C₄×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
194	C₄×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
195	C₄×C₅	C₂×#125	1	1	yes	characteristic
196	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	2	8	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	4	4	no	—
203	C₂²×C₄	—	4	4	no	—
204	C₂²×C₄	—	4	4	no	—
205	C₂²×C₄	—	4	4	no	—
206	C₂²×C₄	—	4	4	no	—
207	C₂²×C₄	C₅×SD₁₆	1	8	yes	—
208	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
209	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
210	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
211	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
213	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
214	C₂²×C₄	C₅×#41	1	2	yes	—
215	C₂²×C₄	C₅×#41	1	2	yes	—
216	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
217	C₂×C₈	—	2	8	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	C₅×C₄⋊C₄	1	16	yes	complemented
221	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
222	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
223	C₂×C₈	C₂×C₅×Q₈	1	4	yes	—
224	C₂×C₈	C₂×C₅×Q₈	1	4	yes	—
225	C₄⋊C₄	—	2	8	no	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	4	4	no	—
230	C₄⋊C₄	—	4	16	no	—
231	C₄⋊C₄	C₂²×C₄×C₅	1	8	yes	—
232	C₄²	—	2	4	no	—
233	C₄²	—	2	2	no	—
234	C₄²	—	2	2	no	—
235	C₂×C₅	#1941	1	8	yes	direct product factor
236	C₂×C₅	C₂×#389	1	1	yes	characteristic
237	C₂×C₅	C₂×C₄×SD₁₆	1	2	yes	—
238	C₂×C₅	C₂×#465	1	2	yes	—
239	C₂×C₅	C₂×#498	1	1	yes	characteristic
240	C₂×C₅	C₂×#499	1	1	yes	characteristic
241	C₂³	C₄×C₅×D₄	1	2	yes	—
242	C₂³	C₅×#121	1	2	yes	—
243	C₂³	C₅×C₄⋊Q₈	1	2	yes	—
244	C₂³	C₂×C₅×SD₁₆	1	4	yes	—
245	C₂³	C₅×#135	1	4	yes	—
246	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
247	C₂×C₄	—	2	16	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	4	16	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	8	8	no	—
258	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
259	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
260	C₂×C₄	C₄×C₅×Q₈	1	4	yes	—
261	C₂×C₄	C₅×#121	1	2	yes	—
262	C₂×C₄	C₅×C₂²⋊Q₈	1	4	yes	—
263	C₂×C₄	C₅×#125	1	4	yes	—
264	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
265	C₂×C₄	C₂×C₅×SD₁₆	1	8	yes	—
266	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
267	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
268	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
269	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
270	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
271	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
272	C₈	—	2	4	no	—
273	C₈	C₂×C₅×C₄⋊C₄	1	8	yes	complemented
274	C₅	C₂×#1941	1	1	yes	characteristic, direct product factor, Sylow
275	C₂²	C₅×#389	1	4	yes	—
276	C₂²	C₄×C₅×SD₁₆	1	8	yes	—
277	C₂²	C₅×#465	1	8	yes	—
278	C₂²	C₅×#498	1	4	yes	—
279	C₂²	C₅×#499	1	4	yes	—
280	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
281	C₂²	C₂×C₅×#121	1	1	yes	characteristic
282	C₂²	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
283	C₂²	C₂²×C₅×SD₁₆	1	2	yes	—
284	C₂²	C₂×C₅×#135	1	2	yes	—
285	C₄	—	2	8	no	—
286	C₄	—	4	8	no	—
287	C₄	—	8	4	no	—
288	C₄	C₅×#389	1	4	yes	—
289	C₄	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
290	C₄	C₂×C₄×C₅×Q₈	1	1	yes	characteristic
291	C₄	C₂×C₅×C₂²⋊Q₈	1	1	yes	characteristic
292	C₄	C₂×C₅×#125	1	1	yes	characteristic
293	C₂	C₅×#1941	1	8	yes	direct product factor
294	C₂	C₂×C₅×#389	1	1	yes	characteristic
295	C₂	C₂×C₄×C₅×SD₁₆	1	2	yes	—
296	C₂	C₂×C₅×#465	1	2	yes	—
297	C₂	C₂×C₅×#498	1	1	yes	characteristic
298	C₂	C₂×C₅×#499	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₅	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₅×#1941	C₂	direct (N × H)
14. C₂×C₅×#106	C₄	semidirect
23. C₂×#1941	C₅	direct (N × H)
51. C₅×#106	C₂×C₄	semidirect
58. #1941	C₂×C₅	direct (N × H)
84. C₂×C₅×C₈	C₄⋊C₄	semidirect
106. C₂×#106	C₄×C₅	semidirect
147. C₅×C₈	C₂×C₄⋊C₄	semidirect
175. #106	C₂×C₄×C₅	semidirect
220. C₂×C₈	C₅×C₄⋊C₄	semidirect
235. C₂×C₅	#1941	direct (N × H)
273. C₈	C₂×C₅×C₄⋊C₄	semidirect
274. C₅	C₂×#1941	direct (N × H)
293. C₂	C₅×#1941	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×#106	2
C₂²×C₄×C₅×C₈	2
C₂×C₅×#389	2
C₅×#1941	2
C₂×C₅×#340	2
C₂×#1941	5

Subgroups — C2×C5×#1941 #11943869

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#1941 #11943869