Subgroups — C₂×C₅×#1841 #11943495

Order 1280.

730 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₅×#1841	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₅×#344	C₂	1	2	yes	maximal
4	C₂×C₅×#339	C₂	1	2	yes	maximal
5	C₂×C₅×#338	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₅×#117	C₂²	1	2	yes	—
11	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₄×C₅×M₁₆	C₄	1	4	yes	complemented
13	C₄×C₅×M₁₆	C₂²	1	4	yes	—
14	C₂×C₅×#97	C₄	1	2	yes	complemented
15	C₂×C₄×C₅×C₈	C₄	1	2	yes	complemented
16	C₅×#344	C₂²	1	8	yes	—
17	C₅×#339	C₂²	1	8	yes	—
18	C₅×#338	C₂²	1	4	yes	—
19	C₅×H(3,ℤ/4)	C₂²	1	4	yes	—
20	C₂×#1841	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂³×C₄×C₅	—	2	1	no	—
22	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
23	C₂×C₅×M₁₆	—	2	2	no	—
24	C₂×C₅×M₁₆	—	2	2	no	—
25	C₂×C₅×M₁₆	—	2	2	no	—
26	C₂×C₅×M₁₆	—	2	2	no	—
27	C₂×C₅×M₁₆	—	2	2	no	—
28	C₂×C₅×M₁₆	—	2	2	no	—
29	C₂×C₅×M₁₆	—	2	2	no	—
30	C₂×C₅×M₁₆	—	2	2	no	—
31	C₂×C₅×M₁₆	—	2	2	no	—
32	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
33	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
34	C₂×C₅×M₁₆	C₂³	1	4	yes	—
35	C₂×C₅×M₁₆	C₂³	1	2	yes	—
36	C₂×C₅×M₁₆	C₂³	1	2	yes	—
37	C₂²×C₅×C₈	—	2	2	no	—
38	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
39	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
40	C₅×#117	—	2	2	no	—
41	C₅×#117	—	2	2	no	—
42	C₅×#117	—	2	2	no	—
43	C₅×#117	C₂³	1	4	yes	—
44	C₂×C₅×C₄⋊C₄	—	2	2	no	—
45	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
46	C₂×C₄²×C₅	—	2	1	no	—
47	C₂×C₄²×C₅	—	2	1	no	—
48	C₂×C₄²×C₅	—	2	1	no	—
49	C₂×C₄²×C₅	—	2	2	no	—
50	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₅	C₂³	1	2	yes	—
54	C₅×#97	C₂×C₄	1	8	yes	complemented
55	C₄×C₅×C₈	C₂×C₄	1	8	yes	complemented
56	C₂×C₄×M₁₆	C₂×C₅	1	1	yes	characteristic
57	#1841	C₂×C₅	1	8	yes	direct product factor
58	C₂×#344	C₂×C₅	1	2	yes	—
59	C₂×#339	C₂×C₅	1	2	yes	—
60	C₂×#338	C₂×C₅	1	1	yes	characteristic
61	C₂×H(3,ℤ/4)	C₂×C₅	1	1	yes	characteristic
62	C₂⁴×C₅	#41	1	1	yes	characteristic
63	C₂²×C₄×C₅	—	2	1	no	—
64	C₂²×C₄×C₅	—	2	2	no	—
65	C₂²×C₄×C₅	—	2	1	no	—
66	C₂²×C₄×C₅	—	2	2	no	—
67	C₂²×C₄×C₅	—	4	4	no	—
68	C₂²×C₄×C₅	—	4	2	no	—
69	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	#41	1	1	yes	characteristic
72	C₂²×C₄×C₅	#41	1	2	yes	—
73	C₂²×C₄×C₅	#41	1	2	yes	—
74	C₂²×C₄×C₅	#41	1	2	yes	—
75	C₂²×C₄×C₅	#41	1	2	yes	—
76	C₂²×C₄×C₅	#41	1	1	yes	characteristic
77	C₂²×C₄×C₅	#41	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
79	C₅×M₁₆	—	2	4	no	—
80	C₅×M₁₆	—	2	4	no	—
81	C₅×M₁₆	—	2	4	no	—
82	C₅×M₁₆	—	2	4	no	—
83	C₅×M₁₆	—	2	4	no	—
84	C₅×M₁₆	—	2	4	no	—
85	C₅×M₁₆	—	4	2	no	—
86	C₅×M₁₆	—	4	2	no	—
87	C₂×C₅×C₈	—	2	2	no	—
88	C₂×C₅×C₈	—	2	2	no	—
89	C₂×C₅×C₈	—	2	2	no	—
90	C₂×C₅×C₈	—	2	2	no	—
91	C₂×C₅×C₈	—	2	4	no	—
92	C₂×C₅×C₈	—	4	2	no	—
93	C₂×C₅×C₈	—	4	2	no	—
94	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
95	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
96	C₅×C₄⋊C₄	—	2	4	no	—
97	C₅×C₄⋊C₄	—	4	2	no	—
98	C₅×C₂²⋊C₄	—	2	4	no	—
99	C₅×C₂²⋊C₄	—	4	2	no	—
100	C₄²×C₅	—	2	1	no	—
101	C₄²×C₅	—	2	2	no	—
102	C₄²×C₅	—	2	1	no	—
103	C₄²×C₅	—	2	2	no	—
104	C₄²×C₅	—	2	4	no	—
105	C₄²×C₅	—	4	2	no	—
106	C₄²×C₅	C₂²×C₄	1	2	yes	—
107	C₄²×C₅	C₂²×C₄	1	2	yes	—
108	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
109	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
110	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
111	C₂×#117	C₂²×C₅	1	2	yes	—
112	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
113	C₄×M₁₆	C₄×C₅	1	4	yes	complemented
114	C₄×M₁₆	C₂²×C₅	1	4	yes	—
115	C₂×#97	C₄×C₅	1	2	yes	complemented
116	C₂×C₄×C₈	C₄×C₅	1	2	yes	complemented
117	#344	C₂²×C₅	1	8	yes	—
118	#339	C₂²×C₅	1	8	yes	—
119	#338	C₂²×C₅	1	4	yes	—
120	H(3,ℤ/4)	C₂²×C₅	1	4	yes	—
121	C₂³×C₅	—	4	2	no	—
122	C₂³×C₅	#117	1	1	yes	characteristic
123	C₂³×C₅	#117	1	1	yes	characteristic
124	C₂³×C₅	#126	1	2	yes	—
125	C₂³×C₅	#126	1	2	yes	—
126	C₂³×C₅	C₂×#41	1	1	yes	characteristic
127	C₂×C₄×C₅	—	2	4	no	—
128	C₂×C₄×C₅	—	2	2	no	—
129	C₂×C₄×C₅	—	2	2	no	—
130	C₂×C₄×C₅	—	4	4	no	—
131	C₂×C₄×C₅	—	4	4	no	—
132	C₂×C₄×C₅	—	4	2	no	—
133	C₂×C₄×C₅	—	4	4	no	—
134	C₂×C₄×C₅	—	8	4	no	—
135	C₂×C₄×C₅	#117	1	1	yes	characteristic
136	C₂×C₄×C₅	#117	1	1	yes	characteristic
137	C₂×C₄×C₅	#117	1	1	yes	characteristic
138	C₂×C₄×C₅	#117	1	2	yes	—
139	C₂×C₄×C₅	#117	1	2	yes	—
140	C₂×C₄×C₅	#117	1	1	yes	characteristic
141	C₂×C₄×C₅	#117	1	1	yes	characteristic
142	C₂×C₄×C₅	#117	1	1	yes	characteristic
143	C₂×C₄×C₅	#126	1	2	yes	—
144	C₂×C₄×C₅	#126	1	2	yes	—
145	C₂×C₄×C₅	#126	1	2	yes	—
146	C₂×C₄×C₅	#126	1	2	yes	—
147	C₂×C₄×C₅	#126	1	2	yes	—
148	C₂×C₄×C₅	#126	1	2	yes	—
149	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂×#41	1	2	yes	—
151	C₂×C₄×C₅	C₂×#41	1	2	yes	—
152	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
153	C₅×C₈	—	2	4	no	—
154	C₅×C₈	—	2	4	no	—
155	C₅×C₈	—	4	4	no	—
156	C₂³×C₄	—	2	1	no	—
157	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
158	C₂×M₁₆	—	2	2	no	—
159	C₂×M₁₆	—	2	2	no	—
160	C₂×M₁₆	—	2	2	no	—
161	C₂×M₁₆	—	2	2	no	—
162	C₂×M₁₆	—	2	2	no	—
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	C₂×C₄×C₅	1	2	yes	—
168	C₂×M₁₆	C₂×C₄×C₅	1	2	yes	—
169	C₂×M₁₆	C₂³×C₅	1	4	yes	—
170	C₂×M₁₆	C₂³×C₅	1	2	yes	—
171	C₂×M₁₆	C₂³×C₅	1	2	yes	—
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
174	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
175	#117	—	2	2	no	—
176	#117	—	2	2	no	—
177	#117	—	2	2	no	—
178	#117	C₂³×C₅	1	4	yes	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₂²⋊C₄	—	2	2	no	—
181	C₂×C₄²	—	2	1	no	—
182	C₂×C₄²	—	2	1	no	—
183	C₂×C₄²	—	2	1	no	—
184	C₂×C₄²	—	2	2	no	—
185	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
186	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
187	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
188	C₂×C₄²	C₂³×C₅	1	2	yes	—
189	#97	C₂×C₄×C₅	1	8	yes	complemented
190	C₄×C₈	C₂×C₄×C₅	1	8	yes	complemented
191	C₂²×C₅	—	2	2	no	—
192	C₂²×C₅	—	2	4	no	—
193	C₂²×C₅	—	4	1	no	—
194	C₂²×C₅	—	4	1	no	—
195	C₂²×C₅	—	4	2	no	—
196	C₂²×C₅	#383	1	1	yes	characteristic
197	C₂²×C₅	#383	1	2	yes	—
198	C₂²×C₅	#383	1	1	yes	characteristic
199	C₂²×C₅	C₂×#117	1	1	yes	characteristic
200	C₂²×C₅	C₂×#126	1	2	yes	—
201	C₄×C₅	—	2	4	no	—
202	C₄×C₅	—	4	2	no	—
203	C₄×C₅	—	8	4	no	—
204	C₄×C₅	#383	1	1	yes	characteristic
205	C₄×C₅	#383	1	2	yes	—
206	C₄×C₅	#383	1	1	yes	characteristic
207	C₄×C₅	C₂×#117	1	1	yes	characteristic
208	C₄×C₅	C₂×#117	1	1	yes	characteristic
209	C₄×C₅	C₂×#126	1	2	yes	—
210	C₂⁴	C₅×#41	1	1	yes	characteristic
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	4	4	no	—
216	C₂²×C₄	—	4	2	no	—
217	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
218	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
219	C₂²×C₄	C₅×#41	1	1	yes	characteristic
220	C₂²×C₄	C₅×#41	1	2	yes	—
221	C₂²×C₄	C₅×#41	1	2	yes	—
222	C₂²×C₄	C₅×#41	1	2	yes	—
223	C₂²×C₄	C₅×#41	1	2	yes	—
224	C₂²×C₄	C₅×#41	1	1	yes	characteristic
225	C₂²×C₄	C₅×#41	1	1	yes	characteristic
226	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
227	M₁₆	—	2	4	no	—
228	M₁₆	—	2	4	no	—
229	M₁₆	—	2	4	no	—
230	M₁₆	—	2	4	no	—
231	M₁₆	—	2	4	no	—
232	M₁₆	—	2	4	no	—
233	M₁₆	—	4	2	no	—
234	M₁₆	—	4	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	—	4	2	no	—
242	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
243	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₂²⋊C₄	—	2	4	no	—
247	C₂²⋊C₄	—	4	2	no	—
248	C₄²	—	2	1	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	1	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	4	no	—
253	C₄²	—	4	2	no	—
254	C₄²	C₂²×C₄×C₅	1	2	yes	—
255	C₄²	C₂²×C₄×C₅	1	2	yes	—
256	C₂×C₅	—	2	1	no	—
257	C₂×C₅	—	2	2	no	—
258	C₂×C₅	—	2	1	no	—
259	C₂×C₅	—	2	2	no	—
260	C₂×C₅	#1841	1	2	yes	direct product factor
261	C₂×C₅	C₂×#383	1	1	yes	characteristic
262	C₂³	—	4	2	no	—
263	C₂³	C₅×#117	1	1	yes	characteristic
264	C₂³	C₅×#117	1	1	yes	characteristic
265	C₂³	C₅×#126	1	2	yes	—
266	C₂³	C₅×#126	1	2	yes	—
267	C₂³	C₂×C₅×#41	1	1	yes	characteristic
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	8	4	no	—
276	C₂×C₄	C₅×#117	1	1	yes	characteristic
277	C₂×C₄	C₅×#117	1	1	yes	characteristic
278	C₂×C₄	C₅×#117	1	1	yes	characteristic
279	C₂×C₄	C₅×#117	1	1	yes	characteristic
280	C₂×C₄	C₅×#117	1	1	yes	characteristic
281	C₂×C₄	C₅×#117	1	1	yes	characteristic
282	C₂×C₄	C₅×#117	1	2	yes	—
283	C₂×C₄	C₅×#117	1	2	yes	—
284	C₂×C₄	C₅×#126	1	2	yes	—
285	C₂×C₄	C₅×#126	1	2	yes	—
286	C₂×C₄	C₅×#126	1	2	yes	—
287	C₂×C₄	C₅×#126	1	2	yes	—
288	C₂×C₄	C₅×#126	1	2	yes	—
289	C₂×C₄	C₅×#126	1	2	yes	—
290	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
291	C₂×C₄	C₂×C₅×#41	1	2	yes	—
292	C₂×C₄	C₂×C₅×#41	1	2	yes	—
293	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
294	C₈	—	2	4	no	—
295	C₈	—	2	4	no	—
296	C₈	—	4	4	no	—
297	C₅	C₂×#1841	1	1	yes	characteristic, direct product factor, Sylow
298	C₂²	—	2	2	no	—
299	C₂²	—	2	4	no	—
300	C₂²	—	4	1	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	4	2	no	—
303	C₂²	C₅×#383	1	1	yes	characteristic
304	C₂²	C₅×#383	1	2	yes	—
305	C₂²	C₅×#383	1	1	yes	characteristic
306	C₂²	C₂×C₅×#117	1	1	yes	characteristic
307	C₂²	C₂×C₅×#126	1	2	yes	—
308	C₄	—	2	4	no	—
309	C₄	—	4	2	no	—
310	C₄	—	8	4	no	—
311	C₄	C₅×#383	1	1	yes	characteristic
312	C₄	C₅×#383	1	1	yes	characteristic
313	C₄	C₅×#383	1	2	yes	—
314	C₄	C₂×C₅×#117	1	1	yes	characteristic
315	C₄	C₂×C₅×#117	1	1	yes	characteristic
316	C₄	C₂×C₅×#126	1	2	yes	—
317	C₂	—	2	1	no	—
318	C₂	—	2	2	no	—
319	C₂	—	2	1	no	—
320	C₂	—	2	2	no	—
321	C₂	C₅×#1841	1	2	yes	direct product factor
322	C₂	C₂×C₅×#383	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	#117
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂²×C₅	#383
prime radical	C₂⁴×C₅	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₅×M₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#1841	C₂	direct (N × H)
12. C₄×C₅×M₁₆	C₄	semidirect
14. C₂×C₅×#97	C₄	semidirect
15. C₂×C₄×C₅×C₈	C₄	semidirect
20. C₂×#1841	C₅	direct (N × H)
54. C₅×#97	C₂×C₄	semidirect
55. C₄×C₅×C₈	C₂×C₄	semidirect
57. #1841	C₂×C₅	direct (N × H)
113. C₄×M₁₆	C₄×C₅	semidirect
115. C₂×#97	C₄×C₅	semidirect
116. C₂×C₄×C₈	C₄×C₅	semidirect
189. #97	C₂×C₄×C₅	semidirect
190. C₄×C₈	C₂×C₄×C₅	semidirect
260. C₂×C₅	#1841	direct (N × H)
297. C₅	C₂×#1841	direct (N × H)
321. C₂	C₅×#1841	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×C₄×C₅×M₁₆	2
C₅×#1841	2
C₂×C₅×#344	2
C₂×C₅×#339	2
C₂×C₅×#338	2
C₂×C₅×H(3,ℤ/4)	2
C₂×#1841	5

Subgroups — C2×C5×#1841 #11943495

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#1841 #11943495