Subgroups — C₄₁×#2405 #19106185

Order 5248.

1116 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₄×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#123×C₄₁	C₂	1	2	yes	maximal, complemented
3	C₂²×C₂²⋊C₄×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#394	C₂	1	4	yes	maximal, complemented
5	C₄₁×#386	C₂	1	2	yes	maximal, complemented
6	C₄₁×#385	C₂	1	2	yes	maximal, complemented
7	C₄₁×#379	C₂	1	2	yes	maximal, complemented
8	C₄₁×#379	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁵×C₄₁	C₂²	1	1	yes	characteristic
10	C₂⁵×C₄₁	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₄₁	—	2	4	no	—
12	C₂²×D₄×C₄₁	—	2	8	no	—
13	C₂²×D₄×C₄₁	C₂²	1	2	yes	—
14	C₂²×D₄×C₄₁	C₂²	1	2	yes	—
15	C₂³×C₄×C₄₁	C₂²	1	1	yes	characteristic
16	C₂³×C₄×C₄₁	C₂²	1	1	yes	characteristic, complemented
17	C₂³×C₄×C₄₁	C₂²	1	1	yes	characteristic, complemented
18	#123×C₄₁	—	2	8	no	—
19	C₂×C₄⋊C₄×C₄₁	C₂²	1	4	yes	complemented
20	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
21	C₂×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	complemented
24	C₂×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	complemented
25	C₂×C₂²⋊C₄×C₄₁	C₂²	1	4	yes	complemented
26	C₂×C₂²⋊C₄×C₄₁	C₂²	1	4	yes	complemented
27	C₂×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	complemented
28	#95×C₄₁	C₂²	1	4	yes	complemented
29	C₂⁴×C₄₁	—	2	4	no	—
30	C₂⁴×C₄₁	—	2	4	no	—
31	C₂⁴×C₄₁	—	2	2	no	—
32	C₂⁴×C₄₁	—	2	4	no	—
33	C₂⁴×C₄₁	—	2	4	no	—
34	C₂⁴×C₄₁	—	4	2	no	—
35	C₂⁴×C₄₁	—	4	2	no	—
36	C₂⁴×C₄₁	D₄	1	4	yes	—
37	C₂⁴×C₄₁	C₂³	1	1	yes	characteristic
38	C₂⁴×C₄₁	C₂³	1	2	yes	—
39	C₂⁴×C₄₁	C₂³	1	1	yes	characteristic
40	C₂⁴×C₄₁	C₂³	1	1	yes	characteristic
41	C₂×D₄×C₄₁	—	2	32	no	—
42	C₂×D₄×C₄₁	—	2	4	no	—
43	C₂×D₄×C₄₁	—	2	4	no	—
44	C₂×D₄×C₄₁	—	2	4	no	—
45	C₂×D₄×C₄₁	—	2	4	no	—
46	C₂×D₄×C₄₁	—	2	4	no	—
47	C₂×D₄×C₄₁	D₄	1	8	yes	—
48	C₂²×C₄×C₄₁	—	2	2	no	—
49	C₂²×C₄×C₄₁	—	2	4	no	—
50	C₂²×C₄×C₄₁	—	2	2	no	—
51	C₂²×C₄×C₄₁	—	2	2	no	—
52	C₂²×C₄×C₄₁	—	2	4	no	—
53	C₂²×C₄×C₄₁	—	4	2	no	—
54	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
55	C₂²×C₄×C₄₁	C₂³	1	4	yes	—
56	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
57	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
58	C₄⋊C₄×C₄₁	—	2	8	no	—
59	C₂²⋊C₄×C₄₁	—	2	4	no	—
60	C₂²⋊C₄×C₄₁	—	2	4	no	—
61	C₂²⋊C₄×C₄₁	—	2	4	no	—
62	C₂²⋊C₄×C₄₁	—	2	8	no	—
63	C₂²⋊C₄×C₄₁	—	2	4	no	—
64	C₂²⋊C₄×C₄₁	—	4	2	no	—
65	C₂²⋊C₄×C₄₁	—	4	4	no	—
66	C₂³×C₄₁	—	2	2	no	—
67	C₂³×C₄₁	—	2	2	no	—
68	C₂³×C₄₁	—	2	2	no	—
69	C₂³×C₄₁	—	2	2	no	—
70	C₂³×C₄₁	—	2	8	no	—
71	C₂³×C₄₁	—	2	8	no	—
72	C₂³×C₄₁	—	2	8	no	—
73	C₂³×C₄₁	—	2	2	no	—
74	C₂³×C₄₁	—	2	2	no	—
75	C₂³×C₄₁	—	2	2	no	—
76	C₂³×C₄₁	—	2	2	no	—
77	C₂³×C₄₁	—	2	4	no	—
78	C₂³×C₄₁	—	2	2	no	—
79	C₂³×C₄₁	—	2	4	no	—
80	C₂³×C₄₁	—	4	4	no	—
81	C₂³×C₄₁	—	4	16	no	—
82	C₂³×C₄₁	—	4	16	no	—
83	C₂³×C₄₁	—	4	4	no	—
84	C₂³×C₄₁	—	4	4	no	—
85	C₂³×C₄₁	C₂×D₄	1	2	yes	—
86	C₂³×C₄₁	C₂×D₄	1	4	yes	—
87	C₂³×C₄₁	C₂×D₄	1	4	yes	—
88	C₂³×C₄₁	C₂×D₄	1	2	yes	—
89	C₂³×C₄₁	C₂×D₄	1	2	yes	—
90	C₂³×C₄₁	#41	1	4	yes	—
91	C₂³×C₄₁	C₂⁴	1	1	yes	characteristic
92	D₄×C₄₁	—	2	16	no	—
93	D₄×C₄₁	—	2	16	no	—
94	C₂×C₄×C₄₁	—	2	8	no	—
95	C₂×C₄×C₄₁	—	2	2	no	—
96	C₂×C₄×C₄₁	—	2	2	no	—
97	C₂×C₄×C₄₁	—	2	4	no	—
98	C₂×C₄×C₄₁	—	2	2	no	—
99	C₂×C₄×C₄₁	—	2	2	no	—
100	C₂×C₄×C₄₁	—	2	4	no	—
101	C₂×C₄×C₄₁	—	2	4	no	—
102	C₂×C₄×C₄₁	—	2	4	no	—
103	C₂×C₄×C₄₁	—	2	4	no	—
104	C₂×C₄×C₄₁	—	4	4	no	—
105	C₂×C₄×C₄₁	—	4	4	no	—
106	C₂×C₄×C₄₁	C₂×D₄	1	4	yes	—
107	C₂²×C₄₁	—	2	2	no	—
108	C₂²×C₄₁	—	2	2	no	—
109	C₂²×C₄₁	—	2	2	no	—
110	C₂²×C₄₁	—	2	2	no	—
111	C₂²×C₄₁	—	2	8	no	—
112	C₂²×C₄₁	—	2	8	no	—
113	C₂²×C₄₁	—	2	2	no	—
114	C₂²×C₄₁	—	2	4	no	—
115	C₂²×C₄₁	—	2	2	no	—
116	C₂²×C₄₁	—	2	2	no	—
117	C₂²×C₄₁	—	2	8	no	—
118	C₂²×C₄₁	—	4	16	no	—
119	C₂²×C₄₁	—	4	16	no	—
120	C₂²×C₄₁	—	4	2	no	—
121	C₂²×C₄₁	—	4	2	no	—
122	C₂²×C₄₁	—	4	4	no	—
123	C₂²×C₄₁	—	4	4	no	—
124	C₂²×C₄₁	#120	1	4	yes	—
125	C₂²×C₄₁	#123	1	4	yes	complemented
126	C₂²×C₄₁	C₂²×D₄	1	1	yes	characteristic
127	C₂²×C₄₁	C₂²×D₄	1	2	yes	—
128	C₂²×C₄₁	C₂×#41	1	2	yes	—
129	C₂²×C₄₁	H(5,𝔽₂)	1	2	yes	—
130	C₄×C₄₁	—	2	4	no	—
131	C₄×C₄₁	—	4	2	no	—
132	C₄×C₄₁	—	4	2	no	—
133	C₄×C₄₁	—	4	4	no	—
134	#2405	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₂×C₄₁	—	2	4	no	—
136	C₂×C₄₁	—	2	4	no	—
137	C₂×C₄₁	—	4	2	no	—
138	C₂×C₄₁	—	4	4	no	—
139	C₂×C₄₁	C₂×#120	1	1	yes	characteristic
140	C₂×C₄₁	C₂×#123	1	1	yes	characteristic
141	C₂×C₄₁	#534	1	1	yes	characteristic
142	C₂×C₄₁	#546	1	4	yes	—
143	C₂³×D₄	C₂×C₄₁	1	1	yes	characteristic
144	C₂×#123	C₂×C₄₁	1	2	yes	complemented
145	C₂²×C₂²⋊C₄	C₂×C₄₁	1	1	yes	characteristic, complemented
146	#394	C₂×C₄₁	1	4	yes	complemented
147	#386	C₂×C₄₁	1	2	yes	complemented
148	#385	C₂×C₄₁	1	2	yes	complemented
149	#379	C₂×C₄₁	1	2	yes	complemented
150	#379	C₂×C₄₁	1	1	yes	characteristic, complemented
151	C₄₁	#2405	1	1	yes	characteristic, direct product factor, Sylow
152	C₂⁵	C₂²×C₄₁	1	1	yes	characteristic
153	C₂⁵	C₂²×C₄₁	1	1	yes	characteristic
154	C₂²×D₄	—	2	4	no	—
155	C₂²×D₄	—	2	8	no	—
156	C₂²×D₄	C₂²×C₄₁	1	2	yes	—
157	C₂²×D₄	C₂²×C₄₁	1	2	yes	—
158	C₂³×C₄	C₂²×C₄₁	1	1	yes	characteristic
159	C₂³×C₄	C₂²×C₄₁	1	1	yes	characteristic, complemented
160	C₂³×C₄	C₂²×C₄₁	1	1	yes	characteristic, complemented
161	#123	—	2	8	no	—
162	C₂×C₄⋊C₄	C₂²×C₄₁	1	4	yes	complemented
163	C₂×C₂²⋊C₄	—	2	4	no	—
164	C₂×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
165	C₂×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
166	C₂×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	complemented
167	C₂×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	complemented
168	C₂×C₂²⋊C₄	C₂²×C₄₁	1	4	yes	complemented
169	C₂×C₂²⋊C₄	C₂²×C₄₁	1	4	yes	complemented
170	C₂×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	complemented
171	#95	C₂²×C₄₁	1	4	yes	complemented
172	C₂⁴	—	2	4	no	—
173	C₂⁴	—	2	4	no	—
174	C₂⁴	—	2	2	no	—
175	C₂⁴	—	2	4	no	—
176	C₂⁴	—	2	4	no	—
177	C₂⁴	—	4	2	no	—
178	C₂⁴	—	4	2	no	—
179	C₂⁴	D₄×C₄₁	1	4	yes	—
180	C₂⁴	C₂³×C₄₁	1	1	yes	characteristic
181	C₂⁴	C₂³×C₄₁	1	2	yes	—
182	C₂⁴	C₂³×C₄₁	1	1	yes	characteristic
183	C₂⁴	C₂³×C₄₁	1	1	yes	characteristic
184	C₂×D₄	—	2	32	no	—
185	C₂×D₄	—	2	4	no	—
186	C₂×D₄	—	2	4	no	—
187	C₂×D₄	—	2	4	no	—
188	C₂×D₄	—	2	4	no	—
189	C₂×D₄	—	2	4	no	—
190	C₂×D₄	D₄×C₄₁	1	8	yes	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
198	C₂²×C₄	C₂³×C₄₁	1	4	yes	—
199	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
200	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
201	C₄⋊C₄	—	2	8	no	—
202	C₂²⋊C₄	—	2	4	no	—
203	C₂²⋊C₄	—	2	4	no	—
204	C₂²⋊C₄	—	2	4	no	—
205	C₂²⋊C₄	—	2	8	no	—
206	C₂²⋊C₄	—	2	4	no	—
207	C₂²⋊C₄	—	4	2	no	—
208	C₂²⋊C₄	—	4	4	no	—
209	C₂³	—	2	2	no	—
210	C₂³	—	2	2	no	—
211	C₂³	—	2	2	no	—
212	C₂³	—	2	2	no	—
213	C₂³	—	2	8	no	—
214	C₂³	—	2	8	no	—
215	C₂³	—	2	8	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	2	2	no	—
218	C₂³	—	2	2	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	4	no	—
221	C₂³	—	2	2	no	—
222	C₂³	—	2	4	no	—
223	C₂³	—	4	4	no	—
224	C₂³	—	4	16	no	—
225	C₂³	—	4	16	no	—
226	C₂³	—	4	4	no	—
227	C₂³	—	4	4	no	—
228	C₂³	C₂×D₄×C₄₁	1	2	yes	—
229	C₂³	C₂×D₄×C₄₁	1	4	yes	—
230	C₂³	C₂×D₄×C₄₁	1	4	yes	—
231	C₂³	C₂×D₄×C₄₁	1	2	yes	—
232	C₂³	C₂×D₄×C₄₁	1	2	yes	—
233	C₂³	#41×C₄₁	1	4	yes	—
234	C₂³	C₂⁴×C₄₁	1	1	yes	characteristic
235	D₄	—	2	16	no	—
236	D₄	—	2	16	no	—
237	C₂×C₄	—	2	8	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	C₂×D₄×C₄₁	1	4	yes	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	8	no	—
255	C₂²	—	2	8	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	4	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	8	no	—
261	C₂²	—	4	16	no	—
262	C₂²	—	4	16	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	4	4	no	—
266	C₂²	—	4	4	no	—
267	C₂²	#120×C₄₁	1	4	yes	—
268	C₂²	#123×C₄₁	1	4	yes	complemented
269	C₂²	C₂²×D₄×C₄₁	1	1	yes	characteristic
270	C₂²	C₂²×D₄×C₄₁	1	2	yes	—
271	C₂²	C₂×#41×C₄₁	1	2	yes	—
272	C₂²	H(5,𝔽₂)×C₄₁	1	2	yes	—
273	C₄	—	2	4	no	—
274	C₄	—	4	2	no	—
275	C₄	—	4	2	no	—
276	C₄	—	4	4	no	—
277	C₂	—	2	4	no	—
278	C₂	—	2	4	no	—
279	C₂	—	4	2	no	—
280	C₂	—	4	4	no	—
281	C₂	C₂×#120×C₄₁	1	1	yes	characteristic
282	C₂	C₂×#123×C₄₁	1	1	yes	characteristic
283	C₂	C₄₁×#534	1	1	yes	characteristic
284	C₂	C₄₁×#546	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₄₁
prime radical	C₂³×D₄×C₄₁	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#123×C₄₁	C₂	semidirect
3. C₂²×C₂²⋊C₄×C₄₁	C₂	semidirect
4. C₄₁×#394	C₂	semidirect
5. C₄₁×#386	C₂	semidirect
6. C₄₁×#385	C₂	semidirect
7. C₄₁×#379	C₂	semidirect
8. C₄₁×#379	C₂	semidirect
16. C₂³×C₄×C₄₁	C₂²	semidirect
17. C₂³×C₄×C₄₁	C₂²	semidirect
19. C₂×C₄⋊C₄×C₄₁	C₂²	semidirect
23. C₂×C₂²⋊C₄×C₄₁	C₂²	semidirect
24. C₂×C₂²⋊C₄×C₄₁	C₂²	semidirect
25. C₂×C₂²⋊C₄×C₄₁	C₂²	semidirect
26. C₂×C₂²⋊C₄×C₄₁	C₂²	semidirect
27. C₂×C₂²⋊C₄×C₄₁	C₂²	semidirect
28. #95×C₄₁	C₂²	semidirect
125. C₂²×C₄₁	#123	semidirect
134. #2405	C₄₁	direct (N × H)
144. C₂×#123	C₂×C₄₁	semidirect
145. C₂²×C₂²⋊C₄	C₂×C₄₁	semidirect
146. #394	C₂×C₄₁	semidirect
147. #386	C₂×C₄₁	semidirect
148. #385	C₂×C₄₁	semidirect
149. #379	C₂×C₄₁	semidirect
150. #379	C₂×C₄₁	semidirect
151. C₄₁	#2405	direct (N × H)
159. C₂³×C₄	C₂²×C₄₁	semidirect
160. C₂³×C₄	C₂²×C₄₁	semidirect
162. C₂×C₄⋊C₄	C₂²×C₄₁	semidirect
166. C₂×C₂²⋊C₄	C₂²×C₄₁	semidirect
167. C₂×C₂²⋊C₄	C₂²×C₄₁	semidirect
168. C₂×C₂²⋊C₄	C₂²×C₄₁	semidirect
169. C₂×C₂²⋊C₄	C₂²×C₄₁	semidirect
170. C₂×C₂²⋊C₄	C₂²×C₄₁	semidirect
171. #95	C₂²×C₄₁	semidirect
268. C₂²	#123×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂³×D₄×C₄₁	2
C₂×#123×C₄₁	2
C₂²×C₂²⋊C₄×C₄₁	2
C₄₁×#394	2
C₄₁×#386	2
C₄₁×#385	2
C₄₁×#379	2
#2405	41

Subgroups — C41×#2405 #19106185

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#2405 #19106185