Subgroups — C₄₁×#33816 #23810017

Order 10496.

502 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3168	C₂	1	2	yes	maximal, complemented
2	C₄₁×#2229	C₂	1	2	yes	maximal, complemented
3	C₄₁×#2228	C₂	1	2	yes	maximal, complemented
4	C₄₁×#2217	C₂	1	2	yes	maximal, complemented
5	C₄₁×#2216	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#2215	C₂	1	1	yes	maximal, characteristic
7	C₄₁×#2211	C₂	1	2	yes	maximal, complemented
8	C₄₁×#2210	C₂	1	2	yes	maximal, complemented
9	C₂×C₄₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₁×#529	—	2	2	no	—
11	C₂×Q₃₂×C₄₁	C₂²	1	1	yes	characteristic
12	C₂×SD₃₂×C₄₁	C₂²	1	2	yes	—
13	C₂×D₁₆×C₄₁	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₄₁	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₄₁	C₂²	1	1	yes	characteristic
16	C₄₁×#499	C₂²	1	2	yes	—
17	C₄₁×#495	C₂²	1	2	yes	—
18	C₄₁×#488	—	2	2	no	—
19	C₄₁×#467	C₂²	1	2	yes	—
20	C₄₁×#466	C₂²	1	2	yes	—
21	C₄₁×#465	—	2	2	no	—
22	C₄×Q₁₆×C₄₁	C₂²	1	2	yes	—
23	C₄×SD₁₆×C₄₁	—	2	2	no	—
24	C₄×D₈×C₄₁	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
26	C₄₁×#367	C₂²	1	2	yes	—
27	C₄₁×#366	C₂²	1	1	yes	characteristic
28	C₄₁×#366	C₂²	1	1	yes	characteristic
29	C₄₁×C₄⋊C₁₆	C₂²	1	2	yes	—
30	C₄₁×C₄⋊C₁₆	C₂²	1	2	yes	—
31	C₄₁×#358	C₂²	1	2	yes	—
32	C₄₁×#358	C₂²	1	2	yes	—
33	C₄₁×#357	C₂²	1	2	yes	—
34	C₄₁×#357	C₂²	1	2	yes	—
35	C₂×Q₁₆×C₄₁	C₂³	1	2	yes	—
36	C₂×SD₁₆×C₄₁	—	2	2	no	—
37	C₂×D₈×C₄₁	C₂³	1	2	yes	—
38	C₂²×C₈×C₄₁	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
40	#126×C₄₁	—	4	2	no	—
41	#125×C₄₁	—	2	2	no	—
42	#124×C₄₁	—	2	2	no	—
43	#123×C₄₁	—	4	2	no	—
44	C₂²⋊Q₈×C₄₁	—	2	2	no	—
45	#121×C₄₁	—	2	2	no	—
46	C₄×Q₈×C₄₁	—	2	2	no	—
47	C₄×D₄×C₄₁	—	2	2	no	—
48	C₄×D₄×C₄₁	—	4	2	no	—
49	#117×C₄₁	—	4	2	no	—
50	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
51	Q₃₂×C₄₁	—	4	1	no	—
52	SD₃₂×C₄₁	—	4	2	no	—
53	D₁₆×C₄₁	—	4	1	no	—
54	C₂×C₁₆×C₄₁	—	4	1	no	—
55	C₂×C₁₆×C₄₁	—	4	1	no	—
56	C₂×C₁₆×C₄₁	C₂³	1	1	yes	characteristic
57	C₂×C₁₆×C₄₁	C₂³	1	2	yes	—
58	C₂×C₁₆×C₄₁	C₂³	1	1	yes	characteristic
59	#107×C₄₁	C₂³	1	2	yes	—
60	#107×C₄₁	C₂³	1	2	yes	—
61	#106×C₄₁	—	2	2	no	—
62	#103×C₄₁	—	2	2	no	—
63	#103×C₄₁	—	2	2	no	—
64	#102×C₄₁	—	2	2	no	—
65	#102×C₄₁	—	2	2	no	—
66	C₄×C₈×C₄₁	D₄	1	2	yes	—
67	C₄×C₈×C₄₁	C₂³	1	2	yes	—
68	C₂×Q₈×C₄₁	—	2	2	no	—
69	C₂×D₄×C₄₁	—	2	2	no	—
70	C₂×D₄×C₄₁	—	4	2	no	—
71	C₂²×C₄×C₄₁	—	4	2	no	—
72	C₂²×C₄×C₄₁	D₈	1	1	yes	characteristic
73	C₂²×C₄×C₄₁	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
75	Q₁₆×C₄₁	—	2	2	no	—
76	Q₁₆×C₄₁	—	2	2	no	—
77	SD₁₆×C₄₁	—	4	4	no	—
78	D₈×C₄₁	—	2	2	no	—
79	D₈×C₄₁	—	2	2	no	—
80	C₂×C₈×C₄₁	—	2	1	no	—
81	C₂×C₈×C₄₁	—	2	1	no	—
82	C₂×C₈×C₄₁	—	4	1	no	—
83	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₄₁	C₂⁴	1	1	yes	characteristic
86	C₄⋊C₄×C₄₁	—	2	2	no	—
87	C₄⋊C₄×C₄₁	—	2	2	no	—
88	C₄⋊C₄×C₄₁	—	4	2	no	—
89	C₄⋊C₄×C₄₁	—	4	2	no	—
90	C₄⋊C₄×C₄₁	—	4	2	no	—
91	C₄⋊C₄×C₄₁	—	4	2	no	—
92	C₂²⋊C₄×C₄₁	—	4	2	no	—
93	C₂²⋊C₄×C₄₁	—	4	2	no	—
94	C₂²⋊C₄×C₄₁	—	4	2	no	—
95	C₂²⋊C₄×C₄₁	—	4	2	no	—
96	C₂²⋊C₄×C₄₁	—	4	2	no	—
97	C₄²×C₄₁	—	4	2	no	—
98	C₄²×C₄₁	D₈	1	2	yes	—
99	C₄²×C₄₁	C₂×D₄	1	2	yes	—
100	C₁₆×C₄₁	—	2	1	no	—
101	C₁₆×C₄₁	—	2	2	no	—
102	C₁₆×C₄₁	—	2	1	no	—
103	C₂³×C₄₁	—	4	2	no	—
104	C₂³×C₄₁	C₂×D₈	1	1	yes	characteristic
105	Q₈×C₄₁	—	4	2	no	—
106	Q₈×C₄₁	—	4	2	no	—
107	D₄×C₄₁	—	4	2	no	—
108	D₄×C₄₁	—	4	2	no	—
109	D₄×C₄₁	—	8	4	no	—
110	C₂×C₄×C₄₁	—	2	2	no	—
111	C₂×C₄×C₄₁	—	2	1	no	—
112	C₂×C₄×C₄₁	—	2	1	no	—
113	C₂×C₄×C₄₁	—	4	1	no	—
114	C₂×C₄×C₄₁	—	4	2	no	—
115	C₂×C₄×C₄₁	—	4	2	no	—
116	C₂×C₄×C₄₁	—	4	2	no	—
117	C₂×C₄×C₄₁	—	8	4	no	—
118	C₂×C₄×C₄₁	C₂×D₈	1	2	yes	—
119	C₂×C₄×C₄₁	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₄₁	C₂×D₈	1	2	yes	—
121	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
122	C₈×C₄₁	—	2	1	no	—
123	C₈×C₄₁	—	2	2	no	—
124	C₈×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
125	C₈×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
126	#33816	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂²×C₄₁	—	2	1	no	—
128	C₂²×C₄₁	—	2	1	no	—
129	C₂²×C₄₁	—	8	2	no	—
130	C₂²×C₄₁	—	8	2	no	—
131	C₂²×C₄₁	—	8	2	no	—
132	C₂²×C₄₁	#508	1	2	yes	—
133	C₂²×C₄₁	C₂²×D₈	1	1	yes	characteristic
134	C₄×C₄₁	—	2	2	no	—
135	C₄×C₄₁	—	2	2	no	—
136	C₄×C₄₁	—	2	1	no	—
137	C₄×C₄₁	—	8	2	no	—
138	C₄×C₄₁	—	8	2	no	—
139	C₄×C₄₁	—	8	2	no	—
140	C₄×C₄₁	#537	1	1	yes	characteristic
141	C₄×C₄₁	#575	1	1	yes	characteristic
142	#3168	C₂×C₄₁	1	2	yes	complemented
143	#2229	C₂×C₄₁	1	2	yes	complemented
144	#2228	C₂×C₄₁	1	2	yes	complemented
145	#2217	C₂×C₄₁	1	2	yes	complemented
146	#2216	C₂×C₄₁	1	1	yes	characteristic, complemented
147	#2215	C₂×C₄₁	1	1	yes	characteristic
148	#2211	C₂×C₄₁	1	2	yes	complemented
149	#2210	C₂×C₄₁	1	2	yes	complemented
150	C₂×C₄⋊C₁₆	C₂×C₄₁	1	1	yes	characteristic, complemented
151	C₂×C₄₁	—	2	2	no	—
152	C₂×C₄₁	—	16	2	no	—
153	C₂×C₄₁	#3245	1	1	yes	characteristic
154	C₂×C₄₁	C₂×#508	1	1	yes	characteristic
155	C₂×C₄₁	#3414	1	1	yes	characteristic
156	#529	—	2	2	no	—
157	C₂×Q₃₂	C₂²×C₄₁	1	1	yes	characteristic
158	C₂×SD₃₂	C₂²×C₄₁	1	2	yes	—
159	C₂×D₁₆	C₂²×C₄₁	1	1	yes	characteristic
160	C₂²×C₁₆	C₂²×C₄₁	1	1	yes	characteristic
161	C₂²×C₁₆	C₂²×C₄₁	1	1	yes	characteristic
162	#499	C₂²×C₄₁	1	2	yes	—
163	#495	C₂²×C₄₁	1	2	yes	—
164	#488	—	2	2	no	—
165	#467	C₂²×C₄₁	1	2	yes	—
166	#466	C₂²×C₄₁	1	2	yes	—
167	#465	—	2	2	no	—
168	C₄×Q₁₆	C₂²×C₄₁	1	2	yes	—
169	C₄×SD₁₆	—	2	2	no	—
170	C₄×D₈	C₂²×C₄₁	1	2	yes	—
171	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
172	#367	C₂²×C₄₁	1	2	yes	—
173	#366	C₂²×C₄₁	1	1	yes	characteristic
174	#366	C₂²×C₄₁	1	1	yes	characteristic
175	C₄⋊C₁₆	C₂²×C₄₁	1	2	yes	—
176	C₄⋊C₁₆	C₂²×C₄₁	1	2	yes	—
177	#358	C₂²×C₄₁	1	2	yes	—
178	#358	C₂²×C₄₁	1	2	yes	—
179	#357	C₂²×C₄₁	1	2	yes	—
180	#357	C₂²×C₄₁	1	2	yes	—
181	C₄₁	#33816	1	1	yes	characteristic, direct product factor, Sylow
182	C₂×Q₁₆	C₂³×C₄₁	1	2	yes	—
183	C₂×SD₁₆	—	2	2	no	—
184	C₂×D₈	C₂³×C₄₁	1	2	yes	—
185	C₂²×C₈	D₄×C₄₁	1	1	yes	characteristic
186	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
187	#126	—	4	2	no	—
188	#125	—	2	2	no	—
189	#124	—	2	2	no	—
190	#123	—	4	2	no	—
191	C₂²⋊Q₈	—	2	2	no	—
192	#121	—	2	2	no	—
193	C₄×Q₈	—	2	2	no	—
194	C₄×D₄	—	2	2	no	—
195	C₄×D₄	—	4	2	no	—
196	#117	—	4	2	no	—
197	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
198	Q₃₂	—	4	1	no	—
199	SD₃₂	—	4	2	no	—
200	D₁₆	—	4	1	no	—
201	C₂×C₁₆	—	4	1	no	—
202	C₂×C₁₆	—	4	1	no	—
203	C₂×C₁₆	C₂³×C₄₁	1	1	yes	characteristic
204	C₂×C₁₆	C₂³×C₄₁	1	2	yes	—
205	C₂×C₁₆	C₂³×C₄₁	1	1	yes	characteristic
206	#107	C₂³×C₄₁	1	2	yes	—
207	#107	C₂³×C₄₁	1	2	yes	—
208	#106	—	2	2	no	—
209	#103	—	2	2	no	—
210	#103	—	2	2	no	—
211	#102	—	2	2	no	—
212	#102	—	2	2	no	—
213	C₄×C₈	D₄×C₄₁	1	2	yes	—
214	C₄×C₈	C₂³×C₄₁	1	2	yes	—
215	C₂×Q₈	—	2	2	no	—
216	C₂×D₄	—	2	2	no	—
217	C₂×D₄	—	4	2	no	—
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	D₈×C₄₁	1	1	yes	characteristic
220	C₂²×C₄	D₈×C₄₁	1	1	yes	characteristic
221	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
222	Q₁₆	—	2	2	no	—
223	Q₁₆	—	2	2	no	—
224	SD₁₆	—	4	4	no	—
225	D₈	—	2	2	no	—
226	D₈	—	2	2	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
231	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
232	C₂×C₈	C₂⁴×C₄₁	1	1	yes	characteristic
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₂²⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	—	4	2	no	—
241	C₂²⋊C₄	—	4	2	no	—
242	C₂²⋊C₄	—	4	2	no	—
243	C₂²⋊C₄	—	4	2	no	—
244	C₄²	—	4	2	no	—
245	C₄²	D₈×C₄₁	1	2	yes	—
246	C₄²	C₂×D₄×C₄₁	1	2	yes	—
247	C₁₆	—	2	1	no	—
248	C₁₆	—	2	2	no	—
249	C₁₆	—	2	1	no	—
250	C₂³	—	4	2	no	—
251	C₂³	C₂×D₈×C₄₁	1	1	yes	characteristic
252	Q₈	—	4	2	no	—
253	Q₈	—	4	2	no	—
254	D₄	—	4	2	no	—
255	D₄	—	4	2	no	—
256	D₄	—	8	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	8	4	no	—
265	C₂×C₄	C₂×D₈×C₄₁	1	2	yes	—
266	C₂×C₄	C₂×D₈×C₄₁	1	1	yes	characteristic
267	C₂×C₄	C₂×D₈×C₄₁	1	2	yes	—
268	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
269	C₈	—	2	1	no	—
270	C₈	—	2	2	no	—
271	C₈	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
272	C₈	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	8	2	no	—
276	C₂²	—	8	2	no	—
277	C₂²	—	8	2	no	—
278	C₂²	C₄₁×#508	1	2	yes	—
279	C₂²	C₂²×D₈×C₄₁	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	1	no	—
283	C₄	—	8	2	no	—
284	C₄	—	8	2	no	—
285	C₄	—	8	2	no	—
286	C₄	C₄₁×#537	1	1	yes	characteristic
287	C₄	C₄₁×#575	1	1	yes	characteristic
288	C₂	—	2	2	no	—
289	C₂	—	16	2	no	—
290	C₂	C₄₁×#3245	1	1	yes	characteristic
291	C₂	C₂×C₄₁×#508	1	1	yes	characteristic
292	C₂	C₄₁×#3414	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₄₁
prime radical	C₄₁×#2215	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₂×C₄²×C₄₁ ◁ C₂×C₄×C₈×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₁×#3168	C₂	semidirect
2. C₄₁×#2229	C₂	semidirect
3. C₄₁×#2228	C₂	semidirect
4. C₄₁×#2217	C₂	semidirect
5. C₄₁×#2216	C₂	semidirect
7. C₄₁×#2211	C₂	semidirect
8. C₄₁×#2210	C₂	semidirect
9. C₂×C₄₁×C₄⋊C₁₆	C₂	semidirect
126. #33816	C₄₁	direct (N × H)
142. #3168	C₂×C₄₁	semidirect
143. #2229	C₂×C₄₁	semidirect
144. #2228	C₂×C₄₁	semidirect
145. #2217	C₂×C₄₁	semidirect
146. #2216	C₂×C₄₁	semidirect
148. #2211	C₂×C₄₁	semidirect
149. #2210	C₂×C₄₁	semidirect
150. C₂×C₄⋊C₁₆	C₂×C₄₁	semidirect
181. C₄₁	#33816	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄₁×#3168	2
C₄₁×#2229	2
C₄₁×#2228	2
C₄₁×#2217	2
C₄₁×#2216	2
C₄₁×#2215	2
C₄₁×#2211	2
C₄₁×#2210	2
C₂×C₄₁×C₄⋊C₁₆	2
#33816	41

Subgroups — C41×#33816 #23810017

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#33816 #23810017