Subgroups — C₄₁×#9721 #23785922

Order 10496.

498 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3169	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#1980	C₂	1	2	yes	maximal, complemented
3	C₄×#106×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#1502	C₂	1	2	yes	maximal, complemented
5	C₄₁×#1451	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#532	C₂²	1	2	yes	—
7	C₄₁×#500	C₄	1	1	yes	characteristic, complemented
8	C₄₁×#493	C₄	1	1	yes	characteristic, complemented
9	C₄₁×#486	C₄	1	2	yes	complemented
10	C₄₁×#466	—	2	1	no	—
11	C₄₁×#466	—	2	1	no	—
12	C₄×D₈×C₄₁	—	2	1	no	—
13	C₄×D₈×C₄₁	—	2	1	no	—
14	C₄₁×#431	C₂²	1	1	yes	characteristic
15	C₂×#106×C₄₁	—	2	1	no	—
16	C₂×#104×C₄₁	—	2	4	no	—
17	C₄×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
19	C₄×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
20	C₄₁×#340	—	2	1	no	—
21	C₄₁×#327	—	2	4	no	—
22	C₄₁×#322	—	2	1	no	—
23	C₄₁×#322	—	2	1	no	—
24	C₂×#41×C₄₁	—	2	2	no	—
25	C₂×D₈×C₄₁	—	2	1	no	—
26	C₂×D₈×C₄₁	—	2	1	no	—
27	C₂×M₁₆×C₄₁	D₄	1	2	yes	—
28	C₂²×C₈×C₄₁	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₄₁	D₄	1	1	yes	characteristic
30	C₄⋊Q₈×C₄₁	C₂×C₄	1	2	yes	—
31	#127×C₄₁	C₂×C₄	1	2	yes	—
32	#124×C₄₁	—	2	2	no	—
33	#121×C₄₁	—	2	2	no	—
34	#121×C₄₁	—	2	2	no	—
35	C₄×D₄×C₄₁	—	2	2	no	—
36	C₄×D₄×C₄₁	—	2	2	no	—
37	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
38	C₂×C₄²×C₄₁	—	2	2	no	—
39	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
40	#107×C₄₁	—	2	1	no	—
41	#107×C₄₁	—	2	1	no	—
42	#106×C₄₁	—	2	4	no	—
43	C₄⋊C₈×C₄₁	—	2	1	no	—
44	#104×C₄₁	—	4	8	no	—
45	#102×C₄₁	—	2	2	no	—
46	#102×C₄₁	—	2	2	no	—
47	C₂²⋊C₈×C₄₁	D₄	1	2	yes	—
48	#97×C₄₁	—	2	1	no	—
49	C₄×C₈×C₄₁	—	2	1	no	—
50	C₄×C₈×C₄₁	—	2	1	no	—
51	C₄×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₄₁	D₄	1	1	yes	characteristic
54	C₄×C₈×C₄₁	D₄	1	1	yes	characteristic
55	#95×C₄₁	—	2	2	no	—
56	#41×C₄₁	—	4	4	no	—
57	#41×C₄₁	—	4	4	no	—
58	C₂×Q₈×C₄₁	—	2	2	no	—
59	C₂×D₄×C₄₁	—	2	2	no	—
60	C₂×D₄×C₄₁	—	2	2	no	—
61	C₂×D₄×C₄₁	—	2	2	no	—
62	C₂×D₄×C₄₁	—	4	2	no	—
63	C₂²×C₄×C₄₁	—	4	2	no	—
64	C₂²×C₄×C₄₁	—	4	2	no	—
65	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
68	D₈×C₄₁	—	4	2	no	—
69	D₈×C₄₁	—	4	2	no	—
70	M₁₆×C₄₁	—	2	4	no	—
71	C₂×C₈×C₄₁	—	2	2	no	—
72	C₂×C₈×C₄₁	—	2	2	no	—
73	C₂×C₈×C₄₁	—	2	2	no	—
74	C₂×C₈×C₄₁	—	4	1	no	—
75	C₂×C₈×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
79	C₄⋊C₄×C₄₁	—	2	4	no	—
80	C₄⋊C₄×C₄₁	—	2	2	no	—
81	C₄⋊C₄×C₄₁	—	2	2	no	—
82	C₄⋊C₄×C₄₁	—	4	2	no	—
83	C₂²⋊C₄×C₄₁	—	4	2	no	—
84	C₂²⋊C₄×C₄₁	—	4	2	no	—
85	C₄²×C₄₁	—	4	4	no	—
86	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₄₁	#41	1	1	yes	characteristic
89	C₄²×C₄₁	#41	1	1	yes	characteristic
90	C₂³×C₄₁	—	4	2	no	—
91	C₂³×C₄₁	#127	1	1	yes	characteristic
92	Q₈×C₄₁	—	4	4	no	—
93	D₄×C₄₁	—	4	2	no	—
94	D₄×C₄₁	—	4	2	no	—
95	D₄×C₄₁	—	4	4	no	—
96	D₄×C₄₁	—	8	2	no	—
97	D₄×C₄₁	—	8	2	no	—
98	D₄×C₄₁	—	8	2	no	—
99	C₂×C₄×C₄₁	—	2	1	no	—
100	C₂×C₄×C₄₁	—	2	1	no	—
101	C₂×C₄×C₄₁	—	4	1	no	—
102	C₂×C₄×C₄₁	—	4	4	no	—
103	C₂×C₄×C₄₁	—	4	2	no	—
104	C₂×C₄×C₄₁	—	8	2	no	—
105	C₂×C₄×C₄₁	—	8	2	no	—
106	C₂×C₄×C₄₁	—	8	4	no	—
107	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₄₁	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₄₁	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
111	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
112	C₂×C₄×C₄₁	#124	1	1	yes	characteristic
113	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	4	yes	—
114	C₈×C₄₁	—	2	1	no	—
115	C₈×C₄₁	—	2	1	no	—
116	C₈×C₄₁	—	2	1	no	—
117	C₈×C₄₁	—	4	2	no	—
118	C₈×C₄₁	—	4	2	no	—
119	C₈×C₄₁	#104	1	2	yes	complemented
120	#9721	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂²×C₄₁	—	2	1	no	—
122	C₂²×C₄₁	—	2	1	no	—
123	C₂²×C₄₁	—	8	2	no	—
124	C₂²×C₄₁	—	8	2	no	—
125	C₂²×C₄₁	—	8	2	no	—
126	C₂²×C₄₁	#390	1	1	yes	characteristic
127	C₂²×C₄₁	#496	1	2	yes	—
128	C₄×C₄₁	—	2	1	no	—
129	C₄×C₄₁	—	2	1	no	—
130	C₄×C₄₁	—	2	1	no	—
131	C₄×C₄₁	—	8	4	no	—
132	C₄×C₄₁	—	8	2	no	—
133	C₄×C₄₁	C₂×#104	1	1	yes	characteristic
134	C₄×C₄₁	#421	1	1	yes	characteristic
135	C₄×C₄₁	#442	1	2	yes	—
136	C₄×C₄₁	#469	1	2	yes	—
137	#3169	C₂×C₄₁	1	1	yes	characteristic
138	#1980	C₂×C₄₁	1	2	yes	complemented
139	C₄×#106	C₂×C₄₁	1	1	yes	characteristic, complemented
140	#1502	C₂×C₄₁	1	2	yes	complemented
141	#1451	C₂×C₄₁	1	1	yes	characteristic, complemented
142	C₂×C₄₁	—	2	2	no	—
143	C₂×C₄₁	—	16	2	no	—
144	C₂×C₄₁	#1972	1	1	yes	characteristic
145	C₂×C₄₁	#1976	1	1	yes	characteristic
146	C₂×C₄₁	#1980	1	1	yes	characteristic
147	#532	C₂²×C₄₁	1	2	yes	—
148	#500	C₄×C₄₁	1	1	yes	characteristic, complemented
149	#493	C₄×C₄₁	1	1	yes	characteristic, complemented
150	#486	C₄×C₄₁	1	2	yes	complemented
151	#466	—	2	1	no	—
152	#466	—	2	1	no	—
153	C₄×D₈	—	2	1	no	—
154	C₄×D₈	—	2	1	no	—
155	#431	C₂²×C₄₁	1	1	yes	characteristic
156	C₂×#106	—	2	1	no	—
157	C₂×#104	—	2	4	no	—
158	C₄×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
159	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
160	C₄×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
161	#340	—	2	1	no	—
162	#327	—	2	4	no	—
163	#322	—	2	1	no	—
164	#322	—	2	1	no	—
165	C₄₁	#9721	1	1	yes	characteristic, direct product factor, Sylow
166	C₂×#41	—	2	2	no	—
167	C₂×D₈	—	2	1	no	—
168	C₂×D₈	—	2	1	no	—
169	C₂×M₁₆	D₄×C₄₁	1	2	yes	—
170	C₂²×C₈	D₄×C₄₁	1	1	yes	characteristic
171	C₂²×C₈	D₄×C₄₁	1	1	yes	characteristic
172	C₄⋊Q₈	C₂×C₄×C₄₁	1	2	yes	—
173	#127	C₂×C₄×C₄₁	1	2	yes	—
174	#124	—	2	2	no	—
175	#121	—	2	2	no	—
176	#121	—	2	2	no	—
177	C₄×D₄	—	2	2	no	—
178	C₄×D₄	—	2	2	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄²	—	2	2	no	—
181	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
182	#107	—	2	1	no	—
183	#107	—	2	1	no	—
184	#106	—	2	4	no	—
185	C₄⋊C₈	—	2	1	no	—
186	#104	—	4	8	no	—
187	#102	—	2	2	no	—
188	#102	—	2	2	no	—
189	C₂²⋊C₈	D₄×C₄₁	1	2	yes	—
190	#97	—	2	1	no	—
191	C₄×C₈	—	2	1	no	—
192	C₄×C₈	—	2	1	no	—
193	C₄×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
194	C₄×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
195	C₄×C₈	D₄×C₄₁	1	1	yes	characteristic
196	C₄×C₈	D₄×C₄₁	1	1	yes	characteristic
197	#95	—	2	2	no	—
198	#41	—	4	4	no	—
199	#41	—	4	4	no	—
200	C₂×Q₈	—	2	2	no	—
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	—	4	2	no	—
205	C₂²×C₄	—	4	2	no	—
206	C₂²×C₄	—	4	2	no	—
207	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
209	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
210	D₈	—	4	2	no	—
211	D₈	—	4	2	no	—
212	M₁₆	—	2	4	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	4	1	no	—
217	C₂×C₈	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
218	C₂×C₈	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
219	C₂×C₈	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
220	C₂×C₈	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	4	2	no	—
225	C₂²⋊C₄	—	4	2	no	—
226	C₂²⋊C₄	—	4	2	no	—
227	C₄²	—	4	4	no	—
228	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
229	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
230	C₄²	#41×C₄₁	1	1	yes	characteristic
231	C₄²	#41×C₄₁	1	1	yes	characteristic
232	C₂³	—	4	2	no	—
233	C₂³	#127×C₄₁	1	1	yes	characteristic
234	Q₈	—	4	4	no	—
235	D₄	—	4	2	no	—
236	D₄	—	4	2	no	—
237	D₄	—	4	4	no	—
238	D₄	—	8	2	no	—
239	D₄	—	8	2	no	—
240	D₄	—	8	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	8	2	no	—
247	C₂×C₄	—	8	2	no	—
248	C₂×C₄	—	8	4	no	—
249	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
250	C₂×C₄	C₄×D₄×C₄₁	1	1	yes	characteristic
251	C₂×C₄	C₄×D₄×C₄₁	1	1	yes	characteristic
252	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
253	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
254	C₂×C₄	#124×C₄₁	1	1	yes	characteristic
255	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	4	yes	—
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	2	1	no	—
259	C₈	—	4	2	no	—
260	C₈	—	4	2	no	—
261	C₈	#104×C₄₁	1	2	yes	complemented
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	8	2	no	—
265	C₂²	—	8	2	no	—
266	C₂²	—	8	2	no	—
267	C₂²	C₄₁×#390	1	1	yes	characteristic
268	C₂²	C₄₁×#496	1	2	yes	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	8	4	no	—
273	C₄	—	8	2	no	—
274	C₄	C₂×#104×C₄₁	1	1	yes	characteristic
275	C₄	C₄₁×#421	1	1	yes	characteristic
276	C₄	C₄₁×#442	1	2	yes	—
277	C₄	C₄₁×#469	1	2	yes	—
278	C₂	—	2	2	no	—
279	C₂	—	16	2	no	—
280	C₂	C₄₁×#1972	1	1	yes	characteristic
281	C₂	C₄₁×#1976	1	1	yes	characteristic
282	C₂	C₄₁×#1980	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₄₁	#121
commutator	C₄²	C₂²×C₄×C₄₁
Frattini, cosocle	C₂×C₄²	C₂³×C₄₁
socle	C₂²×C₄₁	#390
prime radical	C₄₁×#3169	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₁ ◁ C₂×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₁×#1980	C₂	semidirect
3. C₄×#106×C₄₁	C₂	semidirect
4. C₄₁×#1502	C₂	semidirect
5. C₄₁×#1451	C₂	semidirect
7. C₄₁×#500	C₄	semidirect
8. C₄₁×#493	C₄	semidirect
9. C₄₁×#486	C₄	semidirect
119. C₈×C₄₁	#104	semidirect
120. #9721	C₄₁	direct (N × H)
138. #1980	C₂×C₄₁	semidirect
139. C₄×#106	C₂×C₄₁	semidirect
140. #1502	C₂×C₄₁	semidirect
141. #1451	C₂×C₄₁	semidirect
148. #500	C₄×C₄₁	semidirect
149. #493	C₄×C₄₁	semidirect
150. #486	C₄×C₄₁	semidirect
165. C₄₁	#9721	direct (N × H)
261. C₈	#104×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₄₁×#3169	2
C₄₁×#1980	2
C₄×#106×C₄₁	2
C₄₁×#1502	2
C₄₁×#1451	2
#9721	41

Subgroups — C41×#9721 #23785922

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#9721 #23785922