Subgroups — C₄₁×#20701 #23796902

Order 10496.

1470 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	C₂²×#117×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#407	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#2280	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1825	C₂	1	8	yes	maximal, complemented
5	C₄₁×#1736	C₂	1	4	yes	maximal, complemented
6	C₂⁴×C₄×C₄₁	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₄₁	C₂²	1	2	yes	—
8	C₂×#117×C₄₁	—	2	4	no	—
9	C₂×#117×C₄₁	—	2	2	no	—
10	C₂×#117×C₄₁	C₄	1	8	yes	complemented
11	C₂×#117×C₄₁	C₂²	1	2	yes	—
12	C₂×#117×C₄₁	C₂²	1	4	yes	—
13	C₂²×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
14	C₂²×C₂²⋊C₄×C₄₁	—	2	1	no	—
15	C₂²×C₄²×C₄₁	—	2	1	no	—
16	C₄₁×#407	—	2	4	no	—
17	C₂×C₂²⋊C₈×C₄₁	C₂²	1	4	yes	—
18	C₄₁×#381	—	2	1	no	—
19	C₄₁×#380	C₂²	1	4	yes	—
20	C₄₁×#379	—	2	1	no	—
21	C₄×C₂²⋊C₄×C₄₁	—	2	2	no	—
22	C₄₁×#340	C₂²	1	16	yes	complemented
23	C₂⁵×C₄₁	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₄₁	—	2	8	no	—
25	C₂³×C₄×C₄₁	—	2	2	no	—
26	C₂³×C₄×C₄₁	C₂×C₄	1	4	yes	—
27	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₄₁	D₄	1	2	yes	—
29	C₂³×C₄×C₄₁	D₄	1	4	yes	—
30	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
31	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
32	C₂×M₁₆×C₄₁	—	2	4	no	—
33	C₂×M₁₆×C₄₁	—	2	8	no	—
34	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
35	#117×C₄₁	—	2	16	no	—
36	#117×C₄₁	—	2	8	no	—
37	#117×C₄₁	—	2	8	no	—
38	#117×C₄₁	C₂×C₄	1	16	yes	—
39	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
40	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
41	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	8	yes	—
42	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
43	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
44	C₂×C₂²⋊C₄×C₄₁	—	2	8	no	—
45	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
46	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
47	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
48	C₂×C₄²×C₄₁	—	2	8	no	—
49	C₂×C₄²×C₄₁	—	2	4	no	—
50	C₂×C₄²×C₄₁	—	2	2	no	—
51	C₂²⋊C₈×C₄₁	—	2	8	no	—
52	#95×C₄₁	—	2	4	no	—
53	C₂⁴×C₄₁	—	2	4	no	—
54	C₂⁴×C₄₁	—	4	4	no	—
55	C₂⁴×C₄₁	C₂²⋊C₄	1	4	yes	—
56	C₂⁴×C₄₁	C₂×D₄	1	2	yes	—
57	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₁	—	2	4	no	—
59	C₂²×C₄×C₄₁	—	2	16	no	—
60	C₂²×C₄×C₄₁	—	2	8	no	—
61	C₂²×C₄×C₄₁	—	2	16	no	—
62	C₂²×C₄×C₄₁	—	2	8	no	—
63	C₂²×C₄×C₄₁	—	2	8	no	—
64	C₂²×C₄×C₄₁	—	2	4	no	—
65	C₂²×C₄×C₄₁	—	2	4	no	—
66	C₂²×C₄×C₄₁	—	4	8	no	—
67	C₂²×C₄×C₄₁	C₂²⋊C₄	1	4	yes	—
68	C₂²×C₄×C₄₁	C₂²⋊C₄	1	8	yes	—
69	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
70	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
73	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₄₁	#41	1	8	yes	—
76	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
77	M₁₆×C₄₁	—	4	8	no	—
78	C₂×C₈×C₄₁	—	2	4	no	—
79	C₂×C₈×C₄₁	—	2	8	no	—
80	C₄⋊C₄×C₄₁	—	2	8	no	—
81	C₄⋊C₄×C₄₁	—	2	4	no	—
82	C₄⋊C₄×C₄₁	—	2	8	no	—
83	C₄⋊C₄×C₄₁	C₂²×C₄	1	8	yes	—
84	C₂²⋊C₄×C₄₁	—	2	16	no	—
85	C₂²⋊C₄×C₄₁	—	4	8	no	—
86	C₂²⋊C₄×C₄₁	—	4	4	no	—
87	C₄²×C₄₁	—	2	16	no	—
88	C₄²×C₄₁	—	4	4	no	—
89	C₂³×C₄₁	—	2	4	no	—
90	C₂³×C₄₁	—	2	4	no	—
91	C₂³×C₄₁	—	2	4	no	—
92	C₂³×C₄₁	—	2	4	no	—
93	C₂³×C₄₁	—	2	8	no	—
94	C₂³×C₄₁	—	4	8	no	—
95	C₂³×C₄₁	—	4	16	no	—
96	C₂³×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
97	C₂³×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
98	C₂³×C₄₁	#123	1	4	yes	—
99	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
100	C₂×C₄×C₄₁	—	2	4	no	—
101	C₂×C₄×C₄₁	—	2	8	no	—
102	C₂×C₄×C₄₁	—	2	8	no	—
103	C₂×C₄×C₄₁	—	2	8	no	—
104	C₂×C₄×C₄₁	—	2	16	no	—
105	C₂×C₄×C₄₁	—	2	8	no	—
106	C₂×C₄×C₄₁	—	4	16	no	—
107	C₂×C₄×C₄₁	—	4	8	no	—
108	C₂×C₄×C₄₁	—	4	4	no	—
109	C₂×C₄×C₄₁	—	4	8	no	—
110	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
111	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
112	C₂×C₄×C₄₁	#117	1	8	yes	—
113	C₂×C₄×C₄₁	#123	1	8	yes	—
114	C₂×C₄×C₄₁	#123	1	4	yes	—
115	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₄₁	C₂×#41	1	4	yes	—
118	C₈×C₄₁	—	4	4	no	—
119	#20701	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂²×C₄₁	—	2	8	no	—
121	C₂²×C₄₁	—	2	4	no	—
122	C₂²×C₄₁	—	2	4	no	—
123	C₂²×C₄₁	—	2	4	no	—
124	C₂²×C₄₁	—	2	4	no	—
125	C₂²×C₄₁	—	4	16	no	—
126	C₂²×C₄₁	—	4	8	no	—
127	C₂²×C₄₁	#381	1	4	yes	—
128	C₂²×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
129	C₂²×C₄₁	C₂×#123	1	2	yes	—
130	C₂²×C₄₁	#575	1	4	yes	—
131	C₄×C₄₁	—	2	4	no	—
132	C₄×C₄₁	—	4	8	no	—
133	C₄×C₄₁	—	8	4	no	—
134	C₄×C₄₁	#381	1	4	yes	—
135	C₄×C₄₁	C₂×#117	1	2	yes	—
136	C₄×C₄₁	C₂×#123	1	2	yes	—
137	C₂²×#117	C₂×C₄₁	1	1	yes	characteristic
138	C₂×#407	C₂×C₄₁	1	1	yes	characteristic
139	#2280	C₂×C₄₁	1	1	yes	characteristic
140	#1825	C₂×C₄₁	1	8	yes	complemented
141	#1736	C₂×C₄₁	1	4	yes	complemented
142	C₂×C₄₁	—	2	4	no	—
143	C₂×C₄₁	—	4	4	no	—
144	C₂×C₄₁	C₂×#381	1	1	yes	characteristic
145	C₂×C₄₁	#2896	1	2	yes	—
146	C₂×C₄₁	#3093	1	4	yes	—
147	C₂⁴×C₄	C₂²×C₄₁	1	1	yes	characteristic
148	C₂²×M₁₆	C₂²×C₄₁	1	2	yes	—
149	C₂×#117	—	2	4	no	—
150	C₂×#117	—	2	2	no	—
151	C₂×#117	C₄×C₄₁	1	8	yes	complemented
152	C₂×#117	C₂²×C₄₁	1	2	yes	—
153	C₂×#117	C₂²×C₄₁	1	4	yes	—
154	C₂²×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
155	C₂²×C₂²⋊C₄	—	2	1	no	—
156	C₂²×C₄²	—	2	1	no	—
157	#407	—	2	4	no	—
158	C₂×C₂²⋊C₈	C₂²×C₄₁	1	4	yes	—
159	#381	—	2	1	no	—
160	#380	C₂²×C₄₁	1	4	yes	—
161	#379	—	2	1	no	—
162	C₄×C₂²⋊C₄	—	2	2	no	—
163	#340	C₂²×C₄₁	1	16	yes	complemented
164	C₄₁	#20701	1	1	yes	characteristic, direct product factor, Sylow
165	C₂⁵	D₄×C₄₁	1	1	yes	characteristic
166	C₂³×C₄	—	2	8	no	—
167	C₂³×C₄	—	2	2	no	—
168	C₂³×C₄	C₂×C₄×C₄₁	1	4	yes	—
169	C₂³×C₄	D₄×C₄₁	1	2	yes	—
170	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
171	C₂³×C₄	D₄×C₄₁	1	4	yes	—
172	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
173	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
174	C₂×M₁₆	—	2	4	no	—
175	C₂×M₁₆	—	2	8	no	—
176	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
177	#117	—	2	16	no	—
178	#117	—	2	8	no	—
179	#117	—	2	8	no	—
180	#117	C₂×C₄×C₄₁	1	16	yes	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	—	2	4	no	—
183	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	8	yes	—
184	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
185	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
186	C₂×C₂²⋊C₄	—	2	8	no	—
187	C₂×C₂²⋊C₄	—	2	4	no	—
188	C₂×C₂²⋊C₄	—	2	2	no	—
189	C₂×C₂²⋊C₄	—	2	4	no	—
190	C₂×C₄²	—	2	8	no	—
191	C₂×C₄²	—	2	4	no	—
192	C₂×C₄²	—	2	2	no	—
193	C₂²⋊C₈	—	2	8	no	—
194	#95	—	2	4	no	—
195	C₂⁴	—	2	4	no	—
196	C₂⁴	—	4	4	no	—
197	C₂⁴	C₂²⋊C₄×C₄₁	1	4	yes	—
198	C₂⁴	C₂×D₄×C₄₁	1	2	yes	—
199	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	16	no	—
202	C₂²×C₄	—	2	8	no	—
203	C₂²×C₄	—	2	16	no	—
204	C₂²×C₄	—	2	8	no	—
205	C₂²×C₄	—	2	8	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	4	8	no	—
209	C₂²×C₄	C₂²⋊C₄×C₄₁	1	4	yes	—
210	C₂²×C₄	C₂²⋊C₄×C₄₁	1	8	yes	—
211	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
212	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
213	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
214	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
215	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
217	C₂²×C₄	#41×C₄₁	1	8	yes	—
218	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
219	M₁₆	—	4	8	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	8	no	—
222	C₄⋊C₄	—	2	8	no	—
223	C₄⋊C₄	—	2	4	no	—
224	C₄⋊C₄	—	2	8	no	—
225	C₄⋊C₄	C₂²×C₄×C₄₁	1	8	yes	—
226	C₂²⋊C₄	—	2	16	no	—
227	C₂²⋊C₄	—	4	8	no	—
228	C₂²⋊C₄	—	4	4	no	—
229	C₄²	—	2	16	no	—
230	C₄²	—	4	4	no	—
231	C₂³	—	2	4	no	—
232	C₂³	—	2	4	no	—
233	C₂³	—	2	4	no	—
234	C₂³	—	2	4	no	—
235	C₂³	—	2	8	no	—
236	C₂³	—	4	8	no	—
237	C₂³	—	4	16	no	—
238	C₂³	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
239	C₂³	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
240	C₂³	#123×C₄₁	1	4	yes	—
241	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	—	2	8	no	—
246	C₂×C₄	—	2	16	no	—
247	C₂×C₄	—	2	8	no	—
248	C₂×C₄	—	4	16	no	—
249	C₂×C₄	—	4	8	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	8	no	—
252	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
253	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
254	C₂×C₄	#117×C₄₁	1	8	yes	—
255	C₂×C₄	#123×C₄₁	1	8	yes	—
256	C₂×C₄	#123×C₄₁	1	4	yes	—
257	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
258	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
259	C₂×C₄	C₂×#41×C₄₁	1	4	yes	—
260	C₈	—	4	4	no	—
261	C₂²	—	2	8	no	—
262	C₂²	—	2	4	no	—
263	C₂²	—	2	4	no	—
264	C₂²	—	2	4	no	—
265	C₂²	—	2	4	no	—
266	C₂²	—	4	16	no	—
267	C₂²	—	4	8	no	—
268	C₂²	C₄₁×#381	1	4	yes	—
269	C₂²	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
270	C₂²	C₂×#123×C₄₁	1	2	yes	—
271	C₂²	C₄₁×#575	1	4	yes	—
272	C₄	—	2	4	no	—
273	C₄	—	4	8	no	—
274	C₄	—	8	4	no	—
275	C₄	C₄₁×#381	1	4	yes	—
276	C₄	C₂×#117×C₄₁	1	2	yes	—
277	C₄	C₂×#123×C₄₁	1	2	yes	—
278	C₂	—	2	4	no	—
279	C₂	—	4	4	no	—
280	C₂	C₂×C₄₁×#381	1	1	yes	characteristic
281	C₂	C₄₁×#2896	1	2	yes	—
282	C₂	C₄₁×#3093	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₂⁵×C₄₁	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂⁴×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₄₁×#1825	C₂	semidirect
5. C₄₁×#1736	C₂	semidirect
10. C₂×#117×C₄₁	C₄	semidirect
22. C₄₁×#340	C₂²	semidirect
119. #20701	C₄₁	direct (N × H)
140. #1825	C₂×C₄₁	semidirect
141. #1736	C₂×C₄₁	semidirect
151. C₂×#117	C₄×C₄₁	semidirect
163. #340	C₂²×C₄₁	semidirect
164. C₄₁	#20701	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×#117×C₄₁	2
C₂×C₄₁×#407	2
C₄₁×#2280	2
C₄₁×#1825	2
C₄₁×#1736	2
#20701	41

Subgroups — C41×#20701 #23796902

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#20701 #23796902