Subgroups — C₄₁×#33465 #23809666

Order 10496.

642 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	C₂×C₄₁×#508	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2957	C₂	1	2	yes	maximal, complemented
3	C₄₁×#2179	C₂	1	4	yes	maximal, complemented
4	C₄₁×#2178	C₂	1	4	yes	maximal, complemented
5	C₂×C₄₁×#368	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#2143	C₂	1	2	yes	maximal, complemented
7	C₄₁×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135×C₄₁	C₂²	1	2	yes	—
9	C₄₁×#568	—	2	2	no	—
10	C₄₁×#508	C₄	1	2	yes	complemented
11	C₄₁×#508	C₄	1	2	yes	complemented
12	C₄₁×#508	C₂²	1	2	yes	—
13	C₄₁×#508	C₂²	1	2	yes	—
14	C₂×Q₃₂×C₄₁	C₄	1	1	yes	characteristic, complemented
15	C₂×SD₃₂×C₄₁	C₄	1	2	yes	complemented
16	C₂×D₁₆×C₄₁	C₄	1	1	yes	characteristic, complemented
17	C₂×M₃₂×C₄₁	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₄₁	C₂²	1	1	yes	characteristic
19	C₄₁×#444	—	2	4	no	—
20	C₄₁×#443	C₂²	1	8	yes	complemented
21	C₂×#108×C₄₁	C₂²	1	2	yes	—
22	C₄₁×#421	—	2	2	no	—
23	C₄₁×#405	C₂²	1	1	yes	characteristic
24	C₄₁×#368	C₂²	1	2	yes	complemented
25	C₄₁×#368	C₂²	1	2	yes	complemented
26	C₄₁×#359	C₂²	1	8	yes	complemented
27	C₄₁×#346	C₂²	1	2	yes	complemented
28	C₄×C₁₆×C₄₁	C₂²	1	2	yes	complemented
29	C₂×#41×C₄₁	—	2	2	no	—
30	#135×C₄₁	—	2	2	no	—
31	#135×C₄₁	—	2	2	no	—
32	#135×C₄₁	C₂×C₄	1	4	yes	—
33	#135×C₄₁	C₂³	1	4	yes	—
34	C₂×Q₁₆×C₄₁	C₂×C₄	1	2	yes	—
35	C₂×SD₁₆×C₄₁	—	2	2	no	—
36	C₂×D₈×C₄₁	C₂×C₄	1	2	yes	—
37	#131×C₄₁	—	2	8	no	—
38	#131×C₄₁	—	4	4	no	—
39	C₂×M₁₆×C₄₁	—	2	2	no	—
40	C₂×M₁₆×C₄₁	—	4	4	no	—
41	C₂×M₁₆×C₄₁	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
43	#117×C₄₁	D₄	1	1	yes	characteristic
44	Q₃₂×C₄₁	C₂×C₄	1	2	yes	—
45	Q₃₂×C₄₁	C₂×C₄	1	2	yes	—
46	SD₃₂×C₄₁	C₂×C₄	1	4	yes	—
47	SD₃₂×C₄₁	C₂×C₄	1	4	yes	—
48	D₁₆×C₄₁	C₂×C₄	1	2	yes	—
49	D₁₆×C₄₁	C₂×C₄	1	2	yes	—
50	M₃₂×C₄₁	—	2	1	no	—
51	M₃₂×C₄₁	—	2	1	no	—
52	C₂×C₁₆×C₄₁	C₂×C₄	1	1	yes	characteristic
53	C₂×C₁₆×C₄₁	C₂×C₄	1	1	yes	characteristic
54	C₂×C₁₆×C₄₁	C₂×C₄	1	1	yes	characteristic
55	C₂×C₁₆×C₄₁	C₂×C₄	1	1	yes	characteristic
56	C₂×C₁₆×C₄₁	C₂³	1	2	yes	—
57	C₂×C₁₆×C₄₁	C₂³	1	1	yes	characteristic
58	C₂×C₁₆×C₄₁	C₂³	1	1	yes	characteristic
59	#108×C₄₁	—	2	2	no	—
60	#108×C₄₁	C₂³	1	4	yes	—
61	#104×C₄₁	—	2	8	no	—
62	#97×C₄₁	D₄	1	2	yes	—
63	C₄×C₈×C₄₁	C₂³	1	2	yes	—
64	#41×C₄₁	—	2	4	no	—
65	#41×C₄₁	—	2	4	no	—
66	#41×C₄₁	—	4	2	no	—
67	#41×C₄₁	—	4	2	no	—
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	C₂×D₄	1	1	yes	characteristic
73	Q₁₆×C₄₁	—	2	2	no	—
74	Q₁₆×C₄₁	C₂²×C₄	1	4	yes	—
75	SD₁₆×C₄₁	—	2	4	no	—
76	SD₁₆×C₄₁	—	2	4	no	—
77	D₈×C₄₁	—	2	2	no	—
78	D₈×C₄₁	C₂²×C₄	1	4	yes	—
79	M₁₆×C₄₁	—	2	4	no	—
80	M₁₆×C₄₁	—	4	2	no	—
81	M₁₆×C₄₁	—	4	8	no	—
82	M₁₆×C₄₁	D₈	1	2	yes	—
83	M₁₆×C₄₁	D₈	1	2	yes	—
84	C₂×C₈×C₄₁	—	4	2	no	—
85	C₂×C₈×C₄₁	—	4	8	no	—
86	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₄₁	#41	1	1	yes	characteristic
91	C₂×C₈×C₄₁	#41	1	1	yes	characteristic
92	C₂×C₈×C₄₁	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₄₁	—	2	1	no	—
94	C₂²⋊C₄×C₄₁	—	2	1	no	—
95	C₄²×C₄₁	C₂×D₄	1	2	yes	—
96	C₁₆×C₄₁	—	2	2	no	—
97	C₁₆×C₄₁	C₂²×C₄	1	1	yes	characteristic
98	C₁₆×C₄₁	C₂²×C₄	1	1	yes	characteristic
99	C₁₆×C₄₁	C₂²×C₄	1	1	yes	characteristic
100	C₁₆×C₄₁	C₂²×C₄	1	1	yes	characteristic
101	C₂³×C₄₁	—	4	2	no	—
102	C₂³×C₄₁	#135	1	1	yes	characteristic
103	Q₈×C₄₁	—	2	4	no	—
104	Q₈×C₄₁	—	4	2	no	—
105	D₄×C₄₁	—	2	4	no	—
106	D₄×C₄₁	—	4	4	no	—
107	D₄×C₄₁	—	4	4	no	—
108	D₄×C₄₁	—	4	2	no	—
109	C₂×C₄×C₄₁	—	2	2	no	—
110	C₂×C₄×C₄₁	—	4	2	no	—
111	C₂×C₄×C₄₁	—	4	4	no	—
112	C₂×C₄×C₄₁	—	4	4	no	—
113	C₂×C₄×C₄₁	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₄₁	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₄₁	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₄₁	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₄₁	#135	1	1	yes	characteristic
118	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
119	C₈×C₄₁	—	4	4	no	—
120	C₈×C₄₁	C₄×D₄	1	1	yes	characteristic
121	C₈×C₄₁	C₄×D₄	1	1	yes	characteristic
122	C₈×C₄₁	C₂×D₈	1	4	yes	—
123	C₈×C₄₁	C₂³×C₄	1	1	yes	characteristic
124	C₈×C₄₁	C₂×#41	1	1	yes	characteristic
125	#33465	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂²×C₄₁	—	4	1	no	—
127	C₂²×C₄₁	—	4	4	no	—
128	C₂²×C₄₁	—	16	2	no	—
129	C₂²×C₄₁	C₄×D₈	1	1	yes	characteristic
130	C₂²×C₄₁	C₄×D₈	1	1	yes	characteristic
131	C₂²×C₄₁	C₂×#135	1	1	yes	characteristic
132	C₄×C₄₁	—	4	2	no	—
133	C₄×C₄₁	—	4	4	no	—
134	C₄×C₄₁	C₄×D₈	1	1	yes	characteristic
135	C₄×C₄₁	C₄×D₈	1	1	yes	characteristic
136	C₄×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
137	C₄×C₄₁	C₂²×D₈	1	1	yes	characteristic
138	C₂×#508	C₂×C₄₁	1	1	yes	characteristic
139	#2957	C₂×C₄₁	1	2	yes	complemented
140	#2179	C₂×C₄₁	1	4	yes	complemented
141	#2178	C₂×C₄₁	1	4	yes	complemented
142	C₂×#368	C₂×C₄₁	1	1	yes	characteristic, complemented
143	#2143	C₂×C₄₁	1	2	yes	complemented
144	#2108	C₂×C₄₁	1	1	yes	characteristic, complemented
145	C₂×C₄₁	—	2	1	no	—
146	C₂×C₄₁	—	2	1	no	—
147	C₂×C₄₁	—	2	1	no	—
148	C₂×C₄₁	—	8	4	no	—
149	C₂×C₄₁	C₂×C₄×D₈	1	1	yes	characteristic
150	C₂×#135	C₂²×C₄₁	1	2	yes	—
151	#568	—	2	2	no	—
152	#508	C₄×C₄₁	1	2	yes	complemented
153	#508	C₄×C₄₁	1	2	yes	complemented
154	#508	C₂²×C₄₁	1	2	yes	—
155	#508	C₂²×C₄₁	1	2	yes	—
156	C₂×Q₃₂	C₄×C₄₁	1	1	yes	characteristic, complemented
157	C₂×SD₃₂	C₄×C₄₁	1	2	yes	complemented
158	C₂×D₁₆	C₄×C₄₁	1	1	yes	characteristic, complemented
159	C₂×M₃₂	C₂²×C₄₁	1	1	yes	characteristic
160	C₂²×C₁₆	C₂²×C₄₁	1	1	yes	characteristic
161	#444	—	2	4	no	—
162	#443	C₂²×C₄₁	1	8	yes	complemented
163	C₂×#108	C₂²×C₄₁	1	2	yes	—
164	#421	—	2	2	no	—
165	#405	C₂²×C₄₁	1	1	yes	characteristic
166	#368	C₂²×C₄₁	1	2	yes	complemented
167	#368	C₂²×C₄₁	1	2	yes	complemented
168	#359	C₂²×C₄₁	1	8	yes	complemented
169	#346	C₂²×C₄₁	1	2	yes	complemented
170	C₄×C₁₆	C₂²×C₄₁	1	2	yes	complemented
171	C₄₁	#33465	1	1	yes	characteristic, direct product factor, Sylow
172	C₂×#41	—	2	2	no	—
173	#135	—	2	2	no	—
174	#135	—	2	2	no	—
175	#135	C₂×C₄×C₄₁	1	4	yes	—
176	#135	C₂³×C₄₁	1	4	yes	—
177	C₂×Q₁₆	C₂×C₄×C₄₁	1	2	yes	—
178	C₂×SD₁₆	—	2	2	no	—
179	C₂×D₈	C₂×C₄×C₄₁	1	2	yes	—
180	#131	—	2	8	no	—
181	#131	—	4	4	no	—
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	—	4	4	no	—
184	C₂×M₁₆	D₄×C₄₁	1	1	yes	characteristic
185	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
186	#117	D₄×C₄₁	1	1	yes	characteristic
187	Q₃₂	C₂×C₄×C₄₁	1	2	yes	—
188	Q₃₂	C₂×C₄×C₄₁	1	2	yes	—
189	SD₃₂	C₂×C₄×C₄₁	1	4	yes	—
190	SD₃₂	C₂×C₄×C₄₁	1	4	yes	—
191	D₁₆	C₂×C₄×C₄₁	1	2	yes	—
192	D₁₆	C₂×C₄×C₄₁	1	2	yes	—
193	M₃₂	—	2	1	no	—
194	M₃₂	—	2	1	no	—
195	C₂×C₁₆	C₂×C₄×C₄₁	1	1	yes	characteristic
196	C₂×C₁₆	C₂×C₄×C₄₁	1	1	yes	characteristic
197	C₂×C₁₆	C₂×C₄×C₄₁	1	1	yes	characteristic
198	C₂×C₁₆	C₂×C₄×C₄₁	1	1	yes	characteristic
199	C₂×C₁₆	C₂³×C₄₁	1	2	yes	—
200	C₂×C₁₆	C₂³×C₄₁	1	1	yes	characteristic
201	C₂×C₁₆	C₂³×C₄₁	1	1	yes	characteristic
202	#108	—	2	2	no	—
203	#108	C₂³×C₄₁	1	4	yes	—
204	#104	—	2	8	no	—
205	#97	D₄×C₄₁	1	2	yes	—
206	C₄×C₈	C₂³×C₄₁	1	2	yes	—
207	#41	—	2	4	no	—
208	#41	—	2	4	no	—
209	#41	—	4	2	no	—
210	#41	—	4	2	no	—
211	C₂×Q₈	—	2	2	no	—
212	C₂×D₄	—	2	2	no	—
213	C₂×D₄	—	4	2	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
216	Q₁₆	—	2	2	no	—
217	Q₁₆	C₂²×C₄×C₄₁	1	4	yes	—
218	SD₁₆	—	2	4	no	—
219	SD₁₆	—	2	4	no	—
220	D₈	—	2	2	no	—
221	D₈	C₂²×C₄×C₄₁	1	4	yes	—
222	M₁₆	—	2	4	no	—
223	M₁₆	—	4	2	no	—
224	M₁₆	—	4	8	no	—
225	M₁₆	D₈×C₄₁	1	2	yes	—
226	M₁₆	D₈×C₄₁	1	2	yes	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	—	4	8	no	—
229	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
230	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
231	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
232	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
233	C₂×C₈	#41×C₄₁	1	1	yes	characteristic
234	C₂×C₈	#41×C₄₁	1	1	yes	characteristic
235	C₂×C₈	C₂⁴×C₄₁	1	1	yes	characteristic
236	C₄⋊C₄	—	2	1	no	—
237	C₂²⋊C₄	—	2	1	no	—
238	C₄²	C₂×D₄×C₄₁	1	2	yes	—
239	C₁₆	—	2	2	no	—
240	C₁₆	C₂²×C₄×C₄₁	1	1	yes	characteristic
241	C₁₆	C₂²×C₄×C₄₁	1	1	yes	characteristic
242	C₁₆	C₂²×C₄×C₄₁	1	1	yes	characteristic
243	C₁₆	C₂²×C₄×C₄₁	1	1	yes	characteristic
244	C₂³	—	4	2	no	—
245	C₂³	#135×C₄₁	1	1	yes	characteristic
246	Q₈	—	2	4	no	—
247	Q₈	—	4	2	no	—
248	D₄	—	2	4	no	—
249	D₄	—	4	4	no	—
250	D₄	—	4	4	no	—
251	D₄	—	4	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	C₄×D₄×C₄₁	1	1	yes	characteristic
257	C₂×C₄	C₄×D₄×C₄₁	1	1	yes	characteristic
258	C₂×C₄	C₂×D₈×C₄₁	1	1	yes	characteristic
259	C₂×C₄	C₂×D₈×C₄₁	1	1	yes	characteristic
260	C₂×C₄	#135×C₄₁	1	1	yes	characteristic
261	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
262	C₈	—	4	4	no	—
263	C₈	C₄×D₄×C₄₁	1	1	yes	characteristic
264	C₈	C₄×D₄×C₄₁	1	1	yes	characteristic
265	C₈	C₂×D₈×C₄₁	1	4	yes	—
266	C₈	C₂³×C₄×C₄₁	1	1	yes	characteristic
267	C₈	C₂×#41×C₄₁	1	1	yes	characteristic
268	C₂²	—	4	1	no	—
269	C₂²	—	4	4	no	—
270	C₂²	—	16	2	no	—
271	C₂²	C₄×D₈×C₄₁	1	1	yes	characteristic
272	C₂²	C₄×D₈×C₄₁	1	1	yes	characteristic
273	C₂²	C₂×#135×C₄₁	1	1	yes	characteristic
274	C₄	—	4	2	no	—
275	C₄	—	4	4	no	—
276	C₄	C₄×D₈×C₄₁	1	1	yes	characteristic
277	C₄	C₄×D₈×C₄₁	1	1	yes	characteristic
278	C₄	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
279	C₄	C₂²×D₈×C₄₁	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	2	1	no	—
283	C₂	—	8	4	no	—
284	C₂	C₂×C₄×D₈×C₄₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₄₁	C₂²×D₈
commutator	C₈	C₂³×C₄×C₄₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₄₁
socle	C₂×C₄₁	C₂×C₄×D₈
prime radical	C₂×C₄₁×#508	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₄₁ ◁ C₂×M₁₆×C₄₁ ◁ C₄₁×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₁×#2957	C₂	semidirect
3. C₄₁×#2179	C₂	semidirect
4. C₄₁×#2178	C₂	semidirect
5. C₂×C₄₁×#368	C₂	semidirect
6. C₄₁×#2143	C₂	semidirect
7. C₄₁×#2108	C₂	semidirect
10. C₄₁×#508	C₄	semidirect
11. C₄₁×#508	C₄	semidirect
14. C₂×Q₃₂×C₄₁	C₄	semidirect
15. C₂×SD₃₂×C₄₁	C₄	semidirect
16. C₂×D₁₆×C₄₁	C₄	semidirect
20. C₄₁×#443	C₂²	semidirect
24. C₄₁×#368	C₂²	semidirect
25. C₄₁×#368	C₂²	semidirect
26. C₄₁×#359	C₂²	semidirect
27. C₄₁×#346	C₂²	semidirect
28. C₄×C₁₆×C₄₁	C₂²	semidirect
125. #33465	C₄₁	direct (N × H)
139. #2957	C₂×C₄₁	semidirect
140. #2179	C₂×C₄₁	semidirect
141. #2178	C₂×C₄₁	semidirect
142. C₂×#368	C₂×C₄₁	semidirect
143. #2143	C₂×C₄₁	semidirect
144. #2108	C₂×C₄₁	semidirect
152. #508	C₄×C₄₁	semidirect
153. #508	C₄×C₄₁	semidirect
156. C₂×Q₃₂	C₄×C₄₁	semidirect
157. C₂×SD₃₂	C₄×C₄₁	semidirect
158. C₂×D₁₆	C₄×C₄₁	semidirect
162. #443	C₂²×C₄₁	semidirect
166. #368	C₂²×C₄₁	semidirect
167. #368	C₂²×C₄₁	semidirect
168. #359	C₂²×C₄₁	semidirect
169. #346	C₂²×C₄₁	semidirect
170. C₄×C₁₆	C₂²×C₄₁	semidirect
171. C₄₁	#33465	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×C₄₁×#508	2
C₄₁×#2957	2
C₄₁×#2179	2
C₄₁×#2178	2
C₂×C₄₁×#368	2
C₄₁×#2143	2
C₄₁×#2108	2
#33465	41

Subgroups — C41×#33465 #23809666

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#33465 #23809666