Subgroups — C₄₁×#20308 #23796509

Order 10496.

626 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3366	C₂	1	2	yes	maximal, complemented
2	C₄₁×#3333	C₂	1	2	yes	maximal, complemented
3	C₄₁×#1709	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#1698	C₂	1	2	yes	maximal, complemented
5	C₄₁×#1697	C₂	1	2	yes	maximal, complemented
6	C₄₁×#1632	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1621	C₂	1	2	yes	maximal, complemented
8	C₄₁×#1591	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1589	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₁×#554	C₂²	1	2	yes	—
11	C₄₁×#550	C₂²	1	2	yes	—
12	C₄₁×#494	C₂²	1	1	yes	characteristic
13	C₄₁×#493	C₂²	1	1	yes	characteristic
14	C₄₁×#490	—	2	2	no	—
15	C₄₁×#486	C₂²	1	2	yes	—
16	C₄₁×#486	C₂²	1	2	yes	—
17	C₄₁×#480	—	2	2	no	—
18	C₄₁×#478	—	2	2	no	—
19	C₄₁×#477	C₂²	1	2	yes	—
20	C₄₁×#476	C₂²	1	2	yes	—
21	C₄₁×#468	—	2	2	no	—
22	C₄₁×#460	C₂²	1	2	yes	—
23	C₄₁×#459	—	2	2	no	—
24	C₄₁×#459	C₂²	1	2	yes	—
25	C₄₁×#452	—	2	2	no	—
26	C₄₁×#446	C₂²	1	2	yes	—
27	C₄₁×#442	—	2	2	no	—
28	C₄×Q₁₆×C₄₁	C₂²	1	1	yes	characteristic
29	C₄×D₈×C₄₁	C₂²	1	1	yes	characteristic
30	C₄₁×#436	C₂²	1	2	yes	—
31	C₄₁×#417	—	2	2	no	—
32	C₄₁×#335	C₂²	1	1	yes	characteristic
33	C₄₁×#335	C₂²	1	1	yes	characteristic
34	C₄₁×#333	C₂²	1	2	yes	—
35	C₄₁×#331	C₂²	1	2	yes	—
36	C₄₁×Q₈⋊C₈	C₂²	1	2	yes	—
37	C₄₁×D₄⋊C₈	C₂²	1	2	yes	—
38	C₈²×C₄₁	C₂²	1	1	yes	characteristic
39	C₂×#41×C₄₁	—	2	2	no	—
40	C₂×Q₁₆×C₄₁	—	4	1	no	—
41	C₂×Q₁₆×C₄₁	D₄	1	1	yes	characteristic
42	C₂×SD₁₆×C₄₁	—	2	2	no	—
43	C₂×D₈×C₄₁	—	2	2	no	—
44	C₂×D₈×C₄₁	—	4	1	no	—
45	C₂×D₈×C₄₁	D₄	1	1	yes	characteristic
46	C₂×M₁₆×C₄₁	—	2	2	no	—
47	C₄⋊Q₈×C₄₁	C₂³	1	2	yes	—
48	#127×C₄₁	—	2	2	no	—
49	#127×C₄₁	C₂³	1	2	yes	—
50	#125×C₄₁	—	2	2	no	—
51	C₂²⋊Q₈×C₄₁	—	2	2	no	—
52	C₂²⋊Q₈×C₄₁	—	4	2	no	—
53	#121×C₄₁	—	2	2	no	—
54	#121×C₄₁	—	4	2	no	—
55	C₄×Q₈×C₄₁	C₂³	1	2	yes	—
56	C₄×D₄×C₄₁	—	2	2	no	—
57	C₄×D₄×C₄₁	—	2	2	no	—
58	C₄×D₄×C₄₁	C₂³	1	2	yes	—
59	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
60	#107×C₄₁	—	2	2	no	—
61	#107×C₄₁	D₄	1	1	yes	characteristic
62	#107×C₄₁	D₄	1	1	yes	characteristic
63	#106×C₄₁	—	2	2	no	—
64	C₄⋊C₈×C₄₁	—	2	2	no	—
65	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
66	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
67	#103×C₄₁	—	2	2	no	—
68	#103×C₄₁	D₄	1	2	yes	—
69	#102×C₄₁	—	2	2	no	—
70	#102×C₄₁	—	2	2	no	—
71	#102×C₄₁	—	2	2	no	—
72	#102×C₄₁	—	2	2	no	—
73	#102×C₄₁	—	2	2	no	—
74	#102×C₄₁	—	2	2	no	—
75	#102×C₄₁	—	2	2	no	—
76	#102×C₄₁	D₄	1	2	yes	—
77	C₂²⋊C₈×C₄₁	—	2	2	no	—
78	#97×C₄₁	—	2	2	no	—
79	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
82	#41×C₄₁	—	4	4	no	—
83	#41×C₄₁	—	8	2	no	—
84	C₂×Q₈×C₄₁	—	2	2	no	—
85	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
86	C₂×D₄×C₄₁	—	2	2	no	—
87	C₂×D₄×C₄₁	—	2	2	no	—
88	C₂×D₄×C₄₁	—	2	2	no	—
89	C₂×D₄×C₄₁	—	4	2	no	—
90	C₂×D₄×C₄₁	—	4	2	no	—
91	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₄₁	—	2	2	no	—
93	C₂²×C₄×C₄₁	—	4	2	no	—
94	C₂²×C₄×C₄₁	—	4	2	no	—
95	Q₁₆×C₄₁	—	4	1	no	—
96	Q₁₆×C₄₁	—	8	2	no	—
97	SD₁₆×C₄₁	—	8	2	no	—
98	D₈×C₄₁	—	4	1	no	—
99	D₈×C₄₁	—	4	4	no	—
100	D₈×C₄₁	—	8	2	no	—
101	M₁₆×C₄₁	—	8	2	no	—
102	C₂×C₈×C₄₁	—	2	1	no	—
103	C₂×C₈×C₄₁	—	2	2	no	—
104	C₂×C₈×C₄₁	—	2	2	no	—
105	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
109	C₄⋊C₄×C₄₁	—	2	2	no	—
110	C₄⋊C₄×C₄₁	—	2	2	no	—
111	C₄⋊C₄×C₄₁	—	2	2	no	—
112	C₄⋊C₄×C₄₁	—	4	2	no	—
113	C₄⋊C₄×C₄₁	—	4	2	no	—
114	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
115	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
116	C₂²⋊C₄×C₄₁	—	2	2	no	—
117	C₂²⋊C₄×C₄₁	—	4	2	no	—
118	C₂²⋊C₄×C₄₁	—	4	2	no	—
119	C₄²×C₄₁	—	2	2	no	—
120	C₄²×C₄₁	C₂⁴	1	1	yes	characteristic
121	C₂³×C₄₁	—	2	2	no	—
122	C₂³×C₄₁	—	4	2	no	—
123	Q₈×C₄₁	—	2	2	no	—
124	Q₈×C₄₁	—	2	2	no	—
125	Q₈×C₄₁	—	8	2	no	—
126	D₄×C₄₁	—	2	2	no	—
127	D₄×C₄₁	—	2	2	no	—
128	D₄×C₄₁	—	4	2	no	—
129	D₄×C₄₁	—	4	2	no	—
130	D₄×C₄₁	—	4	2	no	—
131	D₄×C₄₁	—	4	2	no	—
132	D₄×C₄₁	—	8	2	no	—
133	D₄×C₄₁	—	8	4	no	—
134	D₄×C₄₁	—	16	2	no	—
135	C₂×C₄×C₄₁	—	2	2	no	—
136	C₂×C₄×C₄₁	—	2	2	no	—
137	C₂×C₄×C₄₁	—	2	2	no	—
138	C₂×C₄×C₄₁	—	4	2	no	—
139	C₂×C₄×C₄₁	—	4	2	no	—
140	C₂×C₄×C₄₁	—	4	2	no	—
141	C₂×C₄×C₄₁	—	8	4	no	—
142	C₂×C₄×C₄₁	—	16	2	no	—
143	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
146	C₈×C₄₁	—	2	2	no	—
147	C₈×C₄₁	—	2	1	no	—
148	C₈×C₄₁	—	2	1	no	—
149	C₈×C₄₁	—	2	1	no	—
150	C₈×C₄₁	—	2	1	no	—
151	C₈×C₄₁	—	4	2	no	—
152	C₈×C₄₁	—	4	2	no	—
153	#20308	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
154	C₂²×C₄₁	—	4	2	no	—
155	C₂²×C₄₁	—	4	2	no	—
156	C₂²×C₄₁	—	4	2	no	—
157	C₂²×C₄₁	—	8	2	no	—
158	C₂²×C₄₁	—	8	2	no	—
159	C₂²×C₄₁	—	8	2	no	—
160	C₂²×C₄₁	D₄²	1	1	yes	characteristic
161	C₄×C₄₁	—	2	1	no	—
162	C₄×C₄₁	—	4	2	no	—
163	C₄×C₄₁	—	4	2	no	—
164	C₄×C₄₁	—	4	2	no	—
165	C₄×C₄₁	—	8	2	no	—
166	C₄×C₄₁	#575	1	1	yes	characteristic
167	C₄×C₄₁	#575	1	1	yes	characteristic
168	C₄×C₄₁	#576	1	1	yes	characteristic
169	C₄×C₄₁	#578	1	1	yes	characteristic
170	#3366	C₂×C₄₁	1	2	yes	complemented
171	#3333	C₂×C₄₁	1	2	yes	complemented
172	#1709	C₂×C₄₁	1	1	yes	characteristic, complemented
173	#1698	C₂×C₄₁	1	2	yes	complemented
174	#1697	C₂×C₄₁	1	2	yes	complemented
175	#1632	C₂×C₄₁	1	2	yes	complemented
176	#1621	C₂×C₄₁	1	2	yes	complemented
177	#1591	C₂×C₄₁	1	1	yes	characteristic, complemented
178	#1589	C₂×C₄₁	1	1	yes	characteristic, complemented
179	C₂×C₄₁	—	8	2	no	—
180	C₂×C₄₁	—	16	2	no	—
181	C₂×C₄₁	#3273	1	1	yes	characteristic
182	C₂×C₄₁	#3277	1	1	yes	characteristic
183	C₂×C₄₁	#3292	1	1	yes	characteristic
184	#554	C₂²×C₄₁	1	2	yes	—
185	#550	C₂²×C₄₁	1	2	yes	—
186	#494	C₂²×C₄₁	1	1	yes	characteristic
187	#493	C₂²×C₄₁	1	1	yes	characteristic
188	#490	—	2	2	no	—
189	#486	C₂²×C₄₁	1	2	yes	—
190	#486	C₂²×C₄₁	1	2	yes	—
191	#480	—	2	2	no	—
192	#478	—	2	2	no	—
193	#477	C₂²×C₄₁	1	2	yes	—
194	#476	C₂²×C₄₁	1	2	yes	—
195	#468	—	2	2	no	—
196	#460	C₂²×C₄₁	1	2	yes	—
197	#459	—	2	2	no	—
198	#459	C₂²×C₄₁	1	2	yes	—
199	#452	—	2	2	no	—
200	#446	C₂²×C₄₁	1	2	yes	—
201	#442	—	2	2	no	—
202	C₄×Q₁₆	C₂²×C₄₁	1	1	yes	characteristic
203	C₄×D₈	C₂²×C₄₁	1	1	yes	characteristic
204	#436	C₂²×C₄₁	1	2	yes	—
205	#417	—	2	2	no	—
206	#335	C₂²×C₄₁	1	1	yes	characteristic
207	#335	C₂²×C₄₁	1	1	yes	characteristic
208	#333	C₂²×C₄₁	1	2	yes	—
209	#331	C₂²×C₄₁	1	2	yes	—
210	Q₈⋊C₈	C₂²×C₄₁	1	2	yes	—
211	D₄⋊C₈	C₂²×C₄₁	1	2	yes	—
212	C₈²	C₂²×C₄₁	1	1	yes	characteristic
213	C₄₁	#20308	1	1	yes	characteristic, direct product factor, Sylow
214	C₂×#41	—	2	2	no	—
215	C₂×Q₁₆	—	4	1	no	—
216	C₂×Q₁₆	D₄×C₄₁	1	1	yes	characteristic
217	C₂×SD₁₆	—	2	2	no	—
218	C₂×D₈	—	2	2	no	—
219	C₂×D₈	—	4	1	no	—
220	C₂×D₈	D₄×C₄₁	1	1	yes	characteristic
221	C₂×M₁₆	—	2	2	no	—
222	C₄⋊Q₈	C₂³×C₄₁	1	2	yes	—
223	#127	—	2	2	no	—
224	#127	C₂³×C₄₁	1	2	yes	—
225	#125	—	2	2	no	—
226	C₂²⋊Q₈	—	2	2	no	—
227	C₂²⋊Q₈	—	4	2	no	—
228	#121	—	2	2	no	—
229	#121	—	4	2	no	—
230	C₄×Q₈	C₂³×C₄₁	1	2	yes	—
231	C₄×D₄	—	2	2	no	—
232	C₄×D₄	—	2	2	no	—
233	C₄×D₄	C₂³×C₄₁	1	2	yes	—
234	C₂×C₄⋊C₄	—	2	2	no	—
235	#107	—	2	2	no	—
236	#107	D₄×C₄₁	1	1	yes	characteristic
237	#107	D₄×C₄₁	1	1	yes	characteristic
238	#106	—	2	2	no	—
239	C₄⋊C₈	—	2	2	no	—
240	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
241	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
242	#103	—	2	2	no	—
243	#103	D₄×C₄₁	1	2	yes	—
244	#102	—	2	2	no	—
245	#102	—	2	2	no	—
246	#102	—	2	2	no	—
247	#102	—	2	2	no	—
248	#102	—	2	2	no	—
249	#102	—	2	2	no	—
250	#102	—	2	2	no	—
251	#102	D₄×C₄₁	1	2	yes	—
252	C₂²⋊C₈	—	2	2	no	—
253	#97	—	2	2	no	—
254	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
255	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
256	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
257	#41	—	4	4	no	—
258	#41	—	8	2	no	—
259	C₂×Q₈	—	2	2	no	—
260	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	—	2	2	no	—
264	C₂×D₄	—	4	2	no	—
265	C₂×D₄	—	4	2	no	—
266	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	4	2	no	—
269	C₂²×C₄	—	4	2	no	—
270	Q₁₆	—	4	1	no	—
271	Q₁₆	—	8	2	no	—
272	SD₁₆	—	8	2	no	—
273	D₈	—	4	1	no	—
274	D₈	—	4	4	no	—
275	D₈	—	8	2	no	—
276	M₁₆	—	8	2	no	—
277	C₂×C₈	—	2	1	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
281	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
282	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
283	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	4	2	no	—
288	C₄⋊C₄	—	4	2	no	—
289	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
290	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
291	C₂²⋊C₄	—	2	2	no	—
292	C₂²⋊C₄	—	4	2	no	—
293	C₂²⋊C₄	—	4	2	no	—
294	C₄²	—	2	2	no	—
295	C₄²	C₂⁴×C₄₁	1	1	yes	characteristic
296	C₂³	—	2	2	no	—
297	C₂³	—	4	2	no	—
298	Q₈	—	2	2	no	—
299	Q₈	—	2	2	no	—
300	Q₈	—	8	2	no	—
301	D₄	—	2	2	no	—
302	D₄	—	2	2	no	—
303	D₄	—	4	2	no	—
304	D₄	—	4	2	no	—
305	D₄	—	4	2	no	—
306	D₄	—	4	2	no	—
307	D₄	—	8	2	no	—
308	D₄	—	8	4	no	—
309	D₄	—	16	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	8	4	no	—
317	C₂×C₄	—	16	2	no	—
318	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
319	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
320	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
321	C₈	—	2	2	no	—
322	C₈	—	2	1	no	—
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	—	2	1	no	—
326	C₈	—	4	2	no	—
327	C₈	—	4	2	no	—
328	C₂²	—	4	2	no	—
329	C₂²	—	4	2	no	—
330	C₂²	—	4	2	no	—
331	C₂²	—	8	2	no	—
332	C₂²	—	8	2	no	—
333	C₂²	—	8	2	no	—
334	C₂²	D₄²×C₄₁	1	1	yes	characteristic
335	C₄	—	2	1	no	—
336	C₄	—	4	2	no	—
337	C₄	—	4	2	no	—
338	C₄	—	4	2	no	—
339	C₄	—	8	2	no	—
340	C₄	C₄₁×#575	1	1	yes	characteristic
341	C₄	C₄₁×#575	1	1	yes	characteristic
342	C₄	C₄₁×#576	1	1	yes	characteristic
343	C₄	C₄₁×#578	1	1	yes	characteristic
344	C₂	—	8	2	no	—
345	C₂	—	16	2	no	—
346	C₂	C₄₁×#3273	1	1	yes	characteristic
347	C₂	C₄₁×#3277	1	1	yes	characteristic
348	C₂	C₄₁×#3292	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₄₁
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₁×#3366	C₂	semidirect
2. C₄₁×#3333	C₂	semidirect
3. C₄₁×#1709	C₂	semidirect
4. C₄₁×#1698	C₂	semidirect
5. C₄₁×#1697	C₂	semidirect
6. C₄₁×#1632	C₂	semidirect
7. C₄₁×#1621	C₂	semidirect
8. C₄₁×#1591	C₂	semidirect
9. C₄₁×#1589	C₂	semidirect
153. #20308	C₄₁	direct (N × H)
170. #3366	C₂×C₄₁	semidirect
171. #3333	C₂×C₄₁	semidirect
172. #1709	C₂×C₄₁	semidirect
173. #1698	C₂×C₄₁	semidirect
174. #1697	C₂×C₄₁	semidirect
175. #1632	C₂×C₄₁	semidirect
176. #1621	C₂×C₄₁	semidirect
177. #1591	C₂×C₄₁	semidirect
178. #1589	C₂×C₄₁	semidirect
213. C₄₁	#20308	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×#3366	2
C₄₁×#3333	2
C₄₁×#1709	2
C₄₁×#1698	2
C₄₁×#1697	2
C₄₁×#1632	2
C₄₁×#1621	2
C₄₁×#1591	2
C₄₁×#1589	2
#20308	41

Subgroups — C41×#20308 #23796509

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#20308 #23796509