Subgroups — C₄₁×#25872 #23802073

Order 10496.

1214 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3497	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#3141	C₂	1	2	yes	maximal, complemented
3	C₄₁×#2568	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#2040	C₂	1	4	yes	maximal, complemented
5	C₄₁×#1980	C₂	1	2	yes	maximal, complemented
6	C₄₁×#1793	C₂	1	4	yes	maximal, complemented
7	C₄×#101×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×2₋¹⁺⁴×C₄₁	C₂²	1	2	yes	—
9	C₄₁×#550	—	2	2	no	—
10	D₄×Q₈×C₄₁	—	2	2	no	—
11	C₄₁×#548	—	2	4	no	—
12	C₄₁×#537	—	2	2	no	—
13	C₄₁×#537	C₂²	1	2	yes	—
14	C₄₁×#532	—	2	2	no	—
15	C₄₁×#532	C₂²	1	2	yes	—
16	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
18	C₄₁×#489	—	2	4	no	—
19	C₄₁×#487	—	2	2	no	—
20	C₄₁×#486	—	2	2	no	—
21	C₂×#104×C₄₁	C₂²	1	8	yes	—
22	C₄₁×#417	C₂²	1	8	yes	—
23	C₂×#101×C₄₁	C₂²	1	2	yes	—
24	C₄×M₁₆×C₄₁	C₂²	1	2	yes	—
25	C₄₁×#400	—	2	2	no	—
26	C₄₁×#391	—	2	2	no	—
27	C₄₁×#391	C₂²	1	2	yes	—
28	C₄₁×#382	C₂²	1	2	yes	—
29	C₄×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
30	C₄₁×#343	C₂²	1	2	yes	—
31	2₋¹⁺⁴×C₄₁	—	2	8	no	—
32	2₋¹⁺⁴×C₄₁	—	4	4	no	—
33	C₂×#41×C₄₁	—	2	4	no	—
34	C₂×#41×C₄₁	—	2	4	no	—
35	C₂×#41×C₄₁	C₂³	1	4	yes	—
36	C₂²×Q₈×C₄₁	—	2	4	no	—
37	C₂²×Q₈×C₄₁	—	2	1	no	—
38	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
39	C₂×M₁₆×C₄₁	C₂³	1	4	yes	—
40	C₄⋊Q₈×C₄₁	—	2	2	no	—
41	C₄⋊Q₈×C₄₁	—	2	2	no	—
42	C₄⋊Q₈×C₄₁	—	2	2	no	—
43	C₄⋊Q₈×C₄₁	D₄	1	2	yes	—
44	#127×C₄₁	—	2	2	no	—
45	#127×C₄₁	D₄	1	2	yes	—
46	#124×C₄₁	—	2	4	no	—
47	#124×C₄₁	D₄	1	4	yes	—
48	C₂²⋊Q₈×C₄₁	—	2	8	no	—
49	C₂²⋊Q₈×C₄₁	—	2	4	no	—
50	#121×C₄₁	—	2	8	no	—
51	#121×C₄₁	—	2	4	no	—
52	#121×C₄₁	—	2	2	no	—
53	#121×C₄₁	—	2	4	no	—
54	#121×C₄₁	—	2	2	no	—
55	C₄×Q₈×C₄₁	—	2	2	no	—
56	C₄×Q₈×C₄₁	—	2	2	no	—
57	C₄×D₄×C₄₁	—	2	4	no	—
58	C₄×D₄×C₄₁	—	2	4	no	—
59	C₄×D₄×C₄₁	—	2	2	no	—
60	C₄×D₄×C₄₁	—	2	2	no	—
61	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
62	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
63	C₂×C₄⋊C₄×C₄₁	D₄	1	2	yes	—
64	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₄₁	D₄	1	2	yes	—
67	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
68	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
69	#104×C₄₁	—	4	8	no	—
70	#103×C₄₁	—	2	8	no	—
71	#102×C₄₁	—	2	8	no	—
72	#101×C₄₁	—	2	4	no	—
73	#97×C₄₁	—	2	2	no	—
74	C₄×C₈×C₄₁	—	2	2	no	—
75	#95×C₄₁	—	2	1	no	—
76	#95×C₄₁	—	2	4	no	—
77	#95×C₄₁	—	2	4	no	—
78	#41×C₄₁	—	2	8	no	—
79	#41×C₄₁	—	2	8	no	—
80	#41×C₄₁	—	4	8	no	—
81	#41×C₄₁	—	4	16	no	—
82	#41×C₄₁	—	4	8	no	—
83	C₂×Q₈×C₄₁	—	2	4	no	—
84	C₂×Q₈×C₄₁	—	2	2	no	—
85	C₂×Q₈×C₄₁	—	2	2	no	—
86	C₂×Q₈×C₄₁	—	4	4	no	—
87	C₂×Q₈×C₄₁	—	4	16	no	—
88	C₂×Q₈×C₄₁	—	4	1	no	—
89	C₂×Q₈×C₄₁	—	4	1	no	—
90	C₂×Q₈×C₄₁	C₂×D₄	1	4	yes	—
91	C₂×Q₈×C₄₁	#41	1	4	yes	—
92	C₂×D₄×C₄₁	—	2	4	no	—
93	C₂×D₄×C₄₁	—	2	4	no	—
94	C₂×D₄×C₄₁	—	2	4	no	—
95	C₂×D₄×C₄₁	—	2	2	no	—
96	C₂×D₄×C₄₁	—	2	2	no	—
97	C₂×D₄×C₄₁	C₂×D₄	1	4	yes	—
98	C₂²×C₄×C₄₁	—	2	8	no	—
99	C₂²×C₄×C₄₁	—	2	4	no	—
100	C₂²×C₄×C₄₁	—	2	2	no	—
101	C₂²×C₄×C₄₁	—	2	2	no	—
102	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
104	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
105	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
107	M₁₆×C₄₁	—	2	4	no	—
108	M₁₆×C₄₁	—	2	4	no	—
109	C₂×C₈×C₄₁	—	2	4	no	—
110	C₄⋊C₄×C₄₁	—	2	4	no	—
111	C₄⋊C₄×C₄₁	—	2	4	no	—
112	C₄⋊C₄×C₄₁	—	2	2	no	—
113	C₄⋊C₄×C₄₁	—	2	2	no	—
114	C₄⋊C₄×C₄₁	—	2	2	no	—
115	C₄⋊C₄×C₄₁	—	2	4	no	—
116	C₄⋊C₄×C₄₁	—	4	2	no	—
117	C₄⋊C₄×C₄₁	—	4	2	no	—
118	C₂²⋊C₄×C₄₁	—	2	8	no	—
119	C₂²⋊C₄×C₄₁	—	2	4	no	—
120	C₂²⋊C₄×C₄₁	—	2	4	no	—
121	C₄²×C₄₁	—	2	2	no	—
122	C₄²×C₄₁	—	2	2	no	—
123	C₄²×C₄₁	—	2	4	no	—
124	C₄²×C₄₁	C₂×D₄	1	2	yes	—
125	C₄²×C₄₁	C₂×D₄	1	2	yes	—
126	C₂³×C₄₁	—	2	4	no	—
127	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
128	Q₈×C₄₁	—	2	8	no	—
129	Q₈×C₄₁	—	2	8	no	—
130	Q₈×C₄₁	—	4	8	no	—
131	Q₈×C₄₁	—	8	2	no	—
132	Q₈×C₄₁	—	8	2	no	—
133	D₄×C₄₁	—	2	8	no	—
134	D₄×C₄₁	—	4	8	no	—
135	D₄×C₄₁	—	4	4	no	—
136	D₄×C₄₁	—	4	4	no	—
137	D₄×C₄₁	—	4	4	no	—
138	D₄×C₄₁	—	8	2	no	—
139	D₄×C₄₁	—	8	4	no	—
140	C₂×C₄×C₄₁	—	2	2	no	—
141	C₂×C₄×C₄₁	—	2	2	no	—
142	C₂×C₄×C₄₁	—	2	2	no	—
143	C₂×C₄×C₄₁	—	2	2	no	—
144	C₂×C₄×C₄₁	—	2	4	no	—
145	C₂×C₄×C₄₁	—	4	1	no	—
146	C₂×C₄×C₄₁	—	4	16	no	—
147	C₂×C₄×C₄₁	—	4	8	no	—
148	C₂×C₄×C₄₁	—	4	2	no	—
149	C₂×C₄×C₄₁	—	4	2	no	—
150	C₂×C₄×C₄₁	—	4	2	no	—
151	C₂×C₄×C₄₁	—	4	4	no	—
152	C₂×C₄×C₄₁	—	4	2	no	—
153	C₂×C₄×C₄₁	—	4	2	no	—
154	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
155	C₂×C₄×C₄₁	#120	1	2	yes	—
156	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
157	C₂×C₄×C₄₁	#124	1	4	yes	—
158	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
159	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
160	C₂×C₄×C₄₁	H(5,𝔽₂)	1	2	yes	—
161	C₈×C₄₁	—	4	4	no	—
162	#25872	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
163	C₂²×C₄₁	—	2	1	no	—
164	C₂²×C₄₁	—	2	1	no	—
165	C₂²×C₄₁	—	4	4	no	—
166	C₂²×C₄₁	—	4	4	no	—
167	C₂²×C₄₁	—	4	4	no	—
168	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
169	C₂²×C₄₁	C₂×#124	1	1	yes	characteristic
170	C₂²×C₄₁	#535	1	1	yes	characteristic
171	C₄×C₄₁	—	2	4	no	—
172	C₄×C₄₁	—	2	1	no	—
173	C₄×C₄₁	—	2	2	no	—
174	C₄×C₄₁	—	4	4	no	—
175	C₄×C₄₁	—	4	2	no	—
176	C₄×C₄₁	—	8	2	no	—
177	C₄×C₄₁	—	8	2	no	—
178	C₄×C₄₁	#454	1	2	yes	—
179	C₄×C₄₁	#546	1	4	yes	—
180	#3497	C₂×C₄₁	1	1	yes	characteristic
181	#3141	C₂×C₄₁	1	2	yes	complemented
182	#2568	C₂×C₄₁	1	1	yes	characteristic, complemented
183	#2040	C₂×C₄₁	1	4	yes	complemented
184	#1980	C₂×C₄₁	1	2	yes	complemented
185	#1793	C₂×C₄₁	1	4	yes	complemented
186	C₄×#101	C₂×C₄₁	1	1	yes	characteristic, complemented
187	C₂×C₄₁	—	2	1	no	—
188	C₂×C₄₁	—	2	1	no	—
189	C₂×C₄₁	—	8	4	no	—
190	C₂×C₄₁	#2435	1	1	yes	characteristic
191	C₂×C₄₁	C₂×#454	1	1	yes	characteristic
192	C₂×C₄₁	#3022	1	1	yes	characteristic
193	C₂×2₋¹⁺⁴	C₂²×C₄₁	1	2	yes	—
194	#550	—	2	2	no	—
195	D₄×Q₈	—	2	2	no	—
196	#548	—	2	4	no	—
197	#537	—	2	2	no	—
198	#537	C₂²×C₄₁	1	2	yes	—
199	#532	—	2	2	no	—
200	#532	C₂²×C₄₁	1	2	yes	—
201	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
202	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
203	#489	—	2	4	no	—
204	#487	—	2	2	no	—
205	#486	—	2	2	no	—
206	C₂×#104	C₂²×C₄₁	1	8	yes	—
207	#417	C₂²×C₄₁	1	8	yes	—
208	C₂×#101	C₂²×C₄₁	1	2	yes	—
209	C₄×M₁₆	C₂²×C₄₁	1	2	yes	—
210	#400	—	2	2	no	—
211	#391	—	2	2	no	—
212	#391	C₂²×C₄₁	1	2	yes	—
213	#382	C₂²×C₄₁	1	2	yes	—
214	C₄×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
215	#343	C₂²×C₄₁	1	2	yes	—
216	C₄₁	#25872	1	1	yes	characteristic, direct product factor, Sylow
217	2₋¹⁺⁴	—	2	8	no	—
218	2₋¹⁺⁴	—	4	4	no	—
219	C₂×#41	—	2	4	no	—
220	C₂×#41	—	2	4	no	—
221	C₂×#41	C₂³×C₄₁	1	4	yes	—
222	C₂²×Q₈	—	2	4	no	—
223	C₂²×Q₈	—	2	1	no	—
224	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
225	C₂×M₁₆	C₂³×C₄₁	1	4	yes	—
226	C₄⋊Q₈	—	2	2	no	—
227	C₄⋊Q₈	—	2	2	no	—
228	C₄⋊Q₈	—	2	2	no	—
229	C₄⋊Q₈	D₄×C₄₁	1	2	yes	—
230	#127	—	2	2	no	—
231	#127	D₄×C₄₁	1	2	yes	—
232	#124	—	2	4	no	—
233	#124	D₄×C₄₁	1	4	yes	—
234	C₂²⋊Q₈	—	2	8	no	—
235	C₂²⋊Q₈	—	2	4	no	—
236	#121	—	2	8	no	—
237	#121	—	2	4	no	—
238	#121	—	2	2	no	—
239	#121	—	2	4	no	—
240	#121	—	2	2	no	—
241	C₄×Q₈	—	2	2	no	—
242	C₄×Q₈	—	2	2	no	—
243	C₄×D₄	—	2	4	no	—
244	C₄×D₄	—	2	4	no	—
245	C₄×D₄	—	2	2	no	—
246	C₄×D₄	—	2	2	no	—
247	C₂×C₄⋊C₄	—	2	1	no	—
248	C₂×C₄⋊C₄	—	2	1	no	—
249	C₂×C₄⋊C₄	D₄×C₄₁	1	2	yes	—
250	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
251	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
252	C₂×C₄²	D₄×C₄₁	1	2	yes	—
253	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
254	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
255	#104	—	4	8	no	—
256	#103	—	2	8	no	—
257	#102	—	2	8	no	—
258	#101	—	2	4	no	—
259	#97	—	2	2	no	—
260	C₄×C₈	—	2	2	no	—
261	#95	—	2	1	no	—
262	#95	—	2	4	no	—
263	#95	—	2	4	no	—
264	#41	—	2	8	no	—
265	#41	—	2	8	no	—
266	#41	—	4	8	no	—
267	#41	—	4	16	no	—
268	#41	—	4	8	no	—
269	C₂×Q₈	—	2	4	no	—
270	C₂×Q₈	—	2	2	no	—
271	C₂×Q₈	—	2	2	no	—
272	C₂×Q₈	—	4	4	no	—
273	C₂×Q₈	—	4	16	no	—
274	C₂×Q₈	—	4	1	no	—
275	C₂×Q₈	—	4	1	no	—
276	C₂×Q₈	C₂×D₄×C₄₁	1	4	yes	—
277	C₂×Q₈	#41×C₄₁	1	4	yes	—
278	C₂×D₄	—	2	4	no	—
279	C₂×D₄	—	2	4	no	—
280	C₂×D₄	—	2	4	no	—
281	C₂×D₄	—	2	2	no	—
282	C₂×D₄	—	2	2	no	—
283	C₂×D₄	C₂×D₄×C₄₁	1	4	yes	—
284	C₂²×C₄	—	2	8	no	—
285	C₂²×C₄	—	2	4	no	—
286	C₂²×C₄	—	2	2	no	—
287	C₂²×C₄	—	2	2	no	—
288	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
289	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
290	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
291	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
292	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
293	M₁₆	—	2	4	no	—
294	M₁₆	—	2	4	no	—
295	C₂×C₈	—	2	4	no	—
296	C₄⋊C₄	—	2	4	no	—
297	C₄⋊C₄	—	2	4	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	2	4	no	—
302	C₄⋊C₄	—	4	2	no	—
303	C₄⋊C₄	—	4	2	no	—
304	C₂²⋊C₄	—	2	8	no	—
305	C₂²⋊C₄	—	2	4	no	—
306	C₂²⋊C₄	—	2	4	no	—
307	C₄²	—	2	2	no	—
308	C₄²	—	2	2	no	—
309	C₄²	—	2	4	no	—
310	C₄²	C₂×D₄×C₄₁	1	2	yes	—
311	C₄²	C₂×D₄×C₄₁	1	2	yes	—
312	C₂³	—	2	4	no	—
313	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
314	Q₈	—	2	8	no	—
315	Q₈	—	2	8	no	—
316	Q₈	—	4	8	no	—
317	Q₈	—	8	2	no	—
318	Q₈	—	8	2	no	—
319	D₄	—	2	8	no	—
320	D₄	—	4	8	no	—
321	D₄	—	4	4	no	—
322	D₄	—	4	4	no	—
323	D₄	—	4	4	no	—
324	D₄	—	8	2	no	—
325	D₄	—	8	4	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	4	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	16	no	—
333	C₂×C₄	—	4	8	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	4	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	—	4	2	no	—
340	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
341	C₂×C₄	#120×C₄₁	1	2	yes	—
342	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
343	C₂×C₄	#124×C₄₁	1	4	yes	—
344	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
345	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
346	C₂×C₄	H(5,𝔽₂)×C₄₁	1	2	yes	—
347	C₈	—	4	4	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	4	4	no	—
351	C₂²	—	4	4	no	—
352	C₂²	—	4	4	no	—
353	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
354	C₂²	C₂×#124×C₄₁	1	1	yes	characteristic
355	C₂²	C₄₁×#535	1	1	yes	characteristic
356	C₄	—	2	4	no	—
357	C₄	—	2	1	no	—
358	C₄	—	2	2	no	—
359	C₄	—	4	4	no	—
360	C₄	—	4	2	no	—
361	C₄	—	8	2	no	—
362	C₄	—	8	2	no	—
363	C₄	C₄₁×#454	1	2	yes	—
364	C₄	C₄₁×#546	1	4	yes	—
365	C₂	—	2	1	no	—
366	C₂	—	2	1	no	—
367	C₂	—	8	4	no	—
368	C₂	C₄₁×#2435	1	1	yes	characteristic
369	C₂	C₂×C₄₁×#454	1	1	yes	characteristic
370	C₂	C₄₁×#3022	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₄₁×#3497	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₂×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₁×#3141	C₂	semidirect
3. C₄₁×#2568	C₂	semidirect
4. C₄₁×#2040	C₂	semidirect
5. C₄₁×#1980	C₂	semidirect
6. C₄₁×#1793	C₂	semidirect
7. C₄×#101×C₄₁	C₂	semidirect
162. #25872	C₄₁	direct (N × H)
181. #3141	C₂×C₄₁	semidirect
182. #2568	C₂×C₄₁	semidirect
183. #2040	C₂×C₄₁	semidirect
184. #1980	C₂×C₄₁	semidirect
185. #1793	C₂×C₄₁	semidirect
186. C₄×#101	C₂×C₄₁	semidirect
216. C₄₁	#25872	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×#3497	2
C₄₁×#3141	2
C₄₁×#2568	2
C₄₁×#2040	2
C₄₁×#1980	2
C₄₁×#1793	2
C₄×#101×C₄₁	2
#25872	41

Subgroups — C41×#25872 #23802073

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#25872 #23802073