Subgroups — C₄₁×#19193 #23795394

Order 10496.

650 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3227	C₂	1	2	yes	maximal
2	C₂×C₄₁×#500	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#2931	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1695	C₂	1	4	yes	maximal, complemented
5	C₄₁×#1675	C₂	1	4	yes	maximal, complemented
6	C₄₁×#1564	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#1548	C₂	1	2	yes	maximal, complemented
8	C₂×#131×C₄₁	—	2	1	no	—
9	C₄₁×#532	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Q₈×C₄₁	C₂²	1	2	yes	—
11	C₄₁×#500	—	2	1	no	—
12	C₄₁×#500	—	2	1	no	—
13	C₄₁×#500	C₂²	1	2	yes	—
14	C₄₁×#500	C₂²	1	2	yes	—
15	C₄₁×#498	—	2	2	no	—
16	C₄₁×#486	—	2	2	no	—
17	C₄₁×#481	—	2	4	no	—
18	C₄₁×#476	C₂²	1	8	yes	—
19	C₄₁×#436	—	2	1	no	—
20	C₄₁×#436	C₂²	1	2	yes	—
21	C₈×D₄×C₄₁	—	2	1	no	—
22	C₈×D₄×C₄₁	C₂²	1	2	yes	—
23	C₄₁×#431	C₂²	1	2	yes	—
24	C₂×#107×C₄₁	—	2	1	no	—
25	C₂×#107×C₄₁	—	2	1	no	—
26	C₄₁×#423	C₂²	1	1	yes	characteristic
27	C₄₁×#408	—	2	1	no	—
28	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
29	C₄₁×#335	C₂²	1	2	yes	—
30	C₄₁×#334	C₂²	1	2	yes	—
31	C₄₁×#333	C₂²	1	8	yes	—
32	C₂×#41×C₄₁	—	2	1	no	—
33	C₂²×Q₈×C₄₁	—	2	2	no	—
34	#131×C₄₁	—	4	4	no	—
35	C₂×M₁₆×C₄₁	—	2	1	no	—
36	C₂×M₁₆×C₄₁	—	2	2	no	—
37	C₂²×C₈×C₄₁	—	2	2	no	—
38	C₂²×C₈×C₄₁	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₄₁	D₄	1	1	yes	characteristic
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	1	yes	characteristic
44	C₄⋊Q₈×C₄₁	C₂³	1	4	yes	—
45	#127×C₄₁	D₄	1	1	yes	characteristic
46	#124×C₄₁	—	2	1	no	—
47	#121×C₄₁	—	2	1	no	—
48	#121×C₄₁	D₄	1	2	yes	—
49	C₄×D₄×C₄₁	—	2	1	no	—
50	C₄×D₄×C₄₁	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
52	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
53	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
54	#107×C₄₁	—	2	2	no	—
55	#107×C₄₁	—	2	2	no	—
56	#107×C₄₁	—	2	2	no	—
57	#107×C₄₁	—	2	2	no	—
58	#106×C₄₁	—	2	8	no	—
59	C₄⋊C₈×C₄₁	—	2	1	no	—
60	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
61	C₄⋊C₈×C₄₁	C₂³	1	4	yes	—
62	#102×C₄₁	—	2	8	no	—
63	C₂²⋊C₈×C₄₁	—	2	2	no	—
64	C₂²⋊C₈×C₄₁	—	2	2	no	—
65	C₂²⋊C₈×C₄₁	—	2	2	no	—
66	C₄×C₈×C₄₁	—	2	2	no	—
67	C₄×C₈×C₄₁	D₄	1	1	yes	characteristic
68	C₄×C₈×C₄₁	D₄	1	1	yes	characteristic
69	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
71	#41×C₄₁	—	4	2	no	—
72	#41×C₄₁	—	4	2	no	—
73	C₂×Q₈×C₄₁	—	2	1	no	—
74	C₂×Q₈×C₄₁	—	2	4	no	—
75	C₂×Q₈×C₄₁	—	4	2	no	—
76	C₂×Q₈×C₄₁	—	8	4	no	—
77	C₂×D₄×C₄₁	—	2	2	no	—
78	C₂×D₄×C₄₁	—	2	1	no	—
79	C₂×D₄×C₄₁	—	2	1	no	—
80	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₄₁	—	2	2	no	—
82	C₂²×C₄×C₄₁	—	4	2	no	—
83	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₄₁	#41	1	1	yes	characteristic
86	M₁₆×C₄₁	—	4	2	no	—
87	M₁₆×C₄₁	—	4	4	no	—
88	C₂×C₈×C₄₁	—	2	2	no	—
89	C₂×C₈×C₄₁	—	2	4	no	—
90	C₂×C₈×C₄₁	—	2	2	no	—
91	C₂×C₈×C₄₁	—	4	4	no	—
92	C₂×C₈×C₄₁	SD₁₆	1	4	yes	—
93	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
97	C₄⋊C₄×C₄₁	—	2	1	no	—
98	C₄⋊C₄×C₄₁	—	2	4	no	—
99	C₄⋊C₄×C₄₁	—	2	4	no	—
100	C₄⋊C₄×C₄₁	—	4	2	no	—
101	C₄⋊C₄×C₄₁	—	4	2	no	—
102	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
103	C₂²⋊C₄×C₄₁	—	2	2	no	—
104	C₂²⋊C₄×C₄₁	—	2	2	no	—
105	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₄₁	#41	1	1	yes	characteristic
108	C₄²×C₄₁	C₂⁴	1	1	yes	characteristic
109	C₂³×C₄₁	—	2	2	no	—
110	C₂³×C₄₁	#121	1	1	yes	characteristic
111	Q₈×C₄₁	—	4	2	no	—
112	Q₈×C₄₁	—	8	4	no	—
113	Q₈×C₄₁	—	8	4	no	—
114	D₄×C₄₁	—	2	4	no	—
115	D₄×C₄₁	—	4	4	no	—
116	D₄×C₄₁	—	4	2	no	—
117	D₄×C₄₁	—	4	2	no	—
118	C₂×C₄×C₄₁	—	2	2	no	—
119	C₂×C₄×C₄₁	—	2	2	no	—
120	C₂×C₄×C₄₁	—	2	2	no	—
121	C₂×C₄×C₄₁	—	4	2	no	—
122	C₂×C₄×C₄₁	—	4	2	no	—
123	C₂×C₄×C₄₁	—	4	4	no	—
124	C₂×C₄×C₄₁	—	8	4	no	—
125	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
126	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
127	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
128	C₂×C₄×C₄₁	C₂×SD₁₆	1	2	yes	—
129	C₂×C₄×C₄₁	#137	1	2	yes	—
130	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
133	C₈×C₄₁	—	2	1	no	—
134	C₈×C₄₁	—	2	1	no	—
135	C₈×C₄₁	—	4	2	no	—
136	C₈×C₄₁	—	4	4	no	—
137	C₈×C₄₁	C₂×SD₁₆	1	2	yes	—
138	C₈×C₄₁	C₂×SD₁₆	1	2	yes	—
139	#19193	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₂²×C₄₁	—	2	1	no	—
141	C₂²×C₄₁	—	2	1	no	—
142	C₂²×C₄₁	—	4	2	no	—
143	C₂²×C₄₁	—	4	2	no	—
144	C₂²×C₄₁	—	4	2	no	—
145	C₂²×C₄₁	#461	1	2	yes	—
146	C₂²×C₄₁	C₂×#121	1	1	yes	characteristic
147	C₄×C₄₁	—	2	1	no	—
148	C₄×C₄₁	—	2	1	no	—
149	C₄×C₄₁	—	4	2	no	—
150	C₄×C₄₁	—	8	4	no	—
151	C₄×C₄₁	#461	1	2	yes	—
152	C₄×C₄₁	#472	1	2	yes	—
153	C₄×C₄₁	C₂²×SD₁₆	1	1	yes	characteristic
154	C₄×C₄₁	C₂×#137	1	1	yes	characteristic
155	C₄×C₄₁	#578	1	2	yes	—
156	#3227	C₂×C₄₁	1	2	yes	—
157	C₂×#500	C₂×C₄₁	1	1	yes	characteristic, complemented
158	#2931	C₂×C₄₁	1	1	yes	characteristic
159	#1695	C₂×C₄₁	1	4	yes	complemented
160	#1675	C₂×C₄₁	1	4	yes	complemented
161	#1564	C₂×C₄₁	1	1	yes	characteristic, complemented
162	#1548	C₂×C₄₁	1	2	yes	complemented
163	C₂×C₄₁	—	2	2	no	—
164	C₂×C₄₁	—	8	2	no	—
165	C₂×C₄₁	C₂×#461	1	1	yes	characteristic
166	C₂×C₄₁	#3059	1	1	yes	characteristic
167	C₂×C₄₁	C₂×#472	1	1	yes	characteristic
168	C₂×#131	—	2	1	no	—
169	#532	C₂²×C₄₁	1	1	yes	characteristic
170	C₂×C₄⋊Q₈	C₂²×C₄₁	1	2	yes	—
171	#500	—	2	1	no	—
172	#500	—	2	1	no	—
173	#500	C₂²×C₄₁	1	2	yes	—
174	#500	C₂²×C₄₁	1	2	yes	—
175	#498	—	2	2	no	—
176	#486	—	2	2	no	—
177	#481	—	2	4	no	—
178	#476	C₂²×C₄₁	1	8	yes	—
179	#436	—	2	1	no	—
180	#436	C₂²×C₄₁	1	2	yes	—
181	C₈×D₄	—	2	1	no	—
182	C₈×D₄	C₂²×C₄₁	1	2	yes	—
183	#431	C₂²×C₄₁	1	2	yes	—
184	C₂×#107	—	2	1	no	—
185	C₂×#107	—	2	1	no	—
186	#423	C₂²×C₄₁	1	1	yes	characteristic
187	#408	—	2	1	no	—
188	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
189	#335	C₂²×C₄₁	1	2	yes	—
190	#334	C₂²×C₄₁	1	2	yes	—
191	#333	C₂²×C₄₁	1	8	yes	—
192	C₄₁	#19193	1	1	yes	characteristic, direct product factor, Sylow
193	C₂×#41	—	2	1	no	—
194	C₂²×Q₈	—	2	2	no	—
195	#131	—	4	4	no	—
196	C₂×M₁₆	—	2	1	no	—
197	C₂×M₁₆	—	2	2	no	—
198	C₂²×C₈	—	2	2	no	—
199	C₂²×C₈	D₄×C₄₁	1	1	yes	characteristic
200	C₂²×C₈	D₄×C₄₁	1	1	yes	characteristic
201	C₄⋊Q₈	—	2	2	no	—
202	C₄⋊Q₈	—	2	2	no	—
203	C₄⋊Q₈	—	2	2	no	—
204	C₄⋊Q₈	D₄×C₄₁	1	1	yes	characteristic
205	C₄⋊Q₈	C₂³×C₄₁	1	4	yes	—
206	#127	D₄×C₄₁	1	1	yes	characteristic
207	#124	—	2	1	no	—
208	#121	—	2	1	no	—
209	#121	D₄×C₄₁	1	2	yes	—
210	C₄×D₄	—	2	1	no	—
211	C₄×D₄	C₂³×C₄₁	1	2	yes	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
215	#107	—	2	2	no	—
216	#107	—	2	2	no	—
217	#107	—	2	2	no	—
218	#107	—	2	2	no	—
219	#106	—	2	8	no	—
220	C₄⋊C₈	—	2	1	no	—
221	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
222	C₄⋊C₈	C₂³×C₄₁	1	4	yes	—
223	#102	—	2	8	no	—
224	C₂²⋊C₈	—	2	2	no	—
225	C₂²⋊C₈	—	2	2	no	—
226	C₂²⋊C₈	—	2	2	no	—
227	C₄×C₈	—	2	2	no	—
228	C₄×C₈	D₄×C₄₁	1	1	yes	characteristic
229	C₄×C₈	D₄×C₄₁	1	1	yes	characteristic
230	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
231	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
232	#41	—	4	2	no	—
233	#41	—	4	2	no	—
234	C₂×Q₈	—	2	1	no	—
235	C₂×Q₈	—	2	4	no	—
236	C₂×Q₈	—	4	2	no	—
237	C₂×Q₈	—	8	4	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	2	1	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	4	2	no	—
244	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
245	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
246	C₂²×C₄	#41×C₄₁	1	1	yes	characteristic
247	M₁₆	—	4	2	no	—
248	M₁₆	—	4	4	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	4	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	4	4	no	—
253	C₂×C₈	SD₁₆×C₄₁	1	4	yes	—
254	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
255	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
256	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
257	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
264	C₂²⋊C₄	—	2	2	no	—
265	C₂²⋊C₄	—	2	2	no	—
266	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
267	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
268	C₄²	#41×C₄₁	1	1	yes	characteristic
269	C₄²	C₂⁴×C₄₁	1	1	yes	characteristic
270	C₂³	—	2	2	no	—
271	C₂³	#121×C₄₁	1	1	yes	characteristic
272	Q₈	—	4	2	no	—
273	Q₈	—	8	4	no	—
274	Q₈	—	8	4	no	—
275	D₄	—	2	4	no	—
276	D₄	—	4	4	no	—
277	D₄	—	4	2	no	—
278	D₄	—	4	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	8	4	no	—
286	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
287	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
288	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
289	C₂×C₄	C₂×SD₁₆×C₄₁	1	2	yes	—
290	C₂×C₄	#137×C₄₁	1	2	yes	—
291	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
292	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
293	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
294	C₈	—	2	1	no	—
295	C₈	—	2	1	no	—
296	C₈	—	4	2	no	—
297	C₈	—	4	4	no	—
298	C₈	C₂×SD₁₆×C₄₁	1	2	yes	—
299	C₈	C₂×SD₁₆×C₄₁	1	2	yes	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	4	2	no	—
303	C₂²	—	4	2	no	—
304	C₂²	—	4	2	no	—
305	C₂²	C₄₁×#461	1	2	yes	—
306	C₂²	C₂×#121×C₄₁	1	1	yes	characteristic
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	4	2	no	—
310	C₄	—	8	4	no	—
311	C₄	C₄₁×#461	1	2	yes	—
312	C₄	C₄₁×#472	1	2	yes	—
313	C₄	C₂²×SD₁₆×C₄₁	1	1	yes	characteristic
314	C₄	C₂×#137×C₄₁	1	1	yes	characteristic
315	C₄	C₄₁×#578	1	2	yes	—
316	C₂	—	2	2	no	—
317	C₂	—	8	2	no	—
318	C₂	C₂×C₄₁×#461	1	1	yes	characteristic
319	C₂	C₄₁×#3059	1	1	yes	characteristic
320	C₂	C₂×C₄₁×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₄₁
prime radical	C₄₁×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ 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. C₂×C₄₁×#500	C₂	semidirect
4. C₄₁×#1695	C₂	semidirect
5. C₄₁×#1675	C₂	semidirect
6. C₄₁×#1564	C₂	semidirect
7. C₄₁×#1548	C₂	semidirect
139. #19193	C₄₁	direct (N × H)
157. C₂×#500	C₂×C₄₁	semidirect
159. #1695	C₂×C₄₁	semidirect
160. #1675	C₂×C₄₁	semidirect
161. #1564	C₂×C₄₁	semidirect
162. #1548	C₂×C₄₁	semidirect
192. C₄₁	#19193	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×#3227	2
C₂×C₄₁×#500	2
C₄₁×#2931	2
C₄₁×#1695	2
C₄₁×#1675	2
C₄₁×#1564	2
C₄₁×#1548	2
#19193	41

Subgroups — C41×#19193 #23795394

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#19193 #23795394