Subgroups — C₄₁×#20289 #23796490

Order 10496.

616 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3369	C₂	1	2	yes	maximal, complemented
2	C₄₁×#3324	C₂	1	2	yes	maximal
3	C₄₁×#1708	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#1703	C₂	1	2	yes	maximal, complemented
5	C₄₁×#1702	C₂	1	2	yes	maximal, complemented
6	C₄₁×#1652	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1629	C₂	1	1	yes	maximal, characteristic
8	C₄₁×#1623	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1590	C₂	1	2	yes	maximal, complemented
10	Q₈²×C₄₁	C₂²	1	1	yes	characteristic, complemented
11	C₄₁×#556	C₂²	1	2	yes	—
12	D₄²×C₄₁	C₂²	1	1	yes	characteristic
13	C₄₁×#498	C₂²	1	1	yes	characteristic, complemented
14	C₄₁×#492	C₂²	1	1	yes	characteristic
15	C₄₁×#489	—	2	2	no	—
16	C₄₁×#487	C₂²	1	2	yes	complemented
17	C₄₁×#486	C₂²	1	2	yes	—
18	C₄₁×#482	—	2	2	no	—
19	C₄₁×#476	C₂²	1	2	yes	—
20	C₄₁×#475	—	2	2	no	—
21	C₄₁×#474	C₂²	1	2	yes	—
22	C₄₁×#468	—	2	2	no	—
23	C₄₁×#464	—	2	2	no	—
24	C₄₁×#462	C₂²	1	2	yes	complemented
25	C₄₁×#460	C₂²	1	2	yes	—
26	C₄₁×#450	—	2	2	no	—
27	C₄₁×#446	C₂²	1	2	yes	complemented
28	C₄₁×#441	—	2	2	no	—
29	C₄×SD₁₆×C₄₁	C₂²	1	2	yes	—
30	C₄₁×#436	C₂²	1	2	yes	—
31	C₄₁×#418	—	2	2	no	—
32	C₄₁×#334	C₂²	1	2	yes	complemented
33	C₄₁×#333	C₂²	1	1	yes	characteristic, complemented
34	C₄₁×#332	C₂²	1	2	yes	complemented
35	C₄₁×#331	C₂²	1	1	yes	characteristic
36	C₄₁×Q₈⋊C₈	C₂²	1	2	yes	complemented
37	C₄₁×D₄⋊C₈	C₂²	1	2	yes	—
38	C₈²×C₄₁	C₂²	1	1	yes	characteristic, complemented
39	C₂²×D₄×C₄₁	—	2	2	no	—
40	C₂×Q₁₆×C₄₁	—	2	2	no	—
41	C₂×SD₁₆×C₄₁	—	2	2	no	—
42	C₂×SD₁₆×C₄₁	—	4	1	no	—
43	C₂×SD₁₆×C₄₁	D₄	1	2	yes	—
44	C₂×M₁₆×C₄₁	—	2	2	no	—
45	C₄⋊Q₈×C₄₁	—	2	2	no	—
46	C₄⋊Q₈×C₄₁	—	4	1	no	—
47	C₄⋊Q₈×C₄₁	C₂³	1	2	yes	—
48	C₄⋊Q₈×C₄₁	C₂³	1	1	yes	characteristic
49	#127×C₄₁	C₂³	1	1	yes	characteristic
50	#126×C₄₁	—	4	2	no	—
51	#124×C₄₁	—	2	2	no	—
52	C₂²⋊Q₈×C₄₁	—	2	2	no	—
53	#121×C₄₁	—	2	2	no	—
54	#120×C₄₁	—	4	1	no	—
55	C₄×Q₈×C₄₁	—	2	2	no	—
56	C₄×Q₈×C₄₁	—	2	2	no	—
57	C₄×Q₈×C₄₁	C₂³	1	2	yes	—
58	C₄×D₄×C₄₁	C₂³	1	2	yes	—
59	#117×C₄₁	—	2	2	no	—
60	#107×C₄₁	—	2	2	no	—
61	#106×C₄₁	—	2	2	no	—
62	#106×C₄₁	—	4	1	no	—
63	#106×C₄₁	D₄	1	2	yes	—
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	—	2	2	no	—
69	#103×C₄₁	—	2	2	no	—
70	#103×C₄₁	D₄	1	2	yes	—
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	2	yes	—
81	C₂⁴×C₄₁	—	4	1	no	—
82	C₂×Q₈×C₄₁	—	2	2	no	—
83	C₂×Q₈×C₄₁	—	2	1	no	—
84	C₂×Q₈×C₄₁	—	4	1	no	—
85	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
86	C₂×D₄×C₄₁	—	2	2	no	—
87	C₂×D₄×C₄₁	—	2	1	no	—
88	C₂×D₄×C₄₁	—	4	2	no	—
89	C₂×D₄×C₄₁	—	4	4	no	—
90	C₂×D₄×C₄₁	—	8	2	no	—
91	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₄₁	—	2	2	no	—
93	Q₁₆×C₄₁	—	4	4	no	—
94	SD₁₆×C₄₁	—	4	2	no	—
95	SD₁₆×C₄₁	—	8	2	no	—
96	SD₁₆×C₄₁	—	8	2	no	—
97	M₁₆×C₄₁	—	8	2	no	—
98	C₂×C₈×C₄₁	—	2	1	no	—
99	C₂×C₈×C₄₁	—	2	2	no	—
100	C₂×C₈×C₄₁	—	2	2	no	—
101	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
103	C₄⋊C₄×C₄₁	—	2	2	no	—
104	C₄⋊C₄×C₄₁	—	2	2	no	—
105	C₄⋊C₄×C₄₁	—	2	2	no	—
106	C₄⋊C₄×C₄₁	—	2	2	no	—
107	C₄⋊C₄×C₄₁	—	4	2	no	—
108	C₄⋊C₄×C₄₁	—	4	2	no	—
109	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
110	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
111	C₂²⋊C₄×C₄₁	—	2	2	no	—
112	C₂²⋊C₄×C₄₁	—	4	1	no	—
113	C₂²⋊C₄×C₄₁	—	4	2	no	—
114	C₂²⋊C₄×C₄₁	—	4	2	no	—
115	C₄²×C₄₁	—	2	2	no	—
116	C₄²×C₄₁	—	4	2	no	—
117	C₄²×C₄₁	—	4	1	no	—
118	C₄²×C₄₁	C₂⁴	1	1	yes	characteristic
119	C₂³×C₄₁	—	2	2	no	—
120	C₂³×C₄₁	—	4	1	no	—
121	C₂³×C₄₁	—	8	4	no	—
122	C₂³×C₄₁	—	16	1	no	—
123	Q₈×C₄₁	—	2	2	no	—
124	Q₈×C₄₁	—	2	2	no	—
125	Q₈×C₄₁	—	4	2	no	—
126	Q₈×C₄₁	—	4	2	no	—
127	Q₈×C₄₁	—	8	1	no	—
128	Q₈×C₄₁	—	16	1	no	—
129	D₄×C₄₁	—	2	2	no	—
130	D₄×C₄₁	—	2	2	no	—
131	D₄×C₄₁	—	4	2	no	—
132	D₄×C₄₁	—	4	2	no	—
133	D₄×C₄₁	—	8	1	no	—
134	D₄×C₄₁	—	8	4	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	—	4	1	no	—
142	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
143	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
144	C₈×C₄₁	—	2	2	no	—
145	C₈×C₄₁	—	2	2	no	—
146	C₈×C₄₁	—	2	2	no	—
147	C₈×C₄₁	—	4	2	no	—
148	C₈×C₄₁	—	4	2	no	—
149	#20289	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₂²×C₄₁	—	4	2	no	—
151	C₂²×C₄₁	—	4	2	no	—
152	C₂²×C₄₁	—	4	2	no	—
153	C₂²×C₄₁	—	8	2	no	—
154	C₂²×C₄₁	—	8	1	no	—
155	C₂²×C₄₁	—	16	4	no	—
156	C₂²×C₄₁	D₄²	1	1	yes	characteristic
157	C₄×C₄₁	—	2	1	no	—
158	C₄×C₄₁	—	4	2	no	—
159	C₄×C₄₁	—	4	2	no	—
160	C₄×C₄₁	—	4	2	no	—
161	C₄×C₄₁	—	8	2	no	—
162	C₄×C₄₁	—	8	1	no	—
163	C₄×C₄₁	#575	1	2	yes	—
164	C₄×C₄₁	#577	1	2	yes	—
165	#3369	C₂×C₄₁	1	2	yes	complemented
166	#3324	C₂×C₄₁	1	2	yes	—
167	#1708	C₂×C₄₁	1	1	yes	characteristic, complemented
168	#1703	C₂×C₄₁	1	2	yes	complemented
169	#1702	C₂×C₄₁	1	2	yes	complemented
170	#1652	C₂×C₄₁	1	2	yes	complemented
171	#1629	C₂×C₄₁	1	1	yes	characteristic
172	#1623	C₂×C₄₁	1	1	yes	characteristic, complemented
173	#1590	C₂×C₄₁	1	2	yes	complemented
174	C₂×C₄₁	—	8	2	no	—
175	C₂×C₄₁	—	16	1	no	—
176	C₂×C₄₁	#3275	1	2	yes	—
177	C₂×C₄₁	#3291	1	1	yes	characteristic
178	Q₈²	C₂²×C₄₁	1	1	yes	characteristic, complemented
179	#556	C₂²×C₄₁	1	2	yes	—
180	D₄²	C₂²×C₄₁	1	1	yes	characteristic
181	#498	C₂²×C₄₁	1	1	yes	characteristic, complemented
182	#492	C₂²×C₄₁	1	1	yes	characteristic
183	#489	—	2	2	no	—
184	#487	C₂²×C₄₁	1	2	yes	complemented
185	#486	C₂²×C₄₁	1	2	yes	—
186	#482	—	2	2	no	—
187	#476	C₂²×C₄₁	1	2	yes	—
188	#475	—	2	2	no	—
189	#474	C₂²×C₄₁	1	2	yes	—
190	#468	—	2	2	no	—
191	#464	—	2	2	no	—
192	#462	C₂²×C₄₁	1	2	yes	complemented
193	#460	C₂²×C₄₁	1	2	yes	—
194	#450	—	2	2	no	—
195	#446	C₂²×C₄₁	1	2	yes	complemented
196	#441	—	2	2	no	—
197	C₄×SD₁₆	C₂²×C₄₁	1	2	yes	—
198	#436	C₂²×C₄₁	1	2	yes	—
199	#418	—	2	2	no	—
200	#334	C₂²×C₄₁	1	2	yes	complemented
201	#333	C₂²×C₄₁	1	1	yes	characteristic, complemented
202	#332	C₂²×C₄₁	1	2	yes	complemented
203	#331	C₂²×C₄₁	1	1	yes	characteristic
204	Q₈⋊C₈	C₂²×C₄₁	1	2	yes	complemented
205	D₄⋊C₈	C₂²×C₄₁	1	2	yes	—
206	C₈²	C₂²×C₄₁	1	1	yes	characteristic, complemented
207	C₄₁	#20289	1	1	yes	characteristic, direct product factor, Sylow
208	C₂²×D₄	—	2	2	no	—
209	C₂×Q₁₆	—	2	2	no	—
210	C₂×SD₁₆	—	2	2	no	—
211	C₂×SD₁₆	—	4	1	no	—
212	C₂×SD₁₆	D₄×C₄₁	1	2	yes	—
213	C₂×M₁₆	—	2	2	no	—
214	C₄⋊Q₈	—	2	2	no	—
215	C₄⋊Q₈	—	4	1	no	—
216	C₄⋊Q₈	C₂³×C₄₁	1	2	yes	—
217	C₄⋊Q₈	C₂³×C₄₁	1	1	yes	characteristic
218	#127	C₂³×C₄₁	1	1	yes	characteristic
219	#126	—	4	2	no	—
220	#124	—	2	2	no	—
221	C₂²⋊Q₈	—	2	2	no	—
222	#121	—	2	2	no	—
223	#120	—	4	1	no	—
224	C₄×Q₈	—	2	2	no	—
225	C₄×Q₈	—	2	2	no	—
226	C₄×Q₈	C₂³×C₄₁	1	2	yes	—
227	C₄×D₄	C₂³×C₄₁	1	2	yes	—
228	#117	—	2	2	no	—
229	#107	—	2	2	no	—
230	#106	—	2	2	no	—
231	#106	—	4	1	no	—
232	#106	D₄×C₄₁	1	2	yes	—
233	C₄⋊C₈	—	2	2	no	—
234	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
235	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
236	#103	—	2	2	no	—
237	#103	—	2	2	no	—
238	#103	—	2	2	no	—
239	#103	D₄×C₄₁	1	2	yes	—
240	#102	—	2	2	no	—
241	#102	—	2	2	no	—
242	#102	—	2	2	no	—
243	#102	—	2	2	no	—
244	#102	—	2	2	no	—
245	#102	D₄×C₄₁	1	2	yes	—
246	C₂²⋊C₈	—	2	2	no	—
247	#97	—	2	2	no	—
248	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
249	C₄×C₈	C₂³×C₄₁	1	2	yes	—
250	C₂⁴	—	4	1	no	—
251	C₂×Q₈	—	2	2	no	—
252	C₂×Q₈	—	2	1	no	—
253	C₂×Q₈	—	4	1	no	—
254	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	2	1	no	—
257	C₂×D₄	—	4	2	no	—
258	C₂×D₄	—	4	4	no	—
259	C₂×D₄	—	8	2	no	—
260	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
261	C₂²×C₄	—	2	2	no	—
262	Q₁₆	—	4	4	no	—
263	SD₁₆	—	4	2	no	—
264	SD₁₆	—	8	2	no	—
265	SD₁₆	—	8	2	no	—
266	M₁₆	—	8	2	no	—
267	C₂×C₈	—	2	1	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
271	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
279	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
280	C₂²⋊C₄	—	2	2	no	—
281	C₂²⋊C₄	—	4	1	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₂²⋊C₄	—	4	2	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	4	2	no	—
286	C₄²	—	4	1	no	—
287	C₄²	C₂⁴×C₄₁	1	1	yes	characteristic
288	C₂³	—	2	2	no	—
289	C₂³	—	4	1	no	—
290	C₂³	—	8	4	no	—
291	C₂³	—	16	1	no	—
292	Q₈	—	2	2	no	—
293	Q₈	—	2	2	no	—
294	Q₈	—	4	2	no	—
295	Q₈	—	4	2	no	—
296	Q₈	—	8	1	no	—
297	Q₈	—	16	1	no	—
298	D₄	—	2	2	no	—
299	D₄	—	2	2	no	—
300	D₄	—	4	2	no	—
301	D₄	—	4	2	no	—
302	D₄	—	8	1	no	—
303	D₄	—	8	4	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
312	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
313	C₈	—	2	2	no	—
314	C₈	—	2	2	no	—
315	C₈	—	2	2	no	—
316	C₈	—	4	2	no	—
317	C₈	—	4	2	no	—
318	C₂²	—	4	2	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	4	2	no	—
321	C₂²	—	8	2	no	—
322	C₂²	—	8	1	no	—
323	C₂²	—	16	4	no	—
324	C₂²	D₄²×C₄₁	1	1	yes	characteristic
325	C₄	—	2	1	no	—
326	C₄	—	4	2	no	—
327	C₄	—	4	2	no	—
328	C₄	—	4	2	no	—
329	C₄	—	8	2	no	—
330	C₄	—	8	1	no	—
331	C₄	C₄₁×#575	1	2	yes	—
332	C₄	C₄₁×#577	1	2	yes	—
333	C₂	—	8	2	no	—
334	C₂	—	16	1	no	—
335	C₂	C₄₁×#3275	1	2	yes	—
336	C₂	C₄₁×#3291	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₄₁
prime radical	D₄²×C₄₁	C₂²
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×#3369	C₂	semidirect
3. C₄₁×#1708	C₂	semidirect
4. C₄₁×#1703	C₂	semidirect
5. C₄₁×#1702	C₂	semidirect
6. C₄₁×#1652	C₂	semidirect
8. C₄₁×#1623	C₂	semidirect
9. C₄₁×#1590	C₂	semidirect
10. Q₈²×C₄₁	C₂²	semidirect
13. C₄₁×#498	C₂²	semidirect
16. C₄₁×#487	C₂²	semidirect
24. C₄₁×#462	C₂²	semidirect
27. C₄₁×#446	C₂²	semidirect
32. C₄₁×#334	C₂²	semidirect
33. C₄₁×#333	C₂²	semidirect
34. C₄₁×#332	C₂²	semidirect
36. C₄₁×Q₈⋊C₈	C₂²	semidirect
38. C₈²×C₄₁	C₂²	semidirect
149. #20289	C₄₁	direct (N × H)
165. #3369	C₂×C₄₁	semidirect
167. #1708	C₂×C₄₁	semidirect
168. #1703	C₂×C₄₁	semidirect
169. #1702	C₂×C₄₁	semidirect
170. #1652	C₂×C₄₁	semidirect
172. #1623	C₂×C₄₁	semidirect
173. #1590	C₂×C₄₁	semidirect
178. Q₈²	C₂²×C₄₁	semidirect
181. #498	C₂²×C₄₁	semidirect
184. #487	C₂²×C₄₁	semidirect
192. #462	C₂²×C₄₁	semidirect
195. #446	C₂²×C₄₁	semidirect
200. #334	C₂²×C₄₁	semidirect
201. #333	C₂²×C₄₁	semidirect
202. #332	C₂²×C₄₁	semidirect
204. Q₈⋊C₈	C₂²×C₄₁	semidirect
206. C₈²	C₂²×C₄₁	semidirect
207. C₄₁	#20289	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×#3369	2
C₄₁×#3324	2
C₄₁×#1708	2
C₄₁×#1703	2
C₄₁×#1702	2
C₄₁×#1652	2
C₄₁×#1629	2
C₄₁×#1623	2
C₄₁×#1590	2
#20289	41

Subgroups — C41×#20289 #23796490

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#20289 #23796490