Subgroups — C₄₁×#33869 #23810070

Order 10496.

546 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3342	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₁×#3283	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#2241	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#2237	C₂	1	2	yes	maximal, complemented
5	C₄₁×#2224	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#2216	C₂	1	2	yes	maximal, complemented
7	C₄₁×#2209	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#2189	C₂	1	2	yes	maximal, complemented
9	C₄×SD₃₂×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
10	D₄×C₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₁×#2138	C₂	1	2	yes	maximal, complemented
12	C₂×#135×C₄₁	C₂²	1	2	yes	—
13	C₄₁×#556	—	2	1	no	—
14	C₄₁×#547	—	2	1	no	—
15	C₂×Q₃₂×C₄₁	C₂²	1	2	yes	—
16	C₂×SD₃₂×C₄₁	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₄₁	C₂²	1	2	yes	—
18	C₄₁×#500	C₂²	1	1	yes	characteristic
19	C₄₁×#494	C₂²	1	1	yes	characteristic
20	C₄₁×#485	—	2	2	no	—
21	C₄₁×#475	—	2	1	no	—
22	C₄₁×#467	C₂²	1	2	yes	—
23	C₄₁×#467	C₂²	1	2	yes	—
24	C₄₁×#461	—	2	1	no	—
25	C₄₁×#452	—	2	2	no	—
26	C₄×Q₁₆×C₄₁	C₂²	1	1	yes	characteristic
27	C₄×D₈×C₄₁	C₂²	1	1	yes	characteristic
28	C₈×D₄×C₄₁	C₂²	1	1	yes	characteristic
29	C₄₁×#427	C₂²	1	2	yes	—
30	C₄₁×#367	C₂²	1	1	yes	characteristic
31	C₄₁×#366	C₂²	1	2	yes	—
32	C₄₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₄₁×#358	C₂²	1	2	yes	—
34	C₄₁×#358	C₂²	1	2	yes	—
35	C₄₁×#358	C₂²	1	1	yes	characteristic
36	C₄₁×#358	C₂²	1	2	yes	—
37	C₄₁×#357	C₂²	1	1	yes	characteristic
38	C₄₁×#357	C₂²	1	2	yes	—
39	C₄₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
40	C₄×C₁₆×C₄₁	C₂²	1	1	yes	characteristic
41	C₂×#41×C₄₁	—	2	2	no	—
42	#135×C₄₁	—	2	4	no	—
43	#135×C₄₁	—	4	2	no	—
44	C₂×Q₁₆×C₄₁	—	2	1	no	—
45	C₂×Q₁₆×C₄₁	C₂³	1	1	yes	characteristic
46	C₂×Q₁₆×C₄₁	C₂³	1	2	yes	—
47	C₂×SD₁₆×C₄₁	—	2	2	no	—
48	C₂×D₈×C₄₁	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
50	C₄⋊Q₈×C₄₁	—	2	1	no	—
51	C₄⋊Q₈×C₄₁	—	2	1	no	—
52	#126×C₄₁	—	4	2	no	—
53	#124×C₄₁	—	4	1	no	—
54	#123×C₄₁	—	4	2	no	—
55	C₂²⋊Q₈×C₄₁	—	2	2	no	—
56	C₂²⋊Q₈×C₄₁	—	2	2	no	—
57	#121×C₄₁	—	4	1	no	—
58	C₄×Q₈×C₄₁	—	2	1	no	—
59	C₄×Q₈×C₄₁	—	4	1	no	—
60	C₄×D₄×C₄₁	—	2	1	no	—
61	C₄×D₄×C₄₁	D₄	1	1	yes	characteristic
62	#117×C₄₁	—	2	2	no	—
63	C₂×C₄⋊C₄×C₄₁	—	4	1	no	—
64	Q₃₂×C₄₁	—	4	2	no	—
65	SD₃₂×C₄₁	—	2	2	no	—
66	C₂×C₁₆×C₄₁	—	4	2	no	—
67	C₂×C₁₆×C₄₁	C₂³	1	1	yes	characteristic
68	C₂×C₁₆×C₄₁	C₂³	1	2	yes	—
69	C₂×C₁₆×C₄₁	C₂³	1	1	yes	characteristic
70	#107×C₄₁	—	2	1	no	—
71	#107×C₄₁	C₂³	1	2	yes	—
72	#107×C₄₁	C₂³	1	1	yes	characteristic
73	#107×C₄₁	C₂³	1	1	yes	characteristic
74	#106×C₄₁	—	2	2	no	—
75	C₄⋊C₈×C₄₁	D₄	1	1	yes	characteristic
76	#103×C₄₁	—	2	1	no	—
77	#103×C₄₁	—	2	2	no	—
78	#103×C₄₁	—	2	2	no	—
79	#102×C₄₁	—	2	1	no	—
80	C₂²⋊C₈×C₄₁	D₄	1	2	yes	—
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	1	no	—
85	C₂×Q₈×C₄₁	—	2	2	no	—
86	C₂×Q₈×C₄₁	—	4	1	no	—
87	C₂×D₄×C₄₁	—	2	1	no	—
88	C₂×D₄×C₄₁	—	4	2	no	—
89	C₂×D₄×C₄₁	D₈	1	1	yes	characteristic
90	C₂²×C₄×C₄₁	—	4	1	no	—
91	C₂²×C₄×C₄₁	—	4	2	no	—
92	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
93	Q₁₆×C₄₁	—	2	1	no	—
94	Q₁₆×C₄₁	—	2	2	no	—
95	Q₁₆×C₄₁	—	2	1	no	—
96	Q₁₆×C₄₁	—	2	2	no	—
97	Q₁₆×C₄₁	—	8	1	no	—
98	SD₁₆×C₄₁	—	4	4	no	—
99	D₈×C₄₁	—	2	1	no	—
100	D₈×C₄₁	#41	1	2	yes	—
101	C₂×C₈×C₄₁	—	2	2	no	—
102	C₂×C₈×C₄₁	—	2	2	no	—
103	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₄₁	C₂⁴	1	1	yes	characteristic
106	C₄⋊C₄×C₄₁	—	2	1	no	—
107	C₄⋊C₄×C₄₁	—	2	1	no	—
108	C₄⋊C₄×C₄₁	—	2	2	no	—
109	C₄⋊C₄×C₄₁	—	4	1	no	—
110	C₄⋊C₄×C₄₁	—	4	2	no	—
111	C₄⋊C₄×C₄₁	—	4	1	no	—
112	C₄⋊C₄×C₄₁	—	4	2	no	—
113	C₄⋊C₄×C₄₁	—	4	1	no	—
114	C₄⋊C₄×C₄₁	—	4	1	no	—
115	C₄⋊C₄×C₄₁	D₈	1	1	yes	characteristic
116	C₂²⋊C₄×C₄₁	—	4	2	no	—
117	C₂²⋊C₄×C₄₁	—	4	2	no	—
118	C₂²⋊C₄×C₄₁	—	4	2	no	—
119	C₂²⋊C₄×C₄₁	—	4	1	no	—
120	C₂²⋊C₄×C₄₁	D₈	1	2	yes	—
121	C₄²×C₄₁	—	4	2	no	—
122	C₄²×C₄₁	—	4	1	no	—
123	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
124	C₁₆×C₄₁	—	2	1	no	—
125	C₁₆×C₄₁	—	2	1	no	—
126	C₁₆×C₄₁	—	2	2	no	—
127	C₂³×C₄₁	—	4	1	no	—
128	C₂³×C₄₁	C₂×D₈	1	2	yes	—
129	Q₈×C₄₁	—	4	2	no	—
130	Q₈×C₄₁	—	4	1	no	—
131	Q₈×C₄₁	—	4	2	no	—
132	Q₈×C₄₁	—	4	1	no	—
133	Q₈×C₄₁	—	8	2	no	—
134	D₄×C₄₁	—	2	2	no	—
135	D₄×C₄₁	—	2	2	no	—
136	D₄×C₄₁	—	4	1	no	—
137	D₄×C₄₁	—	8	4	no	—
138	C₂×C₄×C₄₁	—	2	2	no	—
139	C₂×C₄×C₄₁	—	2	2	no	—
140	C₂×C₄×C₄₁	—	4	1	no	—
141	C₂×C₄×C₄₁	—	4	2	no	—
142	C₂×C₄×C₄₁	—	4	1	no	—
143	C₂×C₄×C₄₁	—	4	1	no	—
144	C₂×C₄×C₄₁	—	4	2	no	—
145	C₂×C₄×C₄₁	—	8	4	no	—
146	C₂×C₄×C₄₁	—	8	2	no	—
147	C₂×C₄×C₄₁	C₂×D₈	1	2	yes	—
148	C₂×C₄×C₄₁	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₄₁	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
151	C₈×C₄₁	—	2	2	no	—
152	C₈×C₄₁	—	2	1	no	—
153	C₈×C₄₁	C₂×#41	1	1	yes	characteristic
154	C₈×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
155	#33869	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
156	C₂²×C₄₁	—	2	2	no	—
157	C₂²×C₄₁	—	2	2	no	—
158	C₂²×C₄₁	—	2	2	no	—
159	C₂²×C₄₁	—	4	2	no	—
160	C₂²×C₄₁	—	8	1	no	—
161	C₂²×C₄₁	—	8	1	no	—
162	C₂²×C₄₁	C₂²×D₈	1	1	yes	characteristic
163	C₄×C₄₁	—	2	1	no	—
164	C₄×C₄₁	—	2	2	no	—
165	C₄×C₄₁	—	8	1	no	—
166	C₄×C₄₁	—	8	2	no	—
167	C₄×C₄₁	—	8	1	no	—
168	C₄×C₄₁	—	8	2	no	—
169	C₄×C₄₁	—	8	1	no	—
170	C₄×C₄₁	#508	1	2	yes	—
171	C₄×C₄₁	#546	1	1	yes	characteristic
172	C₄×C₄₁	#577	1	1	yes	characteristic
173	#3342	C₂×C₄₁	1	1	yes	characteristic, complemented
174	#3283	C₂×C₄₁	1	1	yes	characteristic
175	#2241	C₂×C₄₁	1	1	yes	characteristic, complemented
176	#2237	C₂×C₄₁	1	2	yes	complemented
177	#2224	C₂×C₄₁	1	1	yes	characteristic, complemented
178	#2216	C₂×C₄₁	1	2	yes	complemented
179	#2209	C₂×C₄₁	1	1	yes	characteristic, complemented
180	#2189	C₂×C₄₁	1	2	yes	complemented
181	C₄×SD₃₂	C₂×C₄₁	1	1	yes	characteristic, complemented
182	D₄×C₁₆	C₂×C₄₁	1	1	yes	characteristic, complemented
183	#2138	C₂×C₄₁	1	2	yes	complemented
184	C₂×C₄₁	—	4	2	no	—
185	C₂×C₄₁	—	8	2	no	—
186	C₂×C₄₁	#3294	1	1	yes	characteristic
187	C₂×C₄₁	C₂×#508	1	1	yes	characteristic
188	C₂×C₄₁	#3417	1	1	yes	characteristic
189	C₂×#135	C₂²×C₄₁	1	2	yes	—
190	#556	—	2	1	no	—
191	#547	—	2	1	no	—
192	C₂×Q₃₂	C₂²×C₄₁	1	2	yes	—
193	C₂×SD₃₂	C₂²×C₄₁	1	1	yes	characteristic
194	C₂²×C₁₆	C₂²×C₄₁	1	2	yes	—
195	#500	C₂²×C₄₁	1	1	yes	characteristic
196	#494	C₂²×C₄₁	1	1	yes	characteristic
197	#485	—	2	2	no	—
198	#475	—	2	1	no	—
199	#467	C₂²×C₄₁	1	2	yes	—
200	#467	C₂²×C₄₁	1	2	yes	—
201	#461	—	2	1	no	—
202	#452	—	2	2	no	—
203	C₄×Q₁₆	C₂²×C₄₁	1	1	yes	characteristic
204	C₄×D₈	C₂²×C₄₁	1	1	yes	characteristic
205	C₈×D₄	C₂²×C₄₁	1	1	yes	characteristic
206	#427	C₂²×C₄₁	1	2	yes	—
207	#367	C₂²×C₄₁	1	1	yes	characteristic
208	#366	C₂²×C₄₁	1	2	yes	—
209	C₄⋊C₁₆	C₂²×C₄₁	1	1	yes	characteristic
210	#358	C₂²×C₄₁	1	2	yes	—
211	#358	C₂²×C₄₁	1	2	yes	—
212	#358	C₂²×C₄₁	1	1	yes	characteristic
213	#358	C₂²×C₄₁	1	2	yes	—
214	#357	C₂²×C₄₁	1	1	yes	characteristic
215	#357	C₂²×C₄₁	1	2	yes	—
216	C₂²⋊C₁₆	C₂²×C₄₁	1	2	yes	—
217	C₄×C₁₆	C₂²×C₄₁	1	1	yes	characteristic
218	C₄₁	#33869	1	1	yes	characteristic, direct product factor, Sylow
219	C₂×#41	—	2	2	no	—
220	#135	—	2	4	no	—
221	#135	—	4	2	no	—
222	C₂×Q₁₆	—	2	1	no	—
223	C₂×Q₁₆	C₂³×C₄₁	1	2	yes	—
224	C₂×Q₁₆	C₂³×C₄₁	1	1	yes	characteristic
225	C₂×SD₁₆	—	2	2	no	—
226	C₂×D₈	C₂³×C₄₁	1	1	yes	characteristic
227	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
228	C₄⋊Q₈	—	2	1	no	—
229	C₄⋊Q₈	—	2	1	no	—
230	#126	—	4	2	no	—
231	#124	—	4	1	no	—
232	#123	—	4	2	no	—
233	C₂²⋊Q₈	—	2	2	no	—
234	C₂²⋊Q₈	—	2	2	no	—
235	#121	—	4	1	no	—
236	C₄×Q₈	—	2	1	no	—
237	C₄×Q₈	—	4	1	no	—
238	C₄×D₄	—	2	1	no	—
239	C₄×D₄	D₄×C₄₁	1	1	yes	characteristic
240	#117	—	2	2	no	—
241	C₂×C₄⋊C₄	—	4	1	no	—
242	Q₃₂	—	4	2	no	—
243	SD₃₂	—	2	2	no	—
244	C₂×C₁₆	—	4	2	no	—
245	C₂×C₁₆	C₂³×C₄₁	1	1	yes	characteristic
246	C₂×C₁₆	C₂³×C₄₁	1	1	yes	characteristic
247	C₂×C₁₆	C₂³×C₄₁	1	2	yes	—
248	#107	—	2	1	no	—
249	#107	C₂³×C₄₁	1	2	yes	—
250	#107	C₂³×C₄₁	1	1	yes	characteristic
251	#107	C₂³×C₄₁	1	1	yes	characteristic
252	#106	—	2	2	no	—
253	C₄⋊C₈	D₄×C₄₁	1	1	yes	characteristic
254	#103	—	2	1	no	—
255	#103	—	2	2	no	—
256	#103	—	2	2	no	—
257	#102	—	2	1	no	—
258	C₂²⋊C₈	D₄×C₄₁	1	2	yes	—
259	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
260	#41	—	4	4	no	—
261	#41	—	8	2	no	—
262	C₂×Q₈	—	2	1	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	4	1	no	—
265	C₂×D₄	—	2	1	no	—
266	C₂×D₄	—	4	2	no	—
267	C₂×D₄	D₈×C₄₁	1	1	yes	characteristic
268	C₂²×C₄	—	4	1	no	—
269	C₂²×C₄	—	4	2	no	—
270	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
271	Q₁₆	—	2	1	no	—
272	Q₁₆	—	2	2	no	—
273	Q₁₆	—	2	1	no	—
274	Q₁₆	—	2	2	no	—
275	Q₁₆	—	8	1	no	—
276	SD₁₆	—	4	4	no	—
277	D₈	—	2	1	no	—
278	D₈	#41×C₄₁	1	2	yes	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
282	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
283	C₂×C₈	C₂⁴×C₄₁	1	1	yes	characteristic
284	C₄⋊C₄	—	2	1	no	—
285	C₄⋊C₄	—	2	1	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	4	1	no	—
288	C₄⋊C₄	—	4	2	no	—
289	C₄⋊C₄	—	4	1	no	—
290	C₄⋊C₄	—	4	2	no	—
291	C₄⋊C₄	—	4	1	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	D₈×C₄₁	1	1	yes	characteristic
294	C₂²⋊C₄	—	4	2	no	—
295	C₂²⋊C₄	—	4	2	no	—
296	C₂²⋊C₄	—	4	2	no	—
297	C₂²⋊C₄	—	4	1	no	—
298	C₂²⋊C₄	D₈×C₄₁	1	2	yes	—
299	C₄²	—	4	2	no	—
300	C₄²	—	4	1	no	—
301	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
302	C₁₆	—	2	1	no	—
303	C₁₆	—	2	1	no	—
304	C₁₆	—	2	2	no	—
305	C₂³	—	4	1	no	—
306	C₂³	C₂×D₈×C₄₁	1	2	yes	—
307	Q₈	—	4	2	no	—
308	Q₈	—	4	1	no	—
309	Q₈	—	4	2	no	—
310	Q₈	—	4	1	no	—
311	Q₈	—	8	2	no	—
312	D₄	—	2	2	no	—
313	D₄	—	2	2	no	—
314	D₄	—	4	1	no	—
315	D₄	—	8	4	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	8	4	no	—
324	C₂×C₄	—	8	2	no	—
325	C₂×C₄	C₂×D₈×C₄₁	1	2	yes	—
326	C₂×C₄	C₂×D₈×C₄₁	1	1	yes	characteristic
327	C₂×C₄	C₂×D₈×C₄₁	1	1	yes	characteristic
328	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
329	C₈	—	2	2	no	—
330	C₈	—	2	1	no	—
331	C₈	C₂×#41×C₄₁	1	1	yes	characteristic
332	C₈	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
333	C₂²	—	2	2	no	—
334	C₂²	—	2	2	no	—
335	C₂²	—	2	2	no	—
336	C₂²	—	4	2	no	—
337	C₂²	—	8	1	no	—
338	C₂²	—	8	1	no	—
339	C₂²	C₂²×D₈×C₄₁	1	1	yes	characteristic
340	C₄	—	2	2	no	—
341	C₄	—	2	1	no	—
342	C₄	—	8	2	no	—
343	C₄	—	8	1	no	—
344	C₄	—	8	1	no	—
345	C₄	—	8	2	no	—
346	C₄	—	8	1	no	—
347	C₄	C₄₁×#508	1	2	yes	—
348	C₄	C₄₁×#546	1	1	yes	characteristic
349	C₄	C₄₁×#577	1	1	yes	characteristic
350	C₂	—	4	2	no	—
351	C₂	—	8	2	no	—
352	C₂	C₄₁×#3294	1	1	yes	characteristic
353	C₂	C₂×C₄₁×#508	1	1	yes	characteristic
354	C₂	C₄₁×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₄₁
prime radical	C₄₁×#3283	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₄×D₄×C₄₁ ◁ C₈×D₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₁×#3342	C₂	semidirect
3. C₄₁×#2241	C₂	semidirect
4. C₄₁×#2237	C₂	semidirect
5. C₄₁×#2224	C₂	semidirect
6. C₄₁×#2216	C₂	semidirect
7. C₄₁×#2209	C₂	semidirect
8. C₄₁×#2189	C₂	semidirect
9. C₄×SD₃₂×C₄₁	C₂	semidirect
10. D₄×C₁₆×C₄₁	C₂	semidirect
11. C₄₁×#2138	C₂	semidirect
155. #33869	C₄₁	direct (N × H)
173. #3342	C₂×C₄₁	semidirect
175. #2241	C₂×C₄₁	semidirect
176. #2237	C₂×C₄₁	semidirect
177. #2224	C₂×C₄₁	semidirect
178. #2216	C₂×C₄₁	semidirect
179. #2209	C₂×C₄₁	semidirect
180. #2189	C₂×C₄₁	semidirect
181. C₄×SD₃₂	C₂×C₄₁	semidirect
182. D₄×C₁₆	C₂×C₄₁	semidirect
183. #2138	C₂×C₄₁	semidirect
218. C₄₁	#33869	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×#3342	2
C₄₁×#3283	2
C₄₁×#2241	2
C₄₁×#2237	2
C₄₁×#2224	2
C₄₁×#2216	2
C₄₁×#2209	2
C₄₁×#2189	2
C₄×SD₃₂×C₄₁	2
D₄×C₁₆×C₄₁	2
C₄₁×#2138	2
#33869	41

Subgroups — C41×#33869 #23810070

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#33869 #23810070