Subgroups — C₄₁×#19271 #23795472

Order 10496.

702 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₁×#3039	C₂	1	2	yes	maximal
3	C₈×#41×C₄₁	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1674	C₂	1	4	yes	maximal, complemented
5	C₄₁×#1673	C₂	1	4	yes	maximal, complemented
6	C₄₁×#1567	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#1546	C₂	1	2	yes	maximal, complemented
8	C₂×#137×C₄₁	—	2	2	no	—
9	C₂×C₄⋊Q₈×C₄₁	C₂²	1	2	yes	—
10	C₄×#41×C₄₁	C₂²	1	1	yes	characteristic
11	C₄₁×#500	—	2	1	no	—
12	C₄₁×#500	—	2	1	no	—
13	C₄₁×#500	C₂²	1	4	yes	—
14	C₄₁×#462	—	2	2	no	—
15	C₄₁×#462	C₂²	1	4	yes	—
16	C₄₁×#461	—	2	2	no	—
17	C₄₁×#461	C₂²	1	4	yes	—
18	C₈×Q₈×C₄₁	C₂²	1	2	yes	—
19	C₈×D₄×C₄₁	—	2	2	no	—
20	C₈×D₄×C₄₁	C₂²	1	2	yes	—
21	C₄₁×#431	—	2	1	no	—
22	C₄₁×#431	C₂²	1	1	yes	characteristic
23	C₂×#107×C₄₁	—	2	2	no	—
24	C₄₁×#423	C₂²	1	2	yes	—
25	C₄₁×#419	—	2	2	no	—
26	C₂×C₄×C₈×C₄₁	—	2	1	no	—
27	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
28	C₄₁×#335	C₂²	1	4	yes	—
29	C₄₁×#332	C₂²	1	8	yes	—
30	C₂×#41×C₄₁	D₄	1	1	yes	characteristic
31	C₂²×Q₈×C₄₁	—	2	2	no	—
32	#137×C₄₁	—	4	8	no	—
33	C₂×Q₁₆×C₄₁	—	2	4	no	—
34	C₂×SD₁₆×C₄₁	—	2	4	no	—
35	C₂×M₁₆×C₄₁	—	2	2	no	—
36	C₂²×C₈×C₄₁	—	2	2	no	—
37	C₂²×C₈×C₄₁	D₄	1	2	yes	—
38	C₄⋊Q₈×C₄₁	—	2	2	no	—
39	C₄⋊Q₈×C₄₁	—	2	2	no	—
40	C₄⋊Q₈×C₄₁	—	2	2	no	—
41	C₄⋊Q₈×C₄₁	C₂³	1	4	yes	—
42	C₄×Q₈×C₄₁	D₄	1	1	yes	characteristic
43	C₄×Q₈×C₄₁	C₂³	1	1	yes	characteristic
44	C₄×D₄×C₄₁	—	2	1	no	—
45	C₄×D₄×C₄₁	—	2	1	no	—
46	C₄×D₄×C₄₁	D₄	1	1	yes	characteristic
47	C₄×D₄×C₄₁	C₂³	1	1	yes	characteristic
48	#117×C₄₁	—	2	1	no	—
49	#117×C₄₁	D₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
51	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
52	C₂×C₄²×C₄₁	—	2	1	no	—
53	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
54	#107×C₄₁	—	2	4	no	—
55	#107×C₄₁	—	2	4	no	—
56	C₄⋊C₈×C₄₁	—	2	2	no	—
57	C₄⋊C₈×C₄₁	—	2	2	no	—
58	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
59	C₄⋊C₈×C₄₁	C₂³	1	4	yes	—
60	#103×C₄₁	—	2	4	no	—
61	#103×C₄₁	—	2	4	no	—
62	C₂²⋊C₈×C₄₁	—	2	1	no	—
63	C₂²⋊C₈×C₄₁	—	2	2	no	—
64	C₄×C₈×C₄₁	—	2	2	no	—
65	C₄×C₈×C₄₁	—	2	1	no	—
66	C₄×C₈×C₄₁	D₄	1	1	yes	characteristic
67	C₄×C₈×C₄₁	D₄	1	1	yes	characteristic
68	C₄×C₈×C₄₁	C₂³	1	2	yes	—
69	#41×C₄₁	—	2	2	no	—
70	#41×C₄₁	D₈	1	4	yes	—
71	C₂×Q₈×C₄₁	—	2	4	no	—
72	C₂×Q₈×C₄₁	—	4	2	no	—
73	C₂×Q₈×C₄₁	—	4	4	no	—
74	C₂×Q₈×C₄₁	—	4	4	no	—
75	C₂×Q₈×C₄₁	C₂×D₄	1	1	yes	characteristic
76	C₂×D₄×C₄₁	—	2	1	no	—
77	C₂×D₄×C₄₁	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₁	—	2	1	no	—
79	C₂²×C₄×C₄₁	—	2	1	no	—
80	C₂²×C₄×C₄₁	—	2	1	no	—
81	C₂²×C₄×C₄₁	—	4	2	no	—
82	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₁	#41	1	1	yes	characteristic
85	Q₁₆×C₄₁	—	4	8	no	—
86	SD₁₆×C₄₁	—	4	8	no	—
87	M₁₆×C₄₁	—	4	4	no	—
88	C₂×C₈×C₄₁	—	2	4	no	—
89	C₂×C₈×C₄₁	—	2	2	no	—
90	C₂×C₈×C₄₁	—	2	4	no	—
91	C₂×C₈×C₄₁	—	4	4	no	—
92	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
94	C₄⋊C₄×C₄₁	—	2	1	no	—
95	C₄⋊C₄×C₄₁	—	2	1	no	—
96	C₄⋊C₄×C₄₁	—	2	4	no	—
97	C₄⋊C₄×C₄₁	—	2	4	no	—
98	C₄⋊C₄×C₄₁	—	4	2	no	—
99	C₄⋊C₄×C₄₁	—	4	2	no	—
100	C₄⋊C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
101	C₄⋊C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
102	C₂²⋊C₄×C₄₁	—	2	1	no	—
103	C₂²⋊C₄×C₄₁	—	2	1	no	—
104	C₂²⋊C₄×C₄₁	—	2	1	no	—
105	C₄²×C₄₁	—	2	1	no	—
106	C₄²×C₄₁	—	2	1	no	—
107	C₄²×C₄₁	—	2	1	no	—
108	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₄₁	#41	1	1	yes	characteristic
111	C₄²×C₄₁	C₂⁴	1	1	yes	characteristic
112	C₂³×C₄₁	—	2	1	no	—
113	C₂³×C₄₁	#121	1	1	yes	characteristic
114	Q₈×C₄₁	—	2	1	no	—
115	Q₈×C₄₁	—	4	4	no	—
116	Q₈×C₄₁	—	4	4	no	—
117	Q₈×C₄₁	—	8	4	no	—
118	Q₈×C₄₁	C₂×D₈	1	2	yes	—
119	D₄×C₄₁	—	2	4	no	—
120	D₄×C₄₁	—	2	1	no	—
121	D₄×C₄₁	C₂×D₈	1	2	yes	—
122	C₂×C₄×C₄₁	—	2	1	no	—
123	C₂×C₄×C₄₁	—	2	2	no	—
124	C₂×C₄×C₄₁	—	2	4	no	—
125	C₂×C₄×C₄₁	—	2	2	no	—
126	C₂×C₄×C₄₁	—	2	1	no	—
127	C₂×C₄×C₄₁	—	2	1	no	—
128	C₂×C₄×C₄₁	—	4	2	no	—
129	C₂×C₄×C₄₁	—	4	2	no	—
130	C₂×C₄×C₄₁	—	4	4	no	—
131	C₂×C₄×C₄₁	—	8	4	no	—
132	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
133	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
134	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
135	C₂×C₄×C₄₁	C₂×D₈	1	2	yes	—
136	C₂×C₄×C₄₁	#135	1	2	yes	—
137	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
140	C₈×C₄₁	—	2	2	no	—
141	C₈×C₄₁	—	4	4	no	—
142	C₈×C₄₁	—	4	2	no	—
143	C₈×C₄₁	#137	1	4	yes	complemented
144	#19271	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂²×C₄₁	—	2	1	no	—
146	C₂²×C₄₁	—	2	1	no	—
147	C₂²×C₄₁	—	2	2	no	—
148	C₂²×C₄₁	—	4	1	no	—
149	C₂²×C₄₁	—	4	1	no	—
150	C₂²×C₄₁	#466	1	2	yes	—
151	C₂²×C₄₁	C₂×#121	1	1	yes	characteristic
152	C₄×C₄₁	—	2	1	no	—
153	C₄×C₄₁	—	2	2	no	—
154	C₄×C₄₁	—	2	1	no	—
155	C₄×C₄₁	—	4	1	no	—
156	C₄×C₄₁	—	4	1	no	—
157	C₄×C₄₁	—	8	4	no	—
158	C₄×C₄₁	#466	1	2	yes	—
159	C₄×C₄₁	#473	1	2	yes	—
160	C₄×C₄₁	C₂²×D₈	1	1	yes	characteristic
161	C₄×C₄₁	C₂×#135	1	1	yes	characteristic
162	C₄×C₄₁	C₂×#137	1	2	yes	—
163	C₂×#500	C₂×C₄₁	1	1	yes	characteristic, complemented
164	#3039	C₂×C₄₁	1	2	yes	—
165	C₈×#41	C₂×C₄₁	1	1	yes	characteristic
166	#1674	C₂×C₄₁	1	4	yes	complemented
167	#1673	C₂×C₄₁	1	4	yes	complemented
168	#1567	C₂×C₄₁	1	1	yes	characteristic, complemented
169	#1546	C₂×C₄₁	1	2	yes	complemented
170	C₂×C₄₁	—	2	2	no	—
171	C₂×C₄₁	—	4	2	no	—
172	C₂×C₄₁	#3039	1	1	yes	characteristic
173	C₂×C₄₁	C₂×#466	1	1	yes	characteristic
174	C₂×C₄₁	C₂×#473	1	1	yes	characteristic
175	C₂×#137	—	2	2	no	—
176	C₂×C₄⋊Q₈	C₂²×C₄₁	1	2	yes	—
177	C₄×#41	C₂²×C₄₁	1	1	yes	characteristic
178	#500	—	2	1	no	—
179	#500	—	2	1	no	—
180	#500	C₂²×C₄₁	1	4	yes	—
181	#462	—	2	2	no	—
182	#462	C₂²×C₄₁	1	4	yes	—
183	#461	—	2	2	no	—
184	#461	C₂²×C₄₁	1	4	yes	—
185	C₈×Q₈	C₂²×C₄₁	1	2	yes	—
186	C₈×D₄	—	2	2	no	—
187	C₈×D₄	C₂²×C₄₁	1	2	yes	—
188	#431	—	2	1	no	—
189	#431	C₂²×C₄₁	1	1	yes	characteristic
190	C₂×#107	—	2	2	no	—
191	#423	C₂²×C₄₁	1	2	yes	—
192	#419	—	2	2	no	—
193	C₂×C₄×C₈	—	2	1	no	—
194	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
195	#335	C₂²×C₄₁	1	4	yes	—
196	#332	C₂²×C₄₁	1	8	yes	—
197	C₄₁	#19271	1	1	yes	characteristic, direct product factor, Sylow
198	C₂×#41	D₄×C₄₁	1	1	yes	characteristic
199	C₂²×Q₈	—	2	2	no	—
200	#137	—	4	8	no	—
201	C₂×Q₁₆	—	2	4	no	—
202	C₂×SD₁₆	—	2	4	no	—
203	C₂×M₁₆	—	2	2	no	—
204	C₂²×C₈	—	2	2	no	—
205	C₂²×C₈	D₄×C₄₁	1	2	yes	—
206	C₄⋊Q₈	—	2	2	no	—
207	C₄⋊Q₈	—	2	2	no	—
208	C₄⋊Q₈	—	2	2	no	—
209	C₄⋊Q₈	C₂³×C₄₁	1	4	yes	—
210	C₄×Q₈	D₄×C₄₁	1	1	yes	characteristic
211	C₄×Q₈	C₂³×C₄₁	1	1	yes	characteristic
212	C₄×D₄	—	2	1	no	—
213	C₄×D₄	—	2	1	no	—
214	C₄×D₄	D₄×C₄₁	1	1	yes	characteristic
215	C₄×D₄	C₂³×C₄₁	1	1	yes	characteristic
216	#117	—	2	1	no	—
217	#117	D₄×C₄₁	1	1	yes	characteristic
218	C₂×C₄⋊C₄	—	2	2	no	—
219	C₂×C₄⋊C₄	—	2	2	no	—
220	C₂×C₄²	—	2	1	no	—
221	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
222	#107	—	2	4	no	—
223	#107	—	2	4	no	—
224	C₄⋊C₈	—	2	2	no	—
225	C₄⋊C₈	—	2	2	no	—
226	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
227	C₄⋊C₈	C₂³×C₄₁	1	4	yes	—
228	#103	—	2	4	no	—
229	#103	—	2	4	no	—
230	C₂²⋊C₈	—	2	1	no	—
231	C₂²⋊C₈	—	2	2	no	—
232	C₄×C₈	—	2	2	no	—
233	C₄×C₈	—	2	1	no	—
234	C₄×C₈	D₄×C₄₁	1	1	yes	characteristic
235	C₄×C₈	D₄×C₄₁	1	1	yes	characteristic
236	C₄×C₈	C₂³×C₄₁	1	2	yes	—
237	#41	—	2	2	no	—
238	#41	D₈×C₄₁	1	4	yes	—
239	C₂×Q₈	—	2	4	no	—
240	C₂×Q₈	—	4	2	no	—
241	C₂×Q₈	—	4	4	no	—
242	C₂×Q₈	—	4	4	no	—
243	C₂×Q₈	C₂×D₄×C₄₁	1	1	yes	characteristic
244	C₂×D₄	—	2	1	no	—
245	C₂×D₄	C₂×D₄×C₄₁	1	1	yes	characteristic
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	2	1	no	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	4	2	no	—
250	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
251	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
252	C₂²×C₄	#41×C₄₁	1	1	yes	characteristic
253	Q₁₆	—	4	8	no	—
254	SD₁₆	—	4	8	no	—
255	M₁₆	—	4	4	no	—
256	C₂×C₈	—	2	4	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	4	no	—
259	C₂×C₈	—	4	4	no	—
260	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
261	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
269	C₄⋊C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
270	C₂²⋊C₄	—	2	1	no	—
271	C₂²⋊C₄	—	2	1	no	—
272	C₂²⋊C₄	—	2	1	no	—
273	C₄²	—	2	1	no	—
274	C₄²	—	2	1	no	—
275	C₄²	—	2	1	no	—
276	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
277	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
278	C₄²	#41×C₄₁	1	1	yes	characteristic
279	C₄²	C₂⁴×C₄₁	1	1	yes	characteristic
280	C₂³	—	2	1	no	—
281	C₂³	#121×C₄₁	1	1	yes	characteristic
282	Q₈	—	2	1	no	—
283	Q₈	—	4	4	no	—
284	Q₈	—	4	4	no	—
285	Q₈	—	8	4	no	—
286	Q₈	C₂×D₈×C₄₁	1	2	yes	—
287	D₄	—	2	4	no	—
288	D₄	—	2	1	no	—
289	D₄	C₂×D₈×C₄₁	1	2	yes	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	8	4	no	—
300	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
301	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
302	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
303	C₂×C₄	C₂×D₈×C₄₁	1	2	yes	—
304	C₂×C₄	#135×C₄₁	1	2	yes	—
305	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
306	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
307	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
308	C₈	—	2	2	no	—
309	C₈	—	4	4	no	—
310	C₈	—	4	2	no	—
311	C₈	#137×C₄₁	1	4	yes	complemented
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	C₄₁×#466	1	2	yes	—
318	C₂²	C₂×#121×C₄₁	1	1	yes	characteristic
319	C₄	—	2	1	no	—
320	C₄	—	2	2	no	—
321	C₄	—	2	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	8	4	no	—
325	C₄	C₄₁×#466	1	2	yes	—
326	C₄	C₄₁×#473	1	2	yes	—
327	C₄	C₂²×D₈×C₄₁	1	1	yes	characteristic
328	C₄	C₂×#135×C₄₁	1	1	yes	characteristic
329	C₄	C₂×#137×C₄₁	1	2	yes	—
330	C₂	—	2	2	no	—
331	C₂	—	4	2	no	—
332	C₂	C₄₁×#3039	1	1	yes	characteristic
333	C₂	C₂×C₄₁×#466	1	1	yes	characteristic
334	C₂	C₂×C₄₁×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₄₁
prime radical	C₂×#41×C₄₁	D₄
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
1. C₂×C₄₁×#500	C₂	semidirect
4. C₄₁×#1674	C₂	semidirect
5. C₄₁×#1673	C₂	semidirect
6. C₄₁×#1567	C₂	semidirect
7. C₄₁×#1546	C₂	semidirect
143. C₈×C₄₁	#137	semidirect
144. #19271	C₄₁	direct (N × H)
163. C₂×#500	C₂×C₄₁	semidirect
166. #1674	C₂×C₄₁	semidirect
167. #1673	C₂×C₄₁	semidirect
168. #1567	C₂×C₄₁	semidirect
169. #1546	C₂×C₄₁	semidirect
197. C₄₁	#19271	direct (N × H)
311. C₈	#137×C₄₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄₁×#500	2
C₄₁×#3039	2
C₈×#41×C₄₁	2
C₄₁×#1674	2
C₄₁×#1673	2
C₄₁×#1567	2
C₄₁×#1546	2
#19271	41

Subgroups — C41×#19271 #23795472

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#19271 #23795472