Subgroups — C₄₁×#18409 #23794610

Order 10496.

1134 conjugacy classes of subgroups.

The subgroup lattice has 360 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×#3227	C₂	1	2	yes	maximal, complemented
3	C₄₁×#2568	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#1616	C₂	1	8	yes	maximal, complemented
5	C₄₁×#1502	C₂	1	2	yes	maximal, complemented
6	C₄₁×#1461	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×2₋¹⁺⁴×C₄₁	—	2	1	no	—
8	C₄₁×#550	—	2	2	no	—
9	D₄×Q₈×C₄₁	—	2	2	no	—
10	C₄₁×#548	—	2	4	no	—
11	C₄₁×#537	—	2	1	no	—
12	C₄₁×#537	C₂²	1	4	yes	—
13	C₄₁×#532	—	2	2	no	—
14	C₄₁×#532	C₂²	1	2	yes	—
15	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
17	C₄₁×#498	—	2	2	no	—
18	C₄₁×#486	—	2	2	no	—
19	C₄₁×#481	C₂²	1	8	yes	—
20	C₄₁×#476	—	2	4	no	—
21	C₄₁×#431	C₂²	1	2	yes	—
22	C₄₁×#400	C₂²	1	4	yes	—
23	C₄₁×#391	—	2	1	no	—
24	C₄₁×#391	—	2	2	no	—
25	C₄₁×#382	—	2	1	no	—
26	C₄×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
27	C₄₁×#327	C₂²	1	8	yes	—
28	C₄₁×#324	C₂²	1	4	yes	—
29	2₋¹⁺⁴×C₄₁	—	4	4	no	—
30	2₋¹⁺⁴×C₄₁	—	4	4	no	—
31	C₂×#41×C₄₁	—	2	4	no	—
32	C₂×#41×C₄₁	—	2	4	no	—
33	C₂×#41×C₄₁	—	2	2	no	—
34	C₂²×Q₈×C₄₁	—	2	4	no	—
35	C₂²×Q₈×C₄₁	—	2	1	no	—
36	C₂²×Q₈×C₄₁	D₄	1	1	yes	characteristic
37	C₄⋊Q₈×C₄₁	—	2	2	no	—
38	C₄⋊Q₈×C₄₁	—	2	2	no	—
39	C₄⋊Q₈×C₄₁	—	2	2	no	—
40	C₄⋊Q₈×C₄₁	D₄	1	2	yes	—
41	#127×C₄₁	—	2	2	no	—
42	#127×C₄₁	D₄	1	2	yes	—
43	#124×C₄₁	—	2	4	no	—
44	#124×C₄₁	—	2	2	no	—
45	C₂²⋊Q₈×C₄₁	—	2	4	no	—
46	C₂²⋊Q₈×C₄₁	—	2	8	no	—
47	#121×C₄₁	—	2	4	no	—
48	#121×C₄₁	—	2	8	no	—
49	#121×C₄₁	—	2	2	no	—
50	#121×C₄₁	—	2	4	no	—
51	#121×C₄₁	C₂³	1	4	yes	—
52	C₄×Q₈×C₄₁	—	2	2	no	—
53	C₄×Q₈×C₄₁	—	2	2	no	—
54	C₄×D₄×C₄₁	—	2	4	no	—
55	C₄×D₄×C₄₁	—	2	4	no	—
56	C₄×D₄×C₄₁	—	2	2	no	—
57	C₄×D₄×C₄₁	D₄	1	4	yes	—
58	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
59	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
60	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
63	C₂×C₄²×C₄₁	—	2	1	no	—
64	C₂×C₄²×C₄₁	D₄	1	2	yes	—
65	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
66	#106×C₄₁	—	2	8	no	—
67	C₄⋊C₈×C₄₁	—	2	2	no	—
68	#102×C₄₁	—	2	8	no	—
69	C₂²⋊C₈×C₄₁	C₂³	1	4	yes	—
70	C₄×C₈×C₄₁	—	2	2	no	—
71	#95×C₄₁	—	2	4	no	—
72	#95×C₄₁	—	2	4	no	—
73	#95×C₄₁	C₂³	1	2	yes	—
74	#41×C₄₁	—	4	8	no	—
75	#41×C₄₁	—	4	16	no	—
76	#41×C₄₁	—	4	8	no	—
77	#41×C₄₁	—	4	4	no	—
78	#41×C₄₁	—	4	4	no	—
79	C₂×Q₈×C₄₁	—	2	4	no	—
80	C₂×Q₈×C₄₁	—	2	2	no	—
81	C₂×Q₈×C₄₁	—	2	2	no	—
82	C₂×Q₈×C₄₁	—	2	2	no	—
83	C₂×Q₈×C₄₁	—	4	4	no	—
84	C₂×Q₈×C₄₁	—	4	16	no	—
85	C₂×Q₈×C₄₁	—	4	1	no	—
86	C₂×Q₈×C₄₁	—	4	1	no	—
87	C₂×Q₈×C₄₁	SD₁₆	1	4	yes	—
88	C₂×D₄×C₄₁	—	2	4	no	—
89	C₂×D₄×C₄₁	—	2	4	no	—
90	C₂×D₄×C₄₁	—	2	4	no	—
91	C₂×D₄×C₄₁	—	2	2	no	—
92	C₂×D₄×C₄₁	—	2	2	no	—
93	C₂×D₄×C₄₁	C₂×D₄	1	4	yes	—
94	C₂²×C₄×C₄₁	—	2	8	no	—
95	C₂²×C₄×C₄₁	—	2	4	no	—
96	C₂²×C₄×C₄₁	—	2	2	no	—
97	C₂²×C₄×C₄₁	—	2	2	no	—
98	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
100	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
101	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
103	C₂×C₈×C₄₁	—	2	4	no	—
104	C₄⋊C₄×C₄₁	—	2	4	no	—
105	C₄⋊C₄×C₄₁	—	2	2	no	—
106	C₄⋊C₄×C₄₁	—	2	2	no	—
107	C₄⋊C₄×C₄₁	—	2	4	no	—
108	C₄⋊C₄×C₄₁	—	2	4	no	—
109	C₄⋊C₄×C₄₁	—	4	2	no	—
110	C₄⋊C₄×C₄₁	—	4	2	no	—
111	C₄⋊C₄×C₄₁	C₂×D₄	1	4	yes	—
112	C₂²⋊C₄×C₄₁	—	2	8	no	—
113	C₂²⋊C₄×C₄₁	—	2	4	no	—
114	C₂²⋊C₄×C₄₁	—	2	4	no	—
115	C₄²×C₄₁	—	2	2	no	—
116	C₄²×C₄₁	—	2	2	no	—
117	C₄²×C₄₁	—	4	2	no	—
118	C₄²×C₄₁	C₂×D₄	1	2	yes	—
119	C₄²×C₄₁	C₂×D₄	1	2	yes	—
120	C₂³×C₄₁	—	2	4	no	—
121	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
122	Q₈×C₄₁	—	2	8	no	—
123	Q₈×C₄₁	—	4	8	no	—
124	Q₈×C₄₁	—	4	4	no	—
125	Q₈×C₄₁	—	8	2	no	—
126	Q₈×C₄₁	—	8	2	no	—
127	D₄×C₄₁	—	4	4	no	—
128	D₄×C₄₁	—	4	8	no	—
129	D₄×C₄₁	—	4	4	no	—
130	D₄×C₄₁	—	4	4	no	—
131	D₄×C₄₁	—	4	4	no	—
132	D₄×C₄₁	—	4	4	no	—
133	D₄×C₄₁	—	8	4	no	—
134	C₂×C₄×C₄₁	—	2	2	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	—	2	4	no	—
139	C₂×C₄×C₄₁	—	4	1	no	—
140	C₂×C₄×C₄₁	—	4	16	no	—
141	C₂×C₄×C₄₁	—	4	8	no	—
142	C₂×C₄×C₄₁	—	4	2	no	—
143	C₂×C₄×C₄₁	—	4	2	no	—
144	C₂×C₄×C₄₁	—	4	2	no	—
145	C₂×C₄×C₄₁	—	4	4	no	—
146	C₂×C₄×C₄₁	—	4	2	no	—
147	C₂×C₄×C₄₁	—	4	2	no	—
148	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
149	C₂×C₄×C₄₁	#120	1	2	yes	—
150	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
151	C₂×C₄×C₄₁	C₂×SD₁₆	1	4	yes	—
152	C₂×C₄×C₄₁	C₂×SD₁₆	1	2	yes	—
153	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
154	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
155	C₈×C₄₁	—	4	4	no	—
156	#18409	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
157	C₂²×C₄₁	—	2	1	no	—
158	C₂²×C₄₁	—	2	1	no	—
159	C₂²×C₄₁	—	4	4	no	—
160	C₂²×C₄₁	—	4	4	no	—
161	C₂²×C₄₁	—	4	4	no	—
162	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
163	C₂²×C₄₁	C₂²×SD₁₆	1	1	yes	characteristic
164	C₂²×C₄₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
165	C₄×C₄₁	—	2	4	no	—
166	C₄×C₄₁	—	2	1	no	—
167	C₄×C₄₁	—	2	2	no	—
168	C₄×C₄₁	—	4	4	no	—
169	C₄×C₄₁	—	4	2	no	—
170	C₄×C₄₁	—	8	2	no	—
171	C₄×C₄₁	—	8	2	no	—
172	C₄×C₄₁	#450	1	4	yes	—
173	C₄×C₄₁	#454	1	2	yes	—
174	#3497	C₂×C₄₁	1	1	yes	characteristic
175	#3227	C₂×C₄₁	1	2	yes	complemented
176	#2568	C₂×C₄₁	1	1	yes	characteristic, complemented
177	#1616	C₂×C₄₁	1	8	yes	complemented
178	#1502	C₂×C₄₁	1	2	yes	complemented
179	#1461	C₂×C₄₁	1	1	yes	characteristic, complemented
180	C₂×C₄₁	—	2	1	no	—
181	C₂×C₄₁	—	2	1	no	—
182	C₂×C₄₁	—	8	4	no	—
183	C₂×C₄₁	C₂×#450	1	1	yes	characteristic
184	C₂×C₄₁	C₂×#454	1	1	yes	characteristic
185	C₂×C₄₁	#3028	1	1	yes	characteristic
186	C₂×2₋¹⁺⁴	—	2	1	no	—
187	#550	—	2	2	no	—
188	D₄×Q₈	—	2	2	no	—
189	#548	—	2	4	no	—
190	#537	—	2	1	no	—
191	#537	C₂²×C₄₁	1	4	yes	—
192	#532	—	2	2	no	—
193	#532	C₂²×C₄₁	1	2	yes	—
194	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
195	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
196	#498	—	2	2	no	—
197	#486	—	2	2	no	—
198	#481	C₂²×C₄₁	1	8	yes	—
199	#476	—	2	4	no	—
200	#431	C₂²×C₄₁	1	2	yes	—
201	#400	C₂²×C₄₁	1	4	yes	—
202	#391	—	2	1	no	—
203	#391	—	2	2	no	—
204	#382	—	2	1	no	—
205	C₄×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
206	#327	C₂²×C₄₁	1	8	yes	—
207	#324	C₂²×C₄₁	1	4	yes	—
208	C₄₁	#18409	1	1	yes	characteristic, direct product factor, Sylow
209	2₋¹⁺⁴	—	4	4	no	—
210	2₋¹⁺⁴	—	4	4	no	—
211	C₂×#41	—	2	4	no	—
212	C₂×#41	—	2	4	no	—
213	C₂×#41	—	2	2	no	—
214	C₂²×Q₈	—	2	4	no	—
215	C₂²×Q₈	—	2	1	no	—
216	C₂²×Q₈	D₄×C₄₁	1	1	yes	characteristic
217	C₄⋊Q₈	—	2	2	no	—
218	C₄⋊Q₈	—	2	2	no	—
219	C₄⋊Q₈	—	2	2	no	—
220	C₄⋊Q₈	D₄×C₄₁	1	2	yes	—
221	#127	—	2	2	no	—
222	#127	D₄×C₄₁	1	2	yes	—
223	#124	—	2	4	no	—
224	#124	—	2	2	no	—
225	C₂²⋊Q₈	—	2	4	no	—
226	C₂²⋊Q₈	—	2	8	no	—
227	#121	—	2	4	no	—
228	#121	—	2	8	no	—
229	#121	—	2	2	no	—
230	#121	—	2	4	no	—
231	#121	C₂³×C₄₁	1	4	yes	—
232	C₄×Q₈	—	2	2	no	—
233	C₄×Q₈	—	2	2	no	—
234	C₄×D₄	—	2	4	no	—
235	C₄×D₄	—	2	4	no	—
236	C₄×D₄	—	2	2	no	—
237	C₄×D₄	D₄×C₄₁	1	4	yes	—
238	C₂×C₄⋊C₄	—	2	1	no	—
239	C₂×C₄⋊C₄	—	2	1	no	—
240	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
241	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
242	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
243	C₂×C₄²	—	2	1	no	—
244	C₂×C₄²	D₄×C₄₁	1	2	yes	—
245	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
246	#106	—	2	8	no	—
247	C₄⋊C₈	—	2	2	no	—
248	#102	—	2	8	no	—
249	C₂²⋊C₈	C₂³×C₄₁	1	4	yes	—
250	C₄×C₈	—	2	2	no	—
251	#95	—	2	4	no	—
252	#95	—	2	4	no	—
253	#95	C₂³×C₄₁	1	2	yes	—
254	#41	—	4	8	no	—
255	#41	—	4	16	no	—
256	#41	—	4	8	no	—
257	#41	—	4	4	no	—
258	#41	—	4	4	no	—
259	C₂×Q₈	—	2	4	no	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	4	4	no	—
264	C₂×Q₈	—	4	16	no	—
265	C₂×Q₈	—	4	1	no	—
266	C₂×Q₈	—	4	1	no	—
267	C₂×Q₈	SD₁₆×C₄₁	1	4	yes	—
268	C₂×D₄	—	2	4	no	—
269	C₂×D₄	—	2	4	no	—
270	C₂×D₄	—	2	4	no	—
271	C₂×D₄	—	2	2	no	—
272	C₂×D₄	—	2	2	no	—
273	C₂×D₄	C₂×D₄×C₄₁	1	4	yes	—
274	C₂²×C₄	—	2	8	no	—
275	C₂²×C₄	—	2	4	no	—
276	C₂²×C₄	—	2	2	no	—
277	C₂²×C₄	—	2	2	no	—
278	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
279	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
280	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
281	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
282	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
283	C₂×C₈	—	2	4	no	—
284	C₄⋊C₄	—	2	4	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	2	4	no	—
288	C₄⋊C₄	—	2	4	no	—
289	C₄⋊C₄	—	4	2	no	—
290	C₄⋊C₄	—	4	2	no	—
291	C₄⋊C₄	C₂×D₄×C₄₁	1	4	yes	—
292	C₂²⋊C₄	—	2	8	no	—
293	C₂²⋊C₄	—	2	4	no	—
294	C₂²⋊C₄	—	2	4	no	—
295	C₄²	—	2	2	no	—
296	C₄²	—	2	2	no	—
297	C₄²	—	4	2	no	—
298	C₄²	C₂×D₄×C₄₁	1	2	yes	—
299	C₄²	C₂×D₄×C₄₁	1	2	yes	—
300	C₂³	—	2	4	no	—
301	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
302	Q₈	—	2	8	no	—
303	Q₈	—	4	8	no	—
304	Q₈	—	4	4	no	—
305	Q₈	—	8	2	no	—
306	Q₈	—	8	2	no	—
307	D₄	—	4	4	no	—
308	D₄	—	4	8	no	—
309	D₄	—	4	4	no	—
310	D₄	—	4	4	no	—
311	D₄	—	4	4	no	—
312	D₄	—	4	4	no	—
313	D₄	—	8	4	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	4	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	16	no	—
321	C₂×C₄	—	4	8	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	4	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
329	C₂×C₄	#120×C₄₁	1	2	yes	—
330	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
331	C₂×C₄	C₂×SD₁₆×C₄₁	1	4	yes	—
332	C₂×C₄	C₂×SD₁₆×C₄₁	1	2	yes	—
333	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
334	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
335	C₈	—	4	4	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	4	4	no	—
339	C₂²	—	4	4	no	—
340	C₂²	—	4	4	no	—
341	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
342	C₂²	C₂²×SD₁₆×C₄₁	1	1	yes	characteristic
343	C₂²	C₂×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
344	C₄	—	2	4	no	—
345	C₄	—	2	1	no	—
346	C₄	—	2	2	no	—
347	C₄	—	4	4	no	—
348	C₄	—	4	2	no	—
349	C₄	—	8	2	no	—
350	C₄	—	8	2	no	—
351	C₄	C₄₁×#450	1	4	yes	—
352	C₄	C₄₁×#454	1	2	yes	—
353	C₂	—	2	1	no	—
354	C₂	—	2	1	no	—
355	C₂	—	8	4	no	—
356	C₂	C₂×C₄₁×#450	1	1	yes	characteristic
357	C₂	C₂×C₄₁×#454	1	1	yes	characteristic
358	C₂	C₄₁×#3028	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 ▷ 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. C₄₁×#3227	C₂	semidirect
3. C₄₁×#2568	C₂	semidirect
4. C₄₁×#1616	C₂	semidirect
5. C₄₁×#1502	C₂	semidirect
6. C₄₁×#1461	C₂	semidirect
156. #18409	C₄₁	direct (N × H)
175. #3227	C₂×C₄₁	semidirect
176. #2568	C₂×C₄₁	semidirect
177. #1616	C₂×C₄₁	semidirect
178. #1502	C₂×C₄₁	semidirect
179. #1461	C₂×C₄₁	semidirect
208. C₄₁	#18409	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×#3227	2
C₄₁×#2568	2
C₄₁×#1616	2
C₄₁×#1502	2
C₄₁×#1461	2
#18409	41

Subgroups — C41×#18409 #23794610

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#18409 #23794610