Subgroups — C₄₁×#13004 #23789205

Order 10496.

560 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#466	C₂	1	1	yes	maximal, characteristic
2	C₂²×#108×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄₁×#361	C₂	1	2	yes	maximal, complemented
4	C₄₁×#2112	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#1387	C₂	1	2	yes	maximal, complemented
6	C₂²×D₈×C₄₁	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₄₁	—	2	1	no	—
8	C₂³×C₈×C₄₁	C₂²	1	1	yes	characteristic
9	C₂×#121×C₄₁	—	2	1	no	—
10	C₂×M₃₂×C₄₁	C₂²	1	2	yes	—
11	C₄₁×#466	—	2	1	no	—
12	C₄₁×#466	—	2	1	no	—
13	C₄₁×#466	C₄	1	2	yes	—
14	C₄₁×#466	C₄	1	2	yes	—
15	C₂×#108×C₄₁	—	2	1	no	—
16	C₂×#108×C₄₁	—	2	4	no	—
17	C₂×#108×C₄₁	C₂²	1	2	yes	—
18	C₂×#107×C₄₁	C₂²	1	1	yes	characteristic
19	C₂×#102×C₄₁	—	2	1	no	—
20	C₄₁×#361	—	2	4	no	—
21	C₄₁×C₂²⋊C₁₆	—	2	1	no	—
22	C₄₁×C₂²⋊C₁₆	—	2	1	no	—
23	C₂²×D₄×C₄₁	—	2	1	no	—
24	C₂²×D₄×C₄₁	—	4	1	no	—
25	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
26	C₂×D₈×C₄₁	—	2	2	no	—
27	C₂×D₈×C₄₁	—	2	1	no	—
28	C₂×D₈×C₄₁	—	2	2	no	—
29	C₂×D₈×C₄₁	C₂×C₄	1	2	yes	—
30	C₂×M₁₆×C₄₁	—	2	4	no	—
31	C₂×M₁₆×C₄₁	—	2	2	no	—
32	C₂×M₁₆×C₄₁	—	4	2	no	—
33	C₂×M₁₆×C₄₁	—	4	1	no	—
34	C₂²×C₈×C₄₁	—	2	2	no	—
35	C₂²×C₈×C₄₁	—	2	2	no	—
36	C₂²×C₈×C₄₁	—	4	1	no	—
37	C₂²×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₄₁	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₄₁	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₄₁	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
43	#121×C₄₁	—	2	2	no	—
44	#121×C₄₁	—	2	2	no	—
45	#121×C₄₁	—	4	1	no	—
46	#121×C₄₁	—	4	1	no	—
47	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
48	C₂×C₂²⋊C₄×C₄₁	—	4	1	no	—
49	M₃₂×C₄₁	—	2	2	no	—
50	M₃₂×C₄₁	—	2	2	no	—
51	C₂×C₁₆×C₄₁	—	2	2	no	—
52	#108×C₄₁	—	2	4	no	—
53	#108×C₄₁	—	2	2	no	—
54	#108×C₄₁	D₄	1	4	yes	complemented
55	#107×C₄₁	—	2	1	no	—
56	#107×C₄₁	C₂×C₄	1	2	yes	—
57	#102×C₄₁	—	2	2	no	—
58	#102×C₄₁	—	2	2	no	—
59	C₂⁴×C₄₁	—	4	1	no	—
60	C₂⁴×C₄₁	SD₁₆	1	1	yes	characteristic
61	C₂×D₄×C₄₁	—	2	2	no	—
62	C₂×D₄×C₄₁	—	4	2	no	—
63	C₂×D₄×C₄₁	—	4	2	no	—
64	C₂×D₄×C₄₁	—	4	1	no	—
65	C₂×D₄×C₄₁	—	4	1	no	—
66	C₂×D₄×C₄₁	—	4	1	no	—
67	C₂×D₄×C₄₁	—	4	1	no	—
68	C₂×D₄×C₄₁	—	4	1	no	—
69	C₂×D₄×C₄₁	—	8	4	no	—
70	C₂²×C₄×C₄₁	—	2	1	no	—
71	C₂²×C₄×C₄₁	—	2	1	no	—
72	C₂²×C₄×C₄₁	—	2	1	no	—
73	C₂²×C₄×C₄₁	—	2	1	no	—
74	C₂²×C₄×C₄₁	—	4	1	no	—
75	C₂²×C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₁	D₈	1	1	yes	characteristic
78	C₂²×C₄×C₄₁	D₈	1	1	yes	characteristic
79	C₂²×C₄×C₄₁	SD₁₆	1	1	yes	characteristic
80	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
81	D₈×C₄₁	—	2	4	no	—
82	D₈×C₄₁	—	4	2	no	—
83	M₁₆×C₄₁	—	2	4	no	—
84	M₁₆×C₄₁	—	4	4	no	—
85	M₁₆×C₄₁	—	4	2	no	—
86	C₂×C₈×C₄₁	—	2	1	no	—
87	C₂×C₈×C₄₁	—	2	1	no	—
88	C₂×C₈×C₄₁	—	2	2	no	—
89	C₂×C₈×C₄₁	—	2	2	no	—
90	C₂×C₈×C₄₁	—	2	2	no	—
91	C₂×C₈×C₄₁	—	2	2	no	—
92	C₂×C₈×C₄₁	—	4	4	no	—
93	C₂×C₈×C₄₁	—	4	2	no	—
94	C₂×C₈×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₈×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₄₁	#41	1	2	yes	—
100	C₄⋊C₄×C₄₁	—	2	2	no	—
101	C₄⋊C₄×C₄₁	—	4	1	no	—
102	C₂²⋊C₄×C₄₁	—	4	2	no	—
103	C₂²⋊C₄×C₄₁	—	4	2	no	—
104	C₁₆×C₄₁	—	4	2	no	—
105	C₂³×C₄₁	—	2	2	no	—
106	C₂³×C₄₁	—	2	1	no	—
107	C₂³×C₄₁	—	2	1	no	—
108	C₂³×C₄₁	—	4	1	no	—
109	C₂³×C₄₁	—	4	2	no	—
110	C₂³×C₄₁	—	8	2	no	—
111	C₂³×C₄₁	—	8	2	no	—
112	C₂³×C₄₁	—	8	1	no	—
113	C₂³×C₄₁	—	8	1	no	—
114	C₂³×C₄₁	#102	1	1	yes	characteristic
115	C₂³×C₄₁	#102	1	1	yes	characteristic
116	C₂³×C₄₁	C₂×SD₁₆	1	1	yes	characteristic
117	D₄×C₄₁	—	4	2	no	—
118	D₄×C₄₁	—	4	2	no	—
119	D₄×C₄₁	—	8	4	no	—
120	D₄×C₄₁	—	8	4	no	—
121	D₄×C₄₁	—	8	1	no	—
122	D₄×C₄₁	—	8	1	no	—
123	C₂×C₄×C₄₁	—	2	1	no	—
124	C₂×C₄×C₄₁	—	2	1	no	—
125	C₂×C₄×C₄₁	—	2	1	no	—
126	C₂×C₄×C₄₁	—	2	1	no	—
127	C₂×C₄×C₄₁	—	2	1	no	—
128	C₂×C₄×C₄₁	—	2	1	no	—
129	C₂×C₄×C₄₁	—	2	1	no	—
130	C₂×C₄×C₄₁	—	2	1	no	—
131	C₂×C₄×C₄₁	—	2	1	no	—
132	C₂×C₄×C₄₁	—	2	1	no	—
133	C₂×C₄×C₄₁	—	4	2	no	—
134	C₂×C₄×C₄₁	—	8	2	no	—
135	C₂×C₄×C₄₁	#102	1	1	yes	characteristic
136	C₂×C₄×C₄₁	#102	1	1	yes	characteristic
137	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
139	C₂×C₄×C₄₁	#123	1	1	yes	characteristic
140	C₂×C₄×C₄₁	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₂×C₄×C₄₁	#137	1	1	yes	characteristic
143	C₈×C₄₁	—	2	1	no	—
144	C₈×C₄₁	—	2	1	no	—
145	C₈×C₄₁	—	4	4	no	—
146	C₈×C₄₁	C₄×D₄	1	2	yes	—
147	C₈×C₄₁	#121	1	2	yes	—
148	#13004	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂²×C₄₁	—	2	1	no	—
150	C₂²×C₄₁	—	2	1	no	—
151	C₂²×C₄₁	—	2	1	no	—
152	C₂²×C₄₁	—	2	1	no	—
153	C₂²×C₄₁	—	2	1	no	—
154	C₂²×C₄₁	—	2	1	no	—
155	C₂²×C₄₁	—	2	1	no	—
156	C₂²×C₄₁	—	4	2	no	—
157	C₂²×C₄₁	—	4	2	no	—
158	C₂²×C₄₁	—	8	2	no	—
159	C₂²×C₄₁	—	8	2	no	—
160	C₂²×C₄₁	—	8	2	no	—
161	C₂²×C₄₁	—	16	2	no	—
162	C₂²×C₄₁	—	16	2	no	—
163	C₂²×C₄₁	#361	1	2	yes	complemented
164	C₂²×C₄₁	C₂×#102	1	1	yes	characteristic
165	C₂²×C₄₁	#448	1	1	yes	characteristic
166	C₂²×C₄₁	#481	1	1	yes	characteristic
167	C₄×C₄₁	—	2	1	no	—
168	C₄×C₄₁	—	2	1	no	—
169	C₄×C₄₁	—	8	2	no	—
170	C₄×C₄₁	#386	1	1	yes	characteristic
171	C₄×C₄₁	#417	1	1	yes	characteristic
172	C₄×C₄₁	#447	1	1	yes	characteristic
173	C₄×C₄₁	#480	1	1	yes	characteristic
174	C₂×#466	C₂×C₄₁	1	1	yes	characteristic
175	C₂²×#108	C₂×C₄₁	1	1	yes	characteristic, complemented
176	C₂×#361	C₂×C₄₁	1	2	yes	complemented
177	#2112	C₂×C₄₁	1	1	yes	characteristic, complemented
178	#1387	C₂×C₄₁	1	2	yes	complemented
179	C₂×C₄₁	—	2	1	no	—
180	C₂×C₄₁	—	2	1	no	—
181	C₂×C₄₁	—	2	2	no	—
182	C₂×C₄₁	—	4	1	no	—
183	C₂×C₄₁	—	16	2	no	—
184	C₂×C₄₁	#1874	1	1	yes	characteristic
185	C₂×C₄₁	C₂×#361	1	1	yes	characteristic
186	C₂×C₄₁	#2148	1	1	yes	characteristic
187	C₂²×D₈	C₂²×C₄₁	1	1	yes	characteristic
188	C₂²×M₁₆	—	2	1	no	—
189	C₂³×C₈	C₂²×C₄₁	1	1	yes	characteristic
190	C₂×#121	—	2	1	no	—
191	C₂×M₃₂	C₂²×C₄₁	1	2	yes	—
192	#466	—	2	1	no	—
193	#466	—	2	1	no	—
194	#466	C₄×C₄₁	1	2	yes	—
195	#466	C₄×C₄₁	1	2	yes	—
196	C₂×#108	—	2	1	no	—
197	C₂×#108	—	2	4	no	—
198	C₂×#108	C₂²×C₄₁	1	2	yes	—
199	C₂×#107	C₂²×C₄₁	1	1	yes	characteristic
200	C₂×#102	—	2	1	no	—
201	#361	—	2	4	no	—
202	C₂²⋊C₁₆	—	2	1	no	—
203	C₂²⋊C₁₆	—	2	1	no	—
204	C₄₁	#13004	1	1	yes	characteristic, direct product factor, Sylow
205	C₂²×D₄	—	2	1	no	—
206	C₂²×D₄	—	4	1	no	—
207	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
208	C₂×D₈	—	2	2	no	—
209	C₂×D₈	—	2	1	no	—
210	C₂×D₈	—	2	2	no	—
211	C₂×D₈	C₂×C₄×C₄₁	1	2	yes	—
212	C₂×M₁₆	—	2	4	no	—
213	C₂×M₁₆	—	2	2	no	—
214	C₂×M₁₆	—	4	2	no	—
215	C₂×M₁₆	—	4	1	no	—
216	C₂²×C₈	—	2	2	no	—
217	C₂²×C₈	—	2	2	no	—
218	C₂²×C₈	—	4	1	no	—
219	C₂²×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
220	C₂²×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
221	C₂²×C₈	D₄×C₄₁	1	1	yes	characteristic
222	C₂²×C₈	D₄×C₄₁	1	1	yes	characteristic
223	C₂²×C₈	D₄×C₄₁	1	1	yes	characteristic
224	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
225	#121	—	2	2	no	—
226	#121	—	2	2	no	—
227	#121	—	4	1	no	—
228	#121	—	4	1	no	—
229	C₂×C₄⋊C₄	—	2	1	no	—
230	C₂×C₂²⋊C₄	—	4	1	no	—
231	M₃₂	—	2	2	no	—
232	M₃₂	—	2	2	no	—
233	C₂×C₁₆	—	2	2	no	—
234	#108	—	2	4	no	—
235	#108	—	2	2	no	—
236	#108	D₄×C₄₁	1	4	yes	complemented
237	#107	—	2	1	no	—
238	#107	C₂×C₄×C₄₁	1	2	yes	—
239	#102	—	2	2	no	—
240	#102	—	2	2	no	—
241	C₂⁴	—	4	1	no	—
242	C₂⁴	SD₁₆×C₄₁	1	1	yes	characteristic
243	C₂×D₄	—	2	2	no	—
244	C₂×D₄	—	4	2	no	—
245	C₂×D₄	—	4	2	no	—
246	C₂×D₄	—	4	1	no	—
247	C₂×D₄	—	4	1	no	—
248	C₂×D₄	—	4	1	no	—
249	C₂×D₄	—	4	1	no	—
250	C₂×D₄	—	4	1	no	—
251	C₂×D₄	—	8	4	no	—
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	2	1	no	—
256	C₂²×C₄	—	4	1	no	—
257	C₂²×C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
258	C₂²×C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
259	C₂²×C₄	D₈×C₄₁	1	1	yes	characteristic
260	C₂²×C₄	D₈×C₄₁	1	1	yes	characteristic
261	C₂²×C₄	SD₁₆×C₄₁	1	1	yes	characteristic
262	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
263	D₈	—	2	4	no	—
264	D₈	—	4	2	no	—
265	M₁₆	—	2	4	no	—
266	M₁₆	—	4	4	no	—
267	M₁₆	—	4	2	no	—
268	C₂×C₈	—	2	1	no	—
269	C₂×C₈	—	2	1	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	4	4	no	—
275	C₂×C₈	—	4	2	no	—
276	C₂×C₈	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
277	C₂×C₈	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
278	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
279	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
280	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
281	C₂×C₈	#41×C₄₁	1	2	yes	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₂²⋊C₄	—	4	2	no	—
285	C₂²⋊C₄	—	4	2	no	—
286	C₁₆	—	4	2	no	—
287	C₂³	—	2	2	no	—
288	C₂³	—	2	1	no	—
289	C₂³	—	2	1	no	—
290	C₂³	—	4	1	no	—
291	C₂³	—	4	2	no	—
292	C₂³	—	8	2	no	—
293	C₂³	—	8	2	no	—
294	C₂³	—	8	1	no	—
295	C₂³	—	8	1	no	—
296	C₂³	#102×C₄₁	1	1	yes	characteristic
297	C₂³	#102×C₄₁	1	1	yes	characteristic
298	C₂³	C₂×SD₁₆×C₄₁	1	1	yes	characteristic
299	D₄	—	4	2	no	—
300	D₄	—	4	2	no	—
301	D₄	—	8	4	no	—
302	D₄	—	8	4	no	—
303	D₄	—	8	1	no	—
304	D₄	—	8	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	8	2	no	—
317	C₂×C₄	#102×C₄₁	1	1	yes	characteristic
318	C₂×C₄	#102×C₄₁	1	1	yes	characteristic
319	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
320	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
321	C₂×C₄	#123×C₄₁	1	1	yes	characteristic
322	C₂×C₄	C₂×D₈×C₄₁	1	1	yes	characteristic
323	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
324	C₂×C₄	#137×C₄₁	1	1	yes	characteristic
325	C₈	—	2	1	no	—
326	C₈	—	2	1	no	—
327	C₈	—	4	4	no	—
328	C₈	C₄×D₄×C₄₁	1	2	yes	—
329	C₈	#121×C₄₁	1	2	yes	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	4	2	no	—
338	C₂²	—	4	2	no	—
339	C₂²	—	8	2	no	—
340	C₂²	—	8	2	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	16	2	no	—
343	C₂²	—	16	2	no	—
344	C₂²	C₄₁×#361	1	2	yes	complemented
345	C₂²	C₂×#102×C₄₁	1	1	yes	characteristic
346	C₂²	C₄₁×#448	1	1	yes	characteristic
347	C₂²	C₄₁×#481	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	8	2	no	—
351	C₄	C₄₁×#386	1	1	yes	characteristic
352	C₄	C₄₁×#417	1	1	yes	characteristic
353	C₄	C₄₁×#447	1	1	yes	characteristic
354	C₄	C₄₁×#480	1	1	yes	characteristic
355	C₂	—	2	1	no	—
356	C₂	—	2	1	no	—
357	C₂	—	2	2	no	—
358	C₂	—	4	1	no	—
359	C₂	—	16	2	no	—
360	C₂	C₄₁×#1874	1	1	yes	characteristic
361	C₂	C₂×C₄₁×#361	1	1	yes	characteristic
362	C₂	C₄₁×#2148	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₄₁
Frattini, cosocle	C₂²×C₈	C₂³×C₄₁
prime radical	C₂×C₄₁×#466	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₂³×C₄×C₄₁ ◁ C₂³×C₈×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×#108×C₄₁	C₂	semidirect
3. C₂×C₄₁×#361	C₂	semidirect
4. C₄₁×#2112	C₂	semidirect
5. C₄₁×#1387	C₂	semidirect
54. #108×C₄₁	D₄	semidirect
148. #13004	C₄₁	direct (N × H)
163. C₂²×C₄₁	#361	semidirect
175. C₂²×#108	C₂×C₄₁	semidirect
176. C₂×#361	C₂×C₄₁	semidirect
177. #2112	C₂×C₄₁	semidirect
178. #1387	C₂×C₄₁	semidirect
204. C₄₁	#13004	direct (N × H)
236. #108	D₄×C₄₁	semidirect
344. C₂²	C₄₁×#361	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#466	2
C₂²×#108×C₄₁	2
C₂×C₄₁×#361	2
C₄₁×#2112	2
C₄₁×#1387	2
#13004	41

Subgroups — C41×#13004 #23789205

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#13004 #23789205