Subgroups — C₄₁×#33095 #23809296

Order 10496.

920 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#2854	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₄⋊C₄×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#1926	C₂	1	4	yes	maximal, complemented
4	C₄₁×#1919	C₂	1	4	yes	maximal, complemented
5	C₄₁×#1874	C₂	1	2	yes	maximal
6	C₄₁×#1853	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1817	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#123×C₄₁	—	2	1	no	—
9	C₂×#121×C₄₁	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
11	C₂²×C₄²×C₄₁	—	2	1	no	—
12	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
13	C₂×#106×C₄₁	C₂²	1	4	yes	—
14	C₂×#102×C₄₁	C₂²	1	4	yes	—
15	C₂×C₂²⋊C₈×C₄₁	C₂²	1	2	yes	—
16	C₄₁×#390	C₂²	1	2	yes	—
17	C₄₁×#389	C₂²	1	2	yes	—
18	C₄₁×#388	—	2	2	no	—
19	C₄₁×#386	—	2	1	no	—
20	C₄₁×#380	C₂²	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₄₁	—	2	2	no	—
22	C₄×C₄⋊C₄×C₄₁	C₂²	1	4	yes	—
23	C₂×#95×C₄₁	—	2	1	no	—
24	C₄₁×#340	C₂²	1	4	yes	—
25	C₄₁×#340	C₂²	1	4	yes	—
26	C₄₁×#336	C₂²	1	4	yes	—
27	C₂²×D₄×C₄₁	—	2	1	no	—
28	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
29	C₂³×C₄×C₄₁	—	2	2	no	—
30	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
31	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
32	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
34	#123×C₄₁	—	4	4	no	—
35	#121×C₄₁	—	2	4	no	—
36	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
37	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
38	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
39	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
41	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
42	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
43	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
44	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
45	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
46	C₂×C₄²×C₄₁	—	2	4	no	—
47	C₂×C₄²×C₄₁	—	2	4	no	—
48	C₂×C₄²×C₄₁	—	4	4	no	—
49	C₂×C₄²×C₄₁	D₄	1	2	yes	—
50	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
51	#106×C₄₁	—	2	8	no	—
52	#102×C₄₁	—	2	8	no	—
53	C₂²⋊C₈×C₄₁	—	2	4	no	—
54	#95×C₄₁	—	2	4	no	—
55	#95×C₄₁	—	2	1	no	—
56	#95×C₄₁	—	2	2	no	—
57	C₂⁴×C₄₁	—	2	1	no	—
58	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
59	C₂×D₄×C₄₁	—	2	4	no	—
60	C₂×D₄×C₄₁	—	2	2	no	—
61	C₂×D₄×C₄₁	—	4	2	no	—
62	C₂×D₄×C₄₁	—	4	2	no	—
63	C₂×D₄×C₄₁	—	4	2	no	—
64	C₂²×C₄×C₄₁	—	2	2	no	—
65	C₂²×C₄×C₄₁	—	2	2	no	—
66	C₂²×C₄×C₄₁	—	2	4	no	—
67	C₂²×C₄×C₄₁	—	2	8	no	—
68	C₂²×C₄×C₄₁	—	2	2	no	—
69	C₂²×C₄×C₄₁	—	2	4	no	—
70	C₂²×C₄×C₄₁	—	2	2	no	—
71	C₂²×C₄×C₄₁	—	2	1	no	—
72	C₂²×C₄×C₄₁	—	4	2	no	—
73	C₂²×C₄×C₄₁	—	4	4	no	—
74	C₂²×C₄×C₄₁	—	4	4	no	—
75	C₂²×C₄×C₄₁	SD₁₆	1	4	yes	—
76	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₄₁	#41	1	4	yes	—
80	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
81	C₂×C₈×C₄₁	—	2	4	no	—
82	C₂×C₈×C₄₁	—	2	4	no	—
83	C₂×C₈×C₄₁	—	2	4	no	—
84	C₄⋊C₄×C₄₁	—	2	8	no	—
85	C₄⋊C₄×C₄₁	—	2	4	no	—
86	C₄⋊C₄×C₄₁	—	2	4	no	—
87	C₄⋊C₄×C₄₁	—	2	2	no	—
88	C₄⋊C₄×C₄₁	—	4	4	no	—
89	C₄⋊C₄×C₄₁	#41	1	8	yes	—
90	C₂²⋊C₄×C₄₁	—	4	4	no	—
91	C₂²⋊C₄×C₄₁	—	4	2	no	—
92	C₂²⋊C₄×C₄₁	—	4	2	no	—
93	C₂²⋊C₄×C₄₁	—	4	4	no	—
94	C₄²×C₄₁	—	2	4	no	—
95	C₄²×C₄₁	—	2	4	no	—
96	C₄²×C₄₁	—	4	8	no	—
97	C₂³×C₄₁	—	2	2	no	—
98	C₂³×C₄₁	—	2	2	no	—
99	C₂³×C₄₁	—	4	1	no	—
100	C₂³×C₄₁	—	4	2	no	—
101	C₂³×C₄₁	—	4	1	no	—
102	C₂³×C₄₁	—	4	1	no	—
103	C₂³×C₄₁	—	4	2	no	—
104	C₂³×C₄₁	#123	1	2	yes	—
105	C₂³×C₄₁	C₂×SD₁₆	1	2	yes	—
106	C₂³×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
107	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
108	D₄×C₄₁	—	4	4	no	—
109	D₄×C₄₁	—	8	4	no	—
110	C₂×C₄×C₄₁	—	2	2	no	—
111	C₂×C₄×C₄₁	—	2	2	no	—
112	C₂×C₄×C₄₁	—	2	2	no	—
113	C₂×C₄×C₄₁	—	2	8	no	—
114	C₂×C₄×C₄₁	—	2	8	no	—
115	C₂×C₄×C₄₁	—	2	2	no	—
116	C₂×C₄×C₄₁	—	2	8	no	—
117	C₂×C₄×C₄₁	—	2	2	no	—
118	C₂×C₄×C₄₁	—	2	2	no	—
119	C₂×C₄×C₄₁	—	4	4	no	—
120	C₂×C₄×C₄₁	—	4	16	no	—
121	C₂×C₄×C₄₁	—	4	4	no	—
122	C₂×C₄×C₄₁	—	4	4	no	—
123	C₂×C₄×C₄₁	—	4	1	no	—
124	C₂×C₄×C₄₁	—	4	2	no	—
125	C₂×C₄×C₄₁	—	4	2	no	—
126	C₂×C₄×C₄₁	—	4	1	no	—
127	C₂×C₄×C₄₁	—	4	1	no	—
128	C₂×C₄×C₄₁	—	4	2	no	—
129	C₂×C₄×C₄₁	#123	1	2	yes	—
130	C₂×C₄×C₄₁	C₂×SD₁₆	1	4	yes	—
131	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
132	C₂×C₄×C₄₁	C₂×#41	1	4	yes	—
133	C₈×C₄₁	—	4	4	no	—
134	#33095	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₂²×C₄₁	—	2	2	no	—
136	C₂²×C₄₁	—	2	4	no	—
137	C₂²×C₄₁	—	2	2	no	—
138	C₂²×C₄₁	—	2	4	no	—
139	C₂²×C₄₁	—	8	1	no	—
140	C₂²×C₄₁	—	8	1	no	—
141	C₂²×C₄₁	—	8	2	no	—
142	C₂²×C₄₁	—	8	1	no	—
143	C₂²×C₄₁	—	8	2	no	—
144	C₂²×C₄₁	#481	1	4	yes	—
145	C₂²×C₄₁	C₂×#123	1	1	yes	characteristic
146	C₂²×C₄₁	#525	1	2	yes	—
147	C₂²×C₄₁	C₂²×SD₁₆	1	1	yes	characteristic
148	C₂²×C₄₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₂²×C₄₁	#575	1	2	yes	—
150	C₄×C₄₁	—	2	4	no	—
151	C₄×C₄₁	—	2	2	no	—
152	C₄×C₄₁	—	4	2	no	—
153	C₄×C₄₁	—	4	8	no	—
154	C₄×C₄₁	—	8	1	no	—
155	C₄×C₄₁	—	8	2	no	—
156	C₄×C₄₁	#529	1	4	yes	—
157	#2854	C₂×C₄₁	1	1	yes	characteristic
158	C₂×C₄×C₄⋊C₄	C₂×C₄₁	1	1	yes	characteristic, complemented
159	#1926	C₂×C₄₁	1	4	yes	complemented
160	#1919	C₂×C₄₁	1	4	yes	complemented
161	#1874	C₂×C₄₁	1	2	yes	—
162	#1853	C₂×C₄₁	1	2	yes	complemented
163	#1817	C₂×C₄₁	1	1	yes	characteristic, complemented
164	C₂×C₄₁	—	2	4	no	—
165	C₂×C₄₁	—	16	1	no	—
166	C₂×C₄₁	#2853	1	1	yes	characteristic
167	C₂×C₄₁	C₂×#481	1	1	yes	characteristic
168	C₂×C₄₁	#3093	1	1	yes	characteristic
169	C₂×C₄₁	#3103	1	2	yes	—
170	C₂×C₄₁	#3114	1	2	yes	—
171	C₂×#123	—	2	1	no	—
172	C₂×#121	C₂²×C₄₁	1	1	yes	characteristic
173	C₂²×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
174	C₂²×C₄²	—	2	1	no	—
175	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
176	C₂×#106	C₂²×C₄₁	1	4	yes	—
177	C₂×#102	C₂²×C₄₁	1	4	yes	—
178	C₂×C₂²⋊C₈	C₂²×C₄₁	1	2	yes	—
179	#390	C₂²×C₄₁	1	2	yes	—
180	#389	C₂²×C₄₁	1	2	yes	—
181	#388	—	2	2	no	—
182	#386	—	2	1	no	—
183	#380	C₂²×C₄₁	1	1	yes	characteristic
184	C₄×C₄⋊C₄	—	2	2	no	—
185	C₄×C₄⋊C₄	C₂²×C₄₁	1	4	yes	—
186	C₂×#95	—	2	1	no	—
187	#340	C₂²×C₄₁	1	4	yes	—
188	#340	C₂²×C₄₁	1	4	yes	—
189	#336	C₂²×C₄₁	1	4	yes	—
190	C₄₁	#33095	1	1	yes	characteristic, direct product factor, Sylow
191	C₂²×D₄	—	2	1	no	—
192	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
193	C₂³×C₄	—	2	2	no	—
194	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
195	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
196	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
197	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
198	#123	—	4	4	no	—
199	#121	—	2	4	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	—	2	8	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
204	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
205	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
206	C₂×C₂²⋊C₄	—	2	1	no	—
207	C₂×C₂²⋊C₄	—	2	1	no	—
208	C₂×C₂²⋊C₄	—	2	2	no	—
209	C₂×C₂²⋊C₄	—	2	2	no	—
210	C₂×C₄²	—	2	4	no	—
211	C₂×C₄²	—	2	4	no	—
212	C₂×C₄²	—	4	4	no	—
213	C₂×C₄²	D₄×C₄₁	1	2	yes	—
214	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
215	#106	—	2	8	no	—
216	#102	—	2	8	no	—
217	C₂²⋊C₈	—	2	4	no	—
218	#95	—	2	4	no	—
219	#95	—	2	1	no	—
220	#95	—	2	2	no	—
221	C₂⁴	—	2	1	no	—
222	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	—	4	2	no	—
226	C₂×D₄	—	4	2	no	—
227	C₂×D₄	—	4	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	2	8	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	4	2	no	—
237	C₂²×C₄	—	4	4	no	—
238	C₂²×C₄	—	4	4	no	—
239	C₂²×C₄	SD₁₆×C₄₁	1	4	yes	—
240	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
241	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
243	C₂²×C₄	#41×C₄₁	1	4	yes	—
244	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	2	4	no	—
248	C₄⋊C₄	—	2	8	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	4	4	no	—
253	C₄⋊C₄	#41×C₄₁	1	8	yes	—
254	C₂²⋊C₄	—	4	4	no	—
255	C₂²⋊C₄	—	4	2	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₂²⋊C₄	—	4	4	no	—
258	C₄²	—	2	4	no	—
259	C₄²	—	2	4	no	—
260	C₄²	—	4	8	no	—
261	C₂³	—	2	2	no	—
262	C₂³	—	2	2	no	—
263	C₂³	—	4	1	no	—
264	C₂³	—	4	2	no	—
265	C₂³	—	4	1	no	—
266	C₂³	—	4	1	no	—
267	C₂³	—	4	2	no	—
268	C₂³	#123×C₄₁	1	2	yes	—
269	C₂³	C₂×SD₁₆×C₄₁	1	2	yes	—
270	C₂³	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
271	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
272	D₄	—	4	4	no	—
273	D₄	—	8	4	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	8	no	—
278	C₂×C₄	—	2	8	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	8	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	16	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	#123×C₄₁	1	2	yes	—
294	C₂×C₄	C₂×SD₁₆×C₄₁	1	4	yes	—
295	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
296	C₂×C₄	C₂×#41×C₄₁	1	4	yes	—
297	C₈	—	4	4	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	4	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	4	no	—
302	C₂²	—	8	1	no	—
303	C₂²	—	8	1	no	—
304	C₂²	—	8	2	no	—
305	C₂²	—	8	1	no	—
306	C₂²	—	8	2	no	—
307	C₂²	C₄₁×#481	1	4	yes	—
308	C₂²	C₂×#123×C₄₁	1	1	yes	characteristic
309	C₂²	C₄₁×#525	1	2	yes	—
310	C₂²	C₂²×SD₁₆×C₄₁	1	1	yes	characteristic
311	C₂²	C₂×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
312	C₂²	C₄₁×#575	1	2	yes	—
313	C₄	—	2	4	no	—
314	C₄	—	2	2	no	—
315	C₄	—	4	2	no	—
316	C₄	—	4	8	no	—
317	C₄	—	8	1	no	—
318	C₄	—	8	2	no	—
319	C₄	C₄₁×#529	1	4	yes	—
320	C₂	—	2	4	no	—
321	C₂	—	16	1	no	—
322	C₂	C₄₁×#2853	1	1	yes	characteristic
323	C₂	C₂×C₄₁×#481	1	1	yes	characteristic
324	C₂	C₄₁×#3093	1	1	yes	characteristic
325	C₂	C₄₁×#3103	1	2	yes	—
326	C₂	C₄₁×#3114	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₂×#121×C₄₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂²×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×C₄⋊C₄×C₄₁	C₂	semidirect
3. C₄₁×#1926	C₂	semidirect
4. C₄₁×#1919	C₂	semidirect
6. C₄₁×#1853	C₂	semidirect
7. C₄₁×#1817	C₂	semidirect
134. #33095	C₄₁	direct (N × H)
158. C₂×C₄×C₄⋊C₄	C₂×C₄₁	semidirect
159. #1926	C₂×C₄₁	semidirect
160. #1919	C₂×C₄₁	semidirect
162. #1853	C₂×C₄₁	semidirect
163. #1817	C₂×C₄₁	semidirect
190. C₄₁	#33095	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×#2854	2
C₂×C₄×C₄⋊C₄×C₄₁	2
C₄₁×#1926	2
C₄₁×#1919	2
C₄₁×#1874	2
C₄₁×#1853	2
C₄₁×#1817	2
#33095	41

Subgroups — C41×#33095 #23809296

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#33095 #23809296