Subgroups — C₄₁×#18369 #23794570

Order 10496.

904 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×#3502	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₁×#3215	C₂	1	2	yes	maximal
3	C₄₁×#2632	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1605	C₂	1	4	yes	maximal, complemented
5	C₄₁×#1599	C₂	1	4	yes	maximal
6	C₄₁×#1515	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1467	C₂	1	1	yes	maximal, characteristic, complemented
8	Q₈²×C₄₁	—	2	2	no	—
9	C₄₁×#557	—	2	2	no	—
10	C₄₁×#557	—	2	4	no	—
11	C₄₁×#544	—	2	1	no	—
12	C₄₁×#544	C₂²	1	4	yes	—
13	C₄₁×#533	—	2	2	no	—
14	C₄₁×#533	C₂²	1	2	yes	—
15	C₂×#124×C₄₁	C₂²	1	1	yes	characteristic
16	C₄₁×#519	—	2	1	no	—
17	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
18	C₄₁×#497	—	2	2	no	—
19	C₄₁×#489	—	2	2	no	—
20	C₄₁×#464	—	2	4	no	—
21	C₄₁×#451	C₂²	1	4	yes	—
22	C₄₁×#450	C₂²	1	4	yes	—
23	C₄₁×#432	C₂²	1	2	yes	—
24	C₄₁×#399	—	2	1	no	—
25	C₄₁×#395	C₂²	1	2	yes	—
26	C₄₁×#394	C₂²	1	2	yes	—
27	C₄₁×#393	—	2	1	no	—
28	C₄₁×#388	—	2	2	no	—
29	C₄₁×#383	C₂²	1	1	yes	characteristic
30	C₄₁×#328	C₂²	1	8	yes	—
31	C₄₁×#324	C₂²	1	4	yes	—
32	C₂²×Q₈×C₄₁	D₄	1	1	yes	characteristic
33	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
34	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
35	C₂×Q₁₆×C₄₁	—	2	4	no	—
36	C₂×SD₁₆×C₄₁	—	2	4	no	—
37	C₄⋊Q₈×C₄₁	—	2	2	no	—
38	C₄⋊Q₈×C₄₁	—	2	2	no	—
39	C₄⋊Q₈×C₄₁	—	2	4	no	—
40	C₄⋊Q₈×C₄₁	—	2	8	no	—
41	C₄⋊Q₈×C₄₁	D₄	1	2	yes	—
42	C₄⋊Q₈×C₄₁	D₄	1	2	yes	—
43	#125×C₄₁	—	2	4	no	—
44	#125×C₄₁	—	2	4	no	—
45	#125×C₄₁	—	2	8	no	—
46	#125×C₄₁	—	2	2	no	—
47	#124×C₄₁	—	2	2	no	—
48	#124×C₄₁	—	2	2	no	—
49	C₂²⋊Q₈×C₄₁	—	2	2	no	—
50	C₂²⋊Q₈×C₄₁	—	2	4	no	—
51	C₂²⋊Q₈×C₄₁	C₂³	1	4	yes	—
52	C₄×Q₈×C₄₁	—	2	4	no	—
53	C₄×Q₈×C₄₁	—	2	4	no	—
54	C₄×Q₈×C₄₁	—	2	4	no	—
55	C₄×Q₈×C₄₁	—	2	4	no	—
56	C₄×Q₈×C₄₁	—	2	2	no	—
57	C₄×Q₈×C₄₁	—	2	2	no	—
58	C₄×Q₈×C₄₁	—	2	2	no	—
59	C₄×Q₈×C₄₁	D₄	1	4	yes	—
60	#117×C₄₁	—	2	4	no	—
61	#117×C₄₁	—	2	2	no	—
62	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
63	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
65	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
66	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
67	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
68	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
69	C₂×C₄²×C₄₁	D₄	1	2	yes	—
70	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
71	C₄⋊C₈×C₄₁	—	2	2	no	—
72	#103×C₄₁	—	2	4	no	—
73	#102×C₄₁	—	2	4	no	—
74	C₂²⋊C₈×C₄₁	C₂³	1	4	yes	—
75	#97×C₄₁	—	2	2	no	—
76	#95×C₄₁	—	2	1	no	—
77	#95×C₄₁	—	2	1	no	—
78	#95×C₄₁	—	2	2	no	—
79	#95×C₄₁	C₂³	1	2	yes	—
80	C₂⁴×C₄₁	—	2	1	no	—
81	C₂×Q₈×C₄₁	—	2	4	no	—
82	C₂×Q₈×C₄₁	—	2	2	no	—
83	C₂×Q₈×C₄₁	—	2	2	no	—
84	C₂×Q₈×C₄₁	—	2	4	no	—
85	C₂×Q₈×C₄₁	—	2	2	no	—
86	C₂×Q₈×C₄₁	—	8	1	no	—
87	C₂×Q₈×C₄₁	C₂×D₄	1	4	yes	—
88	C₂×D₄×C₄₁	—	2	2	no	—
89	C₂×D₄×C₄₁	—	8	1	no	—
90	C₂²×C₄×C₄₁	—	2	2	no	—
91	C₂²×C₄×C₄₁	—	2	1	no	—
92	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
94	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
95	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
97	Q₁₆×C₄₁	—	8	4	no	—
98	SD₁₆×C₄₁	—	8	4	no	—
99	C₂×C₈×C₄₁	—	2	4	no	—
100	C₄⋊C₄×C₄₁	—	2	4	no	—
101	C₄⋊C₄×C₄₁	—	2	4	no	—
102	C₄⋊C₄×C₄₁	—	2	4	no	—
103	C₄⋊C₄×C₄₁	—	2	8	no	—
104	C₄⋊C₄×C₄₁	—	2	2	no	—
105	C₄⋊C₄×C₄₁	—	2	2	no	—
106	C₄⋊C₄×C₄₁	—	2	2	no	—
107	C₄⋊C₄×C₄₁	—	2	2	no	—
108	C₄⋊C₄×C₄₁	—	2	4	no	—
109	C₄⋊C₄×C₄₁	—	2	4	no	—
110	C₄⋊C₄×C₄₁	—	2	4	no	—
111	C₄⋊C₄×C₄₁	—	8	2	no	—
112	C₄⋊C₄×C₄₁	C₂×D₄	1	4	yes	—
113	C₂²⋊C₄×C₄₁	—	2	4	no	—
114	C₂²⋊C₄×C₄₁	—	4	1	no	—
115	C₂²⋊C₄×C₄₁	—	4	2	no	—
116	C₂²⋊C₄×C₄₁	—	4	1	no	—
117	C₂²⋊C₄×C₄₁	—	4	4	no	—
118	C₂²⋊C₄×C₄₁	—	8	2	no	—
119	C₄²×C₄₁	—	2	8	no	—
120	C₄²×C₄₁	—	2	2	no	—
121	C₄²×C₄₁	—	2	2	no	—
122	C₄²×C₄₁	—	2	4	no	—
123	C₄²×C₄₁	C₂×D₄	1	2	yes	—
124	C₄²×C₄₁	C₂×D₄	1	2	yes	—
125	C₂³×C₄₁	—	4	2	no	—
126	C₂³×C₄₁	—	4	2	no	—
127	C₂³×C₄₁	—	4	1	no	—
128	C₂³×C₄₁	—	4	2	no	—
129	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
130	Q₈×C₄₁	—	2	8	no	—
131	Q₈×C₄₁	—	4	4	no	—
132	Q₈×C₄₁	—	4	4	no	—
133	Q₈×C₄₁	—	4	4	no	—
134	Q₈×C₄₁	—	8	4	no	—
135	D₄×C₄₁	—	4	4	no	—
136	C₂×C₄×C₄₁	—	2	2	no	—
137	C₂×C₄×C₄₁	—	2	2	no	—
138	C₂×C₄×C₄₁	—	2	2	no	—
139	C₂×C₄×C₄₁	—	2	4	no	—
140	C₂×C₄×C₄₁	—	2	4	no	—
141	C₂×C₄×C₄₁	—	2	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	—	4	2	no	—
146	C₂×C₄×C₄₁	—	4	2	no	—
147	C₂×C₄×C₄₁	—	4	2	no	—
148	C₂×C₄×C₄₁	—	4	1	no	—
149	C₂×C₄×C₄₁	—	4	2	no	—
150	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
151	C₂×C₄×C₄₁	#120	1	2	yes	—
152	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
153	C₂×C₄×C₄₁	#137	1	4	yes	—
154	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
155	C₈×C₄₁	—	4	4	no	—
156	#18369	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	—	8	4	no	—
160	C₂²×C₄₁	—	8	1	no	—
161	C₂²×C₄₁	—	8	1	no	—
162	C₂²×C₄₁	—	8	1	no	—
163	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
164	C₂²×C₄₁	C₂×#137	1	2	yes	—
165	C₄×C₄₁	—	2	2	no	—
166	C₄×C₄₁	—	2	4	no	—
167	C₄×C₄₁	—	2	1	no	—
168	C₄×C₄₁	—	2	1	no	—
169	C₄×C₄₁	—	2	2	no	—
170	C₄×C₄₁	—	4	4	no	—
171	C₄×C₄₁	—	4	2	no	—
172	C₄×C₄₁	—	4	4	no	—
173	C₄×C₄₁	—	8	2	no	—
174	C₄×C₄₁	—	8	1	no	—
175	#3502	C₂×C₄₁	1	1	yes	characteristic, complemented
176	#3215	C₂×C₄₁	1	2	yes	—
177	#2632	C₂×C₄₁	1	1	yes	characteristic
178	#1605	C₂×C₄₁	1	4	yes	complemented
179	#1599	C₂×C₄₁	1	4	yes	—
180	#1515	C₂×C₄₁	1	2	yes	complemented
181	#1467	C₂×C₄₁	1	1	yes	characteristic, complemented
182	C₂×C₄₁	—	2	2	no	—
183	C₂×C₄₁	—	16	1	no	—
184	C₂×C₄₁	#3004	1	1	yes	characteristic
185	C₂×C₄₁	#3020	1	1	yes	characteristic
186	C₂×C₄₁	#3027	1	1	yes	characteristic
187	Q₈²	—	2	2	no	—
188	#557	—	2	2	no	—
189	#557	—	2	4	no	—
190	#544	—	2	1	no	—
191	#544	C₂²×C₄₁	1	4	yes	—
192	#533	—	2	2	no	—
193	#533	C₂²×C₄₁	1	2	yes	—
194	C₂×#124	C₂²×C₄₁	1	1	yes	characteristic
195	#519	—	2	1	no	—
196	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
197	#497	—	2	2	no	—
198	#489	—	2	2	no	—
199	#464	—	2	4	no	—
200	#451	C₂²×C₄₁	1	4	yes	—
201	#450	C₂²×C₄₁	1	4	yes	—
202	#432	C₂²×C₄₁	1	2	yes	—
203	#399	—	2	1	no	—
204	#395	C₂²×C₄₁	1	2	yes	—
205	#394	C₂²×C₄₁	1	2	yes	—
206	#393	—	2	1	no	—
207	#388	—	2	2	no	—
208	#383	C₂²×C₄₁	1	1	yes	characteristic
209	#328	C₂²×C₄₁	1	8	yes	—
210	#324	C₂²×C₄₁	1	4	yes	—
211	C₄₁	#18369	1	1	yes	characteristic, direct product factor, Sylow
212	C₂²×Q₈	D₄×C₄₁	1	1	yes	characteristic
213	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
214	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
215	C₂×Q₁₆	—	2	4	no	—
216	C₂×SD₁₆	—	2	4	no	—
217	C₄⋊Q₈	—	2	2	no	—
218	C₄⋊Q₈	—	2	2	no	—
219	C₄⋊Q₈	—	2	4	no	—
220	C₄⋊Q₈	—	2	8	no	—
221	C₄⋊Q₈	D₄×C₄₁	1	2	yes	—
222	C₄⋊Q₈	D₄×C₄₁	1	2	yes	—
223	#125	—	2	4	no	—
224	#125	—	2	4	no	—
225	#125	—	2	8	no	—
226	#125	—	2	2	no	—
227	#124	—	2	2	no	—
228	#124	—	2	2	no	—
229	C₂²⋊Q₈	—	2	2	no	—
230	C₂²⋊Q₈	—	2	4	no	—
231	C₂²⋊Q₈	C₂³×C₄₁	1	4	yes	—
232	C₄×Q₈	—	2	4	no	—
233	C₄×Q₈	—	2	4	no	—
234	C₄×Q₈	—	2	4	no	—
235	C₄×Q₈	—	2	4	no	—
236	C₄×Q₈	—	2	2	no	—
237	C₄×Q₈	—	2	2	no	—
238	C₄×Q₈	—	2	2	no	—
239	C₄×Q₈	D₄×C₄₁	1	4	yes	—
240	#117	—	2	4	no	—
241	#117	—	2	2	no	—
242	C₂×C₄⋊C₄	—	2	2	no	—
243	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
244	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
245	C₂×C₂²⋊C₄	—	2	1	no	—
246	C₂×C₂²⋊C₄	—	2	1	no	—
247	C₂×C₂²⋊C₄	—	2	2	no	—
248	C₂×C₂²⋊C₄	—	2	2	no	—
249	C₂×C₄²	D₄×C₄₁	1	2	yes	—
250	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
251	C₄⋊C₈	—	2	2	no	—
252	#103	—	2	4	no	—
253	#102	—	2	4	no	—
254	C₂²⋊C₈	C₂³×C₄₁	1	4	yes	—
255	#97	—	2	2	no	—
256	#95	—	2	1	no	—
257	#95	—	2	1	no	—
258	#95	—	2	2	no	—
259	#95	C₂³×C₄₁	1	2	yes	—
260	C₂⁴	—	2	1	no	—
261	C₂×Q₈	—	2	4	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	2	4	no	—
265	C₂×Q₈	—	2	2	no	—
266	C₂×Q₈	—	8	1	no	—
267	C₂×Q₈	C₂×D₄×C₄₁	1	4	yes	—
268	C₂×D₄	—	2	2	no	—
269	C₂×D₄	—	8	1	no	—
270	C₂²×C₄	—	2	2	no	—
271	C₂²×C₄	—	2	1	no	—
272	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
273	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
274	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
275	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
276	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
277	Q₁₆	—	8	4	no	—
278	SD₁₆	—	8	4	no	—
279	C₂×C₈	—	2	4	no	—
280	C₄⋊C₄	—	2	4	no	—
281	C₄⋊C₄	—	2	4	no	—
282	C₄⋊C₄	—	2	4	no	—
283	C₄⋊C₄	—	2	8	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	2	4	no	—
289	C₄⋊C₄	—	2	4	no	—
290	C₄⋊C₄	—	2	4	no	—
291	C₄⋊C₄	—	8	2	no	—
292	C₄⋊C₄	C₂×D₄×C₄₁	1	4	yes	—
293	C₂²⋊C₄	—	2	4	no	—
294	C₂²⋊C₄	—	4	1	no	—
295	C₂²⋊C₄	—	4	2	no	—
296	C₂²⋊C₄	—	4	1	no	—
297	C₂²⋊C₄	—	4	4	no	—
298	C₂²⋊C₄	—	8	2	no	—
299	C₄²	—	2	8	no	—
300	C₄²	—	2	2	no	—
301	C₄²	—	2	2	no	—
302	C₄²	—	2	4	no	—
303	C₄²	C₂×D₄×C₄₁	1	2	yes	—
304	C₄²	C₂×D₄×C₄₁	1	2	yes	—
305	C₂³	—	4	2	no	—
306	C₂³	—	4	2	no	—
307	C₂³	—	4	1	no	—
308	C₂³	—	4	2	no	—
309	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
310	Q₈	—	2	8	no	—
311	Q₈	—	4	4	no	—
312	Q₈	—	4	4	no	—
313	Q₈	—	4	4	no	—
314	Q₈	—	8	4	no	—
315	D₄	—	4	4	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	4	no	—
320	C₂×C₄	—	2	4	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
331	C₂×C₄	#120×C₄₁	1	2	yes	—
332	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
333	C₂×C₄	#137×C₄₁	1	4	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	—	8	4	no	—
339	C₂²	—	8	1	no	—
340	C₂²	—	8	1	no	—
341	C₂²	—	8	1	no	—
342	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
343	C₂²	C₂×#137×C₄₁	1	2	yes	—
344	C₄	—	2	2	no	—
345	C₄	—	2	4	no	—
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	2	no	—
349	C₄	—	4	4	no	—
350	C₄	—	4	2	no	—
351	C₄	—	4	4	no	—
352	C₄	—	8	2	no	—
353	C₄	—	8	1	no	—
354	C₂	—	2	2	no	—
355	C₂	—	16	1	no	—
356	C₂	C₄₁×#3004	1	1	yes	characteristic
357	C₂	C₄₁×#3020	1	1	yes	characteristic
358	C₂	C₄₁×#3027	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₂²×D₄×C₄₁	C₂³
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
1. C₄₁×#3502	C₂	semidirect
4. C₄₁×#1605	C₂	semidirect
6. C₄₁×#1515	C₂	semidirect
7. C₄₁×#1467	C₂	semidirect
156. #18369	C₄₁	direct (N × H)
175. #3502	C₂×C₄₁	semidirect
178. #1605	C₂×C₄₁	semidirect
180. #1515	C₂×C₄₁	semidirect
181. #1467	C₂×C₄₁	semidirect
211. C₄₁	#18369	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×#3502	2
C₄₁×#3215	2
C₄₁×#2632	2
C₄₁×#1605	2
C₄₁×#1599	2
C₄₁×#1515	2
C₄₁×#1467	2
#18369	41

Subgroups — C41×#18369 #23794570

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#18369 #23794570