Subgroups — C₄₁×#18187 #23794388

Order 10496.

1166 conjugacy classes of subgroups.

The subgroup lattice has 378 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×#3112	C₂	1	2	yes	maximal, complemented
3	C₄₁×#2399	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#1610	C₂	1	4	yes	maximal, complemented
5	C₄₁×#1604	C₂	1	4	yes	maximal, complemented
6	C₄₁×#1505	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1467	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×2₋¹⁺⁴×C₄₁	—	2	1	no	—
9	C₄₁×#550	—	2	2	no	—
10	D₄×Q₈×C₄₁	—	2	2	no	—
11	C₄₁×#548	—	2	4	no	—
12	C₄₁×#537	—	2	1	no	—
13	C₄₁×#537	C₂²	1	4	yes	—
14	C₄₁×#532	—	2	2	no	—
15	C₄₁×#532	C₂²	1	2	yes	—
16	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
18	C₄₁×#480	C₂²	1	4	yes	—
19	C₄₁×#477	—	2	2	no	—
20	C₄₁×#460	—	2	2	no	—
21	C₄₁×#448	C₂²	1	4	yes	—
22	C₄₁×#440	—	2	4	no	—
23	C₄₁×#432	C₂²	1	2	yes	—
24	C₄₁×#398	—	2	1	no	—
25	C₄₁×#398	C₂²	1	2	yes	—
26	C₄₁×#395	—	2	1	no	—
27	C₄₁×#395	C₂²	1	2	yes	—
28	C₄×C₄⋊C₄×C₄₁	—	2	2	no	—
29	C₄₁×#376	C₂²	1	1	yes	characteristic
30	C₄₁×#327	C₂²	1	8	yes	—
31	C₄₁×#324	C₂²	1	4	yes	—
32	2₋¹⁺⁴×C₄₁	—	4	8	no	—
33	C₂×#41×C₄₁	—	2	4	no	—
34	C₂×#41×C₄₁	—	2	4	no	—
35	C₂×#41×C₄₁	—	2	2	no	—
36	C₂²×Q₈×C₄₁	—	2	4	no	—
37	C₂²×Q₈×C₄₁	D₄	1	1	yes	characteristic
38	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
39	C₂×SD₁₆×C₄₁	—	2	4	no	—
40	C₄⋊Q₈×C₄₁	—	2	2	no	—
41	C₄⋊Q₈×C₄₁	D₄	1	2	yes	—
42	#127×C₄₁	—	2	2	no	—
43	#127×C₄₁	D₄	1	2	yes	—
44	#124×C₄₁	—	2	4	no	—
45	#124×C₄₁	—	2	2	no	—
46	C₂²⋊Q₈×C₄₁	—	2	4	no	—
47	C₂²⋊Q₈×C₄₁	—	2	8	no	—
48	#121×C₄₁	—	2	4	no	—
49	#121×C₄₁	—	2	8	no	—
50	#121×C₄₁	—	2	2	no	—
51	#121×C₄₁	—	2	4	no	—
52	#121×C₄₁	C₂³	1	4	yes	—
53	C₄×Q₈×C₄₁	—	2	2	no	—
54	C₄×Q₈×C₄₁	—	2	2	no	—
55	C₄×Q₈×C₄₁	—	2	2	no	—
56	C₄×Q₈×C₄₁	—	2	2	no	—
57	C₄×D₄×C₄₁	—	2	4	no	—
58	C₄×D₄×C₄₁	—	2	4	no	—
59	C₄×D₄×C₄₁	—	2	2	no	—
60	C₄×D₄×C₄₁	D₄	1	4	yes	—
61	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
62	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
63	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
64	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
67	C₂×C₄²×C₄₁	—	2	1	no	—
68	C₂×C₄²×C₄₁	—	2	1	no	—
69	C₂×C₄²×C₄₁	D₄	1	2	yes	—
70	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
71	#107×C₄₁	—	2	4	no	—
72	C₄⋊C₈×C₄₁	—	2	2	no	—
73	#103×C₄₁	—	2	4	no	—
74	#102×C₄₁	—	2	4	no	—
75	C₂²⋊C₈×C₄₁	C₂³	1	4	yes	—
76	#97×C₄₁	—	2	2	no	—
77	#95×C₄₁	—	2	1	no	—
78	#95×C₄₁	—	2	2	no	—
79	#95×C₄₁	—	2	2	no	—
80	#95×C₄₁	C₂³	1	2	yes	—
81	#41×C₄₁	—	4	8	no	—
82	#41×C₄₁	—	4	8	no	—
83	#41×C₄₁	—	4	16	no	—
84	#41×C₄₁	—	4	8	no	—
85	C₂×Q₈×C₄₁	—	2	4	no	—
86	C₂×Q₈×C₄₁	—	2	2	no	—
87	C₂×Q₈×C₄₁	—	2	2	no	—
88	C₂×Q₈×C₄₁	—	2	4	no	—
89	C₂×Q₈×C₄₁	—	2	2	no	—
90	C₂×Q₈×C₄₁	—	4	2	no	—
91	C₂×Q₈×C₄₁	—	4	16	no	—
92	C₂×D₄×C₄₁	—	2	4	no	—
93	C₂×D₄×C₄₁	—	2	4	no	—
94	C₂×D₄×C₄₁	—	2	4	no	—
95	C₂×D₄×C₄₁	—	2	2	no	—
96	C₂×D₄×C₄₁	—	2	2	no	—
97	C₂×D₄×C₄₁	C₂×D₄	1	4	yes	—
98	C₂²×C₄×C₄₁	—	2	8	no	—
99	C₂²×C₄×C₄₁	—	2	2	no	—
100	C₂²×C₄×C₄₁	—	2	1	no	—
101	C₂²×C₄×C₄₁	—	2	4	no	—
102	C₂²×C₄×C₄₁	—	2	1	no	—
103	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
106	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
107	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
108	SD₁₆×C₄₁	—	4	8	no	—
109	C₂×C₈×C₄₁	—	2	4	no	—
110	C₄⋊C₄×C₄₁	—	2	4	no	—
111	C₄⋊C₄×C₄₁	—	2	2	no	—
112	C₄⋊C₄×C₄₁	—	2	2	no	—
113	C₄⋊C₄×C₄₁	—	2	4	no	—
114	C₄⋊C₄×C₄₁	—	2	2	no	—
115	C₄⋊C₄×C₄₁	—	4	4	no	—
116	C₄⋊C₄×C₄₁	—	4	2	no	—
117	C₄⋊C₄×C₄₁	—	4	4	no	—
118	C₄⋊C₄×C₄₁	C₂×D₄	1	4	yes	—
119	C₂²⋊C₄×C₄₁	—	2	8	no	—
120	C₂²⋊C₄×C₄₁	—	2	4	no	—
121	C₂²⋊C₄×C₄₁	—	2	4	no	—
122	C₄²×C₄₁	—	2	4	no	—
123	C₄²×C₄₁	—	2	2	no	—
124	C₄²×C₄₁	—	2	2	no	—
125	C₄²×C₄₁	—	4	1	no	—
126	C₄²×C₄₁	—	4	1	no	—
127	C₄²×C₄₁	C₂×D₄	1	2	yes	—
128	C₄²×C₄₁	C₂×D₄	1	2	yes	—
129	C₂³×C₄₁	—	2	4	no	—
130	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
131	Q₈×C₄₁	—	2	8	no	—
132	Q₈×C₄₁	—	2	4	no	—
133	Q₈×C₄₁	—	4	2	no	—
134	Q₈×C₄₁	—	4	4	no	—
135	Q₈×C₄₁	—	4	8	no	—
136	D₄×C₄₁	—	4	4	no	—
137	D₄×C₄₁	—	4	4	no	—
138	D₄×C₄₁	—	4	8	no	—
139	D₄×C₄₁	—	4	8	no	—
140	D₄×C₄₁	—	4	4	no	—
141	D₄×C₄₁	—	8	4	no	—
142	C₂×C₄×C₄₁	—	2	2	no	—
143	C₂×C₄×C₄₁	—	2	2	no	—
144	C₂×C₄×C₄₁	—	2	2	no	—
145	C₂×C₄×C₄₁	—	2	2	no	—
146	C₂×C₄×C₄₁	—	2	4	no	—
147	C₂×C₄×C₄₁	—	2	2	no	—
148	C₂×C₄×C₄₁	—	2	2	no	—
149	C₂×C₄×C₄₁	—	2	4	no	—
150	C₂×C₄×C₄₁	—	4	1	no	—
151	C₂×C₄×C₄₁	—	4	8	no	—
152	C₂×C₄×C₄₁	—	4	16	no	—
153	C₂×C₄×C₄₁	—	4	2	no	—
154	C₂×C₄×C₄₁	—	4	1	no	—
155	C₂×C₄×C₄₁	—	4	1	no	—
156	C₂×C₄×C₄₁	—	4	2	no	—
157	C₂×C₄×C₄₁	—	4	2	no	—
158	C₂×C₄×C₄₁	—	4	1	no	—
159	C₂×C₄×C₄₁	—	4	2	no	—
160	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
161	C₂×C₄×C₄₁	#120	1	2	yes	—
162	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
163	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	4	yes	—
164	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
165	C₈×C₄₁	—	4	4	no	—
166	#18187	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
167	C₂²×C₄₁	—	2	1	no	—
168	C₂²×C₄₁	—	2	1	no	—
169	C₂²×C₄₁	—	4	4	no	—
170	C₂²×C₄₁	—	4	4	no	—
171	C₂²×C₄₁	—	4	4	no	—
172	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
173	C₂²×C₄₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
174	C₄×C₄₁	—	2	2	no	—
175	C₄×C₄₁	—	2	4	no	—
176	C₄×C₄₁	—	2	1	no	—
177	C₄×C₄₁	—	2	2	no	—
178	C₄×C₄₁	—	4	4	no	—
179	C₄×C₄₁	—	4	4	no	—
180	C₄×C₄₁	—	4	2	no	—
181	C₄×C₄₁	—	4	2	no	—
182	C₄×C₄₁	—	8	1	no	—
183	C₄×C₄₁	#456	1	2	yes	—
184	#3497	C₂×C₄₁	1	1	yes	characteristic
185	#3112	C₂×C₄₁	1	2	yes	complemented
186	#2399	C₂×C₄₁	1	1	yes	characteristic, complemented
187	#1610	C₂×C₄₁	1	4	yes	complemented
188	#1604	C₂×C₄₁	1	4	yes	complemented
189	#1505	C₂×C₄₁	1	2	yes	complemented
190	#1467	C₂×C₄₁	1	1	yes	characteristic, complemented
191	C₂×C₄₁	—	2	2	no	—
192	C₂×C₄₁	—	8	4	no	—
193	C₂×C₄₁	#3003	1	1	yes	characteristic
194	C₂×C₄₁	C₂×#456	1	1	yes	characteristic
195	C₂×C₄₁	#3025	1	1	yes	characteristic
196	C₂×2₋¹⁺⁴	—	2	1	no	—
197	#550	—	2	2	no	—
198	D₄×Q₈	—	2	2	no	—
199	#548	—	2	4	no	—
200	#537	—	2	1	no	—
201	#537	C₂²×C₄₁	1	4	yes	—
202	#532	—	2	2	no	—
203	#532	C₂²×C₄₁	1	2	yes	—
204	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
205	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
206	#480	C₂²×C₄₁	1	4	yes	—
207	#477	—	2	2	no	—
208	#460	—	2	2	no	—
209	#448	C₂²×C₄₁	1	4	yes	—
210	#440	—	2	4	no	—
211	#432	C₂²×C₄₁	1	2	yes	—
212	#398	—	2	1	no	—
213	#398	C₂²×C₄₁	1	2	yes	—
214	#395	—	2	1	no	—
215	#395	C₂²×C₄₁	1	2	yes	—
216	C₄×C₄⋊C₄	—	2	2	no	—
217	#376	C₂²×C₄₁	1	1	yes	characteristic
218	#327	C₂²×C₄₁	1	8	yes	—
219	#324	C₂²×C₄₁	1	4	yes	—
220	C₄₁	#18187	1	1	yes	characteristic, direct product factor, Sylow
221	2₋¹⁺⁴	—	4	8	no	—
222	C₂×#41	—	2	4	no	—
223	C₂×#41	—	2	4	no	—
224	C₂×#41	—	2	2	no	—
225	C₂²×Q₈	—	2	4	no	—
226	C₂²×Q₈	D₄×C₄₁	1	1	yes	characteristic
227	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
228	C₂×SD₁₆	—	2	4	no	—
229	C₄⋊Q₈	—	2	2	no	—
230	C₄⋊Q₈	D₄×C₄₁	1	2	yes	—
231	#127	—	2	2	no	—
232	#127	D₄×C₄₁	1	2	yes	—
233	#124	—	2	4	no	—
234	#124	—	2	2	no	—
235	C₂²⋊Q₈	—	2	4	no	—
236	C₂²⋊Q₈	—	2	8	no	—
237	#121	—	2	4	no	—
238	#121	—	2	8	no	—
239	#121	—	2	2	no	—
240	#121	—	2	4	no	—
241	#121	C₂³×C₄₁	1	4	yes	—
242	C₄×Q₈	—	2	2	no	—
243	C₄×Q₈	—	2	2	no	—
244	C₄×Q₈	—	2	2	no	—
245	C₄×Q₈	—	2	2	no	—
246	C₄×D₄	—	2	4	no	—
247	C₄×D₄	—	2	4	no	—
248	C₄×D₄	—	2	2	no	—
249	C₄×D₄	D₄×C₄₁	1	4	yes	—
250	C₂×C₄⋊C₄	—	2	1	no	—
251	C₂×C₄⋊C₄	—	2	2	no	—
252	C₂×C₄⋊C₄	—	2	2	no	—
253	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
254	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
255	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
256	C₂×C₄²	—	2	1	no	—
257	C₂×C₄²	—	2	1	no	—
258	C₂×C₄²	D₄×C₄₁	1	2	yes	—
259	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
260	#107	—	2	4	no	—
261	C₄⋊C₈	—	2	2	no	—
262	#103	—	2	4	no	—
263	#102	—	2	4	no	—
264	C₂²⋊C₈	C₂³×C₄₁	1	4	yes	—
265	#97	—	2	2	no	—
266	#95	—	2	1	no	—
267	#95	—	2	2	no	—
268	#95	—	2	2	no	—
269	#95	C₂³×C₄₁	1	2	yes	—
270	#41	—	4	8	no	—
271	#41	—	4	8	no	—
272	#41	—	4	16	no	—
273	#41	—	4	8	no	—
274	C₂×Q₈	—	2	4	no	—
275	C₂×Q₈	—	2	2	no	—
276	C₂×Q₈	—	2	2	no	—
277	C₂×Q₈	—	2	4	no	—
278	C₂×Q₈	—	2	2	no	—
279	C₂×Q₈	—	4	2	no	—
280	C₂×Q₈	—	4	16	no	—
281	C₂×D₄	—	2	4	no	—
282	C₂×D₄	—	2	4	no	—
283	C₂×D₄	—	2	4	no	—
284	C₂×D₄	—	2	2	no	—
285	C₂×D₄	—	2	2	no	—
286	C₂×D₄	C₂×D₄×C₄₁	1	4	yes	—
287	C₂²×C₄	—	2	8	no	—
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	2	1	no	—
290	C₂²×C₄	—	2	4	no	—
291	C₂²×C₄	—	2	1	no	—
292	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
293	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
294	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
295	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
296	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
297	SD₁₆	—	4	8	no	—
298	C₂×C₈	—	2	4	no	—
299	C₄⋊C₄	—	2	4	no	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	2	4	no	—
303	C₄⋊C₄	—	2	2	no	—
304	C₄⋊C₄	—	4	4	no	—
305	C₄⋊C₄	—	4	2	no	—
306	C₄⋊C₄	—	4	4	no	—
307	C₄⋊C₄	C₂×D₄×C₄₁	1	4	yes	—
308	C₂²⋊C₄	—	2	8	no	—
309	C₂²⋊C₄	—	2	4	no	—
310	C₂²⋊C₄	—	2	4	no	—
311	C₄²	—	2	4	no	—
312	C₄²	—	2	2	no	—
313	C₄²	—	2	2	no	—
314	C₄²	—	4	1	no	—
315	C₄²	—	4	1	no	—
316	C₄²	C₂×D₄×C₄₁	1	2	yes	—
317	C₄²	C₂×D₄×C₄₁	1	2	yes	—
318	C₂³	—	2	4	no	—
319	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
320	Q₈	—	2	8	no	—
321	Q₈	—	2	4	no	—
322	Q₈	—	4	4	no	—
323	Q₈	—	4	8	no	—
324	Q₈	—	4	2	no	—
325	D₄	—	4	4	no	—
326	D₄	—	4	4	no	—
327	D₄	—	4	8	no	—
328	D₄	—	4	8	no	—
329	D₄	—	4	4	no	—
330	D₄	—	8	4	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	4	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	4	no	—
339	C₂×C₄	—	4	1	no	—
340	C₂×C₄	—	4	8	no	—
341	C₂×C₄	—	4	16	no	—
342	C₂×C₄	—	4	2	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	4	1	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
350	C₂×C₄	#120×C₄₁	1	2	yes	—
351	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
352	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	4	yes	—
353	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
354	C₈	—	4	4	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	4	4	no	—
358	C₂²	—	4	4	no	—
359	C₂²	—	4	4	no	—
360	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
361	C₂²	C₂×AGL(1,ℤ/8)×C₄₁	1	2	yes	—
362	C₄	—	2	2	no	—
363	C₄	—	2	4	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	2	no	—
366	C₄	—	4	4	no	—
367	C₄	—	4	4	no	—
368	C₄	—	4	2	no	—
369	C₄	—	4	2	no	—
370	C₄	—	8	1	no	—
371	C₄	C₄₁×#456	1	2	yes	—
372	C₂	—	2	2	no	—
373	C₂	—	8	4	no	—
374	C₂	C₄₁×#3003	1	1	yes	characteristic
375	C₂	C₂×C₄₁×#456	1	1	yes	characteristic
376	C₂	C₄₁×#3025	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×#3112	C₂	semidirect
3. C₄₁×#2399	C₂	semidirect
4. C₄₁×#1610	C₂	semidirect
5. C₄₁×#1604	C₂	semidirect
6. C₄₁×#1505	C₂	semidirect
7. C₄₁×#1467	C₂	semidirect
166. #18187	C₄₁	direct (N × H)
185. #3112	C₂×C₄₁	semidirect
186. #2399	C₂×C₄₁	semidirect
187. #1610	C₂×C₄₁	semidirect
188. #1604	C₂×C₄₁	semidirect
189. #1505	C₂×C₄₁	semidirect
190. #1467	C₂×C₄₁	semidirect
220. C₄₁	#18187	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×#3112	2
C₄₁×#2399	2
C₄₁×#1610	2
C₄₁×#1604	2
C₄₁×#1505	2
C₄₁×#1467	2
#18187	41

Subgroups — C41×#18187 #23794388

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#18187 #23794388