Subgroups — C₄₁×#18691 #23794892

Order 10496.

996 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×#3549	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#3254	C₂	1	2	yes	maximal, complemented
3	C₄₁×#3131	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#1646	C₂	1	4	yes	maximal, complemented
5	C₄₁×#1639	C₂	1	4	yes	maximal, complemented
6	C₄₁×#1560	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#1492	C₂	1	2	yes	maximal, complemented
8	C₄₁×H(3,𝔽₄)	—	2	2	no	—
9	C₄₁×#555	—	2	2	no	—
10	C₄₁×#555	C₂²	1	2	yes	—
11	D₄²×C₄₁	—	2	2	no	—
12	D₄²×C₄₁	C₂²	1	2	yes	—
13	C₄₁×#535	—	2	2	no	—
14	C₄₁×#535	—	2	4	no	—
15	C₄₁×#533	C₂²	1	1	yes	characteristic, complemented
16	C₄₁×#532	C₂²	1	1	yes	characteristic
17	C₄₁×#532	C₂²	1	2	yes	—
18	C₄₁×#488	—	2	2	no	—
19	C₄₁×#487	C₂²	1	2	yes	complemented
20	C₄₁×#486	C₂²	1	2	yes	—
21	C₄₁×#477	—	2	2	no	—
22	C₄₁×#474	C₂²	1	4	yes	complemented
23	C₄₁×#468	—	2	4	no	—
24	C₄₁×#461	C₂²	1	4	yes	complemented
25	C₄₁×#460	—	2	2	no	—
26	C₄₁×#423	C₂²	1	2	yes	complemented
27	C₄₁×#416	—	2	2	no	—
28	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic, complemented
29	C₄₁×#333	C₂²	1	2	yes	complemented
30	C₄₁×#331	C₂²	1	2	yes	—
31	C₄₁×D₄⋊C₈	C₂²	1	8	yes	complemented
32	C₂×#41×C₄₁	—	2	2	no	—
33	C₂×#41×C₄₁	D₄	1	2	yes	—
34	C₂²×D₄×C₄₁	—	2	2	no	—
35	C₂²×D₄×C₄₁	—	2	4	no	—
36	C₂×SD₁₆×C₄₁	—	2	4	no	—
37	C₂×M₁₆×C₄₁	—	2	2	no	—
38	C₂²×C₈×C₄₁	—	2	1	no	—
39	C₄⋊Q₈×C₄₁	D₄	1	2	yes	—
40	C₄⋊Q₈×C₄₁	C₂³	1	2	yes	—
41	C₄⋊Q₈×C₄₁	C₂³	1	1	yes	characteristic
42	#127×C₄₁	—	2	4	no	—
43	#127×C₄₁	D₄	1	2	yes	—
44	#127×C₄₁	C₂³	1	1	yes	characteristic
45	#125×C₄₁	—	2	1	no	—
46	#124×C₄₁	—	2	8	no	—
47	#124×C₄₁	—	2	4	no	—
48	#124×C₄₁	—	2	2	no	—
49	#124×C₄₁	D₄	1	2	yes	—
50	C₂²⋊Q₈×C₄₁	—	2	2	no	—
51	#121×C₄₁	—	2	4	no	—
52	#121×C₄₁	—	2	4	no	—
53	#121×C₄₁	—	2	2	no	—
54	#121×C₄₁	—	2	2	no	—
55	#121×C₄₁	—	2	4	no	—
56	#121×C₄₁	D₄	1	4	yes	—
57	#120×C₄₁	—	2	4	no	—
58	#120×C₄₁	—	2	8	no	—
59	#120×C₄₁	—	2	4	no	—
60	C₄×Q₈×C₄₁	—	2	2	no	—
61	C₄×D₄×C₄₁	—	2	2	no	—
62	C₄×D₄×C₄₁	—	2	2	no	—
63	C₄×D₄×C₄₁	C₂³	1	4	yes	—
64	#117×C₄₁	—	2	2	no	—
65	#117×C₄₁	—	2	4	no	—
66	#117×C₄₁	—	2	1	no	—
67	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
68	#107×C₄₁	—	2	4	no	—
69	C₄⋊C₈×C₄₁	—	2	2	no	—
70	C₄⋊C₈×C₄₁	C₂³	1	4	yes	—
71	#103×C₄₁	—	2	4	no	—
72	#103×C₄₁	—	2	4	no	—
73	#102×C₄₁	—	2	4	no	—
74	#102×C₄₁	—	2	4	no	—
75	C₄×C₈×C₄₁	—	2	1	no	—
76	C₄×C₈×C₄₁	C₂³	1	2	yes	—
77	C₂⁴×C₄₁	—	2	4	no	—
78	#41×C₄₁	—	4	4	no	—
79	#41×C₄₁	—	8	2	no	—
80	#41×C₄₁	—	8	2	no	—
81	C₂×Q₈×C₄₁	—	2	2	no	—
82	C₂×Q₈×C₄₁	—	2	2	no	—
83	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
84	C₂×D₄×C₄₁	—	2	4	no	—
85	C₂×D₄×C₄₁	—	2	8	no	—
86	C₂×D₄×C₄₁	—	2	2	no	—
87	C₂×D₄×C₄₁	—	2	4	no	—
88	C₂×D₄×C₄₁	—	2	2	no	—
89	C₂×D₄×C₄₁	—	2	2	no	—
90	C₂×D₄×C₄₁	—	2	2	no	—
91	C₂×D₄×C₄₁	—	8	4	no	—
92	C₂×D₄×C₄₁	—	8	4	no	—
93	C₂×D₄×C₄₁	—	8	4	no	—
94	C₂×D₄×C₄₁	C₂×D₄	1	4	yes	—
95	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
96	C₂²×C₄×C₄₁	—	2	2	no	—
97	C₂²×C₄×C₄₁	—	2	4	no	—
98	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
100	SD₁₆×C₄₁	—	8	4	no	—
101	M₁₆×C₄₁	—	8	2	no	—
102	C₂×C₈×C₄₁	—	2	2	no	—
103	C₂×C₈×C₄₁	—	2	4	no	—
104	C₂×C₈×C₄₁	—	4	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	2	no	—
109	C₄⋊C₄×C₄₁	—	4	2	no	—
110	C₄⋊C₄×C₄₁	C₂×D₄	1	4	yes	—
111	C₂²⋊C₄×C₄₁	—	2	8	no	—
112	C₂²⋊C₄×C₄₁	—	2	4	no	—
113	C₂²⋊C₄×C₄₁	—	2	4	no	—
114	C₂²⋊C₄×C₄₁	—	2	4	no	—
115	C₂²⋊C₄×C₄₁	—	2	4	no	—
116	C₂²⋊C₄×C₄₁	—	4	1	no	—
117	C₄²×C₄₁	—	2	4	no	—
118	C₄²×C₄₁	—	4	1	no	—
119	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₄₁	C₂×D₄	1	2	yes	—
121	C₄²×C₄₁	C₂⁴	1	1	yes	characteristic
122	C₂³×C₄₁	—	2	4	no	—
123	C₂³×C₄₁	—	2	2	no	—
124	C₂³×C₄₁	—	8	8	no	—
125	C₂³×C₄₁	—	8	4	no	—
126	C₂³×C₄₁	#120	1	1	yes	characteristic
127	Q₈×C₄₁	—	4	2	no	—
128	Q₈×C₄₁	—	4	2	no	—
129	Q₈×C₄₁	—	4	2	no	—
130	Q₈×C₄₁	—	8	2	no	—
131	D₄×C₄₁	—	2	4	no	—
132	D₄×C₄₁	—	2	4	no	—
133	D₄×C₄₁	—	4	2	no	—
134	D₄×C₄₁	—	4	2	no	—
135	D₄×C₄₁	—	4	2	no	—
136	D₄×C₄₁	—	4	4	no	—
137	D₄×C₄₁	—	8	2	no	—
138	D₄×C₄₁	—	8	2	no	—
139	D₄×C₄₁	—	8	4	no	—
140	D₄×C₄₁	—	8	8	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	—	4	4	no	—
148	C₂×C₄×C₄₁	—	4	4	no	—
149	C₂×C₄×C₄₁	—	4	4	no	—
150	C₂×C₄×C₄₁	—	4	2	no	—
151	C₂×C₄×C₄₁	#120	1	2	yes	—
152	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
153	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
155	C₈×C₄₁	—	4	2	no	—
156	C₈×C₄₁	—	4	4	no	—
157	#18691	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
158	C₂²×C₄₁	—	2	2	no	—
159	C₂²×C₄₁	—	2	1	no	—
160	C₂²×C₄₁	—	4	4	no	—
161	C₂²×C₄₁	—	4	4	no	—
162	C₂²×C₄₁	—	4	4	no	—
163	C₂²×C₄₁	—	4	2	no	—
164	C₂²×C₄₁	—	4	4	no	—
165	C₂²×C₄₁	—	16	8	no	—
166	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
167	C₄×C₄₁	—	2	2	no	—
168	C₄×C₄₁	—	2	1	no	—
169	C₄×C₄₁	—	4	4	no	—
170	C₄×C₄₁	—	4	2	no	—
171	C₄×C₄₁	—	8	2	no	—
172	C₄×C₄₁	#455	1	2	yes	—
173	C₄×C₄₁	#576	1	2	yes	—
174	C₄×C₄₁	#577	1	2	yes	—
175	#3549	C₂×C₄₁	1	1	yes	characteristic
176	#3254	C₂×C₄₁	1	2	yes	complemented
177	#3131	C₂×C₄₁	1	1	yes	characteristic, complemented
178	#1646	C₂×C₄₁	1	4	yes	complemented
179	#1639	C₂×C₄₁	1	4	yes	complemented
180	#1560	C₂×C₄₁	1	1	yes	characteristic, complemented
181	#1492	C₂×C₄₁	1	2	yes	complemented
182	C₂×C₄₁	—	4	1	no	—
183	C₂×C₄₁	—	8	4	no	—
184	C₂×C₄₁	—	8	2	no	—
185	C₂×C₄₁	#3012	1	2	yes	—
186	C₂×C₄₁	C₂×#455	1	1	yes	characteristic
187	H(3,𝔽₄)	—	2	2	no	—
188	#555	—	2	2	no	—
189	#555	C₂²×C₄₁	1	2	yes	—
190	D₄²	—	2	2	no	—
191	D₄²	C₂²×C₄₁	1	2	yes	—
192	#535	—	2	2	no	—
193	#535	—	2	4	no	—
194	#533	C₂²×C₄₁	1	1	yes	characteristic, complemented
195	#532	C₂²×C₄₁	1	1	yes	characteristic
196	#532	C₂²×C₄₁	1	2	yes	—
197	#488	—	2	2	no	—
198	#487	C₂²×C₄₁	1	2	yes	complemented
199	#486	C₂²×C₄₁	1	2	yes	—
200	#477	—	2	2	no	—
201	#474	C₂²×C₄₁	1	4	yes	complemented
202	#468	—	2	4	no	—
203	#461	C₂²×C₄₁	1	4	yes	complemented
204	#460	—	2	2	no	—
205	#423	C₂²×C₄₁	1	2	yes	complemented
206	#416	—	2	2	no	—
207	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic, complemented
208	#333	C₂²×C₄₁	1	2	yes	complemented
209	#331	C₂²×C₄₁	1	2	yes	—
210	D₄⋊C₈	C₂²×C₄₁	1	8	yes	complemented
211	C₄₁	#18691	1	1	yes	characteristic, direct product factor, Sylow
212	C₂×#41	—	2	2	no	—
213	C₂×#41	D₄×C₄₁	1	2	yes	—
214	C₂²×D₄	—	2	2	no	—
215	C₂²×D₄	—	2	4	no	—
216	C₂×SD₁₆	—	2	4	no	—
217	C₂×M₁₆	—	2	2	no	—
218	C₂²×C₈	—	2	1	no	—
219	C₄⋊Q₈	D₄×C₄₁	1	2	yes	—
220	C₄⋊Q₈	C₂³×C₄₁	1	2	yes	—
221	C₄⋊Q₈	C₂³×C₄₁	1	1	yes	characteristic
222	#127	—	2	4	no	—
223	#127	D₄×C₄₁	1	2	yes	—
224	#127	C₂³×C₄₁	1	1	yes	characteristic
225	#125	—	2	1	no	—
226	#124	—	2	8	no	—
227	#124	—	2	4	no	—
228	#124	—	2	2	no	—
229	#124	D₄×C₄₁	1	2	yes	—
230	C₂²⋊Q₈	—	2	2	no	—
231	#121	—	2	4	no	—
232	#121	—	2	4	no	—
233	#121	—	2	2	no	—
234	#121	—	2	2	no	—
235	#121	—	2	4	no	—
236	#121	D₄×C₄₁	1	4	yes	—
237	#120	—	2	4	no	—
238	#120	—	2	8	no	—
239	#120	—	2	4	no	—
240	C₄×Q₈	—	2	2	no	—
241	C₄×D₄	—	2	2	no	—
242	C₄×D₄	—	2	2	no	—
243	C₄×D₄	C₂³×C₄₁	1	4	yes	—
244	#117	—	2	2	no	—
245	#117	—	2	4	no	—
246	#117	—	2	1	no	—
247	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
248	#107	—	2	4	no	—
249	C₄⋊C₈	—	2	2	no	—
250	C₄⋊C₈	C₂³×C₄₁	1	4	yes	—
251	#103	—	2	4	no	—
252	#103	—	2	4	no	—
253	#102	—	2	4	no	—
254	#102	—	2	4	no	—
255	C₄×C₈	—	2	1	no	—
256	C₄×C₈	C₂³×C₄₁	1	2	yes	—
257	C₂⁴	—	2	4	no	—
258	#41	—	4	4	no	—
259	#41	—	8	2	no	—
260	#41	—	8	2	no	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
264	C₂×D₄	—	2	4	no	—
265	C₂×D₄	—	2	8	no	—
266	C₂×D₄	—	2	2	no	—
267	C₂×D₄	—	2	4	no	—
268	C₂×D₄	—	2	2	no	—
269	C₂×D₄	—	2	2	no	—
270	C₂×D₄	—	2	2	no	—
271	C₂×D₄	—	8	4	no	—
272	C₂×D₄	—	8	4	no	—
273	C₂×D₄	—	8	4	no	—
274	C₂×D₄	C₂×D₄×C₄₁	1	4	yes	—
275	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
276	C₂²×C₄	—	2	2	no	—
277	C₂²×C₄	—	2	4	no	—
278	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
279	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
280	SD₁₆	—	8	4	no	—
281	M₁₆	—	8	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	4	no	—
284	C₂×C₈	—	4	2	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	4	2	no	—
290	C₄⋊C₄	C₂×D₄×C₄₁	1	4	yes	—
291	C₂²⋊C₄	—	2	8	no	—
292	C₂²⋊C₄	—	2	4	no	—
293	C₂²⋊C₄	—	2	4	no	—
294	C₂²⋊C₄	—	2	4	no	—
295	C₂²⋊C₄	—	2	4	no	—
296	C₂²⋊C₄	—	4	1	no	—
297	C₄²	—	2	4	no	—
298	C₄²	—	4	1	no	—
299	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
300	C₄²	C₂×D₄×C₄₁	1	2	yes	—
301	C₄²	C₂⁴×C₄₁	1	1	yes	characteristic
302	C₂³	—	2	4	no	—
303	C₂³	—	2	2	no	—
304	C₂³	—	8	8	no	—
305	C₂³	—	8	4	no	—
306	C₂³	#120×C₄₁	1	1	yes	characteristic
307	Q₈	—	4	2	no	—
308	Q₈	—	4	2	no	—
309	Q₈	—	4	2	no	—
310	Q₈	—	8	2	no	—
311	D₄	—	2	4	no	—
312	D₄	—	2	4	no	—
313	D₄	—	4	2	no	—
314	D₄	—	4	2	no	—
315	D₄	—	4	2	no	—
316	D₄	—	4	4	no	—
317	D₄	—	8	2	no	—
318	D₄	—	8	2	no	—
319	D₄	—	8	4	no	—
320	D₄	—	8	8	no	—
321	D₄	—	8	4	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	4	no	—
327	C₂×C₄	—	4	4	no	—
328	C₂×C₄	—	4	4	no	—
329	C₂×C₄	—	4	4	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	#120×C₄₁	1	2	yes	—
332	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
333	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
334	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
335	C₈	—	4	2	no	—
336	C₈	—	4	4	no	—
337	C₂²	—	2	2	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	4	4	no	—
340	C₂²	—	4	4	no	—
341	C₂²	—	4	4	no	—
342	C₂²	—	4	2	no	—
343	C₂²	—	4	4	no	—
344	C₂²	—	16	8	no	—
345	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
346	C₄	—	2	2	no	—
347	C₄	—	2	1	no	—
348	C₄	—	4	4	no	—
349	C₄	—	4	2	no	—
350	C₄	—	8	2	no	—
351	C₄	C₄₁×#455	1	2	yes	—
352	C₄	C₄₁×#576	1	2	yes	—
353	C₄	C₄₁×#577	1	2	yes	—
354	C₂	—	4	1	no	—
355	C₂	—	8	4	no	—
356	C₂	—	8	2	no	—
357	C₂	C₄₁×#3012	1	2	yes	—
358	C₂	C₂×C₄₁×#455	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₄₁
prime radical	C₄₁×#3549	C₂
LCS	G ▷ C₄² ▷ 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×#3254	C₂	semidirect
3. C₄₁×#3131	C₂	semidirect
4. C₄₁×#1646	C₂	semidirect
5. C₄₁×#1639	C₂	semidirect
6. C₄₁×#1560	C₂	semidirect
7. C₄₁×#1492	C₂	semidirect
15. C₄₁×#533	C₂²	semidirect
19. C₄₁×#487	C₂²	semidirect
22. C₄₁×#474	C₂²	semidirect
24. C₄₁×#461	C₂²	semidirect
26. C₄₁×#423	C₂²	semidirect
28. C₂×C₄×C₈×C₄₁	C₂²	semidirect
29. C₄₁×#333	C₂²	semidirect
31. C₄₁×D₄⋊C₈	C₂²	semidirect
157. #18691	C₄₁	direct (N × H)
176. #3254	C₂×C₄₁	semidirect
177. #3131	C₂×C₄₁	semidirect
178. #1646	C₂×C₄₁	semidirect
179. #1639	C₂×C₄₁	semidirect
180. #1560	C₂×C₄₁	semidirect
181. #1492	C₂×C₄₁	semidirect
194. #533	C₂²×C₄₁	semidirect
198. #487	C₂²×C₄₁	semidirect
201. #474	C₂²×C₄₁	semidirect
203. #461	C₂²×C₄₁	semidirect
205. #423	C₂²×C₄₁	semidirect
207. C₂×C₄×C₈	C₂²×C₄₁	semidirect
208. #333	C₂²×C₄₁	semidirect
210. D₄⋊C₈	C₂²×C₄₁	semidirect
211. C₄₁	#18691	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×#3549	2
C₄₁×#3254	2
C₄₁×#3131	2
C₄₁×#1646	2
C₄₁×#1639	2
C₄₁×#1560	2
C₄₁×#1492	2
#18691	41

Subgroups — C41×#18691 #23794892

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#18691 #23794892