Subgroups — C₄₁×#20003 #23796204

Order 10496.

638 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3362	C₂	1	2	yes	maximal, complemented
2	C₄₁×#3305	C₂	1	2	yes	maximal, complemented
3	C₄₁×#1709	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#1678	C₂	1	2	yes	maximal, complemented
5	C₄₁×#1677	C₂	1	2	yes	maximal, complemented
6	C₄₁×#1634	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#1633	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1621	C₂	1	2	yes	maximal, complemented
9	C₈×SD₁₆×C₄₁	C₂	1	2	yes	maximal, complemented
10	Q₈²×C₄₁	C₂²	1	1	yes	characteristic
11	C₄₁×#555	C₂²	1	1	yes	characteristic
12	C₄₁×#550	C₂²	1	2	yes	—
13	C₄₁×#494	C₂²	1	1	yes	characteristic
14	C₄₁×#493	C₂²	1	1	yes	characteristic
15	C₄₁×#486	—	2	2	no	—
16	C₄₁×#486	C₂²	1	2	yes	—
17	C₄₁×#486	C₂²	1	2	yes	—
18	C₄₁×#482	—	2	2	no	—
19	C₄₁×#477	C₂²	1	2	yes	—
20	C₄₁×#476	—	2	2	no	—
21	C₄₁×#476	C₂²	1	2	yes	—
22	C₄₁×#466	—	2	2	no	—
23	C₄₁×#460	—	2	2	no	—
24	C₄₁×#460	C₂²	1	2	yes	—
25	C₄₁×#459	C₂²	1	2	yes	—
26	C₄₁×#449	—	2	2	no	—
27	C₈×Q₈×C₄₁	C₂²	1	2	yes	—
28	C₄×SD₁₆×C₄₁	—	2	2	no	—
29	C₄×SD₁₆×C₄₁	C₂²	1	2	yes	—
30	C₈×D₄×C₄₁	C₂²	1	2	yes	—
31	C₄₁×#416	—	2	2	no	—
32	C₄₁×#334	C₂²	1	2	yes	—
33	C₄₁×#333	C₂²	1	1	yes	characteristic
34	C₄₁×#333	C₂²	1	1	yes	characteristic
35	C₄₁×#331	C₂²	1	2	yes	—
36	C₄₁×Q₈⋊C₈	C₂²	1	2	yes	—
37	C₄₁×D₄⋊C₈	C₂²	1	2	yes	—
38	C₈²×C₄₁	C₂²	1	1	yes	characteristic
39	C₂×#41×C₄₁	—	2	2	no	—
40	C₂×Q₁₆×C₄₁	—	4	1	no	—
41	C₂×SD₁₆×C₄₁	—	2	2	no	—
42	C₂×SD₁₆×C₄₁	D₄	1	2	yes	—
43	C₂×D₈×C₄₁	—	2	2	no	—
44	C₂×D₈×C₄₁	—	4	1	no	—
45	C₂²×C₈×C₄₁	—	2	2	no	—
46	C₄⋊Q₈×C₄₁	—	2	2	no	—
47	C₄⋊Q₈×C₄₁	—	4	1	no	—
48	C₄⋊Q₈×C₄₁	C₂³	1	1	yes	characteristic
49	C₄⋊Q₈×C₄₁	C₂³	1	1	yes	characteristic
50	#127×C₄₁	—	2	2	no	—
51	#127×C₄₁	C₂³	1	2	yes	—
52	#124×C₄₁	—	4	1	no	—
53	#121×C₄₁	—	2	2	no	—
54	#121×C₄₁	—	2	2	no	—
55	#121×C₄₁	—	4	2	no	—
56	C₄×Q₈×C₄₁	—	2	2	no	—
57	C₄×Q₈×C₄₁	C₂³	1	2	yes	—
58	C₄×D₄×C₄₁	—	2	2	no	—
59	C₄×D₄×C₄₁	C₂³	1	2	yes	—
60	#117×C₄₁	—	2	2	no	—
61	#107×C₄₁	—	2	2	no	—
62	#106×C₄₁	—	2	2	no	—
63	#106×C₄₁	D₄	1	2	yes	—
64	C₄⋊C₈×C₄₁	—	2	2	no	—
65	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
66	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
67	#103×C₄₁	—	2	2	no	—
68	#103×C₄₁	D₄	1	2	yes	—
69	#102×C₄₁	—	2	2	no	—
70	#102×C₄₁	—	2	2	no	—
71	#102×C₄₁	—	2	2	no	—
72	#102×C₄₁	—	2	2	no	—
73	#102×C₄₁	—	2	2	no	—
74	#102×C₄₁	—	2	2	no	—
75	#102×C₄₁	—	2	2	no	—
76	#102×C₄₁	D₄	1	2	yes	—
77	C₂²⋊C₈×C₄₁	—	2	2	no	—
78	C₄×C₈×C₄₁	—	2	2	no	—
79	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₄₁	C₂³	1	2	yes	—
81	#41×C₄₁	—	4	4	no	—
82	#41×C₄₁	—	8	2	no	—
83	C₂×Q₈×C₄₁	—	2	1	no	—
84	C₂×Q₈×C₄₁	—	2	1	no	—
85	C₂×Q₈×C₄₁	—	4	1	no	—
86	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
87	C₂×D₄×C₄₁	—	2	2	no	—
88	C₂×D₄×C₄₁	—	2	2	no	—
89	C₂×D₄×C₄₁	—	2	2	no	—
90	C₂×D₄×C₄₁	—	4	2	no	—
91	C₂×D₄×C₄₁	—	4	2	no	—
92	C₂×D₄×C₄₁	—	4	1	no	—
93	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
94	C₂²×C₄×C₄₁	—	2	2	no	—
95	C₂²×C₄×C₄₁	—	4	2	no	—
96	Q₁₆×C₄₁	—	8	2	no	—
97	SD₁₆×C₄₁	—	2	4	no	—
98	SD₁₆×C₄₁	—	4	4	no	—
99	D₈×C₄₁	—	8	2	no	—
100	D₈×C₄₁	—	8	2	no	—
101	C₂×C₈×C₄₁	—	2	1	no	—
102	C₂×C₈×C₄₁	—	2	2	no	—
103	C₂×C₈×C₄₁	—	2	2	no	—
104	C₂×C₈×C₄₁	—	8	2	no	—
105	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
107	C₄⋊C₄×C₄₁	—	2	2	no	—
108	C₄⋊C₄×C₄₁	—	2	2	no	—
109	C₄⋊C₄×C₄₁	—	2	2	no	—
110	C₄⋊C₄×C₄₁	—	4	2	no	—
111	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
112	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
113	C₂²⋊C₄×C₄₁	—	2	2	no	—
114	C₂²⋊C₄×C₄₁	—	4	2	no	—
115	C₂²⋊C₄×C₄₁	—	4	2	no	—
116	C₄²×C₄₁	—	2	2	no	—
117	C₄²×C₄₁	—	4	1	no	—
118	C₄²×C₄₁	—	4	1	no	—
119	C₄²×C₄₁	C₂⁴	1	1	yes	characteristic
120	C₂³×C₄₁	—	2	2	no	—
121	C₂³×C₄₁	—	4	2	no	—
122	Q₈×C₄₁	—	2	2	no	—
123	Q₈×C₄₁	—	8	1	no	—
124	Q₈×C₄₁	—	8	1	no	—
125	Q₈×C₄₁	—	16	1	no	—
126	Q₈×C₄₁	#135	1	4	yes	—
127	D₄×C₄₁	—	2	2	no	—
128	D₄×C₄₁	—	2	2	no	—
129	D₄×C₄₁	—	4	2	no	—
130	D₄×C₄₁	—	4	2	no	—
131	D₄×C₄₁	—	4	2	no	—
132	D₄×C₄₁	—	4	2	no	—
133	D₄×C₄₁	—	8	4	no	—
134	D₄×C₄₁	—	8	2	no	—
135	D₄×C₄₁	—	16	2	no	—
136	D₄×C₄₁	—	16	1	no	—
137	C₂×C₄×C₄₁	—	2	2	no	—
138	C₂×C₄×C₄₁	—	2	2	no	—
139	C₂×C₄×C₄₁	—	2	2	no	—
140	C₂×C₄×C₄₁	—	4	2	no	—
141	C₂×C₄×C₄₁	—	4	2	no	—
142	C₂×C₄×C₄₁	—	4	1	no	—
143	C₂×C₄×C₄₁	—	4	1	no	—
144	C₂×C₄×C₄₁	—	8	4	no	—
145	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
146	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
147	C₈×C₄₁	—	2	2	no	—
148	C₈×C₄₁	—	2	2	no	—
149	C₈×C₄₁	—	2	2	no	—
150	C₈×C₄₁	—	4	2	no	—
151	C₈×C₄₁	—	4	2	no	—
152	#20003	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
153	C₂²×C₄₁	—	4	2	no	—
154	C₂²×C₄₁	—	4	2	no	—
155	C₂²×C₄₁	—	4	2	no	—
156	C₂²×C₄₁	—	8	2	no	—
157	C₂²×C₄₁	—	8	2	no	—
158	C₂²×C₄₁	—	8	2	no	—
159	C₂²×C₄₁	D₄²	1	1	yes	characteristic
160	C₄×C₄₁	—	2	4	no	—
161	C₄×C₄₁	—	2	1	no	—
162	C₄×C₄₁	—	4	2	no	—
163	C₄×C₄₁	—	4	2	no	—
164	C₄×C₄₁	—	8	1	no	—
165	C₄×C₄₁	—	8	1	no	—
166	C₄×C₄₁	C₂×#135	1	2	yes	—
167	C₄×C₄₁	#577	1	2	yes	—
168	#3362	C₂×C₄₁	1	2	yes	complemented
169	#3305	C₂×C₄₁	1	2	yes	complemented
170	#1709	C₂×C₄₁	1	1	yes	characteristic, complemented
171	#1678	C₂×C₄₁	1	2	yes	complemented
172	#1677	C₂×C₄₁	1	2	yes	complemented
173	#1634	C₂×C₄₁	1	1	yes	characteristic, complemented
174	#1633	C₂×C₄₁	1	1	yes	characteristic, complemented
175	#1621	C₂×C₄₁	1	2	yes	complemented
176	C₈×SD₁₆	C₂×C₄₁	1	2	yes	complemented
177	C₂×C₄₁	—	8	2	no	—
178	C₂×C₄₁	—	16	2	no	—
179	C₂×C₄₁	#3282	1	2	yes	—
180	C₂×C₄₁	#3292	1	1	yes	characteristic
181	Q₈²	C₂²×C₄₁	1	1	yes	characteristic
182	#555	C₂²×C₄₁	1	1	yes	characteristic
183	#550	C₂²×C₄₁	1	2	yes	—
184	#494	C₂²×C₄₁	1	1	yes	characteristic
185	#493	C₂²×C₄₁	1	1	yes	characteristic
186	#486	—	2	2	no	—
187	#486	C₂²×C₄₁	1	2	yes	—
188	#486	C₂²×C₄₁	1	2	yes	—
189	#482	—	2	2	no	—
190	#477	C₂²×C₄₁	1	2	yes	—
191	#476	—	2	2	no	—
192	#476	C₂²×C₄₁	1	2	yes	—
193	#466	—	2	2	no	—
194	#460	—	2	2	no	—
195	#460	C₂²×C₄₁	1	2	yes	—
196	#459	C₂²×C₄₁	1	2	yes	—
197	#449	—	2	2	no	—
198	C₈×Q₈	C₂²×C₄₁	1	2	yes	—
199	C₄×SD₁₆	—	2	2	no	—
200	C₄×SD₁₆	C₂²×C₄₁	1	2	yes	—
201	C₈×D₄	C₂²×C₄₁	1	2	yes	—
202	#416	—	2	2	no	—
203	#334	C₂²×C₄₁	1	2	yes	—
204	#333	C₂²×C₄₁	1	1	yes	characteristic
205	#333	C₂²×C₄₁	1	1	yes	characteristic
206	#331	C₂²×C₄₁	1	2	yes	—
207	Q₈⋊C₈	C₂²×C₄₁	1	2	yes	—
208	D₄⋊C₈	C₂²×C₄₁	1	2	yes	—
209	C₈²	C₂²×C₄₁	1	1	yes	characteristic
210	C₄₁	#20003	1	1	yes	characteristic, direct product factor, Sylow
211	C₂×#41	—	2	2	no	—
212	C₂×Q₁₆	—	4	1	no	—
213	C₂×SD₁₆	—	2	2	no	—
214	C₂×SD₁₆	D₄×C₄₁	1	2	yes	—
215	C₂×D₈	—	2	2	no	—
216	C₂×D₈	—	4	1	no	—
217	C₂²×C₈	—	2	2	no	—
218	C₄⋊Q₈	—	2	2	no	—
219	C₄⋊Q₈	—	4	1	no	—
220	C₄⋊Q₈	C₂³×C₄₁	1	1	yes	characteristic
221	C₄⋊Q₈	C₂³×C₄₁	1	1	yes	characteristic
222	#127	—	2	2	no	—
223	#127	C₂³×C₄₁	1	2	yes	—
224	#124	—	4	1	no	—
225	#121	—	2	2	no	—
226	#121	—	2	2	no	—
227	#121	—	4	2	no	—
228	C₄×Q₈	—	2	2	no	—
229	C₄×Q₈	C₂³×C₄₁	1	2	yes	—
230	C₄×D₄	—	2	2	no	—
231	C₄×D₄	C₂³×C₄₁	1	2	yes	—
232	#117	—	2	2	no	—
233	#107	—	2	2	no	—
234	#106	—	2	2	no	—
235	#106	D₄×C₄₁	1	2	yes	—
236	C₄⋊C₈	—	2	2	no	—
237	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
238	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
239	#103	—	2	2	no	—
240	#103	D₄×C₄₁	1	2	yes	—
241	#102	—	2	2	no	—
242	#102	—	2	2	no	—
243	#102	—	2	2	no	—
244	#102	—	2	2	no	—
245	#102	—	2	2	no	—
246	#102	—	2	2	no	—
247	#102	—	2	2	no	—
248	#102	D₄×C₄₁	1	2	yes	—
249	C₂²⋊C₈	—	2	2	no	—
250	C₄×C₈	—	2	2	no	—
251	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
252	C₄×C₈	C₂³×C₄₁	1	2	yes	—
253	#41	—	4	4	no	—
254	#41	—	8	2	no	—
255	C₂×Q₈	—	2	1	no	—
256	C₂×Q₈	—	2	1	no	—
257	C₂×Q₈	—	4	1	no	—
258	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	4	2	no	—
263	C₂×D₄	—	4	2	no	—
264	C₂×D₄	—	4	1	no	—
265	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	4	2	no	—
268	Q₁₆	—	8	2	no	—
269	SD₁₆	—	2	4	no	—
270	SD₁₆	—	4	4	no	—
271	D₈	—	8	2	no	—
272	D₈	—	8	2	no	—
273	C₂×C₈	—	2	1	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	8	2	no	—
277	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
278	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	4	2	no	—
283	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
284	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
285	C₂²⋊C₄	—	2	2	no	—
286	C₂²⋊C₄	—	4	2	no	—
287	C₂²⋊C₄	—	4	2	no	—
288	C₄²	—	2	2	no	—
289	C₄²	—	4	1	no	—
290	C₄²	—	4	1	no	—
291	C₄²	C₂⁴×C₄₁	1	1	yes	characteristic
292	C₂³	—	2	2	no	—
293	C₂³	—	4	2	no	—
294	Q₈	—	2	2	no	—
295	Q₈	—	8	1	no	—
296	Q₈	—	8	1	no	—
297	Q₈	—	16	1	no	—
298	Q₈	#135×C₄₁	1	4	yes	—
299	D₄	—	2	2	no	—
300	D₄	—	2	2	no	—
301	D₄	—	4	2	no	—
302	D₄	—	4	2	no	—
303	D₄	—	4	2	no	—
304	D₄	—	4	2	no	—
305	D₄	—	8	4	no	—
306	D₄	—	8	2	no	—
307	D₄	—	16	2	no	—
308	D₄	—	16	1	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	8	4	no	—
317	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
318	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
319	C₈	—	2	2	no	—
320	C₈	—	2	2	no	—
321	C₈	—	2	2	no	—
322	C₈	—	4	2	no	—
323	C₈	—	4	2	no	—
324	C₂²	—	4	2	no	—
325	C₂²	—	4	2	no	—
326	C₂²	—	4	2	no	—
327	C₂²	—	8	2	no	—
328	C₂²	—	8	2	no	—
329	C₂²	—	8	2	no	—
330	C₂²	D₄²×C₄₁	1	1	yes	characteristic
331	C₄	—	2	1	no	—
332	C₄	—	2	4	no	—
333	C₄	—	4	2	no	—
334	C₄	—	4	2	no	—
335	C₄	—	8	1	no	—
336	C₄	—	8	1	no	—
337	C₄	C₂×#135×C₄₁	1	2	yes	—
338	C₄	C₄₁×#577	1	2	yes	—
339	C₂	—	8	2	no	—
340	C₂	—	16	2	no	—
341	C₂	C₄₁×#3282	1	2	yes	—
342	C₂	C₄₁×#3292	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₄₁
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₁×#3362	C₂	semidirect
2. C₄₁×#3305	C₂	semidirect
3. C₄₁×#1709	C₂	semidirect
4. C₄₁×#1678	C₂	semidirect
5. C₄₁×#1677	C₂	semidirect
6. C₄₁×#1634	C₂	semidirect
7. C₄₁×#1633	C₂	semidirect
8. C₄₁×#1621	C₂	semidirect
9. C₈×SD₁₆×C₄₁	C₂	semidirect
152. #20003	C₄₁	direct (N × H)
168. #3362	C₂×C₄₁	semidirect
169. #3305	C₂×C₄₁	semidirect
170. #1709	C₂×C₄₁	semidirect
171. #1678	C₂×C₄₁	semidirect
172. #1677	C₂×C₄₁	semidirect
173. #1634	C₂×C₄₁	semidirect
174. #1633	C₂×C₄₁	semidirect
175. #1621	C₂×C₄₁	semidirect
176. C₈×SD₁₆	C₂×C₄₁	semidirect
210. C₄₁	#20003	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×#3362	2
C₄₁×#3305	2
C₄₁×#1709	2
C₄₁×#1678	2
C₄₁×#1677	2
C₄₁×#1634	2
C₄₁×#1633	2
C₄₁×#1621	2
C₈×SD₁₆×C₄₁	2
#20003	41

Subgroups — C41×#20003 #23796204

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#20003 #23796204