Subgroups — C₄₁×#33771 #23809972

Order 10496.

730 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3169	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄₁×#493	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#2246	C₂	1	2	yes	maximal, complemented
4	C₄₁×#2240	C₂	1	2	yes	maximal, complemented
5	C₄₁×#2206	C₂	1	4	yes	maximal, complemented
6	C₄₁×#2184	C₂	1	4	yes	maximal, complemented
7	C₄₁×#2162	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₄₁	—	2	1	no	—
9	C₂²×D₈×C₄₁	C₂²	1	2	yes	—
10	C₄₁×#532	—	2	1	no	—
11	C₂×#127×C₄₁	—	2	1	no	—
12	C₂×D₁₆×C₄₁	C₂²	1	8	yes	complemented
13	C₄₁×#500	C₂²	1	1	yes	characteristic, complemented
14	C₄₁×#493	—	2	1	no	—
15	C₄₁×#493	—	2	1	no	—
16	C₄₁×#493	C₂²	1	2	yes	—
17	C₄₁×#493	C₂²	1	2	yes	—
18	C₄₁×#493	C₂²	1	1	yes	characteristic, complemented
19	C₄₁×#486	—	2	1	no	—
20	C₄₁×#466	—	2	1	no	—
21	C₄₁×#466	C₂²	1	2	yes	—
22	C₄×D₈×C₄₁	—	2	1	no	—
23	C₄×D₈×C₄₁	C₂²	1	2	yes	complemented
24	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
25	C₄₁×C₄⋊C₁₆	C₂²	1	2	yes	complemented
26	C₄₁×#357	C₂²	1	8	yes	complemented
27	C₄₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
28	C₄×C₁₆×C₄₁	C₂²	1	2	yes	complemented
29	C₂×#41×C₄₁	—	4	1	no	—
30	C₂²×D₄×C₄₁	—	2	2	no	—
31	C₂²×D₄×C₄₁	—	4	1	no	—
32	C₂²×D₄×C₄₁	—	4	1	no	—
33	C₂×D₈×C₄₁	—	2	2	no	—
34	C₂×D₈×C₄₁	—	2	2	no	—
35	C₂×D₈×C₄₁	—	2	2	no	—
36	C₂×D₈×C₄₁	—	2	1	no	—
37	C₂×D₈×C₄₁	—	4	4	no	—
38	C₂×D₈×C₄₁	—	4	4	no	—
39	C₂×D₈×C₄₁	C₂³	1	2	yes	—
40	C₂×D₈×C₄₁	C₂³	1	4	yes	—
41	C₂²×C₈×C₄₁	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
43	C₄⋊Q₈×C₄₁	—	2	1	no	—
44	#127×C₄₁	—	2	2	no	—
45	#127×C₄₁	—	2	2	no	—
46	#127×C₄₁	—	2	1	no	—
47	#127×C₄₁	—	4	1	no	—
48	#127×C₄₁	—	4	1	no	—
49	#124×C₄₁	—	4	1	no	—
50	#121×C₄₁	—	2	2	no	—
51	#121×C₄₁	—	4	1	no	—
52	C₄×D₄×C₄₁	—	2	2	no	—
53	C₄×D₄×C₄₁	—	4	1	no	—
54	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
55	D₁₆×C₄₁	—	4	8	no	—
56	C₂×C₁₆×C₄₁	C₂³	1	4	yes	—
57	#107×C₄₁	—	2	1	no	—
58	#107×C₄₁	C₂³	1	2	yes	—
59	#102×C₄₁	—	2	2	no	—
60	#102×C₄₁	—	2	2	no	—
61	C₄×C₈×C₄₁	D₄	1	1	yes	characteristic
62	C₄×C₈×C₄₁	D₄	1	1	yes	characteristic
63	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
65	C₂⁴×C₄₁	—	4	2	no	—
66	#41×C₄₁	—	8	4	no	—
67	C₂×Q₈×C₄₁	—	4	1	no	—
68	C₂×D₄×C₄₁	—	2	2	no	—
69	C₂×D₄×C₄₁	—	2	4	no	—
70	C₂×D₄×C₄₁	—	4	4	no	—
71	C₂×D₄×C₄₁	—	4	4	no	—
72	C₂×D₄×C₄₁	—	4	2	no	—
73	C₂×D₄×C₄₁	—	4	1	no	—
74	C₂×D₄×C₄₁	—	4	1	no	—
75	C₂×D₄×C₄₁	—	4	2	no	—
76	C₂×D₄×C₄₁	—	4	2	no	—
77	C₂×D₄×C₄₁	—	4	2	no	—
78	C₂×D₄×C₄₁	—	4	2	no	—
79	C₂×D₄×C₄₁	—	4	2	no	—
80	C₂×D₄×C₄₁	—	8	4	no	—
81	C₂×D₄×C₄₁	—	8	4	no	—
82	C₂²×C₄×C₄₁	—	4	2	no	—
83	C₂²×C₄×C₄₁	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₄₁	D₈	1	1	yes	characteristic
85	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
86	D₈×C₄₁	—	2	4	no	—
87	D₈×C₄₁	—	2	8	no	—
88	D₈×C₄₁	—	4	4	no	—
89	D₈×C₄₁	—	4	4	no	—
90	D₈×C₄₁	—	4	2	no	—
91	D₈×C₄₁	—	4	4	no	—
92	C₂×C₈×C₄₁	—	2	2	no	—
93	C₂×C₈×C₄₁	—	2	2	no	—
94	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₄₁	C₂⁴	1	1	yes	characteristic
98	C₄⋊C₄×C₄₁	—	2	2	no	—
99	C₄⋊C₄×C₄₁	—	4	1	no	—
100	C₂²⋊C₄×C₄₁	—	4	2	no	—
101	C₂²⋊C₄×C₄₁	—	4	2	no	—
102	C₄²×C₄₁	D₈	1	1	yes	characteristic
103	C₄²×C₄₁	D₈	1	1	yes	characteristic
104	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
106	C₁₆×C₄₁	—	2	4	no	—
107	C₂³×C₄₁	—	4	2	no	—
108	C₂³×C₄₁	—	4	4	no	—
109	C₂³×C₄₁	—	8	4	no	—
110	C₂³×C₄₁	—	8	4	no	—
111	C₂³×C₄₁	—	8	4	no	—
112	C₂³×C₄₁	C₂×D₈	1	1	yes	characteristic
113	Q₈×C₄₁	—	8	2	no	—
114	D₄×C₄₁	—	4	4	no	—
115	D₄×C₄₁	—	4	2	no	—
116	D₄×C₄₁	—	4	4	no	—
117	D₄×C₄₁	—	4	2	no	—
118	D₄×C₄₁	—	8	4	no	—
119	D₄×C₄₁	—	8	4	no	—
120	D₄×C₄₁	—	8	4	no	—
121	D₄×C₄₁	—	8	4	no	—
122	D₄×C₄₁	—	8	4	no	—
123	D₄×C₄₁	—	8	2	no	—
124	D₄×C₄₁	—	8	2	no	—
125	D₄×C₄₁	—	8	4	no	—
126	C₂×C₄×C₄₁	—	2	1	no	—
127	C₂×C₄×C₄₁	—	2	1	no	—
128	C₂×C₄×C₄₁	—	2	2	no	—
129	C₂×C₄×C₄₁	—	4	2	no	—
130	C₂×C₄×C₄₁	—	8	4	no	—
131	C₂×C₄×C₄₁	D₁₆	1	4	yes	complemented
132	C₂×C₄×C₄₁	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₄₁	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₄₁	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₄₁	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₄₁	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
138	C₈×C₄₁	—	2	1	no	—
139	C₈×C₄₁	—	2	1	no	—
140	C₈×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
141	C₈×C₄₁	H(5,𝔽₂)	1	2	yes	—
142	#33771	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂²×C₄₁	—	2	1	no	—
144	C₂²×C₄₁	—	2	1	no	—
145	C₂²×C₄₁	—	8	2	no	—
146	C₂²×C₄₁	—	8	4	no	—
147	C₂²×C₄₁	—	8	4	no	—
148	C₂²×C₄₁	—	8	2	no	—
149	C₂²×C₄₁	—	8	4	no	—
150	C₂²×C₄₁	—	8	2	no	—
151	C₂²×C₄₁	—	16	8	no	—
152	C₂²×C₄₁	C₂×D₁₆	1	2	yes	—
153	C₂²×C₄₁	C₂²×D₈	1	1	yes	characteristic
154	C₄×C₄₁	—	2	1	no	—
155	C₄×C₄₁	—	2	1	no	—
156	C₄×C₄₁	—	8	2	no	—
157	C₄×C₄₁	C₂×D₁₆	1	2	yes	—
158	C₄×C₄₁	C₂×D₁₆	1	2	yes	—
159	C₄×C₄₁	#509	1	2	yes	—
160	C₄×C₄₁	#535	1	1	yes	characteristic
161	C₄×C₄₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
162	#3169	C₂×C₄₁	1	1	yes	characteristic, complemented
163	C₂×#493	C₂×C₄₁	1	1	yes	characteristic, complemented
164	#2246	C₂×C₄₁	1	2	yes	complemented
165	#2240	C₂×C₄₁	1	2	yes	complemented
166	#2206	C₂×C₄₁	1	4	yes	complemented
167	#2184	C₂×C₄₁	1	4	yes	complemented
168	#2162	C₂×C₄₁	1	1	yes	characteristic, complemented
169	C₂×C₄₁	—	2	2	no	—
170	C₂×C₄₁	—	16	2	no	—
171	C₂×C₄₁	—	16	4	no	—
172	C₂×C₄₁	#3205	1	1	yes	characteristic
173	C₂×C₄₁	C₂²×D₁₆	1	1	yes	characteristic
174	C₂×C₄₁	C₂×#509	1	1	yes	characteristic
175	C₂²×D₈	—	2	1	no	—
176	C₂²×D₈	C₂²×C₄₁	1	2	yes	—
177	#532	—	2	1	no	—
178	C₂×#127	—	2	1	no	—
179	C₂×D₁₆	C₂²×C₄₁	1	8	yes	complemented
180	#500	C₂²×C₄₁	1	1	yes	characteristic, complemented
181	#493	—	2	1	no	—
182	#493	—	2	1	no	—
183	#493	C₂²×C₄₁	1	2	yes	—
184	#493	C₂²×C₄₁	1	2	yes	—
185	#493	C₂²×C₄₁	1	1	yes	characteristic, complemented
186	#486	—	2	1	no	—
187	#466	—	2	1	no	—
188	#466	C₂²×C₄₁	1	2	yes	—
189	C₄×D₈	—	2	1	no	—
190	C₄×D₈	C₂²×C₄₁	1	2	yes	complemented
191	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
192	C₄⋊C₁₆	C₂²×C₄₁	1	2	yes	complemented
193	#357	C₂²×C₄₁	1	8	yes	complemented
194	C₂²⋊C₁₆	C₂²×C₄₁	1	2	yes	—
195	C₄×C₁₆	C₂²×C₄₁	1	2	yes	complemented
196	C₄₁	#33771	1	1	yes	characteristic, direct product factor, Sylow
197	C₂×#41	—	4	1	no	—
198	C₂²×D₄	—	2	2	no	—
199	C₂²×D₄	—	4	1	no	—
200	C₂²×D₄	—	4	1	no	—
201	C₂×D₈	—	2	2	no	—
202	C₂×D₈	—	2	2	no	—
203	C₂×D₈	—	2	2	no	—
204	C₂×D₈	—	2	1	no	—
205	C₂×D₈	—	4	4	no	—
206	C₂×D₈	—	4	4	no	—
207	C₂×D₈	C₂³×C₄₁	1	4	yes	—
208	C₂×D₈	C₂³×C₄₁	1	2	yes	—
209	C₂²×C₈	D₄×C₄₁	1	1	yes	characteristic
210	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
211	C₄⋊Q₈	—	2	1	no	—
212	#127	—	2	2	no	—
213	#127	—	2	2	no	—
214	#127	—	2	1	no	—
215	#127	—	4	1	no	—
216	#127	—	4	1	no	—
217	#124	—	4	1	no	—
218	#121	—	2	2	no	—
219	#121	—	4	1	no	—
220	C₄×D₄	—	2	2	no	—
221	C₄×D₄	—	4	1	no	—
222	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
223	D₁₆	—	4	8	no	—
224	C₂×C₁₆	C₂³×C₄₁	1	4	yes	—
225	#107	—	2	1	no	—
226	#107	C₂³×C₄₁	1	2	yes	—
227	#102	—	2	2	no	—
228	#102	—	2	2	no	—
229	C₄×C₈	D₄×C₄₁	1	1	yes	characteristic
230	C₄×C₈	D₄×C₄₁	1	1	yes	characteristic
231	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
232	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
233	C₂⁴	—	4	2	no	—
234	#41	—	8	4	no	—
235	C₂×Q₈	—	4	1	no	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	—	2	4	no	—
238	C₂×D₄	—	4	4	no	—
239	C₂×D₄	—	4	4	no	—
240	C₂×D₄	—	4	2	no	—
241	C₂×D₄	—	4	1	no	—
242	C₂×D₄	—	4	1	no	—
243	C₂×D₄	—	4	2	no	—
244	C₂×D₄	—	4	2	no	—
245	C₂×D₄	—	4	2	no	—
246	C₂×D₄	—	4	2	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂×D₄	—	8	4	no	—
249	C₂×D₄	—	8	4	no	—
250	C₂²×C₄	—	4	2	no	—
251	C₂²×C₄	D₈×C₄₁	1	1	yes	characteristic
252	C₂²×C₄	D₈×C₄₁	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
254	D₈	—	2	4	no	—
255	D₈	—	2	8	no	—
256	D₈	—	4	4	no	—
257	D₈	—	4	4	no	—
258	D₈	—	4	4	no	—
259	D₈	—	4	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
263	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
264	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
265	C₂×C₈	C₂⁴×C₄₁	1	1	yes	characteristic
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₂²⋊C₄	—	4	2	no	—
269	C₂²⋊C₄	—	4	2	no	—
270	C₄²	D₈×C₄₁	1	1	yes	characteristic
271	C₄²	D₈×C₄₁	1	1	yes	characteristic
272	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
273	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
274	C₁₆	—	2	4	no	—
275	C₂³	—	4	2	no	—
276	C₂³	—	4	4	no	—
277	C₂³	—	8	4	no	—
278	C₂³	—	8	4	no	—
279	C₂³	—	8	4	no	—
280	C₂³	C₂×D₈×C₄₁	1	1	yes	characteristic
281	Q₈	—	8	2	no	—
282	D₄	—	4	4	no	—
283	D₄	—	4	2	no	—
284	D₄	—	4	4	no	—
285	D₄	—	4	2	no	—
286	D₄	—	8	4	no	—
287	D₄	—	8	4	no	—
288	D₄	—	8	4	no	—
289	D₄	—	8	4	no	—
290	D₄	—	8	4	no	—
291	D₄	—	8	2	no	—
292	D₄	—	8	4	no	—
293	D₄	—	8	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	8	4	no	—
299	C₂×C₄	D₁₆×C₄₁	1	4	yes	complemented
300	C₂×C₄	C₂×D₈×C₄₁	1	1	yes	characteristic
301	C₂×C₄	C₂×D₈×C₄₁	1	1	yes	characteristic
302	C₂×C₄	C₂×D₈×C₄₁	1	1	yes	characteristic
303	C₂×C₄	C₂×D₈×C₄₁	1	1	yes	characteristic
304	C₂×C₄	C₂×D₈×C₄₁	1	1	yes	characteristic
305	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₈	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
309	C₈	H(5,𝔽₂)×C₄₁	1	2	yes	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	4	no	—
314	C₂²	—	8	4	no	—
315	C₂²	—	8	2	no	—
316	C₂²	—	8	4	no	—
317	C₂²	—	8	2	no	—
318	C₂²	—	16	8	no	—
319	C₂²	C₂×D₁₆×C₄₁	1	2	yes	—
320	C₂²	C₂²×D₈×C₄₁	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	8	2	no	—
324	C₄	C₂×D₁₆×C₄₁	1	2	yes	—
325	C₄	C₂×D₁₆×C₄₁	1	2	yes	—
326	C₄	C₄₁×#509	1	2	yes	—
327	C₄	C₄₁×#535	1	1	yes	characteristic
328	C₄	C₂×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
329	C₂	—	2	2	no	—
330	C₂	—	16	4	no	—
331	C₂	—	16	2	no	—
332	C₂	C₄₁×#3205	1	1	yes	characteristic
333	C₂	C₂²×D₁₆×C₄₁	1	1	yes	characteristic
334	C₂	C₂×C₄₁×#509	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₄₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₂×C₄²×C₄₁ ◁ C₂×C₄×C₈×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₁×#3169	C₂	semidirect
2. C₂×C₄₁×#493	C₂	semidirect
3. C₄₁×#2246	C₂	semidirect
4. C₄₁×#2240	C₂	semidirect
5. C₄₁×#2206	C₂	semidirect
6. C₄₁×#2184	C₂	semidirect
7. C₄₁×#2162	C₂	semidirect
12. C₂×D₁₆×C₄₁	C₂²	semidirect
13. C₄₁×#500	C₂²	semidirect
18. C₄₁×#493	C₂²	semidirect
23. C₄×D₈×C₄₁	C₂²	semidirect
25. C₄₁×C₄⋊C₁₆	C₂²	semidirect
26. C₄₁×#357	C₂²	semidirect
28. C₄×C₁₆×C₄₁	C₂²	semidirect
131. C₂×C₄×C₄₁	D₁₆	semidirect
142. #33771	C₄₁	direct (N × H)
162. #3169	C₂×C₄₁	semidirect
163. C₂×#493	C₂×C₄₁	semidirect
164. #2246	C₂×C₄₁	semidirect
165. #2240	C₂×C₄₁	semidirect
166. #2206	C₂×C₄₁	semidirect
167. #2184	C₂×C₄₁	semidirect
168. #2162	C₂×C₄₁	semidirect
179. C₂×D₁₆	C₂²×C₄₁	semidirect
180. #500	C₂²×C₄₁	semidirect
185. #493	C₂²×C₄₁	semidirect
190. C₄×D₈	C₂²×C₄₁	semidirect
192. C₄⋊C₁₆	C₂²×C₄₁	semidirect
193. #357	C₂²×C₄₁	semidirect
195. C₄×C₁₆	C₂²×C₄₁	semidirect
196. C₄₁	#33771	direct (N × H)
299. C₂×C₄	D₁₆×C₄₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄₁×#3169	2
C₂×C₄₁×#493	2
C₄₁×#2246	2
C₄₁×#2240	2
C₄₁×#2206	2
C₄₁×#2184	2
C₄₁×#2162	2
#33771	41

Subgroups — C41×#33771 #23809972

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#33771 #23809972