Subgroups — C₄₁×#33784 #23809985

Order 10496.

584 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3168	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#2251	C₂	1	2	yes	maximal, complemented
4	C₄₁×#2245	C₂	1	2	yes	maximal, complemented
5	C₄₁×#2209	C₂	1	2	yes	maximal, complemented
6	C₄₁×#2208	C₂	1	2	yes	maximal, complemented
7	C₄₁×#2188	C₂	1	2	yes	maximal, complemented
8	C₄₁×#2185	C₂	1	2	yes	maximal
9	C₄₁×#2163	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Q₁₆×C₄₁	—	2	1	no	—
11	C₂²×Q₁₆×C₄₁	C₂²	1	2	yes	—
12	C₂×C₄⋊Q₈×C₄₁	—	2	1	no	—
13	C₄₁×#529	—	2	1	no	—
14	C₂×Q₃₂×C₄₁	C₂²	1	4	yes	—
15	C₂×SD₃₂×C₄₁	C₂²	1	4	yes	—
16	C₄₁×#499	C₂²	1	1	yes	characteristic
17	C₄₁×#495	C₂²	1	1	yes	characteristic
18	C₄₁×#494	—	2	1	no	—
19	C₄₁×#494	—	2	1	no	—
20	C₄₁×#494	C₂²	1	2	yes	—
21	C₄₁×#494	C₂²	1	2	yes	—
22	C₄₁×#488	—	2	1	no	—
23	C₄₁×#467	C₂²	1	1	yes	characteristic
24	C₄₁×#466	C₂²	1	1	yes	characteristic
25	C₄₁×#465	—	2	1	no	—
26	C₄×Q₁₆×C₄₁	C₂²	1	1	yes	characteristic
27	C₄×SD₁₆×C₄₁	—	2	1	no	—
28	C₄×D₈×C₄₁	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
30	C₄₁×C₄⋊C₁₆	C₂²	1	2	yes	—
31	C₄₁×#358	C₂²	1	4	yes	—
32	C₄₁×#358	C₂²	1	4	yes	—
33	C₄₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
34	C₄₁×#346	C₂²	1	2	yes	—
35	C₂²×Q₈×C₄₁	—	2	2	no	—
36	C₂×Q₁₆×C₄₁	—	2	2	no	—
37	C₂×Q₁₆×C₄₁	—	2	2	no	—
38	C₂×Q₁₆×C₄₁	—	2	2	no	—
39	C₂×Q₁₆×C₄₁	—	4	4	no	—
40	C₂×Q₁₆×C₄₁	—	4	4	no	—
41	C₂×Q₁₆×C₄₁	C₂³	1	4	yes	—
42	C₂×Q₁₆×C₄₁	C₂³	1	1	yes	characteristic
43	C₂×SD₁₆×C₄₁	—	2	1	no	—
44	C₂×D₈×C₄₁	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₄₁	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
47	C₄⋊Q₈×C₄₁	—	2	2	no	—
48	C₄⋊Q₈×C₄₁	—	2	2	no	—
49	C₄⋊Q₈×C₄₁	—	4	1	no	—
50	C₄⋊Q₈×C₄₁	—	4	1	no	—
51	#126×C₄₁	—	4	1	no	—
52	#125×C₄₁	—	2	1	no	—
53	#124×C₄₁	—	2	1	no	—
54	#123×C₄₁	—	4	1	no	—
55	C₂²⋊Q₈×C₄₁	—	2	1	no	—
56	#121×C₄₁	—	2	1	no	—
57	C₄×Q₈×C₄₁	—	2	1	no	—
58	C₄×D₄×C₄₁	—	2	1	no	—
59	C₄×D₄×C₄₁	—	4	1	no	—
60	#117×C₄₁	—	4	1	no	—
61	C₂×C₄⋊C₄×C₄₁	—	4	1	no	—
62	C₂×C₄⋊C₄×C₄₁	—	4	1	no	—
63	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
64	Q₃₂×C₄₁	—	4	4	no	—
65	SD₃₂×C₄₁	—	4	4	no	—
66	C₂×C₁₆×C₄₁	C₂³	1	4	yes	—
67	#107×C₄₁	C₂³	1	1	yes	characteristic
68	#107×C₄₁	C₂³	1	1	yes	characteristic
69	#106×C₄₁	—	2	1	no	—
70	#103×C₄₁	—	2	1	no	—
71	#103×C₄₁	—	2	1	no	—
72	#102×C₄₁	—	2	1	no	—
73	#102×C₄₁	—	2	1	no	—
74	C₄×C₈×C₄₁	D₄	1	1	yes	characteristic
75	C₄×C₈×C₄₁	D₄	1	1	yes	characteristic
76	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
78	C₂×Q₈×C₄₁	—	2	1	no	—
79	C₂×Q₈×C₄₁	—	2	4	no	—
80	C₂×Q₈×C₄₁	—	4	4	no	—
81	C₂×Q₈×C₄₁	—	4	4	no	—
82	C₂×Q₈×C₄₁	—	4	2	no	—
83	C₂×D₄×C₄₁	—	2	1	no	—
84	C₂×D₄×C₄₁	—	4	1	no	—
85	C₂²×C₄×C₄₁	—	4	2	no	—
86	C₂²×C₄×C₄₁	—	4	1	no	—
87	C₂²×C₄×C₄₁	D₈	1	1	yes	characteristic
88	C₂²×C₄×C₄₁	D₈	1	1	yes	characteristic
89	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
90	Q₁₆×C₄₁	—	2	4	no	—
91	Q₁₆×C₄₁	—	2	2	no	—
92	Q₁₆×C₄₁	—	2	4	no	—
93	Q₁₆×C₄₁	—	4	4	no	—
94	Q₁₆×C₄₁	—	4	4	no	—
95	Q₁₆×C₄₁	—	4	4	no	—
96	SD₁₆×C₄₁	—	4	2	no	—
97	D₈×C₄₁	—	2	2	no	—
98	C₂×C₈×C₄₁	—	2	2	no	—
99	C₂×C₈×C₄₁	—	2	2	no	—
100	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₄₁	C₂⁴	1	1	yes	characteristic
104	C₄⋊C₄×C₄₁	—	2	1	no	—
105	C₄⋊C₄×C₄₁	—	2	1	no	—
106	C₄⋊C₄×C₄₁	—	4	2	no	—
107	C₄⋊C₄×C₄₁	—	4	2	no	—
108	C₄⋊C₄×C₄₁	—	4	2	no	—
109	C₄⋊C₄×C₄₁	—	4	2	no	—
110	C₄⋊C₄×C₄₁	—	4	1	no	—
111	C₄⋊C₄×C₄₁	—	4	1	no	—
112	C₄⋊C₄×C₄₁	—	4	1	no	—
113	C₄⋊C₄×C₄₁	—	4	1	no	—
114	C₂²⋊C₄×C₄₁	—	4	1	no	—
115	C₂²⋊C₄×C₄₁	—	4	1	no	—
116	C₂²⋊C₄×C₄₁	—	4	1	no	—
117	C₂²⋊C₄×C₄₁	—	4	1	no	—
118	C₂²⋊C₄×C₄₁	—	4	1	no	—
119	C₄²×C₄₁	—	4	1	no	—
120	C₄²×C₄₁	D₈	1	1	yes	characteristic
121	C₄²×C₄₁	D₈	1	1	yes	characteristic
122	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
124	C₁₆×C₄₁	—	2	4	no	—
125	C₂³×C₄₁	—	4	1	no	—
126	C₂³×C₄₁	C₂×D₈	1	1	yes	characteristic
127	Q₈×C₄₁	—	4	4	no	—
128	Q₈×C₄₁	—	4	1	no	—
129	Q₈×C₄₁	—	4	1	no	—
130	Q₈×C₄₁	—	4	4	no	—
131	Q₈×C₄₁	—	8	4	no	—
132	D₄×C₄₁	—	4	1	no	—
133	D₄×C₄₁	—	4	1	no	—
134	D₄×C₄₁	—	8	2	no	—
135	C₂×C₄×C₄₁	—	2	2	no	—
136	C₂×C₄×C₄₁	—	2	1	no	—
137	C₂×C₄×C₄₁	—	2	1	no	—
138	C₂×C₄×C₄₁	—	2	2	no	—
139	C₂×C₄×C₄₁	—	4	1	no	—
140	C₂×C₄×C₄₁	—	4	1	no	—
141	C₂×C₄×C₄₁	—	4	1	no	—
142	C₂×C₄×C₄₁	—	4	4	no	—
143	C₂×C₄×C₄₁	—	8	4	no	—
144	C₂×C₄×C₄₁	—	8	2	no	—
145	C₂×C₄×C₄₁	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₄₁	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₄₁	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₄₁	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₄₁	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
151	C₈×C₄₁	—	2	1	no	—
152	C₈×C₄₁	—	2	1	no	—
153	C₈×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
154	C₈×C₄₁	H(5,𝔽₂)	1	2	yes	—
155	#33784	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
156	C₂²×C₄₁	—	2	1	no	—
157	C₂²×C₄₁	—	2	1	no	—
158	C₂²×C₄₁	—	8	1	no	—
159	C₂²×C₄₁	—	8	1	no	—
160	C₂²×C₄₁	—	8	1	no	—
161	C₂²×C₄₁	#510	1	2	yes	—
162	C₂²×C₄₁	C₂²×D₈	1	1	yes	characteristic
163	C₄×C₄₁	—	2	1	no	—
164	C₄×C₄₁	—	2	1	no	—
165	C₄×C₄₁	—	2	1	no	—
166	C₄×C₄₁	—	2	1	no	—
167	C₄×C₄₁	—	8	1	no	—
168	C₄×C₄₁	—	8	1	no	—
169	C₄×C₄₁	—	8	4	no	—
170	C₄×C₄₁	—	8	1	no	—
171	C₄×C₄₁	#510	1	2	yes	—
172	C₄×C₄₁	#535	1	1	yes	characteristic
173	C₄×C₄₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
174	#3168	C₂×C₄₁	1	1	yes	characteristic
175	C₂×#494	C₂×C₄₁	1	1	yes	characteristic, complemented
176	#2251	C₂×C₄₁	1	2	yes	complemented
177	#2245	C₂×C₄₁	1	2	yes	complemented
178	#2209	C₂×C₄₁	1	2	yes	complemented
179	#2208	C₂×C₄₁	1	2	yes	complemented
180	#2188	C₂×C₄₁	1	2	yes	complemented
181	#2185	C₂×C₄₁	1	2	yes	—
182	#2163	C₂×C₄₁	1	1	yes	characteristic, complemented
183	C₂×C₄₁	—	2	2	no	—
184	C₂×C₄₁	—	16	1	no	—
185	C₂×C₄₁	#3205	1	1	yes	characteristic
186	C₂×C₄₁	C₂×#510	1	1	yes	characteristic
187	C₂×C₄₁	C₂×#510	1	1	yes	characteristic
188	C₂²×Q₁₆	—	2	1	no	—
189	C₂²×Q₁₆	C₂²×C₄₁	1	2	yes	—
190	C₂×C₄⋊Q₈	—	2	1	no	—
191	#529	—	2	1	no	—
192	C₂×Q₃₂	C₂²×C₄₁	1	4	yes	—
193	C₂×SD₃₂	C₂²×C₄₁	1	4	yes	—
194	#499	C₂²×C₄₁	1	1	yes	characteristic
195	#495	C₂²×C₄₁	1	1	yes	characteristic
196	#494	—	2	1	no	—
197	#494	—	2	1	no	—
198	#494	C₂²×C₄₁	1	2	yes	—
199	#494	C₂²×C₄₁	1	2	yes	—
200	#488	—	2	1	no	—
201	#467	C₂²×C₄₁	1	1	yes	characteristic
202	#466	C₂²×C₄₁	1	1	yes	characteristic
203	#465	—	2	1	no	—
204	C₄×Q₁₆	C₂²×C₄₁	1	1	yes	characteristic
205	C₄×SD₁₆	—	2	1	no	—
206	C₄×D₈	C₂²×C₄₁	1	1	yes	characteristic
207	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
208	C₄⋊C₁₆	C₂²×C₄₁	1	2	yes	—
209	#358	C₂²×C₄₁	1	4	yes	—
210	#358	C₂²×C₄₁	1	4	yes	—
211	C₂²⋊C₁₆	C₂²×C₄₁	1	2	yes	—
212	#346	C₂²×C₄₁	1	2	yes	—
213	C₄₁	#33784	1	1	yes	characteristic, direct product factor, Sylow
214	C₂²×Q₈	—	2	2	no	—
215	C₂×Q₁₆	—	2	2	no	—
216	C₂×Q₁₆	—	2	2	no	—
217	C₂×Q₁₆	—	2	2	no	—
218	C₂×Q₁₆	—	4	4	no	—
219	C₂×Q₁₆	—	4	4	no	—
220	C₂×Q₁₆	C₂³×C₄₁	1	4	yes	—
221	C₂×Q₁₆	C₂³×C₄₁	1	1	yes	characteristic
222	C₂×SD₁₆	—	2	1	no	—
223	C₂×D₈	C₂³×C₄₁	1	1	yes	characteristic
224	C₂²×C₈	D₄×C₄₁	1	1	yes	characteristic
225	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
226	C₄⋊Q₈	—	2	2	no	—
227	C₄⋊Q₈	—	2	2	no	—
228	C₄⋊Q₈	—	4	1	no	—
229	C₄⋊Q₈	—	4	1	no	—
230	#126	—	4	1	no	—
231	#125	—	2	1	no	—
232	#124	—	2	1	no	—
233	#123	—	4	1	no	—
234	C₂²⋊Q₈	—	2	1	no	—
235	#121	—	2	1	no	—
236	C₄×Q₈	—	2	1	no	—
237	C₄×D₄	—	2	1	no	—
238	C₄×D₄	—	4	1	no	—
239	#117	—	4	1	no	—
240	C₂×C₄⋊C₄	—	4	1	no	—
241	C₂×C₄⋊C₄	—	4	1	no	—
242	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
243	Q₃₂	—	4	4	no	—
244	SD₃₂	—	4	4	no	—
245	C₂×C₁₆	C₂³×C₄₁	1	4	yes	—
246	#107	C₂³×C₄₁	1	1	yes	characteristic
247	#107	C₂³×C₄₁	1	1	yes	characteristic
248	#106	—	2	1	no	—
249	#103	—	2	1	no	—
250	#103	—	2	1	no	—
251	#102	—	2	1	no	—
252	#102	—	2	1	no	—
253	C₄×C₈	D₄×C₄₁	1	1	yes	characteristic
254	C₄×C₈	D₄×C₄₁	1	1	yes	characteristic
255	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
256	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
257	C₂×Q₈	—	2	1	no	—
258	C₂×Q₈	—	2	4	no	—
259	C₂×Q₈	—	4	4	no	—
260	C₂×Q₈	—	4	4	no	—
261	C₂×Q₈	—	4	2	no	—
262	C₂×D₄	—	2	1	no	—
263	C₂×D₄	—	4	1	no	—
264	C₂²×C₄	—	4	2	no	—
265	C₂²×C₄	—	4	1	no	—
266	C₂²×C₄	D₈×C₄₁	1	1	yes	characteristic
267	C₂²×C₄	D₈×C₄₁	1	1	yes	characteristic
268	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
269	Q₁₆	—	2	4	no	—
270	Q₁₆	—	2	2	no	—
271	Q₁₆	—	2	4	no	—
272	Q₁₆	—	4	4	no	—
273	Q₁₆	—	4	4	no	—
274	Q₁₆	—	4	4	no	—
275	SD₁₆	—	4	2	no	—
276	D₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
280	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
281	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
282	C₂×C₈	C₂⁴×C₄₁	1	1	yes	characteristic
283	C₄⋊C₄	—	2	1	no	—
284	C₄⋊C₄	—	2	1	no	—
285	C₄⋊C₄	—	4	2	no	—
286	C₄⋊C₄	—	4	2	no	—
287	C₄⋊C₄	—	4	2	no	—
288	C₄⋊C₄	—	4	2	no	—
289	C₄⋊C₄	—	4	1	no	—
290	C₄⋊C₄	—	4	1	no	—
291	C₄⋊C₄	—	4	1	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₂²⋊C₄	—	4	1	no	—
294	C₂²⋊C₄	—	4	1	no	—
295	C₂²⋊C₄	—	4	1	no	—
296	C₂²⋊C₄	—	4	1	no	—
297	C₂²⋊C₄	—	4	1	no	—
298	C₄²	—	4	1	no	—
299	C₄²	D₈×C₄₁	1	1	yes	characteristic
300	C₄²	D₈×C₄₁	1	1	yes	characteristic
301	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
302	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
303	C₁₆	—	2	4	no	—
304	C₂³	—	4	1	no	—
305	C₂³	C₂×D₈×C₄₁	1	1	yes	characteristic
306	Q₈	—	4	4	no	—
307	Q₈	—	4	1	no	—
308	Q₈	—	4	1	no	—
309	Q₈	—	4	4	no	—
310	Q₈	—	8	4	no	—
311	D₄	—	4	1	no	—
312	D₄	—	4	1	no	—
313	D₄	—	8	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	4	no	—
322	C₂×C₄	—	8	4	no	—
323	C₂×C₄	—	8	2	no	—
324	C₂×C₄	C₂×D₈×C₄₁	1	1	yes	characteristic
325	C₂×C₄	C₂×D₈×C₄₁	1	1	yes	characteristic
326	C₂×C₄	C₂×D₈×C₄₁	1	1	yes	characteristic
327	C₂×C₄	C₂×D₈×C₄₁	1	1	yes	characteristic
328	C₂×C₄	C₂×D₈×C₄₁	1	1	yes	characteristic
329	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
330	C₈	—	2	1	no	—
331	C₈	—	2	1	no	—
332	C₈	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
333	C₈	H(5,𝔽₂)×C₄₁	1	2	yes	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	8	1	no	—
337	C₂²	—	8	1	no	—
338	C₂²	—	8	1	no	—
339	C₂²	C₄₁×#510	1	2	yes	—
340	C₂²	C₂²×D₈×C₄₁	1	1	yes	characteristic
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	—	8	1	no	—
346	C₄	—	8	1	no	—
347	C₄	—	8	4	no	—
348	C₄	—	8	1	no	—
349	C₄	C₄₁×#510	1	2	yes	—
350	C₄	C₄₁×#535	1	1	yes	characteristic
351	C₄	C₂×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
352	C₂	—	2	2	no	—
353	C₂	—	16	1	no	—
354	C₂	C₄₁×#3205	1	1	yes	characteristic
355	C₂	C₂×C₄₁×#510	1	1	yes	characteristic
356	C₂	C₂×C₄₁×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₄₁
prime radical	C₄₁×#466	C₂²
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
2. C₂×C₄₁×#494	C₂	semidirect
3. C₄₁×#2251	C₂	semidirect
4. C₄₁×#2245	C₂	semidirect
5. C₄₁×#2209	C₂	semidirect
6. C₄₁×#2208	C₂	semidirect
7. C₄₁×#2188	C₂	semidirect
9. C₄₁×#2163	C₂	semidirect
155. #33784	C₄₁	direct (N × H)
175. C₂×#494	C₂×C₄₁	semidirect
176. #2251	C₂×C₄₁	semidirect
177. #2245	C₂×C₄₁	semidirect
178. #2209	C₂×C₄₁	semidirect
179. #2208	C₂×C₄₁	semidirect
180. #2188	C₂×C₄₁	semidirect
182. #2163	C₂×C₄₁	semidirect
213. C₄₁	#33784	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×#3168	2
C₂×C₄₁×#494	2
C₄₁×#2251	2
C₄₁×#2245	2
C₄₁×#2209	2
C₄₁×#2208	2
C₄₁×#2188	2
C₄₁×#2185	2
C₄₁×#2163	2
#33784	41

Subgroups — C41×#33784 #23809985

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#33784 #23809985