Subgroups — C₄₁×#20823 #23797024

Order 10496.

898 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#97×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2285	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#2283	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1847	C₂	1	2	yes	maximal, complemented
5	C₄₁×#1836	C₂	1	2	yes	maximal, complemented
6	C₄₁×#1808	C₂	1	2	yes	maximal
7	C₄₁×#1807	C₂	1	2	yes	maximal
8	C₄₁×#1776	C₂	1	4	yes	maximal, complemented
9	C₂³×C₈×C₄₁	C₂²	1	2	yes	—
10	C₂×#117×C₄₁	—	2	1	no	—
11	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
12	C₂×#107×C₄₁	C₂²	1	4	yes	—
13	C₂×#106×C₄₁	C₂²	1	4	yes	—
14	C₂×#97×C₄₁	C₄	1	8	yes	complemented
15	C₂×#97×C₄₁	C₂²	1	2	yes	—
16	C₂×#97×C₄₁	C₂²	1	1	yes	characteristic
17	C₂×#97×C₄₁	C₂²	1	1	yes	characteristic
18	C₄₁×#384	C₂²	1	2	yes	—
19	C₄₁×#383	—	2	1	no	—
20	C₄₁×#382	—	2	1	no	—
21	C₄₁×#382	C₂²	1	2	yes	—
22	C₄₁×#381	—	2	1	no	—
23	C₄₁×#380	—	2	1	no	—
24	C₄₁×#380	C₂²	1	1	yes	characteristic
25	C₄₁×#380	C₂²	1	1	yes	characteristic
26	C₄₁×#380	C₂²	1	2	yes	—
27	C₄×C₄⋊C₄×C₄₁	—	2	1	no	—
28	C₄×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
29	C₄×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
30	C₄×C₂²⋊C₄×C₄₁	—	2	1	no	—
31	C₄₁×#376	—	2	1	no	—
32	C₄₁×#340	C₂²	1	4	yes	—
33	C₄₁×#340	C₂²	1	4	yes	—
34	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
35	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₄₁	—	2	8	no	—
38	C₂²×C₈×C₄₁	C₂×C₄	1	8	yes	—
39	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
40	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
41	#117×C₄₁	—	4	4	no	—
42	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
44	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
47	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
48	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
49	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
50	C₂×C₄²×C₄₁	—	2	2	no	—
51	C₂×C₄²×C₄₁	—	2	2	no	—
52	C₂×C₄²×C₄₁	C₂×C₄	1	4	yes	—
53	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₄₁	Q₈	1	4	yes	—
56	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
58	#107×C₄₁	—	2	8	no	—
59	#106×C₄₁	—	2	8	no	—
60	#97×C₄₁	C₂×C₄	1	4	yes	—
61	#97×C₄₁	C₂×C₄	1	8	yes	—
62	#97×C₄₁	C₂×C₄	1	4	yes	—
63	#95×C₄₁	—	2	1	no	—
64	#95×C₄₁	—	2	1	no	—
65	#95×C₄₁	—	2	2	no	—
66	#95×C₄₁	—	2	2	no	—
67	#95×C₄₁	—	2	2	no	—
68	#95×C₄₁	—	2	2	no	—
69	#95×C₄₁	—	2	2	no	—
70	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₁	—	2	2	no	—
72	C₂²×C₄×C₄₁	—	2	2	no	—
73	C₂²×C₄×C₄₁	—	2	4	no	—
74	C₂²×C₄×C₄₁	—	2	2	no	—
75	C₂²×C₄×C₄₁	—	2	2	no	—
76	C₂²×C₄×C₄₁	—	2	4	no	—
77	C₂²×C₄×C₄₁	—	4	2	no	—
78	C₂²×C₄×C₄₁	C₄⋊C₄	1	4	yes	—
79	C₂²×C₄×C₄₁	C₄⋊C₄	1	4	yes	—
80	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₄₁	C₂×Q₈	1	2	yes	—
87	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
88	C₂×C₈×C₄₁	—	2	4	no	—
89	C₂×C₈×C₄₁	—	2	16	no	—
90	C₂×C₈×C₄₁	C₂²×C₄	1	4	yes	—
91	C₂×C₈×C₄₁	C₂²×C₄	1	4	yes	—
92	C₂×C₈×C₄₁	#41	1	8	yes	—
93	C₄⋊C₄×C₄₁	—	2	4	no	—
94	C₄⋊C₄×C₄₁	—	2	4	no	—
95	C₄⋊C₄×C₄₁	—	2	4	no	—
96	C₄⋊C₄×C₄₁	—	2	4	no	—
97	C₄⋊C₄×C₄₁	—	4	4	no	—
98	C₄⋊C₄×C₄₁	—	4	4	no	—
99	C₂²⋊C₄×C₄₁	—	4	2	no	—
100	C₂²⋊C₄×C₄₁	—	4	4	no	—
101	C₂²⋊C₄×C₄₁	—	4	2	no	—
102	C₄²×C₄₁	—	4	4	no	—
103	C₄²×C₄₁	—	4	4	no	—
104	C₄²×C₄₁	C₄⋊C₄	1	4	yes	—
105	C₄²×C₄₁	C₄⋊C₄	1	4	yes	—
106	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
107	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
108	C₄²×C₄₁	C₂×Q₈	1	2	yes	—
109	C₄²×C₄₁	C₂×Q₈	1	2	yes	—
110	C₂³×C₄₁	—	2	2	no	—
111	C₂³×C₄₁	—	2	2	no	—
112	C₂³×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
113	C₂³×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
114	C₂³×C₄₁	#137	1	2	yes	—
115	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₄₁	—	2	2	no	—
117	C₂×C₄×C₄₁	—	2	1	no	—
118	C₂×C₄×C₄₁	—	2	1	no	—
119	C₂×C₄×C₄₁	—	2	1	no	—
120	C₂×C₄×C₄₁	—	2	1	no	—
121	C₂×C₄×C₄₁	—	2	2	no	—
122	C₂×C₄×C₄₁	—	2	2	no	—
123	C₂×C₄×C₄₁	—	2	8	no	—
124	C₂×C₄×C₄₁	—	2	4	no	—
125	C₂×C₄×C₄₁	—	2	4	no	—
126	C₂×C₄×C₄₁	—	4	4	no	—
127	C₂×C₄×C₄₁	—	4	4	no	—
128	C₂×C₄×C₄₁	—	4	2	no	—
129	C₂×C₄×C₄₁	—	4	2	no	—
130	C₂×C₄×C₄₁	—	4	4	no	—
131	C₂×C₄×C₄₁	—	4	2	no	—
132	C₂×C₄×C₄₁	—	4	2	no	—
133	C₂×C₄×C₄₁	—	4	2	no	—
134	C₂×C₄×C₄₁	—	4	2	no	—
135	C₂×C₄×C₄₁	—	4	4	no	—
136	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
137	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
138	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
139	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
140	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
141	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
142	C₂×C₄×C₄₁	C₂²×Q₈	1	1	yes	characteristic
143	C₂×C₄×C₄₁	C₂×#41	1	4	yes	—
144	C₈×C₄₁	—	2	4	no	—
145	C₈×C₄₁	#117	1	8	yes	complemented
146	#20823	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
147	C₂²×C₄₁	—	2	2	no	—
148	C₂²×C₄₁	—	2	4	no	—
149	C₂²×C₄₁	—	2	4	no	—
150	C₂²×C₄₁	—	2	2	no	—
151	C₂²×C₄₁	#428	1	4	yes	—
152	C₂²×C₄₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
153	C₂²×C₄₁	#532	1	2	yes	—
154	C₂²×C₄₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
155	C₂²×C₄₁	C₂×#137	1	1	yes	characteristic
156	C₂²×C₄₁	#575	1	2	yes	—
157	C₄×C₄₁	—	2	2	no	—
158	C₄×C₄₁	—	2	2	no	—
159	C₄×C₄₁	—	2	2	no	—
160	C₄×C₄₁	—	2	2	no	—
161	C₄×C₄₁	—	2	2	no	—
162	C₄×C₄₁	—	8	4	no	—
163	C₄×C₄₁	—	8	2	no	—
164	C₄×C₄₁	—	8	2	no	—
165	C₄×C₄₁	C₂×#117	1	2	yes	—
166	C₄×C₄₁	#533	1	2	yes	—
167	C₂²×#97	C₂×C₄₁	1	1	yes	characteristic
168	#2285	C₂×C₄₁	1	1	yes	characteristic
169	#2283	C₂×C₄₁	1	1	yes	characteristic
170	#1847	C₂×C₄₁	1	2	yes	complemented
171	#1836	C₂×C₄₁	1	2	yes	complemented
172	#1808	C₂×C₄₁	1	2	yes	—
173	#1807	C₂×C₄₁	1	2	yes	—
174	#1776	C₂×C₄₁	1	4	yes	complemented
175	C₂×C₄₁	—	2	4	no	—
176	C₂×C₄₁	#2285	1	1	yes	characteristic
177	C₂×C₄₁	C₂×#428	1	1	yes	characteristic
178	C₂×C₄₁	#2911	1	1	yes	characteristic
179	C₂×C₄₁	#3180	1	2	yes	—
180	C₂×C₄₁	#3181	1	2	yes	—
181	C₂³×C₈	C₂²×C₄₁	1	2	yes	—
182	C₂×#117	—	2	1	no	—
183	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
184	C₂×#107	C₂²×C₄₁	1	4	yes	—
185	C₂×#106	C₂²×C₄₁	1	4	yes	—
186	C₂×#97	C₄×C₄₁	1	8	yes	complemented
187	C₂×#97	C₂²×C₄₁	1	1	yes	characteristic
188	C₂×#97	C₂²×C₄₁	1	1	yes	characteristic
189	C₂×#97	C₂²×C₄₁	1	2	yes	—
190	#384	C₂²×C₄₁	1	2	yes	—
191	#383	—	2	1	no	—
192	#382	—	2	1	no	—
193	#382	C₂²×C₄₁	1	2	yes	—
194	#381	—	2	1	no	—
195	#380	—	2	1	no	—
196	#380	C₂²×C₄₁	1	1	yes	characteristic
197	#380	C₂²×C₄₁	1	2	yes	—
198	#380	C₂²×C₄₁	1	1	yes	characteristic
199	C₄×C₄⋊C₄	—	2	1	no	—
200	C₄×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
201	C₄×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
202	C₄×C₂²⋊C₄	—	2	1	no	—
203	#376	—	2	1	no	—
204	#340	C₂²×C₄₁	1	4	yes	—
205	#340	C₂²×C₄₁	1	4	yes	—
206	C₄₁	#20823	1	1	yes	characteristic, direct product factor, Sylow
207	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
208	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
209	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
210	C₂²×C₈	—	2	8	no	—
211	C₂²×C₈	C₂×C₄×C₄₁	1	8	yes	—
212	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
213	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
214	#117	—	4	4	no	—
215	C₂×C₄⋊C₄	—	2	2	no	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
218	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
219	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
220	C₂×C₂²⋊C₄	—	2	2	no	—
221	C₂×C₂²⋊C₄	—	2	1	no	—
222	C₂×C₂²⋊C₄	—	2	1	no	—
223	C₂×C₄²	—	2	2	no	—
224	C₂×C₄²	—	2	2	no	—
225	C₂×C₄²	C₂×C₄×C₄₁	1	4	yes	—
226	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
227	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
228	C₂×C₄²	Q₈×C₄₁	1	4	yes	—
229	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
230	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
231	#107	—	2	8	no	—
232	#106	—	2	8	no	—
233	#97	C₂×C₄×C₄₁	1	4	yes	—
234	#97	C₂×C₄×C₄₁	1	8	yes	—
235	#97	C₂×C₄×C₄₁	1	4	yes	—
236	#95	—	2	1	no	—
237	#95	—	2	1	no	—
238	#95	—	2	2	no	—
239	#95	—	2	2	no	—
240	#95	—	2	2	no	—
241	#95	—	2	2	no	—
242	#95	—	2	2	no	—
243	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	4	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	4	no	—
250	C₂²×C₄	—	4	2	no	—
251	C₂²×C₄	C₄⋊C₄×C₄₁	1	4	yes	—
252	C₂²×C₄	C₄⋊C₄×C₄₁	1	4	yes	—
253	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
254	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
258	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
259	C₂²×C₄	C₂×Q₈×C₄₁	1	2	yes	—
260	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
261	C₂×C₈	—	2	4	no	—
262	C₂×C₈	—	2	16	no	—
263	C₂×C₈	C₂²×C₄×C₄₁	1	4	yes	—
264	C₂×C₈	C₂²×C₄×C₄₁	1	4	yes	—
265	C₂×C₈	#41×C₄₁	1	8	yes	—
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	2	4	no	—
270	C₄⋊C₄	—	4	4	no	—
271	C₄⋊C₄	—	4	4	no	—
272	C₂²⋊C₄	—	4	4	no	—
273	C₂²⋊C₄	—	4	2	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₄²	—	4	4	no	—
276	C₄²	—	4	4	no	—
277	C₄²	C₄⋊C₄×C₄₁	1	4	yes	—
278	C₄²	C₄⋊C₄×C₄₁	1	4	yes	—
279	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
280	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
281	C₄²	C₂×Q₈×C₄₁	1	2	yes	—
282	C₄²	C₂×Q₈×C₄₁	1	2	yes	—
283	C₂³	—	2	2	no	—
284	C₂³	—	2	2	no	—
285	C₂³	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
286	C₂³	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
287	C₂³	#137×C₄₁	1	2	yes	—
288	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	8	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	4	no	—
309	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
310	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
311	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
312	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
313	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
314	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
315	C₂×C₄	C₂²×Q₈×C₄₁	1	1	yes	characteristic
316	C₂×C₄	C₂×#41×C₄₁	1	4	yes	—
317	C₈	—	2	4	no	—
318	C₈	#117×C₄₁	1	8	yes	complemented
319	C₂²	—	2	2	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	4	no	—
322	C₂²	—	2	4	no	—
323	C₂²	C₄₁×#428	1	4	yes	—
324	C₂²	C₂²×C₄⋊C₄×C₄₁	1	1	yes	characteristic
325	C₂²	C₄₁×#532	1	2	yes	—
326	C₂²	C₂×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
327	C₂²	C₂×#137×C₄₁	1	1	yes	characteristic
328	C₂²	C₄₁×#575	1	2	yes	—
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	8	4	no	—
335	C₄	—	8	2	no	—
336	C₄	—	8	2	no	—
337	C₄	C₂×#117×C₄₁	1	2	yes	—
338	C₄	C₄₁×#533	1	2	yes	—
339	C₂	—	2	4	no	—
340	C₂	C₄₁×#2285	1	1	yes	characteristic
341	C₂	C₂×C₄₁×#428	1	1	yes	characteristic
342	C₂	C₄₁×#2911	1	1	yes	characteristic
343	C₂	C₄₁×#3180	1	2	yes	—
344	C₂	C₄₁×#3181	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₂⁴×C₄₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂²×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₄₁×#1847	C₂	semidirect
5. C₄₁×#1836	C₂	semidirect
8. C₄₁×#1776	C₂	semidirect
14. C₂×#97×C₄₁	C₄	semidirect
145. C₈×C₄₁	#117	semidirect
146. #20823	C₄₁	direct (N × H)
170. #1847	C₂×C₄₁	semidirect
171. #1836	C₂×C₄₁	semidirect
174. #1776	C₂×C₄₁	semidirect
186. C₂×#97	C₄×C₄₁	semidirect
206. C₄₁	#20823	direct (N × H)
318. C₈	#117×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#97×C₄₁	2
C₄₁×#2285	2
C₄₁×#2283	2
C₄₁×#1847	2
C₄₁×#1836	2
C₄₁×#1808	2
C₄₁×#1807	2
C₄₁×#1776	2
#20823	41

Subgroups — C41×#20823 #23797024

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#20823 #23797024