Subgroups — C₄₁×#20575 #23796776

Order 10496.

798 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#446	C₂	1	2	yes	maximal
2	C₄₁×#2295	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#1771	C₂	1	2	yes	maximal
4	C₄₁×#1771	C₂	1	2	yes	maximal
5	C₈×C₄⋊C₄×C₄₁	C₂	1	2	yes	maximal
6	C₄₁×#1768	C₂	1	2	yes	maximal
7	C₄×C₄⋊C₈×C₄₁	C₂	1	1	yes	maximal, characteristic
8	C₄₁×#1746	C₂	1	2	yes	maximal
9	C₄₁×#1745	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
11	C₄₁×#446	C₄	1	16	yes	complemented
12	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
14	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
15	C₂×#97×C₄₁	C₂²	1	2	yes	—
16	C₂×#97×C₄₁	C₂²	1	2	yes	—
17	C₂×#97×C₄₁	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₄₁	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₄₁	C₂²	1	2	yes	—
22	C₄₁×#391	C₄	1	2	yes	—
23	C₄₁×#389	C₄	1	2	yes	—
24	C₄₁×#387	C₄	1	4	yes	—
25	C₄×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
26	C₄×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
27	C₄₁×#376	C₂²	1	2	yes	—
28	C₄³×C₄₁	C₂²	1	1	yes	characteristic
29	C₄₁×#336	C₂²	1	2	yes	—
30	C₄₁×#336	C₂²	1	1	yes	characteristic
31	C₄₁×#336	C₂²	1	2	yes	—
32	C₄₁×#336	C₂²	1	2	yes	—
33	C₄₁×#336	C₂²	1	2	yes	—
34	C₄₁×#336	C₂²	1	1	yes	characteristic
35	C₂²×Q₈×C₄₁	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
37	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
38	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
39	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
40	C₄×Q₈×C₄₁	C₂×C₄	1	8	yes	—
41	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
42	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
43	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
44	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
50	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
52	C₄⋊C₈×C₄₁	C₂×C₄	1	8	yes	—
53	C₄⋊C₈×C₄₁	C₂×C₄	1	8	yes	—
54	C₄⋊C₈×C₄₁	C₂×C₄	1	8	yes	—
55	#97×C₄₁	—	4	2	no	—
56	#97×C₄₁	C₂×C₄	1	8	yes	—
57	#97×C₄₁	C₂×C₄	1	8	yes	—
58	C₄×C₈×C₄₁	—	4	1	no	—
59	C₄×C₈×C₄₁	—	4	1	no	—
60	C₄×C₈×C₄₁	—	4	2	no	—
61	C₄×C₈×C₄₁	C₂×C₄	1	8	yes	—
62	#95×C₄₁	C₂×C₄	1	4	yes	—
63	#95×C₄₁	C₂×C₄	1	4	yes	—
64	#95×C₄₁	C₂×C₄	1	2	yes	—
65	#95×C₄₁	C₂×C₄	1	4	yes	—
66	C₂×Q₈×C₄₁	—	4	2	no	—
67	C₂×Q₈×C₄₁	C₄²	1	4	yes	—
68	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
70	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
76	C₂×C₈×C₄₁	—	2	2	no	—
77	C₂×C₈×C₄₁	—	2	2	no	—
78	C₂×C₈×C₄₁	—	2	2	no	—
79	C₂×C₈×C₄₁	—	2	2	no	—
80	C₂×C₈×C₄₁	—	2	2	no	—
81	C₂×C₈×C₄₁	—	2	2	no	—
82	C₂×C₈×C₄₁	—	2	2	no	—
83	C₂×C₈×C₄₁	—	2	2	no	—
84	C₂×C₈×C₄₁	C₂²×C₄	1	4	yes	—
85	C₂×C₈×C₄₁	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₄₁	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₄₁	C₂²×C₄	1	4	yes	—
88	C₄⋊C₄×C₄₁	—	2	8	no	—
89	C₄⋊C₄×C₄₁	—	4	2	no	—
90	C₄⋊C₄×C₄₁	C₄²	1	8	yes	—
91	C₄⋊C₄×C₄₁	C₄²	1	4	yes	—
92	C₄²×C₄₁	—	2	4	no	—
93	C₄²×C₄₁	—	2	1	no	—
94	C₄²×C₄₁	—	2	2	no	—
95	C₄²×C₄₁	—	2	1	no	—
96	C₄²×C₄₁	—	2	2	no	—
97	C₄²×C₄₁	—	2	2	no	—
98	C₄²×C₄₁	—	4	2	no	—
99	C₄²×C₄₁	C₂²×C₄	1	4	yes	—
100	C₄²×C₄₁	C₂²×C₄	1	4	yes	—
101	C₄²×C₄₁	C₂²×C₄	1	4	yes	—
102	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
103	Q₈×C₄₁	—	4	4	no	—
104	C₂×C₄×C₄₁	—	2	2	no	—
105	C₂×C₄×C₄₁	—	2	2	no	—
106	C₂×C₄×C₄₁	—	2	4	no	—
107	C₂×C₄×C₄₁	—	2	2	no	—
108	C₂×C₄×C₄₁	—	2	2	no	—
109	C₂×C₄×C₄₁	—	2	4	no	—
110	C₂×C₄×C₄₁	—	2	2	no	—
111	C₂×C₄×C₄₁	—	2	4	no	—
112	C₂×C₄×C₄₁	—	2	2	no	—
113	C₂×C₄×C₄₁	—	2	4	no	—
114	C₂×C₄×C₄₁	—	2	2	no	—
115	C₂×C₄×C₄₁	—	2	4	no	—
116	C₂×C₄×C₄₁	C₂×C₄²	1	4	yes	—
117	C₂×C₄×C₄₁	C₂×C₄²	1	2	yes	—
118	C₂×C₄×C₄₁	C₂×C₄²	1	4	yes	—
119	C₂×C₄×C₄₁	C₂×C₄²	1	2	yes	—
120	C₂×C₄×C₄₁	#131	1	4	yes	—
121	C₂×C₄×C₄₁	C₂³×C₄	1	2	yes	—
122	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
123	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
124	C₂×C₄×C₄₁	2₋¹⁺⁴	1	2	yes	—
125	C₈×C₄₁	—	2	4	no	—
126	C₈×C₄₁	—	2	4	no	—
127	C₈×C₄₁	—	2	4	no	—
128	C₈×C₄₁	—	2	4	no	—
129	#20575	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂²×C₄₁	C₂²×C₄²	1	1	yes	characteristic
131	C₂²×C₄₁	#519	1	1	yes	characteristic
132	C₂²×C₄₁	#520	1	1	yes	characteristic
133	C₂²×C₄₁	C₂×#131	1	2	yes	—
134	C₂²×C₄₁	#568	1	2	yes	—
135	C₄×C₄₁	—	2	2	no	—
136	C₄×C₄₁	—	2	4	no	—
137	C₄×C₄₁	—	2	2	no	—
138	C₄×C₄₁	—	2	4	no	—
139	C₄×C₄₁	—	2	4	no	—
140	C₄×C₄₁	—	4	2	no	—
141	C₄×C₄₁	—	4	4	no	—
142	C₄×C₄₁	#519	1	2	yes	—
143	C₄×C₄₁	#520	1	2	yes	—
144	C₂×#446	C₂×C₄₁	1	2	yes	—
145	#2295	C₂×C₄₁	1	1	yes	characteristic
146	#1771	C₂×C₄₁	1	2	yes	—
147	#1771	C₂×C₄₁	1	2	yes	—
148	C₈×C₄⋊C₄	C₂×C₄₁	1	2	yes	—
149	#1768	C₂×C₄₁	1	2	yes	—
150	C₄×C₄⋊C₈	C₂×C₄₁	1	1	yes	characteristic
151	#1746	C₂×C₄₁	1	2	yes	—
152	#1745	C₂×C₄₁	1	1	yes	characteristic
153	C₂×C₄₁	#2276	1	1	yes	characteristic
154	C₂×C₄₁	#2873	1	1	yes	characteristic
155	C₂×C₄₁	C₄×#131	1	1	yes	characteristic
156	C₂×C₄₁	#2985	1	2	yes	—
157	C₂×C₄₁	#2992	1	2	yes	—
158	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
159	#446	C₄×C₄₁	1	16	yes	complemented
160	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
161	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
162	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
163	C₂×#97	C₂²×C₄₁	1	2	yes	—
164	C₂×#97	C₂²×C₄₁	1	2	yes	—
165	C₂×#97	C₂²×C₄₁	1	2	yes	—
166	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
167	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
168	C₂×C₄×C₈	C₂²×C₄₁	1	2	yes	—
169	C₂×C₄×C₈	C₂²×C₄₁	1	2	yes	—
170	#391	C₄×C₄₁	1	2	yes	—
171	#389	C₄×C₄₁	1	2	yes	—
172	#387	C₄×C₄₁	1	4	yes	—
173	C₄×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
174	C₄×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
175	#376	C₂²×C₄₁	1	2	yes	—
176	C₄³	C₂²×C₄₁	1	1	yes	characteristic
177	#336	C₂²×C₄₁	1	1	yes	characteristic
178	#336	C₂²×C₄₁	1	1	yes	characteristic
179	#336	C₂²×C₄₁	1	2	yes	—
180	#336	C₂²×C₄₁	1	2	yes	—
181	#336	C₂²×C₄₁	1	2	yes	—
182	#336	C₂²×C₄₁	1	2	yes	—
183	C₄₁	#20575	1	1	yes	characteristic, direct product factor, Sylow
184	C₂²×Q₈	C₂×C₄×C₄₁	1	1	yes	characteristic
185	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
186	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
187	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
188	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
189	C₄×Q₈	C₂×C₄×C₄₁	1	8	yes	—
190	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
191	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
192	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
193	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
194	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
195	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
196	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
197	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
198	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
199	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
200	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
201	C₄⋊C₈	C₂×C₄×C₄₁	1	8	yes	—
202	C₄⋊C₈	C₂×C₄×C₄₁	1	8	yes	—
203	C₄⋊C₈	C₂×C₄×C₄₁	1	8	yes	—
204	#97	—	4	2	no	—
205	#97	C₂×C₄×C₄₁	1	8	yes	—
206	#97	C₂×C₄×C₄₁	1	8	yes	—
207	C₄×C₈	—	4	1	no	—
208	C₄×C₈	—	4	1	no	—
209	C₄×C₈	—	4	2	no	—
210	C₄×C₈	C₂×C₄×C₄₁	1	8	yes	—
211	#95	C₂×C₄×C₄₁	1	4	yes	—
212	#95	C₂×C₄×C₄₁	1	4	yes	—
213	#95	C₂×C₄×C₄₁	1	2	yes	—
214	#95	C₂×C₄×C₄₁	1	4	yes	—
215	C₂×Q₈	—	4	2	no	—
216	C₂×Q₈	C₄²×C₄₁	1	4	yes	—
217	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
218	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
219	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
220	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
221	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
222	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
223	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
224	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	C₂²×C₄×C₄₁	1	4	yes	—
234	C₂×C₈	C₂²×C₄×C₄₁	1	4	yes	—
235	C₂×C₈	C₂²×C₄×C₄₁	1	4	yes	—
236	C₂×C₈	C₂²×C₄×C₄₁	1	4	yes	—
237	C₄⋊C₄	—	2	8	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₄⋊C₄	C₄²×C₄₁	1	8	yes	—
240	C₄⋊C₄	C₄²×C₄₁	1	4	yes	—
241	C₄²	—	2	4	no	—
242	C₄²	—	2	1	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	2	1	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	4	2	no	—
248	C₄²	C₂²×C₄×C₄₁	1	4	yes	—
249	C₄²	C₂²×C₄×C₄₁	1	4	yes	—
250	C₄²	C₂²×C₄×C₄₁	1	4	yes	—
251	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
252	Q₈	—	4	4	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	C₂×C₄²×C₄₁	1	4	yes	—
266	C₂×C₄	C₂×C₄²×C₄₁	1	2	yes	—
267	C₂×C₄	C₂×C₄²×C₄₁	1	4	yes	—
268	C₂×C₄	C₂×C₄²×C₄₁	1	2	yes	—
269	C₂×C₄	#131×C₄₁	1	4	yes	—
270	C₂×C₄	C₂³×C₄×C₄₁	1	2	yes	—
271	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
272	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
273	C₂×C₄	2₋¹⁺⁴×C₄₁	1	2	yes	—
274	C₈	—	2	4	no	—
275	C₈	—	2	4	no	—
276	C₈	—	2	4	no	—
277	C₈	—	2	4	no	—
278	C₂²	C₂²×C₄²×C₄₁	1	1	yes	characteristic
279	C₂²	C₄₁×#519	1	1	yes	characteristic
280	C₂²	C₄₁×#520	1	1	yes	characteristic
281	C₂²	C₂×#131×C₄₁	1	2	yes	—
282	C₂²	C₄₁×#568	1	2	yes	—
283	C₄	—	2	2	no	—
284	C₄	—	2	4	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	4	no	—
287	C₄	—	2	4	no	—
288	C₄	—	4	2	no	—
289	C₄	—	4	4	no	—
290	C₄	C₄₁×#519	1	2	yes	—
291	C₄	C₄₁×#520	1	2	yes	—
292	C₂	C₄₁×#2276	1	1	yes	characteristic
293	C₂	C₄₁×#2873	1	1	yes	characteristic
294	C₂	C₄×#131×C₄₁	1	1	yes	characteristic
295	C₂	C₄₁×#2985	1	2	yes	—
296	C₂	C₄₁×#2992	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
11. C₄₁×#446	C₄	semidirect
129. #20575	C₄₁	direct (N × H)
159. #446	C₄×C₄₁	semidirect
183. C₄₁	#20575	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#446	2
C₄₁×#2295	2
C₄₁×#1771	2
C₈×C₄⋊C₄×C₄₁	2
C₄₁×#1768	2
C₄×C₄⋊C₈×C₄₁	2
C₄₁×#1746	2
C₄₁×#1745	2
#20575	41

Subgroups — C41×#20575 #23796776

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#20575 #23796776