Subgroups — C₄₁×#20532 #23796733

Order 10496.

1106 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#408	C₂	1	2	yes	maximal
2	C₄₁×#2297	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#1753	C₂	1	4	yes	maximal, complemented
4	C₄₁×#1752	C₂	1	4	yes	maximal, complemented
5	C₄₁×#1750	C₂	1	2	yes	maximal, complemented
6	C₄₁×#1749	C₂	1	2	yes	maximal, complemented
7	C₂²×#41×C₄₁	C₂²	1	1	yes	characteristic
8	C₂²×M₁₆×C₄₁	C₂²	1	2	yes	—
9	C₂³×C₈×C₄₁	C₂²	1	2	yes	—
10	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
12	C₄₁×#408	C₄	1	16	yes	complemented
13	C₂×C₂²⋊C₈×C₄₁	C₂²	1	4	yes	—
14	C₂×C₂²⋊C₈×C₄₁	C₂²	1	4	yes	—
15	C₂×#97×C₄₁	C₂²	1	4	yes	—
16	C₂×#97×C₄₁	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₄₁	C₂²	1	2	yes	—
18	C₄₁×#391	C₄	1	2	yes	—
19	C₄₁×#390	C₄	1	2	yes	—
20	C₄₁×#386	C₄	1	4	yes	—
21	C₄×C₂²⋊C₄×C₄₁	C₂²	1	4	yes	—
22	C₄₁×#336	C₂²	1	2	yes	—
23	C₄₁×#336	C₂²	1	2	yes	—
24	C₄₁×#336	C₂²	1	4	yes	—
25	C₂×#41×C₄₁	—	4	2	no	—
26	C₂×#41×C₄₁	—	4	1	no	—
27	C₂×#41×C₄₁	—	4	1	no	—
28	C₂×#41×C₄₁	C₂×C₄	1	8	yes	—
29	C₂²×Q₈×C₄₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×D₄×C₄₁	C₂×C₄	1	2	yes	—
31	C₂²×D₄×C₄₁	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₄₁	C₂×C₄	1	2	yes	—
33	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
34	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
35	C₂×M₁₆×C₄₁	—	4	2	no	—
36	C₂×M₁₆×C₄₁	—	4	2	no	—
37	C₂×M₁₆×C₄₁	C₂×C₄	1	8	yes	complemented
38	C₂²×C₈×C₄₁	—	4	2	no	—
39	C₂²×C₈×C₄₁	—	4	2	no	—
40	C₂²×C₈×C₄₁	C₂×C₄	1	8	yes	complemented
41	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
42	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
43	#117×C₄₁	—	4	1	no	—
44	#117×C₄₁	—	4	1	no	—
45	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
46	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
47	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	8	yes	—
48	C₂×C₄²×C₄₁	—	4	2	no	—
49	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
51	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
52	C₂²⋊C₈×C₄₁	C₂×C₄	1	16	yes	—
53	C₂²⋊C₈×C₄₁	C₂×C₄	1	16	yes	complemented
54	#97×C₄₁	—	4	4	no	—
55	#97×C₄₁	—	4	2	no	—
56	C₄×C₈×C₄₁	—	4	2	no	—
57	#95×C₄₁	C₂×C₄	1	8	yes	—
58	C₂⁴×C₄₁	C₂²×C₄	1	2	yes	—
59	C₂⁴×C₄₁	C₂²×C₄	1	1	yes	characteristic
60	#41×C₄₁	—	4	8	no	—
61	#41×C₄₁	—	4	8	no	—
62	C₂×Q₈×C₄₁	—	4	1	no	—
63	C₂×Q₈×C₄₁	—	4	1	no	—
64	C₂×Q₈×C₄₁	C₄²	1	4	yes	—
65	C₂×D₄×C₄₁	—	4	1	no	—
66	C₂×D₄×C₄₁	—	4	1	no	—
67	C₂×D₄×C₄₁	—	4	2	no	—
68	C₂×D₄×C₄₁	—	4	2	no	—
69	C₂×D₄×C₄₁	C₄²	1	8	yes	—
70	C₂×D₄×C₄₁	C₄²	1	4	yes	—
71	C₂²×C₄×C₄₁	—	2	2	no	—
72	C₂²×C₄×C₄₁	—	2	1	no	—
73	C₂²×C₄×C₄₁	—	2	1	no	—
74	C₂²×C₄×C₄₁	—	2	2	no	—
75	C₂²×C₄×C₄₁	—	2	2	no	—
76	C₂²×C₄×C₄₁	—	2	2	no	—
77	C₂²×C₄×C₄₁	—	2	2	no	—
78	C₂²×C₄×C₄₁	—	4	2	no	—
79	C₂²×C₄×C₄₁	—	4	4	no	—
80	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
81	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
82	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
85	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
86	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
87	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
88	M₁₆×C₄₁	—	4	8	no	—
89	C₂×C₈×C₄₁	—	2	4	no	—
90	C₂×C₈×C₄₁	—	2	4	no	—
91	C₂×C₈×C₄₁	—	2	4	no	—
92	C₂×C₈×C₄₁	—	2	4	no	—
93	C₂×C₈×C₄₁	—	4	8	no	—
94	C₂×C₈×C₄₁	C₂²×C₄	1	8	yes	—
95	C₂×C₈×C₄₁	C₂²×C₄	1	8	yes	—
96	C₄⋊C₄×C₄₁	—	4	2	no	—
97	C₂²⋊C₄×C₄₁	—	4	8	no	—
98	C₂²⋊C₄×C₄₁	—	4	2	no	—
99	C₄²×C₄₁	—	2	2	no	—
100	C₄²×C₄₁	—	2	4	no	—
101	C₄²×C₄₁	—	2	2	no	—
102	C₄²×C₄₁	—	4	2	no	—
103	C₂³×C₄₁	—	2	1	no	—
104	C₂³×C₄₁	—	2	1	no	—
105	C₂³×C₄₁	—	2	2	no	—
106	C₂³×C₄₁	—	2	2	no	—
107	C₂³×C₄₁	—	2	8	no	—
108	C₂³×C₄₁	—	2	4	no	—
109	C₂³×C₄₁	C₂×C₄²	1	2	yes	—
110	C₂³×C₄₁	C₂×C₄²	1	4	yes	—
111	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
112	Q₈×C₄₁	—	4	4	no	—
113	D₄×C₄₁	—	4	4	no	—
114	D₄×C₄₁	—	4	8	no	—
115	C₂×C₄×C₄₁	—	2	1	no	—
116	C₂×C₄×C₄₁	—	2	1	no	—
117	C₂×C₄×C₄₁	—	2	2	no	—
118	C₂×C₄×C₄₁	—	2	2	no	—
119	C₂×C₄×C₄₁	—	2	4	no	—
120	C₂×C₄×C₄₁	—	2	4	no	—
121	C₂×C₄×C₄₁	—	2	4	no	—
122	C₂×C₄×C₄₁	—	2	4	no	—
123	C₂×C₄×C₄₁	—	2	4	no	—
124	C₂×C₄×C₄₁	—	2	4	no	—
125	C₂×C₄×C₄₁	—	2	4	no	—
126	C₂×C₄×C₄₁	—	2	4	no	—
127	C₂×C₄×C₄₁	—	2	4	no	—
128	C₂×C₄×C₄₁	—	2	8	no	—
129	C₂×C₄×C₄₁	—	2	4	no	—
130	C₂×C₄×C₄₁	—	4	8	no	—
131	C₂×C₄×C₄₁	C₂×C₄²	1	4	yes	—
132	C₂×C₄×C₄₁	C₂×C₄²	1	2	yes	—
133	C₂×C₄×C₄₁	C₂³×C₄	1	2	yes	—
134	C₂×C₄×C₄₁	H(5,𝔽₂)	1	2	yes	—
135	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
136	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
137	C₈×C₄₁	—	2	8	no	—
138	C₈×C₄₁	—	2	8	no	—
139	#20532	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₂²×C₄₁	—	2	8	no	—
141	C₂²×C₄₁	—	2	4	no	—
142	C₂²×C₄₁	—	2	4	no	—
143	C₂²×C₄₁	—	2	2	no	—
144	C₂²×C₄₁	—	4	2	no	—
145	C₂²×C₄₁	—	4	4	no	—
146	C₂²×C₄₁	—	4	2	no	—
147	C₂²×C₄₁	—	4	4	no	—
148	C₂²×C₄₁	C₂²×C₄²	1	1	yes	characteristic
149	C₂²×C₄₁	#518	1	1	yes	characteristic
150	C₂²×C₄₁	#518	1	1	yes	characteristic
151	C₂²×C₄₁	#568	1	4	yes	—
152	C₄×C₄₁	—	2	2	no	—
153	C₄×C₄₁	—	2	4	no	—
154	C₄×C₄₁	—	4	4	no	—
155	C₄×C₄₁	—	4	4	no	—
156	C₄×C₄₁	#518	1	2	yes	—
157	C₄×C₄₁	#518	1	2	yes	—
158	C₂×#408	C₂×C₄₁	1	2	yes	—
159	#2297	C₂×C₄₁	1	1	yes	characteristic
160	#1753	C₂×C₄₁	1	4	yes	complemented
161	#1752	C₂×C₄₁	1	4	yes	complemented
162	#1750	C₂×C₄₁	1	2	yes	complemented
163	#1749	C₂×C₄₁	1	2	yes	complemented
164	C₂×C₄₁	—	4	2	no	—
165	C₂×C₄₁	—	4	4	no	—
166	C₂×C₄₁	#2273	1	1	yes	characteristic
167	C₂×C₄₁	#2875	1	2	yes	—
168	C₂×C₄₁	#2978	1	2	yes	—
169	C₂×C₄₁	#2980	1	2	yes	—
170	C₂²×#41	C₂²×C₄₁	1	1	yes	characteristic
171	C₂²×M₁₆	C₂²×C₄₁	1	2	yes	—
172	C₂³×C₈	C₂²×C₄₁	1	2	yes	—
173	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
174	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
175	#408	C₄×C₄₁	1	16	yes	complemented
176	C₂×C₂²⋊C₈	C₂²×C₄₁	1	4	yes	—
177	C₂×C₂²⋊C₈	C₂²×C₄₁	1	4	yes	—
178	C₂×#97	C₂²×C₄₁	1	4	yes	—
179	C₂×#97	C₂²×C₄₁	1	2	yes	—
180	C₂×C₄×C₈	C₂²×C₄₁	1	2	yes	—
181	#391	C₄×C₄₁	1	2	yes	—
182	#390	C₄×C₄₁	1	2	yes	—
183	#386	C₄×C₄₁	1	4	yes	—
184	C₄×C₂²⋊C₄	C₂²×C₄₁	1	4	yes	—
185	#336	C₂²×C₄₁	1	2	yes	—
186	#336	C₂²×C₄₁	1	2	yes	—
187	#336	C₂²×C₄₁	1	4	yes	—
188	C₄₁	#20532	1	1	yes	characteristic, direct product factor, Sylow
189	C₂×#41	—	4	2	no	—
190	C₂×#41	—	4	1	no	—
191	C₂×#41	—	4	1	no	—
192	C₂×#41	C₂×C₄×C₄₁	1	8	yes	—
193	C₂²×Q₈	C₂×C₄×C₄₁	1	1	yes	characteristic
194	C₂²×D₄	C₂×C₄×C₄₁	1	2	yes	—
195	C₂²×D₄	C₂×C₄×C₄₁	1	1	yes	characteristic
196	C₂³×C₄	C₂×C₄×C₄₁	1	2	yes	—
197	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
198	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
199	C₂×M₁₆	—	4	2	no	—
200	C₂×M₁₆	—	4	2	no	—
201	C₂×M₁₆	C₂×C₄×C₄₁	1	8	yes	complemented
202	C₂²×C₈	—	4	2	no	—
203	C₂²×C₈	—	4	2	no	—
204	C₂²×C₈	C₂×C₄×C₄₁	1	8	yes	complemented
205	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
206	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
207	#117	—	4	1	no	—
208	#117	—	4	1	no	—
209	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
210	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
211	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	8	yes	—
212	C₂×C₄²	—	4	2	no	—
213	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
214	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
215	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
216	C₂²⋊C₈	C₂×C₄×C₄₁	1	16	yes	—
217	C₂²⋊C₈	C₂×C₄×C₄₁	1	16	yes	complemented
218	#97	—	4	4	no	—
219	#97	—	4	2	no	—
220	C₄×C₈	—	4	2	no	—
221	#95	C₂×C₄×C₄₁	1	8	yes	—
222	C₂⁴	C₂²×C₄×C₄₁	1	2	yes	—
223	C₂⁴	C₂²×C₄×C₄₁	1	1	yes	characteristic
224	#41	—	4	8	no	—
225	#41	—	4	8	no	—
226	C₂×Q₈	—	4	1	no	—
227	C₂×Q₈	—	4	1	no	—
228	C₂×Q₈	C₄²×C₄₁	1	4	yes	—
229	C₂×D₄	—	4	1	no	—
230	C₂×D₄	—	4	1	no	—
231	C₂×D₄	—	4	2	no	—
232	C₂×D₄	—	4	2	no	—
233	C₂×D₄	C₄²×C₄₁	1	8	yes	—
234	C₂×D₄	C₄²×C₄₁	1	4	yes	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	4	2	no	—
243	C₂²×C₄	—	4	4	no	—
244	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
245	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
246	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
247	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
248	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
249	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
250	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
251	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
252	M₁₆	—	4	8	no	—
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	—	2	4	no	—
255	C₂×C₈	—	2	4	no	—
256	C₂×C₈	—	2	4	no	—
257	C₂×C₈	—	4	8	no	—
258	C₂×C₈	C₂²×C₄×C₄₁	1	8	yes	—
259	C₂×C₈	C₂²×C₄×C₄₁	1	8	yes	—
260	C₄⋊C₄	—	4	2	no	—
261	C₂²⋊C₄	—	4	2	no	—
262	C₂²⋊C₄	—	4	8	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	2	4	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	4	2	no	—
267	C₂³	—	2	1	no	—
268	C₂³	—	2	1	no	—
269	C₂³	—	2	2	no	—
270	C₂³	—	2	2	no	—
271	C₂³	—	2	8	no	—
272	C₂³	—	2	4	no	—
273	C₂³	C₂×C₄²×C₄₁	1	2	yes	—
274	C₂³	C₂×C₄²×C₄₁	1	4	yes	—
275	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
276	Q₈	—	4	4	no	—
277	D₄	—	4	4	no	—
278	D₄	—	4	8	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	8	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	4	8	no	—
295	C₂×C₄	C₂×C₄²×C₄₁	1	4	yes	—
296	C₂×C₄	C₂×C₄²×C₄₁	1	2	yes	—
297	C₂×C₄	C₂³×C₄×C₄₁	1	2	yes	—
298	C₂×C₄	H(5,𝔽₂)×C₄₁	1	2	yes	—
299	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
300	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
301	C₈	—	2	8	no	—
302	C₈	—	2	8	no	—
303	C₂²	—	2	4	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	8	no	—
306	C₂²	—	2	4	no	—
307	C₂²	—	4	2	no	—
308	C₂²	—	4	4	no	—
309	C₂²	—	4	2	no	—
310	C₂²	—	4	4	no	—
311	C₂²	C₂²×C₄²×C₄₁	1	1	yes	characteristic
312	C₂²	C₄₁×#518	1	1	yes	characteristic
313	C₂²	C₄₁×#518	1	1	yes	characteristic
314	C₂²	C₄₁×#568	1	4	yes	—
315	C₄	—	2	2	no	—
316	C₄	—	2	4	no	—
317	C₄	—	4	4	no	—
318	C₄	—	4	4	no	—
319	C₄	C₄₁×#518	1	2	yes	—
320	C₄	C₄₁×#518	1	2	yes	—
321	C₂	—	4	2	no	—
322	C₂	—	4	4	no	—
323	C₂	C₄₁×#2273	1	1	yes	characteristic
324	C₂	C₄₁×#2875	1	2	yes	—
325	C₂	C₄₁×#2978	1	2	yes	—
326	C₂	C₄₁×#2980	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₄₁	C₂⁴
commutator	C₂²	C₂²×C₄²×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
socle	C₂³×C₄₁	C₂³×C₄
prime radical	C₂²×#41×C₄₁	C₂²
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
3. C₄₁×#1753	C₂	semidirect
4. C₄₁×#1752	C₂	semidirect
5. C₄₁×#1750	C₂	semidirect
6. C₄₁×#1749	C₂	semidirect
12. C₄₁×#408	C₄	semidirect
37. C₂×M₁₆×C₄₁	C₂×C₄	semidirect
40. C₂²×C₈×C₄₁	C₂×C₄	semidirect
53. C₂²⋊C₈×C₄₁	C₂×C₄	semidirect
139. #20532	C₄₁	direct (N × H)
160. #1753	C₂×C₄₁	semidirect
161. #1752	C₂×C₄₁	semidirect
162. #1750	C₂×C₄₁	semidirect
163. #1749	C₂×C₄₁	semidirect
175. #408	C₄×C₄₁	semidirect
188. C₄₁	#20532	direct (N × H)
201. C₂×M₁₆	C₂×C₄×C₄₁	semidirect
204. C₂²×C₈	C₂×C₄×C₄₁	semidirect
217. C₂²⋊C₈	C₂×C₄×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#408	2
C₄₁×#2297	2
C₄₁×#1753	2
C₄₁×#1752	2
C₄₁×#1750	2
C₄₁×#1749	2
#20532	41

Subgroups — C41×#20532 #23796733

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#20532 #23796733