Subgroups — C₄₁×#12563 #23788764

Order 10496.

570 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×#108×C₄₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₁×#427	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1383	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₄₁	—	2	1	no	—
6	C₂³×C₈×C₄₁	C₂²	1	1	yes	characteristic
7	C₂×#117×C₄₁	—	2	1	no	—
8	C₂×M₃₂×C₄₁	—	2	2	no	—
9	C₂×M₃₂×C₄₁	—	2	2	no	—
10	C₂×M₃₂×C₄₁	C₄	1	2	yes	complemented
11	C₂×M₃₂×C₄₁	C₂²	1	2	yes	—
12	C₂²×C₁₆×C₄₁	C₄	1	2	yes	complemented
13	C₂×#108×C₄₁	—	2	1	no	—
14	C₂×#108×C₄₁	—	2	2	no	—
15	C₂×#108×C₄₁	C₄	1	4	yes	complemented
16	C₂×#108×C₄₁	C₂²	1	2	yes	—
17	C₄₁×#427	—	2	2	no	—
18	C₄₁×#427	C₄	1	4	yes	—
19	C₂×#107×C₄₁	—	2	1	no	—
20	C₂×#106×C₄₁	C₂²	1	2	yes	—
21	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
22	C₂×M₁₆×C₄₁	—	2	4	no	—
23	C₂×M₁₆×C₄₁	—	2	2	no	—
24	C₂×M₁₆×C₄₁	—	4	2	no	—
25	C₂×M₁₆×C₄₁	—	4	1	no	—
26	C₂²×C₈×C₄₁	—	4	1	no	—
27	C₂²×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₄₁	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₄₁	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₄₁	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₄₁	D₄	1	2	yes	—
33	C₂²×C₈×C₄₁	D₄	1	2	yes	—
34	C₂²×C₈×C₄₁	Q₈	1	1	yes	characteristic
35	C₂²×C₈×C₄₁	Q₈	1	1	yes	characteristic
36	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
37	#117×C₄₁	—	2	4	no	—
38	#117×C₄₁	—	4	2	no	—
39	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
40	C₂×C₂²⋊C₄×C₄₁	—	4	1	no	—
41	C₂×C₄²×C₄₁	—	4	1	no	—
42	M₃₂×C₄₁	—	2	4	no	—
43	M₃₂×C₄₁	—	2	2	no	—
44	M₃₂×C₄₁	—	2	2	no	—
45	C₂×C₁₆×C₄₁	—	2	4	no	—
46	C₂×C₁₆×C₄₁	C₂×C₄	1	4	yes	—
47	#108×C₄₁	—	2	4	no	—
48	#108×C₄₁	—	2	2	no	—
49	#108×C₄₁	C₂×C₄	1	4	yes	—
50	#107×C₄₁	—	2	4	no	—
51	#106×C₄₁	—	2	2	no	—
52	#106×C₄₁	C₂×C₄	1	4	yes	—
53	C₂⁴×C₄₁	SD₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₄₁	—	4	2	no	—
55	C₂²×C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₄₁	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₁	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₁	D₈	1	2	yes	—
60	C₂²×C₄×C₄₁	SD₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₄₁	SD₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₄₁	SD₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₄₁	Q₁₆	1	2	yes	—
64	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
65	M₁₆×C₄₁	—	2	4	no	—
66	M₁₆×C₄₁	—	4	4	no	—
67	M₁₆×C₄₁	—	4	2	no	—
68	C₂×C₈×C₄₁	—	4	4	no	—
69	C₂×C₈×C₄₁	—	4	2	no	—
70	C₂×C₈×C₄₁	C₄²	1	2	yes	—
71	C₂×C₈×C₄₁	C₄²	1	2	yes	—
72	C₂×C₈×C₄₁	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₄₁	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₄₁	C₂²⋊C₄	1	2	yes	—
75	C₂×C₈×C₄₁	C₂²⋊C₄	1	2	yes	—
76	C₂×C₈×C₄₁	C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₄₁	C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₄₁	C₄⋊C₄	1	2	yes	—
79	C₂×C₈×C₄₁	C₄⋊C₄	1	2	yes	—
80	C₂×C₈×C₄₁	C₄⋊C₄	1	2	yes	—
81	C₂×C₈×C₄₁	C₄⋊C₄	1	2	yes	—
82	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₄₁	C₂×Q₈	1	1	yes	characteristic
88	C₄⋊C₄×C₄₁	—	2	4	no	—
89	C₄⋊C₄×C₄₁	—	4	2	no	—
90	C₂²⋊C₄×C₄₁	—	4	4	no	—
91	C₄²×C₄₁	—	4	4	no	—
92	C₁₆×C₄₁	—	2	4	no	—
93	C₂³×C₄₁	—	4	2	no	—
94	C₂³×C₄₁	#102	1	2	yes	—
95	C₂³×C₄₁	#103	1	2	yes	—
96	C₂³×C₄₁	#106	1	1	yes	characteristic
97	C₂³×C₄₁	#106	1	1	yes	characteristic
98	C₂³×C₄₁	C₂×SD₁₆	1	1	yes	characteristic
99	C₂×C₄×C₄₁	—	4	4	no	—
100	C₂×C₄×C₄₁	—	8	4	no	—
101	C₂×C₄×C₄₁	#95	1	2	yes	—
102	C₂×C₄×C₄₁	#95	1	2	yes	—
103	C₂×C₄×C₄₁	#102	1	2	yes	—
104	C₂×C₄×C₄₁	#102	1	2	yes	—
105	C₂×C₄×C₄₁	#102	1	2	yes	—
106	C₂×C₄×C₄₁	#103	1	2	yes	—
107	C₂×C₄×C₄₁	#103	1	2	yes	—
108	C₂×C₄×C₄₁	#103	1	2	yes	—
109	C₂×C₄×C₄₁	#106	1	1	yes	characteristic
110	C₂×C₄×C₄₁	#106	1	1	yes	characteristic
111	C₂×C₄×C₄₁	#107	1	2	yes	—
112	C₂×C₄×C₄₁	#107	1	2	yes	—
113	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₄₁	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₄₁	C₂×SD₁₆	1	1	yes	characteristic
118	C₂×C₄×C₄₁	C₂×Q₁₆	1	1	yes	characteristic
119	C₈×C₄₁	—	4	4	no	—
120	C₈×C₄₁	#95	1	2	yes	—
121	C₈×C₄₁	#95	1	2	yes	—
122	C₈×C₄₁	C₂×C₄²	1	1	yes	characteristic
123	C₈×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₈×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₈×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
126	#12563	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂²×C₄₁	—	2	2	no	—
128	C₂²×C₄₁	—	2	2	no	—
129	C₂²×C₄₁	—	2	2	no	—
130	C₂²×C₄₁	—	4	2	no	—
131	C₂²×C₄₁	—	4	2	no	—
132	C₂²×C₄₁	#340	1	2	yes	—
133	C₂²×C₄₁	#340	1	2	yes	—
134	C₂²×C₄₁	C₂×#102	1	1	yes	characteristic
135	C₂²×C₄₁	C₂×#103	1	1	yes	characteristic
136	C₂²×C₄₁	C₂×#106	1	1	yes	characteristic
137	C₄×C₄₁	—	8	4	no	—
138	C₄×C₄₁	#340	1	2	yes	—
139	C₄×C₄₁	#340	1	2	yes	—
140	C₄×C₄₁	C₂×#95	1	1	yes	characteristic
141	C₄×C₄₁	C₂×#102	1	1	yes	characteristic
142	C₄×C₄₁	C₂×#103	1	1	yes	characteristic
143	C₄×C₄₁	C₂×#107	1	1	yes	characteristic
144	C₂²×M₃₂	C₂×C₄₁	1	1	yes	characteristic
145	C₂²×#108	C₂×C₄₁	1	1	yes	characteristic
146	C₂×#427	C₂×C₄₁	1	1	yes	characteristic
147	#1383	C₂×C₄₁	1	4	yes	complemented
148	C₂×C₄₁	—	2	1	no	—
149	C₂×C₄₁	—	2	2	no	—
150	C₂×C₄₁	—	2	2	no	—
151	C₂×C₄₁	—	2	1	no	—
152	C₂×C₄₁	C₂×#340	1	1	yes	characteristic
153	C₂×C₄₁	#2155	1	2	yes	—
154	C₂²×M₁₆	—	2	1	no	—
155	C₂³×C₈	C₂²×C₄₁	1	1	yes	characteristic
156	C₂×#117	—	2	1	no	—
157	C₂×M₃₂	—	2	2	no	—
158	C₂×M₃₂	—	2	2	no	—
159	C₂×M₃₂	C₄×C₄₁	1	2	yes	complemented
160	C₂×M₃₂	C₂²×C₄₁	1	2	yes	—
161	C₂²×C₁₆	C₄×C₄₁	1	2	yes	complemented
162	C₂×#108	—	2	1	no	—
163	C₂×#108	—	2	2	no	—
164	C₂×#108	C₄×C₄₁	1	4	yes	complemented
165	C₂×#108	C₂²×C₄₁	1	2	yes	—
166	#427	—	2	2	no	—
167	#427	C₄×C₄₁	1	4	yes	—
168	C₂×#107	—	2	1	no	—
169	C₂×#106	C₂²×C₄₁	1	2	yes	—
170	C₄₁	#12563	1	1	yes	characteristic, direct product factor, Sylow
171	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
172	C₂×M₁₆	—	2	4	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	4	2	no	—
175	C₂×M₁₆	—	4	1	no	—
176	C₂²×C₈	—	4	1	no	—
177	C₂²×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
178	C₂²×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
179	C₂²×C₈	C₂×C₄×C₄₁	1	2	yes	—
180	C₂²×C₈	C₂×C₄×C₄₁	1	2	yes	—
181	C₂²×C₈	D₄×C₄₁	1	1	yes	characteristic
182	C₂²×C₈	D₄×C₄₁	1	2	yes	—
183	C₂²×C₈	D₄×C₄₁	1	2	yes	—
184	C₂²×C₈	Q₈×C₄₁	1	1	yes	characteristic
185	C₂²×C₈	Q₈×C₄₁	1	1	yes	characteristic
186	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
187	#117	—	2	4	no	—
188	#117	—	4	2	no	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₂²⋊C₄	—	4	1	no	—
191	C₂×C₄²	—	4	1	no	—
192	M₃₂	—	2	4	no	—
193	M₃₂	—	2	2	no	—
194	M₃₂	—	2	2	no	—
195	C₂×C₁₆	—	2	4	no	—
196	C₂×C₁₆	C₂×C₄×C₄₁	1	4	yes	—
197	#108	—	2	4	no	—
198	#108	—	2	2	no	—
199	#108	C₂×C₄×C₄₁	1	4	yes	—
200	#107	—	2	4	no	—
201	#106	—	2	2	no	—
202	#106	C₂×C₄×C₄₁	1	4	yes	—
203	C₂⁴	SD₁₆×C₄₁	1	1	yes	characteristic
204	C₂²×C₄	—	4	2	no	—
205	C₂²×C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
206	C₂²×C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
207	C₂²×C₄	C₄⋊C₄×C₄₁	1	1	yes	characteristic
208	C₂²×C₄	C₄⋊C₄×C₄₁	1	1	yes	characteristic
209	C₂²×C₄	D₈×C₄₁	1	2	yes	—
210	C₂²×C₄	SD₁₆×C₄₁	1	1	yes	characteristic
211	C₂²×C₄	SD₁₆×C₄₁	1	1	yes	characteristic
212	C₂²×C₄	SD₁₆×C₄₁	1	1	yes	characteristic
213	C₂²×C₄	Q₁₆×C₄₁	1	2	yes	—
214	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
215	M₁₆	—	2	4	no	—
216	M₁₆	—	4	4	no	—
217	M₁₆	—	4	2	no	—
218	C₂×C₈	—	4	4	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	C₄²×C₄₁	1	2	yes	—
221	C₂×C₈	C₄²×C₄₁	1	2	yes	—
222	C₂×C₈	C₂²⋊C₄×C₄₁	1	2	yes	—
223	C₂×C₈	C₂²⋊C₄×C₄₁	1	2	yes	—
224	C₂×C₈	C₂²⋊C₄×C₄₁	1	2	yes	—
225	C₂×C₈	C₂²⋊C₄×C₄₁	1	2	yes	—
226	C₂×C₈	C₄⋊C₄×C₄₁	1	1	yes	characteristic
227	C₂×C₈	C₄⋊C₄×C₄₁	1	1	yes	characteristic
228	C₂×C₈	C₄⋊C₄×C₄₁	1	2	yes	—
229	C₂×C₈	C₄⋊C₄×C₄₁	1	2	yes	—
230	C₂×C₈	C₄⋊C₄×C₄₁	1	2	yes	—
231	C₂×C₈	C₄⋊C₄×C₄₁	1	2	yes	—
232	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
233	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
234	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
235	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
236	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
237	C₂×C₈	C₂×Q₈×C₄₁	1	1	yes	characteristic
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	—	4	4	no	—
241	C₄²	—	4	4	no	—
242	C₁₆	—	2	4	no	—
243	C₂³	—	4	2	no	—
244	C₂³	#102×C₄₁	1	2	yes	—
245	C₂³	#103×C₄₁	1	2	yes	—
246	C₂³	#106×C₄₁	1	1	yes	characteristic
247	C₂³	#106×C₄₁	1	1	yes	characteristic
248	C₂³	C₂×SD₁₆×C₄₁	1	1	yes	characteristic
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	8	4	no	—
251	C₂×C₄	#95×C₄₁	1	2	yes	—
252	C₂×C₄	#95×C₄₁	1	2	yes	—
253	C₂×C₄	#102×C₄₁	1	2	yes	—
254	C₂×C₄	#102×C₄₁	1	2	yes	—
255	C₂×C₄	#102×C₄₁	1	2	yes	—
256	C₂×C₄	#103×C₄₁	1	2	yes	—
257	C₂×C₄	#103×C₄₁	1	2	yes	—
258	C₂×C₄	#103×C₄₁	1	2	yes	—
259	C₂×C₄	#106×C₄₁	1	1	yes	characteristic
260	C₂×C₄	#106×C₄₁	1	1	yes	characteristic
261	C₂×C₄	#107×C₄₁	1	2	yes	—
262	C₂×C₄	#107×C₄₁	1	2	yes	—
263	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
264	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
265	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
266	C₂×C₄	C₂×D₈×C₄₁	1	1	yes	characteristic
267	C₂×C₄	C₂×SD₁₆×C₄₁	1	1	yes	characteristic
268	C₂×C₄	C₂×Q₁₆×C₄₁	1	1	yes	characteristic
269	C₈	—	4	4	no	—
270	C₈	#95×C₄₁	1	2	yes	—
271	C₈	#95×C₄₁	1	2	yes	—
272	C₈	C₂×C₄²×C₄₁	1	1	yes	characteristic
273	C₈	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
274	C₈	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
275	C₈	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	C₄₁×#340	1	2	yes	—
282	C₂²	C₄₁×#340	1	2	yes	—
283	C₂²	C₂×#102×C₄₁	1	1	yes	characteristic
284	C₂²	C₂×#103×C₄₁	1	1	yes	characteristic
285	C₂²	C₂×#106×C₄₁	1	1	yes	characteristic
286	C₄	—	8	4	no	—
287	C₄	C₄₁×#340	1	2	yes	—
288	C₄	C₄₁×#340	1	2	yes	—
289	C₄	C₂×#95×C₄₁	1	1	yes	characteristic
290	C₄	C₂×#102×C₄₁	1	1	yes	characteristic
291	C₄	C₂×#103×C₄₁	1	1	yes	characteristic
292	C₄	C₂×#107×C₄₁	1	1	yes	characteristic
293	C₂	—	2	1	no	—
294	C₂	—	2	2	no	—
295	C₂	—	2	2	no	—
296	C₂	—	2	1	no	—
297	C₂	C₂×C₄₁×#340	1	1	yes	characteristic
298	C₂	C₄₁×#2155	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₄₁	#106
commutator	C₈	C₂×C₄²×C₄₁
Frattini, cosocle	C₂²×C₈	C₂³×C₄₁
socle	C₂²×C₄₁	C₂×#106
prime radical	C₂⁴×C₄₁	SD₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₁ ◁ C₂³×C₄×C₄₁ ◁ C₂³×C₈×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₄₁×#1383	C₂	semidirect
10. C₂×M₃₂×C₄₁	C₄	semidirect
12. C₂²×C₁₆×C₄₁	C₄	semidirect
15. C₂×#108×C₄₁	C₄	semidirect
126. #12563	C₄₁	direct (N × H)
147. #1383	C₂×C₄₁	semidirect
159. C₂×M₃₂	C₄×C₄₁	semidirect
161. C₂²×C₁₆	C₄×C₄₁	semidirect
164. C₂×#108	C₄×C₄₁	semidirect
170. C₄₁	#12563	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×M₃₂×C₄₁	2
C₂²×#108×C₄₁	2
C₂×C₄₁×#427	2
C₄₁×#1383	2
#12563	41

Subgroups — C41×#12563 #23788764

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#12563 #23788764