Subgroups — C₄₁×#12567 #23788768

Order 10496.

586 conjugacy classes of subgroups.

The subgroup lattice has 318 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	C₂×C₄₁×#427	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₁×#427	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1386	C₂	1	4	yes	maximal, complemented
5	C₂³×C₈×C₄₁	C₂²	1	1	yes	characteristic
6	C₂×#117×C₄₁	—	2	1	no	—
7	C₂×#117×C₄₁	—	2	1	no	—
8	C₂×M₃₂×C₄₁	—	2	2	no	—
9	C₂×M₃₂×C₄₁	—	2	1	no	—
10	C₂×M₃₂×C₄₁	C₄	1	4	yes	complemented
11	C₂×M₃₂×C₄₁	C₂²	1	2	yes	—
12	C₂²×C₁₆×C₄₁	—	2	1	no	—
13	C₄₁×#427	—	2	1	no	—
14	C₄₁×#427	—	2	1	no	—
15	C₄₁×#427	—	2	2	no	—
16	C₄₁×#427	C₄	1	2	yes	complemented
17	C₄₁×#427	C₄	1	4	yes	complemented
18	C₄₁×#427	C₂²	1	2	yes	—
19	C₂×#107×C₄₁	—	2	1	no	—
20	C₂×#107×C₄₁	C₄	1	2	yes	complemented
21	C₂×#106×C₄₁	—	2	1	no	—
22	C₂×#106×C₄₁	C₂²	1	2	yes	—
23	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₄₁	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₄₁	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₄₁	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₄₁	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₄₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₄₁	D₄	1	2	yes	—
32	C₂²×C₈×C₄₁	Q₈	1	2	yes	—
33	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
34	#117×C₄₁	—	2	4	no	—
35	#117×C₄₁	—	2	2	no	—
36	#117×C₄₁	—	2	2	no	—
37	#117×C₄₁	—	4	1	no	—
38	#117×C₄₁	—	4	2	no	—
39	#117×C₄₁	—	4	1	no	—
40	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
41	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
42	C₂×C₂²⋊C₄×C₄₁	—	4	1	no	—
43	C₂×C₂²⋊C₄×C₄₁	—	4	1	no	—
44	C₂×C₄²×C₄₁	—	4	1	no	—
45	C₂×C₄²×C₄₁	—	4	1	no	—
46	M₃₂×C₄₁	—	2	4	no	—
47	M₃₂×C₄₁	—	2	2	no	—
48	M₃₂×C₄₁	C₂×C₄	1	4	yes	complemented
49	C₂×C₁₆×C₄₁	—	2	4	no	—
50	C₂×C₁₆×C₄₁	—	2	2	no	—
51	#107×C₄₁	—	2	4	no	—
52	#107×C₄₁	—	2	2	no	—
53	#107×C₄₁	C₂×C₄	1	4	yes	—
54	#106×C₄₁	—	2	2	no	—
55	#106×C₄₁	—	2	2	no	—
56	#106×C₄₁	—	2	2	no	—
57	#106×C₄₁	C₂×C₄	1	4	yes	complemented
58	C₂⁴×C₄₁	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₄₁	—	4	2	no	—
60	C₂²×C₄×C₄₁	—	4	2	no	—
61	C₂²×C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₁	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₄₁	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₄₁	SD₁₆	1	2	yes	—
67	C₂²×C₄×C₄₁	SD₁₆	1	1	yes	characteristic
68	C₂²×C₄×C₄₁	SD₁₆	1	1	yes	characteristic
69	C₂²×C₄×C₄₁	Q₁₆	1	2	yes	—
70	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₄₁	C₄²	1	2	yes	—
72	C₂×C₈×C₄₁	C₄²	1	2	yes	—
73	C₂×C₈×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₄₁	C₂²⋊C₄	1	2	yes	—
75	C₂×C₈×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₄₁	C₂²⋊C₄	1	2	yes	—
77	C₂×C₈×C₄₁	C₂²⋊C₄	1	2	yes	—
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	C₄⋊C₄	1	2	yes	—
83	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₄₁	C₂×Q₈	1	1	yes	characteristic
89	C₄⋊C₄×C₄₁	—	2	4	no	—
90	C₄⋊C₄×C₄₁	—	2	4	no	—
91	C₄⋊C₄×C₄₁	—	4	2	no	—
92	C₄⋊C₄×C₄₁	—	4	2	no	—
93	C₂²⋊C₄×C₄₁	—	4	2	no	—
94	C₂²⋊C₄×C₄₁	—	4	4	no	—
95	C₂²⋊C₄×C₄₁	—	4	2	no	—
96	C₄²×C₄₁	—	4	4	no	—
97	C₄²×C₄₁	—	4	2	no	—
98	C₄²×C₄₁	—	4	2	no	—
99	C₁₆×C₄₁	—	2	4	no	—
100	C₂³×C₄₁	—	4	2	no	—
101	C₂³×C₄₁	#102	1	1	yes	characteristic
102	C₂³×C₄₁	#102	1	1	yes	characteristic
103	C₂³×C₄₁	#102	1	2	yes	—
104	C₂³×C₄₁	#107	1	2	yes	—
105	C₂³×C₄₁	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₄₁	—	4	4	no	—
107	C₂×C₄×C₄₁	—	4	4	no	—
108	C₂×C₄×C₄₁	—	8	4	no	—
109	C₂×C₄×C₄₁	—	8	4	no	—
110	C₂×C₄×C₄₁	#95	1	2	yes	—
111	C₂×C₄×C₄₁	#95	1	2	yes	—
112	C₂×C₄×C₄₁	#102	1	2	yes	—
113	C₂×C₄×C₄₁	#102	1	1	yes	characteristic
114	C₂×C₄×C₄₁	#102	1	1	yes	characteristic
115	C₂×C₄×C₄₁	#103	1	2	yes	—
116	C₂×C₄×C₄₁	#103	1	2	yes	—
117	C₂×C₄×C₄₁	#103	1	2	yes	—
118	C₂×C₄×C₄₁	#103	1	2	yes	—
119	C₂×C₄×C₄₁	#106	1	2	yes	—
120	C₂×C₄×C₄₁	#106	1	2	yes	—
121	C₂×C₄×C₄₁	#107	1	2	yes	—
122	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₄₁	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₄×C₄₁	C₂×SD₁₆	1	1	yes	characteristic
127	C₂×C₄×C₄₁	C₂×Q₁₆	1	1	yes	characteristic
128	C₈×C₄₁	#95	1	2	yes	—
129	C₈×C₄₁	#95	1	2	yes	—
130	C₈×C₄₁	C₂×C₄²	1	1	yes	characteristic
131	C₈×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₈×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₈×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
134	#12567	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₂²×C₄₁	—	2	2	no	—
136	C₂²×C₄₁	—	2	2	no	—
137	C₂²×C₄₁	—	2	2	no	—
138	C₂²×C₄₁	—	4	2	no	—
139	C₂²×C₄₁	—	4	2	no	—
140	C₂²×C₄₁	#340	1	2	yes	—
141	C₂²×C₄₁	#340	1	2	yes	—
142	C₂²×C₄₁	C₂×#102	1	1	yes	characteristic
143	C₂²×C₄₁	C₂×#102	1	1	yes	characteristic
144	C₂²×C₄₁	C₂×#107	1	1	yes	characteristic
145	C₄×C₄₁	—	8	4	no	—
146	C₄×C₄₁	—	8	4	no	—
147	C₄×C₄₁	#340	1	2	yes	—
148	C₄×C₄₁	#340	1	2	yes	—
149	C₄×C₄₁	C₂×#95	1	1	yes	characteristic
150	C₄×C₄₁	C₂×#103	1	1	yes	characteristic
151	C₄×C₄₁	C₂×#103	1	1	yes	characteristic
152	C₄×C₄₁	C₂×#106	1	1	yes	characteristic
153	C₂²×M₃₂	C₂×C₄₁	1	1	yes	characteristic
154	C₂×#427	C₂×C₄₁	1	1	yes	characteristic
155	C₂×#427	C₂×C₄₁	1	1	yes	characteristic
156	#1386	C₂×C₄₁	1	4	yes	complemented
157	C₂×C₄₁	—	2	1	no	—
158	C₂×C₄₁	—	2	2	no	—
159	C₂×C₄₁	—	2	2	no	—
160	C₂×C₄₁	—	2	1	no	—
161	C₂×C₄₁	C₂×#340	1	1	yes	characteristic
162	C₂×C₄₁	#2145	1	2	yes	—
163	C₂³×C₈	C₂²×C₄₁	1	1	yes	characteristic
164	C₂×#117	—	2	1	no	—
165	C₂×#117	—	2	1	no	—
166	C₂×M₃₂	—	2	2	no	—
167	C₂×M₃₂	—	2	1	no	—
168	C₂×M₃₂	C₄×C₄₁	1	4	yes	complemented
169	C₂×M₃₂	C₂²×C₄₁	1	2	yes	—
170	C₂²×C₁₆	—	2	1	no	—
171	#427	—	2	1	no	—
172	#427	—	2	1	no	—
173	#427	—	2	2	no	—
174	#427	C₄×C₄₁	1	2	yes	complemented
175	#427	C₄×C₄₁	1	4	yes	complemented
176	#427	C₂²×C₄₁	1	2	yes	—
177	C₂×#107	—	2	1	no	—
178	C₂×#107	C₄×C₄₁	1	2	yes	complemented
179	C₂×#106	—	2	1	no	—
180	C₂×#106	C₂²×C₄₁	1	2	yes	—
181	C₄₁	#12567	1	1	yes	characteristic, direct product factor, Sylow
182	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
183	C₂²×C₈	C₂×C₄×C₄₁	1	2	yes	—
184	C₂²×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
185	C₂²×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
186	C₂²×C₈	C₂×C₄×C₄₁	1	2	yes	—
187	C₂²×C₈	D₄×C₄₁	1	1	yes	characteristic
188	C₂²×C₈	D₄×C₄₁	1	1	yes	characteristic
189	C₂²×C₈	D₄×C₄₁	1	1	yes	characteristic
190	C₂²×C₈	D₄×C₄₁	1	2	yes	—
191	C₂²×C₈	Q₈×C₄₁	1	2	yes	—
192	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
193	#117	—	2	2	no	—
194	#117	—	2	4	no	—
195	#117	—	2	2	no	—
196	#117	—	4	1	no	—
197	#117	—	4	2	no	—
198	#117	—	4	1	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₂²⋊C₄	—	4	1	no	—
202	C₂×C₂²⋊C₄	—	4	1	no	—
203	C₂×C₄²	—	4	1	no	—
204	C₂×C₄²	—	4	1	no	—
205	M₃₂	—	2	4	no	—
206	M₃₂	—	2	2	no	—
207	M₃₂	C₂×C₄×C₄₁	1	4	yes	complemented
208	C₂×C₁₆	—	2	4	no	—
209	C₂×C₁₆	—	2	2	no	—
210	#107	—	2	4	no	—
211	#107	—	2	2	no	—
212	#107	C₂×C₄×C₄₁	1	4	yes	—
213	#106	—	2	2	no	—
214	#106	—	2	2	no	—
215	#106	—	2	2	no	—
216	#106	C₂×C₄×C₄₁	1	4	yes	complemented
217	C₂⁴	D₈×C₄₁	1	1	yes	characteristic
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	—	4	2	no	—
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	C₄⋊C₄×C₄₁	1	2	yes	—
224	C₂²×C₄	D₈×C₄₁	1	1	yes	characteristic
225	C₂²×C₄	SD₁₆×C₄₁	1	1	yes	characteristic
226	C₂²×C₄	SD₁₆×C₄₁	1	2	yes	—
227	C₂²×C₄	SD₁₆×C₄₁	1	1	yes	characteristic
228	C₂²×C₄	Q₁₆×C₄₁	1	2	yes	—
229	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
230	C₂×C₈	C₄²×C₄₁	1	2	yes	—
231	C₂×C₈	C₄²×C₄₁	1	2	yes	—
232	C₂×C₈	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
233	C₂×C₈	C₂²⋊C₄×C₄₁	1	2	yes	—
234	C₂×C₈	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
235	C₂×C₈	C₂²⋊C₄×C₄₁	1	2	yes	—
236	C₂×C₈	C₂²⋊C₄×C₄₁	1	2	yes	—
237	C₂×C₈	C₄⋊C₄×C₄₁	1	2	yes	—
238	C₂×C₈	C₄⋊C₄×C₄₁	1	2	yes	—
239	C₂×C₈	C₄⋊C₄×C₄₁	1	2	yes	—
240	C₂×C₈	C₄⋊C₄×C₄₁	1	2	yes	—
241	C₂×C₈	C₄⋊C₄×C₄₁	1	2	yes	—
242	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
243	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
244	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
245	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
246	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
247	C₂×C₈	C₂×Q₈×C₄₁	1	1	yes	characteristic
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₂²⋊C₄	—	4	2	no	—
253	C₂²⋊C₄	—	4	4	no	—
254	C₂²⋊C₄	—	4	2	no	—
255	C₄²	—	4	4	no	—
256	C₄²	—	4	2	no	—
257	C₄²	—	4	2	no	—
258	C₁₆	—	2	4	no	—
259	C₂³	—	4	2	no	—
260	C₂³	#102×C₄₁	1	1	yes	characteristic
261	C₂³	#102×C₄₁	1	1	yes	characteristic
262	C₂³	#102×C₄₁	1	2	yes	—
263	C₂³	#107×C₄₁	1	2	yes	—
264	C₂³	C₂×D₈×C₄₁	1	1	yes	characteristic
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	8	4	no	—
268	C₂×C₄	—	8	4	no	—
269	C₂×C₄	#95×C₄₁	1	2	yes	—
270	C₂×C₄	#95×C₄₁	1	2	yes	—
271	C₂×C₄	#102×C₄₁	1	2	yes	—
272	C₂×C₄	#102×C₄₁	1	1	yes	characteristic
273	C₂×C₄	#102×C₄₁	1	1	yes	characteristic
274	C₂×C₄	#103×C₄₁	1	2	yes	—
275	C₂×C₄	#103×C₄₁	1	2	yes	—
276	C₂×C₄	#103×C₄₁	1	2	yes	—
277	C₂×C₄	#103×C₄₁	1	2	yes	—
278	C₂×C₄	#106×C₄₁	1	2	yes	—
279	C₂×C₄	#106×C₄₁	1	2	yes	—
280	C₂×C₄	#107×C₄₁	1	2	yes	—
281	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
282	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
283	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
284	C₂×C₄	C₂×SD₁₆×C₄₁	1	1	yes	characteristic
285	C₂×C₄	C₂×SD₁₆×C₄₁	1	1	yes	characteristic
286	C₂×C₄	C₂×Q₁₆×C₄₁	1	1	yes	characteristic
287	C₈	#95×C₄₁	1	2	yes	—
288	C₈	#95×C₄₁	1	2	yes	—
289	C₈	C₂×C₄²×C₄₁	1	1	yes	characteristic
290	C₈	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
291	C₈	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
292	C₈	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	4	2	no	—
297	C₂²	—	4	2	no	—
298	C₂²	C₄₁×#340	1	2	yes	—
299	C₂²	C₄₁×#340	1	2	yes	—
300	C₂²	C₂×#102×C₄₁	1	1	yes	characteristic
301	C₂²	C₂×#102×C₄₁	1	1	yes	characteristic
302	C₂²	C₂×#107×C₄₁	1	1	yes	characteristic
303	C₄	—	8	4	no	—
304	C₄	—	8	4	no	—
305	C₄	C₄₁×#340	1	2	yes	—
306	C₄	C₄₁×#340	1	2	yes	—
307	C₄	C₂×#95×C₄₁	1	1	yes	characteristic
308	C₄	C₂×#103×C₄₁	1	1	yes	characteristic
309	C₄	C₂×#103×C₄₁	1	1	yes	characteristic
310	C₄	C₂×#106×C₄₁	1	1	yes	characteristic
311	C₂	—	2	1	no	—
312	C₂	—	2	2	no	—
313	C₂	—	2	2	no	—
314	C₂	—	2	1	no	—
315	C₂	C₂×C₄₁×#340	1	1	yes	characteristic
316	C₂	C₄₁×#2145	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₄₁	#102
commutator	C₈	C₂×C₄²×C₄₁
Frattini, cosocle	C₂²×C₈	C₂³×C₄₁
socle	C₂²×C₄₁	C₂×#102
prime radical	C₂⁴×C₄₁	D₈
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×#1386	C₂	semidirect
10. C₂×M₃₂×C₄₁	C₄	semidirect
16. C₄₁×#427	C₄	semidirect
17. C₄₁×#427	C₄	semidirect
20. C₂×#107×C₄₁	C₄	semidirect
48. M₃₂×C₄₁	C₂×C₄	semidirect
57. #106×C₄₁	C₂×C₄	semidirect
134. #12567	C₄₁	direct (N × H)
156. #1386	C₂×C₄₁	semidirect
168. C₂×M₃₂	C₄×C₄₁	semidirect
174. #427	C₄×C₄₁	semidirect
175. #427	C₄×C₄₁	semidirect
178. C₂×#107	C₄×C₄₁	semidirect
181. C₄₁	#12567	direct (N × H)
207. M₃₂	C₂×C₄×C₄₁	semidirect
216. #106	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
C₂²×M₃₂×C₄₁	2
C₂×C₄₁×#427	2
C₄₁×#1386	2
#12567	41

Subgroups — C41×#12567 #23788768

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#12567 #23788768