Subgroups — C₄₁×#17825 #23794026

Order 10496.

572 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	C₄₁×#2989	C₂	1	2	yes	maximal
2	C₄₁×#2922	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#1574	C₂	1	2	yes	maximal, complemented
4	C₄₁×#1573	C₂	1	2	yes	maximal, complemented
5	C₄₁×#1572	C₂	1	2	yes	maximal, complemented
6	C₄₁×#1571	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1569	C₂	1	2	yes	maximal, complemented
8	C₄₁×#1568	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1567	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#41×C₄₁	C₂²	1	1	yes	characteristic
11	C₄₁×#446	—	2	2	no	—
12	C₄₁×#436	—	2	2	no	—
13	C₄₁×#435	—	2	2	no	—
14	C₄₁×#435	—	2	2	no	—
15	C₄₁×#433	C₄	1	4	yes	—
16	C₄₁×#432	C₂²	1	2	yes	—
17	C₄₁×#432	C₂²	1	2	yes	—
18	C₄₁×#431	C₂²	1	2	yes	—
19	C₄₁×#431	C₂²	1	2	yes	—
20	C₄₁×#431	C₂²	1	2	yes	—
21	C₄₁×#424	C₄	1	2	yes	—
22	C₄₁×#423	C₂²	1	2	yes	—
23	C₄₁×#423	C₂²	1	2	yes	—
24	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
25	C₄₁×#408	C₄	1	2	yes	—
26	C₄×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
28	C₄₁×#335	C₂²	1	2	yes	—
29	C₄₁×#335	C₂²	1	1	yes	characteristic
30	C₄₁×#335	C₂²	1	1	yes	characteristic
31	C₄₁×#335	C₂²	1	1	yes	characteristic
32	C₄₁×#335	C₂²	1	1	yes	characteristic
33	C₄₁×#335	C₂²	1	2	yes	—
34	C₄₁×#335	C₂²	1	2	yes	—
35	C₄₁×#334	C₂²	1	2	yes	—
36	C₄₁×#334	C₂²	1	2	yes	—
37	C₄₁×#334	C₂²	1	2	yes	—
38	C₂×#41×C₄₁	C₂×C₄	1	1	yes	characteristic
39	C₂×M₁₆×C₄₁	—	2	2	no	—
40	C₂×M₁₆×C₄₁	—	2	2	no	—
41	C₂×M₁₆×C₄₁	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₄₁	—	2	2	no	—
43	C₂²×C₈×C₄₁	C₂×C₄	1	2	yes	—
44	C₄×Q₈×C₄₁	D₄	1	1	yes	characteristic
45	C₄×Q₈×C₄₁	Q₈	1	1	yes	characteristic
46	C₄×D₄×C₄₁	D₄	1	1	yes	characteristic
47	C₄×D₄×C₄₁	D₄	1	2	yes	—
48	C₄×D₄×C₄₁	Q₈	1	1	yes	characteristic
49	C₄×D₄×C₄₁	Q₈	1	2	yes	—
50	#117×C₄₁	C₂×C₄	1	1	yes	characteristic
51	#117×C₄₁	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
54	C₄⋊C₈×C₄₁	—	2	2	no	—
55	C₄⋊C₈×C₄₁	—	2	2	no	—
56	C₄⋊C₈×C₄₁	—	2	2	no	—
57	C₄⋊C₈×C₄₁	C₂×C₄	1	4	yes	—
58	C₄⋊C₈×C₄₁	C₂×C₄	1	4	yes	—
59	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
60	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
61	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
62	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
63	C₂²⋊C₈×C₄₁	—	2	2	no	—
64	C₂²⋊C₈×C₄₁	—	2	2	no	—
65	C₂²⋊C₈×C₄₁	—	2	2	no	—
66	C₂²⋊C₈×C₄₁	C₂×C₄	1	4	yes	—
67	C₂²⋊C₈×C₄₁	C₂×C₄	1	4	yes	—
68	#97×C₄₁	—	2	2	no	—
69	#97×C₄₁	—	2	2	no	—
70	#97×C₄₁	C₂×C₄	1	2	yes	—
71	C₄×C₈×C₄₁	—	2	2	no	—
72	C₄×C₈×C₄₁	C₂×C₄	1	2	yes	—
73	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₄₁	C₂³	1	2	yes	—
75	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
76	#41×C₄₁	—	4	2	no	—
77	C₂×Q₈×C₄₁	C₄⋊C₄	1	1	yes	characteristic
78	C₂×D₄×C₄₁	C₄⋊C₄	1	1	yes	characteristic
79	C₂×D₄×C₄₁	C₄⋊C₄	1	2	yes	—
80	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₁	C₂×Q₈	1	1	yes	characteristic
85	C₂²×C₄×C₄₁	C₂×Q₈	1	2	yes	—
86	M₁₆×C₄₁	—	4	2	no	—
87	M₁₆×C₄₁	—	8	2	no	—
88	M₁₆×C₄₁	—	8	2	no	—
89	C₂×C₈×C₄₁	—	2	2	no	—
90	C₂×C₈×C₄₁	—	2	2	no	—
91	C₂×C₈×C₄₁	—	2	2	no	—
92	C₂×C₈×C₄₁	—	2	2	no	—
93	C₂×C₈×C₄₁	—	4	2	no	—
94	C₂×C₈×C₄₁	—	8	2	no	—
95	C₂×C₈×C₄₁	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₄₁	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₄₁	C₂²×C₄	1	4	yes	—
98	C₄⋊C₄×C₄₁	C₄⋊C₄	1	1	yes	characteristic
99	C₄⋊C₄×C₄₁	C₄⋊C₄	1	2	yes	—
100	C₄⋊C₄×C₄₁	C₄⋊C₄	1	1	yes	characteristic
101	C₄⋊C₄×C₄₁	C₄⋊C₄	1	2	yes	—
102	C₂²⋊C₄×C₄₁	C₄⋊C₄	1	2	yes	—
103	C₂²⋊C₄×C₄₁	C₄⋊C₄	1	2	yes	—
104	C₂²⋊C₄×C₄₁	C₄⋊C₄	1	2	yes	—
105	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
107	C₄²×C₄₁	C₂×D₄	1	2	yes	—
108	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₄₁	C₂×Q₈	1	1	yes	characteristic
110	C₄²×C₄₁	C₂×Q₈	1	2	yes	—
111	C₄²×C₄₁	C₂⁴	1	1	yes	characteristic
112	C₂³×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
113	C₂³×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
114	Q₈×C₄₁	—	4	1	no	—
115	D₄×C₄₁	—	4	1	no	—
116	D₄×C₄₁	—	4	2	no	—
117	C₂×C₄×C₄₁	—	2	2	no	—
118	C₂×C₄×C₄₁	—	2	4	no	—
119	C₂×C₄×C₄₁	—	2	2	no	—
120	C₂×C₄×C₄₁	—	2	1	no	—
121	C₂×C₄×C₄₁	—	2	2	no	—
122	C₂×C₄×C₄₁	—	2	1	no	—
123	C₂×C₄×C₄₁	—	4	1	no	—
124	C₂×C₄×C₄₁	—	4	2	no	—
125	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
130	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₄₁	C₂²×Q₈	1	1	yes	characteristic
134	C₈×C₄₁	—	2	2	no	—
135	C₈×C₄₁	—	2	2	no	—
136	C₈×C₄₁	—	2	4	no	—
137	C₈×C₄₁	—	4	2	no	—
138	C₈×C₄₁	—	4	2	no	—
139	C₈×C₄₁	—	4	2	no	—
140	C₈×C₄₁	—	4	2	no	—
141	#17825	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₂²×C₄₁	—	2	1	no	—
143	C₂²×C₄₁	—	2	2	no	—
144	C₂²×C₄₁	—	2	2	no	—
145	C₂²×C₄₁	—	2	1	no	—
146	C₂²×C₄₁	—	2	2	no	—
147	C₂²×C₄₁	—	2	1	no	—
148	C₂²×C₄₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
149	C₄×C₄₁	—	2	1	no	—
150	C₄×C₄₁	—	2	2	no	—
151	C₄×C₄₁	—	2	2	no	—
152	C₄×C₄₁	—	2	1	no	—
153	C₄×C₄₁	—	2	1	no	—
154	C₄×C₄₁	—	2	1	no	—
155	C₄×C₄₁	—	2	2	no	—
156	C₄×C₄₁	#568	1	2	yes	—
157	C₄×C₄₁	#576	1	1	yes	characteristic
158	C₄×C₄₁	#578	1	1	yes	characteristic
159	#2989	C₂×C₄₁	1	2	yes	—
160	#2922	C₂×C₄₁	1	1	yes	characteristic
161	#1574	C₂×C₄₁	1	2	yes	complemented
162	#1573	C₂×C₄₁	1	2	yes	complemented
163	#1572	C₂×C₄₁	1	2	yes	complemented
164	#1571	C₂×C₄₁	1	2	yes	complemented
165	#1569	C₂×C₄₁	1	2	yes	complemented
166	#1568	C₂×C₄₁	1	1	yes	characteristic, complemented
167	#1567	C₂×C₄₁	1	1	yes	characteristic, complemented
168	C₂×C₄₁	—	4	1	no	—
169	C₂×C₄₁	—	4	2	no	—
170	C₂×C₄₁	#2905	1	1	yes	characteristic
171	C₂×C₄₁	#2913	1	1	yes	characteristic
172	C₂×C₄₁	#2916	1	1	yes	characteristic
173	C₄×#41	C₂²×C₄₁	1	1	yes	characteristic
174	#446	—	2	2	no	—
175	#436	—	2	2	no	—
176	#435	—	2	2	no	—
177	#435	—	2	2	no	—
178	#433	C₄×C₄₁	1	4	yes	—
179	#432	C₂²×C₄₁	1	2	yes	—
180	#432	C₂²×C₄₁	1	2	yes	—
181	#431	C₂²×C₄₁	1	2	yes	—
182	#431	C₂²×C₄₁	1	2	yes	—
183	#431	C₂²×C₄₁	1	2	yes	—
184	#424	C₄×C₄₁	1	2	yes	—
185	#423	C₂²×C₄₁	1	2	yes	—
186	#423	C₂²×C₄₁	1	2	yes	—
187	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
188	#408	C₄×C₄₁	1	2	yes	—
189	C₄×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
190	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
191	#335	C₂²×C₄₁	1	2	yes	—
192	#335	C₂²×C₄₁	1	1	yes	characteristic
193	#335	C₂²×C₄₁	1	1	yes	characteristic
194	#335	C₂²×C₄₁	1	1	yes	characteristic
195	#335	C₂²×C₄₁	1	1	yes	characteristic
196	#335	C₂²×C₄₁	1	2	yes	—
197	#335	C₂²×C₄₁	1	2	yes	—
198	#334	C₂²×C₄₁	1	2	yes	—
199	#334	C₂²×C₄₁	1	2	yes	—
200	#334	C₂²×C₄₁	1	2	yes	—
201	C₄₁	#17825	1	1	yes	characteristic, direct product factor, Sylow
202	C₂×#41	C₂×C₄×C₄₁	1	1	yes	characteristic
203	C₂×M₁₆	—	2	2	no	—
204	C₂×M₁₆	—	2	2	no	—
205	C₂×M₁₆	C₂×C₄×C₄₁	1	2	yes	—
206	C₂²×C₈	—	2	2	no	—
207	C₂²×C₈	C₂×C₄×C₄₁	1	2	yes	—
208	C₄×Q₈	D₄×C₄₁	1	1	yes	characteristic
209	C₄×Q₈	Q₈×C₄₁	1	1	yes	characteristic
210	C₄×D₄	D₄×C₄₁	1	1	yes	characteristic
211	C₄×D₄	D₄×C₄₁	1	2	yes	—
212	C₄×D₄	Q₈×C₄₁	1	1	yes	characteristic
213	C₄×D₄	Q₈×C₄₁	1	2	yes	—
214	#117	C₂×C₄×C₄₁	1	1	yes	characteristic
215	#117	C₂×C₄×C₄₁	1	2	yes	—
216	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
217	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
218	C₄⋊C₈	—	2	2	no	—
219	C₄⋊C₈	—	2	2	no	—
220	C₄⋊C₈	—	2	2	no	—
221	C₄⋊C₈	C₂×C₄×C₄₁	1	4	yes	—
222	C₄⋊C₈	C₂×C₄×C₄₁	1	4	yes	—
223	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
224	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
225	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
226	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
227	C₂²⋊C₈	—	2	2	no	—
228	C₂²⋊C₈	—	2	2	no	—
229	C₂²⋊C₈	—	2	2	no	—
230	C₂²⋊C₈	C₂×C₄×C₄₁	1	4	yes	—
231	C₂²⋊C₈	C₂×C₄×C₄₁	1	4	yes	—
232	#97	—	2	2	no	—
233	#97	—	2	2	no	—
234	#97	C₂×C₄×C₄₁	1	2	yes	—
235	C₄×C₈	—	2	2	no	—
236	C₄×C₈	C₂×C₄×C₄₁	1	2	yes	—
237	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
238	C₄×C₈	C₂³×C₄₁	1	2	yes	—
239	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
240	#41	—	4	2	no	—
241	C₂×Q₈	C₄⋊C₄×C₄₁	1	1	yes	characteristic
242	C₂×D₄	C₄⋊C₄×C₄₁	1	1	yes	characteristic
243	C₂×D₄	C₄⋊C₄×C₄₁	1	2	yes	—
244	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
245	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
246	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
247	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
248	C₂²×C₄	C₂×Q₈×C₄₁	1	1	yes	characteristic
249	C₂²×C₄	C₂×Q₈×C₄₁	1	2	yes	—
250	M₁₆	—	4	2	no	—
251	M₁₆	—	8	2	no	—
252	M₁₆	—	8	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	4	2	no	—
258	C₂×C₈	—	8	2	no	—
259	C₂×C₈	C₂²×C₄×C₄₁	1	2	yes	—
260	C₂×C₈	C₂²×C₄×C₄₁	1	2	yes	—
261	C₂×C₈	C₂²×C₄×C₄₁	1	4	yes	—
262	C₄⋊C₄	C₄⋊C₄×C₄₁	1	1	yes	characteristic
263	C₄⋊C₄	C₄⋊C₄×C₄₁	1	2	yes	—
264	C₄⋊C₄	C₄⋊C₄×C₄₁	1	1	yes	characteristic
265	C₄⋊C₄	C₄⋊C₄×C₄₁	1	2	yes	—
266	C₂²⋊C₄	C₄⋊C₄×C₄₁	1	2	yes	—
267	C₂²⋊C₄	C₄⋊C₄×C₄₁	1	2	yes	—
268	C₂²⋊C₄	C₄⋊C₄×C₄₁	1	2	yes	—
269	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
270	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
271	C₄²	C₂×D₄×C₄₁	1	2	yes	—
272	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
273	C₄²	C₂×Q₈×C₄₁	1	1	yes	characteristic
274	C₄²	C₂×Q₈×C₄₁	1	2	yes	—
275	C₄²	C₂⁴×C₄₁	1	1	yes	characteristic
276	C₂³	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
277	C₂³	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
278	Q₈	—	4	1	no	—
279	D₄	—	4	1	no	—
280	D₄	—	4	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
290	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
291	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
292	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
293	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
294	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
295	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
296	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
297	C₂×C₄	C₂²×Q₈×C₄₁	1	1	yes	characteristic
298	C₈	—	2	2	no	—
299	C₈	—	2	2	no	—
300	C₈	—	2	4	no	—
301	C₈	—	4	2	no	—
302	C₈	—	4	2	no	—
303	C₈	—	4	2	no	—
304	C₈	—	4	2	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	1	no	—
311	C₂²	C₂²×C₄⋊C₄×C₄₁	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	2	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	2	no	—
319	C₄	C₄₁×#568	1	2	yes	—
320	C₄	C₄₁×#576	1	1	yes	characteristic
321	C₄	C₄₁×#578	1	1	yes	characteristic
322	C₂	—	4	1	no	—
323	C₂	—	4	2	no	—
324	C₂	C₄₁×#2905	1	1	yes	characteristic
325	C₂	C₄₁×#2913	1	1	yes	characteristic
326	C₂	C₄₁×#2916	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₄₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₄₁
Frattini, cosocle	C₄²	C₂⁴×C₄₁
socle	C₂²×C₄₁	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₄₁	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₁ ◁ C₄×#41×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄₁×#1574	C₂	semidirect
4. C₄₁×#1573	C₂	semidirect
5. C₄₁×#1572	C₂	semidirect
6. C₄₁×#1571	C₂	semidirect
7. C₄₁×#1569	C₂	semidirect
8. C₄₁×#1568	C₂	semidirect
9. C₄₁×#1567	C₂	semidirect
141. #17825	C₄₁	direct (N × H)
161. #1574	C₂×C₄₁	semidirect
162. #1573	C₂×C₄₁	semidirect
163. #1572	C₂×C₄₁	semidirect
164. #1571	C₂×C₄₁	semidirect
165. #1569	C₂×C₄₁	semidirect
166. #1568	C₂×C₄₁	semidirect
167. #1567	C₂×C₄₁	semidirect
201. C₄₁	#17825	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄₁×#2989	2
C₄₁×#2922	2
C₄₁×#1574	2
C₄₁×#1573	2
C₄₁×#1572	2
C₄₁×#1571	2
C₄₁×#1569	2
C₄₁×#1568	2
C₄₁×#1567	2
#17825	41

Subgroups — C41×#17825 #23794026

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#17825 #23794026