Subgroups — C₄₁×#8452 #23784653

Order 10496.

868 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#555	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#1956	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#1802	C₂	1	2	yes	maximal, complemented
4	C₂×C₄₁×#333	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄₁×D₄⋊C₈	C₂	1	2	yes	maximal, complemented
6	C₄₁×#555	—	2	2	no	—
7	C₄₁×#555	—	2	4	no	—
8	C₄₁×#555	C₄	1	4	yes	complemented
9	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
10	C₂×#124×C₄₁	—	2	2	no	—
11	C₂×#121×C₄₁	—	2	2	no	—
12	C₂×C₄×D₄×C₄₁	C₂²	1	2	yes	—
13	C₂×#117×C₄₁	—	2	2	no	—
14	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
15	C₂×#103×C₄₁	—	2	2	no	—
16	C₂×#102×C₄₁	—	2	2	no	—
17	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
18	C₄₁×#384	C₂²	1	1	yes	characteristic
19	C₄₁×#340	—	2	2	no	—
20	C₄₁×#333	—	2	2	no	—
21	C₄₁×D₄⋊C₈	—	2	4	no	—
22	C₂²×Q₈×C₄₁	D₄	1	2	yes	—
23	C₂²×D₄×C₄₁	—	2	2	no	—
24	C₂²×D₄×C₄₁	D₄	1	2	yes	—
25	C₂³×C₄×C₄₁	—	2	2	no	—
26	C₂²×C₈×C₄₁	—	2	1	no	—
27	C₂²×C₈×C₄₁	—	2	2	no	—
28	C₄⋊Q₈×C₄₁	—	2	2	no	—
29	C₄⋊Q₈×C₄₁	C₂×C₄	1	2	yes	—
30	C₄⋊Q₈×C₄₁	D₄	1	2	yes	—
31	#124×C₄₁	—	2	8	no	—
32	#124×C₄₁	—	2	8	no	—
33	#121×C₄₁	—	2	8	no	—
34	#121×C₄₁	—	2	8	no	—
35	C₄×D₄×C₄₁	—	2	2	no	—
36	C₄×D₄×C₄₁	—	2	2	no	—
37	C₄×D₄×C₄₁	C₂×C₄	1	4	yes	—
38	C₄×D₄×C₄₁	D₄	1	4	yes	—
39	#117×C₄₁	—	2	8	no	—
40	#117×C₄₁	—	2	8	no	—
41	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
42	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
44	C₂×C₄⋊C₄×C₄₁	D₄	1	2	yes	—
45	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
46	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
47	C₂×C₄²×C₄₁	—	2	2	no	—
48	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
49	C₄⋊C₈×C₄₁	—	2	2	no	—
50	C₄⋊C₈×C₄₁	—	2	2	no	—
51	#103×C₄₁	—	4	4	no	—
52	#102×C₄₁	—	4	4	no	—
53	C₄×C₈×C₄₁	—	2	2	no	—
54	C₂⁴×C₄₁	—	2	2	no	—
55	C₂×Q₈×C₄₁	—	2	2	no	—
56	C₂×Q₈×C₄₁	—	4	2	no	—
57	C₂×Q₈×C₄₁	—	4	2	no	—
58	C₂×Q₈×C₄₁	C₂²⋊C₄	1	4	yes	—
59	C₂×D₄×C₄₁	—	2	8	no	—
60	C₂×D₄×C₄₁	—	2	2	no	—
61	C₂×D₄×C₄₁	—	2	2	no	—
62	C₂×D₄×C₄₁	—	2	2	no	—
63	C₂×D₄×C₄₁	—	2	2	no	—
64	C₂×D₄×C₄₁	—	2	2	no	—
65	C₂×D₄×C₄₁	—	8	2	no	—
66	C₂×D₄×C₄₁	—	8	2	no	—
67	C₂×D₄×C₄₁	C₂²⋊C₄	1	4	yes	—
68	C₂²×C₄×C₄₁	—	2	2	no	—
69	C₂²×C₄×C₄₁	—	2	2	no	—
70	C₂²×C₄×C₄₁	—	2	4	no	—
71	C₂²×C₄×C₄₁	—	2	4	no	—
72	C₂²×C₄×C₄₁	—	4	2	no	—
73	C₂²×C₄×C₄₁	—	4	2	no	—
74	C₂²×C₄×C₄₁	—	4	2	no	—
75	C₂²×C₄×C₄₁	—	4	2	no	—
76	C₂²×C₄×C₄₁	—	4	1	no	—
77	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₄₁	—	2	2	no	—
80	C₂×C₈×C₄₁	—	4	1	no	—
81	C₂×C₈×C₄₁	—	4	1	no	—
82	C₂×C₈×C₄₁	—	4	2	no	—
83	C₂×C₈×C₄₁	—	4	2	no	—
84	C₂×C₈×C₄₁	—	4	2	no	—
85	C₄⋊C₄×C₄₁	—	2	4	no	—
86	C₄⋊C₄×C₄₁	—	2	4	no	—
87	C₄⋊C₄×C₄₁	—	2	2	no	—
88	C₄⋊C₄×C₄₁	—	4	2	no	—
89	C₄⋊C₄×C₄₁	—	4	2	no	—
90	C₄⋊C₄×C₄₁	—	4	2	no	—
91	C₄⋊C₄×C₄₁	C₂²⋊C₄	1	4	yes	—
92	C₂²⋊C₄×C₄₁	—	2	16	no	—
93	C₂²⋊C₄×C₄₁	—	2	4	no	—
94	C₂²⋊C₄×C₄₁	—	2	4	no	—
95	C₄²×C₄₁	—	2	8	no	—
96	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₄₁	C₂×D₄	1	2	yes	—
98	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
99	C₂³×C₄₁	—	2	4	no	—
100	C₂³×C₄₁	—	4	2	no	—
101	C₂³×C₄₁	—	4	2	no	—
102	C₂³×C₄₁	—	4	2	no	—
103	C₂³×C₄₁	—	4	2	no	—
104	C₂³×C₄₁	—	4	2	no	—
105	C₂³×C₄₁	—	4	2	no	—
106	C₂³×C₄₁	#120	1	1	yes	characteristic
107	Q₈×C₄₁	—	4	4	no	—
108	Q₈×C₄₁	—	8	2	no	—
109	D₄×C₄₁	—	2	4	no	—
110	D₄×C₄₁	—	2	4	no	—
111	D₄×C₄₁	—	4	2	no	—
112	D₄×C₄₁	—	4	2	no	—
113	D₄×C₄₁	—	8	8	no	—
114	C₂×C₄×C₄₁	—	2	1	no	—
115	C₂×C₄×C₄₁	—	2	1	no	—
116	C₂×C₄×C₄₁	—	2	4	no	—
117	C₂×C₄×C₄₁	—	2	4	no	—
118	C₂×C₄×C₄₁	—	4	2	no	—
119	C₂×C₄×C₄₁	—	4	2	no	—
120	C₂×C₄×C₄₁	—	4	2	no	—
121	C₂×C₄×C₄₁	—	4	2	no	—
122	C₂×C₄×C₄₁	—	4	4	no	—
123	C₂×C₄×C₄₁	—	4	4	no	—
124	C₂×C₄×C₄₁	—	4	4	no	—
125	C₂×C₄×C₄₁	—	4	4	no	—
126	C₂×C₄×C₄₁	—	8	2	no	—
127	C₂×C₄×C₄₁	—	8	1	no	—
128	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
130	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
131	C₂×C₄×C₄₁	#120	1	2	yes	—
132	C₂×C₄×C₄₁	#135	1	2	yes	—
133	C₂×C₄×C₄₁	#135	1	2	yes	—
134	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
135	C₂×C₄×C₄₁	#137	1	2	yes	—
136	C₈×C₄₁	—	4	2	no	—
137	C₈×C₄₁	—	8	2	no	—
138	#8452	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₂²×C₄₁	—	4	4	no	—
140	C₂²×C₄₁	—	4	4	no	—
141	C₂²×C₄₁	—	4	4	no	—
142	C₂²×C₄₁	—	8	2	no	—
143	C₂²×C₄₁	—	8	2	no	—
144	C₂²×C₄₁	—	8	2	no	—
145	C₂²×C₄₁	—	8	2	no	—
146	C₂²×C₄₁	#379	1	1	yes	characteristic
147	C₂²×C₄₁	#449	1	2	yes	—
148	C₂²×C₄₁	#452	1	2	yes	—
149	C₂²×C₄₁	#453	1	1	yes	characteristic
150	C₂²×C₄₁	#455	1	1	yes	characteristic
151	C₄×C₄₁	—	2	1	no	—
152	C₄×C₄₁	—	2	1	no	—
153	C₄×C₄₁	—	4	4	no	—
154	C₄×C₄₁	—	4	4	no	—
155	C₄×C₄₁	—	16	1	no	—
156	C₄×C₄₁	#416	1	2	yes	—
157	C₄×C₄₁	#417	1	2	yes	—
158	C₄×C₄₁	#449	1	2	yes	—
159	C₄×C₄₁	#452	1	2	yes	—
160	C₂×#555	C₂×C₄₁	1	1	yes	characteristic
161	#1956	C₂×C₄₁	1	1	yes	characteristic, complemented
162	#1802	C₂×C₄₁	1	2	yes	complemented
163	C₂×#333	C₂×C₄₁	1	1	yes	characteristic, complemented
164	C₂×D₄⋊C₈	C₂×C₄₁	1	2	yes	complemented
165	C₂×C₄₁	—	8	4	no	—
166	C₂×C₄₁	#1624	1	1	yes	characteristic
167	C₂×C₄₁	#1634	1	1	yes	characteristic
168	C₂×C₄₁	#1646	1	2	yes	—
169	C₂×C₄₁	#1788	1	2	yes	—
170	C₂×C₄₁	#1792	1	1	yes	characteristic
171	#555	—	2	2	no	—
172	#555	—	2	4	no	—
173	#555	C₄×C₄₁	1	4	yes	complemented
174	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
175	C₂×#124	—	2	2	no	—
176	C₂×#121	—	2	2	no	—
177	C₂×C₄×D₄	C₂²×C₄₁	1	2	yes	—
178	C₂×#117	—	2	2	no	—
179	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
180	C₂×#103	—	2	2	no	—
181	C₂×#102	—	2	2	no	—
182	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
183	#384	C₂²×C₄₁	1	1	yes	characteristic
184	#340	—	2	2	no	—
185	#333	—	2	2	no	—
186	D₄⋊C₈	—	2	4	no	—
187	C₄₁	#8452	1	1	yes	characteristic, direct product factor, Sylow
188	C₂²×Q₈	D₄×C₄₁	1	2	yes	—
189	C₂²×D₄	—	2	2	no	—
190	C₂²×D₄	D₄×C₄₁	1	2	yes	—
191	C₂³×C₄	—	2	2	no	—
192	C₂²×C₈	—	2	1	no	—
193	C₂²×C₈	—	2	2	no	—
194	C₄⋊Q₈	—	2	2	no	—
195	C₄⋊Q₈	C₂×C₄×C₄₁	1	2	yes	—
196	C₄⋊Q₈	D₄×C₄₁	1	2	yes	—
197	#124	—	2	8	no	—
198	#124	—	2	8	no	—
199	#121	—	2	8	no	—
200	#121	—	2	8	no	—
201	C₄×D₄	—	2	2	no	—
202	C₄×D₄	—	2	2	no	—
203	C₄×D₄	C₂×C₄×C₄₁	1	4	yes	—
204	C₄×D₄	D₄×C₄₁	1	4	yes	—
205	#117	—	2	8	no	—
206	#117	—	2	8	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	2	1	no	—
210	C₂×C₄⋊C₄	D₄×C₄₁	1	2	yes	—
211	C₂×C₂²⋊C₄	—	2	4	no	—
212	C₂×C₂²⋊C₄	—	2	2	no	—
213	C₂×C₄²	—	2	2	no	—
214	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
215	C₄⋊C₈	—	2	2	no	—
216	C₄⋊C₈	—	2	2	no	—
217	#103	—	4	4	no	—
218	#102	—	4	4	no	—
219	C₄×C₈	—	2	2	no	—
220	C₂⁴	—	2	2	no	—
221	C₂×Q₈	—	2	2	no	—
222	C₂×Q₈	—	4	2	no	—
223	C₂×Q₈	—	4	2	no	—
224	C₂×Q₈	C₂²⋊C₄×C₄₁	1	4	yes	—
225	C₂×D₄	—	2	8	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	—	2	2	no	—
230	C₂×D₄	—	2	2	no	—
231	C₂×D₄	—	8	2	no	—
232	C₂×D₄	—	8	2	no	—
233	C₂×D₄	C₂²⋊C₄×C₄₁	1	4	yes	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	4	2	no	—
239	C₂²×C₄	—	4	2	no	—
240	C₂²×C₄	—	4	2	no	—
241	C₂²×C₄	—	4	2	no	—
242	C₂²×C₄	—	4	1	no	—
243	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	4	1	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	—	4	2	no	—
249	C₂×C₈	—	4	2	no	—
250	C₂×C₈	—	4	2	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	C₂²⋊C₄×C₄₁	1	4	yes	—
258	C₂²⋊C₄	—	2	16	no	—
259	C₂²⋊C₄	—	2	4	no	—
260	C₂²⋊C₄	—	2	4	no	—
261	C₄²	—	2	8	no	—
262	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
263	C₄²	C₂×D₄×C₄₁	1	2	yes	—
264	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
265	C₂³	—	2	4	no	—
266	C₂³	—	4	2	no	—
267	C₂³	—	4	2	no	—
268	C₂³	—	4	2	no	—
269	C₂³	—	4	2	no	—
270	C₂³	—	4	2	no	—
271	C₂³	—	4	2	no	—
272	C₂³	#120×C₄₁	1	1	yes	characteristic
273	Q₈	—	4	4	no	—
274	Q₈	—	8	2	no	—
275	D₄	—	2	4	no	—
276	D₄	—	2	4	no	—
277	D₄	—	4	2	no	—
278	D₄	—	4	2	no	—
279	D₄	—	8	8	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	4	4	no	—
292	C₂×C₄	—	8	2	no	—
293	C₂×C₄	—	8	1	no	—
294	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
295	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
296	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
297	C₂×C₄	#120×C₄₁	1	2	yes	—
298	C₂×C₄	#135×C₄₁	1	2	yes	—
299	C₂×C₄	#135×C₄₁	1	2	yes	—
300	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
301	C₂×C₄	#137×C₄₁	1	2	yes	—
302	C₈	—	4	2	no	—
303	C₈	—	8	2	no	—
304	C₂²	—	4	4	no	—
305	C₂²	—	4	4	no	—
306	C₂²	—	4	4	no	—
307	C₂²	—	8	2	no	—
308	C₂²	—	8	2	no	—
309	C₂²	—	8	2	no	—
310	C₂²	—	8	2	no	—
311	C₂²	C₄₁×#379	1	1	yes	characteristic
312	C₂²	C₄₁×#449	1	2	yes	—
313	C₂²	C₄₁×#452	1	2	yes	—
314	C₂²	C₄₁×#453	1	1	yes	characteristic
315	C₂²	C₄₁×#455	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	4	4	no	—
319	C₄	—	4	4	no	—
320	C₄	—	16	1	no	—
321	C₄	C₄₁×#416	1	2	yes	—
322	C₄	C₄₁×#417	1	2	yes	—
323	C₄	C₄₁×#449	1	2	yes	—
324	C₄	C₄₁×#452	1	2	yes	—
325	C₂	—	8	4	no	—
326	C₂	C₄₁×#1624	1	1	yes	characteristic
327	C₂	C₄₁×#1634	1	1	yes	characteristic
328	C₂	C₄₁×#1646	1	2	yes	—
329	C₂	C₄₁×#1788	1	2	yes	—
330	C₂	C₄₁×#1792	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₁	#120
commutator	C₄²	C₂²×C₄×C₄₁
Frattini, cosocle	C₂×C₄²	C₂³×C₄₁
prime radical	C₂×C₄₁×#555	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₁×#1956	C₂	semidirect
3. C₄₁×#1802	C₂	semidirect
4. C₂×C₄₁×#333	C₂	semidirect
5. C₂×C₄₁×D₄⋊C₈	C₂	semidirect
8. C₄₁×#555	C₄	semidirect
138. #8452	C₄₁	direct (N × H)
161. #1956	C₂×C₄₁	semidirect
162. #1802	C₂×C₄₁	semidirect
163. C₂×#333	C₂×C₄₁	semidirect
164. C₂×D₄⋊C₈	C₂×C₄₁	semidirect
173. #555	C₄×C₄₁	semidirect
187. C₄₁	#8452	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×#555	2
C₄₁×#1956	2
C₄₁×#1802	2
C₂×C₄₁×#333	2
C₂×C₄₁×D₄⋊C₈	2
#8452	41

Subgroups — C41×#8452 #23784653

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#8452 #23784653