Subgroups — C₄₁×#18068 #23794269

Order 10496.

586 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3170	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2962	C₂	1	2	yes	maximal
3	C₄₁×#1591	C₂	1	2	yes	maximal, complemented
4	C₄₁×#1584	C₂	1	4	yes	maximal, complemented
5	C₈×Q₁₆×C₄₁	C₂	1	2	yes	maximal, complemented
6	C₂×C₄₁×#335	C₂	1	1	yes	maximal, characteristic
7	C₄₁×#1481	C₂	1	2	yes	maximal
8	C₈×M₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic
9	C₄₁×#533	C₂²	1	2	yes	—
10	C₄₁×#500	C₄	1	1	yes	characteristic
11	C₄₁×#494	C₄	1	1	yes	characteristic
12	C₄₁×#487	C₄	1	2	yes	—
13	C₄₁×#467	C₄	1	2	yes	—
14	C₄₁×#467	C₄	1	2	yes	—
15	C₄₁×#446	—	2	4	no	—
16	C₄₁×#446	C₂²	1	4	yes	—
17	C₈×Q₈×C₄₁	C₂²	1	4	yes	—
18	C₄×Q₁₆×C₄₁	C₂²	1	2	yes	—
19	C₄×Q₁₆×C₄₁	C₂²	1	2	yes	—
20	C₄₁×#431	C₂²	1	1	yes	characteristic
21	C₄₁×#424	—	2	2	no	—
22	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
23	C₄₁×#405	—	2	2	no	—
24	C₄×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
26	C₄₁×#335	C₂²	1	1	yes	characteristic
27	C₄₁×#335	C₂²	1	2	yes	—
28	C₄₁×#335	C₂²	1	1	yes	characteristic
29	C₄₁×Q₈⋊C₈	C₂²	1	4	yes	—
30	C₄₁×Q₈⋊C₈	C₂²	1	4	yes	—
31	C₄₁×#322	C₂²	1	2	yes	—
32	C₈²×C₄₁	C₂²	1	2	yes	—
33	C₂×Q₁₆×C₄₁	C₂×C₄	1	2	yes	—
34	C₂×Q₁₆×C₄₁	C₂×C₄	1	2	yes	—
35	C₂×M₁₆×C₄₁	D₄	1	1	yes	characteristic
36	C₂×M₁₆×C₄₁	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₄₁	—	2	2	no	—
38	C₂²×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
40	C₄⋊Q₈×C₄₁	C₂×C₄	1	2	yes	—
41	C₄⋊Q₈×C₄₁	C₂×C₄	1	2	yes	—
42	#125×C₄₁	—	2	2	no	—
43	C₂²⋊Q₈×C₄₁	—	2	2	no	—
44	C₂²⋊Q₈×C₄₁	C₂×C₄	1	4	yes	—
45	C₄×Q₈×C₄₁	—	2	2	no	—
46	C₄×Q₈×C₄₁	C₂³	1	4	yes	—
47	#117×C₄₁	—	2	2	no	—
48	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
49	#107×C₄₁	C₂×C₄	1	2	yes	—
50	#107×C₄₁	C₂×C₄	1	2	yes	—
51	C₄⋊C₈×C₄₁	—	2	4	no	—
52	C₄⋊C₈×C₄₁	—	2	4	no	—
53	C₄⋊C₈×C₄₁	—	2	2	no	—
54	C₄⋊C₈×C₄₁	—	2	1	no	—
55	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
56	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
57	#103×C₄₁	C₂×C₄	1	4	yes	—
58	#103×C₄₁	C₂×C₄	1	4	yes	—
59	C₂²⋊C₈×C₄₁	—	2	1	no	—
60	#97×C₄₁	—	2	4	no	—
61	#97×C₄₁	D₄	1	2	yes	—
62	C₄×C₈×C₄₁	—	2	4	no	—
63	C₄×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₄₁	C₂³	1	2	yes	—
66	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₄₁	C₂³	1	2	yes	—
69	C₂×Q₈×C₄₁	—	2	2	no	—
70	C₂×Q₈×C₄₁	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₁	#41	1	1	yes	characteristic
74	Q₁₆×C₄₁	—	2	4	no	—
75	Q₁₆×C₄₁	—	4	2	no	—
76	M₁₆×C₄₁	—	2	2	no	—
77	M₁₆×C₄₁	Q₁₆	1	4	yes	complemented
78	C₂×C₈×C₄₁	—	2	2	no	—
79	C₂×C₈×C₄₁	—	2	2	no	—
80	C₂×C₈×C₄₁	—	2	2	no	—
81	C₂×C₈×C₄₁	—	2	2	no	—
82	C₂×C₈×C₄₁	—	4	1	no	—
83	C₂×C₈×C₄₁	—	4	4	no	—
84	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
90	C₄⋊C₄×C₄₁	—	2	2	no	—
91	C₄⋊C₄×C₄₁	—	2	2	no	—
92	C₄⋊C₄×C₄₁	—	2	2	no	—
93	C₄⋊C₄×C₄₁	—	2	4	no	—
94	C₄⋊C₄×C₄₁	—	2	4	no	—
95	C₄⋊C₄×C₄₁	C₂²×C₄	1	4	yes	—
96	C₂²⋊C₄×C₄₁	—	2	4	no	—
97	C₄²×C₄₁	—	2	4	no	—
98	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₄₁	#41	1	1	yes	characteristic
101	C₄²×C₄₁	C₂⁴	1	1	yes	characteristic
102	C₂³×C₄₁	C₄×D₄	1	1	yes	characteristic
103	Q₈×C₄₁	—	2	4	no	—
104	Q₈×C₄₁	—	2	4	no	—
105	Q₈×C₄₁	—	8	2	no	—
106	C₂×C₄×C₄₁	—	2	1	no	—
107	C₂×C₄×C₄₁	—	2	1	no	—
108	C₂×C₄×C₄₁	—	2	4	no	—
109	C₂×C₄×C₄₁	—	2	4	no	—
110	C₂×C₄×C₄₁	—	4	1	no	—
111	C₂×C₄×C₄₁	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₄₁	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₁	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₄₁	C₂×Q₁₆	1	2	yes	—
115	C₂×C₄×C₄₁	#135	1	2	yes	—
116	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
119	C₈×C₄₁	—	2	1	no	—
120	C₈×C₄₁	—	2	1	no	—
121	C₈×C₄₁	—	2	1	no	—
122	C₈×C₄₁	—	2	2	no	—
123	C₈×C₄₁	—	4	2	no	—
124	C₈×C₄₁	—	4	2	no	—
125	C₈×C₄₁	—	4	2	no	—
126	C₈×C₄₁	#131	1	2	yes	—
127	C₈×C₄₁	C₂×Q₁₆	1	4	yes	—
128	#18068	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂²×C₄₁	—	2	1	no	—
130	C₂²×C₄₁	—	2	1	no	—
131	C₂²×C₄₁	C₄×Q₁₆	1	2	yes	—
132	C₂²×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
133	C₄×C₄₁	—	2	1	no	—
134	C₄×C₄₁	—	2	1	no	—
135	C₄×C₄₁	—	2	1	no	—
136	C₄×C₄₁	—	4	4	no	—
137	C₄×C₄₁	—	4	4	no	—
138	C₄×C₄₁	C₄×Q₁₆	1	2	yes	—
139	C₄×C₄₁	C₂×#131	1	1	yes	characteristic
140	C₄×C₄₁	#568	1	1	yes	characteristic
141	C₄×C₄₁	C₂²×Q₁₆	1	1	yes	characteristic
142	C₄×C₄₁	C₂×#135	1	1	yes	characteristic
143	#3170	C₂×C₄₁	1	1	yes	characteristic
144	#2962	C₂×C₄₁	1	2	yes	—
145	#1591	C₂×C₄₁	1	2	yes	complemented
146	#1584	C₂×C₄₁	1	4	yes	complemented
147	C₈×Q₁₆	C₂×C₄₁	1	2	yes	complemented
148	C₂×#335	C₂×C₄₁	1	1	yes	characteristic
149	#1481	C₂×C₄₁	1	2	yes	—
150	C₈×M₁₆	C₂×C₄₁	1	1	yes	characteristic
151	C₂×C₄₁	—	2	2	no	—
152	C₂×C₄₁	#2935	1	1	yes	characteristic
153	C₂×C₄₁	C₂×C₄×Q₁₆	1	1	yes	characteristic
154	C₂×C₄₁	#2957	1	1	yes	characteristic
155	#533	C₂²×C₄₁	1	2	yes	—
156	#500	C₄×C₄₁	1	1	yes	characteristic
157	#494	C₄×C₄₁	1	1	yes	characteristic
158	#487	C₄×C₄₁	1	2	yes	—
159	#467	C₄×C₄₁	1	2	yes	—
160	#467	C₄×C₄₁	1	2	yes	—
161	#446	—	2	4	no	—
162	#446	C₂²×C₄₁	1	4	yes	—
163	C₈×Q₈	C₂²×C₄₁	1	4	yes	—
164	C₄×Q₁₆	C₂²×C₄₁	1	2	yes	—
165	C₄×Q₁₆	C₂²×C₄₁	1	2	yes	—
166	#431	C₂²×C₄₁	1	1	yes	characteristic
167	#424	—	2	2	no	—
168	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
169	#405	—	2	2	no	—
170	C₄×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
171	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
172	#335	C₂²×C₄₁	1	1	yes	characteristic
173	#335	C₂²×C₄₁	1	2	yes	—
174	#335	C₂²×C₄₁	1	1	yes	characteristic
175	Q₈⋊C₈	C₂²×C₄₁	1	4	yes	—
176	Q₈⋊C₈	C₂²×C₄₁	1	4	yes	—
177	#322	C₂²×C₄₁	1	2	yes	—
178	C₈²	C₂²×C₄₁	1	2	yes	—
179	C₄₁	#18068	1	1	yes	characteristic, direct product factor, Sylow
180	C₂×Q₁₆	C₂×C₄×C₄₁	1	2	yes	—
181	C₂×Q₁₆	C₂×C₄×C₄₁	1	2	yes	—
182	C₂×M₁₆	D₄×C₄₁	1	1	yes	characteristic
183	C₂×M₁₆	D₄×C₄₁	1	1	yes	characteristic
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
186	C₂²×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
187	C₄⋊Q₈	C₂×C₄×C₄₁	1	2	yes	—
188	C₄⋊Q₈	C₂×C₄×C₄₁	1	2	yes	—
189	#125	—	2	2	no	—
190	C₂²⋊Q₈	—	2	2	no	—
191	C₂²⋊Q₈	C₂×C₄×C₄₁	1	4	yes	—
192	C₄×Q₈	—	2	2	no	—
193	C₄×Q₈	C₂³×C₄₁	1	4	yes	—
194	#117	—	2	2	no	—
195	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
196	#107	C₂×C₄×C₄₁	1	2	yes	—
197	#107	C₂×C₄×C₄₁	1	2	yes	—
198	C₄⋊C₈	—	2	4	no	—
199	C₄⋊C₈	—	2	4	no	—
200	C₄⋊C₈	—	2	2	no	—
201	C₄⋊C₈	—	2	1	no	—
202	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
203	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
204	#103	C₂×C₄×C₄₁	1	4	yes	—
205	#103	C₂×C₄×C₄₁	1	4	yes	—
206	C₂²⋊C₈	—	2	1	no	—
207	#97	—	2	4	no	—
208	#97	D₄×C₄₁	1	2	yes	—
209	C₄×C₈	—	2	4	no	—
210	C₄×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
211	C₄×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
212	C₄×C₈	C₂³×C₄₁	1	2	yes	—
213	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
214	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
215	C₄×C₈	C₂³×C₄₁	1	2	yes	—
216	C₂×Q₈	—	2	2	no	—
217	C₂×Q₈	C₂²×C₄×C₄₁	1	4	yes	—
218	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
220	C₂²×C₄	#41×C₄₁	1	1	yes	characteristic
221	Q₁₆	—	2	4	no	—
222	Q₁₆	—	4	2	no	—
223	M₁₆	—	2	2	no	—
224	M₁₆	Q₁₆×C₄₁	1	4	yes	complemented
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	4	4	no	—
231	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
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	2	yes	—
236	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	C₂²×C₄×C₄₁	1	4	yes	—
243	C₂²⋊C₄	—	2	4	no	—
244	C₄²	—	2	4	no	—
245	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
246	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
247	C₄²	#41×C₄₁	1	1	yes	characteristic
248	C₄²	C₂⁴×C₄₁	1	1	yes	characteristic
249	C₂³	C₄×D₄×C₄₁	1	1	yes	characteristic
250	Q₈	—	2	4	no	—
251	Q₈	—	2	4	no	—
252	Q₈	—	8	2	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	C₄×D₄×C₄₁	1	1	yes	characteristic
259	C₂×C₄	C₄×D₄×C₄₁	1	1	yes	characteristic
260	C₂×C₄	C₄×D₄×C₄₁	1	1	yes	characteristic
261	C₂×C₄	C₂×Q₁₆×C₄₁	1	2	yes	—
262	C₂×C₄	#135×C₄₁	1	2	yes	—
263	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
264	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
265	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
266	C₈	—	2	1	no	—
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	2	2	no	—
270	C₈	—	4	2	no	—
271	C₈	—	4	2	no	—
272	C₈	—	4	2	no	—
273	C₈	#131×C₄₁	1	2	yes	—
274	C₈	C₂×Q₁₆×C₄₁	1	4	yes	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	C₄×Q₁₆×C₄₁	1	2	yes	—
278	C₂²	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	4	4	no	—
283	C₄	—	4	4	no	—
284	C₄	C₄×Q₁₆×C₄₁	1	2	yes	—
285	C₄	C₂×#131×C₄₁	1	1	yes	characteristic
286	C₄	C₄₁×#568	1	1	yes	characteristic
287	C₄	C₂²×Q₁₆×C₄₁	1	1	yes	characteristic
288	C₄	C₂×#135×C₄₁	1	1	yes	characteristic
289	C₂	—	2	2	no	—
290	C₂	C₄₁×#2935	1	1	yes	characteristic
291	C₂	C₂×C₄×Q₁₆×C₄₁	1	1	yes	characteristic
292	C₂	C₄₁×#2957	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	C₂×C₄×D₄
prime radical	C₂³×C₄₁	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₁ ◁ C₄×M₁₆×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄₁×#1591	C₂	semidirect
4. C₄₁×#1584	C₂	semidirect
5. C₈×Q₁₆×C₄₁	C₂	semidirect
77. M₁₆×C₄₁	Q₁₆	semidirect
128. #18068	C₄₁	direct (N × H)
145. #1591	C₂×C₄₁	semidirect
146. #1584	C₂×C₄₁	semidirect
147. C₈×Q₁₆	C₂×C₄₁	semidirect
179. C₄₁	#18068	direct (N × H)
224. M₁₆	Q₁₆×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₄₁×#3170	2
C₄₁×#2962	2
C₄₁×#1591	2
C₄₁×#1584	2
C₈×Q₁₆×C₄₁	2
C₂×C₄₁×#335	2
C₄₁×#1481	2
C₈×M₁₆×C₄₁	2
#18068	41

Subgroups — C41×#18068 #23794269

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#18068 #23794269