Subgroups — C₄₁×#20818 #23797019

Order 10496.

894 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₈×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2283	C₂	1	2	yes	maximal
3	C₄₁×#1846	C₂	1	2	yes	maximal, complemented
4	C₄₁×#1835	C₂	1	2	yes	maximal, complemented
5	C₄₁×#1808	C₂	1	2	yes	maximal
6	C₄₁×#1807	C₂	1	2	yes	maximal
7	C₄×#107×C₄₁	C₂	1	2	yes	maximal, complemented
8	C₄×#106×C₄₁	C₂	1	2	yes	maximal, complemented
9	C₂³×C₈×C₄₁	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₄₁	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
12	C₂×#107×C₄₁	C₂²	1	4	yes	—
13	C₂×#106×C₄₁	C₂²	1	4	yes	—
14	C₂×C₄×C₈×C₄₁	C₄	1	8	yes	complemented
15	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
19	C₄₁×#383	—	2	2	no	—
20	C₄₁×#382	C₂²	1	4	yes	—
21	C₄₁×#381	—	2	2	no	—
22	C₄₁×#380	C₂²	1	4	yes	—
23	C₄×C₄⋊C₄×C₄₁	C₂²	1	4	yes	—
24	C₄×C₂²⋊C₄×C₄₁	—	2	2	no	—
25	C₄₁×#376	—	2	2	no	—
26	C₄₁×#340	C₂²	1	4	yes	—
27	C₄₁×#340	C₂²	1	4	yes	—
28	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₄₁	—	2	4	no	—
32	C₂²×C₈×C₄₁	—	2	4	no	—
33	C₂²×C₈×C₄₁	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₄₁	C₂×C₄	1	4	yes	—
35	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₄₁	C₂³	1	8	yes	—
40	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
41	C₂×C₄²×C₄₁	—	2	4	no	—
42	C₂×C₄²×C₄₁	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₁	Q₈	1	4	yes	—
46	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
48	#107×C₄₁	—	2	8	no	—
49	#106×C₄₁	—	2	8	no	—
50	C₄×C₈×C₄₁	C₂×C₄	1	4	yes	—
51	C₄×C₈×C₄₁	C₂×C₄	1	4	yes	—
52	C₄×C₈×C₄₁	C₂×C₄	1	4	yes	—
53	C₄×C₈×C₄₁	C₂×C₄	1	4	yes	—
54	#95×C₄₁	—	2	4	no	—
55	#95×C₄₁	—	2	4	no	—
56	#95×C₄₁	—	2	4	no	—
57	#95×C₄₁	—	2	4	no	—
58	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₁	—	2	4	no	—
60	C₂²×C₄×C₄₁	—	2	4	no	—
61	C₂²×C₄×C₄₁	—	2	2	no	—
62	C₂²×C₄×C₄₁	—	2	2	no	—
63	C₂²×C₄×C₄₁	—	2	8	no	—
64	C₂²×C₄×C₄₁	C₄⋊C₄	1	4	yes	—
65	C₂²×C₄×C₄₁	C₄⋊C₄	1	4	yes	—
66	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₁	C₂×Q₈	1	2	yes	—
73	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
74	C₂×C₈×C₄₁	—	2	2	no	—
75	C₂×C₈×C₄₁	—	2	2	no	—
76	C₂×C₈×C₄₁	—	2	8	no	—
77	C₂×C₈×C₄₁	—	2	8	no	—
78	C₂×C₈×C₄₁	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₄₁	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₄₁	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₄₁	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₄₁	#41	1	4	yes	—
83	C₂×C₈×C₄₁	#41	1	4	yes	—
84	C₄⋊C₄×C₄₁	—	2	8	no	—
85	C₄⋊C₄×C₄₁	—	2	8	no	—
86	C₂²⋊C₄×C₄₁	—	4	8	no	—
87	C₄²×C₄₁	—	4	8	no	—
88	C₄²×C₄₁	C₄⋊C₄	1	4	yes	—
89	C₄²×C₄₁	C₄⋊C₄	1	4	yes	—
90	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
91	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
92	C₄²×C₄₁	C₂×Q₈	1	2	yes	—
93	C₄²×C₄₁	C₂×Q₈	1	2	yes	—
94	C₂³×C₄₁	—	2	2	no	—
95	C₂³×C₄₁	—	2	2	no	—
96	C₂³×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
97	C₂³×C₄₁	#135	1	4	yes	—
98	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₄₁	—	2	2	no	—
100	C₂×C₄×C₄₁	—	2	2	no	—
101	C₂×C₄×C₄₁	—	2	2	no	—
102	C₂×C₄×C₄₁	—	2	2	no	—
103	C₂×C₄×C₄₁	—	2	8	no	—
104	C₂×C₄×C₄₁	—	2	2	no	—
105	C₂×C₄×C₄₁	—	2	2	no	—
106	C₂×C₄×C₄₁	—	2	4	no	—
107	C₂×C₄×C₄₁	—	2	8	no	—
108	C₂×C₄×C₄₁	—	4	8	no	—
109	C₂×C₄×C₄₁	—	4	8	no	—
110	C₂×C₄×C₄₁	—	4	8	no	—
111	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₄₁	#135	1	4	yes	—
117	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₄₁	C₂²×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
120	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
121	C₈×C₄₁	—	2	2	no	—
122	C₈×C₄₁	—	2	2	no	—
123	C₈×C₄₁	#117	1	4	yes	—
124	C₈×C₄₁	#117	1	4	yes	—
125	#20818	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂²×C₄₁	—	2	2	no	—
127	C₂²×C₄₁	—	2	8	no	—
128	C₂²×C₄₁	—	2	2	no	—
129	C₂²×C₄₁	#427	1	4	yes	—
130	C₂²×C₄₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
131	C₂²×C₄₁	#532	1	2	yes	—
132	C₂²×C₄₁	C₂×#135	1	2	yes	—
133	C₂²×C₄₁	C₂×#135	1	2	yes	—
134	C₄×C₄₁	—	2	2	no	—
135	C₄×C₄₁	—	2	2	no	—
136	C₄×C₄₁	—	2	2	no	—
137	C₄×C₄₁	—	2	2	no	—
138	C₄×C₄₁	—	8	8	no	—
139	C₄×C₄₁	#427	1	4	yes	—
140	C₄×C₄₁	C₂×#117	1	1	yes	characteristic
141	C₄×C₄₁	C₂×#117	1	1	yes	characteristic
142	C₄×C₄₁	#533	1	2	yes	—
143	C₂²×C₄×C₈	C₂×C₄₁	1	1	yes	characteristic
144	#2283	C₂×C₄₁	1	2	yes	—
145	#1846	C₂×C₄₁	1	2	yes	complemented
146	#1835	C₂×C₄₁	1	2	yes	complemented
147	#1808	C₂×C₄₁	1	2	yes	—
148	#1807	C₂×C₄₁	1	2	yes	—
149	C₄×#107	C₂×C₄₁	1	2	yes	complemented
150	C₄×#106	C₂×C₄₁	1	2	yes	complemented
151	C₂×C₄₁	—	2	4	no	—
152	C₂×C₄₁	#2285	1	1	yes	characteristic
153	C₂×C₄₁	C₂×#427	1	1	yes	characteristic
154	C₂×C₄₁	C₂×#427	1	1	yes	characteristic
155	C₂×C₄₁	#3168	1	4	yes	—
156	C₂³×C₈	C₂²×C₄₁	1	1	yes	characteristic
157	C₂³×C₈	C₂²×C₄₁	1	1	yes	characteristic
158	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
159	C₂×#107	C₂²×C₄₁	1	4	yes	—
160	C₂×#106	C₂²×C₄₁	1	4	yes	—
161	C₂×C₄×C₈	C₄×C₄₁	1	8	yes	complemented
162	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
163	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
164	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
165	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
166	#383	—	2	2	no	—
167	#382	C₂²×C₄₁	1	4	yes	—
168	#381	—	2	2	no	—
169	#380	C₂²×C₄₁	1	4	yes	—
170	C₄×C₄⋊C₄	C₂²×C₄₁	1	4	yes	—
171	C₄×C₂²⋊C₄	—	2	2	no	—
172	#376	—	2	2	no	—
173	#340	C₂²×C₄₁	1	4	yes	—
174	#340	C₂²×C₄₁	1	4	yes	—
175	C₄₁	#20818	1	1	yes	characteristic, direct product factor, Sylow
176	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
177	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
178	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
179	C₂²×C₈	—	2	4	no	—
180	C₂²×C₈	—	2	4	no	—
181	C₂²×C₈	C₂×C₄×C₄₁	1	4	yes	—
182	C₂²×C₈	C₂×C₄×C₄₁	1	4	yes	—
183	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
184	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
185	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
186	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
187	C₂×C₄⋊C₄	C₂³×C₄₁	1	8	yes	—
188	C₂×C₂²⋊C₄	—	2	4	no	—
189	C₂×C₄²	—	2	4	no	—
190	C₂×C₄²	C₂×C₄×C₄₁	1	4	yes	—
191	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
192	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
193	C₂×C₄²	Q₈×C₄₁	1	4	yes	—
194	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
195	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
196	#107	—	2	8	no	—
197	#106	—	2	8	no	—
198	C₄×C₈	C₂×C₄×C₄₁	1	4	yes	—
199	C₄×C₈	C₂×C₄×C₄₁	1	4	yes	—
200	C₄×C₈	C₂×C₄×C₄₁	1	4	yes	—
201	C₄×C₈	C₂×C₄×C₄₁	1	4	yes	—
202	#95	—	2	4	no	—
203	#95	—	2	4	no	—
204	#95	—	2	4	no	—
205	#95	—	2	4	no	—
206	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	8	no	—
212	C₂²×C₄	C₄⋊C₄×C₄₁	1	4	yes	—
213	C₂²×C₄	C₄⋊C₄×C₄₁	1	4	yes	—
214	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
215	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
220	C₂²×C₄	C₂×Q₈×C₄₁	1	2	yes	—
221	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	8	no	—
225	C₂×C₈	—	2	8	no	—
226	C₂×C₈	C₂²×C₄×C₄₁	1	2	yes	—
227	C₂×C₈	C₂²×C₄×C₄₁	1	2	yes	—
228	C₂×C₈	C₂²×C₄×C₄₁	1	2	yes	—
229	C₂×C₈	C₂²×C₄×C₄₁	1	2	yes	—
230	C₂×C₈	#41×C₄₁	1	4	yes	—
231	C₂×C₈	#41×C₄₁	1	4	yes	—
232	C₄⋊C₄	—	2	8	no	—
233	C₄⋊C₄	—	2	8	no	—
234	C₂²⋊C₄	—	4	8	no	—
235	C₄²	—	4	8	no	—
236	C₄²	C₄⋊C₄×C₄₁	1	4	yes	—
237	C₄²	C₄⋊C₄×C₄₁	1	4	yes	—
238	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
239	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
240	C₄²	C₂×Q₈×C₄₁	1	2	yes	—
241	C₄²	C₂×Q₈×C₄₁	1	2	yes	—
242	C₂³	—	2	2	no	—
243	C₂³	—	2	2	no	—
244	C₂³	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
245	C₂³	#135×C₄₁	1	4	yes	—
246	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	8	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	8	no	—
256	C₂×C₄	—	4	8	no	—
257	C₂×C₄	—	4	8	no	—
258	C₂×C₄	—	4	8	no	—
259	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
260	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
261	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
262	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
263	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
264	C₂×C₄	#135×C₄₁	1	4	yes	—
265	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
266	C₂×C₄	C₂²×Q₈×C₄₁	1	1	yes	characteristic
267	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
268	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
269	C₈	—	2	2	no	—
270	C₈	—	2	2	no	—
271	C₈	#117×C₄₁	1	4	yes	—
272	C₈	#117×C₄₁	1	4	yes	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	8	no	—
276	C₂²	C₄₁×#427	1	4	yes	—
277	C₂²	C₂²×C₄⋊C₄×C₄₁	1	1	yes	characteristic
278	C₂²	C₄₁×#532	1	2	yes	—
279	C₂²	C₂×#135×C₄₁	1	2	yes	—
280	C₂²	C₂×#135×C₄₁	1	2	yes	—
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	2	2	no	—
285	C₄	—	8	8	no	—
286	C₄	C₄₁×#427	1	4	yes	—
287	C₄	C₂×#117×C₄₁	1	1	yes	characteristic
288	C₄	C₂×#117×C₄₁	1	1	yes	characteristic
289	C₄	C₄₁×#533	1	2	yes	—
290	C₂	—	2	4	no	—
291	C₂	C₄₁×#2285	1	1	yes	characteristic
292	C₂	C₂×C₄₁×#427	1	1	yes	characteristic
293	C₂	C₂×C₄₁×#427	1	1	yes	characteristic
294	C₂	C₄₁×#3168	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₄₁×#1846	C₂	semidirect
4. C₄₁×#1835	C₂	semidirect
7. C₄×#107×C₄₁	C₂	semidirect
8. C₄×#106×C₄₁	C₂	semidirect
14. C₂×C₄×C₈×C₄₁	C₄	semidirect
125. #20818	C₄₁	direct (N × H)
145. #1846	C₂×C₄₁	semidirect
146. #1835	C₂×C₄₁	semidirect
149. C₄×#107	C₂×C₄₁	semidirect
150. C₄×#106	C₂×C₄₁	semidirect
161. C₂×C₄×C₈	C₄×C₄₁	semidirect
175. C₄₁	#20818	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×C₄×C₈×C₄₁	2
C₄₁×#2283	2
C₄₁×#1846	2
C₄₁×#1835	2
C₄₁×#1808	2
C₄₁×#1807	2
C₄×#107×C₄₁	2
C₄×#106×C₄₁	2
#20818	41

Subgroups — C41×#20818 #23797019

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#20818 #23797019