Subgroups — C₄₁×#21920 #23798121

Order 10496.

694 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#2285	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#1917	C₂	1	4	yes	maximal, complemented
3	C₄₁×#1916	C₂	1	4	yes	maximal, complemented
4	C₄₁×#1817	C₂	1	2	yes	maximal
5	C₄₁×#1798	C₂	1	2	yes	maximal
6	C₄₁×#1750	C₂	1	2	yes	maximal
7	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
8	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
12	C₂×C₂²⋊C₈×C₄₁	C₂²	1	4	yes	—
13	C₂×#97×C₄₁	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₄₁	C₂²	1	2	yes	—
15	C₄₁×#384	C₄	1	1	yes	characteristic
16	C₄₁×#384	C₄	1	1	yes	characteristic
17	C₄₁×#382	C₄	1	1	yes	characteristic
18	C₄₁×#382	C₄	1	1	yes	characteristic
19	C₄₁×#380	C₄	1	4	yes	—
20	C₄×C₄⋊C₄×C₄₁	C₂²	1	4	yes	—
21	C₄₁×#336	C₂²	1	4	yes	—
22	C₄₁×#336	C₂²	1	2	yes	—
23	C₄₁×#336	C₂²	1	4	yes	—
24	C₄₁×#336	C₂²	1	2	yes	—
25	C₄₁×#336	C₂²	1	4	yes	—
26	C₂³×C₄×C₄₁	C₂×C₄	1	2	yes	—
27	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
28	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
29	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
30	#117×C₄₁	—	4	1	no	—
31	#117×C₄₁	—	4	1	no	—
32	#117×C₄₁	—	4	1	no	—
33	#117×C₄₁	—	4	1	no	—
34	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
35	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
40	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
41	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₄₁	—	2	4	no	—
43	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
46	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
47	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
48	C₄⋊C₈×C₄₁	—	4	2	no	—
49	C₄⋊C₈×C₄₁	—	4	2	no	—
50	C₂²⋊C₈×C₄₁	—	2	8	no	—
51	#97×C₄₁	—	4	2	no	—
52	C₄×C₈×C₄₁	—	4	2	no	—
53	#95×C₄₁	C₂×C₄	1	4	yes	—
54	#95×C₄₁	C₂×C₄	1	4	yes	—
55	C₂⁴×C₄₁	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₁	—	2	1	no	—
57	C₂²×C₄×C₄₁	—	2	1	no	—
58	C₂²×C₄×C₄₁	—	2	2	no	—
59	C₂²×C₄×C₄₁	—	2	4	no	—
60	C₂²×C₄×C₄₁	—	2	4	no	—
61	C₂²×C₄×C₄₁	—	4	2	no	—
62	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₄₁	#41	1	4	yes	—
70	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
71	C₂×C₈×C₄₁	—	2	4	no	—
72	C₂×C₈×C₄₁	—	2	4	no	—
73	C₂×C₈×C₄₁	—	2	4	no	—
74	C₂×C₈×C₄₁	—	2	4	no	—
75	C₂×C₈×C₄₁	—	2	4	no	—
76	C₂×C₈×C₄₁	—	2	4	no	—
77	C₄⋊C₄×C₄₁	—	2	8	no	—
78	C₄⋊C₄×C₄₁	—	4	4	no	—
79	C₄⋊C₄×C₄₁	—	4	1	no	—
80	C₄⋊C₄×C₄₁	—	4	1	no	—
81	C₄⋊C₄×C₄₁	—	4	1	no	—
82	C₄⋊C₄×C₄₁	—	4	1	no	—
83	C₂²⋊C₄×C₄₁	—	4	2	no	—
84	C₂²⋊C₄×C₄₁	—	4	2	no	—
85	C₄²×C₄₁	—	2	2	no	—
86	C₄²×C₄₁	—	2	2	no	—
87	C₄²×C₄₁	—	2	4	no	—
88	C₄²×C₄₁	—	2	2	no	—
89	C₄²×C₄₁	—	2	2	no	—
90	C₄²×C₄₁	—	2	2	no	—
91	C₄²×C₄₁	—	2	2	no	—
92	C₂³×C₄₁	—	2	1	no	—
93	C₂³×C₄₁	—	2	1	no	—
94	C₂³×C₄₁	—	2	2	no	—
95	C₂³×C₄₁	—	2	2	no	—
96	C₂³×C₄₁	#117	1	2	yes	—
97	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₄₁	—	2	2	no	—
99	C₂×C₄×C₄₁	—	2	1	no	—
100	C₂×C₄×C₄₁	—	2	1	no	—
101	C₂×C₄×C₄₁	—	2	2	no	—
102	C₂×C₄×C₄₁	—	2	4	no	—
103	C₂×C₄×C₄₁	—	2	4	no	—
104	C₂×C₄×C₄₁	—	2	2	no	—
105	C₂×C₄×C₄₁	—	2	2	no	—
106	C₂×C₄×C₄₁	—	2	2	no	—
107	C₂×C₄×C₄₁	—	2	2	no	—
108	C₂×C₄×C₄₁	—	2	2	no	—
109	C₂×C₄×C₄₁	—	2	4	no	—
110	C₂×C₄×C₄₁	—	2	4	no	—
111	C₂×C₄×C₄₁	—	2	2	no	—
112	C₂×C₄×C₄₁	—	2	4	no	—
113	C₂×C₄×C₄₁	—	2	4	no	—
114	C₂×C₄×C₄₁	—	2	2	no	—
115	C₂×C₄×C₄₁	—	4	8	no	—
116	C₂×C₄×C₄₁	#117	1	2	yes	—
117	C₂×C₄×C₄₁	#131	1	4	yes	—
118	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
119	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
120	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
121	C₂×C₄×C₄₁	2₋¹⁺⁴	1	2	yes	—
122	C₈×C₄₁	—	4	4	no	—
123	C₈×C₄₁	—	4	4	no	—
124	#21920	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂²×C₄₁	—	2	2	no	—
126	C₂²×C₄₁	—	2	4	no	—
127	C₂²×C₄₁	—	4	1	no	—
128	C₂²×C₄₁	—	4	1	no	—
129	C₂²×C₄₁	—	4	2	no	—
130	C₂²×C₄₁	C₂×#117	1	1	yes	characteristic
131	C₂²×C₄₁	#518	1	1	yes	characteristic
132	C₂²×C₄₁	#519	1	1	yes	characteristic
133	C₂²×C₄₁	C₂×#131	1	2	yes	—
134	C₂²×C₄₁	#568	1	2	yes	—
135	C₄×C₄₁	—	2	2	no	—
136	C₄×C₄₁	—	2	4	no	—
137	C₄×C₄₁	—	4	2	no	—
138	C₄×C₄₁	—	4	2	no	—
139	C₄×C₄₁	—	4	2	no	—
140	C₄×C₄₁	—	4	2	no	—
141	C₄×C₄₁	—	4	2	no	—
142	C₄×C₄₁	#539	1	2	yes	—
143	C₄×C₄₁	#539	1	2	yes	—
144	#2285	C₂×C₄₁	1	1	yes	characteristic
145	#1917	C₂×C₄₁	1	4	yes	complemented
146	#1916	C₂×C₄₁	1	4	yes	complemented
147	#1817	C₂×C₄₁	1	2	yes	—
148	#1798	C₂×C₄₁	1	2	yes	—
149	#1750	C₂×C₄₁	1	2	yes	—
150	C₂×C₄₁	—	4	2	no	—
151	C₂×C₄₁	#2313	1	1	yes	characteristic
152	C₂×C₄₁	#2924	1	2	yes	—
153	C₂×C₄₁	#2979	1	1	yes	characteristic
154	C₂×C₄₁	#2980	1	1	yes	characteristic
155	C₂×C₄₁	#2985	1	2	yes	—
156	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
157	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
158	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
159	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
160	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
161	C₂×C₂²⋊C₈	C₂²×C₄₁	1	4	yes	—
162	C₂×#97	C₂²×C₄₁	1	2	yes	—
163	C₂×C₄×C₈	C₂²×C₄₁	1	2	yes	—
164	#384	C₄×C₄₁	1	1	yes	characteristic
165	#384	C₄×C₄₁	1	1	yes	characteristic
166	#382	C₄×C₄₁	1	1	yes	characteristic
167	#382	C₄×C₄₁	1	1	yes	characteristic
168	#380	C₄×C₄₁	1	4	yes	—
169	C₄×C₄⋊C₄	C₂²×C₄₁	1	4	yes	—
170	#336	C₂²×C₄₁	1	4	yes	—
171	#336	C₂²×C₄₁	1	2	yes	—
172	#336	C₂²×C₄₁	1	4	yes	—
173	#336	C₂²×C₄₁	1	2	yes	—
174	#336	C₂²×C₄₁	1	4	yes	—
175	C₄₁	#21920	1	1	yes	characteristic, direct product factor, Sylow
176	C₂³×C₄	C₂×C₄×C₄₁	1	2	yes	—
177	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
178	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
179	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
180	#117	—	4	1	no	—
181	#117	—	4	1	no	—
182	#117	—	4	1	no	—
183	#117	—	4	1	no	—
184	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
185	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
186	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
187	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
188	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
189	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
190	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
191	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
192	C₂×C₄²	—	2	4	no	—
193	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
194	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
195	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
196	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
197	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
198	C₄⋊C₈	—	4	2	no	—
199	C₄⋊C₈	—	4	2	no	—
200	C₂²⋊C₈	—	2	8	no	—
201	#97	—	4	2	no	—
202	C₄×C₈	—	4	2	no	—
203	#95	C₂×C₄×C₄₁	1	4	yes	—
204	#95	C₂×C₄×C₄₁	1	4	yes	—
205	C₂⁴	C₂²×C₄×C₄₁	1	1	yes	characteristic
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	4	2	no	—
212	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
213	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
214	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
215	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
216	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
217	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
218	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
219	C₂²×C₄	#41×C₄₁	1	4	yes	—
220	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	4	no	—
227	C₄⋊C₄	—	2	8	no	—
228	C₄⋊C₄	—	4	1	no	—
229	C₄⋊C₄	—	4	1	no	—
230	C₄⋊C₄	—	4	1	no	—
231	C₄⋊C₄	—	4	1	no	—
232	C₄⋊C₄	—	4	4	no	—
233	C₂²⋊C₄	—	4	2	no	—
234	C₂²⋊C₄	—	4	2	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	2	2	no	—
237	C₄²	—	2	4	no	—
238	C₄²	—	2	2	no	—
239	C₄²	—	2	2	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₂³	—	2	1	no	—
243	C₂³	—	2	1	no	—
244	C₂³	—	2	2	no	—
245	C₂³	—	2	2	no	—
246	C₂³	#117×C₄₁	1	2	yes	—
247	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	4	8	no	—
266	C₂×C₄	#117×C₄₁	1	2	yes	—
267	C₂×C₄	#131×C₄₁	1	4	yes	—
268	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
269	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
270	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
271	C₂×C₄	2₋¹⁺⁴×C₄₁	1	2	yes	—
272	C₈	—	4	4	no	—
273	C₈	—	4	4	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	4	no	—
276	C₂²	—	4	1	no	—
277	C₂²	—	4	1	no	—
278	C₂²	—	4	2	no	—
279	C₂²	C₂×#117×C₄₁	1	1	yes	characteristic
280	C₂²	C₄₁×#518	1	1	yes	characteristic
281	C₂²	C₄₁×#519	1	1	yes	characteristic
282	C₂²	C₂×#131×C₄₁	1	2	yes	—
283	C₂²	C₄₁×#568	1	2	yes	—
284	C₄	—	2	2	no	—
285	C₄	—	2	4	no	—
286	C₄	—	4	2	no	—
287	C₄	—	4	2	no	—
288	C₄	—	4	2	no	—
289	C₄	—	4	2	no	—
290	C₄	—	4	2	no	—
291	C₄	C₄₁×#539	1	2	yes	—
292	C₄	C₄₁×#539	1	2	yes	—
293	C₂	—	4	2	no	—
294	C₂	C₄₁×#2313	1	1	yes	characteristic
295	C₂	C₄₁×#2924	1	2	yes	—
296	C₂	C₄₁×#2979	1	1	yes	characteristic
297	C₂	C₄₁×#2980	1	1	yes	characteristic
298	C₂	C₄₁×#2985	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₄₁×#1917	C₂	semidirect
3. C₄₁×#1916	C₂	semidirect
124. #21920	C₄₁	direct (N × H)
145. #1917	C₂×C₄₁	semidirect
146. #1916	C₂×C₄₁	semidirect
175. C₄₁	#21920	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×#2285	2
C₄₁×#1917	2
C₄₁×#1916	2
C₄₁×#1817	2
C₄₁×#1798	2
C₄₁×#1750	2
#21920	41

Subgroups — C41×#21920 #23798121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#21920 #23798121