Subgroups — C₄₁×#22013 #23798214

Order 10496.

722 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	C₄₁×#2293	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#1954	C₂	1	2	yes	maximal
3	C₄₁×#1939	C₂	1	2	yes	maximal, complemented
4	C₄₁×#1924	C₂	1	4	yes	maximal, complemented
5	C₄₁×#1818	C₂	1	2	yes	maximal, complemented
6	C₄₁×#1798	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1767	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1744	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
10	C₂×#117×C₄₁	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈×C₄₁	C₂²	1	4	yes	—
13	C₂×C₂²⋊C₈×C₄₁	C₂²	1	4	yes	—
14	C₂×C₂²⋊C₈×C₄₁	C₂²	1	4	yes	—
15	C₂×#97×C₄₁	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₄₁	C₂²	1	2	yes	—
17	C₄₁×#390	C₄	1	2	yes	—
18	C₄₁×#389	C₄	1	2	yes	—
19	C₄₁×#388	C₄	1	4	yes	—
20	C₄×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
21	C₄×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
22	C₄³×C₄₁	C₂²	1	1	yes	characteristic
23	C₄₁×#336	C₂²	1	2	yes	—
24	C₄₁×#336	C₂²	1	1	yes	characteristic
25	C₄₁×#336	C₂²	1	1	yes	characteristic
26	C₄₁×#336	C₂²	1	4	yes	—
27	C₄₁×#336	C₂²	1	2	yes	—
28	C₂²×D₄×C₄₁	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
30	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
31	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
32	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
33	C₄×D₄×C₄₁	—	2	4	no	—
34	#117×C₄₁	—	4	4	no	—
35	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
38	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
39	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
40	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
45	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₄₁	—	2	4	no	—
48	C₄⋊C₈×C₄₁	—	4	4	no	—
49	C₂²⋊C₈×C₄₁	—	4	4	no	—
50	C₂²⋊C₈×C₄₁	—	4	4	no	—
51	#97×C₄₁	—	4	2	no	—
52	C₄×C₈×C₄₁	—	2	4	no	—
53	#95×C₄₁	C₂×C₄	1	4	yes	—
54	#95×C₄₁	C₂×C₄	1	2	yes	—
55	C₂⁴×C₄₁	C₂²×C₄	1	2	yes	—
56	C₂×D₄×C₄₁	—	2	2	no	—
57	C₂×D₄×C₄₁	—	2	4	no	—
58	C₂²×C₄×C₄₁	—	2	4	no	—
59	C₂²×C₄×C₄₁	—	2	2	no	—
60	C₂²×C₄×C₄₁	—	2	4	no	—
61	C₂²×C₄×C₄₁	—	2	2	no	—
62	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₄₁	C₂²×C₄	1	4	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	1	yes	characteristic
67	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
69	C₂×C₈×C₄₁	—	2	2	no	—
70	C₂×C₈×C₄₁	—	2	2	no	—
71	C₂×C₈×C₄₁	—	2	4	no	—
72	C₂×C₈×C₄₁	—	2	4	no	—
73	C₂×C₈×C₄₁	—	2	2	no	—
74	C₂×C₈×C₄₁	—	2	2	no	—
75	C₂×C₈×C₄₁	—	2	4	no	—
76	C₂×C₈×C₄₁	—	2	2	no	—
77	C₂×C₈×C₄₁	—	2	2	no	—
78	C₄⋊C₄×C₄₁	—	2	2	no	—
79	C₄⋊C₄×C₄₁	—	2	4	no	—
80	C₄⋊C₄×C₄₁	—	4	4	no	—
81	C₂²⋊C₄×C₄₁	—	2	4	no	—
82	C₂²⋊C₄×C₄₁	—	4	4	no	—
83	C₂²⋊C₄×C₄₁	—	4	4	no	—
84	C₄²×C₄₁	—	2	4	no	—
85	C₄²×C₄₁	—	2	1	no	—
86	C₄²×C₄₁	—	2	4	no	—
87	C₄²×C₄₁	—	2	2	no	—
88	C₄²×C₄₁	—	2	1	no	—
89	C₄²×C₄₁	—	4	2	no	—
90	C₄²×C₄₁	#41	1	4	yes	—
91	C₂³×C₄₁	—	2	2	no	—
92	C₂³×C₄₁	—	2	2	no	—
93	C₂³×C₄₁	—	2	2	no	—
94	C₂³×C₄₁	—	2	4	no	—
95	C₂³×C₄₁	—	2	4	no	—
96	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
97	D₄×C₄₁	—	4	4	no	—
98	C₂×C₄×C₄₁	—	2	2	no	—
99	C₂×C₄×C₄₁	—	2	2	no	—
100	C₂×C₄×C₄₁	—	2	2	no	—
101	C₂×C₄×C₄₁	—	2	4	no	—
102	C₂×C₄×C₄₁	—	2	2	no	—
103	C₂×C₄×C₄₁	—	2	2	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	4	no	—
108	C₂×C₄×C₄₁	—	2	2	no	—
109	C₂×C₄×C₄₁	—	2	2	no	—
110	C₂×C₄×C₄₁	—	2	4	no	—
111	C₂×C₄×C₄₁	—	4	4	no	—
112	C₂×C₄×C₄₁	—	4	4	no	—
113	C₂×C₄×C₄₁	#117	1	2	yes	—
114	C₂×C₄×C₄₁	#117	1	2	yes	—
115	C₂×C₄×C₄₁	#131	1	4	yes	—
116	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
117	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
118	C₂×C₄×C₄₁	H(5,𝔽₂)	1	2	yes	—
119	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
120	C₈×C₄₁	—	4	2	no	—
121	C₈×C₄₁	—	4	4	no	—
122	C₈×C₄₁	—	4	2	no	—
123	#22013	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂²×C₄₁	—	4	2	no	—
125	C₂²×C₄₁	—	4	2	no	—
126	C₂²×C₄₁	—	4	2	no	—
127	C₂²×C₄₁	—	4	4	no	—
128	C₂²×C₄₁	—	4	4	no	—
129	C₂²×C₄₁	C₂×#117	1	1	yes	characteristic
130	C₂²×C₄₁	#518	1	1	yes	characteristic
131	C₂²×C₄₁	#520	1	1	yes	characteristic
132	C₂²×C₄₁	C₂×#131	1	2	yes	—
133	C₂²×C₄₁	#568	1	2	yes	—
134	C₄×C₄₁	—	2	2	no	—
135	C₄×C₄₁	—	4	2	no	—
136	C₄×C₄₁	—	4	2	no	—
137	C₄×C₄₁	—	4	4	no	—
138	C₄×C₄₁	—	4	2	no	—
139	C₄×C₄₁	#433	1	4	yes	—
140	C₄×C₄₁	#540	1	2	yes	—
141	C₄×C₄₁	#542	1	2	yes	—
142	#2293	C₂×C₄₁	1	1	yes	characteristic
143	#1954	C₂×C₄₁	1	2	yes	—
144	#1939	C₂×C₄₁	1	2	yes	complemented
145	#1924	C₂×C₄₁	1	4	yes	complemented
146	#1818	C₂×C₄₁	1	2	yes	complemented
147	#1798	C₂×C₄₁	1	2	yes	complemented
148	#1767	C₂×C₄₁	1	1	yes	characteristic, complemented
149	#1744	C₂×C₄₁	1	1	yes	characteristic, complemented
150	C₂×C₄₁	—	8	2	no	—
151	C₂×C₄₁	#2319	1	1	yes	characteristic
152	C₂×C₄₁	C₂×#433	1	1	yes	characteristic
153	C₂×C₄₁	#2923	1	1	yes	characteristic
154	C₂×C₄₁	#2981	1	2	yes	—
155	C₂×C₄₁	#2993	1	2	yes	—
156	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
157	C₂×#117	C₂²×C₄₁	1	2	yes	—
158	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
159	C₂×C₄⋊C₈	C₂²×C₄₁	1	4	yes	—
160	C₂×C₂²⋊C₈	C₂²×C₄₁	1	4	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	#390	C₄×C₄₁	1	2	yes	—
165	#389	C₄×C₄₁	1	2	yes	—
166	#388	C₄×C₄₁	1	4	yes	—
167	C₄×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
168	C₄×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
169	C₄³	C₂²×C₄₁	1	1	yes	characteristic
170	#336	C₂²×C₄₁	1	1	yes	characteristic
171	#336	C₂²×C₄₁	1	2	yes	—
172	#336	C₂²×C₄₁	1	1	yes	characteristic
173	#336	C₂²×C₄₁	1	4	yes	—
174	#336	C₂²×C₄₁	1	2	yes	—
175	C₄₁	#22013	1	1	yes	characteristic, direct product factor, Sylow
176	C₂²×D₄	C₂×C₄×C₄₁	1	1	yes	characteristic
177	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
178	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
179	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
180	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
181	C₄×D₄	—	2	4	no	—
182	#117	—	4	4	no	—
183	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
184	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
185	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
186	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
187	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
188	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
189	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
190	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
191	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
192	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
193	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
194	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
195	C₄⋊C₈	—	2	4	no	—
196	C₄⋊C₈	—	4	4	no	—
197	C₂²⋊C₈	—	4	4	no	—
198	C₂²⋊C₈	—	4	4	no	—
199	#97	—	4	2	no	—
200	C₄×C₈	—	2	4	no	—
201	#95	C₂×C₄×C₄₁	1	4	yes	—
202	#95	C₂×C₄×C₄₁	1	2	yes	—
203	C₂⁴	C₂²×C₄×C₄₁	1	2	yes	—
204	C₂×D₄	—	2	2	no	—
205	C₂×D₄	—	2	4	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
211	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
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	1	yes	characteristic
215	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
216	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	4	4	no	—
229	C₂²⋊C₄	—	2	4	no	—
230	C₂²⋊C₄	—	4	4	no	—
231	C₂²⋊C₄	—	4	4	no	—
232	C₄²	—	2	4	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	2	4	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	2	1	no	—
237	C₄²	—	4	2	no	—
238	C₄²	#41×C₄₁	1	4	yes	—
239	C₂³	—	2	2	no	—
240	C₂³	—	2	2	no	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	4	no	—
243	C₂³	—	2	4	no	—
244	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
245	D₄	—	4	4	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	#117×C₄₁	1	2	yes	—
262	C₂×C₄	#117×C₄₁	1	2	yes	—
263	C₂×C₄	#131×C₄₁	1	4	yes	—
264	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
265	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
266	C₂×C₄	H(5,𝔽₂)×C₄₁	1	2	yes	—
267	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
268	C₈	—	4	2	no	—
269	C₈	—	4	4	no	—
270	C₈	—	4	2	no	—
271	C₂²	—	4	2	no	—
272	C₂²	—	4	2	no	—
273	C₂²	—	4	2	no	—
274	C₂²	—	4	4	no	—
275	C₂²	—	4	4	no	—
276	C₂²	C₂×#117×C₄₁	1	1	yes	characteristic
277	C₂²	C₄₁×#518	1	1	yes	characteristic
278	C₂²	C₄₁×#520	1	1	yes	characteristic
279	C₂²	C₂×#131×C₄₁	1	2	yes	—
280	C₂²	C₄₁×#568	1	2	yes	—
281	C₄	—	2	2	no	—
282	C₄	—	4	2	no	—
283	C₄	—	4	2	no	—
284	C₄	—	4	4	no	—
285	C₄	—	4	2	no	—
286	C₄	C₄₁×#433	1	4	yes	—
287	C₄	C₄₁×#540	1	2	yes	—
288	C₄	C₄₁×#542	1	2	yes	—
289	C₂	—	8	2	no	—
290	C₂	C₄₁×#2319	1	1	yes	characteristic
291	C₂	C₂×C₄₁×#433	1	1	yes	characteristic
292	C₂	C₄₁×#2923	1	1	yes	characteristic
293	C₂	C₄₁×#2981	1	2	yes	—
294	C₂	C₄₁×#2993	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	C₂²×D₄×C₄₁	C₂×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
3. C₄₁×#1939	C₂	semidirect
4. C₄₁×#1924	C₂	semidirect
5. C₄₁×#1818	C₂	semidirect
6. C₄₁×#1798	C₂	semidirect
7. C₄₁×#1767	C₂	semidirect
8. C₄₁×#1744	C₂	semidirect
123. #22013	C₄₁	direct (N × H)
144. #1939	C₂×C₄₁	semidirect
145. #1924	C₂×C₄₁	semidirect
146. #1818	C₂×C₄₁	semidirect
147. #1798	C₂×C₄₁	semidirect
148. #1767	C₂×C₄₁	semidirect
149. #1744	C₂×C₄₁	semidirect
175. C₄₁	#22013	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×#2293	2
C₄₁×#1954	2
C₄₁×#1939	2
C₄₁×#1924	2
C₄₁×#1818	2
C₄₁×#1798	2
C₄₁×#1767	2
C₄₁×#1744	2
#22013	41

Subgroups — C41×#22013 #23798214

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#22013 #23798214