Subgroups — C₄₁×#20547 #23796748

Order 10496.

994 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#431	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₈×C₄₁	C₂	1	1	yes	maximal, characteristic
3	C₄×#117×C₄₁	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1770	C₂	1	4	yes	maximal, complemented
5	C₄₁×#1767	C₂	1	4	yes	maximal, complemented
6	C₄₁×#1752	C₂	1	4	yes	maximal
7	C₂³×C₈×C₄₁	C₂²	1	2	yes	—
8	C₂×#117×C₄₁	C₄	1	2	yes	—
9	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
10	C₄₁×#431	C₄	1	8	yes	complemented
11	C₂×C₄⋊C₈×C₄₁	C₄	1	8	yes	complemented
12	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
14	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
15	C₂×C₂²⋊C₈×C₄₁	C₂²	1	2	yes	—
16	C₂×#97×C₄₁	C₂²	1	8	yes	—
17	C₂×C₄×C₈×C₄₁	C₂²	1	2	yes	—
18	C₄×C₄⋊C₄×C₄₁	C₄	1	2	yes	—
19	C₄×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
20	C₄×C₂²⋊C₄×C₄₁	C₄	1	2	yes	—
21	C₄×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
22	C₄₁×#376	C₄	1	2	yes	—
23	C₄³×C₄₁	C₂²	1	2	yes	—
24	C₄₁×#336	C₂²	1	8	yes	—
25	C₂³×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
28	C₂²×C₈×C₄₁	—	4	4	no	—
29	C₂²×C₈×C₄₁	C₂×C₄	1	8	yes	—
30	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
31	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
32	#117×C₄₁	—	2	8	no	—
33	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
34	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₄₁	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₄₁	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₄₁	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₄₁	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₄₁	C₂³	1	4	yes	—
42	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₄₁	C₂×C₄	1	8	yes	—
45	C₄⋊C₈×C₄₁	C₂×C₄	1	8	yes	—
46	C₄⋊C₈×C₄₁	C₂×C₄	1	8	yes	—
47	C₂²⋊C₈×C₄₁	C₂×C₄	1	8	yes	—
48	#97×C₄₁	—	4	8	no	—
49	C₄×C₈×C₄₁	C₂×C₄	1	8	yes	—
50	#95×C₄₁	C₂×C₄	1	4	yes	—
51	#95×C₄₁	C₂×C₄	1	4	yes	—
52	C₂⁴×C₄₁	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₁	—	2	2	no	—
54	C₂²×C₄×C₄₁	—	2	2	no	—
55	C₂²×C₄×C₄₁	—	4	2	no	—
56	C₂²×C₄×C₄₁	C₄²	1	4	yes	—
57	C₂²×C₄×C₄₁	C₄²	1	4	yes	—
58	C₂²×C₄×C₄₁	M₁₆	1	8	yes	—
59	C₂²×C₄×C₄₁	C₂²×C₄	1	8	yes	—
60	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
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	1	yes	characteristic
66	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
68	C₂×C₈×C₄₁	—	2	4	no	—
69	C₂×C₈×C₄₁	—	2	8	no	—
70	C₂×C₈×C₄₁	—	2	4	no	—
71	C₂×C₈×C₄₁	—	4	8	no	—
72	C₂×C₈×C₄₁	C₂²×C₄	1	8	yes	—
73	C₂×C₈×C₄₁	C₂²×C₄	1	8	yes	—
74	C₄⋊C₄×C₄₁	—	2	8	no	—
75	C₂²⋊C₄×C₄₁	—	2	8	no	—
76	C₄²×C₄₁	—	2	8	no	—
77	C₄²×C₄₁	—	2	4	no	—
78	C₄²×C₄₁	—	2	4	no	—
79	C₄²×C₄₁	—	4	4	no	—
80	C₄²×C₄₁	C₄²	1	4	yes	—
81	C₄²×C₄₁	C₄²	1	4	yes	—
82	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
83	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
84	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
85	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
86	C₂³×C₄₁	—	2	2	no	—
87	C₂³×C₄₁	—	2	2	no	—
88	C₂³×C₄₁	C₂×C₄²	1	2	yes	—
89	C₂³×C₄₁	C₂×M₁₆	1	4	yes	—
90	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₄₁	—	2	2	no	—
92	C₂×C₄×C₄₁	—	2	2	no	—
93	C₂×C₄×C₄₁	—	2	2	no	—
94	C₂×C₄×C₄₁	—	2	2	no	—
95	C₂×C₄×C₄₁	—	2	2	no	—
96	C₂×C₄×C₄₁	—	2	4	no	—
97	C₂×C₄×C₄₁	—	2	8	no	—
98	C₂×C₄×C₄₁	—	2	8	no	—
99	C₂×C₄×C₄₁	—	2	8	no	—
100	C₂×C₄×C₄₁	—	4	4	no	—
101	C₂×C₄×C₄₁	#97	1	16	yes	complemented
102	C₂×C₄×C₄₁	C₂×C₄²	1	2	yes	—
103	C₂×C₄×C₄₁	C₂×C₄²	1	2	yes	—
104	C₂×C₄×C₄₁	C₂×C₄²	1	2	yes	—
105	C₂×C₄×C₄₁	C₂×C₄²	1	2	yes	—
106	C₂×C₄×C₄₁	C₂×C₄²	1	2	yes	—
107	C₂×C₄×C₄₁	C₂×M₁₆	1	4	yes	—
108	C₂×C₄×C₄₁	C₂×M₁₆	1	4	yes	—
109	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₁	H(5,𝔽₂)	1	2	yes	—
112	C₂×C₄×C₄₁	2₋¹⁺⁴	1	2	yes	—
113	C₈×C₄₁	—	2	8	no	—
114	C₈×C₄₁	—	2	8	no	—
115	#20547	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂²×C₄₁	—	2	2	no	—
117	C₂²×C₄₁	—	2	8	no	—
118	C₂²×C₄₁	—	2	2	no	—
119	C₂²×C₄₁	C₂×#97	1	4	yes	—
120	C₂²×C₄₁	C₂²×C₄²	1	1	yes	characteristic
121	C₂²×C₄₁	#520	1	2	yes	—
122	C₂²×C₄₁	C₂²×M₁₆	1	2	yes	—
123	C₂²×C₄₁	#568	1	2	yes	—
124	C₄×C₄₁	—	2	2	no	—
125	C₄×C₄₁	—	2	2	no	—
126	C₄×C₄₁	—	2	2	no	—
127	C₄×C₄₁	—	4	8	no	—
128	C₄×C₄₁	C₂×#97	1	4	yes	—
129	C₄×C₄₁	C₂×#97	1	4	yes	—
130	C₄×C₄₁	#518	1	1	yes	characteristic
131	C₄×C₄₁	#519	1	1	yes	characteristic
132	C₄×C₄₁	#520	1	2	yes	—
133	C₂×#431	C₂×C₄₁	1	1	yes	characteristic
134	C₂²×C₄⋊C₈	C₂×C₄₁	1	1	yes	characteristic
135	C₄×#117	C₂×C₄₁	1	1	yes	characteristic
136	#1770	C₂×C₄₁	1	4	yes	complemented
137	#1767	C₂×C₄₁	1	4	yes	complemented
138	#1752	C₂×C₄₁	1	4	yes	—
139	C₂×C₄₁	—	2	4	no	—
140	C₂×C₄₁	#2274	1	1	yes	characteristic
141	C₂×C₄₁	C₂²×#97	1	1	yes	characteristic
142	C₂×C₄₁	#2873	1	1	yes	characteristic
143	C₂×C₄₁	#2989	1	4	yes	—
144	C₂³×C₈	C₂²×C₄₁	1	2	yes	—
145	C₂×#117	C₄×C₄₁	1	2	yes	—
146	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
147	#431	C₄×C₄₁	1	8	yes	complemented
148	C₂×C₄⋊C₈	C₄×C₄₁	1	8	yes	complemented
149	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
150	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
151	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
152	C₂×C₂²⋊C₈	C₂²×C₄₁	1	2	yes	—
153	C₂×#97	C₂²×C₄₁	1	8	yes	—
154	C₂×C₄×C₈	C₂²×C₄₁	1	2	yes	—
155	C₄×C₄⋊C₄	C₄×C₄₁	1	2	yes	—
156	C₄×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
157	C₄×C₂²⋊C₄	C₄×C₄₁	1	2	yes	—
158	C₄×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
159	#376	C₄×C₄₁	1	2	yes	—
160	C₄³	C₂²×C₄₁	1	2	yes	—
161	#336	C₂²×C₄₁	1	8	yes	—
162	C₄₁	#20547	1	1	yes	characteristic, direct product factor, Sylow
163	C₂³×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
164	C₂³×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
165	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
166	C₂²×C₈	—	4	4	no	—
167	C₂²×C₈	C₂×C₄×C₄₁	1	8	yes	—
168	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
169	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
170	#117	—	2	8	no	—
171	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
172	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
173	C₂×C₄²	C₂×C₄×C₄₁	1	4	yes	—
174	C₂×C₄²	C₂×C₄×C₄₁	1	4	yes	—
175	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
176	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
177	C₂×C₄²	C₂×C₄×C₄₁	1	4	yes	—
178	C₂×C₄²	C₂×C₄×C₄₁	1	4	yes	—
179	C₂×C₄²	C₂³×C₄₁	1	4	yes	—
180	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
181	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
182	C₄⋊C₈	C₂×C₄×C₄₁	1	8	yes	—
183	C₄⋊C₈	C₂×C₄×C₄₁	1	8	yes	—
184	C₄⋊C₈	C₂×C₄×C₄₁	1	8	yes	—
185	C₂²⋊C₈	C₂×C₄×C₄₁	1	8	yes	—
186	#97	—	4	8	no	—
187	C₄×C₈	C₂×C₄×C₄₁	1	8	yes	—
188	#95	C₂×C₄×C₄₁	1	4	yes	—
189	#95	C₂×C₄×C₄₁	1	4	yes	—
190	C₂⁴	C₂²×C₄×C₄₁	1	1	yes	characteristic
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	C₄²×C₄₁	1	4	yes	—
195	C₂²×C₄	C₄²×C₄₁	1	4	yes	—
196	C₂²×C₄	M₁₆×C₄₁	1	8	yes	—
197	C₂²×C₄	C₂²×C₄×C₄₁	1	8	yes	—
198	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
199	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
200	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
201	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
202	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
203	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
204	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
205	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	2	8	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	4	8	no	—
210	C₂×C₈	C₂²×C₄×C₄₁	1	8	yes	—
211	C₂×C₈	C₂²×C₄×C₄₁	1	8	yes	—
212	C₄⋊C₄	—	2	8	no	—
213	C₂²⋊C₄	—	2	8	no	—
214	C₄²	—	2	8	no	—
215	C₄²	—	2	4	no	—
216	C₄²	—	2	4	no	—
217	C₄²	—	4	4	no	—
218	C₄²	C₄²×C₄₁	1	4	yes	—
219	C₄²	C₄²×C₄₁	1	4	yes	—
220	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
221	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
222	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
223	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
224	C₂³	—	2	2	no	—
225	C₂³	—	2	2	no	—
226	C₂³	C₂×C₄²×C₄₁	1	2	yes	—
227	C₂³	C₂×M₁₆×C₄₁	1	4	yes	—
228	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	8	no	—
236	C₂×C₄	—	2	8	no	—
237	C₂×C₄	—	2	8	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	#97×C₄₁	1	16	yes	complemented
240	C₂×C₄	C₂×C₄²×C₄₁	1	2	yes	—
241	C₂×C₄	C₂×C₄²×C₄₁	1	2	yes	—
242	C₂×C₄	C₂×C₄²×C₄₁	1	2	yes	—
243	C₂×C₄	C₂×C₄²×C₄₁	1	2	yes	—
244	C₂×C₄	C₂×C₄²×C₄₁	1	2	yes	—
245	C₂×C₄	C₂×M₁₆×C₄₁	1	4	yes	—
246	C₂×C₄	C₂×M₁₆×C₄₁	1	4	yes	—
247	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
248	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
249	C₂×C₄	H(5,𝔽₂)×C₄₁	1	2	yes	—
250	C₂×C₄	2₋¹⁺⁴×C₄₁	1	2	yes	—
251	C₈	—	2	8	no	—
252	C₈	—	2	8	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	8	no	—
256	C₂²	C₂×#97×C₄₁	1	4	yes	—
257	C₂²	C₂²×C₄²×C₄₁	1	1	yes	characteristic
258	C₂²	C₄₁×#520	1	2	yes	—
259	C₂²	C₂²×M₁₆×C₄₁	1	2	yes	—
260	C₂²	C₄₁×#568	1	2	yes	—
261	C₄	—	2	2	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	4	8	no	—
265	C₄	C₂×#97×C₄₁	1	4	yes	—
266	C₄	C₂×#97×C₄₁	1	4	yes	—
267	C₄	C₄₁×#518	1	1	yes	characteristic
268	C₄	C₄₁×#519	1	1	yes	characteristic
269	C₄	C₄₁×#520	1	2	yes	—
270	C₂	—	2	4	no	—
271	C₂	C₄₁×#2274	1	1	yes	characteristic
272	C₂	C₂²×#97×C₄₁	1	1	yes	characteristic
273	C₂	C₄₁×#2873	1	1	yes	characteristic
274	C₂	C₄₁×#2989	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₄₁×#1770	C₂	semidirect
5. C₄₁×#1767	C₂	semidirect
10. C₄₁×#431	C₄	semidirect
11. C₂×C₄⋊C₈×C₄₁	C₄	semidirect
101. C₂×C₄×C₄₁	#97	semidirect
115. #20547	C₄₁	direct (N × H)
136. #1770	C₂×C₄₁	semidirect
137. #1767	C₂×C₄₁	semidirect
147. #431	C₄×C₄₁	semidirect
148. C₂×C₄⋊C₈	C₄×C₄₁	semidirect
162. C₄₁	#20547	direct (N × H)
239. C₂×C₄	#97×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#431	2
C₂²×C₄⋊C₈×C₄₁	2
C₄×#117×C₄₁	2
C₄₁×#1770	2
C₄₁×#1767	2
C₄₁×#1752	2
#20547	41

Subgroups — C41×#20547 #23796748

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#20547 #23796748