Subgroups — C₄₁×#20710 #23796911

Order 10496.

1276 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3428	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#410	C₂	1	2	yes	maximal
3	C₄₁×#1828	C₂	1	8	yes	maximal, complemented
4	C₄₁×#1740	C₂	1	4	yes	maximal, complemented
5	C₂²×#41×C₄₁	C₂²	1	1	yes	characteristic
6	C₄₁×#520	—	2	4	no	—
7	C₄₁×#519	C₄	1	2	yes	complemented
8	C₄₁×#518	C₄	1	2	yes	complemented
9	C₄₁×#518	C₂²	1	4	yes	—
10	C₄×#41×C₄₁	—	2	4	no	—
11	C₂×#117×C₄₁	C₄	1	4	yes	complemented
12	C₂×#117×C₄₁	C₂²	1	4	yes	—
13	C₂×#117×C₄₁	C₂²	1	2	yes	—
14	C₄₁×#410	—	2	4	no	—
15	C₄₁×#410	—	2	4	no	—
16	C₂×#99×C₄₁	C₂²	1	8	yes	—
17	C₄₁×#342	C₂²	1	16	yes	—
18	C₂×#41×C₄₁	—	2	4	no	—
19	C₂×#41×C₄₁	—	2	4	no	—
20	C₂×#41×C₄₁	D₄	1	4	yes	—
21	C₂×#41×C₄₁	D₄	1	4	yes	—
22	C₂²×Q₈×C₄₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×D₄×C₄₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×D₄×C₄₁	C₂³	1	2	yes	—
25	C₂³×C₄×C₄₁	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
27	C₄×Q₈×C₄₁	—	2	8	no	—
28	C₄×D₄×C₄₁	—	2	8	no	—
29	C₄×D₄×C₄₁	—	2	16	no	—
30	#117×C₄₁	—	2	4	no	—
31	#117×C₄₁	—	2	4	no	—
32	#117×C₄₁	—	2	2	no	—
33	#117×C₄₁	—	2	2	no	—
34	#117×C₄₁	—	2	4	no	—
35	#117×C₄₁	—	2	4	no	—
36	#117×C₄₁	—	2	4	no	—
37	#117×C₄₁	—	2	4	no	—
38	#117×C₄₁	C₂×C₄	1	4	yes	—
39	#117×C₄₁	C₂×C₄	1	8	yes	—
40	#117×C₄₁	C₂×C₄	1	4	yes	—
41	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
42	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
44	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	8	yes	—
45	C₂×C₂²⋊C₄×C₄₁	C₂³	1	8	yes	—
46	C₂×C₂²⋊C₄×C₄₁	C₂³	1	4	yes	—
47	C₂×C₄²×C₄₁	—	2	4	no	—
48	C₂×C₄²×C₄₁	—	2	2	no	—
49	C₂×C₄²×C₄₁	—	2	4	no	—
50	#99×C₄₁	—	2	16	no	—
51	C₂⁴×C₄₁	C₂²×C₄	1	2	yes	—
52	C₂⁴×C₄₁	C₂⁴	1	1	yes	characteristic
53	#41×C₄₁	—	2	8	no	—
54	#41×C₄₁	—	2	8	no	—
55	#41×C₄₁	—	2	8	no	—
56	#41×C₄₁	—	2	8	no	—
57	C₂×Q₈×C₄₁	—	2	4	no	—
58	C₂×Q₈×C₄₁	C₂²⋊C₄	1	4	yes	—
59	C₂×D₄×C₄₁	—	2	8	no	—
60	C₂×D₄×C₄₁	—	2	4	no	—
61	C₂×D₄×C₄₁	C₂²⋊C₄	1	4	yes	—
62	C₂×D₄×C₄₁	C₂×D₄	1	8	yes	—
63	C₂²×C₄×C₄₁	—	2	4	no	—
64	C₂²×C₄×C₄₁	—	2	4	no	—
65	C₂²×C₄×C₄₁	—	2	8	no	—
66	C₂²×C₄×C₄₁	—	2	4	no	—
67	C₂²×C₄×C₄₁	—	2	8	no	—
68	C₂²×C₄×C₄₁	C₂²⋊C₄	1	4	yes	—
69	C₂²×C₄×C₄₁	C₂²⋊C₄	1	4	yes	—
70	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₄₁	#41	1	4	yes	—
76	C₂²×C₄×C₄₁	#41	1	4	yes	—
77	C₄⋊C₄×C₄₁	—	2	4	no	—
78	C₄⋊C₄×C₄₁	—	2	8	no	—
79	C₄⋊C₄×C₄₁	—	2	4	no	—
80	C₄⋊C₄×C₄₁	—	4	4	no	—
81	C₄⋊C₄×C₄₁	—	4	4	no	—
82	C₂²⋊C₄×C₄₁	—	2	8	no	—
83	C₂²⋊C₄×C₄₁	—	2	4	no	—
84	C₂²⋊C₄×C₄₁	—	2	8	no	—
85	C₂²⋊C₄×C₄₁	—	2	8	no	—
86	C₂²⋊C₄×C₄₁	C₂²×C₄	1	8	yes	—
87	C₄²×C₄₁	—	2	4	no	—
88	C₄²×C₄₁	—	2	8	no	—
89	C₄²×C₄₁	—	2	4	no	—
90	C₄²×C₄₁	—	4	4	no	—
91	C₄²×C₄₁	—	4	4	no	—
92	C₂³×C₄₁	—	2	4	no	—
93	C₂³×C₄₁	—	8	1	no	—
94	C₂³×C₄₁	—	8	2	no	—
95	C₂³×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
96	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
97	C₂³×C₄₁	C₂²×D₄	1	2	yes	—
98	C₂³×C₄₁	C₂×#41	1	4	yes	—
99	Q₈×C₄₁	—	2	8	no	—
100	D₄×C₄₁	—	2	16	no	—
101	D₄×C₄₁	—	2	8	no	—
102	C₂×C₄×C₄₁	—	2	4	no	—
103	C₂×C₄×C₄₁	—	2	4	no	—
104	C₂×C₄×C₄₁	—	2	8	no	—
105	C₂×C₄×C₄₁	—	2	4	no	—
106	C₂×C₄×C₄₁	—	2	8	no	—
107	C₂×C₄×C₄₁	—	4	8	no	—
108	C₂×C₄×C₄₁	—	8	4	no	—
109	C₂×C₄×C₄₁	—	8	8	no	—
110	C₂×C₄×C₄₁	—	8	8	no	—
111	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
112	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
113	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₄₁	#117	1	2	yes	—
116	C₂×C₄×C₄₁	#117	1	4	yes	—
117	C₂×C₄×C₄₁	#117	1	2	yes	—
118	C₂×C₄×C₄₁	#123	1	4	yes	—
119	C₂×C₄×C₄₁	#123	1	4	yes	—
120	C₂×C₄×C₄₁	#123	1	4	yes	—
121	C₂×C₄×C₄₁	#123	1	4	yes	—
122	#20710	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂²×C₄₁	—	2	4	no	—
124	C₂²×C₄₁	—	4	2	no	—
125	C₂²×C₄₁	—	4	4	no	—
126	C₂²×C₄₁	—	4	1	no	—
127	C₂²×C₄₁	—	4	4	no	—
128	C₂²×C₄₁	—	8	4	no	—
129	C₂²×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
130	C₂²×C₄₁	C₂×#117	1	2	yes	—
131	C₂²×C₄₁	C₂×#123	1	4	yes	—
132	C₄×C₄₁	—	2	4	no	—
133	C₄×C₄₁	—	4	8	no	—
134	C₄×C₄₁	—	8	8	no	—
135	C₄×C₄₁	#381	1	2	yes	—
136	C₄×C₄₁	#381	1	1	yes	characteristic
137	C₄×C₄₁	#381	1	4	yes	—
138	C₄×C₄₁	#381	1	1	yes	characteristic
139	#3428	C₂×C₄₁	1	1	yes	characteristic
140	C₂×#410	C₂×C₄₁	1	2	yes	—
141	#1828	C₂×C₄₁	1	8	yes	complemented
142	#1740	C₂×C₄₁	1	4	yes	complemented
143	C₂×C₄₁	—	2	1	no	—
144	C₂×C₄₁	—	2	4	no	—
145	C₂×C₄₁	—	2	2	no	—
146	C₂×C₄₁	—	4	4	no	—
147	C₂×C₄₁	C₂×#381	1	1	yes	characteristic
148	C₂²×#41	C₂²×C₄₁	1	1	yes	characteristic
149	#520	—	2	4	no	—
150	#519	C₄×C₄₁	1	2	yes	complemented
151	#518	C₄×C₄₁	1	2	yes	complemented
152	#518	C₂²×C₄₁	1	4	yes	—
153	C₄×#41	—	2	4	no	—
154	C₂×#117	C₄×C₄₁	1	4	yes	complemented
155	C₂×#117	C₂²×C₄₁	1	4	yes	—
156	C₂×#117	C₂²×C₄₁	1	2	yes	—
157	#410	—	2	4	no	—
158	#410	—	2	4	no	—
159	C₂×#99	C₂²×C₄₁	1	8	yes	—
160	#342	C₂²×C₄₁	1	16	yes	—
161	C₄₁	#20710	1	1	yes	characteristic, direct product factor, Sylow
162	C₂×#41	—	2	4	no	—
163	C₂×#41	—	2	4	no	—
164	C₂×#41	D₄×C₄₁	1	4	yes	—
165	C₂×#41	D₄×C₄₁	1	4	yes	—
166	C₂²×Q₈	C₂×C₄×C₄₁	1	1	yes	characteristic
167	C₂²×D₄	C₂×C₄×C₄₁	1	1	yes	characteristic
168	C₂²×D₄	C₂³×C₄₁	1	2	yes	—
169	C₂³×C₄	C₂×C₄×C₄₁	1	2	yes	—
170	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
171	C₄×Q₈	—	2	8	no	—
172	C₄×D₄	—	2	8	no	—
173	C₄×D₄	—	2	16	no	—
174	#117	—	2	4	no	—
175	#117	—	2	4	no	—
176	#117	—	2	2	no	—
177	#117	—	2	2	no	—
178	#117	—	2	4	no	—
179	#117	—	2	4	no	—
180	#117	—	2	4	no	—
181	#117	—	2	4	no	—
182	#117	C₂×C₄×C₄₁	1	4	yes	—
183	#117	C₂×C₄×C₄₁	1	8	yes	—
184	#117	C₂×C₄×C₄₁	1	4	yes	—
185	C₂×C₄⋊C₄	—	2	4	no	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	—	2	4	no	—
188	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	8	yes	—
189	C₂×C₂²⋊C₄	C₂³×C₄₁	1	8	yes	—
190	C₂×C₂²⋊C₄	C₂³×C₄₁	1	4	yes	—
191	C₂×C₄²	—	2	4	no	—
192	C₂×C₄²	—	2	2	no	—
193	C₂×C₄²	—	2	4	no	—
194	#99	—	2	16	no	—
195	C₂⁴	C₂²×C₄×C₄₁	1	2	yes	—
196	C₂⁴	C₂⁴×C₄₁	1	1	yes	characteristic
197	#41	—	2	8	no	—
198	#41	—	2	8	no	—
199	#41	—	2	8	no	—
200	#41	—	2	8	no	—
201	C₂×Q₈	—	2	4	no	—
202	C₂×Q₈	C₂²⋊C₄×C₄₁	1	4	yes	—
203	C₂×D₄	—	2	8	no	—
204	C₂×D₄	—	2	4	no	—
205	C₂×D₄	C₂²⋊C₄×C₄₁	1	4	yes	—
206	C₂×D₄	C₂×D₄×C₄₁	1	8	yes	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	8	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	8	no	—
212	C₂²×C₄	C₂²⋊C₄×C₄₁	1	4	yes	—
213	C₂²×C₄	C₂²⋊C₄×C₄₁	1	4	yes	—
214	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
215	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
216	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
218	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
219	C₂²×C₄	#41×C₄₁	1	4	yes	—
220	C₂²×C₄	#41×C₄₁	1	4	yes	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	2	8	no	—
223	C₄⋊C₄	—	2	4	no	—
224	C₄⋊C₄	—	4	4	no	—
225	C₄⋊C₄	—	4	4	no	—
226	C₂²⋊C₄	—	2	8	no	—
227	C₂²⋊C₄	—	2	4	no	—
228	C₂²⋊C₄	—	2	8	no	—
229	C₂²⋊C₄	—	2	8	no	—
230	C₂²⋊C₄	C₂²×C₄×C₄₁	1	8	yes	—
231	C₄²	—	2	4	no	—
232	C₄²	—	2	8	no	—
233	C₄²	—	2	4	no	—
234	C₄²	—	4	4	no	—
235	C₄²	—	4	4	no	—
236	C₂³	—	2	4	no	—
237	C₂³	—	8	1	no	—
238	C₂³	—	8	2	no	—
239	C₂³	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
240	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
241	C₂³	C₂²×D₄×C₄₁	1	2	yes	—
242	C₂³	C₂×#41×C₄₁	1	4	yes	—
243	Q₈	—	2	8	no	—
244	D₄	—	2	16	no	—
245	D₄	—	2	8	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	8	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	4	8	no	—
252	C₂×C₄	—	8	4	no	—
253	C₂×C₄	—	8	8	no	—
254	C₂×C₄	—	8	8	no	—
255	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
256	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
257	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
258	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
259	C₂×C₄	#117×C₄₁	1	2	yes	—
260	C₂×C₄	#117×C₄₁	1	2	yes	—
261	C₂×C₄	#117×C₄₁	1	4	yes	—
262	C₂×C₄	#123×C₄₁	1	4	yes	—
263	C₂×C₄	#123×C₄₁	1	4	yes	—
264	C₂×C₄	#123×C₄₁	1	4	yes	—
265	C₂×C₄	#123×C₄₁	1	4	yes	—
266	C₂²	—	2	4	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	4	no	—
269	C₂²	—	4	1	no	—
270	C₂²	—	4	4	no	—
271	C₂²	—	8	4	no	—
272	C₂²	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
273	C₂²	C₂×#117×C₄₁	1	2	yes	—
274	C₂²	C₂×#123×C₄₁	1	4	yes	—
275	C₄	—	2	4	no	—
276	C₄	—	4	8	no	—
277	C₄	—	8	8	no	—
278	C₄	C₄₁×#381	1	1	yes	characteristic
279	C₄	C₄₁×#381	1	2	yes	—
280	C₄	C₄₁×#381	1	1	yes	characteristic
281	C₄	C₄₁×#381	1	4	yes	—
282	C₂	—	2	1	no	—
283	C₂	—	2	4	no	—
284	C₂	—	2	2	no	—
285	C₂	—	4	4	no	—
286	C₂	C₂×C₄₁×#381	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₄₁	#381
commutator	C₂³	C₂³×C₄×C₄₁
Frattini, cosocle	C₂⁴	C₂⁴×C₄₁
socle	C₂×C₄₁	C₂×#381
prime radical	C₂²×#41×C₄₁	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₄₁ ◁ C₂³×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄₁×#1828	C₂	semidirect
4. C₄₁×#1740	C₂	semidirect
7. C₄₁×#519	C₄	semidirect
8. C₄₁×#518	C₄	semidirect
11. C₂×#117×C₄₁	C₄	semidirect
122. #20710	C₄₁	direct (N × H)
141. #1828	C₂×C₄₁	semidirect
142. #1740	C₂×C₄₁	semidirect
150. #519	C₄×C₄₁	semidirect
151. #518	C₄×C₄₁	semidirect
154. C₂×#117	C₄×C₄₁	semidirect
161. C₄₁	#20710	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×#3428	2
C₂×C₄₁×#410	2
C₄₁×#1828	2
C₄₁×#1740	2
#20710	41

Subgroups — C41×#20710 #23796911

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#20710 #23796911