Subgroups — C₄₁×#3136 #19106916

Order 5248.

404 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#544	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₁×#535	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#490	C₂	1	2	yes	maximal, complemented
4	C₄₁×#489	C₂	1	2	yes	maximal, complemented
5	C₄₁×#488	C₂	1	2	yes	maximal, complemented
6	C₄₁×#486	C₂	1	2	yes	maximal, complemented
7	C₄₁×#418	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#417	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#416	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#102×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₁×#405	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₄₁	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₄₁	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₄₁	C₂²	1	1	yes	characteristic
16	C₄⋊Q₈×C₄₁	—	2	1	no	—
17	C₄⋊Q₈×C₄₁	C₂²	1	2	yes	complemented
18	#127×C₄₁	C₂²	1	2	yes	—
19	#125×C₄₁	—	2	1	no	—
20	#125×C₄₁	C₂²	1	2	yes	complemented
21	#124×C₄₁	C₂²	1	2	yes	—
22	C₂²⋊Q₈×C₄₁	—	2	2	no	—
23	#121×C₄₁	—	2	1	no	—
24	#121×C₄₁	—	2	1	no	—
25	#120×C₄₁	—	2	2	no	—
26	#117×C₄₁	C₂²	1	1	yes	characteristic
27	#117×C₄₁	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
29	#103×C₄₁	C₂²	1	1	yes	characteristic, complemented
30	#103×C₄₁	C₂²	1	1	yes	characteristic, complemented
31	#103×C₄₁	C₂²	1	2	yes	complemented
32	#102×C₄₁	C₂²	1	2	yes	—
33	#102×C₄₁	C₂²	1	2	yes	—
34	#102×C₄₁	C₂²	1	2	yes	—
35	#102×C₄₁	C₂²	1	2	yes	—
36	#102×C₄₁	C₂²	1	1	yes	characteristic, complemented
37	#102×C₄₁	C₂²	1	1	yes	characteristic, complemented
38	#102×C₄₁	C₂²	1	2	yes	complemented
39	#97×C₄₁	C₂²	1	2	yes	complemented
40	C₄×C₈×C₄₁	C₂²	1	2	yes	complemented
41	C₂⁴×C₄₁	—	2	1	no	—
42	#41×C₄₁	—	2	2	no	—
43	#41×C₄₁	—	2	2	no	—
44	C₂×Q₈×C₄₁	—	2	2	no	—
45	C₂×Q₈×C₄₁	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₄₁	—	2	1	no	—
47	C₂×D₄×C₄₁	—	2	1	no	—
48	C₂×D₄×C₄₁	—	2	2	no	—
49	C₂×D₄×C₄₁	—	2	2	no	—
50	C₂×D₄×C₄₁	—	2	2	no	—
51	C₂×D₄×C₄₁	—	2	2	no	—
52	C₂×D₄×C₄₁	C₂³	1	2	yes	—
53	C₂×D₄×C₄₁	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₄₁	—	2	1	no	—
55	C₂²×C₄×C₄₁	—	2	1	no	—
56	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
57	M₁₆×C₄₁	—	2	2	no	—
58	C₂×C₈×C₄₁	—	2	2	no	—
59	C₂×C₈×C₄₁	C₂³	1	1	yes	characteristic
60	C₂×C₈×C₄₁	C₂³	1	2	yes	—
61	C₂×C₈×C₄₁	C₂³	1	1	yes	characteristic
62	C₄⋊C₄×C₄₁	—	2	1	no	—
63	C₄⋊C₄×C₄₁	—	2	2	no	—
64	C₄⋊C₄×C₄₁	—	2	2	no	—
65	C₄⋊C₄×C₄₁	—	2	2	no	—
66	C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
67	C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
68	C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
69	C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
71	C₂²⋊C₄×C₄₁	—	2	1	no	—
72	C₂²⋊C₄×C₄₁	—	2	2	no	—
73	C₂²⋊C₄×C₄₁	—	2	2	no	—
74	C₂²⋊C₄×C₄₁	D₄	1	2	yes	—
75	C₄²×C₄₁	—	2	1	no	—
76	C₄²×C₄₁	C₂³	1	2	yes	—
77	C₂³×C₄₁	—	2	2	no	—
78	C₂³×C₄₁	—	2	1	no	—
79	C₂³×C₄₁	—	4	2	no	—
80	C₂³×C₄₁	—	8	1	no	—
81	C₂³×C₄₁	—	8	1	no	—
82	C₂³×C₄₁	C₂×D₄	1	1	yes	characteristic
83	Q₈×C₄₁	—	2	1	no	—
84	Q₈×C₄₁	—	2	1	no	—
85	Q₈×C₄₁	—	8	2	no	—
86	D₄×C₄₁	—	2	2	no	—
87	D₄×C₄₁	—	2	2	no	—
88	D₄×C₄₁	—	2	1	no	—
89	D₄×C₄₁	—	2	1	no	—
90	D₄×C₄₁	—	4	2	no	—
91	D₄×C₄₁	—	4	2	no	—
92	D₄×C₄₁	—	8	2	no	—
93	D₄×C₄₁	—	8	2	no	—
94	C₂×C₄×C₄₁	—	2	1	no	—
95	C₂×C₄×C₄₁	—	2	1	no	—
96	C₂×C₄×C₄₁	—	2	1	no	—
97	C₂×C₄×C₄₁	—	2	2	no	—
98	C₂×C₄×C₄₁	—	4	2	no	—
99	C₂×C₄×C₄₁	—	8	1	no	—
100	C₂×C₄×C₄₁	C₂×D₄	1	2	yes	—
101	C₂×C₄×C₄₁	C₂×D₄	1	2	yes	—
102	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₄₁	#41	1	2	yes	—
104	C₂×C₄×C₄₁	#41	1	2	yes	—
105	C₂×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
106	C₈×C₄₁	—	2	1	no	—
107	C₈×C₄₁	—	2	1	no	—
108	C₈×C₄₁	—	2	2	no	—
109	C₂²×C₄₁	—	2	1	no	—
110	C₂²×C₄₁	—	2	1	no	—
111	C₂²×C₄₁	—	4	1	no	—
112	C₂²×C₄₁	—	4	2	no	—
113	C₂²×C₄₁	—	4	1	no	—
114	C₂²×C₄₁	—	4	2	no	—
115	C₂²×C₄₁	—	4	1	no	—
116	C₂²×C₄₁	—	4	2	no	—
117	C₂²×C₄₁	—	8	4	no	—
118	C₂²×C₄₁	#124	1	2	yes	—
119	C₂²×C₄₁	C₂²×D₄	1	1	yes	characteristic
120	C₄×C₄₁	—	2	2	no	—
121	C₄×C₄₁	—	2	2	no	—
122	C₄×C₄₁	—	4	1	no	—
123	C₄×C₄₁	—	4	2	no	—
124	C₄×C₄₁	—	4	1	no	—
125	C₄×C₄₁	—	4	1	no	—
126	C₄×C₄₁	#124	1	2	yes	—
127	C₄×C₄₁	C₂×#41	1	1	yes	characteristic
128	C₄×C₄₁	C₂×#41	1	1	yes	characteristic
129	#3136	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂×C₄₁	—	2	2	no	—
131	C₂×C₄₁	—	8	1	no	—
132	C₂×C₄₁	—	8	2	no	—
133	C₂×C₄₁	C₂×#124	1	1	yes	characteristic
134	C₂×C₄₁	#576	1	1	yes	characteristic
135	C₂×C₄₁	#577	1	1	yes	characteristic
136	#544	C₂×C₄₁	1	1	yes	characteristic, complemented
137	#535	C₂×C₄₁	1	1	yes	characteristic
138	#490	C₂×C₄₁	1	2	yes	complemented
139	#489	C₂×C₄₁	1	2	yes	complemented
140	#488	C₂×C₄₁	1	2	yes	complemented
141	#486	C₂×C₄₁	1	2	yes	complemented
142	#418	C₂×C₄₁	1	1	yes	characteristic, complemented
143	#417	C₂×C₄₁	1	1	yes	characteristic, complemented
144	#416	C₂×C₄₁	1	1	yes	characteristic, complemented
145	C₂×#102	C₂×C₄₁	1	1	yes	characteristic, complemented
146	#405	C₂×C₄₁	1	1	yes	characteristic, complemented
147	C₄₁	#3136	1	1	yes	characteristic, direct product factor, Sylow
148	C₂×#41	C₂²×C₄₁	1	1	yes	characteristic
149	C₂²×D₄	C₂²×C₄₁	1	1	yes	characteristic
150	C₂×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
151	C₂²×C₈	C₂²×C₄₁	1	1	yes	characteristic
152	C₄⋊Q₈	—	2	1	no	—
153	C₄⋊Q₈	C₂²×C₄₁	1	2	yes	complemented
154	#127	C₂²×C₄₁	1	2	yes	—
155	#125	—	2	1	no	—
156	#125	C₂²×C₄₁	1	2	yes	complemented
157	#124	C₂²×C₄₁	1	2	yes	—
158	C₂²⋊Q₈	—	2	2	no	—
159	#121	—	2	1	no	—
160	#121	—	2	1	no	—
161	#120	—	2	2	no	—
162	#117	C₂²×C₄₁	1	1	yes	characteristic
163	#117	C₂²×C₄₁	1	1	yes	characteristic
164	C₂×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
165	#103	C₂²×C₄₁	1	1	yes	characteristic, complemented
166	#103	C₂²×C₄₁	1	1	yes	characteristic, complemented
167	#103	C₂²×C₄₁	1	2	yes	complemented
168	#102	C₂²×C₄₁	1	2	yes	—
169	#102	C₂²×C₄₁	1	2	yes	—
170	#102	C₂²×C₄₁	1	2	yes	—
171	#102	C₂²×C₄₁	1	2	yes	—
172	#102	C₂²×C₄₁	1	1	yes	characteristic, complemented
173	#102	C₂²×C₄₁	1	1	yes	characteristic, complemented
174	#102	C₂²×C₄₁	1	2	yes	complemented
175	#97	C₂²×C₄₁	1	2	yes	complemented
176	C₄×C₈	C₂²×C₄₁	1	2	yes	complemented
177	C₂⁴	—	2	1	no	—
178	#41	—	2	2	no	—
179	#41	—	2	2	no	—
180	C₂×Q₈	—	2	2	no	—
181	C₂×Q₈	C₂³×C₄₁	1	1	yes	characteristic
182	C₂×D₄	—	2	1	no	—
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	—	2	2	no	—
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	C₂³×C₄₁	1	2	yes	—
189	C₂×D₄	C₂³×C₄₁	1	1	yes	characteristic
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
193	M₁₆	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	C₂³×C₄₁	1	1	yes	characteristic
196	C₂×C₈	C₂³×C₄₁	1	2	yes	—
197	C₂×C₈	C₂³×C₄₁	1	1	yes	characteristic
198	C₄⋊C₄	—	2	1	no	—
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
203	C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
204	C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
205	C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
206	C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
207	C₂²⋊C₄	—	2	1	no	—
208	C₂²⋊C₄	—	2	2	no	—
209	C₂²⋊C₄	—	2	2	no	—
210	C₂²⋊C₄	D₄×C₄₁	1	2	yes	—
211	C₄²	—	2	1	no	—
212	C₄²	C₂³×C₄₁	1	2	yes	—
213	C₂³	—	2	2	no	—
214	C₂³	—	2	1	no	—
215	C₂³	—	4	2	no	—
216	C₂³	—	8	1	no	—
217	C₂³	—	8	1	no	—
218	C₂³	C₂×D₄×C₄₁	1	1	yes	characteristic
219	Q₈	—	2	1	no	—
220	Q₈	—	2	1	no	—
221	Q₈	—	8	2	no	—
222	D₄	—	2	2	no	—
223	D₄	—	2	2	no	—
224	D₄	—	2	1	no	—
225	D₄	—	2	1	no	—
226	D₄	—	4	2	no	—
227	D₄	—	4	2	no	—
228	D₄	—	8	2	no	—
229	D₄	—	8	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	8	1	no	—
236	C₂×C₄	C₂×D₄×C₄₁	1	2	yes	—
237	C₂×C₄	C₂×D₄×C₄₁	1	2	yes	—
238	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
239	C₂×C₄	#41×C₄₁	1	2	yes	—
240	C₂×C₄	#41×C₄₁	1	2	yes	—
241	C₂×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
242	C₈	—	2	1	no	—
243	C₈	—	2	1	no	—
244	C₈	—	2	2	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	4	1	no	—
248	C₂²	—	4	2	no	—
249	C₂²	—	4	1	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	1	no	—
252	C₂²	—	4	2	no	—
253	C₂²	—	8	4	no	—
254	C₂²	#124×C₄₁	1	2	yes	—
255	C₂²	C₂²×D₄×C₄₁	1	1	yes	characteristic
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	4	1	no	—
259	C₄	—	4	2	no	—
260	C₄	—	4	1	no	—
261	C₄	—	4	1	no	—
262	C₄	#124×C₄₁	1	2	yes	—
263	C₄	C₂×#41×C₄₁	1	1	yes	characteristic
264	C₄	C₂×#41×C₄₁	1	1	yes	characteristic
265	C₂	—	2	2	no	—
266	C₂	—	8	2	no	—
267	C₂	—	8	1	no	—
268	C₂	C₂×#124×C₄₁	1	1	yes	characteristic
269	C₂	C₄₁×#576	1	1	yes	characteristic
270	C₂	C₄₁×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₄₁
prime radical	C₄₁×#535	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ #117×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₁×#544	C₂	semidirect
3. C₄₁×#490	C₂	semidirect
4. C₄₁×#489	C₂	semidirect
5. C₄₁×#488	C₂	semidirect
6. C₄₁×#486	C₂	semidirect
7. C₄₁×#418	C₂	semidirect
8. C₄₁×#417	C₂	semidirect
9. C₄₁×#416	C₂	semidirect
10. C₂×#102×C₄₁	C₂	semidirect
11. C₄₁×#405	C₂	semidirect
17. C₄⋊Q₈×C₄₁	C₂²	semidirect
20. #125×C₄₁	C₂²	semidirect
29. #103×C₄₁	C₂²	semidirect
30. #103×C₄₁	C₂²	semidirect
31. #103×C₄₁	C₂²	semidirect
36. #102×C₄₁	C₂²	semidirect
37. #102×C₄₁	C₂²	semidirect
38. #102×C₄₁	C₂²	semidirect
39. #97×C₄₁	C₂²	semidirect
40. C₄×C₈×C₄₁	C₂²	semidirect
129. #3136	C₄₁	direct (N × H)
136. #544	C₂×C₄₁	semidirect
138. #490	C₂×C₄₁	semidirect
139. #489	C₂×C₄₁	semidirect
140. #488	C₂×C₄₁	semidirect
141. #486	C₂×C₄₁	semidirect
142. #418	C₂×C₄₁	semidirect
143. #417	C₂×C₄₁	semidirect
144. #416	C₂×C₄₁	semidirect
145. C₂×#102	C₂×C₄₁	semidirect
146. #405	C₂×C₄₁	semidirect
147. C₄₁	#3136	direct (N × H)
153. C₄⋊Q₈	C₂²×C₄₁	semidirect
156. #125	C₂²×C₄₁	semidirect
165. #103	C₂²×C₄₁	semidirect
166. #103	C₂²×C₄₁	semidirect
167. #103	C₂²×C₄₁	semidirect
172. #102	C₂²×C₄₁	semidirect
173. #102	C₂²×C₄₁	semidirect
174. #102	C₂²×C₄₁	semidirect
175. #97	C₂²×C₄₁	semidirect
176. C₄×C₈	C₂²×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₄₁×#544	2
C₄₁×#535	2
C₄₁×#490	2
C₄₁×#489	2
C₄₁×#488	2
C₄₁×#486	2
C₄₁×#418	2
C₄₁×#417	2
C₄₁×#416	2
C₂×#102×C₄₁	2
C₄₁×#405	2
#3136	41

Subgroups — C41×#3136 #19106916

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#3136 #19106916