Subgroups — C₄₁×#3035 #19106815

Order 5248.

496 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#135×C₄₁	C₂	1	2	yes	maximal, complemented
2	C₄₁×#532	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#41×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#464	C₂	1	2	yes	maximal, complemented
5	C₄₁×#463	C₂	1	2	yes	maximal, complemented
6	C₄₁×#462	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#461	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#460	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#459	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄⋊C₈×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₁×#416	C₂	1	2	yes	maximal, complemented
12	C₂×#41×C₄₁	C₂²	1	2	yes	—
13	C₂×#41×C₄₁	C₂²	1	1	yes	characteristic
14	#135×C₄₁	—	2	8	no	—
15	C₂×Q₁₆×C₄₁	C₂²	1	2	yes	—
16	C₂×SD₁₆×C₄₁	C₂²	1	2	yes	—
17	C₂×SD₁₆×C₄₁	C₂²	1	2	yes	—
18	C₂×D₈×C₄₁	C₂²	1	2	yes	—
19	C₂²×C₈×C₄₁	C₂²	1	2	yes	—
20	C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
21	#127×C₄₁	C₂²	1	1	yes	characteristic
22	#124×C₄₁	C₂²	1	2	yes	—
23	#121×C₄₁	—	2	1	no	—
24	#121×C₄₁	—	2	1	no	—
25	C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
26	C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₄₁	—	2	1	no	—
28	C₄×D₄×C₄₁	—	2	1	no	—
29	C₄×D₄×C₄₁	—	2	1	no	—
30	C₄×D₄×C₄₁	—	2	1	no	—
31	C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
32	C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
33	#117×C₄₁	—	2	1	no	—
34	#117×C₄₁	C₂²	1	1	yes	characteristic
35	C₂×C₄²×C₄₁	—	2	1	no	—
36	C₂×C₄²×C₄₁	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
39	C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
40	#103×C₄₁	C₂²	1	2	yes	—
41	#103×C₄₁	C₂²	1	2	yes	—
42	#102×C₄₁	C₂²	1	2	yes	—
43	#102×C₄₁	C₂²	1	2	yes	—
44	#41×C₄₁	—	2	2	no	—
45	#41×C₄₁	—	2	4	no	—
46	#41×C₄₁	—	2	4	no	—
47	#41×C₄₁	D₄	1	4	yes	complemented
48	C₂×Q₈×C₄₁	C₂³	1	2	yes	—
49	C₂×Q₈×C₄₁	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₄₁	—	2	1	no	—
51	C₂×D₄×C₄₁	—	2	2	no	—
52	C₂×D₄×C₄₁	—	2	1	no	—
53	C₂×D₄×C₄₁	—	2	1	no	—
54	C₂×D₄×C₄₁	C₂³	1	2	yes	—
55	C₂×D₄×C₄₁	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₄₁	—	2	1	no	—
57	C₂²×C₄×C₄₁	—	2	1	no	—
58	C₂²×C₄×C₄₁	—	2	1	no	—
59	C₂²×C₄×C₄₁	—	2	2	no	—
60	C₂²×C₄×C₄₁	D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₁	D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
63	Q₁₆×C₄₁	—	2	4	no	—
64	SD₁₆×C₄₁	—	2	4	no	—
65	SD₁₆×C₄₁	—	2	4	no	—
66	D₈×C₄₁	—	2	4	no	—
67	C₂×C₈×C₄₁	—	2	4	no	—
68	C₂×C₈×C₄₁	C₂³	1	2	yes	—
69	C₂×C₈×C₄₁	C₂³	1	2	yes	—
70	C₄⋊C₄×C₄₁	—	2	1	no	—
71	C₄⋊C₄×C₄₁	—	2	1	no	—
72	C₄⋊C₄×C₄₁	—	2	1	no	—
73	C₄⋊C₄×C₄₁	—	2	1	no	—
74	C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
76	C₂²⋊C₄×C₄₁	—	2	1	no	—
77	C₂²⋊C₄×C₄₁	—	2	1	no	—
78	C₂²⋊C₄×C₄₁	—	2	1	no	—
79	C₂²⋊C₄×C₄₁	—	2	2	no	—
80	C₂²⋊C₄×C₄₁	—	2	2	no	—
81	C₄²×C₄₁	—	2	1	no	—
82	C₄²×C₄₁	—	2	1	no	—
83	C₄²×C₄₁	—	2	1	no	—
84	C₄²×C₄₁	D₄	1	1	yes	characteristic
85	C₄²×C₄₁	D₄	1	1	yes	characteristic
86	C₄²×C₄₁	C₂³	1	1	yes	characteristic
87	C₄²×C₄₁	C₂³	1	1	yes	characteristic
88	C₂³×C₄₁	—	2	1	no	—
89	C₂³×C₄₁	—	2	2	no	—
90	C₂³×C₄₁	C₂×D₄	1	1	yes	characteristic
91	Q₈×C₄₁	—	2	1	no	—
92	Q₈×C₄₁	—	2	2	no	—
93	Q₈×C₄₁	—	2	2	no	—
94	Q₈×C₄₁	C₂×D₄	1	2	yes	—
95	D₄×C₄₁	—	2	4	no	—
96	D₄×C₄₁	—	2	2	no	—
97	D₄×C₄₁	—	2	1	no	—
98	D₄×C₄₁	—	2	2	no	—
99	D₄×C₄₁	—	4	2	no	—
100	D₄×C₄₁	—	4	4	no	—
101	D₄×C₄₁	—	4	2	no	—
102	D₄×C₄₁	C₂×D₄	1	2	yes	—
103	C₂×C₄×C₄₁	—	2	1	no	—
104	C₂×C₄×C₄₁	—	2	4	no	—
105	C₂×C₄×C₄₁	—	2	2	no	—
106	C₂×C₄×C₄₁	—	2	1	no	—
107	C₂×C₄×C₄₁	—	2	1	no	—
108	C₂×C₄×C₄₁	—	2	2	no	—
109	C₂×C₄×C₄₁	—	4	1	no	—
110	C₂×C₄×C₄₁	—	4	2	no	—
111	C₂×C₄×C₄₁	—	4	4	no	—
112	C₂×C₄×C₄₁	—	4	2	no	—
113	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₄₁	C₂×D₄	1	2	yes	—
117	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₄₁	#41	1	2	yes	—
120	C₂×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
121	C₈×C₄₁	—	2	2	no	—
122	C₈×C₄₁	—	2	2	no	—
123	C₂²×C₄₁	—	2	1	no	—
124	C₂²×C₄₁	—	2	1	no	—
125	C₂²×C₄₁	—	2	2	no	—
126	C₂²×C₄₁	—	4	1	no	—
127	C₂²×C₄₁	—	4	1	no	—
128	C₂²×C₄₁	—	4	2	no	—
129	C₂²×C₄₁	—	4	2	no	—
130	C₂²×C₄₁	—	4	2	no	—
131	C₂²×C₄₁	#121	1	2	yes	—
132	C₂²×C₄₁	C₂²×D₄	1	1	yes	characteristic
133	C₄×C₄₁	—	2	2	no	—
134	C₄×C₄₁	—	2	1	no	—
135	C₄×C₄₁	—	2	1	no	—
136	C₄×C₄₁	—	2	1	no	—
137	C₄×C₄₁	—	4	2	no	—
138	C₄×C₄₁	—	4	1	no	—
139	C₄×C₄₁	—	4	1	no	—
140	C₄×C₄₁	#121	1	2	yes	—
141	C₄×C₄₁	#135	1	2	yes	complemented
142	C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
143	C₄×C₄₁	C₂×#41	1	1	yes	characteristic
144	#3035	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂×C₄₁	—	2	2	no	—
146	C₂×C₄₁	—	4	2	no	—
147	C₂×C₄₁	—	8	2	no	—
148	C₂×C₄₁	C₂×#121	1	1	yes	characteristic
149	C₂×C₄₁	C₂×#135	1	1	yes	characteristic
150	C₂×C₄₁	#575	1	1	yes	characteristic
151	C₂×#135	C₂×C₄₁	1	2	yes	complemented
152	#532	C₂×C₄₁	1	1	yes	characteristic, complemented
153	C₄×#41	C₂×C₄₁	1	1	yes	characteristic, complemented
154	#464	C₂×C₄₁	1	2	yes	complemented
155	#463	C₂×C₄₁	1	2	yes	complemented
156	#462	C₂×C₄₁	1	1	yes	characteristic, complemented
157	#461	C₂×C₄₁	1	1	yes	characteristic, complemented
158	#460	C₂×C₄₁	1	1	yes	characteristic, complemented
159	#459	C₂×C₄₁	1	1	yes	characteristic, complemented
160	C₂×C₄⋊C₈	C₂×C₄₁	1	1	yes	characteristic, complemented
161	#416	C₂×C₄₁	1	2	yes	complemented
162	C₄₁	#3035	1	1	yes	characteristic, direct product factor, Sylow
163	C₂×#41	C₂²×C₄₁	1	2	yes	—
164	C₂×#41	C₂²×C₄₁	1	1	yes	characteristic
165	#135	—	2	8	no	—
166	C₂×Q₁₆	C₂²×C₄₁	1	2	yes	—
167	C₂×SD₁₆	C₂²×C₄₁	1	2	yes	—
168	C₂×SD₁₆	C₂²×C₄₁	1	2	yes	—
169	C₂×D₈	C₂²×C₄₁	1	2	yes	—
170	C₂²×C₈	C₂²×C₄₁	1	2	yes	—
171	C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
172	#127	C₂²×C₄₁	1	1	yes	characteristic
173	#124	C₂²×C₄₁	1	2	yes	—
174	#121	—	2	1	no	—
175	#121	—	2	1	no	—
176	C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
177	C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
178	C₄×D₄	—	2	1	no	—
179	C₄×D₄	—	2	1	no	—
180	C₄×D₄	—	2	1	no	—
181	C₄×D₄	—	2	1	no	—
182	C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
183	C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
184	#117	—	2	1	no	—
185	#117	C₂²×C₄₁	1	1	yes	characteristic
186	C₂×C₄²	—	2	1	no	—
187	C₂×C₄²	C₂²×C₄₁	1	1	yes	characteristic
188	C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
189	C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
190	C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
191	#103	C₂²×C₄₁	1	2	yes	—
192	#103	C₂²×C₄₁	1	2	yes	—
193	#102	C₂²×C₄₁	1	2	yes	—
194	#102	C₂²×C₄₁	1	2	yes	—
195	#41	—	2	2	no	—
196	#41	—	2	4	no	—
197	#41	—	2	4	no	—
198	#41	D₄×C₄₁	1	4	yes	complemented
199	C₂×Q₈	C₂³×C₄₁	1	2	yes	—
200	C₂×Q₈	C₂³×C₄₁	1	1	yes	characteristic
201	C₂×D₄	—	2	1	no	—
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	—	2	1	no	—
204	C₂×D₄	—	2	1	no	—
205	C₂×D₄	C₂³×C₄₁	1	2	yes	—
206	C₂×D₄	C₂³×C₄₁	1	1	yes	characteristic
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	D₄×C₄₁	1	1	yes	characteristic
212	C₂²×C₄	D₄×C₄₁	1	1	yes	characteristic
213	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
214	Q₁₆	—	2	4	no	—
215	SD₁₆	—	2	4	no	—
216	SD₁₆	—	2	4	no	—
217	D₈	—	2	4	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	C₂³×C₄₁	1	2	yes	—
220	C₂×C₈	C₂³×C₄₁	1	2	yes	—
221	C₄⋊C₄	—	2	1	no	—
222	C₄⋊C₄	—	2	1	no	—
223	C₄⋊C₄	—	2	1	no	—
224	C₄⋊C₄	—	2	1	no	—
225	C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
226	C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
227	C₂²⋊C₄	—	2	1	no	—
228	C₂²⋊C₄	—	2	1	no	—
229	C₂²⋊C₄	—	2	1	no	—
230	C₂²⋊C₄	—	2	2	no	—
231	C₂²⋊C₄	—	2	2	no	—
232	C₄²	—	2	1	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	2	1	no	—
235	C₄²	D₄×C₄₁	1	1	yes	characteristic
236	C₄²	D₄×C₄₁	1	1	yes	characteristic
237	C₄²	C₂³×C₄₁	1	1	yes	characteristic
238	C₄²	C₂³×C₄₁	1	1	yes	characteristic
239	C₂³	—	2	1	no	—
240	C₂³	—	2	2	no	—
241	C₂³	C₂×D₄×C₄₁	1	1	yes	characteristic
242	Q₈	—	2	1	no	—
243	Q₈	—	2	2	no	—
244	Q₈	—	2	2	no	—
245	Q₈	C₂×D₄×C₄₁	1	2	yes	—
246	D₄	—	2	4	no	—
247	D₄	—	2	2	no	—
248	D₄	—	2	1	no	—
249	D₄	—	2	2	no	—
250	D₄	—	4	2	no	—
251	D₄	—	4	4	no	—
252	D₄	—	4	2	no	—
253	D₄	C₂×D₄×C₄₁	1	2	yes	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
265	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
266	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
267	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
268	C₂×C₄	C₂×D₄×C₄₁	1	2	yes	—
269	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
270	C₂×C₄	#41×C₄₁	1	2	yes	—
271	C₂×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
272	C₈	—	2	2	no	—
273	C₈	—	2	2	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	4	1	no	—
278	C₂²	—	4	1	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	#121×C₄₁	1	2	yes	—
283	C₂²	C₂²×D₄×C₄₁	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	4	2	no	—
289	C₄	—	4	1	no	—
290	C₄	—	4	1	no	—
291	C₄	#121×C₄₁	1	2	yes	—
292	C₄	#135×C₄₁	1	2	yes	complemented
293	C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
294	C₄	C₂×#41×C₄₁	1	1	yes	characteristic
295	C₂	—	2	2	no	—
296	C₂	—	4	2	no	—
297	C₂	—	8	2	no	—
298	C₂	C₂×#121×C₄₁	1	1	yes	characteristic
299	C₂	C₂×#135×C₄₁	1	1	yes	characteristic
300	C₂	C₄₁×#575	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₄₁	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₄₁
socle	C₂²×C₄₁	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₁ ◁ C₂×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#135×C₄₁	C₂	semidirect
2. C₄₁×#532	C₂	semidirect
3. C₄×#41×C₄₁	C₂	semidirect
4. C₄₁×#464	C₂	semidirect
5. C₄₁×#463	C₂	semidirect
6. C₄₁×#462	C₂	semidirect
7. C₄₁×#461	C₂	semidirect
8. C₄₁×#460	C₂	semidirect
9. C₄₁×#459	C₂	semidirect
10. C₂×C₄⋊C₈×C₄₁	C₂	semidirect
11. C₄₁×#416	C₂	semidirect
47. #41×C₄₁	D₄	semidirect
141. C₄×C₄₁	#135	semidirect
144. #3035	C₄₁	direct (N × H)
151. C₂×#135	C₂×C₄₁	semidirect
152. #532	C₂×C₄₁	semidirect
153. C₄×#41	C₂×C₄₁	semidirect
154. #464	C₂×C₄₁	semidirect
155. #463	C₂×C₄₁	semidirect
156. #462	C₂×C₄₁	semidirect
157. #461	C₂×C₄₁	semidirect
158. #460	C₂×C₄₁	semidirect
159. #459	C₂×C₄₁	semidirect
160. C₂×C₄⋊C₈	C₂×C₄₁	semidirect
161. #416	C₂×C₄₁	semidirect
162. C₄₁	#3035	direct (N × H)
198. #41	D₄×C₄₁	semidirect
292. C₄	#135×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#135×C₄₁	2
C₄₁×#532	2
C₄×#41×C₄₁	2
C₄₁×#464	2
C₄₁×#463	2
C₄₁×#462	2
C₄₁×#461	2
C₄₁×#460	2
C₄₁×#459	2
C₂×C₄⋊C₈×C₄₁	2
C₄₁×#416	2
#3035	41

Subgroups — C41×#3035 #19106815

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#3035 #19106815