Subgroups — C₄₁×#33826 #23810027

Order 10496.

488 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3177	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₁×#3176	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#2229	C₂	1	2	yes	maximal, complemented
4	C₄₁×#2228	C₂	1	2	yes	maximal, complemented
5	C₄₁×#2220	C₂	1	2	yes	maximal, complemented
6	C₄₁×#2217	C₂	1	2	yes	maximal, complemented
7	C₄₁×#2208	C₂	1	2	yes	maximal, complemented
8	C₄₁×#2207	C₂	1	2	yes	maximal, complemented
9	C₄₁×#2151	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₁×#541	—	2	1	no	—
11	C₄₁×#540	—	2	1	no	—
12	C₂×SD₃₂×C₄₁	C₂²	1	2	yes	—
13	C₂×SD₃₂×C₄₁	C₂²	1	2	yes	—
14	C₂×M₃₂×C₄₁	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₄₁	C₂²	1	1	yes	characteristic
16	C₄₁×#499	C₂²	1	1	yes	characteristic
17	C₄₁×#499	C₂²	1	1	yes	characteristic
18	C₄₁×#494	C₂²	1	1	yes	characteristic
19	C₄₁×#493	C₂²	1	1	yes	characteristic
20	C₄₁×#491	—	2	1	no	—
21	C₄₁×#490	—	2	1	no	—
22	C₄₁×#470	—	2	1	no	—
23	C₄₁×#469	—	2	1	no	—
24	C₄₁×#467	C₂²	1	2	yes	—
25	C₄₁×#466	C₂²	1	2	yes	—
26	C₄₁×#442	—	2	1	no	—
27	C₄₁×#441	—	2	1	no	—
28	C₄×Q₁₆×C₄₁	C₂²	1	2	yes	—
29	C₄×D₈×C₄₁	C₂²	1	2	yes	—
30	C₄₁×#405	C₂²	1	1	yes	characteristic
31	C₄₁×#367	C₂²	1	2	yes	—
32	C₄₁×#366	C₂²	1	2	yes	—
33	C₄₁×C₄⋊C₁₆	C₂²	1	2	yes	—
34	C₄₁×C₄⋊C₁₆	C₂²	1	2	yes	—
35	C₄₁×#358	C₂²	1	2	yes	—
36	C₄₁×#358	C₂²	1	2	yes	—
37	C₄₁×#357	C₂²	1	2	yes	—
38	C₄₁×#357	C₂²	1	2	yes	—
39	C₂×Q₁₆×C₄₁	—	2	1	no	—
40	C₂×Q₁₆×C₄₁	C₂³	1	2	yes	—
41	C₂×D₈×C₄₁	—	2	1	no	—
42	C₂×D₈×C₄₁	C₂³	1	2	yes	—
43	C₂×M₁₆×C₄₁	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
45	C₄⋊Q₈×C₄₁	—	2	1	no	—
46	#127×C₄₁	—	2	1	no	—
47	#126×C₄₁	—	4	2	no	—
48	#125×C₄₁	—	2	1	no	—
49	#125×C₄₁	—	2	1	no	—
50	#125×C₄₁	—	4	2	no	—
51	#123×C₄₁	—	4	2	no	—
52	C₂²⋊Q₈×C₄₁	—	2	2	no	—
53	#121×C₄₁	—	2	2	no	—
54	#121×C₄₁	—	4	2	no	—
55	C₄×Q₈×C₄₁	—	2	2	no	—
56	C₄×D₄×C₄₁	—	2	2	no	—
57	#117×C₄₁	D₄	1	1	yes	characteristic
58	SD₃₂×C₄₁	—	4	2	no	—
59	SD₃₂×C₄₁	—	4	2	no	—
60	M₃₂×C₄₁	—	4	1	no	—
61	C₂×C₁₆×C₄₁	—	4	1	no	—
62	C₂×C₁₆×C₄₁	C₂³	1	2	yes	—
63	C₂×C₁₆×C₄₁	C₂³	1	2	yes	—
64	#107×C₄₁	C₂³	1	2	yes	—
65	#107×C₄₁	C₂³	1	2	yes	—
66	#106×C₄₁	—	2	1	no	—
67	#106×C₄₁	—	2	1	no	—
68	#103×C₄₁	—	2	2	no	—
69	#103×C₄₁	—	2	2	no	—
70	#102×C₄₁	—	2	2	no	—
71	#102×C₄₁	—	2	2	no	—
72	#97×C₄₁	D₄	1	1	yes	characteristic
73	#97×C₄₁	D₄	1	1	yes	characteristic
74	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
76	C₂×Q₈×C₄₁	—	2	2	no	—
77	C₂×D₄×C₄₁	—	2	2	no	—
78	C₂×D₄×C₄₁	—	4	2	no	—
79	C₂×D₄×C₄₁	—	4	2	no	—
80	C₂²×C₄×C₄₁	—	4	2	no	—
81	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
82	Q₁₆×C₄₁	—	2	2	no	—
83	Q₁₆×C₄₁	—	2	2	no	—
84	Q₁₆×C₄₁	—	4	2	no	—
85	D₈×C₄₁	—	2	2	no	—
86	D₈×C₄₁	—	2	2	no	—
87	D₈×C₄₁	—	4	2	no	—
88	M₁₆×C₄₁	—	4	1	no	—
89	C₂×C₈×C₄₁	—	2	1	no	—
90	C₂×C₈×C₄₁	—	2	1	no	—
91	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₄₁	C₂⁴	1	1	yes	characteristic
95	C₄⋊C₄×C₄₁	—	2	2	no	—
96	C₄⋊C₄×C₄₁	—	2	2	no	—
97	C₄⋊C₄×C₄₁	—	4	2	no	—
98	C₄⋊C₄×C₄₁	—	4	2	no	—
99	C₄⋊C₄×C₄₁	—	4	2	no	—
100	C₄⋊C₄×C₄₁	—	4	2	no	—
101	C₄⋊C₄×C₄₁	—	4	2	no	—
102	C₄⋊C₄×C₄₁	D₈	1	2	yes	—
103	C₂²⋊C₄×C₄₁	—	4	2	no	—
104	C₂²⋊C₄×C₄₁	—	4	2	no	—
105	C₂²⋊C₄×C₄₁	—	4	2	no	—
106	C₂²⋊C₄×C₄₁	D₈	1	2	yes	—
107	C₄²×C₄₁	—	4	2	no	—
108	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
110	C₁₆×C₄₁	—	2	2	no	—
111	C₁₆×C₄₁	—	2	2	no	—
112	C₂³×C₄₁	—	4	2	no	—
113	C₂³×C₄₁	C₂×D₈	1	1	yes	characteristic
114	Q₈×C₄₁	—	4	2	no	—
115	Q₈×C₄₁	—	4	2	no	—
116	D₄×C₄₁	—	4	2	no	—
117	D₄×C₄₁	—	4	2	no	—
118	D₄×C₄₁	—	16	2	no	—
119	D₄×C₄₁	—	16	2	no	—
120	C₂×C₄×C₄₁	—	2	1	no	—
121	C₂×C₄×C₄₁	—	2	1	no	—
122	C₂×C₄×C₄₁	—	4	2	no	—
123	C₂×C₄×C₄₁	—	4	2	no	—
124	C₂×C₄×C₄₁	—	4	2	no	—
125	C₂×C₄×C₄₁	—	16	2	no	—
126	C₂×C₄×C₄₁	C₂×D₈	1	2	yes	—
127	C₂×C₄×C₄₁	C₂×D₈	1	2	yes	—
128	C₂×C₄×C₄₁	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
130	C₈×C₄₁	—	2	1	no	—
131	C₈×C₄₁	—	2	1	no	—
132	C₈×C₄₁	—	2	1	no	—
133	C₈×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
134	C₈×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
135	#33826	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂²×C₄₁	—	2	2	no	—
137	C₂²×C₄₁	—	2	1	no	—
138	C₂²×C₄₁	—	8	2	no	—
139	C₂²×C₄₁	—	8	2	no	—
140	C₂²×C₄₁	—	8	2	no	—
141	C₂²×C₄₁	C₂²×D₈	1	1	yes	characteristic
142	C₄×C₄₁	—	2	2	no	—
143	C₄×C₄₁	—	2	2	no	—
144	C₄×C₄₁	—	2	1	no	—
145	C₄×C₄₁	—	8	2	no	—
146	C₄×C₄₁	—	8	2	no	—
147	C₄×C₄₁	—	8	2	no	—
148	C₄×C₄₁	#537	1	1	yes	characteristic
149	C₄×C₄₁	#575	1	1	yes	characteristic
150	#3177	C₂×C₄₁	1	1	yes	characteristic, complemented
151	#3176	C₂×C₄₁	1	1	yes	characteristic
152	#2229	C₂×C₄₁	1	2	yes	complemented
153	#2228	C₂×C₄₁	1	2	yes	complemented
154	#2220	C₂×C₄₁	1	2	yes	complemented
155	#2217	C₂×C₄₁	1	2	yes	complemented
156	#2208	C₂×C₄₁	1	2	yes	complemented
157	#2207	C₂×C₄₁	1	2	yes	complemented
158	#2151	C₂×C₄₁	1	1	yes	characteristic, complemented
159	C₂×C₄₁	—	4	1	no	—
160	C₂×C₄₁	—	16	2	no	—
161	C₂×C₄₁	#3245	1	1	yes	characteristic
162	C₂×C₄₁	#3416	1	2	yes	—
163	#541	—	2	1	no	—
164	#540	—	2	1	no	—
165	C₂×SD₃₂	C₂²×C₄₁	1	2	yes	—
166	C₂×SD₃₂	C₂²×C₄₁	1	2	yes	—
167	C₂×M₃₂	C₂²×C₄₁	1	1	yes	characteristic
168	C₂²×C₁₆	C₂²×C₄₁	1	1	yes	characteristic
169	#499	C₂²×C₄₁	1	1	yes	characteristic
170	#499	C₂²×C₄₁	1	1	yes	characteristic
171	#494	C₂²×C₄₁	1	1	yes	characteristic
172	#493	C₂²×C₄₁	1	1	yes	characteristic
173	#491	—	2	1	no	—
174	#490	—	2	1	no	—
175	#470	—	2	1	no	—
176	#469	—	2	1	no	—
177	#467	C₂²×C₄₁	1	2	yes	—
178	#466	C₂²×C₄₁	1	2	yes	—
179	#442	—	2	1	no	—
180	#441	—	2	1	no	—
181	C₄×Q₁₆	C₂²×C₄₁	1	2	yes	—
182	C₄×D₈	C₂²×C₄₁	1	2	yes	—
183	#405	C₂²×C₄₁	1	1	yes	characteristic
184	#367	C₂²×C₄₁	1	2	yes	—
185	#366	C₂²×C₄₁	1	2	yes	—
186	C₄⋊C₁₆	C₂²×C₄₁	1	2	yes	—
187	C₄⋊C₁₆	C₂²×C₄₁	1	2	yes	—
188	#358	C₂²×C₄₁	1	2	yes	—
189	#358	C₂²×C₄₁	1	2	yes	—
190	#357	C₂²×C₄₁	1	2	yes	—
191	#357	C₂²×C₄₁	1	2	yes	—
192	C₄₁	#33826	1	1	yes	characteristic, direct product factor, Sylow
193	C₂×Q₁₆	—	2	1	no	—
194	C₂×Q₁₆	C₂³×C₄₁	1	2	yes	—
195	C₂×D₈	—	2	1	no	—
196	C₂×D₈	C₂³×C₄₁	1	2	yes	—
197	C₂×M₁₆	D₄×C₄₁	1	1	yes	characteristic
198	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
199	C₄⋊Q₈	—	2	1	no	—
200	#127	—	2	1	no	—
201	#126	—	4	2	no	—
202	#125	—	2	1	no	—
203	#125	—	2	1	no	—
204	#125	—	4	2	no	—
205	#123	—	4	2	no	—
206	C₂²⋊Q₈	—	2	2	no	—
207	#121	—	2	2	no	—
208	#121	—	4	2	no	—
209	C₄×Q₈	—	2	2	no	—
210	C₄×D₄	—	2	2	no	—
211	#117	D₄×C₄₁	1	1	yes	characteristic
212	SD₃₂	—	4	2	no	—
213	SD₃₂	—	4	2	no	—
214	M₃₂	—	4	1	no	—
215	C₂×C₁₆	—	4	1	no	—
216	C₂×C₁₆	C₂³×C₄₁	1	2	yes	—
217	C₂×C₁₆	C₂³×C₄₁	1	2	yes	—
218	#107	C₂³×C₄₁	1	2	yes	—
219	#107	C₂³×C₄₁	1	2	yes	—
220	#106	—	2	1	no	—
221	#106	—	2	1	no	—
222	#103	—	2	2	no	—
223	#103	—	2	2	no	—
224	#102	—	2	2	no	—
225	#102	—	2	2	no	—
226	#97	D₄×C₄₁	1	1	yes	characteristic
227	#97	D₄×C₄₁	1	1	yes	characteristic
228	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
229	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
230	C₂×Q₈	—	2	2	no	—
231	C₂×D₄	—	2	2	no	—
232	C₂×D₄	—	4	2	no	—
233	C₂×D₄	—	4	2	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
236	Q₁₆	—	2	2	no	—
237	Q₁₆	—	2	2	no	—
238	Q₁₆	—	4	2	no	—
239	D₈	—	2	2	no	—
240	D₈	—	2	2	no	—
241	D₈	—	4	2	no	—
242	M₁₆	—	4	1	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
246	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
247	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
248	C₂×C₈	C₂⁴×C₄₁	1	1	yes	characteristic
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	D₈×C₄₁	1	2	yes	—
257	C₂²⋊C₄	—	4	2	no	—
258	C₂²⋊C₄	—	4	2	no	—
259	C₂²⋊C₄	—	4	2	no	—
260	C₂²⋊C₄	D₈×C₄₁	1	2	yes	—
261	C₄²	—	4	2	no	—
262	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
263	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
264	C₁₆	—	2	2	no	—
265	C₁₆	—	2	2	no	—
266	C₂³	—	4	2	no	—
267	C₂³	C₂×D₈×C₄₁	1	1	yes	characteristic
268	Q₈	—	4	2	no	—
269	Q₈	—	4	2	no	—
270	D₄	—	4	2	no	—
271	D₄	—	4	2	no	—
272	D₄	—	16	2	no	—
273	D₄	—	16	2	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	16	2	no	—
280	C₂×C₄	C₂×D₈×C₄₁	1	2	yes	—
281	C₂×C₄	C₂×D₈×C₄₁	1	2	yes	—
282	C₂×C₄	C₂×D₈×C₄₁	1	1	yes	characteristic
283	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	2	1	no	—
287	C₈	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
288	C₈	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
289	C₂²	—	2	2	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	8	2	no	—
292	C₂²	—	8	2	no	—
293	C₂²	—	8	2	no	—
294	C₂²	C₂²×D₈×C₄₁	1	1	yes	characteristic
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	2	1	no	—
298	C₄	—	8	2	no	—
299	C₄	—	8	2	no	—
300	C₄	—	8	2	no	—
301	C₄	C₄₁×#537	1	1	yes	characteristic
302	C₄	C₄₁×#575	1	1	yes	characteristic
303	C₂	—	4	1	no	—
304	C₂	—	16	2	no	—
305	C₂	C₄₁×#3245	1	1	yes	characteristic
306	C₂	C₄₁×#3416	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₄₁
prime radical	C₄₁×#3176	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ #117×C₄₁ ◁ C₄₁×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₁×#3177	C₂	semidirect
3. C₄₁×#2229	C₂	semidirect
4. C₄₁×#2228	C₂	semidirect
5. C₄₁×#2220	C₂	semidirect
6. C₄₁×#2217	C₂	semidirect
7. C₄₁×#2208	C₂	semidirect
8. C₄₁×#2207	C₂	semidirect
9. C₄₁×#2151	C₂	semidirect
135. #33826	C₄₁	direct (N × H)
150. #3177	C₂×C₄₁	semidirect
152. #2229	C₂×C₄₁	semidirect
153. #2228	C₂×C₄₁	semidirect
154. #2220	C₂×C₄₁	semidirect
155. #2217	C₂×C₄₁	semidirect
156. #2208	C₂×C₄₁	semidirect
157. #2207	C₂×C₄₁	semidirect
158. #2151	C₂×C₄₁	semidirect
192. C₄₁	#33826	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×#3177	2
C₄₁×#3176	2
C₄₁×#2229	2
C₄₁×#2228	2
C₄₁×#2220	2
C₄₁×#2217	2
C₄₁×#2208	2
C₄₁×#2207	2
C₄₁×#2151	2
#33826	41

Subgroups — C41×#33826 #23810027

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#33826 #23810027