Subgroups — C₄₁×#33829 #23810030

Order 10496.

488 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×#3175	C₂	1	2	yes	maximal, complemented
2	C₄₁×#2226	C₂	1	2	yes	maximal, complemented
3	C₄₁×#2225	C₂	1	2	yes	maximal, complemented
4	C₄₁×#2219	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#2218	C₂	1	1	yes	maximal, characteristic
6	C₄₁×#2217	C₂	1	2	yes	maximal, complemented
7	C₄₁×#2211	C₂	1	2	yes	maximal, complemented
8	C₄₁×#2210	C₂	1	2	yes	maximal, complemented
9	C₄₁×#2151	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₁×#542	—	2	2	no	—
11	C₂×Q₃₂×C₄₁	C₂²	1	1	yes	characteristic
12	C₂×SD₃₂×C₄₁	C₂²	1	2	yes	—
13	C₂×D₁₆×C₄₁	C₂²	1	1	yes	characteristic
14	C₂×M₃₂×C₄₁	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₄₁	C₂²	1	1	yes	characteristic
16	C₄₁×#500	C₂²	1	2	yes	—
17	C₄₁×#495	C₂²	1	2	yes	—
18	C₄₁×#489	—	2	2	no	—
19	C₄₁×#468	—	2	2	no	—
20	C₄₁×#467	C₂²	1	2	yes	—
21	C₄₁×#466	C₂²	1	2	yes	—
22	C₄₁×#440	—	2	2	no	—
23	C₄×Q₁₆×C₄₁	C₂²	1	2	yes	—
24	C₄×D₈×C₄₁	C₂²	1	2	yes	—
25	C₄₁×#405	C₂²	1	1	yes	characteristic
26	C₄₁×#367	C₂²	1	2	yes	—
27	C₄₁×#367	C₂²	1	1	yes	characteristic
28	C₄₁×#367	C₂²	1	1	yes	characteristic
29	C₄₁×C₄⋊C₁₆	C₂²	1	2	yes	—
30	C₄₁×C₄⋊C₁₆	C₂²	1	2	yes	—
31	C₄₁×#358	C₂²	1	2	yes	—
32	C₄₁×#358	C₂²	1	2	yes	—
33	C₄₁×#357	C₂²	1	2	yes	—
34	C₄₁×#357	C₂²	1	2	yes	—
35	C₂×Q₁₆×C₄₁	C₂³	1	2	yes	—
36	C₂×SD₁₆×C₄₁	—	2	2	no	—
37	C₂×D₈×C₄₁	C₂³	1	2	yes	—
38	C₂×M₁₆×C₄₁	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
40	C₄⋊Q₈×C₄₁	—	2	2	no	—
41	#126×C₄₁	—	4	2	no	—
42	#124×C₄₁	—	2	2	no	—
43	#124×C₄₁	—	4	2	no	—
44	#123×C₄₁	—	4	2	no	—
45	C₂²⋊Q₈×C₄₁	—	2	2	no	—
46	C₂²⋊Q₈×C₄₁	—	4	2	no	—
47	#121×C₄₁	—	2	2	no	—
48	C₄×Q₈×C₄₁	—	2	2	no	—
49	C₄×D₄×C₄₁	—	2	2	no	—
50	#117×C₄₁	D₄	1	1	yes	characteristic
51	Q₃₂×C₄₁	—	4	1	no	—
52	SD₃₂×C₄₁	—	4	2	no	—
53	D₁₆×C₄₁	—	4	1	no	—
54	M₃₂×C₄₁	—	4	1	no	—
55	C₂×C₁₆×C₄₁	—	4	1	no	—
56	C₂×C₁₆×C₄₁	C₂³	1	2	yes	—
57	C₂×C₁₆×C₄₁	C₂³	1	1	yes	characteristic
58	C₂×C₁₆×C₄₁	C₂³	1	1	yes	characteristic
59	#107×C₄₁	—	2	2	no	—
60	#107×C₄₁	C₂³	1	2	yes	—
61	#107×C₄₁	C₂³	1	2	yes	—
62	#103×C₄₁	—	2	2	no	—
63	#103×C₄₁	—	2	2	no	—
64	#102×C₄₁	—	2	2	no	—
65	#102×C₄₁	—	2	2	no	—
66	#97×C₄₁	D₄	1	2	yes	—
67	C₄×C₈×C₄₁	C₂³	1	2	yes	—
68	C₂×Q₈×C₄₁	—	2	2	no	—
69	C₂×Q₈×C₄₁	—	4	2	no	—
70	C₂×D₄×C₄₁	—	2	2	no	—
71	C₂×D₄×C₄₁	—	4	2	no	—
72	C₂²×C₄×C₄₁	—	4	2	no	—
73	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
74	Q₁₆×C₄₁	—	2	2	no	—
75	Q₁₆×C₄₁	—	2	2	no	—
76	SD₁₆×C₄₁	—	4	4	no	—
77	D₈×C₄₁	—	2	2	no	—
78	D₈×C₄₁	—	2	2	no	—
79	M₁₆×C₄₁	—	4	1	no	—
80	C₂×C₈×C₄₁	—	2	1	no	—
81	C₂×C₈×C₄₁	—	2	1	no	—
82	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₄₁	C₂⁴	1	1	yes	characteristic
85	C₄⋊C₄×C₄₁	—	2	2	no	—
86	C₄⋊C₄×C₄₁	—	2	2	no	—
87	C₄⋊C₄×C₄₁	—	4	2	no	—
88	C₄⋊C₄×C₄₁	—	4	2	no	—
89	C₄⋊C₄×C₄₁	—	4	2	no	—
90	C₄⋊C₄×C₄₁	D₈	1	2	yes	—
91	C₂²⋊C₄×C₄₁	—	4	2	no	—
92	C₂²⋊C₄×C₄₁	—	4	2	no	—
93	C₂²⋊C₄×C₄₁	—	4	2	no	—
94	C₂²⋊C₄×C₄₁	—	4	2	no	—
95	C₂²⋊C₄×C₄₁	—	4	2	no	—
96	C₂²⋊C₄×C₄₁	D₈	1	1	yes	characteristic
97	C₂²⋊C₄×C₄₁	D₈	1	1	yes	characteristic
98	C₄²×C₄₁	—	4	2	no	—
99	C₄²×C₄₁	C₂×D₄	1	2	yes	—
100	C₁₆×C₄₁	—	2	2	no	—
101	C₁₆×C₄₁	—	2	1	no	—
102	C₁₆×C₄₁	—	2	1	no	—
103	C₂³×C₄₁	—	4	2	no	—
104	C₂³×C₄₁	C₂×D₈	1	1	yes	characteristic
105	Q₈×C₄₁	—	4	2	no	—
106	Q₈×C₄₁	—	4	2	no	—
107	Q₈×C₄₁	—	16	2	no	—
108	D₄×C₄₁	—	4	2	no	—
109	D₄×C₄₁	—	4	2	no	—
110	D₄×C₄₁	—	16	2	no	—
111	C₂×C₄×C₄₁	—	2	1	no	—
112	C₂×C₄×C₄₁	—	2	1	no	—
113	C₂×C₄×C₄₁	—	4	2	no	—
114	C₂×C₄×C₄₁	—	4	2	no	—
115	C₂×C₄×C₄₁	—	4	2	no	—
116	C₂×C₄×C₄₁	—	16	2	no	—
117	C₂×C₄×C₄₁	C₂×D₈	1	2	yes	—
118	C₂×C₄×C₄₁	C₂×D₈	1	2	yes	—
119	C₂×C₄×C₄₁	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
121	C₈×C₄₁	—	2	1	no	—
122	C₈×C₄₁	—	2	2	no	—
123	C₈×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
124	C₈×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
125	#33829	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂²×C₄₁	—	2	2	no	—
127	C₂²×C₄₁	—	2	1	no	—
128	C₂²×C₄₁	—	8	2	no	—
129	C₂²×C₄₁	—	8	2	no	—
130	C₂²×C₄₁	—	8	2	no	—
131	C₂²×C₄₁	C₂²×D₈	1	1	yes	characteristic
132	C₄×C₄₁	—	2	2	no	—
133	C₄×C₄₁	—	2	2	no	—
134	C₄×C₄₁	—	2	1	no	—
135	C₄×C₄₁	—	8	2	no	—
136	C₄×C₄₁	—	8	2	no	—
137	C₄×C₄₁	—	8	2	no	—
138	C₄×C₄₁	#537	1	1	yes	characteristic
139	C₄×C₄₁	#575	1	1	yes	characteristic
140	#3175	C₂×C₄₁	1	2	yes	complemented
141	#2226	C₂×C₄₁	1	2	yes	complemented
142	#2225	C₂×C₄₁	1	2	yes	complemented
143	#2219	C₂×C₄₁	1	1	yes	characteristic, complemented
144	#2218	C₂×C₄₁	1	1	yes	characteristic
145	#2217	C₂×C₄₁	1	2	yes	complemented
146	#2211	C₂×C₄₁	1	2	yes	complemented
147	#2210	C₂×C₄₁	1	2	yes	complemented
148	#2151	C₂×C₄₁	1	1	yes	characteristic, complemented
149	C₂×C₄₁	—	4	1	no	—
150	C₂×C₄₁	—	16	2	no	—
151	C₂×C₄₁	#3245	1	1	yes	characteristic
152	C₂×C₄₁	#3416	1	2	yes	—
153	#542	—	2	2	no	—
154	C₂×Q₃₂	C₂²×C₄₁	1	1	yes	characteristic
155	C₂×SD₃₂	C₂²×C₄₁	1	2	yes	—
156	C₂×D₁₆	C₂²×C₄₁	1	1	yes	characteristic
157	C₂×M₃₂	C₂²×C₄₁	1	1	yes	characteristic
158	C₂²×C₁₆	C₂²×C₄₁	1	1	yes	characteristic
159	#500	C₂²×C₄₁	1	2	yes	—
160	#495	C₂²×C₄₁	1	2	yes	—
161	#489	—	2	2	no	—
162	#468	—	2	2	no	—
163	#467	C₂²×C₄₁	1	2	yes	—
164	#466	C₂²×C₄₁	1	2	yes	—
165	#440	—	2	2	no	—
166	C₄×Q₁₆	C₂²×C₄₁	1	2	yes	—
167	C₄×D₈	C₂²×C₄₁	1	2	yes	—
168	#405	C₂²×C₄₁	1	1	yes	characteristic
169	#367	C₂²×C₄₁	1	2	yes	—
170	#367	C₂²×C₄₁	1	1	yes	characteristic
171	#367	C₂²×C₄₁	1	1	yes	characteristic
172	C₄⋊C₁₆	C₂²×C₄₁	1	2	yes	—
173	C₄⋊C₁₆	C₂²×C₄₁	1	2	yes	—
174	#358	C₂²×C₄₁	1	2	yes	—
175	#358	C₂²×C₄₁	1	2	yes	—
176	#357	C₂²×C₄₁	1	2	yes	—
177	#357	C₂²×C₄₁	1	2	yes	—
178	C₄₁	#33829	1	1	yes	characteristic, direct product factor, Sylow
179	C₂×Q₁₆	C₂³×C₄₁	1	2	yes	—
180	C₂×SD₁₆	—	2	2	no	—
181	C₂×D₈	C₂³×C₄₁	1	2	yes	—
182	C₂×M₁₆	D₄×C₄₁	1	1	yes	characteristic
183	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
184	C₄⋊Q₈	—	2	2	no	—
185	#126	—	4	2	no	—
186	#124	—	2	2	no	—
187	#124	—	4	2	no	—
188	#123	—	4	2	no	—
189	C₂²⋊Q₈	—	2	2	no	—
190	C₂²⋊Q₈	—	4	2	no	—
191	#121	—	2	2	no	—
192	C₄×Q₈	—	2	2	no	—
193	C₄×D₄	—	2	2	no	—
194	#117	D₄×C₄₁	1	1	yes	characteristic
195	Q₃₂	—	4	1	no	—
196	SD₃₂	—	4	2	no	—
197	D₁₆	—	4	1	no	—
198	M₃₂	—	4	1	no	—
199	C₂×C₁₆	—	4	1	no	—
200	C₂×C₁₆	C₂³×C₄₁	1	1	yes	characteristic
201	C₂×C₁₆	C₂³×C₄₁	1	2	yes	—
202	C₂×C₁₆	C₂³×C₄₁	1	1	yes	characteristic
203	#107	—	2	2	no	—
204	#107	C₂³×C₄₁	1	2	yes	—
205	#107	C₂³×C₄₁	1	2	yes	—
206	#103	—	2	2	no	—
207	#103	—	2	2	no	—
208	#102	—	2	2	no	—
209	#102	—	2	2	no	—
210	#97	D₄×C₄₁	1	2	yes	—
211	C₄×C₈	C₂³×C₄₁	1	2	yes	—
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	—	4	2	no	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	—	4	2	no	—
216	C₂²×C₄	—	4	2	no	—
217	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
218	Q₁₆	—	2	2	no	—
219	Q₁₆	—	2	2	no	—
220	SD₁₆	—	4	4	no	—
221	D₈	—	2	2	no	—
222	D₈	—	2	2	no	—
223	M₁₆	—	4	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
227	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
228	C₂×C₈	C₂⁴×C₄₁	1	1	yes	characteristic
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	4	2	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄⋊C₄	D₈×C₄₁	1	2	yes	—
235	C₂²⋊C₄	—	4	2	no	—
236	C₂²⋊C₄	—	4	2	no	—
237	C₂²⋊C₄	—	4	2	no	—
238	C₂²⋊C₄	—	4	2	no	—
239	C₂²⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	D₈×C₄₁	1	1	yes	characteristic
241	C₂²⋊C₄	D₈×C₄₁	1	1	yes	characteristic
242	C₄²	—	4	2	no	—
243	C₄²	C₂×D₄×C₄₁	1	2	yes	—
244	C₁₆	—	2	2	no	—
245	C₁₆	—	2	1	no	—
246	C₁₆	—	2	1	no	—
247	C₂³	—	4	2	no	—
248	C₂³	C₂×D₈×C₄₁	1	1	yes	characteristic
249	Q₈	—	4	2	no	—
250	Q₈	—	4	2	no	—
251	Q₈	—	16	2	no	—
252	D₄	—	4	2	no	—
253	D₄	—	4	2	no	—
254	D₄	—	16	2	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	16	2	no	—
261	C₂×C₄	C₂×D₈×C₄₁	1	2	yes	—
262	C₂×C₄	C₂×D₈×C₄₁	1	2	yes	—
263	C₂×C₄	C₂×D₈×C₄₁	1	1	yes	characteristic
264	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
265	C₈	—	2	1	no	—
266	C₈	—	2	2	no	—
267	C₈	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
268	C₈	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
269	C₂²	—	2	2	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	8	2	no	—
272	C₂²	—	8	2	no	—
273	C₂²	—	8	2	no	—
274	C₂²	C₂²×D₈×C₄₁	1	1	yes	characteristic
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	2	1	no	—
278	C₄	—	8	2	no	—
279	C₄	—	8	2	no	—
280	C₄	—	8	2	no	—
281	C₄	C₄₁×#537	1	1	yes	characteristic
282	C₄	C₄₁×#575	1	1	yes	characteristic
283	C₂	—	4	1	no	—
284	C₂	—	16	2	no	—
285	C₂	C₄₁×#3245	1	1	yes	characteristic
286	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×#2218	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×#3175	C₂	semidirect
2. C₄₁×#2226	C₂	semidirect
3. C₄₁×#2225	C₂	semidirect
4. C₄₁×#2219	C₂	semidirect
6. C₄₁×#2217	C₂	semidirect
7. C₄₁×#2211	C₂	semidirect
8. C₄₁×#2210	C₂	semidirect
9. C₄₁×#2151	C₂	semidirect
125. #33829	C₄₁	direct (N × H)
140. #3175	C₂×C₄₁	semidirect
141. #2226	C₂×C₄₁	semidirect
142. #2225	C₂×C₄₁	semidirect
143. #2219	C₂×C₄₁	semidirect
145. #2217	C₂×C₄₁	semidirect
146. #2211	C₂×C₄₁	semidirect
147. #2210	C₂×C₄₁	semidirect
148. #2151	C₂×C₄₁	semidirect
178. C₄₁	#33829	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×#3175	2
C₄₁×#2226	2
C₄₁×#2225	2
C₄₁×#2219	2
C₄₁×#2218	2
C₄₁×#2217	2
C₄₁×#2211	2
C₄₁×#2210	2
C₄₁×#2151	2
#33829	41

Subgroups — C41×#33829 #23810030

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#33829 #23810030