Subgroups — C₄₁×#12833 #23789034

Order 10496.

358 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3407	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2155	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄₁×#365	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#1815	C₂	1	1	yes	maximal, characteristic
5	C₄₁×#1814	C₂	1	1	yes	maximal, characteristic
6	C₄₁×#1388	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#1386	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#131×C₄₁	C₂²	1	1	yes	characteristic
9	C₄₁×#504	—	2	2	no	—
10	C₄₁×#504	C₄	1	4	yes	complemented
11	C₂×M₃₂×C₄₁	—	2	2	no	—
12	C₂×M₃₂×C₄₁	C₂²	1	1	yes	characteristic
13	C₂×M₃₂×C₄₁	C₂²	1	1	yes	characteristic
14	C₂×#108×C₄₁	C₂²	1	1	yes	characteristic
15	C₂×#108×C₄₁	C₂²	1	1	yes	characteristic
16	C₄₁×#427	C₂²	1	1	yes	characteristic
17	C₂×#106×C₄₁	C₂²	1	1	yes	characteristic
18	C₄₁×#421	—	2	1	no	—
19	C₄₁×#417	—	2	1	no	—
20	C₄₁×#365	—	2	2	no	—
21	C₄₁×#365	—	2	2	no	—
22	C₄₁×#344	—	2	1	no	—
23	C₄₁×#343	—	2	1	no	—
24	C₂×#41×C₄₁	D₄	1	1	yes	characteristic
25	#131×C₄₁	—	2	1	no	—
26	#131×C₄₁	—	2	1	no	—
27	#131×C₄₁	C₂×C₄	1	2	yes	—
28	#131×C₄₁	Q₈	1	2	yes	—
29	C₂×M₁₆×C₄₁	—	2	1	no	—
30	C₂×M₁₆×C₄₁	—	2	1	no	—
31	C₂×M₁₆×C₄₁	—	2	1	no	—
32	C₂×M₁₆×C₄₁	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₄₁	—	2	1	no	—
34	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
35	#117×C₄₁	—	2	1	no	—
36	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
37	M₃₂×C₄₁	—	2	1	no	—
38	M₃₂×C₄₁	—	2	4	no	—
39	M₃₂×C₄₁	C₂×C₄	1	2	yes	—
40	M₃₂×C₄₁	D₄	1	2	yes	—
41	M₃₂×C₄₁	D₄	1	2	yes	—
42	C₂×C₁₆×C₄₁	—	2	1	no	—
43	C₂×C₁₆×C₄₁	—	2	4	no	—
44	C₂×C₁₆×C₄₁	C₂×C₄	1	2	yes	—
45	#108×C₄₁	—	2	1	no	—
46	#108×C₄₁	—	2	1	no	—
47	#108×C₄₁	—	2	1	no	—
48	#108×C₄₁	—	2	1	no	—
49	#107×C₄₁	—	2	1	no	—
50	#106×C₄₁	—	2	1	no	—
51	#106×C₄₁	—	2	1	no	—
52	#106×C₄₁	—	2	1	no	—
53	#104×C₄₁	—	4	2	no	—
54	#103×C₄₁	—	4	1	no	—
55	#102×C₄₁	—	4	1	no	—
56	#41×C₄₁	—	2	1	no	—
57	#41×C₄₁	—	2	1	no	—
58	#41×C₄₁	C₄⋊C₄	1	2	yes	—
59	#41×C₄₁	SD₁₆	1	2	yes	—
60	C₂×Q₈×C₄₁	SD₁₆	1	1	yes	characteristic
61	C₂×D₄×C₄₁	—	2	1	no	—
62	C₂×D₄×C₄₁	SD₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₄₁	—	2	1	no	—
64	C₂²×C₄×C₄₁	—	4	1	no	—
65	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
66	M₁₆×C₄₁	—	2	1	no	—
67	M₁₆×C₄₁	—	2	2	no	—
68	M₁₆×C₄₁	—	2	2	no	—
69	M₁₆×C₄₁	—	4	1	no	—
70	M₁₆×C₄₁	—	4	1	no	—
71	M₁₆×C₄₁	—	4	1	no	—
72	M₁₆×C₄₁	—	4	1	no	—
73	M₁₆×C₄₁	C₄⋊C₄	1	2	yes	—
74	C₂×C₈×C₄₁	—	2	1	no	—
75	C₂×C₈×C₄₁	—	2	2	no	—
76	C₂×C₈×C₄₁	—	2	2	no	—
77	C₂×C₈×C₄₁	—	4	1	no	—
78	C₂×C₈×C₄₁	—	4	1	no	—
79	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₄₁	C₂×Q₈	1	1	yes	characteristic
83	C₂×C₈×C₄₁	#41	1	1	yes	characteristic
84	C₂×C₈×C₄₁	#41	1	1	yes	characteristic
85	C₄⋊C₄×C₄₁	—	4	1	no	—
86	C₄⋊C₄×C₄₁	—	4	1	no	—
87	C₄⋊C₄×C₄₁	—	4	1	no	—
88	C₂²⋊C₄×C₄₁	—	4	1	no	—
89	C₄²×C₄₁	—	4	1	no	—
90	C₁₆×C₄₁	—	2	2	no	—
91	C₁₆×C₄₁	C₂²⋊C₄	1	4	yes	—
92	C₂³×C₄₁	—	2	1	no	—
93	C₂³×C₄₁	#135	1	1	yes	characteristic
94	Q₈×C₄₁	—	2	1	no	—
95	Q₈×C₄₁	#106	1	2	yes	—
96	D₄×C₄₁	—	2	2	no	—
97	D₄×C₄₁	—	2	2	no	—
98	D₄×C₄₁	—	2	1	no	—
99	D₄×C₄₁	#106	1	2	yes	—
100	C₂×C₄×C₄₁	—	2	1	no	—
101	C₂×C₄×C₄₁	—	2	2	no	—
102	C₂×C₄×C₄₁	—	2	2	no	—
103	C₂×C₄×C₄₁	—	8	1	no	—
104	C₂×C₄×C₄₁	—	8	1	no	—
105	C₂×C₄×C₄₁	—	16	1	no	—
106	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
108	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
109	C₂×C₄×C₄₁	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₄×C₄₁	C₂×SD₁₆	1	1	yes	characteristic
111	C₂×C₄×C₄₁	#135	1	1	yes	characteristic
112	C₈×C₄₁	—	2	2	no	—
113	C₈×C₄₁	—	8	1	no	—
114	C₈×C₄₁	—	8	1	no	—
115	C₈×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₈×C₄₁	#117	1	1	yes	characteristic
117	C₈×C₄₁	C₂²⋊Q₈	1	1	yes	characteristic
118	C₈×C₄₁	C₂²⋊Q₈	1	1	yes	characteristic
119	#12833	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂²×C₄₁	—	2	2	no	—
121	C₂²×C₄₁	—	4	1	no	—
122	C₂²×C₄₁	—	8	1	no	—
123	C₂²×C₄₁	#427	1	1	yes	characteristic
124	C₂²×C₄₁	#465	1	1	yes	characteristic
125	C₂²×C₄₁	#465	1	1	yes	characteristic
126	C₄×C₄₁	—	2	2	no	—
127	C₄×C₄₁	—	16	1	no	—
128	C₄×C₄₁	—	16	1	no	—
129	C₄×C₄₁	#380	1	1	yes	characteristic
130	C₄×C₄₁	C₂×#106	1	1	yes	characteristic
131	C₄×C₄₁	#465	1	1	yes	characteristic
132	C₄×C₄₁	#465	1	1	yes	characteristic
133	#3407	C₂×C₄₁	1	1	yes	characteristic
134	#2155	C₂×C₄₁	1	1	yes	characteristic, complemented
135	C₂×#365	C₂×C₄₁	1	1	yes	characteristic, complemented
136	#1815	C₂×C₄₁	1	1	yes	characteristic
137	#1814	C₂×C₄₁	1	1	yes	characteristic
138	#1388	C₂×C₄₁	1	1	yes	characteristic, complemented
139	#1386	C₂×C₄₁	1	1	yes	characteristic, complemented
140	C₂×C₄₁	—	2	1	no	—
141	C₂×C₄₁	—	2	1	no	—
142	C₂×C₄₁	—	2	1	no	—
143	C₂×C₄₁	—	4	2	no	—
144	C₂×C₄₁	#1807	1	1	yes	characteristic
145	C₂×#131	C₂²×C₄₁	1	1	yes	characteristic
146	#504	—	2	2	no	—
147	#504	C₄×C₄₁	1	4	yes	complemented
148	C₂×M₃₂	—	2	2	no	—
149	C₂×M₃₂	C₂²×C₄₁	1	1	yes	characteristic
150	C₂×M₃₂	C₂²×C₄₁	1	1	yes	characteristic
151	C₂×#108	C₂²×C₄₁	1	1	yes	characteristic
152	C₂×#108	C₂²×C₄₁	1	1	yes	characteristic
153	#427	C₂²×C₄₁	1	1	yes	characteristic
154	C₂×#106	C₂²×C₄₁	1	1	yes	characteristic
155	#421	—	2	1	no	—
156	#417	—	2	1	no	—
157	#365	—	2	2	no	—
158	#365	—	2	2	no	—
159	#344	—	2	1	no	—
160	#343	—	2	1	no	—
161	C₄₁	#12833	1	1	yes	characteristic, direct product factor, Sylow
162	C₂×#41	D₄×C₄₁	1	1	yes	characteristic
163	#131	—	2	1	no	—
164	#131	—	2	1	no	—
165	#131	C₂×C₄×C₄₁	1	2	yes	—
166	#131	Q₈×C₄₁	1	2	yes	—
167	C₂×M₁₆	—	2	1	no	—
168	C₂×M₁₆	—	2	1	no	—
169	C₂×M₁₆	—	2	1	no	—
170	C₂×M₁₆	D₄×C₄₁	1	1	yes	characteristic
171	C₂²×C₈	—	2	1	no	—
172	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
173	#117	—	2	1	no	—
174	C₂×C₄⋊C₄	—	2	1	no	—
175	M₃₂	—	2	1	no	—
176	M₃₂	—	2	4	no	—
177	M₃₂	C₂×C₄×C₄₁	1	2	yes	—
178	M₃₂	D₄×C₄₁	1	2	yes	—
179	M₃₂	D₄×C₄₁	1	2	yes	—
180	C₂×C₁₆	—	2	1	no	—
181	C₂×C₁₆	—	2	4	no	—
182	C₂×C₁₆	C₂×C₄×C₄₁	1	2	yes	—
183	#108	—	2	1	no	—
184	#108	—	2	1	no	—
185	#108	—	2	1	no	—
186	#108	—	2	1	no	—
187	#107	—	2	1	no	—
188	#106	—	2	1	no	—
189	#106	—	2	1	no	—
190	#106	—	2	1	no	—
191	#104	—	4	2	no	—
192	#103	—	4	1	no	—
193	#102	—	4	1	no	—
194	#41	—	2	1	no	—
195	#41	—	2	1	no	—
196	#41	C₄⋊C₄×C₄₁	1	2	yes	—
197	#41	SD₁₆×C₄₁	1	2	yes	—
198	C₂×Q₈	SD₁₆×C₄₁	1	1	yes	characteristic
199	C₂×D₄	—	2	1	no	—
200	C₂×D₄	SD₁₆×C₄₁	1	1	yes	characteristic
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	4	1	no	—
203	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
204	M₁₆	—	2	1	no	—
205	M₁₆	—	2	2	no	—
206	M₁₆	—	2	2	no	—
207	M₁₆	—	4	1	no	—
208	M₁₆	—	4	1	no	—
209	M₁₆	—	4	1	no	—
210	M₁₆	—	4	1	no	—
211	M₁₆	C₄⋊C₄×C₄₁	1	2	yes	—
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	4	1	no	—
216	C₂×C₈	—	4	1	no	—
217	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
218	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
219	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
220	C₂×C₈	C₂×Q₈×C₄₁	1	1	yes	characteristic
221	C₂×C₈	#41×C₄₁	1	1	yes	characteristic
222	C₂×C₈	#41×C₄₁	1	1	yes	characteristic
223	C₄⋊C₄	—	4	1	no	—
224	C₄⋊C₄	—	4	1	no	—
225	C₄⋊C₄	—	4	1	no	—
226	C₂²⋊C₄	—	4	1	no	—
227	C₄²	—	4	1	no	—
228	C₁₆	—	2	2	no	—
229	C₁₆	C₂²⋊C₄×C₄₁	1	4	yes	—
230	C₂³	—	2	1	no	—
231	C₂³	#135×C₄₁	1	1	yes	characteristic
232	Q₈	—	2	1	no	—
233	Q₈	#106×C₄₁	1	2	yes	—
234	D₄	—	2	2	no	—
235	D₄	—	2	2	no	—
236	D₄	—	2	1	no	—
237	D₄	#106×C₄₁	1	2	yes	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	8	1	no	—
242	C₂×C₄	—	8	1	no	—
243	C₂×C₄	—	16	1	no	—
244	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
245	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
246	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
247	C₂×C₄	C₂×SD₁₆×C₄₁	1	1	yes	characteristic
248	C₂×C₄	C₂×SD₁₆×C₄₁	1	1	yes	characteristic
249	C₂×C₄	#135×C₄₁	1	1	yes	characteristic
250	C₈	—	2	2	no	—
251	C₈	—	8	1	no	—
252	C₈	—	8	1	no	—
253	C₈	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
254	C₈	#117×C₄₁	1	1	yes	characteristic
255	C₈	C₂²⋊Q₈×C₄₁	1	1	yes	characteristic
256	C₈	C₂²⋊Q₈×C₄₁	1	1	yes	characteristic
257	C₂²	—	2	2	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	8	1	no	—
260	C₂²	C₄₁×#427	1	1	yes	characteristic
261	C₂²	C₄₁×#465	1	1	yes	characteristic
262	C₂²	C₄₁×#465	1	1	yes	characteristic
263	C₄	—	2	2	no	—
264	C₄	—	16	1	no	—
265	C₄	—	16	1	no	—
266	C₄	C₄₁×#380	1	1	yes	characteristic
267	C₄	C₂×#106×C₄₁	1	1	yes	characteristic
268	C₄	C₄₁×#465	1	1	yes	characteristic
269	C₄	C₄₁×#465	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	2	1	no	—
272	C₂	—	2	1	no	—
273	C₂	—	4	2	no	—
274	C₂	C₄₁×#1807	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₄₁	#1807
commutator	C₂×C₈	C₂²×C₄×C₄₁
Frattini, cosocle	C₂²×C₈	C₂³×C₄₁
prime radical	C₂×#41×C₄₁	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄₁ ◁ C₂²×C₄×C₄₁ ◁ C₂×#131×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₁×#2155	C₂	semidirect
3. C₂×C₄₁×#365	C₂	semidirect
6. C₄₁×#1388	C₂	semidirect
7. C₄₁×#1386	C₂	semidirect
10. C₄₁×#504	C₄	semidirect
119. #12833	C₄₁	direct (N × H)
134. #2155	C₂×C₄₁	semidirect
135. C₂×#365	C₂×C₄₁	semidirect
138. #1388	C₂×C₄₁	semidirect
139. #1386	C₂×C₄₁	semidirect
147. #504	C₄×C₄₁	semidirect
161. C₄₁	#12833	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×#3407	2
C₄₁×#2155	2
C₂×C₄₁×#365	2
C₄₁×#1815	2
C₄₁×#1814	2
C₄₁×#1388	2
C₄₁×#1386	2
#12833	41

Subgroups — C41×#12833 #23789034

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#12833 #23789034