Subgroups — C₄₁×#12956 #23789157

Order 10496.

460 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#442	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#365	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄₁×#361	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#2140	C₂	1	1	yes	maximal, characteristic
5	C₄₁×#1946	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#1386	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#1375	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₄₁	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₄×C₄₁	—	2	1	no	—
10	C₂×M₃₂×C₄₁	C₂²	1	1	yes	characteristic
11	C₂×M₃₂×C₄₁	C₂²	1	1	yes	characteristic
12	C₄₁×#442	—	2	2	no	—
13	C₄₁×#442	—	2	2	no	—
14	C₂×#108×C₄₁	C₂²	1	1	yes	characteristic
15	C₄₁×#427	C₂²	1	1	yes	characteristic
16	C₂×#106×C₄₁	C₂²	1	1	yes	characteristic
17	C₄₁×#424	—	2	1	no	—
18	C₂×#102×C₄₁	—	2	1	no	—
19	C₂×#97×C₄₁	C₂²	1	1	yes	characteristic
20	C₄₁×#365	—	2	2	no	—
21	C₄₁×#361	—	2	2	no	—
22	C₄₁×#357	C₄	1	4	yes	complemented
23	C₄₁×#339	—	2	1	no	—
24	C₂²×D₄×C₄₁	—	2	1	no	—
25	C₂³×C₄×C₄₁	—	4	1	no	—
26	C₂×D₈×C₄₁	—	2	1	no	—
27	C₂×D₈×C₄₁	—	2	1	no	—
28	C₂×D₈×C₄₁	—	2	1	no	—
29	C₂×D₈×C₄₁	C₂×C₄	1	2	yes	—
30	C₂×D₈×C₄₁	D₄	1	2	yes	—
31	C₂×D₈×C₄₁	Q₈	1	2	yes	—
32	C₂×M₁₆×C₄₁	—	2	1	no	—
33	C₂²×C₈×C₄₁	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
35	C₄×D₄×C₄₁	—	2	4	no	—
36	C₄×D₄×C₄₁	—	4	2	no	—
37	#117×C₄₁	—	2	1	no	—
38	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
39	C₂×C₂²⋊C₄×C₄₁	—	4	1	no	—
40	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
41	M₃₂×C₄₁	—	2	1	no	—
42	M₃₂×C₄₁	—	2	1	no	—
43	M₃₂×C₄₁	—	2	1	no	—
44	M₃₂×C₄₁	—	2	1	no	—
45	C₂×C₁₆×C₄₁	—	2	1	no	—
46	C₂×C₁₆×C₄₁	C₂×C₄	1	2	yes	complemented
47	#108×C₄₁	—	2	1	no	—
48	#108×C₄₁	—	2	1	no	—
49	#107×C₄₁	C₂×C₄	1	2	yes	complemented
50	#106×C₄₁	—	2	1	no	—
51	#106×C₄₁	—	2	1	no	—
52	#106×C₄₁	D₄	1	2	yes	—
53	C₄⋊C₈×C₄₁	—	4	2	no	—
54	#102×C₄₁	—	2	4	no	—
55	#97×C₄₁	—	2	1	no	—
56	#97×C₄₁	—	2	1	no	—
57	C₂⁴×C₄₁	—	4	1	no	—
58	C₂×D₄×C₄₁	—	2	2	no	—
59	C₂×D₄×C₄₁	—	2	2	no	—
60	C₂×D₄×C₄₁	—	2	2	no	—
61	C₂×D₄×C₄₁	—	4	1	no	—
62	C₂×D₄×C₄₁	—	4	1	no	—
63	C₂×D₄×C₄₁	—	4	1	no	—
64	C₂²×C₄×C₄₁	—	2	1	no	—
65	C₂²×C₄×C₄₁	—	4	4	no	—
66	C₂²×C₄×C₄₁	—	4	1	no	—
67	C₂²×C₄×C₄₁	—	8	4	no	—
68	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
69	D₈×C₄₁	—	2	2	no	—
70	D₈×C₄₁	—	2	2	no	—
71	D₈×C₄₁	—	2	2	no	—
72	D₈×C₄₁	C₄⋊C₄	1	4	yes	—
73	M₁₆×C₄₁	—	4	1	no	—
74	M₁₆×C₄₁	—	4	1	no	—
75	C₂×C₈×C₄₁	—	2	1	no	—
76	C₂×C₈×C₄₁	—	2	1	no	—
77	C₂×C₈×C₄₁	—	2	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	—	2	2	no	—
86	C₄⋊C₄×C₄₁	—	2	2	no	—
87	C₄⋊C₄×C₄₁	—	4	1	no	—
88	C₂²⋊C₄×C₄₁	—	4	4	no	—
89	C₂²⋊C₄×C₄₁	—	4	1	no	—
90	C₄²×C₄₁	—	2	1	no	—
91	C₄²×C₄₁	—	4	1	no	—
92	C₄²×C₄₁	SD₁₆	1	2	yes	—
93	C₁₆×C₄₁	—	2	2	no	—
94	C₁₆×C₄₁	—	4	1	no	—
95	C₂³×C₄₁	—	4	2	no	—
96	C₂³×C₄₁	—	4	2	no	—
97	C₂³×C₄₁	—	4	2	no	—
98	C₂³×C₄₁	—	8	2	no	—
99	C₂³×C₄₁	—	8	2	no	—
100	C₂³×C₄₁	AGL(1,ℤ/8)	1	1	yes	characteristic
101	D₄×C₄₁	—	2	4	no	—
102	D₄×C₄₁	—	4	2	no	—
103	D₄×C₄₁	—	4	2	no	—
104	D₄×C₄₁	—	4	2	no	—
105	C₂×C₄×C₄₁	—	2	2	no	—
106	C₂×C₄×C₄₁	—	4	2	no	—
107	C₂×C₄×C₄₁	—	4	2	no	—
108	C₂×C₄×C₄₁	—	4	2	no	—
109	C₂×C₄×C₄₁	—	8	8	no	—
110	C₂×C₄×C₄₁	—	8	2	no	—
111	C₂×C₄×C₄₁	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
113	C₂×C₄×C₄₁	C₂²⋊Q₈	1	1	yes	characteristic
114	C₂×C₄×C₄₁	C₂×SD₁₆	1	1	yes	characteristic
115	C₂×C₄×C₄₁	#135	1	1	yes	characteristic
116	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	1	yes	characteristic
117	C₈×C₄₁	—	4	1	no	—
118	C₈×C₄₁	—	8	1	no	—
119	C₈×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₈×C₄₁	C₄×D₄	1	1	yes	characteristic
121	C₈×C₄₁	C₂²⋊Q₈	1	1	yes	characteristic
122	C₈×C₄₁	#123	1	1	yes	characteristic
123	#12956	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂²×C₄₁	—	2	2	no	—
125	C₂²×C₄₁	—	4	4	no	—
126	C₂²×C₄₁	—	8	2	no	—
127	C₂²×C₄₁	—	8	2	no	—
128	C₂²×C₄₁	—	8	4	no	—
129	C₂²×C₄₁	—	8	4	no	—
130	C₂²×C₄₁	#442	1	1	yes	characteristic
131	C₂²×C₄₁	#450	1	1	yes	characteristic
132	C₂²×C₄₁	#478	1	1	yes	characteristic
133	C₄×C₄₁	—	4	2	no	—
134	C₄×C₄₁	—	8	2	no	—
135	C₄×C₄₁	—	8	2	no	—
136	C₄×C₄₁	#385	1	1	yes	characteristic
137	C₄×C₄₁	C₄×SD₁₆	1	1	yes	characteristic
138	C₄×C₄₁	#449	1	1	yes	characteristic
139	C₄×C₄₁	#476	1	1	yes	characteristic
140	C₂×#442	C₂×C₄₁	1	1	yes	characteristic
141	C₂×#365	C₂×C₄₁	1	1	yes	characteristic, complemented
142	C₂×#361	C₂×C₄₁	1	1	yes	characteristic
143	#2140	C₂×C₄₁	1	1	yes	characteristic
144	#1946	C₂×C₄₁	1	1	yes	characteristic, complemented
145	#1386	C₂×C₄₁	1	1	yes	characteristic, complemented
146	#1375	C₂×C₄₁	1	1	yes	characteristic, complemented
147	C₂×C₄₁	—	2	1	no	—
148	C₂×C₄₁	—	2	1	no	—
149	C₂×C₄₁	—	8	4	no	—
150	C₂×C₄₁	#1864	1	1	yes	characteristic
151	C₂×C₄₁	AGL(1,ℤ/16)	1	2	yes	—
152	C₂²×D₈	C₂²×C₄₁	1	1	yes	characteristic
153	C₂×C₄×D₄	—	2	1	no	—
154	C₂×M₃₂	C₂²×C₄₁	1	1	yes	characteristic
155	C₂×M₃₂	C₂²×C₄₁	1	1	yes	characteristic
156	#442	—	2	2	no	—
157	#442	—	2	2	no	—
158	C₂×#108	C₂²×C₄₁	1	1	yes	characteristic
159	#427	C₂²×C₄₁	1	1	yes	characteristic
160	C₂×#106	C₂²×C₄₁	1	1	yes	characteristic
161	#424	—	2	1	no	—
162	C₂×#102	—	2	1	no	—
163	C₂×#97	C₂²×C₄₁	1	1	yes	characteristic
164	#365	—	2	2	no	—
165	#361	—	2	2	no	—
166	#357	C₄×C₄₁	1	4	yes	complemented
167	#339	—	2	1	no	—
168	C₄₁	#12956	1	1	yes	characteristic, direct product factor, Sylow
169	C₂²×D₄	—	2	1	no	—
170	C₂³×C₄	—	4	1	no	—
171	C₂×D₈	—	2	1	no	—
172	C₂×D₈	—	2	1	no	—
173	C₂×D₈	—	2	1	no	—
174	C₂×D₈	C₂×C₄×C₄₁	1	2	yes	—
175	C₂×D₈	D₄×C₄₁	1	2	yes	—
176	C₂×D₈	Q₈×C₄₁	1	2	yes	—
177	C₂×M₁₆	—	2	1	no	—
178	C₂²×C₈	D₄×C₄₁	1	1	yes	characteristic
179	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
180	C₄×D₄	—	2	4	no	—
181	C₄×D₄	—	4	2	no	—
182	#117	—	2	1	no	—
183	C₂×C₄⋊C₄	—	2	1	no	—
184	C₂×C₂²⋊C₄	—	4	1	no	—
185	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
186	M₃₂	—	2	1	no	—
187	M₃₂	—	2	1	no	—
188	M₃₂	—	2	1	no	—
189	M₃₂	—	2	1	no	—
190	C₂×C₁₆	—	2	1	no	—
191	C₂×C₁₆	C₂×C₄×C₄₁	1	2	yes	complemented
192	#108	—	2	1	no	—
193	#108	—	2	1	no	—
194	#107	C₂×C₄×C₄₁	1	2	yes	complemented
195	#106	—	2	1	no	—
196	#106	—	2	1	no	—
197	#106	D₄×C₄₁	1	2	yes	—
198	C₄⋊C₈	—	4	2	no	—
199	#102	—	2	4	no	—
200	#97	—	2	1	no	—
201	#97	—	2	1	no	—
202	C₂⁴	—	4	1	no	—
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	—	2	2	no	—
205	C₂×D₄	—	2	2	no	—
206	C₂×D₄	—	4	1	no	—
207	C₂×D₄	—	4	1	no	—
208	C₂×D₄	—	4	1	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	4	4	no	—
211	C₂²×C₄	—	4	1	no	—
212	C₂²×C₄	—	8	4	no	—
213	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
214	D₈	—	2	2	no	—
215	D₈	—	2	2	no	—
216	D₈	—	2	2	no	—
217	D₈	C₄⋊C₄×C₄₁	1	4	yes	—
218	M₁₆	—	4	1	no	—
219	M₁₆	—	4	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
225	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
226	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
227	C₂×C₈	C₂×Q₈×C₄₁	1	1	yes	characteristic
228	C₂×C₈	#41×C₄₁	1	1	yes	characteristic
229	C₂×C₈	#41×C₄₁	1	1	yes	characteristic
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	4	1	no	—
233	C₂²⋊C₄	—	4	4	no	—
234	C₂²⋊C₄	—	4	1	no	—
235	C₄²	—	2	1	no	—
236	C₄²	—	4	1	no	—
237	C₄²	SD₁₆×C₄₁	1	2	yes	—
238	C₁₆	—	2	2	no	—
239	C₁₆	—	4	1	no	—
240	C₂³	—	4	2	no	—
241	C₂³	—	4	2	no	—
242	C₂³	—	4	2	no	—
243	C₂³	—	8	2	no	—
244	C₂³	—	8	2	no	—
245	C₂³	AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
246	D₄	—	2	4	no	—
247	D₄	—	4	2	no	—
248	D₄	—	4	2	no	—
249	D₄	—	4	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	8	8	no	—
255	C₂×C₄	—	8	2	no	—
256	C₂×C₄	C₄×D₄×C₄₁	1	1	yes	characteristic
257	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
258	C₂×C₄	C₂²⋊Q₈×C₄₁	1	1	yes	characteristic
259	C₂×C₄	C₂×SD₁₆×C₄₁	1	1	yes	characteristic
260	C₂×C₄	#135×C₄₁	1	1	yes	characteristic
261	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
262	C₈	—	4	1	no	—
263	C₈	—	8	1	no	—
264	C₈	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
265	C₈	C₄×D₄×C₄₁	1	1	yes	characteristic
266	C₈	C₂²⋊Q₈×C₄₁	1	1	yes	characteristic
267	C₈	#123×C₄₁	1	1	yes	characteristic
268	C₂²	—	2	2	no	—
269	C₂²	—	4	4	no	—
270	C₂²	—	8	2	no	—
271	C₂²	—	8	2	no	—
272	C₂²	—	8	4	no	—
273	C₂²	—	8	4	no	—
274	C₂²	C₄₁×#442	1	1	yes	characteristic
275	C₂²	C₄₁×#450	1	1	yes	characteristic
276	C₂²	C₄₁×#478	1	1	yes	characteristic
277	C₄	—	4	2	no	—
278	C₄	—	8	2	no	—
279	C₄	—	8	2	no	—
280	C₄	C₄₁×#385	1	1	yes	characteristic
281	C₄	C₄×SD₁₆×C₄₁	1	1	yes	characteristic
282	C₄	C₄₁×#449	1	1	yes	characteristic
283	C₄	C₄₁×#476	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	—	8	4	no	—
287	C₂	C₄₁×#1864	1	1	yes	characteristic
288	C₂	C₄₁×AGL(1,ℤ/16)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₄₁	#442
commutator	C₂×C₈	C₂²×C₄×C₄₁
Frattini, cosocle	C₂²×C₈	C₂³×C₄₁
prime radical	C₂²×D₈×C₄₁	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₂²×C₄×C₄₁ ◁ C₂×#97×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₁×#365	C₂	semidirect
5. C₄₁×#1946	C₂	semidirect
6. C₄₁×#1386	C₂	semidirect
7. C₄₁×#1375	C₂	semidirect
22. C₄₁×#357	C₄	semidirect
46. C₂×C₁₆×C₄₁	C₂×C₄	semidirect
49. #107×C₄₁	C₂×C₄	semidirect
123. #12956	C₄₁	direct (N × H)
141. C₂×#365	C₂×C₄₁	semidirect
144. #1946	C₂×C₄₁	semidirect
145. #1386	C₂×C₄₁	semidirect
146. #1375	C₂×C₄₁	semidirect
166. #357	C₄×C₄₁	semidirect
168. C₄₁	#12956	direct (N × H)
191. C₂×C₁₆	C₂×C₄×C₄₁	semidirect
194. #107	C₂×C₄×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#442	2
C₂×C₄₁×#365	2
C₂×C₄₁×#361	2
C₄₁×#2140	2
C₄₁×#1946	2
C₄₁×#1386	2
C₄₁×#1375	2
#12956	41

Subgroups — C41×#12956 #23789157

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#12956 #23789157