Subgroups — C₄₁×#3247 #19107027

Order 5248.

398 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#532	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₁×#529	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#478	C₂	1	2	yes	maximal, complemented
4	C₄₁×#477	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#476	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#466	C₂	1	2	yes	maximal, complemented
7	C₄₁×#465	C₂	1	2	yes	maximal, complemented
8	C₄₁×#463	C₂	1	2	yes	maximal, complemented
9	C₄₁×#460	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₁×#459	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄⋊C₈×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₄₁	—	2	1	no	—
13	C₂×SD₁₆×C₄₁	C₂²	1	2	yes	—
14	C₂×D₈×C₄₁	C₂²	1	2	yes	—
15	C₂²×C₈×C₄₁	C₂²	1	2	yes	—
16	C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
17	#127×C₄₁	C₂²	1	1	yes	characteristic
18	#126×C₄₁	—	2	1	no	—
19	#125×C₄₁	C₂²	1	1	yes	characteristic
20	#124×C₄₁	—	2	1	no	—
21	#124×C₄₁	C₂²	1	1	yes	characteristic
22	#123×C₄₁	—	2	1	no	—
23	C₂²⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
24	#121×C₄₁	—	2	1	no	—
25	#121×C₄₁	C₂²	1	1	yes	characteristic
26	#121×C₄₁	C₂²	1	2	yes	—
27	C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
28	C₄×D₄×C₄₁	—	2	1	no	—
29	C₄×D₄×C₄₁	—	2	1	no	—
30	C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
31	C₄×D₄×C₄₁	C₂²	1	2	yes	—
32	#117×C₄₁	—	2	1	no	—
33	C₂×C₄²×C₄₁	C₂²	1	1	yes	characteristic
34	#107×C₄₁	C₂²	1	2	yes	—
35	#106×C₄₁	C₂²	1	2	yes	—
36	C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
39	#103×C₄₁	C₂²	1	2	yes	—
40	#102×C₄₁	C₂²	1	2	yes	—
41	#102×C₄₁	C₂²	1	2	yes	—
42	#102×C₄₁	C₂²	1	2	yes	—
43	#41×C₄₁	—	4	4	no	—
44	C₂×Q₈×C₄₁	—	2	1	no	—
45	C₂×Q₈×C₄₁	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₄₁	—	2	1	no	—
47	C₂×D₄×C₄₁	—	2	2	no	—
48	C₂×D₄×C₄₁	—	2	1	no	—
49	C₂×D₄×C₄₁	—	2	1	no	—
50	C₂×D₄×C₄₁	C₂³	1	1	yes	characteristic
51	C₂×D₄×C₄₁	C₂³	1	2	yes	—
52	C₂²×C₄×C₄₁	—	2	1	no	—
53	C₂²×C₄×C₄₁	—	2	2	no	—
54	C₂²×C₄×C₄₁	D₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₁	D₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
57	SD₁₆×C₄₁	—	4	2	no	—
58	D₈×C₄₁	—	4	2	no	—
59	C₂×C₈×C₄₁	—	4	2	no	—
60	C₂×C₈×C₄₁	C₂³	1	2	yes	—
61	C₂×C₈×C₄₁	C₂³	1	2	yes	—
62	C₄⋊C₄×C₄₁	—	2	1	no	—
63	C₄⋊C₄×C₄₁	—	2	1	no	—
64	C₄⋊C₄×C₄₁	—	2	1	no	—
65	C₄⋊C₄×C₄₁	—	2	1	no	—
66	C₄⋊C₄×C₄₁	—	2	1	no	—
67	C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
70	C₂²⋊C₄×C₄₁	—	2	1	no	—
71	C₂²⋊C₄×C₄₁	—	2	1	no	—
72	C₂²⋊C₄×C₄₁	—	2	1	no	—
73	C₂²⋊C₄×C₄₁	—	2	2	no	—
74	C₂²⋊C₄×C₄₁	—	2	1	no	—
75	C₂²⋊C₄×C₄₁	—	2	2	no	—
76	C₂²⋊C₄×C₄₁	—	2	1	no	—
77	C₄²×C₄₁	—	2	1	no	—
78	C₄²×C₄₁	D₄	1	1	yes	characteristic
79	C₄²×C₄₁	D₄	1	1	yes	characteristic
80	C₄²×C₄₁	C₂³	1	1	yes	characteristic
81	C₄²×C₄₁	C₂³	1	1	yes	characteristic
82	C₂³×C₄₁	—	2	1	no	—
83	C₂³×C₄₁	—	2	2	no	—
84	C₂³×C₄₁	C₂×D₄	1	1	yes	characteristic
85	Q₈×C₄₁	—	2	1	no	—
86	Q₈×C₄₁	—	2	1	no	—
87	Q₈×C₄₁	—	4	2	no	—
88	D₄×C₄₁	—	2	1	no	—
89	D₄×C₄₁	—	2	2	no	—
90	D₄×C₄₁	—	2	1	no	—
91	D₄×C₄₁	—	2	2	no	—
92	D₄×C₄₁	—	4	2	no	—
93	D₄×C₄₁	—	4	2	no	—
94	D₄×C₄₁	—	4	4	no	—
95	D₄×C₄₁	—	4	2	no	—
96	C₂×C₄×C₄₁	—	2	2	no	—
97	C₂×C₄×C₄₁	—	2	2	no	—
98	C₂×C₄×C₄₁	—	2	1	no	—
99	C₂×C₄×C₄₁	—	2	1	no	—
100	C₂×C₄×C₄₁	—	2	1	no	—
101	C₂×C₄×C₄₁	—	2	1	no	—
102	C₂×C₄×C₄₁	—	2	1	no	—
103	C₂×C₄×C₄₁	—	2	2	no	—
104	C₂×C₄×C₄₁	—	4	4	no	—
105	C₂×C₄×C₄₁	—	4	2	no	—
106	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
112	C₈×C₄₁	—	2	2	no	—
113	C₈×C₄₁	—	2	2	no	—
114	C₂²×C₄₁	—	2	1	no	—
115	C₂²×C₄₁	—	2	1	no	—
116	C₂²×C₄₁	—	4	1	no	—
117	C₂²×C₄₁	—	4	2	no	—
118	C₂²×C₄₁	—	4	1	no	—
119	C₂²×C₄₁	—	4	2	no	—
120	C₂²×C₄₁	—	4	2	no	—
121	C₂²×C₄₁	—	4	1	no	—
122	C₂²×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
123	C₂²×C₄₁	C₂²×D₄	1	1	yes	characteristic
124	C₄×C₄₁	—	2	1	no	—
125	C₄×C₄₁	—	2	1	no	—
126	C₄×C₄₁	—	2	1	no	—
127	C₄×C₄₁	—	2	1	no	—
128	C₄×C₄₁	—	4	1	no	—
129	C₄×C₄₁	—	4	1	no	—
130	C₄×C₄₁	—	4	2	no	—
131	C₄×C₄₁	—	4	1	no	—
132	C₄×C₄₁	#135	1	2	yes	—
133	C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
134	C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
135	#3247	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂×C₄₁	—	2	2	no	—
137	C₂×C₄₁	—	8	1	no	—
138	C₂×C₄₁	—	8	2	no	—
139	C₂×C₄₁	#537	1	1	yes	characteristic
140	C₂×C₄₁	C₂×#135	1	1	yes	characteristic
141	C₂×C₄₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
142	#532	C₂×C₄₁	1	1	yes	characteristic, complemented
143	#529	C₂×C₄₁	1	1	yes	characteristic, complemented
144	#478	C₂×C₄₁	1	2	yes	complemented
145	#477	C₂×C₄₁	1	1	yes	characteristic, complemented
146	#476	C₂×C₄₁	1	1	yes	characteristic, complemented
147	#466	C₂×C₄₁	1	2	yes	complemented
148	#465	C₂×C₄₁	1	2	yes	complemented
149	#463	C₂×C₄₁	1	2	yes	complemented
150	#460	C₂×C₄₁	1	1	yes	characteristic, complemented
151	#459	C₂×C₄₁	1	1	yes	characteristic, complemented
152	C₂×C₄⋊C₈	C₂×C₄₁	1	1	yes	characteristic, complemented
153	C₄₁	#3247	1	1	yes	characteristic, direct product factor, Sylow
154	C₂×#41	—	2	1	no	—
155	C₂×SD₁₆	C₂²×C₄₁	1	2	yes	—
156	C₂×D₈	C₂²×C₄₁	1	2	yes	—
157	C₂²×C₈	C₂²×C₄₁	1	2	yes	—
158	C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
159	#127	C₂²×C₄₁	1	1	yes	characteristic
160	#126	—	2	1	no	—
161	#125	C₂²×C₄₁	1	1	yes	characteristic
162	#124	—	2	1	no	—
163	#124	C₂²×C₄₁	1	1	yes	characteristic
164	#123	—	2	1	no	—
165	C₂²⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
166	#121	—	2	1	no	—
167	#121	C₂²×C₄₁	1	1	yes	characteristic
168	#121	C₂²×C₄₁	1	2	yes	—
169	C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
170	C₄×D₄	—	2	1	no	—
171	C₄×D₄	—	2	1	no	—
172	C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
173	C₄×D₄	C₂²×C₄₁	1	2	yes	—
174	#117	—	2	1	no	—
175	C₂×C₄²	C₂²×C₄₁	1	1	yes	characteristic
176	#107	C₂²×C₄₁	1	2	yes	—
177	#106	C₂²×C₄₁	1	2	yes	—
178	C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
179	C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
180	C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
181	#103	C₂²×C₄₁	1	2	yes	—
182	#102	C₂²×C₄₁	1	2	yes	—
183	#102	C₂²×C₄₁	1	2	yes	—
184	#102	C₂²×C₄₁	1	2	yes	—
185	#41	—	4	4	no	—
186	C₂×Q₈	—	2	1	no	—
187	C₂×Q₈	C₂³×C₄₁	1	1	yes	characteristic
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	—	2	2	no	—
190	C₂×D₄	—	2	1	no	—
191	C₂×D₄	—	2	1	no	—
192	C₂×D₄	C₂³×C₄₁	1	1	yes	characteristic
193	C₂×D₄	C₂³×C₄₁	1	2	yes	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	D₄×C₄₁	1	1	yes	characteristic
197	C₂²×C₄	D₄×C₄₁	1	1	yes	characteristic
198	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
199	SD₁₆	—	4	2	no	—
200	D₈	—	4	2	no	—
201	C₂×C₈	—	4	2	no	—
202	C₂×C₈	C₂³×C₄₁	1	2	yes	—
203	C₂×C₈	C₂³×C₄₁	1	2	yes	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	—	2	1	no	—
207	C₄⋊C₄	—	2	1	no	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
210	C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
211	C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
212	C₂²⋊C₄	—	2	1	no	—
213	C₂²⋊C₄	—	2	1	no	—
214	C₂²⋊C₄	—	2	1	no	—
215	C₂²⋊C₄	—	2	2	no	—
216	C₂²⋊C₄	—	2	1	no	—
217	C₂²⋊C₄	—	2	2	no	—
218	C₂²⋊C₄	—	2	1	no	—
219	C₄²	—	2	1	no	—
220	C₄²	D₄×C₄₁	1	1	yes	characteristic
221	C₄²	D₄×C₄₁	1	1	yes	characteristic
222	C₄²	C₂³×C₄₁	1	1	yes	characteristic
223	C₄²	C₂³×C₄₁	1	1	yes	characteristic
224	C₂³	—	2	1	no	—
225	C₂³	—	2	2	no	—
226	C₂³	C₂×D₄×C₄₁	1	1	yes	characteristic
227	Q₈	—	2	1	no	—
228	Q₈	—	2	1	no	—
229	Q₈	—	4	2	no	—
230	D₄	—	2	1	no	—
231	D₄	—	2	2	no	—
232	D₄	—	2	1	no	—
233	D₄	—	2	2	no	—
234	D₄	—	4	2	no	—
235	D₄	—	4	4	no	—
236	D₄	—	4	2	no	—
237	D₄	—	4	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
249	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
250	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
251	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
252	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
253	C₂×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
254	C₈	—	2	2	no	—
255	C₈	—	2	2	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	2	1	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	4	2	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	2	no	—
262	C₂²	—	4	2	no	—
263	C₂²	—	4	1	no	—
264	C₂²	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
265	C₂²	C₂²×D₄×C₄₁	1	1	yes	characteristic
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	4	1	no	—
271	C₄	—	4	1	no	—
272	C₄	—	4	2	no	—
273	C₄	—	4	1	no	—
274	C₄	#135×C₄₁	1	2	yes	—
275	C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
276	C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
277	C₂	—	2	2	no	—
278	C₂	—	8	1	no	—
279	C₂	—	8	2	no	—
280	C₂	C₄₁×#537	1	1	yes	characteristic
281	C₂	C₂×#135×C₄₁	1	1	yes	characteristic
282	C₂	C₂×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₄₁×#532	C₂	semidirect
2. C₄₁×#529	C₂	semidirect
3. C₄₁×#478	C₂	semidirect
4. C₄₁×#477	C₂	semidirect
5. C₄₁×#476	C₂	semidirect
6. C₄₁×#466	C₂	semidirect
7. C₄₁×#465	C₂	semidirect
8. C₄₁×#463	C₂	semidirect
9. C₄₁×#460	C₂	semidirect
10. C₄₁×#459	C₂	semidirect
11. C₂×C₄⋊C₈×C₄₁	C₂	semidirect
135. #3247	C₄₁	direct (N × H)
142. #532	C₂×C₄₁	semidirect
143. #529	C₂×C₄₁	semidirect
144. #478	C₂×C₄₁	semidirect
145. #477	C₂×C₄₁	semidirect
146. #476	C₂×C₄₁	semidirect
147. #466	C₂×C₄₁	semidirect
148. #465	C₂×C₄₁	semidirect
149. #463	C₂×C₄₁	semidirect
150. #460	C₂×C₄₁	semidirect
151. #459	C₂×C₄₁	semidirect
152. C₂×C₄⋊C₈	C₂×C₄₁	semidirect
153. C₄₁	#3247	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×#532	2
C₄₁×#529	2
C₄₁×#478	2
C₄₁×#477	2
C₄₁×#476	2
C₄₁×#466	2
C₄₁×#465	2
C₄₁×#463	2
C₄₁×#460	2
C₄₁×#459	2
C₂×C₄⋊C₈×C₄₁	2
#3247	41

Subgroups — C41×#3247 #19107027

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#3247 #19107027