Subgroups — C₄₁×#3332 #19107112

Order 5248.

430 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#550	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₁×#547	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#486	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#481	C₂	1	2	yes	maximal, complemented
5	C₄₁×#475	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#469	C₂	1	2	yes	maximal, complemented
7	C₄₁×#459	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#449	C₂	1	2	yes	maximal, complemented
9	C₄×SD₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₁×#436	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₁×#417	C₂	1	2	yes	maximal, complemented
12	C₂×#41×C₄₁	—	2	1	no	—
13	C₂×#41×C₄₁	C₂²	1	2	yes	—
14	C₂×SD₁₆×C₄₁	C₂²	1	1	yes	characteristic
15	C₂×D₈×C₄₁	C₂²	1	2	yes	—
16	C₂×M₁₆×C₄₁	C₂²	1	2	yes	—
17	C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
18	#127×C₄₁	—	2	1	no	—
19	#127×C₄₁	C₂²	1	1	yes	characteristic
20	#123×C₄₁	—	2	2	no	—
21	C₂²⋊Q₈×C₄₁	—	2	1	no	—
22	#121×C₄₁	—	2	2	no	—
23	#121×C₄₁	C₂²	1	2	yes	—
24	#121×C₄₁	C₂²	1	2	yes	—
25	C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₄₁	—	2	1	no	—
27	C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
28	C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
30	#106×C₄₁	C₂²	1	1	yes	characteristic
31	#106×C₄₁	C₂²	1	2	yes	—
32	C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
33	#103×C₄₁	C₂²	1	1	yes	characteristic
34	#103×C₄₁	C₂²	1	2	yes	—
35	#102×C₄₁	C₂²	1	2	yes	—
36	#102×C₄₁	C₂²	1	2	yes	—
37	#102×C₄₁	C₂²	1	1	yes	characteristic
38	#102×C₄₁	C₂²	1	2	yes	—
39	C₂²⋊C₈×C₄₁	C₂²	1	2	yes	—
40	C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
41	#41×C₄₁	—	2	4	no	—
42	#41×C₄₁	—	4	2	no	—
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	2	no	—
47	C₂×D₄×C₄₁	—	2	2	no	—
48	C₂×D₄×C₄₁	—	2	2	no	—
49	C₂×D₄×C₄₁	—	2	1	no	—
50	C₂×D₄×C₄₁	—	2	1	no	—
51	C₂×D₄×C₄₁	D₄	1	1	yes	characteristic
52	C₂×D₄×C₄₁	C₂³	1	2	yes	—
53	C₂×D₄×C₄₁	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₄₁	—	2	2	no	—
55	C₂²×C₄×C₄₁	—	2	2	no	—
56	C₂²×C₄×C₄₁	—	2	1	no	—
57	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
58	SD₁₆×C₄₁	—	2	2	no	—
59	D₈×C₄₁	—	4	2	no	—
60	M₁₆×C₄₁	—	4	2	no	—
61	C₂×C₈×C₄₁	C₂³	1	1	yes	characteristic
62	C₂×C₈×C₄₁	C₂³	1	1	yes	characteristic
63	C₂×C₈×C₄₁	C₂³	1	2	yes	—
64	C₄⋊C₄×C₄₁	—	2	2	no	—
65	C₄⋊C₄×C₄₁	—	2	1	no	—
66	C₄⋊C₄×C₄₁	—	2	1	no	—
67	C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
68	C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
70	C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
71	C₂²⋊C₄×C₄₁	—	2	2	no	—
72	C₂²⋊C₄×C₄₁	—	2	1	no	—
73	C₂²⋊C₄×C₄₁	—	2	2	no	—
74	C₂²⋊C₄×C₄₁	D₄	1	2	yes	—
75	C₄²×C₄₁	—	2	1	no	—
76	C₄²×C₄₁	C₂³	1	1	yes	characteristic
77	C₂³×C₄₁	—	2	1	no	—
78	C₂³×C₄₁	—	2	2	no	—
79	C₂³×C₄₁	C₂×D₄	1	2	yes	—
80	Q₈×C₄₁	—	2	1	no	—
81	Q₈×C₄₁	—	4	2	no	—
82	Q₈×C₄₁	#41	1	2	yes	complemented
83	D₄×C₄₁	—	2	2	no	—
84	D₄×C₄₁	—	2	1	no	—
85	D₄×C₄₁	—	2	2	no	—
86	D₄×C₄₁	—	2	1	no	—
87	D₄×C₄₁	—	2	2	no	—
88	D₄×C₄₁	—	4	4	no	—
89	D₄×C₄₁	—	4	4	no	—
90	D₄×C₄₁	—	4	2	no	—
91	D₄×C₄₁	—	8	1	no	—
92	D₄×C₄₁	—	8	2	no	—
93	C₂×C₄×C₄₁	—	2	1	no	—
94	C₂×C₄×C₄₁	—	2	2	no	—
95	C₂×C₄×C₄₁	—	2	2	no	—
96	C₂×C₄×C₄₁	—	2	1	no	—
97	C₂×C₄×C₄₁	—	2	1	no	—
98	C₂×C₄×C₄₁	—	2	2	no	—
99	C₂×C₄×C₄₁	—	4	4	no	—
100	C₂×C₄×C₄₁	—	4	4	no	—
101	C₂×C₄×C₄₁	—	4	2	no	—
102	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₄₁	C₂×D₄	1	2	yes	—
104	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
106	C₈×C₄₁	—	2	1	no	—
107	C₈×C₄₁	—	2	1	no	—
108	C₈×C₄₁	—	2	2	no	—
109	C₂²×C₄₁	—	2	2	no	—
110	C₂²×C₄₁	—	2	2	no	—
111	C₂²×C₄₁	—	2	2	no	—
112	C₂²×C₄₁	—	4	2	no	—
113	C₂²×C₄₁	—	4	1	no	—
114	C₂²×C₄₁	—	4	1	no	—
115	C₂²×C₄₁	—	4	2	no	—
116	C₂²×C₄₁	—	4	2	no	—
117	C₂²×C₄₁	—	4	1	no	—
118	C₂²×C₄₁	C₂²×D₄	1	1	yes	characteristic
119	C₄×C₄₁	—	2	2	no	—
120	C₄×C₄₁	—	2	2	no	—
121	C₄×C₄₁	—	2	1	no	—
122	C₄×C₄₁	—	4	1	no	—
123	C₄×C₄₁	—	4	2	no	—
124	C₄×C₄₁	—	4	1	no	—
125	C₄×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
126	C₄×C₄₁	C₂×#41	1	1	yes	characteristic
127	C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
128	#3332	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂×C₄₁	—	4	2	no	—
130	C₂×C₄₁	—	8	2	no	—
131	C₂×C₄₁	—	8	1	no	—
132	C₂×C₄₁	#546	1	1	yes	characteristic
133	C₂×C₄₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₂×C₄₁	#577	1	1	yes	characteristic
135	#550	C₂×C₄₁	1	1	yes	characteristic, complemented
136	#547	C₂×C₄₁	1	1	yes	characteristic, complemented
137	#486	C₂×C₄₁	1	1	yes	characteristic, complemented
138	#481	C₂×C₄₁	1	2	yes	complemented
139	#475	C₂×C₄₁	1	1	yes	characteristic, complemented
140	#469	C₂×C₄₁	1	2	yes	complemented
141	#459	C₂×C₄₁	1	1	yes	characteristic, complemented
142	#449	C₂×C₄₁	1	2	yes	complemented
143	C₄×SD₁₆	C₂×C₄₁	1	1	yes	characteristic, complemented
144	#436	C₂×C₄₁	1	1	yes	characteristic, complemented
145	#417	C₂×C₄₁	1	2	yes	complemented
146	C₄₁	#3332	1	1	yes	characteristic, direct product factor, Sylow
147	C₂×#41	—	2	1	no	—
148	C₂×#41	C₂²×C₄₁	1	2	yes	—
149	C₂×SD₁₆	C₂²×C₄₁	1	1	yes	characteristic
150	C₂×D₈	C₂²×C₄₁	1	2	yes	—
151	C₂×M₁₆	C₂²×C₄₁	1	2	yes	—
152	C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
153	#127	—	2	1	no	—
154	#127	C₂²×C₄₁	1	1	yes	characteristic
155	#123	—	2	2	no	—
156	C₂²⋊Q₈	—	2	1	no	—
157	#121	—	2	2	no	—
158	#121	C₂²×C₄₁	1	2	yes	—
159	#121	C₂²×C₄₁	1	2	yes	—
160	C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
161	C₄×D₄	—	2	1	no	—
162	C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
163	C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
164	C₂×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
165	#106	C₂²×C₄₁	1	1	yes	characteristic
166	#106	C₂²×C₄₁	1	2	yes	—
167	C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
168	#103	C₂²×C₄₁	1	1	yes	characteristic
169	#103	C₂²×C₄₁	1	2	yes	—
170	#102	C₂²×C₄₁	1	2	yes	—
171	#102	C₂²×C₄₁	1	2	yes	—
172	#102	C₂²×C₄₁	1	1	yes	characteristic
173	#102	C₂²×C₄₁	1	2	yes	—
174	C₂²⋊C₈	C₂²×C₄₁	1	2	yes	—
175	C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
176	#41	—	2	4	no	—
177	#41	—	4	2	no	—
178	#41	—	4	4	no	—
179	C₂×Q₈	—	2	1	no	—
180	C₂×Q₈	C₂³×C₄₁	1	1	yes	characteristic
181	C₂×D₄	—	2	2	no	—
182	C₂×D₄	—	2	2	no	—
183	C₂×D₄	—	2	2	no	—
184	C₂×D₄	—	2	1	no	—
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	D₄×C₄₁	1	1	yes	characteristic
187	C₂×D₄	C₂³×C₄₁	1	2	yes	—
188	C₂×D₄	C₂³×C₄₁	1	1	yes	characteristic
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
193	SD₁₆	—	2	2	no	—
194	D₈	—	4	2	no	—
195	M₁₆	—	4	2	no	—
196	C₂×C₈	C₂³×C₄₁	1	1	yes	characteristic
197	C₂×C₈	C₂³×C₄₁	1	1	yes	characteristic
198	C₂×C₈	C₂³×C₄₁	1	2	yes	—
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	2	1	no	—
201	C₄⋊C₄	—	2	1	no	—
202	C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
203	C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
204	C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
205	C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
206	C₂²⋊C₄	—	2	2	no	—
207	C₂²⋊C₄	—	2	1	no	—
208	C₂²⋊C₄	—	2	2	no	—
209	C₂²⋊C₄	D₄×C₄₁	1	2	yes	—
210	C₄²	—	2	1	no	—
211	C₄²	C₂³×C₄₁	1	1	yes	characteristic
212	C₂³	—	2	1	no	—
213	C₂³	—	2	2	no	—
214	C₂³	C₂×D₄×C₄₁	1	2	yes	—
215	Q₈	—	2	1	no	—
216	Q₈	—	4	2	no	—
217	Q₈	#41×C₄₁	1	2	yes	complemented
218	D₄	—	2	2	no	—
219	D₄	—	2	1	no	—
220	D₄	—	2	2	no	—
221	D₄	—	2	1	no	—
222	D₄	—	2	2	no	—
223	D₄	—	4	4	no	—
224	D₄	—	4	4	no	—
225	D₄	—	4	2	no	—
226	D₄	—	8	1	no	—
227	D₄	—	8	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
238	C₂×C₄	C₂×D₄×C₄₁	1	2	yes	—
239	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
240	C₂×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
241	C₈	—	2	1	no	—
242	C₈	—	2	1	no	—
243	C₈	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	4	2	no	—
248	C₂²	—	4	1	no	—
249	C₂²	—	4	1	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	2	no	—
252	C₂²	—	4	1	no	—
253	C₂²	C₂²×D₄×C₄₁	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	1	no	—
257	C₄	—	4	1	no	—
258	C₄	—	4	2	no	—
259	C₄	—	4	1	no	—
260	C₄	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
261	C₄	C₂×#41×C₄₁	1	1	yes	characteristic
262	C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
263	C₂	—	4	2	no	—
264	C₂	—	8	2	no	—
265	C₂	—	8	1	no	—
266	C₂	C₄₁×#546	1	1	yes	characteristic
267	C₂	C₂×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
268	C₂	C₄₁×#577	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 ◁ C₄×D₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₁×#550	C₂	semidirect
2. C₄₁×#547	C₂	semidirect
3. C₄₁×#486	C₂	semidirect
4. C₄₁×#481	C₂	semidirect
5. C₄₁×#475	C₂	semidirect
6. C₄₁×#469	C₂	semidirect
7. C₄₁×#459	C₂	semidirect
8. C₄₁×#449	C₂	semidirect
9. C₄×SD₁₆×C₄₁	C₂	semidirect
10. C₄₁×#436	C₂	semidirect
11. C₄₁×#417	C₂	semidirect
82. Q₈×C₄₁	#41	semidirect
128. #3332	C₄₁	direct (N × H)
135. #550	C₂×C₄₁	semidirect
136. #547	C₂×C₄₁	semidirect
137. #486	C₂×C₄₁	semidirect
138. #481	C₂×C₄₁	semidirect
139. #475	C₂×C₄₁	semidirect
140. #469	C₂×C₄₁	semidirect
141. #459	C₂×C₄₁	semidirect
142. #449	C₂×C₄₁	semidirect
143. C₄×SD₁₆	C₂×C₄₁	semidirect
144. #436	C₂×C₄₁	semidirect
145. #417	C₂×C₄₁	semidirect
146. C₄₁	#3332	direct (N × H)
217. Q₈	#41×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₄₁×#550	2
C₄₁×#547	2
C₄₁×#486	2
C₄₁×#481	2
C₄₁×#475	2
C₄₁×#469	2
C₄₁×#459	2
C₄₁×#449	2
C₄×SD₁₆×C₄₁	2
C₄₁×#436	2
C₄₁×#417	2
#3332	41

Subgroups — C41×#3332 #19107112

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#3332 #19107112