Subgroups — C₄₁×#3357 #19107137

Order 5248.

354 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#555	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#554	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#500	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#485	C₂	1	2	yes	maximal, complemented
5	C₄₁×#480	C₂	1	2	yes	maximal, complemented
6	C₄₁×#477	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#476	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#468	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×#428	C₂	1	2	yes	maximal, complemented
12	C₂×SD₁₆×C₄₁	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₄₁	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
16	#125×C₄₁	—	2	1	no	—
17	#124×C₄₁	—	2	2	no	—
18	C₂²⋊Q₈×C₄₁	—	2	2	no	—
19	C₂²⋊Q₈×C₄₁	C₂²	1	2	yes	—
20	#121×C₄₁	—	2	1	no	—
21	#121×C₄₁	C₂²	1	2	yes	—
22	C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₄₁	—	2	1	no	—
24	C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
25	C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
26	#117×C₄₁	—	2	1	no	—
27	#117×C₄₁	C₂²	1	2	yes	—
28	C₂×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
29	#107×C₄₁	C₂²	1	2	yes	—
30	#107×C₄₁	C₂²	1	2	yes	—
31	#107×C₄₁	C₂²	1	2	yes	—
32	#106×C₄₁	C₂²	1	1	yes	characteristic
33	#106×C₄₁	C₂²	1	2	yes	—
34	C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
35	#103×C₄₁	C₂²	1	1	yes	characteristic
36	#103×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	C₂×Q₈×C₄₁	—	2	1	no	—
42	C₂×Q₈×C₄₁	—	2	1	no	—
43	C₂×Q₈×C₄₁	C₂³	1	1	yes	characteristic
44	C₂×D₄×C₄₁	—	2	2	no	—
45	C₂×D₄×C₄₁	D₄	1	1	yes	characteristic
46	C₂×D₄×C₄₁	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₄₁	—	2	2	no	—
48	C₂²×C₄×C₄₁	—	2	1	no	—
49	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
50	SD₁₆×C₄₁	—	2	2	no	—
51	M₁₆×C₄₁	—	2	4	no	—
52	C₂×C₈×C₄₁	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₄₁	C₂³	1	2	yes	—
54	C₂×C₈×C₄₁	C₂³	1	1	yes	characteristic
55	C₄⋊C₄×C₄₁	—	2	2	no	—
56	C₄⋊C₄×C₄₁	—	2	1	no	—
57	C₄⋊C₄×C₄₁	—	2	1	no	—
58	C₄⋊C₄×C₄₁	—	2	1	no	—
59	C₄⋊C₄×C₄₁	—	2	2	no	—
60	C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
62	C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
63	C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
64	C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
65	C₂²⋊C₄×C₄₁	—	2	2	no	—
66	C₂²⋊C₄×C₄₁	—	2	2	no	—
67	C₂²⋊C₄×C₄₁	—	2	2	no	—
68	C₂²⋊C₄×C₄₁	—	2	1	no	—
69	C₂²⋊C₄×C₄₁	D₄	1	2	yes	—
70	C₄²×C₄₁	—	2	2	no	—
71	C₄²×C₄₁	—	2	1	no	—
72	C₄²×C₄₁	C₂³	1	1	yes	characteristic
73	C₂³×C₄₁	—	2	1	no	—
74	C₂³×C₄₁	C₂×D₄	1	2	yes	—
75	Q₈×C₄₁	—	2	1	no	—
76	Q₈×C₄₁	—	2	1	no	—
77	Q₈×C₄₁	—	8	1	no	—
78	Q₈×C₄₁	—	8	1	no	—
79	D₄×C₄₁	—	2	2	no	—
80	D₄×C₄₁	—	2	1	no	—
81	D₄×C₄₁	—	2	1	no	—
82	D₄×C₄₁	—	8	2	no	—
83	C₂×C₄×C₄₁	—	2	2	no	—
84	C₂×C₄×C₄₁	—	2	2	no	—
85	C₂×C₄×C₄₁	—	2	1	no	—
86	C₂×C₄×C₄₁	—	2	1	no	—
87	C₂×C₄×C₄₁	—	2	2	no	—
88	C₂×C₄×C₄₁	—	2	1	no	—
89	C₂×C₄×C₄₁	—	2	2	no	—
90	C₂×C₄×C₄₁	—	4	2	no	—
91	C₂×C₄×C₄₁	—	8	2	no	—
92	C₂×C₄×C₄₁	C₂×D₄	1	2	yes	—
93	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
96	C₈×C₄₁	—	2	2	no	—
97	C₈×C₄₁	—	2	1	no	—
98	C₈×C₄₁	#41	1	2	yes	—
99	C₂²×C₄₁	—	2	2	no	—
100	C₂²×C₄₁	—	2	2	no	—
101	C₂²×C₄₁	—	2	2	no	—
102	C₂²×C₄₁	—	4	1	no	—
103	C₂²×C₄₁	—	4	1	no	—
104	C₂²×C₄₁	—	4	1	no	—
105	C₂²×C₄₁	C₂²×D₄	1	1	yes	characteristic
106	C₄×C₄₁	—	2	2	no	—
107	C₄×C₄₁	—	2	1	no	—
108	C₄×C₄₁	—	4	1	no	—
109	C₄×C₄₁	—	4	1	no	—
110	C₄×C₄₁	—	4	1	no	—
111	C₄×C₄₁	—	4	2	no	—
112	C₄×C₄₁	—	4	2	no	—
113	C₄×C₄₁	#137	1	2	yes	—
114	C₄×C₄₁	C₂×#41	1	1	yes	characteristic
115	C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
116	#3357	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂×C₄₁	—	4	2	no	—
118	C₂×C₄₁	—	8	1	no	—
119	C₂×C₄₁	#547	1	1	yes	characteristic
120	C₂×C₄₁	C₂×#137	1	1	yes	characteristic
121	C₂×C₄₁	#576	1	1	yes	characteristic
122	#555	C₂×C₄₁	1	1	yes	characteristic
123	#554	C₂×C₄₁	1	1	yes	characteristic, complemented
124	#500	C₂×C₄₁	1	1	yes	characteristic, complemented
125	#485	C₂×C₄₁	1	2	yes	complemented
126	#480	C₂×C₄₁	1	2	yes	complemented
127	#477	C₂×C₄₁	1	1	yes	characteristic, complemented
128	#476	C₂×C₄₁	1	1	yes	characteristic, complemented
129	#468	C₂×C₄₁	1	2	yes	complemented
130	C₄×SD₁₆	C₂×C₄₁	1	1	yes	characteristic, complemented
131	#436	C₂×C₄₁	1	1	yes	characteristic, complemented
132	#428	C₂×C₄₁	1	2	yes	complemented
133	C₄₁	#3357	1	1	yes	characteristic, direct product factor, Sylow
134	C₂×SD₁₆	C₂²×C₄₁	1	1	yes	characteristic
135	C₂×M₁₆	C₂²×C₄₁	1	2	yes	—
136	C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
137	C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
138	#125	—	2	1	no	—
139	#124	—	2	2	no	—
140	C₂²⋊Q₈	—	2	2	no	—
141	C₂²⋊Q₈	C₂²×C₄₁	1	2	yes	—
142	#121	—	2	1	no	—
143	#121	C₂²×C₄₁	1	2	yes	—
144	C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
145	C₄×D₄	—	2	1	no	—
146	C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
147	C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
148	#117	—	2	1	no	—
149	#117	C₂²×C₄₁	1	2	yes	—
150	C₂×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
151	#107	C₂²×C₄₁	1	2	yes	—
152	#107	C₂²×C₄₁	1	2	yes	—
153	#107	C₂²×C₄₁	1	2	yes	—
154	#106	C₂²×C₄₁	1	1	yes	characteristic
155	#106	C₂²×C₄₁	1	2	yes	—
156	C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
157	#103	C₂²×C₄₁	1	1	yes	characteristic
158	#103	C₂²×C₄₁	1	2	yes	—
159	#102	C₂²×C₄₁	1	1	yes	characteristic
160	#102	C₂²×C₄₁	1	2	yes	—
161	C₂²⋊C₈	C₂²×C₄₁	1	2	yes	—
162	C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
163	C₂×Q₈	—	2	1	no	—
164	C₂×Q₈	—	2	1	no	—
165	C₂×Q₈	C₂³×C₄₁	1	1	yes	characteristic
166	C₂×D₄	—	2	2	no	—
167	C₂×D₄	D₄×C₄₁	1	1	yes	characteristic
168	C₂×D₄	C₂³×C₄₁	1	1	yes	characteristic
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
172	SD₁₆	—	2	2	no	—
173	M₁₆	—	2	4	no	—
174	C₂×C₈	C₂³×C₄₁	1	1	yes	characteristic
175	C₂×C₈	C₂³×C₄₁	1	2	yes	—
176	C₂×C₈	C₂³×C₄₁	1	1	yes	characteristic
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	1	no	—
179	C₄⋊C₄	—	2	1	no	—
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
183	C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
184	C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
185	C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
186	C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
187	C₂²⋊C₄	—	2	2	no	—
188	C₂²⋊C₄	—	2	2	no	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	2	1	no	—
191	C₂²⋊C₄	D₄×C₄₁	1	2	yes	—
192	C₄²	—	2	2	no	—
193	C₄²	—	2	1	no	—
194	C₄²	C₂³×C₄₁	1	1	yes	characteristic
195	C₂³	—	2	1	no	—
196	C₂³	C₂×D₄×C₄₁	1	2	yes	—
197	Q₈	—	2	1	no	—
198	Q₈	—	2	1	no	—
199	Q₈	—	8	1	no	—
200	Q₈	—	8	1	no	—
201	D₄	—	2	2	no	—
202	D₄	—	2	1	no	—
203	D₄	—	2	1	no	—
204	D₄	—	8	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	8	2	no	—
214	C₂×C₄	C₂×D₄×C₄₁	1	2	yes	—
215	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
216	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
217	C₂×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
218	C₈	—	2	2	no	—
219	C₈	—	2	1	no	—
220	C₈	#41×C₄₁	1	2	yes	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	4	1	no	—
225	C₂²	—	4	1	no	—
226	C₂²	—	4	1	no	—
227	C₂²	C₂²×D₄×C₄₁	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	2	1	no	—
230	C₄	—	4	1	no	—
231	C₄	—	4	1	no	—
232	C₄	—	4	1	no	—
233	C₄	—	4	2	no	—
234	C₄	—	4	2	no	—
235	C₄	#137×C₄₁	1	2	yes	—
236	C₄	C₂×#41×C₄₁	1	1	yes	characteristic
237	C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
238	C₂	—	4	2	no	—
239	C₂	—	8	1	no	—
240	C₂	C₄₁×#547	1	1	yes	characteristic
241	C₂	C₂×#137×C₄₁	1	1	yes	characteristic
242	C₂	C₄₁×#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₄₁
prime radical	C₄₁×#555	C₂
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
2. C₄₁×#554	C₂	semidirect
3. C₄₁×#500	C₂	semidirect
4. C₄₁×#485	C₂	semidirect
5. C₄₁×#480	C₂	semidirect
6. C₄₁×#477	C₂	semidirect
7. C₄₁×#476	C₂	semidirect
8. C₄₁×#468	C₂	semidirect
9. C₄×SD₁₆×C₄₁	C₂	semidirect
10. C₄₁×#436	C₂	semidirect
11. C₄₁×#428	C₂	semidirect
116. #3357	C₄₁	direct (N × H)
123. #554	C₂×C₄₁	semidirect
124. #500	C₂×C₄₁	semidirect
125. #485	C₂×C₄₁	semidirect
126. #480	C₂×C₄₁	semidirect
127. #477	C₂×C₄₁	semidirect
128. #476	C₂×C₄₁	semidirect
129. #468	C₂×C₄₁	semidirect
130. C₄×SD₁₆	C₂×C₄₁	semidirect
131. #436	C₂×C₄₁	semidirect
132. #428	C₂×C₄₁	semidirect
133. C₄₁	#3357	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×#555	2
C₄₁×#554	2
C₄₁×#500	2
C₄₁×#485	2
C₄₁×#480	2
C₄₁×#477	2
C₄₁×#476	2
C₄₁×#468	2
C₄×SD₁₆×C₄₁	2
C₄₁×#436	2
C₄₁×#428	2
#3357	41

Subgroups — C41×#3357 #19107137

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#3357 #19107137