Subgroups — C₄₁×#13289 #23789490

Order 10496.

342 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#499	C₂	1	1	yes	maximal, characteristic
2	C₄×M₃₂×C₄₁	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#1848	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1387	C₂	1	4	yes	maximal
5	C₂×#125×C₄₁	—	2	1	no	—
6	C₂×M₃₂×C₄₁	C₂²	1	2	yes	—
7	C₄₁×#499	—	2	1	no	—
8	C₄₁×#499	—	2	1	no	—
9	C₄₁×#499	C₄	1	2	yes	—
10	C₄₁×#499	C₄	1	2	yes	—
11	C₂×#108×C₄₁	—	2	1	no	—
12	C₂×#108×C₄₁	C₂²	1	2	yes	—
13	C₂×#107×C₄₁	C₂²	1	2	yes	—
14	C₂×#106×C₄₁	—	2	1	no	—
15	C₄₁×#423	—	2	1	no	—
16	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
17	C₄₁×#346	—	2	1	no	—
18	C₄×C₁₆×C₄₁	—	2	1	no	—
19	C₂×M₁₆×C₄₁	—	2	2	no	—
20	C₂²×C₈×C₄₁	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
22	#125×C₄₁	—	2	2	no	—
23	#125×C₄₁	—	2	2	no	—
24	#125×C₄₁	—	4	2	no	—
25	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
26	C₂×C₄⋊C₄×C₄₁	—	4	2	no	—
27	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
28	M₃₂×C₄₁	—	2	2	no	—
29	M₃₂×C₄₁	Q₈	1	4	yes	—
30	C₂×C₁₆×C₄₁	—	2	2	no	—
31	#108×C₄₁	—	2	4	no	—
32	#108×C₄₁	—	4	2	no	—
33	#107×C₄₁	—	2	2	no	—
34	#107×C₄₁	C₂×C₄	1	4	yes	—
35	#106×C₄₁	—	2	2	no	—
36	#106×C₄₁	—	2	2	no	—
37	C₄⋊C₈×C₄₁	—	4	2	no	—
38	C₄×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₄₁	D₄	1	1	yes	characteristic
41	C₄×C₈×C₄₁	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₁	—	4	2	no	—
43	C₂²×C₄×C₄₁	SD₁₆	1	2	yes	—
44	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
45	M₁₆×C₄₁	—	4	2	no	—
46	M₁₆×C₄₁	—	4	2	no	—
47	C₂×C₈×C₄₁	—	4	2	no	—
48	C₂×C₈×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₄₁	SD₁₆	1	2	yes	—
51	C₂×C₈×C₄₁	Q₁₆	1	2	yes	—
52	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₄₁	C₂×Q₈	1	2	yes	—
55	C₂×C₈×C₄₁	#41	1	2	yes	—
56	C₄⋊C₄×C₄₁	—	2	4	no	—
57	C₄⋊C₄×C₄₁	—	4	4	no	—
58	C₄⋊C₄×C₄₁	—	4	4	no	—
59	C₄⋊C₄×C₄₁	—	4	2	no	—
60	C₄²×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
61	C₄²×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₄₁	D₈	1	2	yes	—
63	C₁₆×C₄₁	—	2	4	no	—
64	C₂³×C₄₁	C₂×SD₁₆	1	1	yes	characteristic
65	C₂×C₄×C₄₁	—	4	2	no	—
66	C₂×C₄×C₄₁	—	4	4	no	—
67	C₂×C₄×C₄₁	—	8	4	no	—
68	C₂×C₄×C₄₁	#102	1	2	yes	—
69	C₂×C₄×C₄₁	#102	1	2	yes	—
70	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₄₁	#124	1	1	yes	characteristic
72	C₂×C₄×C₄₁	C₄⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₄₁	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄×C₄₁	C₂×SD₁₆	1	1	yes	characteristic
75	C₂×C₄×C₄₁	C₂×Q₁₆	1	1	yes	characteristic
76	C₈×C₄₁	—	8	2	no	—
77	C₈×C₄₁	#103	1	2	yes	—
78	C₈×C₄₁	#103	1	2	yes	—
79	C₈×C₄₁	C₄×Q₈	1	2	yes	—
80	C₈×C₄₁	C₂²⋊Q₈	1	2	yes	—
81	#13289	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂²×C₄₁	—	2	2	no	—
83	C₂²×C₄₁	C₂×#102	1	1	yes	characteristic
84	C₂²×C₄₁	#487	1	1	yes	characteristic
85	C₂²×C₄₁	#498	1	1	yes	characteristic
86	C₄×C₄₁	—	2	2	no	—
87	C₄×C₄₁	—	2	2	no	—
88	C₄×C₄₁	—	8	4	no	—
89	C₄×C₄₁	#391	1	1	yes	characteristic
90	C₄×C₄₁	C₂×#103	1	1	yes	characteristic
91	C₄×C₄₁	#486	1	1	yes	characteristic
92	C₄×C₄₁	#500	1	1	yes	characteristic
93	C₂×#499	C₂×C₄₁	1	1	yes	characteristic
94	C₄×M₃₂	C₂×C₄₁	1	1	yes	characteristic
95	#1848	C₂×C₄₁	1	1	yes	characteristic
96	#1387	C₂×C₄₁	1	4	yes	—
97	C₂×C₄₁	—	2	1	no	—
98	C₂×C₄₁	—	2	1	no	—
99	C₂×C₄₁	#1990	1	1	yes	characteristic
100	C₂×C₄₁	#2148	1	2	yes	—
101	C₂×#125	—	2	1	no	—
102	C₂×M₃₂	C₂²×C₄₁	1	2	yes	—
103	#499	—	2	1	no	—
104	#499	—	2	1	no	—
105	#499	C₄×C₄₁	1	2	yes	—
106	#499	C₄×C₄₁	1	2	yes	—
107	C₂×#108	—	2	1	no	—
108	C₂×#108	C₂²×C₄₁	1	2	yes	—
109	C₂×#107	C₂²×C₄₁	1	2	yes	—
110	C₂×#106	—	2	1	no	—
111	#423	—	2	1	no	—
112	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
113	#346	—	2	1	no	—
114	C₄×C₁₆	—	2	1	no	—
115	C₄₁	#13289	1	1	yes	characteristic, direct product factor, Sylow
116	C₂×M₁₆	—	2	2	no	—
117	C₂²×C₈	D₄×C₄₁	1	1	yes	characteristic
118	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
119	#125	—	2	2	no	—
120	#125	—	2	2	no	—
121	#125	—	4	2	no	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₄⋊C₄	—	4	2	no	—
124	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
125	M₃₂	—	2	2	no	—
126	M₃₂	Q₈×C₄₁	1	4	yes	—
127	C₂×C₁₆	—	2	2	no	—
128	#108	—	2	4	no	—
129	#108	—	4	2	no	—
130	#107	—	2	2	no	—
131	#107	C₂×C₄×C₄₁	1	4	yes	—
132	#106	—	2	2	no	—
133	#106	—	2	2	no	—
134	C₄⋊C₈	—	4	2	no	—
135	C₄×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
136	C₄×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
137	C₄×C₈	D₄×C₄₁	1	1	yes	characteristic
138	C₄×C₈	D₄×C₄₁	1	1	yes	characteristic
139	C₂²×C₄	—	4	2	no	—
140	C₂²×C₄	SD₁₆×C₄₁	1	2	yes	—
141	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
142	M₁₆	—	4	2	no	—
143	M₁₆	—	4	2	no	—
144	C₂×C₈	—	4	2	no	—
145	C₂×C₈	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
146	C₂×C₈	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
147	C₂×C₈	SD₁₆×C₄₁	1	2	yes	—
148	C₂×C₈	Q₁₆×C₄₁	1	2	yes	—
149	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
150	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
151	C₂×C₈	C₂×Q₈×C₄₁	1	2	yes	—
152	C₂×C₈	#41×C₄₁	1	2	yes	—
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	4	4	no	—
155	C₄⋊C₄	—	4	4	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₄²	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
158	C₄²	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
159	C₄²	D₈×C₄₁	1	2	yes	—
160	C₁₆	—	2	4	no	—
161	C₂³	C₂×SD₁₆×C₄₁	1	1	yes	characteristic
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	4	no	—
164	C₂×C₄	—	8	4	no	—
165	C₂×C₄	#102×C₄₁	1	2	yes	—
166	C₂×C₄	#102×C₄₁	1	2	yes	—
167	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
168	C₂×C₄	#124×C₄₁	1	1	yes	characteristic
169	C₂×C₄	C₄⋊Q₈×C₄₁	1	1	yes	characteristic
170	C₂×C₄	C₂×D₈×C₄₁	1	1	yes	characteristic
171	C₂×C₄	C₂×SD₁₆×C₄₁	1	1	yes	characteristic
172	C₂×C₄	C₂×Q₁₆×C₄₁	1	1	yes	characteristic
173	C₈	—	8	2	no	—
174	C₈	#103×C₄₁	1	2	yes	—
175	C₈	#103×C₄₁	1	2	yes	—
176	C₈	C₄×Q₈×C₄₁	1	2	yes	—
177	C₈	C₂²⋊Q₈×C₄₁	1	2	yes	—
178	C₂²	—	2	2	no	—
179	C₂²	C₂×#102×C₄₁	1	1	yes	characteristic
180	C₂²	C₄₁×#487	1	1	yes	characteristic
181	C₂²	C₄₁×#498	1	1	yes	characteristic
182	C₄	—	2	2	no	—
183	C₄	—	2	2	no	—
184	C₄	—	8	4	no	—
185	C₄	C₄₁×#391	1	1	yes	characteristic
186	C₄	C₂×#103×C₄₁	1	1	yes	characteristic
187	C₄	C₄₁×#486	1	1	yes	characteristic
188	C₄	C₄₁×#500	1	1	yes	characteristic
189	C₂	—	2	1	no	—
190	C₂	—	2	1	no	—
191	C₂	C₄₁×#1990	1	1	yes	characteristic
192	C₂	C₄₁×#2148	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₄₁
Frattini, cosocle	C₂²×C₈	C₂³×C₄₁
prime radical	C₂³×C₄₁	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₂×C₄²×C₄₁ ◁ C₂×C₄×C₈×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
81. #13289	C₄₁	direct (N × H)
115. C₄₁	#13289	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#499	2
C₄×M₃₂×C₄₁	2
C₄₁×#1848	2
C₄₁×#1387	2
#13289	41

Subgroups — C41×#13289 #23789490

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#13289 #23789490