Subgroups — C₄₁×#23263 #23799464

Order 10496.

692 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#125×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#1916	C₂	1	4	yes	maximal
3	C₄₁×#1916	C₂	1	4	yes	maximal
4	C₈×C₄⋊C₄×C₄₁	C₂	1	4	yes	maximal
5	C₄₁×#1744	C₂	1	2	yes	maximal
6	C₂×#125×C₄₁	C₄	1	1	yes	characteristic
7	C₂×C₄⋊C₈×C₄₁	C₂²	1	4	yes	—
8	C₂×C₄×C₈×C₄₁	C₂²	1	4	yes	—
9	C₂×C₄×C₈×C₄₁	C₂²	1	4	yes	—
10	C₄₁×#389	C₄	1	2	yes	—
11	C₄₁×#387	C₄	1	4	yes	—
12	C₄₁×#382	C₄	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₄₁	C₂²	1	4	yes	—
14	C₄×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
15	C₄³×C₄₁	C₂²	1	1	yes	characteristic
16	C₄₁×#336	C₂²	1	2	yes	—
17	C₄₁×#336	C₂²	1	4	yes	—
18	C₄₁×#336	C₂²	1	4	yes	—
19	C₄₁×#336	C₂²	1	4	yes	—
20	C₄₁×#336	C₂²	1	2	yes	—
21	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
22	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
23	#125×C₄₁	C₈	1	8	yes	complemented
24	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
25	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
26	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
27	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
33	C₂×C₄²×C₄₁	C₂³	1	4	yes	—
34	C₄⋊C₈×C₄₁	—	2	8	no	—
35	C₄×C₈×C₄₁	—	2	8	no	—
36	C₄×C₈×C₄₁	—	2	8	no	—
37	#95×C₄₁	C₂×C₄	1	4	yes	—
38	#95×C₄₁	C₂×C₄	1	4	yes	—
39	#95×C₄₁	C₂×C₄	1	4	yes	—
40	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
42	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
43	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
44	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
46	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
47	C₂×C₈×C₄₁	—	2	4	no	—
48	C₂×C₈×C₄₁	—	2	4	no	—
49	C₂×C₈×C₄₁	—	2	4	no	—
50	C₂×C₈×C₄₁	—	2	4	no	—
51	C₂×C₈×C₄₁	—	2	4	no	—
52	C₂×C₈×C₄₁	#41	1	8	yes	—
53	C₄⋊C₄×C₄₁	—	2	8	no	—
54	C₄⋊C₄×C₄₁	—	4	2	no	—
55	C₄⋊C₄×C₄₁	C₂×C₈	1	16	yes	—
56	C₄⋊C₄×C₄₁	C₂×C₈	1	8	yes	—
57	C₄²×C₄₁	—	2	4	no	—
58	C₄²×C₄₁	—	2	4	no	—
59	C₄²×C₄₁	—	2	4	no	—
60	C₄²×C₄₁	—	2	4	no	—
61	C₄²×C₄₁	—	2	2	no	—
62	C₄²×C₄₁	—	4	1	no	—
63	C₄²×C₄₁	C₂×C₈	1	4	yes	—
64	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
65	C₂×C₄×C₄₁	—	2	2	no	—
66	C₂×C₄×C₄₁	—	2	4	no	—
67	C₂×C₄×C₄₁	—	2	2	no	—
68	C₂×C₄×C₄₁	—	2	1	no	—
69	C₂×C₄×C₄₁	—	2	2	no	—
70	C₂×C₄×C₄₁	—	2	4	no	—
71	C₂×C₄×C₄₁	—	2	1	no	—
72	C₂×C₄×C₄₁	—	2	2	no	—
73	C₂×C₄×C₄₁	—	2	4	no	—
74	C₂×C₄×C₄₁	—	2	1	no	—
75	C₂×C₄×C₄₁	—	2	2	no	—
76	C₂×C₄×C₄₁	—	2	4	no	—
77	C₂×C₄×C₄₁	—	2	4	no	—
78	C₂×C₄×C₄₁	C₂²×C₈	1	4	yes	—
79	C₂×C₄×C₄₁	C₂²×C₈	1	2	yes	—
80	C₂×C₄×C₄₁	C₂²×C₈	1	8	yes	—
81	C₂×C₄×C₄₁	#131	1	4	yes	—
82	C₂×C₄×C₄₁	C₂×#41	1	4	yes	—
83	C₂×C₄×C₄₁	2₋¹⁺⁴	1	2	yes	—
84	C₈×C₄₁	—	2	8	no	—
85	C₈×C₄₁	—	4	4	no	—
86	#23263	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂²×C₄₁	C₄×#41	1	2	yes	—
88	C₂²×C₄₁	#519	1	1	yes	characteristic
89	C₂²×C₄₁	C₂³×C₈	1	1	yes	characteristic
90	C₂²×C₄₁	C₂×#131	1	2	yes	—
91	C₂²×C₄₁	#568	1	1	yes	characteristic
92	C₄×C₄₁	—	2	4	no	—
93	C₄×C₄₁	—	2	8	no	—
94	C₄×C₄₁	—	2	4	no	—
95	C₄×C₄₁	—	2	2	no	—
96	C₄×C₄₁	—	4	4	no	—
97	C₄×C₄₁	—	4	1	no	—
98	C₄×C₄₁	#552	1	4	yes	—
99	C₄×#125	C₂×C₄₁	1	1	yes	characteristic
100	#1916	C₂×C₄₁	1	4	yes	—
101	#1916	C₂×C₄₁	1	4	yes	—
102	C₈×C₄⋊C₄	C₂×C₄₁	1	4	yes	—
103	#1744	C₂×C₄₁	1	2	yes	—
104	C₂×C₄₁	#2371	1	1	yes	characteristic
105	C₂×C₄₁	C₈×#41	1	2	yes	—
106	C₂×C₄₁	#2968	1	2	yes	—
107	C₂×C₄₁	#2982	1	1	yes	characteristic
108	C₂×C₄₁	#2986	1	1	yes	characteristic
109	C₂×#125	C₄×C₄₁	1	1	yes	characteristic
110	C₂×C₄⋊C₈	C₂²×C₄₁	1	4	yes	—
111	C₂×C₄×C₈	C₂²×C₄₁	1	4	yes	—
112	C₂×C₄×C₈	C₂²×C₄₁	1	4	yes	—
113	#389	C₄×C₄₁	1	2	yes	—
114	#387	C₄×C₄₁	1	4	yes	—
115	#382	C₄×C₄₁	1	1	yes	characteristic
116	C₄×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
117	C₄×C₄⋊C₄	C₂²×C₄₁	1	4	yes	—
118	C₄³	C₂²×C₄₁	1	1	yes	characteristic
119	#336	C₂²×C₄₁	1	4	yes	—
120	#336	C₂²×C₄₁	1	4	yes	—
121	#336	C₂²×C₄₁	1	2	yes	—
122	#336	C₂²×C₄₁	1	4	yes	—
123	#336	C₂²×C₄₁	1	2	yes	—
124	C₄₁	#23263	1	1	yes	characteristic, direct product factor, Sylow
125	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
126	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
127	#125	C₈×C₄₁	1	8	yes	complemented
128	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
129	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
130	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
131	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
132	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
133	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
134	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
135	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
136	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
137	C₂×C₄²	C₂³×C₄₁	1	4	yes	—
138	C₄⋊C₈	—	2	8	no	—
139	C₄×C₈	—	2	8	no	—
140	C₄×C₈	—	2	8	no	—
141	#95	C₂×C₄×C₄₁	1	4	yes	—
142	#95	C₂×C₄×C₄₁	1	4	yes	—
143	#95	C₂×C₄×C₄₁	1	4	yes	—
144	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
145	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
146	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
147	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
148	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
149	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
150	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	—	2	4	no	—
155	C₂×C₈	—	2	4	no	—
156	C₂×C₈	#41×C₄₁	1	8	yes	—
157	C₄⋊C₄	—	2	8	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	C₂×C₈×C₄₁	1	16	yes	—
160	C₄⋊C₄	C₂×C₈×C₄₁	1	8	yes	—
161	C₄²	—	2	4	no	—
162	C₄²	—	2	4	no	—
163	C₄²	—	2	4	no	—
164	C₄²	—	2	4	no	—
165	C₄²	—	2	2	no	—
166	C₄²	—	4	1	no	—
167	C₄²	C₂×C₈×C₄₁	1	4	yes	—
168	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	C₂²×C₈×C₄₁	1	4	yes	—
183	C₂×C₄	C₂²×C₈×C₄₁	1	2	yes	—
184	C₂×C₄	C₂²×C₈×C₄₁	1	8	yes	—
185	C₂×C₄	#131×C₄₁	1	4	yes	—
186	C₂×C₄	C₂×#41×C₄₁	1	4	yes	—
187	C₂×C₄	2₋¹⁺⁴×C₄₁	1	2	yes	—
188	C₈	—	2	8	no	—
189	C₈	—	4	4	no	—
190	C₂²	C₄×#41×C₄₁	1	2	yes	—
191	C₂²	C₄₁×#519	1	1	yes	characteristic
192	C₂²	C₂³×C₈×C₄₁	1	1	yes	characteristic
193	C₂²	C₂×#131×C₄₁	1	2	yes	—
194	C₂²	C₄₁×#568	1	1	yes	characteristic
195	C₄	—	2	4	no	—
196	C₄	—	2	8	no	—
197	C₄	—	2	4	no	—
198	C₄	—	2	2	no	—
199	C₄	—	4	4	no	—
200	C₄	—	4	1	no	—
201	C₄	C₄₁×#552	1	4	yes	—
202	C₂	C₄₁×#2371	1	1	yes	characteristic
203	C₂	C₈×#41×C₄₁	1	2	yes	—
204	C₂	C₄₁×#2968	1	2	yes	—
205	C₂	C₄₁×#2982	1	1	yes	characteristic
206	C₂	C₄₁×#2986	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₄₁	C₂⁴
commutator	C₂²	C₂³×C₈×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
socle, prime radical	C₂³×C₄₁	C₂³×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
23. #125×C₄₁	C₈	semidirect
86. #23263	C₄₁	direct (N × H)
124. C₄₁	#23263	direct (N × H)
127. #125	C₈×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×#125×C₄₁	2
C₄₁×#1916	2
C₈×C₄⋊C₄×C₄₁	2
C₄₁×#1744	2
#23263	41

Subgroups — C41×#23263 #23799464

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#23263 #23799464