Subgroups — C₄₁×#2565 #19106345

Order 5248.

452 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×#125×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈×C₄₁	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#398	C₂	1	4	yes	maximal
4	C₄₁×#391	C₂	1	1	yes	maximal, characteristic
5	C₄₁×#391	C₂	1	1	yes	maximal, characteristic
6	C₄₁×#387	C₂	1	4	yes	maximal
7	C₄₁×#384	C₂	1	1	yes	maximal, characteristic
8	C₄₁×#382	C₂	1	1	yes	maximal, characteristic
9	C₄×C₄⋊C₄×C₄₁	C₂	1	1	yes	maximal, characteristic
10	C₂²×Q₈×C₄₁	C₂²	1	1	yes	characteristic
11	#125×C₄₁	—	2	4	no	—
12	C₄×Q₈×C₄₁	—	2	4	no	—
13	C₂×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₄₁	C₂²	1	4	yes	—
21	C₂×C₄²×C₄₁	C₂²	1	2	yes	—
22	C₂×C₄²×C₄₁	C₂²	1	1	yes	characteristic
23	C₂×C₄²×C₄₁	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₄₁	C₂²	1	1	yes	characteristic
25	#95×C₄₁	C₂²	1	2	yes	—
26	#95×C₄₁	C₂²	1	4	yes	—
27	#95×C₄₁	C₂²	1	4	yes	—
28	#95×C₄₁	C₂²	1	4	yes	—
29	C₂×Q₈×C₄₁	—	2	2	no	—
30	C₂×Q₈×C₄₁	—	2	2	no	—
31	C₂×Q₈×C₄₁	Q₈	1	4	yes	—
32	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
33	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
34	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
35	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
36	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
39	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
40	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₄₁	—	2	4	no	—
42	C₄⋊C₄×C₄₁	—	2	2	no	—
43	C₄⋊C₄×C₄₁	—	2	4	no	—
44	C₄⋊C₄×C₄₁	—	2	4	no	—
45	C₄⋊C₄×C₄₁	—	2	2	no	—
46	C₄⋊C₄×C₄₁	—	2	2	no	—
47	C₄⋊C₄×C₄₁	—	2	4	no	—
48	C₄⋊C₄×C₄₁	—	2	8	no	—
49	C₄²×C₄₁	—	2	4	no	—
50	C₄²×C₄₁	—	2	2	no	—
51	C₄²×C₄₁	—	2	2	no	—
52	C₄²×C₄₁	D₄	1	4	yes	—
53	C₂³×C₄₁	C₂⁴	1	1	yes	characteristic
54	Q₈×C₄₁	—	2	8	no	—
55	C₂×C₄×C₄₁	—	2	2	no	—
56	C₂×C₄×C₄₁	—	2	2	no	—
57	C₂×C₄×C₄₁	—	2	2	no	—
58	C₂×C₄×C₄₁	—	2	2	no	—
59	C₂×C₄×C₄₁	—	2	2	no	—
60	C₂×C₄×C₄₁	—	2	2	no	—
61	C₂×C₄×C₄₁	—	2	2	no	—
62	C₂×C₄×C₄₁	—	2	2	no	—
63	C₂×C₄×C₄₁	—	2	2	no	—
64	C₂×C₄×C₄₁	—	2	2	no	—
65	C₂×C₄×C₄₁	—	2	4	no	—
66	C₂×C₄×C₄₁	—	2	4	no	—
67	C₂×C₄×C₄₁	—	2	2	no	—
68	C₂×C₄×C₄₁	—	2	4	no	—
69	C₂×C₄×C₄₁	—	2	2	no	—
70	C₂×C₄×C₄₁	C₂×D₄	1	2	yes	—
71	C₂×C₄×C₄₁	C₂×D₄	1	2	yes	—
72	C₂×C₄×C₄₁	C₂×D₄	1	2	yes	—
73	C₂×C₄×C₄₁	C₂×Q₈	1	4	yes	—
74	C₂×C₄×C₄₁	C₂×Q₈	1	2	yes	—
75	C₂×C₄×C₄₁	#41	1	4	yes	—
76	C₂×C₄×C₄₁	#41	1	2	yes	—
77	C₂²×C₄₁	C₂²×D₄	1	1	yes	characteristic
78	C₂²×C₄₁	C₂²×Q₈	1	1	yes	characteristic
79	C₂²×C₄₁	C₂×#41	1	1	yes	characteristic
80	C₂²×C₄₁	C₂×#41	1	2	yes	—
81	C₂²×C₄₁	2₋¹⁺⁴	1	2	yes	—
82	C₄×C₄₁	—	2	4	no	—
83	C₄×C₄₁	—	2	2	no	—
84	C₄×C₄₁	—	2	2	no	—
85	C₄×C₄₁	—	2	4	no	—
86	C₄×C₄₁	—	4	2	no	—
87	C₄×C₄₁	—	4	2	no	—
88	C₄×C₄₁	—	4	2	no	—
89	C₄×C₄₁	—	4	2	no	—
90	C₄×C₄₁	C₂²⋊Q₈	1	4	yes	—
91	#2565	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂×C₄₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
93	C₂×C₄₁	#532	1	1	yes	characteristic
94	C₂×C₄₁	#536	1	1	yes	characteristic
95	C₂×C₄₁	#552	1	2	yes	—
96	C₂×C₄₁	#557	1	2	yes	—
97	C₂×#125	C₂×C₄₁	1	1	yes	characteristic
98	C₂×C₄×Q₈	C₂×C₄₁	1	1	yes	characteristic
99	#398	C₂×C₄₁	1	4	yes	—
100	#391	C₂×C₄₁	1	1	yes	characteristic
101	#391	C₂×C₄₁	1	1	yes	characteristic
102	#387	C₂×C₄₁	1	4	yes	—
103	#384	C₂×C₄₁	1	1	yes	characteristic
104	#382	C₂×C₄₁	1	1	yes	characteristic
105	C₄×C₄⋊C₄	C₂×C₄₁	1	1	yes	characteristic
106	C₄₁	#2565	1	1	yes	characteristic, direct product factor, Sylow
107	C₂²×Q₈	C₂²×C₄₁	1	1	yes	characteristic
108	#125	—	2	4	no	—
109	C₄×Q₈	—	2	4	no	—
110	C₂×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
111	C₂×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
112	C₂×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
113	C₂×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
114	C₂×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
115	C₂×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
116	C₂×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
117	C₂×C₄⋊C₄	C₂²×C₄₁	1	4	yes	—
118	C₂×C₄²	C₂²×C₄₁	1	2	yes	—
119	C₂×C₄²	C₂²×C₄₁	1	1	yes	characteristic
120	C₂×C₄²	C₂²×C₄₁	1	1	yes	characteristic
121	C₂×C₄²	C₂²×C₄₁	1	1	yes	characteristic
122	#95	C₂²×C₄₁	1	2	yes	—
123	#95	C₂²×C₄₁	1	4	yes	—
124	#95	C₂²×C₄₁	1	4	yes	—
125	#95	C₂²×C₄₁	1	4	yes	—
126	C₂×Q₈	—	2	2	no	—
127	C₂×Q₈	—	2	2	no	—
128	C₂×Q₈	Q₈×C₄₁	1	4	yes	—
129	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
130	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
131	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
132	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
133	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
134	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
135	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
136	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
137	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
138	C₄⋊C₄	—	2	4	no	—
139	C₄⋊C₄	—	2	2	no	—
140	C₄⋊C₄	—	2	4	no	—
141	C₄⋊C₄	—	2	4	no	—
142	C₄⋊C₄	—	2	2	no	—
143	C₄⋊C₄	—	2	2	no	—
144	C₄⋊C₄	—	2	4	no	—
145	C₄⋊C₄	—	2	8	no	—
146	C₄²	—	2	4	no	—
147	C₄²	—	2	2	no	—
148	C₄²	—	2	2	no	—
149	C₄²	D₄×C₄₁	1	4	yes	—
150	C₂³	C₂⁴×C₄₁	1	1	yes	characteristic
151	Q₈	—	2	8	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	C₂×D₄×C₄₁	1	2	yes	—
168	C₂×C₄	C₂×D₄×C₄₁	1	2	yes	—
169	C₂×C₄	C₂×D₄×C₄₁	1	2	yes	—
170	C₂×C₄	C₂×Q₈×C₄₁	1	4	yes	—
171	C₂×C₄	C₂×Q₈×C₄₁	1	2	yes	—
172	C₂×C₄	#41×C₄₁	1	4	yes	—
173	C₂×C₄	#41×C₄₁	1	2	yes	—
174	C₂²	C₂²×D₄×C₄₁	1	1	yes	characteristic
175	C₂²	C₂²×Q₈×C₄₁	1	1	yes	characteristic
176	C₂²	C₂×#41×C₄₁	1	1	yes	characteristic
177	C₂²	C₂×#41×C₄₁	1	2	yes	—
178	C₂²	2₋¹⁺⁴×C₄₁	1	2	yes	—
179	C₄	—	2	4	no	—
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	2	4	no	—
183	C₄	—	4	2	no	—
184	C₄	—	4	2	no	—
185	C₄	—	4	2	no	—
186	C₄	—	4	2	no	—
187	C₄	C₂²⋊Q₈×C₄₁	1	4	yes	—
188	C₂	C₂×C₂²⋊Q₈×C₄₁	1	1	yes	characteristic
189	C₂	C₄₁×#532	1	1	yes	characteristic
190	C₂	C₄₁×#536	1	1	yes	characteristic
191	C₂	C₄₁×#552	1	2	yes	—
192	C₂	C₄₁×#557	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
91. #2565	C₄₁	direct (N × H)
106. C₄₁	#2565	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×#125×C₄₁	2
C₂×C₄×Q₈×C₄₁	2
C₄₁×#398	2
C₄₁×#391	2
C₄₁×#387	2
C₄₁×#384	2
C₄₁×#382	2
C₄×C₄⋊C₄×C₄₁	2
#2565	41

Subgroups — C41×#2565 #19106345

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#2565 #19106345