Subgroups — C₄×C₄₁×M₆₄ #23789749

Order 10496.

250 conjugacy classes of subgroups.

The subgroup lattice has 138 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×M₆₄	C₂	1	2	yes	maximal
2	C₂×C₄×C₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#1397	C₂	1	2	yes	maximal, complemented
4	C₄×C₃₂×C₄₁	C₂	1	2	yes	maximal, complemented
5	C₂²×C₁₆×C₄₁	C₄	1	1	yes	characteristic
6	C₂²×C₁₆×C₄₁	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₈×C₄₁	C₄	1	1	yes	characteristic
8	C₄₁×M₆₄	C₄	1	8	yes	direct product factor
9	C₂×C₃₂×C₄₁	C₂²	1	4	yes	—
10	C₄×C₁₆×C₄₁	C₄	1	1	yes	characteristic
11	C₄×C₁₆×C₄₁	C₄	1	1	yes	characteristic
12	C₄×C₁₆×C₄₁	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₄₁	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₄₁	C₈	1	1	yes	characteristic
15	C₂²×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₄₁	C₈	1	1	yes	characteristic
17	C₂×C₁₆×C₄₁	C₈	1	4	yes	—
18	C₂×C₁₆×C₄₁	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₆×C₄₁	C₂×C₄	1	4	yes	—
20	C₂×C₁₆×C₄₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆×C₄₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₄₁	C₂³	1	1	yes	characteristic
23	C₄×C₈×C₄₁	C₈	1	1	yes	characteristic
24	C₄×C₈×C₄₁	C₈	1	1	yes	characteristic
25	C₄×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
27	C₃₂×C₄₁	C₂×C₄	1	8	yes	complemented
28	C₂²×C₄×C₄₁	C₁₆	1	2	yes	—
29	C₂²×C₄×C₄₁	C₂×C₈	1	1	yes	characteristic
30	C₂×C₈×C₄₁	C₁₆	1	4	yes	—
31	C₂×C₈×C₄₁	C₄²	1	2	yes	—
32	C₂×C₈×C₄₁	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₄₁	C₂×C₈	1	2	yes	—
34	C₂×C₈×C₄₁	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₄₁	C₂×C₈	1	1	yes	characteristic
36	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
37	C₄²×C₄₁	C₁₆	1	2	yes	—
38	C₄²×C₄₁	C₂×C₈	1	1	yes	characteristic
39	C₄²×C₄₁	C₂×C₈	1	1	yes	characteristic
40	C₁₆×C₄₁	C₄²	1	2	yes	—
41	C₁₆×C₄₁	C₂×C₈	1	2	yes	—
42	C₁₆×C₄₁	C₂×C₈	1	2	yes	—
43	C₁₆×C₄₁	C₂²×C₄	1	2	yes	—
44	C₂³×C₄₁	C₂×C₁₆	1	1	yes	characteristic
45	C₂×C₄×C₄₁	—	2	4	no	—
46	C₂×C₄×C₄₁	C₄×C₈	1	2	yes	—
47	C₂×C₄×C₄₁	C₂×C₁₆	1	2	yes	—
48	C₂×C₄×C₄₁	C₂×C₁₆	1	1	yes	characteristic
49	C₂×C₄×C₄₁	C₂×C₁₆	1	2	yes	—
50	C₂×C₄×C₄₁	C₂×C₁₆	1	2	yes	—
51	C₂×C₄×C₄₁	C₂²×C₈	1	1	yes	characteristic
52	C₈×C₄₁	C₄×C₈	1	2	yes	—
53	C₈×C₄₁	C₂×C₁₆	1	2	yes	—
54	C₈×C₄₁	C₂×C₁₆	1	2	yes	—
55	C₈×C₄₁	C₂×C₄²	1	1	yes	characteristic
56	C₈×C₄₁	C₂²×C₈	1	1	yes	characteristic
57	C₄×M₆₄	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂²×C₄₁	—	2	2	no	—
59	C₂²×C₄₁	C₄×C₁₆	1	2	yes	—
60	C₂²×C₄₁	C₂²×C₁₆	1	1	yes	characteristic
61	C₄×C₄₁	—	2	2	no	—
62	C₄×C₄₁	C₄×C₁₆	1	2	yes	—
63	C₄×C₄₁	M₆₄	1	4	yes	direct product factor
64	C₄×C₄₁	C₂×C₄×C₈	1	1	yes	characteristic
65	C₄×C₄₁	C₂²×C₁₆	1	1	yes	characteristic
66	C₂×M₆₄	C₂×C₄₁	1	2	yes	—
67	C₂×C₄×C₁₆	C₂×C₄₁	1	1	yes	characteristic
68	#1397	C₂×C₄₁	1	2	yes	complemented
69	C₄×C₃₂	C₂×C₄₁	1	2	yes	complemented
70	C₂×C₄₁	—	2	2	no	—
71	C₂×C₄₁	C₂×C₄×C₁₆	1	1	yes	characteristic
72	C₂×C₄₁	C₂×M₆₄	1	2	yes	—
73	C₂²×C₁₆	C₄×C₄₁	1	1	yes	characteristic
74	C₂²×C₁₆	C₂²×C₄₁	1	1	yes	characteristic
75	C₂×C₄×C₈	C₄×C₄₁	1	1	yes	characteristic
76	M₆₄	C₄×C₄₁	1	8	yes	direct product factor
77	C₂×C₃₂	C₂²×C₄₁	1	4	yes	—
78	C₄×C₁₆	C₄×C₄₁	1	1	yes	characteristic
79	C₄×C₁₆	C₄×C₄₁	1	1	yes	characteristic
80	C₄×C₁₆	C₂²×C₄₁	1	1	yes	characteristic
81	C₄×C₁₆	C₂²×C₄₁	1	1	yes	characteristic
82	C₄₁	C₄×M₆₄	1	1	yes	characteristic, direct product factor, Sylow
83	C₂²×C₈	C₈×C₄₁	1	1	yes	characteristic
84	C₂²×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
85	C₂×C₄²	C₈×C₄₁	1	1	yes	characteristic
86	C₂×C₁₆	C₈×C₄₁	1	4	yes	—
87	C₂×C₁₆	C₂×C₄×C₄₁	1	1	yes	characteristic
88	C₂×C₁₆	C₂×C₄×C₄₁	1	4	yes	—
89	C₂×C₁₆	C₂×C₄×C₄₁	1	1	yes	characteristic
90	C₂×C₁₆	C₂×C₄×C₄₁	1	1	yes	characteristic
91	C₂×C₁₆	C₂³×C₄₁	1	1	yes	characteristic
92	C₄×C₈	C₈×C₄₁	1	1	yes	characteristic
93	C₄×C₈	C₈×C₄₁	1	1	yes	characteristic
94	C₄×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
95	C₄×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
96	C₃₂	C₂×C₄×C₄₁	1	8	yes	complemented
97	C₂²×C₄	C₁₆×C₄₁	1	2	yes	—
98	C₂²×C₄	C₂×C₈×C₄₁	1	1	yes	characteristic
99	C₂×C₈	C₁₆×C₄₁	1	4	yes	—
100	C₂×C₈	C₄²×C₄₁	1	2	yes	—
101	C₂×C₈	C₂×C₈×C₄₁	1	1	yes	characteristic
102	C₂×C₈	C₂×C₈×C₄₁	1	2	yes	—
103	C₂×C₈	C₂×C₈×C₄₁	1	1	yes	characteristic
104	C₂×C₈	C₂×C₈×C₄₁	1	1	yes	characteristic
105	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
106	C₄²	C₁₆×C₄₁	1	2	yes	—
107	C₄²	C₂×C₈×C₄₁	1	1	yes	characteristic
108	C₄²	C₂×C₈×C₄₁	1	1	yes	characteristic
109	C₁₆	C₄²×C₄₁	1	2	yes	—
110	C₁₆	C₂×C₈×C₄₁	1	2	yes	—
111	C₁₆	C₂×C₈×C₄₁	1	2	yes	—
112	C₁₆	C₂²×C₄×C₄₁	1	2	yes	—
113	C₂³	C₂×C₁₆×C₄₁	1	1	yes	characteristic
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	C₄×C₈×C₄₁	1	2	yes	—
116	C₂×C₄	C₂×C₁₆×C₄₁	1	2	yes	—
117	C₂×C₄	C₂×C₁₆×C₄₁	1	2	yes	—
118	C₂×C₄	C₂×C₁₆×C₄₁	1	1	yes	characteristic
119	C₂×C₄	C₂×C₁₆×C₄₁	1	2	yes	—
120	C₂×C₄	C₂²×C₈×C₄₁	1	1	yes	characteristic
121	C₈	C₄×C₈×C₄₁	1	2	yes	—
122	C₈	C₂×C₁₆×C₄₁	1	2	yes	—
123	C₈	C₂×C₁₆×C₄₁	1	2	yes	—
124	C₈	C₂×C₄²×C₄₁	1	1	yes	characteristic
125	C₈	C₂²×C₈×C₄₁	1	1	yes	characteristic
126	C₂²	—	2	2	no	—
127	C₂²	C₄×C₁₆×C₄₁	1	2	yes	—
128	C₂²	C₂²×C₁₆×C₄₁	1	1	yes	characteristic
129	C₄	—	2	2	no	—
130	C₄	C₄×C₁₆×C₄₁	1	2	yes	—
131	C₄	C₄₁×M₆₄	1	4	yes	direct product factor
132	C₄	C₂×C₄×C₈×C₄₁	1	1	yes	characteristic
133	C₄	C₂²×C₁₆×C₄₁	1	1	yes	characteristic
134	C₂	—	2	2	no	—
135	C₂	C₂×C₄×C₁₆×C₄₁	1	1	yes	characteristic
136	C₂	C₂×C₄₁×M₆₄	1	2	yes	—

	subgroup	G/H
center	C₄×C₁₆×C₄₁	C₂²
commutator	C₂	C₂×C₄×C₁₆×C₄₁
Frattini, cosocle	C₂×C₁₆	C₂³×C₄₁
socle	C₂²×C₄₁	C₂²×C₁₆
prime radical	C₂³×C₄₁	C₂×C₁₆
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₆×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄₁×#1397	C₂	semidirect
4. C₄×C₃₂×C₄₁	C₂	semidirect
8. C₄₁×M₆₄	C₄	direct (N × H)
27. C₃₂×C₄₁	C₂×C₄	semidirect
57. C₄×M₆₄	C₄₁	direct (N × H)
63. C₄×C₄₁	M₆₄	direct (N × H)
68. #1397	C₂×C₄₁	semidirect
69. C₄×C₃₂	C₂×C₄₁	semidirect
76. M₆₄	C₄×C₄₁	direct (N × H)
82. C₄₁	C₄×M₆₄	direct (N × H)
96. C₃₂	C₂×C₄×C₄₁	semidirect
131. C₄	C₄₁×M₆₄	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×M₆₄	2
C₂×C₄×C₁₆×C₄₁	2
C₄₁×#1397	2
C₄×C₃₂×C₄₁	2
C₄×M₆₄	41

Subgroups — C4×C41×M64 #23789749

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₄₁×M₆₄ #23789749