Subgroups — C₂×C₄×C₄₁×C₆₄ #72198565

Order 20992.

324 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄₁×C₆₄	C₂	1	2	yes	maximal
2	C₂×C₄×C₃₂×C₄₁	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄₁×C₆₄	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₃₂×C₄₁	C₄	1	1	yes	characteristic
5	C₂²×C₃₂×C₄₁	C₂²	1	1	yes	characteristic
6	C₂×C₄×C₁₆×C₄₁	C₄	1	1	yes	characteristic
7	C₂×C₄₁×C₆₄	C₄	1	8	yes	direct product factor
8	C₂×C₄₁×C₆₄	C₂²	1	4	yes	—
9	C₄×C₃₂×C₄₁	C₄	1	2	yes	—
10	C₄×C₃₂×C₄₁	C₂²	1	2	yes	—
11	C₂²×C₁₆×C₄₁	C₈	1	1	yes	characteristic
12	C₂²×C₁₆×C₄₁	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₄₁	C₈	1	1	yes	characteristic
14	C₂×C₃₂×C₄₁	C₈	1	4	yes	—
15	C₂×C₃₂×C₄₁	C₂×C₄	1	1	yes	characteristic
16	C₂×C₃₂×C₄₁	C₂×C₄	1	2	yes	—
17	C₂×C₃₂×C₄₁	C₂×C₄	1	4	yes	—
18	C₂×C₃₂×C₄₁	C₂³	1	1	yes	characteristic
19	C₄×C₁₆×C₄₁	C₈	1	2	yes	—
20	C₄×C₁₆×C₄₁	C₂×C₄	1	2	yes	—
21	C₄₁×C₆₄	C₂×C₄	1	8	yes	direct product factor
22	C₂²×C₈×C₄₁	C₁₆	1	1	yes	characteristic
23	C₂²×C₈×C₄₁	C₂×C₈	1	1	yes	characteristic
24	C₂×C₄²×C₄₁	C₁₆	1	1	yes	characteristic
25	C₂×C₁₆×C₄₁	C₁₆	1	4	yes	—
26	C₂×C₁₆×C₄₁	C₄²	1	2	yes	—
27	C₂×C₁₆×C₄₁	C₂×C₈	1	1	yes	characteristic
28	C₂×C₁₆×C₄₁	C₂×C₈	1	2	yes	—
29	C₂×C₁₆×C₄₁	C₂×C₈	1	2	yes	—
30	C₂×C₁₆×C₄₁	C₂²×C₄	1	1	yes	characteristic
31	C₄×C₈×C₄₁	C₁₆	1	2	yes	—
32	C₄×C₈×C₄₁	C₂×C₈	1	2	yes	—
33	C₃₂×C₄₁	C₄²	1	2	yes	—
34	C₃₂×C₄₁	C₂×C₈	1	4	yes	—
35	C₃₂×C₄₁	C₂²×C₄	1	2	yes	—
36	C₂²×C₄×C₄₁	C₃₂	1	2	yes	—
37	C₂²×C₄×C₄₁	C₂×C₁₆	1	1	yes	characteristic
38	C₂×C₈×C₄₁	C₃₂	1	4	yes	—
39	C₂×C₈×C₄₁	C₄×C₈	1	2	yes	—
40	C₂×C₈×C₄₁	C₂×C₁₆	1	1	yes	characteristic
41	C₂×C₈×C₄₁	C₂×C₁₆	1	2	yes	—
42	C₂×C₈×C₄₁	C₂×C₁₆	1	2	yes	—
43	C₂×C₈×C₄₁	C₂²×C₈	1	1	yes	characteristic
44	C₄²×C₄₁	C₃₂	1	2	yes	—
45	C₄²×C₄₁	C₂×C₁₆	1	2	yes	—
46	C₁₆×C₄₁	C₄×C₈	1	2	yes	—
47	C₁₆×C₄₁	C₂×C₁₆	1	4	yes	—
48	C₁₆×C₄₁	C₂×C₄²	1	1	yes	characteristic
49	C₁₆×C₄₁	C₂²×C₈	1	1	yes	characteristic
50	C₂×C₄×C₆₄	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₂³×C₄₁	C₂×C₃₂	1	1	yes	characteristic
52	C₂×C₄×C₄₁	C₆₄	1	8	yes	direct product factor
53	C₂×C₄×C₄₁	C₄×C₁₆	1	2	yes	—
54	C₂×C₄×C₄₁	C₂×C₃₂	1	1	yes	characteristic
55	C₂×C₄×C₄₁	C₂×C₃₂	1	2	yes	—
56	C₂×C₄×C₄₁	C₂×C₃₂	1	4	yes	—
57	C₂×C₄×C₄₁	C₂²×C₁₆	1	1	yes	characteristic
58	C₈×C₄₁	C₄×C₁₆	1	2	yes	—
59	C₈×C₄₁	C₂×C₃₂	1	4	yes	—
60	C₈×C₄₁	C₂×C₄×C₈	1	1	yes	characteristic
61	C₈×C₄₁	C₂²×C₁₆	1	1	yes	characteristic
62	C₂²×C₆₄	C₂×C₄₁	1	2	yes	—
63	C₂×C₄×C₃₂	C₂×C₄₁	1	1	yes	characteristic
64	C₄×C₆₄	C₂×C₄₁	1	4	yes	direct product factor
65	C₂²×C₄₁	C₄×C₃₂	1	2	yes	—
66	C₂²×C₄₁	C₂×C₆₄	1	4	yes	—
67	C₂²×C₄₁	C₂²×C₃₂	1	1	yes	characteristic
68	C₄×C₄₁	C₄×C₃₂	1	2	yes	—
69	C₄×C₄₁	C₂×C₆₄	1	8	yes	direct product factor
70	C₄×C₄₁	C₂×C₄×C₁₆	1	1	yes	characteristic
71	C₄×C₄₁	C₂²×C₃₂	1	1	yes	characteristic
72	C₂²×C₃₂	C₄×C₄₁	1	1	yes	characteristic
73	C₂²×C₃₂	C₂²×C₄₁	1	1	yes	characteristic
74	C₂×C₄×C₁₆	C₄×C₄₁	1	1	yes	characteristic
75	C₂×C₆₄	C₄×C₄₁	1	8	yes	direct product factor
76	C₂×C₆₄	C₂²×C₄₁	1	4	yes	—
77	C₄×C₃₂	C₄×C₄₁	1	2	yes	—
78	C₄×C₃₂	C₂²×C₄₁	1	2	yes	—
79	C₂×C₄₁	C₄×C₆₄	1	4	yes	direct product factor
80	C₂×C₄₁	C₂×C₄×C₃₂	1	1	yes	characteristic
81	C₂×C₄₁	C₂²×C₆₄	1	2	yes	—
82	C₂²×C₁₆	C₈×C₄₁	1	1	yes	characteristic
83	C₂²×C₁₆	C₂×C₄×C₄₁	1	1	yes	characteristic
84	C₂×C₄×C₈	C₈×C₄₁	1	1	yes	characteristic
85	C₂×C₃₂	C₈×C₄₁	1	4	yes	—
86	C₂×C₃₂	C₂×C₄×C₄₁	1	1	yes	characteristic
87	C₂×C₃₂	C₂×C₄×C₄₁	1	2	yes	—
88	C₂×C₃₂	C₂×C₄×C₄₁	1	4	yes	—
89	C₂×C₃₂	C₂³×C₄₁	1	1	yes	characteristic
90	C₄×C₁₆	C₈×C₄₁	1	2	yes	—
91	C₄×C₁₆	C₂×C₄×C₄₁	1	2	yes	—
92	C₆₄	C₂×C₄×C₄₁	1	8	yes	direct product factor
93	C₄₁	C₂×C₄×C₆₄	1	1	yes	characteristic, direct product factor, Sylow
94	C₂²×C₈	C₁₆×C₄₁	1	1	yes	characteristic
95	C₂²×C₈	C₂×C₈×C₄₁	1	1	yes	characteristic
96	C₂×C₄²	C₁₆×C₄₁	1	1	yes	characteristic
97	C₂×C₁₆	C₁₆×C₄₁	1	4	yes	—
98	C₂×C₁₆	C₄²×C₄₁	1	2	yes	—
99	C₂×C₁₆	C₂×C₈×C₄₁	1	1	yes	characteristic
100	C₂×C₁₆	C₂×C₈×C₄₁	1	2	yes	—
101	C₂×C₁₆	C₂×C₈×C₄₁	1	2	yes	—
102	C₂×C₁₆	C₂²×C₄×C₄₁	1	1	yes	characteristic
103	C₄×C₈	C₁₆×C₄₁	1	2	yes	—
104	C₄×C₈	C₂×C₈×C₄₁	1	2	yes	—
105	C₃₂	C₄²×C₄₁	1	2	yes	—
106	C₃₂	C₂×C₈×C₄₁	1	4	yes	—
107	C₃₂	C₂²×C₄×C₄₁	1	2	yes	—
108	C₂²×C₄	C₃₂×C₄₁	1	2	yes	—
109	C₂²×C₄	C₂×C₁₆×C₄₁	1	1	yes	characteristic
110	C₂×C₈	C₃₂×C₄₁	1	4	yes	—
111	C₂×C₈	C₄×C₈×C₄₁	1	2	yes	—
112	C₂×C₈	C₂×C₁₆×C₄₁	1	1	yes	characteristic
113	C₂×C₈	C₂×C₁₆×C₄₁	1	2	yes	—
114	C₂×C₈	C₂×C₁₆×C₄₁	1	2	yes	—
115	C₂×C₈	C₂²×C₈×C₄₁	1	1	yes	characteristic
116	C₄²	C₃₂×C₄₁	1	2	yes	—
117	C₄²	C₂×C₁₆×C₄₁	1	2	yes	—
118	C₁₆	C₄×C₈×C₄₁	1	2	yes	—
119	C₁₆	C₂×C₁₆×C₄₁	1	4	yes	—
120	C₁₆	C₂×C₄²×C₄₁	1	1	yes	characteristic
121	C₁₆	C₂²×C₈×C₄₁	1	1	yes	characteristic
122	C₂³	C₂×C₃₂×C₄₁	1	1	yes	characteristic
123	C₂×C₄	C₄₁×C₆₄	1	8	yes	direct product factor
124	C₂×C₄	C₄×C₁₆×C₄₁	1	2	yes	—
125	C₂×C₄	C₂×C₃₂×C₄₁	1	1	yes	characteristic
126	C₂×C₄	C₂×C₃₂×C₄₁	1	2	yes	—
127	C₂×C₄	C₂×C₃₂×C₄₁	1	4	yes	—
128	C₂×C₄	C₂²×C₁₆×C₄₁	1	1	yes	characteristic
129	C₈	C₄×C₁₆×C₄₁	1	2	yes	—
130	C₈	C₂×C₃₂×C₄₁	1	4	yes	—
131	C₈	C₂×C₄×C₈×C₄₁	1	1	yes	characteristic
132	C₈	C₂²×C₁₆×C₄₁	1	1	yes	characteristic
133	C₂²	C₄×C₃₂×C₄₁	1	2	yes	—
134	C₂²	C₂×C₄₁×C₆₄	1	4	yes	—
135	C₂²	C₂²×C₃₂×C₄₁	1	1	yes	characteristic
136	C₄	C₄×C₃₂×C₄₁	1	2	yes	—
137	C₄	C₂×C₄₁×C₆₄	1	8	yes	direct product factor
138	C₄	C₂×C₄×C₁₆×C₄₁	1	1	yes	characteristic
139	C₄	C₂²×C₃₂×C₄₁	1	1	yes	characteristic
140	C₂	C₄×C₄₁×C₆₄	1	4	yes	direct product factor
141	C₂	C₂×C₄×C₃₂×C₄₁	1	1	yes	characteristic
142	C₂	C₂²×C₄₁×C₆₄	1	2	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₃₂	C₂³×C₄₁
socle, prime radical	C₂³×C₄₁	C₂×C₃₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄×C₄₁×C₆₄	C₂	direct (N × H)
7. C₂×C₄₁×C₆₄	C₄	direct (N × H)
21. C₄₁×C₆₄	C₂×C₄	direct (N × H)
50. C₂×C₄×C₆₄	C₄₁	direct (N × H)
52. C₂×C₄×C₄₁	C₆₄	direct (N × H)
64. C₄×C₆₄	C₂×C₄₁	direct (N × H)
69. C₄×C₄₁	C₂×C₆₄	direct (N × H)
75. C₂×C₆₄	C₄×C₄₁	direct (N × H)
79. C₂×C₄₁	C₄×C₆₄	direct (N × H)
92. C₆₄	C₂×C₄×C₄₁	direct (N × H)
93. C₄₁	C₂×C₄×C₆₄	direct (N × H)
123. C₂×C₄	C₄₁×C₆₄	direct (N × H)
137. C₄	C₂×C₄₁×C₆₄	direct (N × H)
140. C₂	C₄×C₄₁×C₆₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄₁×C₆₄	2
C₂×C₄×C₃₂×C₄₁	2
C₄×C₄₁×C₆₄	2
C₂×C₄×C₆₄	41

Subgroups — C2×C4×C41×C64 #72198565

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₄₁×C₆₄ #72198565