Subgroups — C₂²×C₄×C₁₆×C₁₃₁ #45719204

Order 33536.

1022 conjugacy classes of subgroups.

The subgroup lattice has 132 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₆×C₁₃₁	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₁₆×C₁₃₁	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₈×C₁₃₁	C₄	1	1	yes	characteristic
5	C₂³×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
6	C₂²×C₄²×C₁₃₁	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₁₃₁	C₄	1	16	yes	direct product factor
8	C₂²×C₁₆×C₁₃₁	C₂²	1	12	yes	—
9	C₂×C₄×C₈×C₁₃₁	C₄	1	6	yes	—
10	C₂×C₄×C₈×C₁₃₁	C₂²	1	6	yes	—
11	C₄×C₁₆×C₁₃₁	C₂²	1	16	yes	direct product factor
12	C₂³×C₄×C₁₃₁	C₈	1	2	yes	—
13	C₂³×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₈×C₁₃₁	C₈	1	8	yes	—
15	C₂²×C₈×C₁₃₁	C₂×C₄	1	3	yes	—
16	C₂²×C₈×C₁₃₁	C₂×C₄	1	6	yes	—
17	C₂²×C₈×C₁₃₁	C₂×C₄	1	8	yes	—
18	C₂²×C₈×C₁₃₁	C₂³	1	3	yes	—
19	C₂×C₄²×C₁₃₁	C₈	1	6	yes	—
20	C₂×C₄²×C₁₃₁	C₂×C₄	1	6	yes	—
21	C₂×C₁₆×C₁₃₁	C₂×C₄	1	48	yes	direct product factor
22	C₂×C₁₆×C₁₃₁	C₂³	1	8	yes	—
23	C₄×C₈×C₁₃₁	C₂×C₄	1	12	yes	—
24	C₄×C₈×C₁₃₁	C₂³	1	4	yes	—
25	C₂⁴×C₁₃₁	C₂×C₈	1	1	yes	characteristic
26	C₂²×C₄×C₁₃₁	C₁₆	1	16	yes	direct product factor
27	C₂²×C₄×C₁₃₁	C₄²	1	4	yes	—
28	C₂²×C₄×C₁₃₁	C₂×C₈	1	3	yes	—
29	C₂²×C₄×C₁₃₁	C₂×C₈	1	4	yes	—
30	C₂²×C₄×C₁₃₁	C₂×C₈	1	12	yes	—
31	C₂²×C₄×C₁₃₁	C₂²×C₄	1	3	yes	—
32	C₂×C₈×C₁₃₁	C₄²	1	12	yes	—
33	C₂×C₈×C₁₃₁	C₂×C₈	1	24	yes	—
34	C₂×C₈×C₁₃₁	C₂²×C₄	1	3	yes	—
35	C₂×C₈×C₁₃₁	C₂²×C₄	1	4	yes	—
36	C₂×C₈×C₁₃₁	C₂²×C₄	1	12	yes	—
37	C₂×C₈×C₁₃₁	C₂⁴	1	1	yes	characteristic
38	C₄²×C₁₃₁	C₂×C₈	1	12	yes	—
39	C₄²×C₁₃₁	C₂²×C₄	1	4	yes	—
40	C₁₆×C₁₃₁	C₂²×C₄	1	16	yes	direct product factor
41	C₂³×C₁₃₁	C₄×C₈	1	4	yes	—
42	C₂³×C₁₃₁	C₂×C₁₆	1	8	yes	—
43	C₂³×C₁₃₁	C₂²×C₈	1	3	yes	—
44	C₂×C₄×C₁₃₁	C₄×C₈	1	12	yes	—
45	C₂×C₄×C₁₃₁	C₂×C₁₆	1	48	yes	direct product factor
46	C₂×C₄×C₁₃₁	C₂×C₄²	1	6	yes	—
47	C₂×C₄×C₁₃₁	C₂²×C₈	1	3	yes	—
48	C₂×C₄×C₁₃₁	C₂²×C₈	1	6	yes	—
49	C₂×C₄×C₁₃₁	C₂²×C₈	1	8	yes	—
50	C₂×C₄×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
51	C₈×C₁₃₁	C₂×C₄²	1	6	yes	—
52	C₈×C₁₃₁	C₂²×C₈	1	8	yes	—
53	C₈×C₁₃₁	C₂³×C₄	1	2	yes	—
54	C₂²×C₁₃₁	C₄×C₁₆	1	16	yes	direct product factor
55	C₂²×C₁₃₁	C₂×C₄×C₈	1	6	yes	—
56	C₂²×C₁₃₁	C₂²×C₁₆	1	12	yes	—
57	C₂²×C₁₃₁	C₂³×C₈	1	1	yes	characteristic
58	C₄×C₁₃₁	C₂×C₄×C₈	1	6	yes	—
59	C₄×C₁₃₁	C₂²×C₁₆	1	16	yes	direct product factor
60	C₄×C₁₃₁	C₂²×C₄²	1	1	yes	characteristic
61	C₄×C₁₃₁	C₂³×C₈	1	1	yes	characteristic
62	C₂×C₁₃₁	C₂×C₄×C₁₆	1	12	yes	direct product factor
63	C₂×C₁₃₁	C₂²×C₄×C₈	1	1	yes	characteristic
64	C₂×C₁₃₁	C₂³×C₁₆	1	2	yes	—
65	C₂²×C₄×C₁₆	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₁₃₁	C₂²×C₄×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
67	C₂³×C₁₆	C₂×C₁₃₁	1	2	yes	—
68	C₂²×C₄×C₈	C₂×C₁₃₁	1	1	yes	characteristic
69	C₂×C₄×C₁₆	C₂×C₁₃₁	1	12	yes	direct product factor
70	C₂³×C₈	C₄×C₁₃₁	1	1	yes	characteristic
71	C₂³×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
72	C₂²×C₄²	C₄×C₁₃₁	1	1	yes	characteristic
73	C₂²×C₁₆	C₄×C₁₃₁	1	16	yes	direct product factor
74	C₂²×C₁₆	C₂²×C₁₃₁	1	12	yes	—
75	C₂×C₄×C₈	C₄×C₁₃₁	1	6	yes	—
76	C₂×C₄×C₈	C₂²×C₁₃₁	1	6	yes	—
77	C₄×C₁₆	C₂²×C₁₃₁	1	16	yes	direct product factor
78	C₂³×C₄	C₈×C₁₃₁	1	2	yes	—
79	C₂³×C₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
80	C₂²×C₈	C₈×C₁₃₁	1	8	yes	—
81	C₂²×C₈	C₂×C₄×C₁₃₁	1	3	yes	—
82	C₂²×C₈	C₂×C₄×C₁₃₁	1	6	yes	—
83	C₂²×C₈	C₂×C₄×C₁₃₁	1	8	yes	—
84	C₂²×C₈	C₂³×C₁₃₁	1	3	yes	—
85	C₂×C₄²	C₈×C₁₃₁	1	6	yes	—
86	C₂×C₄²	C₂×C₄×C₁₃₁	1	6	yes	—
87	C₂×C₁₆	C₂×C₄×C₁₃₁	1	48	yes	direct product factor
88	C₂×C₁₆	C₂³×C₁₃₁	1	8	yes	—
89	C₄×C₈	C₂×C₄×C₁₃₁	1	12	yes	—
90	C₄×C₈	C₂³×C₁₃₁	1	4	yes	—
91	C₂⁴	C₂×C₈×C₁₃₁	1	1	yes	characteristic
92	C₂²×C₄	C₁₆×C₁₃₁	1	16	yes	direct product factor
93	C₂²×C₄	C₄²×C₁₃₁	1	4	yes	—
94	C₂²×C₄	C₂×C₈×C₁₃₁	1	3	yes	—
95	C₂²×C₄	C₂×C₈×C₁₃₁	1	4	yes	—
96	C₂²×C₄	C₂×C₈×C₁₃₁	1	12	yes	—
97	C₂²×C₄	C₂²×C₄×C₁₃₁	1	3	yes	—
98	C₂×C₈	C₄²×C₁₃₁	1	12	yes	—
99	C₂×C₈	C₂×C₈×C₁₃₁	1	24	yes	—
100	C₂×C₈	C₂²×C₄×C₁₃₁	1	3	yes	—
101	C₂×C₈	C₂²×C₄×C₁₃₁	1	4	yes	—
102	C₂×C₈	C₂²×C₄×C₁₃₁	1	12	yes	—
103	C₂×C₈	C₂⁴×C₁₃₁	1	1	yes	characteristic
104	C₄²	C₂×C₈×C₁₃₁	1	12	yes	—
105	C₄²	C₂²×C₄×C₁₃₁	1	4	yes	—
106	C₁₆	C₂²×C₄×C₁₃₁	1	16	yes	direct product factor
107	C₂³	C₄×C₈×C₁₃₁	1	4	yes	—
108	C₂³	C₂×C₁₆×C₁₃₁	1	8	yes	—
109	C₂³	C₂²×C₈×C₁₃₁	1	3	yes	—
110	C₂×C₄	C₄×C₈×C₁₃₁	1	12	yes	—
111	C₂×C₄	C₂×C₁₆×C₁₃₁	1	48	yes	direct product factor
112	C₂×C₄	C₂×C₄²×C₁₃₁	1	6	yes	—
113	C₂×C₄	C₂²×C₈×C₁₃₁	1	3	yes	—
114	C₂×C₄	C₂²×C₈×C₁₃₁	1	6	yes	—
115	C₂×C₄	C₂²×C₈×C₁₃₁	1	8	yes	—
116	C₂×C₄	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
117	C₈	C₂×C₄²×C₁₃₁	1	6	yes	—
118	C₈	C₂²×C₈×C₁₃₁	1	8	yes	—
119	C₈	C₂³×C₄×C₁₃₁	1	2	yes	—
120	C₂²	C₄×C₁₆×C₁₃₁	1	16	yes	direct product factor
121	C₂²	C₂×C₄×C₈×C₁₃₁	1	6	yes	—
122	C₂²	C₂²×C₁₆×C₁₃₁	1	12	yes	—
123	C₂²	C₂³×C₈×C₁₃₁	1	1	yes	characteristic
124	C₄	C₂×C₄×C₈×C₁₃₁	1	6	yes	—
125	C₄	C₂²×C₁₆×C₁₃₁	1	16	yes	direct product factor
126	C₄	C₂²×C₄²×C₁₃₁	1	1	yes	characteristic
127	C₄	C₂³×C₈×C₁₃₁	1	1	yes	characteristic
128	C₂	C₂×C₄×C₁₆×C₁₃₁	1	12	yes	direct product factor
129	C₂	C₂²×C₄×C₈×C₁₃₁	1	1	yes	characteristic
130	C₂	C₂³×C₁₆×C₁₃₁	1	2	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₁
socle, prime radical	C₂⁴×C₁₃₁	C₂×C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×C₁₆×C₁₃₁	C₂	direct (N × H)
7. C₂²×C₁₆×C₁₃₁	C₄	direct (N × H)
11. C₄×C₁₆×C₁₃₁	C₂²	direct (N × H)
21. C₂×C₁₆×C₁₃₁	C₂×C₄	direct (N × H)
26. C₂²×C₄×C₁₃₁	C₁₆	direct (N × H)
40. C₁₆×C₁₃₁	C₂²×C₄	direct (N × H)
45. C₂×C₄×C₁₃₁	C₂×C₁₆	direct (N × H)
54. C₂²×C₁₃₁	C₄×C₁₆	direct (N × H)
59. C₄×C₁₃₁	C₂²×C₁₆	direct (N × H)
62. C₂×C₁₃₁	C₂×C₄×C₁₆	direct (N × H)
65. C₂²×C₄×C₁₆	C₁₃₁	direct (N × H)
66. C₁₃₁	C₂²×C₄×C₁₆	direct (N × H)
69. C₂×C₄×C₁₆	C₂×C₁₃₁	direct (N × H)
73. C₂²×C₁₆	C₄×C₁₃₁	direct (N × H)
77. C₄×C₁₆	C₂²×C₁₃₁	direct (N × H)
87. C₂×C₁₆	C₂×C₄×C₁₃₁	direct (N × H)
92. C₂²×C₄	C₁₆×C₁₃₁	direct (N × H)
106. C₁₆	C₂²×C₄×C₁₃₁	direct (N × H)
111. C₂×C₄	C₂×C₁₆×C₁₃₁	direct (N × H)
120. C₂²	C₄×C₁₆×C₁₃₁	direct (N × H)
125. C₄	C₂²×C₁₆×C₁₃₁	direct (N × H)
128. C₂	C₂×C₄×C₁₆×C₁₃₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₁₆×C₁₃₁	2
C₂²×C₄×C₈×C₁₃₁	2
C₂×C₄×C₁₆×C₁₃₁	2
C₂²×C₄×C₁₆	131

Subgroups — C22×C4×C16×C131 #45719204

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×C₁₆×C₁₃₁ #45719204