Subgroups — C₄×#131×C₂₅₁ #44547461

Order 32128.

572 conjugacy classes of subgroups.

The subgroup lattice has 108 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#131×C₂₅₁	C₂	1	2	yes	maximal
2	C₄×#41×C₂₅₁	C₂	1	1	yes	maximal, characteristic
3	#405×C₂₅₁	C₂	1	6	yes	maximal, complemented
4	C₄×M₁₆×C₂₅₁	C₂	1	3	yes	maximal, complemented
5	C₂×C₄×C₈×C₂₅₁	C₂	1	3	yes	maximal, complemented
6	C₂×#41×C₂₅₁	C₂²	1	1	yes	characteristic
7	#131×C₂₅₁	C₄	1	16	yes	direct product factor
8	C₂×M₁₆×C₂₅₁	C₂²	1	6	yes	—
9	C₂²×C₈×C₂₅₁	C₂²	1	6	yes	—
10	C₄×Q₈×C₂₅₁	C₄	1	2	yes	—
11	C₄×D₄×C₂₅₁	C₄	1	6	yes	—
12	#117×C₂₅₁	C₂²	1	3	yes	—
13	C₂×C₄²×C₂₅₁	C₂²	1	3	yes	—
14	#97×C₂₅₁	C₂²	1	6	yes	—
15	C₄×C₈×C₂₅₁	C₂²	1	3	yes	—
16	C₄×C₈×C₂₅₁	C₂²	1	6	yes	—
17	C₄×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
18	#41×C₂₅₁	C₂×C₄	1	8	yes	—
19	C₂×Q₈×C₂₅₁	C₂×C₄	1	1	yes	characteristic
20	C₂×D₄×C₂₅₁	C₂×C₄	1	3	yes	—
21	C₂²×C₄×C₂₅₁	C₂×C₄	1	6	yes	—
22	C₂²×C₄×C₂₅₁	C₂³	1	3	yes	—
23	M₁₆×C₂₅₁	C₂×C₄	1	24	yes	complemented
24	C₂×C₈×C₂₅₁	C₂×C₄	1	24	yes	complemented
25	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
26	C₂×C₈×C₂₅₁	C₂³	1	6	yes	—
27	C₄⋊C₄×C₂₅₁	C₂×C₄	1	6	yes	—
28	C₂²⋊C₄×C₂₅₁	C₂×C₄	1	6	yes	—
29	C₄²×C₂₅₁	C₂×C₄	1	6	yes	—
30	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
31	C₄²×C₂₅₁	C₂³	1	3	yes	—
32	C₂³×C₂₅₁	C₂²×C₄	1	3	yes	—
33	Q₈×C₂₅₁	C₄²	1	4	yes	—
34	D₄×C₂₅₁	C₄²	1	12	yes	—
35	C₂×C₄×C₂₅₁	—	2	12	no	—
36	C₂×C₄×C₂₅₁	C₂²×C₄	1	6	yes	—
37	C₂×C₄×C₂₅₁	C₂²×C₄	1	3	yes	—
38	C₂×C₄×C₂₅₁	C₂²×C₄	1	12	yes	—
39	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
40	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
41	C₈×C₂₅₁	C₂²×C₄	1	4	yes	—
42	C₈×C₂₅₁	C₂²×C₄	1	12	yes	—
43	C₂²×C₂₅₁	—	2	6	no	—
44	C₂²×C₂₅₁	C₂×C₄²	1	6	yes	—
45	C₂²×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
46	C₄×C₂₅₁	—	2	6	no	—
47	C₄×C₂₅₁	C₂×C₄²	1	6	yes	—
48	C₄×C₂₅₁	#131	1	4	yes	direct product factor
49	C₄×C₂₅₁	C₂³×C₄	1	2	yes	—
50	C₂×C₂₅₁	—	2	6	no	—
51	C₂×C₂₅₁	C₂²×C₄²	1	1	yes	characteristic
52	C₂×C₂₅₁	C₂×#131	1	2	yes	—
53	C₂₅₁	C₄×#131	1	1	yes	characteristic, direct product factor, Sylow
54	C₄×#131	C₂₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂×#131	C₂×C₂₅₁	1	2	yes	—
56	C₄×#41	C₂×C₂₅₁	1	1	yes	characteristic
57	#405	C₂×C₂₅₁	1	6	yes	complemented
58	C₄×M₁₆	C₂×C₂₅₁	1	3	yes	complemented
59	C₂×C₄×C₈	C₂×C₂₅₁	1	3	yes	complemented
60	C₂×#41	C₂²×C₂₅₁	1	1	yes	characteristic
61	#131	C₄×C₂₅₁	1	16	yes	direct product factor
62	C₂×M₁₆	C₂²×C₂₅₁	1	6	yes	—
63	C₂²×C₈	C₂²×C₂₅₁	1	6	yes	—
64	C₄×Q₈	C₄×C₂₅₁	1	2	yes	—
65	C₄×D₄	C₄×C₂₅₁	1	6	yes	—
66	#117	C₂²×C₂₅₁	1	3	yes	—
67	C₂×C₄²	C₂²×C₂₅₁	1	3	yes	—
68	#97	C₂²×C₂₅₁	1	6	yes	—
69	C₄×C₈	C₂²×C₂₅₁	1	3	yes	—
70	C₄×C₈	C₂²×C₂₅₁	1	6	yes	—
71	C₄×C₈	C₂²×C₂₅₁	1	1	yes	characteristic
72	#41	C₂×C₄×C₂₅₁	1	8	yes	—
73	C₂×Q₈	C₂×C₄×C₂₅₁	1	1	yes	characteristic
74	C₂×D₄	C₂×C₄×C₂₅₁	1	3	yes	—
75	C₂²×C₄	C₂×C₄×C₂₅₁	1	6	yes	—
76	C₂²×C₄	C₂³×C₂₅₁	1	3	yes	—
77	M₁₆	C₂×C₄×C₂₅₁	1	24	yes	complemented
78	C₂×C₈	C₂×C₄×C₂₅₁	1	24	yes	complemented
79	C₂×C₈	C₂³×C₂₅₁	1	2	yes	—
80	C₂×C₈	C₂³×C₂₅₁	1	6	yes	—
81	C₄⋊C₄	C₂×C₄×C₂₅₁	1	6	yes	—
82	C₂²⋊C₄	C₂×C₄×C₂₅₁	1	6	yes	—
83	C₄²	C₂×C₄×C₂₅₁	1	6	yes	—
84	C₄²	C₂³×C₂₅₁	1	1	yes	characteristic
85	C₄²	C₂³×C₂₅₁	1	3	yes	—
86	C₂³	C₂²×C₄×C₂₅₁	1	3	yes	—
87	Q₈	C₄²×C₂₅₁	1	4	yes	—
88	D₄	C₄²×C₂₅₁	1	12	yes	—
89	C₂×C₄	—	2	12	no	—
90	C₂×C₄	C₂²×C₄×C₂₅₁	1	6	yes	—
91	C₂×C₄	C₂²×C₄×C₂₅₁	1	3	yes	—
92	C₂×C₄	C₂²×C₄×C₂₅₁	1	12	yes	—
93	C₂×C₄	C₂²×C₄×C₂₅₁	1	2	yes	—
94	C₂×C₄	C₂⁴×C₂₅₁	1	1	yes	characteristic
95	C₈	C₂²×C₄×C₂₅₁	1	4	yes	—
96	C₈	C₂²×C₄×C₂₅₁	1	12	yes	—
97	C₂²	—	2	6	no	—
98	C₂²	C₂×C₄²×C₂₅₁	1	6	yes	—
99	C₂²	C₂³×C₄×C₂₅₁	1	1	yes	characteristic
100	C₄	—	2	6	no	—
101	C₄	C₂×C₄²×C₂₅₁	1	6	yes	—
102	C₄	#131×C₂₅₁	1	4	yes	direct product factor
103	C₄	C₂³×C₄×C₂₅₁	1	2	yes	—
104	C₂	—	2	6	no	—
105	C₂	C₂²×C₄²×C₂₅₁	1	1	yes	characteristic
106	C₂	C₂×#131×C₂₅₁	1	2	yes	—

	subgroup	G/H
center	C₄×C₈×C₂₅₁	C₂²
commutator	C₂	C₂²×C₄²×C₂₅₁
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₅₁
socle	C₂²×C₂₅₁	C₂³×C₄
prime radical	C₂×#41×C₂₅₁	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈×C₂₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. #405×C₂₅₁	C₂	semidirect
4. C₄×M₁₆×C₂₅₁	C₂	semidirect
5. C₂×C₄×C₈×C₂₅₁	C₂	semidirect
7. #131×C₂₅₁	C₄	direct (N × H)
23. M₁₆×C₂₅₁	C₂×C₄	semidirect
24. C₂×C₈×C₂₅₁	C₂×C₄	semidirect
48. C₄×C₂₅₁	#131	direct (N × H)
53. C₂₅₁	C₄×#131	direct (N × H)
54. C₄×#131	C₂₅₁	direct (N × H)
57. #405	C₂×C₂₅₁	semidirect
58. C₄×M₁₆	C₂×C₂₅₁	semidirect
59. C₂×C₄×C₈	C₂×C₂₅₁	semidirect
61. #131	C₄×C₂₅₁	direct (N × H)
77. M₁₆	C₂×C₄×C₂₅₁	semidirect
78. C₂×C₈	C₂×C₄×C₂₅₁	semidirect
102. C₄	#131×C₂₅₁	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×#131×C₂₅₁	2
C₄×#41×C₂₅₁	2
#405×C₂₅₁	2
C₄×M₁₆×C₂₅₁	2
C₂×C₄×C₈×C₂₅₁	2
C₄×#131	251

Subgroups — C4×#131×C251 #44547461

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#131×C₂₅₁ #44547461