Subgroups — C₂×C₄×M₁₆×C₂₅₁ #44547458

Order 32128.

656 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	C₂²×M₁₆×C₂₅₁	C₂	1	2	yes	maximal
2	C₂²×C₄²×C₂₅₁	C₂	1	1	yes	maximal, characteristic
3	C₄×M₁₆×C₂₅₁	C₂	1	8	yes	maximal, direct product factor
4	C₂×#97×C₂₅₁	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₈×C₂₅₁	C₂	1	2	yes	maximal, complemented
6	C₂³×C₄×C₂₅₁	C₄	1	2	yes	—
7	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
8	C₂×M₁₆×C₂₅₁	C₄	1	16	yes	direct product factor
9	C₂×M₁₆×C₂₅₁	C₂²	1	8	yes	—
10	C₂²×C₈×C₂₅₁	C₂²	1	4	yes	—
11	C₂×C₄²×C₂₅₁	C₄	1	2	yes	—
12	C₂×C₄²×C₂₅₁	C₄	1	4	yes	—
13	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₂₅₁	C₂²	1	4	yes	—
15	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
16	#97×C₂₅₁	C₂²	1	8	yes	complemented
17	C₄×C₈×C₂₅₁	C₂²	1	8	yes	complemented
18	C₂⁴×C₂₅₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₂₅₁	—	2	8	no	—
20	C₂²×C₄×C₂₅₁	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₂₅₁	C₂×C₄	1	8	yes	—
22	C₂²×C₄×C₂₅₁	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₂₅₁	C₂×C₄	1	8	yes	—
25	C₂²×C₄×C₂₅₁	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
27	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
28	M₁₆×C₂₅₁	C₂×C₄	1	32	yes	direct product factor
29	C₂×C₈×C₂₅₁	C₂×C₄	1	16	yes	complemented
30	C₂×C₈×C₂₅₁	C₂³	1	8	yes	—
31	C₄²×C₂₅₁	C₂×C₄	1	2	yes	—
32	C₄²×C₂₅₁	C₂×C₄	1	8	yes	—
33	C₄²×C₂₅₁	C₂×C₄	1	2	yes	—
34	C₄²×C₂₅₁	C₂³	1	2	yes	—
35	C₄²×C₂₅₁	C₂³	1	2	yes	—
36	C₂³×C₂₅₁	—	2	4	no	—
37	C₂³×C₂₅₁	C₄²	1	4	yes	—
38	C₂³×C₂₅₁	C₂²×C₄	1	2	yes	—
39	C₂³×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₂₅₁	—	2	4	no	—
41	C₂×C₄×C₂₅₁	—	2	16	no	—
42	C₂×C₄×C₂₅₁	C₄²	1	8	yes	—
43	C₂×C₄×C₂₅₁	C₄²	1	4	yes	—
44	C₂×C₄×C₂₅₁	M₁₆	1	8	yes	direct product factor
45	C₂×C₄×C₂₅₁	C₂²×C₄	1	4	yes	—
46	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₂₅₁	C₂²×C₄	1	4	yes	—
49	C₂×C₄×C₂₅₁	C₂²×C₄	1	8	yes	—
50	C₂×C₄×C₂₅₁	C₂²×C₄	1	4	yes	—
51	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
52	C₈×C₂₅₁	C₂²×C₄	1	16	yes	complemented
53	C₂²×C₂₅₁	—	2	4	no	—
54	C₂²×C₂₅₁	—	2	8	no	—
55	C₂²×C₂₅₁	C₂×C₄²	1	4	yes	—
56	C₂²×C₂₅₁	C₂×C₄²	1	2	yes	—
57	C₂²×C₂₅₁	C₂×M₁₆	1	4	yes	—
58	C₂²×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
59	C₄×C₂₅₁	—	2	4	no	—
60	C₄×C₂₅₁	C₂×C₄²	1	2	yes	—
61	C₄×C₂₅₁	C₂×C₄²	1	4	yes	—
62	C₄×C₂₅₁	C₂×M₁₆	1	8	yes	direct product factor
63	C₄×C₂₅₁	C₂³×C₄	1	2	yes	—
64	C₂×C₂₅₁	—	2	4	no	—
65	C₂×C₂₅₁	C₄×M₁₆	1	4	yes	direct product factor
66	C₂×C₂₅₁	C₂²×C₄²	1	1	yes	characteristic
67	C₂×C₂₅₁	C₂²×M₁₆	1	2	yes	—
68	C₂₅₁	C₂×C₄×M₁₆	1	1	yes	characteristic, direct product factor, Sylow
69	C₂×C₄×M₁₆	C₂₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂²×M₁₆	C₂×C₂₅₁	1	2	yes	—
71	C₂²×C₄²	C₂×C₂₅₁	1	1	yes	characteristic
72	C₄×M₁₆	C₂×C₂₅₁	1	8	yes	direct product factor
73	C₂×#97	C₂×C₂₅₁	1	2	yes	complemented
74	C₂×C₄×C₈	C₂×C₂₅₁	1	2	yes	complemented
75	C₂³×C₄	C₄×C₂₅₁	1	2	yes	—
76	C₂³×C₄	C₂²×C₂₅₁	1	1	yes	characteristic
77	C₂×M₁₆	C₄×C₂₅₁	1	16	yes	direct product factor
78	C₂×M₁₆	C₂²×C₂₅₁	1	8	yes	—
79	C₂²×C₈	C₂²×C₂₅₁	1	4	yes	—
80	C₂×C₄²	C₄×C₂₅₁	1	4	yes	—
81	C₂×C₄²	C₄×C₂₅₁	1	2	yes	—
82	C₂×C₄²	C₂²×C₂₅₁	1	1	yes	characteristic
83	C₂×C₄²	C₂²×C₂₅₁	1	4	yes	—
84	C₂×C₄²	C₂²×C₂₅₁	1	1	yes	characteristic
85	#97	C₂²×C₂₅₁	1	8	yes	complemented
86	C₄×C₈	C₂²×C₂₅₁	1	8	yes	complemented
87	C₂⁴	C₂×C₄×C₂₅₁	1	1	yes	characteristic
88	C₂²×C₄	—	2	8	no	—
89	C₂²×C₄	C₂×C₄×C₂₅₁	1	2	yes	—
90	C₂²×C₄	C₂×C₄×C₂₅₁	1	8	yes	—
91	C₂²×C₄	C₂×C₄×C₂₅₁	1	2	yes	—
92	C₂²×C₄	C₂×C₄×C₂₅₁	1	1	yes	characteristic
93	C₂²×C₄	C₂×C₄×C₂₅₁	1	8	yes	—
94	C₂²×C₄	C₂×C₄×C₂₅₁	1	2	yes	—
95	C₂²×C₄	C₂³×C₂₅₁	1	1	yes	characteristic
96	C₂²×C₄	C₂³×C₂₅₁	1	2	yes	—
97	M₁₆	C₂×C₄×C₂₅₁	1	32	yes	direct product factor
98	C₂×C₈	C₂×C₄×C₂₅₁	1	16	yes	complemented
99	C₂×C₈	C₂³×C₂₅₁	1	8	yes	—
100	C₄²	C₂×C₄×C₂₅₁	1	2	yes	—
101	C₄²	C₂×C₄×C₂₅₁	1	8	yes	—
102	C₄²	C₂×C₄×C₂₅₁	1	2	yes	—
103	C₄²	C₂³×C₂₅₁	1	2	yes	—
104	C₄²	C₂³×C₂₅₁	1	2	yes	—
105	C₂³	—	2	4	no	—
106	C₂³	C₄²×C₂₅₁	1	4	yes	—
107	C₂³	C₂²×C₄×C₂₅₁	1	2	yes	—
108	C₂³	C₂²×C₄×C₂₅₁	1	1	yes	characteristic
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	2	16	no	—
111	C₂×C₄	C₄²×C₂₅₁	1	8	yes	—
112	C₂×C₄	C₄²×C₂₅₁	1	4	yes	—
113	C₂×C₄	M₁₆×C₂₅₁	1	8	yes	direct product factor
114	C₂×C₄	C₂²×C₄×C₂₅₁	1	4	yes	—
115	C₂×C₄	C₂²×C₄×C₂₅₁	1	1	yes	characteristic
116	C₂×C₄	C₂²×C₄×C₂₅₁	1	2	yes	—
117	C₂×C₄	C₂²×C₄×C₂₅₁	1	4	yes	—
118	C₂×C₄	C₂²×C₄×C₂₅₁	1	8	yes	—
119	C₂×C₄	C₂²×C₄×C₂₅₁	1	4	yes	—
120	C₂×C₄	C₂⁴×C₂₅₁	1	1	yes	characteristic
121	C₈	C₂²×C₄×C₂₅₁	1	16	yes	complemented
122	C₂²	—	2	8	no	—
123	C₂²	—	2	4	no	—
124	C₂²	C₂×C₄²×C₂₅₁	1	4	yes	—
125	C₂²	C₂×C₄²×C₂₅₁	1	2	yes	—
126	C₂²	C₂×M₁₆×C₂₅₁	1	4	yes	—
127	C₂²	C₂³×C₄×C₂₅₁	1	1	yes	characteristic
128	C₄	—	2	4	no	—
129	C₄	C₂×C₄²×C₂₅₁	1	2	yes	—
130	C₄	C₂×C₄²×C₂₅₁	1	4	yes	—
131	C₄	C₂×M₁₆×C₂₅₁	1	8	yes	direct product factor
132	C₄	C₂³×C₄×C₂₅₁	1	2	yes	—
133	C₂	—	2	4	no	—
134	C₂	C₄×M₁₆×C₂₅₁	1	4	yes	direct product factor
135	C₂	C₂²×C₄²×C₂₅₁	1	1	yes	characteristic
136	C₂	C₂²×M₁₆×C₂₅₁	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₄×M₁₆×C₂₅₁	C₂	direct (N × H)
4. C₂×#97×C₂₅₁	C₂	semidirect
5. C₂×C₄×C₈×C₂₅₁	C₂	semidirect
8. C₂×M₁₆×C₂₅₁	C₄	direct (N × H)
16. #97×C₂₅₁	C₂²	semidirect
17. C₄×C₈×C₂₅₁	C₂²	semidirect
28. M₁₆×C₂₅₁	C₂×C₄	direct (N × H)
29. C₂×C₈×C₂₅₁	C₂×C₄	semidirect
44. C₂×C₄×C₂₅₁	M₁₆	direct (N × H)
52. C₈×C₂₅₁	C₂²×C₄	semidirect
62. C₄×C₂₅₁	C₂×M₁₆	direct (N × H)
65. C₂×C₂₅₁	C₄×M₁₆	direct (N × H)
68. C₂₅₁	C₂×C₄×M₁₆	direct (N × H)
69. C₂×C₄×M₁₆	C₂₅₁	direct (N × H)
72. C₄×M₁₆	C₂×C₂₅₁	direct (N × H)
73. C₂×#97	C₂×C₂₅₁	semidirect
74. C₂×C₄×C₈	C₂×C₂₅₁	semidirect
77. C₂×M₁₆	C₄×C₂₅₁	direct (N × H)
85. #97	C₂²×C₂₅₁	semidirect
86. C₄×C₈	C₂²×C₂₅₁	semidirect
97. M₁₆	C₂×C₄×C₂₅₁	direct (N × H)
98. C₂×C₈	C₂×C₄×C₂₅₁	semidirect
113. C₂×C₄	M₁₆×C₂₅₁	direct (N × H)
121. C₈	C₂²×C₄×C₂₅₁	semidirect
131. C₄	C₂×M₁₆×C₂₅₁	direct (N × H)
134. C₂	C₄×M₁₆×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
C₂²×M₁₆×C₂₅₁	2
C₂²×C₄²×C₂₅₁	2
C₄×M₁₆×C₂₅₁	2
C₂×#97×C₂₅₁	2
C₂×C₄×C₈×C₂₅₁	2
C₂×C₄×M₁₆	251

Subgroups — C2×C4×M16×C251 #44547458

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×M₁₆×C₂₅₁ #44547458