Subgroups — C₂²×M₁₆×D₂₅₁ #44565065

Order 32128.

1720 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×D₂₅₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×#16860794	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#16860793	C₂	1	2	yes	maximal, complemented
4	C₂×M₁₆×D₂₅₁	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₈×D₂₅₁	C₂	1	2	yes	maximal, complemented
6	C₂²×M₁₆×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁴×D₂₅₁	C₄	1	1	yes	characteristic
8	C₂³×Dic₂₅₁	C₄	1	1	yes	characteristic
9	C₂²×C₄×D₂₅₁	C₄	1	12	yes	—
10	C₂²×C₄×D₂₅₁	C₄	1	1	yes	characteristic
11	C₂²×C₄×D₂₅₁	C₄	1	1	yes	characteristic
12	C₂²×C₄×D₂₅₁	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₂₅₁	C₂²	1	12	yes	—
14	C₂²×C₄×D₂₅₁	C₂²	1	1	yes	characteristic
15	C₂×#16860794	C₂²	1	12	yes	complemented
16	C₂×#16860793	C₂²	1	24	yes	complemented
17	M₁₆×D₂₅₁	C₂²	1	64	yes	direct product factor
18	C₂²×C₂₅₁⋊C₈	C₂²	1	2	yes	complemented
19	C₂×C₈×D₂₅₁	C₂²	1	24	yes	complemented
20	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
21	C₂×M₁₆×C₂₅₁	C₂²	1	12	yes	complemented
22	C₂²×C₈×C₂₅₁	C₂²	1	2	yes	complemented
23	C₂³×D₂₅₁	—	2	8	no	—
24	C₂³×D₂₅₁	C₂×C₄	1	12	yes	—
25	C₂³×D₂₅₁	C₂×C₄	1	1	yes	characteristic
26	C₂³×D₂₅₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×Dic₂₅₁	C₂×C₄	1	12	yes	—
28	C₂²×Dic₂₅₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×Dic₂₅₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×D₂₅₁	C₂×C₄	1	48	yes	—
31	C₂×C₄×D₂₅₁	C₂×C₄	1	12	yes	—
32	C₂×C₄×D₂₅₁	C₂×C₄	1	6	yes	—
33	C₂×C₄×D₂₅₁	C₂×C₄	1	6	yes	—
34	C₂×C₄×D₂₅₁	C₂×C₄	1	12	yes	—
35	C₂×C₄×D₂₅₁	C₂³	1	6	yes	—
36	C₂×C₄×D₂₅₁	C₂³	1	16	yes	—
37	C₂×C₄×D₂₅₁	C₂³	1	6	yes	—
38	#16860794	C₂³	1	16	yes	complemented
39	#16860793	C₂³	1	32	yes	complemented
40	C₂×C₂₅₁⋊C₈	C₂³	1	12	yes	complemented
41	C₈×D₂₅₁	C₂³	1	32	yes	complemented
42	C₂⁴×C₂₅₁	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₂₅₁	C₂×C₄	1	6	yes	—
44	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₂₅₁	C₂³	1	6	yes	—
46	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
47	M₁₆×C₂₅₁	C₂³	1	16	yes	complemented
48	C₂×C₈×C₂₅₁	C₂³	1	12	yes	complemented
49	C₂²×D₂₅₁	—	2	4	no	—
50	C₂²×D₂₅₁	—	2	48	no	—
51	C₂²×D₂₅₁	M₁₆	1	8	yes	direct product factor
52	C₂²×D₂₅₁	C₂²×C₄	1	6	yes	—
53	C₂²×D₂₅₁	C₂²×C₄	1	16	yes	—
54	C₂²×D₂₅₁	C₂²×C₄	1	6	yes	—
55	C₂×Dic₂₅₁	C₂²×C₄	1	16	yes	—
56	C₂×Dic₂₅₁	C₂²×C₄	1	6	yes	—
57	C₂×Dic₂₅₁	C₂²×C₄	1	6	yes	—
58	C₄×D₂₅₁	C₂²×C₄	1	16	yes	—
59	C₄×D₂₅₁	C₂²×C₄	1	12	yes	—
60	C₄×D₂₅₁	C₂²×C₄	1	12	yes	—
61	C₄×D₂₅₁	C₂²×C₄	1	16	yes	—
62	C₄×D₂₅₁	C₂⁴	1	4	yes	—
63	C₄×D₂₅₁	C₂⁴	1	4	yes	—
64	C₂₅₁⋊C₈	C₂⁴	1	8	yes	complemented
65	C₂³×C₂₅₁	—	2	4	no	—
66	C₂³×C₂₅₁	C₂²×C₄	1	6	yes	—
67	C₂³×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₂₅₁	C₂²×C₄	1	12	yes	—
69	C₂×C₄×C₂₅₁	C₂²×C₄	1	6	yes	—
70	C₂×C₄×C₂₅₁	C₂²×C₄	1	3	yes	—
71	C₂×C₄×C₂₅₁	C₂⁴	1	3	yes	—
72	C₂×C₄×C₂₅₁	C₂⁴	1	4	yes	—
73	C₈×C₂₅₁	C₂⁴	1	8	yes	complemented
74	C₂²×C₂₅₁	—	2	12	no	—
75	C₂²×C₂₅₁	C₂×M₁₆	1	4	yes	complemented
76	C₂²×C₂₅₁	C₂³×C₄	1	4	yes	—
77	C₂²×C₂₅₁	C₂³×C₄	1	3	yes	—
78	C₂×D₂₅₁	—	2	12	no	—
79	C₂×D₂₅₁	—	2	32	no	—
80	C₂×D₂₅₁	C₂×M₁₆	1	24	yes	direct product factor
81	C₂×D₂₅₁	C₂³×C₄	1	4	yes	—
82	C₂×D₂₅₁	C₂³×C₄	1	4	yes	—
83	C₄×C₂₅₁	C₂³×C₄	1	3	yes	—
84	C₄×C₂₅₁	C₂³×C₄	1	4	yes	—
85	C₄×C₂₅₁	C₂⁵	1	1	yes	characteristic
86	Dic₂₅₁	C₂³×C₄	1	4	yes	—
87	Dic₂₅₁	C₂³×C₄	1	4	yes	—
88	C₂×C₂₅₁	—	2	4	no	—
89	C₂×C₂₅₁	C₂²×M₁₆	1	6	yes	complemented
90	C₂×C₂₅₁	C₂⁴×C₄	1	1	yes	characteristic
91	D₂₅₁	—	2	4	no	—
92	D₂₅₁	C₂²×M₁₆	1	8	yes	direct product factor
93	C₂₅₁	C₂³×M₁₆	1	1	yes	characteristic, complemented, Sylow
94	C₂³×M₁₆	—	251	1	no	maximal, Sylow
95	C₂⁴×C₄	—	251	1	no	—
96	C₂²×M₁₆	—	251	1	no	—
97	C₂²×M₁₆	—	251	24	no	—
98	C₂²×M₁₆	—	251	2	no	—
99	C₂²×M₁₆	D₂₅₁	1	1	yes	characteristic, direct product factor
100	C₂³×C₈	—	251	2	no	—
101	C₂⁵	—	251	1	no	—
102	C₂³×C₄	—	251	1	no	—
103	C₂³×C₄	—	251	12	no	—
104	C₂³×C₄	—	251	1	no	—
105	C₂³×C₄	—	251	1	no	—
106	C₂³×C₄	—	251	12	no	—
107	C₂³×C₄	—	251	1	no	—
108	C₂³×C₄	—	251	1	no	—
109	C₂³×C₄	C₂×D₂₅₁	1	1	yes	characteristic
110	C₂×M₁₆	—	251	12	no	—
111	C₂×M₁₆	—	251	64	no	—
112	C₂×M₁₆	—	251	24	no	—
113	C₂×M₁₆	C₂×D₂₅₁	1	12	yes	direct product factor
114	C₂²×C₈	—	251	2	no	—
115	C₂²×C₈	—	251	24	no	—
116	C₂²×C₈	C₂×D₂₅₁	1	2	yes	complemented
117	C₂⁴	—	502	8	no	—
118	C₂⁴	—	251	12	no	—
119	C₂⁴	—	251	1	no	—
120	C₂⁴	—	251	1	no	—
121	C₂⁴	C₄×D₂₅₁	1	1	yes	characteristic
122	C₂²×C₄	—	251	12	no	—
123	C₂²×C₄	—	251	1	no	—
124	C₂²×C₄	—	251	1	no	—
125	C₂²×C₄	—	251	48	no	—
126	C₂²×C₄	—	251	12	no	—
127	C₂²×C₄	—	251	12	no	—
128	C₂²×C₄	—	251	6	no	—
129	C₂²×C₄	—	251	6	no	—
130	C₂²×C₄	—	251	6	no	—
131	C₂²×C₄	—	251	16	no	—
132	C₂²×C₄	—	251	6	no	—
133	C₂²×C₄	C₄×D₂₅₁	1	1	yes	characteristic
134	C₂²×C₄	C₄×D₂₅₁	1	6	yes	—
135	C₂²×C₄	C₂²×D₂₅₁	1	6	yes	—
136	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
137	M₁₆	—	251	16	no	—
138	M₁₆	—	251	32	no	—
139	M₁₆	C₂²×D₂₅₁	1	16	yes	direct product factor
140	C₂×C₈	—	251	12	no	—
141	C₂×C₈	—	251	32	no	—
142	C₂×C₈	C₂²×D₂₅₁	1	12	yes	complemented
143	C₂³	—	2	4	no	—
144	C₂³	—	502	48	no	—
145	C₂³	—	502	4	no	—
146	C₂³	—	251	16	no	—
147	C₂³	—	251	6	no	—
148	C₂³	—	251	6	no	—
149	C₂³	—	251	8	no	—
150	C₂³	C₂×C₄×D₂₅₁	1	6	yes	—
151	C₂³	C₂×C₄×D₂₅₁	1	1	yes	characteristic
152	C₂×C₄	—	251	16	no	—
153	C₂×C₄	—	251	6	no	—
154	C₂×C₄	—	251	6	no	—
155	C₂×C₄	—	251	16	no	—
156	C₂×C₄	—	251	12	no	—
157	C₂×C₄	—	251	12	no	—
158	C₂×C₄	—	251	16	no	—
159	C₂×C₄	—	251	4	no	—
160	C₂×C₄	—	251	4	no	—
161	C₂×C₄	C₂×C₄×D₂₅₁	1	12	yes	—
162	C₂×C₄	C₂×C₄×D₂₅₁	1	6	yes	—
163	C₂×C₄	C₂×C₄×D₂₅₁	1	3	yes	—
164	C₂×C₄	C₂³×D₂₅₁	1	3	yes	—
165	C₂×C₄	C₂³×D₂₅₁	1	4	yes	—
166	C₈	—	251	8	no	—
167	C₈	C₂³×D₂₅₁	1	8	yes	complemented
168	C₂²	—	2	12	no	—
169	C₂²	—	502	32	no	—
170	C₂²	—	502	12	no	—
171	C₂²	—	251	4	no	—
172	C₂²	—	251	4	no	—
173	C₂²	—	251	24	no	—
174	C₂²	M₁₆×D₂₅₁	1	4	yes	direct product factor
175	C₂²	C₂²×C₄×D₂₅₁	1	4	yes	—
176	C₂²	C₂²×C₄×D₂₅₁	1	3	yes	—
177	C₄	—	251	4	no	—
178	C₄	—	251	4	no	—
179	C₄	C₂²×C₄×D₂₅₁	1	4	yes	—
180	C₄	C₂²×C₄×D₂₅₁	1	3	yes	—
181	C₄	C₂⁴×D₂₅₁	1	1	yes	characteristic
182	C₂	—	2	4	no	—
183	C₂	—	502	4	no	—
184	C₂	—	251	8	no	—
185	C₂	C₂×M₁₆×D₂₅₁	1	6	yes	direct product factor
186	C₂	C₂³×C₄×D₂₅₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₂₅₁
commutator	C₂×C₂₅₁	C₂⁴×C₄
Frattini	C₄	C₂⁴×D₂₅₁
Fitting	C₂²×M₁₆×C₂₅₁	C₂
socle	C₂³×C₂₅₁	C₂²×C₄
prime radical	C₂⁴×D₂₅₁	C₄
cosocle	C₄×C₂₅₁	C₂⁵
hypercenter	C₂²×M₁₆	D₂₅₁
hypocenter	C₂₅₁	C₂³×M₁₆
LCS	G ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂²×#16860794	C₂	semidirect
3. C₂²×#16860793	C₂	semidirect
4. C₂×M₁₆×D₂₅₁	C₂	direct (N × H)
5. C₂²×C₈×D₂₅₁	C₂	semidirect
6. C₂²×M₁₆×C₂₅₁	C₂	semidirect
15. C₂×#16860794	C₂²	semidirect
16. C₂×#16860793	C₂²	semidirect
17. M₁₆×D₂₅₁	C₂²	direct (N × H)
18. C₂²×C₂₅₁⋊C₈	C₂²	semidirect
19. C₂×C₈×D₂₅₁	C₂²	semidirect
21. C₂×M₁₆×C₂₅₁	C₂²	semidirect
22. C₂²×C₈×C₂₅₁	C₂²	semidirect
38. #16860794	C₂³	semidirect
39. #16860793	C₂³	semidirect
40. C₂×C₂₅₁⋊C₈	C₂³	semidirect
41. C₈×D₂₅₁	C₂³	semidirect
47. M₁₆×C₂₅₁	C₂³	semidirect
48. C₂×C₈×C₂₅₁	C₂³	semidirect
51. C₂²×D₂₅₁	M₁₆	direct (N × H)
64. C₂₅₁⋊C₈	C₂⁴	semidirect
73. C₈×C₂₅₁	C₂⁴	semidirect
75. C₂²×C₂₅₁	C₂×M₁₆	semidirect
80. C₂×D₂₅₁	C₂×M₁₆	direct (N × H)
89. C₂×C₂₅₁	C₂²×M₁₆	semidirect
92. D₂₅₁	C₂²×M₁₆	direct (N × H)
93. C₂₅₁	C₂³×M₁₆	semidirect
99. C₂²×M₁₆	D₂₅₁	direct (N × H)
113. C₂×M₁₆	C₂×D₂₅₁	direct (N × H)
116. C₂²×C₈	C₂×D₂₅₁	semidirect
139. M₁₆	C₂²×D₂₅₁	direct (N × H)
142. C₂×C₈	C₂²×D₂₅₁	semidirect
167. C₈	C₂³×D₂₅₁	semidirect
174. C₂²	M₁₆×D₂₅₁	direct (N × H)
185. C₂	C₂×M₁₆×D₂₅₁	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×C₄×D₂₅₁	2
C₂²×#16860794	2
C₂²×#16860793	2
C₂×M₁₆×D₂₅₁	2
C₂²×C₈×D₂₅₁	2
C₂²×M₁₆×C₂₅₁	2
C₂³×M₁₆	251

Subgroups — C22×M16×D251 #44565065

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×M₁₆×D₂₅₁ #44565065