Subgroups — C₂×C₂²⋊C₈×D₂₅₁ #44558795

Order 32128.

1072 conjugacy classes of subgroups.

The subgroup lattice has 248 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×C₈×D₂₅₁	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊(C₂₅₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×(C₂×D₂₅₁)⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₈×D₂₅₁	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₂²⋊C₈×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	—	2	8	no	—
10	C₂²×C₄×D₂₅₁	C₄	1	1	yes	characteristic
11	C₂²×C₄×D₂₅₁	C₄	1	1	yes	characteristic
12	C₂²×C₄×D₂₅₁	C₄	1	4	yes	—
13	C₂²×C₄×D₂₅₁	C₂²	1	4	yes	—
14	C₂²×C₄×D₂₅₁	C₂²	1	1	yes	characteristic
15	C₂²×C₄×D₂₅₁	C₂²	1	1	yes	characteristic
16	C₂²×C₂₅₁⋊C₈	C₂²	1	2	yes	complemented
17	C₂×C₈×D₂₅₁	—	2	8	no	—
18	C₂×C₈×D₂₅₁	C₂²	1	8	yes	complemented
19	C₂²⋊(C₂₅₁⋊C₈)	C₂²	1	4	yes	complemented
20	(C₂×D₂₅₁)⋊C₈	C₂²	1	8	yes	complemented
21	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
22	C₂²×C₈×C₂₅₁	C₂²	1	2	yes	complemented
23	C₂²⋊C₈×C₂₅₁	C₂²	1	4	yes	complemented
24	C₂³×D₂₅₁	—	2	4	no	—
25	C₂³×D₂₅₁	—	2	4	no	—
26	C₂³×D₂₅₁	C₈	1	8	yes	complemented
27	C₂³×D₂₅₁	C₂×C₄	1	4	yes	—
28	C₂³×D₂₅₁	C₂×C₄	1	1	yes	characteristic
29	C₂³×D₂₅₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×Dic₂₅₁	—	2	4	no	—
31	C₂²×Dic₂₅₁	C₂×C₄	1	4	yes	—
32	C₂²×Dic₂₅₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×Dic₂₅₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×D₂₅₁	—	2	4	no	—
35	C₂×C₄×D₂₅₁	—	2	4	no	—
36	C₂×C₄×D₂₅₁	—	2	16	no	—
37	C₂×C₄×D₂₅₁	—	2	16	no	—
38	C₂×C₄×D₂₅₁	—	2	4	no	—
39	C₂×C₄×D₂₅₁	C₂×C₄	1	4	yes	—
40	C₂×C₄×D₂₅₁	C₂×C₄	1	4	yes	—
41	C₂×C₄×D₂₅₁	C₂×C₄	1	2	yes	—
42	C₂×C₄×D₂₅₁	C₂×C₄	1	2	yes	—
43	C₂×C₄×D₂₅₁	D₄	1	8	yes	—
44	C₂×C₄×D₂₅₁	C₂³	1	2	yes	—
45	C₂×C₄×D₂₅₁	C₂³	1	2	yes	—
46	C₂×C₂₅₁⋊C₈	—	2	4	no	—
47	C₂×C₂₅₁⋊C₈	C₂³	1	4	yes	complemented
48	C₈×D₂₅₁	—	2	16	no	—
49	C₂⁴×C₂₅₁	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₂₅₁	—	2	4	no	—
51	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₂₅₁	C₂×C₄	1	2	yes	—
53	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
54	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
55	C₂×C₈×C₂₅₁	—	2	4	no	—
56	C₂×C₈×C₂₅₁	C₂³	1	4	yes	complemented
57	C₂²×D₂₅₁	—	2	2	no	—
58	C₂²×D₂₅₁	—	2	2	no	—
59	C₂²×D₂₅₁	—	2	32	no	—
60	C₂²×D₂₅₁	—	2	8	no	—
61	C₂²×D₂₅₁	—	2	8	no	—
62	C₂²×D₂₅₁	C₂²⋊C₄	1	4	yes	—
63	C₂²×D₂₅₁	C₂×C₈	1	4	yes	—
64	C₂²×D₂₅₁	C₂×C₈	1	16	yes	complemented
65	C₂²×D₂₅₁	C₂×C₈	1	4	yes	complemented
66	C₂²×D₂₅₁	M₁₆	1	4	yes	—
67	C₂²×D₂₅₁	C₂²×C₄	1	2	yes	—
68	C₂²×D₂₅₁	C₂²×C₄	1	2	yes	—
69	C₂×Dic₂₅₁	—	2	2	no	—
70	C₂×Dic₂₅₁	—	2	8	no	—
71	C₂×Dic₂₅₁	C₂²⋊C₄	1	4	yes	—
72	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
73	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
74	C₄×D₂₅₁	—	2	8	no	—
75	C₄×D₂₅₁	—	2	4	no	—
76	C₄×D₂₅₁	—	2	4	no	—
77	C₄×D₂₅₁	—	2	8	no	—
78	C₄×D₂₅₁	C₂²⋊C₄	1	8	yes	—
79	C₄×D₂₅₁	C₂×D₄	1	8	yes	—
80	C₂₅₁⋊C₈	—	2	4	no	—
81	C₂³×C₂₅₁	—	2	2	no	—
82	C₂³×C₂₅₁	—	2	2	no	—
83	C₂³×C₂₅₁	C₂×C₈	1	4	yes	complemented
84	C₂³×C₂₅₁	C₂²×C₄	1	2	yes	—
85	C₂³×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₂₅₁	—	2	2	no	—
87	C₂×C₄×C₂₅₁	—	2	4	no	—
88	C₂×C₄×C₂₅₁	—	2	4	no	—
89	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₂₅₁	C₂×D₄	1	4	yes	—
92	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
93	C₈×C₂₅₁	—	2	4	no	—
94	C₂²×C₂₅₁	—	2	2	no	—
95	C₂²×C₂₅₁	—	2	2	no	—
96	C₂²×C₂₅₁	—	2	8	no	—
97	C₂²×C₂₅₁	C₂×C₂²⋊C₄	1	2	yes	—
98	C₂²×C₂₅₁	C₂²×C₈	1	2	yes	—
99	C₂²×C₂₅₁	C₂²×C₈	1	4	yes	complemented
100	C₂²×C₂₅₁	C₂×M₁₆	1	2	yes	—
101	C₂²×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
102	C₂×D₂₅₁	—	2	2	no	—
103	C₂×D₂₅₁	—	2	2	no	—
104	C₂×D₂₅₁	—	2	8	no	—
105	C₂×D₂₅₁	—	2	32	no	—
106	C₂×D₂₅₁	C₂²⋊C₈	1	16	yes	direct product factor
107	C₂×D₂₅₁	C₂×C₂²⋊C₄	1	4	yes	—
108	C₂×D₂₅₁	C₂²×C₈	1	4	yes	—
109	C₂×D₂₅₁	C₂²×C₈	1	4	yes	complemented
110	C₂×D₂₅₁	C₂×M₁₆	1	4	yes	—
111	C₄×C₂₅₁	—	2	2	no	—
112	C₄×C₂₅₁	C₂×C₂²⋊C₄	1	2	yes	—
113	C₄×C₂₅₁	C₂²×D₄	1	2	yes	—
114	Dic₂₅₁	—	2	2	no	—
115	Dic₂₅₁	C₂×C₂²⋊C₄	1	4	yes	—
116	C₂×C₂₅₁	—	2	4	no	—
117	C₂×C₂₅₁	C₂×C₂²⋊C₈	1	4	yes	complemented
118	C₂×C₂₅₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₂₅₁	C₂³×C₈	1	1	yes	characteristic
120	C₂×C₂₅₁	C₂²×M₁₆	1	1	yes	characteristic
121	D₂₅₁	—	2	4	no	—
122	D₂₅₁	C₂×C₂²⋊C₈	1	8	yes	direct product factor
123	C₂₅₁	C₂²×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
124	C₂²×C₂²⋊C₈	—	251	1	no	maximal, Sylow
125	C₂⁴×C₄	—	251	1	no	—
126	C₂³×C₈	—	251	2	no	—
127	C₂×C₂²⋊C₈	—	251	1	no	—
128	C₂×C₂²⋊C₈	—	251	2	no	—
129	C₂×C₂²⋊C₈	—	251	8	no	—
130	C₂×C₂²⋊C₈	D₂₅₁	1	1	yes	characteristic, direct product factor
131	C₂⁵	—	251	1	no	—
132	C₂³×C₄	—	502	8	no	—
133	C₂³×C₄	—	251	1	no	—
134	C₂³×C₄	—	251	1	no	—
135	C₂³×C₄	—	251	4	no	—
136	C₂³×C₄	—	251	1	no	—
137	C₂³×C₄	—	251	4	no	—
138	C₂³×C₄	—	251	1	no	—
139	C₂³×C₄	—	251	1	no	—
140	C₂³×C₄	C₂×D₂₅₁	1	1	yes	characteristic
141	C₂²×C₈	—	502	8	no	—
142	C₂²×C₈	—	251	2	no	—
143	C₂²×C₈	—	251	8	no	—
144	C₂²×C₈	C₂×D₂₅₁	1	2	yes	complemented
145	C₂²⋊C₈	—	251	4	no	—
146	C₂²⋊C₈	—	251	8	no	—
147	C₂²⋊C₈	C₂×D₂₅₁	1	4	yes	direct product factor
148	C₂⁴	—	502	4	no	—
149	C₂⁴	—	502	4	no	—
150	C₂⁴	—	251	8	no	—
151	C₂⁴	—	251	4	no	—
152	C₂⁴	—	251	1	no	—
153	C₂⁴	—	251	1	no	—
154	C₂⁴	C₄×D₂₅₁	1	1	yes	characteristic
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	—	502	4	no	—
157	C₂²×C₄	—	502	4	no	—
158	C₂²×C₄	—	502	16	no	—
159	C₂²×C₄	—	502	16	no	—
160	C₂²×C₄	—	502	4	no	—
161	C₂²×C₄	—	502	4	no	—
162	C₂²×C₄	—	251	1	no	—
163	C₂²×C₄	—	251	4	no	—
164	C₂²×C₄	—	251	4	no	—
165	C₂²×C₄	—	251	4	no	—
166	C₂²×C₄	—	251	2	no	—
167	C₂²×C₄	—	251	2	no	—
168	C₂²×C₄	—	251	1	no	—
169	C₂²×C₄	—	251	2	no	—
170	C₂²×C₄	—	251	8	no	—
171	C₂²×C₄	—	251	2	no	—
172	C₂²×C₄	C₄×D₂₅₁	1	1	yes	characteristic
173	C₂²×C₄	C₄×D₂₅₁	1	2	yes	—
174	C₂²×C₄	C₂²×D₂₅₁	1	2	yes	—
175	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	—	502	16	no	—
178	C₂×C₈	—	502	4	no	—
179	C₂×C₈	—	251	4	no	—
180	C₂×C₈	C₂²×D₂₅₁	1	4	yes	complemented
181	C₂³	—	2	2	no	—
182	C₂³	—	2	2	no	—
183	C₂³	—	502	8	no	—
184	C₂³	—	502	32	no	—
185	C₂³	—	502	8	no	—
186	C₂³	—	502	2	no	—
187	C₂³	—	502	2	no	—
188	C₂³	—	251	2	no	—
189	C₂³	—	251	4	no	—
190	C₂³	—	251	16	no	—
191	C₂³	—	251	2	no	—
192	C₂³	—	251	4	no	—
193	C₂³	—	251	4	no	—
194	C₂³	—	251	4	no	—
195	C₂³	C₈×D₂₅₁	1	4	yes	complemented
196	C₂³	C₂×C₄×D₂₅₁	1	2	yes	—
197	C₂³	C₂×C₄×D₂₅₁	1	1	yes	characteristic
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	502	8	no	—
202	C₂×C₄	—	502	8	no	—
203	C₂×C₄	—	502	2	no	—
204	C₂×C₄	—	502	8	no	—
205	C₂×C₄	—	502	4	no	—
206	C₂×C₄	—	502	4	no	—
207	C₂×C₄	—	251	2	no	—
208	C₂×C₄	—	251	2	no	—
209	C₂×C₄	—	251	4	no	—
210	C₂×C₄	—	251	8	no	—
211	C₂×C₄	—	251	8	no	—
212	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
213	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
214	C₂×C₄	D₄×D₂₅₁	1	4	yes	—
215	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
216	C₈	—	2	4	no	—
217	C₈	—	502	4	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	2	8	no	—
220	C₂²	—	2	2	no	—
221	C₂²	—	502	32	no	—
222	C₂²	—	502	2	no	—
223	C₂²	—	502	8	no	—
224	C₂²	—	502	2	no	—
225	C₂²	—	251	4	no	—
226	C₂²	—	251	4	no	—
227	C₂²	—	251	16	no	—
228	C₂²	—	251	4	no	—
229	C₂²	—	251	4	no	—
230	C₂²	C₂²⋊C₄×D₂₅₁	1	2	yes	—
231	C₂²	C₂×C₈×D₂₅₁	1	2	yes	—
232	C₂²	C₂×C₈×D₂₅₁	1	4	yes	complemented
233	C₂²	M₁₆×D₂₅₁	1	2	yes	—
234	C₂²	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	502	2	no	—
237	C₄	—	251	4	no	—
238	C₄	C₂²⋊C₄×D₂₅₁	1	2	yes	—
239	C₄	C₂×D₄×D₂₅₁	1	2	yes	—
240	C₂	—	2	4	no	—
241	C₂	—	502	4	no	—
242	C₂	—	251	8	no	—
243	C₂	C₂²⋊C₈×D₂₅₁	1	4	yes	direct product factor
244	C₂	C₂×C₂²⋊C₄×D₂₅₁	1	1	yes	characteristic
245	C₂	C₂²×C₈×D₂₅₁	1	1	yes	characteristic
246	C₂	C₂×M₁₆×D₂₅₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₂₅₁
commutator	C₂×C₂₅₁	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	C₂×C₂²⋊C₈×C₂₅₁	C₂
socle	C₂³×C₂₅₁	C₂²×C₄
prime radical	C₂⁴×D₂₅₁	C₄
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	C₂×C₂²⋊C₈	D₂₅₁
hypocenter	C₂₅₁	C₂²×C₂²⋊C₈
LCS	G ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₂²×C₈×D₂₅₁	C₂	semidirect
3. C₂×C₂²⋊(C₂₅₁⋊C₈)	C₂	semidirect
4. C₂×(C₂×D₂₅₁)⋊C₈	C₂	semidirect
5. C₂²⋊C₈×D₂₅₁	C₂	direct (N × H)
6. C₂×C₂²⋊C₈×C₂₅₁	C₂	semidirect
16. C₂²×C₂₅₁⋊C₈	C₂²	semidirect
18. C₂×C₈×D₂₅₁	C₂²	semidirect
19. C₂²⋊(C₂₅₁⋊C₈)	C₂²	semidirect
20. (C₂×D₂₅₁)⋊C₈	C₂²	semidirect
22. C₂²×C₈×C₂₅₁	C₂²	semidirect
23. C₂²⋊C₈×C₂₅₁	C₂²	semidirect
26. C₂³×D₂₅₁	C₈	semidirect
47. C₂×C₂₅₁⋊C₈	C₂³	semidirect
56. C₂×C₈×C₂₅₁	C₂³	semidirect
64. C₂²×D₂₅₁	C₂×C₈	semidirect
65. C₂²×D₂₅₁	C₂×C₈	semidirect
83. C₂³×C₂₅₁	C₂×C₈	semidirect
99. C₂²×C₂₅₁	C₂²×C₈	semidirect
106. C₂×D₂₅₁	C₂²⋊C₈	direct (N × H)
109. C₂×D₂₅₁	C₂²×C₈	semidirect
117. C₂×C₂₅₁	C₂×C₂²⋊C₈	semidirect
122. D₂₅₁	C₂×C₂²⋊C₈	direct (N × H)
123. C₂₅₁	C₂²×C₂²⋊C₈	semidirect
130. C₂×C₂²⋊C₈	D₂₅₁	direct (N × H)
144. C₂²×C₈	C₂×D₂₅₁	semidirect
147. C₂²⋊C₈	C₂×D₂₅₁	direct (N × H)
180. C₂×C₈	C₂²×D₂₅₁	semidirect
195. C₂³	C₈×D₂₅₁	semidirect
232. C₂²	C₂×C₈×D₂₅₁	semidirect
243. C₂	C₂²⋊C₈×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×C₈×D₂₅₁	2
C₂×C₂²⋊(C₂₅₁⋊C₈)	2
C₂×(C₂×D₂₅₁)⋊C₈	2
C₂²⋊C₈×D₂₅₁	2
C₂×C₂²⋊C₈×C₂₅₁	2
C₂²×C₂²⋊C₈	251

Subgroups — C2×C22⋊C8×D251 #44558795

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂²⋊C₈×D₂₅₁ #44558795