Subgroups — C₂×#103×D₂₅₁ #44558853

Order 32128.

816 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×D₂₅₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×D₂₅₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₄×D₂₅₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#22548640	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#22548639	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#22548638	C₂	1	1	yes	maximal, characteristic, complemented
7	#103×D₂₅₁	C₂	1	8	yes	maximal, direct product factor
8	C₂×#103×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Dic₅₀₂	—	2	1	no	—
10	C₂²×Dic₅₀₂	C₂²	1	1	yes	characteristic
11	C₂×Q₈×D₂₅₁	—	2	4	no	—
12	C₂×Q₈×D₂₅₁	—	2	2	no	—
13	C₂×Q₈×D₂₅₁	C₄	1	8	yes	complemented
14	C₂×Q₈×D₂₅₁	C₂²	1	4	yes	—
15	C₂²×C₄×D₂₅₁	—	2	1	no	—
16	C₂²×C₄×D₂₅₁	—	2	1	no	—
17	C₂²×C₄×D₂₅₁	C₂²	1	1	yes	characteristic
18	C₂²×C₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₂₅₁	—	2	4	no	—
20	C₂×C₈×D₂₅₁	C₂²	1	4	yes	—
21	C₂×C₄⋊Dic₂₅₁	C₂²	1	1	yes	characteristic
22	C₂×Dic₂₅₁⋊C₄	—	2	1	no	—
23	C₄⋊C₄×D₂₅₁	—	2	2	no	—
24	C₄⋊C₄×D₂₅₁	C₂²	1	4	yes	—
25	#22548640	C₂²	1	4	yes	complemented
26	#22548639	C₂²	1	4	yes	complemented
27	#22548638	C₂²	1	4	yes	complemented
28	C₂²×Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
29	C₂²×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
31	#103×C₂₅₁	C₂²	1	4	yes	complemented
32	C₂³×D₂₅₁	D₄	1	1	yes	characteristic
33	C₂×Dic₅₀₂	—	2	8	no	—
34	C₂×Dic₅₀₂	—	2	4	no	—
35	C₂×Dic₅₀₂	—	2	1	no	—
36	C₂×Dic₅₀₂	—	2	2	no	—
37	C₂×Dic₅₀₂	C₂×C₄	1	4	yes	complemented
38	C₂×Dic₅₀₂	C₂³	1	2	yes	—
39	Q₈×D₂₅₁	—	2	16	no	—
40	Q₈×D₂₅₁	—	2	8	no	—
41	Q₈×D₂₅₁	C₂×C₄	1	16	yes	complemented
42	C₂²×Dic₂₅₁	—	2	1	no	—
43	C₂²×Dic₂₅₁	—	2	1	no	—
44	C₂²×Dic₂₅₁	D₄	1	1	yes	characteristic
45	C₂×C₄×D₂₅₁	—	2	8	no	—
46	C₂×C₄×D₂₅₁	—	2	4	no	—
47	C₂×C₄×D₂₅₁	—	2	4	no	—
48	C₂×C₄×D₂₅₁	—	4	4	no	—
49	C₂×C₄×D₂₅₁	C₂×C₄	1	4	yes	—
50	C₂×C₄×D₂₅₁	D₄	1	2	yes	—
51	C₂×C₄×D₂₅₁	D₄	1	4	yes	—
52	C₂×C₄×D₂₅₁	C₂³	1	2	yes	—
53	C₂×C₂₅₁⋊C₈	—	2	2	no	—
54	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
55	C₈×D₂₅₁	—	2	8	no	—
56	C₄⋊Dic₂₅₁	—	2	1	no	—
57	C₄⋊Dic₂₅₁	C₂³	1	2	yes	—
58	Dic₂₅₁⋊C₄	—	2	4	no	—
59	C₂×Q₈×C₂₅₁	—	2	2	no	—
60	C₂×Q₈×C₂₅₁	—	2	1	no	—
61	C₂×Q₈×C₂₅₁	C₂×C₄	1	4	yes	complemented
62	C₂×Q₈×C₂₅₁	C₂³	1	2	yes	—
63	C₂²×C₄×C₂₅₁	—	2	1	no	—
64	C₂²×C₄×C₂₅₁	—	2	1	no	—
65	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
66	C₂×C₈×C₂₅₁	—	2	2	no	—
67	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
68	C₄⋊C₄×C₂₅₁	—	2	1	no	—
69	C₄⋊C₄×C₂₅₁	C₂³	1	2	yes	—
70	C₂²×D₂₅₁	C₂²⋊C₄	1	4	yes	—
71	C₂²×D₂₅₁	SD₁₆	1	4	yes	—
72	C₂²×D₂₅₁	Q₁₆	1	4	yes	—
73	C₂²×D₂₅₁	C₂×D₄	1	2	yes	—
74	Dic₅₀₂	—	2	4	no	—
75	Dic₅₀₂	—	2	2	no	—
76	Dic₅₀₂	—	2	16	no	—
77	Dic₅₀₂	C₂²×C₄	1	4	yes	complemented
78	C₂×Dic₂₅₁	—	2	4	no	—
79	C₂×Dic₂₅₁	—	2	2	no	—
80	C₂×Dic₂₅₁	—	2	2	no	—
81	C₂×Dic₂₅₁	—	4	2	no	—
82	C₂×Dic₂₅₁	C₂²⋊C₄	1	4	yes	—
83	C₂×Dic₂₅₁	C₂×D₄	1	2	yes	—
84	C₄×D₂₅₁	—	2	16	no	—
85	C₄×D₂₅₁	—	4	8	no	—
86	C₄×D₂₅₁	C₂²⋊C₄	1	8	yes	—
87	C₄×D₂₅₁	C₂²×C₄	1	4	yes	—
88	C₄×D₂₅₁	C₂×D₄	1	4	yes	—
89	C₂₅₁⋊C₈	—	2	2	no	—
90	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
91	Q₈×C₂₅₁	—	2	4	no	—
92	Q₈×C₂₅₁	—	2	2	no	—
93	Q₈×C₂₅₁	C₂²×C₄	1	4	yes	complemented
94	C₂×C₄×C₂₅₁	—	2	4	no	—
95	C₂×C₄×C₂₅₁	—	2	2	no	—
96	C₂×C₄×C₂₅₁	—	2	2	no	—
97	C₂×C₄×C₂₅₁	—	4	2	no	—
98	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
99	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
101	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
102	C₈×C₂₅₁	—	2	2	no	—
103	C₂²×C₂₅₁	C₂×C₂²⋊C₄	1	2	yes	—
104	C₂²×C₂₅₁	C₂×SD₁₆	1	2	yes	—
105	C₂²×C₂₅₁	C₂×Q₁₆	1	2	yes	—
106	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
107	C₂×D₂₅₁	#103	1	16	yes	direct product factor
108	C₂×D₂₅₁	C₂×C₂²⋊C₄	1	4	yes	—
109	C₂×D₂₅₁	C₂×SD₁₆	1	4	yes	—
110	C₂×D₂₅₁	C₂×Q₁₆	1	4	yes	—
111	C₄×C₂₅₁	—	2	4	no	—
112	C₄×C₂₅₁	—	4	2	no	—
113	C₄×C₂₅₁	C₂×C₂²⋊C₄	1	2	yes	—
114	C₄×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
115	C₄×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
116	Dic₂₅₁	—	2	4	no	—
117	Dic₂₅₁	—	4	2	no	—
118	Dic₂₅₁	C₂×C₂²⋊C₄	1	4	yes	—
119	C₂×C₂₅₁	C₂×#103	1	4	yes	complemented
120	C₂×C₂₅₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₂₅₁	C₂²×SD₁₆	1	1	yes	characteristic
122	C₂×C₂₅₁	C₂²×Q₁₆	1	1	yes	characteristic
123	D₂₅₁	C₂×#103	1	8	yes	direct product factor
124	C₂₅₁	C₂²×#103	1	1	yes	characteristic, complemented, Sylow
125	C₂²×#103	—	251	1	no	maximal, Sylow
126	C₂³×Q₈	—	251	1	no	—
127	C₂³×C₈	—	251	1	no	—
128	C₂²×C₄⋊C₄	—	251	1	no	—
129	C₂×#103	—	251	1	no	—
130	C₂×#103	—	251	1	no	—
131	C₂×#103	—	251	1	no	—
132	C₂×#103	—	251	8	no	—
133	C₂×#103	D₂₅₁	1	1	yes	characteristic, direct product factor
134	C₂²×Q₈	—	502	1	no	—
135	C₂²×Q₈	—	502	2	no	—
136	C₂²×Q₈	—	502	4	no	—
137	C₂²×Q₈	—	251	8	no	—
138	C₂²×Q₈	—	251	4	no	—
139	C₂²×Q₈	—	251	1	no	—
140	C₂²×Q₈	C₂×D₂₅₁	1	1	yes	characteristic
141	C₂³×C₄	—	502	1	no	—
142	C₂³×C₄	—	502	1	no	—
143	C₂³×C₄	—	251	1	no	—
144	C₂²×C₈	—	502	4	no	—
145	C₂²×C₈	—	251	1	no	—
146	C₂²×C₈	—	251	4	no	—
147	C₂²×C₈	C₂×D₂₅₁	1	1	yes	characteristic
148	C₂×C₄⋊C₄	—	502	2	no	—
149	C₂×C₄⋊C₄	—	502	1	no	—
150	C₂×C₄⋊C₄	—	251	1	no	—
151	C₂×C₄⋊C₄	—	251	4	no	—
152	C₂×C₄⋊C₄	C₂×D₂₅₁	1	1	yes	characteristic
153	#103	—	251	4	no	—
154	#103	—	251	4	no	—
155	#103	—	251	4	no	—
156	#103	C₂×D₂₅₁	1	4	yes	direct product factor
157	C₂⁴	—	251	1	no	—
158	C₂×Q₈	—	2	1	no	—
159	C₂×Q₈	—	2	2	no	—
160	C₂×Q₈	—	502	4	no	—
161	C₂×Q₈	—	502	8	no	—
162	C₂×Q₈	—	502	8	no	—
163	C₂×Q₈	—	502	16	no	—
164	C₂×Q₈	—	502	1	no	—
165	C₂×Q₈	—	502	2	no	—
166	C₂×Q₈	—	251	16	no	—
167	C₂×Q₈	—	251	4	no	—
168	C₂×Q₈	—	251	2	no	—
169	C₂×Q₈	C₄×D₂₅₁	1	4	yes	complemented
170	C₂×Q₈	C₂²×D₂₅₁	1	2	yes	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	1004	4	no	—
174	C₂²×C₄	—	502	4	no	—
175	C₂²×C₄	—	502	4	no	—
176	C₂²×C₄	—	502	8	no	—
177	C₂²×C₄	—	502	1	no	—
178	C₂²×C₄	—	502	1	no	—
179	C₂²×C₄	—	251	1	no	—
180	C₂²×C₄	—	251	4	no	—
181	C₂²×C₄	—	251	2	no	—
182	C₂²×C₄	—	251	4	no	—
183	C₂²×C₄	—	251	2	no	—
184	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	502	8	no	—
187	C₂×C₈	—	502	2	no	—
188	C₂×C₈	—	251	2	no	—
189	C₂×C₈	C₂²×D₂₅₁	1	2	yes	—
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	—	502	4	no	—
192	C₄⋊C₄	—	502	1	no	—
193	C₄⋊C₄	—	251	2	no	—
194	C₄⋊C₄	C₂²×D₂₅₁	1	2	yes	—
195	C₂³	—	251	2	no	—
196	C₂³	—	251	4	no	—
197	C₂³	—	251	4	no	—
198	C₂³	—	251	4	no	—
199	C₂³	D₄×D₂₅₁	1	1	yes	characteristic
200	Q₈	—	2	2	no	—
201	Q₈	—	2	4	no	—
202	Q₈	—	502	16	no	—
203	Q₈	—	502	2	no	—
204	Q₈	—	502	4	no	—
205	Q₈	—	251	4	no	—
206	Q₈	C₂×C₄×D₂₅₁	1	4	yes	complemented
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	1004	8	no	—
212	C₂×C₄	—	1004	2	no	—
213	C₂×C₄	—	502	16	no	—
214	C₂×C₄	—	502	2	no	—
215	C₂×C₄	—	502	2	no	—
216	C₂×C₄	—	502	4	no	—
217	C₂×C₄	—	251	2	no	—
218	C₂×C₄	—	251	4	no	—
219	C₂×C₄	—	251	4	no	—
220	C₂×C₄	—	251	4	no	—
221	C₂×C₄	—	251	8	no	—
222	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
223	C₂×C₄	D₄×D₂₅₁	1	2	yes	—
224	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
225	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
226	C₈	—	2	2	no	—
227	C₈	—	502	2	no	—
228	C₂²	—	251	4	no	—
229	C₂²	—	251	4	no	—
230	C₂²	—	251	4	no	—
231	C₂²	—	251	16	no	—
232	C₂²	C₂²⋊C₄×D₂₅₁	1	2	yes	—
233	C₂²	SD₁₆×D₂₅₁	1	2	yes	—
234	C₂²	Q₁₆×D₂₅₁	1	2	yes	—
235	C₂²	C₂×D₄×D₂₅₁	1	1	yes	characteristic
236	C₄	—	2	4	no	—
237	C₄	—	4	2	no	—
238	C₄	—	1004	2	no	—
239	C₄	—	502	4	no	—
240	C₄	—	251	4	no	—
241	C₄	C₂²⋊C₄×D₂₅₁	1	2	yes	—
242	C₄	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
243	C₄	C₂×D₄×D₂₅₁	1	1	yes	characteristic
244	C₂	—	251	8	no	—
245	C₂	#103×D₂₅₁	1	4	yes	direct product factor
246	C₂	C₂×C₂²⋊C₄×D₂₅₁	1	1	yes	characteristic
247	C₂	C₂×SD₁₆×D₂₅₁	1	1	yes	characteristic
248	C₂	C₂×Q₁₆×D₂₅₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×D₂₅₁
commutator	C₄×C₂₅₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	C₂×#103×C₂₅₁	C₂
socle	C₂³×C₂₅₁	C₂×D₄
prime radical	C₂³×D₂₅₁	D₄
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	C₂×#103	D₂₅₁
hypocenter	C₂₅₁	C₂²×#103
LCS	G ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#103

Complements

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

N	H ≅ G/N	type
4. C₂×#22548640	C₂	semidirect
5. C₂×#22548639	C₂	semidirect
6. C₂×#22548638	C₂	semidirect
7. #103×D₂₅₁	C₂	direct (N × H)
8. C₂×#103×C₂₅₁	C₂	semidirect
13. C₂×Q₈×D₂₅₁	C₄	semidirect
25. #22548640	C₂²	semidirect
26. #22548639	C₂²	semidirect
27. #22548638	C₂²	semidirect
31. #103×C₂₅₁	C₂²	semidirect
37. C₂×Dic₅₀₂	C₂×C₄	semidirect
41. Q₈×D₂₅₁	C₂×C₄	semidirect
61. C₂×Q₈×C₂₅₁	C₂×C₄	semidirect
77. Dic₅₀₂	C₂²×C₄	semidirect
93. Q₈×C₂₅₁	C₂²×C₄	semidirect
107. C₂×D₂₅₁	#103	direct (N × H)
119. C₂×C₂₅₁	C₂×#103	semidirect
123. D₂₅₁	C₂×#103	direct (N × H)
124. C₂₅₁	C₂²×#103	semidirect
133. C₂×#103	D₂₅₁	direct (N × H)
156. #103	C₂×D₂₅₁	direct (N × H)
169. C₂×Q₈	C₄×D₂₅₁	semidirect
206. Q₈	C₂×C₄×D₂₅₁	semidirect
245. C₂	#103×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
C₂²×Q₈×D₂₅₁	2
C₂²×C₈×D₂₅₁	2
C₂×C₄⋊C₄×D₂₅₁	2
C₂×#22548640	2
C₂×#22548639	2
C₂×#22548638	2
#103×D₂₅₁	2
C₂×#103×C₂₅₁	2
C₂²×#103	251

Subgroups — C2×#103×D251 #44558853

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#103×D₂₅₁ #44558853