Subgroups — #423×D₂₅₁ #44558917

Order 32128.

576 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×D₂₅₁	C₂	1	2	yes	maximal
2	C₂×C₄²×D₂₅₁	C₂	1	1	yes	maximal, characteristic
3	#30631711	C₂	1	1	yes	maximal, characteristic, complemented
4	#30631710	C₂	1	2	yes	maximal, complemented
5	#30631709	C₂	1	4	yes	maximal, complemented
6	C₄⋊C₈×D₂₅₁	C₂	1	4	yes	maximal, complemented
7	#423×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄×D₂₅₁	C₄	1	2	yes	—
9	C₂²×C₄×D₂₅₁	C₂²	1	1	yes	characteristic
10	C₂×#16860794	C₂²	1	2	yes	—
11	C₂×#16860793	C₂²	1	4	yes	—
12	M₁₆×D₂₅₁	—	2	8	no	—
13	C₂×C₈×D₂₅₁	C₂²	1	4	yes	—
14	C₂×C₄×Dic₂₅₁	C₄	1	2	yes	—
15	C₂×C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
16	C₄²×D₂₅₁	C₄	1	1	yes	characteristic
17	C₄²×D₂₅₁	C₄	1	1	yes	characteristic
18	C₄²×D₂₅₁	C₄	1	1	yes	characteristic
19	C₄²×D₂₅₁	C₄	1	1	yes	characteristic
20	C₄²×D₂₅₁	C₂²	1	2	yes	—
21	C₄²×D₂₅₁	C₂²	1	2	yes	—
22	C₄⋊(C₂₅₁⋊C₈)	C₂²	1	4	yes	complemented
23	Dic₂₅₁⋊C₈	C₂²	1	8	yes	complemented
24	C₂×M₁₆×C₂₅₁	C₂²	1	2	yes	—
25	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
26	C₄⋊C₈×C₂₅₁	C₂²	1	4	yes	complemented
27	C₂³×D₂₅₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×Dic₂₅₁	C₂×C₄	1	2	yes	—
29	C₂²×Dic₂₅₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×D₂₅₁	—	2	8	no	—
31	C₂×C₄×D₂₅₁	C₂×C₄	1	2	yes	—
32	C₂×C₄×D₂₅₁	C₂×C₄	1	2	yes	—
33	C₂×C₄×D₂₅₁	C₂×C₄	1	2	yes	—
34	C₂×C₄×D₂₅₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×D₂₅₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×D₂₅₁	C₂×C₄	1	2	yes	—
37	C₂×C₄×D₂₅₁	D₄	1	4	yes	—
38	C₂×C₄×D₂₅₁	Q₈	1	4	yes	—
39	C₂×C₄×D₂₅₁	C₂³	1	1	yes	characteristic
40	C₂×C₄×D₂₅₁	C₂³	1	1	yes	characteristic
41	#16860794	—	2	4	no	—
42	#16860793	—	2	8	no	—
43	C₂×C₂₅₁⋊C₈	C₂³	1	4	yes	—
44	C₈×D₂₅₁	—	2	8	no	—
45	C₄×Dic₂₅₁	C₂×C₄	1	1	yes	characteristic
46	C₄×Dic₂₅₁	C₂×C₄	1	1	yes	characteristic
47	C₄×Dic₂₅₁	C₂×C₄	1	2	yes	—
48	C₄×Dic₂₅₁	C₂×C₄	1	2	yes	—
49	C₄×Dic₂₅₁	C₂×C₄	1	2	yes	—
50	C₄×Dic₂₅₁	C₂×C₄	1	2	yes	—
51	C₄×Dic₂₅₁	C₂³	1	2	yes	—
52	C₂²×C₄×C₂₅₁	C₂×C₄	1	2	yes	—
53	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
54	M₁₆×C₂₅₁	—	2	4	no	—
55	C₂×C₈×C₂₅₁	C₂³	1	4	yes	—
56	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
57	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
58	C₄²×C₂₅₁	C₂³	1	2	yes	—
59	C₂²×D₂₅₁	—	2	4	no	—
60	C₂²×D₂₅₁	C₄⋊C₄	1	4	yes	—
61	C₂²×D₂₅₁	C₂²×C₄	1	1	yes	characteristic
62	C₂²×D₂₅₁	C₂²×C₄	1	1	yes	characteristic
63	C₂×Dic₂₅₁	—	2	4	no	—
64	C₂×Dic₂₅₁	C₄⋊C₄	1	4	yes	—
65	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
66	C₂×Dic₂₅₁	C₂²×C₄	1	1	yes	characteristic
67	C₂×Dic₂₅₁	C₂²×C₄	1	1	yes	characteristic
68	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
69	C₄×D₂₅₁	—	2	4	no	—
70	C₄×D₂₅₁	—	2	4	no	—
71	C₄×D₂₅₁	—	2	4	no	—
72	C₄×D₂₅₁	C₄⋊C₄	1	4	yes	—
73	C₄×D₂₅₁	C₄⋊C₄	1	4	yes	—
74	C₄×D₂₅₁	M₁₆	1	8	yes	complemented
75	C₄×D₂₅₁	C₂×D₄	1	2	yes	—
76	C₄×D₂₅₁	C₂×D₄	1	2	yes	—
77	C₄×D₂₅₁	C₂×Q₈	1	2	yes	—
78	C₄×D₂₅₁	C₂×Q₈	1	2	yes	—
79	C₂₅₁⋊C₈	—	2	4	no	—
80	C₂³×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₂₅₁	—	2	4	no	—
82	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
86	C₂×C₄×C₂₅₁	C₂×Q₈	1	2	yes	—
87	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
88	C₈×C₂₅₁	—	2	4	no	—
89	C₂²×C₂₅₁	—	2	2	no	—
90	C₂²×C₂₅₁	C₂×C₄⋊C₄	1	2	yes	—
91	C₂²×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
92	C₂×D₂₅₁	—	2	2	no	—
93	C₂×D₂₅₁	—	2	8	no	—
94	C₂×D₂₅₁	C₂×C₄⋊C₄	1	2	yes	—
95	C₂×D₂₅₁	C₂×C₄⋊C₄	1	2	yes	—
96	C₂×D₂₅₁	C₂×M₁₆	1	4	yes	—
97	C₄×C₂₅₁	—	2	2	no	—
98	C₄×C₂₅₁	C₂×C₄⋊C₄	1	2	yes	—
99	C₄×C₂₅₁	C₂×M₁₆	1	4	yes	complemented
100	C₄×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
101	C₄×C₂₅₁	C₂²×Q₈	1	1	yes	characteristic
102	Dic₂₅₁	—	2	2	no	—
103	Dic₂₅₁	C₂×C₄⋊C₄	1	2	yes	—
104	Dic₂₅₁	C₂×C₄⋊C₄	1	2	yes	—
105	Dic₂₅₁	C₂×M₁₆	1	4	yes	complemented
106	C₂×C₂₅₁	—	2	2	no	—
107	C₂×C₂₅₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₂₅₁	C₂²×M₁₆	1	2	yes	—
109	D₂₅₁	—	2	2	no	—
110	D₂₅₁	#423	1	4	yes	direct product factor
111	C₂₅₁	C₂×#423	1	1	yes	characteristic, complemented, Sylow
112	C₂×#423	—	251	1	no	maximal, Sylow
113	C₂²×M₁₆	—	251	2	no	—
114	C₂²×C₄²	—	251	1	no	—
115	#423	—	251	2	no	—
116	#423	—	251	1	no	—
117	#423	—	251	4	no	—
118	#423	D₂₅₁	1	1	yes	characteristic, direct product factor
119	C₂×C₄⋊C₈	—	251	4	no	—
120	C₂³×C₄	—	251	2	no	—
121	C₂³×C₄	—	251	1	no	—
122	C₂×M₁₆	—	502	8	no	—
123	C₂×M₁₆	—	251	2	no	—
124	C₂×M₁₆	—	251	4	no	—
125	C₂×M₁₆	C₂×D₂₅₁	1	2	yes	—
126	C₂²×C₈	—	251	4	no	—
127	C₂×C₄²	—	251	2	no	—
128	C₂×C₄²	—	251	1	no	—
129	C₂×C₄²	—	251	1	no	—
130	C₂×C₄²	—	251	1	no	—
131	C₂×C₄²	—	251	1	no	—
132	C₂×C₄²	—	251	1	no	—
133	C₂×C₄²	—	251	2	no	—
134	C₂×C₄²	—	251	2	no	—
135	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
136	C₄⋊C₈	—	251	8	no	—
137	C₄⋊C₈	—	251	4	no	—
138	C₄⋊C₈	C₂×D₂₅₁	1	4	yes	complemented
139	C₂⁴	—	251	1	no	—
140	C₂²×C₄	—	502	8	no	—
141	C₂²×C₄	—	251	2	no	—
142	C₂²×C₄	—	251	2	no	—
143	C₂²×C₄	—	251	2	no	—
144	C₂²×C₄	—	251	2	no	—
145	C₂²×C₄	—	251	1	no	—
146	C₂²×C₄	—	251	1	no	—
147	C₂²×C₄	—	251	1	no	—
148	C₂²×C₄	—	251	2	no	—
149	C₂²×C₄	—	251	4	no	—
150	C₂²×C₄	—	251	4	no	—
151	C₂²×C₄	—	251	1	no	—
152	C₂²×C₄	—	251	1	no	—
153	C₂²×C₄	C₄×D₂₅₁	1	2	yes	—
154	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
155	M₁₆	—	2	4	no	—
156	M₁₆	—	502	8	no	—
157	M₁₆	—	502	4	no	—
158	C₂×C₈	—	502	8	no	—
159	C₂×C₈	—	251	4	no	—
160	C₂×C₈	C₂²×D₂₅₁	1	4	yes	—
161	C₄²	—	251	1	no	—
162	C₄²	—	251	2	no	—
163	C₄²	—	251	2	no	—
164	C₄²	—	251	2	no	—
165	C₄²	—	251	1	no	—
166	C₄²	—	251	2	no	—
167	C₄²	—	251	2	no	—
168	C₄²	C₄×D₂₅₁	1	1	yes	characteristic
169	C₄²	C₄×D₂₅₁	1	1	yes	characteristic
170	C₄²	C₂²×D₂₅₁	1	2	yes	—
171	C₂³	—	502	4	no	—
172	C₂³	—	251	1	no	—
173	C₂³	—	251	4	no	—
174	C₂³	—	251	1	no	—
175	C₂³	C₂×C₄×D₂₅₁	1	1	yes	characteristic
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	502	4	no	—
178	C₂×C₄	—	502	4	no	—
179	C₂×C₄	—	502	4	no	—
180	C₂×C₄	—	502	4	no	—
181	C₂×C₄	—	251	2	no	—
182	C₂×C₄	—	251	1	no	—
183	C₂×C₄	—	251	1	no	—
184	C₂×C₄	—	251	2	no	—
185	C₂×C₄	—	251	4	no	—
186	C₂×C₄	—	251	2	no	—
187	C₂×C₄	—	251	2	no	—
188	C₂×C₄	—	251	2	no	—
189	C₂×C₄	—	251	2	no	—
190	C₂×C₄	—	251	4	no	—
191	C₂×C₄	—	251	4	no	—
192	C₂×C₄	—	251	8	no	—
193	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
194	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
195	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
196	C₂×C₄	D₄×D₂₅₁	1	2	yes	—
197	C₂×C₄	Q₈×D₂₅₁	1	2	yes	—
198	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
199	C₈	—	2	4	no	—
200	C₈	—	502	4	no	—
201	C₂²	—	2	2	no	—
202	C₂²	—	502	8	no	—
203	C₂²	—	502	2	no	—
204	C₂²	—	251	2	no	—
205	C₂²	—	251	2	no	—
206	C₂²	—	251	4	no	—
207	C₂²	C₄⋊C₄×D₂₅₁	1	2	yes	—
208	C₂²	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
209	C₄	—	2	2	no	—
210	C₄	—	502	2	no	—
211	C₄	—	251	2	no	—
212	C₄	—	251	2	no	—
213	C₄	—	251	4	no	—
214	C₄	C₄⋊C₄×D₂₅₁	1	2	yes	—
215	C₄	M₁₆×D₂₅₁	1	4	yes	complemented
216	C₄	C₂×D₄×D₂₅₁	1	1	yes	characteristic
217	C₄	C₂×Q₈×D₂₅₁	1	1	yes	characteristic
218	C₂	—	2	2	no	—
219	C₂	—	502	2	no	—
220	C₂	—	251	4	no	—
221	C₂	C₂×C₄⋊C₄×D₂₅₁	1	1	yes	characteristic
222	C₂	C₂×M₁₆×D₂₅₁	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₅₁
commutator, socle	C₂²×C₂₅₁	C₂³×C₄
Fitting	#423×C₂₅₁	C₂
prime radical	C₂³×D₂₅₁	C₂×C₄
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	#423	D₂₅₁
hypocenter	C₂₅₁	C₂×#423
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂×C₄ ◁ #423

Complements

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

N	H ≅ G/N	type
3. #30631711	C₂	semidirect
4. #30631710	C₂	semidirect
5. #30631709	C₂	semidirect
6. C₄⋊C₈×D₂₅₁	C₂	semidirect
7. #423×C₂₅₁	C₂	semidirect
22. C₄⋊(C₂₅₁⋊C₈)	C₂²	semidirect
23. Dic₂₅₁⋊C₈	C₂²	semidirect
26. C₄⋊C₈×C₂₅₁	C₂²	semidirect
74. C₄×D₂₅₁	M₁₆	semidirect
99. C₄×C₂₅₁	C₂×M₁₆	semidirect
105. Dic₂₅₁	C₂×M₁₆	semidirect
110. D₂₅₁	#423	direct (N × H)
111. C₂₅₁	C₂×#423	semidirect
118. #423	D₂₅₁	direct (N × H)
138. C₄⋊C₈	C₂×D₂₅₁	semidirect
215. C₄	M₁₆×D₂₅₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×M₁₆×D₂₅₁	2
C₂×C₄²×D₂₅₁	2
#30631711	2
#30631710	2
#30631709	2
C₄⋊C₈×D₂₅₁	2
#423×C₂₅₁	2
C₂×#423	251

Subgroups — #423×D251 #44558917

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #423×D₂₅₁ #44558917