Subgroups — D₄×M₁₆×C₂₅₁ #44547521

Order 32128.

544 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	C₂²×M₁₆×C₂₅₁	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₄×C₂₅₁	C₂	1	1	yes	maximal, characteristic
3	#436×C₂₅₁	C₂	1	2	yes	maximal, complemented
4	#435×C₂₅₁	C₂	1	4	yes	maximal, complemented
5	C₈×D₄×C₂₅₁	C₂	1	2	yes	maximal, complemented
6	#423×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#407×C₂₅₁	C₂	1	2	yes	maximal, complemented
8	C₄×M₁₆×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₄×C₂₅₁	C₄	1	1	yes	characteristic
10	C₂³×C₄×C₂₅₁	C₂²	1	2	yes	—
11	C₂×M₁₆×C₂₅₁	—	2	8	no	—
12	C₂×M₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₂₅₁	C₂²	1	2	yes	—
14	C₂×M₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
15	C₂×M₁₆×C₂₅₁	C₂²	1	4	yes	complemented
16	C₂²×C₈×C₂₅₁	C₂²	1	4	yes	complemented
17	C₄×D₄×C₂₅₁	C₄	1	2	yes	—
18	C₄×D₄×C₂₅₁	C₄	1	1	yes	characteristic
19	C₄×D₄×C₂₅₁	C₄	1	1	yes	characteristic
20	C₄×D₄×C₂₅₁	C₂²	1	1	yes	characteristic
21	C₄×D₄×C₂₅₁	C₂²	1	2	yes	—
22	C₄×D₄×C₂₅₁	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₂₅₁	C₄	1	1	yes	characteristic
24	C₂×C₂²⋊C₄×C₂₅₁	C₄	1	2	yes	—
25	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
26	C₄⋊C₈×C₂₅₁	C₂²	1	2	yes	complemented
27	C₄⋊C₈×C₂₅₁	C₂²	1	2	yes	complemented
28	C₂²⋊C₈×C₂₅₁	C₂²	1	4	yes	complemented
29	C₂²⋊C₈×C₂₅₁	C₂²	1	4	yes	complemented
30	#97×C₂₅₁	C₂²	1	2	yes	complemented
31	C₄×C₈×C₂₅₁	C₂²	1	2	yes	complemented
32	C₂⁴×C₂₅₁	C₂×C₄	1	2	yes	—
33	C₂×D₄×C₂₅₁	—	2	4	no	—
34	C₂×D₄×C₂₅₁	C₂×C₄	1	2	yes	—
35	C₂×D₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
36	C₂×D₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₂₅₁	—	2	4	no	—
38	C₂²×C₄×C₂₅₁	—	2	4	no	—
39	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₂₅₁	C₂×C₄	1	2	yes	—
41	C₂²×C₄×C₂₅₁	C₂×C₄	1	2	yes	—
42	C₂²×C₄×C₂₅₁	C₂×C₄	1	2	yes	—
43	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
45	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
46	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
47	M₁₆×C₂₅₁	—	2	4	no	—
48	M₁₆×C₂₅₁	—	2	2	no	—
49	M₁₆×C₂₅₁	—	2	8	no	—
50	M₁₆×C₂₅₁	D₄	1	4	yes	direct product factor
51	C₂×C₈×C₂₅₁	—	2	8	no	—
52	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
53	C₂×C₈×C₂₅₁	C₂³	1	4	yes	—
54	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
55	C₄⋊C₄×C₂₅₁	C₂×C₄	1	2	yes	—
56	C₄⋊C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
57	C₄⋊C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
58	C₂²⋊C₄×C₂₅₁	C₂×C₄	1	2	yes	—
59	C₂²⋊C₄×C₂₅₁	C₂×C₄	1	2	yes	—
60	C₂²⋊C₄×C₂₅₁	C₂×C₄	1	2	yes	—
61	C₂²⋊C₄×C₂₅₁	C₂×C₄	1	2	yes	—
62	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
63	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
64	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
65	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
66	C₂³×C₂₅₁	—	2	4	no	—
67	C₂³×C₂₅₁	—	2	4	no	—
68	C₂³×C₂₅₁	—	4	2	no	—
69	C₂³×C₂₅₁	C₂²×C₄	1	2	yes	—
70	C₂³×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
71	C₂³×C₂₅₁	C₂²×C₄	1	2	yes	—
72	D₄×C₂₅₁	—	2	4	no	—
73	D₄×C₂₅₁	—	2	2	no	—
74	D₄×C₂₅₁	M₁₆	1	4	yes	direct product factor
75	C₂×C₄×C₂₅₁	—	2	4	no	—
76	C₂×C₄×C₂₅₁	—	2	4	no	—
77	C₂×C₄×C₂₅₁	—	2	4	no	—
78	C₂×C₄×C₂₅₁	—	2	2	no	—
79	C₂×C₄×C₂₅₁	—	2	2	no	—
80	C₂×C₄×C₂₅₁	—	2	2	no	—
81	C₂×C₄×C₂₅₁	—	2	2	no	—
82	C₂×C₄×C₂₅₁	—	2	2	no	—
83	C₂×C₄×C₂₅₁	—	4	1	no	—
84	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
89	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₂₅₁	#41	1	2	yes	—
93	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
94	C₈×C₂₅₁	—	2	4	no	—
95	C₈×C₂₅₁	—	2	2	no	—
96	C₈×C₂₅₁	C₂×D₄	1	4	yes	complemented
97	C₂²×C₂₅₁	—	2	2	no	—
98	C₂²×C₂₅₁	—	2	1	no	—
99	C₂²×C₂₅₁	—	2	1	no	—
100	C₂²×C₂₅₁	—	2	2	no	—
101	C₂²×C₂₅₁	—	2	2	no	—
102	C₂²×C₂₅₁	—	2	2	no	—
103	C₂²×C₂₅₁	—	2	2	no	—
104	C₂²×C₂₅₁	—	4	8	no	—
105	C₂²×C₂₅₁	C₄×D₄	1	2	yes	—
106	C₂²×C₂₅₁	C₂×M₁₆	1	4	yes	complemented
107	C₂²×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
108	C₄×C₂₅₁	—	2	2	no	—
109	C₄×C₂₅₁	—	2	2	no	—
110	C₄×C₂₅₁	—	2	1	no	—
111	C₄×C₂₅₁	—	2	1	no	—
112	C₄×C₂₅₁	—	2	1	no	—
113	C₄×C₂₅₁	C₄×D₄	1	2	yes	—
114	C₄×C₂₅₁	C₂×M₁₆	1	2	yes	complemented
115	C₄×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
116	C₄×C₂₅₁	C₂×#41	1	1	yes	characteristic
117	C₂×C₂₅₁	—	2	2	no	—
118	C₂×C₂₅₁	—	2	4	no	—
119	C₂×C₂₅₁	—	4	2	no	—
120	C₂×C₂₅₁	C₂×C₄×D₄	1	1	yes	characteristic
121	C₂×C₂₅₁	C₂²×M₁₆	1	1	yes	characteristic
122	C₂×C₂₅₁	#568	1	1	yes	characteristic
123	C₂₅₁	D₄×M₁₆	1	1	yes	characteristic, direct product factor, Sylow
124	D₄×M₁₆	C₂₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂²×M₁₆	C₂×C₂₅₁	1	2	yes	complemented
126	C₂×C₄×D₄	C₂×C₂₅₁	1	1	yes	characteristic
127	#436	C₂×C₂₅₁	1	2	yes	complemented
128	#435	C₂×C₂₅₁	1	4	yes	complemented
129	C₈×D₄	C₂×C₂₅₁	1	2	yes	complemented
130	#423	C₂×C₂₅₁	1	1	yes	characteristic, complemented
131	#407	C₂×C₂₅₁	1	2	yes	complemented
132	C₄×M₁₆	C₂×C₂₅₁	1	1	yes	characteristic, complemented
133	C₂²×D₄	C₄×C₂₅₁	1	1	yes	characteristic
134	C₂³×C₄	C₂²×C₂₅₁	1	2	yes	—
135	C₂×M₁₆	—	2	8	no	—
136	C₂×M₁₆	C₂²×C₂₅₁	1	2	yes	—
137	C₂×M₁₆	C₂²×C₂₅₁	1	1	yes	characteristic
138	C₂×M₁₆	C₂²×C₂₅₁	1	1	yes	characteristic
139	C₂×M₁₆	C₂²×C₂₅₁	1	4	yes	complemented
140	C₂²×C₈	C₂²×C₂₅₁	1	4	yes	complemented
141	C₄×D₄	C₄×C₂₅₁	1	2	yes	—
142	C₄×D₄	C₄×C₂₅₁	1	1	yes	characteristic
143	C₄×D₄	C₄×C₂₅₁	1	1	yes	characteristic
144	C₄×D₄	C₂²×C₂₅₁	1	1	yes	characteristic
145	C₄×D₄	C₂²×C₂₅₁	1	2	yes	—
146	C₄×D₄	C₂²×C₂₅₁	1	1	yes	characteristic
147	C₂×C₄⋊C₄	C₄×C₂₅₁	1	1	yes	characteristic
148	C₂×C₂²⋊C₄	C₄×C₂₅₁	1	2	yes	—
149	C₂×C₄²	C₂²×C₂₅₁	1	1	yes	characteristic
150	C₄⋊C₈	C₂²×C₂₅₁	1	2	yes	complemented
151	C₄⋊C₈	C₂²×C₂₅₁	1	2	yes	complemented
152	C₂²⋊C₈	C₂²×C₂₅₁	1	4	yes	complemented
153	C₂²⋊C₈	C₂²×C₂₅₁	1	4	yes	complemented
154	#97	C₂²×C₂₅₁	1	2	yes	complemented
155	C₄×C₈	C₂²×C₂₅₁	1	2	yes	complemented
156	C₂⁴	C₂×C₄×C₂₅₁	1	2	yes	—
157	C₂×D₄	—	2	4	no	—
158	C₂×D₄	C₂×C₄×C₂₅₁	1	2	yes	—
159	C₂×D₄	C₂×C₄×C₂₅₁	1	1	yes	characteristic
160	C₂×D₄	C₂×C₄×C₂₅₁	1	1	yes	characteristic
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	C₂×C₄×C₂₅₁	1	1	yes	characteristic
164	C₂²×C₄	C₂×C₄×C₂₅₁	1	2	yes	—
165	C₂²×C₄	C₂×C₄×C₂₅₁	1	2	yes	—
166	C₂²×C₄	C₂×C₄×C₂₅₁	1	2	yes	—
167	C₂²×C₄	C₂×C₄×C₂₅₁	1	1	yes	characteristic
168	C₂²×C₄	C₂³×C₂₅₁	1	2	yes	—
169	C₂²×C₄	C₂³×C₂₅₁	1	2	yes	—
170	C₂²×C₄	C₂³×C₂₅₁	1	1	yes	characteristic
171	M₁₆	—	2	4	no	—
172	M₁₆	—	2	2	no	—
173	M₁₆	—	2	8	no	—
174	M₁₆	D₄×C₂₅₁	1	4	yes	direct product factor
175	C₂×C₈	—	2	8	no	—
176	C₂×C₈	C₂³×C₂₅₁	1	2	yes	—
177	C₂×C₈	C₂³×C₂₅₁	1	4	yes	—
178	C₂×C₈	C₂³×C₂₅₁	1	2	yes	—
179	C₄⋊C₄	C₂×C₄×C₂₅₁	1	2	yes	—
180	C₄⋊C₄	C₂×C₄×C₂₅₁	1	1	yes	characteristic
181	C₄⋊C₄	C₂×C₄×C₂₅₁	1	1	yes	characteristic
182	C₂²⋊C₄	C₂×C₄×C₂₅₁	1	2	yes	—
183	C₂²⋊C₄	C₂×C₄×C₂₅₁	1	2	yes	—
184	C₂²⋊C₄	C₂×C₄×C₂₅₁	1	2	yes	—
185	C₂²⋊C₄	C₂×C₄×C₂₅₁	1	2	yes	—
186	C₄²	C₂×C₄×C₂₅₁	1	1	yes	characteristic
187	C₄²	C₂×C₄×C₂₅₁	1	1	yes	characteristic
188	C₄²	C₂³×C₂₅₁	1	1	yes	characteristic
189	C₄²	C₂³×C₂₅₁	1	1	yes	characteristic
190	C₂³	—	2	4	no	—
191	C₂³	—	2	4	no	—
192	C₂³	—	4	2	no	—
193	C₂³	C₂²×C₄×C₂₅₁	1	2	yes	—
194	C₂³	C₂²×C₄×C₂₅₁	1	1	yes	characteristic
195	C₂³	C₂²×C₄×C₂₅₁	1	2	yes	—
196	D₄	—	2	4	no	—
197	D₄	—	2	2	no	—
198	D₄	M₁₆×C₂₅₁	1	4	yes	direct product factor
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	4	1	no	—
208	C₂×C₄	C₂²×C₄×C₂₅₁	1	1	yes	characteristic
209	C₂×C₄	C₂²×C₄×C₂₅₁	1	1	yes	characteristic
210	C₂×C₄	C₂²×C₄×C₂₅₁	1	1	yes	characteristic
211	C₂×C₄	C₂²×C₄×C₂₅₁	1	2	yes	—
212	C₂×C₄	C₂²×C₄×C₂₅₁	1	2	yes	—
213	C₂×C₄	C₂²×C₄×C₂₅₁	1	1	yes	characteristic
214	C₂×C₄	C₂²×C₄×C₂₅₁	1	1	yes	characteristic
215	C₂×C₄	C₂×D₄×C₂₅₁	1	2	yes	—
216	C₂×C₄	#41×C₂₅₁	1	2	yes	—
217	C₂×C₄	C₂⁴×C₂₅₁	1	1	yes	characteristic
218	C₈	—	2	4	no	—
219	C₈	—	2	2	no	—
220	C₈	C₂×D₄×C₂₅₁	1	4	yes	complemented
221	C₂²	—	2	2	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	4	8	no	—
229	C₂²	C₄×D₄×C₂₅₁	1	2	yes	—
230	C₂²	C₂×M₁₆×C₂₅₁	1	4	yes	complemented
231	C₂²	C₂³×C₄×C₂₅₁	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	2	1	no	—
237	C₄	C₄×D₄×C₂₅₁	1	2	yes	—
238	C₄	C₂×M₁₆×C₂₅₁	1	2	yes	complemented
239	C₄	C₂²×D₄×C₂₅₁	1	1	yes	characteristic
240	C₄	C₂×#41×C₂₅₁	1	1	yes	characteristic
241	C₂	—	2	2	no	—
242	C₂	—	2	4	no	—
243	C₂	—	4	2	no	—
244	C₂	C₂×C₄×D₄×C₂₅₁	1	1	yes	characteristic
245	C₂	C₂²×M₁₆×C₂₅₁	1	1	yes	characteristic
246	C₂	#568×C₂₅₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₅₁	C₂⁴
commutator	C₂²	C₂³×C₄×C₂₅₁
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₅₁
socle	C₂²×C₂₅₁	C₂³×C₄
prime radical	C₂²×D₄×C₂₅₁	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×M₁₆×C₂₅₁	C₂	semidirect
3. #436×C₂₅₁	C₂	semidirect
4. #435×C₂₅₁	C₂	semidirect
5. C₈×D₄×C₂₅₁	C₂	semidirect
6. #423×C₂₅₁	C₂	semidirect
7. #407×C₂₅₁	C₂	semidirect
8. C₄×M₁₆×C₂₅₁	C₂	semidirect
15. C₂×M₁₆×C₂₅₁	C₂²	semidirect
16. C₂²×C₈×C₂₅₁	C₂²	semidirect
26. C₄⋊C₈×C₂₅₁	C₂²	semidirect
27. C₄⋊C₈×C₂₅₁	C₂²	semidirect
28. C₂²⋊C₈×C₂₅₁	C₂²	semidirect
29. C₂²⋊C₈×C₂₅₁	C₂²	semidirect
30. #97×C₂₅₁	C₂²	semidirect
31. C₄×C₈×C₂₅₁	C₂²	semidirect
50. M₁₆×C₂₅₁	D₄	direct (N × H)
74. D₄×C₂₅₁	M₁₆	direct (N × H)
96. C₈×C₂₅₁	C₂×D₄	semidirect
106. C₂²×C₂₅₁	C₂×M₁₆	semidirect
114. C₄×C₂₅₁	C₂×M₁₆	semidirect
123. C₂₅₁	D₄×M₁₆	direct (N × H)
124. D₄×M₁₆	C₂₅₁	direct (N × H)
125. C₂²×M₁₆	C₂×C₂₅₁	semidirect
127. #436	C₂×C₂₅₁	semidirect
128. #435	C₂×C₂₅₁	semidirect
129. C₈×D₄	C₂×C₂₅₁	semidirect
130. #423	C₂×C₂₅₁	semidirect
131. #407	C₂×C₂₅₁	semidirect
132. C₄×M₁₆	C₂×C₂₅₁	semidirect
139. C₂×M₁₆	C₂²×C₂₅₁	semidirect
140. C₂²×C₈	C₂²×C₂₅₁	semidirect
150. C₄⋊C₈	C₂²×C₂₅₁	semidirect
151. C₄⋊C₈	C₂²×C₂₅₁	semidirect
152. C₂²⋊C₈	C₂²×C₂₅₁	semidirect
153. C₂²⋊C₈	C₂²×C₂₅₁	semidirect
154. #97	C₂²×C₂₅₁	semidirect
155. C₄×C₈	C₂²×C₂₅₁	semidirect
174. M₁₆	D₄×C₂₅₁	direct (N × H)
198. D₄	M₁₆×C₂₅₁	direct (N × H)
220. C₈	C₂×D₄×C₂₅₁	semidirect
230. C₂²	C₂×M₁₆×C₂₅₁	semidirect
238. C₄	C₂×M₁₆×C₂₅₁	semidirect

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
C₂×C₄×D₄×C₂₅₁	2
#436×C₂₅₁	2
#435×C₂₅₁	2
C₈×D₄×C₂₅₁	2
#423×C₂₅₁	2
#407×C₂₅₁	2
C₄×M₁₆×C₂₅₁	2
D₄×M₁₆	251

Subgroups — D4×M16×C251 #44547521

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×M₁₆×C₂₅₁ #44547521