Subgroups — D₅²×M₁₆ #13105795

Order 1600.

664 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₅²	C₂	1	1	yes	maximal, characteristic
2	M₁₆×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×D₅×M₁₆	C₂	1	2	yes	maximal, complemented
4	#11716936	C₂	1	1	yes	maximal, characteristic, complemented
5	D₅×#681	C₂	1	2	yes	maximal, complemented
6	#11716909	C₂	1	2	yes	maximal, complemented
7	D₅×#676	C₂	1	4	yes	maximal, complemented
8	C₈×D₅²	C₂	1	2	yes	maximal, complemented
9	C₂²×D₅²	C₄	1	1	yes	characteristic
10	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₅×D₅	C₂²	1	2	yes	—
12	C₂×#6036	C₄	1	1	yes	characteristic
13	C₂×D₅×Dic₅	C₄	1	2	yes	—
14	C₄×D₅²	C₄	1	1	yes	characteristic
15	C₄×D₅²	C₄	1	2	yes	—
16	C₄×D₅²	C₄	1	1	yes	characteristic
17	C₄×D₅²	C₂²	1	1	yes	characteristic
18	C₄×D₅²	C₂²	1	2	yes	—
19	C₄×D₅²	C₂²	1	1	yes	characteristic
20	C₅²×M₁₆	C₂²	1	1	yes	characteristic, complemented
21	#87804	C₂²	1	1	yes	characteristic, complemented
22	#87799	C₂²	1	2	yes	complemented
23	C₈×C₅⋊D₅	C₂²	1	2	yes	complemented
24	C₅×#681	C₂²	1	2	yes	complemented
25	C₅×#676	C₂²	1	4	yes	complemented
26	C₅×C₈×D₅	C₂²	1	4	yes	complemented
27	#87769	C₂²	1	2	yes	complemented
28	#87768	C₂²	1	4	yes	complemented
29	#87767	C₂²	1	2	yes	complemented
30	D₅×#168	C₂²	1	4	yes	complemented
31	C₂×D₅×M₁₆	—	5	2	no	maximal
32	C₂²×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₅×D₅	C₂×C₄	1	2	yes	—
34	C₂×D₅²	—	2	4	no	—
35	C₂×D₅²	C₂×C₄	1	2	yes	—
36	C₂×D₅²	C₂×C₄	1	1	yes	characteristic
37	C₂×D₅²	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
39	C₂×#1039	C₂×C₄	1	1	yes	characteristic
40	C₄×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
41	C₄×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
42	C₄×C₅⋊D₅	C₂³	1	1	yes	characteristic
43	C₄×C₅⋊D₅	C₂³	1	1	yes	characteristic
44	C₂×C₅×Dic₅	C₂×C₄	1	2	yes	—
45	C₄×C₅×D₅	C₂×C₄	1	2	yes	—
46	C₄×C₅×D₅	C₂×C₄	1	2	yes	—
47	C₄×C₅×D₅	C₂³	1	2	yes	—
48	C₄×C₅×D₅	C₂³	1	2	yes	—
49	#6036	C₂×C₄	1	2	yes	—
50	#6036	C₂×C₄	1	1	yes	characteristic
51	#6036	C₂×C₄	1	1	yes	characteristic
52	D₅×Dic₅	C₂×C₄	1	2	yes	—
53	D₅×Dic₅	C₂×C₄	1	2	yes	—
54	D₅×Dic₅	C₂×C₄	1	2	yes	—
55	D₅×Dic₅	C₂×C₄	1	2	yes	—
56	C₅²×C₈	C₂³	1	2	yes	complemented
57	#6029	C₂³	1	2	yes	complemented
58	C₅×#168	C₂³	1	4	yes	complemented
59	C₂²×C₄×D₅	—	5	2	no	—
60	C₂×C₅×M₁₆	—	5	2	no	—
61	C₂×#681	—	5	2	no	—
62	D₅×M₁₆	—	5	2	no	—
63	D₅×M₁₆	—	5	2	no	—
64	D₅×M₁₆	—	5	2	no	—
65	D₅×M₁₆	—	5	4	no	—
66	D₅×M₁₆	—	5	4	no	—
67	D₅×M₁₆	—	10	2	no	—
68	D₅×M₁₆	D₅	1	2	yes	direct product factor
69	C₂×#676	—	5	4	no	—
70	C₂×C₈×D₅	—	5	4	no	—
71	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₅⋊D₅	—	2	2	no	—
73	C₂×C₅⋊D₅	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₅⋊D₅	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₅×D₅	—	2	4	no	—
76	C₂×C₅×D₅	C₂²×C₄	1	2	yes	—
77	C₂×C₅×D₅	C₂²×C₄	1	2	yes	—
78	D₅²	—	2	4	no	—
79	D₅²	—	2	2	no	—
80	D₅²	M₁₆	1	4	yes	direct product factor
81	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
82	C₄×C₅²	C₂⁴	1	1	yes	characteristic
83	#1039	C₂²×C₄	1	1	yes	characteristic
84	#1039	C₂²×C₄	1	1	yes	characteristic
85	C₅×Dic₅	C₂²×C₄	1	2	yes	—
86	C₅×Dic₅	C₂²×C₄	1	2	yes	—
87	C₂³×D₅	—	5	2	no	—
88	C₂²×C₄×C₅	—	5	2	no	—
89	C₂²×Dic₅	—	5	2	no	—
90	C₂×C₄×D₅	—	5	2	no	—
91	C₂×C₄×D₅	—	5	2	no	—
92	C₂×C₄×D₅	—	5	2	no	—
93	C₂×C₄×D₅	—	5	2	no	—
94	C₂×C₄×D₅	—	5	2	no	—
95	C₂×C₄×D₅	—	5	2	no	—
96	C₂×C₄×D₅	—	5	2	no	—
97	C₂×C₄×D₅	—	5	2	no	—
98	C₂×C₄×D₅	—	5	2	no	—
99	C₂×C₄×D₅	—	5	2	no	—
100	C₂×C₄×D₅	—	5	2	no	—
101	C₂×C₄×D₅	—	10	2	no	—
102	C₂×C₄×D₅	C₂×D₅	1	2	yes	—
103	C₅×M₁₆	—	2	2	no	—
104	C₅×M₁₆	—	5	2	no	—
105	C₅×M₁₆	—	5	4	no	—
106	C₅×M₁₆	C₂×D₅	1	2	yes	complemented
107	C₂×C₅×C₈	—	5	4	no	—
108	#681	—	5	2	no	—
109	#681	—	5	4	no	—
110	#681	—	10	2	no	—
111	#681	C₂×D₅	1	2	yes	complemented
112	C₂×#168	—	5	4	no	—
113	#676	—	5	4	no	—
114	#676	—	5	4	no	—
115	#676	—	5	4	no	—
116	#676	—	10	4	no	—
117	#676	C₂×D₅	1	4	yes	complemented
118	C₈×D₅	—	5	4	no	—
119	C₈×D₅	—	5	4	no	—
120	C₈×D₅	—	5	4	no	—
121	C₈×D₅	—	10	4	no	—
122	C₈×D₅	C₂×D₅	1	4	yes	complemented
123	C₂²×M₁₆	—	25	1	no	Sylow
124	C₂×C₅²	—	2	1	no	—
125	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
126	C₅⋊D₅	—	2	1	no	—
127	C₅⋊D₅	C₂×M₁₆	1	2	yes	complemented
128	C₅×D₅	—	2	2	no	—
129	C₅×D₅	C₂×M₁₆	1	4	yes	complemented
130	C₂³×C₅	—	5	2	no	—
131	C₂²×D₅	—	5	2	no	—
132	C₂²×D₅	—	5	2	no	—
133	C₂²×D₅	—	5	2	no	—
134	C₂²×D₅	—	5	2	no	—
135	C₂²×D₅	—	5	2	no	—
136	C₂²×D₅	—	10	2	no	—
137	C₂²×D₅	—	10	8	no	—
138	C₂²×D₅	C₄×D₅	1	2	yes	—
139	C₂×C₄×C₅	—	2	2	no	—
140	C₂×C₄×C₅	—	5	2	no	—
141	C₂×C₄×C₅	—	5	2	no	—
142	C₂×C₄×C₅	—	5	2	no	—
143	C₂×C₄×C₅	—	5	2	no	—
144	C₂×C₄×C₅	—	5	2	no	—
145	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
146	C₂×Dic₅	—	5	2	no	—
147	C₂×Dic₅	—	5	2	no	—
148	C₂×Dic₅	—	5	2	no	—
149	C₂×Dic₅	—	5	2	no	—
150	C₂×Dic₅	—	5	2	no	—
151	C₂×Dic₅	—	10	2	no	—
152	C₂×Dic₅	C₄×D₅	1	2	yes	—
153	C₄×D₅	—	5	2	no	—
154	C₄×D₅	—	5	2	no	—
155	C₄×D₅	—	5	2	no	—
156	C₄×D₅	—	5	2	no	—
157	C₄×D₅	—	5	2	no	—
158	C₄×D₅	—	5	2	no	—
159	C₄×D₅	—	5	2	no	—
160	C₄×D₅	—	5	2	no	—
161	C₄×D₅	—	5	2	no	—
162	C₄×D₅	—	5	2	no	—
163	C₄×D₅	—	5	2	no	—
164	C₄×D₅	—	5	2	no	—
165	C₄×D₅	—	10	2	no	—
166	C₄×D₅	—	10	2	no	—
167	C₄×D₅	—	10	2	no	—
168	C₄×D₅	—	10	2	no	—
169	C₄×D₅	C₄×D₅	1	2	yes	—
170	C₄×D₅	C₄×D₅	1	2	yes	—
171	C₄×D₅	C₂²×D₅	1	2	yes	—
172	C₄×D₅	C₂²×D₅	1	2	yes	—
173	C₅×C₈	—	2	4	no	—
174	C₅×C₈	—	5	4	no	—
175	C₅×C₈	C₂²×D₅	1	4	yes	complemented
176	#168	—	5	4	no	—
177	#168	—	10	4	no	—
178	#168	C₂²×D₅	1	4	yes	complemented
179	C₂³×C₄	—	25	1	no	—
180	C₂×M₁₆	—	5	2	no	—
181	C₂×M₁₆	—	25	4	no	—
182	C₂×M₁₆	—	25	1	no	—
183	C₂×M₁₆	—	25	2	no	—
184	C₂×M₁₆	—	25	2	no	—
185	C₂×M₁₆	—	25	1	no	—
186	C₂²×C₈	—	25	2	no	—
187	C₅²	C₂²×M₁₆	1	1	yes	characteristic, complemented, Sylow
188	C₂²×C₅	—	2	2	no	—
189	C₂²×C₅	—	5	2	no	—
190	C₂²×C₅	—	5	2	no	—
191	C₂²×C₅	—	10	4	no	—
192	C₂²×C₅	C₂×C₄×D₅	1	2	yes	—
193	C₂×D₅	—	2	4	no	—
194	C₂×D₅	—	5	2	no	—
195	C₂×D₅	—	5	8	no	—
196	C₂×D₅	—	5	2	no	—
197	C₂×D₅	—	10	2	no	—
198	C₂×D₅	—	10	2	no	—
199	C₂×D₅	—	10	4	no	—
200	C₂×D₅	—	10	4	no	—
201	C₂×D₅	—	10	4	no	—
202	C₂×D₅	—	10	4	no	—
203	C₂×D₅	—	20	4	no	—
204	C₂×D₅	C₂×C₄×D₅	1	2	yes	—
205	C₂×D₅	C₂×C₄×D₅	1	2	yes	—
206	C₄×C₅	—	2	2	no	—
207	C₄×C₅	—	2	2	no	—
208	C₄×C₅	—	5	2	no	—
209	C₄×C₅	—	5	2	no	—
210	C₄×C₅	C₂×C₄×D₅	1	2	yes	—
211	C₄×C₅	C₂³×D₅	1	2	yes	—
212	Dic₅	—	5	2	no	—
213	Dic₅	—	5	2	no	—
214	Dic₅	—	10	2	no	—
215	Dic₅	—	10	2	no	—
216	Dic₅	C₂×C₄×D₅	1	2	yes	—
217	Dic₅	C₂×C₄×D₅	1	2	yes	—
218	C₂⁴	—	25	1	no	—
219	C₂²×C₄	—	5	2	no	—
220	C₂²×C₄	—	25	1	no	—
221	C₂²×C₄	—	25	2	no	—
222	C₂²×C₄	—	25	1	no	—
223	C₂²×C₄	—	25	1	no	—
224	C₂²×C₄	—	25	1	no	—
225	C₂²×C₄	—	25	2	no	—
226	C₂²×C₄	—	25	1	no	—
227	C₂²×C₄	—	25	2	no	—
228	C₂²×C₄	—	25	1	no	—
229	M₁₆	—	5	2	no	—
230	M₁₆	—	5	4	no	—
231	M₁₆	—	25	2	no	—
232	M₁₆	—	25	4	no	—
233	M₁₆	—	25	1	no	—
234	M₁₆	—	25	2	no	—
235	M₁₆	D₅²	1	1	yes	characteristic, direct product factor
236	C₂×C₈	—	5	4	no	—
237	C₂×C₈	—	25	2	no	—
238	C₂×C₈	—	25	4	no	—
239	C₂×C₈	—	25	2	no	—
240	C₂×C₅	—	2	2	no	—
241	C₂×C₅	—	2	2	no	—
242	C₂×C₅	—	4	2	no	—
243	C₂×C₅	—	5	4	no	—
244	C₂×C₅	—	10	2	no	—
245	C₂×C₅	C₂²×C₄×D₅	1	2	yes	—
246	D₅	—	2	2	no	—
247	D₅	—	5	4	no	—
248	D₅	—	10	4	no	—
249	D₅	—	10	2	no	—
250	D₅	—	20	2	no	—
251	D₅	D₅×M₁₆	1	4	yes	direct product factor
252	C₂³	—	5	2	no	—
253	C₂³	—	25	1	no	—
254	C₂³	—	25	2	no	—
255	C₂³	—	25	1	no	—
256	C₂³	—	25	1	no	—
257	C₂³	—	50	4	no	—
258	C₂×C₄	—	5	2	no	—
259	C₂×C₄	—	5	2	no	—
260	C₂×C₄	—	5	2	no	—
261	C₂×C₄	—	5	2	no	—
262	C₂×C₄	—	5	2	no	—
263	C₂×C₄	—	25	1	no	—
264	C₂×C₄	—	25	1	no	—
265	C₂×C₄	—	25	2	no	—
266	C₂×C₄	—	25	2	no	—
267	C₂×C₄	—	25	2	no	—
268	C₂×C₄	—	25	2	no	—
269	C₂×C₄	—	25	2	no	—
270	C₂×C₄	—	25	1	no	—
271	C₂×C₄	—	25	1	no	—
272	C₂×C₄	—	25	1	no	—
273	C₂×C₄	—	25	1	no	—
274	C₂×C₄	—	25	1	no	—
275	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
276	C₈	—	5	4	no	—
277	C₈	—	25	2	no	—
278	C₈	C₂×D₅²	1	2	yes	complemented
279	C₅	—	2	2	no	—
280	C₅	C₂×D₅×M₁₆	1	2	yes	complemented
281	C₂²	—	5	2	no	—
282	C₂²	—	5	2	no	—
283	C₂²	—	10	4	no	—
284	C₂²	—	25	4	no	—
285	C₂²	—	25	1	no	—
286	C₂²	—	25	1	no	—
287	C₂²	—	50	2	no	—
288	C₂²	—	50	4	no	—
289	C₂²	—	50	2	no	—
290	C₂²	C₄×D₅²	1	1	yes	characteristic
291	C₄	—	5	2	no	—
292	C₄	—	5	2	no	—
293	C₄	—	25	1	no	—
294	C₄	—	25	1	no	—
295	C₄	C₄×D₅²	1	1	yes	characteristic
296	C₄	C₂²×D₅²	1	1	yes	characteristic
297	C₂	—	2	1	no	—
298	C₂	—	5	4	no	—
299	C₂	—	10	2	no	—
300	C₂	—	25	2	no	—
301	C₂	—	50	1	no	—
302	C₂	C₂×C₄×D₅²	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂²×D₅²
commutator, socle	C₂×C₅²	C₂³×C₄
Fitting	C₅²×M₁₆	C₂²
prime radical	C₂²×D₅²	C₄
cosocle	C₄×C₅²	C₂⁴
hypercenter	M₁₆	D₅²
hypocenter	C₅²	C₂²×M₁₆
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
2. M₁₆×C₅⋊D₅	C₂	semidirect
3. C₅×D₅×M₁₆	C₂	semidirect
4. #11716936	C₂	semidirect
5. D₅×#681	C₂	semidirect
6. #11716909	C₂	semidirect
7. D₅×#676	C₂	semidirect
8. C₈×D₅²	C₂	semidirect
20. C₅²×M₁₆	C₂²	semidirect
21. #87804	C₂²	semidirect
22. #87799	C₂²	semidirect
23. C₈×C₅⋊D₅	C₂²	semidirect
24. C₅×#681	C₂²	semidirect
25. C₅×#676	C₂²	semidirect
26. C₅×C₈×D₅	C₂²	semidirect
27. #87769	C₂²	semidirect
28. #87768	C₂²	semidirect
29. #87767	C₂²	semidirect
30. D₅×#168	C₂²	semidirect
56. C₅²×C₈	C₂³	semidirect
57. #6029	C₂³	semidirect
58. C₅×#168	C₂³	semidirect
68. D₅×M₁₆	D₅	direct (N × H)
80. D₅²	M₁₆	direct (N × H)
106. C₅×M₁₆	C₂×D₅	semidirect
111. #681	C₂×D₅	semidirect
117. #676	C₂×D₅	semidirect
122. C₈×D₅	C₂×D₅	semidirect
127. C₅⋊D₅	C₂×M₁₆	semidirect
129. C₅×D₅	C₂×M₁₆	semidirect
175. C₅×C₈	C₂²×D₅	semidirect
178. #168	C₂²×D₅	semidirect
187. C₅²	C₂²×M₁₆	semidirect
235. M₁₆	D₅²	direct (N × H)
251. D₅	D₅×M₁₆	direct (N × H)
278. C₈	C₂×D₅²	semidirect
280. C₅	C₂×D₅×M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₅²	2
M₁₆×C₅⋊D₅	2
C₅×D₅×M₁₆	2
#11716936	2
D₅×#681	2
#11716909	2
D₅×#676	2
C₈×D₅²	2
C₂×D₅×M₁₆	5

Subgroups — D52×M16 #13105795

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅²×M₁₆ #13105795