Subgroups — C₅×D₃×ΓL(1,𝔽₂₅) #13087062

Order 1440.

292 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×D₃²	C₂	1	1	yes	maximal, characteristic
2	C₅×#3743	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×D₃×C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#3712	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#3711	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×D₃×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×D₃×M₁₆	—	3	1	no	maximal
9	C₂×C₅×ΓL(1,𝔽₂₅)	—	3	1	no	maximal
10	C₂×C₅×D₃²	C₄	1	1	yes	characteristic
11	C₄×C₅×C₃⋊D₃	C₂²	1	1	yes	characteristic
12	C₃×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
13	C₃×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
14	C₅×#624	C₄	1	1	yes	characteristic
15	C₅×D₃×Dic₃	C₄	1	1	yes	characteristic
16	C₅×D₃×Dic₃	C₄	1	1	yes	characteristic
17	C₃²×C₅×C₈	C₂²	1	1	yes	characteristic, complemented
18	C₅×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic, complemented
19	C₃×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
20	C₃×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
21	D₃×ΓL(1,𝔽₂₅)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂×C₄×C₅×D₃	—	3	1	no	—
23	C₂×C₄×C₅×D₃	—	3	1	no	—
24	C₃×C₅×M₁₆	—	3	1	no	—
25	C₂×C₃×C₅×C₈	—	3	1	no	—
26	C₅×#208	—	3	1	no	—
27	C₂×C₅×C₃⋊C₈	—	3	1	no	—
28	C₅×ΓL(1,𝔽₂₅)	—	3	1	no	—
29	C₅×ΓL(1,𝔽₂₅)	—	3	1	no	—
30	C₅×ΓL(1,𝔽₂₅)	—	3	1	no	—
31	C₅×ΓL(1,𝔽₂₅)	—	3	1	no	—
32	C₅×ΓL(1,𝔽₂₅)	—	3	1	no	—
33	C₅×ΓL(1,𝔽₂₅)	—	6	1	no	—
34	C₅×ΓL(1,𝔽₂₅)	D₃	1	1	yes	characteristic, direct product factor
35	C₅×D₃×C₈	—	3	1	no	—
36	C₅×D₃×C₈	D₃	1	1	yes	characteristic, complemented
37	C₂×C₅×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×C₅×D₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×C₅×D₃	C₂×C₄	1	1	yes	characteristic
40	C₅×D₃²	—	2	2	no	—
41	C₃²×C₄×C₅	C₂³	1	1	yes	characteristic
42	C₅×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
43	C₃×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
44	C₃×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₅×M₁₆	—	9	1	no	Hall
46	C₄×D₃²	C₂×C₅	1	1	yes	characteristic
47	#3743	C₂×C₅	1	1	yes	characteristic, complemented
48	C₃×ΓL(1,𝔽₂₅)	C₂×C₅	1	1	yes	characteristic, complemented
49	C₃×D₃×C₈	C₂×C₅	1	1	yes	characteristic, complemented
50	#3712	C₂×C₅	1	1	yes	characteristic, complemented
51	#3711	C₂×C₅	1	1	yes	characteristic, complemented
52	D₃×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
53	C₂²×C₅×D₃	—	3	1	no	—
54	C₂²×C₅×D₃	—	3	1	no	—
55	C₂×C₃×C₄×C₅	—	3	1	no	—
56	C₂×C₃×C₄×C₅	—	3	1	no	—
57	C₂×C₅×Dic₃	—	3	1	no	—
58	C₂×C₅×Dic₃	—	3	1	no	—
59	C₄×C₅×D₃	—	3	1	no	—
60	C₄×C₅×D₃	—	3	1	no	—
61	C₄×C₅×D₃	—	3	1	no	—
62	C₄×C₅×D₃	—	3	1	no	—
63	C₄×C₅×D₃	—	3	1	no	—
64	C₄×C₅×D₃	—	3	1	no	—
65	C₄×C₅×D₃	—	6	1	no	—
66	C₄×C₅×D₃	C₂×D₃	1	1	yes	characteristic
67	C₄×C₅×D₃	C₂×D₃	1	1	yes	characteristic
68	C₃×C₅×C₈	—	2	1	no	—
69	C₃×C₅×C₈	—	3	1	no	—
70	C₃×C₅×C₈	—	3	1	no	—
71	C₃×C₅×C₈	C₂×D₃	1	1	yes	characteristic, complemented
72	C₃×C₅×C₈	C₂×D₃	1	1	yes	characteristic, complemented
73	C₅×C₃⋊C₈	—	3	1	no	—
74	C₅×C₃⋊C₈	—	3	1	no	—
75	C₅×C₃⋊C₈	—	6	1	no	—
76	C₅×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic, complemented
77	C₅×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic, complemented
78	D₃×M₁₆	—	3	1	no	—
79	C₂×ΓL(1,𝔽₂₅)	—	3	1	no	—
80	C₂×C₃²×C₅	C₂²×C₄	1	1	yes	characteristic
81	C₅×C₃⋊D₃	—	2	1	no	—
82	C₃×C₅×D₃	—	2	1	no	—
83	C₃×C₅×D₃	M₁₆	1	2	yes	complemented
84	C₂²×C₄×C₅	—	9	1	no	—
85	C₅×M₁₆	—	3	1	no	—
86	C₅×M₁₆	—	9	1	no	—
87	C₅×M₁₆	—	9	1	no	—
88	C₅×M₁₆	—	9	1	no	—
89	C₂×C₅×C₈	—	3	1	no	—
90	C₂×C₅×C₈	—	9	1	no	—
91	C₂×D₃²	C₄×C₅	1	1	yes	characteristic
92	C₄×C₃⋊D₃	C₂²×C₅	1	1	yes	characteristic
93	C₃×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
94	C₃×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
95	#624	C₄×C₅	1	1	yes	characteristic
96	D₃×Dic₃	C₄×C₅	1	1	yes	characteristic
97	D₃×Dic₃	C₄×C₅	1	1	yes	characteristic
98	C₃²×C₈	C₂²×C₅	1	1	yes	characteristic, complemented
99	C₃⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic, complemented
100	C₃×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic, complemented
101	C₃×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic, complemented
102	C₂²×C₃×C₅	—	3	1	no	—
103	C₂²×C₃×C₅	—	3	1	no	—
104	C₂×C₅×D₃	—	3	1	no	—
105	C₂×C₅×D₃	—	3	1	no	—
106	C₂×C₅×D₃	—	6	1	no	—
107	C₂×C₅×D₃	—	6	2	no	—
108	C₂×C₅×D₃	—	6	2	no	—
109	C₂×C₅×D₃	C₄×D₃	1	1	yes	characteristic
110	C₂×C₅×D₃	C₄×D₃	1	1	yes	characteristic
111	C₃×C₄×C₅	—	2	1	no	—
112	C₃×C₄×C₅	—	3	1	no	—
113	C₃×C₄×C₅	—	3	1	no	—
114	C₃×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
115	C₃×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
116	C₅×Dic₃	—	3	1	no	—
117	C₅×Dic₃	—	3	1	no	—
118	C₅×Dic₃	—	6	1	no	—
119	C₅×Dic₃	C₄×D₃	1	1	yes	characteristic
120	C₅×Dic₃	C₄×D₃	1	1	yes	characteristic
121	C₂×C₄×D₃	—	3	1	no	—
122	C₂×C₄×D₃	—	3	1	no	—
123	C₃×M₁₆	—	3	1	no	—
124	C₂×C₃×C₈	—	3	1	no	—
125	#208	—	3	1	no	—
126	C₂×C₃⋊C₈	—	3	1	no	—
127	ΓL(1,𝔽₂₅)	—	3	1	no	—
128	ΓL(1,𝔽₂₅)	—	3	1	no	—
129	ΓL(1,𝔽₂₅)	—	3	1	no	—
130	ΓL(1,𝔽₂₅)	—	3	1	no	—
131	ΓL(1,𝔽₂₅)	—	3	1	no	—
132	ΓL(1,𝔽₂₅)	—	6	1	no	—
133	ΓL(1,𝔽₂₅)	C₅×D₃	1	1	yes	characteristic, direct product factor
134	D₃×C₈	—	3	1	no	—
135	D₃×C₈	C₅×D₃	1	1	yes	characteristic, complemented
136	C₃²×C₅	C₂×M₁₆	1	1	yes	characteristic, complemented, Hall
137	C₂³×C₅	—	9	1	no	—
138	C₂×C₄×C₅	—	3	1	no	—
139	C₂×C₄×C₅	—	3	1	no	—
140	C₂×C₄×C₅	—	9	1	no	—
141	C₂×C₄×C₅	—	9	1	no	—
142	C₂×C₄×C₅	—	9	1	no	—
143	C₂×C₄×C₅	—	9	1	no	—
144	C₅×C₈	—	3	1	no	—
145	C₅×C₈	—	3	1	no	—
146	C₅×C₈	—	9	1	no	—
147	C₅×C₈	D₃²	1	1	yes	characteristic, complemented
148	C₂×C₃⋊D₃	C₂×C₄×C₅	1	1	yes	characteristic
149	C₂×C₃×D₃	C₂×C₄×C₅	1	1	yes	characteristic
150	C₂×C₃×D₃	C₂×C₄×C₅	1	1	yes	characteristic
151	D₃²	—	2	2	no	—
152	C₃²×C₄	C₂³×C₅	1	1	yes	characteristic
153	C₃⋊Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
154	C₃×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
155	C₃×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
156	C₂×M₁₆	—	9	1	no	Sylow
157	C₂×C₃×C₅	—	2	1	no	—
158	C₂×C₃×C₅	—	3	2	no	—
159	C₂×C₃×C₅	—	6	1	no	—
160	C₂×C₃×C₅	C₂×C₄×D₃	1	1	yes	characteristic
161	C₂×C₃×C₅	C₂×C₄×D₃	1	1	yes	characteristic
162	C₅×D₃	—	2	1	no	—
163	C₅×D₃	—	6	1	no	—
164	C₅×D₃	—	6	1	no	—
165	C₅×D₃	—	12	1	no	—
166	C₅×D₃	ΓL(1,𝔽₂₅)	1	2	yes	direct product factor
167	C₂²×D₃	—	3	1	no	—
168	C₂²×D₃	—	3	1	no	—
169	C₂×C₃×C₄	—	3	1	no	—
170	C₂×C₃×C₄	—	3	1	no	—
171	C₂×Dic₃	—	3	1	no	—
172	C₂×Dic₃	—	3	1	no	—
173	C₄×D₃	—	3	1	no	—
174	C₄×D₃	—	3	1	no	—
175	C₄×D₃	—	3	1	no	—
176	C₄×D₃	—	3	1	no	—
177	C₄×D₃	—	3	1	no	—
178	C₄×D₃	—	3	1	no	—
179	C₄×D₃	—	6	1	no	—
180	C₄×D₃	C₂×C₅×D₃	1	1	yes	characteristic
181	C₄×D₃	C₂×C₅×D₃	1	1	yes	characteristic
182	C₃×C₈	—	2	1	no	—
183	C₃×C₈	—	3	1	no	—
184	C₃×C₈	—	3	1	no	—
185	C₃×C₈	C₂×C₅×D₃	1	1	yes	characteristic, complemented
186	C₃×C₈	C₂×C₅×D₃	1	1	yes	characteristic, complemented
187	C₃⋊C₈	—	3	1	no	—
188	C₃⋊C₈	—	3	1	no	—
189	C₃⋊C₈	—	6	1	no	—
190	C₃⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic, complemented
191	C₃⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic, complemented
192	C₂²×C₅	—	3	1	no	—
193	C₂²×C₅	—	3	1	no	—
194	C₂²×C₅	—	9	1	no	—
195	C₂²×C₅	—	18	2	no	—
196	C₄×C₅	—	3	1	no	—
197	C₄×C₅	—	3	1	no	—
198	C₄×C₅	—	9	1	no	—
199	C₄×C₅	C₂×D₃²	1	1	yes	characteristic
200	C₂×C₃²	C₂²×C₄×C₅	1	1	yes	characteristic
201	C₃⋊D₃	—	2	1	no	—
202	C₃×D₃	—	2	1	no	—
203	C₃×D₃	C₅×M₁₆	1	2	yes	complemented
204	C₂²×C₄	—	9	1	no	—
205	M₁₆	—	3	1	no	—
206	M₁₆	—	9	1	no	—
207	M₁₆	—	9	1	no	—
208	M₁₆	—	9	1	no	—
209	C₂×C₈	—	3	1	no	—
210	C₂×C₈	—	9	1	no	—
211	C₃×C₅	—	2	1	no	—
212	C₃×C₅	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic, complemented
213	C₃×C₅	D₃×M₁₆	1	1	yes	characteristic, complemented
214	C₂²×C₃	—	3	1	no	—
215	C₂²×C₃	—	3	1	no	—
216	C₂×D₃	—	3	1	no	—
217	C₂×D₃	—	3	1	no	—
218	C₂×D₃	—	6	1	no	—
219	C₂×D₃	—	6	2	no	—
220	C₂×D₃	—	6	2	no	—
221	C₂×D₃	C₄×C₅×D₃	1	1	yes	characteristic
222	C₂×D₃	C₄×C₅×D₃	1	1	yes	characteristic
223	C₃×C₄	—	2	1	no	—
224	C₃×C₄	—	3	1	no	—
225	C₃×C₄	—	3	1	no	—
226	C₃×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
227	C₃×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
228	Dic₃	—	3	1	no	—
229	Dic₃	—	3	1	no	—
230	Dic₃	—	6	1	no	—
231	Dic₃	C₄×C₅×D₃	1	1	yes	characteristic
232	Dic₃	C₄×C₅×D₃	1	1	yes	characteristic
233	C₂×C₅	—	3	2	no	—
234	C₂×C₅	—	6	1	no	—
235	C₂×C₅	—	18	1	no	—
236	C₂×C₅	C₄×D₃²	1	1	yes	characteristic
237	C₃²	C₂×C₅×M₁₆	1	1	yes	characteristic, complemented, Sylow
238	C₂³	—	9	1	no	—
239	C₂×C₄	—	3	1	no	—
240	C₂×C₄	—	3	1	no	—
241	C₂×C₄	—	9	1	no	—
242	C₂×C₄	—	9	1	no	—
243	C₂×C₄	—	9	1	no	—
244	C₂×C₄	—	9	1	no	—
245	C₈	—	3	1	no	—
246	C₈	—	3	1	no	—
247	C₈	—	9	1	no	—
248	C₈	C₅×D₃²	1	1	yes	characteristic, complemented
249	C₂×C₃	—	2	1	no	—
250	C₂×C₃	—	3	2	no	—
251	C₂×C₃	—	6	1	no	—
252	C₂×C₃	C₂×C₄×C₅×D₃	1	1	yes	characteristic
253	C₂×C₃	C₂×C₄×C₅×D₃	1	1	yes	characteristic
254	D₃	—	2	1	no	—
255	D₃	—	6	1	no	—
256	D₃	—	6	1	no	—
257	D₃	—	12	1	no	—
258	D₃	C₅×ΓL(1,𝔽₂₅)	1	2	yes	direct product factor
259	C₅	D₃×ΓL(1,𝔽₂₅)	1	1	yes	characteristic, direct product factor, Sylow
260	C₂²	—	3	1	no	—
261	C₂²	—	3	1	no	—
262	C₂²	—	9	1	no	—
263	C₂²	—	18	2	no	—
264	C₄	—	3	1	no	—
265	C₄	—	3	1	no	—
266	C₄	—	9	1	no	—
267	C₄	C₂×C₅×D₃²	1	1	yes	characteristic
268	C₃	—	2	1	no	—
269	C₃	C₂×C₅×ΓL(1,𝔽₂₅)	1	1	yes	characteristic, complemented
270	C₃	C₅×D₃×M₁₆	1	1	yes	characteristic, complemented
271	C₂	—	3	2	no	—
272	C₂	—	6	1	no	—
273	C₂	—	18	1	no	—
274	C₂	C₄×C₅×D₃²	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	C₂×D₃²
commutator	C₂×C₃²	C₂²×C₄×C₅
Frattini	C₄	C₂×C₅×D₃²
Fitting	C₃²×C₅×C₈	C₂²
socle	C₂×C₃²×C₅	C₂²×C₄
prime radical	C₂×C₅×D₃²	C₄
cosocle	C₃²×C₄	C₂³×C₅
hypercenter	C₅×C₈	D₃²
hypocenter	C₃²	C₂×C₅×M₁₆
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄×C₅ ◁ C₅×C₈

Complements

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

N	H ≅ G/N	type
2. C₅×#3743	C₂	semidirect
3. C₃×C₅×ΓL(1,𝔽₂₅)	C₂	semidirect
4. C₃×C₅×D₃×C₈	C₂	semidirect
5. C₅×#3712	C₂	semidirect
6. C₅×#3711	C₂	semidirect
7. C₅×D₃×C₃⋊C₈	C₂	semidirect
17. C₃²×C₅×C₈	C₂²	semidirect
18. C₅×C₃⋊(C₃⋊C₈)	C₂²	semidirect
19. C₃×C₅×C₃⋊C₈	C₂²	semidirect
20. C₃×C₅×C₃⋊C₈	C₂²	semidirect
21. D₃×ΓL(1,𝔽₂₅)	C₅	direct (N × H)
34. C₅×ΓL(1,𝔽₂₅)	D₃	direct (N × H)
36. C₅×D₃×C₈	D₃	semidirect
47. #3743	C₂×C₅	semidirect
48. C₃×ΓL(1,𝔽₂₅)	C₂×C₅	semidirect
49. C₃×D₃×C₈	C₂×C₅	semidirect
50. #3712	C₂×C₅	semidirect
51. #3711	C₂×C₅	semidirect
52. D₃×C₃⋊C₈	C₂×C₅	semidirect
71. C₃×C₅×C₈	C₂×D₃	semidirect
72. C₃×C₅×C₈	C₂×D₃	semidirect
76. C₅×C₃⋊C₈	C₂×D₃	semidirect
77. C₅×C₃⋊C₈	C₂×D₃	semidirect
83. C₃×C₅×D₃	M₁₆	semidirect
98. C₃²×C₈	C₂²×C₅	semidirect
99. C₃⋊(C₃⋊C₈)	C₂²×C₅	semidirect
100. C₃×C₃⋊C₈	C₂²×C₅	semidirect
101. C₃×C₃⋊C₈	C₂²×C₅	semidirect
133. ΓL(1,𝔽₂₅)	C₅×D₃	direct (N × H)
135. D₃×C₈	C₅×D₃	semidirect
136. C₃²×C₅	C₂×M₁₆	semidirect
147. C₅×C₈	D₃²	semidirect
166. C₅×D₃	ΓL(1,𝔽₂₅)	direct (N × H)
185. C₃×C₈	C₂×C₅×D₃	semidirect
186. C₃×C₈	C₂×C₅×D₃	semidirect
190. C₃⋊C₈	C₂×C₅×D₃	semidirect
191. C₃⋊C₈	C₂×C₅×D₃	semidirect
203. C₃×D₃	C₅×M₁₆	semidirect
212. C₃×C₅	C₂×ΓL(1,𝔽₂₅)	semidirect
213. C₃×C₅	D₃×M₁₆	semidirect
237. C₃²	C₂×C₅×M₁₆	semidirect
248. C₈	C₅×D₃²	semidirect
258. D₃	C₅×ΓL(1,𝔽₂₅)	direct (N × H)
259. C₅	D₃×ΓL(1,𝔽₂₅)	direct (N × H)
269. C₃	C₂×C₅×ΓL(1,𝔽₂₅)	semidirect
270. C₃	C₅×D₃×M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×D₃²	2
C₅×#3743	2
C₃×C₅×ΓL(1,𝔽₂₅)	2
C₃×C₅×D₃×C₈	2
C₅×#3712	2
C₅×#3711	2
C₅×D₃×C₃⋊C₈	2
C₅×D₃×M₁₆	3
C₂×C₅×ΓL(1,𝔽₂₅)	3
D₃×ΓL(1,𝔽₂₅)	5

Subgroups — C5×D3×ΓL(1,𝔽25) #13087062

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×D₃×ΓL(1,𝔽₂₅) #13087062