Subgroups — C₅×#63804 #13087076

Order 1440.

268 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3798	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×#3711	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#3710	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×D₃×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#907	—	3	1	no	maximal
9	C₅×#901	—	3	1	no	maximal
10	C₄×C₅×C₃⋊D₃	C₂²	1	1	yes	characteristic
11	C₃×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
12	C₃×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
13	C₅×#627	C₄	1	1	yes	characteristic
14	C₅×#626	C₄	1	1	yes	characteristic
15	C₅×#626	C₄	1	1	yes	characteristic
16	C₅×#625	C₄	1	1	yes	characteristic
17	C₃²×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₅×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
19	C₃×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₃×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
21	#63804	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₅×#235	—	3	1	no	—
23	C₅×#235	—	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)	—	6	1	no	—
32	C₅×ΓL(1,𝔽₂₅)	D₃	1	1	yes	characteristic, complemented
33	C₅×D₃×C₈	—	3	1	no	—
34	C₅×D₃×C₈	—	3	1	no	—
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	C₃²×C₄×C₅	C₂³	1	1	yes	characteristic
41	C₅×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
42	C₃×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
43	C₃×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
44	C₅×#131	—	9	1	no	Hall
45	#3798	C₂×C₅	1	1	yes	characteristic
46	#3743	C₂×C₅	1	1	yes	characteristic, complemented
47	C₃×ΓL(1,𝔽₂₅)	C₂×C₅	1	1	yes	characteristic, complemented
48	C₃×D₃×C₈	C₂×C₅	1	1	yes	characteristic, complemented
49	#3711	C₂×C₅	1	1	yes	characteristic, complemented
50	#3710	C₂×C₅	1	1	yes	characteristic, complemented
51	D₃×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
52	C₂×C₃×C₄×C₅	—	3	1	no	—
53	C₂×C₃×C₄×C₅	—	3	1	no	—
54	C₅×#67	—	3	1	no	—
55	C₅×#67	—	3	1	no	—
56	C₅×#67	—	3	1	no	—
57	C₅×#67	—	3	1	no	—
58	C₅×D₁₂	—	3	1	no	—
59	C₅×D₁₂	—	3	1	no	—
60	C₄×C₅×D₃	—	3	1	no	—
61	C₄×C₅×D₃	—	3	1	no	—
62	C₄×C₅×D₃	—	6	1	no	—
63	C₄×C₅×D₃	C₂×D₃	1	1	yes	characteristic
64	C₄×C₅×D₃	C₂×D₃	1	1	yes	characteristic
65	C₅×Dic₆	—	3	1	no	—
66	C₅×Dic₆	—	3	1	no	—
67	C₃×C₅×C₈	—	2	1	no	—
68	C₃×C₅×C₈	—	3	1	no	—
69	C₃×C₅×C₈	—	3	1	no	—
70	C₃×C₅×C₈	C₂×D₃	1	1	yes	characteristic
71	C₃×C₅×C₈	C₂×D₃	1	1	yes	characteristic
72	C₅×C₃⋊C₈	—	3	1	no	—
73	C₅×C₃⋊C₈	—	3	1	no	—
74	C₅×C₃⋊C₈	—	6	1	no	—
75	C₅×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
76	C₅×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
77	#907	—	3	1	no	—
78	#901	—	3	1	no	—
79	C₂×C₃²×C₅	C₂²×C₄	1	1	yes	characteristic
80	C₅×C₃⋊D₃	—	2	1	no	—
81	C₃×C₅×D₃	—	2	1	no	—
82	C₃×C₅×D₃	—	2	1	no	—
83	C₅×#41	—	9	1	no	—
84	C₅×M₁₆	—	3	1	no	—
85	C₅×M₁₆	—	9	1	no	—
86	C₅×M₁₆	—	9	1	no	—
87	C₂×C₅×C₈	—	3	1	no	—
88	C₂×C₅×C₈	—	9	1	no	—
89	C₂×C₅×C₈	—	9	1	no	—
90	C₄×C₃⋊D₃	C₂²×C₅	1	1	yes	characteristic
91	C₃×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
92	C₃×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
93	#627	C₄×C₅	1	1	yes	characteristic
94	#626	C₄×C₅	1	1	yes	characteristic
95	#626	C₄×C₅	1	1	yes	characteristic
96	#625	C₄×C₅	1	1	yes	characteristic
97	C₃²×C₈	C₂²×C₅	1	1	yes	characteristic
98	C₃⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
99	C₃×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
100	C₃×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
101	C₂²×C₃×C₅	—	3	1	no	—
102	C₂²×C₃×C₅	—	3	1	no	—
103	C₂×C₅×D₃	—	3	1	no	—
104	C₂×C₅×D₃	—	3	1	no	—
105	C₂×C₅×D₃	—	6	1	no	—
106	C₂×C₅×D₃	C₄×D₃	1	1	yes	characteristic
107	C₂×C₅×D₃	C₄×D₃	1	1	yes	characteristic
108	C₃×C₄×C₅	—	2	1	no	—
109	C₃×C₄×C₅	—	3	1	no	—
110	C₃×C₄×C₅	—	3	1	no	—
111	C₃×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
112	C₃×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
113	C₅×Dic₃	—	3	1	no	—
114	C₅×Dic₃	—	3	1	no	—
115	C₅×Dic₃	—	6	1	no	—
116	C₅×Dic₃	C₄×D₃	1	1	yes	characteristic
117	C₅×Dic₃	C₄×D₃	1	1	yes	characteristic
118	#235	—	3	1	no	—
119	#235	—	3	1	no	—
120	C₃×M₁₆	—	3	1	no	—
121	C₂×C₃×C₈	—	3	1	no	—
122	#208	—	3	1	no	—
123	C₂×C₃⋊C₈	—	3	1	no	—
124	ΓL(1,𝔽₂₅)	—	3	1	no	—
125	ΓL(1,𝔽₂₅)	—	3	1	no	—
126	ΓL(1,𝔽₂₅)	—	3	1	no	—
127	ΓL(1,𝔽₂₅)	—	6	1	no	—
128	ΓL(1,𝔽₂₅)	C₅×D₃	1	1	yes	characteristic, complemented
129	D₃×C₈	—	3	1	no	—
130	D₃×C₈	—	3	1	no	—
131	D₃×C₈	—	3	1	no	—
132	D₃×C₈	C₅×D₃	1	1	yes	characteristic, complemented
133	C₃²×C₅	#131	1	1	yes	characteristic, complemented, Hall
134	C₅×Q₈	—	9	1	no	—
135	C₅×D₄	—	9	1	no	—
136	C₅×D₄	—	9	1	no	—
137	C₅×D₄	—	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	C₅×C₈	—	3	1	no	—
142	C₅×C₈	—	3	1	no	—
143	C₅×C₈	—	9	1	no	—
144	C₅×C₈	D₃²	1	1	yes	characteristic
145	C₂×C₃⋊D₃	C₂×C₄×C₅	1	1	yes	characteristic
146	C₂×C₃×D₃	C₂×C₄×C₅	1	1	yes	characteristic
147	C₂×C₃×D₃	C₂×C₄×C₅	1	1	yes	characteristic
148	C₃²×C₄	C₂³×C₅	1	1	yes	characteristic
149	C₃⋊Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
150	C₃×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
151	C₃×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
152	#131	—	9	1	no	Sylow
153	C₂×C₃×C₅	—	2	1	no	—
154	C₂×C₃×C₅	—	6	1	no	—
155	C₂×C₃×C₅	—	6	1	no	—
156	C₂×C₃×C₅	C₂×C₄×D₃	1	1	yes	characteristic
157	C₂×C₃×C₅	C₂×C₄×D₃	1	1	yes	characteristic
158	C₅×D₃	—	2	1	no	—
159	C₅×D₃	—	2	1	no	—
160	C₅×D₃	—	6	1	no	—
161	C₅×D₃	—	6	1	no	—
162	C₅×D₃	—	12	1	no	—
163	C₂×C₃×C₄	—	3	1	no	—
164	C₂×C₃×C₄	—	3	1	no	—
165	#67	—	3	1	no	—
166	#67	—	3	1	no	—
167	#67	—	3	1	no	—
168	#67	—	3	1	no	—
169	D₁₂	—	3	1	no	—
170	D₁₂	—	3	1	no	—
171	C₄×D₃	—	3	1	no	—
172	C₄×D₃	—	3	1	no	—
173	C₄×D₃	—	6	1	no	—
174	C₄×D₃	C₂×C₅×D₃	1	1	yes	characteristic
175	C₄×D₃	C₂×C₅×D₃	1	1	yes	characteristic
176	Dic₆	—	3	1	no	—
177	Dic₆	—	3	1	no	—
178	C₃×C₈	—	2	1	no	—
179	C₃×C₈	—	3	1	no	—
180	C₃×C₈	—	3	1	no	—
181	C₃×C₈	C₂×C₅×D₃	1	1	yes	characteristic
182	C₃×C₈	C₂×C₅×D₃	1	1	yes	characteristic
183	C₃⋊C₈	—	3	1	no	—
184	C₃⋊C₈	—	3	1	no	—
185	C₃⋊C₈	—	6	1	no	—
186	C₃⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
187	C₃⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
188	C₂²×C₅	—	3	1	no	—
189	C₂²×C₅	—	3	1	no	—
190	C₂²×C₅	—	9	1	no	—
191	C₄×C₅	—	3	1	no	—
192	C₄×C₅	—	3	1	no	—
193	C₄×C₅	—	9	1	no	—
194	C₄×C₅	C₂×D₃²	1	1	yes	characteristic
195	C₂×C₃²	C₂²×C₄×C₅	1	1	yes	characteristic
196	C₃⋊D₃	—	2	1	no	—
197	C₃×D₃	—	2	1	no	—
198	C₃×D₃	—	2	1	no	—
199	#41	—	9	1	no	—
200	M₁₆	—	3	1	no	—
201	M₁₆	—	9	1	no	—
202	M₁₆	—	9	1	no	—
203	C₂×C₈	—	3	1	no	—
204	C₂×C₈	—	9	1	no	—
205	C₂×C₈	—	9	1	no	—
206	C₃×C₅	—	2	1	no	—
207	C₃×C₅	#901	1	1	yes	characteristic, complemented
208	C₃×C₅	#907	1	1	yes	characteristic, complemented
209	C₂²×C₃	—	3	1	no	—
210	C₂²×C₃	—	3	1	no	—
211	C₂×D₃	—	3	1	no	—
212	C₂×D₃	—	3	1	no	—
213	C₂×D₃	—	6	1	no	—
214	C₂×D₃	C₄×C₅×D₃	1	1	yes	characteristic
215	C₂×D₃	C₄×C₅×D₃	1	1	yes	characteristic
216	C₃×C₄	—	2	1	no	—
217	C₃×C₄	—	3	1	no	—
218	C₃×C₄	—	3	1	no	—
219	C₃×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
220	C₃×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
221	Dic₃	—	3	1	no	—
222	Dic₃	—	3	1	no	—
223	Dic₃	—	6	1	no	—
224	Dic₃	C₄×C₅×D₃	1	1	yes	characteristic
225	Dic₃	C₄×C₅×D₃	1	1	yes	characteristic
226	C₂×C₅	—	6	1	no	—
227	C₂×C₅	—	6	1	no	—
228	C₂×C₅	—	18	1	no	—
229	C₂×C₅	C₄×D₃²	1	1	yes	characteristic
230	C₃²	C₅×#131	1	1	yes	characteristic, complemented, Sylow
231	Q₈	—	9	1	no	—
232	D₄	—	9	1	no	—
233	D₄	—	9	1	no	—
234	D₄	—	9	1	no	—
235	C₂×C₄	—	3	1	no	—
236	C₂×C₄	—	3	1	no	—
237	C₂×C₄	—	9	1	no	—
238	C₈	—	3	1	no	—
239	C₈	—	3	1	no	—
240	C₈	—	9	1	no	—
241	C₈	C₅×D₃²	1	1	yes	characteristic
242	C₂×C₃	—	2	1	no	—
243	C₂×C₃	—	6	1	no	—
244	C₂×C₃	—	6	1	no	—
245	C₂×C₃	C₂×C₄×C₅×D₃	1	1	yes	characteristic
246	C₂×C₃	C₂×C₄×C₅×D₃	1	1	yes	characteristic
247	D₃	—	2	1	no	—
248	D₃	—	2	1	no	—
249	D₃	—	6	1	no	—
250	D₃	—	6	1	no	—
251	D₃	—	12	1	no	—
252	C₅	#63804	1	1	yes	characteristic, direct product factor, Sylow
253	C₂²	—	3	1	no	—
254	C₂²	—	3	1	no	—
255	C₂²	—	9	1	no	—
256	C₄	—	3	1	no	—
257	C₄	—	3	1	no	—
258	C₄	—	9	1	no	—
259	C₄	C₂×C₅×D₃²	1	1	yes	characteristic
260	C₃	—	2	1	no	—
261	C₃	C₅×#901	1	1	yes	characteristic, complemented
262	C₃	C₅×#907	1	1	yes	characteristic, complemented
263	C₂	—	6	1	no	—
264	C₂	—	6	1	no	—
265	C₂	—	18	1	no	—
266	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	C₅×#3798	C₂
cosocle	C₃²×C₄	C₂³×C₅
hypercenter	C₅×C₈	D₃²
hypocenter	C₃²	C₅×#131
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×#3711	C₂	semidirect
6. C₅×#3710	C₂	semidirect
7. C₅×D₃×C₃⋊C₈	C₂	semidirect
21. #63804	C₅	direct (N × H)
32. C₅×ΓL(1,𝔽₂₅)	D₃	semidirect
36. C₅×D₃×C₈	D₃	semidirect
46. #3743	C₂×C₅	semidirect
47. C₃×ΓL(1,𝔽₂₅)	C₂×C₅	semidirect
48. C₃×D₃×C₈	C₂×C₅	semidirect
49. #3711	C₂×C₅	semidirect
50. #3710	C₂×C₅	semidirect
51. D₃×C₃⋊C₈	C₂×C₅	semidirect
128. ΓL(1,𝔽₂₅)	C₅×D₃	semidirect
132. D₃×C₈	C₅×D₃	semidirect
133. C₃²×C₅	#131	semidirect
207. C₃×C₅	#901	semidirect
208. C₃×C₅	#907	semidirect
230. C₃²	C₅×#131	semidirect
252. C₅	#63804	direct (N × H)
261. C₃	C₅×#901	semidirect
262. C₃	C₅×#907	semidirect

Maximal subgroups

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

subgroup	index
C₅×#3798	2
C₅×#3743	2
C₃×C₅×ΓL(1,𝔽₂₅)	2
C₃×C₅×D₃×C₈	2
C₅×#3711	2
C₅×#3710	2
C₅×D₃×C₃⋊C₈	2
C₅×#907	3
C₅×#901	3
#63804	5

Subgroups — C5×#63804 #13087076

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#63804 #13087076