Subgroups — C₅×#68069 #14280481

Order 1920.

416 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	C₄×C₅×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic
2	C₅×#4685	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×D₃×C₈	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#322	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4455	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4455	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4454	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1451	—	3	1	no	maximal, Hall
9	C₂×C₅×ΓL(1,𝔽₂₅)	C₄	1	2	yes	—
10	C₂×C₅×D₃×C₈	C₄	1	2	yes	—
11	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
12	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₅×(C₂×D₃)⋊C₈	C₄	1	2	yes	—
16	C₅×#815	C₄	1	2	yes	—
17	C₅×Dic₃⋊C₈	C₄	1	2	yes	—
18	C₅×C₈×Dic₃	C₄	1	2	yes	—
19	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
22	#68069	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₅×#431	—	3	1	no	—
24	C₄×C₅×M₁₆	—	3	1	no	—
25	C₂×C₄×C₅×C₈	—	3	1	no	—
26	C₅×#322	—	3	1	no	—
27	C₅×#322	—	3	1	no	—
28	C₅×#322	—	3	1	no	—
29	C₅×#322	D₃	1	1	yes	characteristic, complemented
30	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
34	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
35	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
36	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
37	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
38	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
39	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
41	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
42	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
43	C₅×ΓL(1,𝔽₂₅)	—	2	4	no	—
44	C₅×D₃×C₈	—	2	4	no	—
45	C₄×ΓL(1,𝔽₂₅)	C₂×C₅	1	1	yes	characteristic
46	#4685	C₂×C₅	1	1	yes	characteristic
47	C₄×D₃×C₈	C₂×C₅	1	1	yes	characteristic
48	C₃×#322	C₂×C₅	1	1	yes	characteristic, complemented
49	#4455	C₂×C₅	1	1	yes	characteristic, complemented
50	#4455	C₂×C₅	1	1	yes	characteristic, complemented
51	#4454	C₂×C₅	1	1	yes	characteristic, complemented
52	C₂×C₅×M₁₆	—	3	2	no	—
53	C₂²×C₅×C₈	—	3	2	no	—
54	C₂×C₄²×C₅	—	3	1	no	—
55	C₅×C₄⋊C₈	—	3	2	no	—
56	C₅×C₂²⋊C₈	—	3	2	no	—
57	C₅×#97	—	3	2	no	—
58	C₄×C₅×C₈	—	3	2	no	—
59	C₄×C₅×C₈	—	3	1	no	—
60	C₄×C₅×C₈	—	3	1	no	—
61	C₄×C₅×C₈	—	3	1	no	—
62	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
63	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
64	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
65	#1451	—	3	1	no	Sylow
66	C₂²×C₅×D₃	C₄²	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₅×Dic₃	C₄²	1	1	yes	characteristic
71	C₂×C₅×Dic₃	C₄²	1	1	yes	characteristic
72	C₂×C₅×Dic₃	C₄²	1	1	yes	characteristic
73	C₄×C₅×D₃	—	2	2	no	—
74	C₄×C₅×D₃	—	2	2	no	—
75	C₄×C₅×D₃	M₁₆	1	4	yes	—
76	C₃×C₅×C₈	—	2	2	no	—
77	C₃×C₅×C₈	—	2	2	no	—
78	C₃×C₅×C₈	M₁₆	1	4	yes	complemented
79	C₅×C₃⋊C₈	—	2	2	no	—
80	C₅×C₃⋊C₈	—	2	2	no	—
81	C₅×C₃⋊C₈	—	2	2	no	—
82	C₂×ΓL(1,𝔽₂₅)	C₄×C₅	1	2	yes	—
83	C₂×D₃×C₈	C₄×C₅	1	2	yes	—
84	C₄²×D₃	C₂²×C₅	1	1	yes	characteristic
85	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
86	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
87	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
88	(C₂×D₃)⋊C₈	C₄×C₅	1	2	yes	—
89	#815	C₄×C₅	1	2	yes	—
90	Dic₃⋊C₈	C₄×C₅	1	2	yes	—
91	C₈×Dic₃	C₄×C₅	1	2	yes	—
92	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
93	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
94	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
95	C₂²×C₄×C₅	—	3	1	no	—
96	C₂²×C₄×C₅	—	3	1	no	—
97	C₂²×C₄×C₅	—	3	1	no	—
98	C₅×M₁₆	—	6	4	no	—
99	C₂×C₅×C₈	—	3	2	no	—
100	C₂×C₅×C₈	—	3	2	no	—
101	C₂×C₅×C₈	—	3	2	no	—
102	C₂×C₅×C₈	—	6	4	no	—
103	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
104	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
105	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
106	C₄²×C₅	—	3	1	no	—
107	C₄²×C₅	—	3	1	no	—
108	C₄²×C₅	—	3	1	no	—
109	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
110	#431	—	3	1	no	—
111	C₄×M₁₆	—	3	1	no	—
112	C₂×C₄×C₈	—	3	1	no	—
113	#322	—	3	1	no	—
114	#322	—	3	1	no	—
115	#322	—	3	1	no	—
116	#322	C₅×D₃	1	1	yes	characteristic, complemented
117	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
118	C₂×C₅×D₃	—	2	1	no	—
119	C₂×C₅×D₃	—	2	1	no	—
120	C₂×C₅×D₃	#97	1	2	yes	—
121	C₃×C₄×C₅	C₂×M₁₆	1	2	yes	—
122	C₃×C₄×C₅	C₂×M₁₆	1	2	yes	—
123	C₃×C₄×C₅	#131	1	2	yes	—
124	C₅×Dic₃	—	2	1	no	—
125	C₅×Dic₃	—	2	1	no	—
126	C₅×Dic₃	#97	1	2	yes	—
127	C₂×C₄×D₃	C₂×C₄×C₅	1	1	yes	characteristic
128	C₂×C₄×D₃	C₂×C₄×C₅	1	1	yes	characteristic
129	C₂×C₄×D₃	C₂×C₄×C₅	1	1	yes	characteristic
130	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
131	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
132	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
133	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
134	C₄×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
135	C₄×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
136	C₄×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
137	C₂×C₃⋊C₈	C₂×C₄×C₅	1	2	yes	—
138	C₂×C₃⋊C₈	C₂×C₄×C₅	1	2	yes	—
139	C₂×C₃⋊C₈	C₂×C₄×C₅	1	2	yes	—
140	ΓL(1,𝔽₂₅)	—	2	4	no	—
141	D₃×C₈	—	2	4	no	—
142	C₂³×C₅	—	3	1	no	—
143	C₂×C₄×C₅	—	3	1	no	—
144	C₂×C₄×C₅	—	3	4	no	—
145	C₂×C₄×C₅	—	3	1	no	—
146	C₂×C₄×C₅	—	3	1	no	—
147	C₂×C₄×C₅	—	6	2	no	—
148	C₂×C₄×C₅	—	6	2	no	—
149	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
152	C₅×C₈	—	2	2	no	—
153	C₅×C₈	—	2	2	no	—
154	C₅×C₈	—	6	2	no	—
155	C₅×C₈	—	6	2	no	—
156	C₅×C₈	—	6	2	no	—
157	C₅×C₈	ΓL(1,𝔽₂₅)	1	4	yes	complemented
158	C₂×M₁₆	—	3	2	no	—
159	C₂²×C₈	—	3	2	no	—
160	C₂×C₄²	—	3	1	no	—
161	C₄⋊C₈	—	3	2	no	—
162	C₂²⋊C₈	—	3	2	no	—
163	#97	—	3	2	no	—
164	C₄×C₈	—	3	2	no	—
165	C₄×C₈	—	3	1	no	—
166	C₄×C₈	—	3	1	no	—
167	C₄×C₈	—	3	1	no	—
168	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
169	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
170	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
171	C₂×C₃×C₅	C₂×#97	1	1	yes	characteristic
172	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
173	C₂×C₃×C₅	#405	1	1	yes	characteristic
174	C₅×D₃	—	2	2	no	—
175	C₂²×D₃	C₄²×C₅	1	1	yes	characteristic
176	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
179	C₂×Dic₃	C₄²×C₅	1	1	yes	characteristic
180	C₂×Dic₃	C₄²×C₅	1	1	yes	characteristic
181	C₂×Dic₃	C₄²×C₅	1	1	yes	characteristic
182	C₄×D₃	—	2	2	no	—
183	C₄×D₃	—	2	2	no	—
184	C₄×D₃	C₅×M₁₆	1	4	yes	—
185	C₃×C₈	—	2	2	no	—
186	C₃×C₈	—	2	2	no	—
187	C₃×C₈	C₅×M₁₆	1	4	yes	complemented
188	C₃⋊C₈	—	2	2	no	—
189	C₃⋊C₈	—	2	2	no	—
190	C₃⋊C₈	—	2	2	no	—
191	C₂²×C₅	—	3	2	no	—
192	C₂²×C₅	—	6	1	no	—
193	C₂²×C₅	—	6	1	no	—
194	C₂²×C₅	C₄²×D₃	1	1	yes	characteristic
195	C₄×C₅	—	3	2	no	—
196	C₄×C₅	—	6	1	no	—
197	C₄×C₅	—	6	1	no	—
198	C₄×C₅	C₂×ΓL(1,𝔽₂₅)	1	2	yes	—
199	C₄×C₅	D₃×M₁₆	1	2	yes	—
200	C₄×C₅	#907	1	2	yes	—
201	C₂²×C₄	—	3	1	no	—
202	C₂²×C₄	—	3	1	no	—
203	C₂²×C₄	—	3	1	no	—
204	M₁₆	—	6	4	no	—
205	C₂×C₈	—	3	2	no	—
206	C₂×C₈	—	3	2	no	—
207	C₂×C₈	—	3	2	no	—
208	C₂×C₈	—	6	4	no	—
209	C₂×C₈	C₄×C₅×D₃	1	2	yes	—
210	C₂×C₈	C₄×C₅×D₃	1	2	yes	—
211	C₂×C₈	C₄×C₅×D₃	1	2	yes	—
212	C₄²	—	3	1	no	—
213	C₄²	—	3	1	no	—
214	C₄²	—	3	1	no	—
215	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
216	C₃×C₅	#1451	1	1	yes	characteristic, complemented, Hall
217	C₂²×C₃	C₂×C₄²×C₅	1	1	yes	characteristic
218	C₂×D₃	—	2	1	no	—
219	C₂×D₃	—	2	1	no	—
220	C₂×D₃	C₅×#97	1	2	yes	—
221	C₃×C₄	C₂×C₅×M₁₆	1	2	yes	—
222	C₃×C₄	C₂×C₅×M₁₆	1	2	yes	—
223	C₃×C₄	C₅×#131	1	2	yes	—
224	Dic₃	—	2	1	no	—
225	Dic₃	—	2	1	no	—
226	Dic₃	C₅×#97	1	2	yes	—
227	C₂×C₅	—	6	2	no	—
228	C₂×C₅	C₄×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
229	C₂×C₅	D₃×#97	1	1	yes	characteristic
230	C₂×C₅	#4708	1	1	yes	characteristic
231	C₂³	—	3	1	no	—
232	C₂×C₄	—	3	1	no	—
233	C₂×C₄	—	3	4	no	—
234	C₂×C₄	—	3	1	no	—
235	C₂×C₄	—	3	1	no	—
236	C₂×C₄	—	6	2	no	—
237	C₂×C₄	—	6	2	no	—
238	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
239	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
240	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
241	C₈	—	2	2	no	—
242	C₈	—	2	2	no	—
243	C₈	—	6	2	no	—
244	C₈	—	6	2	no	—
245	C₈	—	6	2	no	—
246	C₈	C₅×ΓL(1,𝔽₂₅)	1	4	yes	complemented
247	C₂×C₃	C₂×C₅×#97	1	1	yes	characteristic
248	C₂×C₃	C₄×C₅×M₁₆	1	1	yes	characteristic
249	C₂×C₃	C₅×#405	1	1	yes	characteristic
250	D₃	—	2	2	no	—
251	C₅	#68069	1	1	yes	characteristic, direct product factor, Sylow
252	C₂²	—	3	2	no	—
253	C₂²	—	6	1	no	—
254	C₂²	—	6	1	no	—
255	C₂²	C₄²×C₅×D₃	1	1	yes	characteristic
256	C₄	—	3	2	no	—
257	C₄	—	6	1	no	—
258	C₄	—	6	1	no	—
259	C₄	C₂×C₅×ΓL(1,𝔽₂₅)	1	2	yes	—
260	C₄	C₅×D₃×M₁₆	1	2	yes	—
261	C₄	C₅×#907	1	2	yes	—
262	C₃	C₅×#1451	1	1	yes	characteristic, complemented, Sylow
263	C₂	—	6	2	no	—
264	C₂	C₄×C₅×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
265	C₂	C₅×D₃×#97	1	1	yes	characteristic
266	C₂	C₅×#4708	1	1	yes	characteristic

	subgroup	G/H
center	C₄²×C₅	C₂²×D₃
commutator	C₂²×C₃	C₂×C₄²×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#322	C₂
socle	C₂²×C₃×C₅	C₂×C₄²
prime radical	C₂²×C₅×D₃	C₄²
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₅×#322	D₃
hypocenter	C₃	C₅×#1451
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₄²×C₅ ◁ C₅×#322

Complements

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

N	H ≅ G/N	type
4. C₃×C₅×#322	C₂	semidirect
5. C₅×#4455	C₂	semidirect
6. C₅×#4455	C₂	semidirect
7. C₅×#4454	C₂	semidirect
22. #68069	C₅	direct (N × H)
29. C₅×#322	D₃	semidirect
48. C₃×#322	C₂×C₅	semidirect
49. #4455	C₂×C₅	semidirect
50. #4455	C₂×C₅	semidirect
51. #4454	C₂×C₅	semidirect
78. C₃×C₅×C₈	M₁₆	semidirect
116. #322	C₅×D₃	semidirect
157. C₅×C₈	ΓL(1,𝔽₂₅)	semidirect
187. C₃×C₈	C₅×M₁₆	semidirect
216. C₃×C₅	#1451	semidirect
246. C₈	C₅×ΓL(1,𝔽₂₅)	semidirect
251. C₅	#68069	direct (N × H)
262. C₃	C₅×#1451	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×ΓL(1,𝔽₂₅)	2
C₅×#4685	2
C₄×C₅×D₃×C₈	2
C₃×C₅×#322	2
C₅×#4455	2
C₅×#4454	2
C₅×#1451	3
#68069	5

Subgroups — C5×#68069 #14280481

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68069 #14280481