Subgroups — C₅×#68569 #14285675

Order 1920.

368 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#4837	C₂	1	2	yes	maximal
2	C₄×C₅×D₃×C₈	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#335	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4479	C₂	1	2	yes	maximal, complemented
5	C₅×#4458	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1567	—	3	1	no	maximal, Hall
7	C₂×C₅×ΓL(1,𝔽₂₅)	—	2	2	no	—
8	C₂×C₅×D₃×C₈	C₄	1	1	yes	characteristic
9	C₂×C₅×D₃×C₈	C₄	1	1	yes	characteristic
10	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
11	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
12	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₅×Dic₃⋊C₈	—	2	2	no	—
14	C₅×C₈×Dic₃	C₄	1	1	yes	characteristic
15	C₅×C₈×Dic₃	C₄	1	1	yes	characteristic
16	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
17	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
18	#68569	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₅×#423	—	3	2	no	—
20	C₂×C₄×C₅×C₈	—	3	1	no	—
21	C₅×#335	—	3	2	no	—
22	C₅×#335	—	3	1	no	—
23	C₅×#335	D₃	1	1	yes	characteristic, complemented
24	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
26	C₂×C₄×C₅×D₃	Q₈	1	1	yes	characteristic
27	C₂×C₃×C₅×C₈	—	2	2	no	—
28	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
30	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
31	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
33	C₄×C₅×Dic₃	Q₈	1	1	yes	characteristic
34	C₂×C₅×C₃⋊C₈	—	2	2	no	—
35	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
37	C₅×ΓL(1,𝔽₂₅)	—	4	4	no	—
38	C₅×D₃×C₈	—	2	2	no	—
39	C₅×D₃×C₈	—	2	2	no	—
40	#4837	C₂×C₅	1	2	yes	—
41	C₄×D₃×C₈	C₂×C₅	1	1	yes	characteristic
42	C₃×#335	C₂×C₅	1	1	yes	characteristic, complemented
43	#4479	C₂×C₅	1	2	yes	complemented
44	#4458	C₂×C₅	1	1	yes	characteristic, complemented
45	C₂×C₅×M₁₆	—	6	2	no	—
46	C₂²×C₅×C₈	—	3	1	no	—
47	C₂²×C₅×C₈	—	3	1	no	—
48	C₂×C₄²×C₅	—	3	1	no	—
49	C₅×C₄⋊C₈	—	3	2	no	—
50	C₅×C₄⋊C₈	—	6	2	no	—
51	C₅×C₄⋊C₈	C₂×D₃	1	2	yes	—
52	C₄×C₅×C₈	—	3	1	no	—
53	C₄×C₅×C₈	—	3	1	no	—
54	C₄×C₅×C₈	—	3	1	no	—
55	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
56	#1567	—	3	1	no	Sylow
57	C₂²×C₅×D₃	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₅×Dic₃	C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₅×Dic₃	C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₅×Dic₃	C₄⋊C₄	1	1	yes	characteristic
64	C₄×C₅×D₃	—	2	1	no	—
65	C₄×C₅×D₃	—	2	1	no	—
66	C₄×C₅×D₃	—	2	2	no	—
67	C₄×C₅×D₃	D₈	1	2	yes	—
68	C₄×C₅×D₃	Q₁₆	1	2	yes	—
69	C₃×C₅×C₈	—	2	1	no	—
70	C₃×C₅×C₈	—	4	2	no	—
71	C₃×C₅×C₈	M₁₆	1	2	yes	complemented
72	C₅×C₃⋊C₈	—	2	1	no	—
73	C₅×C₃⋊C₈	—	4	2	no	—
74	C₅×C₃⋊C₈	M₁₆	1	2	yes	complemented
75	C₂×ΓL(1,𝔽₂₅)	—	2	2	no	—
76	C₂×D₃×C₈	C₄×C₅	1	1	yes	characteristic
77	C₂×D₃×C₈	C₄×C₅	1	1	yes	characteristic
78	C₄²×D₃	C₂²×C₅	1	1	yes	characteristic
79	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
80	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
81	Dic₃⋊C₈	—	2	2	no	—
82	C₈×Dic₃	C₄×C₅	1	1	yes	characteristic
83	C₈×Dic₃	C₄×C₅	1	1	yes	characteristic
84	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	2	yes	—
85	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
86	C₂²×C₄×C₅	—	3	1	no	—
87	C₂²×C₄×C₅	—	3	1	no	—
88	C₂²×C₄×C₅	—	3	1	no	—
89	C₅×M₁₆	—	12	4	no	—
90	C₂×C₅×C₈	—	2	2	no	—
91	C₂×C₅×C₈	—	3	1	no	—
92	C₂×C₅×C₈	—	3	1	no	—
93	C₂×C₅×C₈	—	6	2	no	—
94	C₂×C₅×C₈	—	6	2	no	—
95	C₂×C₅×C₈	—	6	2	no	—
96	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
97	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
98	C₄²×C₅	—	3	1	no	—
99	C₄²×C₅	—	3	1	no	—
100	C₄²×C₅	—	3	1	no	—
101	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
102	#423	—	3	2	no	—
103	C₂×C₄×C₈	—	3	1	no	—
104	#335	—	3	2	no	—
105	#335	—	3	1	no	—
106	#335	C₅×D₃	1	1	yes	characteristic, complemented
107	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₅×D₃	—	2	1	no	—
109	C₂×C₅×D₃	—	2	1	no	—
110	C₂×C₅×D₃	#107	1	2	yes	—
111	C₃×C₄×C₅	—	2	1	no	—
112	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
113	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
114	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
115	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
116	C₅×Dic₃	—	2	1	no	—
117	C₅×Dic₃	#107	1	2	yes	—
118	C₅×Dic₃	#108	1	2	yes	—
119	C₂×C₄×D₃	C₂×C₄×C₅	1	1	yes	characteristic
120	C₂×C₄×D₃	C₅×D₄	1	1	yes	characteristic
121	C₂×C₄×D₃	C₅×Q₈	1	1	yes	characteristic
122	C₂×C₃×C₈	—	2	2	no	—
123	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
124	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
125	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
126	C₄×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
127	C₄×Dic₃	C₅×D₄	1	1	yes	characteristic
128	C₄×Dic₃	C₅×Q₈	1	1	yes	characteristic
129	C₂×C₃⋊C₈	—	2	2	no	—
130	C₂×C₃⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
131	C₂×C₃⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
132	ΓL(1,𝔽₂₅)	—	4	4	no	—
133	D₃×C₈	—	2	2	no	—
134	D₃×C₈	—	2	2	no	—
135	C₂³×C₅	—	3	1	no	—
136	C₂×C₄×C₅	—	3	1	no	—
137	C₂×C₄×C₅	—	3	1	no	—
138	C₂×C₄×C₅	—	3	2	no	—
139	C₂×C₄×C₅	—	3	2	no	—
140	C₂×C₄×C₅	—	3	1	no	—
141	C₂×C₄×C₅	—	6	2	no	—
142	C₂×C₄×C₅	—	6	1	no	—
143	C₂×C₄×C₅	—	6	1	no	—
144	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
145	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
146	C₂×C₄×C₅	D₃×Q₈	1	1	yes	characteristic
147	C₅×C₈	—	2	1	no	—
148	C₅×C₈	—	3	2	no	—
149	C₅×C₈	—	4	2	no	—
150	C₅×C₈	—	6	1	no	—
151	C₅×C₈	—	12	2	no	—
152	C₅×C₈	ΓL(1,𝔽₂₅)	1	2	yes	complemented
153	C₂×M₁₆	—	6	2	no	—
154	C₂²×C₈	—	3	1	no	—
155	C₂²×C₈	—	3	1	no	—
156	C₂×C₄²	—	3	1	no	—
157	C₄⋊C₈	—	3	2	no	—
158	C₄⋊C₈	—	6	2	no	—
159	C₄⋊C₈	C₂×C₅×D₃	1	2	yes	—
160	C₄×C₈	—	3	1	no	—
161	C₄×C₈	—	3	1	no	—
162	C₄×C₈	—	3	1	no	—
163	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
164	C₂×C₃×C₅	#423	1	1	yes	characteristic
165	C₂×C₃×C₅	C₂×#107	1	1	yes	characteristic
166	C₂×C₃×C₅	C₂×#108	1	1	yes	characteristic
167	C₅×D₃	—	2	2	no	—
168	C₂²×D₃	C₅×C₄⋊C₄	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
172	C₂×Dic₃	C₅×C₄⋊C₄	1	1	yes	characteristic
173	C₂×Dic₃	C₅×C₄⋊C₄	1	1	yes	characteristic
174	C₂×Dic₃	C₅×C₄⋊C₄	1	1	yes	characteristic
175	C₄×D₃	—	2	1	no	—
176	C₄×D₃	—	2	1	no	—
177	C₄×D₃	—	2	2	no	—
178	C₄×D₃	C₅×D₈	1	2	yes	—
179	C₄×D₃	C₅×Q₁₆	1	2	yes	—
180	C₃×C₈	—	2	1	no	—
181	C₃×C₈	—	4	2	no	—
182	C₃×C₈	C₅×M₁₆	1	2	yes	complemented
183	C₃⋊C₈	—	2	1	no	—
184	C₃⋊C₈	—	4	2	no	—
185	C₃⋊C₈	C₅×M₁₆	1	2	yes	complemented
186	C₂²×C₅	—	3	2	no	—
187	C₂²×C₅	—	6	1	no	—
188	C₂²×C₅	—	6	1	no	—
189	C₂²×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
190	C₄×C₅	—	2	1	no	—
191	C₄×C₅	—	3	2	no	—
192	C₄×C₅	—	3	2	no	—
193	C₄×C₅	—	6	1	no	—
194	C₄×C₅	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
195	C₄×C₅	D₃×M₁₆	1	1	yes	characteristic
196	C₄×C₅	D₃×D₈	1	1	yes	characteristic
197	C₄×C₅	D₃×Q₁₆	1	1	yes	characteristic
198	C₂²×C₄	—	3	1	no	—
199	C₂²×C₄	—	3	1	no	—
200	C₂²×C₄	—	3	1	no	—
201	M₁₆	—	12	4	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	3	1	no	—
204	C₂×C₈	—	3	1	no	—
205	C₂×C₈	—	6	2	no	—
206	C₂×C₈	—	6	2	no	—
207	C₂×C₈	—	6	2	no	—
208	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
209	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
210	C₄²	—	3	1	no	—
211	C₄²	—	3	1	no	—
212	C₄²	—	3	1	no	—
213	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
214	C₃×C₅	#1567	1	1	yes	characteristic, complemented, Hall
215	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
216	C₂×D₃	—	2	1	no	—
217	C₂×D₃	—	2	1	no	—
218	C₂×D₃	C₅×#107	1	2	yes	—
219	C₃×C₄	—	2	1	no	—
220	C₃×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
221	C₃×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
222	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
223	C₃×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
224	Dic₃	—	2	1	no	—
225	Dic₃	C₅×#107	1	2	yes	—
226	Dic₃	C₅×#108	1	2	yes	—
227	C₂×C₅	—	6	2	no	—
228	C₂×C₅	#4837	1	1	yes	characteristic
229	C₂×C₅	D₃×#107	1	1	yes	characteristic
230	C₂×C₅	D₃×#108	1	1	yes	characteristic
231	C₂³	—	3	1	no	—
232	C₂×C₄	—	3	1	no	—
233	C₂×C₄	—	3	1	no	—
234	C₂×C₄	—	3	2	no	—
235	C₂×C₄	—	3	2	no	—
236	C₂×C₄	—	3	1	no	—
237	C₂×C₄	—	6	2	no	—
238	C₂×C₄	—	6	1	no	—
239	C₂×C₄	—	6	1	no	—
240	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
241	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
242	C₂×C₄	C₅×D₃×Q₈	1	1	yes	characteristic
243	C₈	—	2	1	no	—
244	C₈	—	3	2	no	—
245	C₈	—	4	2	no	—
246	C₈	—	6	1	no	—
247	C₈	—	12	2	no	—
248	C₈	C₅×ΓL(1,𝔽₂₅)	1	2	yes	complemented
249	C₂×C₃	C₅×#423	1	1	yes	characteristic
250	C₂×C₃	C₂×C₅×#107	1	1	yes	characteristic
251	C₂×C₃	C₂×C₅×#108	1	1	yes	characteristic
252	D₃	—	2	2	no	—
253	C₅	#68569	1	1	yes	characteristic, direct product factor, Sylow
254	C₂²	—	3	2	no	—
255	C₂²	—	6	1	no	—
256	C₂²	—	6	1	no	—
257	C₂²	C₅×D₃×C₄⋊C₄	1	1	yes	characteristic
258	C₄	—	2	1	no	—
259	C₄	—	3	2	no	—
260	C₄	—	3	2	no	—
261	C₄	—	6	1	no	—
262	C₄	C₂×C₅×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
263	C₄	C₅×D₃×M₁₆	1	1	yes	characteristic
264	C₄	C₅×D₃×D₈	1	1	yes	characteristic
265	C₄	C₅×D₃×Q₁₆	1	1	yes	characteristic
266	C₃	C₅×#1567	1	1	yes	characteristic, complemented, Sylow
267	C₂	—	6	2	no	—
268	C₂	C₅×#4837	1	1	yes	characteristic
269	C₂	C₅×D₃×#107	1	1	yes	characteristic
270	C₂	C₅×D₃×#108	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	D₃×D₄
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#335	C₂
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂²×C₅×D₃	C₄⋊C₄
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₅×#335	D₃
hypocenter	C₃	C₅×#1567
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₄²×C₅ ◁ C₅×#335

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×#335	C₂	semidirect
4. C₅×#4479	C₂	semidirect
5. C₅×#4458	C₂	semidirect
18. #68569	C₅	direct (N × H)
23. C₅×#335	D₃	semidirect
42. C₃×#335	C₂×C₅	semidirect
43. #4479	C₂×C₅	semidirect
44. #4458	C₂×C₅	semidirect
71. C₃×C₅×C₈	M₁₆	semidirect
74. C₅×C₃⋊C₈	M₁₆	semidirect
106. #335	C₅×D₃	semidirect
152. C₅×C₈	ΓL(1,𝔽₂₅)	semidirect
182. C₃×C₈	C₅×M₁₆	semidirect
185. C₃⋊C₈	C₅×M₁₆	semidirect
214. C₃×C₅	#1567	semidirect
248. C₈	C₅×ΓL(1,𝔽₂₅)	semidirect
253. C₅	#68569	direct (N × H)
266. C₃	C₅×#1567	semidirect

Maximal subgroups

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

subgroup	index
C₅×#4837	2
C₄×C₅×D₃×C₈	2
C₃×C₅×#335	2
C₅×#4479	2
C₅×#4458	2
C₅×#1567	3
#68569	5

Subgroups — C5×#68569 #14285675

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68569 #14285675