Subgroups — D₅×#890 #11753497

Order 960.

452 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#91779	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#890	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×D₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	D₅×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
5	D₅×C₄⋊Dic₃	C₂	1	2	yes	maximal
6	D₅×Dic₃⋊C₄	C₂	1	2	yes	maximal
7	C₄×D₅×Dic₃	C₂	1	1	yes	maximal, characteristic
8	#91384	C₂	1	1	yes	maximal, characteristic, complemented
9	#91381	C₂	1	1	yes	maximal, characteristic, complemented
10	#91335	C₂	1	2	yes	maximal, complemented
11	#91334	C₂	1	2	yes	maximal, complemented
12	D₅×#125	—	3	1	no	maximal, Hall
13	C₂×C₃×C₄×D₅	C₂²	1	2	yes	—
14	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
15	C₂×D₅×Dic₃	C₂²	1	1	yes	characteristic
16	C₂×D₅×Dic₃	C₂²	1	2	yes	—
17	C₂×D₅×Dic₃	C₂²	1	1	yes	characteristic
18	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₄⋊Dic₁₅	C₂²	1	2	yes	—
20	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
21	#6712	C₂²	1	2	yes	—
22	C₄×Dic₁₅	C₂²	1	1	yes	characteristic
23	C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
24	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
25	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
26	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
27	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
28	C₃×#683	C₂²	1	2	yes	—
29	Dic₃⋊Dic₅	C₂²	1	2	yes	—
30	#6669	C₂²	1	2	yes	—
31	#6668	C₂²	1	2	yes	—
32	#6668	C₂²	1	2	yes	—
33	#6668	C₂²	1	2	yes	—
34	Dic₃×Dic₅	C₂²	1	1	yes	characteristic
35	Dic₃×Dic₅	C₂²	1	1	yes	characteristic
36	C₂×#890	—	5	1	no	maximal, Hall
37	C₅×#125	—	3	1	no	—
38	D₅×C₄⋊C₄	—	3	2	no	—
39	D₅×C₄⋊C₄	—	3	2	no	—
40	D₅×C₄⋊C₄	—	3	1	no	—
41	D₅×C₄⋊C₄	D₃	1	1	yes	characteristic
42	#4036	—	3	1	no	—
43	#4036	—	3	1	no	—
44	#4035	—	3	2	no	—
45	#4035	—	3	2	no	—
46	C₄²×D₅	—	3	1	no	—
47	#4016	—	3	1	no	—
48	C₂²×C₃×D₅	C₂³	1	1	yes	characteristic
49	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
50	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
51	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
52	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
53	C₂×Dic₁₅	C₂³	1	2	yes	—
54	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
55	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
56	C₂×C₅×Dic₃	C₂³	1	2	yes	—
57	C₂×C₃×Dic₅	C₂³	1	2	yes	—
58	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
59	C₃×C₄×D₅	—	2	4	no	—
60	C₃×C₄×D₅	Q₈	1	4	yes	—
61	D₅×Dic₃	—	2	2	no	—
62	D₅×Dic₃	—	2	4	no	—
63	D₅×Dic₃	—	2	2	no	—
64	C₂×C₃×C₄⋊C₄	—	5	1	no	—
65	C₂×C₄⋊Dic₃	—	5	2	no	—
66	C₂×C₄⋊Dic₃	—	5	1	no	—
67	C₂×Dic₃⋊C₄	—	5	2	no	—
68	C₂×C₄×Dic₃	—	5	1	no	—
69	#890	—	5	1	no	—
70	#890	—	5	2	no	—
71	#890	—	5	1	no	—
72	#890	—	5	2	no	—
73	#890	—	5	1	no	—
74	#890	D₅	1	1	yes	characteristic, direct product factor
75	C₂×C₄×D₅	—	3	1	no	—
76	C₂×C₄×D₅	—	3	1	no	—
77	C₂×C₄×D₅	—	3	2	no	—
78	C₂×C₄×D₅	C₂×D₃	1	2	yes	—
79	C₂×C₄×D₅	C₂×D₃	1	1	yes	characteristic
80	C₅×C₄⋊C₄	—	3	2	no	—
81	C₅×C₄⋊C₄	—	3	2	no	—
82	C₅×C₄⋊C₄	—	3	1	no	—
83	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
84	C₄²×C₅	—	3	1	no	—
85	C₄⋊Dic₅	—	3	1	no	—
86	C₄⋊Dic₅	—	3	2	no	—
87	C₄⋊Dic₅	—	3	2	no	—
88	C₄⋊Dic₅	C₂×D₃	1	1	yes	characteristic
89	#683	—	3	2	no	—
90	#683	—	3	2	no	—
91	#683	—	3	2	no	—
92	#683	—	3	2	no	—
93	#683	—	3	2	no	—
94	#683	C₂×D₃	1	2	yes	—
95	C₄×Dic₅	—	3	1	no	—
96	C₄×Dic₅	—	3	1	no	—
97	C₄×Dic₅	—	3	1	no	—
98	C₂×#125	—	15	1	no	Sylow
99	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
100	C₂×C₃×D₅	C₂×Q₈	1	2	yes	—
101	C₂×C₃×D₅	#41	1	2	yes	—
102	C₂×C₃×D₅	#41	1	2	yes	—
103	C₃×C₄×C₅	—	2	2	no	—
104	C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
105	Dic₁₅	—	2	1	no	—
106	Dic₁₅	—	2	1	no	—
107	Dic₁₅	—	2	2	no	—
108	C₃×Dic₅	—	2	2	no	—
109	C₃×Dic₅	C₂×Q₈	1	2	yes	—
110	C₅×Dic₃	—	2	1	no	—
111	C₅×Dic₃	—	2	1	no	—
112	C₅×Dic₃	—	2	2	no	—
113	C₂²×C₃×C₄	—	5	1	no	—
114	C₂²×C₃×C₄	—	5	2	no	—
115	C₂²×Dic₃	—	5	1	no	—
116	C₂²×Dic₃	—	5	1	no	—
117	C₂²×Dic₃	—	5	2	no	—
118	C₃×C₄⋊C₄	—	5	2	no	—
119	C₃×C₄⋊C₄	—	5	1	no	—
120	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
121	C₄⋊Dic₃	—	5	2	no	—
122	C₄⋊Dic₃	—	5	1	no	—
123	C₄⋊Dic₃	—	5	2	no	—
124	C₄⋊Dic₃	—	5	2	no	—
125	C₄⋊Dic₃	—	5	2	no	—
126	C₄⋊Dic₃	C₂×D₅	1	2	yes	—
127	C₄⋊Dic₃	C₂×D₅	1	1	yes	characteristic
128	Dic₃⋊C₄	—	5	2	no	—
129	Dic₃⋊C₄	—	5	2	no	—
130	Dic₃⋊C₄	—	5	2	no	—
131	Dic₃⋊C₄	C₂×D₅	1	2	yes	—
132	C₄×Dic₃	—	5	1	no	—
133	C₄×Dic₃	—	5	1	no	—
134	C₄×Dic₃	—	5	1	no	—
135	C₄×Dic₃	C₂×D₅	1	1	yes	characteristic
136	C₂²×D₅	C₂²×D₃	1	1	yes	characteristic
137	C₂×C₄×C₅	—	3	1	no	—
138	C₂×C₄×C₅	—	3	1	no	—
139	C₂×C₄×C₅	—	3	2	no	—
140	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
141	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
142	C₂×Dic₅	—	3	1	no	—
143	C₂×Dic₅	—	3	1	no	—
144	C₂×Dic₅	—	3	2	no	—
145	C₂×Dic₅	C₂²×D₃	1	2	yes	—
146	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
147	C₄×D₅	—	2	4	no	—
148	C₄×D₅	—	6	2	no	—
149	C₄×D₅	—	6	2	no	—
150	C₄×D₅	—	6	4	no	—
151	C₄×D₅	Dic₆	1	4	yes	—
152	#125	—	3	1	no	—
153	#125	—	15	1	no	—
154	#125	—	15	2	no	—
155	#125	—	15	2	no	—
156	#125	—	15	1	no	—
157	#125	—	15	1	no	—
158	C₂×C₄⋊C₄	—	5	1	no	—
159	C₂×C₄⋊C₄	—	15	2	no	—
160	C₂×C₄⋊C₄	—	15	1	no	—
161	C₂×C₄⋊C₄	—	15	2	no	—
162	C₂×C₄²	—	15	1	no	—
163	C₂×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
164	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
165	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
166	C₃×D₅	#125	1	4	yes	complemented
167	C₂³×C₃	—	5	1	no	—
168	C₂×C₃×C₄	—	5	1	no	—
169	C₂×C₃×C₄	—	5	4	no	—
170	C₂×C₃×C₄	—	5	2	no	—
171	C₂×C₃×C₄	—	10	4	no	—
172	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
173	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
174	C₂×Dic₃	—	5	1	no	—
175	C₂×Dic₃	—	5	1	no	—
176	C₂×Dic₃	—	5	2	no	—
177	C₂×Dic₃	—	10	2	no	—
178	C₂×Dic₃	—	10	4	no	—
179	C₂×Dic₃	—	10	2	no	—
180	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
181	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
182	C₂×Dic₃	C₂²×D₅	1	2	yes	—
183	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
184	C₂×D₅	C₂×Dic₆	1	2	yes	—
185	C₂×D₅	#237	1	2	yes	—
186	C₂×D₅	#239	1	2	yes	—
187	C₄×C₅	—	2	2	no	—
188	C₄×C₅	—	6	1	no	—
189	C₄×C₅	—	6	1	no	—
190	C₄×C₅	—	6	2	no	—
191	C₄×C₅	C₂×Dic₆	1	2	yes	—
192	Dic₅	—	2	2	no	—
193	Dic₅	—	6	1	no	—
194	Dic₅	—	6	1	no	—
195	Dic₅	—	6	2	no	—
196	Dic₅	C₂×Dic₆	1	2	yes	—
197	C₂²×C₄	—	5	1	no	—
198	C₂²×C₄	—	5	2	no	—
199	C₂²×C₄	—	15	1	no	—
200	C₂²×C₄	—	15	1	no	—
201	C₂²×C₄	—	15	2	no	—
202	C₄⋊C₄	—	3	2	no	—
203	C₄⋊C₄	—	3	2	no	—
204	C₄⋊C₄	—	3	1	no	—
205	C₄⋊C₄	—	5	2	no	—
206	C₄⋊C₄	—	5	1	no	—
207	C₄⋊C₄	—	15	2	no	—
208	C₄⋊C₄	—	15	2	no	—
209	C₄⋊C₄	—	15	2	no	—
210	C₄⋊C₄	—	15	2	no	—
211	C₄⋊C₄	—	15	2	no	—
212	C₄⋊C₄	—	15	2	no	—
213	C₄⋊C₄	—	15	2	no	—
214	C₄⋊C₄	—	15	1	no	—
215	C₄⋊C₄	D₃×D₅	1	1	yes	characteristic
216	C₄²	—	3	1	no	—
217	C₄²	—	15	1	no	—
218	C₄²	—	15	1	no	—
219	C₄²	—	15	1	no	—
220	C₃×C₅	C₂×#125	1	1	yes	characteristic, complemented, Hall
221	C₂²×C₃	—	5	2	no	—
222	C₂²×C₃	—	5	2	no	—
223	C₂²×C₃	—	5	2	no	—
224	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
225	C₃×C₄	—	2	2	no	—
226	C₃×C₄	—	5	2	no	—
227	C₃×C₄	—	10	2	no	—
228	C₃×C₄	Q₈×D₅	1	2	yes	—
229	Dic₃	—	2	1	no	—
230	Dic₃	—	2	1	no	—
231	Dic₃	—	2	2	no	—
232	Dic₃	—	10	1	no	—
233	Dic₃	—	10	1	no	—
234	Dic₃	—	10	2	no	—
235	C₂×C₅	C₂²×Dic₆	1	1	yes	characteristic
236	C₂×C₅	C₂×#237	1	1	yes	characteristic
237	C₂×C₅	C₂×#239	1	1	yes	characteristic
238	D₅	#890	1	4	yes	direct product factor
239	C₂³	—	5	1	no	—
240	C₂×C₄	—	3	1	no	—
241	C₂×C₄	—	3	1	no	—
242	C₂×C₄	—	3	2	no	—
243	C₂×C₄	—	5	1	no	—
244	C₂×C₄	—	5	4	no	—
245	C₂×C₄	—	5	2	no	—
246	C₂×C₄	—	10	4	no	—
247	C₂×C₄	—	15	1	no	—
248	C₂×C₄	—	15	1	no	—
249	C₂×C₄	—	15	2	no	—
250	C₂×C₄	—	30	2	no	—
251	C₂×C₄	—	30	2	no	—
252	C₂×C₄	—	30	4	no	—
253	C₂×C₄	C₂×D₃×D₅	1	2	yes	—
254	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
255	C₂×C₃	—	5	4	no	—
256	C₂×C₃	C₂×Q₈×D₅	1	1	yes	characteristic
257	C₂×C₃	D₅×#41	1	1	yes	characteristic
258	C₂×C₃	D₅×#41	1	1	yes	characteristic
259	C₅	C₂×#890	1	1	yes	characteristic, complemented, Sylow
260	C₂²	—	5	2	no	—
261	C₂²	—	5	2	no	—
262	C₂²	—	5	2	no	—
263	C₂²	C₂²×D₃×D₅	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	5	2	no	—
266	C₄	—	6	1	no	—
267	C₄	—	6	1	no	—
268	C₄	—	6	2	no	—
269	C₄	—	10	2	no	—
270	C₄	—	30	1	no	—
271	C₄	—	30	1	no	—
272	C₄	—	30	2	no	—
273	C₄	D₅×Dic₆	1	2	yes	—
274	C₃	D₅×#125	1	1	yes	characteristic, complemented, Sylow
275	C₂	—	5	4	no	—
276	C₂	C₂×D₅×Dic₆	1	1	yes	characteristic
277	C₂	D₅×#237	1	1	yes	characteristic
278	C₂	D₅×#239	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×D₅
commutator, socle, cosocle	C₂²×C₃×C₅	C₂⁴
Fitting	C₃×C₅×C₄⋊C₄	C₂²
prime radical	C₂²×C₃×D₅	C₂³
hypercenter	C₄⋊C₄	D₃×D₅
hypocenter	C₃×C₅	C₂×#125
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. #91779	C₂	semidirect
2. C₅×#890	C₂	semidirect
8. #91384	C₂	semidirect
9. #91381	C₂	semidirect
10. #91335	C₂	semidirect
11. #91334	C₂	semidirect
74. #890	D₅	direct (N × H)
166. C₃×D₅	#125	semidirect
220. C₃×C₅	C₂×#125	semidirect
238. D₅	#890	direct (N × H)
259. C₅	C₂×#890	semidirect
274. C₃	D₅×#125	semidirect

Maximal subgroups

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

subgroup	index
#91779	2
C₅×#890	2
C₃×D₅×C₄⋊C₄	2
D₅×C₄⋊Dic₃	2
D₅×Dic₃⋊C₄	2
C₄×D₅×Dic₃	2
#91384	2
#91381	2
#91335	2
#91334	2
D₅×#125	3
C₂×#890	5

Subgroups — D5×#890 #11753497

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅×#890 #11753497