Subgroups — D₅×#1381 #12014185

Order 1280.

574 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₅×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×D₅×#108	C₂	1	2	yes	maximal
3	C₅×#1381	C₂	1	1	yes	maximal, characteristic, complemented
4	#10598299	C₂	1	1	yes	maximal, characteristic, complemented
5	#10598219	C₂	1	2	yes	maximal, complemented
6	C₂×D₅×M₁₆	—	2	2	no	—
7	C₂²×C₈×D₅	C₂²	1	1	yes	characteristic
8	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₅×#108	C₂²	1	2	yes	—
10	C₂×#3954	—	2	2	no	—
11	C₂×#3951	C₂²	1	2	yes	—
12	C₂²×#672	C₂²	1	1	yes	characteristic
13	C₂×D₅×C₁₆	—	2	4	no	—
14	C₂×D₅×C₁₆	C₄	1	2	yes	—
15	C₂×D₅×C₁₆	C₄	1	2	yes	—
16	D₅×#108	—	2	4	no	—
17	D₅×#108	C₄	1	8	yes	—
18	C₂×#1381	—	5	1	no	maximal, Sylow
19	C₂²×C₄×D₅	D₄	1	1	yes	characteristic
20	C₂×C₅×M₁₆	—	2	2	no	—
21	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
22	C₂×#681	—	2	2	no	—
23	C₂²×#168	D₄	1	1	yes	characteristic
24	D₅×M₁₆	—	2	8	no	—
25	D₅×M₁₆	—	4	4	no	—
26	C₂×#676	—	4	2	no	—
27	C₂×C₈×D₅	—	4	2	no	—
28	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×D₅	C₂×C₄	1	4	yes	—
31	C₂×C₈×D₅	D₄	1	4	yes	—
32	C₂×C₈×D₅	Q₈	1	1	yes	characteristic
33	C₂×C₈×D₅	Q₈	1	1	yes	characteristic
34	C₂×C₅×C₁₆	—	2	2	no	—
35	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
36	C₅×#108	—	2	2	no	—
37	C₅×#108	C₂×C₄	1	4	yes	—
38	#3954	—	2	8	no	—
39	#3951	—	2	2	no	—
40	#3951	C₂×C₄	1	4	yes	—
41	C₂×#672	—	2	2	no	—
42	C₂×#672	C₂×C₄	1	2	yes	—
43	D₅×C₁₆	—	2	4	no	—
44	D₅×C₁₆	—	2	4	no	—
45	C₂³×C₁₆	—	5	1	no	—
46	C₂²×#108	—	5	2	no	—
47	#1381	—	5	1	no	—
48	#1381	—	5	2	no	—
49	#1381	D₅	1	1	yes	characteristic, direct product factor
50	C₂³×D₅	Q₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
52	C₂²×Dic₅	Q₁₆	1	1	yes	characteristic
53	C₂×C₄×D₅	C₂²⋊C₄	1	4	yes	—
54	C₂×C₄×D₅	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×D₅	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×D₅	D₈	1	1	yes	characteristic
57	C₂×C₄×D₅	D₈	1	1	yes	characteristic
58	C₂×C₄×D₅	SD₁₆	1	4	yes	—
59	C₅×M₁₆	—	2	4	no	—
60	C₅×M₁₆	—	4	2	no	—
61	C₂×C₅×C₈	—	4	2	no	—
62	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
64	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
65	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
66	#681	—	2	4	no	—
67	#681	—	4	2	no	—
68	C₂×#168	—	4	2	no	—
69	C₂×#168	C₂²⋊C₄	1	4	yes	—
70	C₂×#168	C₄⋊C₄	1	1	yes	characteristic
71	C₂×#168	C₄⋊C₄	1	1	yes	characteristic
72	#676	—	4	8	no	—
73	C₈×D₅	—	4	8	no	—
74	C₈×D₅	C₄²	1	2	yes	—
75	C₈×D₅	C₄²	1	2	yes	—
76	C₈×D₅	C₂²⋊C₄	1	2	yes	—
77	C₈×D₅	C₂²⋊C₄	1	2	yes	—
78	C₈×D₅	C₄⋊C₄	1	4	yes	—
79	C₈×D₅	C₄⋊C₄	1	4	yes	—
80	C₅×C₁₆	—	2	2	no	—
81	#672	—	2	2	no	—
82	C₂²×M₁₆	—	10	2	no	—
83	C₂³×C₈	—	5	1	no	—
84	C₂²×C₁₆	—	5	1	no	—
85	C₂²×C₁₆	—	5	2	no	—
86	C₂²×C₁₆	—	5	2	no	—
87	C₂²×C₁₆	—	10	4	no	—
88	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
89	C₂×#108	—	5	2	no	—
90	C₂×#108	—	5	8	no	—
91	C₂×#108	—	10	4	no	—
92	C₂×#108	—	10	2	no	—
93	C₂×#108	C₂×D₅	1	2	yes	—
94	C₂³×C₅	C₂×Q₁₆	1	1	yes	characteristic
95	C₂²×D₅	—	2	4	no	—
96	C₂²×D₅	#103	1	4	yes	—
97	C₂²×D₅	#107	1	1	yes	characteristic
98	C₂²×D₅	#107	1	1	yes	characteristic
99	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
100	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
103	C₂×Dic₅	#103	1	4	yes	—
104	C₂×Dic₅	#107	1	1	yes	characteristic
105	C₂×Dic₅	#107	1	1	yes	characteristic
106	C₄×D₅	#95	1	2	yes	—
107	C₄×D₅	#95	1	2	yes	—
108	C₄×D₅	#102	1	4	yes	—
109	C₄×D₅	#102	1	4	yes	—
110	C₄×D₅	#106	1	2	yes	—
111	C₄×D₅	#106	1	2	yes	—
112	C₅×C₈	—	4	4	no	—
113	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
114	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₅×C₈	C₂×C₄⋊C₄	1	2	yes	—
116	#168	—	4	4	no	—
117	#168	#95	1	2	yes	—
118	#168	#95	1	2	yes	—
119	C₂³×C₄	—	5	1	no	—
120	C₂×M₁₆	—	2	2	no	—
121	C₂×M₁₆	—	10	8	no	—
122	C₂×M₁₆	—	10	2	no	—
123	C₂×M₁₆	—	20	2	no	—
124	C₂×M₁₆	—	20	4	no	—
125	C₂²×C₈	—	5	1	no	—
126	C₂²×C₈	—	5	4	no	—
127	C₂²×C₈	—	5	1	no	—
128	C₂²×C₈	—	5	1	no	—
129	C₂²×C₈	—	5	4	no	—
130	C₂²×C₈	—	5	1	no	—
131	C₂²×C₈	—	5	1	no	—
132	C₂²×C₈	—	20	2	no	—
133	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
134	C₂×C₁₆	—	2	2	no	—
135	C₂×C₁₆	—	5	2	no	—
136	C₂×C₁₆	—	10	4	no	—
137	C₂×C₁₆	—	10	4	no	—
138	C₂×C₁₆	—	10	2	no	—
139	C₂×C₁₆	C₄×D₅	1	2	yes	—
140	#108	—	2	2	no	—
141	#108	—	5	4	no	—
142	#108	—	10	8	no	—
143	#108	—	10	2	no	—
144	#108	C₄×D₅	1	4	yes	—
145	C₂²×C₅	—	2	2	no	—
146	C₂²×C₅	C₂×#103	1	2	yes	—
147	C₂²×C₅	C₂×#107	1	1	yes	characteristic
148	C₂×D₅	—	2	2	no	—
149	C₂×D₅	—	2	8	no	—
150	C₂×D₅	#340	1	2	yes	—
151	C₂×D₅	#340	1	2	yes	—
152	C₂×D₅	#368	1	4	yes	—
153	C₄×C₅	C₂×#95	1	1	yes	characteristic
154	C₄×C₅	C₂×#102	1	2	yes	—
155	C₄×C₅	C₂×#106	1	1	yes	characteristic
156	Dic₅	#340	1	2	yes	—
157	Dic₅	#340	1	2	yes	—
158	C₂⁴	—	5	1	no	—
159	C₂²×C₄	—	5	1	no	—
160	C₂²×C₄	—	5	4	no	—
161	C₂²×C₄	—	5	1	no	—
162	C₂²×C₄	—	5	1	no	—
163	C₂²×C₄	—	5	1	no	—
164	C₂²×C₄	—	5	1	no	—
165	C₂²×C₄	—	5	4	no	—
166	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
167	M₁₆	—	2	4	no	—
168	M₁₆	—	4	2	no	—
169	M₁₆	—	10	4	no	—
170	M₁₆	—	20	8	no	—
171	M₁₆	—	20	2	no	—
172	C₂×C₈	—	4	2	no	—
173	C₂×C₈	—	5	4	no	—
174	C₂×C₈	—	5	1	no	—
175	C₂×C₈	—	5	1	no	—
176	C₂×C₈	—	5	2	no	—
177	C₂×C₈	—	5	2	no	—
178	C₂×C₈	—	5	2	no	—
179	C₂×C₈	—	5	2	no	—
180	C₂×C₈	—	5	4	no	—
181	C₂×C₈	—	5	4	no	—
182	C₂×C₈	—	20	8	no	—
183	C₂×C₈	—	20	2	no	—
184	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
185	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
186	C₂×C₈	D₄×D₅	1	2	yes	—
187	C₂×C₈	Q₈×D₅	1	1	yes	characteristic
188	C₁₆	—	2	2	no	—
189	C₁₆	—	10	2	no	—
190	C₂×C₅	—	2	2	no	—
191	C₂×C₅	C₂×#340	1	1	yes	characteristic
192	C₂×C₅	C₂×#368	1	2	yes	—
193	D₅	—	2	2	no	—
194	D₅	#1381	1	4	yes	direct product factor
195	C₂³	—	5	1	no	—
196	C₂³	—	5	4	no	—
197	C₂³	—	5	1	no	—
198	C₂³	—	10	4	no	—
199	C₂³	D₅×Q₁₆	1	1	yes	characteristic
200	C₂×C₄	—	5	4	no	—
201	C₂×C₄	—	5	1	no	—
202	C₂×C₄	—	5	1	no	—
203	C₂×C₄	—	5	4	no	—
204	C₂×C₄	—	5	4	no	—
205	C₂×C₄	—	5	2	no	—
206	C₂×C₄	—	5	2	no	—
207	C₂×C₄	—	5	2	no	—
208	C₂×C₄	—	5	2	no	—
209	C₂×C₄	D₅×C₂²⋊C₄	1	2	yes	—
210	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
211	C₂×C₄	D₅×D₈	1	1	yes	characteristic
212	C₂×C₄	D₅×SD₁₆	1	2	yes	—
213	C₈	—	4	4	no	—
214	C₈	—	5	2	no	—
215	C₈	—	5	2	no	—
216	C₈	—	20	4	no	—
217	C₈	C₄²×D₅	1	1	yes	characteristic
218	C₈	D₅×C₂²⋊C₄	1	1	yes	characteristic
219	C₈	D₅×C₄⋊C₄	1	2	yes	—
220	C₅	C₂×#1381	1	1	yes	characteristic, complemented, Sylow
221	C₂²	—	2	2	no	—
222	C₂²	—	5	4	no	—
223	C₂²	—	5	2	no	—
224	C₂²	—	5	2	no	—
225	C₂²	—	10	8	no	—
226	C₂²	—	10	2	no	—
227	C₂²	D₅×#103	1	2	yes	—
228	C₂²	D₅×#107	1	1	yes	characteristic
229	C₄	—	5	2	no	—
230	C₄	—	5	2	no	—
231	C₄	D₅×#95	1	1	yes	characteristic
232	C₄	D₅×#102	1	2	yes	—
233	C₄	D₅×#106	1	1	yes	characteristic
234	C₂	—	2	2	no	—
235	C₂	—	5	4	no	—
236	C₂	—	10	2	no	—
237	C₂	D₅×#340	1	1	yes	characteristic
238	C₂	D₅×#368	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₅×D₈
commutator	C₅×C₈	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₅×#1381	C₂
socle	C₂²×C₅	C₂×#107
prime radical	C₂³×D₅	Q₁₆
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#1381	D₅
hypocenter	C₅	C₂×#1381
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

Complements

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

N	H ≅ G/N	type
3. C₅×#1381	C₂	semidirect
4. #10598299	C₂	semidirect
5. #10598219	C₂	semidirect
49. #1381	D₅	direct (N × H)
194. D₅	#1381	direct (N × H)
220. C₅	C₂×#1381	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₅×C₁₆	2
C₂×D₅×#108	2
C₅×#1381	2
#10598299	2
#10598219	2
C₂×#1381	5

Subgroups — D5×#1381 #12014185

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅×#1381 #12014185