Subgroups — D₅×#1455 #12065004

Order 1280.

994 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₅×M₁₆	C₂	1	3	yes	maximal
2	C₅×#1455	C₂	1	1	yes	maximal, characteristic, complemented
3	#10600642	C₂	1	3	yes	maximal, complemented
4	#10598868	C₂	1	3	yes	maximal, complemented
5	D₅×#322	C₂	1	3	yes	maximal, complemented
6	#10598843	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈²×D₅	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₅×M₁₆	C₄	1	6	yes	—
9	C₂×C₄²×D₅	C₂²	1	1	yes	characteristic
10	C₄×C₅×M₁₆	C₂²	1	3	yes	—
11	M₁₆×Dic₅	C₄	1	6	yes	—
12	C₄×#681	C₂²	1	3	yes	—
13	#64996	C₄	1	6	yes	—
14	D₅×#97	C₄	1	6	yes	—
15	C₄×#676	C₂²	1	3	yes	—
16	C₄×#676	C₂²	1	3	yes	—
17	C₄×C₈×D₅	C₂²	1	3	yes	—
18	C₄×C₈×D₅	C₂²	1	3	yes	—
19	C₅×#322	C₂²	1	3	yes	complemented
20	C₅×C₈²	C₂²	1	1	yes	characteristic, complemented
21	#64675	C₂²	1	3	yes	complemented
22	#64674	C₂²	1	6	yes	complemented
23	C₈×#168	C₂²	1	3	yes	complemented
24	C₂×#1455	—	5	1	no	maximal, Sylow
25	C₂²×C₄×D₅	C₂×C₄	1	3	yes	—
26	C₂×C₅×M₁₆	C₂×C₄	1	6	yes	—
27	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
28	C₂×C₄×Dic₅	C₂×C₄	1	3	yes	—
29	C₂×#681	C₂×C₄	1	6	yes	—
30	D₅×M₁₆	—	2	24	no	—
31	C₂×#676	C₂×C₄	1	6	yes	—
32	C₂×#676	C₂×C₄	1	6	yes	—
33	C₂×C₈×D₅	C₂×C₄	1	6	yes	—
34	C₂×C₈×D₅	C₂×C₄	1	6	yes	—
35	C₄²×D₅	C₂×C₄	1	3	yes	—
36	C₄²×D₅	C₂×C₄	1	3	yes	—
37	C₄²×D₅	C₂³	1	1	yes	characteristic
38	C₄²×D₅	C₂³	1	1	yes	characteristic
39	C₅×#97	C₂×C₄	1	6	yes	—
40	C₄×C₅×C₈	C₂³	1	3	yes	—
41	C₄×C₅×C₈	C₂³	1	3	yes	—
42	#3947	C₂×C₄	1	6	yes	—
43	#3947	C₂×C₄	1	6	yes	—
44	C₈×Dic₅	C₂×C₄	1	6	yes	—
45	C₈×Dic₅	C₂×C₄	1	6	yes	—
46	#3935	C₂×C₄	1	6	yes	—
47	C₄×#168	C₂³	1	3	yes	—
48	C₄×#168	C₂³	1	3	yes	—
49	C₂×C₄×M₁₆	—	5	3	no	—
50	#1455	—	5	3	no	—
51	#1455	—	5	3	no	—
52	#1455	—	5	1	no	—
53	#1455	D₅	1	1	yes	characteristic, direct product factor
54	C₂×#322	—	5	3	no	—
55	C₂×C₈²	—	5	1	no	—
56	C₂³×D₅	C₄²	1	1	yes	characteristic
57	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
58	C₂²×Dic₅	C₄²	1	3	yes	—
59	C₂×C₄×D₅	—	2	12	no	—
60	C₂×C₄×D₅	C₄²	1	3	yes	—
61	C₂×C₄×D₅	C₄²	1	3	yes	—
62	C₂×C₄×D₅	C₂²×C₄	1	3	yes	—
63	C₂×C₄×D₅	C₂²×C₄	1	3	yes	—
64	C₅×M₁₆	—	2	12	no	—
65	C₂×C₅×C₈	C₂²×C₄	1	6	yes	—
66	C₂×C₅×C₈	C₂²×C₄	1	6	yes	—
67	C₄²×C₅	C₂²×C₄	1	3	yes	—
68	C₄²×C₅	C₂⁴	1	1	yes	characteristic
69	C₄×Dic₅	C₄²	1	6	yes	—
70	C₄×Dic₅	C₂²×C₄	1	3	yes	—
71	C₄×Dic₅	C₂²×C₄	1	3	yes	—
72	#681	—	2	12	no	—
73	C₂×#168	C₂²×C₄	1	6	yes	—
74	C₂×#168	C₂²×C₄	1	6	yes	—
75	#676	—	2	12	no	—
76	#676	—	2	12	no	—
77	C₈×D₅	—	2	12	no	—
78	C₈×D₅	M₁₆	1	24	yes	complemented
79	C₂²×M₁₆	—	5	6	no	—
80	C₂²×C₄²	—	5	1	no	—
81	C₄×M₁₆	—	5	6	no	—
82	C₄×M₁₆	—	5	3	no	—
83	C₄×M₁₆	—	5	6	no	—
84	C₄×M₁₆	—	5	3	no	—
85	C₄×M₁₆	—	5	3	no	—
86	C₄×M₁₆	C₂×D₅	1	3	yes	—
87	C₂×#97	—	5	6	no	—
88	C₂×C₄×C₈	—	5	3	no	—
89	C₂×C₄×C₈	—	5	3	no	—
90	#322	—	5	6	no	—
91	#322	—	5	3	no	—
92	#322	C₂×D₅	1	3	yes	complemented
93	C₈²	—	5	3	no	—
94	C₈²	C₂×D₅	1	1	yes	characteristic, complemented
95	C₂³×C₅	C₂×C₄²	1	1	yes	characteristic
96	C₂²×D₅	—	2	6	no	—
97	C₂²×D₅	C₂×C₄²	1	1	yes	characteristic
98	C₂²×D₅	C₂×C₄²	1	1	yes	characteristic
99	C₂×C₄×C₅	—	2	6	no	—
100	C₂×C₄×C₅	C₂×C₄²	1	3	yes	—
101	C₂×C₄×C₅	C₂³×C₄	1	3	yes	—
102	C₂×Dic₅	—	2	6	no	—
103	C₂×Dic₅	C₂×C₄²	1	3	yes	—
104	C₂×Dic₅	C₂×C₄²	1	3	yes	—
105	C₄×D₅	—	2	6	no	—
106	C₄×D₅	—	2	6	no	—
107	C₄×D₅	—	2	6	no	—
108	C₄×D₅	C₂×M₁₆	1	12	yes	—
109	C₅×C₈	—	2	6	no	—
110	C₅×C₈	C₂×M₁₆	1	12	yes	complemented
111	#168	—	2	6	no	—
112	#168	C₂×M₁₆	1	12	yes	complemented
113	C₂³×C₄	—	5	3	no	—
114	C₂×M₁₆	—	5	6	no	—
115	C₂×M₁₆	—	5	6	no	—
116	C₂×M₁₆	—	5	6	no	—
117	C₂×M₁₆	—	10	24	no	—
118	C₂×M₁₆	C₄×D₅	1	6	yes	—
119	C₂²×C₈	—	5	6	no	—
120	C₂²×C₈	—	5	6	no	—
121	C₂×C₄²	—	5	3	no	—
122	C₂×C₄²	—	5	3	no	—
123	C₂×C₄²	—	5	3	no	—
124	C₂×C₄²	—	5	1	no	—
125	C₂×C₄²	—	5	1	no	—
126	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
127	#97	—	5	6	no	—
128	#97	—	5	6	no	—
129	#97	—	5	6	no	—
130	#97	C₄×D₅	1	6	yes	—
131	C₄×C₈	—	5	6	no	—
132	C₄×C₈	—	5	6	no	—
133	C₄×C₈	—	5	3	no	—
134	C₄×C₈	—	5	3	no	—
135	C₄×C₈	C₂²×D₅	1	3	yes	—
136	C₄×C₈	C₂²×D₅	1	3	yes	—
137	C₂²×C₅	—	2	3	no	—
138	C₂²×C₅	C₂²×C₄²	1	1	yes	characteristic
139	C₂×D₅	—	2	3	no	—
140	C₂×D₅	—	4	4	no	—
141	C₂×D₅	C₄×M₁₆	1	6	yes	—
142	C₄×C₅	—	2	3	no	—
143	C₄×C₅	C₂²×M₁₆	1	6	yes	—
144	Dic₅	—	2	3	no	—
145	Dic₅	C₄×M₁₆	1	6	yes	—
146	C₂⁴	—	5	1	no	—
147	C₂²×C₄	—	5	3	no	—
148	C₂²×C₄	—	5	3	no	—
149	C₂²×C₄	—	5	3	no	—
150	C₂²×C₄	—	5	3	no	—
151	C₂²×C₄	—	5	3	no	—
152	C₂²×C₄	—	10	12	no	—
153	C₂²×C₄	C₂×C₄×D₅	1	3	yes	—
154	M₁₆	—	2	12	no	—
155	M₁₆	—	10	12	no	—
156	M₁₆	—	10	12	no	—
157	M₁₆	—	10	12	no	—
158	C₂×C₈	—	5	6	no	—
159	C₂×C₈	—	5	24	no	—
160	C₂×C₈	—	5	6	no	—
161	C₂×C₈	—	10	12	no	—
162	C₂×C₈	C₂×C₄×D₅	1	6	yes	—
163	C₂×C₈	C₂×C₄×D₅	1	6	yes	—
164	C₄²	—	5	3	no	—
165	C₄²	—	5	3	no	—
166	C₄²	—	5	6	no	—
167	C₄²	C₂×C₄×D₅	1	3	yes	—
168	C₄²	C₂³×D₅	1	1	yes	characteristic
169	C₂×C₅	—	4	1	no	—
170	C₂×C₅	C₂×C₄×M₁₆	1	3	yes	—
171	D₅	—	4	1	no	—
172	D₅	#1455	1	4	yes	direct product factor
173	C₂³	—	5	1	no	—
174	C₂³	—	5	1	no	—
175	C₂³	—	10	6	no	—
176	C₂³	C₄²×D₅	1	1	yes	characteristic
177	C₂×C₄	—	2	6	no	—
178	C₂×C₄	—	5	3	no	—
179	C₂×C₄	—	5	12	no	—
180	C₂×C₄	—	5	3	no	—
181	C₂×C₄	—	10	6	no	—
182	C₂×C₄	—	10	6	no	—
183	C₂×C₄	—	10	6	no	—
184	C₂×C₄	—	10	6	no	—
185	C₂×C₄	C₄²×D₅	1	3	yes	—
186	C₂×C₄	C₂²×C₄×D₅	1	3	yes	—
187	C₈	—	2	6	no	—
188	C₈	—	5	12	no	—
189	C₈	—	10	6	no	—
190	C₈	D₅×M₁₆	1	12	yes	complemented
191	C₅	C₂×#1455	1	1	yes	characteristic, complemented, Sylow
192	C₂²	—	2	3	no	—
193	C₂²	—	5	6	no	—
194	C₂²	—	10	3	no	—
195	C₂²	—	20	4	no	—
196	C₂²	C₂×C₄²×D₅	1	1	yes	characteristic
197	C₄	—	2	3	no	—
198	C₄	—	5	6	no	—
199	C₄	—	10	3	no	—
200	C₄	C₂×D₅×M₁₆	1	6	yes	—
201	C₂	—	4	1	no	—
202	C₂	—	5	4	no	—
203	C₂	—	20	1	no	—
204	C₂	C₄×D₅×M₁₆	1	3	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₅
commutator, socle	C₂²×C₅	C₂²×C₄²
Fitting	C₅×#1455	C₂
prime radical	C₂³×D₅	C₄²
cosocle	C₄²×C₅	C₂⁴
hypercenter	#1455	D₅
hypocenter	C₅	C₂×#1455
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₄² ◁ #1455

Complements

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

N	H ≅ G/N	type
2. C₅×#1455	C₂	semidirect
3. #10600642	C₂	semidirect
4. #10598868	C₂	semidirect
5. D₅×#322	C₂	semidirect
6. #10598843	C₂	semidirect
7. C₈²×D₅	C₂	semidirect
19. C₅×#322	C₂²	semidirect
20. C₅×C₈²	C₂²	semidirect
21. #64675	C₂²	semidirect
22. #64674	C₂²	semidirect
23. C₈×#168	C₂²	semidirect
53. #1455	D₅	direct (N × H)
78. C₈×D₅	M₁₆	semidirect
92. #322	C₂×D₅	semidirect
94. C₈²	C₂×D₅	semidirect
110. C₅×C₈	C₂×M₁₆	semidirect
112. #168	C₂×M₁₆	semidirect
172. D₅	#1455	direct (N × H)
190. C₈	D₅×M₁₆	semidirect
191. C₅	C₂×#1455	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₅×M₁₆	2
C₅×#1455	2
#10600642	2
#10598868	2
D₅×#322	2
#10598843	2
C₈²×D₅	2
C₂×#1455	5

Subgroups — D5×#1455 #12065004

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅×#1455 #12065004