Subgroups — Dic₅×#367 #12014705

Order 1280.

366 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#367	C₂	1	1	yes	maximal, characteristic
2	C₂×#64725	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×Dic₅	C₂	1	1	yes	maximal, characteristic
4	Dic₅×#107	C₂	1	2	yes	maximal
5	#10598213	C₂	1	2	yes	maximal
6	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₅×#107	C₂²	1	2	yes	—
8	C₂×#3950	C₂²	1	2	yes	—
9	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
10	C₂²×#672	C₂²	1	1	yes	characteristic
11	C₄⋊C₄×Dic₅	—	2	2	no	—
12	C₅×#367	C₄	1	4	yes	complemented
13	#64749	—	2	2	no	—
14	#64725	C₄	1	4	yes	complemented
15	C₁₆×Dic₅	C₄	1	4	yes	complemented
16	C₄×#367	—	5	1	no	maximal, Sylow
17	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
18	C₂×C₅×C₄⋊C₄	—	2	2	no	—
19	C₂×C₄⋊Dic₅	—	2	2	no	—
20	C₂×C₄×Dic₅	—	4	2	no	—
21	C₂×C₄×Dic₅	D₄	1	1	yes	characteristic
22	C₂²×#168	D₄	1	1	yes	characteristic
23	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
25	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
26	C₅×#107	—	2	2	no	—
27	C₅×#107	C₂×C₄	1	4	yes	—
28	#3963	—	4	2	no	—
29	#3950	—	2	2	no	—
30	#3950	C₂×C₄	1	4	yes	—
31	C₈×Dic₅	C₂×C₄	1	2	yes	—
32	C₈×Dic₅	Q₈	1	2	yes	—
33	C₂×#672	—	2	1	no	—
34	C₂×#672	—	2	1	no	—
35	C₂×#672	C₂×C₄	1	2	yes	—
36	C₂×#367	—	5	1	no	—
37	C₂×#367	D₅	1	1	yes	characteristic
38	C₂×C₄×C₁₆	—	5	1	no	—
39	C₄×#107	—	5	2	no	—
40	#1380	—	5	2	no	—
41	C₂²×C₄×C₅	—	4	2	no	—
42	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₅	—	4	2	no	—
44	C₂²×Dic₅	D₈	1	1	yes	characteristic
45	C₂²×Dic₅	D₈	1	1	yes	characteristic
46	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₅×C₈	#41	1	1	yes	characteristic
51	C₂×C₅×C₈	#41	1	1	yes	characteristic
52	C₅×C₄⋊C₄	—	2	4	no	—
53	C₅×C₄⋊C₄	—	4	2	no	—
54	C₄⋊Dic₅	—	2	4	no	—
55	C₄⋊Dic₅	—	4	2	no	—
56	C₄×Dic₅	—	8	4	no	—
57	C₄×Dic₅	C₄⋊C₄	1	2	yes	—
58	C₄×Dic₅	Q₁₆	1	2	yes	—
59	C₂×#168	—	2	1	no	—
60	C₂×#168	—	2	1	no	—
61	C₂×#168	C₄⋊C₄	1	2	yes	—
62	C₅×C₁₆	C₄²	1	4	yes	complemented
63	#672	—	2	2	no	—
64	C₂²×C₁₆	—	5	1	no	—
65	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
66	C₂×#107	—	5	2	no	—
67	C₂×#107	C₂×D₅	1	2	yes	—
68	C₂×C₄×C₈	—	5	1	no	—
69	C₄×C₄⋊C₄	—	10	2	no	—
70	#367	—	5	4	no	—
71	#367	Dic₅	1	4	yes	direct product factor
72	C₄×C₁₆	—	5	4	no	—
73	#340	—	10	2	no	—
74	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₅	—	4	4	no	—
76	C₂×C₄×C₅	—	8	2	no	—
77	C₂×C₄×C₅	—	8	2	no	—
78	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
80	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
81	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
82	C₂×C₄×C₅	#135	1	1	yes	characteristic
83	C₂×C₄×C₅	#135	1	1	yes	characteristic
84	C₂×Dic₅	—	2	2	no	—
85	C₂×Dic₅	—	4	4	no	—
86	C₂×Dic₅	—	8	2	no	—
87	C₂×Dic₅	—	8	2	no	—
88	C₂×Dic₅	#107	1	2	yes	—
89	C₂×Dic₅	#107	1	2	yes	—
90	C₂×Dic₅	SD₃₂	1	4	yes	—
91	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
92	C₅×C₈	#117	1	1	yes	characteristic
93	C₅×C₈	C₄×Q₈	1	1	yes	characteristic
94	C₅×C₈	C₄×Q₈	1	1	yes	characteristic
95	#168	—	2	2	no	—
96	C₂²×C₈	—	5	1	no	—
97	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
98	C₂×C₄⋊C₄	—	2	2	no	—
99	C₂×C₄⋊C₄	—	10	2	no	—
100	C₂×C₄²	—	5	1	no	—
101	C₂×C₄²	—	20	2	no	—
102	C₂×C₁₆	—	5	2	no	—
103	C₂×C₁₆	—	10	1	no	—
104	C₂×C₁₆	—	10	1	no	—
105	C₂×C₁₆	C₄×D₅	1	2	yes	—
106	C₂×C₁₆	C₄×D₅	1	2	yes	—
107	C₂×C₁₆	C₂×Dic₅	1	2	yes	—
108	#107	—	2	2	no	—
109	#107	—	5	4	no	—
110	#107	—	10	2	no	—
111	#107	C₂×Dic₅	1	4	yes	—
112	C₄×C₈	—	5	2	no	—
113	C₄×C₈	—	5	2	no	—
114	#95	—	20	2	no	—
115	C₂²×C₅	C₂×#107	1	1	yes	characteristic
116	C₂²×C₅	C₄×D₈	1	1	yes	characteristic
117	C₂²×C₅	C₄×D₈	1	1	yes	characteristic
118	C₂²×C₅	C₂×SD₃₂	1	2	yes	—
119	C₂²×C₅	#508	1	2	yes	—
120	C₄×C₅	—	8	4	no	—
121	C₄×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
122	C₄×C₅	#427	1	1	yes	characteristic
123	C₄×C₅	C₄×Q₁₆	1	1	yes	characteristic
124	C₄×C₅	C₄×Q₁₆	1	1	yes	characteristic
125	Dic₅	—	2	2	no	—
126	Dic₅	—	8	4	no	—
127	Dic₅	#367	1	4	yes	direct product factor
128	C₂²×C₄	—	4	2	no	—
129	C₂²×C₄	—	5	1	no	—
130	C₂²×C₄	—	5	1	no	—
131	C₂²×C₄	—	20	2	no	—
132	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
133	C₂×C₈	—	5	2	no	—
134	C₂×C₈	—	10	1	no	—
135	C₂×C₈	—	10	1	no	—
136	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
137	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
138	C₂×C₈	#711	1	1	yes	characteristic
139	C₂×C₈	Q₈×D₅	1	1	yes	characteristic
140	C₂×C₈	#713	1	1	yes	characteristic
141	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
142	C₄⋊C₄	—	2	4	no	—
143	C₄⋊C₄	—	4	2	no	—
144	C₄⋊C₄	—	10	4	no	—
145	C₄⋊C₄	—	20	2	no	—
146	C₄²	—	5	2	no	—
147	C₄²	—	5	2	no	—
148	C₄²	—	40	4	no	—
149	C₁₆	—	10	2	no	—
150	C₁₆	C₄×Dic₅	1	4	yes	complemented
151	C₂×C₅	C₄×#107	1	1	yes	characteristic
152	C₂×C₅	C₂×#367	1	1	yes	characteristic
153	C₂×C₅	#2158	1	1	yes	characteristic
154	C₂×C₅	C₄×SD₃₂	1	2	yes	—
155	C₂×C₅	C₄×SD₃₂	1	2	yes	—
156	C₂³	D₅×D₈	1	1	yes	characteristic
157	C₂×C₄	—	4	4	no	—
158	C₂×C₄	—	5	2	no	—
159	C₂×C₄	—	5	4	no	—
160	C₂×C₄	—	5	2	no	—
161	C₂×C₄	—	8	2	no	—
162	C₂×C₄	—	8	2	no	—
163	C₂×C₄	—	10	2	no	—
164	C₂×C₄	—	20	4	no	—
165	C₂×C₄	—	40	2	no	—
166	C₂×C₄	—	40	2	no	—
167	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
168	C₂×C₄	#4039	1	1	yes	characteristic
169	C₂×C₄	#4058	1	1	yes	characteristic
170	C₂×C₄	D₅×Q₁₆	1	1	yes	characteristic
171	C₂×C₄	#4065	1	1	yes	characteristic
172	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
173	C₈	—	10	2	no	—
174	C₈	#4033	1	1	yes	characteristic
175	C₈	#4038	1	1	yes	characteristic
176	C₈	C₂×C₄×Dic₅	1	1	yes	characteristic
177	C₈	Q₈×Dic₅	1	1	yes	characteristic
178	C₅	C₄×#367	1	1	yes	characteristic, complemented, Sylow
179	C₂²	#65161	1	1	yes	characteristic
180	C₂²	D₅×#107	1	1	yes	characteristic
181	C₂²	D₅×SD₃₂	1	2	yes	—
182	C₂²	#65205	1	2	yes	—
183	C₂²	D₈×Dic₅	1	1	yes	characteristic
184	C₄	—	5	4	no	—
185	C₄	—	8	4	no	—
186	C₄	—	10	2	no	—
187	C₄	—	40	4	no	—
188	C₄	#65162	1	1	yes	characteristic
189	C₄	#65169	1	1	yes	characteristic
190	C₄	C₄⋊C₄×Dic₅	1	1	yes	characteristic
191	C₄	Q₁₆×Dic₅	1	1	yes	characteristic
192	C₂	#10600130	1	2	yes	—
193	C₂	D₅×#367	1	1	yes	characteristic
194	C₂	#10600138	1	1	yes	characteristic
195	C₂	Dic₅×#107	1	1	yes	characteristic
196	C₂	Dic₅×SD₃₂	1	2	yes	—

	subgroup	G/H
center	C₂³	D₅×D₈
commutator	C₅×C₈	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂×C₅×#367	C₂
socle, prime radical	C₂³×C₅	C₂×D₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×#367	D₅
hypocenter	C₅	C₄×#367
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

Complements

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

N	H ≅ G/N	type
12. C₅×#367	C₄	semidirect
14. #64725	C₄	semidirect
15. C₁₆×Dic₅	C₄	semidirect
62. C₅×C₁₆	C₄²	semidirect
71. #367	Dic₅	direct (N × H)
127. Dic₅	#367	direct (N × H)
150. C₁₆	C₄×Dic₅	semidirect
178. C₅	C₄×#367	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#367	2
C₂×#64725	2
C₂×C₁₆×Dic₅	2
Dic₅×#107	2
#10598213	2
C₄×#367	5

Subgroups — Dic5×#367 #12014705

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₅×#367 #12014705