Subgroups — Dic₅×#812 #14312606

Order 1920.

392 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#91096	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#812	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic
4	#11745873	C₂	1	2	yes	maximal, complemented
5	Dic₅×C₃⋊C₁₆	C₂	1	2	yes	maximal, complemented
6	Dic₅×M₃₂	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₄×Dic₅	C₄	1	1	yes	characteristic
9	C₂²×C₃×#168	C₄	1	1	yes	characteristic
10	#91096	C₄	1	4	yes	complemented
11	C₂×C₃⋊#672	C₂²	1	2	yes	—
12	C₅×#812	C₄	1	4	yes	complemented
13	C₂×C₅×C₃⋊C₁₆	C₂²	1	2	yes	—
14	C₃×C₈×Dic₅	C₄	1	1	yes	characteristic
15	C₃×C₈×Dic₅	C₄	1	1	yes	characteristic
16	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
17	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
18	C₄×#812	—	5	1	no	maximal, Hall
19	C₂×C₅×M₃₂	—	3	1	no	—
20	C₂×C₈×Dic₅	D₃	1	1	yes	characteristic
21	C₂×#3944	—	3	1	no	—
22	#64722	—	3	2	no	—
23	C₁₆×Dic₅	—	3	2	no	—
24	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃×Dic₅	C₈	1	2	yes	—
26	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
30	C₃×C₄×Dic₅	C₈	1	2	yes	—
31	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
32	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×#168	C₈	1	4	yes	—
34	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
36	C₃⋊#672	C₂×C₄	1	4	yes	complemented
37	C₅×C₃⋊C₁₆	C₂×C₄	1	4	yes	complemented
38	C₂×C₃×C₄×C₈	—	5	1	no	—
39	C₂×#812	—	5	1	no	—
40	C₂×#812	D₅	1	1	yes	characteristic
41	#4461	—	5	2	no	—
42	C₄×C₃⋊C₁₆	—	5	2	no	—
43	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
44	C₂×C₄×Dic₅	Dic₃	1	1	yes	characteristic
45	C₂²×#168	Dic₃	1	1	yes	characteristic
46	C₅×M₃₂	—	3	4	no	—
47	C₂×C₅×C₁₆	—	3	2	no	—
48	C₈×Dic₅	Dic₃	1	1	yes	characteristic
49	C₈×Dic₅	Dic₃	1	1	yes	characteristic
50	C₈×Dic₅	C₂×D₃	1	1	yes	characteristic
51	C₈×Dic₅	C₂×D₃	1	1	yes	characteristic
52	#3944	—	3	4	no	—
53	C₂×#672	—	3	2	no	—
54	C₄×M₃₂	—	15	1	no	Sylow
55	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
57	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
59	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₃×Dic₅	—	2	4	no	—
61	C₂×C₃×Dic₅	C₂×C₈	1	2	yes	—
62	C₂×C₃×Dic₅	C₂×C₈	1	2	yes	—
63	C₃×C₅×C₈	C₄²	1	2	yes	—
64	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
65	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
66	C₃×#168	C₂×C₈	1	2	yes	—
67	C₃×#168	C₂×C₈	1	2	yes	—
68	C₂²×C₃×C₈	—	5	1	no	—
69	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
70	C₂×C₃×C₄²	—	5	1	no	—
71	C₃×C₄×C₈	—	5	1	no	—
72	C₃×C₄×C₈	—	5	1	no	—
73	C₃×C₄×C₈	—	5	1	no	—
74	C₃×C₄×C₈	—	5	1	no	—
75	#812	—	5	4	no	—
76	#812	Dic₅	1	4	yes	direct product factor
77	C₂×C₃⋊C₁₆	—	5	2	no	—
78	C₂×C₃⋊C₁₆	C₂×D₅	1	2	yes	—
79	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
80	C₂²×Dic₅	C₃⋊C₈	1	2	yes	—
81	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
82	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
83	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
84	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
85	C₄×Dic₅	C₃⋊C₈	1	2	yes	—
86	C₄×Dic₅	C₂×Dic₃	1	1	yes	characteristic
87	C₄×Dic₅	C₂×Dic₃	1	1	yes	characteristic
88	C₂×#168	C₃⋊C₈	1	4	yes	—
89	C₂×#168	C₂×Dic₃	1	1	yes	characteristic
90	C₂×#168	C₂×Dic₃	1	1	yes	characteristic
91	C₅×C₁₆	—	3	4	no	—
92	#672	—	3	4	no	—
93	C₂×M₃₂	—	3	1	no	—
94	C₂×M₃₂	—	15	1	no	—
95	C₂×C₄×C₈	—	5	1	no	—
96	#346	—	15	2	no	—
97	C₄×C₁₆	—	15	2	no	—
98	C₂²×C₃×C₅	—	2	2	no	—
99	C₂²×C₃×C₅	C₄×C₈	1	2	yes	—
100	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
101	C₃×C₄×C₅	C₄×C₈	1	2	yes	—
102	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
103	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
104	C₃×Dic₅	—	2	2	no	—
105	C₃×Dic₅	M₃₂	1	4	yes	complemented
106	C₂²×C₃×C₄	—	5	2	no	—
107	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
108	C₂×C₃×C₈	—	5	4	no	—
109	C₂×C₃×C₈	—	5	1	no	—
110	C₂×C₃×C₈	—	5	1	no	—
111	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
112	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
113	C₂×C₃×C₈	C₂×Dic₅	1	2	yes	—
114	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
115	C₃×C₄²	—	5	2	no	—
116	C₃×C₄²	—	5	1	no	—
117	C₃×C₄²	—	5	1	no	—
118	C₃⋊C₁₆	—	5	4	no	—
119	C₃⋊C₁₆	C₂×Dic₅	1	4	yes	complemented
120	C₂³×C₅	C₂×C₃⋊C₈	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂×C₃⋊C₈	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂×C₃⋊C₈	1	2	yes	—
123	C₂×C₄×C₅	C₄×Dic₃	1	2	yes	—
124	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
125	C₂×Dic₅	—	2	4	no	—
126	C₂×Dic₅	C₂×C₃⋊C₈	1	2	yes	—
127	C₂×Dic₅	C₂×C₃⋊C₈	1	2	yes	—
128	C₅×C₈	C₄×Dic₃	1	2	yes	—
129	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
130	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
131	#168	C₂×C₃⋊C₈	1	2	yes	—
132	#168	C₂×C₃⋊C₈	1	2	yes	—
133	C₂²×C₈	—	5	1	no	—
134	C₂²×C₈	D₃×D₅	1	1	yes	characteristic
135	C₂×C₄²	—	5	1	no	—
136	M₃₂	—	3	4	no	—
137	M₃₂	—	15	4	no	—
138	C₂×C₁₆	—	3	2	no	—
139	C₂×C₁₆	—	15	2	no	—
140	C₄×C₈	—	5	1	no	—
141	C₄×C₈	—	5	1	no	—
142	C₄×C₈	—	5	1	no	—
143	C₄×C₈	—	5	1	no	—
144	C₂×C₃×C₅	—	2	2	no	—
145	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
146	C₂×C₃×C₅	C₂×M₃₂	1	2	yes	—
147	C₂³×C₃	C₈×D₅	1	1	yes	characteristic
148	C₂×C₃×C₄	—	5	2	no	—
149	C₂×C₃×C₄	—	5	2	no	—
150	C₂×C₃×C₄	—	10	4	no	—
151	C₂×C₃×C₄	C₈×D₅	1	1	yes	characteristic
152	C₂×C₃×C₄	C₈×D₅	1	2	yes	—
153	C₂×C₃×C₄	C₄×Dic₅	1	2	yes	—
154	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
155	C₃×C₈	—	5	2	no	—
156	C₃×C₈	—	5	2	no	—
157	C₃×C₈	C₄×Dic₅	1	2	yes	—
158	C₃×C₈	C₂×C₄×D₅	1	1	yes	characteristic
159	C₃×C₈	C₂²×Dic₅	1	1	yes	characteristic
160	C₂²×C₅	—	2	2	no	—
161	C₂²×C₅	C₄×C₃⋊C₈	1	2	yes	—
162	C₂²×C₅	C₂²×C₃⋊C₈	1	1	yes	characteristic
163	C₄×C₅	C₄×C₃⋊C₈	1	2	yes	—
164	C₄×C₅	C₂²×C₃⋊C₈	1	1	yes	characteristic
165	C₄×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
166	Dic₅	—	2	2	no	—
167	Dic₅	#812	1	4	yes	direct product factor
168	C₂²×C₄	—	5	2	no	—
169	C₂²×C₄	D₅×Dic₃	1	1	yes	characteristic
170	C₂×C₈	—	5	1	no	—
171	C₂×C₈	—	5	1	no	—
172	C₂×C₈	—	5	4	no	—
173	C₂×C₈	D₅×Dic₃	1	1	yes	characteristic
174	C₂×C₈	D₃×Dic₅	1	2	yes	—
175	C₂×C₈	#1200	1	2	yes	—
176	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
177	C₄²	—	5	1	no	—
178	C₄²	—	5	1	no	—
179	C₄²	—	5	2	no	—
180	C₁₆	—	3	4	no	—
181	C₁₆	—	15	4	no	—
182	C₃×C₅	C₄×M₃₂	1	1	yes	characteristic, complemented, Hall
183	C₂²×C₃	—	2	2	no	—
184	C₂²×C₃	C₈×Dic₅	1	2	yes	—
185	C₂²×C₃	C₂×C₈×D₅	1	1	yes	characteristic
186	C₃×C₄	—	5	4	no	—
187	C₃×C₄	—	10	2	no	—
188	C₃×C₄	C₈×Dic₅	1	2	yes	—
189	C₃×C₄	C₂×C₈×D₅	1	1	yes	characteristic
190	C₃×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
191	C₂×C₅	—	2	2	no	—
192	C₂×C₅	C₂×C₄×C₃⋊C₈	1	1	yes	characteristic
193	C₂×C₅	C₂×#812	1	2	yes	—
194	C₂³	D₅×C₃⋊C₈	1	1	yes	characteristic
195	C₂×C₄	—	5	2	no	—
196	C₂×C₄	—	5	2	no	—
197	C₂×C₄	—	10	4	no	—
198	C₂×C₄	D₅×C₃⋊C₈	1	1	yes	characteristic
199	C₂×C₄	D₅×C₃⋊C₈	1	2	yes	—
200	C₂×C₄	Dic₃×Dic₅	1	2	yes	—
201	C₂×C₄	C₂×D₅×Dic₃	1	1	yes	characteristic
202	C₈	—	5	2	no	—
203	C₈	—	5	2	no	—
204	C₈	Dic₃×Dic₅	1	2	yes	—
205	C₈	C₂×D₃×Dic₅	1	1	yes	characteristic
206	C₈	C₂×#1200	1	1	yes	characteristic
207	C₂×C₃	—	2	2	no	—
208	C₂×C₃	D₅×M₃₂	1	2	yes	—
209	C₂×C₃	C₂×C₈×Dic₅	1	1	yes	characteristic
210	C₅	C₄×#812	1	1	yes	characteristic, complemented, Sylow
211	C₂²	—	2	2	no	—
212	C₂²	Dic₅×C₃⋊C₈	1	2	yes	—
213	C₂²	C₂×D₅×C₃⋊C₈	1	1	yes	characteristic
214	C₄	—	5	4	no	—
215	C₄	—	10	2	no	—
216	C₄	Dic₅×C₃⋊C₈	1	2	yes	—
217	C₄	C₂×D₅×C₃⋊C₈	1	1	yes	characteristic
218	C₄	C₂×Dic₃×Dic₅	1	1	yes	characteristic
219	C₃	Dic₅×M₃₂	1	1	yes	characteristic, complemented, Sylow
220	C₂	—	2	2	no	—
221	C₂	D₅×#812	1	2	yes	—
222	C₂	C₂×Dic₅×C₃⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×D₃×D₅
commutator	C₂×C₃×C₅	C₂×C₄×C₈
Fitting	C₂²×C₃×C₅×C₈	C₂²
socle	C₂²×C₃×C₅	C₂²×C₈
prime radical	C₂³×C₃×C₅	C₂×C₈
cosocle	C₂×C₃×C₅×C₈	C₂³
hypercenter	C₂²×C₈	D₃×D₅
hypocenter	C₃×C₅	C₄×M₃₂
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
4. #11745873	C₂	semidirect
5. Dic₅×C₃⋊C₁₆	C₂	semidirect
10. #91096	C₄	semidirect
12. C₅×#812	C₄	semidirect
36. C₃⋊#672	C₂×C₄	semidirect
37. C₅×C₃⋊C₁₆	C₂×C₄	semidirect
76. #812	Dic₅	direct (N × H)
105. C₃×Dic₅	M₃₂	semidirect
119. C₃⋊C₁₆	C₂×Dic₅	semidirect
167. Dic₅	#812	direct (N × H)
182. C₃×C₅	C₄×M₃₂	semidirect
210. C₅	C₄×#812	semidirect
219. C₃	Dic₅×M₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#91096	2
C₂×C₅×#812	2
C₂×C₃×C₈×Dic₅	2
#11745873	2
Dic₅×C₃⋊C₁₆	2
Dic₅×M₃₂	3
C₄×#812	5

Subgroups — Dic5×#812 #14312606

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₅×#812 #14312606