Subgroups — C₃×C₄²×Dic₁₀ #14319614

Order 1920.

1368 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×Dic₁₀	C₂	1	3	yes	maximal
2	C₃×C₄³×C₅	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×C₄⋊Dic₅	C₂	1	3	yes	maximal
4	C₃×C₄×#683	C₂	1	6	yes	maximal
5	C₃×C₄²×Dic₅	C₂	1	2	yes	maximal
6	C₄²×Dic₁₀	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₄²×C₅	C₂²	1	3	yes	—
9	C₂×C₃×C₄²×C₅	C₂²	1	3	yes	—
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄⋊Dic₅	C₂²	1	3	yes	—
12	C₂×C₃×#683	C₂²	1	6	yes	—
13	C₂×C₃×C₄×Dic₅	C₂²	1	6	yes	—
14	C₂×C₃×C₄×Dic₅	C₂²	1	6	yes	—
15	C₃×C₄×Dic₁₀	C₄	1	24	yes	direct product factor
16	C₃×#3963	C₂²	1	6	yes	—
17	C₃×C₄²×Q₈	—	5	1	no	maximal, Hall
18	C₂×C₄×Dic₁₀	C₂×C₃	1	3	yes	—
19	C₄³×C₅	C₂×C₃	1	1	yes	characteristic
20	C₄×C₄⋊Dic₅	C₂×C₃	1	3	yes	—
21	C₄×#683	C₂×C₃	1	6	yes	—
22	C₄²×Dic₅	C₂×C₃	1	2	yes	—
23	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
24	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
26	C₂²×C₃×Dic₅	C₂³	1	6	yes	—
27	C₂²×C₃×Dic₅	C₂³	1	2	yes	—
28	C₂×C₃×Dic₁₀	C₂×C₄	1	12	yes	—
29	C₃×C₄²×C₅	—	2	6	no	—
30	C₃×C₄²×C₅	C₂×C₄	1	12	yes	—
31	C₃×C₄²×C₅	Q₈	1	4	yes	complemented
32	C₃×C₄⋊Dic₅	C₂×C₄	1	12	yes	—
33	C₃×#683	C₂×C₄	1	24	yes	—
34	C₃×C₄×Dic₅	—	2	12	no	—
35	C₃×C₄×Dic₅	C₂×C₄	1	24	yes	—
36	C₂×C₃×C₄×Q₈	—	5	3	no	—
37	C₃×C₄×C₄⋊C₄	—	5	6	no	—
38	C₃×C₄×C₄⋊C₄	—	5	3	no	—
39	C₃×C₄³	—	5	2	no	—
40	C₃×C₄³	D₅	1	1	yes	characteristic
41	C₂²×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
42	C₂×C₄²×C₅	C₂²×C₃	1	3	yes	—
43	C₂×C₄²×C₅	C₂²×C₃	1	3	yes	—
44	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
45	C₂×C₄⋊Dic₅	C₂²×C₃	1	3	yes	—
46	C₂×#683	C₂²×C₃	1	6	yes	—
47	C₂×C₄×Dic₅	C₂²×C₃	1	6	yes	—
48	C₂×C₄×Dic₅	C₂²×C₃	1	6	yes	—
49	C₄×Dic₁₀	C₃×C₄	1	24	yes	direct product factor
50	#3963	C₂²×C₃	1	6	yes	—
51	C₄²×Q₈	—	5	1	no	Sylow
52	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	—	2	6	no	—
54	C₂×C₃×C₄×C₅	C₂²×C₄	1	6	yes	—
55	C₂×C₃×C₄×C₅	C₂²×C₄	1	6	yes	—
56	C₂×C₃×C₄×C₅	C₂²×C₄	1	6	yes	—
57	C₂×C₃×C₄×C₅	C₂×Q₈	1	6	yes	—
58	C₂×C₃×C₄×C₅	#41	1	6	yes	—
59	C₂×C₃×Dic₅	—	2	12	no	—
60	C₂×C₃×Dic₅	C₂²×C₄	1	12	yes	—
61	C₂×C₃×Dic₅	C₂²×C₄	1	12	yes	—
62	C₃×Dic₁₀	C₄²	1	16	yes	direct product factor
63	C₂²×C₃×Q₈	—	5	1	no	—
64	C₃×C₄×Q₈	—	5	24	no	—
65	C₂×C₃×C₄⋊C₄	—	5	6	no	—
66	C₂×C₃×C₄⋊C₄	—	5	3	no	—
67	C₂×C₃×C₄²	—	5	6	no	—
68	C₂×C₃×C₄²	—	5	6	no	—
69	C₂×C₃×C₄²	C₂×D₅	1	3	yes	—
70	C₂×C₃×C₄²	C₂×D₅	1	3	yes	—
71	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
72	C₃×#95	—	5	6	no	—
73	C₂²×C₄×C₅	C₂³×C₃	1	3	yes	—
74	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂³×C₃	1	3	yes	—
76	C₂²×Dic₅	C₂³×C₃	1	6	yes	—
77	C₂²×Dic₅	C₂³×C₃	1	2	yes	—
78	C₂×Dic₁₀	C₂×C₃×C₄	1	12	yes	—
79	C₄²×C₅	—	2	6	no	—
80	C₄²×C₅	C₂×C₃×C₄	1	12	yes	—
81	C₄²×C₅	C₃×Q₈	1	4	yes	complemented
82	C₄⋊Dic₅	C₂×C₃×C₄	1	12	yes	—
83	#683	C₂×C₃×C₄	1	24	yes	—
84	C₄×Dic₅	—	2	12	no	—
85	C₄×Dic₅	C₂×C₃×C₄	1	24	yes	—
86	C₂×C₄×Q₈	—	5	3	no	—
87	C₄×C₄⋊C₄	—	5	6	no	—
88	C₄×C₄⋊C₄	—	5	3	no	—
89	C₄³	—	5	2	no	—
90	C₄³	C₃×D₅	1	1	yes	characteristic
91	C₂²×C₃×C₅	C₂³×C₄	1	3	yes	—
92	C₂²×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
93	C₂²×C₃×C₅	C₂×#41	1	3	yes	—
94	C₃×C₄×C₅	—	2	6	no	—
95	C₃×C₄×C₅	C₂×C₄²	1	4	yes	—
96	C₃×C₄×C₅	C₄×Q₈	1	12	yes	complemented
97	C₃×Dic₅	—	2	12	no	—
98	C₃×Dic₅	C₂×C₄²	1	8	yes	—
99	C₂×C₃×Q₈	—	5	12	no	—
100	C₂²×C₃×C₄	—	5	2	no	—
101	C₂²×C₃×C₄	—	5	6	no	—
102	C₂²×C₃×C₄	C₂²×D₅	1	3	yes	—
103	C₂²×C₃×C₄	C₂²×D₅	1	3	yes	—
104	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
105	C₃×C₄⋊C₄	—	5	24	no	—
106	C₃×C₄⋊C₄	—	5	12	no	—
107	C₃×C₄²	—	2	6	no	—
108	C₃×C₄²	—	5	24	no	—
109	C₃×C₄²	—	10	12	no	—
110	C₃×C₄²	Dic₁₀	1	4	yes	direct product factor
111	C₃×C₄²	C₄×D₅	1	12	yes	—
112	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
113	C₂×C₄×C₅	—	2	6	no	—
114	C₂×C₄×C₅	C₂²×C₃×C₄	1	6	yes	—
115	C₂×C₄×C₅	C₂²×C₃×C₄	1	6	yes	—
116	C₂×C₄×C₅	C₂²×C₃×C₄	1	6	yes	—
117	C₂×C₄×C₅	C₂×C₃×Q₈	1	6	yes	—
118	C₂×C₄×C₅	C₃×#41	1	6	yes	—
119	C₂×Dic₅	—	2	12	no	—
120	C₂×Dic₅	C₂²×C₃×C₄	1	12	yes	—
121	C₂×Dic₅	C₂²×C₃×C₄	1	12	yes	—
122	Dic₁₀	C₃×C₄²	1	16	yes	direct product factor
123	C₂²×Q₈	—	5	1	no	—
124	C₄×Q₈	—	5	24	no	—
125	C₂×C₄⋊C₄	—	5	6	no	—
126	C₂×C₄⋊C₄	—	5	3	no	—
127	C₂×C₄²	—	5	6	no	—
128	C₂×C₄²	—	5	6	no	—
129	C₂×C₄²	C₂×C₃×D₅	1	3	yes	—
130	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
131	C₂×C₄²	C₂×C₃×D₅	1	3	yes	—
132	#95	—	5	6	no	—
133	C₂×C₃×C₅	C₂²×C₄²	1	1	yes	characteristic
134	C₂×C₃×C₅	C₂×C₄×Q₈	1	3	yes	—
135	C₂×C₃×C₅	C₄×#41	1	3	yes	—
136	C₂³×C₃	C₂³×D₅	1	1	yes	characteristic
137	C₃×Q₈	—	5	16	no	—
138	C₂×C₃×C₄	—	2	6	no	—
139	C₂×C₃×C₄	—	5	12	no	—
140	C₂×C₃×C₄	—	5	12	no	—
141	C₂×C₃×C₄	—	10	12	no	—
142	C₂×C₃×C₄	C₂×Dic₁₀	1	6	yes	—
143	C₂×C₃×C₄	C₂×C₄×D₅	1	6	yes	—
144	C₂×C₃×C₄	C₂×C₄×D₅	1	6	yes	—
145	C₂×C₃×C₄	C₂×C₄×D₅	1	6	yes	—
146	C₂×C₃×C₄	#709	1	6	yes	—
147	C₂²×C₅	C₂³×C₃×C₄	1	3	yes	—
148	C₂²×C₅	C₂²×C₃×Q₈	1	1	yes	characteristic
149	C₂²×C₅	C₂×C₃×#41	1	3	yes	—
150	C₄×C₅	—	2	6	no	—
151	C₄×C₅	C₂×C₃×C₄²	1	4	yes	—
152	C₄×C₅	C₃×C₄×Q₈	1	12	yes	complemented
153	Dic₅	—	2	12	no	—
154	Dic₅	C₂×C₃×C₄²	1	8	yes	—
155	C₂×Q₈	—	5	12	no	—
156	C₂²×C₄	—	5	2	no	—
157	C₂²×C₄	—	5	6	no	—
158	C₂²×C₄	C₂²×C₃×D₅	1	3	yes	—
159	C₂²×C₄	C₂²×C₃×D₅	1	3	yes	—
160	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
161	C₄⋊C₄	—	5	24	no	—
162	C₄⋊C₄	—	5	12	no	—
163	C₄²	—	2	6	no	—
164	C₄²	—	5	24	no	—
165	C₄²	—	10	12	no	—
166	C₄²	C₃×Dic₁₀	1	4	yes	direct product factor
167	C₄²	C₃×C₄×D₅	1	12	yes	—
168	C₃×C₅	C₄²×Q₈	1	1	yes	characteristic, complemented, Hall
169	C₂²×C₃	C₂²×Dic₁₀	1	1	yes	characteristic
170	C₂²×C₃	C₂²×C₄×D₅	1	3	yes	—
171	C₂²×C₃	C₂×#709	1	3	yes	—
172	C₃×C₄	—	2	6	no	—
173	C₃×C₄	—	5	8	no	—
174	C₃×C₄	—	10	12	no	—
175	C₃×C₄	C₄×Dic₁₀	1	12	yes	direct product factor
176	C₃×C₄	C₄²×D₅	1	4	yes	—
177	C₂×C₅	C₂²×C₃×C₄²	1	1	yes	characteristic
178	C₂×C₅	C₂×C₃×C₄×Q₈	1	3	yes	—
179	C₂×C₅	C₃×C₄×#41	1	3	yes	—
180	C₂³	C₂³×C₃×D₅	1	1	yes	characteristic
181	Q₈	—	5	16	no	—
182	C₂×C₄	—	2	6	no	—
183	C₂×C₄	—	5	12	no	—
184	C₂×C₄	—	5	12	no	—
185	C₂×C₄	—	10	12	no	—
186	C₂×C₄	C₂×C₃×Dic₁₀	1	6	yes	—
187	C₂×C₄	C₂×C₃×C₄×D₅	1	6	yes	—
188	C₂×C₄	C₂×C₃×C₄×D₅	1	6	yes	—
189	C₂×C₄	C₂×C₃×C₄×D₅	1	6	yes	—
190	C₂×C₄	C₃×#709	1	6	yes	—
191	C₂×C₃	C₂×C₄×Dic₁₀	1	3	yes	—
192	C₂×C₃	C₂×C₄²×D₅	1	1	yes	characteristic
193	C₂×C₃	C₄×#709	1	3	yes	—
194	C₅	C₃×C₄²×Q₈	1	1	yes	characteristic, complemented, Sylow
195	C₂²	C₂²×C₃×Dic₁₀	1	1	yes	characteristic
196	C₂²	C₂²×C₃×C₄×D₅	1	3	yes	—
197	C₂²	C₂×C₃×#709	1	3	yes	—
198	C₄	—	2	6	no	—
199	C₄	—	5	8	no	—
200	C₄	—	10	12	no	—
201	C₄	C₃×C₄×Dic₁₀	1	12	yes	direct product factor
202	C₄	C₃×C₄²×D₅	1	4	yes	—
203	C₃	C₄²×Dic₁₀	1	1	yes	characteristic, direct product factor, Sylow
204	C₂	C₂×C₃×C₄×Dic₁₀	1	3	yes	—
205	C₂	C₂×C₃×C₄²×D₅	1	1	yes	characteristic
206	C₂	C₃×C₄×#709	1	3	yes	—

	subgroup	G/H
center	C₂×C₃×C₄²	C₂×D₅
commutator	C₂×C₅	C₂²×C₃×C₄²
Frattini	C₂³	C₂³×C₃×D₅
Fitting	C₃×C₄³×C₅	C₂
socle, prime radical	C₂³×C₃×C₅	C₂⁴
cosocle	C₂³×C₅	C₂⁴×C₃
hypercenter	C₃×C₄³	D₅
hypocenter	C₅	C₃×C₄²×Q₈
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄² ◁ C₃×C₄³

Complements

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

N	H ≅ G/N	type
6. C₄²×Dic₁₀	C₃	direct (N × H)
15. C₃×C₄×Dic₁₀	C₄	direct (N × H)
31. C₃×C₄²×C₅	Q₈	semidirect
49. C₄×Dic₁₀	C₃×C₄	direct (N × H)
62. C₃×Dic₁₀	C₄²	direct (N × H)
81. C₄²×C₅	C₃×Q₈	semidirect
96. C₃×C₄×C₅	C₄×Q₈	semidirect
110. C₃×C₄²	Dic₁₀	direct (N × H)
122. Dic₁₀	C₃×C₄²	direct (N × H)
152. C₄×C₅	C₃×C₄×Q₈	semidirect
166. C₄²	C₃×Dic₁₀	direct (N × H)
168. C₃×C₅	C₄²×Q₈	semidirect
175. C₃×C₄	C₄×Dic₁₀	direct (N × H)
194. C₅	C₃×C₄²×Q₈	semidirect
201. C₄	C₃×C₄×Dic₁₀	direct (N × H)
203. C₃	C₄²×Dic₁₀	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×Dic₁₀	2
C₃×C₄³×C₅	2
C₃×C₄×C₄⋊Dic₅	2
C₃×C₄×#683	2
C₃×C₄²×Dic₅	2
C₄²×Dic₁₀	3
C₃×C₄²×Q₈	5

Subgroups — C3×C42×Dic10 #14319614

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄²×Dic₁₀ #14319614