Subgroups — C₃×#10606050 #14322243

Order 1920.

1848 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#4039	C₂	1	6	yes	maximal, complemented
2	C₂×C₃×#4020	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#384	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65277	C₂	1	6	yes	maximal, complemented
5	C₃×C₄²×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
6	#10606050	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×D₂₀	C₂²	1	6	yes	—
8	C₂²×C₃×C₄×D₅	C₂²	1	4	yes	—
9	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	6	yes	—
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₃×#685	C₂²	1	12	yes	—
12	C₂×C₃×C₄×Dic₅	C₂²	1	3	yes	—
13	C₂×C₃×C₄×Dic₅	C₂²	1	3	yes	—
14	C₃×#4039	—	2	24	no	—
15	C₃×#4020	C₄	1	8	yes	complemented
16	C₃×C₄×#127	—	5	1	no	maximal, Hall
17	C₂×#4039	C₂×C₃	1	6	yes	complemented
18	C₂×#4020	C₂×C₃	1	1	yes	characteristic
19	C₅×#384	C₂×C₃	1	1	yes	characteristic, complemented
20	#65277	C₂×C₃	1	6	yes	complemented
21	C₄²×Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
22	C₂³×C₃×D₅	C₂³	1	4	yes	—
23	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
24	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
25	C₂²×C₃×Dic₅	C₂³	1	3	yes	—
26	C₂²×C₃×Dic₅	C₂³	1	1	yes	characteristic
27	C₂×C₃×D₂₀	—	2	12	no	—
28	C₂×C₃×D₂₀	—	2	12	no	—
29	C₂×C₃×D₂₀	C₂×C₄	1	24	yes	complemented
30	C₂×C₃×C₄×D₅	—	2	24	no	—
31	C₃×C₅×C₄⋊C₄	—	2	12	no	—
32	C₃×C₄²×C₅	C₂×C₄	1	4	yes	complemented
33	C₃×#685	—	2	24	no	—
34	C₃×C₄×Dic₅	—	2	6	no	—
35	C₃×C₄×Dic₅	D₄	1	12	yes	complemented
36	C₂×C₃×#127	—	5	1	no	—
37	C₂×C₃×C₄×D₄	—	5	6	no	—
38	C₃×#390	—	5	6	no	—
39	C₃×#384	D₅	1	1	yes	characteristic, complemented
40	C₃×C₄³	—	5	1	no	—
41	C₂²×D₂₀	C₂²×C₃	1	6	yes	—
42	C₂²×C₄×D₅	C₂²×C₃	1	4	yes	—
43	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	6	yes	—
44	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
45	C₂×#685	C₂²×C₃	1	12	yes	—
46	C₂×C₄×Dic₅	C₂²×C₃	1	3	yes	—
47	C₂×C₄×Dic₅	C₂²×C₃	1	3	yes	—
48	#4039	—	2	24	no	—
49	#4020	C₃×C₄	1	8	yes	complemented
50	C₄×#127	—	5	1	no	Sylow
51	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
52	C₂²×C₃×D₅	—	2	12	no	—
53	C₂²×C₃×D₅	—	2	12	no	—
54	C₂²×C₃×D₅	C₂²×C₄	1	8	yes	—
55	C₂×C₃×C₄×C₅	—	2	12	no	—
56	C₂×C₃×C₄×C₅	C₂²×C₄	1	6	yes	—
57	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
58	C₂×C₃×C₄×C₅	#41	1	6	yes	—
59	C₂×C₃×Dic₅	—	2	6	no	—
60	C₂×C₃×Dic₅	C₂×D₄	1	6	yes	—
61	C₂×C₃×Dic₅	C₂×D₄	1	6	yes	—
62	C₃×D₂₀	—	2	48	no	—
63	C₃×C₄×D₅	—	4	16	no	—
64	C₂²×C₃×D₄	—	5	6	no	—
65	C₂³×C₃×C₄	—	5	4	no	—
66	C₃×#127	—	5	8	no	—
67	C₃×C₄×D₄	—	10	24	no	—
68	C₂×C₃×C₄⋊C₄	C₂×D₅	1	6	yes	—
69	C₂×C₃×C₂²⋊C₄	—	5	12	no	—
70	C₂×C₃×C₄²	—	5	3	no	—
71	C₂×C₃×C₄²	—	5	3	no	—
72	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
73	C₂³×D₅	C₂³×C₃	1	4	yes	—
74	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
75	C₂²×C₄×C₅	C₂³×C₃	1	3	yes	—
76	C₂²×Dic₅	C₂³×C₃	1	3	yes	—
77	C₂²×Dic₅	C₂³×C₃	1	1	yes	characteristic
78	C₂×D₂₀	—	2	12	no	—
79	C₂×D₂₀	—	2	12	no	—
80	C₂×D₂₀	C₂×C₃×C₄	1	24	yes	complemented
81	C₂×C₄×D₅	—	2	24	no	—
82	C₅×C₄⋊C₄	—	2	12	no	—
83	C₄²×C₅	C₂×C₃×C₄	1	4	yes	complemented
84	#685	—	2	24	no	—
85	C₄×Dic₅	—	2	6	no	—
86	C₄×Dic₅	C₃×D₄	1	12	yes	complemented
87	C₂×#127	—	5	1	no	—
88	C₂×C₄×D₄	—	5	6	no	—
89	#390	—	5	6	no	—
90	#384	C₃×D₅	1	1	yes	characteristic, complemented
91	C₄³	—	5	1	no	—
92	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
93	C₂²×C₃×C₅	C₂²×D₄	1	3	yes	—
94	C₂²×C₃×C₅	C₂×#41	1	3	yes	—
95	C₂×C₃×D₅	—	2	24	no	—
96	C₂×C₃×D₅	—	4	4	no	—
97	C₂×C₃×D₅	—	4	12	no	—
98	C₃×C₄×C₅	—	4	4	no	—
99	C₃×C₄×C₅	C₄×D₄	1	12	yes	complemented
100	C₃×Dic₅	—	2	6	no	—
101	C₃×Dic₅	#127	1	4	yes	complemented
102	C₂⁴×C₃	—	5	4	no	—
103	C₂×C₃×D₄	—	5	24	no	—
104	C₂×C₃×D₄	—	10	12	no	—
105	C₂×C₃×D₄	—	10	12	no	—
106	C₂²×C₃×C₄	—	5	1	no	—
107	C₂²×C₃×C₄	—	5	3	no	—
108	C₂²×C₃×C₄	—	10	24	no	—
109	C₂²×C₃×C₄	C₂²×D₅	1	4	yes	—
110	C₂²×C₃×C₄	C₂²×D₅	1	3	yes	—
111	C₃×C₄⋊C₄	—	2	12	no	—
112	C₃×C₂²⋊C₄	—	10	24	no	—
113	C₃×C₄²	—	5	12	no	—
114	C₃×C₄²	—	10	6	no	—
115	C₃×C₄²	C₄×D₅	1	4	yes	complemented
116	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
117	C₂²×D₅	—	2	12	no	—
118	C₂²×D₅	—	2	12	no	—
119	C₂²×D₅	C₂²×C₃×C₄	1	8	yes	—
120	C₂×C₄×C₅	—	2	12	no	—
121	C₂×C₄×C₅	C₂²×C₃×C₄	1	6	yes	—
122	C₂×C₄×C₅	C₂×C₃×D₄	1	6	yes	—
123	C₂×C₄×C₅	C₃×#41	1	6	yes	—
124	C₂×Dic₅	—	2	6	no	—
125	C₂×Dic₅	C₂×C₃×D₄	1	6	yes	—
126	C₂×Dic₅	C₂×C₃×D₄	1	6	yes	—
127	D₂₀	—	2	48	no	—
128	C₄×D₅	—	4	16	no	—
129	C₂²×D₄	—	5	6	no	—
130	C₂³×C₄	—	5	4	no	—
131	#127	—	5	8	no	—
132	C₄×D₄	—	10	24	no	—
133	C₂×C₄⋊C₄	C₂×C₃×D₅	1	6	yes	—
134	C₂×C₂²⋊C₄	—	5	12	no	—
135	C₂×C₄²	—	5	3	no	—
136	C₂×C₄²	—	5	3	no	—
137	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
138	C₂×C₃×C₅	C₂×C₄×D₄	1	3	yes	—
139	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
140	C₂×C₃×C₅	#532	1	3	yes	—
141	C₃×D₅	—	4	8	no	—
142	C₂³×C₃	—	5	8	no	—
143	C₂³×C₃	—	10	12	no	—
144	C₂³×C₃	—	10	12	no	—
145	C₂³×C₃	C₂³×D₅	1	1	yes	characteristic
146	C₃×D₄	—	10	48	no	—
147	C₂×C₃×C₄	—	2	12	no	—
148	C₂×C₃×C₄	—	5	6	no	—
149	C₂×C₃×C₄	—	5	6	no	—
150	C₂×C₃×C₄	—	10	6	no	—
151	C₂×C₃×C₄	—	20	16	no	—
152	C₂×C₃×C₄	C₂×C₄×D₅	1	6	yes	—
153	C₂×C₃×C₄	D₄×D₅	1	6	yes	—
154	C₂×C₃×C₄	#713	1	6	yes	—
155	C₂²×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
156	C₂²×C₅	C₂²×C₃×D₄	1	3	yes	—
157	C₂²×C₅	C₂×C₃×#41	1	3	yes	—
158	C₂×D₅	—	2	24	no	—
159	C₂×D₅	—	4	4	no	—
160	C₂×D₅	—	4	12	no	—
161	C₄×C₅	—	4	4	no	—
162	C₄×C₅	C₃×C₄×D₄	1	12	yes	complemented
163	Dic₅	—	2	6	no	—
164	Dic₅	C₃×#127	1	4	yes	complemented
165	C₂⁴	—	5	4	no	—
166	C₂×D₄	—	5	24	no	—
167	C₂×D₄	—	10	12	no	—
168	C₂×D₄	—	10	12	no	—
169	C₂²×C₄	—	5	1	no	—
170	C₂²×C₄	—	5	3	no	—
171	C₂²×C₄	—	10	24	no	—
172	C₂²×C₄	C₂²×C₃×D₅	1	4	yes	—
173	C₂²×C₄	C₂²×C₃×D₅	1	3	yes	—
174	C₄⋊C₄	—	2	12	no	—
175	C₂²⋊C₄	—	10	24	no	—
176	C₄²	—	5	12	no	—
177	C₄²	—	10	6	no	—
178	C₄²	C₃×C₄×D₅	1	4	yes	complemented
179	C₃×C₅	C₄×#127	1	1	yes	characteristic, complemented, Hall
180	C₂²×C₃	—	10	24	no	—
181	C₂²×C₃	—	20	4	no	—
182	C₂²×C₃	—	20	12	no	—
183	C₂²×C₃	C₂²×C₄×D₅	1	1	yes	characteristic
184	C₂²×C₃	C₂×D₄×D₅	1	3	yes	—
185	C₂²×C₃	C₂×#713	1	3	yes	—
186	C₃×C₄	—	4	4	no	—
187	C₃×C₄	—	5	4	no	—
188	C₃×C₄	—	10	6	no	—
189	C₃×C₄	#4039	1	12	yes	complemented
190	C₂×C₅	C₂×C₃×C₄×D₄	1	3	yes	—
191	C₂×C₅	C₂×C₃×#127	1	1	yes	characteristic
192	C₂×C₅	C₃×#532	1	3	yes	—
193	D₅	—	4	8	no	—
194	C₂³	—	5	8	no	—
195	C₂³	—	10	12	no	—
196	C₂³	—	10	12	no	—
197	C₂³	C₂³×C₃×D₅	1	1	yes	characteristic
198	D₄	—	10	48	no	—
199	C₂×C₄	—	2	12	no	—
200	C₂×C₄	—	5	6	no	—
201	C₂×C₄	—	5	6	no	—
202	C₂×C₄	—	10	6	no	—
203	C₂×C₄	—	20	16	no	—
204	C₂×C₄	C₂×C₃×C₄×D₅	1	6	yes	—
205	C₂×C₄	C₃×D₄×D₅	1	6	yes	—
206	C₂×C₄	C₃×#713	1	6	yes	—
207	C₂×C₃	—	20	8	no	—
208	C₂×C₃	C₂×#4039	1	3	yes	—
209	C₂×C₃	D₅×#127	1	1	yes	characteristic
210	C₂×C₃	#66059	1	3	yes	—
211	C₅	C₃×C₄×#127	1	1	yes	characteristic, complemented, Sylow
212	C₂²	—	10	24	no	—
213	C₂²	—	20	4	no	—
214	C₂²	—	20	12	no	—
215	C₂²	C₂²×C₃×C₄×D₅	1	1	yes	characteristic
216	C₂²	C₂×C₃×D₄×D₅	1	3	yes	—
217	C₂²	C₂×C₃×#713	1	3	yes	—
218	C₄	—	4	4	no	—
219	C₄	—	5	4	no	—
220	C₄	—	10	6	no	—
221	C₄	C₃×#4039	1	12	yes	complemented
222	C₃	#10606050	1	1	yes	characteristic, direct product factor, Sylow
223	C₂	—	20	8	no	—
224	C₂	C₂×C₃×#4039	1	3	yes	—
225	C₂	C₃×D₅×#127	1	1	yes	characteristic
226	C₂	C₃×#66059	1	3	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂³×D₅
commutator	C₂²×C₅	C₂³×C₃×C₄
Frattini	C₂³	C₂³×C₃×D₅
Fitting	C₃×C₅×#384	C₂
socle	C₂³×C₃×C₅	C₂⁴
prime radical	C₂×C₃×#4020	C₂
cosocle	C₂³×C₅	C₂⁴×C₃
hypercenter	C₃×#384	D₅
hypocenter	C₅	C₃×C₄×#127
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₃×#384

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#4039	C₂	semidirect
3. C₃×C₅×#384	C₂	semidirect
4. C₃×#65277	C₂	semidirect
5. C₃×C₄²×Dic₅	C₂	semidirect
6. #10606050	C₃	direct (N × H)
15. C₃×#4020	C₄	semidirect
17. C₂×#4039	C₂×C₃	semidirect
19. C₅×#384	C₂×C₃	semidirect
20. #65277	C₂×C₃	semidirect
21. C₄²×Dic₅	C₂×C₃	semidirect
29. C₂×C₃×D₂₀	C₂×C₄	semidirect
32. C₃×C₄²×C₅	C₂×C₄	semidirect
35. C₃×C₄×Dic₅	D₄	semidirect
39. C₃×#384	D₅	semidirect
49. #4020	C₃×C₄	semidirect
80. C₂×D₂₀	C₂×C₃×C₄	semidirect
83. C₄²×C₅	C₂×C₃×C₄	semidirect
86. C₄×Dic₅	C₃×D₄	semidirect
90. #384	C₃×D₅	semidirect
99. C₃×C₄×C₅	C₄×D₄	semidirect
101. C₃×Dic₅	#127	semidirect
115. C₃×C₄²	C₄×D₅	semidirect
162. C₄×C₅	C₃×C₄×D₄	semidirect
164. Dic₅	C₃×#127	semidirect
178. C₄²	C₃×C₄×D₅	semidirect
179. C₃×C₅	C₄×#127	semidirect
189. C₃×C₄	#4039	semidirect
211. C₅	C₃×C₄×#127	semidirect
221. C₄	C₃×#4039	semidirect
222. C₃	#10606050	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×#4039	2
C₂×C₃×#4020	2
C₃×C₅×#384	2
C₃×#65277	2
C₃×C₄²×Dic₅	2
#10606050	3
C₃×C₄×#127	5

Subgroups — C3×#10606050 #14322243

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10606050 #14322243