Subgroups — C₃×#10610230 #14439082

Order 1920.

1128 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#4050	C₂	1	4	yes	maximal, complemented
2	C₃×#65811	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×C₅×#97	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65004	C₂	1	4	yes	maximal, complemented
5	C₃×#65001	C₂	1	4	yes	maximal, complemented
6	#10610230	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×#709	C₂²	1	4	yes	—
8	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₃×#4076	—	2	4	no	—
11	C₂×C₃×Dic₂₀	C₂²	1	4	yes	—
12	C₃×#4050	—	2	16	no	—
13	C₂×C₃×D₄₀	C₂²	1	4	yes	—
14	C₂×C₃×#677	C₂²	1	8	yes	—
15	C₃×#4021	C₂²	1	4	yes	—
16	C₃×#4020	C₂²	1	2	yes	—
17	C₃×C₄×D₂₀	—	2	4	no	—
18	C₃×#4015	C₂²	1	2	yes	—
19	C₃×C₅×#97	C₂²	1	4	yes	—
20	C₃×#3150	—	5	1	no	maximal, Hall
21	C₂×#4050	C₂×C₃	1	4	yes	complemented
22	#65811	C₂×C₃	1	2	yes	complemented
23	C₂×C₅×#97	C₂×C₃	1	1	yes	characteristic, complemented
24	#65004	C₂×C₃	1	4	yes	complemented
25	#65001	C₂×C₃	1	4	yes	complemented
26	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
27	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂×C₃×C₂²⋊D₅	—	2	4	no	—
29	C₃×#709	—	2	16	no	—
30	C₂×C₃×D₂₀	—	2	4	no	—
31	C₂×C₃×D₂₀	C₂³	1	4	yes	—
32	C₂×C₃×C₄×D₅	—	2	4	no	—
33	C₂×C₃×Dic₁₀	C₂³	1	4	yes	—
34	C₂×C₃×C₅×C₈	D₄	1	8	yes	complemented
35	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
36	C₃×C₄²×C₅	D₄	1	2	yes	—
37	C₃×C₄²×C₅	C₂³	1	2	yes	—
38	C₃×#685	—	2	8	no	—
39	C₃×C₄⋊Dic₅	—	2	4	no	—
40	C₃×Dic₂₀	—	2	8	no	—
41	C₃×D₄₀	—	2	8	no	—
42	C₃×#677	—	2	16	no	—
43	C₂×C₃×#135	—	5	4	no	—
44	C₃×#532	—	5	2	no	—
45	C₃×#497	—	5	4	no	—
46	C₃×#496	—	5	4	no	—
47	C₂×C₃×#97	D₅	1	1	yes	characteristic, complemented
48	C₂×#709	C₂²×C₃	1	4	yes	—
49	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
50	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
51	#4076	—	2	4	no	—
52	C₂×Dic₂₀	C₂²×C₃	1	4	yes	—
53	#4050	—	2	16	no	—
54	C₂×D₄₀	C₂²×C₃	1	4	yes	—
55	C₂×#677	C₂²×C₃	1	8	yes	—
56	#4021	C₂²×C₃	1	4	yes	—
57	#4020	C₂²×C₃	1	2	yes	—
58	C₄×D₂₀	—	2	4	no	—
59	#4015	C₂²×C₃	1	2	yes	—
60	C₅×#97	C₂²×C₃	1	4	yes	—
61	#3150	—	5	1	no	Sylow
62	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₃×D₅	—	2	4	no	—
64	C₂×C₃×C₄×C₅	—	4	2	no	—
65	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
66	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
68	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
69	C₃×C₂²⋊D₅	—	4	8	no	—
70	C₂×C₃×Dic₅	—	2	4	no	—
71	C₃×D₂₀	—	2	8	no	—
72	C₃×D₂₀	—	4	8	no	—
73	C₃×C₄×D₅	—	4	8	no	—
74	C₃×Dic₁₀	—	2	8	no	—
75	C₃×C₅×C₈	C₂×D₄	1	8	yes	—
76	C₂×C₃×#41	—	5	4	no	—
77	C₃×#135	—	10	16	no	—
78	C₂×C₃×Q₁₆	—	5	4	no	—
79	C₂×C₃×SD₁₆	—	5	8	no	—
80	C₂×C₃×D₈	—	5	4	no	—
81	C₂²×C₃×C₈	C₂×D₅	1	2	yes	—
82	C₃×C₄⋊Q₈	—	5	2	no	—
83	C₃×#127	—	5	2	no	—
84	C₃×#124	—	5	4	no	—
85	C₃×#121	—	10	4	no	—
86	C₃×C₄×D₄	—	10	4	no	—
87	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
88	C₃×#97	C₂×D₅	1	4	yes	—
89	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
90	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
91	C₂×C₂²⋊D₅	—	2	4	no	—
92	#709	—	2	16	no	—
93	C₂×D₂₀	—	2	4	no	—
94	C₂×D₂₀	C₂³×C₃	1	4	yes	—
95	C₂×C₄×D₅	—	2	4	no	—
96	C₂×Dic₁₀	C₂³×C₃	1	4	yes	—
97	C₂×C₅×C₈	C₃×D₄	1	8	yes	complemented
98	C₂×C₅×C₈	C₂³×C₃	1	4	yes	—
99	C₄²×C₅	C₃×D₄	1	2	yes	—
100	C₄²×C₅	C₂³×C₃	1	2	yes	—
101	#685	—	2	8	no	—
102	C₄⋊Dic₅	—	2	4	no	—
103	Dic₂₀	—	2	8	no	—
104	D₄₀	—	2	8	no	—
105	#677	—	2	16	no	—
106	C₂×#135	—	5	4	no	—
107	#532	—	5	2	no	—
108	#497	—	5	4	no	—
109	#496	—	5	4	no	—
110	C₂×#97	C₃×D₅	1	1	yes	characteristic, complemented
111	C₂²×C₃×C₅	—	2	2	no	—
112	C₂²×C₃×C₅	#127	1	2	yes	—
113	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₃×D₅	—	4	4	no	—
115	C₂×C₃×D₅	—	4	8	no	—
116	C₃×C₄×C₅	—	2	4	no	—
117	C₃×C₄×C₅	#127	1	2	yes	—
118	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
119	C₃×Dic₅	—	4	4	no	—
120	C₃×#41	—	10	16	no	—
121	C₂×C₃×Q₈	—	5	4	no	—
122	C₂×C₃×D₄	—	5	4	no	—
123	C₂×C₃×D₄	—	10	4	no	—
124	C₂×C₃×D₄	—	10	4	no	—
125	C₂²×C₃×C₄	—	10	4	no	—
126	C₂²×C₃×C₄	D₂₀	1	2	yes	—
127	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
128	C₃×Q₁₆	—	10	8	no	—
129	C₃×SD₁₆	—	10	16	no	—
130	C₃×D₈	—	10	8	no	—
131	C₂×C₃×C₈	D₂₀	1	8	yes	complemented
132	C₂×C₃×C₈	C₂²×D₅	1	4	yes	—
133	C₃×C₄⋊C₄	—	10	4	no	—
134	C₃×C₂²⋊C₄	—	10	8	no	—
135	C₃×C₄²	D₂₀	1	2	yes	—
136	C₃×C₄²	C₂²×D₅	1	2	yes	—
137	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
138	C₂²×D₅	—	2	4	no	—
139	C₂×C₄×C₅	—	4	2	no	—
140	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
141	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
143	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
144	C₂²⋊D₅	—	4	8	no	—
145	C₂×Dic₅	—	2	4	no	—
146	D₂₀	—	2	8	no	—
147	D₂₀	—	4	8	no	—
148	C₄×D₅	—	4	8	no	—
149	Dic₁₀	—	2	8	no	—
150	C₅×C₈	C₂×C₃×D₄	1	8	yes	—
151	C₂×#41	—	5	4	no	—
152	#135	—	10	16	no	—
153	C₂×Q₁₆	—	5	4	no	—
154	C₂×SD₁₆	—	5	8	no	—
155	C₂×D₈	—	5	4	no	—
156	C₂²×C₈	C₂×C₃×D₅	1	2	yes	—
157	C₄⋊Q₈	—	5	2	no	—
158	#127	—	5	2	no	—
159	#124	—	5	4	no	—
160	#121	—	10	4	no	—
161	C₄×D₄	—	10	4	no	—
162	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
163	#97	C₂×C₃×D₅	1	4	yes	—
164	C₂×C₃×C₅	—	2	2	no	—
165	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
166	C₂×C₃×C₅	#575	1	2	yes	—
167	C₃×D₅	—	8	4	no	—
168	C₂³×C₃	—	10	4	no	—
169	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
170	C₃×Q₈	—	10	8	no	—
171	C₃×D₄	—	10	8	no	—
172	C₃×D₄	—	20	8	no	—
173	C₃×D₄	—	20	8	no	—
174	C₂×C₃×C₄	—	4	2	no	—
175	C₂×C₃×C₄	—	10	4	no	—
176	C₂×C₃×C₄	—	20	8	no	—
177	C₂×C₃×C₄	C₂×D₂₀	1	4	yes	—
178	C₂×C₃×C₄	C₂×D₂₀	1	4	yes	—
179	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
181	C₃×C₈	C₂×D₂₀	1	8	yes	—
182	C₂²×C₅	—	2	2	no	—
183	C₂²×C₅	C₃×#127	1	2	yes	—
184	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
185	C₂×D₅	—	4	8	no	—
186	C₂×D₅	—	4	4	no	—
187	C₄×C₅	—	2	4	no	—
188	C₄×C₅	C₃×#127	1	2	yes	—
189	C₄×C₅	C₂²×C₃×D₄	1	2	yes	—
190	Dic₅	—	4	4	no	—
191	#41	—	10	16	no	—
192	C₂×Q₈	—	5	4	no	—
193	C₂×D₄	—	5	4	no	—
194	C₂×D₄	—	10	4	no	—
195	C₂×D₄	—	10	4	no	—
196	C₂²×C₄	—	10	4	no	—
197	C₂²×C₄	C₃×D₂₀	1	2	yes	—
198	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
199	Q₁₆	—	10	8	no	—
200	SD₁₆	—	10	16	no	—
201	D₈	—	10	8	no	—
202	C₂×C₈	C₃×D₂₀	1	8	yes	complemented
203	C₂×C₈	C₂²×C₃×D₅	1	4	yes	—
204	C₄⋊C₄	—	10	4	no	—
205	C₂²⋊C₄	—	10	8	no	—
206	C₄²	C₃×D₂₀	1	2	yes	—
207	C₄²	C₂²×C₃×D₅	1	2	yes	—
208	C₃×C₅	#3150	1	1	yes	characteristic, complemented, Hall
209	C₂²×C₃	—	2	2	no	—
210	C₂²×C₃	—	20	4	no	—
211	C₂²×C₃	—	20	8	no	—
212	C₂²×C₃	#4020	1	2	yes	—
213	C₂²×C₃	C₂²×D₂₀	1	1	yes	characteristic
214	C₃×C₄	—	2	4	no	—
215	C₃×C₄	—	20	4	no	—
216	C₃×C₄	#4020	1	2	yes	—
217	C₃×C₄	C₂²×D₂₀	1	2	yes	—
218	C₂×C₅	—	2	2	no	—
219	C₂×C₅	C₂×C₃×#127	1	1	yes	characteristic
220	C₂×C₅	C₃×#575	1	2	yes	—
221	D₅	—	8	4	no	—
222	C₂³	—	10	4	no	—
223	C₂³	C₂×C₃×D₂₀	1	1	yes	characteristic
224	Q₈	—	10	8	no	—
225	D₄	—	10	8	no	—
226	D₄	—	20	8	no	—
227	D₄	—	20	8	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	10	4	no	—
230	C₂×C₄	—	20	8	no	—
231	C₂×C₄	C₂×C₃×D₂₀	1	4	yes	—
232	C₂×C₄	C₂×C₃×D₂₀	1	4	yes	—
233	C₂×C₄	C₂×C₃×D₂₀	1	1	yes	characteristic
234	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
235	C₈	C₂×C₃×D₂₀	1	8	yes	—
236	C₂×C₃	—	2	2	no	—
237	C₂×C₃	—	40	4	no	—
238	C₂×C₃	C₂×#4020	1	1	yes	characteristic
239	C₂×C₃	#66082	1	2	yes	—
240	C₅	C₃×#3150	1	1	yes	characteristic, complemented, Sylow
241	C₂²	—	2	2	no	—
242	C₂²	—	20	4	no	—
243	C₂²	—	20	8	no	—
244	C₂²	C₃×#4020	1	2	yes	—
245	C₂²	C₂²×C₃×D₂₀	1	1	yes	characteristic
246	C₄	—	2	4	no	—
247	C₄	—	20	4	no	—
248	C₄	C₃×#4020	1	2	yes	—
249	C₄	C₂²×C₃×D₂₀	1	2	yes	—
250	C₃	#10610230	1	1	yes	characteristic, direct product factor, Sylow
251	C₂	—	2	2	no	—
252	C₂	—	40	4	no	—
253	C₂	C₂×C₃×#4020	1	1	yes	characteristic
254	C₂	C₃×#66082	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₂₀
commutator, cosocle	C₂×C₄×C₅	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₅
Fitting	C₂×C₃×C₅×#97	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
hypercenter	C₂×C₃×#97	D₅
hypocenter	C₅	C₃×#3150
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₂×C₃×#97

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#4050	C₂	semidirect
2. C₃×#65811	C₂	semidirect
3. C₂×C₃×C₅×#97	C₂	semidirect
4. C₃×#65004	C₂	semidirect
5. C₃×#65001	C₂	semidirect
6. #10610230	C₃	direct (N × H)
21. C₂×#4050	C₂×C₃	semidirect
22. #65811	C₂×C₃	semidirect
23. C₂×C₅×#97	C₂×C₃	semidirect
24. #65004	C₂×C₃	semidirect
25. #65001	C₂×C₃	semidirect
34. C₂×C₃×C₅×C₈	D₄	semidirect
47. C₂×C₃×#97	D₅	semidirect
97. C₂×C₅×C₈	C₃×D₄	semidirect
110. C₂×#97	C₃×D₅	semidirect
131. C₂×C₃×C₈	D₂₀	semidirect
202. C₂×C₈	C₃×D₂₀	semidirect
208. C₃×C₅	#3150	semidirect
240. C₅	C₃×#3150	semidirect
250. C₃	#10610230	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×#4050	2
C₃×#65811	2
C₂×C₃×C₅×#97	2
C₃×#65004	2
C₃×#65001	2
#10610230	3
C₃×#3150	5

Subgroups — C3×#10610230 #14439082

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10610230 #14439082