Subgroups — C₃×#10600616 #14305959

Order 1920.

824 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#4020	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×#97	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#3953	C₂	1	4	yes	maximal, complemented
4	C₃×#65225	C₂	1	1	yes	maximal, characteristic, complemented
5	#10600616	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×D₂₀	—	2	2	no	—
7	C₂²×C₃×D₂₀	C₂²	1	2	yes	—
8	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
11	C₃×#4020	—	2	2	no	—
12	C₃×#4020	C₄	1	4	yes	complemented
13	C₃×C₅×#97	—	2	2	no	—
14	C₃×#3963	—	2	2	no	—
15	C₃×#3953	—	2	8	no	—
16	C₃×#1961	—	5	1	no	maximal, Hall
17	C₂×#4020	C₂×C₃	1	1	yes	characteristic
18	C₂×C₅×#97	C₂×C₃	1	1	yes	characteristic, complemented
19	C₂×#3953	C₂×C₃	1	4	yes	complemented
20	#65225	C₂×C₃	1	1	yes	characteristic, complemented
21	C₂³×C₃×D₅	—	2	2	no	—
22	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
23	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
24	C₂²×C₃×Dic₅	—	2	2	no	—
25	C₂×C₃×D₂₀	—	2	4	no	—
26	C₂×C₃×D₂₀	—	2	4	no	—
27	C₂×C₃×D₂₀	—	2	8	no	—
28	C₂×C₃×D₂₀	—	2	2	no	—
29	C₂×C₃×D₂₀	—	4	8	no	—
30	C₂×C₃×D₂₀	C₂×C₄	1	4	yes	—
31	C₂×C₃×C₅×C₈	—	2	2	no	—
32	C₂×C₃×C₅×C₈	—	2	2	no	—
33	C₂×C₃×C₅×C₈	D₄	1	4	yes	complemented
34	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
35	C₃×C₄²×C₅	D₄	1	2	yes	—
36	C₃×C₄⋊Dic₅	—	2	4	no	—
37	C₂×C₃×#127	—	5	1	no	—
38	C₂×C₃×#102	—	5	4	no	—
39	C₂×C₃×#97	D₅	1	1	yes	characteristic, complemented
40	C₃×#382	—	5	1	no	—
41	C₂²×D₂₀	—	2	2	no	—
42	C₂²×D₂₀	C₂²×C₃	1	2	yes	—
43	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
44	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
45	C₂×C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
46	#4020	—	2	2	no	—
47	#4020	C₃×C₄	1	4	yes	complemented
48	C₅×#97	—	2	2	no	—
49	#3963	—	2	2	no	—
50	#3953	—	2	8	no	—
51	#1961	—	5	1	no	Sylow
52	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₃×D₅	—	2	4	no	—
54	C₂²×C₃×D₅	—	4	2	no	—
55	C₂²×C₃×D₅	—	4	2	no	—
56	C₂²×C₃×D₅	—	4	8	no	—
57	C₂×C₃×C₄×C₅	—	2	4	no	—
58	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
59	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
61	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	#41	1	2	yes	—
63	C₂×C₃×Dic₅	—	4	2	no	—
64	C₂×C₃×Dic₅	—	4	4	no	—
65	C₃×D₂₀	—	2	8	no	—
66	C₃×D₂₀	—	4	16	no	—
67	C₃×D₂₀	—	4	4	no	—
68	C₃×C₅×C₈	—	2	4	no	—
69	C₂²×C₃×D₄	—	5	2	no	—
70	C₂²×C₃×D₄	—	10	2	no	—
71	C₂²×C₃×C₈	C₂×D₅	1	2	yes	—
72	C₃×#127	—	5	4	no	—
73	C₃×#127	—	10	2	no	—
74	C₂×C₃×C₄⋊C₄	—	5	2	no	—
75	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
76	C₃×#102	—	10	8	no	—
77	C₃×#97	—	2	2	no	—
78	C₃×#95	—	10	2	no	—
79	C₂³×D₅	—	2	2	no	—
80	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
81	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
82	C₂²×Dic₅	—	2	2	no	—
83	C₂×D₂₀	—	2	4	no	—
84	C₂×D₂₀	—	2	4	no	—
85	C₂×D₂₀	—	2	8	no	—
86	C₂×D₂₀	—	2	2	no	—
87	C₂×D₂₀	—	4	8	no	—
88	C₂×D₂₀	C₂×C₃×C₄	1	4	yes	—
89	C₂×C₅×C₈	—	2	2	no	—
90	C₂×C₅×C₈	—	2	2	no	—
91	C₂×C₅×C₈	C₃×D₄	1	4	yes	complemented
92	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
93	C₄²×C₅	C₃×D₄	1	2	yes	—
94	C₄⋊Dic₅	—	2	4	no	—
95	C₂×#127	—	5	1	no	—
96	C₂×#102	—	5	4	no	—
97	C₂×#97	C₃×D₅	1	1	yes	characteristic, complemented
98	#382	—	5	1	no	—
99	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂²×C₃×C₅	#124	1	1	yes	characteristic
101	C₂²×C₃×C₅	#127	1	1	yes	characteristic
102	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	4	yes	—
103	C₂×C₃×D₅	—	4	4	no	—
104	C₂×C₃×D₅	—	4	8	no	—
105	C₂×C₃×D₅	—	8	4	no	—
106	C₂×C₃×D₅	—	8	4	no	—
107	C₃×C₄×C₅	—	2	4	no	—
108	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
109	C₃×C₄×C₅	#121	1	2	yes	—
110	C₃×Dic₅	—	8	2	no	—
111	C₂⁴×C₃	—	10	2	no	—
112	C₂×C₃×D₄	—	5	4	no	—
113	C₂×C₃×D₄	—	10	8	no	—
114	C₂×C₃×D₄	—	10	4	no	—
115	C₂×C₃×D₄	—	10	4	no	—
116	C₂×C₃×D₄	—	10	2	no	—
117	C₂×C₃×D₄	—	20	8	no	—
118	C₂²×C₃×C₄	—	10	2	no	—
119	C₂²×C₃×C₄	D₂₀	1	2	yes	—
120	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
121	C₂×C₃×C₈	—	2	2	no	—
122	C₂×C₃×C₈	—	2	2	no	—
123	C₂×C₃×C₈	D₂₀	1	4	yes	complemented
124	C₃×C₄⋊C₄	—	10	4	no	—
125	C₃×C₄²	C₄×D₅	1	2	yes	—
126	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
127	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
128	C₂²×D₅	—	2	4	no	—
129	C₂²×D₅	—	4	2	no	—
130	C₂²×D₅	—	4	2	no	—
131	C₂²×D₅	—	4	8	no	—
132	C₂×C₄×C₅	—	2	4	no	—
133	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	4	yes	—
134	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
136	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
137	C₂×C₄×C₅	C₃×#41	1	2	yes	—
138	C₂×Dic₅	—	4	2	no	—
139	C₂×Dic₅	—	4	4	no	—
140	D₂₀	—	2	8	no	—
141	D₂₀	—	4	16	no	—
142	D₂₀	—	4	4	no	—
143	C₅×C₈	—	2	4	no	—
144	C₂²×D₄	—	5	2	no	—
145	C₂²×D₄	—	10	2	no	—
146	C₂²×C₈	C₂×C₃×D₅	1	2	yes	—
147	#127	—	5	4	no	—
148	#127	—	10	2	no	—
149	C₂×C₄⋊C₄	—	5	2	no	—
150	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
151	#102	—	10	8	no	—
152	#97	—	2	2	no	—
153	#95	—	10	2	no	—
154	C₂×C₃×C₅	#390	1	1	yes	characteristic
155	C₂×C₃×C₅	#418	1	2	yes	—
156	C₂×C₃×C₅	#490	1	2	yes	—
157	C₂×C₃×C₅	#496	1	2	yes	—
158	C₃×D₅	—	8	4	no	—
159	C₂³×C₃	—	10	4	no	—
160	C₂³×C₃	—	20	2	no	—
161	C₂³×C₃	—	20	2	no	—
162	C₂³×C₃	—	20	8	no	—
163	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
164	C₃×D₄	—	10	8	no	—
165	C₃×D₄	—	20	16	no	—
166	C₃×D₄	—	20	4	no	—
167	C₂×C₃×C₄	—	2	4	no	—
168	C₂×C₃×C₄	—	20	2	no	—
169	C₂×C₃×C₄	—	20	4	no	—
170	C₂×C₃×C₄	#685	1	4	yes	—
171	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
172	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
173	C₂×C₃×C₄	#709	1	2	yes	—
174	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
175	C₃×C₈	—	2	4	no	—
176	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
177	C₂²×C₅	C₃×#124	1	1	yes	characteristic
178	C₂²×C₅	C₃×#127	1	1	yes	characteristic
179	C₂²×C₅	C₃×AGL(1,ℤ/8)	1	4	yes	—
180	C₂×D₅	—	4	4	no	—
181	C₂×D₅	—	4	8	no	—
182	C₂×D₅	—	8	4	no	—
183	C₂×D₅	—	8	4	no	—
184	C₄×C₅	—	2	4	no	—
185	C₄×C₅	C₃×C₄×D₄	1	2	yes	—
186	C₄×C₅	C₃×#121	1	2	yes	—
187	Dic₅	—	8	2	no	—
188	C₂⁴	—	10	2	no	—
189	C₂×D₄	—	5	4	no	—
190	C₂×D₄	—	10	8	no	—
191	C₂×D₄	—	10	4	no	—
192	C₂×D₄	—	10	4	no	—
193	C₂×D₄	—	10	2	no	—
194	C₂×D₄	—	20	8	no	—
195	C₂²×C₄	—	10	2	no	—
196	C₂²×C₄	C₃×D₂₀	1	2	yes	—
197	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	C₃×D₂₀	1	4	yes	complemented
201	C₄⋊C₄	—	10	4	no	—
202	C₄²	C₃×C₄×D₅	1	2	yes	—
203	C₄²	C₃×C₂²⋊D₅	1	2	yes	—
204	C₃×C₅	#1961	1	1	yes	characteristic, complemented, Hall
205	C₂²×C₃	—	20	8	no	—
206	C₂²×C₃	—	20	4	no	—
207	C₂²×C₃	—	40	4	no	—
208	C₂²×C₃	—	40	4	no	—
209	C₂²×C₃	#4020	1	1	yes	characteristic
210	C₂²×C₃	#4021	1	1	yes	characteristic
211	C₂²×C₃	#4054	1	4	yes	—
212	C₂²×C₃	C₂×#685	1	1	yes	characteristic
213	C₃×C₄	—	2	4	no	—
214	C₃×C₄	—	40	2	no	—
215	C₃×C₄	C₄×D₂₀	1	2	yes	—
216	C₃×C₄	#4076	1	2	yes	—
217	C₂×C₅	C₃×#390	1	1	yes	characteristic
218	C₂×C₅	C₃×#418	1	2	yes	—
219	C₂×C₅	C₃×#490	1	2	yes	—
220	C₂×C₅	C₃×#496	1	2	yes	—
221	D₅	—	8	4	no	—
222	C₂³	—	10	4	no	—
223	C₂³	—	20	2	no	—
224	C₂³	—	20	2	no	—
225	C₂³	—	20	8	no	—
226	C₂³	C₂×C₃×D₂₀	1	1	yes	characteristic
227	D₄	—	10	8	no	—
228	D₄	—	20	16	no	—
229	D₄	—	20	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	20	2	no	—
232	C₂×C₄	—	20	4	no	—
233	C₂×C₄	C₃×#685	1	4	yes	—
234	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
235	C₂×C₄	C₂×C₃×D₂₀	1	2	yes	—
236	C₂×C₄	C₃×#709	1	2	yes	—
237	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
238	C₈	—	2	4	no	—
239	C₂×C₃	—	40	4	no	—
240	C₂×C₃	#65001	1	2	yes	—
241	C₂×C₃	#65002	1	2	yes	—
242	C₂×C₃	#65228	1	1	yes	characteristic
243	C₂×C₃	#65420	1	2	yes	—
244	C₅	C₃×#1961	1	1	yes	characteristic, complemented, Sylow
245	C₂²	—	20	8	no	—
246	C₂²	—	20	4	no	—
247	C₂²	—	40	4	no	—
248	C₂²	—	40	4	no	—
249	C₂²	C₃×#4020	1	1	yes	characteristic
250	C₂²	C₃×#4021	1	1	yes	characteristic
251	C₂²	C₃×#4054	1	4	yes	—
252	C₂²	C₂×C₃×#685	1	1	yes	characteristic
253	C₄	—	2	4	no	—
254	C₄	—	40	2	no	—
255	C₄	C₃×C₄×D₂₀	1	2	yes	—
256	C₄	C₃×#4076	1	2	yes	—
257	C₃	#10600616	1	1	yes	characteristic, direct product factor, Sylow
258	C₂	—	40	4	no	—
259	C₂	C₃×#65001	1	2	yes	—
260	C₂	C₃×#65002	1	2	yes	—
261	C₂	C₃×#65228	1	1	yes	characteristic
262	C₂	C₃×#65420	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂×D₂₀
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₂×C₃×C₅×#97	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂×C₃×#4020	C₂
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₂×C₃×#97	D₅
hypocenter	C₅	C₃×#1961
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×#97

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×#97	C₂	semidirect
3. C₂×C₃×#3953	C₂	semidirect
4. C₃×#65225	C₂	semidirect
5. #10600616	C₃	direct (N × H)
12. C₃×#4020	C₄	semidirect
18. C₂×C₅×#97	C₂×C₃	semidirect
19. C₂×#3953	C₂×C₃	semidirect
20. #65225	C₂×C₃	semidirect
33. C₂×C₃×C₅×C₈	D₄	semidirect
39. C₂×C₃×#97	D₅	semidirect
47. #4020	C₃×C₄	semidirect
91. C₂×C₅×C₈	C₃×D₄	semidirect
97. C₂×#97	C₃×D₅	semidirect
123. C₂×C₃×C₈	D₂₀	semidirect
200. C₂×C₈	C₃×D₂₀	semidirect
204. C₃×C₅	#1961	semidirect
244. C₅	C₃×#1961	semidirect
257. C₃	#10600616	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×#4020	2
C₂×C₃×C₅×#97	2
C₂×C₃×#3953	2
C₃×#65225	2
#10600616	3
C₃×#1961	5

Subgroups — C3×#10600616 #14305959

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10600616 #14305959