Subgroups — C₃×C₅×#1821 #14274855

Order 1920.

632 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#407	C₂	1	2	yes	maximal
3	C₃×C₅×#343	C₂	1	4	yes	maximal, complemented
4	C₅×#1821	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂³×C₃×C₄×C₅	C₄	1	2	yes	—
6	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₂×C₃×C₅×M₁₆	C₂²	1	4	yes	—
8	C₂×C₃×C₅×C₄⋊C₄	—	2	2	no	—
9	C₂×C₃×C₅×C₄⋊C₄	—	2	2	no	—
10	C₂×C₃×C₅×C₄⋊C₄	C₄	1	2	yes	—
11	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
12	C₃×C₅×C₂²⋊C₈	—	2	2	no	—
13	C₃×C₅×C₂²⋊C₈	—	2	2	no	—
14	C₃×#1821	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂²×C₅×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
16	C₅×#407	C₂×C₃	1	2	yes	—
17	C₅×#343	C₂×C₃	1	4	yes	complemented
18	C₂⁴×C₃×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₃×C₄×C₅	—	2	4	no	—
20	C₂²×C₃×C₄×C₅	—	2	2	no	—
21	C₂²×C₃×C₄×C₅	—	2	2	no	—
22	C₂²×C₃×C₄×C₅	—	4	4	no	—
23	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃×C₄×C₅	C₂×C₄	1	4	yes	—
25	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
26	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
27	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₃×C₅×M₁₆	—	2	8	no	—
29	C₂×C₃×C₅×C₈	—	2	4	no	—
30	C₃×C₅×C₄⋊C₄	—	2	4	no	—
31	C₃×C₅×C₄⋊C₄	—	2	4	no	—
32	C₂²×C₃×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic
33	C₃×#407	C₂×C₅	1	2	yes	—
34	C₃×#343	C₂×C₅	1	4	yes	complemented
35	C₂³×C₄×C₅	C₃×C₄	1	2	yes	—
36	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
37	C₂×C₅×M₁₆	C₂²×C₃	1	4	yes	—
38	C₂×C₅×C₄⋊C₄	—	2	2	no	—
39	C₂×C₅×C₄⋊C₄	—	2	2	no	—
40	C₂×C₅×C₄⋊C₄	C₃×C₄	1	2	yes	—
41	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
42	C₅×C₂²⋊C₈	—	2	2	no	—
43	C₅×C₂²⋊C₈	—	2	2	no	—
44	#1821	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
45	C₂³×C₃×C₅	—	2	2	no	—
46	C₂³×C₃×C₅	—	2	2	no	—
47	C₂³×C₃×C₅	—	2	2	no	—
48	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
49	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
50	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	—	2	2	no	—
52	C₂×C₃×C₄×C₅	—	2	4	no	—
53	C₂×C₃×C₄×C₅	—	2	4	no	—
54	C₂×C₃×C₄×C₅	—	2	4	no	—
55	C₂×C₃×C₄×C₅	—	4	4	no	—
56	C₂×C₃×C₄×C₅	—	4	4	no	—
57	C₂×C₃×C₄×C₅	—	4	4	no	—
58	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
59	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
60	C₂×C₃×C₄×C₅	#41	1	4	yes	—
61	C₃×C₅×C₈	—	4	4	no	—
62	C₂³×C₃×C₄	C₄×C₅	1	2	yes	—
63	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
64	C₂×C₃×M₁₆	C₂²×C₅	1	4	yes	—
65	C₂×C₃×C₄⋊C₄	—	2	2	no	—
66	C₂×C₃×C₄⋊C₄	—	2	2	no	—
67	C₂×C₃×C₄⋊C₄	C₄×C₅	1	2	yes	—
68	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
69	C₃×C₂²⋊C₈	—	2	2	no	—
70	C₃×C₂²⋊C₈	—	2	2	no	—
71	C₂⁴×C₅	C₂×C₃×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	—	2	4	no	—
73	C₂²×C₄×C₅	—	2	2	no	—
74	C₂²×C₄×C₅	—	2	2	no	—
75	C₂²×C₄×C₅	—	4	4	no	—
76	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₂×C₃×C₄	1	4	yes	—
78	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
79	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
80	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
81	C₅×M₁₆	—	2	8	no	—
82	C₂×C₅×C₈	—	2	4	no	—
83	C₅×C₄⋊C₄	—	2	4	no	—
84	C₅×C₄⋊C₄	—	2	4	no	—
85	C₂²×C₄⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic
86	#407	C₂×C₃×C₅	1	2	yes	—
87	#343	C₂×C₃×C₅	1	4	yes	complemented
88	C₂²×C₃×C₅	—	2	8	no	—
89	C₂²×C₃×C₅	—	2	1	no	—
90	C₂²×C₃×C₅	—	2	1	no	—
91	C₂²×C₃×C₅	—	2	1	no	—
92	C₂²×C₃×C₅	—	2	1	no	—
93	C₂²×C₃×C₅	—	2	1	no	—
94	C₂²×C₃×C₅	—	2	1	no	—
95	C₂²×C₃×C₅	#100	1	2	yes	—
96	C₂²×C₃×C₅	#101	1	2	yes	—
97	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂²×C₃×C₅	#117	1	2	yes	—
99	C₃×C₄×C₅	—	2	2	no	—
100	C₃×C₄×C₅	—	4	4	no	—
101	C₃×C₄×C₅	#123	1	4	yes	—
102	C₂⁴×C₃	C₂×C₄×C₅	1	1	yes	characteristic
103	C₂²×C₃×C₄	—	2	4	no	—
104	C₂²×C₃×C₄	—	2	2	no	—
105	C₂²×C₃×C₄	—	2	2	no	—
106	C₂²×C₃×C₄	—	4	4	no	—
107	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
108	C₂²×C₃×C₄	C₂×C₄×C₅	1	4	yes	—
109	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
110	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
111	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
112	C₃×M₁₆	—	2	8	no	—
113	C₂×C₃×C₈	—	2	4	no	—
114	C₃×C₄⋊C₄	—	2	4	no	—
115	C₃×C₄⋊C₄	—	2	4	no	—
116	C₂³×C₅	—	2	2	no	—
117	C₂³×C₅	—	2	2	no	—
118	C₂³×C₅	—	2	2	no	—
119	C₂³×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
120	C₂³×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
121	C₂³×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
122	C₂×C₄×C₅	—	2	2	no	—
123	C₂×C₄×C₅	—	2	4	no	—
124	C₂×C₄×C₅	—	2	4	no	—
125	C₂×C₄×C₅	—	2	4	no	—
126	C₂×C₄×C₅	—	4	4	no	—
127	C₂×C₄×C₅	—	4	4	no	—
128	C₂×C₄×C₅	—	4	4	no	—
129	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
130	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
131	C₂×C₄×C₅	C₃×#41	1	4	yes	—
132	C₅×C₈	—	4	4	no	—
133	C₂³×C₄	C₃×C₄×C₅	1	2	yes	—
134	C₂³×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
135	C₂×M₁₆	C₂²×C₃×C₅	1	4	yes	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	2	2	no	—
138	C₂×C₄⋊C₄	C₃×C₄×C₅	1	2	yes	—
139	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
140	C₂²⋊C₈	—	2	2	no	—
141	C₂²⋊C₈	—	2	2	no	—
142	C₂×C₃×C₅	—	2	2	no	—
143	C₂×C₃×C₅	—	2	2	no	—
144	C₂×C₃×C₅	—	2	2	no	—
145	C₂×C₃×C₅	#381	1	1	yes	characteristic
146	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
147	C₂×C₃×C₅	C₂×#101	1	1	yes	characteristic
148	C₂³×C₃	—	2	2	no	—
149	C₂³×C₃	—	2	2	no	—
150	C₂³×C₃	—	2	2	no	—
151	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
152	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
153	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
154	C₂×C₃×C₄	—	2	2	no	—
155	C₂×C₃×C₄	—	2	4	no	—
156	C₂×C₃×C₄	—	2	4	no	—
157	C₂×C₃×C₄	—	2	4	no	—
158	C₂×C₃×C₄	—	4	4	no	—
159	C₂×C₃×C₄	—	4	4	no	—
160	C₂×C₃×C₄	—	4	4	no	—
161	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
162	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
163	C₂×C₃×C₄	C₅×#41	1	4	yes	—
164	C₃×C₈	—	4	4	no	—
165	C₂²×C₅	—	2	8	no	—
166	C₂²×C₅	—	2	1	no	—
167	C₂²×C₅	—	2	1	no	—
168	C₂²×C₅	—	2	1	no	—
169	C₂²×C₅	—	2	1	no	—
170	C₂²×C₅	—	2	1	no	—
171	C₂²×C₅	—	2	1	no	—
172	C₂²×C₅	C₃×#100	1	2	yes	—
173	C₂²×C₅	C₃×#101	1	2	yes	—
174	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
175	C₂²×C₅	C₃×#117	1	2	yes	—
176	C₄×C₅	—	2	2	no	—
177	C₄×C₅	—	4	4	no	—
178	C₄×C₅	C₃×#123	1	4	yes	—
179	C₂⁴	C₂×C₃×C₄×C₅	1	1	yes	characteristic
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	4	4	no	—
184	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₃×C₄×C₅	1	4	yes	—
186	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
187	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
188	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
189	M₁₆	—	2	8	no	—
190	C₂×C₈	—	2	4	no	—
191	C₄⋊C₄	—	2	4	no	—
192	C₄⋊C₄	—	2	4	no	—
193	C₃×C₅	#1821	1	1	yes	characteristic, direct product factor, Hall
194	C₂²×C₃	—	2	8	no	—
195	C₂²×C₃	—	2	1	no	—
196	C₂²×C₃	—	2	1	no	—
197	C₂²×C₃	—	2	1	no	—
198	C₂²×C₃	—	2	1	no	—
199	C₂²×C₃	—	2	1	no	—
200	C₂²×C₃	—	2	1	no	—
201	C₂²×C₃	C₅×#100	1	2	yes	—
202	C₂²×C₃	C₅×#101	1	2	yes	—
203	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
204	C₂²×C₃	C₅×#117	1	2	yes	—
205	C₃×C₄	—	2	2	no	—
206	C₃×C₄	—	4	4	no	—
207	C₃×C₄	C₅×#123	1	4	yes	—
208	C₂×C₅	—	2	2	no	—
209	C₂×C₅	—	2	2	no	—
210	C₂×C₅	—	2	2	no	—
211	C₂×C₅	C₃×#381	1	1	yes	characteristic
212	C₂×C₅	C₂×C₃×#100	1	1	yes	characteristic
213	C₂×C₅	C₂×C₃×#101	1	1	yes	characteristic
214	C₂³	—	2	2	no	—
215	C₂³	—	2	2	no	—
216	C₂³	—	2	2	no	—
217	C₂³	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
218	C₂³	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
219	C₂³	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
228	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
229	C₂×C₄	C₃×C₅×#41	1	4	yes	—
230	C₈	—	4	4	no	—
231	C₂×C₃	—	2	2	no	—
232	C₂×C₃	—	2	2	no	—
233	C₂×C₃	—	2	2	no	—
234	C₂×C₃	C₅×#381	1	1	yes	characteristic
235	C₂×C₃	C₂×C₅×#100	1	1	yes	characteristic
236	C₂×C₃	C₂×C₅×#101	1	1	yes	characteristic
237	C₅	C₃×#1821	1	1	yes	characteristic, direct product factor, Sylow
238	C₂²	—	2	8	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	C₃×C₅×#100	1	2	yes	—
246	C₂²	C₃×C₅×#101	1	2	yes	—
247	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
248	C₂²	C₃×C₅×#117	1	2	yes	—
249	C₄	—	2	2	no	—
250	C₄	—	4	4	no	—
251	C₄	C₃×C₅×#123	1	4	yes	—
252	C₃	C₅×#1821	1	1	yes	characteristic, direct product factor, Sylow
253	C₂	—	2	2	no	—
254	C₂	—	2	2	no	—
255	C₂	—	2	2	no	—
256	C₂	C₃×C₅×#381	1	1	yes	characteristic
257	C₂	C₂×C₃×C₅×#100	1	1	yes	characteristic
258	C₂	C₂×C₃×C₅×#101	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂³×C₃×C₅
prime radical	C₂⁴×C₃×C₅	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₂³×C₃×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×#343	C₂	semidirect
4. C₅×#1821	C₃	direct (N × H)
14. C₃×#1821	C₅	direct (N × H)
17. C₅×#343	C₂×C₃	semidirect
34. C₃×#343	C₂×C₅	semidirect
44. #1821	C₃×C₅	direct (N × H)
87. #343	C₂×C₃×C₅	semidirect
193. C₃×C₅	#1821	direct (N × H)
237. C₅	C₃×#1821	direct (N × H)
252. C₃	C₅×#1821	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₅×C₄⋊C₄	2
C₃×C₅×#407	2
C₃×C₅×#343	2
C₅×#1821	3
C₃×#1821	5

Subgroups — C3×C5×#1821 #14274855

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1821 #14274855