Subgroups — C₃×C₅×#1811 #14274845

Order 1920.

544 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×#108	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#407	C₂	1	2	yes	maximal
4	C₃×C₅×#341	C₂	1	2	yes	maximal, complemented
5	C₅×#1811	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₂×C₃×C₅×M₁₆	—	2	4	no	—
8	C₂×C₃×C₅×M₁₆	C₄	1	4	yes	—
9	C₂×C₃×C₅×M₁₆	C₂²	1	4	yes	—
10	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₃×C₅×#108	—	2	4	no	—
13	C₃×C₅×C₂²⋊C₈	—	2	4	no	—
14	C₃×#1811	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂²×C₅×M₁₆	C₂×C₃	1	1	yes	characteristic
16	C₂×C₅×#108	C₂×C₃	1	2	yes	complemented
17	C₅×#407	C₂×C₃	1	2	yes	—
18	C₅×#341	C₂×C₃	1	2	yes	complemented
19	C₂⁴×C₃×C₅	Q₈	1	1	yes	characteristic
20	C₂²×C₃×C₄×C₅	—	2	2	no	—
21	C₂²×C₃×C₄×C₅	—	2	2	no	—
22	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
23	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
24	C₂²×C₃×C₄×C₅	Q₈	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₃×C₅×M₁₆	—	2	8	no	—
27	C₃×C₅×M₁₆	—	2	4	no	—
28	C₃×C₅×M₁₆	—	2	4	no	—
29	C₂×C₃×C₅×C₈	—	2	2	no	—
30	C₂×C₃×C₅×C₈	—	2	4	no	—
31	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
32	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
33	C₂×C₃×C₅×C₈	D₄	1	4	yes	—
34	C₂²×C₃×M₁₆	C₂×C₅	1	1	yes	characteristic
35	C₂×C₃×#108	C₂×C₅	1	2	yes	complemented
36	C₃×#407	C₂×C₅	1	2	yes	—
37	C₃×#341	C₂×C₅	1	2	yes	complemented
38	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
39	C₂×C₅×M₁₆	—	2	4	no	—
40	C₂×C₅×M₁₆	C₃×C₄	1	4	yes	—
41	C₂×C₅×M₁₆	C₂²×C₃	1	4	yes	—
42	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
43	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
44	C₅×#108	—	2	4	no	—
45	C₅×C₂²⋊C₈	—	2	4	no	—
46	#1811	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
47	C₂³×C₃×C₅	—	2	2	no	—
48	C₂³×C₃×C₅	—	4	2	no	—
49	C₂³×C₃×C₅	C₄⋊C₄	1	2	yes	—
50	C₂³×C₃×C₅	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	—	2	2	no	—
52	C₂×C₃×C₄×C₅	—	2	2	no	—
53	C₂×C₃×C₄×C₅	—	2	2	no	—
54	C₂×C₃×C₄×C₅	—	2	4	no	—
55	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
56	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
58	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	#41	1	2	yes	—
60	C₃×C₅×C₈	—	2	2	no	—
61	C₃×C₅×C₈	—	4	4	no	—
62	C₃×C₅×C₈	C₂²⋊C₄	1	4	yes	—
63	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
64	C₂×C₃×M₁₆	—	2	4	no	—
65	C₂×C₃×M₁₆	C₄×C₅	1	4	yes	—
66	C₂×C₃×M₁₆	C₂²×C₅	1	4	yes	—
67	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
68	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
69	C₃×#108	—	2	4	no	—
70	C₃×C₂²⋊C₈	—	2	4	no	—
71	C₂⁴×C₅	C₃×Q₈	1	1	yes	characteristic
72	C₂²×C₄×C₅	—	2	2	no	—
73	C₂²×C₄×C₅	—	2	2	no	—
74	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
75	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
76	C₂²×C₄×C₅	C₃×Q₈	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
78	C₅×M₁₆	—	2	8	no	—
79	C₅×M₁₆	—	2	4	no	—
80	C₅×M₁₆	—	2	4	no	—
81	C₂×C₅×C₈	—	2	2	no	—
82	C₂×C₅×C₈	—	2	4	no	—
83	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
84	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
85	C₂×C₅×C₈	C₃×D₄	1	4	yes	—
86	C₂²×M₁₆	C₂×C₃×C₅	1	1	yes	characteristic
87	C₂×#108	C₂×C₃×C₅	1	2	yes	complemented
88	#407	C₂×C₃×C₅	1	2	yes	—
89	#341	C₂×C₃×C₅	1	2	yes	complemented
90	C₂²×C₃×C₅	—	2	2	no	—
91	C₂²×C₃×C₅	—	2	2	no	—
92	C₂²×C₃×C₅	—	2	4	no	—
93	C₂²×C₃×C₅	—	4	4	no	—
94	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂²×C₃×C₅	C₂²⋊Q₈	1	2	yes	—
96	C₃×C₄×C₅	—	2	2	no	—
97	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₃×C₄×C₅	#117	1	1	yes	characteristic
99	C₃×C₄×C₅	#121	1	2	yes	—
100	C₂⁴×C₃	C₅×Q₈	1	1	yes	characteristic
101	C₂²×C₃×C₄	—	2	2	no	—
102	C₂²×C₃×C₄	—	2	2	no	—
103	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
104	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
105	C₂²×C₃×C₄	C₅×Q₈	1	1	yes	characteristic
106	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
107	C₃×M₁₆	—	2	8	no	—
108	C₃×M₁₆	—	2	4	no	—
109	C₃×M₁₆	—	2	4	no	—
110	C₂×C₃×C₈	—	2	2	no	—
111	C₂×C₃×C₈	—	2	4	no	—
112	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
113	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
114	C₂×C₃×C₈	C₅×D₄	1	4	yes	—
115	C₂³×C₅	—	2	2	no	—
116	C₂³×C₅	—	4	2	no	—
117	C₂³×C₅	C₃×C₄⋊C₄	1	2	yes	—
118	C₂³×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₅	—	2	2	no	—
120	C₂×C₄×C₅	—	2	2	no	—
121	C₂×C₄×C₅	—	2	2	no	—
122	C₂×C₄×C₅	—	2	4	no	—
123	C₂×C₄×C₅	C₃×C₄⋊C₄	1	2	yes	—
124	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
126	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
127	C₂×C₄×C₅	C₃×#41	1	2	yes	—
128	C₅×C₈	—	2	2	no	—
129	C₅×C₈	—	4	4	no	—
130	C₅×C₈	C₃×C₂²⋊C₄	1	4	yes	—
131	C₂³×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
132	C₂×M₁₆	—	2	4	no	—
133	C₂×M₁₆	C₃×C₄×C₅	1	4	yes	—
134	C₂×M₁₆	C₂²×C₃×C₅	1	4	yes	—
135	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
136	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
137	#108	—	2	4	no	—
138	C₂²⋊C₈	—	2	4	no	—
139	C₂×C₃×C₅	—	2	2	no	—
140	C₂×C₃×C₅	—	4	2	no	—
141	C₂×C₃×C₅	#380	1	1	yes	characteristic
142	C₂×C₃×C₅	#430	1	2	yes	—
143	C₂³×C₃	—	2	2	no	—
144	C₂³×C₃	—	4	2	no	—
145	C₂³×C₃	C₅×C₄⋊C₄	1	2	yes	—
146	C₂³×C₃	C₂×C₅×Q₈	1	1	yes	characteristic
147	C₂×C₃×C₄	—	2	2	no	—
148	C₂×C₃×C₄	—	2	2	no	—
149	C₂×C₃×C₄	—	2	2	no	—
150	C₂×C₃×C₄	—	2	4	no	—
151	C₂×C₃×C₄	C₅×C₄⋊C₄	1	2	yes	—
152	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
153	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
154	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
155	C₂×C₃×C₄	C₅×#41	1	2	yes	—
156	C₃×C₈	—	2	2	no	—
157	C₃×C₈	—	4	4	no	—
158	C₃×C₈	C₅×C₂²⋊C₄	1	4	yes	—
159	C₂²×C₅	—	2	2	no	—
160	C₂²×C₅	—	2	2	no	—
161	C₂²×C₅	—	2	4	no	—
162	C₂²×C₅	—	4	4	no	—
163	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
164	C₂²×C₅	C₃×C₂²⋊Q₈	1	2	yes	—
165	C₄×C₅	—	2	2	no	—
166	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
167	C₄×C₅	C₃×#117	1	1	yes	characteristic
168	C₄×C₅	C₃×#121	1	2	yes	—
169	C₂⁴	C₃×C₅×Q₈	1	1	yes	characteristic
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	C₂×C₃×C₄×C₅	1	2	yes	—
173	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
174	C₂²×C₄	C₃×C₅×Q₈	1	1	yes	characteristic
175	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
176	M₁₆	—	2	8	no	—
177	M₁₆	—	2	4	no	—
178	M₁₆	—	2	4	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	4	no	—
181	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
182	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
183	C₂×C₈	C₃×C₅×D₄	1	4	yes	—
184	C₃×C₅	#1811	1	1	yes	characteristic, direct product factor, Hall
185	C₂²×C₃	—	2	2	no	—
186	C₂²×C₃	—	2	2	no	—
187	C₂²×C₃	—	2	4	no	—
188	C₂²×C₃	—	4	4	no	—
189	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
190	C₂²×C₃	C₅×C₂²⋊Q₈	1	2	yes	—
191	C₃×C₄	—	2	2	no	—
192	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
193	C₃×C₄	C₅×#117	1	1	yes	characteristic
194	C₃×C₄	C₅×#121	1	2	yes	—
195	C₂×C₅	—	2	2	no	—
196	C₂×C₅	—	4	2	no	—
197	C₂×C₅	C₃×#380	1	1	yes	characteristic
198	C₂×C₅	C₃×#430	1	2	yes	—
199	C₂³	—	2	2	no	—
200	C₂³	—	4	2	no	—
201	C₂³	C₃×C₅×C₄⋊C₄	1	2	yes	—
202	C₂³	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	C₃×C₅×C₄⋊C₄	1	2	yes	—
208	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
209	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
210	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
211	C₂×C₄	C₃×C₅×#41	1	2	yes	—
212	C₈	—	2	2	no	—
213	C₈	—	4	4	no	—
214	C₈	C₃×C₅×C₂²⋊C₄	1	4	yes	—
215	C₂×C₃	—	2	2	no	—
216	C₂×C₃	—	4	2	no	—
217	C₂×C₃	C₅×#380	1	1	yes	characteristic
218	C₂×C₃	C₅×#430	1	2	yes	—
219	C₅	C₃×#1811	1	1	yes	characteristic, direct product factor, Sylow
220	C₂²	—	2	2	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	4	no	—
223	C₂²	—	4	4	no	—
224	C₂²	C₂×C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
225	C₂²	C₃×C₅×C₂²⋊Q₈	1	2	yes	—
226	C₄	—	2	2	no	—
227	C₄	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
228	C₄	C₃×C₅×#117	1	1	yes	characteristic
229	C₄	C₃×C₅×#121	1	2	yes	—
230	C₃	C₅×#1811	1	1	yes	characteristic, direct product factor, Sylow
231	C₂	—	2	2	no	—
232	C₂	—	4	2	no	—
233	C₂	C₃×C₅×#380	1	1	yes	characteristic
234	C₂	C₃×C₅×#430	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄×C₅	C₂×D₄
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂³×C₃×C₅
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂⁴×C₃×C₅	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄×C₅ ◁ C₂³×C₃×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×#108	C₂	semidirect
4. C₃×C₅×#341	C₂	semidirect
5. C₅×#1811	C₃	direct (N × H)
14. C₃×#1811	C₅	direct (N × H)
16. C₂×C₅×#108	C₂×C₃	semidirect
18. C₅×#341	C₂×C₃	semidirect
35. C₂×C₃×#108	C₂×C₅	semidirect
37. C₃×#341	C₂×C₅	semidirect
46. #1811	C₃×C₅	direct (N × H)
87. C₂×#108	C₂×C₃×C₅	semidirect
89. #341	C₂×C₃×C₅	semidirect
184. C₃×C₅	#1811	direct (N × H)
219. C₅	C₃×#1811	direct (N × H)
230. C₃	C₅×#1811	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×M₁₆	2
C₂×C₃×C₅×#108	2
C₃×C₅×#407	2
C₃×C₅×#341	2
C₅×#1811	3
C₃×#1811	5

Subgroups — C3×C5×#1811 #14274845

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1811 #14274845