Subgroups — C₃×C₅×#1818 #14274852

Order 1920.

584 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#117	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×C₂²⋊C₈	C₂	1	2	yes	maximal
3	C₃×C₅×#336	C₂	1	4	yes	maximal, complemented
4	C₅×#1818	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
6	C₂²×C₃×C₅×C₈	C₂²	1	4	yes	—
7	C₃×C₅×#117	—	2	4	no	—
8	C₂×C₃×C₅×C₄⋊C₄	C₄	1	2	yes	—
9	C₂×C₃×C₅×C₂²⋊C₄	C₄	1	2	yes	—
10	C₂×C₃×C₄²×C₅	C₂²	1	2	yes	—
11	C₃×C₅×C₂²⋊C₈	—	2	4	no	—
12	C₃×#1818	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₅×#117	C₂×C₃	1	1	yes	characteristic
14	C₂×C₅×C₂²⋊C₈	C₂×C₃	1	2	yes	—
15	C₅×#336	C₂×C₃	1	4	yes	complemented
16	C₂⁴×C₃×C₅	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₃×C₄×C₅	—	2	4	no	—
18	C₂²×C₃×C₄×C₅	C₂×C₄	1	4	yes	—
19	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₃×C₄×C₅	D₄	1	4	yes	—
21	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
22	C₂×C₃×C₅×C₈	—	2	4	no	—
23	C₂×C₃×C₅×C₈	—	2	8	no	—
24	C₃×C₅×C₄⋊C₄	—	2	4	no	—
25	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
26	C₃×C₄²×C₅	—	2	4	no	—
27	C₂×C₃×#117	C₂×C₅	1	1	yes	characteristic
28	C₂×C₃×C₂²⋊C₈	C₂×C₅	1	2	yes	—
29	C₃×#336	C₂×C₅	1	4	yes	complemented
30	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
31	C₂²×C₅×C₈	C₂²×C₃	1	4	yes	—
32	C₅×#117	—	2	4	no	—
33	C₂×C₅×C₄⋊C₄	C₃×C₄	1	2	yes	—
34	C₂×C₅×C₂²⋊C₄	C₃×C₄	1	2	yes	—
35	C₂×C₄²×C₅	C₂²×C₃	1	2	yes	—
36	C₅×C₂²⋊C₈	—	2	4	no	—
37	#1818	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
38	C₂³×C₃×C₅	—	2	2	no	—
39	C₂³×C₃×C₅	—	2	4	no	—
40	C₂³×C₃×C₅	C₂²⋊C₄	1	2	yes	—
41	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₃×C₄×C₅	—	2	2	no	—
43	C₂×C₃×C₄×C₅	—	2	8	no	—
44	C₂×C₃×C₄×C₅	—	2	4	no	—
45	C₂×C₃×C₄×C₅	—	2	8	no	—
46	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
47	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
48	C₂×C₃×C₄×C₅	#41	1	4	yes	—
49	C₃×C₅×C₈	—	4	4	no	—
50	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
51	C₂²×C₃×C₈	C₂²×C₅	1	4	yes	—
52	C₃×#117	—	2	4	no	—
53	C₂×C₃×C₄⋊C₄	C₄×C₅	1	2	yes	—
54	C₂×C₃×C₂²⋊C₄	C₄×C₅	1	2	yes	—
55	C₂×C₃×C₄²	C₂²×C₅	1	2	yes	—
56	C₃×C₂²⋊C₈	—	2	4	no	—
57	C₂⁴×C₅	C₂×C₃×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅	—	2	4	no	—
59	C₂²×C₄×C₅	C₂×C₃×C₄	1	4	yes	—
60	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₃×D₄	1	4	yes	—
62	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
63	C₂×C₅×C₈	—	2	4	no	—
64	C₂×C₅×C₈	—	2	8	no	—
65	C₅×C₄⋊C₄	—	2	4	no	—
66	C₅×C₂²⋊C₄	—	2	4	no	—
67	C₄²×C₅	—	2	4	no	—
68	C₂×#117	C₂×C₃×C₅	1	1	yes	characteristic
69	C₂×C₂²⋊C₈	C₂×C₃×C₅	1	2	yes	—
70	#336	C₂×C₃×C₅	1	4	yes	complemented
71	C₂²×C₃×C₅	—	2	2	no	—
72	C₂²×C₃×C₅	—	2	4	no	—
73	C₂²×C₃×C₅	—	4	4	no	—
74	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₃×C₅	#117	1	2	yes	—
76	C₂²×C₃×C₅	#131	1	4	yes	—
77	C₃×C₄×C₅	—	2	2	no	—
78	C₃×C₄×C₅	—	4	4	no	—
79	C₃×C₄×C₅	#123	1	4	yes	—
80	C₂⁴×C₃	C₂×C₄×C₅	1	1	yes	characteristic
81	C₂²×C₃×C₄	—	2	4	no	—
82	C₂²×C₃×C₄	C₂×C₄×C₅	1	4	yes	—
83	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
84	C₂²×C₃×C₄	C₅×D₄	1	4	yes	—
85	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
86	C₂×C₃×C₈	—	2	4	no	—
87	C₂×C₃×C₈	—	2	8	no	—
88	C₃×C₄⋊C₄	—	2	4	no	—
89	C₃×C₂²⋊C₄	—	2	4	no	—
90	C₃×C₄²	—	2	4	no	—
91	C₂³×C₅	—	2	2	no	—
92	C₂³×C₅	—	2	4	no	—
93	C₂³×C₅	C₃×C₂²⋊C₄	1	2	yes	—
94	C₂³×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
95	C₂×C₄×C₅	—	2	2	no	—
96	C₂×C₄×C₅	—	2	8	no	—
97	C₂×C₄×C₅	—	2	4	no	—
98	C₂×C₄×C₅	—	2	8	no	—
99	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
100	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
101	C₂×C₄×C₅	C₃×#41	1	4	yes	—
102	C₅×C₈	—	4	4	no	—
103	C₂³×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
104	C₂²×C₈	C₂²×C₃×C₅	1	4	yes	—
105	#117	—	2	4	no	—
106	C₂×C₄⋊C₄	C₃×C₄×C₅	1	2	yes	—
107	C₂×C₂²⋊C₄	C₃×C₄×C₅	1	2	yes	—
108	C₂×C₄²	C₂²×C₃×C₅	1	2	yes	—
109	C₂²⋊C₈	—	2	4	no	—
110	C₂×C₃×C₅	—	4	2	no	—
111	C₂×C₃×C₅	#381	1	1	yes	characteristic
112	C₂×C₃×C₅	#408	1	2	yes	—
113	C₂×C₃×C₅	#433	1	4	yes	—
114	C₂³×C₃	—	2	2	no	—
115	C₂³×C₃	—	2	4	no	—
116	C₂³×C₃	C₅×C₂²⋊C₄	1	2	yes	—
117	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
118	C₂×C₃×C₄	—	2	2	no	—
119	C₂×C₃×C₄	—	2	8	no	—
120	C₂×C₃×C₄	—	2	4	no	—
121	C₂×C₃×C₄	—	2	8	no	—
122	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
123	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
124	C₂×C₃×C₄	C₅×#41	1	4	yes	—
125	C₃×C₈	—	4	4	no	—
126	C₂²×C₅	—	2	2	no	—
127	C₂²×C₅	—	2	4	no	—
128	C₂²×C₅	—	4	4	no	—
129	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
130	C₂²×C₅	C₃×#117	1	2	yes	—
131	C₂²×C₅	C₃×#131	1	4	yes	—
132	C₄×C₅	—	2	2	no	—
133	C₄×C₅	—	4	4	no	—
134	C₄×C₅	C₃×#123	1	4	yes	—
135	C₂⁴	C₂×C₃×C₄×C₅	1	1	yes	characteristic
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	C₂×C₃×C₄×C₅	1	4	yes	—
138	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
139	C₂²×C₄	C₃×C₅×D₄	1	4	yes	—
140	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	—	2	8	no	—
143	C₄⋊C₄	—	2	4	no	—
144	C₂²⋊C₄	—	2	4	no	—
145	C₄²	—	2	4	no	—
146	C₃×C₅	#1818	1	1	yes	characteristic, direct product factor, Hall
147	C₂²×C₃	—	2	2	no	—
148	C₂²×C₃	—	2	4	no	—
149	C₂²×C₃	—	4	4	no	—
150	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
151	C₂²×C₃	C₅×#117	1	2	yes	—
152	C₂²×C₃	C₅×#131	1	4	yes	—
153	C₃×C₄	—	2	2	no	—
154	C₃×C₄	—	4	4	no	—
155	C₃×C₄	C₅×#123	1	4	yes	—
156	C₂×C₅	—	4	2	no	—
157	C₂×C₅	C₃×#381	1	1	yes	characteristic
158	C₂×C₅	C₃×#408	1	2	yes	—
159	C₂×C₅	C₃×#433	1	4	yes	—
160	C₂³	—	2	2	no	—
161	C₂³	—	2	4	no	—
162	C₂³	C₃×C₅×C₂²⋊C₄	1	2	yes	—
163	C₂³	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	8	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	8	no	—
168	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
169	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
170	C₂×C₄	C₃×C₅×#41	1	4	yes	—
171	C₈	—	4	4	no	—
172	C₂×C₃	—	4	2	no	—
173	C₂×C₃	C₅×#381	1	1	yes	characteristic
174	C₂×C₃	C₅×#408	1	2	yes	—
175	C₂×C₃	C₅×#433	1	4	yes	—
176	C₅	C₃×#1818	1	1	yes	characteristic, direct product factor, Sylow
177	C₂²	—	2	2	no	—
178	C₂²	—	2	4	no	—
179	C₂²	—	4	4	no	—
180	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
181	C₂²	C₃×C₅×#117	1	2	yes	—
182	C₂²	C₃×C₅×#131	1	4	yes	—
183	C₄	—	2	2	no	—
184	C₄	—	4	4	no	—
185	C₄	C₃×C₅×#123	1	4	yes	—
186	C₃	C₅×#1818	1	1	yes	characteristic, direct product factor, Sylow
187	C₂	—	4	2	no	—
188	C₂	C₃×C₅×#381	1	1	yes	characteristic
189	C₂	C₃×C₅×#408	1	2	yes	—
190	C₂	C₃×C₅×#433	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×#336	C₂	semidirect
4. C₅×#1818	C₃	direct (N × H)
12. C₃×#1818	C₅	direct (N × H)
15. C₅×#336	C₂×C₃	semidirect
29. C₃×#336	C₂×C₅	semidirect
37. #1818	C₃×C₅	direct (N × H)
70. #336	C₂×C₃×C₅	semidirect
146. C₃×C₅	#1818	direct (N × H)
176. C₅	C₃×#1818	direct (N × H)
186. C₃	C₅×#1818	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×C₅×#117	2
C₂×C₃×C₅×C₂²⋊C₈	2
C₃×C₅×#336	2
C₅×#1818	3
C₃×#1818	5

Subgroups — C3×C5×#1818 #14274852

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1818 #14274852