Subgroups — C₃×C₅×#1555 #14274589

Order 1920.

664 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#127	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#432	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#331	C₂	1	4	yes	maximal, complemented
4	C₅×#1555	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂²×C₃×C₅×D₄	—	2	2	no	—
6	C₂²×C₃×C₅×D₄	C₄	1	2	yes	—
7	C₃×C₅×#127	—	2	2	no	—
8	C₃×C₅×#127	C₄	1	2	yes	—
9	C₃×C₅×#127	C₂²	1	2	yes	—
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₃×C₅×C₄⋊C₈	C₂²	1	4	yes	complemented
12	C₃×C₅×C₂²⋊C₈	—	2	2	no	—
13	C₃×C₅×#97	—	2	2	no	—
14	C₃×#1555	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂×C₅×#127	C₂×C₃	1	1	yes	characteristic
16	C₅×#432	C₂×C₃	1	2	yes	complemented
17	C₅×#331	C₂×C₃	1	4	yes	complemented
18	C₂⁴×C₃×C₅	—	2	2	no	—
19	C₂×C₃×C₅×D₄	—	2	4	no	—
20	C₂×C₃×C₅×D₄	—	2	2	no	—
21	C₂×C₃×C₅×D₄	—	2	4	no	—
22	C₂×C₃×C₅×D₄	—	4	8	no	—
23	C₂×C₃×C₅×D₄	—	4	4	no	—
24	C₂×C₃×C₅×D₄	C₂×C₄	1	4	yes	—
25	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
27	C₂×C₃×C₅×C₈	—	2	4	no	—
28	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
29	C₃×C₄²×C₅	D₄	1	2	yes	—
30	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
31	C₂×C₃×#127	C₂×C₅	1	1	yes	characteristic
32	C₃×#432	C₂×C₅	1	2	yes	complemented
33	C₃×#331	C₂×C₅	1	4	yes	complemented
34	C₂²×C₅×D₄	—	2	2	no	—
35	C₂²×C₅×D₄	C₃×C₄	1	2	yes	—
36	C₅×#127	—	2	2	no	—
37	C₅×#127	C₃×C₄	1	2	yes	—
38	C₅×#127	C₂²×C₃	1	2	yes	—
39	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
40	C₅×C₄⋊C₈	C₂²×C₃	1	4	yes	complemented
41	C₅×C₂²⋊C₈	—	2	2	no	—
42	C₅×#97	—	2	2	no	—
43	#1555	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
44	C₂³×C₃×C₅	—	2	4	no	—
45	C₂³×C₃×C₅	—	4	8	no	—
46	C₂³×C₃×C₅	—	4	4	no	—
47	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
48	C₃×C₅×D₄	—	4	4	no	—
49	C₃×C₅×D₄	—	4	8	no	—
50	C₃×C₅×D₄	—	4	4	no	—
51	C₃×C₅×D₄	—	4	8	no	—
52	C₂×C₃×C₄×C₅	—	2	2	no	—
53	C₂×C₃×C₄×C₅	—	2	4	no	—
54	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
55	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
58	C₃×C₅×C₈	—	4	4	no	—
59	C₂²×C₃×D₄	—	2	2	no	—
60	C₂²×C₃×D₄	C₄×C₅	1	2	yes	—
61	C₃×#127	—	2	2	no	—
62	C₃×#127	C₄×C₅	1	2	yes	—
63	C₃×#127	C₂²×C₅	1	2	yes	—
64	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
65	C₃×C₄⋊C₈	C₂²×C₅	1	4	yes	complemented
66	C₃×C₂²⋊C₈	—	2	2	no	—
67	C₃×#97	—	2	2	no	—
68	C₂⁴×C₅	—	2	2	no	—
69	C₂×C₅×D₄	—	2	2	no	—
70	C₂×C₅×D₄	—	2	4	no	—
71	C₂×C₅×D₄	—	2	4	no	—
72	C₂×C₅×D₄	—	4	4	no	—
73	C₂×C₅×D₄	—	4	8	no	—
74	C₂×C₅×D₄	C₂×C₃×C₄	1	4	yes	—
75	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
77	C₂×C₅×C₈	—	2	4	no	—
78	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
79	C₄²×C₅	C₃×D₄	1	2	yes	—
80	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
81	C₂×#127	C₂×C₃×C₅	1	1	yes	characteristic
82	#432	C₂×C₃×C₅	1	2	yes	complemented
83	#331	C₂×C₃×C₅	1	4	yes	complemented
84	C₂²×C₃×C₅	—	2	2	no	—
85	C₂²×C₃×C₅	—	4	4	no	—
86	C₂²×C₃×C₅	—	4	8	no	—
87	C₂²×C₃×C₅	—	8	8	no	—
88	C₂²×C₃×C₅	#100	1	2	yes	—
89	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₃×C₄×C₅	—	2	1	no	—
91	C₃×C₄×C₅	—	2	1	no	—
92	C₃×C₄×C₅	—	2	2	no	—
93	C₃×C₄×C₅	AGL(1,ℤ/8)	1	4	yes	—
94	C₂⁴×C₃	—	2	2	no	—
95	C₂×C₃×D₄	—	2	2	no	—
96	C₂×C₃×D₄	—	2	4	no	—
97	C₂×C₃×D₄	—	2	4	no	—
98	C₂×C₃×D₄	—	4	4	no	—
99	C₂×C₃×D₄	—	4	8	no	—
100	C₂×C₃×D₄	C₂×C₄×C₅	1	4	yes	—
101	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
102	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
103	C₂×C₃×C₈	—	2	4	no	—
104	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
105	C₃×C₄²	C₅×D₄	1	2	yes	—
106	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
107	C₂³×C₅	—	2	4	no	—
108	C₂³×C₅	—	4	8	no	—
109	C₂³×C₅	—	4	4	no	—
110	C₂³×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
111	C₅×D₄	—	4	4	no	—
112	C₅×D₄	—	4	4	no	—
113	C₅×D₄	—	4	8	no	—
114	C₅×D₄	—	4	8	no	—
115	C₂×C₄×C₅	—	2	2	no	—
116	C₂×C₄×C₅	—	2	4	no	—
117	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
118	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
120	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
121	C₅×C₈	—	4	4	no	—
122	C₂²×D₄	—	2	2	no	—
123	C₂²×D₄	C₃×C₄×C₅	1	2	yes	—
124	#127	—	2	2	no	—
125	#127	C₃×C₄×C₅	1	2	yes	—
126	#127	C₂²×C₃×C₅	1	2	yes	—
127	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
128	C₄⋊C₈	C₂²×C₃×C₅	1	4	yes	complemented
129	C₂²⋊C₈	—	2	2	no	—
130	#97	—	2	2	no	—
131	C₂×C₃×C₅	—	2	2	no	—
132	C₂×C₃×C₅	—	8	4	no	—
133	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
134	C₂×C₃×C₅	#418	1	2	yes	—
135	C₂³×C₃	—	2	4	no	—
136	C₂³×C₃	—	4	8	no	—
137	C₂³×C₃	—	4	4	no	—
138	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
139	C₃×D₄	—	4	4	no	—
140	C₃×D₄	—	4	4	no	—
141	C₃×D₄	—	4	8	no	—
142	C₃×D₄	—	4	8	no	—
143	C₂×C₃×C₄	—	2	2	no	—
144	C₂×C₃×C₄	—	2	4	no	—
145	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
146	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
147	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
148	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
149	C₃×C₈	—	4	4	no	—
150	C₂²×C₅	—	2	2	no	—
151	C₂²×C₅	—	4	4	no	—
152	C₂²×C₅	—	4	8	no	—
153	C₂²×C₅	—	8	8	no	—
154	C₂²×C₅	C₃×#100	1	2	yes	—
155	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
156	C₄×C₅	—	2	1	no	—
157	C₄×C₅	—	2	1	no	—
158	C₄×C₅	—	2	2	no	—
159	C₄×C₅	C₃×AGL(1,ℤ/8)	1	4	yes	—
160	C₂⁴	—	2	2	no	—
161	C₂×D₄	—	2	2	no	—
162	C₂×D₄	—	2	4	no	—
163	C₂×D₄	—	2	4	no	—
164	C₂×D₄	—	4	4	no	—
165	C₂×D₄	—	4	8	no	—
166	C₂×D₄	C₂×C₃×C₄×C₅	1	4	yes	—
167	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
168	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
169	C₂×C₈	—	2	4	no	—
170	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
171	C₄²	C₃×C₅×D₄	1	2	yes	—
172	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
173	C₃×C₅	#1555	1	1	yes	characteristic, direct product factor, Hall
174	C₂²×C₃	—	2	2	no	—
175	C₂²×C₃	—	4	4	no	—
176	C₂²×C₃	—	4	8	no	—
177	C₂²×C₃	—	8	8	no	—
178	C₂²×C₃	C₅×#100	1	2	yes	—
179	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
180	C₃×C₄	—	2	1	no	—
181	C₃×C₄	—	2	1	no	—
182	C₃×C₄	—	2	2	no	—
183	C₃×C₄	C₅×AGL(1,ℤ/8)	1	4	yes	—
184	C₂×C₅	—	2	2	no	—
185	C₂×C₅	—	8	4	no	—
186	C₂×C₅	C₂×C₃×#100	1	1	yes	characteristic
187	C₂×C₅	C₃×#418	1	2	yes	—
188	C₂³	—	2	4	no	—
189	C₂³	—	4	8	no	—
190	C₂³	—	4	4	no	—
191	C₂³	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
192	D₄	—	4	4	no	—
193	D₄	—	4	4	no	—
194	D₄	—	4	8	no	—
195	D₄	—	4	8	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
199	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
200	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
201	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
202	C₈	—	4	4	no	—
203	C₂×C₃	—	2	2	no	—
204	C₂×C₃	—	8	4	no	—
205	C₂×C₃	C₂×C₅×#100	1	1	yes	characteristic
206	C₂×C₃	C₅×#418	1	2	yes	—
207	C₅	C₃×#1555	1	1	yes	characteristic, direct product factor, Sylow
208	C₂²	—	2	2	no	—
209	C₂²	—	4	4	no	—
210	C₂²	—	4	8	no	—
211	C₂²	—	8	8	no	—
212	C₂²	C₃×C₅×#100	1	2	yes	—
213	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	2	no	—
217	C₄	C₃×C₅×AGL(1,ℤ/8)	1	4	yes	—
218	C₃	C₅×#1555	1	1	yes	characteristic, direct product factor, Sylow
219	C₂	—	2	2	no	—
220	C₂	—	8	4	no	—
221	C₂	C₂×C₃×C₅×#100	1	1	yes	characteristic
222	C₂	C₃×C₅×#418	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₄²	C₂³×C₃×C₅
prime radical	C₂×C₃×C₅×#127	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₂×C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#432	C₂	semidirect
3. C₃×C₅×#331	C₂	semidirect
4. C₅×#1555	C₃	direct (N × H)
11. C₃×C₅×C₄⋊C₈	C₂²	semidirect
14. C₃×#1555	C₅	direct (N × H)
16. C₅×#432	C₂×C₃	semidirect
17. C₅×#331	C₂×C₃	semidirect
32. C₃×#432	C₂×C₅	semidirect
33. C₃×#331	C₂×C₅	semidirect
40. C₅×C₄⋊C₈	C₂²×C₃	semidirect
43. #1555	C₃×C₅	direct (N × H)
65. C₃×C₄⋊C₈	C₂²×C₅	semidirect
82. #432	C₂×C₃×C₅	semidirect
83. #331	C₂×C₃×C₅	semidirect
128. C₄⋊C₈	C₂²×C₃×C₅	semidirect
173. C₃×C₅	#1555	direct (N × H)
207. C₅	C₃×#1555	direct (N × H)
218. C₃	C₅×#1555	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×#127	2
C₃×C₅×#432	2
C₃×C₅×#331	2
C₅×#1555	3
C₃×#1555	5

Subgroups — C3×C5×#1555 #14274589

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1555 #14274589