Subgroups — C₃×C₅×#1561 #14274595

Order 1920.

420 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#533	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#423	C₂	1	2	yes	maximal
3	C₃×C₅×#332	C₂	1	4	yes	maximal, complemented
4	C₅×#1561	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂×C₃×C₅×M₁₆	—	2	2	no	—
6	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₃×C₅×#125	C₄	1	2	yes	—
8	C₃×C₅×C₂²⋊Q₈	—	2	2	no	—
9	C₃×C₄×C₅×Q₈	—	2	2	no	—
10	C₃×C₅×#117	C₄	1	2	yes	—
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₃×C₅×C₄⋊C₈	—	2	2	no	—
13	C₃×C₅×C₄⋊C₈	C₂²	1	4	yes	—
14	C₃×#1561	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₅×#533	C₂×C₃	1	1	yes	characteristic
16	C₅×#423	C₂×C₃	1	2	yes	—
17	C₅×#332	C₂×C₃	1	4	yes	complemented
18	C₂×C₃×C₅×Q₈	—	2	2	no	—
19	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
21	C₃×C₅×M₁₆	—	4	4	no	—
22	C₂×C₃×C₅×C₈	—	2	4	no	—
23	C₃×C₅×C₄⋊C₄	—	2	4	no	—
24	C₃×C₅×C₄⋊C₄	—	2	2	no	—
25	C₃×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
26	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
27	C₃×C₄²×C₅	—	2	2	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	C₃×#533	C₂×C₅	1	1	yes	characteristic
32	C₃×#423	C₂×C₅	1	2	yes	—
33	C₃×#332	C₂×C₅	1	4	yes	complemented
34	C₂×C₅×M₁₆	—	2	2	no	—
35	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
36	C₅×#125	C₃×C₄	1	2	yes	—
37	C₅×C₂²⋊Q₈	—	2	2	no	—
38	C₄×C₅×Q₈	—	2	2	no	—
39	C₅×#117	C₃×C₄	1	2	yes	—
40	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
41	C₅×C₄⋊C₈	—	2	2	no	—
42	C₅×C₄⋊C₈	C₂²×C₃	1	4	yes	—
43	#1561	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
44	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
45	C₃×C₅×Q₈	—	4	2	no	—
46	C₃×C₅×Q₈	—	4	2	no	—
47	C₂×C₃×C₄×C₅	—	2	2	no	—
48	C₂×C₃×C₄×C₅	—	2	2	no	—
49	C₂×C₃×C₄×C₅	—	2	2	no	—
50	C₂×C₃×C₄×C₅	—	2	4	no	—
51	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
52	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
55	C₃×C₅×C₈	—	4	4	no	—
56	C₂×C₃×M₁₆	—	2	2	no	—
57	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
58	C₃×#125	C₄×C₅	1	2	yes	—
59	C₃×C₂²⋊Q₈	—	2	2	no	—
60	C₃×C₄×Q₈	—	2	2	no	—
61	C₃×#117	C₄×C₅	1	2	yes	—
62	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
63	C₃×C₄⋊C₈	—	2	2	no	—
64	C₃×C₄⋊C₈	C₂²×C₅	1	4	yes	—
65	C₂×C₅×Q₈	—	2	2	no	—
66	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
68	C₅×M₁₆	—	4	4	no	—
69	C₂×C₅×C₈	—	2	4	no	—
70	C₅×C₄⋊C₄	—	2	4	no	—
71	C₅×C₄⋊C₄	—	2	2	no	—
72	C₅×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
73	C₅×C₂²⋊C₄	—	2	2	no	—
74	C₄²×C₅	—	2	2	no	—
75	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
76	C₄²×C₅	C₃×D₄	1	2	yes	—
77	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
78	#533	C₂×C₃×C₅	1	1	yes	characteristic
79	#423	C₂×C₃×C₅	1	2	yes	—
80	#332	C₂×C₃×C₅	1	4	yes	complemented
81	C₂²×C₃×C₅	—	2	1	no	—
82	C₂²×C₃×C₅	—	2	2	no	—
83	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₃×C₄×C₅	—	2	1	no	—
85	C₃×C₄×C₅	—	2	2	no	—
86	C₃×C₄×C₅	—	4	4	no	—
87	C₃×C₄×C₅	#101	1	2	yes	—
88	C₃×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
89	C₃×C₄×C₅	#137	1	2	yes	—
90	C₂×C₃×Q₈	—	2	2	no	—
91	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
92	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
93	C₃×M₁₆	—	4	4	no	—
94	C₂×C₃×C₈	—	2	4	no	—
95	C₃×C₄⋊C₄	—	2	4	no	—
96	C₃×C₄⋊C₄	—	2	2	no	—
97	C₃×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
98	C₃×C₂²⋊C₄	—	2	2	no	—
99	C₃×C₄²	—	2	2	no	—
100	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
101	C₃×C₄²	C₅×D₄	1	2	yes	—
102	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
103	C₂³×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
104	C₅×Q₈	—	4	2	no	—
105	C₅×Q₈	—	4	2	no	—
106	C₂×C₄×C₅	—	2	2	no	—
107	C₂×C₄×C₅	—	2	2	no	—
108	C₂×C₄×C₅	—	2	2	no	—
109	C₂×C₄×C₅	—	2	4	no	—
110	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
111	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
113	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
114	C₅×C₈	—	4	4	no	—
115	C₂×M₁₆	—	2	2	no	—
116	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
117	#125	C₃×C₄×C₅	1	2	yes	—
118	C₂²⋊Q₈	—	2	2	no	—
119	C₄×Q₈	—	2	2	no	—
120	#117	C₃×C₄×C₅	1	2	yes	—
121	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
122	C₄⋊C₈	—	2	2	no	—
123	C₄⋊C₈	C₂²×C₃×C₅	1	4	yes	—
124	C₂×C₃×C₅	—	4	1	no	—
125	C₂×C₃×C₅	C₂×#101	1	1	yes	characteristic
126	C₂×C₃×C₅	#417	1	2	yes	—
127	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
128	C₃×Q₈	—	4	2	no	—
129	C₃×Q₈	—	4	2	no	—
130	C₂×C₃×C₄	—	2	2	no	—
131	C₂×C₃×C₄	—	2	2	no	—
132	C₂×C₃×C₄	—	2	2	no	—
133	C₂×C₃×C₄	—	2	4	no	—
134	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
135	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
137	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
138	C₃×C₈	—	4	4	no	—
139	C₂²×C₅	—	2	1	no	—
140	C₂²×C₅	—	2	2	no	—
141	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
142	C₄×C₅	—	2	1	no	—
143	C₄×C₅	—	2	2	no	—
144	C₄×C₅	—	4	4	no	—
145	C₄×C₅	C₃×#101	1	2	yes	—
146	C₄×C₅	C₃×AGL(1,ℤ/8)	1	2	yes	—
147	C₄×C₅	C₃×#137	1	2	yes	—
148	C₂×Q₈	—	2	2	no	—
149	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
150	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
151	M₁₆	—	4	4	no	—
152	C₂×C₈	—	2	4	no	—
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	C₂×C₃×C₄×C₅	1	4	yes	—
156	C₂²⋊C₄	—	2	2	no	—
157	C₄²	—	2	2	no	—
158	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
159	C₄²	C₃×C₅×D₄	1	2	yes	—
160	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
161	C₃×C₅	#1561	1	1	yes	characteristic, direct product factor, Hall
162	C₂²×C₃	—	2	1	no	—
163	C₂²×C₃	—	2	2	no	—
164	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
165	C₃×C₄	—	2	1	no	—
166	C₃×C₄	—	2	2	no	—
167	C₃×C₄	—	4	4	no	—
168	C₃×C₄	C₅×#101	1	2	yes	—
169	C₃×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
170	C₃×C₄	C₅×#137	1	2	yes	—
171	C₂×C₅	—	4	1	no	—
172	C₂×C₅	C₂×C₃×#101	1	1	yes	characteristic
173	C₂×C₅	C₃×#417	1	2	yes	—
174	C₂³	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
175	Q₈	—	4	2	no	—
176	Q₈	—	4	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
182	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
183	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
184	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
185	C₈	—	4	4	no	—
186	C₂×C₃	—	4	1	no	—
187	C₂×C₃	C₂×C₅×#101	1	1	yes	characteristic
188	C₂×C₃	C₅×#417	1	2	yes	—
189	C₅	C₃×#1561	1	1	yes	characteristic, direct product factor, Sylow
190	C₂²	—	2	1	no	—
191	C₂²	—	2	2	no	—
192	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
193	C₄	—	2	1	no	—
194	C₄	—	2	2	no	—
195	C₄	—	4	4	no	—
196	C₄	C₃×C₅×#101	1	2	yes	—
197	C₄	C₃×C₅×AGL(1,ℤ/8)	1	2	yes	—
198	C₄	C₃×C₅×#137	1	2	yes	—
199	C₃	C₅×#1561	1	1	yes	characteristic, direct product factor, Sylow
200	C₂	—	4	1	no	—
201	C₂	C₂×C₃×C₅×#101	1	1	yes	characteristic
202	C₂	C₃×C₅×#417	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	C₂³×C₃×C₅	C₂²⋊C₄
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
3. C₃×C₅×#332	C₂	semidirect
4. C₅×#1561	C₃	direct (N × H)
14. C₃×#1561	C₅	direct (N × H)
17. C₅×#332	C₂×C₃	semidirect
33. C₃×#332	C₂×C₅	semidirect
43. #1561	C₃×C₅	direct (N × H)
80. #332	C₂×C₃×C₅	semidirect
161. C₃×C₅	#1561	direct (N × H)
189. C₅	C₃×#1561	direct (N × H)
199. C₃	C₅×#1561	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅×#533	2
C₃×C₅×#423	2
C₃×C₅×#332	2
C₅×#1561	3
C₃×#1561	5

Subgroups — C3×C5×#1561 #14274595

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1561 #14274595