Subgroups — C₃×C₅×#1544 #14274578

Order 1920.

404 conjugacy classes of subgroups.

The subgroup lattice has 188 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×#431	C₂	1	2	yes	maximal
3	C₃×C₅×#332	C₂	1	4	yes	maximal, complemented
4	C₅×#1544	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
6	C₃×C₅×#125	C₄	1	2	yes	—
7	C₃×C₅×C₂²⋊Q₈	—	2	2	no	—
8	C₃×C₄×C₅×Q₈	—	2	2	no	—
9	C₃×C₅×#117	C₄	1	2	yes	—
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₃×C₅×C₄⋊C₈	C₂²	1	4	yes	—
12	C₃×C₅×C₂²⋊C₈	—	2	2	no	—
13	C₃×C₄×C₅×C₈	—	2	2	no	—
14	C₃×#1544	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₅×#533	C₂×C₃	1	1	yes	characteristic
16	C₅×#431	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	C₂×C₃×C₅×C₈	—	2	4	no	—
22	C₃×C₅×C₄⋊C₄	—	2	4	no	—
23	C₃×C₅×C₄⋊C₄	—	2	2	no	—
24	C₃×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
25	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
26	C₃×C₄²×C₅	—	2	2	no	—
27	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
28	C₃×C₄²×C₅	D₄	1	2	yes	—
29	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
30	C₃×#533	C₂×C₅	1	1	yes	characteristic
31	C₃×#431	C₂×C₅	1	2	yes	—
32	C₃×#332	C₂×C₅	1	4	yes	complemented
33	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
34	C₅×#125	C₃×C₄	1	2	yes	—
35	C₅×C₂²⋊Q₈	—	2	2	no	—
36	C₄×C₅×Q₈	—	2	2	no	—
37	C₅×#117	C₃×C₄	1	2	yes	—
38	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
39	C₅×C₄⋊C₈	C₂²×C₃	1	4	yes	—
40	C₅×C₂²⋊C₈	—	2	2	no	—
41	C₄×C₅×C₈	—	2	2	no	—
42	#1544	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
43	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
44	C₃×C₅×Q₈	—	4	4	no	—
45	C₂×C₃×C₄×C₅	—	2	2	no	—
46	C₂×C₃×C₄×C₅	—	2	4	no	—
47	C₂×C₃×C₄×C₅	—	2	4	no	—
48	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
49	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
52	C₃×C₅×C₈	—	4	4	no	—
53	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
54	C₃×#125	C₄×C₅	1	2	yes	—
55	C₃×C₂²⋊Q₈	—	2	2	no	—
56	C₃×C₄×Q₈	—	2	2	no	—
57	C₃×#117	C₄×C₅	1	2	yes	—
58	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
59	C₃×C₄⋊C₈	C₂²×C₅	1	4	yes	—
60	C₃×C₂²⋊C₈	—	2	2	no	—
61	C₃×C₄×C₈	—	2	2	no	—
62	C₂×C₅×Q₈	—	2	2	no	—
63	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
65	C₂×C₅×C₈	—	2	4	no	—
66	C₅×C₄⋊C₄	—	2	4	no	—
67	C₅×C₄⋊C₄	—	2	2	no	—
68	C₅×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
69	C₅×C₂²⋊C₄	—	2	2	no	—
70	C₄²×C₅	—	2	2	no	—
71	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
72	C₄²×C₅	C₃×D₄	1	2	yes	—
73	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
74	#533	C₂×C₃×C₅	1	1	yes	characteristic
75	#431	C₂×C₃×C₅	1	2	yes	—
76	#332	C₂×C₃×C₅	1	4	yes	complemented
77	C₂²×C₃×C₅	—	2	1	no	—
78	C₂²×C₃×C₅	—	2	2	no	—
79	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₃×C₄×C₅	—	2	1	no	—
81	C₃×C₄×C₅	—	2	2	no	—
82	C₃×C₄×C₅	—	4	4	no	—
83	C₃×C₄×C₅	#101	1	2	yes	—
84	C₃×C₄×C₅	#135	1	4	yes	—
85	C₂×C₃×Q₈	—	2	2	no	—
86	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
87	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
88	C₂×C₃×C₈	—	2	4	no	—
89	C₃×C₄⋊C₄	—	2	4	no	—
90	C₃×C₄⋊C₄	—	2	2	no	—
91	C₃×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
92	C₃×C₂²⋊C₄	—	2	2	no	—
93	C₃×C₄²	—	2	2	no	—
94	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
95	C₃×C₄²	C₅×D₄	1	2	yes	—
96	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
97	C₂³×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
98	C₅×Q₈	—	4	4	no	—
99	C₂×C₄×C₅	—	2	2	no	—
100	C₂×C₄×C₅	—	2	4	no	—
101	C₂×C₄×C₅	—	2	4	no	—
102	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
103	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
105	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
106	C₅×C₈	—	4	4	no	—
107	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
108	#125	C₃×C₄×C₅	1	2	yes	—
109	C₂²⋊Q₈	—	2	2	no	—
110	C₄×Q₈	—	2	2	no	—
111	#117	C₃×C₄×C₅	1	2	yes	—
112	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
113	C₄⋊C₈	C₂²×C₃×C₅	1	4	yes	—
114	C₂²⋊C₈	—	2	2	no	—
115	C₄×C₈	—	2	2	no	—
116	C₂×C₃×C₅	—	4	1	no	—
117	C₂×C₃×C₅	C₂×#101	1	1	yes	characteristic
118	C₂×C₃×C₅	#416	1	2	yes	—
119	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
120	C₃×Q₈	—	4	4	no	—
121	C₂×C₃×C₄	—	2	4	no	—
122	C₂×C₃×C₄	—	2	2	no	—
123	C₂×C₃×C₄	—	2	4	no	—
124	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
125	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
127	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
128	C₃×C₈	—	4	4	no	—
129	C₂²×C₅	—	2	1	no	—
130	C₂²×C₅	—	2	2	no	—
131	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
132	C₄×C₅	—	2	1	no	—
133	C₄×C₅	—	2	2	no	—
134	C₄×C₅	—	4	4	no	—
135	C₄×C₅	C₃×#101	1	2	yes	—
136	C₄×C₅	C₃×#135	1	4	yes	—
137	C₂×Q₈	—	2	2	no	—
138	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
139	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
140	C₂×C₈	—	2	4	no	—
141	C₄⋊C₄	—	2	4	no	—
142	C₄⋊C₄	—	2	2	no	—
143	C₄⋊C₄	C₂×C₃×C₄×C₅	1	4	yes	—
144	C₂²⋊C₄	—	2	2	no	—
145	C₄²	—	2	2	no	—
146	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
147	C₄²	C₃×C₅×D₄	1	2	yes	—
148	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
149	C₃×C₅	#1544	1	1	yes	characteristic, direct product factor, Hall
150	C₂²×C₃	—	2	1	no	—
151	C₂²×C₃	—	2	2	no	—
152	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
153	C₃×C₄	—	2	1	no	—
154	C₃×C₄	—	2	2	no	—
155	C₃×C₄	—	4	4	no	—
156	C₃×C₄	C₅×#101	1	2	yes	—
157	C₃×C₄	C₅×#135	1	4	yes	—
158	C₂×C₅	—	4	1	no	—
159	C₂×C₅	C₂×C₃×#101	1	1	yes	characteristic
160	C₂×C₅	C₃×#416	1	2	yes	—
161	C₂³	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
162	Q₈	—	4	4	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
167	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
168	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
169	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
170	C₈	—	4	4	no	—
171	C₂×C₃	—	4	1	no	—
172	C₂×C₃	C₂×C₅×#101	1	1	yes	characteristic
173	C₂×C₃	C₅×#416	1	2	yes	—
174	C₅	C₃×#1544	1	1	yes	characteristic, direct product factor, Sylow
175	C₂²	—	2	1	no	—
176	C₂²	—	2	2	no	—
177	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
178	C₄	—	2	1	no	—
179	C₄	—	2	2	no	—
180	C₄	—	4	4	no	—
181	C₄	C₃×C₅×#101	1	2	yes	—
182	C₄	C₃×C₅×#135	1	4	yes	—
183	C₃	C₅×#1544	1	1	yes	characteristic, direct product factor, Sylow
184	C₂	—	4	1	no	—
185	C₂	C₂×C₃×C₅×#101	1	1	yes	characteristic
186	C₂	C₃×C₅×#416	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄×C₅	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₄²	C₂³×C₃×C₅
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂³×C₃×C₅	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄×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×#1544	C₃	direct (N × H)
14. C₃×#1544	C₅	direct (N × H)
17. C₅×#332	C₂×C₃	semidirect
32. C₃×#332	C₂×C₅	semidirect
42. #1544	C₃×C₅	direct (N × H)
76. #332	C₂×C₃×C₅	semidirect
149. C₃×C₅	#1544	direct (N × H)
174. C₅	C₃×#1544	direct (N × H)
183. C₃	C₅×#1544	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×#431	2
C₃×C₅×#332	2
C₅×#1544	3
C₃×#1544	5

Subgroups — C3×C5×#1544 #14274578

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1544 #14274578