Subgroups — C₃×C₅×#1848 #14274882

Order 1920.

440 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#108	C₂	1	4	yes	maximal
2	C₃×C₅×#423	C₂	1	2	yes	maximal
3	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
4	C₅×#1848	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂×C₃×C₅×M₁₆	C₂²	1	4	yes	—
6	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₃×C₅×#108	—	2	8	no	—
9	C₃×C₅×C₄⋊C₈	—	2	4	no	—
10	C₃×C₄×C₅×C₈	C₄	1	2	yes	—
11	C₃×C₄×C₅×C₈	C₄	1	2	yes	—
12	C₃×#1848	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₅×#108	C₂×C₃	1	4	yes	—
14	C₅×#423	C₂×C₃	1	2	yes	—
15	C₂×C₄×C₅×C₈	C₂×C₃	1	1	yes	characteristic
16	C₂²×C₃×C₄×C₅	Q₈	1	2	yes	—
17	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
18	C₃×C₅×M₁₆	—	2	8	no	—
19	C₂×C₃×C₅×C₈	—	2	4	no	—
20	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	—
21	C₂×C₃×C₅×C₈	D₄	1	4	yes	—
22	C₂×C₃×C₅×C₈	Q₈	1	4	yes	—
23	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
24	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
25	C₃×C₄²×C₅	D₄	1	2	yes	—
26	C₂×C₃×#108	C₂×C₅	1	4	yes	—
27	C₃×#423	C₂×C₅	1	2	yes	—
28	C₂×C₃×C₄×C₈	C₂×C₅	1	1	yes	characteristic
29	C₂×C₅×M₁₆	C₂²×C₃	1	4	yes	—
30	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
31	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
32	C₅×#108	—	2	8	no	—
33	C₅×C₄⋊C₈	—	2	4	no	—
34	C₄×C₅×C₈	C₃×C₄	1	2	yes	—
35	C₄×C₅×C₈	C₃×C₄	1	2	yes	—
36	#1848	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
37	C₂³×C₃×C₅	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₃×C₄×C₅	—	2	4	no	—
39	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
40	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
41	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
43	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
44	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
45	C₃×C₅×C₈	—	4	4	no	—
46	C₃×C₅×C₈	C₄⋊C₄	1	8	yes	—
47	C₂×C₃×M₁₆	C₂²×C₅	1	4	yes	—
48	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
49	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
50	C₃×#108	—	2	8	no	—
51	C₃×C₄⋊C₈	—	2	4	no	—
52	C₃×C₄×C₈	C₄×C₅	1	2	yes	—
53	C₃×C₄×C₈	C₄×C₅	1	2	yes	—
54	C₂²×C₄×C₅	C₃×Q₈	1	2	yes	—
55	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
56	C₅×M₁₆	—	2	8	no	—
57	C₂×C₅×C₈	—	2	4	no	—
58	C₂×C₅×C₈	C₂×C₃×C₄	1	4	yes	—
59	C₂×C₅×C₈	C₃×D₄	1	4	yes	—
60	C₂×C₅×C₈	C₃×Q₈	1	4	yes	—
61	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
62	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
63	C₄²×C₅	C₃×D₄	1	2	yes	—
64	C₂×#108	C₂×C₃×C₅	1	4	yes	—
65	#423	C₂×C₃×C₅	1	2	yes	—
66	C₂×C₄×C₈	C₂×C₃×C₅	1	1	yes	characteristic
67	C₂²×C₃×C₅	—	2	2	no	—
68	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₃×C₅	C₄⋊Q₈	1	2	yes	—
70	C₃×C₄×C₅	—	2	2	no	—
71	C₃×C₄×C₅	#108	1	4	yes	complemented
72	C₃×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
73	C₃×C₄×C₅	#127	1	1	yes	characteristic
74	C₃×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
75	C₂²×C₃×C₄	C₅×Q₈	1	2	yes	—
76	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
77	C₃×M₁₆	—	2	8	no	—
78	C₂×C₃×C₈	—	2	4	no	—
79	C₂×C₃×C₈	C₂×C₄×C₅	1	4	yes	—
80	C₂×C₃×C₈	C₅×D₄	1	4	yes	—
81	C₂×C₃×C₈	C₅×Q₈	1	4	yes	—
82	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
83	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
84	C₃×C₄²	C₅×D₄	1	2	yes	—
85	C₂³×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
86	C₂×C₄×C₅	—	2	4	no	—
87	C₂×C₄×C₅	C₃×C₄⋊C₄	1	2	yes	—
88	C₂×C₄×C₅	C₃×C₄⋊C₄	1	2	yes	—
89	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
90	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
91	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
92	C₂×C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
93	C₅×C₈	—	4	4	no	—
94	C₅×C₈	C₃×C₄⋊C₄	1	8	yes	—
95	C₂×M₁₆	C₂²×C₃×C₅	1	4	yes	—
96	C₂²×C₈	C₂²×C₃×C₅	1	2	yes	—
97	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
98	#108	—	2	8	no	—
99	C₄⋊C₈	—	2	4	no	—
100	C₄×C₈	C₃×C₄×C₅	1	2	yes	—
101	C₄×C₈	C₃×C₄×C₅	1	2	yes	—
102	C₂×C₃×C₅	—	2	2	no	—
103	C₂×C₃×C₅	#384	1	1	yes	characteristic
104	C₂×C₃×C₅	C₂×#108	1	2	yes	—
105	C₂³×C₃	C₂×C₅×Q₈	1	1	yes	characteristic
106	C₂×C₃×C₄	—	2	4	no	—
107	C₂×C₃×C₄	C₅×C₄⋊C₄	1	2	yes	—
108	C₂×C₃×C₄	C₅×C₄⋊C₄	1	2	yes	—
109	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
110	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
111	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
112	C₂×C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
113	C₃×C₈	—	4	4	no	—
114	C₃×C₈	C₅×C₄⋊C₄	1	8	yes	—
115	C₂²×C₅	—	2	2	no	—
116	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
117	C₂²×C₅	C₃×C₄⋊Q₈	1	2	yes	—
118	C₄×C₅	—	2	2	no	—
119	C₄×C₅	C₃×#108	1	4	yes	complemented
120	C₄×C₅	C₂×C₃×C₄⋊C₄	1	2	yes	—
121	C₄×C₅	C₃×#127	1	1	yes	characteristic
122	C₄×C₅	C₃×C₄⋊Q₈	1	1	yes	characteristic
123	C₂²×C₄	C₃×C₅×Q₈	1	2	yes	—
124	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
125	M₁₆	—	2	8	no	—
126	C₂×C₈	—	2	4	no	—
127	C₂×C₈	C₂×C₃×C₄×C₅	1	4	yes	—
128	C₂×C₈	C₃×C₅×D₄	1	4	yes	—
129	C₂×C₈	C₃×C₅×Q₈	1	4	yes	—
130	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
131	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
132	C₄²	C₃×C₅×D₄	1	2	yes	—
133	C₃×C₅	#1848	1	1	yes	characteristic, direct product factor, Hall
134	C₂²×C₃	—	2	2	no	—
135	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
136	C₂²×C₃	C₅×C₄⋊Q₈	1	2	yes	—
137	C₃×C₄	—	2	2	no	—
138	C₃×C₄	C₅×#108	1	4	yes	complemented
139	C₃×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
140	C₃×C₄	C₅×#127	1	1	yes	characteristic
141	C₃×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
142	C₂×C₅	—	2	2	no	—
143	C₂×C₅	C₃×#384	1	1	yes	characteristic
144	C₂×C₅	C₂×C₃×#108	1	2	yes	—
145	C₂³	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	C₃×C₅×C₄⋊C₄	1	2	yes	—
148	C₂×C₄	C₃×C₅×C₄⋊C₄	1	2	yes	—
149	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
150	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
151	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
152	C₂×C₄	C₂×C₃×C₅×Q₈	1	2	yes	—
153	C₈	—	4	4	no	—
154	C₈	C₃×C₅×C₄⋊C₄	1	8	yes	—
155	C₂×C₃	—	2	2	no	—
156	C₂×C₃	C₅×#384	1	1	yes	characteristic
157	C₂×C₃	C₂×C₅×#108	1	2	yes	—
158	C₅	C₃×#1848	1	1	yes	characteristic, direct product factor, Sylow
159	C₂²	—	2	2	no	—
160	C₂²	C₂×C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
161	C₂²	C₃×C₅×C₄⋊Q₈	1	2	yes	—
162	C₄	—	2	2	no	—
163	C₄	C₃×C₅×#108	1	4	yes	complemented
164	C₄	C₂×C₃×C₅×C₄⋊C₄	1	2	yes	—
165	C₄	C₃×C₅×#127	1	1	yes	characteristic
166	C₄	C₃×C₅×C₄⋊Q₈	1	1	yes	characteristic
167	C₃	C₅×#1848	1	1	yes	characteristic, direct product factor, Sylow
168	C₂	—	2	2	no	—
169	C₂	C₃×C₅×#384	1	1	yes	characteristic
170	C₂	C₂×C₃×C₅×#108	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	C₂³×C₃×C₅	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ 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
4. C₅×#1848	C₃	direct (N × H)
12. C₃×#1848	C₅	direct (N × H)
36. #1848	C₃×C₅	direct (N × H)
71. C₃×C₄×C₅	#108	semidirect
119. C₄×C₅	C₃×#108	semidirect
133. C₃×C₅	#1848	direct (N × H)
138. C₃×C₄	C₅×#108	semidirect
158. C₅	C₃×#1848	direct (N × H)
163. C₄	C₃×C₅×#108	semidirect
167. C₃	C₅×#1848	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×#108	2
C₃×C₅×#423	2
C₂×C₃×C₄×C₅×C₈	2
C₅×#1848	3
C₃×#1848	5

Subgroups — C3×C5×#1848 #14274882

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1848 #14274882