Subgroups — C₃×C₅×#2110 #14275144

Order 1920.

320 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×M₃₂	C₂	1	2	yes	maximal
2	C₃×C₅×#405	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#347	C₂	1	4	yes	maximal, complemented
4	C₅×#2110	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂×C₃×C₅×M₁₆	C₄	1	1	yes	characteristic
6	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₃×C₅×#117	C₄	1	1	yes	characteristic
8	C₃×C₅×M₃₂	C₄	1	4	yes	complemented
9	C₃×C₅×M₃₂	C₂²	1	4	yes	—
10	C₂×C₃×C₅×C₁₆	C₄	1	4	yes	complemented
11	C₃×C₅×#97	C₂²	1	2	yes	—
12	C₃×C₄×C₅×C₈	C₄	1	2	yes	—
13	C₃×#2110	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂×C₅×M₃₂	C₂×C₃	1	2	yes	—
15	C₅×#405	C₂×C₃	1	1	yes	characteristic
16	C₅×#347	C₂×C₃	1	4	yes	complemented
17	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
18	C₃×C₅×M₁₆	—	2	2	no	—
19	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
20	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
22	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
23	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
24	C₃×C₅×C₄⋊C₄	—	2	1	no	—
25	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
26	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
27	C₃×C₅×C₁₆	C₂×C₄	1	8	yes	—
28	C₂×C₃×M₃₂	C₂×C₅	1	2	yes	—
29	C₃×#405	C₂×C₅	1	1	yes	characteristic
30	C₃×#347	C₂×C₅	1	4	yes	complemented
31	C₂×C₅×M₁₆	C₃×C₄	1	1	yes	characteristic
32	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
33	C₅×#117	C₃×C₄	1	1	yes	characteristic
34	C₅×M₃₂	C₃×C₄	1	4	yes	complemented
35	C₅×M₃₂	C₂²×C₃	1	4	yes	—
36	C₂×C₅×C₁₆	C₃×C₄	1	4	yes	complemented
37	C₅×#97	C₂²×C₃	1	2	yes	—
38	C₄×C₅×C₈	C₃×C₄	1	2	yes	—
39	#2110	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
40	C₂³×C₃×C₅	M₁₆	1	1	yes	characteristic
41	C₂×C₃×C₄×C₅	—	2	2	no	—
42	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
43	C₂×C₃×C₄×C₅	M₁₆	1	1	yes	characteristic
44	C₂×C₃×C₄×C₅	M₁₆	1	2	yes	—
45	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
46	C₃×C₅×C₈	—	2	2	no	—
47	C₃×C₅×C₈	C₄²	1	1	yes	characteristic
48	C₃×C₅×C₈	C₄²	1	1	yes	characteristic
49	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
50	C₂×C₃×M₁₆	C₄×C₅	1	1	yes	characteristic
51	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
52	C₃×#117	C₄×C₅	1	1	yes	characteristic
53	C₃×M₃₂	C₄×C₅	1	4	yes	complemented
54	C₃×M₃₂	C₂²×C₅	1	4	yes	—
55	C₂×C₃×C₁₆	C₄×C₅	1	4	yes	complemented
56	C₃×#97	C₂²×C₅	1	2	yes	—
57	C₃×C₄×C₈	C₄×C₅	1	2	yes	—
58	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
59	C₅×M₁₆	—	2	2	no	—
60	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
61	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
62	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
63	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
64	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
65	C₅×C₄⋊C₄	—	2	1	no	—
66	C₅×C₂²⋊C₄	—	2	1	no	—
67	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
68	C₅×C₁₆	C₂×C₃×C₄	1	8	yes	—
69	C₂×M₃₂	C₂×C₃×C₅	1	2	yes	—
70	#405	C₂×C₃×C₅	1	1	yes	characteristic
71	#347	C₂×C₃×C₅	1	4	yes	complemented
72	C₂²×C₃×C₅	—	4	1	no	—
73	C₂²×C₃×C₅	#97	1	2	yes	—
74	C₂²×C₃×C₅	C₂×M₁₆	1	1	yes	characteristic
75	C₃×C₄×C₅	—	4	2	no	—
76	C₃×C₄×C₅	#97	1	2	yes	—
77	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
78	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
79	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
80	C₃×M₁₆	—	2	2	no	—
81	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
82	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
83	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
84	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
85	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
86	C₃×C₄⋊C₄	—	2	1	no	—
87	C₃×C₂²⋊C₄	—	2	1	no	—
88	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
89	C₃×C₁₆	C₂×C₄×C₅	1	8	yes	—
90	C₂³×C₅	C₃×M₁₆	1	1	yes	characteristic
91	C₂×C₄×C₅	—	2	2	no	—
92	C₂×C₄×C₅	C₃×C₄²	1	2	yes	—
93	C₂×C₄×C₅	C₃×M₁₆	1	1	yes	characteristic
94	C₂×C₄×C₅	C₃×M₁₆	1	2	yes	—
95	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
96	C₅×C₈	—	2	2	no	—
97	C₅×C₈	C₃×C₄²	1	1	yes	characteristic
98	C₅×C₈	C₃×C₄²	1	1	yes	characteristic
99	C₅×C₈	C₂²×C₃×C₄	1	2	yes	—
100	C₂×M₁₆	C₃×C₄×C₅	1	1	yes	characteristic
101	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
102	#117	C₃×C₄×C₅	1	1	yes	characteristic
103	M₃₂	C₃×C₄×C₅	1	4	yes	complemented
104	M₃₂	C₂²×C₃×C₅	1	4	yes	—
105	C₂×C₁₆	C₃×C₄×C₅	1	4	yes	complemented
106	#97	C₂²×C₃×C₅	1	2	yes	—
107	C₄×C₈	C₃×C₄×C₅	1	2	yes	—
108	C₂×C₃×C₅	—	2	1	no	—
109	C₂×C₃×C₅	—	2	2	no	—
110	C₂×C₃×C₅	C₂×#97	1	1	yes	characteristic
111	C₂³×C₃	C₅×M₁₆	1	1	yes	characteristic
112	C₂×C₃×C₄	—	2	2	no	—
113	C₂×C₃×C₄	C₄²×C₅	1	2	yes	—
114	C₂×C₃×C₄	C₅×M₁₆	1	2	yes	—
115	C₂×C₃×C₄	C₅×M₁₆	1	1	yes	characteristic
116	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
117	C₃×C₈	—	2	2	no	—
118	C₃×C₈	C₄²×C₅	1	1	yes	characteristic
119	C₃×C₈	C₄²×C₅	1	1	yes	characteristic
120	C₃×C₈	C₂²×C₄×C₅	1	2	yes	—
121	C₂²×C₅	—	4	1	no	—
122	C₂²×C₅	C₃×#97	1	2	yes	—
123	C₂²×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
124	C₄×C₅	—	4	2	no	—
125	C₄×C₅	C₃×#97	1	2	yes	—
126	C₄×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
127	C₄×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
128	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
129	M₁₆	—	2	2	no	—
130	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
131	C₂×C₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
132	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
133	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
134	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
135	C₄⋊C₄	—	2	1	no	—
136	C₂²⋊C₄	—	2	1	no	—
137	C₄²	C₂×C₃×C₄×C₅	1	2	yes	—
138	C₁₆	C₂×C₃×C₄×C₅	1	8	yes	—
139	C₃×C₅	#2110	1	1	yes	characteristic, direct product factor, Hall
140	C₂²×C₃	—	4	1	no	—
141	C₂²×C₃	C₅×#97	1	2	yes	—
142	C₂²×C₃	C₂×C₅×M₁₆	1	1	yes	characteristic
143	C₃×C₄	—	4	2	no	—
144	C₃×C₄	C₅×#97	1	2	yes	—
145	C₃×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
146	C₃×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
147	C₂×C₅	—	2	1	no	—
148	C₂×C₅	—	2	2	no	—
149	C₂×C₅	C₂×C₃×#97	1	1	yes	characteristic
150	C₂³	C₃×C₅×M₁₆	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	C₃×C₄²×C₅	1	2	yes	—
153	C₂×C₄	C₃×C₅×M₁₆	1	2	yes	—
154	C₂×C₄	C₃×C₅×M₁₆	1	1	yes	characteristic
155	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
156	C₈	—	2	2	no	—
157	C₈	C₃×C₄²×C₅	1	1	yes	characteristic
158	C₈	C₃×C₄²×C₅	1	1	yes	characteristic
159	C₈	C₂²×C₃×C₄×C₅	1	2	yes	—
160	C₂×C₃	—	2	1	no	—
161	C₂×C₃	—	2	2	no	—
162	C₂×C₃	C₂×C₅×#97	1	1	yes	characteristic
163	C₅	C₃×#2110	1	1	yes	characteristic, direct product factor, Sylow
164	C₂²	—	4	1	no	—
165	C₂²	C₃×C₅×#97	1	2	yes	—
166	C₂²	C₂×C₃×C₅×M₁₆	1	1	yes	characteristic
167	C₄	—	4	2	no	—
168	C₄	C₃×C₅×#97	1	2	yes	—
169	C₄	C₂×C₃×C₄²×C₅	1	1	yes	characteristic
170	C₄	C₂×C₃×C₅×M₁₆	1	1	yes	characteristic
171	C₃	C₅×#2110	1	1	yes	characteristic, direct product factor, Sylow
172	C₂	—	2	1	no	—
173	C₂	—	2	2	no	—
174	C₂	C₂×C₃×C₅×#97	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₅×C₈	C₄²
commutator	C₄	C₂×C₃×C₄²×C₅
Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
socle	C₂×C₃×C₅	C₂×#97
prime radical	C₂³×C₃×C₅	M₁₆
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₃×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×#347	C₂	semidirect
4. C₅×#2110	C₃	direct (N × H)
8. C₃×C₅×M₃₂	C₄	semidirect
10. C₂×C₃×C₅×C₁₆	C₄	semidirect
13. C₃×#2110	C₅	direct (N × H)
16. C₅×#347	C₂×C₃	semidirect
30. C₃×#347	C₂×C₅	semidirect
34. C₅×M₃₂	C₃×C₄	semidirect
36. C₂×C₅×C₁₆	C₃×C₄	semidirect
39. #2110	C₃×C₅	direct (N × H)
53. C₃×M₃₂	C₄×C₅	semidirect
55. C₂×C₃×C₁₆	C₄×C₅	semidirect
71. #347	C₂×C₃×C₅	semidirect
103. M₃₂	C₃×C₄×C₅	semidirect
105. C₂×C₁₆	C₃×C₄×C₅	semidirect
139. C₃×C₅	#2110	direct (N × H)
163. C₅	C₃×#2110	direct (N × H)
171. C₃	C₅×#2110	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×M₃₂	2
C₃×C₅×#405	2
C₃×C₅×#347	2
C₅×#2110	3
C₃×#2110	5

Subgroups — C3×C5×#2110 #14275144

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2110 #14275144