Subgroups — C₃×C₅×#2165 #14275199

Order 1920.

256 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#364	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#349	C₂	1	2	yes	maximal
4	C₃×C₅×#347	C₂	1	2	yes	maximal, complemented
5	C₅×#2165	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₃×C₅×M₁₆	C₄	1	1	yes	characteristic
7	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₄²×C₅	C₄	1	1	yes	characteristic
9	C₃×C₅×M₃₂	C₂²	1	4	yes	—
10	C₃×C₅×#97	C₄	1	1	yes	characteristic
11	C₃×C₅×#97	C₂²	1	1	yes	characteristic
12	C₃×C₄×C₅×C₈	C₄	1	1	yes	characteristic
13	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₃×#2165	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₄×C₅×M₁₆	C₂×C₃	1	1	yes	characteristic
16	C₅×#364	C₂×C₃	1	2	yes	complemented
17	C₅×#349	C₂×C₃	1	2	yes	—
18	C₅×#347	C₂×C₃	1	2	yes	complemented
19	C₂²×C₃×C₄×C₅	—	2	1	no	—
20	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
21	C₃×C₅×M₁₆	—	2	2	no	—
22	C₃×C₅×M₁₆	—	2	2	no	—
23	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
27	C₃×C₄²×C₅	—	2	1	no	—
28	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
29	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
30	C₃×C₅×C₁₆	—	2	4	no	—
31	C₃×C₄×M₁₆	C₂×C₅	1	1	yes	characteristic
32	C₃×#364	C₂×C₅	1	2	yes	complemented
33	C₃×#349	C₂×C₅	1	2	yes	—
34	C₃×#347	C₂×C₅	1	2	yes	complemented
35	C₂×C₅×M₁₆	C₃×C₄	1	1	yes	characteristic
36	C₂×C₅×M₁₆	C₂²×C₃	1	1	yes	characteristic
37	C₂×C₄²×C₅	C₃×C₄	1	1	yes	characteristic
38	C₅×M₃₂	C₂²×C₃	1	4	yes	—
39	C₅×#97	C₃×C₄	1	1	yes	characteristic
40	C₅×#97	C₂²×C₃	1	1	yes	characteristic
41	C₄×C₅×C₈	C₃×C₄	1	1	yes	characteristic
42	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
43	#2165	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
44	C₂³×C₃×C₅	M₁₆	1	1	yes	characteristic
45	C₂×C₃×C₄×C₅	—	2	1	no	—
46	C₂×C₃×C₄×C₅	—	2	1	no	—
47	C₂×C₃×C₄×C₅	—	2	1	no	—
48	C₂×C₃×C₄×C₅	—	2	1	no	—
49	C₂×C₃×C₄×C₅	—	4	2	no	—
50	C₂×C₃×C₄×C₅	M₁₆	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
52	C₃×C₅×C₈	—	2	1	no	—
53	C₃×C₅×C₈	—	2	1	no	—
54	C₃×C₅×C₈	M₁₆	1	2	yes	—
55	C₃×C₅×C₈	#41	1	2	yes	—
56	C₂×C₃×M₁₆	C₄×C₅	1	1	yes	characteristic
57	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
58	C₂×C₃×C₄²	C₄×C₅	1	1	yes	characteristic
59	C₃×M₃₂	C₂²×C₅	1	4	yes	—
60	C₃×#97	C₄×C₅	1	1	yes	characteristic
61	C₃×#97	C₂²×C₅	1	1	yes	characteristic
62	C₃×C₄×C₈	C₄×C₅	1	1	yes	characteristic
63	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
64	C₂²×C₄×C₅	—	2	1	no	—
65	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
66	C₅×M₁₆	—	2	2	no	—
67	C₅×M₁₆	—	2	2	no	—
68	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
69	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
70	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
71	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
72	C₄²×C₅	—	2	1	no	—
73	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
74	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
75	C₅×C₁₆	—	2	4	no	—
76	C₄×M₁₆	C₂×C₃×C₅	1	1	yes	characteristic
77	#364	C₂×C₃×C₅	1	2	yes	complemented
78	#349	C₂×C₃×C₅	1	2	yes	—
79	#347	C₂×C₃×C₅	1	2	yes	complemented
80	C₂²×C₃×C₅	—	2	1	no	—
81	C₂²×C₃×C₅	—	4	1	no	—
82	C₂²×C₃×C₅	C₂×M₁₆	1	1	yes	characteristic
83	C₃×C₄×C₅	—	2	2	no	—
84	C₃×C₄×C₅	—	2	1	no	—
85	C₃×C₄×C₅	—	4	1	no	—
86	C₃×C₄×C₅	#117	1	1	yes	characteristic
87	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
88	C₂²×C₃×C₄	—	2	1	no	—
89	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
90	C₃×M₁₆	—	2	2	no	—
91	C₃×M₁₆	—	2	2	no	—
92	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
93	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
94	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
95	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
96	C₃×C₄²	—	2	1	no	—
97	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
98	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
99	C₃×C₁₆	—	2	4	no	—
100	C₂³×C₅	C₃×M₁₆	1	1	yes	characteristic
101	C₂×C₄×C₅	—	2	1	no	—
102	C₂×C₄×C₅	—	2	1	no	—
103	C₂×C₄×C₅	—	2	1	no	—
104	C₂×C₄×C₅	—	2	1	no	—
105	C₂×C₄×C₅	—	4	2	no	—
106	C₂×C₄×C₅	C₃×M₁₆	1	1	yes	characteristic
107	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
108	C₅×C₈	—	2	1	no	—
109	C₅×C₈	—	2	1	no	—
110	C₅×C₈	C₃×M₁₆	1	2	yes	—
111	C₅×C₈	C₃×#41	1	2	yes	—
112	C₂×M₁₆	C₃×C₄×C₅	1	1	yes	characteristic
113	C₂×M₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
114	C₂×C₄²	C₃×C₄×C₅	1	1	yes	characteristic
115	M₃₂	C₂²×C₃×C₅	1	4	yes	—
116	#97	C₃×C₄×C₅	1	1	yes	characteristic
117	#97	C₂²×C₃×C₅	1	1	yes	characteristic
118	C₄×C₈	C₃×C₄×C₅	1	1	yes	characteristic
119	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
120	C₂×C₃×C₅	—	2	1	no	—
121	C₂×C₃×C₅	—	4	1	no	—
122	C₂×C₃×C₅	#432	1	1	yes	characteristic
123	C₂³×C₃	C₅×M₁₆	1	1	yes	characteristic
124	C₂×C₃×C₄	—	2	1	no	—
125	C₂×C₃×C₄	—	2	1	no	—
126	C₂×C₃×C₄	—	2	1	no	—
127	C₂×C₃×C₄	—	2	1	no	—
128	C₂×C₃×C₄	—	4	2	no	—
129	C₂×C₃×C₄	C₅×M₁₆	1	1	yes	characteristic
130	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
131	C₃×C₈	—	2	1	no	—
132	C₃×C₈	—	2	1	no	—
133	C₃×C₈	C₅×M₁₆	1	2	yes	—
134	C₃×C₈	C₅×#41	1	2	yes	—
135	C₂²×C₅	—	2	1	no	—
136	C₂²×C₅	—	4	1	no	—
137	C₂²×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
138	C₄×C₅	—	2	2	no	—
139	C₄×C₅	—	2	1	no	—
140	C₄×C₅	—	4	1	no	—
141	C₄×C₅	C₃×#117	1	1	yes	characteristic
142	C₄×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
143	C₂²×C₄	—	2	1	no	—
144	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
145	M₁₆	—	2	2	no	—
146	M₁₆	—	2	2	no	—
147	C₂×C₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
148	C₂×C₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
149	C₂×C₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
150	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
151	C₄²	—	2	1	no	—
152	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
153	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
154	C₁₆	—	2	4	no	—
155	C₃×C₅	#2165	1	1	yes	characteristic, direct product factor, Hall
156	C₂²×C₃	—	2	1	no	—
157	C₂²×C₃	—	4	1	no	—
158	C₂²×C₃	C₂×C₅×M₁₆	1	1	yes	characteristic
159	C₃×C₄	—	2	2	no	—
160	C₃×C₄	—	2	1	no	—
161	C₃×C₄	—	4	1	no	—
162	C₃×C₄	C₅×#117	1	1	yes	characteristic
163	C₃×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
164	C₂×C₅	—	2	1	no	—
165	C₂×C₅	—	4	1	no	—
166	C₂×C₅	C₃×#432	1	1	yes	characteristic
167	C₂³	C₃×C₅×M₁₆	1	1	yes	characteristic
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	C₃×C₅×M₁₆	1	1	yes	characteristic
174	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	C₃×C₅×M₁₆	1	2	yes	—
178	C₈	C₃×C₅×#41	1	2	yes	—
179	C₂×C₃	—	2	1	no	—
180	C₂×C₃	—	4	1	no	—
181	C₂×C₃	C₅×#432	1	1	yes	characteristic
182	C₅	C₃×#2165	1	1	yes	characteristic, direct product factor, Sylow
183	C₂²	—	2	1	no	—
184	C₂²	—	4	1	no	—
185	C₂²	C₂×C₃×C₅×M₁₆	1	1	yes	characteristic
186	C₄	—	2	2	no	—
187	C₄	—	2	1	no	—
188	C₄	—	4	1	no	—
189	C₄	C₃×C₅×#117	1	1	yes	characteristic
190	C₄	C₂×C₃×C₅×M₁₆	1	1	yes	characteristic
191	C₃	C₅×#2165	1	1	yes	characteristic, direct product factor, Sylow
192	C₂	—	2	1	no	—
193	C₂	—	4	1	no	—
194	C₂	C₃×C₅×#432	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄×C₅	#117
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
socle	C₂×C₃×C₅	#432
prime radical	C₂³×C₃×C₅	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄×C₅ ◁ C₃×C₅×#97 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#364	C₂	semidirect
4. C₃×C₅×#347	C₂	semidirect
5. C₅×#2165	C₃	direct (N × H)
14. C₃×#2165	C₅	direct (N × H)
16. C₅×#364	C₂×C₃	semidirect
18. C₅×#347	C₂×C₃	semidirect
32. C₃×#364	C₂×C₅	semidirect
34. C₃×#347	C₂×C₅	semidirect
43. #2165	C₃×C₅	direct (N × H)
77. #364	C₂×C₃×C₅	semidirect
79. #347	C₂×C₃×C₅	semidirect
155. C₃×C₅	#2165	direct (N × H)
182. C₅	C₃×#2165	direct (N × H)
191. C₃	C₅×#2165	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×C₄×C₅×M₁₆	2
C₃×C₅×#364	2
C₃×C₅×#349	2
C₃×C₅×#347	2
C₅×#2165	3
C₃×#2165	5

Subgroups — C3×C5×#2165 #14275199

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2165 #14275199