Subgroups — C₃×#10602304 #14315630

Order 1920.

292 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#508	C₂	1	1	yes	maximal, characteristic
2	C₃×#65485	C₂	1	2	yes	maximal, complemented
3	C₃×#64787	C₂	1	2	yes	maximal, complemented
4	C₃×#64726	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₁₆×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
6	#10602304	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₃×C₅×#135	C₂²	1	2	yes	—
8	C₃×#4094	—	2	2	no	—
9	C₃×C₅×Q₃₂	C₄	1	1	yes	characteristic, complemented
10	C₃×C₅×SD₃₂	C₄	1	2	yes	complemented
11	C₃×C₅×D₁₆	C₄	1	1	yes	characteristic, complemented
12	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
13	C₃×#3969	—	2	2	no	—
14	C₃×#3951	C₂²	1	2	yes	—
15	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
16	C₂×C₃×#672	C₂²	1	1	yes	characteristic
17	C₃×#2178	—	5	1	no	maximal, Hall
18	C₅×#508	C₂×C₃	1	1	yes	characteristic
19	#65485	C₂×C₃	1	2	yes	complemented
20	#64787	C₂×C₃	1	2	yes	complemented
21	#64726	C₂×C₃	1	1	yes	characteristic, complemented
22	C₁₆×Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
23	C₃×C₅×#41	—	2	2	no	—
24	C₃×C₅×Q₁₆	C₂×C₄	1	2	yes	—
25	C₃×C₅×SD₁₆	—	2	2	no	—
26	C₃×C₅×D₈	C₂×C₄	1	2	yes	—
27	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
28	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
29	C₃×#681	—	2	2	no	—
30	C₃×#681	—	4	2	no	—
31	C₂×C₃×#168	—	4	2	no	—
32	C₂×C₃×#168	D₄	1	1	yes	characteristic
33	C₃×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
34	C₃×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
35	C₃×#672	—	2	1	no	—
36	C₃×#508	D₅	1	1	yes	characteristic
37	C₃×#443	—	5	2	no	—
38	C₃×#368	—	5	1	no	—
39	C₃×#359	—	5	2	no	—
40	C₃×C₄×C₁₆	—	5	1	no	—
41	C₅×#135	C₂²×C₃	1	2	yes	—
42	#4094	—	2	2	no	—
43	C₅×Q₃₂	C₃×C₄	1	1	yes	characteristic, complemented
44	C₅×SD₃₂	C₃×C₄	1	2	yes	complemented
45	C₅×D₁₆	C₃×C₄	1	1	yes	characteristic, complemented
46	C₂×C₅×C₁₆	C₂²×C₃	1	1	yes	characteristic
47	#3969	—	2	2	no	—
48	#3951	C₂²×C₃	1	2	yes	—
49	C₈×Dic₅	C₂²×C₃	1	1	yes	characteristic
50	C₂×#672	C₂²×C₃	1	1	yes	characteristic
51	#2178	—	5	1	no	Sylow
52	C₃×C₅×Q₈	—	2	2	no	—
53	C₃×C₅×D₄	—	2	2	no	—
54	C₃×C₅×D₄	—	4	2	no	—
55	C₂×C₃×C₄×C₅	—	4	2	no	—
56	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
57	C₂×C₃×Dic₅	—	2	1	no	—
58	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
59	C₃×C₅×C₈	#41	1	1	yes	characteristic
60	C₃×#168	—	4	2	no	—
61	C₃×#168	D₈	1	1	yes	characteristic
62	C₃×#168	D₈	1	1	yes	characteristic
63	C₃×#135	C₂×D₅	1	2	yes	—
64	C₃×#131	—	10	2	no	—
65	C₃×Q₃₂	Dic₅	1	1	yes	characteristic, complemented
66	C₃×SD₃₂	Dic₅	1	2	yes	complemented
67	C₃×D₁₆	Dic₅	1	1	yes	characteristic, complemented
68	C₂×C₃×C₁₆	—	5	1	no	—
69	C₂×C₃×C₁₆	C₂×D₅	1	1	yes	characteristic
70	C₃×#108	—	5	2	no	—
71	C₃×#104	—	10	2	no	—
72	C₃×C₄×C₈	—	5	1	no	—
73	C₅×#41	—	2	2	no	—
74	C₅×Q₁₆	C₂×C₃×C₄	1	2	yes	—
75	C₅×SD₁₆	—	2	2	no	—
76	C₅×D₈	C₂×C₃×C₄	1	2	yes	—
77	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
78	C₄×Dic₅	C₃×D₄	1	1	yes	characteristic
79	#681	—	2	2	no	—
80	#681	—	4	2	no	—
81	C₂×#168	—	4	2	no	—
82	C₂×#168	C₃×D₄	1	1	yes	characteristic
83	C₅×C₁₆	C₂×C₃×C₄	1	1	yes	characteristic
84	C₅×C₁₆	C₂×C₃×C₄	1	1	yes	characteristic
85	#672	—	2	1	no	—
86	#508	C₃×D₅	1	1	yes	characteristic
87	#443	—	5	2	no	—
88	#368	—	5	1	no	—
89	#359	—	5	2	no	—
90	C₄×C₁₆	—	5	1	no	—
91	C₂²×C₃×C₅	—	4	2	no	—
92	C₂²×C₃×C₅	#135	1	1	yes	characteristic
93	C₃×C₄×C₅	—	4	2	no	—
94	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
95	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
96	C₃×Dic₅	—	2	2	no	—
97	C₃×#41	—	2	2	no	—
98	C₃×Q₁₆	C₂×Dic₅	1	2	yes	—
99	C₃×SD₁₆	—	2	2	no	—
100	C₃×D₈	C₂×Dic₅	1	2	yes	—
101	C₃×M₁₆	—	10	2	no	—
102	C₃×M₁₆	—	20	2	no	—
103	C₂×C₃×C₈	—	5	1	no	—
104	C₂×C₃×C₈	—	20	2	no	—
105	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
106	C₃×C₄²	—	5	1	no	—
107	C₃×C₁₆	—	10	1	no	—
108	C₃×C₁₆	C₂×Dic₅	1	1	yes	characteristic
109	C₃×C₁₆	C₂×Dic₅	1	1	yes	characteristic
110	C₅×Q₈	—	2	2	no	—
111	C₅×D₄	—	2	2	no	—
112	C₅×D₄	—	4	2	no	—
113	C₂×C₄×C₅	—	4	2	no	—
114	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
115	C₂×Dic₅	—	2	1	no	—
116	C₅×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
117	C₅×C₈	C₃×#41	1	1	yes	characteristic
118	#168	—	4	2	no	—
119	#168	C₃×D₈	1	1	yes	characteristic
120	#168	C₃×D₈	1	1	yes	characteristic
121	#135	C₂×C₃×D₅	1	2	yes	—
122	#131	—	10	2	no	—
123	Q₃₂	C₃×Dic₅	1	1	yes	characteristic, complemented
124	SD₃₂	C₃×Dic₅	1	2	yes	complemented
125	D₁₆	C₃×Dic₅	1	1	yes	characteristic, complemented
126	C₂×C₁₆	—	5	1	no	—
127	C₂×C₁₆	C₂×C₃×D₅	1	1	yes	characteristic
128	#108	—	5	2	no	—
129	#104	—	10	2	no	—
130	C₄×C₈	—	5	1	no	—
131	C₂×C₃×C₅	—	2	1	no	—
132	C₂×C₃×C₅	—	8	2	no	—
133	C₂×C₃×C₅	C₄×D₈	1	1	yes	characteristic
134	C₃×Q₈	—	2	2	no	—
135	C₃×D₄	—	2	2	no	—
136	C₃×D₄	—	4	2	no	—
137	C₂×C₃×C₄	—	4	2	no	—
138	C₂×C₃×C₄	—	10	1	no	—
139	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
140	C₃×C₈	—	5	1	no	—
141	C₃×C₈	—	5	1	no	—
142	C₃×C₈	—	20	2	no	—
143	C₃×C₈	#711	1	1	yes	characteristic
144	C₃×C₈	C₂²×Dic₅	1	1	yes	characteristic
145	C₂²×C₅	—	4	2	no	—
146	C₂²×C₅	C₃×#135	1	1	yes	characteristic
147	C₄×C₅	—	4	2	no	—
148	C₄×C₅	C₃×C₄×D₄	1	1	yes	characteristic
149	C₄×C₅	C₂×C₃×D₈	1	1	yes	characteristic
150	Dic₅	—	2	2	no	—
151	#41	—	2	2	no	—
152	Q₁₆	C₂×C₃×Dic₅	1	2	yes	—
153	SD₁₆	—	2	2	no	—
154	D₈	C₂×C₃×Dic₅	1	2	yes	—
155	M₁₆	—	10	2	no	—
156	M₁₆	—	20	2	no	—
157	C₂×C₈	—	5	1	no	—
158	C₂×C₈	—	20	2	no	—
159	C₂×C₈	C₂²×C₃×D₅	1	1	yes	characteristic
160	C₄²	—	5	1	no	—
161	C₁₆	—	10	1	no	—
162	C₁₆	C₂×C₃×Dic₅	1	1	yes	characteristic
163	C₁₆	C₂×C₃×Dic₅	1	1	yes	characteristic
164	C₃×C₅	#2178	1	1	yes	characteristic, complemented, Hall
165	C₂²×C₃	—	4	2	no	—
166	C₂²×C₃	#4058	1	1	yes	characteristic
167	C₃×C₄	—	4	2	no	—
168	C₃×C₄	—	10	2	no	—
169	C₃×C₄	D₅×D₈	1	1	yes	characteristic
170	C₃×C₄	D₄×Dic₅	1	1	yes	characteristic
171	C₂×C₅	—	2	1	no	—
172	C₂×C₅	—	8	2	no	—
173	C₂×C₅	C₃×C₄×D₈	1	1	yes	characteristic
174	Q₈	—	2	2	no	—
175	D₄	—	2	2	no	—
176	D₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	10	1	no	—
179	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
180	C₈	—	5	1	no	—
181	C₈	—	5	1	no	—
182	C₈	—	20	2	no	—
183	C₈	C₃×#711	1	1	yes	characteristic
184	C₈	C₂²×C₃×Dic₅	1	1	yes	characteristic
185	C₂×C₃	—	2	1	no	—
186	C₂×C₃	—	8	2	no	—
187	C₂×C₃	D₈×Dic₅	1	1	yes	characteristic
188	C₅	C₃×#2178	1	1	yes	characteristic, complemented, Sylow
189	C₂²	—	4	2	no	—
190	C₂²	C₃×#4058	1	1	yes	characteristic
191	C₄	—	4	2	no	—
192	C₄	—	10	2	no	—
193	C₄	C₃×D₅×D₈	1	1	yes	characteristic
194	C₄	C₃×D₄×Dic₅	1	1	yes	characteristic
195	C₃	#10602304	1	1	yes	characteristic, direct product factor, Sylow
196	C₂	—	2	1	no	—
197	C₂	—	8	2	no	—
198	C₂	C₃×D₈×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	D₅×D₈
commutator	C₅×C₈	C₂²×C₃×C₄
Frattini	C₂×C₈	C₂²×C₃×D₅
Fitting, prime radical	C₃×C₅×#508	C₂
socle	C₂×C₃×C₅	C₄×D₈
cosocle	C₂×C₅×C₈	C₂³×C₃
hypercenter	C₃×#508	D₅
hypocenter	C₅	C₃×#2178
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₈ ◁ C₃×#508

Complements

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

N	H ≅ G/N	type
2. C₃×#65485	C₂	semidirect
3. C₃×#64787	C₂	semidirect
4. C₃×#64726	C₂	semidirect
5. C₃×C₁₆×Dic₅	C₂	semidirect
6. #10602304	C₃	direct (N × H)
9. C₃×C₅×Q₃₂	C₄	semidirect
10. C₃×C₅×SD₃₂	C₄	semidirect
11. C₃×C₅×D₁₆	C₄	semidirect
19. #65485	C₂×C₃	semidirect
20. #64787	C₂×C₃	semidirect
21. #64726	C₂×C₃	semidirect
22. C₁₆×Dic₅	C₂×C₃	semidirect
43. C₅×Q₃₂	C₃×C₄	semidirect
44. C₅×SD₃₂	C₃×C₄	semidirect
45. C₅×D₁₆	C₃×C₄	semidirect
65. C₃×Q₃₂	Dic₅	semidirect
66. C₃×SD₃₂	Dic₅	semidirect
67. C₃×D₁₆	Dic₅	semidirect
123. Q₃₂	C₃×Dic₅	semidirect
124. SD₃₂	C₃×Dic₅	semidirect
125. D₁₆	C₃×Dic₅	semidirect
164. C₃×C₅	#2178	semidirect
188. C₅	C₃×#2178	semidirect
195. C₃	#10602304	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×#508	2
C₃×#65485	2
C₃×#64787	2
C₃×#64726	2
C₃×C₁₆×Dic₅	2
#10602304	3
C₃×#2178	5

Subgroups — C3×#10602304 #14315630

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10602304 #14315630