Subgroups — C₃×#10600642 #14280707

Order 1920.

428 conjugacy classes of subgroups.

The subgroup lattice has 192 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×C₄×#681	C₂	1	2	yes	maximal
3	C₃×#64675	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#64674	C₂	1	2	yes	maximal, complemented
5	C₃×C₈×#168	C₂	1	1	yes	maximal, characteristic, complemented
6	#10600642	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×M₁₆	C₄	1	2	yes	—
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₃×#681	C₄	1	4	yes	—
10	C₃×C₅×#97	C₄	1	2	yes	—
11	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₃×#3935	C₄	1	4	yes	—
14	C₃×C₄×#168	C₂²	1	2	yes	—
15	C₃×C₄×#168	C₂²	1	2	yes	—
16	C₃×#1455	—	5	1	no	maximal, Hall
17	C₄×C₅×M₁₆	C₂×C₃	1	1	yes	characteristic
18	C₄×#681	C₂×C₃	1	2	yes	—
19	#64675	C₂×C₃	1	1	yes	characteristic, complemented
20	#64674	C₂×C₃	1	2	yes	complemented
21	C₈×#168	C₂×C₃	1	1	yes	characteristic, complemented
22	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
23	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
24	C₃×C₅×M₁₆	—	2	4	no	—
25	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
26	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
27	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
28	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
29	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
30	C₃×#681	—	2	8	no	—
31	C₂×C₃×#168	C₂×C₄	1	4	yes	—
32	C₂×C₃×#168	C₂×C₄	1	4	yes	—
33	C₃×C₄×M₁₆	—	5	2	no	—
34	C₃×C₄×M₁₆	D₅	1	1	yes	characteristic
35	C₃×#322	—	5	2	no	—
36	C₃×#322	—	5	1	no	—
37	C₃×C₈²	—	5	1	no	—
38	C₂×C₅×M₁₆	C₃×C₄	1	2	yes	—
39	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
40	C₂×#681	C₃×C₄	1	4	yes	—
41	C₅×#97	C₃×C₄	1	2	yes	—
42	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
43	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
44	#3935	C₃×C₄	1	4	yes	—
45	C₄×#168	C₂²×C₃	1	2	yes	—
46	C₄×#168	C₂²×C₃	1	2	yes	—
47	#1455	—	5	1	no	Sylow
48	C₂³×C₃×C₅	C₄²	1	1	yes	characteristic
49	C₂×C₃×C₄×C₅	—	2	4	no	—
50	C₂×C₃×C₄×C₅	—	2	2	no	—
51	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
53	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
54	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
55	C₃×C₅×C₈	—	2	2	no	—
56	C₃×C₅×C₈	M₁₆	1	4	yes	complemented
57	C₃×#168	—	2	4	no	—
58	C₃×#168	M₁₆	1	8	yes	complemented
59	C₂×C₃×M₁₆	—	5	4	no	—
60	C₂×C₃×M₁₆	Dic₅	1	2	yes	—
61	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
62	C₃×#97	—	5	4	no	—
63	C₃×#97	Dic₅	1	2	yes	—
64	C₃×C₄×C₈	—	5	2	no	—
65	C₃×C₄×C₈	—	5	2	no	—
66	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
67	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
69	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
70	C₅×M₁₆	—	2	4	no	—
71	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
72	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
73	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
74	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
75	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
76	#681	—	2	8	no	—
77	C₂×#168	C₂×C₃×C₄	1	4	yes	—
78	C₂×#168	C₂×C₃×C₄	1	4	yes	—
79	C₄×M₁₆	—	5	2	no	—
80	C₄×M₁₆	C₃×D₅	1	1	yes	characteristic
81	#322	—	5	2	no	—
82	#322	—	5	1	no	—
83	C₈²	—	5	1	no	—
84	C₂²×C₃×C₅	—	2	2	no	—
85	C₂²×C₃×C₅	—	2	1	no	—
86	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
87	C₃×C₄×C₅	—	2	2	no	—
88	C₃×C₄×C₅	—	2	1	no	—
89	C₃×C₄×C₅	C₂×M₁₆	1	4	yes	—
90	C₃×C₄×C₅	C₂×M₁₆	1	2	yes	—
91	C₂²×C₃×C₄	C₄×D₅	1	2	yes	—
92	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
93	C₃×M₁₆	—	2	4	no	—
94	C₃×M₁₆	—	10	8	no	—
95	C₂×C₃×C₈	—	5	4	no	—
96	C₂×C₃×C₈	—	5	4	no	—
97	C₂×C₃×C₈	C₂×Dic₅	1	2	yes	—
98	C₂×C₃×C₈	C₂×Dic₅	1	2	yes	—
99	C₃×C₄²	C₄×D₅	1	2	yes	—
100	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
101	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
102	C₂³×C₅	C₃×C₄²	1	1	yes	characteristic
103	C₂×C₄×C₅	—	2	4	no	—
104	C₂×C₄×C₅	—	2	2	no	—
105	C₂×C₄×C₅	C₃×C₄²	1	1	yes	characteristic
106	C₂×C₄×C₅	C₃×C₄²	1	2	yes	—
107	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
108	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
109	C₅×C₈	—	2	2	no	—
110	C₅×C₈	C₃×M₁₆	1	4	yes	complemented
111	#168	—	2	4	no	—
112	#168	C₃×M₁₆	1	8	yes	complemented
113	C₂×M₁₆	—	5	4	no	—
114	C₂×M₁₆	C₃×Dic₅	1	2	yes	—
115	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
116	#97	—	5	4	no	—
117	#97	C₃×Dic₅	1	2	yes	—
118	C₄×C₈	—	5	2	no	—
119	C₄×C₈	—	5	2	no	—
120	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic
121	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic
122	C₂×C₃×C₅	—	4	1	no	—
123	C₂×C₃×C₅	C₄×M₁₆	1	2	yes	—
124	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
125	C₂³×C₃	C₄×Dic₅	1	1	yes	characteristic
126	C₂×C₃×C₄	—	2	4	no	—
127	C₂×C₃×C₄	—	2	2	no	—
128	C₂×C₃×C₄	C₄×Dic₅	1	1	yes	characteristic
129	C₂×C₃×C₄	C₄×Dic₅	1	2	yes	—
130	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
131	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
132	C₃×C₈	—	2	2	no	—
133	C₃×C₈	—	5	8	no	—
134	C₃×C₈	—	10	4	no	—
135	C₃×C₈	#681	1	4	yes	complemented
136	C₂²×C₅	—	2	2	no	—
137	C₂²×C₅	—	2	1	no	—
138	C₂²×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
139	C₄×C₅	—	2	2	no	—
140	C₄×C₅	—	2	1	no	—
141	C₄×C₅	C₂×C₃×M₁₆	1	4	yes	—
142	C₄×C₅	C₂×C₃×M₁₆	1	2	yes	—
143	C₂²×C₄	C₃×C₄×D₅	1	2	yes	—
144	C₂²×C₄	C₂×C₃×Dic₅	1	1	yes	characteristic
145	M₁₆	—	2	4	no	—
146	M₁₆	—	10	8	no	—
147	C₂×C₈	—	5	4	no	—
148	C₂×C₈	—	5	4	no	—
149	C₂×C₈	C₂×C₃×Dic₅	1	2	yes	—
150	C₂×C₈	C₂×C₃×Dic₅	1	2	yes	—
151	C₄²	C₃×C₄×D₅	1	2	yes	—
152	C₄²	C₂×C₃×Dic₅	1	1	yes	characteristic
153	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
154	C₃×C₅	#1455	1	1	yes	characteristic, complemented, Hall
155	C₂²×C₃	—	2	2	no	—
156	C₂²×C₃	—	2	1	no	—
157	C₂²×C₃	C₂×C₄×Dic₅	1	1	yes	characteristic
158	C₃×C₄	—	2	2	no	—
159	C₃×C₄	—	2	1	no	—
160	C₃×C₄	D₅×M₁₆	1	4	yes	—
161	C₃×C₄	C₂×#681	1	2	yes	—
162	C₂×C₅	—	4	1	no	—
163	C₂×C₅	C₃×C₄×M₁₆	1	2	yes	—
164	C₂×C₅	C₃×C₄×M₁₆	1	1	yes	characteristic
165	C₂³	C₃×C₄×Dic₅	1	1	yes	characteristic
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	C₃×C₄×Dic₅	1	1	yes	characteristic
169	C₂×C₄	C₃×C₄×Dic₅	1	2	yes	—
170	C₂×C₄	C₂×C₃×C₄×D₅	1	2	yes	—
171	C₂×C₄	C₂²×C₃×Dic₅	1	1	yes	characteristic
172	C₈	—	2	2	no	—
173	C₈	—	5	8	no	—
174	C₈	—	10	4	no	—
175	C₈	C₃×#681	1	4	yes	complemented
176	C₂×C₃	—	4	1	no	—
177	C₂×C₃	C₄×#681	1	1	yes	characteristic
178	C₂×C₃	M₁₆×Dic₅	1	2	yes	—
179	C₅	C₃×#1455	1	1	yes	characteristic, complemented, Sylow
180	C₂²	—	2	2	no	—
181	C₂²	—	2	1	no	—
182	C₂²	C₂×C₃×C₄×Dic₅	1	1	yes	characteristic
183	C₄	—	2	2	no	—
184	C₄	—	2	1	no	—
185	C₄	C₃×D₅×M₁₆	1	4	yes	—
186	C₄	C₂×C₃×#681	1	2	yes	—
187	C₃	#10600642	1	1	yes	characteristic, direct product factor, Sylow
188	C₂	—	4	1	no	—
189	C₂	C₃×C₄×#681	1	1	yes	characteristic
190	C₂	C₃×M₁₆×Dic₅	1	2	yes	—

	subgroup	G/H
center	C₃×C₄²	C₂²×D₅
commutator	C₂²×C₅	C₂×C₃×C₄²
Frattini	C₄²	C₂²×C₃×D₅
Fitting	C₃×C₄×C₅×M₁₆	C₂
socle	C₂²×C₃×C₅	C₂×C₄²
prime radical	C₂³×C₃×C₅	C₄²
cosocle	C₄²×C₅	C₂³×C₃
hypercenter	C₃×C₄×M₁₆	D₅
hypocenter	C₅	C₃×#1455
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₃×C₄² ◁ C₃×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. C₃×#64675	C₂	semidirect
4. C₃×#64674	C₂	semidirect
5. C₃×C₈×#168	C₂	semidirect
6. #10600642	C₃	direct (N × H)
19. #64675	C₂×C₃	semidirect
20. #64674	C₂×C₃	semidirect
21. C₈×#168	C₂×C₃	semidirect
56. C₃×C₅×C₈	M₁₆	semidirect
58. C₃×#168	M₁₆	semidirect
110. C₅×C₈	C₃×M₁₆	semidirect
112. #168	C₃×M₁₆	semidirect
135. C₃×C₈	#681	semidirect
154. C₃×C₅	#1455	semidirect
175. C₈	C₃×#681	semidirect
179. C₅	C₃×#1455	semidirect
187. C₃	#10600642	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×C₄×#681	2
C₃×#64675	2
C₃×#64674	2
C₃×C₈×#168	2
#10600642	3
C₃×#1455	5

Subgroups — C3×#10600642 #14280707

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10600642 #14280707