Subgroups — C₃×#10600072 #14315128

Order 1920.

324 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#3930	C₂	1	2	yes	maximal, complemented
2	C₃×C₈×D₂₀	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#64727	C₂	1	2	yes	maximal, complemented
5	C₃×C₄×#672	C₂	1	1	yes	maximal, characteristic, complemented
6	#10600072	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₈×D₅	C₂²	1	2	yes	—
8	C₃×C₄×D₂₀	C₄	1	1	yes	characteristic
9	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
10	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₃×#3952	C₄	1	2	yes	—
12	C₂×C₃×#672	C₂²	1	1	yes	characteristic
13	C₂×C₃×#672	C₂²	1	1	yes	characteristic
14	C₃×C₄⋊#168	C₄	1	1	yes	characteristic
15	C₃×#3930	—	2	4	no	—
16	C₃×#2169	—	5	1	no	maximal, Hall
17	C₂×#3930	C₂×C₃	1	2	yes	complemented
18	C₈×D₂₀	C₂×C₃	1	1	yes	characteristic
19	C₅×C₄⋊C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
20	#64727	C₂×C₃	1	2	yes	complemented
21	C₄×#672	C₂×C₃	1	1	yes	characteristic, complemented
22	C₂×C₃×D₂₀	C₈	1	1	yes	characteristic
23	C₂×C₃×C₄×D₅	C₂×C₄	1	2	yes	—
24	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
26	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
27	C₃×#685	C₈	1	2	yes	—
28	C₃×C₄⋊Dic₅	C₈	1	1	yes	characteristic
29	C₂×C₃×#168	C₂×C₄	1	2	yes	—
30	C₃×C₈×D₅	—	2	2	no	—
31	C₃×C₈×D₅	—	2	2	no	—
32	C₃×C₅×C₁₆	—	2	2	no	—
33	C₃×#672	—	2	1	no	—
34	C₃×#672	D₄	1	2	yes	complemented
35	C₂×C₃×M₃₂	—	5	2	no	—
36	C₃×C₈×D₄	—	5	1	no	—
37	C₃×C₄⋊C₁₆	D₅	1	1	yes	characteristic, complemented
38	C₃×C₂²⋊C₁₆	—	5	2	no	—
39	C₃×C₄×C₁₆	—	5	1	no	—
40	C₂×C₈×D₅	C₂²×C₃	1	2	yes	—
41	C₄×D₂₀	C₃×C₄	1	1	yes	characteristic
42	C₂×C₅×C₁₆	C₂²×C₃	1	2	yes	—
43	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
44	#3952	C₃×C₄	1	2	yes	—
45	C₂×#672	C₂²×C₃	1	1	yes	characteristic
46	C₂×#672	C₂²×C₃	1	1	yes	characteristic
47	C₄⋊#168	C₃×C₄	1	1	yes	characteristic
48	#3930	—	2	4	no	—
49	#2169	—	5	1	no	Sylow
50	C₂²×C₃×D₅	C₂×C₈	1	2	yes	—
51	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₃×Dic₅	C₂×C₈	1	2	yes	—
55	C₃×D₂₀	—	2	2	no	—
56	C₃×C₄×D₅	—	2	2	no	—
57	C₃×C₄×D₅	—	2	2	no	—
58	C₃×C₅×C₈	—	2	1	no	—
59	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
60	C₃×C₅×C₈	#41	1	1	yes	characteristic
61	C₃×#168	—	2	2	no	—
62	C₂²×C₃×C₈	—	5	2	no	—
63	C₃×C₄×D₄	—	5	1	no	—
64	C₃×M₃₂	—	10	4	no	—
65	C₂×C₃×C₁₆	—	5	1	no	—
66	C₂×C₃×C₁₆	—	5	1	no	—
67	C₂×C₃×C₁₆	C₂×D₅	1	2	yes	—
68	C₃×C₄⋊C₈	—	5	1	no	—
69	C₃×C₂²⋊C₈	—	5	2	no	—
70	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
71	C₂×D₂₀	C₃×C₈	1	1	yes	characteristic
72	C₂×C₄×D₅	C₂×C₃×C₄	1	2	yes	—
73	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
74	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
75	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
76	#685	C₃×C₈	1	2	yes	—
77	C₄⋊Dic₅	C₃×C₈	1	1	yes	characteristic
78	C₂×#168	C₂×C₃×C₄	1	2	yes	—
79	C₈×D₅	—	2	2	no	—
80	C₈×D₅	—	2	2	no	—
81	C₅×C₁₆	—	2	2	no	—
82	#672	—	2	1	no	—
83	#672	C₃×D₄	1	2	yes	complemented
84	C₂×M₃₂	—	5	2	no	—
85	C₈×D₄	—	5	1	no	—
86	C₄⋊C₁₆	C₃×D₅	1	1	yes	characteristic, complemented
87	C₂²⋊C₁₆	—	5	2	no	—
88	C₄×C₁₆	—	5	1	no	—
89	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₃×D₅	—	2	2	no	—
91	C₂×C₃×D₅	—	2	2	no	—
92	C₂×C₃×D₅	—	2	2	no	—
93	C₃×C₄×C₅	—	2	1	no	—
94	C₃×C₄×C₅	M₃₂	1	2	yes	complemented
95	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
96	C₃×C₄×C₅	#131	1	1	yes	characteristic
97	C₃×Dic₅	—	2	2	no	—
98	C₂×C₃×D₄	—	5	1	no	—
99	C₂²×C₃×C₄	—	5	2	no	—
100	C₂×C₃×C₈	—	5	2	no	—
101	C₂×C₃×C₈	—	10	2	no	—
102	C₂×C₃×C₈	—	10	2	no	—
103	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
104	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
105	C₃×C₄⋊C₄	—	5	1	no	—
106	C₃×C₂²⋊C₄	—	5	2	no	—
107	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
108	C₃×C₁₆	—	2	2	no	—
109	C₃×C₁₆	—	5	2	no	—
110	C₃×C₁₆	—	10	1	no	—
111	C₂²×D₅	C₂×C₃×C₈	1	2	yes	—
112	C₂×C₄×C₅	C₂×C₃×C₈	1	1	yes	characteristic
113	C₂×C₄×C₅	C₂×C₃×C₈	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
115	C₂×Dic₅	C₂×C₃×C₈	1	2	yes	—
116	D₂₀	—	2	2	no	—
117	C₄×D₅	—	2	2	no	—
118	C₄×D₅	—	2	2	no	—
119	C₅×C₈	—	2	1	no	—
120	C₅×C₈	C₂×C₃×D₄	1	1	yes	characteristic
121	C₅×C₈	C₃×#41	1	1	yes	characteristic
122	#168	—	2	2	no	—
123	C₂²×C₈	—	5	2	no	—
124	C₄×D₄	—	5	1	no	—
125	M₃₂	—	10	4	no	—
126	C₂×C₁₆	—	5	1	no	—
127	C₂×C₁₆	—	5	1	no	—
128	C₂×C₁₆	C₂×C₃×D₅	1	2	yes	—
129	C₄⋊C₈	—	5	1	no	—
130	C₂²⋊C₈	—	5	2	no	—
131	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic
132	C₂×C₃×C₅	C₈×D₄	1	1	yes	characteristic
133	C₂×C₃×C₅	C₂×M₃₂	1	1	yes	characteristic
134	C₂×C₃×C₅	#504	1	1	yes	characteristic
135	C₃×D₅	—	4	2	no	—
136	C₂³×C₃	—	5	2	no	—
137	C₃×D₄	—	10	2	no	—
138	C₂×C₃×C₄	—	5	2	no	—
139	C₂×C₃×C₄	—	10	2	no	—
140	C₂×C₃×C₄	—	10	2	no	—
141	C₂×C₃×C₄	C₈×D₅	1	1	yes	characteristic
142	C₂×C₃×C₄	C₈×D₅	1	1	yes	characteristic
143	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
144	C₃×C₈	—	2	1	no	—
145	C₃×C₈	—	10	2	no	—
146	C₃×C₈	D₄×D₅	1	1	yes	characteristic
147	C₃×C₈	#713	1	1	yes	characteristic
148	C₂²×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
149	C₂×D₅	—	2	2	no	—
150	C₂×D₅	—	2	2	no	—
151	C₂×D₅	—	2	2	no	—
152	C₄×C₅	—	2	1	no	—
153	C₄×C₅	C₃×M₃₂	1	2	yes	complemented
154	C₄×C₅	C₃×C₄×D₄	1	1	yes	characteristic
155	C₄×C₅	C₃×#131	1	1	yes	characteristic
156	Dic₅	—	2	2	no	—
157	C₂×D₄	—	5	1	no	—
158	C₂²×C₄	—	5	2	no	—
159	C₂×C₈	—	5	2	no	—
160	C₂×C₈	—	10	2	no	—
161	C₂×C₈	—	10	2	no	—
162	C₂×C₈	C₃×C₄×D₅	1	1	yes	characteristic
163	C₂×C₈	C₂²×C₃×D₅	1	1	yes	characteristic
164	C₄⋊C₄	—	5	1	no	—
165	C₂²⋊C₄	—	5	2	no	—
166	C₄²	C₃×C₄×D₅	1	1	yes	characteristic
167	C₁₆	—	2	2	no	—
168	C₁₆	—	5	2	no	—
169	C₁₆	—	10	1	no	—
170	C₃×C₅	#2169	1	1	yes	characteristic, complemented, Hall
171	C₂²×C₃	—	10	2	no	—
172	C₂²×C₃	—	10	2	no	—
173	C₂²×C₃	—	10	2	no	—
174	C₂²×C₃	C₂×C₈×D₅	1	1	yes	characteristic
175	C₃×C₄	—	2	1	no	—
176	C₃×C₄	—	10	2	no	—
177	C₃×C₄	#3930	1	2	yes	complemented
178	C₃×C₄	#4039	1	1	yes	characteristic
179	C₃×C₄	#4053	1	1	yes	characteristic
180	C₂×C₅	C₃×C₈×D₄	1	1	yes	characteristic
181	C₂×C₅	C₂×C₃×M₃₂	1	1	yes	characteristic
182	C₂×C₅	C₃×#504	1	1	yes	characteristic
183	D₅	—	4	2	no	—
184	C₂³	—	5	2	no	—
185	D₄	—	10	2	no	—
186	C₂×C₄	—	5	2	no	—
187	C₂×C₄	—	10	2	no	—
188	C₂×C₄	—	10	2	no	—
189	C₂×C₄	C₃×C₈×D₅	1	1	yes	characteristic
190	C₂×C₄	C₃×C₈×D₅	1	1	yes	characteristic
191	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
192	C₈	—	2	1	no	—
193	C₈	—	10	2	no	—
194	C₈	C₃×D₄×D₅	1	1	yes	characteristic
195	C₈	C₃×#713	1	1	yes	characteristic
196	C₂×C₃	—	20	2	no	—
197	C₂×C₃	#65123	1	1	yes	characteristic
198	C₂×C₃	C₂×#3930	1	1	yes	characteristic
199	C₂×C₃	#65196	1	1	yes	characteristic
200	C₅	C₃×#2169	1	1	yes	characteristic, complemented, Sylow
201	C₂²	—	10	2	no	—
202	C₂²	—	10	2	no	—
203	C₂²	—	10	2	no	—
204	C₂²	C₂×C₃×C₈×D₅	1	1	yes	characteristic
205	C₄	—	2	1	no	—
206	C₄	—	10	2	no	—
207	C₄	C₃×#3930	1	2	yes	complemented
208	C₄	C₃×#4039	1	1	yes	characteristic
209	C₄	C₃×#4053	1	1	yes	characteristic
210	C₃	#10600072	1	1	yes	characteristic, direct product factor, Sylow
211	C₂	—	20	2	no	—
212	C₂	C₃×#65123	1	1	yes	characteristic
213	C₂	C₂×C₃×#3930	1	1	yes	characteristic
214	C₂	C₃×#65196	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₈	C₂²×D₅
commutator	C₂²×C₅	C₂²×C₃×C₈
Frattini	C₂×C₈	C₂²×C₃×D₅
Fitting	C₃×C₅×C₄⋊C₁₆	C₂
socle	C₂²×C₃×C₅	C₂²×C₈
prime radical	C₂×C₃×D₂₀	C₈
cosocle	C₂×C₅×C₈	C₂³×C₃
hypercenter	C₃×C₄⋊C₁₆	D₅
hypocenter	C₅	C₃×#2169
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₈ ◁ C₃×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#3930	C₂	semidirect
3. C₃×C₅×C₄⋊C₁₆	C₂	semidirect
4. C₃×#64727	C₂	semidirect
5. C₃×C₄×#672	C₂	semidirect
6. #10600072	C₃	direct (N × H)
17. C₂×#3930	C₂×C₃	semidirect
19. C₅×C₄⋊C₁₆	C₂×C₃	semidirect
20. #64727	C₂×C₃	semidirect
21. C₄×#672	C₂×C₃	semidirect
34. C₃×#672	D₄	semidirect
37. C₃×C₄⋊C₁₆	D₅	semidirect
83. #672	C₃×D₄	semidirect
86. C₄⋊C₁₆	C₃×D₅	semidirect
94. C₃×C₄×C₅	M₃₂	semidirect
153. C₄×C₅	C₃×M₃₂	semidirect
170. C₃×C₅	#2169	semidirect
177. C₃×C₄	#3930	semidirect
200. C₅	C₃×#2169	semidirect
207. C₄	C₃×#3930	semidirect
210. C₃	#10600072	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×#3930	2
C₃×C₈×D₂₀	2
C₃×C₅×C₄⋊C₁₆	2
C₃×#64727	2
C₃×C₄×#672	2
#10600072	3
C₃×#2169	5

Subgroups — C3×#10600072 #14315128

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10600072 #14315128