Subgroups — C₃×#10599880 #14312747

Order 1920.

320 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×D₅×M₃₂	C₂	1	2	yes	maximal
2	C₃×#64997	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#347	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#64732	C₂	1	2	yes	maximal, complemented
5	C₃×#64686	C₂	1	1	yes	maximal, characteristic, complemented
6	#10599880	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×#676	C₄	1	1	yes	characteristic
8	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
9	C₃×#4018	C₄	1	1	yes	characteristic
10	C₃×C₅×M₃₂	C₂²	1	2	yes	—
11	C₃×C₅×#97	C₂²	1	1	yes	characteristic
12	C₃×#3947	C₂²	1	1	yes	characteristic
13	C₃×C₈×Dic₅	C₄	1	1	yes	characteristic
14	C₃×#3944	C₂²	1	2	yes	—
15	C₃×C₄×#168	C₄	1	1	yes	characteristic
16	C₃×#3930	C₄	1	4	yes	complemented
17	C₃×D₅×C₁₆	C₄	1	4	yes	complemented
18	C₃×#2110	—	5	1	no	maximal, Hall
19	D₅×M₃₂	C₂×C₃	1	2	yes	—
20	#64997	C₂×C₃	1	1	yes	characteristic
21	C₅×#347	C₂×C₃	1	1	yes	characteristic, complemented
22	#64732	C₂×C₃	1	2	yes	complemented
23	#64686	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂×C₃×C₄×D₅	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	C₂×C₄	1	1	yes	characteristic
28	C₃×#685	—	2	1	no	—
29	C₃×#683	—	2	1	no	—
30	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
33	C₃×#676	—	2	2	no	—
34	C₃×C₈×D₅	C₂×C₄	1	2	yes	—
35	C₃×C₈×D₅	C₂×C₄	1	2	yes	—
36	C₃×C₅×C₁₆	C₂×C₄	1	4	yes	—
37	C₃×#672	C₂×C₄	1	4	yes	—
38	C₂×C₃×M₃₂	—	5	2	no	—
39	C₃×#405	—	5	1	no	—
40	C₃×#347	—	5	2	no	—
41	C₃×#347	—	5	1	no	—
42	C₃×#347	D₅	1	1	yes	characteristic, complemented
43	C₂×#676	C₃×C₄	1	1	yes	characteristic
44	C₂×C₈×D₅	C₂²×C₃	1	1	yes	characteristic
45	#4018	C₃×C₄	1	1	yes	characteristic
46	C₅×M₃₂	C₂²×C₃	1	2	yes	—
47	C₅×#97	C₂²×C₃	1	1	yes	characteristic
48	#3947	C₂²×C₃	1	1	yes	characteristic
49	C₈×Dic₅	C₃×C₄	1	1	yes	characteristic
50	#3944	C₂²×C₃	1	2	yes	—
51	C₄×#168	C₃×C₄	1	1	yes	characteristic
52	#3930	C₃×C₄	1	4	yes	complemented
53	D₅×C₁₆	C₃×C₄	1	4	yes	complemented
54	#2110	—	5	1	no	Sylow
55	C₂²×C₃×D₅	M₁₆	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	—	2	1	no	—
57	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₃×Dic₅	—	2	1	no	—
59	C₂×C₃×Dic₅	M₁₆	1	1	yes	characteristic
60	C₃×C₄×D₅	C₄²	1	2	yes	—
61	C₃×C₄×D₅	M₁₆	1	2	yes	—
62	C₃×C₅×C₈	—	2	1	no	—
63	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
64	C₃×#168	—	2	1	no	—
65	C₃×#168	C₄²	1	1	yes	characteristic
66	C₃×#168	C₄²	1	1	yes	characteristic
67	C₂×C₃×M₁₆	—	5	1	no	—
68	C₂²×C₃×C₈	—	5	1	no	—
69	C₃×#117	—	5	1	no	—
70	C₃×M₃₂	—	5	2	no	—
71	C₃×M₃₂	—	5	4	no	—
72	C₃×M₃₂	C₂×D₅	1	2	yes	—
73	C₂×C₃×C₁₆	—	5	4	no	—
74	C₃×#97	—	5	1	no	—
75	C₃×#97	C₂×D₅	1	1	yes	characteristic
76	C₃×C₄×C₈	—	5	1	no	—
77	C₃×C₄×C₈	—	5	1	no	—
78	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
79	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
80	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
81	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
82	#685	—	2	1	no	—
83	#683	—	2	1	no	—
84	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
85	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
86	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
87	#676	—	2	2	no	—
88	C₈×D₅	C₂×C₃×C₄	1	2	yes	—
89	C₈×D₅	C₂×C₃×C₄	1	2	yes	—
90	C₅×C₁₆	C₂×C₃×C₄	1	4	yes	—
91	#672	C₂×C₃×C₄	1	4	yes	—
92	C₂×M₃₂	—	5	2	no	—
93	#405	—	5	1	no	—
94	#347	—	5	2	no	—
95	#347	—	5	1	no	—
96	#347	C₃×D₅	1	1	yes	characteristic, complemented
97	C₂²×C₃×C₅	C₂×M₁₆	1	1	yes	characteristic
98	C₂×C₃×D₅	—	4	1	no	—
99	C₂×C₃×D₅	#97	1	2	yes	—
100	C₃×C₄×C₅	—	4	1	no	—
101	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
102	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
103	C₃×Dic₅	—	4	1	no	—
104	C₃×Dic₅	#97	1	2	yes	—
105	C₂²×C₃×C₄	—	5	1	no	—
106	C₃×M₁₆	—	10	2	no	—
107	C₂×C₃×C₈	—	5	1	no	—
108	C₂×C₃×C₈	—	5	2	no	—
109	C₂×C₃×C₈	—	5	2	no	—
110	C₂×C₃×C₈	—	5	1	no	—
111	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
112	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
113	C₃×C₄⋊C₄	—	10	1	no	—
114	C₃×C₂²⋊C₄	—	10	1	no	—
115	C₃×C₄²	—	5	1	no	—
116	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
117	C₃×C₁₆	—	5	4	no	—
118	C₃×C₁₆	C₄×D₅	1	4	yes	—
119	C₂²×D₅	C₃×M₁₆	1	1	yes	characteristic
120	C₂×C₄×C₅	—	2	1	no	—
121	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
122	C₂×Dic₅	—	2	1	no	—
123	C₂×Dic₅	C₃×M₁₆	1	1	yes	characteristic
124	C₄×D₅	C₃×C₄²	1	2	yes	—
125	C₄×D₅	C₃×M₁₆	1	2	yes	—
126	C₅×C₈	—	2	1	no	—
127	C₅×C₈	C₂²×C₃×C₄	1	2	yes	—
128	#168	—	2	1	no	—
129	#168	C₃×C₄²	1	1	yes	characteristic
130	#168	C₃×C₄²	1	1	yes	characteristic
131	C₂×M₁₆	—	5	1	no	—
132	C₂²×C₈	—	5	1	no	—
133	#117	—	5	1	no	—
134	M₃₂	—	5	2	no	—
135	M₃₂	—	5	4	no	—
136	M₃₂	C₂×C₃×D₅	1	2	yes	—
137	C₂×C₁₆	—	5	4	no	—
138	#97	—	5	1	no	—
139	#97	C₂×C₃×D₅	1	1	yes	characteristic
140	C₄×C₈	—	5	1	no	—
141	C₄×C₈	—	5	1	no	—
142	C₂×C₃×C₅	—	2	1	no	—
143	C₂×C₃×C₅	C₂×#97	1	1	yes	characteristic
144	C₃×D₅	—	2	2	no	—
145	C₂³×C₃	—	5	1	no	—
146	C₂×C₃×C₄	—	2	1	no	—
147	C₂×C₃×C₄	—	5	1	no	—
148	C₂×C₃×C₄	—	5	2	no	—
149	C₂×C₃×C₄	—	5	2	no	—
150	C₂×C₃×C₄	—	10	1	no	—
151	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
152	C₃×C₈	—	2	1	no	—
153	C₃×C₈	—	5	1	no	—
154	C₃×C₈	—	5	1	no	—
155	C₃×C₈	—	10	1	no	—
156	C₃×C₈	C₂×C₄×D₅	1	2	yes	—
157	C₂²×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
158	C₂×D₅	—	4	1	no	—
159	C₂×D₅	C₃×#97	1	2	yes	—
160	C₄×C₅	—	4	1	no	—
161	C₄×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
162	C₄×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
163	Dic₅	—	4	1	no	—
164	Dic₅	C₃×#97	1	2	yes	—
165	C₂²×C₄	—	5	1	no	—
166	M₁₆	—	10	2	no	—
167	C₂×C₈	—	5	1	no	—
168	C₂×C₈	—	5	2	no	—
169	C₂×C₈	—	5	2	no	—
170	C₂×C₈	—	5	1	no	—
171	C₂×C₈	C₃×C₄×D₅	1	1	yes	characteristic
172	C₂×C₈	C₂²×C₃×D₅	1	1	yes	characteristic
173	C₄⋊C₄	—	10	1	no	—
174	C₂²⋊C₄	—	10	1	no	—
175	C₄²	—	5	1	no	—
176	C₄²	C₃×C₄×D₅	1	1	yes	characteristic
177	C₁₆	—	5	4	no	—
178	C₁₆	C₃×C₄×D₅	1	4	yes	—
179	C₃×C₅	#2110	1	1	yes	characteristic, complemented, Hall
180	C₂²×C₃	—	5	2	no	—
181	C₂²×C₃	—	20	1	no	—
182	C₂²×C₃	D₅×M₁₆	1	1	yes	characteristic
183	C₃×C₄	—	4	1	no	—
184	C₃×C₄	—	5	2	no	—
185	C₃×C₄	—	20	1	no	—
186	C₃×C₄	C₄²×D₅	1	1	yes	characteristic
187	C₃×C₄	D₅×M₁₆	1	1	yes	characteristic
188	C₂×C₅	—	2	1	no	—
189	C₂×C₅	C₂×C₃×#97	1	1	yes	characteristic
190	D₅	—	2	2	no	—
191	C₂³	—	5	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	5	1	no	—
194	C₂×C₄	—	5	2	no	—
195	C₂×C₄	—	5	2	no	—
196	C₂×C₄	—	10	1	no	—
197	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
198	C₈	—	2	1	no	—
199	C₈	—	5	1	no	—
200	C₈	—	5	1	no	—
201	C₈	—	10	1	no	—
202	C₈	C₂×C₃×C₄×D₅	1	2	yes	—
203	C₂×C₃	—	2	1	no	—
204	C₂×C₃	—	10	2	no	—
205	C₂×C₃	D₅×#97	1	1	yes	characteristic
206	C₅	C₃×#2110	1	1	yes	characteristic, complemented, Sylow
207	C₂²	—	5	2	no	—
208	C₂²	—	20	1	no	—
209	C₂²	C₃×D₅×M₁₆	1	1	yes	characteristic
210	C₄	—	4	1	no	—
211	C₄	—	5	2	no	—
212	C₄	—	20	1	no	—
213	C₄	C₃×C₄²×D₅	1	1	yes	characteristic
214	C₄	C₃×D₅×M₁₆	1	1	yes	characteristic
215	C₃	#10599880	1	1	yes	characteristic, direct product factor, Sylow
216	C₂	—	2	1	no	—
217	C₂	—	10	2	no	—
218	C₂	C₃×D₅×#97	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	C₄²×D₅
commutator	C₄×C₅	C₂×C₃×C₄²
Frattini	C₂×C₈	C₂²×C₃×D₅
Fitting	C₃×C₅×#347	C₂
socle	C₂×C₃×C₅	C₂×#97
prime radical	C₂²×C₃×D₅	M₁₆
cosocle	C₂×C₅×C₈	C₂³×C₃
hypercenter	C₃×#347	D₅
hypocenter	C₅	C₃×#2110
LCS	G ▷ C₄×C₅ ▷ C₅
UCS	C₁ ◁ C₃×C₄ ◁ C₃×#347

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×#347	C₂	semidirect
4. C₃×#64732	C₂	semidirect
5. C₃×#64686	C₂	semidirect
6. #10599880	C₃	direct (N × H)
16. C₃×#3930	C₄	semidirect
17. C₃×D₅×C₁₆	C₄	semidirect
21. C₅×#347	C₂×C₃	semidirect
22. #64732	C₂×C₃	semidirect
23. #64686	C₂×C₃	semidirect
42. C₃×#347	D₅	semidirect
52. #3930	C₃×C₄	semidirect
53. D₅×C₁₆	C₃×C₄	semidirect
96. #347	C₃×D₅	semidirect
179. C₃×C₅	#2110	semidirect
206. C₅	C₃×#2110	semidirect
215. C₃	#10599880	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×D₅×M₃₂	2
C₃×#64997	2
C₃×C₅×#347	2
C₃×#64732	2
C₃×#64686	2
#10599880	3
C₃×#2110	5

Subgroups — C3×#10599880 #14312747

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10599880 #14312747