Subgroups — C₃×#10602598 #14284709

Order 1920.

492 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#532	C₂	1	1	yes	maximal, characteristic
2	C₃×#65214	C₂	1	2	yes	maximal, complemented
3	C₃×#64766	C₂	1	2	yes	maximal, complemented
4	C₃×#64764	C₂	1	2	yes	maximal, complemented
5	#10602598	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₃×C₅×#41	C₄	1	2	yes	—
7	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
8	C₃×C₅×#127	C₂²	1	1	yes	characteristic
9	C₃×C₅×#124	C₄	1	2	yes	—
10	C₃×C₅×#121	—	2	2	no	—
11	C₃×C₄×C₅×D₄	—	2	2	no	—
12	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₃×C₂²⋊#168	—	2	2	no	—
14	C₃×C₄⋊#168	C₂²	1	4	yes	—
15	C₃×C₄×#168	—	2	2	no	—
16	C₃×#1543	—	5	1	no	maximal, Hall
17	C₅×#532	C₂×C₃	1	1	yes	characteristic
18	#65214	C₂×C₃	1	2	yes	complemented
19	#64766	C₂×C₃	1	2	yes	complemented
20	#64764	C₂×C₃	1	2	yes	complemented
21	C₃×C₅×#41	—	4	4	no	—
22	C₂×C₃×C₅×Q₈	C₂×C₄	1	2	yes	—
23	C₂×C₃×C₅×D₄	—	2	2	no	—
24	C₂×C₃×C₅×D₄	—	2	2	no	—
25	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
26	C₂²×C₃×C₄×C₅	—	2	2	no	—
27	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
29	C₃×C₅×C₄⋊C₄	—	2	2	no	—
30	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
31	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
32	C₃×C₄²×C₅	D₄	1	2	yes	—
33	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
34	C₂×C₃×#168	—	2	4	no	—
35	C₃×#532	D₅	1	1	yes	characteristic
36	C₃×#431	—	5	2	no	—
37	C₃×#333	—	5	2	no	—
38	C₃×#331	—	5	2	no	—
39	C₂×C₅×#41	C₃×C₄	1	2	yes	—
40	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
41	C₅×#127	C₂²×C₃	1	1	yes	characteristic
42	C₅×#124	C₃×C₄	1	2	yes	—
43	C₅×#121	—	2	2	no	—
44	C₄×C₅×D₄	—	2	2	no	—
45	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
46	C₂²⋊#168	—	2	2	no	—
47	C₄⋊#168	C₂²×C₃	1	4	yes	—
48	C₄×#168	—	2	2	no	—
49	#1543	—	5	1	no	Sylow
50	C₂³×C₃×C₅	—	2	2	no	—
51	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
52	C₃×C₅×Q₈	—	4	2	no	—
53	C₃×C₅×D₄	—	4	2	no	—
54	C₃×C₅×D₄	—	4	4	no	—
55	C₃×C₅×D₄	—	4	4	no	—
56	C₂×C₃×C₄×C₅	—	2	2	no	—
57	C₂×C₃×C₄×C₅	—	2	4	no	—
58	C₂×C₃×C₄×C₅	—	2	2	no	—
59	C₂×C₃×C₄×C₅	—	4	4	no	—
60	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
61	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
64	C₃×#168	—	4	4	no	—
65	C₂×C₃×#41	Dic₅	1	2	yes	—
66	C₃×C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic
67	C₃×#127	C₂×D₅	1	1	yes	characteristic
68	C₃×#124	Dic₅	1	2	yes	—
69	C₃×#121	—	2	2	no	—
70	C₃×C₄×D₄	—	2	2	no	—
71	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
72	C₃×C₄⋊C₈	—	5	4	no	—
73	C₃×C₂²⋊C₈	—	10	2	no	—
74	C₃×C₄×C₈	—	10	2	no	—
75	C₅×#41	—	4	4	no	—
76	C₂×C₅×Q₈	C₂×C₃×C₄	1	2	yes	—
77	C₂×C₅×D₄	—	2	2	no	—
78	C₂×C₅×D₄	—	2	2	no	—
79	C₂×C₅×D₄	C₂×C₃×C₄	1	2	yes	—
80	C₂²×C₄×C₅	—	2	2	no	—
81	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
83	C₅×C₄⋊C₄	—	2	2	no	—
84	C₅×C₂²⋊C₄	—	2	4	no	—
85	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
86	C₄²×C₅	C₃×D₄	1	2	yes	—
87	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
88	C₂×#168	—	2	4	no	—
89	#532	C₃×D₅	1	1	yes	characteristic
90	#431	—	5	2	no	—
91	#333	—	5	2	no	—
92	#331	—	5	2	no	—
93	C₂²×C₃×C₅	—	2	1	no	—
94	C₂²×C₃×C₅	—	2	2	no	—
95	C₂²×C₃×C₅	—	4	2	no	—
96	C₂²×C₃×C₅	—	4	4	no	—
97	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₃×C₄×C₅	—	2	1	no	—
99	C₃×C₄×C₅	—	2	2	no	—
100	C₃×C₄×C₅	—	4	2	no	—
101	C₃×C₄×C₅	#100	1	2	yes	—
102	C₃×C₄×C₅	#135	1	4	yes	—
103	C₃×#41	—	4	4	no	—
104	C₂×C₃×Q₈	C₂×Dic₅	1	2	yes	—
105	C₂×C₃×D₄	—	2	2	no	—
106	C₂×C₃×D₄	—	2	2	no	—
107	C₂×C₃×D₄	C₂×Dic₅	1	2	yes	—
108	C₂²×C₃×C₄	—	2	2	no	—
109	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
110	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
111	C₂×C₃×C₈	—	10	4	no	—
112	C₃×C₄⋊C₄	—	2	2	no	—
113	C₃×C₂²⋊C₄	—	2	4	no	—
114	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
115	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
116	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
117	C₂³×C₅	—	2	2	no	—
118	C₂³×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
119	C₅×Q₈	—	4	2	no	—
120	C₅×D₄	—	4	2	no	—
121	C₅×D₄	—	4	4	no	—
122	C₅×D₄	—	4	4	no	—
123	C₂×C₄×C₅	—	2	2	no	—
124	C₂×C₄×C₅	—	2	4	no	—
125	C₂×C₄×C₅	—	2	2	no	—
126	C₂×C₄×C₅	—	4	4	no	—
127	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
128	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
131	#168	—	4	4	no	—
132	C₂×#41	C₃×Dic₅	1	2	yes	—
133	C₄⋊Q₈	C₂×C₃×D₅	1	1	yes	characteristic
134	#127	C₂×C₃×D₅	1	1	yes	characteristic
135	#124	C₃×Dic₅	1	2	yes	—
136	#121	—	2	2	no	—
137	C₄×D₄	—	2	2	no	—
138	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
139	C₄⋊C₈	—	5	4	no	—
140	C₂²⋊C₈	—	10	2	no	—
141	C₄×C₈	—	10	2	no	—
142	C₂×C₃×C₅	—	4	1	no	—
143	C₂×C₃×C₅	—	8	2	no	—
144	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
145	C₂×C₃×C₅	#416	1	2	yes	—
146	C₂³×C₃	—	2	2	no	—
147	C₂³×C₃	C₂²⋊Dic₅	1	1	yes	characteristic
148	C₃×Q₈	—	4	2	no	—
149	C₃×D₄	—	4	2	no	—
150	C₃×D₄	—	4	4	no	—
151	C₃×D₄	—	4	4	no	—
152	C₂×C₃×C₄	—	2	4	no	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	—	2	2	no	—
155	C₂×C₃×C₄	—	4	4	no	—
156	C₂×C₃×C₄	C₂²⋊Dic₅	1	2	yes	—
157	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
158	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
159	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
160	C₃×C₈	—	20	4	no	—
161	C₂²×C₅	—	2	1	no	—
162	C₂²×C₅	—	2	2	no	—
163	C₂²×C₅	—	4	2	no	—
164	C₂²×C₅	—	4	4	no	—
165	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
166	C₄×C₅	—	2	1	no	—
167	C₄×C₅	—	2	2	no	—
168	C₄×C₅	—	4	2	no	—
169	C₄×C₅	C₃×#100	1	2	yes	—
170	C₄×C₅	C₃×#135	1	4	yes	—
171	#41	—	4	4	no	—
172	C₂×Q₈	C₂×C₃×Dic₅	1	2	yes	—
173	C₂×D₄	—	2	2	no	—
174	C₂×D₄	—	2	2	no	—
175	C₂×D₄	C₂×C₃×Dic₅	1	2	yes	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	C₂×C₃×Dic₅	1	1	yes	characteristic
178	C₂²×C₄	C₃×C₂²⋊D₅	1	2	yes	—
179	C₂×C₈	—	10	4	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₂²⋊C₄	—	2	4	no	—
182	C₄²	C₂×C₃×Dic₅	1	1	yes	characteristic
183	C₄²	C₃×C₂²⋊D₅	1	2	yes	—
184	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
185	C₃×C₅	#1543	1	1	yes	characteristic, complemented, Hall
186	C₂²×C₃	—	2	1	no	—
187	C₂²×C₃	—	2	2	no	—
188	C₂²×C₃	—	4	2	no	—
189	C₂²×C₃	—	4	4	no	—
190	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
191	C₃×C₄	—	2	1	no	—
192	C₃×C₄	—	2	2	no	—
193	C₃×C₄	—	4	2	no	—
194	C₃×C₄	#3965	1	2	yes	—
195	C₃×C₄	#4096	1	4	yes	—
196	C₂×C₅	—	4	1	no	—
197	C₂×C₅	—	8	2	no	—
198	C₂×C₅	C₂×C₃×#100	1	1	yes	characteristic
199	C₂×C₅	C₃×#416	1	2	yes	—
200	C₂³	—	2	2	no	—
201	C₂³	C₃×C₂²⋊Dic₅	1	1	yes	characteristic
202	Q₈	—	4	2	no	—
203	D₄	—	4	2	no	—
204	D₄	—	4	4	no	—
205	D₄	—	4	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	C₃×C₂²⋊Dic₅	1	2	yes	—
211	C₂×C₄	C₃×C₂²⋊Dic₅	1	1	yes	characteristic
212	C₂×C₄	C₂²×C₃×Dic₅	1	1	yes	characteristic
213	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	2	yes	—
214	C₈	—	20	4	no	—
215	C₂×C₃	—	4	1	no	—
216	C₂×C₃	—	8	2	no	—
217	C₂×C₃	C₂×#3965	1	1	yes	characteristic
218	C₂×C₃	#65522	1	2	yes	—
219	C₅	C₃×#1543	1	1	yes	characteristic, complemented, Sylow
220	C₂²	—	2	1	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	4	2	no	—
223	C₂²	—	4	4	no	—
224	C₂²	C₂×C₃×C₂²⋊Dic₅	1	1	yes	characteristic
225	C₄	—	2	1	no	—
226	C₄	—	2	2	no	—
227	C₄	—	4	2	no	—
228	C₄	C₃×#3965	1	2	yes	—
229	C₄	C₃×#4096	1	4	yes	—
230	C₃	#10602598	1	1	yes	characteristic, direct product factor, Sylow
231	C₂	—	4	1	no	—
232	C₂	—	8	2	no	—
233	C₂	C₂×C₃×#3965	1	1	yes	characteristic
234	C₂	C₃×#65522	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	C₂²⋊Dic₅
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₄²	C₂²×C₃×D₅
Fitting, prime radical	C₃×C₅×#532	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
cosocle	C₄²×C₅	C₂³×C₃
hypercenter	C₃×#532	D₅
hypocenter	C₅	C₃×#1543
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₃×#532

Complements

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

N	H ≅ G/N	type
2. C₃×#65214	C₂	semidirect
3. C₃×#64766	C₂	semidirect
4. C₃×#64764	C₂	semidirect
5. #10602598	C₃	direct (N × H)
18. #65214	C₂×C₃	semidirect
19. #64766	C₂×C₃	semidirect
20. #64764	C₂×C₃	semidirect
185. C₃×C₅	#1543	semidirect
219. C₅	C₃×#1543	semidirect
230. C₃	#10602598	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×#532	2
C₃×#65214	2
C₃×#64766	2
C₃×#64764	2
#10602598	3
C₃×#1543	5

Subgroups — C3×#10602598 #14284709

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10602598 #14284709