Subgroups — C₃×#10602512 #14284336

Order 1920.

496 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#124	C₂	1	1	yes	maximal, characteristic
2	C₃×#65215	C₂	1	2	yes	maximal, complemented
3	C₃×#64760	C₂	1	4	yes	maximal, complemented
4	#10602512	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂²×C₃×C₅×Q₈	C₄	1	1	yes	characteristic
6	C₂²×C₃×C₅×D₄	C₄	1	1	yes	characteristic
7	C₃×C₅×#124	—	2	2	no	—
8	C₃×C₅×#124	C₄	1	2	yes	—
9	C₃×C₅×#124	C₂²	1	2	yes	—
10	C₂×C₃×C₅×C₂²⋊C₄	—	2	2	no	—
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₃×C₂²⋊#168	—	2	2	no	—
13	C₃×C₄⋊#168	—	2	2	no	—
14	C₃×#3935	C₂²	1	4	yes	complemented
15	C₃×#1533	—	5	1	no	maximal, Hall
16	C₂×C₅×#124	C₂×C₃	1	1	yes	characteristic
17	#65215	C₂×C₃	1	2	yes	complemented
18	#64760	C₂×C₃	1	4	yes	complemented
19	C₂⁴×C₃×C₅	—	2	1	no	—
20	C₂×C₃×C₅×Q₈	—	2	1	no	—
21	C₂×C₃×C₅×Q₈	—	4	2	no	—
22	C₂×C₃×C₅×Q₈	C₂×C₄	1	2	yes	—
23	C₂×C₃×C₅×D₄	—	2	1	no	—
24	C₂×C₃×C₅×D₄	—	4	2	no	—
25	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
26	C₂²×C₃×C₄×C₅	—	2	1	no	—
27	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
29	C₃×C₅×C₂²⋊C₄	—	2	4	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	C₂×C₃×#124	D₅	1	1	yes	characteristic
36	C₃×#432	—	5	2	no	—
37	C₃×#329	—	5	4	no	—
38	C₂²×C₅×Q₈	C₃×C₄	1	1	yes	characteristic
39	C₂²×C₅×D₄	C₃×C₄	1	1	yes	characteristic
40	C₅×#124	—	2	2	no	—
41	C₅×#124	C₃×C₄	1	2	yes	—
42	C₅×#124	C₂²×C₃	1	2	yes	—
43	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
44	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
45	C₂²⋊#168	—	2	2	no	—
46	C₄⋊#168	—	2	2	no	—
47	#3935	C₂²×C₃	1	4	yes	complemented
48	#1533	—	5	1	no	Sylow
49	C₂³×C₃×C₅	—	2	2	no	—
50	C₂³×C₃×C₅	—	4	4	no	—
51	C₂³×C₃×C₅	—	4	2	no	—
52	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
53	C₃×C₅×Q₈	—	4	2	no	—
54	C₃×C₅×Q₈	—	4	2	no	—
55	C₃×C₅×D₄	—	4	2	no	—
56	C₃×C₅×D₄	—	4	2	no	—
57	C₂×C₃×C₄×C₅	—	2	2	no	—
58	C₂×C₃×C₄×C₅	—	2	2	no	—
59	C₂×C₃×C₄×C₅	—	4	2	no	—
60	C₂×C₃×C₄×C₅	—	4	2	no	—
61	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
62	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
65	C₃×#168	—	4	4	no	—
66	C₂²×C₃×Q₈	Dic₅	1	1	yes	characteristic
67	C₂²×C₃×D₄	Dic₅	1	1	yes	characteristic
68	C₃×#124	—	2	2	no	—
69	C₃×#124	Dic₅	1	2	yes	—
70	C₃×#124	C₂×D₅	1	2	yes	—
71	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
72	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
73	C₃×C₄⋊C₈	—	10	2	no	—
74	C₃×C₂²⋊C₈	—	10	2	no	—
75	C₃×#97	—	5	4	no	—
76	C₂⁴×C₅	—	2	1	no	—
77	C₂×C₅×Q₈	—	2	1	no	—
78	C₂×C₅×Q₈	—	4	2	no	—
79	C₂×C₅×Q₈	C₂×C₃×C₄	1	2	yes	—
80	C₂×C₅×D₄	—	2	1	no	—
81	C₂×C₅×D₄	—	4	2	no	—
82	C₂×C₅×D₄	C₂×C₃×C₄	1	2	yes	—
83	C₂²×C₄×C₅	—	2	1	no	—
84	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
86	C₅×C₂²⋊C₄	—	2	4	no	—
87	C₅×C₂²⋊C₄	—	2	4	no	—
88	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
89	C₄²×C₅	C₃×D₄	1	2	yes	—
90	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
91	C₂×#168	—	2	4	no	—
92	C₂×#124	C₃×D₅	1	1	yes	characteristic
93	#432	—	5	2	no	—
94	#329	—	5	4	no	—
95	C₂²×C₃×C₅	—	2	2	no	—
96	C₂²×C₃×C₅	—	4	2	no	—
97	C₂²×C₃×C₅	—	4	4	no	—
98	C₂²×C₃×C₅	—	8	4	no	—
99	C₂²×C₃×C₅	#100	1	2	yes	—
100	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₃×C₄×C₅	—	2	1	no	—
102	C₃×C₄×C₅	—	2	1	no	—
103	C₃×C₄×C₅	—	2	2	no	—
104	C₃×C₄×C₅	—	2	2	no	—
105	C₃×C₄×C₅	—	4	2	no	—
106	C₂⁴×C₃	—	2	1	no	—
107	C₂×C₃×Q₈	—	2	1	no	—
108	C₂×C₃×Q₈	—	4	2	no	—
109	C₂×C₃×Q₈	C₂×Dic₅	1	2	yes	—
110	C₂×C₃×D₄	—	2	1	no	—
111	C₂×C₃×D₄	—	4	2	no	—
112	C₂×C₃×D₄	C₂×Dic₅	1	2	yes	—
113	C₂²×C₃×C₄	—	2	1	no	—
114	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
115	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
116	C₂×C₃×C₈	—	10	4	no	—
117	C₃×C₂²⋊C₄	—	2	4	no	—
118	C₃×C₂²⋊C₄	—	2	4	no	—
119	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
120	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
121	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
122	C₂³×C₅	—	2	2	no	—
123	C₂³×C₅	—	4	4	no	—
124	C₂³×C₅	—	4	2	no	—
125	C₂³×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
126	C₅×Q₈	—	4	2	no	—
127	C₅×Q₈	—	4	2	no	—
128	C₅×D₄	—	4	2	no	—
129	C₅×D₄	—	4	2	no	—
130	C₂×C₄×C₅	—	2	2	no	—
131	C₂×C₄×C₅	—	2	2	no	—
132	C₂×C₄×C₅	—	4	2	no	—
133	C₂×C₄×C₅	—	4	2	no	—
134	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
135	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
138	#168	—	4	4	no	—
139	C₂²×Q₈	C₃×Dic₅	1	1	yes	characteristic
140	C₂²×D₄	C₃×Dic₅	1	1	yes	characteristic
141	#124	—	2	2	no	—
142	#124	C₃×Dic₅	1	2	yes	—
143	#124	C₂×C₃×D₅	1	2	yes	—
144	C₂×C₂²⋊C₄	—	2	2	no	—
145	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
146	C₄⋊C₈	—	10	2	no	—
147	C₂²⋊C₈	—	10	2	no	—
148	#97	—	5	4	no	—
149	C₂×C₃×C₅	—	2	2	no	—
150	C₂×C₃×C₅	—	8	2	no	—
151	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
152	C₂×C₃×C₅	#421	1	2	yes	—
153	C₂³×C₃	—	2	2	no	—
154	C₂³×C₃	—	4	4	no	—
155	C₂³×C₃	—	4	2	no	—
156	C₂³×C₃	C₂²⋊Dic₅	1	1	yes	characteristic
157	C₃×Q₈	—	4	2	no	—
158	C₃×Q₈	—	4	2	no	—
159	C₃×D₄	—	4	2	no	—
160	C₃×D₄	—	4	2	no	—
161	C₂×C₃×C₄	—	2	2	no	—
162	C₂×C₃×C₄	—	2	2	no	—
163	C₂×C₃×C₄	—	4	2	no	—
164	C₂×C₃×C₄	—	4	2	no	—
165	C₂×C₃×C₄	C₂²⋊Dic₅	1	2	yes	—
166	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
167	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
169	C₃×C₈	—	20	4	no	—
170	C₂²×C₅	—	2	2	no	—
171	C₂²×C₅	—	4	2	no	—
172	C₂²×C₅	—	4	4	no	—
173	C₂²×C₅	—	8	4	no	—
174	C₂²×C₅	C₃×#100	1	2	yes	—
175	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
176	C₄×C₅	—	2	1	no	—
177	C₄×C₅	—	2	1	no	—
178	C₄×C₅	—	2	2	no	—
179	C₄×C₅	—	2	2	no	—
180	C₄×C₅	—	4	2	no	—
181	C₂⁴	—	2	1	no	—
182	C₂×Q₈	—	2	1	no	—
183	C₂×Q₈	—	4	2	no	—
184	C₂×Q₈	C₂×C₃×Dic₅	1	2	yes	—
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	—	4	2	no	—
187	C₂×D₄	C₂×C₃×Dic₅	1	2	yes	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	C₂×C₃×Dic₅	1	1	yes	characteristic
190	C₂²×C₄	C₃×C₂²⋊D₅	1	2	yes	—
191	C₂×C₈	—	10	4	no	—
192	C₂²⋊C₄	—	2	4	no	—
193	C₂²⋊C₄	—	2	4	no	—
194	C₄²	C₂×C₃×Dic₅	1	1	yes	characteristic
195	C₄²	C₃×C₂²⋊D₅	1	2	yes	—
196	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
197	C₃×C₅	#1533	1	1	yes	characteristic, complemented, Hall
198	C₂²×C₃	—	2	2	no	—
199	C₂²×C₃	—	4	2	no	—
200	C₂²×C₃	—	4	4	no	—
201	C₂²×C₃	—	8	4	no	—
202	C₂²×C₃	#3965	1	2	yes	—
203	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
204	C₃×C₄	—	2	1	no	—
205	C₃×C₄	—	2	1	no	—
206	C₃×C₄	—	2	2	no	—
207	C₃×C₄	—	2	2	no	—
208	C₃×C₄	—	4	2	no	—
209	C₂×C₅	—	2	2	no	—
210	C₂×C₅	—	8	2	no	—
211	C₂×C₅	C₂×C₃×#100	1	1	yes	characteristic
212	C₂×C₅	C₃×#421	1	2	yes	—
213	C₂³	—	2	2	no	—
214	C₂³	—	4	4	no	—
215	C₂³	—	4	2	no	—
216	C₂³	C₃×C₂²⋊Dic₅	1	1	yes	characteristic
217	Q₈	—	4	2	no	—
218	Q₈	—	4	2	no	—
219	D₄	—	4	2	no	—
220	D₄	—	4	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	C₃×C₂²⋊Dic₅	1	2	yes	—
226	C₂×C₄	C₃×C₂²⋊Dic₅	1	1	yes	characteristic
227	C₂×C₄	C₂²×C₃×Dic₅	1	1	yes	characteristic
228	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	2	yes	—
229	C₈	—	20	4	no	—
230	C₂×C₃	—	2	2	no	—
231	C₂×C₃	—	8	2	no	—
232	C₂×C₃	C₂×#3965	1	1	yes	characteristic
233	C₂×C₃	#65525	1	2	yes	—
234	C₅	C₃×#1533	1	1	yes	characteristic, complemented, Sylow
235	C₂²	—	2	2	no	—
236	C₂²	—	4	2	no	—
237	C₂²	—	4	4	no	—
238	C₂²	—	8	4	no	—
239	C₂²	C₃×#3965	1	2	yes	—
240	C₂²	C₂×C₃×C₂²⋊Dic₅	1	1	yes	characteristic
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	2	2	no	—
244	C₄	—	2	2	no	—
245	C₄	—	4	2	no	—
246	C₃	#10602512	1	1	yes	characteristic, direct product factor, Sylow
247	C₂	—	2	2	no	—
248	C₂	—	8	2	no	—
249	C₂	C₂×C₃×#3965	1	1	yes	characteristic
250	C₂	C₃×#65525	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂×C₂²⋊Dic₅
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₄²	C₂²×C₃×D₅
Fitting	C₂×C₃×C₅×#124	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×C₃×C₅×D₄	C₄
cosocle	C₄²×C₅	C₂³×C₃
hypercenter	C₂×C₃×#124	D₅
hypocenter	C₅	C₃×#1533
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×#124

Complements

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

N	H ≅ G/N	type
2. C₃×#65215	C₂	semidirect
3. C₃×#64760	C₂	semidirect
4. #10602512	C₃	direct (N × H)
14. C₃×#3935	C₂²	semidirect
17. #65215	C₂×C₃	semidirect
18. #64760	C₂×C₃	semidirect
47. #3935	C₂²×C₃	semidirect
197. C₃×C₅	#1533	semidirect
234. C₅	C₃×#1533	semidirect
246. C₃	#10602512	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×C₅×#124	2
C₃×#65215	2
C₃×#64760	2
#10602512	3
C₃×#1533	5

Subgroups — C3×#10602512 #14284336

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10602512 #14284336