Subgroups — C₃×C₅×#1536 #14274570

Order 1920.

476 conjugacy classes of subgroups.

The subgroup lattice has 232 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×C₅×#432	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#329	C₂	1	4	yes	maximal, complemented
4	C₅×#1536	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂×C₃×C₅×#41	C₄	1	2	yes	—
6	C₃×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic
7	C₃×C₅×#127	C₄	1	1	yes	characteristic
8	C₃×C₅×#124	C₂²	1	2	yes	—
9	C₃×C₅×#121	—	2	2	no	—
10	C₃×C₄×C₅×D₄	—	2	2	no	—
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₃×C₅×C₄⋊C₈	—	2	2	no	—
13	C₃×C₅×C₂²⋊C₈	—	2	2	no	—
14	C₃×C₅×#97	C₂²	1	4	yes	—
15	C₃×#1536	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₅×#532	C₂×C₃	1	1	yes	characteristic
17	C₅×#432	C₂×C₃	1	2	yes	complemented
18	C₅×#329	C₂×C₃	1	4	yes	complemented
19	C₃×C₅×#41	—	4	4	no	—
20	C₂×C₃×C₅×Q₈	C₂×C₄	1	2	yes	—
21	C₂×C₃×C₅×D₄	—	2	2	no	—
22	C₂×C₃×C₅×D₄	—	2	2	no	—
23	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
24	C₂²×C₃×C₄×C₅	—	2	2	no	—
25	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
27	C₂×C₃×C₅×C₈	—	2	4	no	—
28	C₃×C₅×C₄⋊C₄	—	2	2	no	—
29	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
30	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
31	C₃×C₄²×C₅	D₄	1	2	yes	—
32	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
33	C₃×#532	C₂×C₅	1	1	yes	characteristic
34	C₃×#432	C₂×C₅	1	2	yes	complemented
35	C₃×#329	C₂×C₅	1	4	yes	complemented
36	C₂×C₅×#41	C₃×C₄	1	2	yes	—
37	C₅×C₄⋊Q₈	C₃×C₄	1	1	yes	characteristic
38	C₅×#127	C₃×C₄	1	1	yes	characteristic
39	C₅×#124	C₂²×C₃	1	2	yes	—
40	C₅×#121	—	2	2	no	—
41	C₄×C₅×D₄	—	2	2	no	—
42	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
43	C₅×C₄⋊C₈	—	2	2	no	—
44	C₅×C₂²⋊C₈	—	2	2	no	—
45	C₅×#97	C₂²×C₃	1	4	yes	—
46	#1536	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
47	C₂³×C₃×C₅	—	2	2	no	—
48	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
49	C₃×C₅×Q₈	—	4	2	no	—
50	C₃×C₅×D₄	—	4	4	no	—
51	C₃×C₅×D₄	—	4	2	no	—
52	C₃×C₅×D₄	—	4	4	no	—
53	C₂×C₃×C₄×C₅	—	2	2	no	—
54	C₂×C₃×C₄×C₅	—	2	2	no	—
55	C₂×C₃×C₄×C₅	—	4	2	no	—
56	C₂×C₃×C₄×C₅	—	4	4	no	—
57	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
58	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
61	C₃×C₅×C₈	—	4	4	no	—
62	C₂×C₃×#41	C₄×C₅	1	2	yes	—
63	C₃×C₄⋊Q₈	C₄×C₅	1	1	yes	characteristic
64	C₃×#127	C₄×C₅	1	1	yes	characteristic
65	C₃×#124	C₂²×C₅	1	2	yes	—
66	C₃×#121	—	2	2	no	—
67	C₃×C₄×D₄	—	2	2	no	—
68	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
69	C₃×C₄⋊C₈	—	2	2	no	—
70	C₃×C₂²⋊C₈	—	2	2	no	—
71	C₃×#97	C₂²×C₅	1	4	yes	—
72	C₅×#41	—	4	4	no	—
73	C₂×C₅×Q₈	C₂×C₃×C₄	1	2	yes	—
74	C₂×C₅×D₄	—	2	2	no	—
75	C₂×C₅×D₄	—	2	2	no	—
76	C₂×C₅×D₄	C₂×C₃×C₄	1	2	yes	—
77	C₂²×C₄×C₅	—	2	2	no	—
78	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
80	C₂×C₅×C₈	—	2	4	no	—
81	C₅×C₄⋊C₄	—	2	2	no	—
82	C₅×C₂²⋊C₄	—	2	4	no	—
83	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
84	C₄²×C₅	C₃×D₄	1	2	yes	—
85	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
86	#532	C₂×C₃×C₅	1	1	yes	characteristic
87	#432	C₂×C₃×C₅	1	2	yes	complemented
88	#329	C₂×C₃×C₅	1	4	yes	complemented
89	C₂²×C₃×C₅	—	2	1	no	—
90	C₂²×C₃×C₅	—	2	2	no	—
91	C₂²×C₃×C₅	—	4	2	no	—
92	C₂²×C₃×C₅	—	4	4	no	—
93	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₃×C₄×C₅	—	2	1	no	—
95	C₃×C₄×C₅	—	2	2	no	—
96	C₃×C₄×C₅	—	2	2	no	—
97	C₃×C₄×C₅	—	4	2	no	—
98	C₃×C₄×C₅	#100	1	2	yes	—
99	C₃×#41	—	4	4	no	—
100	C₂×C₃×Q₈	C₂×C₄×C₅	1	2	yes	—
101	C₂×C₃×D₄	—	2	2	no	—
102	C₂×C₃×D₄	—	2	2	no	—
103	C₂×C₃×D₄	C₂×C₄×C₅	1	2	yes	—
104	C₂²×C₃×C₄	—	2	2	no	—
105	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
106	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
107	C₂×C₃×C₈	—	2	4	no	—
108	C₃×C₄⋊C₄	—	2	2	no	—
109	C₃×C₂²⋊C₄	—	2	4	no	—
110	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
111	C₃×C₄²	C₅×D₄	1	2	yes	—
112	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
113	C₂³×C₅	—	2	2	no	—
114	C₂³×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
115	C₅×Q₈	—	4	2	no	—
116	C₅×D₄	—	4	2	no	—
117	C₅×D₄	—	4	4	no	—
118	C₅×D₄	—	4	4	no	—
119	C₂×C₄×C₅	—	2	2	no	—
120	C₂×C₄×C₅	—	2	2	no	—
121	C₂×C₄×C₅	—	4	2	no	—
122	C₂×C₄×C₅	—	4	4	no	—
123	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
124	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
126	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
127	C₅×C₈	—	4	4	no	—
128	C₂×#41	C₃×C₄×C₅	1	2	yes	—
129	C₄⋊Q₈	C₃×C₄×C₅	1	1	yes	characteristic
130	#127	C₃×C₄×C₅	1	1	yes	characteristic
131	#124	C₂²×C₃×C₅	1	2	yes	—
132	#121	—	2	2	no	—
133	C₄×D₄	—	2	2	no	—
134	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
135	C₄⋊C₈	—	2	2	no	—
136	C₂²⋊C₈	—	2	2	no	—
137	#97	C₂²×C₃×C₅	1	4	yes	—
138	C₂×C₃×C₅	—	4	1	no	—
139	C₂×C₃×C₅	—	8	2	no	—
140	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
141	C₂×C₃×C₅	#421	1	2	yes	—
142	C₂³×C₃	—	2	2	no	—
143	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
144	C₃×Q₈	—	4	2	no	—
145	C₃×D₄	—	4	2	no	—
146	C₃×D₄	—	4	4	no	—
147	C₃×D₄	—	4	4	no	—
148	C₂×C₃×C₄	—	2	2	no	—
149	C₂×C₃×C₄	—	2	2	no	—
150	C₂×C₃×C₄	—	4	2	no	—
151	C₂×C₃×C₄	—	4	4	no	—
152	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
153	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
156	C₃×C₈	—	4	4	no	—
157	C₂²×C₅	—	2	1	no	—
158	C₂²×C₅	—	2	2	no	—
159	C₂²×C₅	—	4	2	no	—
160	C₂²×C₅	—	4	4	no	—
161	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
162	C₄×C₅	—	2	1	no	—
163	C₄×C₅	—	2	2	no	—
164	C₄×C₅	—	2	2	no	—
165	C₄×C₅	—	4	2	no	—
166	C₄×C₅	C₃×#100	1	2	yes	—
167	#41	—	4	4	no	—
168	C₂×Q₈	C₂×C₃×C₄×C₅	1	2	yes	—
169	C₂×D₄	—	2	2	no	—
170	C₂×D₄	—	2	2	no	—
171	C₂×D₄	C₂×C₃×C₄×C₅	1	2	yes	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
174	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
175	C₂×C₈	—	2	4	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₂²⋊C₄	—	2	4	no	—
178	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
179	C₄²	C₃×C₅×D₄	1	2	yes	—
180	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
181	C₃×C₅	#1536	1	1	yes	characteristic, direct product factor, Hall
182	C₂²×C₃	—	2	1	no	—
183	C₂²×C₃	—	2	2	no	—
184	C₂²×C₃	—	4	2	no	—
185	C₂²×C₃	—	4	4	no	—
186	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
187	C₃×C₄	—	2	1	no	—
188	C₃×C₄	—	2	2	no	—
189	C₃×C₄	—	2	2	no	—
190	C₃×C₄	—	4	2	no	—
191	C₃×C₄	C₅×#100	1	2	yes	—
192	C₂×C₅	—	4	1	no	—
193	C₂×C₅	—	8	2	no	—
194	C₂×C₅	C₂×C₃×#100	1	1	yes	characteristic
195	C₂×C₅	C₃×#421	1	2	yes	—
196	C₂³	—	2	2	no	—
197	C₂³	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
198	Q₈	—	4	2	no	—
199	D₄	—	4	2	no	—
200	D₄	—	4	4	no	—
201	D₄	—	4	4	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	4	4	no	—
206	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
207	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
208	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
209	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
210	C₈	—	4	4	no	—
211	C₂×C₃	—	4	1	no	—
212	C₂×C₃	—	8	2	no	—
213	C₂×C₃	C₂×C₅×#100	1	1	yes	characteristic
214	C₂×C₃	C₅×#421	1	2	yes	—
215	C₅	C₃×#1536	1	1	yes	characteristic, direct product factor, Sylow
216	C₂²	—	2	1	no	—
217	C₂²	—	2	2	no	—
218	C₂²	—	4	2	no	—
219	C₂²	—	4	4	no	—
220	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
221	C₄	—	2	1	no	—
222	C₄	—	2	2	no	—
223	C₄	—	2	2	no	—
224	C₄	—	4	2	no	—
225	C₄	C₃×C₅×#100	1	2	yes	—
226	C₃	C₅×#1536	1	1	yes	characteristic, direct product factor, Sylow
227	C₂	—	4	1	no	—
228	C₂	—	8	2	no	—
229	C₂	C₂×C₃×C₅×#100	1	1	yes	characteristic
230	C₂	C₃×C₅×#421	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#432	C₂	semidirect
3. C₃×C₅×#329	C₂	semidirect
4. C₅×#1536	C₃	direct (N × H)
15. C₃×#1536	C₅	direct (N × H)
17. C₅×#432	C₂×C₃	semidirect
18. C₅×#329	C₂×C₃	semidirect
34. C₃×#432	C₂×C₅	semidirect
35. C₃×#329	C₂×C₅	semidirect
46. #1536	C₃×C₅	direct (N × H)
87. #432	C₂×C₃×C₅	semidirect
88. #329	C₂×C₃×C₅	semidirect
181. C₃×C₅	#1536	direct (N × H)
215. C₅	C₃×#1536	direct (N × H)
226. C₃	C₅×#1536	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×C₅×#432	2
C₃×C₅×#329	2
C₅×#1536	3
C₃×#1536	5

Subgroups — C3×C5×#1536 #14274570

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1536 #14274570