Subgroups — C₃×C₅×#2232 #14275266

Order 1920.

300 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#466	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×#107	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#366	C₂	1	2	yes	maximal, complemented
4	C₃×C₅×#357	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#2232	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×D₈	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₃×C₅×#121	—	2	1	no	—
10	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
11	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
12	C₃×C₅×#107	—	2	1	no	—
13	C₃×C₅×#107	C₂²	1	1	yes	characteristic
14	C₃×C₅×#107	C₂²	1	2	yes	—
15	C₃×C₅×#102	—	2	1	no	—
16	C₃×#2232	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₅×#466	C₂×C₃	1	1	yes	characteristic
18	C₂×C₅×#107	C₂×C₃	1	1	yes	characteristic, complemented
19	C₅×#366	C₂×C₃	1	2	yes	complemented
20	C₅×#357	C₂×C₃	1	2	yes	complemented
21	C₅×C₂²⋊C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
22	C₂×C₃×C₅×D₄	—	2	1	no	—
23	C₂×C₃×C₅×D₄	—	4	1	no	—
24	C₂²×C₃×C₄×C₅	—	4	1	no	—
25	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
26	C₃×C₅×D₈	—	2	2	no	—
27	C₂×C₃×C₅×C₈	—	2	2	no	—
28	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
29	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
30	C₃×C₅×C₄⋊C₄	—	2	2	no	—
31	C₃×C₅×C₄⋊C₄	—	2	1	no	—
32	C₃×C₅×C₄⋊C₄	—	4	1	no	—
33	C₃×C₅×C₂²⋊C₄	—	4	1	no	—
34	C₃×C₅×C₁₆	—	2	2	no	—
35	C₃×#466	C₂×C₅	1	1	yes	characteristic
36	C₂×C₃×#107	C₂×C₅	1	1	yes	characteristic, complemented
37	C₃×#366	C₂×C₅	1	2	yes	complemented
38	C₃×#357	C₂×C₅	1	2	yes	complemented
39	C₃×C₂²⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
40	C₂×C₅×D₈	C₂²×C₃	1	1	yes	characteristic
41	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
42	C₅×#121	—	2	1	no	—
43	C₂×C₅×C₄⋊C₄	—	2	1	no	—
44	C₂×C₅×C₁₆	C₂²×C₃	1	2	yes	—
45	C₅×#107	—	2	1	no	—
46	C₅×#107	C₂²×C₃	1	2	yes	—
47	C₅×#107	C₂²×C₃	1	1	yes	characteristic
48	C₅×#102	—	2	1	no	—
49	#2232	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
50	C₂³×C₃×C₅	—	4	1	no	—
51	C₂³×C₃×C₅	D₈	1	1	yes	characteristic
52	C₃×C₅×D₄	—	4	1	no	—
53	C₃×C₅×D₄	—	4	1	no	—
54	C₃×C₅×D₄	—	8	2	no	—
55	C₂×C₃×C₄×C₅	—	2	1	no	—
56	C₂×C₃×C₄×C₅	—	2	1	no	—
57	C₂×C₃×C₄×C₅	—	4	1	no	—
58	C₂×C₃×C₄×C₅	—	4	2	no	—
59	C₂×C₃×C₄×C₅	—	8	2	no	—
60	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₃×C₅×C₈	—	2	1	no	—
63	C₃×C₅×C₈	#41	1	2	yes	—
64	C₂×C₃×D₈	C₂²×C₅	1	1	yes	characteristic
65	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
66	C₃×#121	—	2	1	no	—
67	C₂×C₃×C₄⋊C₄	—	2	1	no	—
68	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	—
69	C₃×#107	—	2	1	no	—
70	C₃×#107	C₂²×C₅	1	1	yes	characteristic
71	C₃×#107	C₂²×C₅	1	2	yes	—
72	C₃×#102	—	2	1	no	—
73	C₂×C₅×D₄	—	2	1	no	—
74	C₂×C₅×D₄	—	4	1	no	—
75	C₂²×C₄×C₅	—	4	1	no	—
76	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
77	C₅×D₈	—	2	2	no	—
78	C₂×C₅×C₈	—	2	2	no	—
79	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
80	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
81	C₅×C₄⋊C₄	—	2	2	no	—
82	C₅×C₄⋊C₄	—	2	1	no	—
83	C₅×C₄⋊C₄	—	4	1	no	—
84	C₅×C₂²⋊C₄	—	4	1	no	—
85	C₅×C₁₆	—	2	2	no	—
86	#466	C₂×C₃×C₅	1	1	yes	characteristic
87	C₂×#107	C₂×C₃×C₅	1	1	yes	characteristic, complemented
88	#366	C₂×C₃×C₅	1	2	yes	complemented
89	#357	C₂×C₃×C₅	1	2	yes	complemented
90	C₂²⋊C₁₆	C₂×C₃×C₅	1	1	yes	characteristic, complemented
91	C₂²×C₃×C₅	—	2	1	no	—
92	C₂²×C₃×C₅	—	2	1	no	—
93	C₂²×C₃×C₅	—	8	1	no	—
94	C₂²×C₃×C₅	—	8	1	no	—
95	C₂²×C₃×C₅	—	8	1	no	—
96	C₂²×C₃×C₅	D₁₆	1	2	yes	—
97	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
98	C₃×C₄×C₅	—	2	1	no	—
99	C₃×C₄×C₅	—	8	2	no	—
100	C₃×C₄×C₅	—	8	1	no	—
101	C₃×C₄×C₅	#123	1	1	yes	characteristic
102	C₃×C₄×C₅	#137	1	1	yes	characteristic
103	C₂×C₃×D₄	—	2	1	no	—
104	C₂×C₃×D₄	—	4	1	no	—
105	C₂²×C₃×C₄	—	4	1	no	—
106	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
107	C₃×D₈	—	2	2	no	—
108	C₂×C₃×C₈	—	2	2	no	—
109	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
110	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
111	C₃×C₄⋊C₄	—	2	1	no	—
112	C₃×C₄⋊C₄	—	2	2	no	—
113	C₃×C₄⋊C₄	—	4	1	no	—
114	C₃×C₂²⋊C₄	—	4	1	no	—
115	C₃×C₁₆	—	2	2	no	—
116	C₂³×C₅	—	4	1	no	—
117	C₂³×C₅	C₃×D₈	1	1	yes	characteristic
118	C₅×D₄	—	4	1	no	—
119	C₅×D₄	—	4	1	no	—
120	C₅×D₄	—	8	2	no	—
121	C₂×C₄×C₅	—	2	1	no	—
122	C₂×C₄×C₅	—	2	1	no	—
123	C₂×C₄×C₅	—	4	1	no	—
124	C₂×C₄×C₅	—	4	2	no	—
125	C₂×C₄×C₅	—	8	2	no	—
126	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
127	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
128	C₅×C₈	—	2	1	no	—
129	C₅×C₈	C₃×#41	1	2	yes	—
130	C₂×D₈	C₂²×C₃×C₅	1	1	yes	characteristic
131	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
132	#121	—	2	1	no	—
133	C₂×C₄⋊C₄	—	2	1	no	—
134	C₂×C₁₆	C₂²×C₃×C₅	1	2	yes	—
135	#107	—	2	1	no	—
136	#107	C₂²×C₃×C₅	1	2	yes	—
137	#107	C₂²×C₃×C₅	1	1	yes	characteristic
138	#102	—	2	1	no	—
139	C₂×C₃×C₅	—	2	2	no	—
140	C₂×C₃×C₅	—	16	1	no	—
141	C₂×C₃×C₅	#480	1	1	yes	characteristic
142	C₂×C₃×C₅	C₂×D₁₆	1	1	yes	characteristic
143	C₂×C₃×C₅	#510	1	1	yes	characteristic
144	C₂³×C₃	—	4	1	no	—
145	C₂³×C₃	C₅×D₈	1	1	yes	characteristic
146	C₃×D₄	—	4	1	no	—
147	C₃×D₄	—	4	1	no	—
148	C₃×D₄	—	8	2	no	—
149	C₂×C₃×C₄	—	2	1	no	—
150	C₂×C₃×C₄	—	2	1	no	—
151	C₂×C₃×C₄	—	4	1	no	—
152	C₂×C₃×C₄	—	4	2	no	—
153	C₂×C₃×C₄	—	8	2	no	—
154	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
156	C₃×C₈	—	2	1	no	—
157	C₃×C₈	C₅×#41	1	2	yes	—
158	C₂²×C₅	—	2	1	no	—
159	C₂²×C₅	—	2	1	no	—
160	C₂²×C₅	—	8	1	no	—
161	C₂²×C₅	—	8	1	no	—
162	C₂²×C₅	—	8	1	no	—
163	C₂²×C₅	C₃×D₁₆	1	2	yes	—
164	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
165	C₄×C₅	—	2	1	no	—
166	C₄×C₅	—	8	2	no	—
167	C₄×C₅	—	8	1	no	—
168	C₄×C₅	C₃×#123	1	1	yes	characteristic
169	C₄×C₅	C₃×#137	1	1	yes	characteristic
170	C₂×D₄	—	2	1	no	—
171	C₂×D₄	—	4	1	no	—
172	C₂²×C₄	—	4	1	no	—
173	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
174	D₈	—	2	2	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
177	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
178	C₄⋊C₄	—	2	1	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	4	1	no	—
181	C₂²⋊C₄	—	4	1	no	—
182	C₁₆	—	2	2	no	—
183	C₃×C₅	#2232	1	1	yes	characteristic, direct product factor, Hall
184	C₂²×C₃	—	2	1	no	—
185	C₂²×C₃	—	2	1	no	—
186	C₂²×C₃	—	8	1	no	—
187	C₂²×C₃	—	8	1	no	—
188	C₂²×C₃	—	8	1	no	—
189	C₂²×C₃	C₅×D₁₆	1	2	yes	—
190	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
191	C₃×C₄	—	2	1	no	—
192	C₃×C₄	—	8	2	no	—
193	C₃×C₄	—	8	1	no	—
194	C₃×C₄	C₅×#123	1	1	yes	characteristic
195	C₃×C₄	C₅×#137	1	1	yes	characteristic
196	C₂×C₅	—	2	2	no	—
197	C₂×C₅	—	16	1	no	—
198	C₂×C₅	C₃×#480	1	1	yes	characteristic
199	C₂×C₅	C₂×C₃×D₁₆	1	1	yes	characteristic
200	C₂×C₅	C₃×#510	1	1	yes	characteristic
201	C₂³	—	4	1	no	—
202	C₂³	C₃×C₅×D₈	1	1	yes	characteristic
203	D₄	—	4	1	no	—
204	D₄	—	4	1	no	—
205	D₄	—	8	2	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	4	1	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	8	2	no	—
211	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
212	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
213	C₈	—	2	1	no	—
214	C₈	C₃×C₅×#41	1	2	yes	—
215	C₂×C₃	—	2	2	no	—
216	C₂×C₃	—	16	1	no	—
217	C₂×C₃	C₅×#480	1	1	yes	characteristic
218	C₂×C₃	C₂×C₅×D₁₆	1	1	yes	characteristic
219	C₂×C₃	C₅×#510	1	1	yes	characteristic
220	C₅	C₃×#2232	1	1	yes	characteristic, direct product factor, Sylow
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	8	1	no	—
224	C₂²	—	8	1	no	—
225	C₂²	—	8	1	no	—
226	C₂²	C₃×C₅×D₁₆	1	2	yes	—
227	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
228	C₄	—	2	1	no	—
229	C₄	—	8	2	no	—
230	C₄	—	8	1	no	—
231	C₄	C₃×C₅×#123	1	1	yes	characteristic
232	C₄	C₃×C₅×#137	1	1	yes	characteristic
233	C₃	C₅×#2232	1	1	yes	characteristic, direct product factor, Sylow
234	C₂	—	2	2	no	—
235	C₂	—	16	1	no	—
236	C₂	C₃×C₅×#480	1	1	yes	characteristic
237	C₂	C₂×C₃×C₅×D₁₆	1	1	yes	characteristic
238	C₂	C₃×C₅×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
prime radical	C₃×C₅×#466	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₂²×C₃×C₄×C₅ ◁ C₂²×C₃×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×#107	C₂	semidirect
3. C₃×C₅×#366	C₂	semidirect
4. C₃×C₅×#357	C₂	semidirect
5. C₃×C₅×C₂²⋊C₁₆	C₂	semidirect
6. C₅×#2232	C₃	direct (N × H)
16. C₃×#2232	C₅	direct (N × H)
18. C₂×C₅×#107	C₂×C₃	semidirect
19. C₅×#366	C₂×C₃	semidirect
20. C₅×#357	C₂×C₃	semidirect
21. C₅×C₂²⋊C₁₆	C₂×C₃	semidirect
36. C₂×C₃×#107	C₂×C₅	semidirect
37. C₃×#366	C₂×C₅	semidirect
38. C₃×#357	C₂×C₅	semidirect
39. C₃×C₂²⋊C₁₆	C₂×C₅	semidirect
49. #2232	C₃×C₅	direct (N × H)
87. C₂×#107	C₂×C₃×C₅	semidirect
88. #366	C₂×C₃×C₅	semidirect
89. #357	C₂×C₃×C₅	semidirect
90. C₂²⋊C₁₆	C₂×C₃×C₅	semidirect
183. C₃×C₅	#2232	direct (N × H)
220. C₅	C₃×#2232	direct (N × H)
233. C₃	C₅×#2232	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×#466	2
C₂×C₃×C₅×#107	2
C₃×C₅×#366	2
C₃×C₅×#357	2
C₃×C₅×C₂²⋊C₁₆	2
C₅×#2232	3
C₃×#2232	5

Subgroups — C3×C5×#2232 #14275266

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2232 #14275266