Subgroups — C₃×C₅×#924 #13087935

Order 1440.

350 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	C₂²×C₃²×C₄×C₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#217	C₂	1	2	yes	maximal
4	C₃×C₅×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×Dic₃⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₃×C₅×C₂²⋊Q₈	—	3	1	no	maximal
7	C₅×#924	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂³×C₃²×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₃²×C₄×C₅	—	2	2	no	—
10	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₅×Dic₃	C₂²	1	2	yes	—
13	C₂×C₃×C₅×Dic₃	C₂²	1	2	yes	—
14	C₃×C₅×Dic₆	—	2	2	no	—
15	C₃×#924	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂×C₃×C₅×Q₈	—	3	1	no	—
17	C₂²×C₃×C₄×C₅	—	2	1	no	—
18	C₂²×C₃×C₄×C₅	D₃	1	1	yes	characteristic
19	C₂²×C₃×C₄×C₅	C₂×C₃	1	1	yes	characteristic
20	C₂×C₅×Dic₆	C₂×C₃	1	1	yes	characteristic, complemented
21	C₃×C₅×C₄⋊C₄	—	3	2	no	—
22	C₃×C₅×C₄⋊C₄	—	3	1	no	—
23	C₃×C₅×C₂²⋊C₄	—	3	2	no	—
24	C₅×#217	C₂×C₃	1	2	yes	—
25	C₅×C₄⋊Dic₃	C₂×C₃	1	1	yes	characteristic, complemented
26	C₅×Dic₃⋊C₄	C₂×C₃	1	2	yes	complemented
27	C₂²×C₃²×C₅	—	2	1	no	—
28	C₂²×C₃²×C₅	—	2	1	no	—
29	C₂²×C₃²×C₅	Q₈	1	2	yes	complemented
30	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
31	C₃²×C₄×C₅	—	2	1	no	—
32	C₃²×C₄×C₅	D₄	1	2	yes	—
33	C₃×C₅×Dic₃	—	2	2	no	—
34	C₃×C₅×Dic₃	—	2	2	no	—
35	C₅×C₂²⋊Q₈	—	3	1	no	Hall
36	C₂²×C₃²×C₄	C₂×C₅	1	1	yes	characteristic
37	C₂×C₃×Dic₆	C₂×C₅	1	1	yes	characteristic, complemented
38	C₃×#217	C₂×C₅	1	2	yes	—
39	C₃×C₄⋊Dic₃	C₂×C₅	1	1	yes	characteristic, complemented
40	C₃×Dic₃⋊C₄	C₂×C₅	1	2	yes	complemented
41	C₂³×C₃×C₅	—	2	1	no	—
42	C₂³×C₃×C₅	C₂×D₃	1	1	yes	characteristic
43	C₂³×C₃×C₅	C₂²×C₃	1	1	yes	characteristic
44	C₃×C₅×Q₈	—	6	2	no	—
45	C₂×C₃×C₄×C₅	—	2	4	no	—
46	C₂×C₃×C₄×C₅	—	2	1	no	—
47	C₂×C₃×C₄×C₅	—	2	1	no	—
48	C₂×C₃×C₄×C₅	—	2	2	no	—
49	C₂×C₃×C₄×C₅	—	2	2	no	—
50	C₂×C₃×C₄×C₅	—	3	2	no	—
51	C₂×C₃×C₄×C₅	—	3	2	no	—
52	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
56	C₂×C₅×Dic₃	C₂²×C₃	1	2	yes	—
57	C₂×C₅×Dic₃	C₂²×C₃	1	2	yes	—
58	C₅×Dic₆	—	2	2	no	—
59	C₃×C₂²⋊Q₈	—	3	1	no	—
60	#924	C₃×C₅	1	1	yes	characteristic, direct product factor
61	C₂×C₃²×C₅	—	2	2	no	—
62	C₂×C₃²×C₅	C₂×D₄	1	1	yes	characteristic
63	C₂×C₃²×C₅	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₃²×C₅	#41	1	1	yes	characteristic
65	C₂×C₅×Q₈	—	3	1	no	—
66	C₂²×C₄×C₅	C₃×D₃	1	1	yes	characteristic
67	C₅×C₄⋊C₄	—	3	2	no	—
68	C₅×C₄⋊C₄	—	3	1	no	—
69	C₅×C₂²⋊C₄	—	3	2	no	—
70	C₂³×C₃²	C₂²×C₅	1	1	yes	characteristic
71	C₂×C₃²×C₄	—	2	2	no	—
72	C₂×C₃²×C₄	C₂²×C₅	1	1	yes	characteristic
73	C₂×C₃²×C₄	C₂²×C₅	1	1	yes	characteristic
74	C₂×C₃×Dic₃	C₂²×C₅	1	2	yes	—
75	C₂×C₃×Dic₃	C₂²×C₅	1	2	yes	—
76	C₃×Dic₆	—	2	2	no	—
77	C₂²×C₃×C₅	—	2	2	no	—
78	C₂²×C₃×C₅	—	2	2	no	—
79	C₂²×C₃×C₅	—	2	1	no	—
80	C₂²×C₃×C₅	—	2	1	no	—
81	C₂²×C₃×C₅	—	2	1	no	—
82	C₂²×C₃×C₅	—	2	1	no	—
83	C₂²×C₃×C₅	—	2	2	no	—
84	C₂²×C₃×C₅	—	2	1	no	—
85	C₂²×C₃×C₅	Dic₆	1	2	yes	complemented
86	C₂²×C₃×C₅	C₃×Q₈	1	2	yes	complemented
87	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
88	C₂²×C₃×C₅	C₂³×C₃	1	1	yes	characteristic
89	C₃×C₄×C₅	—	2	2	no	—
90	C₃×C₄×C₅	—	2	1	no	—
91	C₃×C₄×C₅	—	2	1	no	—
92	C₃×C₄×C₅	—	2	2	no	—
93	C₃×C₄×C₅	—	6	2	no	—
94	C₃×C₄×C₅	—	6	2	no	—
95	C₃×C₄×C₅	#67	1	2	yes	—
96	C₃×C₄×C₅	C₃×D₄	1	2	yes	—
97	C₅×Dic₃	—	2	2	no	—
98	C₅×Dic₃	—	2	2	no	—
99	C₂×C₃×Q₈	—	3	1	no	—
100	C₂²×C₃×C₄	—	2	1	no	—
101	C₂²×C₃×C₄	C₅×D₃	1	1	yes	characteristic
102	C₂²×C₃×C₄	C₂×C₃×C₅	1	1	yes	characteristic
103	C₂×Dic₆	C₂×C₃×C₅	1	1	yes	characteristic, complemented
104	C₃×C₄⋊C₄	—	3	2	no	—
105	C₃×C₄⋊C₄	—	3	1	no	—
106	C₃×C₂²⋊C₄	—	3	2	no	—
107	#217	C₂×C₃×C₅	1	2	yes	—
108	C₄⋊Dic₃	C₂×C₃×C₅	1	1	yes	characteristic, complemented
109	Dic₃⋊C₄	C₂×C₃×C₅	1	2	yes	complemented
110	C₃²×C₅	C₂²⋊Q₈	1	1	yes	characteristic, complemented, Hall
111	C₂³×C₅	C₂×C₃×D₃	1	1	yes	characteristic
112	C₅×Q₈	—	6	2	no	—
113	C₂×C₄×C₅	—	2	2	no	—
114	C₂×C₄×C₅	—	3	2	no	—
115	C₂×C₄×C₅	—	3	2	no	—
116	C₂×C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic
117	C₂×C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic
118	C₂²×C₃²	—	2	1	no	—
119	C₂²×C₃²	—	2	1	no	—
120	C₂²×C₃²	C₅×Q₈	1	2	yes	complemented
121	C₂²×C₃²	C₂³×C₅	1	1	yes	characteristic
122	C₃²×C₄	—	2	1	no	—
123	C₃²×C₄	C₅×D₄	1	2	yes	—
124	C₃×Dic₃	—	2	2	no	—
125	C₃×Dic₃	—	2	2	no	—
126	C₂²⋊Q₈	—	3	1	no	Sylow
127	C₂×C₃×C₅	—	2	4	no	—
128	C₂×C₃×C₅	—	2	2	no	—
129	C₂×C₃×C₅	—	2	2	no	—
130	C₂×C₃×C₅	—	2	1	no	—
131	C₂×C₃×C₅	—	2	1	no	—
132	C₂×C₃×C₅	—	2	1	no	—
133	C₂×C₃×C₅	C₂×Dic₆	1	1	yes	characteristic
134	C₂×C₃×C₅	#235	1	1	yes	characteristic
135	C₂×C₃×C₅	C₂×#67	1	1	yes	characteristic
136	C₂×C₃×C₅	C₂×C₃×D₄	1	1	yes	characteristic
137	C₂×C₃×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
138	C₂×C₃×C₅	C₃×#41	1	1	yes	characteristic
139	C₂³×C₃	—	2	1	no	—
140	C₂³×C₃	C₂×C₅×D₃	1	1	yes	characteristic
141	C₂³×C₃	C₂²×C₃×C₅	1	1	yes	characteristic
142	C₃×Q₈	—	6	2	no	—
143	C₂×C₃×C₄	—	2	4	no	—
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	2	no	—
146	C₂×C₃×C₄	—	2	1	no	—
147	C₂×C₃×C₄	—	2	1	no	—
148	C₂×C₃×C₄	—	3	2	no	—
149	C₂×C₃×C₄	—	3	2	no	—
150	C₂×C₃×C₄	C₂×C₅×D₃	1	1	yes	characteristic
151	C₂×C₃×C₄	C₂×C₅×D₃	1	1	yes	characteristic
152	C₂×C₃×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
153	C₂×C₃×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
154	C₂×Dic₃	C₂²×C₃×C₅	1	2	yes	—
155	C₂×Dic₃	C₂²×C₃×C₅	1	2	yes	—
156	Dic₆	—	2	2	no	—
157	C₂²×C₅	—	2	1	no	—
158	C₂²×C₅	—	2	1	no	—
159	C₂²×C₅	C₃×Dic₆	1	2	yes	complemented
160	C₂²×C₅	C₂²×C₃×D₃	1	1	yes	characteristic
161	C₄×C₅	—	2	1	no	—
162	C₄×C₅	—	6	2	no	—
163	C₄×C₅	—	6	2	no	—
164	C₄×C₅	C₃×#67	1	2	yes	—
165	C₂×C₃²	—	2	2	no	—
166	C₂×C₃²	C₂×C₅×D₄	1	1	yes	characteristic
167	C₂×C₃²	C₂×C₅×Q₈	1	1	yes	characteristic
168	C₂×C₃²	C₅×#41	1	1	yes	characteristic
169	C₂×Q₈	—	3	1	no	—
170	C₂²×C₄	C₃×C₅×D₃	1	1	yes	characteristic
171	C₄⋊C₄	—	3	2	no	—
172	C₄⋊C₄	—	3	1	no	—
173	C₂²⋊C₄	—	3	2	no	—
174	C₃×C₅	—	2	1	no	—
175	C₃×C₅	#924	1	1	yes	characteristic, direct product factor
176	C₃×C₅	C₃×C₂²⋊Q₈	1	1	yes	characteristic, complemented
177	C₂²×C₃	—	2	2	no	—
178	C₂²×C₃	—	2	2	no	—
179	C₂²×C₃	—	2	1	no	—
180	C₂²×C₃	—	2	1	no	—
181	C₂²×C₃	—	2	1	no	—
182	C₂²×C₃	—	2	1	no	—
183	C₂²×C₃	—	2	2	no	—
184	C₂²×C₃	—	2	1	no	—
185	C₂²×C₃	C₅×Dic₆	1	2	yes	complemented
186	C₂²×C₃	C₃×C₅×Q₈	1	2	yes	complemented
187	C₂²×C₃	C₂²×C₅×D₃	1	1	yes	characteristic
188	C₂²×C₃	C₂³×C₃×C₅	1	1	yes	characteristic
189	C₃×C₄	—	2	2	no	—
190	C₃×C₄	—	2	1	no	—
191	C₃×C₄	—	2	1	no	—
192	C₃×C₄	—	2	2	no	—
193	C₃×C₄	—	6	2	no	—
194	C₃×C₄	—	6	2	no	—
195	C₃×C₄	C₅×#67	1	2	yes	—
196	C₃×C₄	C₃×C₅×D₄	1	2	yes	—
197	Dic₃	—	2	2	no	—
198	Dic₃	—	2	2	no	—
199	C₂×C₅	—	2	2	no	—
200	C₂×C₅	C₂×C₃×Dic₆	1	1	yes	characteristic
201	C₂×C₅	C₃×#235	1	1	yes	characteristic
202	C₂×C₅	C₂×C₃×#67	1	1	yes	characteristic
203	C₃²	C₅×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
204	C₂³	C₂×C₃×C₅×D₃	1	1	yes	characteristic
205	Q₈	—	6	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	3	2	no	—
208	C₂×C₄	—	3	2	no	—
209	C₂×C₄	C₂×C₃×C₅×D₃	1	1	yes	characteristic
210	C₂×C₄	C₂×C₃×C₅×D₃	1	1	yes	characteristic
211	C₂×C₃	—	2	4	no	—
212	C₂×C₃	—	2	2	no	—
213	C₂×C₃	—	2	2	no	—
214	C₂×C₃	—	2	1	no	—
215	C₂×C₃	—	2	1	no	—
216	C₂×C₃	—	2	1	no	—
217	C₂×C₃	C₂×C₅×Dic₆	1	1	yes	characteristic
218	C₂×C₃	C₅×#235	1	1	yes	characteristic
219	C₂×C₃	C₂×C₅×#67	1	1	yes	characteristic
220	C₂×C₃	C₂×C₃×C₅×D₄	1	1	yes	characteristic
221	C₂×C₃	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
222	C₂×C₃	C₃×C₅×#41	1	1	yes	characteristic
223	C₅	C₃×#924	1	1	yes	characteristic, direct product factor, Sylow
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	C₃×C₅×Dic₆	1	2	yes	complemented
227	C₂²	C₂²×C₃×C₅×D₃	1	1	yes	characteristic
228	C₄	—	2	1	no	—
229	C₄	—	6	2	no	—
230	C₄	—	6	2	no	—
231	C₄	C₃×C₅×#67	1	2	yes	—
232	C₃	—	2	1	no	—
233	C₃	C₅×#924	1	1	yes	characteristic, direct product factor
234	C₃	C₃×C₅×C₂²⋊Q₈	1	1	yes	characteristic, complemented
235	C₂	—	2	2	no	—
236	C₂	C₂×C₃×C₅×Dic₆	1	1	yes	characteristic
237	C₂	C₃×C₅×#235	1	1	yes	characteristic
238	C₂	C₂×C₃×C₅×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₅	C₂²×D₃
commutator, cosocle	C₂²×C₃	C₂³×C₃×C₅
Frattini	C₂²	C₂²×C₃×C₅×D₃
Fitting	C₂²×C₃²×C₄×C₅	C₂
socle	C₂²×C₃²×C₅	C₂³
prime radical	C₂³×C₃²×C₅	C₂²
hypercenter	C₂²×C₃×C₄×C₅	D₃
hypocenter	C₃	C₃×C₅×C₂²⋊Q₈
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₂²×C₃×C₄×C₅

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×Dic₆	C₂	semidirect
4. C₃×C₅×C₄⋊Dic₃	C₂	semidirect
5. C₃×C₅×Dic₃⋊C₄	C₂	semidirect
7. C₅×#924	C₃	direct (N × H)
15. C₃×#924	C₅	direct (N × H)
20. C₂×C₅×Dic₆	C₂×C₃	semidirect
25. C₅×C₄⋊Dic₃	C₂×C₃	semidirect
26. C₅×Dic₃⋊C₄	C₂×C₃	semidirect
29. C₂²×C₃²×C₅	Q₈	semidirect
37. C₂×C₃×Dic₆	C₂×C₅	semidirect
39. C₃×C₄⋊Dic₃	C₂×C₅	semidirect
40. C₃×Dic₃⋊C₄	C₂×C₅	semidirect
60. #924	C₃×C₅	direct (N × H)
85. C₂²×C₃×C₅	Dic₆	semidirect
86. C₂²×C₃×C₅	C₃×Q₈	semidirect
103. C₂×Dic₆	C₂×C₃×C₅	semidirect
108. C₄⋊Dic₃	C₂×C₃×C₅	semidirect
109. Dic₃⋊C₄	C₂×C₃×C₅	semidirect
110. C₃²×C₅	C₂²⋊Q₈	semidirect
120. C₂²×C₃²	C₅×Q₈	semidirect
159. C₂²×C₅	C₃×Dic₆	semidirect
175. C₃×C₅	#924	direct (N × H)
176. C₃×C₅	C₃×C₂²⋊Q₈	semidirect
185. C₂²×C₃	C₅×Dic₆	semidirect
186. C₂²×C₃	C₃×C₅×Q₈	semidirect
203. C₃²	C₅×C₂²⋊Q₈	semidirect
223. C₅	C₃×#924	direct (N × H)
226. C₂²	C₃×C₅×Dic₆	semidirect
233. C₃	C₅×#924	direct (N × H)
234. C₃	C₃×C₅×C₂²⋊Q₈	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃²×C₄×C₅	2
C₂×C₃×C₅×Dic₆	2
C₃×C₅×#217	2
C₃×C₅×C₄⋊Dic₃	2
C₃×C₅×Dic₃⋊C₄	2
C₃×C₅×C₂²⋊Q₈	3
C₅×#924	3
C₃×#924	5

Subgroups — C3×C5×#924 #13087935

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#924 #13087935