Subgroups — C₃×C₅×#946 #13087957

Order 1440.

334 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₄×C₅×D₃	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×(C₂×D₃)⋊C₄	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×#946	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃×C₅×D₃	C₂²	1	1	yes	characteristic
9	C₃²×C₅×Q₈	—	2	2	no	—
10	C₂×C₃²×C₄×C₅	C₂²	1	2	yes	—
11	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₅×Dic₃	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₅×Dic₃	C₂²	1	2	yes	—
14	C₃×C₄×C₅×D₃	—	2	2	no	—
15	C₃×#946	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂×C₃×C₅×Q₈	—	2	1	no	—
17	C₂×C₃×C₅×Q₈	D₃	1	1	yes	characteristic, complemented
18	C₂×C₃×C₅×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
19	C₂²×C₃×C₄×C₅	—	3	1	no	—
20	C₂×C₄×C₅×D₃	C₂×C₃	1	1	yes	characteristic
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×(C₂×D₃)⋊C₄	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	C₂³	1	1	yes	characteristic
28	C₂×C₃×C₅×D₃	—	2	1	no	—
29	C₂×C₃×C₅×D₃	—	2	1	no	—
30	C₂×C₃×C₅×D₃	Q₈	1	2	yes	complemented
31	C₃²×C₄×C₅	—	2	2	no	—
32	C₃²×C₄×C₅	D₄	1	2	yes	—
33	C₃×C₅×Dic₃	—	2	1	no	—
34	C₃×C₅×Dic₃	—	2	2	no	—
35	C₅×C₂²⋊Q₈	—	3	1	no	Hall
36	C₂×C₃²×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
37	C₂×C₃×C₄×D₃	C₂×C₅	1	1	yes	characteristic
38	C₃×(C₂×D₃)⋊C₄	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	—	3	1	no	—
42	C₂²×C₅×D₃	C₂²×C₃	1	1	yes	characteristic
43	C₃×C₅×Q₈	—	2	4	no	—
44	C₃×C₅×Q₈	—	2	2	no	—
45	C₃×C₅×Q₈	—	2	2	no	—
46	C₂×C₃×C₄×C₅	—	2	2	no	—
47	C₂×C₃×C₄×C₅	—	2	1	no	—
48	C₂×C₃×C₄×C₅	—	3	2	no	—
49	C₂×C₃×C₄×C₅	—	3	1	no	—
50	C₂×C₃×C₄×C₅	—	6	2	no	—
51	C₂×C₃×C₄×C₅	C₂×D₃	1	2	yes	—
52	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂²×C₃	1	2	yes	—
54	C₂×C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
55	C₂×C₅×Dic₃	C₂²×C₃	1	1	yes	characteristic
56	C₂×C₅×Dic₃	C₂²×C₃	1	2	yes	—
57	C₄×C₅×D₃	—	2	2	no	—
58	C₃×C₂²⋊Q₈	—	3	1	no	—
59	#946	C₃×C₅	1	1	yes	characteristic, direct product factor
60	C₂×C₃²×C₅	C₂×D₄	1	1	yes	characteristic
61	C₂×C₃²×C₅	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₃²×C₅	#41	1	1	yes	characteristic
63	C₃×C₅×D₃	—	2	2	no	—
64	C₂×C₅×Q₈	C₃×D₃	1	1	yes	characteristic, complemented
65	C₂²×C₄×C₅	—	3	1	no	—
66	C₅×C₄⋊C₄	—	3	2	no	—
67	C₅×C₄⋊C₄	—	3	1	no	—
68	C₅×C₂²⋊C₄	—	3	2	no	—
69	C₂²×C₃×D₃	C₂²×C₅	1	1	yes	characteristic
70	C₃²×Q₈	—	2	2	no	—
71	C₂×C₃²×C₄	C₂²×C₅	1	2	yes	—
72	C₂×C₃²×C₄	C₂²×C₅	1	1	yes	characteristic
73	C₂×C₃×Dic₃	C₂²×C₅	1	1	yes	characteristic
74	C₂×C₃×Dic₃	C₂²×C₅	1	2	yes	—
75	C₃×C₄×D₃	—	2	2	no	—
76	C₂²×C₃×C₅	—	2	1	no	—
77	C₂²×C₃×C₅	—	3	2	no	—
78	C₂²×C₃×C₅	—	6	1	no	—
79	C₂²×C₃×C₅	—	6	1	no	—
80	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
81	C₂²×C₃×C₅	C₂³×C₃	1	1	yes	characteristic
82	C₂×C₅×D₃	—	2	1	no	—
83	C₂×C₅×D₃	—	2	1	no	—
84	C₂×C₅×D₃	C₃×Q₈	1	2	yes	complemented
85	C₃×C₄×C₅	—	2	4	no	—
86	C₃×C₄×C₅	—	2	2	no	—
87	C₃×C₄×C₅	—	2	2	no	—
88	C₃×C₄×C₅	—	2	2	no	—
89	C₃×C₄×C₅	—	6	2	no	—
90	C₃×C₄×C₅	—	6	1	no	—
91	C₃×C₄×C₅	#67	1	2	yes	—
92	C₃×C₄×C₅	C₃×D₄	1	2	yes	—
93	C₅×Dic₃	—	2	1	no	—
94	C₅×Dic₃	—	2	2	no	—
95	C₂×C₃×Q₈	—	2	1	no	—
96	C₂×C₃×Q₈	C₅×D₃	1	1	yes	characteristic, complemented
97	C₂×C₃×Q₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
98	C₂²×C₃×C₄	—	3	1	no	—
99	C₂×C₄×D₃	C₂×C₃×C₅	1	1	yes	characteristic
100	C₃×C₄⋊C₄	—	3	2	no	—
101	C₃×C₄⋊C₄	—	3	1	no	—
102	C₃×C₂²⋊C₄	—	3	2	no	—
103	(C₂×D₃)⋊C₄	C₂×C₃×C₅	1	2	yes	—
104	C₄⋊Dic₃	C₂×C₃×C₅	1	1	yes	characteristic, complemented
105	Dic₃⋊C₄	C₂×C₃×C₅	1	2	yes	complemented
106	C₃²×C₅	C₂²⋊Q₈	1	1	yes	characteristic, complemented, Hall
107	C₂³×C₅	—	3	1	no	—
108	C₅×Q₈	—	2	2	no	—
109	C₂×C₄×C₅	—	3	2	no	—
110	C₂×C₄×C₅	—	3	1	no	—
111	C₂×C₄×C₅	—	6	2	no	—
112	C₂×C₄×C₅	C₂×C₃×D₃	1	2	yes	—
113	C₂×C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic
114	C₂²×C₃²	C₂³×C₅	1	1	yes	characteristic
115	C₂×C₃×D₃	—	2	1	no	—
116	C₂×C₃×D₃	—	2	1	no	—
117	C₂×C₃×D₃	C₅×Q₈	1	2	yes	complemented
118	C₃²×C₄	—	2	2	no	—
119	C₃²×C₄	C₅×D₄	1	2	yes	—
120	C₃×Dic₃	—	2	1	no	—
121	C₃×Dic₃	—	2	2	no	—
122	C₂²⋊Q₈	—	3	1	no	Sylow
123	C₂×C₃×C₅	—	2	1	no	—
124	C₂×C₃×C₅	—	2	1	no	—
125	C₂×C₃×C₅	—	2	1	no	—
126	C₂×C₃×C₅	—	6	2	no	—
127	C₂×C₃×C₅	D₃×Q₈	1	1	yes	characteristic
128	C₂×C₃×C₅	#239	1	1	yes	characteristic
129	C₂×C₃×C₅	C₂×#67	1	1	yes	characteristic
130	C₂×C₃×C₅	C₂×C₃×D₄	1	1	yes	characteristic
131	C₂×C₃×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
132	C₂×C₃×C₅	C₃×#41	1	1	yes	characteristic
133	C₅×D₃	—	2	2	no	—
134	C₂³×C₃	—	3	1	no	—
135	C₂²×D₃	C₂²×C₃×C₅	1	1	yes	characteristic
136	C₃×Q₈	—	2	4	no	—
137	C₃×Q₈	—	2	2	no	—
138	C₃×Q₈	—	2	2	no	—
139	C₂×C₃×C₄	—	2	2	no	—
140	C₂×C₃×C₄	—	2	1	no	—
141	C₂×C₃×C₄	—	3	2	no	—
142	C₂×C₃×C₄	—	3	1	no	—
143	C₂×C₃×C₄	—	6	2	no	—
144	C₂×C₃×C₄	C₂×C₅×D₃	1	2	yes	—
145	C₂×C₃×C₄	C₂×C₅×D₃	1	1	yes	characteristic
146	C₂×C₃×C₄	C₂²×C₃×C₅	1	2	yes	—
147	C₂×C₃×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
148	C₂×Dic₃	C₂²×C₃×C₅	1	1	yes	characteristic
149	C₂×Dic₃	C₂²×C₃×C₅	1	2	yes	—
150	C₄×D₃	—	2	2	no	—
151	C₂²×C₅	—	3	2	no	—
152	C₂²×C₅	—	6	1	no	—
153	C₂²×C₅	—	6	1	no	—
154	C₂²×C₅	C₂²×C₃×D₃	1	1	yes	characteristic
155	C₄×C₅	—	2	2	no	—
156	C₄×C₅	—	6	2	no	—
157	C₄×C₅	—	6	1	no	—
158	C₄×C₅	C₃×#67	1	2	yes	—
159	C₂×C₃²	C₂×C₅×D₄	1	1	yes	characteristic
160	C₂×C₃²	C₂×C₅×Q₈	1	1	yes	characteristic
161	C₂×C₃²	C₅×#41	1	1	yes	characteristic
162	C₃×D₃	—	2	2	no	—
163	C₂×Q₈	C₃×C₅×D₃	1	1	yes	characteristic, complemented
164	C₂²×C₄	—	3	1	no	—
165	C₄⋊C₄	—	3	2	no	—
166	C₄⋊C₄	—	3	1	no	—
167	C₂²⋊C₄	—	3	2	no	—
168	C₃×C₅	—	2	1	no	—
169	C₃×C₅	#946	1	1	yes	characteristic, direct product factor
170	C₃×C₅	C₃×C₂²⋊Q₈	1	1	yes	characteristic, complemented
171	C₂²×C₃	—	2	1	no	—
172	C₂²×C₃	—	3	2	no	—
173	C₂²×C₃	—	6	1	no	—
174	C₂²×C₃	—	6	1	no	—
175	C₂²×C₃	C₂²×C₅×D₃	1	1	yes	characteristic
176	C₂²×C₃	C₂³×C₃×C₅	1	1	yes	characteristic
177	C₂×D₃	—	2	1	no	—
178	C₂×D₃	—	2	1	no	—
179	C₂×D₃	C₃×C₅×Q₈	1	2	yes	complemented
180	C₃×C₄	—	2	4	no	—
181	C₃×C₄	—	2	2	no	—
182	C₃×C₄	—	2	2	no	—
183	C₃×C₄	—	2	2	no	—
184	C₃×C₄	—	6	2	no	—
185	C₃×C₄	—	6	1	no	—
186	C₃×C₄	C₅×#67	1	2	yes	—
187	C₃×C₄	C₃×C₅×D₄	1	2	yes	—
188	Dic₃	—	2	1	no	—
189	Dic₃	—	2	2	no	—
190	C₂×C₅	—	6	2	no	—
191	C₂×C₅	C₃×D₃×Q₈	1	1	yes	characteristic
192	C₂×C₅	C₃×#239	1	1	yes	characteristic
193	C₂×C₅	C₂×C₃×#67	1	1	yes	characteristic
194	C₃²	C₅×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
195	C₂³	—	3	1	no	—
196	Q₈	—	2	2	no	—
197	C₂×C₄	—	3	2	no	—
198	C₂×C₄	—	3	1	no	—
199	C₂×C₄	—	6	2	no	—
200	C₂×C₄	C₂×C₃×C₅×D₃	1	2	yes	—
201	C₂×C₄	C₂×C₃×C₅×D₃	1	1	yes	characteristic
202	C₂×C₃	—	2	1	no	—
203	C₂×C₃	—	2	1	no	—
204	C₂×C₃	—	2	1	no	—
205	C₂×C₃	—	6	2	no	—
206	C₂×C₃	C₅×D₃×Q₈	1	1	yes	characteristic
207	C₂×C₃	C₅×#239	1	1	yes	characteristic
208	C₂×C₃	C₂×C₅×#67	1	1	yes	characteristic
209	C₂×C₃	C₂×C₃×C₅×D₄	1	1	yes	characteristic
210	C₂×C₃	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
211	C₂×C₃	C₃×C₅×#41	1	1	yes	characteristic
212	D₃	—	2	2	no	—
213	C₅	C₃×#946	1	1	yes	characteristic, direct product factor, Sylow
214	C₂²	—	3	2	no	—
215	C₂²	—	6	1	no	—
216	C₂²	—	6	1	no	—
217	C₂²	C₂²×C₃×C₅×D₃	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	6	2	no	—
220	C₄	—	6	1	no	—
221	C₄	C₃×C₅×#67	1	2	yes	—
222	C₃	—	2	1	no	—
223	C₃	C₅×#946	1	1	yes	characteristic, direct product factor
224	C₃	C₃×C₅×C₂²⋊Q₈	1	1	yes	characteristic, complemented
225	C₂	—	6	2	no	—
226	C₂	C₃×C₅×D₃×Q₈	1	1	yes	characteristic
227	C₂	C₃×C₅×#239	1	1	yes	characteristic
228	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	C₂×C₃²×C₅×Q₈	C₂
socle	C₂²×C₃²×C₅	C₂³
prime radical	C₂²×C₃×C₅×D₃	C₂²
hypercenter	C₂×C₃×C₅×Q₈	D₃
hypocenter	C₃	C₃×C₅×C₂²⋊Q₈
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₂×C₃×C₅×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×C₅×Q₈	C₂	semidirect
4. C₃×C₅×C₄⋊Dic₃	C₂	semidirect
5. C₃×C₅×Dic₃⋊C₄	C₂	semidirect
7. C₅×#946	C₃	direct (N × H)
15. C₃×#946	C₅	direct (N × H)
17. C₂×C₃×C₅×Q₈	D₃	semidirect
18. C₂×C₃×C₅×Q₈	C₂×C₃	semidirect
25. C₅×C₄⋊Dic₃	C₂×C₃	semidirect
26. C₅×Dic₃⋊C₄	C₂×C₃	semidirect
30. C₂×C₃×C₅×D₃	Q₈	semidirect
36. C₂×C₃²×Q₈	C₂×C₅	semidirect
39. C₃×C₄⋊Dic₃	C₂×C₅	semidirect
40. C₃×Dic₃⋊C₄	C₂×C₅	semidirect
59. #946	C₃×C₅	direct (N × H)
64. C₂×C₅×Q₈	C₃×D₃	semidirect
84. C₂×C₅×D₃	C₃×Q₈	semidirect
96. C₂×C₃×Q₈	C₅×D₃	semidirect
97. C₂×C₃×Q₈	C₂×C₃×C₅	semidirect
104. C₄⋊Dic₃	C₂×C₃×C₅	semidirect
105. Dic₃⋊C₄	C₂×C₃×C₅	semidirect
106. C₃²×C₅	C₂²⋊Q₈	semidirect
117. C₂×C₃×D₃	C₅×Q₈	semidirect
163. C₂×Q₈	C₃×C₅×D₃	semidirect
169. C₃×C₅	#946	direct (N × H)
170. C₃×C₅	C₃×C₂²⋊Q₈	semidirect
179. C₂×D₃	C₃×C₅×Q₈	semidirect
194. C₃²	C₅×C₂²⋊Q₈	semidirect
213. C₅	C₃×#946	direct (N × H)
223. C₃	C₅×#946	direct (N × H)
224. 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
C₂×C₃²×C₅×Q₈	2
C₂×C₃×C₄×C₅×D₃	2
C₃×C₅×(C₂×D₃)⋊C₄	2
C₃×C₅×C₄⋊Dic₃	2
C₃×C₅×Dic₃⋊C₄	2
C₃×C₅×C₂²⋊Q₈	3
C₅×#946	3
C₃×#946	5

Subgroups — C3×C5×#946 #13087957

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#946 #13087957