Subgroups — C₂×C₃×C₅×Dic₃⋊C₄ #13087934

Order 1440.

422 conjugacy classes of subgroups.

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

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

Complements

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

N	H ≅ G/N	type
4. C₃×C₅×Dic₃⋊C₄	C₂	direct (N × H)
6. C₂×C₅×Dic₃⋊C₄	C₃	direct (N × H)
11. C₂×C₃×C₅×Dic₃	C₄	semidirect
14. C₂×C₃×Dic₃⋊C₄	C₅	direct (N × H)
23. C₅×Dic₃⋊C₄	C₂×C₃	direct (N × H)
30. C₃×C₅×Dic₃	C₂×C₄	semidirect
35. C₃×Dic₃⋊C₄	C₂×C₅	direct (N × H)
50. C₂×C₅×Dic₃	C₃×C₄	semidirect
54. C₂×Dic₃⋊C₄	C₃×C₅	direct (N × H)
55. C₂×C₃²×C₅	C₄⋊C₄	semidirect
67. C₂×C₃×Dic₃	C₄×C₅	semidirect
88. C₅×Dic₃	C₂×C₃×C₄	semidirect
97. Dic₃⋊C₄	C₂×C₃×C₅	direct (N × H)
98. C₃²×C₅	C₂×C₄⋊C₄	semidirect
112. C₃×Dic₃	C₂×C₄×C₅	semidirect
118. C₂×C₃×C₅	Dic₃⋊C₄	direct (N × H)
119. C₂×C₃×C₅	C₃×C₄⋊C₄	semidirect
140. C₂×Dic₃	C₃×C₄×C₅	semidirect
150. C₂×C₃²	C₅×C₄⋊C₄	semidirect
159. C₃×C₅	C₂×Dic₃⋊C₄	direct (N × H)
160. C₃×C₅	C₂×C₃×C₄⋊C₄	semidirect
179. Dic₃	C₂×C₃×C₄×C₅	semidirect
180. C₂×C₅	C₃×Dic₃⋊C₄	direct (N × H)
184. C₃²	C₂×C₅×C₄⋊C₄	semidirect
196. C₂×C₃	C₅×Dic₃⋊C₄	direct (N × H)
197. C₂×C₃	C₃×C₅×C₄⋊C₄	semidirect
204. C₅	C₂×C₃×Dic₃⋊C₄	direct (N × H)
213. C₃	C₂×C₅×Dic₃⋊C₄	direct (N × H)
214. C₃	C₂×C₃×C₅×C₄⋊C₄	semidirect
215. C₂	C₃×C₅×Dic₃⋊C₄	direct (N × H)

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
C₂²×C₃×C₅×Dic₃	2
C₃×C₅×Dic₃⋊C₄	2
C₂×C₃×C₅×C₄⋊C₄	3
C₂×C₅×Dic₃⋊C₄	3
C₂×C₃×Dic₃⋊C₄	5

Subgroups — C2×C3×C5×Dic3⋊C4 #13087934

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×Dic₃⋊C₄ #13087934