Subgroups — C₂×C₃×#6719 #13088049

Order 1440.

662 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃²×C₅	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×Dic₁₅	C₂	1	2	yes	maximal, complemented
3	C₃×#6719	C₂	1	4	yes	maximal, direct product factor
4	C₂×#6719	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₂×C₃×C₂²⋊Dic₅	—	3	1	no	maximal
6	C₂³×C₃²×C₅	—	2	4	no	—
7	C₂³×C₃²×C₅	C₄	1	4	yes	complemented
8	C₂³×C₃²×C₅	C₂²	1	2	yes	—
9	C₂³×C₃²×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₃×Dic₁₅	—	2	4	no	—
11	C₂×C₃×Dic₁₅	C₂²	1	4	yes	complemented
12	C₂×C₃×#217	—	5	1	no	maximal, Hall
13	C₂⁴×C₃×C₅	—	2	1	no	—
14	C₂⁴×C₃×C₅	D₃	1	1	yes	characteristic
15	C₂⁴×C₃×C₅	C₂×C₃	1	1	yes	characteristic
16	C₂²×Dic₁₅	C₂×C₃	1	2	yes	complemented
17	C₂²×C₃×Dic₅	—	3	2	no	—
18	#6719	C₂×C₃	1	4	yes	direct product factor
19	C₃×C₂²⋊Dic₅	—	3	4	no	—
20	C₂²×C₃²×C₅	—	2	8	no	—
21	C₂²×C₃²×C₅	—	2	4	no	—
22	C₂²×C₃²×C₅	C₂×C₄	1	2	yes	—
23	C₂²×C₃²×C₅	C₂×C₄	1	4	yes	complemented
24	C₂²×C₃²×C₅	D₄	1	4	yes	—
25	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
26	C₃×Dic₁₅	—	2	4	no	—
27	C₂×C₂²⋊Dic₅	—	3	1	no	Hall
28	C₂⁴×C₃²	D₅	1	1	yes	characteristic
29	C₂²×C₃×Dic₃	—	5	2	no	—
30	C₃×#217	—	5	4	no	—
31	C₂³×C₃×C₅	—	2	8	no	—
32	C₂³×C₃×C₅	—	2	4	no	—
33	C₂³×C₃×C₅	—	2	2	no	—
34	C₂³×C₃×C₅	—	2	1	no	—
35	C₂³×C₃×C₅	—	2	4	no	—
36	C₂³×C₃×C₅	—	2	4	no	—
37	C₂³×C₃×C₅	Dic₃	1	4	yes	complemented
38	C₂³×C₃×C₅	C₃×C₄	1	4	yes	complemented
39	C₂³×C₃×C₅	C₂×D₃	1	2	yes	—
40	C₂³×C₃×C₅	C₂×D₃	1	1	yes	characteristic
41	C₂³×C₃×C₅	C₂²×C₃	1	2	yes	—
42	C₂³×C₃×C₅	C₂²×C₃	1	1	yes	characteristic
43	C₂×Dic₁₅	—	2	4	no	—
44	C₂×Dic₁₅	C₂²×C₃	1	4	yes	complemented
45	C₂×C₃×Dic₅	—	3	4	no	—
46	C₂×C₃×Dic₅	—	6	4	no	—
47	C₂×C₃×C₂²⋊C₄	—	15	1	no	—
48	C₂×#217	—	5	1	no	—
49	C₂×C₃²×C₅	—	2	4	no	—
50	C₂×C₃²×C₅	C₂²⋊C₄	1	4	yes	complemented
51	C₂×C₃²×C₅	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₃²×C₅	C₂×D₄	1	2	yes	—
53	C₂⁴×C₅	C₃×D₃	1	1	yes	characteristic
54	C₂²×Dic₅	—	3	2	no	—
55	C₂²⋊Dic₅	—	3	4	no	—
56	C₂³×C₃²	—	2	4	no	—
57	C₂³×C₃²	Dic₅	1	4	yes	complemented
58	C₂³×C₃²	C₂×D₅	1	2	yes	—
59	C₂³×C₃²	C₂×D₅	1	1	yes	characteristic
60	C₂×C₃×Dic₃	—	5	4	no	—
61	C₂×C₃×Dic₃	—	10	4	no	—
62	C₂²×C₃×C₅	—	2	16	no	—
63	C₂²×C₃×C₅	—	2	8	no	—
64	C₂²×C₃×C₅	—	2	8	no	—
65	C₂²×C₃×C₅	—	2	8	no	—
66	C₂²×C₃×C₅	—	2	4	no	—
67	C₂²×C₃×C₅	—	2	4	no	—
68	C₂²×C₃×C₅	—	2	2	no	—
69	C₂²×C₃×C₅	—	2	4	no	—
70	C₂²×C₃×C₅	—	2	4	no	—
71	C₂²×C₃×C₅	—	2	1	no	—
72	C₂²×C₃×C₅	C₂×Dic₃	1	2	yes	—
73	C₂²×C₃×C₅	C₂×Dic₃	1	4	yes	complemented
74	C₂²×C₃×C₅	#67	1	4	yes	—
75	C₂²×C₃×C₅	C₂×C₃×C₄	1	2	yes	—
76	C₂²×C₃×C₅	C₂×C₃×C₄	1	4	yes	complemented
77	C₂²×C₃×C₅	C₃×D₄	1	4	yes	—
78	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
79	C₂²×C₃×C₅	C₂³×C₃	1	1	yes	characteristic
80	Dic₁₅	—	2	4	no	—
81	C₃×Dic₅	—	6	4	no	—
82	C₂⁴×C₃	—	2	1	no	—
83	C₂⁴×C₃	C₃×D₅	1	1	yes	characteristic
84	C₂⁴×C₃	D₁₅	1	1	yes	characteristic
85	C₂²×C₃×C₄	—	15	2	no	—
86	C₂²×Dic₃	—	5	2	no	—
87	C₃×C₂²⋊C₄	—	15	4	no	—
88	#217	—	5	4	no	—
89	C₃²×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic, complemented, Hall
90	C₂³×C₅	—	2	4	no	—
91	C₂³×C₅	C₃×Dic₃	1	4	yes	complemented
92	C₂³×C₅	C₂×C₃×D₃	1	2	yes	—
93	C₂³×C₅	C₂×C₃×D₃	1	1	yes	characteristic
94	C₂×Dic₅	—	3	4	no	—
95	C₂×Dic₅	—	6	4	no	—
96	C₂²×C₃²	—	2	8	no	—
97	C₂²×C₃²	—	2	4	no	—
98	C₂²×C₃²	C₂×Dic₅	1	2	yes	—
99	C₂²×C₃²	C₂×Dic₅	1	4	yes	complemented
100	C₂²×C₃²	C₂²⋊D₅	1	4	yes	—
101	C₂²×C₃²	C₂²×D₅	1	1	yes	characteristic
102	C₃×Dic₃	—	10	4	no	—
103	C₂×C₂²⋊C₄	—	15	1	no	Sylow
104	C₂×C₃×C₅	—	2	8	no	—
105	C₂×C₃×C₅	—	2	4	no	—
106	C₂×C₃×C₅	—	2	4	no	—
107	C₂×C₃×C₅	—	2	4	no	—
108	C₂×C₃×C₅	—	2	1	no	—
109	C₂×C₃×C₅	—	2	2	no	—
110	C₂×C₃×C₅	#217	1	4	yes	complemented
111	C₂×C₃×C₅	C₃×C₂²⋊C₄	1	4	yes	complemented
112	C₂×C₃×C₅	C₂²×Dic₃	1	1	yes	characteristic
113	C₂×C₃×C₅	C₂×#67	1	2	yes	—
114	C₂×C₃×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
115	C₂×C₃×C₅	C₂×C₃×D₄	1	2	yes	—
116	C₂³×C₃	—	2	8	no	—
117	C₂³×C₃	—	2	4	no	—
118	C₂³×C₃	—	2	4	no	—
119	C₂³×C₃	—	2	4	no	—
120	C₂³×C₃	—	2	2	no	—
121	C₂³×C₃	—	2	1	no	—
122	C₂³×C₃	C₃×Dic₅	1	4	yes	complemented
123	C₂³×C₃	Dic₁₅	1	4	yes	complemented
124	C₂³×C₃	C₂×C₃×D₅	1	2	yes	—
125	C₂³×C₃	C₂×C₃×D₅	1	1	yes	characteristic
126	C₂³×C₃	C₂×D₁₅	1	2	yes	—
127	C₂³×C₃	C₂×D₁₅	1	1	yes	characteristic
128	C₂×C₃×C₄	—	15	4	no	—
129	C₂×C₃×C₄	—	30	4	no	—
130	C₂×Dic₃	—	5	4	no	—
131	C₂×Dic₃	—	10	4	no	—
132	C₂²×C₅	—	2	8	no	—
133	C₂²×C₅	—	2	4	no	—
134	C₂²×C₅	C₂×C₃×Dic₃	1	2	yes	—
135	C₂²×C₅	C₂×C₃×Dic₃	1	4	yes	complemented
136	C₂²×C₅	C₃×#67	1	4	yes	—
137	C₂²×C₅	C₂²×C₃×D₃	1	1	yes	characteristic
138	Dic₅	—	6	4	no	—
139	C₂×C₃²	—	2	4	no	—
140	C₂×C₃²	C₂²⋊Dic₅	1	4	yes	complemented
141	C₂×C₃²	C₂²×Dic₅	1	1	yes	characteristic
142	C₂×C₃²	C₂×C₂²⋊D₅	1	2	yes	—
143	C₂⁴	C₃×D₁₅	1	1	yes	characteristic
144	C₂²×C₄	—	15	2	no	—
145	C₂²⋊C₄	—	15	4	no	—
146	C₃×C₅	—	2	1	no	—
147	C₃×C₅	C₂×#217	1	1	yes	characteristic, complemented
148	C₃×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic, complemented
149	C₂²×C₃	—	2	8	no	—
150	C₂²×C₃	—	2	16	no	—
151	C₂²×C₃	—	2	8	no	—
152	C₂²×C₃	—	2	8	no	—
153	C₂²×C₃	—	2	4	no	—
154	C₂²×C₃	—	2	4	no	—
155	C₂²×C₃	—	2	2	no	—
156	C₂²×C₃	—	2	4	no	—
157	C₂²×C₃	—	2	4	no	—
158	C₂²×C₃	—	2	1	no	—
159	C₂²×C₃	C₂×C₃×Dic₅	1	2	yes	—
160	C₂²×C₃	C₂×C₃×Dic₅	1	4	yes	complemented
161	C₂²×C₃	C₃×C₂²⋊D₅	1	4	yes	—
162	C₂²×C₃	C₂×Dic₁₅	1	2	yes	—
163	C₂²×C₃	C₂×Dic₁₅	1	4	yes	complemented
164	C₂²×C₃	#1220	1	4	yes	—
165	C₂²×C₃	C₂²×C₃×D₅	1	1	yes	characteristic
166	C₂²×C₃	C₂²×D₁₅	1	1	yes	characteristic
167	C₃×C₄	—	30	4	no	—
168	Dic₃	—	10	4	no	—
169	C₂×C₅	—	2	4	no	—
170	C₂×C₅	C₃×#217	1	4	yes	complemented
171	C₂×C₅	C₂²×C₃×Dic₃	1	1	yes	characteristic
172	C₂×C₅	C₂×C₃×#67	1	2	yes	—
173	C₃²	C₂×C₂²⋊Dic₅	1	1	yes	characteristic, complemented, Sylow
174	C₂³	—	2	4	no	—
175	C₂³	C₃×Dic₁₅	1	4	yes	complemented
176	C₂³	C₂×C₃×D₁₅	1	2	yes	—
177	C₂³	C₂×C₃×D₁₅	1	1	yes	characteristic
178	C₂×C₄	—	15	4	no	—
179	C₂×C₄	—	30	4	no	—
180	C₂×C₃	—	2	8	no	—
181	C₂×C₃	—	2	4	no	—
182	C₂×C₃	—	2	4	no	—
183	C₂×C₃	—	2	1	no	—
184	C₂×C₃	—	2	2	no	—
185	C₂×C₃	—	2	4	no	—
186	C₂×C₃	C₃×C₂²⋊Dic₅	1	4	yes	complemented
187	C₂×C₃	#6719	1	4	yes	direct product factor
188	C₂×C₃	C₂²×C₃×Dic₅	1	1	yes	characteristic
189	C₂×C₃	C₂×C₃×C₂²⋊D₅	1	2	yes	—
190	C₂×C₃	C₂²×Dic₁₅	1	1	yes	characteristic
191	C₂×C₃	C₂×#1220	1	2	yes	—
192	C₅	C₂×C₃×#217	1	1	yes	characteristic, complemented, Sylow
193	C₂²	—	2	4	no	—
194	C₂²	—	2	8	no	—
195	C₂²	C₂×C₃×Dic₁₅	1	2	yes	—
196	C₂²	C₂×C₃×Dic₁₅	1	4	yes	complemented
197	C₂²	C₃×#1220	1	4	yes	—
198	C₂²	C₂²×C₃×D₁₅	1	1	yes	characteristic
199	C₄	—	30	4	no	—
200	C₃	—	2	1	no	—
201	C₃	C₂×C₃×C₂²⋊Dic₅	1	1	yes	characteristic, complemented
202	C₃	C₂×#6719	1	1	yes	characteristic, direct product factor
203	C₂	—	2	4	no	—
204	C₂	C₃×#6719	1	4	yes	direct product factor
205	C₂	C₂²×C₃×Dic₁₅	1	1	yes	characteristic
206	C₂	C₂×C₃×#1220	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂×D₁₅
commutator	C₂×C₃×C₅	C₂²×C₃×C₄
Frattini	C₂²	C₂²×C₃×D₁₅
Fitting, prime radical	C₂⁴×C₃²×C₅	C₂
socle	C₂³×C₃²×C₅	C₂²
cosocle	C₂²×C₃×C₅	C₂³×C₃
hypercenter	C₂⁴×C₃	D₁₅
hypocenter	C₃×C₅	C₂×C₃×C₂²⋊C₄
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂⁴×C₃

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×Dic₁₅	C₂	semidirect
3. C₃×#6719	C₂	direct (N × H)
4. C₂×#6719	C₃	direct (N × H)
7. C₂³×C₃²×C₅	C₄	semidirect
11. C₂×C₃×Dic₁₅	C₂²	semidirect
16. C₂²×Dic₁₅	C₂×C₃	semidirect
18. #6719	C₂×C₃	direct (N × H)
23. C₂²×C₃²×C₅	C₂×C₄	semidirect
37. C₂³×C₃×C₅	Dic₃	semidirect
38. C₂³×C₃×C₅	C₃×C₄	semidirect
44. C₂×Dic₁₅	C₂²×C₃	semidirect
50. C₂×C₃²×C₅	C₂²⋊C₄	semidirect
57. C₂³×C₃²	Dic₅	semidirect
73. C₂²×C₃×C₅	C₂×Dic₃	semidirect
76. C₂²×C₃×C₅	C₂×C₃×C₄	semidirect
89. C₃²×C₅	C₂×C₂²⋊C₄	semidirect
91. C₂³×C₅	C₃×Dic₃	semidirect
99. C₂²×C₃²	C₂×Dic₅	semidirect
110. C₂×C₃×C₅	#217	semidirect
111. C₂×C₃×C₅	C₃×C₂²⋊C₄	semidirect
122. C₂³×C₃	C₃×Dic₅	semidirect
123. C₂³×C₃	Dic₁₅	semidirect
135. C₂²×C₅	C₂×C₃×Dic₃	semidirect
140. C₂×C₃²	C₂²⋊Dic₅	semidirect
147. C₃×C₅	C₂×#217	semidirect
148. C₃×C₅	C₂×C₃×C₂²⋊C₄	semidirect
160. C₂²×C₃	C₂×C₃×Dic₅	semidirect
163. C₂²×C₃	C₂×Dic₁₅	semidirect
170. C₂×C₅	C₃×#217	semidirect
173. C₃²	C₂×C₂²⋊Dic₅	semidirect
175. C₂³	C₃×Dic₁₅	semidirect
186. C₂×C₃	C₃×C₂²⋊Dic₅	semidirect
187. C₂×C₃	#6719	direct (N × H)
192. C₅	C₂×C₃×#217	semidirect
196. C₂²	C₂×C₃×Dic₁₅	semidirect
201. C₃	C₂×C₃×C₂²⋊Dic₅	semidirect
202. C₃	C₂×#6719	direct (N × H)
204. C₂	C₃×#6719	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₃²×C₅	2
C₂²×C₃×Dic₁₅	2
C₃×#6719	2
C₂×#6719	3
C₂×C₃×C₂²⋊Dic₅	3
C₂×C₃×#217	5

Subgroups — C2×C3×#6719 #13088049

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#6719 #13088049