Subgroups — C₂×C₃×D₅×C₃⋊C₈ #13086480

Order 1440.

358 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	C₂×C₃²×C₄×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₃⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×D₅×C₃⋊C₈	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₃×C₈×D₅	—	3	1	no	maximal
6	C₂×D₅×C₃⋊C₈	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂²×C₃²×D₅	C₄	1	1	yes	characteristic
8	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₃²×Dic₅	C₄	1	1	yes	characteristic
10	C₃²×C₄×D₅	C₄	1	2	yes	—
11	C₃²×C₄×D₅	C₂²	1	2	yes	—
12	C₃×C₃⋊#168	C₂²	1	2	yes	complemented
13	C₃×C₅×C₃⋊C₈	C₂²	1	2	yes	complemented
14	C₂²×C₃×C₃⋊C₈	—	5	1	no	maximal, Hall
15	C₂×C₃×C₄×D₅	—	2	1	no	—
16	C₂×C₃×C₄×D₅	D₃	1	1	yes	characteristic
17	C₂×C₃×C₄×D₅	C₂×C₃	1	1	yes	characteristic
18	C₂×C₃×C₅×C₈	—	3	1	no	—
19	C₂×C₃⋊#168	C₂×C₃	1	1	yes	characteristic, complemented
20	C₂×C₅×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
21	C₂×C₃×#168	—	3	1	no	—
22	C₃×C₈×D₅	—	3	4	no	—
23	D₅×C₃⋊C₈	C₂×C₃	1	4	yes	direct product factor
24	C₂²×C₃²×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃²×D₅	C₈	1	4	yes	complemented
26	C₂×C₃²×D₅	C₂×C₄	1	2	yes	—
27	C₃²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
28	C₃²×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₃²×Dic₅	C₂×C₄	1	2	yes	—
30	C₂×C₈×D₅	—	3	1	no	Hall
31	C₂²×C₃²×C₄	—	5	1	no	—
32	C₂×C₃×C₃⋊C₈	—	5	1	no	—
33	C₂×C₃×C₃⋊C₈	—	5	4	no	—
34	C₂×C₃×C₃⋊C₈	D₅	1	1	yes	characteristic, direct product factor
35	C₂²×C₃×D₅	—	2	1	no	—
36	C₂²×C₃×D₅	Dic₃	1	1	yes	characteristic
37	C₂²×C₃×D₅	C₃×C₄	1	1	yes	characteristic
38	C₂×C₃×C₄×C₅	—	2	1	no	—
39	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
40	C₂×C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
41	C₂×C₃×Dic₅	—	2	1	no	—
42	C₂×C₃×Dic₅	Dic₃	1	1	yes	characteristic
43	C₂×C₃×Dic₅	C₃×C₄	1	1	yes	characteristic
44	C₃×C₄×D₅	—	2	2	no	—
45	C₃×C₄×D₅	—	2	2	no	—
46	C₃×C₄×D₅	Dic₃	1	2	yes	—
47	C₃×C₄×D₅	C₃×C₄	1	2	yes	—
48	C₃×C₄×D₅	C₂×D₃	1	2	yes	—
49	C₃×C₄×D₅	C₂²×C₃	1	2	yes	—
50	C₃×C₅×C₈	—	3	2	no	—
51	C₃⋊#168	C₂²×C₃	1	2	yes	complemented
52	C₃×#168	—	3	2	no	—
53	C₅×C₃⋊C₈	C₂²×C₃	1	2	yes	complemented
54	C₂²×C₃×C₈	—	15	1	no	—
55	C₂²×C₃⋊C₈	—	5	1	no	—
56	C₂×C₃²×C₅	C₂×C₈	1	2	yes	complemented
57	C₂×C₃²×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₃²×D₅	C₂×C₈	1	4	yes	complemented
59	C₂×C₄×D₅	C₃×D₃	1	1	yes	characteristic
60	C₂×C₅×C₈	—	3	1	no	—
61	C₂×#168	—	3	1	no	—
62	C₈×D₅	—	3	4	no	—
63	C₂³×C₃²	—	5	1	no	—
64	C₂×C₃²×C₄	—	5	1	no	—
65	C₂×C₃²×C₄	—	5	2	no	—
66	C₂×C₃²×C₄	—	5	2	no	—
67	C₂×C₃²×C₄	C₂×D₅	1	1	yes	characteristic
68	C₃×C₃⋊C₈	—	5	2	no	—
69	C₃×C₃⋊C₈	C₂×D₅	1	2	yes	direct product factor
70	C₂²×C₃×C₅	—	2	1	no	—
71	C₂²×C₃×C₅	C₂×Dic₃	1	1	yes	characteristic
72	C₂²×C₃×C₅	C₂×C₃×C₄	1	1	yes	characteristic
73	C₂×C₃×D₅	—	2	4	no	—
74	C₂×C₃×D₅	—	2	2	no	—
75	C₂×C₃×D₅	C₃⋊C₈	1	4	yes	direct product factor
76	C₂×C₃×D₅	C₃×C₈	1	4	yes	complemented
77	C₂×C₃×D₅	C₂×Dic₃	1	2	yes	—
78	C₂×C₃×D₅	C₂×C₃×C₄	1	2	yes	—
79	C₃×C₄×C₅	—	2	1	no	—
80	C₃×C₄×C₅	—	2	1	no	—
81	C₃×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
82	C₃×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
83	C₃×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
84	C₃×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
85	C₃×Dic₅	—	2	2	no	—
86	C₃×Dic₅	C₂×Dic₃	1	2	yes	—
87	C₃×Dic₅	C₂×C₃×C₄	1	2	yes	—
88	C₂²×C₃×C₄	—	5	1	no	—
89	C₂²×C₃×C₄	—	5	1	no	—
90	C₂²×C₃×C₄	—	10	1	no	—
91	C₂×C₃×C₈	—	3	1	no	—
92	C₂×C₃×C₈	—	15	4	no	—
93	C₂×C₃×C₈	—	15	1	no	—
94	C₂×C₃⋊C₈	—	5	1	no	—
95	C₂×C₃⋊C₈	—	5	4	no	—
96	C₂×C₃⋊C₈	C₃×D₅	1	1	yes	characteristic, direct product factor
97	C₃²×C₅	C₂²×C₈	1	1	yes	characteristic, complemented, Hall
98	C₂²×D₅	C₃×Dic₃	1	1	yes	characteristic
99	C₂×C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic
100	C₂×Dic₅	C₃×Dic₃	1	1	yes	characteristic
101	C₄×D₅	C₃×Dic₃	1	2	yes	—
102	C₄×D₅	C₂×C₃×D₃	1	2	yes	—
103	C₅×C₈	—	3	2	no	—
104	#168	—	3	2	no	—
105	C₂²×C₃²	—	5	4	no	—
106	C₂²×C₃²	—	5	2	no	—
107	C₂²×C₃²	C₄×D₅	1	1	yes	characteristic
108	C₃²×C₄	—	5	2	no	—
109	C₃²×C₄	C₄×D₅	1	1	yes	characteristic
110	C₃²×C₄	C₂²×D₅	1	1	yes	characteristic
111	C₂²×C₈	—	15	1	no	Sylow
112	C₂×C₃×C₅	—	2	2	no	—
113	C₂×C₃×C₅	—	2	1	no	—
114	C₂×C₃×C₅	C₂×C₃⋊C₈	1	2	yes	complemented
115	C₂×C₃×C₅	C₂×C₃×C₈	1	2	yes	complemented
116	C₂×C₃×C₅	C₂²×Dic₃	1	1	yes	characteristic
117	C₂×C₃×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
118	C₃×D₅	—	2	4	no	—
119	C₃×D₅	C₂×C₃⋊C₈	1	4	yes	direct product factor
120	C₃×D₅	C₂×C₃×C₈	1	4	yes	complemented
121	C₂³×C₃	—	5	1	no	—
122	C₂³×C₃	—	5	1	no	—
123	C₂³×C₃	—	10	1	no	—
124	C₂×C₃×C₄	—	2	1	no	—
125	C₂×C₃×C₄	—	5	1	no	—
126	C₂×C₃×C₄	—	5	1	no	—
127	C₂×C₃×C₄	—	5	2	no	—
128	C₂×C₃×C₄	—	5	2	no	—
129	C₂×C₃×C₄	—	5	2	no	—
130	C₂×C₃×C₄	—	5	2	no	—
131	C₂×C₃×C₄	—	10	2	no	—
132	C₂×C₃×C₄	—	10	2	no	—
133	C₂×C₃×C₄	—	10	1	no	—
134	C₂×C₃×C₄	D₃×D₅	1	1	yes	characteristic
135	C₂×C₃×C₄	C₂×C₃×D₅	1	1	yes	characteristic
136	C₃×C₈	—	3	2	no	—
137	C₃×C₈	—	15	2	no	—
138	C₃⋊C₈	—	5	2	no	—
139	C₃⋊C₈	C₂×C₃×D₅	1	2	yes	direct product factor
140	C₂²×C₅	C₂×C₃×Dic₃	1	1	yes	characteristic
141	C₂×D₅	C₃×C₃⋊C₈	1	4	yes	direct product factor
142	C₂×D₅	C₂×C₃×Dic₃	1	2	yes	—
143	C₄×C₅	C₂×C₃×Dic₃	1	1	yes	characteristic
144	C₄×C₅	C₂²×C₃×D₃	1	1	yes	characteristic
145	Dic₅	C₂×C₃×Dic₃	1	2	yes	—
146	C₂×C₃²	—	5	4	no	—
147	C₂×C₃²	C₈×D₅	1	2	yes	complemented
148	C₂×C₃²	C₂×C₄×D₅	1	1	yes	characteristic
149	C₂²×C₄	—	5	1	no	—
150	C₂×C₈	—	3	1	no	—
151	C₂×C₈	—	15	4	no	—
152	C₂×C₈	—	15	1	no	—
153	C₃×C₅	—	2	1	no	—
154	C₃×C₅	C₂²×C₃⋊C₈	1	1	yes	characteristic, complemented
155	C₃×C₅	C₂²×C₃×C₈	1	1	yes	characteristic, complemented
156	C₂²×C₃	—	2	1	no	—
157	C₂²×C₃	—	5	4	no	—
158	C₂²×C₃	—	5	2	no	—
159	C₂²×C₃	—	5	2	no	—
160	C₂²×C₃	—	5	4	no	—
161	C₂²×C₃	—	10	2	no	—
162	C₂²×C₃	—	10	4	no	—
163	C₂²×C₃	D₅×Dic₃	1	1	yes	characteristic
164	C₂²×C₃	C₃×C₄×D₅	1	1	yes	characteristic
165	C₃×C₄	—	2	1	no	—
166	C₃×C₄	—	2	1	no	—
167	C₃×C₄	—	5	2	no	—
168	C₃×C₄	—	5	2	no	—
169	C₃×C₄	—	10	2	no	—
170	C₃×C₄	D₅×Dic₃	1	1	yes	characteristic
171	C₃×C₄	C₃×C₄×D₅	1	1	yes	characteristic
172	C₃×C₄	C₂×D₃×D₅	1	1	yes	characteristic
173	C₃×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
174	C₂×C₅	C₂×C₃×C₃⋊C₈	1	2	yes	complemented
175	C₂×C₅	C₂²×C₃×Dic₃	1	1	yes	characteristic
176	D₅	C₂×C₃×C₃⋊C₈	1	4	yes	direct product factor
177	C₃²	C₂×C₈×D₅	1	1	yes	characteristic, complemented, Sylow
178	C₂³	—	5	1	no	—
179	C₂×C₄	—	5	1	no	—
180	C₂×C₄	—	5	2	no	—
181	C₂×C₄	—	5	2	no	—
182	C₂×C₄	C₃×D₃×D₅	1	1	yes	characteristic
183	C₈	—	3	2	no	—
184	C₈	—	15	2	no	—
185	C₂×C₃	—	2	1	no	—
186	C₂×C₃	—	2	2	no	—
187	C₂×C₃	—	5	4	no	—
188	C₂×C₃	—	5	4	no	—
189	C₂×C₃	—	10	4	no	—
190	C₂×C₃	D₅×C₃⋊C₈	1	2	yes	direct product factor
191	C₂×C₃	C₃×C₈×D₅	1	2	yes	complemented
192	C₂×C₃	C₂×D₅×Dic₃	1	1	yes	characteristic
193	C₂×C₃	C₂×C₃×C₄×D₅	1	1	yes	characteristic
194	C₅	C₂²×C₃×C₃⋊C₈	1	1	yes	characteristic, complemented, Sylow
195	C₂²	—	5	2	no	—
196	C₂²	—	5	4	no	—
197	C₂²	C₃×D₅×Dic₃	1	1	yes	characteristic
198	C₄	—	5	2	no	—
199	C₄	C₃×D₅×Dic₃	1	1	yes	characteristic
200	C₄	C₂×C₃×D₃×D₅	1	1	yes	characteristic
201	C₃	—	2	1	no	—
202	C₃	C₂×D₅×C₃⋊C₈	1	1	yes	characteristic, direct product factor
203	C₃	C₂×C₃×C₈×D₅	1	1	yes	characteristic, complemented
204	C₂	—	5	4	no	—
205	C₂	C₃×D₅×C₃⋊C₈	1	2	yes	direct product factor
206	C₂	C₂×C₃×D₅×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃×C₄	D₃×D₅
commutator, hypocenter	C₃×C₅	C₂²×C₃×C₈
Frattini	C₄	C₂×C₃×D₃×D₅
Fitting	C₂×C₃²×C₄×C₅	C₂²
socle	C₂²×C₃²×C₅	C₂×C₄
prime radical	C₂²×C₃²×D₅	C₄
cosocle	C₃×C₄×C₅	C₂³×C₃
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₃×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₃⋊#168	C₂	semidirect
3. C₂×C₃×C₅×C₃⋊C₈	C₂	semidirect
4. C₃×D₅×C₃⋊C₈	C₂	direct (N × H)
6. C₂×D₅×C₃⋊C₈	C₃	direct (N × H)
12. C₃×C₃⋊#168	C₂²	semidirect
13. C₃×C₅×C₃⋊C₈	C₂²	semidirect
19. C₂×C₃⋊#168	C₂×C₃	semidirect
20. C₂×C₅×C₃⋊C₈	C₂×C₃	semidirect
23. D₅×C₃⋊C₈	C₂×C₃	direct (N × H)
25. C₂×C₃²×D₅	C₈	semidirect
34. C₂×C₃×C₃⋊C₈	D₅	direct (N × H)
51. C₃⋊#168	C₂²×C₃	semidirect
53. C₅×C₃⋊C₈	C₂²×C₃	semidirect
56. C₂×C₃²×C₅	C₂×C₈	semidirect
58. C₃²×D₅	C₂×C₈	semidirect
69. C₃×C₃⋊C₈	C₂×D₅	direct (N × H)
75. C₂×C₃×D₅	C₃⋊C₈	direct (N × H)
76. C₂×C₃×D₅	C₃×C₈	semidirect
96. C₂×C₃⋊C₈	C₃×D₅	direct (N × H)
97. C₃²×C₅	C₂²×C₈	semidirect
114. C₂×C₃×C₅	C₂×C₃⋊C₈	semidirect
115. C₂×C₃×C₅	C₂×C₃×C₈	semidirect
119. C₃×D₅	C₂×C₃⋊C₈	direct (N × H)
120. C₃×D₅	C₂×C₃×C₈	semidirect
139. C₃⋊C₈	C₂×C₃×D₅	direct (N × H)
141. C₂×D₅	C₃×C₃⋊C₈	direct (N × H)
147. C₂×C₃²	C₈×D₅	semidirect
154. C₃×C₅	C₂²×C₃⋊C₈	semidirect
155. C₃×C₅	C₂²×C₃×C₈	semidirect
174. C₂×C₅	C₂×C₃×C₃⋊C₈	semidirect
176. D₅	C₂×C₃×C₃⋊C₈	direct (N × H)
177. C₃²	C₂×C₈×D₅	semidirect
190. C₂×C₃	D₅×C₃⋊C₈	direct (N × H)
191. C₂×C₃	C₃×C₈×D₅	semidirect
194. C₅	C₂²×C₃×C₃⋊C₈	semidirect
202. C₃	C₂×D₅×C₃⋊C₈	direct (N × H)
203. C₃	C₂×C₃×C₈×D₅	semidirect
205. C₂	C₃×D₅×C₃⋊C₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₄×D₅	2
C₂×C₃×C₃⋊#168	2
C₂×C₃×C₅×C₃⋊C₈	2
C₃×D₅×C₃⋊C₈	2
C₂×C₃×C₈×D₅	3
C₂×D₅×C₃⋊C₈	3
C₂²×C₃×C₃⋊C₈	5

Subgroups — C2×C3×D5×C3⋊C8 #13086480

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×D₅×C₃⋊C₈ #13086480