Subgroups — C₂×C₃×C₄⋊Dic₁₅ #13088022

Order 1440.

422 conjugacy classes of subgroups.

The subgroup lattice has 188 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×Dic₁₅	C₂	1	2	yes	maximal
3	C₃×C₄⋊Dic₁₅	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄⋊Dic₁₅	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₂×C₃×C₄⋊Dic₅	—	3	1	no	maximal
6	C₂³×C₃²×C₅	C₂²	1	1	yes	characteristic
7	C₂×C₃²×C₄×C₅	C₄	1	4	yes	complemented
8	C₂×C₃²×C₄×C₅	C₂²	1	2	yes	—
9	C₂×C₃×Dic₁₅	—	2	4	no	—
10	C₂×C₃×Dic₁₅	C₂²	1	4	yes	—
11	C₂×C₃×C₄⋊Dic₃	—	5	1	no	maximal, Hall
12	C₂²×C₃×C₄×C₅	—	2	1	no	—
13	C₂²×C₃×C₄×C₅	D₃	1	1	yes	characteristic
14	C₂²×C₃×C₄×C₅	C₂×C₃	1	1	yes	characteristic
15	C₂²×Dic₁₅	C₂×C₃	1	2	yes	—
16	C₂²×C₃×Dic₅	—	3	2	no	—
17	C₄⋊Dic₁₅	C₂×C₃	1	4	yes	direct product factor
18	C₃×C₄⋊Dic₅	—	3	4	no	—
19	C₂²×C₃²×C₅	C₂×C₄	1	2	yes	—
20	C₂²×C₃²×C₅	D₄	1	2	yes	—
21	C₂²×C₃²×C₅	Q₈	1	2	yes	—
22	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
23	C₃²×C₄×C₅	C₂×C₄	1	4	yes	complemented
24	C₃×Dic₁₅	—	2	4	no	—
25	C₂×C₄⋊Dic₅	—	3	1	no	Hall
26	C₂²×C₃²×C₄	D₅	1	1	yes	characteristic
27	C₂²×C₃×Dic₃	—	5	2	no	—
28	C₃×C₄⋊Dic₃	—	5	4	no	—
29	C₂³×C₃×C₅	—	2	1	no	—
30	C₂³×C₃×C₅	C₂×D₃	1	1	yes	characteristic
31	C₂³×C₃×C₅	C₂²×C₃	1	1	yes	characteristic
32	C₂×C₃×C₄×C₅	—	2	4	no	—
33	C₂×C₃×C₄×C₅	—	2	2	no	—
34	C₂×C₃×C₄×C₅	Dic₃	1	4	yes	complemented
35	C₂×C₃×C₄×C₅	C₃×C₄	1	4	yes	complemented
36	C₂×C₃×C₄×C₅	C₂×D₃	1	2	yes	—
37	C₂×C₃×C₄×C₅	C₂²×C₃	1	2	yes	—
38	C₂×Dic₁₅	—	2	4	no	—
39	C₂×Dic₁₅	C₂²×C₃	1	4	yes	—
40	C₂×C₃×Dic₅	—	3	4	no	—
41	C₂×C₃×Dic₅	—	6	4	no	—
42	C₂×C₃×C₄⋊C₄	—	15	1	no	—
43	C₂×C₄⋊Dic₃	—	5	1	no	—
44	C₂×C₃²×C₅	C₄⋊C₄	1	4	yes	complemented
45	C₂×C₃²×C₅	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₃²×C₅	C₂×D₄	1	1	yes	characteristic
47	C₂×C₃²×C₅	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₄×C₅	C₃×D₃	1	1	yes	characteristic
49	C₂²×Dic₅	—	3	2	no	—
50	C₄⋊Dic₅	—	3	4	no	—
51	C₂³×C₃²	C₂×D₅	1	1	yes	characteristic
52	C₂×C₃²×C₄	Dic₅	1	4	yes	complemented
53	C₂×C₃²×C₄	C₂×D₅	1	2	yes	—
54	C₂×C₃×Dic₃	—	5	4	no	—
55	C₂×C₃×Dic₃	—	10	4	no	—
56	C₂²×C₃×C₅	—	2	2	no	—
57	C₂²×C₃×C₅	—	2	2	no	—
58	C₂²×C₃×C₅	—	2	2	no	—
59	C₂²×C₃×C₅	—	2	1	no	—
60	C₂²×C₃×C₅	Dic₆	1	2	yes	—
61	C₂²×C₃×C₅	D₁₂	1	2	yes	—
62	C₂²×C₃×C₅	C₂×Dic₃	1	2	yes	—
63	C₂²×C₃×C₅	C₂×C₃×C₄	1	2	yes	—
64	C₂²×C₃×C₅	C₃×D₄	1	2	yes	—
65	C₂²×C₃×C₅	C₃×Q₈	1	2	yes	—
66	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
67	C₂²×C₃×C₅	C₂³×C₃	1	1	yes	characteristic
68	C₃×C₄×C₅	—	2	4	no	—
69	C₃×C₄×C₅	C₂×Dic₃	1	4	yes	complemented
70	C₃×C₄×C₅	C₂×C₃×C₄	1	4	yes	complemented
71	Dic₁₅	—	2	4	no	—
72	C₃×Dic₅	—	6	4	no	—
73	C₂²×C₃×C₄	—	2	1	no	—
74	C₂²×C₃×C₄	—	15	2	no	—
75	C₂²×C₃×C₄	C₃×D₅	1	1	yes	characteristic
76	C₂²×C₃×C₄	D₁₅	1	1	yes	characteristic
77	C₂²×Dic₃	—	5	2	no	—
78	C₃×C₄⋊C₄	—	15	4	no	—
79	C₄⋊Dic₃	—	5	4	no	—
80	C₃²×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic, complemented, Hall
81	C₂³×C₅	C₂×C₃×D₃	1	1	yes	characteristic
82	C₂×C₄×C₅	C₃×Dic₃	1	4	yes	complemented
83	C₂×C₄×C₅	C₂×C₃×D₃	1	2	yes	—
84	C₂×Dic₅	—	3	4	no	—
85	C₂×Dic₅	—	6	4	no	—
86	C₂²×C₃²	Dic₁₀	1	2	yes	—
87	C₂²×C₃²	D₂₀	1	2	yes	—
88	C₂²×C₃²	C₂×Dic₅	1	2	yes	—
89	C₂²×C₃²	C₂²×D₅	1	1	yes	characteristic
90	C₃²×C₄	C₂×Dic₅	1	4	yes	complemented
91	C₃×Dic₃	—	10	4	no	—
92	C₂×C₄⋊C₄	—	15	1	no	Sylow
93	C₂×C₃×C₅	—	2	4	no	—
94	C₂×C₃×C₅	—	2	1	no	—
95	C₂×C₃×C₅	—	2	1	no	—
96	C₂×C₃×C₅	—	2	1	no	—
97	C₂×C₃×C₅	C₄⋊Dic₃	1	4	yes	complemented
98	C₂×C₃×C₅	C₃×C₄⋊C₄	1	4	yes	complemented
99	C₂×C₃×C₅	C₂×Dic₆	1	1	yes	characteristic
100	C₂×C₃×C₅	C₂×D₁₂	1	1	yes	characteristic
101	C₂×C₃×C₅	C₂²×Dic₃	1	1	yes	characteristic
102	C₂×C₃×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
103	C₂×C₃×C₅	C₂×C₃×D₄	1	1	yes	characteristic
104	C₂×C₃×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
105	C₂³×C₃	—	2	1	no	—
106	C₂³×C₃	C₂×C₃×D₅	1	1	yes	characteristic
107	C₂³×C₃	C₂×D₁₅	1	1	yes	characteristic
108	C₂×C₃×C₄	—	2	4	no	—
109	C₂×C₃×C₄	—	2	2	no	—
110	C₂×C₃×C₄	—	15	4	no	—
111	C₂×C₃×C₄	—	30	4	no	—
112	C₂×C₃×C₄	C₃×Dic₅	1	4	yes	complemented
113	C₂×C₃×C₄	Dic₁₅	1	4	yes	complemented
114	C₂×C₃×C₄	C₂×C₃×D₅	1	2	yes	—
115	C₂×C₃×C₄	C₂×D₁₅	1	2	yes	—
116	C₂×Dic₃	—	5	4	no	—
117	C₂×Dic₃	—	10	4	no	—
118	C₂²×C₅	C₃×Dic₆	1	2	yes	—
119	C₂²×C₅	C₃×D₁₂	1	2	yes	—
120	C₂²×C₅	C₂×C₃×Dic₃	1	2	yes	—
121	C₂²×C₅	C₂²×C₃×D₃	1	1	yes	characteristic
122	C₄×C₅	C₂×C₃×Dic₃	1	4	yes	complemented
123	Dic₅	—	6	4	no	—
124	C₂×C₃²	C₄⋊Dic₅	1	4	yes	complemented
125	C₂×C₃²	C₂×Dic₁₀	1	1	yes	characteristic
126	C₂×C₃²	C₂×D₂₀	1	1	yes	characteristic
127	C₂×C₃²	C₂²×Dic₅	1	1	yes	characteristic
128	C₂²×C₄	—	15	2	no	—
129	C₂²×C₄	C₃×D₁₅	1	1	yes	characteristic
130	C₄⋊C₄	—	15	4	no	—
131	C₃×C₅	—	2	1	no	—
132	C₃×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic, complemented
133	C₃×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic, complemented
134	C₂²×C₃	—	2	2	no	—
135	C₂²×C₃	—	2	2	no	—
136	C₂²×C₃	—	2	2	no	—
137	C₂²×C₃	—	2	1	no	—
138	C₂²×C₃	C₃×Dic₁₀	1	2	yes	—
139	C₂²×C₃	C₃×D₂₀	1	2	yes	—
140	C₂²×C₃	C₂×C₃×Dic₅	1	2	yes	—
141	C₂²×C₃	Dic₃₀	1	2	yes	—
142	C₂²×C₃	D₆₀	1	2	yes	—
143	C₂²×C₃	C₂×Dic₁₅	1	2	yes	—
144	C₂²×C₃	C₂²×C₃×D₅	1	1	yes	characteristic
145	C₂²×C₃	C₂²×D₁₅	1	1	yes	characteristic
146	C₃×C₄	—	2	4	no	—
147	C₃×C₄	—	30	4	no	—
148	C₃×C₄	C₂×C₃×Dic₅	1	4	yes	complemented
149	C₃×C₄	C₂×Dic₁₅	1	4	yes	complemented
150	Dic₃	—	10	4	no	—
151	C₂×C₅	C₃×C₄⋊Dic₃	1	4	yes	complemented
152	C₂×C₅	C₂×C₃×Dic₆	1	1	yes	characteristic
153	C₂×C₅	C₂×C₃×D₁₂	1	1	yes	characteristic
154	C₂×C₅	C₂²×C₃×Dic₃	1	1	yes	characteristic
155	C₃²	C₂×C₄⋊Dic₅	1	1	yes	characteristic, complemented, Sylow
156	C₂³	C₂×C₃×D₁₅	1	1	yes	characteristic
157	C₂×C₄	—	15	4	no	—
158	C₂×C₄	—	30	4	no	—
159	C₂×C₄	C₃×Dic₁₅	1	4	yes	complemented
160	C₂×C₄	C₂×C₃×D₁₅	1	2	yes	—
161	C₂×C₃	—	2	1	no	—
162	C₂×C₃	—	2	1	no	—
163	C₂×C₃	—	2	1	no	—
164	C₂×C₃	—	2	4	no	—
165	C₂×C₃	C₃×C₄⋊Dic₅	1	4	yes	complemented
166	C₂×C₃	C₄⋊Dic₁₅	1	4	yes	direct product factor
167	C₂×C₃	C₂×C₃×Dic₁₀	1	1	yes	characteristic
168	C₂×C₃	C₂×C₃×D₂₀	1	1	yes	characteristic
169	C₂×C₃	C₂²×C₃×Dic₅	1	1	yes	characteristic
170	C₂×C₃	C₂×Dic₃₀	1	1	yes	characteristic
171	C₂×C₃	C₂×D₆₀	1	1	yes	characteristic
172	C₂×C₃	C₂²×Dic₁₅	1	1	yes	characteristic
173	C₅	C₂×C₃×C₄⋊Dic₃	1	1	yes	characteristic, complemented, Sylow
174	C₂²	C₃×Dic₃₀	1	2	yes	—
175	C₂²	C₃×D₆₀	1	2	yes	—
176	C₂²	C₂×C₃×Dic₁₅	1	2	yes	—
177	C₂²	C₂²×C₃×D₁₅	1	1	yes	characteristic
178	C₄	—	30	4	no	—
179	C₄	C₂×C₃×Dic₁₅	1	4	yes	complemented
180	C₃	—	2	1	no	—
181	C₃	C₂×C₃×C₄⋊Dic₅	1	1	yes	characteristic, complemented
182	C₃	C₂×C₄⋊Dic₁₅	1	1	yes	characteristic, direct product factor
183	C₂	C₃×C₄⋊Dic₁₅	1	4	yes	direct product factor
184	C₂	C₂×C₃×Dic₃₀	1	1	yes	characteristic
185	C₂	C₂×C₃×D₆₀	1	1	yes	characteristic
186	C₂	C₂²×C₃×Dic₁₅	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₃×C₄⋊Dic₁₅	C₂	direct (N × H)
4. C₂×C₄⋊Dic₁₅	C₃	direct (N × H)
7. C₂×C₃²×C₄×C₅	C₄	semidirect
17. C₄⋊Dic₁₅	C₂×C₃	direct (N × H)
23. C₃²×C₄×C₅	C₂×C₄	semidirect
34. C₂×C₃×C₄×C₅	Dic₃	semidirect
35. C₂×C₃×C₄×C₅	C₃×C₄	semidirect
44. C₂×C₃²×C₅	C₄⋊C₄	semidirect
52. C₂×C₃²×C₄	Dic₅	semidirect
69. C₃×C₄×C₅	C₂×Dic₃	semidirect
70. C₃×C₄×C₅	C₂×C₃×C₄	semidirect
80. C₃²×C₅	C₂×C₄⋊C₄	semidirect
82. C₂×C₄×C₅	C₃×Dic₃	semidirect
90. C₃²×C₄	C₂×Dic₅	semidirect
97. C₂×C₃×C₅	C₄⋊Dic₃	semidirect
98. C₂×C₃×C₅	C₃×C₄⋊C₄	semidirect
112. C₂×C₃×C₄	C₃×Dic₅	semidirect
113. C₂×C₃×C₄	Dic₁₅	semidirect
122. C₄×C₅	C₂×C₃×Dic₃	semidirect
124. C₂×C₃²	C₄⋊Dic₅	semidirect
132. C₃×C₅	C₂×C₄⋊Dic₃	semidirect
133. C₃×C₅	C₂×C₃×C₄⋊C₄	semidirect
148. C₃×C₄	C₂×C₃×Dic₅	semidirect
149. C₃×C₄	C₂×Dic₁₅	semidirect
151. C₂×C₅	C₃×C₄⋊Dic₃	semidirect
155. C₃²	C₂×C₄⋊Dic₅	semidirect
159. C₂×C₄	C₃×Dic₁₅	semidirect
165. C₂×C₃	C₃×C₄⋊Dic₅	semidirect
166. C₂×C₃	C₄⋊Dic₁₅	direct (N × H)
173. C₅	C₂×C₃×C₄⋊Dic₃	semidirect
179. C₄	C₂×C₃×Dic₁₅	semidirect
181. C₃	C₂×C₃×C₄⋊Dic₅	semidirect
182. C₃	C₂×C₄⋊Dic₁₅	direct (N × H)
183. C₂	C₃×C₄⋊Dic₁₅	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×Dic₁₅	2
C₃×C₄⋊Dic₁₅	2
C₂×C₄⋊Dic₁₅	3
C₂×C₃×C₄⋊Dic₅	3
C₂×C₃×C₄⋊Dic₃	5

Subgroups — C2×C3×C4⋊Dic15 #13088022

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₄⋊Dic₁₅ #13088022