Subgroups — C₄×D₅⋊(C₃⋊C₁₆) #14312519

Order 1920.

432 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₅⋊(C₃⋊C₁₆)	C₂	1	2	yes	maximal
2	C₃×C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic
3	#11746756	C₂	1	2	yes	maximal, complemented
4	C₄×C₃⋊#674	C₂	1	2	yes	maximal, complemented
5	C₄×D₅⋊C₁₆	—	3	1	no	maximal, Hall
6	C₂×C₃×C₈×D₅	C₄	1	1	yes	characteristic
7	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
8	C₃×C₄²×D₅	C₄	1	1	yes	characteristic
9	C₂×C₃⋊#674	C₂²	1	4	yes	—
10	D₅⋊(C₃⋊C₁₆)	C₄	1	8	yes	direct product factor
11	C₃×C₄×C₅×C₈	C₄	1	1	yes	characteristic
12	C₃×C₈×Dic₅	C₄	1	1	yes	characteristic
13	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
14	C₃×C₄×#168	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₃⋊C₁₆	—	5	1	no	maximal, Hall
16	C₂×D₅⋊C₁₆	—	3	2	no	—
17	C₄×C₈×D₅	D₃	1	1	yes	characteristic
18	#64885	—	3	2	no	—
19	C₄×#674	—	3	2	no	—
20	C₂×C₃×C₄×D₅	C₈	1	2	yes	—
21	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
24	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
25	C₃×C₄×Dic₅	C₈	1	2	yes	—
26	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×#168	C₂³	1	1	yes	characteristic
29	C₃×C₈×D₅	C₈	1	4	yes	—
30	C₃×C₈×D₅	C₂×C₄	1	4	yes	—
31	C₃⋊#674	C₂×C₄	1	8	yes	complemented
32	C₂×C₃×C₄×C₈	—	5	1	no	—
33	C₂²×C₃⋊C₁₆	—	5	2	no	—
34	C₄×C₃⋊C₁₆	—	5	2	no	—
35	C₄×C₃⋊C₁₆	—	5	2	no	—
36	C₂×C₈×D₅	Dic₃	1	1	yes	characteristic
37	C₂×C₈×D₅	C₂×D₃	1	1	yes	characteristic
38	C₄²×D₅	Dic₃	1	1	yes	characteristic
39	C₂×#674	—	3	4	no	—
40	D₅⋊C₁₆	—	3	8	no	—
41	C₄×C₅×C₈	Dic₃	1	1	yes	characteristic
42	C₈×Dic₅	Dic₃	1	1	yes	characteristic
43	C₈×Dic₅	C₂×D₃	1	1	yes	characteristic
44	C₄×#168	C₂×D₃	1	1	yes	characteristic
45	C₂×C₄×C₁₆	—	15	1	no	Sylow
46	C₂²×C₃×D₅	C₂×C₈	1	1	yes	characteristic
47	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
48	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₃×Dic₅	C₂×C₈	1	1	yes	characteristic
50	C₂×C₃×Dic₅	C₂×C₈	1	2	yes	—
51	C₃×C₄×D₅	C₁₆	1	8	yes	complemented
52	C₃×C₄×D₅	C₄²	1	2	yes	—
53	C₃×C₄×D₅	C₂×C₈	1	2	yes	—
54	C₃×C₅×C₈	C₄²	1	2	yes	—
55	C₃×C₅×C₈	C₂×C₈	1	2	yes	—
56	C₃×#168	C₂×C₈	1	2	yes	—
57	C₃×#168	C₂²×C₄	1	2	yes	—
58	C₂²×C₃×C₈	—	5	1	no	—
59	C₂²×C₃×C₈	—	5	1	no	—
60	C₂×C₃×C₄²	—	5	1	no	—
61	C₃×C₄×C₈	—	5	1	no	—
62	C₃×C₄×C₈	—	5	1	no	—
63	C₃×C₄×C₈	—	5	1	no	—
64	C₃×C₄×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
65	C₂×C₃⋊C₁₆	—	5	8	no	—
66	C₂×C₃⋊C₁₆	—	5	4	no	—
67	C₂×C₄×D₅	C₃⋊C₈	1	2	yes	—
68	C₂×C₄×D₅	C₂×Dic₃	1	1	yes	characteristic
69	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
70	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
71	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
72	C₄×Dic₅	C₃⋊C₈	1	2	yes	—
73	C₄×Dic₅	C₂×Dic₃	1	1	yes	characteristic
74	C₂×#168	C₂×Dic₃	1	1	yes	characteristic
75	C₂×#168	C₂²×D₃	1	1	yes	characteristic
76	C₈×D₅	C₃⋊C₈	1	4	yes	—
77	C₈×D₅	C₄×D₃	1	4	yes	—
78	#674	—	3	8	no	—
79	C₂²×C₁₆	—	15	2	no	—
80	C₂×C₄×C₈	—	5	1	no	—
81	C₄×C₁₆	—	15	2	no	—
82	C₄×C₁₆	—	15	2	no	—
83	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₃×D₅	C₄×C₈	1	2	yes	—
85	C₂×C₃×D₅	C₂×C₁₆	1	4	yes	—
86	C₃×C₄×C₅	C₂×C₁₆	1	4	yes	complemented
87	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
88	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
89	C₃×Dic₅	C₄×C₈	1	2	yes	—
90	C₃×Dic₅	C₂×C₁₆	1	4	yes	complemented
91	C₂²×C₃×C₄	—	5	2	no	—
92	C₂²×C₃×C₄	—	5	1	no	—
93	C₂×C₃×C₈	—	5	1	no	—
94	C₂×C₃×C₈	—	5	1	no	—
95	C₂×C₃×C₈	—	5	4	no	—
96	C₂×C₃×C₈	—	5	4	no	—
97	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
98	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
99	C₃×C₄²	—	5	2	no	—
100	C₃×C₄²	—	5	1	no	—
101	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
102	C₃⋊C₁₆	—	5	8	no	—
103	C₂²×D₅	C₂×C₃⋊C₈	1	1	yes	characteristic
104	C₂×C₄×C₅	C₂×C₃⋊C₈	1	2	yes	—
105	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
106	C₂×Dic₅	C₂×C₃⋊C₈	1	1	yes	characteristic
107	C₂×Dic₅	C₂×C₃⋊C₈	1	2	yes	—
108	C₄×D₅	C₃⋊C₁₆	1	8	yes	complemented
109	C₄×D₅	C₂×C₃⋊C₈	1	2	yes	—
110	C₄×D₅	C₄×Dic₃	1	2	yes	—
111	C₅×C₈	C₂×C₃⋊C₈	1	2	yes	—
112	C₅×C₈	C₄×Dic₃	1	2	yes	—
113	#168	C₂×C₃⋊C₈	1	2	yes	—
114	#168	C₂×C₄×D₃	1	2	yes	—
115	C₂²×C₈	—	5	1	no	—
116	C₂²×C₈	—	5	1	no	—
117	C₂×C₄²	—	5	1	no	—
118	C₂×C₁₆	—	15	4	no	—
119	C₂×C₁₆	—	15	8	no	—
120	C₄×C₈	—	5	1	no	—
121	C₄×C₈	—	5	1	no	—
122	C₄×C₈	—	5	1	no	—
123	C₄×C₈	ΓL(1,𝔽₁₆)	1	1	yes	characteristic
124	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
125	C₂×C₃×C₅	C₂²×C₁₆	1	2	yes	—
126	C₃×D₅	C₄×C₁₆	1	4	yes	complemented
127	C₂³×C₃	—	5	1	no	—
128	C₂×C₃×C₄	—	5	1	no	—
129	C₂×C₃×C₄	—	5	2	no	—
130	C₂×C₃×C₄	—	5	2	no	—
131	C₂×C₃×C₄	—	5	2	no	—
132	C₂×C₃×C₄	—	5	8	no	—
133	C₂×C₃×C₄	D₅⋊C₈	1	2	yes	—
134	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
135	C₃×C₈	—	5	2	no	—
136	C₃×C₈	—	5	2	no	—
137	C₃×C₈	D₅⋊C₈	1	2	yes	—
138	C₃×C₈	C₄×AGL(1,𝔽₅)	1	2	yes	—
139	C₂²×C₅	C₂²×C₃⋊C₈	1	1	yes	characteristic
140	C₂×D₅	C₄×C₃⋊C₈	1	2	yes	—
141	C₂×D₅	C₂×C₃⋊C₁₆	1	4	yes	—
142	C₄×C₅	C₂×C₃⋊C₁₆	1	4	yes	complemented
143	C₄×C₅	C₂²×C₃⋊C₈	1	1	yes	characteristic
144	C₄×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
145	Dic₅	C₄×C₃⋊C₈	1	2	yes	—
146	Dic₅	C₂×C₃⋊C₁₆	1	4	yes	complemented
147	C₂²×C₄	—	5	2	no	—
148	C₂²×C₄	—	5	1	no	—
149	C₂×C₈	—	5	1	no	—
150	C₂×C₈	—	5	1	no	—
151	C₂×C₈	—	5	4	no	—
152	C₂×C₈	—	5	4	no	—
153	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
154	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
155	C₄²	—	5	2	no	—
156	C₄²	—	5	1	no	—
157	C₄²	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
158	C₁₆	—	15	8	no	—
159	C₃×C₅	C₂×C₄×C₁₆	1	1	yes	characteristic, complemented, Hall
160	C₂²×C₃	—	5	2	no	—
161	C₂²×C₃	—	5	4	no	—
162	C₂²×C₃	C₂×D₅⋊C₈	1	1	yes	characteristic
163	C₃×C₄	—	5	2	no	—
164	C₃×C₄	—	5	4	no	—
165	C₃×C₄	D₅⋊C₁₆	1	4	yes	complemented
166	C₃×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
167	C₃×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
168	C₂×C₅	C₂×C₄×C₃⋊C₈	1	1	yes	characteristic
169	C₂×C₅	C₂²×C₃⋊C₁₆	1	2	yes	—
170	D₅	C₄×C₃⋊C₁₆	1	4	yes	complemented
171	C₂³	—	5	1	no	—
172	C₂×C₄	—	5	1	no	—
173	C₂×C₄	—	5	2	no	—
174	C₂×C₄	—	5	2	no	—
175	C₂×C₄	—	5	8	no	—
176	C₂×C₄	—	5	2	no	—
177	C₂×C₄	D₅⋊(C₃⋊C₈)	1	2	yes	—
178	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
179	C₈	—	5	2	no	—
180	C₈	—	5	2	no	—
181	C₈	D₅⋊(C₃⋊C₈)	1	2	yes	—
182	C₈	C₄×ΓL(1,𝔽₁₆)	1	2	yes	—
183	C₂×C₃	—	5	4	no	—
184	C₂×C₃	C₄×D₅⋊C₈	1	1	yes	characteristic
185	C₂×C₃	C₂×D₅⋊C₁₆	1	2	yes	—
186	C₅	C₂×C₄×C₃⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
187	C₂²	—	5	2	no	—
188	C₂²	—	5	4	no	—
189	C₂²	C₂×D₅⋊(C₃⋊C₈)	1	1	yes	characteristic
190	C₄	—	5	2	no	—
191	C₄	—	5	4	no	—
192	C₄	D₅⋊(C₃⋊C₁₆)	1	4	yes	direct product factor
193	C₄	C₂×D₅⋊(C₃⋊C₈)	1	1	yes	characteristic
194	C₄	C₂×C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
195	C₃	C₄×D₅⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
196	C₂	—	5	4	no	—
197	C₂	C₄×D₅⋊(C₃⋊C₈)	1	1	yes	characteristic
198	C₂	C₂×D₅⋊(C₃⋊C₁₆)	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₄×C₈	ΓL(1,𝔽₁₆)
commutator, hypocenter	C₃×C₅	C₂×C₄×C₁₆
Frattini	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)
Fitting	C₃×C₄×C₅×C₈	C₄
socle	C₂²×C₃×C₅	C₂²×C₈
prime radical	C₂²×C₃×D₅	C₂×C₈
cosocle	C₂×C₃×#168	C₂³
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #11746756	C₂	semidirect
4. C₄×C₃⋊#674	C₂	semidirect
10. D₅⋊(C₃⋊C₁₆)	C₄	direct (N × H)
31. C₃⋊#674	C₂×C₄	semidirect
51. C₃×C₄×D₅	C₁₆	semidirect
86. C₃×C₄×C₅	C₂×C₁₆	semidirect
90. C₃×Dic₅	C₂×C₁₆	semidirect
108. C₄×D₅	C₃⋊C₁₆	semidirect
126. C₃×D₅	C₄×C₁₆	semidirect
142. C₄×C₅	C₂×C₃⋊C₁₆	semidirect
146. Dic₅	C₂×C₃⋊C₁₆	semidirect
159. C₃×C₅	C₂×C₄×C₁₆	semidirect
165. C₃×C₄	D₅⋊C₁₆	semidirect
170. D₅	C₄×C₃⋊C₁₆	semidirect
186. C₅	C₂×C₄×C₃⋊C₁₆	semidirect
192. C₄	D₅⋊(C₃⋊C₁₆)	direct (N × H)
195. C₃	C₄×D₅⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₅⋊(C₃⋊C₁₆)	2
C₃×C₄×C₈×D₅	2
#11746756	2
C₄×C₃⋊#674	2
C₄×D₅⋊C₁₆	3
C₂×C₄×C₃⋊C₁₆	5

Subgroups — C4×D5⋊(C3⋊C16) #14312519

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×D₅⋊(C₃⋊C₁₆) #14312519