Subgroups — C₄×C₈×ΓL(1,𝔽₁₆) #14294189

Order 1920.

816 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×C₈×ΓL(1,𝔽₁₆)	C₂	1	4	yes	maximal
2	C₄²×ΓL(1,𝔽₁₆)	C₂	1	1	yes	maximal, characteristic
3	C₄×D₅⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	C₃×C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic
5	C₄×C₈×AGL(1,𝔽₅)	—	3	1	no	maximal, Hall
6	C₂×C₄×ΓL(1,𝔽₁₆)	C₄	1	4	yes	—
7	C₂×C₄×ΓL(1,𝔽₁₆)	C₂²	1	2	yes	—
8	C₂×D₅⋊(C₃⋊C₈)	C₂²	1	2	yes	—
9	C₂×C₃×C₈×D₅	C₂²	1	2	yes	—
10	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
11	C₄×C₃⋊#170	C₄	1	4	yes	complemented
12	C₈×ΓL(1,𝔽₁₆)	C₄	1	16	yes	direct product factor
13	C₃×C₄×C₅×C₈	C₄	1	1	yes	characteristic, complemented
14	C₃×C₈×Dic₅	C₄	1	2	yes	complemented
15	C₃×C₄×#168	C₄	1	1	yes	characteristic, complemented
16	C₄×C₈×Dic₃	—	5	1	no	maximal, Hall
17	C₂×C₈×AGL(1,𝔽₅)	—	3	4	no	—
18	C₄²×AGL(1,𝔽₅)	—	3	1	no	—
19	C₄×D₅⋊C₈	—	3	1	no	—
20	C₄×C₈×D₅	D₃	1	1	yes	characteristic
21	C₂²×ΓL(1,𝔽₁₆)	C₂×C₄	1	4	yes	—
22	C₂×C₃×C₄×D₅	C₂×C₄	1	2	yes	—
23	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
24	C₂×C₃⋊#170	C₂×C₄	1	4	yes	—
25	C₄×ΓL(1,𝔽₁₆)	C₈	1	16	yes	direct product factor
26	C₄×ΓL(1,𝔽₁₆)	C₂×C₄	1	8	yes	—
27	D₅⋊(C₃⋊C₈)	C₂×C₄	1	8	yes	—
28	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
29	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
30	C₃×C₄×Dic₅	C₂×C₄	1	2	yes	—
31	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×#168	C₂×C₄	1	2	yes	—
33	C₃×C₈×D₅	C₂×C₄	1	8	yes	—
34	C₂×C₃×C₄×C₈	—	5	1	no	—
35	C₂×C₈×Dic₃	—	5	4	no	—
36	C₄²×Dic₃	—	5	1	no	—
37	C₂×C₄×C₃⋊C₈	—	5	1	no	—
38	C₂×C₄×AGL(1,𝔽₅)	—	3	4	no	—
39	C₂×C₄×AGL(1,𝔽₅)	—	3	2	no	—
40	C₂×D₅⋊C₈	—	3	2	no	—
41	C₂×C₈×D₅	C₂×D₃	1	2	yes	—
42	C₄²×D₅	C₂×D₃	1	1	yes	characteristic
43	C₄×#170	—	3	4	no	—
44	C₈×AGL(1,𝔽₅)	—	3	16	no	—
45	C₄×C₅×C₈	Dic₃	1	1	yes	characteristic, complemented
46	C₈×Dic₅	Dic₃	1	2	yes	complemented
47	C₄×#168	Dic₃	1	1	yes	characteristic, complemented
48	C₄²×C₈	—	15	1	no	Sylow
49	C₂²×C₃×D₅	C₂²×C₄	1	1	yes	characteristic
50	C₂×ΓL(1,𝔽₁₆)	C₄²	1	8	yes	—
51	C₂×ΓL(1,𝔽₁₆)	C₂×C₈	1	16	yes	—
52	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
53	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₃×Dic₅	C₄²	1	2	yes	—
55	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
56	C₃×C₄×D₅	C₂×C₈	1	8	yes	—
57	C₃×C₄×D₅	C₂²×C₄	1	4	yes	—
58	C₃⋊#170	C₄²	1	8	yes	complemented
59	C₃×C₅×C₈	C₄²	1	4	yes	complemented
60	C₃×#168	C₄²	1	4	yes	complemented
61	C₂²×C₃×C₈	—	5	2	no	—
62	C₂×C₃×C₄²	—	5	1	no	—
63	C₂×C₄×Dic₃	—	5	4	no	—
64	C₂×C₄×Dic₃	—	5	2	no	—
65	C₂²×C₃⋊C₈	—	5	2	no	—
66	C₃×C₄×C₈	—	5	2	no	—
67	C₃×C₄×C₈	—	5	1	no	—
68	C₃×C₄×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
69	C₈×Dic₃	—	5	16	no	—
70	C₄×C₃⋊C₈	—	5	4	no	—
71	C₂²×AGL(1,𝔽₅)	—	3	4	no	—
72	C₂×C₄×D₅	C₄×D₃	1	2	yes	—
73	C₂×C₄×D₅	C₂²×D₃	1	1	yes	characteristic
74	C₂×#170	—	3	4	no	—
75	C₄×AGL(1,𝔽₅)	—	3	16	no	—
76	C₄×AGL(1,𝔽₅)	—	3	8	no	—
77	D₅⋊C₈	—	3	8	no	—
78	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
79	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
80	C₄×Dic₅	C₄×D₃	1	2	yes	—
81	C₄×Dic₅	C₂×Dic₃	1	1	yes	characteristic
82	C₂×#168	C₂×Dic₃	1	2	yes	—
83	C₈×D₅	C₄×D₃	1	8	yes	—
84	C₂×C₄×C₈	—	5	1	no	—
85	C₂×C₄×C₈	—	15	1	no	—
86	C₂×C₄×C₈	—	15	4	no	—
87	C₄³	—	15	1	no	—
88	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
89	C₂×C₃×D₅	C₂×C₄²	1	2	yes	—
90	C₂×C₃×D₅	C₂²×C₈	1	4	yes	—
91	ΓL(1,𝔽₁₆)	C₄×C₈	1	16	yes	direct product factor
92	C₃×C₄×C₅	C₄×C₈	1	4	yes	complemented
93	C₃×C₄×C₅	C₂×C₄²	1	2	yes	—
94	C₃×Dic₅	C₄×C₈	1	4	yes	complemented
95	C₃×Dic₅	C₂×C₄²	1	2	yes	—
96	C₂²×C₃×C₄	—	5	2	no	—
97	C₂²×C₃×C₄	—	5	1	no	—
98	C₂²×Dic₃	—	5	4	no	—
99	C₂×C₃×C₈	—	5	2	no	—
100	C₂×C₃×C₈	—	5	8	no	—
101	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
102	C₃×C₄²	—	5	2	no	—
103	C₃×C₄²	—	5	1	no	—
104	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
105	C₄×Dic₃	—	5	8	no	—
106	C₄×Dic₃	—	5	16	no	—
107	C₂×C₃⋊C₈	—	5	4	no	—
108	C₂×C₃⋊C₈	—	5	8	no	—
109	C₂²×D₅	C₂×C₄×D₃	1	1	yes	characteristic
110	C₂×AGL(1,𝔽₅)	—	3	8	no	—
111	C₂×AGL(1,𝔽₅)	—	3	16	no	—
112	C₂×C₄×C₅	C₄×Dic₃	1	2	yes	—
113	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
114	C₂×Dic₅	C₄×Dic₃	1	2	yes	—
115	C₂×Dic₅	C₂×C₄×D₃	1	1	yes	characteristic
116	C₄×D₅	D₃×C₈	1	8	yes	—
117	C₄×D₅	C₂×C₄×D₃	1	4	yes	—
118	#170	—	3	8	no	—
119	C₅×C₈	C₄×Dic₃	1	4	yes	complemented
120	#168	C₄×Dic₃	1	4	yes	complemented
121	C₂²×C₈	—	5	2	no	—
122	C₂²×C₈	—	15	2	no	—
123	C₂×C₄²	—	5	1	no	—
124	C₂×C₄²	—	15	2	no	—
125	C₂×C₄²	—	15	4	no	—
126	C₄×C₈	—	5	2	no	—
127	C₄×C₈	—	5	1	no	—
128	C₄×C₈	—	15	4	no	—
129	C₄×C₈	—	15	16	no	—
130	C₄×C₈	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, direct product factor
131	C₂×C₃×C₅	C₄³	1	1	yes	characteristic
132	C₂×C₃×C₅	C₂×C₄×C₈	1	2	yes	—
133	C₃×D₅	C₂×C₄×C₈	1	4	yes	—
134	C₂³×C₃	—	5	1	no	—
135	C₂×C₃×C₄	—	5	2	no	—
136	C₂×C₃×C₄	—	5	1	no	—
137	C₂×C₃×C₄	—	5	4	no	—
138	C₂×C₃×C₄	—	5	8	no	—
139	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	2	yes	—
140	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
141	C₂×Dic₃	—	5	8	no	—
142	C₂×Dic₃	—	5	16	no	—
143	C₃×C₈	—	5	4	no	—
144	C₃×C₈	C₄×AGL(1,𝔽₅)	1	4	yes	complemented
145	C₃⋊C₈	—	5	8	no	—
146	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
147	C₂×D₅	C₄²×D₃	1	2	yes	—
148	C₂×D₅	C₂×D₃×C₈	1	4	yes	—
149	AGL(1,𝔽₅)	—	3	16	no	—
150	C₄×C₅	C₈×Dic₃	1	4	yes	complemented
151	C₄×C₅	C₂×C₄×Dic₃	1	2	yes	—
152	Dic₅	C₈×Dic₃	1	4	yes	complemented
153	Dic₅	C₄²×D₃	1	2	yes	—
154	C₂²×C₄	—	5	2	no	—
155	C₂²×C₄	—	5	1	no	—
156	C₂²×C₄	—	15	4	no	—
157	C₂×C₈	—	5	2	no	—
158	C₂×C₈	—	5	8	no	—
159	C₂×C₈	—	15	8	no	—
160	C₂×C₈	—	15	4	no	—
161	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	2	yes	—
162	C₄²	—	5	2	no	—
163	C₄²	—	5	1	no	—
164	C₄²	—	15	8	no	—
165	C₄²	—	15	16	no	—
166	C₄²	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
167	C₃×C₅	C₄²×C₈	1	1	yes	characteristic, complemented, Hall
168	C₂²×C₃	—	5	2	no	—
169	C₂²×C₃	—	5	4	no	—
170	C₂²×C₃	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
171	C₃×C₄	—	5	2	no	—
172	C₃×C₄	—	5	4	no	—
173	C₃×C₄	C₈×AGL(1,𝔽₅)	1	4	yes	complemented
174	C₃×C₄	C₂×C₄×AGL(1,𝔽₅)	1	2	yes	—
175	Dic₃	—	5	16	no	—
176	C₂×C₅	C₄²×Dic₃	1	1	yes	characteristic
177	C₂×C₅	C₂×C₈×Dic₃	1	2	yes	—
178	D₅	C₄×D₃×C₈	1	4	yes	—
179	C₂³	—	5	1	no	—
180	C₂×C₄	—	5	1	no	—
181	C₂×C₄	—	5	2	no	—
182	C₂×C₄	—	5	4	no	—
183	C₂×C₄	—	5	8	no	—
184	C₂×C₄	—	15	8	no	—
185	C₂×C₄	—	15	16	no	—
186	C₂×C₄	C₄×ΓL(1,𝔽₁₆)	1	2	yes	—
187	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
188	C₈	—	5	4	no	—
189	C₈	—	15	8	no	—
190	C₈	C₄×ΓL(1,𝔽₁₆)	1	4	yes	direct product factor
191	C₂×C₃	—	5	4	no	—
192	C₂×C₃	C₄²×AGL(1,𝔽₅)	1	1	yes	characteristic
193	C₂×C₃	C₂×C₈×AGL(1,𝔽₅)	1	2	yes	—
194	C₅	C₄×C₈×Dic₃	1	1	yes	characteristic, complemented, Sylow
195	C₂²	—	5	2	no	—
196	C₂²	—	5	4	no	—
197	C₂²	C₂×C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
198	C₄	—	5	2	no	—
199	C₄	—	5	4	no	—
200	C₄	—	15	16	no	—
201	C₄	C₈×ΓL(1,𝔽₁₆)	1	4	yes	direct product factor
202	C₄	C₂×C₄×ΓL(1,𝔽₁₆)	1	2	yes	—
203	C₃	C₄×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
204	C₂	—	5	4	no	—
205	C₂	C₄²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
206	C₂	C₂×C₈×ΓL(1,𝔽₁₆)	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₄×C₈	ΓL(1,𝔽₁₆)
commutator, hypocenter	C₃×C₅	C₄²×C₈
Frattini	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)
Fitting	C₃×C₄×C₅×C₈	C₄
socle	C₂²×C₃×C₅	C₂×C₄²
prime radical	C₂²×C₃×D₅	C₂²×C₄
cosocle	C₂×C₃×C₄×D₅	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
11. C₄×C₃⋊#170	C₄	semidirect
12. C₈×ΓL(1,𝔽₁₆)	C₄	direct (N × H)
13. C₃×C₄×C₅×C₈	C₄	semidirect
14. C₃×C₈×Dic₅	C₄	semidirect
15. C₃×C₄×#168	C₄	semidirect
25. C₄×ΓL(1,𝔽₁₆)	C₈	direct (N × H)
45. C₄×C₅×C₈	Dic₃	semidirect
46. C₈×Dic₅	Dic₃	semidirect
47. C₄×#168	Dic₃	semidirect
58. C₃⋊#170	C₄²	semidirect
59. C₃×C₅×C₈	C₄²	semidirect
60. C₃×#168	C₄²	semidirect
68. C₃×C₄×C₈	AGL(1,𝔽₅)	semidirect
91. ΓL(1,𝔽₁₆)	C₄×C₈	direct (N × H)
92. C₃×C₄×C₅	C₄×C₈	semidirect
94. C₃×Dic₅	C₄×C₈	semidirect
119. C₅×C₈	C₄×Dic₃	semidirect
120. #168	C₄×Dic₃	semidirect
130. C₄×C₈	ΓL(1,𝔽₁₆)	direct (N × H)
144. C₃×C₈	C₄×AGL(1,𝔽₅)	semidirect
150. C₄×C₅	C₈×Dic₃	semidirect
152. Dic₅	C₈×Dic₃	semidirect
167. C₃×C₅	C₄²×C₈	semidirect
173. C₃×C₄	C₈×AGL(1,𝔽₅)	semidirect
190. C₈	C₄×ΓL(1,𝔽₁₆)	direct (N × H)
194. C₅	C₄×C₈×Dic₃	semidirect
201. C₄	C₈×ΓL(1,𝔽₁₆)	direct (N × H)
203. C₃	C₄×C₈×AGL(1,𝔽₅)	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×ΓL(1,𝔽₁₆)	2
C₄²×ΓL(1,𝔽₁₆)	2
C₄×D₅⋊(C₃⋊C₈)	2
C₃×C₄×C₈×D₅	2
C₄×C₈×AGL(1,𝔽₅)	3
C₄×C₈×Dic₃	5

Subgroups — C4×C8×ΓL(1,𝔽16) #14294189

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₈×ΓL(1,𝔽₁₆) #14294189