Subgroups — C₂²×C₈×ΓL(1,𝔽₁₆) #14411167

Order 1920.

1520 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×ΓL(1,𝔽₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₂²×D₅⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×C₈×D₅	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×ΓL(1,𝔽₁₆)	C₂	1	12	yes	maximal, direct product factor
5	C₂²×C₈×AGL(1,𝔽₅)	—	3	1	no	maximal, Hall
6	C₂³×ΓL(1,𝔽₁₆)	C₄	1	2	yes	—
7	C₂²×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
8	C₂²×C₃⋊#170	C₄	1	2	yes	complemented
9	C₂×C₄×ΓL(1,𝔽₁₆)	C₄	1	6	yes	—
10	C₂×C₄×ΓL(1,𝔽₁₆)	C₂²	1	6	yes	—
11	C₂×D₅⋊(C₃⋊C₈)	C₄	1	6	yes	complemented
12	C₂×D₅⋊(C₃⋊C₈)	C₂²	1	6	yes	—
13	C₂²×C₃×C₅×C₈	C₄	1	1	yes	characteristic, complemented
14	C₂²×C₃×#168	C₄	1	1	yes	characteristic, complemented
15	C₂×C₃×C₈×D₅	C₄	1	6	yes	complemented
16	C₂×C₃×C₈×D₅	C₂²	1	6	yes	—
17	C₈×ΓL(1,𝔽₁₆)	C₂²	1	16	yes	direct product factor
18	C₂²×C₈×Dic₃	—	5	1	no	maximal, Hall
19	C₂²×C₄×AGL(1,𝔽₅)	—	3	1	no	—
20	C₂²×D₅⋊C₈	—	3	1	no	—
21	C₂²×C₈×D₅	D₃	1	1	yes	characteristic
22	C₂×C₈×AGL(1,𝔽₅)	—	3	12	no	—
23	C₂³×C₃×D₅	C₂×C₄	1	1	yes	characteristic
24	C₂²×ΓL(1,𝔽₁₆)	C₈	1	16	yes	direct product factor
25	C₂²×ΓL(1,𝔽₁₆)	C₂×C₄	1	12	yes	—
26	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×C₄×D₅	C₂×C₄	1	3	yes	—
29	C₂×C₃×C₄×D₅	C₂×C₄	1	3	yes	—
30	C₂×C₃×C₄×D₅	C₂×C₄	1	3	yes	—
31	C₂×C₃×C₄×D₅	C₂³	1	3	yes	—
32	C₂×C₃⋊#170	C₂×C₄	1	12	yes	complemented
33	C₄×ΓL(1,𝔽₁₆)	C₂×C₄	1	12	yes	—
34	C₄×ΓL(1,𝔽₁₆)	C₂³	1	4	yes	—
35	D₅⋊(C₃⋊C₈)	C₂×C₄	1	12	yes	complemented
36	D₅⋊(C₃⋊C₈)	C₂³	1	4	yes	—
37	C₂×C₃×C₅×C₈	C₂×C₄	1	6	yes	complemented
38	C₂×C₃×#168	C₂×C₄	1	6	yes	complemented
39	C₃×C₈×D₅	C₂×C₄	1	12	yes	complemented
40	C₃×C₈×D₅	C₂³	1	4	yes	—
41	C₂³×C₃×C₈	—	5	1	no	—
42	C₂²×C₄×Dic₃	—	5	1	no	—
43	C₂³×C₃⋊C₈	—	5	1	no	—
44	C₂×C₈×Dic₃	—	5	12	no	—
45	C₂³×AGL(1,𝔽₅)	—	3	2	no	—
46	C₂²×C₄×D₅	C₂×D₃	1	1	yes	characteristic
47	C₂²×#170	—	3	2	no	—
48	C₂×C₄×AGL(1,𝔽₅)	—	3	6	no	—
49	C₂×C₄×AGL(1,𝔽₅)	—	3	6	no	—
50	C₂×D₅⋊C₈	—	3	6	no	—
51	C₂×D₅⋊C₈	—	3	6	no	—
52	C₂²×C₅×C₈	Dic₃	1	1	yes	characteristic, complemented
53	C₂²×#168	Dic₃	1	1	yes	characteristic, complemented
54	C₂×C₈×D₅	Dic₃	1	6	yes	complemented
55	C₂×C₈×D₅	C₂×D₃	1	6	yes	—
56	C₈×AGL(1,𝔽₅)	—	3	16	no	—
57	C₂²×C₄×C₈	—	15	1	no	Sylow
58	C₂³×C₃×C₅	C₄²	1	1	yes	characteristic
59	C₂²×C₃×D₅	C₄²	1	3	yes	—
60	C₂²×C₃×D₅	C₂×C₈	1	8	yes	—
61	C₂²×C₃×D₅	C₂²×C₄	1	3	yes	—
62	C₂×ΓL(1,𝔽₁₆)	C₂×C₈	1	48	yes	direct product factor
63	C₂×ΓL(1,𝔽₁₆)	C₂²×C₄	1	8	yes	—
64	C₂×C₃×C₄×C₅	C₄²	1	3	yes	—
65	C₂×C₃×C₄×C₅	C₂²×C₄	1	3	yes	—
66	C₂×C₃×Dic₅	C₄²	1	3	yes	—
67	C₂×C₃×Dic₅	C₂²×C₄	1	3	yes	—
68	C₃×C₄×D₅	C₄²	1	6	yes	—
69	C₃×C₄×D₅	C₂²×C₄	1	3	yes	—
70	C₃×C₄×D₅	C₂²×C₄	1	3	yes	—
71	C₃×C₄×D₅	C₂²×C₄	1	3	yes	—
72	C₃×C₄×D₅	C₂⁴	1	1	yes	characteristic
73	C₃⋊#170	C₂²×C₄	1	8	yes	complemented
74	C₃×C₅×C₈	C₂²×C₄	1	4	yes	complemented
75	C₃×#168	C₂²×C₄	1	4	yes	complemented
76	C₂³×C₃×C₄	—	5	1	no	—
77	C₂³×Dic₃	—	5	2	no	—
78	C₂²×C₃×C₈	—	5	1	no	—
79	C₂²×C₃×C₈	—	5	6	no	—
80	C₂²×C₃×C₈	—	5	6	no	—
81	C₂²×C₃×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
82	C₂×C₄×Dic₃	—	5	6	no	—
83	C₂×C₄×Dic₃	—	5	6	no	—
84	C₂²×C₃⋊C₈	—	5	6	no	—
85	C₂²×C₃⋊C₈	—	5	6	no	—
86	C₂²×C₃⋊C₈	—	5	2	no	—
87	C₈×Dic₃	—	5	16	no	—
88	C₂³×D₅	C₄×D₃	1	1	yes	characteristic
89	C₂²×AGL(1,𝔽₅)	—	3	16	no	—
90	C₂²×AGL(1,𝔽₅)	—	3	12	no	—
91	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
92	C₂²×Dic₅	C₄×D₃	1	1	yes	characteristic
93	C₂×C₄×D₅	C₄×D₃	1	3	yes	—
94	C₂×C₄×D₅	C₄×D₃	1	3	yes	—
95	C₂×C₄×D₅	C₂×Dic₃	1	3	yes	—
96	C₂×C₄×D₅	C₂²×D₃	1	3	yes	—
97	C₂×#170	—	3	12	no	—
98	C₄×AGL(1,𝔽₅)	—	3	12	no	—
99	C₄×AGL(1,𝔽₅)	—	3	4	no	—
100	D₅⋊C₈	—	3	12	no	—
101	D₅⋊C₈	—	3	4	no	—
102	C₂×C₅×C₈	C₂×Dic₃	1	6	yes	complemented
103	C₂×#168	C₂×Dic₃	1	6	yes	complemented
104	C₈×D₅	C₂×Dic₃	1	12	yes	complemented
105	C₈×D₅	C₂²×D₃	1	4	yes	—
106	C₂³×C₈	—	5	1	no	—
107	C₂³×C₈	—	15	1	no	—
108	C₂²×C₄²	—	15	1	no	—
109	C₂×C₄×C₈	—	15	12	no	—
110	C₂²×C₃×C₅	C₄×C₈	1	4	yes	complemented
111	C₂²×C₃×C₅	C₂×C₄²	1	3	yes	—
112	C₂×C₃×D₅	C₄×C₈	1	12	yes	complemented
113	C₂×C₃×D₅	C₂×C₄²	1	3	yes	—
114	C₂×C₃×D₅	C₂²×C₈	1	12	yes	—
115	C₂×C₃×D₅	C₂³×C₄	1	1	yes	characteristic
116	ΓL(1,𝔽₁₆)	C₂²×C₈	1	16	yes	direct product factor
117	C₃×C₄×C₅	C₂×C₄²	1	3	yes	—
118	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
119	C₃×Dic₅	C₂×C₄²	1	3	yes	—
120	C₃×Dic₅	C₂³×C₄	1	1	yes	characteristic
121	C₂⁴×C₃	—	5	1	no	—
122	C₂²×C₃×C₄	—	5	1	no	—
123	C₂²×C₃×C₄	—	5	3	no	—
124	C₂²×C₃×C₄	—	5	3	no	—
125	C₂²×C₃×C₄	—	5	3	no	—
126	C₂²×C₃×C₄	—	5	3	no	—
127	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
128	C₂²×Dic₃	—	5	12	no	—
129	C₂²×Dic₃	—	5	16	no	—
130	C₂×C₃×C₈	—	5	6	no	—
131	C₂×C₃×C₈	—	5	12	no	—
132	C₂×C₃×C₈	—	5	4	no	—
133	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	6	yes	complemented
134	C₄×Dic₃	—	5	4	no	—
135	C₄×Dic₃	—	5	12	no	—
136	C₂×C₃⋊C₈	—	5	4	no	—
137	C₂×C₃⋊C₈	—	5	12	no	—
138	C₂×C₃⋊C₈	—	5	12	no	—
139	C₂³×C₅	C₄×Dic₃	1	1	yes	characteristic
140	C₂²×D₅	D₃×C₈	1	8	yes	—
141	C₂²×D₅	C₄×Dic₃	1	3	yes	—
142	C₂²×D₅	C₂×C₄×D₃	1	3	yes	—
143	C₂×AGL(1,𝔽₅)	—	3	48	no	—
144	C₂×AGL(1,𝔽₅)	—	3	8	no	—
145	C₂×C₄×C₅	C₄×Dic₃	1	3	yes	—
146	C₂×C₄×C₅	C₂²×Dic₃	1	3	yes	—
147	C₂×Dic₅	C₄×Dic₃	1	3	yes	—
148	C₂×Dic₅	C₂×C₄×D₃	1	3	yes	—
149	C₄×D₅	C₄×Dic₃	1	6	yes	—
150	C₄×D₅	C₂×C₄×D₃	1	3	yes	—
151	C₄×D₅	C₂×C₄×D₃	1	3	yes	—
152	C₄×D₅	C₂²×Dic₃	1	3	yes	—
153	C₄×D₅	C₂³×D₃	1	1	yes	characteristic
154	#170	—	3	8	no	—
155	C₅×C₈	C₂²×Dic₃	1	4	yes	complemented
156	#168	C₂²×Dic₃	1	4	yes	complemented
157	C₂³×C₄	—	5	1	no	—
158	C₂³×C₄	—	15	2	no	—
159	C₂²×C₈	—	5	1	no	—
160	C₂²×C₈	—	5	6	no	—
161	C₂²×C₈	—	5	6	no	—
162	C₂²×C₈	—	15	6	no	—
163	C₂²×C₈	—	15	6	no	—
164	C₂²×C₈	—	15	2	no	—
165	C₂²×C₈	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, direct product factor
166	C₂×C₄²	—	15	6	no	—
167	C₂×C₄²	—	15	6	no	—
168	C₄×C₈	—	15	16	no	—
169	C₂×C₃×C₅	C₂×C₄×C₈	1	6	yes	complemented
170	C₂×C₃×C₅	C₂²×C₄²	1	1	yes	characteristic
171	C₃×D₅	C₂×C₄×C₈	1	6	yes	complemented
172	C₃×D₅	C₂³×C₈	1	2	yes	—
173	C₂³×C₃	—	5	3	no	—
174	C₂³×C₃	—	5	3	no	—
175	C₂³×C₃	—	5	8	no	—
176	C₂³×C₃	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
177	C₂×C₃×C₄	—	5	3	no	—
178	C₂×C₃×C₄	—	5	3	no	—
179	C₂×C₃×C₄	—	5	6	no	—
180	C₂×C₃×C₄	—	5	3	no	—
181	C₂×C₃×C₄	—	5	3	no	—
182	C₂×C₃×C₄	—	5	3	no	—
183	C₂×C₃×C₄	—	5	1	no	—
184	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	3	yes	—
185	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	3	yes	—
186	C₂×Dic₃	—	5	48	no	—
187	C₂×Dic₃	—	5	8	no	—
188	C₃×C₈	—	5	4	no	—
189	C₃×C₈	C₂²×AGL(1,𝔽₅)	1	4	yes	complemented
190	C₃⋊C₈	—	5	8	no	—
191	C₂²×C₅	C₈×Dic₃	1	4	yes	complemented
192	C₂²×C₅	C₂×C₄×Dic₃	1	3	yes	—
193	C₂×D₅	C₈×Dic₃	1	12	yes	complemented
194	C₂×D₅	C₂×D₃×C₈	1	12	yes	—
195	C₂×D₅	C₂×C₄×Dic₃	1	3	yes	—
196	C₂×D₅	C₂²×C₄×D₃	1	1	yes	characteristic
197	AGL(1,𝔽₅)	—	3	16	no	—
198	C₄×C₅	C₂×C₄×Dic₃	1	3	yes	—
199	C₄×C₅	C₂³×Dic₃	1	1	yes	characteristic
200	Dic₅	C₂×C₄×Dic₃	1	3	yes	—
201	Dic₅	C₂²×C₄×D₃	1	1	yes	characteristic
202	C₂⁴	—	5	1	no	—
203	C₂²×C₄	—	5	1	no	—
204	C₂²×C₄	—	5	3	no	—
205	C₂²×C₄	—	5	3	no	—
206	C₂²×C₄	—	5	3	no	—
207	C₂²×C₄	—	5	3	no	—
208	C₂²×C₄	—	15	12	no	—
209	C₂²×C₄	—	15	16	no	—
210	C₂²×C₄	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
211	C₂×C₈	—	5	6	no	—
212	C₂×C₈	—	5	4	no	—
213	C₂×C₈	—	5	12	no	—
214	C₂×C₈	—	15	4	no	—
215	C₂×C₈	—	15	12	no	—
216	C₂×C₈	—	15	12	no	—
217	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	6	yes	direct product factor
218	C₄²	—	15	4	no	—
219	C₄²	—	15	12	no	—
220	C₃×C₅	C₂²×C₄×C₈	1	1	yes	characteristic, complemented, Hall
221	C₂²×C₃	—	5	12	no	—
222	C₂²×C₃	—	5	12	no	—
223	C₂²×C₃	—	5	3	no	—
224	C₂²×C₃	—	5	1	no	—
225	C₂²×C₃	C₈×AGL(1,𝔽₅)	1	4	yes	complemented
226	C₂²×C₃	C₂×C₄×AGL(1,𝔽₅)	1	3	yes	—
227	C₃×C₄	—	5	3	no	—
228	C₃×C₄	—	5	1	no	—
229	C₃×C₄	C₂×C₄×AGL(1,𝔽₅)	1	3	yes	—
230	C₃×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
231	Dic₃	—	5	16	no	—
232	C₂×C₅	C₂×C₈×Dic₃	1	6	yes	complemented
233	C₂×C₅	C₂²×C₄×Dic₃	1	1	yes	characteristic
234	D₅	C₂×C₈×Dic₃	1	6	yes	complemented
235	D₅	C₂²×D₃×C₈	1	2	yes	—
236	C₂³	—	5	3	no	—
237	C₂³	—	5	3	no	—
238	C₂³	—	5	8	no	—
239	C₂³	C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
240	C₂×C₄	—	5	3	no	—
241	C₂×C₄	—	5	3	no	—
242	C₂×C₄	—	5	3	no	—
243	C₂×C₄	—	5	3	no	—
244	C₂×C₄	—	5	6	no	—
245	C₂×C₄	—	5	1	no	—
246	C₂×C₄	—	5	3	no	—
247	C₂×C₄	—	15	8	no	—
248	C₂×C₄	—	15	48	no	—
249	C₂×C₄	C₄×ΓL(1,𝔽₁₆)	1	3	yes	—
250	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	3	yes	—
251	C₈	—	5	4	no	—
252	C₈	—	15	8	no	—
253	C₈	C₂²×ΓL(1,𝔽₁₆)	1	4	yes	direct product factor
254	C₂×C₃	—	5	2	no	—
255	C₂×C₃	—	5	6	no	—
256	C₂×C₃	C₂×C₈×AGL(1,𝔽₅)	1	6	yes	complemented
257	C₂×C₃	C₂²×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
258	C₅	C₂²×C₈×Dic₃	1	1	yes	characteristic, complemented, Sylow
259	C₂²	—	5	1	no	—
260	C₂²	—	5	3	no	—
261	C₂²	—	5	12	no	—
262	C₂²	—	5	12	no	—
263	C₂²	C₈×ΓL(1,𝔽₁₆)	1	4	yes	direct product factor
264	C₂²	C₂×C₄×ΓL(1,𝔽₁₆)	1	3	yes	—
265	C₄	—	5	1	no	—
266	C₄	—	5	3	no	—
267	C₄	—	15	16	no	—
268	C₄	C₂×C₄×ΓL(1,𝔽₁₆)	1	3	yes	—
269	C₄	C₂³×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
270	C₃	C₂²×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
271	C₂	—	5	2	no	—
272	C₂	—	5	6	no	—
273	C₂	C₂×C₈×ΓL(1,𝔽₁₆)	1	6	yes	direct product factor
274	C₂	C₂²×C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₈	ΓL(1,𝔽₁₆)
commutator, hypocenter	C₃×C₅	C₂²×C₄×C₈
Frattini	C₄	C₂³×ΓL(1,𝔽₁₆)
Fitting	C₂²×C₃×C₅×C₈	C₄
socle	C₂³×C₃×C₅	C₄²
prime radical	C₂³×C₃×D₅	C₂×C₄
cosocle	C₃×C₄×D₅	C₂⁴
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
4. C₂×C₈×ΓL(1,𝔽₁₆)	C₂	direct (N × H)
8. C₂²×C₃⋊#170	C₄	semidirect
11. C₂×D₅⋊(C₃⋊C₈)	C₄	semidirect
13. C₂²×C₃×C₅×C₈	C₄	semidirect
14. C₂²×C₃×#168	C₄	semidirect
15. C₂×C₃×C₈×D₅	C₄	semidirect
17. C₈×ΓL(1,𝔽₁₆)	C₂²	direct (N × H)
24. C₂²×ΓL(1,𝔽₁₆)	C₈	direct (N × H)
32. C₂×C₃⋊#170	C₂×C₄	semidirect
35. D₅⋊(C₃⋊C₈)	C₂×C₄	semidirect
37. C₂×C₃×C₅×C₈	C₂×C₄	semidirect
38. C₂×C₃×#168	C₂×C₄	semidirect
39. C₃×C₈×D₅	C₂×C₄	semidirect
52. C₂²×C₅×C₈	Dic₃	semidirect
53. C₂²×#168	Dic₃	semidirect
54. C₂×C₈×D₅	Dic₃	semidirect
62. C₂×ΓL(1,𝔽₁₆)	C₂×C₈	direct (N × H)
73. C₃⋊#170	C₂²×C₄	semidirect
74. C₃×C₅×C₈	C₂²×C₄	semidirect
75. C₃×#168	C₂²×C₄	semidirect
81. C₂²×C₃×C₈	AGL(1,𝔽₅)	semidirect
102. C₂×C₅×C₈	C₂×Dic₃	semidirect
103. C₂×#168	C₂×Dic₃	semidirect
104. C₈×D₅	C₂×Dic₃	semidirect
110. C₂²×C₃×C₅	C₄×C₈	semidirect
112. C₂×C₃×D₅	C₄×C₈	semidirect
116. ΓL(1,𝔽₁₆)	C₂²×C₈	direct (N × H)
133. C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	semidirect
155. C₅×C₈	C₂²×Dic₃	semidirect
156. #168	C₂²×Dic₃	semidirect
165. C₂²×C₈	ΓL(1,𝔽₁₆)	direct (N × H)
169. C₂×C₃×C₅	C₂×C₄×C₈	semidirect
171. C₃×D₅	C₂×C₄×C₈	semidirect
189. C₃×C₈	C₂²×AGL(1,𝔽₅)	semidirect
191. C₂²×C₅	C₈×Dic₃	semidirect
193. C₂×D₅	C₈×Dic₃	semidirect
217. C₂×C₈	C₂×ΓL(1,𝔽₁₆)	direct (N × H)
220. C₃×C₅	C₂²×C₄×C₈	semidirect
225. C₂²×C₃	C₈×AGL(1,𝔽₅)	semidirect
232. C₂×C₅	C₂×C₈×Dic₃	semidirect
234. D₅	C₂×C₈×Dic₃	semidirect
253. C₈	C₂²×ΓL(1,𝔽₁₆)	direct (N × H)
256. C₂×C₃	C₂×C₈×AGL(1,𝔽₅)	semidirect
258. C₅	C₂²×C₈×Dic₃	semidirect
263. C₂²	C₈×ΓL(1,𝔽₁₆)	direct (N × H)
270. C₃	C₂²×C₈×AGL(1,𝔽₅)	semidirect
273. C₂	C₂×C₈×ΓL(1,𝔽₁₆)	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×C₄×ΓL(1,𝔽₁₆)	2
C₂²×D₅⋊(C₃⋊C₈)	2
C₂²×C₃×C₈×D₅	2
C₂×C₈×ΓL(1,𝔽₁₆)	2
C₂²×C₈×AGL(1,𝔽₅)	3
C₂²×C₈×Dic₃	5

Subgroups — C22×C8×ΓL(1,𝔽16) #14411167

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₈×ΓL(1,𝔽₁₆) #14411167