Subgroups — #14294871

Order 1920.

592 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×ΓL(1,𝔽₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₄×D₅⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic
4	#11746763	C₂	1	4	yes	maximal
5	#10603839	—	3	1	no	maximal, Hall
6	C₂×C₄×ΓL(1,𝔽₁₆)	C₄	1	2	yes	—
7	C₂×C₄×ΓL(1,𝔽₁₆)	C₄	1	2	yes	—
8	C₂×C₄×ΓL(1,𝔽₁₆)	C₂²	1	2	yes	—
9	C₂×D₅⋊(C₃⋊C₈)	C₂²	1	2	yes	—
10	C₂×C₃×C₈×D₅	C₂²	1	2	yes	—
11	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
12	C₄×C₃⋊#170	C₄	1	2	yes	complemented
13	C₄×C₃⋊#170	C₄	1	2	yes	complemented
14	C₃×C₄×C₅×C₈	C₄	1	1	yes	characteristic, complemented
15	C₃×C₈×Dic₅	C₄	1	2	yes	complemented
16	C₃×C₄×#168	C₄	1	1	yes	characteristic, complemented
17	#69750	—	5	1	no	maximal, Hall
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	(C₂×AGL(1,𝔽₅))⋊C₈	—	3	4	no	—
22	C₂²×ΓL(1,𝔽₁₆)	C₂×C₄	1	4	yes	—
23	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
26	C₂×C₃⋊#170	C₂×C₄	1	4	yes	—
27	C₄×ΓL(1,𝔽₁₆)	—	2	4	no	—
28	C₄×ΓL(1,𝔽₁₆)	—	2	4	no	—
29	C₄×ΓL(1,𝔽₁₆)	C₈	1	8	yes	complemented
30	D₅⋊(C₃⋊C₈)	—	2	4	no	—
31	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
32	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
33	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
34	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
35	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×#168	C₂×C₄	1	2	yes	—
37	C₃×C₈×D₅	—	2	4	no	—
38	C₂×C₃×C₄×C₈	—	5	1	no	—
39	C₄²×Dic₃	—	5	1	no	—
40	C₂×C₄×C₃⋊C₈	—	5	1	no	—
41	#4550	—	5	4	no	—
42	C₂×C₄×AGL(1,𝔽₅)	—	3	2	no	—
43	C₂×C₄×AGL(1,𝔽₅)	—	3	2	no	—
44	C₂×C₄×AGL(1,𝔽₅)	—	3	2	no	—
45	C₂×D₅⋊C₈	—	3	2	no	—
46	C₂×C₈×D₅	C₂×D₃	1	2	yes	—
47	C₄²×D₅	C₂×D₃	1	1	yes	characteristic
48	C₄×#170	—	3	2	no	—
49	C₄×#170	—	3	2	no	—
50	C₄×C₅×C₈	Dic₃	1	1	yes	characteristic, complemented
51	C₈×Dic₅	Dic₃	1	2	yes	complemented
52	C₄×#168	Dic₃	1	1	yes	characteristic, complemented
53	#1744	—	15	1	no	Sylow
54	C₂²×C₃×D₅	C₂²×C₄	1	1	yes	characteristic
55	C₂×ΓL(1,𝔽₁₆)	—	2	4	no	—
56	C₂×ΓL(1,𝔽₁₆)	—	2	4	no	—
57	C₂×ΓL(1,𝔽₁₆)	C₂×C₈	1	8	yes	—
58	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₃×Dic₅	C₄²	1	1	yes	characteristic
62	C₂×C₃×Dic₅	C₄²	1	1	yes	characteristic
63	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
64	C₃×C₄×D₅	C₂×C₈	1	4	yes	—
65	C₃×C₄×D₅	M₁₆	1	4	yes	—
66	C₃×C₄×D₅	#41	1	4	yes	—
67	C₃⋊#170	—	2	4	no	—
68	C₃×C₅×C₈	—	2	2	no	—
69	C₃×#168	—	2	2	no	—
70	C₂²×C₃×C₈	—	5	2	no	—
71	C₂×C₃×C₄²	—	5	1	no	—
72	C₂×C₄×Dic₃	—	5	2	no	—
73	C₂×C₄×Dic₃	—	5	2	no	—
74	C₂×C₄×Dic₃	—	5	2	no	—
75	C₂²×C₃⋊C₈	—	5	2	no	—
76	C₃×C₄×C₈	—	5	2	no	—
77	C₃×C₄×C₈	—	5	1	no	—
78	C₃×C₄×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
79	C₄×C₃⋊C₈	—	5	2	no	—
80	C₄×C₃⋊C₈	—	5	2	no	—
81	C₂²×AGL(1,𝔽₅)	—	3	4	no	—
82	C₂×C₄×D₅	C₄×D₃	1	1	yes	characteristic
83	C₂×C₄×D₅	C₄×D₃	1	1	yes	characteristic
84	C₂×C₄×D₅	C₂²×D₃	1	1	yes	characteristic
85	C₂×#170	—	3	4	no	—
86	C₄×AGL(1,𝔽₅)	—	3	8	no	—
87	C₄×AGL(1,𝔽₅)	—	6	4	no	—
88	C₄×AGL(1,𝔽₅)	—	6	4	no	—
89	D₅⋊C₈	—	6	4	no	—
90	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
91	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
92	C₄×Dic₅	C₄×D₃	1	1	yes	characteristic
93	C₄×Dic₅	C₄×D₃	1	1	yes	characteristic
94	C₄×Dic₅	C₂×Dic₃	1	1	yes	characteristic
95	C₂×#168	C₂×Dic₃	1	2	yes	—
96	C₈×D₅	—	2	4	no	—
97	C₂×C₄×C₈	—	5	1	no	—
98	C₂×C₄×C₈	—	15	1	no	—
99	C₄³	—	15	1	no	—
100	#336	—	15	4	no	—
101	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₃×D₅	#117	1	2	yes	—
103	C₂×C₃×D₅	C₂²×C₈	1	2	yes	—
104	C₂×C₃×D₅	C₂×M₁₆	1	2	yes	—
105	ΓL(1,𝔽₁₆)	—	2	8	no	—
106	C₃×C₄×C₅	C₄×C₈	1	2	yes	—
107	C₃×C₄×C₅	#97	1	2	yes	—
108	C₃×C₄×C₅	#117	1	2	yes	—
109	C₃×Dic₅	C₄×C₈	1	2	yes	—
110	C₃×Dic₅	#97	1	2	yes	—
111	C₃×Dic₅	#117	1	2	yes	—
112	C₂²×C₃×C₄	—	5	1	no	—
113	C₂²×C₃×C₄	—	5	1	no	—
114	C₂²×C₃×C₄	—	5	1	no	—
115	C₂²×Dic₃	—	5	4	no	—
116	C₂×C₃×C₈	—	5	2	no	—
117	C₂×C₃×C₈	—	10	4	no	—
118	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
119	C₃×C₄²	—	5	1	no	—
120	C₃×C₄²	—	5	1	no	—
121	C₃×C₄²	—	5	1	no	—
122	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
123	C₄×Dic₃	—	5	8	no	—
124	C₄×Dic₃	—	10	4	no	—
125	C₄×Dic₃	—	10	4	no	—
126	C₂×C₃⋊C₈	—	5	4	no	—
127	C₂×C₃⋊C₈	—	10	4	no	—
128	C₂²×D₅	C₂×C₄×D₃	1	1	yes	characteristic
129	C₂×AGL(1,𝔽₅)	—	3	8	no	—
130	C₂×AGL(1,𝔽₅)	—	6	4	no	—
131	C₂×AGL(1,𝔽₅)	—	6	4	no	—
132	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
133	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
135	C₂×Dic₅	C₄×Dic₃	1	1	yes	characteristic
136	C₂×Dic₅	C₄×Dic₃	1	1	yes	characteristic
137	C₂×Dic₅	C₂×C₄×D₃	1	1	yes	characteristic
138	C₄×D₅	D₃×C₈	1	4	yes	—
139	C₄×D₅	ΓL(1,𝔽₂₅)	1	4	yes	—
140	C₄×D₅	#235	1	4	yes	—
141	#170	—	6	4	no	—
142	C₅×C₈	—	2	2	no	—
143	#168	—	2	2	no	—
144	C₂²×C₈	—	5	2	no	—
145	C₂²×C₈	—	15	2	no	—
146	C₂×C₄²	—	5	1	no	—
147	C₂×C₄²	—	15	2	no	—
148	C₂×C₄²	—	15	2	no	—
149	C₂×C₄²	—	15	2	no	—
150	C₄×C₈	—	5	2	no	—
151	C₄×C₈	—	5	1	no	—
152	C₄×C₈	—	15	2	no	—
153	C₄×C₈	—	15	2	no	—
154	C₄×C₈	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
155	C₂×C₃×C₅	#376	1	1	yes	characteristic
156	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
157	C₂×C₃×C₅	C₂×#97	1	1	yes	characteristic
158	C₃×D₅	#431	1	4	yes	—
159	C₂³×C₃	—	5	1	no	—
160	C₂×C₃×C₄	—	5	1	no	—
161	C₂×C₃×C₄	—	5	1	no	—
162	C₂×C₃×C₄	—	5	1	no	—
163	C₂×C₃×C₄	—	5	4	no	—
164	C₂×C₃×C₄	—	5	4	no	—
165	C₂×C₃×C₄	—	5	4	no	—
166	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
167	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
169	C₂×Dic₃	—	5	8	no	—
170	C₂×Dic₃	—	10	4	no	—
171	C₂×Dic₃	—	10	4	no	—
172	C₃×C₈	—	2	2	no	—
173	C₃×C₈	—	10	2	no	—
174	C₃⋊C₈	—	10	4	no	—
175	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
176	C₂×D₅	#872	1	2	yes	—
177	C₂×D₅	C₂×D₃×C₈	1	2	yes	—
178	C₂×D₅	C₂×ΓL(1,𝔽₂₅)	1	2	yes	—
179	AGL(1,𝔽₅)	—	6	8	no	—
180	C₄×C₅	C₈×Dic₃	1	2	yes	—
181	C₄×C₅	#815	1	2	yes	—
182	C₄×C₅	#926	1	2	yes	—
183	Dic₅	C₈×Dic₃	1	2	yes	—
184	Dic₅	#815	1	2	yes	—
185	Dic₅	#872	1	2	yes	—
186	C₂²×C₄	—	5	1	no	—
187	C₂²×C₄	—	5	1	no	—
188	C₂²×C₄	—	5	1	no	—
189	C₂²×C₄	—	15	4	no	—
190	C₂×C₈	—	5	2	no	—
191	C₂×C₈	—	10	4	no	—
192	C₂×C₈	—	15	4	no	—
193	C₂×C₈	—	30	4	no	—
194	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	2	yes	—
195	C₄²	—	5	1	no	—
196	C₄²	—	5	1	no	—
197	C₄²	—	5	1	no	—
198	C₄²	—	15	8	no	—
199	C₄²	—	30	4	no	—
200	C₄²	—	30	4	no	—
201	C₄²	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
202	C₃×C₅	#1744	1	1	yes	characteristic, complemented, Hall
203	C₂²×C₃	—	5	2	no	—
204	C₂²×C₃	—	5	2	no	—
205	C₂²×C₃	—	5	2	no	—
206	C₂²×C₃	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
207	C₃×C₄	—	5	2	no	—
208	C₃×C₄	—	5	2	no	—
209	C₃×C₄	—	5	2	no	—
210	C₃×C₄	C₈×AGL(1,𝔽₅)	1	2	yes	—
211	C₃×C₄	#3992	1	2	yes	—
212	C₃×C₄	#4130	1	2	yes	—
213	Dic₃	—	10	8	no	—
214	C₂×C₅	#4932	1	1	yes	characteristic
215	C₂×C₅	C₂×C₈×Dic₃	1	1	yes	characteristic
216	C₂×C₅	C₂×#815	1	1	yes	characteristic
217	D₅	#4685	1	4	yes	—
218	C₂³	—	5	1	no	—
219	C₂×C₄	—	5	1	no	—
220	C₂×C₄	—	5	1	no	—
221	C₂×C₄	—	5	1	no	—
222	C₂×C₄	—	5	4	no	—
223	C₂×C₄	—	5	4	no	—
224	C₂×C₄	—	5	4	no	—
225	C₂×C₄	—	15	8	no	—
226	C₂×C₄	—	30	4	no	—
227	C₂×C₄	—	30	4	no	—
228	C₂×C₄	C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
229	C₂×C₄	C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
230	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
231	C₈	—	2	2	no	—
232	C₈	—	10	2	no	—
233	C₈	—	30	4	no	—
234	C₂×C₃	—	5	4	no	—
235	C₂×C₃	#65686	1	1	yes	characteristic
236	C₂×C₃	C₂×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
237	C₂×C₃	C₂×#3992	1	1	yes	characteristic
238	C₅	#69750	1	1	yes	characteristic, complemented, Sylow
239	C₂²	—	5	2	no	—
240	C₂²	—	5	2	no	—
241	C₂²	—	5	2	no	—
242	C₂²	C₂×C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
243	C₄	—	5	2	no	—
244	C₄	—	5	2	no	—
245	C₄	—	5	2	no	—
246	C₄	—	30	8	no	—
247	C₄	C₈×ΓL(1,𝔽₁₆)	1	2	yes	—
248	C₄	#91221	1	2	yes	—
249	C₄	#91989	1	2	yes	—
250	C₃	#10603839	1	1	yes	characteristic, complemented, Sylow
251	C₂	—	5	4	no	—
252	C₂	#11752345	1	1	yes	characteristic
253	C₂	C₂×C₈×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
254	C₂	C₂×#91221	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×ΓL(1,𝔽₁₆)
commutator	C₂×C₃×C₅	C₂×C₄×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
hypercenter	C₄×C₈	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	#1744
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
12. C₄×C₃⋊#170	C₄	semidirect
13. C₄×C₃⋊#170	C₄	semidirect
14. C₃×C₄×C₅×C₈	C₄	semidirect
15. C₃×C₈×Dic₅	C₄	semidirect
16. C₃×C₄×#168	C₄	semidirect
29. C₄×ΓL(1,𝔽₁₆)	C₈	semidirect
50. C₄×C₅×C₈	Dic₃	semidirect
51. C₈×Dic₅	Dic₃	semidirect
52. C₄×#168	Dic₃	semidirect
78. C₃×C₄×C₈	AGL(1,𝔽₅)	semidirect
154. C₄×C₈	ΓL(1,𝔽₁₆)	semidirect
202. C₃×C₅	#1744	semidirect
238. C₅	#69750	semidirect
250. C₃	#10603839	semidirect

Maximal subgroups

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

subgroup	index
C₄²×ΓL(1,𝔽₁₆)	2
C₄×D₅⋊(C₃⋊C₈)	2
C₃×C₄×C₈×D₅	2
#11746763	2
#10603839	3
#69750	5