Subgroups — #14276797

Order 1920.

416 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊ΓL(1,𝔽₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₄×D₅⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₃×D₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#10598589	—	3	1	no	maximal, Hall
5	C₂×C₄⋊ΓL(1,𝔽₁₆)	C₄	1	2	yes	—
6	C₂×C₄×ΓL(1,𝔽₁₆)	—	2	1	no	—
7	C₂×D₅⋊(C₃⋊C₈)	—	2	1	no	—
8	C₂×C₃×C₈×D₅	—	2	1	no	—
9	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
10	#91235	—	2	1	no	—
11	C₄×C₃⋊#170	—	2	1	no	—
12	C₄×C₃⋊#170	C₄	1	2	yes	complemented
13	C₃×C₅×C₄⋊C₈	C₄	1	1	yes	characteristic, complemented
14	C₃×#3946	—	2	1	no	—
15	C₃×C₄⋊#168	C₄	1	1	yes	characteristic, complemented
16	#67704	—	5	1	no	maximal, Hall
17	C₄×C₄⋊AGL(1,𝔽₅)	—	3	1	no	—
18	C₄×D₅⋊C₈	—	3	1	no	—
19	D₅×C₄⋊C₈	D₃	1	1	yes	characteristic
20	C₂²×ΓL(1,𝔽₁₆)	—	2	2	no	—
21	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
23	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
24	C₂×C₃⋊#170	—	2	2	no	—
25	C₄⋊ΓL(1,𝔽₁₆)	—	2	2	no	—
26	C₄⋊ΓL(1,𝔽₁₆)	C₈	1	4	yes	complemented
27	C₄×ΓL(1,𝔽₁₆)	—	4	1	no	—
28	C₄×ΓL(1,𝔽₁₆)	—	4	1	no	—
29	D₅⋊(C₃⋊C₈)	—	2	4	no	—
30	C₂×C₃×C₅×C₈	—	2	1	no	—
31	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
32	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
33	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
34	C₃×C₄×Dic₅	Q₈	1	1	yes	characteristic
35	C₂×C₃×#168	—	2	1	no	—
36	C₃×C₈×D₅	—	4	1	no	—
37	C₃×C₈×D₅	—	4	1	no	—
38	C₂×C₃×C₄⋊C₈	—	5	1	no	—
39	C₄×C₄⋊Dic₃	—	5	1	no	—
40	C₂×C₄×C₃⋊C₈	—	5	1	no	—
41	C₂×C₄⋊AGL(1,𝔽₅)	—	3	2	no	—
42	C₂×C₄×AGL(1,𝔽₅)	—	6	1	no	—
43	C₂×D₅⋊C₈	—	6	1	no	—
44	C₂×C₈×D₅	—	2	1	no	—
45	C₄²×D₅	C₂×D₃	1	1	yes	characteristic
46	(C₂×AGL(1,𝔽₅))⋊C₄	—	6	1	no	—
47	C₄×#170	—	3	2	no	—
48	C₄×#170	—	6	1	no	—
49	C₅×C₄⋊C₈	Dic₃	1	1	yes	characteristic, complemented
50	#3946	—	2	1	no	—
51	C₄⋊#168	Dic₃	1	1	yes	characteristic, complemented
52	#1279	—	15	1	no	Sylow
53	C₂²×C₃×D₅	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×ΓL(1,𝔽₁₆)	—	2	4	no	—
55	C₂×ΓL(1,𝔽₁₆)	—	4	2	no	—
56	C₂×ΓL(1,𝔽₁₆)	—	4	2	no	—
57	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₃×Dic₅	C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₃×Dic₅	C₄⋊C₄	1	1	yes	characteristic
63	C₃×C₄×D₅	—	2	2	no	—
64	C₃×C₄×D₅	C₂×C₈	1	2	yes	—
65	C₃×C₄×D₅	M₁₆	1	2	yes	—
66	C₃×C₄×D₅	D₈	1	1	yes	characteristic
67	C₃×C₄×D₅	SD₁₆	1	1	yes	characteristic
68	C₃×C₄×D₅	SD₁₆	1	1	yes	characteristic
69	C₃×C₄×D₅	Q₁₆	1	1	yes	characteristic
70	C₃⋊#170	—	2	4	no	—
71	C₃×C₅×C₈	—	4	1	no	—
72	C₃×#168	—	4	1	no	—
73	C₂²×C₃×C₈	—	10	1	no	—
74	C₂×C₃×C₄²	—	5	1	no	—
75	C₂×C₄⋊Dic₃	—	5	2	no	—
76	C₂×C₄×Dic₃	—	10	1	no	—
77	C₂²×C₃⋊C₈	—	10	1	no	—
78	C₃×C₄⋊C₈	—	5	1	no	—
79	C₃×C₄⋊C₈	—	10	1	no	—
80	C₃×C₄⋊C₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
81	C₃⋊#95	—	10	1	no	—
82	C₄×C₃⋊C₈	—	5	2	no	—
83	C₄×C₃⋊C₈	—	10	1	no	—
84	C₂²×AGL(1,𝔽₅)	—	6	2	no	—
85	C₂×C₄×D₅	C₄×D₃	1	1	yes	characteristic
86	C₂×C₄×D₅	D₁₂	1	1	yes	characteristic
87	C₂×C₄×D₅	#67	1	1	yes	characteristic
88	C₂×#170	—	6	2	no	—
89	C₄⋊AGL(1,𝔽₅)	—	3	4	no	—
90	C₄⋊AGL(1,𝔽₅)	—	6	2	no	—
91	C₄×AGL(1,𝔽₅)	—	12	1	no	—
92	C₄×AGL(1,𝔽₅)	—	12	1	no	—
93	D₅⋊C₈	—	6	4	no	—
94	C₂×C₅×C₈	—	2	1	no	—
95	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
96	C₄×Dic₅	Dic₆	1	1	yes	characteristic
97	C₄×Dic₅	C₄×D₃	1	1	yes	characteristic
98	C₄×Dic₅	#67	1	1	yes	characteristic
99	C₂×#168	—	2	1	no	—
100	C₈×D₅	—	4	1	no	—
101	C₈×D₅	—	4	1	no	—
102	C₂×C₄⋊C₈	—	5	1	no	—
103	C₂×C₄×C₈	—	15	1	no	—
104	C₄×C₄⋊C₄	—	15	1	no	—
105	C₂²×C₃×C₅	#95	1	1	yes	characteristic
106	C₂×C₃×D₅	C₂²⋊C₈	1	2	yes	—
107	C₂×C₃×D₅	#102	1	1	yes	characteristic
108	C₂×C₃×D₅	#103	1	1	yes	characteristic
109	C₂×C₃×D₅	#104	1	2	yes	—
110	ΓL(1,𝔽₁₆)	—	4	4	no	—
111	C₃×C₄×C₅	—	2	1	no	—
112	C₃×C₄×C₅	C₄×C₈	1	1	yes	characteristic
113	C₃×C₄×C₅	#97	1	1	yes	characteristic
114	C₃×C₄×C₅	#102	1	1	yes	characteristic
115	C₃×C₄×C₅	#103	1	1	yes	characteristic
116	C₃×Dic₅	—	2	1	no	—
117	C₃×Dic₅	C₄⋊C₈	1	2	yes	—
118	C₃×Dic₅	#106	1	1	yes	characteristic
119	C₃×Dic₅	#107	1	1	yes	characteristic
120	C₂²×C₃×C₄	—	5	1	no	—
121	C₂²×C₃×C₄	—	5	1	no	—
122	C₂²×C₃×C₄	—	5	1	no	—
123	C₂²×Dic₃	—	10	2	no	—
124	C₂×C₃×C₈	—	2	1	no	—
125	C₂×C₃×C₈	—	10	1	no	—
126	C₂×C₃×C₈	—	20	1	no	—
127	C₂×C₃×C₈	—	20	1	no	—
128	C₃×C₄²	—	5	1	no	—
129	C₃×C₄²	—	5	1	no	—
130	C₃×C₄²	—	5	1	no	—
131	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
132	C₄⋊Dic₃	—	5	4	no	—
133	C₄⋊Dic₃	—	10	2	no	—
134	C₄×Dic₃	—	20	1	no	—
135	C₄×Dic₃	—	20	1	no	—
136	C₂×C₃⋊C₈	—	10	2	no	—
137	C₂×C₃⋊C₈	—	10	4	no	—
138	C₂²×D₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
139	C₂×AGL(1,𝔽₅)	—	6	4	no	—
140	C₂×AGL(1,𝔽₅)	—	12	2	no	—
141	C₂×AGL(1,𝔽₅)	—	12	2	no	—
142	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
143	C₂×C₄×C₅	C₄⋊Dic₃	1	1	yes	characteristic
144	C₂×C₄×C₅	#217	1	1	yes	characteristic
145	C₂×Dic₅	Dic₃⋊C₄	1	1	yes	characteristic
146	C₂×Dic₅	Dic₃⋊C₄	1	1	yes	characteristic
147	C₂×Dic₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
148	C₄×D₅	—	2	2	no	—
149	C₄×D₅	D₃×C₈	1	2	yes	—
150	C₄×D₅	ΓL(1,𝔽₂₅)	1	2	yes	—
151	C₄×D₅	#213	1	1	yes	characteristic
152	C₄×D₅	#214	1	1	yes	characteristic
153	C₄×D₅	#215	1	1	yes	characteristic
154	C₄×D₅	#216	1	1	yes	characteristic
155	#170	—	6	4	no	—
156	C₅×C₈	—	4	1	no	—
157	#168	—	4	1	no	—
158	C₂²×C₈	—	10	1	no	—
159	C₂²×C₈	—	30	1	no	—
160	C₂×C₄⋊C₄	—	15	2	no	—
161	C₂×C₄²	—	5	1	no	—
162	C₂×C₄²	—	30	1	no	—
163	C₄⋊C₈	—	5	1	no	—
164	C₄⋊C₈	—	10	1	no	—
165	C₄⋊C₈	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
166	C₄×C₈	—	15	2	no	—
167	C₄×C₈	—	30	1	no	—
168	#95	—	30	1	no	—
169	C₂×C₃×C₅	#336	1	1	yes	characteristic
170	C₂×C₃×C₅	#339	1	1	yes	characteristic
171	C₂×C₃×C₅	#340	1	1	yes	characteristic
172	C₃×D₅	D₄⋊C₈	1	2	yes	—
173	C₃×D₅	Q₈⋊C₈	1	2	yes	—
174	C₂³×C₃	—	5	1	no	—
175	C₂×C₃×C₄	—	5	1	no	—
176	C₂×C₃×C₄	—	5	1	no	—
177	C₂×C₃×C₄	—	5	2	no	—
178	C₂×C₃×C₄	—	5	2	no	—
179	C₂×C₃×C₄	—	5	1	no	—
180	C₂×C₃×C₄	—	5	1	no	—
181	C₂×C₃×C₄	—	5	1	no	—
182	C₂×C₃×C₄	—	5	1	no	—
183	C₂×C₃×C₄	—	5	1	no	—
184	C₂×C₃×C₄	—	10	2	no	—
185	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
186	C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
188	C₂×Dic₃	—	10	4	no	—
189	C₂×Dic₃	—	20	2	no	—
190	C₂×Dic₃	—	20	2	no	—
191	C₃×C₈	—	4	1	no	—
192	C₃×C₈	—	20	1	no	—
193	C₃⋊C₈	—	10	4	no	—
194	C₂²×C₅	C₃⋊#95	1	1	yes	characteristic
195	C₂×D₅	#809	1	1	yes	characteristic
196	C₂×D₅	#810	1	1	yes	characteristic
197	C₂×D₅	(C₂×D₃)⋊C₈	1	2	yes	—
198	C₂×D₅	#825	1	2	yes	—
199	AGL(1,𝔽₅)	—	12	4	no	—
200	C₄×C₅	—	2	1	no	—
201	C₄×C₅	C₈×Dic₃	1	1	yes	characteristic
202	C₄×C₅	#815	1	1	yes	characteristic
203	C₄×C₅	#832	1	1	yes	characteristic
204	C₄×C₅	#835	1	1	yes	characteristic
205	Dic₅	—	2	1	no	—
206	Dic₅	#807	1	1	yes	characteristic
207	Dic₅	#808	1	1	yes	characteristic
208	Dic₅	Dic₃⋊C₈	1	2	yes	—
209	C₂²×C₄	—	5	1	no	—
210	C₂²×C₄	—	5	1	no	—
211	C₂²×C₄	—	5	1	no	—
212	C₂²×C₄	—	30	2	no	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	10	1	no	—
215	C₂×C₈	—	20	1	no	—
216	C₂×C₈	—	20	1	no	—
217	C₂×C₈	—	30	2	no	—
218	C₂×C₈	—	30	4	no	—
219	C₄⋊C₄	—	15	4	no	—
220	C₄⋊C₄	—	30	2	no	—
221	C₄²	—	5	1	no	—
222	C₄²	—	5	1	no	—
223	C₄²	—	5	1	no	—
224	C₄²	—	60	1	no	—
225	C₄²	—	60	1	no	—
226	C₄²	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
227	C₃×C₅	#1279	1	1	yes	characteristic, complemented, Hall
228	C₂²×C₃	—	5	1	no	—
229	C₂²×C₃	—	5	1	no	—
230	C₂²×C₃	—	5	2	no	—
231	C₂²×C₃	—	5	2	no	—
232	C₂²×C₃	(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic
233	C₃×C₄	—	2	1	no	—
234	C₃×C₄	—	5	1	no	—
235	C₃×C₄	—	5	1	no	—
236	C₃×C₄	—	5	2	no	—
237	C₃×C₄	—	10	1	no	—
238	C₃×C₄	C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
239	C₃×C₄	#3992	1	1	yes	characteristic
240	C₃×C₄	#4007	1	1	yes	characteristic
241	C₃×C₄	#4009	1	1	yes	characteristic
242	Dic₃	—	20	4	no	—
243	C₂×C₅	#4529	1	1	yes	characteristic
244	C₂×C₅	#4550	1	1	yes	characteristic
245	C₂×C₅	#4555	1	1	yes	characteristic
246	D₅	#4483	1	2	yes	—
247	D₅	#4484	1	2	yes	—
248	C₂³	—	5	1	no	—
249	C₂×C₄	—	5	1	no	—
250	C₂×C₄	—	5	1	no	—
251	C₂×C₄	—	5	2	no	—
252	C₂×C₄	—	5	2	no	—
253	C₂×C₄	—	5	1	no	—
254	C₂×C₄	—	5	1	no	—
255	C₂×C₄	—	5	1	no	—
256	C₂×C₄	—	5	1	no	—
257	C₂×C₄	—	5	1	no	—
258	C₂×C₄	—	10	2	no	—
259	C₂×C₄	—	30	4	no	—
260	C₂×C₄	—	60	2	no	—
261	C₂×C₄	—	60	2	no	—
262	C₂×C₄	C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
263	C₂×C₄	C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
264	C₂×C₄	#6763	1	1	yes	characteristic
265	C₈	—	4	1	no	—
266	C₈	—	20	1	no	—
267	C₈	—	30	4	no	—
268	C₂×C₃	—	5	2	no	—
269	C₂×C₃	—	5	2	no	—
270	C₂×C₃	#64874	1	1	yes	characteristic
271	C₂×C₃	(C₂×AGL(1,𝔽₅))⋊C₈	1	1	yes	characteristic
272	C₂×C₃	#64900	1	1	yes	characteristic
273	C₅	#67704	1	1	yes	characteristic, complemented, Sylow
274	C₂²	—	5	1	no	—
275	C₂²	—	5	1	no	—
276	C₂²	—	5	2	no	—
277	C₂²	—	5	2	no	—
278	C₂²	#91235	1	1	yes	characteristic
279	C₄	—	2	1	no	—
280	C₄	—	5	1	no	—
281	C₄	—	5	1	no	—
282	C₄	—	5	2	no	—
283	C₄	—	10	1	no	—
284	C₄	—	60	4	no	—
285	C₄	C₈×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
286	C₄	#91221	1	1	yes	characteristic
287	C₄	#91236	1	1	yes	characteristic
288	C₄	#91238	1	1	yes	characteristic
289	C₃	#10598589	1	1	yes	characteristic, complemented, Sylow
290	C₂	—	5	2	no	—
291	C₂	—	5	2	no	—
292	C₂	#11746745	1	1	yes	characteristic
293	C₂	#11746763	1	1	yes	characteristic
294	C₂	#11746771	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6763
commutator	C₃×C₄×C₅	C₄×C₈
Frattini	C₄²	C₂×ΓL(1,𝔽₁₆)
Fitting	C₃×C₅×C₄⋊C₈	C₄
socle	C₂²×C₃×C₅	#95
prime radical	C₂²×C₃×D₅	C₂²⋊C₄
cosocle	C₃×C₄²×D₅	C₂²
hypercenter	C₄⋊C₈	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	#1279
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ 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. C₃×C₅×C₄⋊C₈	C₄	semidirect
15. C₃×C₄⋊#168	C₄	semidirect
26. C₄⋊ΓL(1,𝔽₁₆)	C₈	semidirect
49. C₅×C₄⋊C₈	Dic₃	semidirect
51. C₄⋊#168	Dic₃	semidirect
80. C₃×C₄⋊C₈	AGL(1,𝔽₅)	semidirect
165. C₄⋊C₈	ΓL(1,𝔽₁₆)	semidirect
227. C₃×C₅	#1279	semidirect
273. C₅	#67704	semidirect
289. C₃	#10598589	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊ΓL(1,𝔽₁₆)	2
C₄×D₅⋊(C₃⋊C₈)	2
C₃×D₅×C₄⋊C₈	2
#10598589	3
#67704	5