Subgroups — #14312680

Order 1920.

368 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11751655	C₂	1	1	yes	maximal, characteristic
2	D₅⋊#798	C₂	1	1	yes	maximal, characteristic
3	D₃×D₅×C₁₆	C₂	1	1	yes	maximal, characteristic
4	#11746715	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64844	C₂	1	1	yes	maximal, characteristic, complemented
6	ΓL(1,𝔽₁₆)⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	AGL(1,𝔽₅)×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	#10604098	—	3	1	no	maximal, Hall
9	D₅⋊ΓL(1,𝔽₂₅)	C₄	1	1	yes	characteristic
10	#91909	C₄	1	1	yes	characteristic
11	D₃×C₈×D₅	C₂²	1	1	yes	characteristic
12	C₈×ΓL(1,𝔽₁₆)	C₂²	1	1	yes	characteristic
13	D₅⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
14	C₃×C₈×AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
15	C₃×D₅⋊C₁₆	C₂²	1	1	yes	characteristic
16	#91166	C₄	1	2	yes	complemented
17	#91165	C₄	1	2	yes	complemented
18	#91150	C₄	1	1	yes	characteristic
19	#91149	C₄	1	1	yes	characteristic
20	C₁₆×D₁₅	C₄	1	1	yes	characteristic, complemented
21	C₅×D₃×C₁₆	C₄	1	1	yes	characteristic, complemented
22	C₃×D₅×C₁₆	C₂²	1	1	yes	characteristic
23	#90933	C₄	1	1	yes	characteristic, complemented
24	D₃×#672	C₄	1	1	yes	characteristic, complemented
25	D₅×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
26	#69072	—	5	1	no	maximal, Hall
27	#65718	—	3	1	no	—
28	D₅⋊M₃₂	—	3	1	no	—
29	C₂×D₅×C₁₆	—	3	1	no	—
30	#64844	—	3	1	no	—
31	#64844	D₃	1	1	yes	characteristic, complemented
32	C₁₆×AGL(1,𝔽₅)	—	3	1	no	—
33	C₁₆×AGL(1,𝔽₅)	—	3	1	no	—
34	C₄×D₃×D₅	C₂×C₄	1	1	yes	characteristic
35	C₄×ΓL(1,𝔽₁₆)	C₂×C₄	1	1	yes	characteristic
36	D₅⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
37	C₃×C₄×AGL(1,𝔽₅)	C₂×C₄	1	1	yes	characteristic
38	C₃×D₅⋊C₈	C₂×C₄	1	1	yes	characteristic
39	#6740	C₈	1	2	yes	—
40	#6739	C₈	1	2	yes	—
41	Dic₃⋊AGL(1,𝔽₅)	C₈	1	2	yes	—
42	(C₂×D₃)⋊AGL(1,𝔽₅)	C₈	1	2	yes	—
43	C₈×D₁₅	C₂×C₄	1	1	yes	characteristic
44	C₅×D₃×C₈	C₂×C₄	1	1	yes	characteristic
45	C₃×C₈×D₅	C₂³	1	1	yes	characteristic
46	#6648	C₂×C₄	1	1	yes	characteristic
47	D₃×#168	C₂×C₄	1	1	yes	characteristic
48	D₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
49	C₃⋊#674	C₂×C₄	1	2	yes	—
50	C₃×#674	C₂×C₄	1	2	yes	—
51	C₃×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
52	C₃⋊#672	C₂×C₄	1	1	yes	characteristic
53	C₃×#672	C₂×C₄	1	1	yes	characteristic
54	C₅×C₃⋊C₁₆	C₂×C₄	1	1	yes	characteristic
55	C₂×#798	—	5	1	no	—
56	C₂×D₃×C₁₆	—	5	1	no	—
57	#4689	—	5	1	no	—
58	C₃×#346	—	5	1	no	—
59	#4502	—	5	1	no	—
60	Dic₃×C₁₆	—	5	1	no	—
61	C₄×C₃⋊C₁₆	—	5	1	no	—
62	#4130	—	3	1	no	—
63	D₅⋊M₁₆	—	3	1	no	—
64	C₂×C₈×D₅	—	3	1	no	—
65	#3998	—	3	2	no	—
66	#3992	—	3	1	no	—
67	#3992	—	3	1	no	—
68	C₈×AGL(1,𝔽₅)	—	3	1	no	—
69	C₈×AGL(1,𝔽₅)	C₂×D₃	1	1	yes	characteristic
70	#3990	—	3	2	no	—
71	D₅⋊C₁₆	—	3	1	no	—
72	D₅⋊C₁₆	C₂×D₃	1	1	yes	characteristic
73	C₂×C₅×C₁₆	—	3	1	no	—
74	C₂×#672	—	3	1	no	—
75	D₅×C₁₆	—	3	1	no	—
76	D₅×C₁₆	—	3	1	no	—
77	D₅×C₁₆	—	3	1	no	—
78	D₅×C₁₆	C₂×D₃	1	1	yes	characteristic
79	#2108	—	15	1	no	Sylow
80	C₂×D₃×D₅	C₂×C₈	1	1	yes	characteristic
81	C₂×ΓL(1,𝔽₁₆)	C₂×C₈	1	2	yes	—
82	C₂×C₃×AGL(1,𝔽₅)	C₂×C₈	1	2	yes	—
83	C₄×D₁₅	C₄²	1	1	yes	characteristic
84	C₄×C₅×D₃	C₄²	1	1	yes	characteristic
85	C₃×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
86	#1200	C₂×C₈	1	1	yes	characteristic
87	D₃×Dic₅	C₂×C₈	1	1	yes	characteristic
88	D₅×Dic₃	C₂×C₈	1	1	yes	characteristic
89	C₃⋊#170	C₂×C₈	1	2	yes	—
90	C₃×#170	C₂×C₈	1	2	yes	—
91	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
92	C₃⋊#168	C₄²	1	1	yes	characteristic
93	C₃×#168	C₂²×C₄	1	1	yes	characteristic
94	C₅×C₃⋊C₈	C₄²	1	1	yes	characteristic
95	C₂×ΓL(1,𝔽₂₅)	—	5	1	no	—
96	C₂×D₃×C₈	—	5	1	no	—
97	#872	—	5	1	no	—
98	C₂×C₃×C₁₆	—	5	1	no	—
99	C₂×C₃×C₁₆	—	5	1	no	—
100	C₃×C₄×C₈	—	5	1	no	—
101	#815	—	5	1	no	—
102	C₈×Dic₃	—	5	1	no	—
103	C₂×C₃⋊C₁₆	—	5	1	no	—
104	C₂×C₃⋊C₁₆	—	5	1	no	—
105	#803	—	5	1	no	—
106	#798	—	5	2	no	—
107	#798	—	5	2	no	—
108	D₃×C₁₆	—	5	1	no	—
109	D₃×C₁₆	—	5	1	no	—
110	D₃×C₁₆	—	5	1	no	—
111	D₃×C₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
112	C₂×C₄×D₅	—	3	1	no	—
113	C₂²⋊AGL(1,𝔽₅)	—	3	2	no	—
114	#704	—	3	2	no	—
115	C₄⋊AGL(1,𝔽₅)	—	3	2	no	—
116	C₄×AGL(1,𝔽₅)	—	3	1	no	—
117	C₄×AGL(1,𝔽₅)	C₄×D₃	1	1	yes	characteristic
118	#700	—	3	2	no	—
119	D₅⋊C₈	—	3	1	no	—
120	D₅⋊C₈	C₄×D₃	1	1	yes	characteristic
121	C₂×C₅×C₈	—	3	1	no	—
122	C₂×#168	—	3	1	no	—
123	C₈×D₅	—	3	1	no	—
124	C₈×D₅	—	3	1	no	—
125	C₈×D₅	—	3	1	no	—
126	C₈×D₅	C₂²×D₃	1	1	yes	characteristic
127	#674	—	3	2	no	—
128	#674	C₄×D₃	1	2	yes	—
129	C₅×C₁₆	—	3	1	no	—
130	C₅×C₁₆	C₄×D₃	1	1	yes	characteristic
131	#672	—	3	1	no	—
132	#672	C₄×D₃	1	1	yes	characteristic
133	C₂×M₃₂	—	15	1	no	—
134	C₂²×C₁₆	—	15	1	no	—
135	#405	—	15	1	no	—
136	#346	—	5	1	no	—
137	#346	—	15	1	no	—
138	C₄×C₁₆	—	15	1	no	—
139	C₄×C₁₆	—	15	1	no	—
140	C₂×D₁₅	C₄×C₈	1	1	yes	characteristic
141	C₂×C₅×D₃	C₄×C₈	1	1	yes	characteristic
142	C₂×C₃×D₅	C₂²×C₈	1	1	yes	characteristic
143	D₃×D₅	—	2	2	no	—
144	ΓL(1,𝔽₁₆)	—	2	2	no	—
145	C₃×AGL(1,𝔽₅)	—	2	2	no	—
146	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
147	Dic₁₅	C₄×C₈	1	1	yes	characteristic
148	C₃×Dic₅	C₂²×C₈	1	1	yes	characteristic
149	C₅×Dic₃	C₄×C₈	1	1	yes	characteristic
150	C₂×C₄×D₃	—	5	1	no	—
151	C₂×C₃×C₈	—	5	1	no	—
152	C₂×C₃×C₈	—	5	1	no	—
153	C₃×C₄²	—	5	1	no	—
154	(C₂×D₃)⋊C₄	—	5	2	no	—
155	Dic₃⋊C₄	—	5	2	no	—
156	C₄×Dic₃	—	5	1	no	—
157	C₂×C₃⋊C₈	—	5	1	no	—
158	C₂×C₃⋊C₈	—	5	1	no	—
159	ΓL(1,𝔽₂₅)	—	5	2	no	—
160	ΓL(1,𝔽₂₅)	—	5	2	no	—
161	D₃×C₈	—	5	1	no	—
162	D₃×C₈	—	5	1	no	—
163	D₃×C₈	—	5	1	no	—
164	D₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
165	C₃×C₁₆	—	5	2	no	—
166	C₃×C₁₆	—	5	1	no	—
167	C₃×C₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
168	C₃⋊C₁₆	—	5	2	no	—
169	C₃⋊C₁₆	—	5	1	no	—
170	C₃⋊C₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
171	C₂²×D₅	—	3	1	no	—
172	C₂×AGL(1,𝔽₅)	—	3	2	no	—
173	C₂×AGL(1,𝔽₅)	D₃×C₈	1	2	yes	—
174	C₂×C₄×C₅	—	3	1	no	—
175	C₂×Dic₅	—	3	1	no	—
176	C₄×D₅	—	3	1	no	—
177	C₄×D₅	—	3	1	no	—
178	C₄×D₅	—	3	1	no	—
179	C₄×D₅	C₂×C₄×D₃	1	1	yes	characteristic
180	#170	—	3	2	no	—
181	#170	D₃×C₈	1	2	yes	—
182	C₅×C₈	—	3	1	no	—
183	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
184	#168	—	3	1	no	—
185	#168	C₂×C₄×D₃	1	1	yes	characteristic
186	C₂×M₁₆	—	15	1	no	—
187	C₂²×C₈	—	15	1	no	—
188	#117	—	15	1	no	—
189	M₃₂	—	15	2	no	—
190	M₃₂	—	15	2	no	—
191	C₂×C₁₆	—	3	1	no	—
192	C₂×C₁₆	—	5	1	no	—
193	C₂×C₁₆	—	5	1	no	—
194	C₂×C₁₆	—	15	1	no	—
195	C₂×C₁₆	—	15	1	no	—
196	C₂×C₁₆	—	15	1	no	—
197	C₂×C₁₆	—	15	1	no	—
198	C₂×C₁₆	—	15	1	no	—
199	#97	—	15	1	no	—
200	#97	—	15	1	no	—
201	C₄×C₈	—	5	1	no	—
202	C₄×C₈	—	15	1	no	—
203	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
204	D₁₅	—	2	1	no	—
205	C₃×D₅	#504	1	2	yes	—
206	C₅×D₃	—	2	1	no	—
207	C₂²×D₃	—	5	1	no	—
208	C₂×C₃×C₄	—	5	1	no	—
209	C₂×C₃×C₄	—	5	2	no	—
210	C₂×Dic₃	—	5	1	no	—
211	C₂×Dic₃	—	5	2	no	—
212	C₄×D₃	—	5	1	no	—
213	C₄×D₃	—	5	1	no	—
214	C₄×D₃	—	5	1	no	—
215	C₄×D₃	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
216	C₃×C₈	—	5	2	no	—
217	C₃×C₈	—	5	1	no	—
218	C₃×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
219	C₃⋊C₈	—	5	2	no	—
220	C₃⋊C₈	—	5	1	no	—
221	C₃⋊C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
222	C₂²×C₅	—	3	1	no	—
223	C₂×D₅	—	3	1	no	—
224	C₂×D₅	—	6	2	no	—
225	C₂×D₅	C₂×D₃×C₈	1	1	yes	characteristic
226	AGL(1,𝔽₅)	—	2	2	no	—
227	AGL(1,𝔽₅)	—	6	2	no	—
228	C₄×C₅	—	3	1	no	—
229	C₄×C₅	C₄²×D₃	1	1	yes	characteristic
230	Dic₅	—	3	1	no	—
231	Dic₅	C₂×D₃×C₈	1	1	yes	characteristic
232	C₂²×C₄	—	15	1	no	—
233	M₁₆	—	15	2	no	—
234	M₁₆	—	15	2	no	—
235	C₂×C₈	—	3	1	no	—
236	C₂×C₈	—	5	1	no	—
237	C₂×C₈	—	5	1	no	—
238	C₂×C₈	—	15	1	no	—
239	C₂×C₈	—	15	1	no	—
240	C₂×C₈	—	15	1	no	—
241	C₂×C₈	—	15	1	no	—
242	C₂×C₈	—	15	1	no	—
243	C₄⋊C₄	—	15	2	no	—
244	C₂²⋊C₄	—	15	2	no	—
245	C₄²	—	5	1	no	—
246	C₄²	—	15	1	no	—
247	C₁₆	—	3	1	no	—
248	C₁₆	—	5	2	no	—
249	C₁₆	—	5	1	no	—
250	C₁₆	—	15	2	no	—
251	C₁₆	—	15	1	no	—
252	C₁₆	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
253	C₃×C₅	#2108	1	1	yes	characteristic, complemented, Hall
254	C₂²×C₃	—	5	1	no	—
255	C₂×D₃	—	5	1	no	—
256	C₂×D₃	—	10	2	no	—
257	C₂×D₃	C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
258	C₃×C₄	—	5	1	no	—
259	C₃×C₄	—	10	2	no	—
260	C₃×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
261	Dic₃	—	5	1	no	—
262	Dic₃	—	10	2	no	—
263	Dic₃	C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
264	C₂×C₅	—	6	1	no	—
265	C₂×C₅	C₄×D₃×C₈	1	1	yes	characteristic
266	D₅	—	6	1	no	—
267	D₅	#4907	1	2	yes	—
268	C₂³	—	15	1	no	—
269	C₂×C₄	—	3	1	no	—
270	C₂×C₄	—	5	1	no	—
271	C₂×C₄	—	5	2	no	—
272	C₂×C₄	—	15	1	no	—
273	C₂×C₄	—	15	1	no	—
274	C₂×C₄	—	15	1	no	—
275	C₂×C₄	—	15	1	no	—
276	C₂×C₄	—	15	2	no	—
277	C₈	—	3	1	no	—
278	C₈	—	5	2	no	—
279	C₈	—	5	1	no	—
280	C₈	—	15	2	no	—
281	C₈	—	15	1	no	—
282	C₈	C₂×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
283	C₂×C₃	—	5	2	no	—
284	C₂×C₃	C₂×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
285	D₃	—	2	1	no	—
286	D₃	—	10	1	no	—
287	C₅	#69072	1	1	yes	characteristic, complemented, Sylow
288	C₂²	—	3	1	no	—
289	C₂²	—	5	1	no	—
290	C₂²	—	15	1	no	—
291	C₂²	—	30	2	no	—
292	C₄	—	3	1	no	—
293	C₄	—	5	1	no	—
294	C₄	—	10	2	no	—
295	C₄	—	15	1	no	—
296	C₄	—	30	2	no	—
297	C₄	C₄×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
298	C₃	#10604098	1	1	yes	characteristic, complemented, Sylow
299	C₂	—	5	2	no	—
300	C₂	—	6	1	no	—
301	C₂	—	30	1	no	—
302	C₂	D₃×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂×D₃×AGL(1,𝔽₅)
commutator, socle	C₂×C₃×C₅	C₂×C₄×C₈
Fitting	C₃×C₅×C₁₆	C₂×C₄
prime radical	C₂×D₃×D₅	C₂×C₈
cosocle	C₃×C₈×D₅	C₂³
hypercenter	C₁₆	D₃×AGL(1,𝔽₅)
hypocenter	C₃×C₅	#2108
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₈ ◁ C₁₆

Complements

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

N	H ≅ G/N	type
4. #11746715	C₂	semidirect
5. C₃×#64844	C₂	semidirect
6. ΓL(1,𝔽₁₆)⋊C₁₆	C₂	semidirect
7. AGL(1,𝔽₅)×C₃⋊C₁₆	C₂	semidirect
16. #91166	C₄	semidirect
17. #91165	C₄	semidirect
20. C₁₆×D₁₅	C₄	semidirect
21. C₅×D₃×C₁₆	C₄	semidirect
23. #90933	C₄	semidirect
24. D₃×#672	C₄	semidirect
31. #64844	D₃	semidirect
111. D₃×C₁₆	AGL(1,𝔽₅)	semidirect
253. C₃×C₅	#2108	semidirect
287. C₅	#69072	semidirect
298. C₃	#10604098	semidirect

Maximal subgroups

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

subgroup	index
#11751655	2
D₅⋊#798	2
D₃×D₅×C₁₆	2
#11746715	2
C₃×#64844	2
ΓL(1,𝔽₁₆)⋊C₁₆	2
AGL(1,𝔽₅)×C₃⋊C₁₆	2
#10604098	3
#69072	5