Subgroups — #14277178

Order 1920.

376 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11752347	C₂	1	1	yes	maximal, characteristic
2	D₅⋊(C₄⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic
3	C₃×D₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#10598586	—	3	1	no	maximal, Hall
5	C₂×C₄⋊ΓL(1,𝔽₁₆)	—	2	1	no	—
6	C₂×C₄⋊ΓL(1,𝔽₁₆)	—	2	1	no	—
7	C₂×C₄⋊ΓL(1,𝔽₁₆)	C₄	1	2	yes	—
8	C₂×D₅⋊(C₃⋊C₈)	—	2	1	no	—
9	C₂×C₃×C₈×D₅	—	2	1	no	—
10	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
11	#91233	C₄	1	2	yes	complemented
12	C₄⋊(C₃⋊#170)	—	2	1	no	—
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	#67702	—	5	1	no	maximal, Hall
17	#65688	—	3	1	no	—
18	D₅⋊(C₄⋊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)	—	4	2	no	—
26	C₄⋊ΓL(1,𝔽₁₆)	—	4	1	no	—
27	C₄⋊ΓL(1,𝔽₁₆)	—	4	1	no	—
28	C₄⋊ΓL(1,𝔽₁₆)	—	4	1	no	—
29	C₄⋊ΓL(1,𝔽₁₆)	—	4	1	no	—
30	D₅⋊(C₃⋊C₈)	—	4	1	no	—
31	D₅⋊(C₃⋊C₈)	—	4	1	no	—
32	C₂×C₃×C₅×C₈	—	2	1	no	—
33	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
34	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
35	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
36	C₃×C₄×Dic₅	Q₈	1	1	yes	characteristic
37	C₂×C₃×#168	—	2	1	no	—
38	C₃×C₈×D₅	—	4	1	no	—
39	C₃×C₈×D₅	—	4	1	no	—
40	C₂×C₃×C₄⋊C₈	—	5	1	no	—
41	#4935	—	5	1	no	—
42	C₂×C₄⋊(C₃⋊C₈)	—	5	1	no	—
43	C₂×C₄⋊AGL(1,𝔽₅)	—	3	2	no	—
44	C₂×C₄⋊AGL(1,𝔽₅)	—	6	1	no	—
45	C₂×C₄⋊AGL(1,𝔽₅)	—	6	1	no	—
46	C₂×D₅⋊C₈	—	6	1	no	—
47	C₂×C₈×D₅	—	2	1	no	—
48	C₄²×D₅	C₂×D₃	1	1	yes	characteristic
49	#4004	—	3	2	no	—
50	C₄⋊#170	—	6	1	no	—
51	C₅×C₄⋊C₈	Dic₃	1	1	yes	characteristic, complemented
52	#3946	—	2	1	no	—
53	C₄⋊#168	Dic₃	1	1	yes	characteristic, complemented
54	#1296	—	15	1	no	Sylow
55	C₂²×C₃×D₅	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×ΓL(1,𝔽₁₆)	—	4	2	no	—
57	C₂×ΓL(1,𝔽₁₆)	—	4	2	no	—
58	C₂×ΓL(1,𝔽₁₆)	—	4	2	no	—
59	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₃×Dic₅	C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₃×Dic₅	C₄⋊C₄	1	1	yes	characteristic
65	C₃×C₄×D₅	—	2	1	no	—
66	C₃×C₄×D₅	—	2	1	no	—
67	C₃×C₄×D₅	D₈	1	1	yes	characteristic
68	C₃×C₄×D₅	D₈	1	1	yes	characteristic
69	C₃×C₄×D₅	SD₁₆	1	1	yes	characteristic
70	C₃×C₄×D₅	SD₁₆	1	1	yes	characteristic
71	C₃×C₄×D₅	SD₁₆	1	1	yes	characteristic
72	C₃×C₄×D₅	SD₁₆	1	1	yes	characteristic
73	C₃×C₄×D₅	Q₁₆	1	1	yes	characteristic
74	C₃×C₄×D₅	Q₁₆	1	1	yes	characteristic
75	C₃⋊#170	—	4	2	no	—
76	C₃×C₅×C₈	—	4	1	no	—
77	C₃×#168	—	4	1	no	—
78	C₂²×C₃×C₈	—	10	1	no	—
79	C₂×C₃×C₄²	—	5	1	no	—
80	C₂×C₄⋊Dic₃	—	5	2	no	—
81	C₂×C₄⋊Dic₃	—	10	1	no	—
82	C₂×C₄⋊Dic₃	—	10	1	no	—
83	C₂²×C₃⋊C₈	—	10	1	no	—
84	C₃×C₄⋊C₈	—	5	1	no	—
85	C₃×C₄⋊C₈	—	10	1	no	—
86	C₃×C₄⋊C₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
87	C₄⋊(C₃⋊C₈)	—	5	2	no	—
88	C₄⋊(C₃⋊C₈)	—	10	1	no	—
89	C₂²×AGL(1,𝔽₅)	—	6	2	no	—
90	C₂×C₄×D₅	C₄×D₃	1	1	yes	characteristic
91	C₂×C₄×D₅	D₁₂	1	1	yes	characteristic
92	C₂×C₄×D₅	#67	1	1	yes	characteristic
93	C₂×#170	—	6	2	no	—
94	C₄⋊AGL(1,𝔽₅)	—	12	2	no	—
95	C₄⋊AGL(1,𝔽₅)	—	12	1	no	—
96	C₄⋊AGL(1,𝔽₅)	—	12	1	no	—
97	C₄⋊AGL(1,𝔽₅)	—	12	1	no	—
98	C₄⋊AGL(1,𝔽₅)	—	12	1	no	—
99	D₅⋊C₈	—	12	1	no	—
100	D₅⋊C₈	—	12	1	no	—
101	C₂×C₅×C₈	—	2	1	no	—
102	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
103	C₄×Dic₅	Dic₆	1	1	yes	characteristic
104	C₄×Dic₅	C₄×D₃	1	1	yes	characteristic
105	C₄×Dic₅	#67	1	1	yes	characteristic
106	C₂×#168	—	2	1	no	—
107	C₈×D₅	—	4	1	no	—
108	C₈×D₅	—	4	1	no	—
109	C₂×C₄⋊C₈	—	5	1	no	—
110	C₂×C₄⋊C₈	—	15	1	no	—
111	#384	—	15	1	no	—
112	C₂²×C₃×C₅	#95	1	1	yes	characteristic
113	C₂×C₃×D₅	#100	1	1	yes	characteristic
114	C₂×C₃×D₅	#101	1	1	yes	characteristic
115	C₂×C₃×D₅	#102	1	1	yes	characteristic
116	C₂×C₃×D₅	#102	1	1	yes	characteristic
117	C₂×C₃×D₅	#103	1	1	yes	characteristic
118	C₂×C₃×D₅	#103	1	1	yes	characteristic
119	ΓL(1,𝔽₁₆)	—	8	2	no	—
120	C₃×C₄×C₅	—	2	1	no	—
121	C₃×C₄×C₅	#102	1	1	yes	characteristic
122	C₃×C₄×C₅	#103	1	1	yes	characteristic
123	C₃×C₄×C₅	#106	1	1	yes	characteristic
124	C₃×C₄×C₅	#107	1	1	yes	characteristic
125	C₃×Dic₅	—	2	1	no	—
126	C₃×Dic₅	#102	1	1	yes	characteristic
127	C₃×Dic₅	#103	1	1	yes	characteristic
128	C₃×Dic₅	#106	1	1	yes	characteristic
129	C₃×Dic₅	#107	1	1	yes	characteristic
130	C₂²×C₃×C₄	—	5	1	no	—
131	C₂²×C₃×C₄	—	5	1	no	—
132	C₂²×C₃×C₄	—	5	1	no	—
133	C₂²×Dic₃	—	10	2	no	—
134	C₂×C₃×C₈	—	2	1	no	—
135	C₂×C₃×C₈	—	10	1	no	—
136	C₂×C₃×C₈	—	20	1	no	—
137	C₂×C₃×C₈	—	20	1	no	—
138	C₃×C₄²	—	5	1	no	—
139	C₃×C₄²	—	5	1	no	—
140	C₃×C₄²	—	5	1	no	—
141	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
142	C₄⋊Dic₃	—	20	1	no	—
143	C₄⋊Dic₃	—	20	1	no	—
144	C₄⋊Dic₃	—	20	1	no	—
145	C₄⋊Dic₃	—	20	1	no	—
146	C₄⋊Dic₃	—	20	2	no	—
147	C₂×C₃⋊C₈	—	10	2	no	—
148	C₂×C₃⋊C₈	—	20	1	no	—
149	C₂×C₃⋊C₈	—	20	1	no	—
150	C₂²×D₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
151	C₂×AGL(1,𝔽₅)	—	12	2	no	—
152	C₂×AGL(1,𝔽₅)	—	12	2	no	—
153	C₂×AGL(1,𝔽₅)	—	12	2	no	—
154	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
155	C₂×C₄×C₅	C₄⋊Dic₃	1	1	yes	characteristic
156	C₂×C₄×C₅	#217	1	1	yes	characteristic
157	C₂×Dic₅	Dic₃⋊C₄	1	1	yes	characteristic
158	C₂×Dic₅	Dic₃⋊C₄	1	1	yes	characteristic
159	C₂×Dic₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
160	C₄×D₅	—	2	1	no	—
161	C₄×D₅	—	2	1	no	—
162	C₄×D₅	#204	1	1	yes	characteristic
163	C₄×D₅	#204	1	1	yes	characteristic
164	C₄×D₅	D₂₄	1	1	yes	characteristic
165	C₄×D₅	Dic₁₂	1	1	yes	characteristic
166	C₄×D₅	#213	1	1	yes	characteristic
167	C₄×D₅	#214	1	1	yes	characteristic
168	C₄×D₅	#215	1	1	yes	characteristic
169	C₄×D₅	#216	1	1	yes	characteristic
170	#170	—	12	2	no	—
171	C₅×C₈	—	4	1	no	—
172	#168	—	4	1	no	—
173	C₂²×C₈	—	10	1	no	—
174	C₂²×C₈	—	30	1	no	—
175	C₂×C₄⋊C₄	—	15	2	no	—
176	C₂×C₄⋊C₄	—	30	1	no	—
177	C₂×C₄⋊C₄	—	30	1	no	—
178	C₂×C₄²	—	5	1	no	—
179	C₄⋊C₈	—	5	1	no	—
180	C₄⋊C₈	—	10	1	no	—
181	C₄⋊C₈	—	15	2	no	—
182	C₄⋊C₈	—	30	1	no	—
183	C₄⋊C₈	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
184	C₂×C₃×C₅	#340	1	1	yes	characteristic
185	C₂×C₃×C₅	#340	1	1	yes	characteristic
186	C₂×C₃×C₅	#343	1	1	yes	characteristic
187	C₃×D₅	#331	1	1	yes	characteristic
188	C₃×D₅	#332	1	1	yes	characteristic
189	C₃×D₅	#332	1	1	yes	characteristic
190	C₃×D₅	#333	1	1	yes	characteristic
191	C₂³×C₃	—	5	1	no	—
192	C₂×C₃×C₄	—	5	1	no	—
193	C₂×C₃×C₄	—	5	1	no	—
194	C₂×C₃×C₄	—	5	1	no	—
195	C₂×C₃×C₄	—	5	1	no	—
196	C₂×C₃×C₄	—	5	1	no	—
197	C₂×C₃×C₄	—	5	1	no	—
198	C₂×C₃×C₄	—	5	1	no	—
199	C₂×C₃×C₄	—	5	1	no	—
200	C₂×C₃×C₄	—	5	1	no	—
201	C₂×C₃×C₄	—	5	1	no	—
202	C₂×C₃×C₄	—	5	1	no	—
203	C₂×C₃×C₄	—	10	1	no	—
204	C₂×C₃×C₄	—	10	1	no	—
205	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
206	C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
207	C₂×C₃×C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
208	C₂×Dic₃	—	20	2	no	—
209	C₂×Dic₃	—	20	2	no	—
210	C₂×Dic₃	—	20	2	no	—
211	C₃×C₈	—	4	1	no	—
212	C₃×C₈	—	20	1	no	—
213	C₃⋊C₈	—	20	2	no	—
214	C₂²×C₅	C₃⋊#95	1	1	yes	characteristic
215	C₂×D₅	#809	1	1	yes	characteristic
216	C₂×D₅	#810	1	1	yes	characteristic
217	C₂×D₅	#816	1	1	yes	characteristic
218	C₂×D₅	#821	1	1	yes	characteristic
219	C₂×D₅	#823	1	1	yes	characteristic
220	C₂×D₅	#824	1	1	yes	characteristic
221	AGL(1,𝔽₅)	—	24	2	no	—
222	C₄×C₅	—	2	1	no	—
223	C₄×C₅	#817	1	1	yes	characteristic
224	C₄×C₅	#818	1	1	yes	characteristic
225	C₄×C₅	#832	1	1	yes	characteristic
226	C₄×C₅	#835	1	1	yes	characteristic
227	Dic₅	—	2	1	no	—
228	Dic₅	#807	1	1	yes	characteristic
229	Dic₅	#808	1	1	yes	characteristic
230	Dic₅	#816	1	1	yes	characteristic
231	Dic₅	#821	1	1	yes	characteristic
232	C₂²×C₄	—	5	1	no	—
233	C₂²×C₄	—	5	1	no	—
234	C₂²×C₄	—	5	1	no	—
235	C₂²×C₄	—	30	2	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	10	1	no	—
238	C₂×C₈	—	20	1	no	—
239	C₂×C₈	—	20	1	no	—
240	C₂×C₈	—	30	2	no	—
241	C₂×C₈	—	60	1	no	—
242	C₂×C₈	—	60	1	no	—
243	C₄⋊C₄	—	60	1	no	—
244	C₄⋊C₄	—	60	1	no	—
245	C₄⋊C₄	—	60	1	no	—
246	C₄⋊C₄	—	60	1	no	—
247	C₄⋊C₄	—	60	2	no	—
248	C₄²	—	5	1	no	—
249	C₄²	—	5	1	no	—
250	C₄²	—	5	1	no	—
251	C₄²	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
252	C₃×C₅	#1296	1	1	yes	characteristic, complemented, Hall
253	C₂²×C₃	—	5	1	no	—
254	C₂²×C₃	—	5	1	no	—
255	C₂²×C₃	—	5	1	no	—
256	C₂²×C₃	—	5	1	no	—
257	C₂²×C₃	—	5	1	no	—
258	C₂²×C₃	—	5	1	no	—
259	C₂²×C₃	(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic
260	C₃×C₄	—	2	1	no	—
261	C₃×C₄	—	5	1	no	—
262	C₃×C₄	—	5	1	no	—
263	C₃×C₄	—	5	1	no	—
264	C₃×C₄	—	5	1	no	—
265	C₃×C₄	—	10	1	no	—
266	C₃×C₄	#3993	1	1	yes	characteristic
267	C₃×C₄	#3994	1	1	yes	characteristic
268	C₃×C₄	#4007	1	1	yes	characteristic
269	C₃×C₄	#4009	1	1	yes	characteristic
270	Dic₃	—	40	2	no	—
271	C₂×C₅	#4529	1	1	yes	characteristic
272	C₂×C₅	#4551	1	1	yes	characteristic
273	C₂×C₅	#4554	1	1	yes	characteristic
274	D₅	#4481	1	1	yes	characteristic
275	D₅	#4482	1	1	yes	characteristic
276	D₅	#4485	1	1	yes	characteristic
277	D₅	#4486	1	1	yes	characteristic
278	C₂³	—	5	1	no	—
279	C₂×C₄	—	5	1	no	—
280	C₂×C₄	—	5	1	no	—
281	C₂×C₄	—	5	1	no	—
282	C₂×C₄	—	5	1	no	—
283	C₂×C₄	—	5	1	no	—
284	C₂×C₄	—	5	1	no	—
285	C₂×C₄	—	5	1	no	—
286	C₂×C₄	—	5	1	no	—
287	C₂×C₄	—	5	1	no	—
288	C₂×C₄	—	5	1	no	—
289	C₂×C₄	—	5	1	no	—
290	C₂×C₄	—	10	1	no	—
291	C₂×C₄	—	10	1	no	—
292	C₂×C₄	—	60	2	no	—
293	C₂×C₄	—	60	2	no	—
294	C₂×C₄	—	60	2	no	—
295	C₂×C₄	C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
296	C₂×C₄	C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
297	C₂×C₄	#6763	1	1	yes	characteristic
298	C₈	—	4	1	no	—
299	C₈	—	20	1	no	—
300	C₈	—	60	2	no	—
301	C₂×C₃	—	5	1	no	—
302	C₂×C₃	—	5	1	no	—
303	C₂×C₃	—	5	1	no	—
304	C₂×C₃	—	5	1	no	—
305	C₂×C₃	#64874	1	1	yes	characteristic
306	C₂×C₃	#64893	1	1	yes	characteristic
307	C₂×C₃	#64899	1	1	yes	characteristic
308	C₅	#67702	1	1	yes	characteristic, complemented, Sylow
309	C₂²	—	5	1	no	—
310	C₂²	—	5	1	no	—
311	C₂²	—	5	1	no	—
312	C₂²	—	5	1	no	—
313	C₂²	—	5	1	no	—
314	C₂²	—	5	1	no	—
315	C₂²	#91235	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	5	1	no	—
318	C₄	—	5	1	no	—
319	C₄	—	5	1	no	—
320	C₄	—	5	1	no	—
321	C₄	—	10	1	no	—
322	C₄	—	120	2	no	—
323	C₄	#91222	1	1	yes	characteristic
324	C₄	#91223	1	1	yes	characteristic
325	C₄	#91236	1	1	yes	characteristic
326	C₄	#91238	1	1	yes	characteristic
327	C₃	#10598586	1	1	yes	characteristic, complemented, Sylow
328	C₂	—	5	1	no	—
329	C₂	—	5	1	no	—
330	C₂	—	5	1	no	—
331	C₂	—	5	1	no	—
332	C₂	#11746745	1	1	yes	characteristic
333	C₂	#11746764	1	1	yes	characteristic
334	C₂	#11746770	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#91235
commutator	C₂×C₃×C₄×C₅	C₄²
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	#1296
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
11. #91233	C₄	semidirect
13. C₃×C₅×C₄⋊C₈	C₄	semidirect
15. C₃×C₄⋊#168	C₄	semidirect
51. C₅×C₄⋊C₈	Dic₃	semidirect
53. C₄⋊#168	Dic₃	semidirect
86. C₃×C₄⋊C₈	AGL(1,𝔽₅)	semidirect
183. C₄⋊C₈	ΓL(1,𝔽₁₆)	semidirect
252. C₃×C₅	#1296	semidirect
308. C₅	#67702	semidirect
327. C₃	#10598586	semidirect

Maximal subgroups

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

subgroup	index
#11752347	2
D₅⋊(C₄⋊(C₃⋊C₈))	2
C₃×D₅×C₄⋊C₈	2
#10598586	3
#67702	5