Subgroups — #14296148

Order 1920.

584 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	C₂×#91221	C₂	1	2	yes	maximal
2	C₄²×ΓL(1,𝔽₁₆)	C₂	1	1	yes	maximal, characteristic
3	D₅⋊(C₄⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic
4	C₃×D₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#11746763	C₂	1	2	yes	maximal
6	#10604028	—	3	1	no	maximal, Hall
7	C₂×C₄×ΓL(1,𝔽₁₆)	C₄	1	2	yes	—
8	C₂×C₄×ΓL(1,𝔽₁₆)	C₄	1	2	yes	—
9	C₂×C₄×ΓL(1,𝔽₁₆)	C₂²	1	1	yes	characteristic
10	C₂×C₄×ΓL(1,𝔽₁₆)	C₂²	1	1	yes	characteristic
11	C₂×D₅⋊(C₃⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₃×C₈×D₅	C₂²	1	2	yes	—
13	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
14	#91233	C₄	1	2	yes	complemented
15	C₄⋊(C₃⋊#170)	C₄	1	2	yes	complemented
16	#91221	—	2	4	no	—
17	C₃×C₅×C₄⋊C₈	C₄	1	1	yes	characteristic, complemented
18	C₃×#3946	C₄	1	2	yes	complemented
19	C₃×C₄⋊#168	C₄	1	1	yes	characteristic, complemented
20	#70441	—	5	1	no	maximal, Hall
21	C₂×#3992	—	3	2	no	—
22	C₄²×AGL(1,𝔽₅)	—	3	1	no	—
23	D₅⋊(C₄⋊C₈)	—	3	1	no	—
24	D₅×C₄⋊C₈	D₃	1	1	yes	characteristic
25	(C₂×AGL(1,𝔽₅))⋊C₈	—	3	2	no	—
26	C₂²×ΓL(1,𝔽₁₆)	C₂×C₄	1	2	yes	—
27	C₂²×ΓL(1,𝔽₁₆)	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
31	C₂×C₃⋊#170	C₂×C₄	1	4	yes	—
32	C₄×ΓL(1,𝔽₁₆)	—	2	4	no	—
33	C₄×ΓL(1,𝔽₁₆)	—	2	4	no	—
34	C₄×ΓL(1,𝔽₁₆)	—	2	1	no	—
35	C₄×ΓL(1,𝔽₁₆)	—	2	1	no	—
36	C₄×ΓL(1,𝔽₁₆)	D₄	1	2	yes	—
37	C₄×ΓL(1,𝔽₁₆)	Q₈	1	2	yes	—
38	D₅⋊(C₃⋊C₈)	—	2	2	no	—
39	D₅⋊(C₃⋊C₈)	—	2	2	no	—
40	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
41	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
42	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
43	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
44	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
45	C₂×C₃×#168	C₂×C₄	1	2	yes	—
46	C₃×C₈×D₅	—	2	2	no	—
47	C₃×C₈×D₅	—	2	2	no	—
48	C₂×C₃×C₄⋊C₈	—	5	1	no	—
49	C₂×#815	—	5	2	no	—
50	C₄²×Dic₃	—	5	1	no	—
51	C₂×C₄⋊(C₃⋊C₈)	—	5	1	no	—
52	#4550	—	5	2	no	—
53	C₂×C₄×AGL(1,𝔽₅)	—	3	1	no	—
54	C₂×C₄×AGL(1,𝔽₅)	—	3	2	no	—
55	C₂×C₄×AGL(1,𝔽₅)	—	3	2	no	—
56	C₂×C₄×AGL(1,𝔽₅)	—	3	1	no	—
57	C₂×D₅⋊C₈	—	3	2	no	—
58	C₂×C₈×D₅	C₂×D₃	1	2	yes	—
59	C₄²×D₅	C₂×D₃	1	1	yes	characteristic
60	#4004	—	3	2	no	—
61	C₄⋊#170	—	3	2	no	—
62	#3992	—	6	4	no	—
63	C₅×C₄⋊C₈	Dic₃	1	1	yes	characteristic, complemented
64	#3946	Dic₃	1	2	yes	complemented
65	C₄⋊#168	Dic₃	1	1	yes	characteristic, complemented
66	#1767	—	15	1	no	Sylow
67	C₂²×C₃×D₅	C₂²×C₄	1	1	yes	characteristic
68	C₂×ΓL(1,𝔽₁₆)	—	2	2	no	—
69	C₂×ΓL(1,𝔽₁₆)	—	2	2	no	—
70	C₂×ΓL(1,𝔽₁₆)	—	2	2	no	—
71	C₂×ΓL(1,𝔽₁₆)	—	2	2	no	—
72	C₂×ΓL(1,𝔽₁₆)	C₄⋊C₄	1	4	yes	—
73	C₂×ΓL(1,𝔽₁₆)	M₁₆	1	4	yes	—
74	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₃×Dic₅	C₄²	1	1	yes	characteristic
78	C₂×C₃×Dic₅	C₄²	1	1	yes	characteristic
79	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
80	C₃×C₄×D₅	—	2	2	no	—
81	C₃×C₄×D₅	M₁₆	1	4	yes	—
82	C₃×C₄×D₅	C₂×D₄	1	1	yes	characteristic
83	C₃×C₄×D₅	C₂×Q₈	1	1	yes	characteristic
84	C₃×C₄×D₅	#41	1	1	yes	characteristic
85	C₃×C₄×D₅	#41	1	1	yes	characteristic
86	C₃⋊#170	—	2	4	no	—
87	C₃×C₅×C₈	—	2	2	no	—
88	C₃×#168	—	2	2	no	—
89	C₂²×C₃×C₈	—	5	2	no	—
90	C₂×C₃×C₄²	—	5	1	no	—
91	C₂×C₄×Dic₃	—	5	1	no	—
92	C₂×C₄×Dic₃	—	5	2	no	—
93	C₂×C₄×Dic₃	—	5	2	no	—
94	C₂×C₄×Dic₃	—	5	1	no	—
95	C₂²×C₃⋊C₈	—	5	2	no	—
96	C₃×C₄⋊C₈	—	5	2	no	—
97	C₃×C₄⋊C₈	—	5	1	no	—
98	C₃×C₄⋊C₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
99	#815	—	10	4	no	—
100	C₄⋊(C₃⋊C₈)	—	5	2	no	—
101	C₄⋊(C₃⋊C₈)	—	5	2	no	—
102	C₂²×AGL(1,𝔽₅)	—	3	2	no	—
103	C₂²×AGL(1,𝔽₅)	—	3	2	no	—
104	C₂×C₄×D₅	C₄×D₃	1	1	yes	characteristic
105	C₂×C₄×D₅	C₄×D₃	1	1	yes	characteristic
106	C₂×C₄×D₅	C₂²×D₃	1	1	yes	characteristic
107	C₂×#170	—	3	4	no	—
108	C₄×AGL(1,𝔽₅)	—	3	2	no	—
109	C₄×AGL(1,𝔽₅)	—	3	2	no	—
110	C₄×AGL(1,𝔽₅)	—	6	4	no	—
111	C₄×AGL(1,𝔽₅)	—	6	4	no	—
112	C₄×AGL(1,𝔽₅)	—	6	1	no	—
113	C₄×AGL(1,𝔽₅)	—	6	1	no	—
114	D₅⋊C₈	—	6	2	no	—
115	D₅⋊C₈	—	6	2	no	—
116	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
117	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
118	C₄×Dic₅	C₄×D₃	1	1	yes	characteristic
119	C₄×Dic₅	C₄×D₃	1	1	yes	characteristic
120	C₄×Dic₅	C₂×Dic₃	1	1	yes	characteristic
121	C₂×#168	C₂×Dic₃	1	2	yes	—
122	C₈×D₅	—	2	2	no	—
123	C₈×D₅	—	2	2	no	—
124	C₂×C₄⋊C₈	—	5	1	no	—
125	C₂×C₄⋊C₈	—	15	1	no	—
126	C₂×#97	—	15	2	no	—
127	C₄³	—	15	1	no	—
128	#336	—	15	2	no	—
129	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
130	C₂×C₃×D₅	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₃×D₅	#117	1	1	yes	characteristic
132	C₂×C₃×D₅	C₂×M₁₆	1	2	yes	—
133	C₂×C₃×D₅	C₂×M₁₆	1	2	yes	—
134	ΓL(1,𝔽₁₆)	—	2	4	no	—
135	ΓL(1,𝔽₁₆)	—	2	4	no	—
136	C₃×C₄×C₅	—	2	1	no	—
137	C₃×C₄×C₅	#97	1	2	yes	complemented
138	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
139	C₃×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
140	C₃×Dic₅	—	2	1	no	—
141	C₃×Dic₅	#97	1	2	yes	complemented
142	C₃×Dic₅	C₄×D₄	1	1	yes	characteristic
143	C₃×Dic₅	C₄×Q₈	1	1	yes	characteristic
144	C₂²×C₃×C₄	—	5	1	no	—
145	C₂²×C₃×C₄	—	5	1	no	—
146	C₂²×C₃×C₄	—	5	1	no	—
147	C₂²×Dic₃	—	5	2	no	—
148	C₂²×Dic₃	—	5	2	no	—
149	C₂×C₃×C₈	—	5	2	no	—
150	C₂×C₃×C₈	—	10	2	no	—
151	C₂×C₃×C₈	—	10	2	no	—
152	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
153	C₃×C₄²	—	5	1	no	—
154	C₃×C₄²	—	5	1	no	—
155	C₃×C₄²	—	5	1	no	—
156	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
157	C₄×Dic₃	—	5	2	no	—
158	C₄×Dic₃	—	5	2	no	—
159	C₄×Dic₃	—	10	4	no	—
160	C₄×Dic₃	—	10	1	no	—
161	C₄×Dic₃	—	10	1	no	—
162	C₄×Dic₃	—	10	4	no	—
163	C₂×C₃⋊C₈	—	5	4	no	—
164	C₂×C₃⋊C₈	—	10	2	no	—
165	C₂×C₃⋊C₈	—	10	2	no	—
166	C₂²×D₅	C₂×C₄×D₃	1	1	yes	characteristic
167	C₂×AGL(1,𝔽₅)	—	3	4	no	—
168	C₂×AGL(1,𝔽₅)	—	3	4	no	—
169	C₂×AGL(1,𝔽₅)	—	6	2	no	—
170	C₂×AGL(1,𝔽₅)	—	6	2	no	—
171	C₂×AGL(1,𝔽₅)	—	6	2	no	—
172	C₂×AGL(1,𝔽₅)	—	6	2	no	—
173	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
174	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
175	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
176	C₂×Dic₅	C₄×Dic₃	1	1	yes	characteristic
177	C₂×Dic₅	C₄×Dic₃	1	1	yes	characteristic
178	C₂×Dic₅	C₂×C₄×D₃	1	1	yes	characteristic
179	C₄×D₅	—	2	2	no	—
180	C₄×D₅	ΓL(1,𝔽₂₅)	1	4	yes	—
181	C₄×D₅	D₃×D₄	1	1	yes	characteristic
182	C₄×D₅	#237	1	1	yes	characteristic
183	C₄×D₅	D₃×Q₈	1	1	yes	characteristic
184	C₄×D₅	#239	1	1	yes	characteristic
185	#170	—	6	4	no	—
186	C₅×C₈	—	2	2	no	—
187	#168	—	2	2	no	—
188	C₂²×C₈	—	5	2	no	—
189	C₂²×C₈	—	15	2	no	—
190	C₂×C₄²	—	5	1	no	—
191	C₂×C₄²	—	15	1	no	—
192	C₂×C₄²	—	15	1	no	—
193	C₂×C₄²	—	15	2	no	—
194	C₂×C₄²	—	15	2	no	—
195	C₄⋊C₈	—	5	2	no	—
196	C₄⋊C₈	—	5	1	no	—
197	C₄⋊C₈	—	15	2	no	—
198	C₄⋊C₈	—	15	2	no	—
199	C₄⋊C₈	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
200	#97	—	30	4	no	—
201	C₂×C₃×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
202	C₂×C₃×C₅	C₂×#97	1	1	yes	characteristic
203	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
204	C₃×D₅	#423	1	2	yes	—
205	C₃×D₅	#432	1	2	yes	—
206	C₂³×C₃	—	5	1	no	—
207	C₂×C₃×C₄	—	5	1	no	—
208	C₂×C₃×C₄	—	5	1	no	—
209	C₂×C₃×C₄	—	5	1	no	—
210	C₂×C₃×C₄	—	5	1	no	—
211	C₂×C₃×C₄	—	5	1	no	—
212	C₂×C₃×C₄	—	5	1	no	—
213	C₂×C₃×C₄	—	5	4	no	—
214	C₂×C₃×C₄	—	5	1	no	—
215	C₂×C₃×C₄	—	10	2	no	—
216	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
217	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
218	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
219	C₂×Dic₃	—	5	4	no	—
220	C₂×Dic₃	—	5	4	no	—
221	C₂×Dic₃	—	10	2	no	—
222	C₂×Dic₃	—	10	2	no	—
223	C₂×Dic₃	—	10	2	no	—
224	C₂×Dic₃	—	10	2	no	—
225	C₃×C₈	—	2	2	no	—
226	C₃×C₈	—	10	2	no	—
227	C₃⋊C₈	—	10	4	no	—
228	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
229	C₂×D₅	D₃×C₄⋊C₄	1	1	yes	characteristic
230	C₂×D₅	#892	1	1	yes	characteristic
231	C₂×D₅	C₂×ΓL(1,𝔽₂₅)	1	2	yes	—
232	C₂×D₅	D₃×M₁₆	1	2	yes	—
233	AGL(1,𝔽₅)	—	6	4	no	—
234	AGL(1,𝔽₅)	—	6	4	no	—
235	C₄×C₅	—	2	1	no	—
236	C₄×C₅	#815	1	2	yes	complemented
237	C₄×C₅	D₄×Dic₃	1	1	yes	characteristic
238	C₄×C₅	Q₈×Dic₃	1	1	yes	characteristic
239	Dic₅	—	2	1	no	—
240	Dic₅	#815	1	2	yes	complemented
241	Dic₅	#887	1	1	yes	characteristic
242	Dic₅	#893	1	1	yes	characteristic
243	C₂²×C₄	—	5	1	no	—
244	C₂²×C₄	—	5	1	no	—
245	C₂²×C₄	—	5	1	no	—
246	C₂²×C₄	—	15	2	no	—
247	C₂²×C₄	—	15	2	no	—
248	C₂×C₈	—	5	2	no	—
249	C₂×C₈	—	10	2	no	—
250	C₂×C₈	—	10	2	no	—
251	C₂×C₈	—	15	4	no	—
252	C₂×C₈	—	30	2	no	—
253	C₂×C₈	—	30	2	no	—
254	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	2	yes	—
255	C₄²	—	5	1	no	—
256	C₄²	—	5	1	no	—
257	C₄²	—	5	1	no	—
258	C₄²	—	15	2	no	—
259	C₄²	—	15	2	no	—
260	C₄²	—	30	4	no	—
261	C₄²	—	30	1	no	—
262	C₄²	—	30	1	no	—
263	C₄²	—	30	4	no	—
264	C₄²	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
265	C₃×C₅	#1767	1	1	yes	characteristic, complemented, Hall
266	C₂²×C₃	—	5	1	no	—
267	C₂²×C₃	—	5	1	no	—
268	C₂²×C₃	—	5	2	no	—
269	C₂²×C₃	—	5	2	no	—
270	C₂²×C₃	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
271	C₃×C₄	—	2	1	no	—
272	C₃×C₄	—	5	1	no	—
273	C₃×C₄	—	5	1	no	—
274	C₃×C₄	—	5	2	no	—
275	C₃×C₄	—	10	1	no	—
276	C₃×C₄	#3992	1	2	yes	complemented
277	C₃×C₄	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
278	C₃×C₄	Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
279	Dic₃	—	10	4	no	—
280	Dic₃	—	10	4	no	—
281	C₂×C₅	Dic₃×C₄⋊C₄	1	1	yes	characteristic
282	C₂×C₅	C₂×#815	1	1	yes	characteristic
283	C₂×C₅	Dic₃×M₁₆	1	1	yes	characteristic
284	D₅	#4835	1	2	yes	—
285	D₅	#4837	1	2	yes	—
286	C₂³	—	5	1	no	—
287	C₂×C₄	—	5	1	no	—
288	C₂×C₄	—	5	1	no	—
289	C₂×C₄	—	5	4	no	—
290	C₂×C₄	—	5	1	no	—
291	C₂×C₄	—	5	1	no	—
292	C₂×C₄	—	5	1	no	—
293	C₂×C₄	—	5	1	no	—
294	C₂×C₄	—	5	1	no	—
295	C₂×C₄	—	10	2	no	—
296	C₂×C₄	—	15	4	no	—
297	C₂×C₄	—	15	4	no	—
298	C₂×C₄	—	30	2	no	—
299	C₂×C₄	—	30	2	no	—
300	C₂×C₄	—	30	2	no	—
301	C₂×C₄	—	30	2	no	—
302	C₂×C₄	C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
303	C₂×C₄	C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
304	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
305	C₈	—	2	2	no	—
306	C₈	—	10	2	no	—
307	C₈	—	30	4	no	—
308	C₂×C₃	—	5	2	no	—
309	C₂×C₃	—	5	2	no	—
310	C₂×C₃	C₄⋊C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
311	C₂×C₃	C₂×#3992	1	1	yes	characteristic
312	C₂×C₃	M₁₆×AGL(1,𝔽₅)	1	1	yes	characteristic
313	C₅	#70441	1	1	yes	characteristic, complemented, Sylow
314	C₂²	—	5	1	no	—
315	C₂²	—	5	1	no	—
316	C₂²	—	5	2	no	—
317	C₂²	—	5	2	no	—
318	C₂²	C₂×C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
319	C₄	—	2	1	no	—
320	C₄	—	5	1	no	—
321	C₄	—	5	1	no	—
322	C₄	—	5	2	no	—
323	C₄	—	10	1	no	—
324	C₄	—	30	4	no	—
325	C₄	—	30	4	no	—
326	C₄	#91221	1	2	yes	complemented
327	C₄	D₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
328	C₄	Q₈×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
329	C₃	#10604028	1	1	yes	characteristic, complemented, Sylow
330	C₂	—	5	2	no	—
331	C₂	—	5	2	no	—
332	C₂	C₄⋊C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
333	C₂	C₂×#91221	1	1	yes	characteristic
334	C₂	M₁₆×ΓL(1,𝔽₁₆)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)
commutator, socle	C₂²×C₃×C₅	C₂×C₄²
Fitting	C₃×C₅×C₄⋊C₈	C₄
prime radical	C₂²×C₃×D₅	C₂²×C₄
cosocle	C₂×C₃×C₄×D₅	C₂³
hypercenter	C₄⋊C₈	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	#1767
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
14. #91233	C₄	semidirect
15. C₄⋊(C₃⋊#170)	C₄	semidirect
17. C₃×C₅×C₄⋊C₈	C₄	semidirect
18. C₃×#3946	C₄	semidirect
19. C₃×C₄⋊#168	C₄	semidirect
63. C₅×C₄⋊C₈	Dic₃	semidirect
64. #3946	Dic₃	semidirect
65. C₄⋊#168	Dic₃	semidirect
98. C₃×C₄⋊C₈	AGL(1,𝔽₅)	semidirect
137. C₃×C₄×C₅	#97	semidirect
141. C₃×Dic₅	#97	semidirect
199. C₄⋊C₈	ΓL(1,𝔽₁₆)	semidirect
236. C₄×C₅	#815	semidirect
240. Dic₅	#815	semidirect
265. C₃×C₅	#1767	semidirect
276. C₃×C₄	#3992	semidirect
313. C₅	#70441	semidirect
326. C₄	#91221	semidirect
329. C₃	#10604028	semidirect

Maximal subgroups

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

subgroup	index
C₂×#91221	2
C₄²×ΓL(1,𝔽₁₆)	2
D₅⋊(C₄⋊(C₃⋊C₈))	2
C₃×D₅×C₄⋊C₈	2
#11746763	2
#10604028	3
#70441	5