Subgroups — #14296668

Order 1920.

544 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#91223	C₂	1	1	yes	maximal, characteristic
2	C₂×#91222	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊ΓL(1,𝔽₁₆)	C₂	1	2	yes	maximal
4	C₃×D₅×#97	C₂	1	1	yes	maximal, characteristic
5	#11746764	C₂	1	2	yes	maximal
6	#10603881	—	3	1	no	maximal, Hall
7	C₂×C₄⋊ΓL(1,𝔽₁₆)	C₂²	1	2	yes	—
8	C₂×C₄⋊ΓL(1,𝔽₁₆)	C₂²	1	2	yes	—
9	C₂×C₄×ΓL(1,𝔽₁₆)	—	2	2	no	—
10	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
12	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
13	#91235	—	2	2	no	—
14	#91223	C₄	1	4	yes	complemented
15	#91222	C₄	1	4	yes	complemented
16	C₃×C₅×#97	C₄	1	1	yes	characteristic, complemented
17	C₃×#3947	C₄	1	1	yes	characteristic, complemented
18	C₃×#3947	C₄	1	1	yes	characteristic, complemented
19	C₃×#3935	C₄	1	1	yes	characteristic, complemented
20	#69814	—	5	1	no	maximal, Hall
21	C₂×#3994	—	3	1	no	—
22	C₂×#3993	—	3	1	no	—
23	C₄×C₄⋊AGL(1,𝔽₅)	—	3	2	no	—
24	D₅×#97	D₃	1	1	yes	characteristic
25	#64893	—	3	2	no	—
26	C₂²×ΓL(1,𝔽₁₆)	—	2	2	no	—
27	C₂²×ΓL(1,𝔽₁₆)	—	2	2	no	—
28	C₂×C₃×C₄×D₅	D₄	1	2	yes	—
29	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
30	C₄⋊ΓL(1,𝔽₁₆)	—	2	2	no	—
31	C₄⋊ΓL(1,𝔽₁₆)	—	2	2	no	—
32	C₄⋊ΓL(1,𝔽₁₆)	C₂×C₄	1	4	yes	—
33	C₄⋊ΓL(1,𝔽₁₆)	C₂×C₄	1	4	yes	—
34	C₄×ΓL(1,𝔽₁₆)	—	4	4	no	—
35	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
37	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
38	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
39	C₃×C₄×Dic₅	Q₈	1	2	yes	—
40	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
42	C₃×C₈×D₅	—	2	1	no	—
43	C₃×C₈×D₅	—	2	1	no	—
44	C₃×C₈×D₅	C₂×C₄	1	2	yes	—
45	C₃×C₈×D₅	C₂×C₄	1	2	yes	—
46	C₂×C₃×#97	—	5	1	no	—
47	C₂×#818	—	5	1	no	—
48	C₂×#817	—	5	1	no	—
49	C₄×C₄⋊Dic₃	—	5	2	no	—
50	#4551	—	5	2	no	—
51	C₂×C₄⋊AGL(1,𝔽₅)	—	3	2	no	—
52	C₂×C₄⋊AGL(1,𝔽₅)	—	3	2	no	—
53	C₂×C₄×AGL(1,𝔽₅)	—	6	2	no	—
54	C₂×C₈×D₅	C₂×D₃	1	1	yes	characteristic
55	C₂×C₈×D₅	C₂×D₃	1	1	yes	characteristic
56	C₄²×D₅	C₂×D₃	1	1	yes	characteristic
57	(C₂×AGL(1,𝔽₅))⋊C₄	—	6	2	no	—
58	#3994	—	3	4	no	—
59	#3993	—	3	4	no	—
60	C₅×#97	Dic₃	1	1	yes	characteristic, complemented
61	#3947	Dic₃	1	1	yes	characteristic, complemented
62	#3947	Dic₃	1	1	yes	characteristic, complemented
63	#3935	Dic₃	1	1	yes	characteristic, complemented
64	#1776	—	15	1	no	Sylow
65	C₂²×C₃×D₅	C₂×D₄	1	1	yes	characteristic
66	C₂×ΓL(1,𝔽₁₆)	—	2	4	no	—
67	C₂×ΓL(1,𝔽₁₆)	—	2	4	no	—
68	C₂×ΓL(1,𝔽₁₆)	—	4	2	no	—
69	C₂×ΓL(1,𝔽₁₆)	—	4	2	no	—
70	C₂×ΓL(1,𝔽₁₆)	—	4	2	no	—
71	C₂×ΓL(1,𝔽₁₆)	—	4	2	no	—
72	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
73	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₃×Dic₅	C₄⋊C₄	1	2	yes	—
75	C₂×C₃×Dic₅	C₂×Q₈	1	1	yes	characteristic
76	C₃×C₄×D₅	—	2	2	no	—
77	C₃×C₄×D₅	—	2	2	no	—
78	C₃×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
79	C₃×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
80	C₃×C₄×D₅	#41	1	1	yes	characteristic
81	C₃×C₄×D₅	#41	1	1	yes	characteristic
82	C₃×C₅×C₈	—	2	1	no	—
83	C₃×C₅×C₈	C₄²	1	2	yes	complemented
84	C₃×#168	—	2	1	no	—
85	C₃×#168	C₄²	1	2	yes	complemented
86	C₂²×C₃×C₈	—	5	1	no	—
87	C₂²×C₃×C₈	—	5	1	no	—
88	C₂×C₃×C₄²	—	5	1	no	—
89	C₂×C₄⋊Dic₃	—	5	2	no	—
90	C₂×C₄⋊Dic₃	—	5	2	no	—
91	C₂×C₄×Dic₃	—	10	2	no	—
92	C₃×#97	—	5	1	no	—
93	C₃×#97	—	5	1	no	—
94	C₃×#97	—	5	1	no	—
95	C₃×#97	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
96	C₃⋊#95	—	10	2	no	—
97	#818	—	5	4	no	—
98	#817	—	5	4	no	—
99	C₂²×AGL(1,𝔽₅)	—	6	2	no	—
100	C₂²×AGL(1,𝔽₅)	—	6	2	no	—
101	C₂×C₄×D₅	D₁₂	1	2	yes	—
102	C₂×C₄×D₅	C₂²×D₃	1	1	yes	characteristic
103	C₄⋊AGL(1,𝔽₅)	—	3	4	no	—
104	C₄⋊AGL(1,𝔽₅)	—	3	4	no	—
105	C₄⋊AGL(1,𝔽₅)	—	6	2	no	—
106	C₄⋊AGL(1,𝔽₅)	—	6	2	no	—
107	C₄×AGL(1,𝔽₅)	—	12	4	no	—
108	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
109	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
110	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
111	C₄×Dic₅	Dic₆	1	2	yes	—
112	C₄×Dic₅	C₂×Dic₃	1	1	yes	characteristic
113	C₂×#168	C₂×Dic₃	1	1	yes	characteristic
114	C₂×#168	C₂×Dic₃	1	1	yes	characteristic
115	C₈×D₅	—	2	1	no	—
116	C₈×D₅	—	2	1	no	—
117	C₈×D₅	C₄×D₃	1	2	yes	—
118	C₈×D₅	C₄×D₃	1	2	yes	—
119	C₂×#107	—	15	1	no	—
120	C₂×#106	—	15	1	no	—
121	C₂×#97	—	5	1	no	—
122	C₄×C₄⋊C₄	—	15	2	no	—
123	#340	—	15	2	no	—
124	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₃×D₅	C₄×D₄	1	1	yes	characteristic
126	C₂×C₃×D₅	C₄×D₄	1	1	yes	characteristic
127	C₂×C₃×D₅	AGL(1,ℤ/8)	1	2	yes	—
128	C₂×C₃×D₅	#137	1	2	yes	—
129	ΓL(1,𝔽₁₆)	—	4	4	no	—
130	ΓL(1,𝔽₁₆)	—	4	4	no	—
131	C₃×C₄×C₅	—	2	2	no	—
132	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
133	C₃×C₄×C₅	#117	1	1	yes	characteristic
134	C₃×Dic₅	—	2	2	no	—
135	C₃×Dic₅	C₄×Q₈	1	1	yes	characteristic
136	C₃×Dic₅	C₄×Q₈	1	1	yes	characteristic
137	C₂²×C₃×C₄	—	5	1	no	—
138	C₂²×C₃×C₄	—	5	2	no	—
139	C₂²×Dic₃	—	10	2	no	—
140	C₂²×Dic₃	—	10	2	no	—
141	C₂×C₃×C₈	—	5	1	no	—
142	C₂×C₃×C₈	—	5	2	no	—
143	C₂×C₃×C₈	—	5	2	no	—
144	C₂×C₃×C₈	—	5	1	no	—
145	C₂×C₃×C₈	—	10	1	no	—
146	C₂×C₃×C₈	—	10	1	no	—
147	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
148	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
149	C₃×C₄²	—	5	2	no	—
150	C₃×C₄²	—	5	1	no	—
151	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
152	C₄⋊Dic₃	—	5	4	no	—
153	C₄⋊Dic₃	—	5	4	no	—
154	C₄⋊Dic₃	—	10	2	no	—
155	C₄⋊Dic₃	—	10	2	no	—
156	C₄×Dic₃	—	20	4	no	—
157	C₂²×D₅	C₂×D₁₂	1	1	yes	characteristic
158	C₂×AGL(1,𝔽₅)	—	6	4	no	—
159	C₂×AGL(1,𝔽₅)	—	6	4	no	—
160	C₂×AGL(1,𝔽₅)	—	12	2	no	—
161	C₂×AGL(1,𝔽₅)	—	12	2	no	—
162	C₂×AGL(1,𝔽₅)	—	12	2	no	—
163	C₂×AGL(1,𝔽₅)	—	12	2	no	—
164	C₂×C₄×C₅	C₄⋊Dic₃	1	2	yes	—
165	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
166	C₂×Dic₅	C₄⋊Dic₃	1	2	yes	—
167	C₂×Dic₅	C₂×Dic₆	1	1	yes	characteristic
168	C₄×D₅	—	2	2	no	—
169	C₄×D₅	—	2	2	no	—
170	C₄×D₅	C₂×C₄×D₃	1	1	yes	characteristic
171	C₄×D₅	C₂×C₄×D₃	1	1	yes	characteristic
172	C₄×D₅	#235	1	1	yes	characteristic
173	C₄×D₅	#235	1	1	yes	characteristic
174	C₅×C₈	—	2	1	no	—
175	C₅×C₈	C₄×Dic₃	1	2	yes	complemented
176	#168	—	2	1	no	—
177	#168	C₄×Dic₃	1	2	yes	complemented
178	C₂²×C₈	—	5	1	no	—
179	C₂²×C₈	—	5	1	no	—
180	C₂×C₄⋊C₄	—	15	2	no	—
181	C₂×C₄⋊C₄	—	15	2	no	—
182	C₂×C₄²	—	5	1	no	—
183	C₂×C₄²	—	30	2	no	—
184	#107	—	15	4	no	—
185	#106	—	15	4	no	—
186	#97	—	5	1	no	—
187	#97	—	5	1	no	—
188	#97	—	5	1	no	—
189	#97	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
190	#95	—	30	2	no	—
191	C₂×C₃×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
192	C₂×C₃×C₅	#428	1	2	yes	—
193	C₃×D₅	#440	1	2	yes	—
194	C₃×D₅	#441	1	1	yes	characteristic
195	C₃×D₅	#442	1	1	yes	characteristic
196	C₂³×C₃	—	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	2	no	—
203	C₂×C₃×C₄	—	10	2	no	—
204	C₂×C₃×C₄	—	10	2	no	—
205	C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
206	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
207	C₂×Dic₃	—	10	4	no	—
208	C₂×Dic₃	—	10	4	no	—
209	C₂×Dic₃	—	20	2	no	—
210	C₂×Dic₃	—	20	2	no	—
211	C₂×Dic₃	—	20	2	no	—
212	C₂×Dic₃	—	20	2	no	—
213	C₃×C₈	—	2	1	no	—
214	C₃×C₈	—	5	2	no	—
215	C₃×C₈	—	10	1	no	—
216	C₃×C₈	C₄×AGL(1,𝔽₅)	1	2	yes	complemented
217	C₂²×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
218	C₂×D₅	C₄×D₁₂	1	1	yes	characteristic
219	C₂×D₅	C₄×D₁₂	1	1	yes	characteristic
220	C₂×D₅	#908	1	2	yes	—
221	C₂×D₅	#909	1	2	yes	—
222	AGL(1,𝔽₅)	—	12	4	no	—
223	AGL(1,𝔽₅)	—	12	4	no	—
224	C₄×C₅	—	2	2	no	—
225	C₄×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
226	C₄×C₅	#926	1	1	yes	characteristic
227	Dic₅	—	2	2	no	—
228	Dic₅	C₄×Dic₆	1	1	yes	characteristic
229	Dic₅	C₄×Dic₆	1	1	yes	characteristic
230	C₂²×C₄	—	5	1	no	—
231	C₂²×C₄	—	5	2	no	—
232	C₂²×C₄	—	30	2	no	—
233	C₂²×C₄	—	30	2	no	—
234	C₂×C₈	—	5	1	no	—
235	C₂×C₈	—	5	2	no	—
236	C₂×C₈	—	5	2	no	—
237	C₂×C₈	—	5	1	no	—
238	C₂×C₈	—	10	1	no	—
239	C₂×C₈	—	10	1	no	—
240	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
241	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
242	C₄⋊C₄	—	15	4	no	—
243	C₄⋊C₄	—	15	4	no	—
244	C₄⋊C₄	—	30	2	no	—
245	C₄⋊C₄	—	30	2	no	—
246	C₄²	—	5	2	no	—
247	C₄²	—	5	1	no	—
248	C₄²	—	60	4	no	—
249	C₄²	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
250	C₃×C₅	#1776	1	1	yes	characteristic, complemented, Hall
251	C₂²×C₃	—	5	1	no	—
252	C₂²×C₃	—	5	1	no	—
253	C₂²×C₃	—	5	2	no	—
254	C₂²×C₃	—	5	2	no	—
255	C₂²×C₃	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
256	C₃×C₄	—	2	2	no	—
257	C₃×C₄	—	5	1	no	—
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	C₃×C₄	#4130	1	1	yes	characteristic
262	Dic₃	—	20	4	no	—
263	Dic₃	—	20	4	no	—
264	C₂×C₅	C₄×C₄⋊Dic₃	1	1	yes	characteristic
265	C₂×C₅	#5121	1	2	yes	—
266	D₅	#4710	1	2	yes	—
267	D₅	#4711	1	1	yes	characteristic
268	D₅	#4716	1	1	yes	characteristic
269	C₂³	—	5	1	no	—
270	C₂×C₄	—	5	1	no	—
271	C₂×C₄	—	5	1	no	—
272	C₂×C₄	—	5	1	no	—
273	C₂×C₄	—	5	1	no	—
274	C₂×C₄	—	5	1	no	—
275	C₂×C₄	—	5	2	no	—
276	C₂×C₄	—	10	2	no	—
277	C₂×C₄	—	10	2	no	—
278	C₂×C₄	—	30	4	no	—
279	C₂×C₄	—	30	4	no	—
280	C₂×C₄	—	60	2	no	—
281	C₂×C₄	—	60	2	no	—
282	C₂×C₄	—	60	2	no	—
283	C₂×C₄	—	60	2	no	—
284	C₂×C₄	C₄⋊ΓL(1,𝔽₁₆)	1	2	yes	—
285	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
286	C₈	—	2	1	no	—
287	C₈	—	5	2	no	—
288	C₈	—	10	1	no	—
289	C₈	C₄×ΓL(1,𝔽₁₆)	1	2	yes	complemented
290	C₂×C₃	—	5	1	no	—
291	C₂×C₃	—	5	2	no	—
292	C₂×C₃	—	5	1	no	—
293	C₂×C₃	C₄×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
294	C₂×C₃	#65727	1	2	yes	—
295	C₅	#69814	1	1	yes	characteristic, complemented, Sylow
296	C₂²	—	5	1	no	—
297	C₂²	—	5	1	no	—
298	C₂²	—	5	2	no	—
299	C₂²	—	5	2	no	—
300	C₂²	C₂×C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
301	C₄	—	2	2	no	—
302	C₄	—	5	1	no	—
303	C₄	—	5	1	no	—
304	C₄	—	10	2	no	—
305	C₄	—	60	4	no	—
306	C₄	—	60	4	no	—
307	C₄	C₂×C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
308	C₄	#91989	1	1	yes	characteristic
309	C₃	#10603881	1	1	yes	characteristic, complemented, Sylow
310	C₂	—	5	1	no	—
311	C₂	—	5	2	no	—
312	C₂	—	5	1	no	—
313	C₂	C₄×C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
314	C₂	#11752386	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₄⋊ΓL(1,𝔽₁₆)
commutator	C₃×C₄×C₅	C₂×C₄²
Frattini	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)
Fitting	C₃×C₅×#97	C₄
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂²×C₃×D₅	C₂×D₄
cosocle	C₂×C₃×C₄×D₅	C₂³
hypercenter	#97	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	#1776
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
14. #91223	C₄	semidirect
15. #91222	C₄	semidirect
16. C₃×C₅×#97	C₄	semidirect
17. C₃×#3947	C₄	semidirect
18. C₃×#3947	C₄	semidirect
19. C₃×#3935	C₄	semidirect
60. C₅×#97	Dic₃	semidirect
61. #3947	Dic₃	semidirect
62. #3947	Dic₃	semidirect
63. #3935	Dic₃	semidirect
83. C₃×C₅×C₈	C₄²	semidirect
85. C₃×#168	C₄²	semidirect
95. C₃×#97	AGL(1,𝔽₅)	semidirect
175. C₅×C₈	C₄×Dic₃	semidirect
177. #168	C₄×Dic₃	semidirect
189. #97	ΓL(1,𝔽₁₆)	semidirect
216. C₃×C₈	C₄×AGL(1,𝔽₅)	semidirect
250. C₃×C₅	#1776	semidirect
289. C₈	C₄×ΓL(1,𝔽₁₆)	semidirect
295. C₅	#69814	semidirect
309. C₃	#10603881	semidirect

Maximal subgroups

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

subgroup	index
C₂×#91223	2
C₂×#91222	2
C₄×C₄⋊ΓL(1,𝔽₁₆)	2
C₃×D₅×#97	2
#11746764	2
#10603881	3
#69814	5