Subgroups — #14296645

Order 1920.

560 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Dic₃×C₄⋊AGL(1,𝔽₅)	C₂	1	2	yes	maximal
2	C₂×#91160	C₂	1	2	yes	maximal
3	C₈×D₅×Dic₃	C₂	1	1	yes	maximal, characteristic
4	#11746764	C₂	1	1	yes	maximal, characteristic
5	C₃×#64893	C₂	1	1	yes	maximal, characteristic
6	C₄×#3994	—	3	1	no	maximal, Hall
7	C₂×C₄⋊ΓL(1,𝔽₁₆)	C₂²	1	2	yes	—
8	C₂×C₃×C₄⋊AGL(1,𝔽₅)	C₂²	1	2	yes	—
9	C₂×Dic₃×AGL(1,𝔽₅)	—	2	2	no	—
10	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
11	C₄×D₅×Dic₃	C₂²	1	1	yes	characteristic
12	C₂×D₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	(C₂×AGL(1,𝔽₅))⋊Dic₃	—	2	2	no	—
14	#91160	C₄	1	8	yes	complemented
15	C₈×Dic₁₅	C₄	1	1	yes	characteristic, complemented
16	C₅×C₈×Dic₃	C₄	1	1	yes	characteristic, complemented
17	#90951	C₄	1	1	yes	characteristic, complemented
18	Dic₃×#168	C₄	1	1	yes	characteristic, complemented
19	#68785	—	5	1	no	maximal, Hall
20	C₂×#3994	—	3	2	no	—
21	C₄×C₄⋊AGL(1,𝔽₅)	—	3	2	no	—
22	C₄×C₈×D₅	—	3	1	no	—
23	#64893	—	3	1	no	—
24	#64893	D₃	1	1	yes	characteristic
25	C₂²×ΓL(1,𝔽₁₆)	—	2	2	no	—
26	C₂²×C₃×AGL(1,𝔽₅)	—	2	2	no	—
27	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
28	C₂×D₅×Dic₃	D₄	1	1	yes	characteristic
29	C₂×D₅×Dic₃	D₄	1	1	yes	characteristic
30	C₄⋊ΓL(1,𝔽₁₆)	—	2	2	no	—
31	C₄⋊ΓL(1,𝔽₁₆)	C₂×C₄	1	4	yes	—
32	C₃×C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
33	C₃×C₄⋊AGL(1,𝔽₅)	C₂×C₄	1	4	yes	—
34	Dic₃×AGL(1,𝔽₅)	—	4	4	no	—
35	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
36	C₄×Dic₁₅	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
38	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
41	C₃×C₈×D₅	—	2	2	no	—
42	Dic₃×Dic₅	Q₈	1	1	yes	characteristic
43	Dic₃×Dic₅	Q₈	1	1	yes	characteristic
44	D₅×C₃⋊C₈	C₂×C₄	1	4	yes	—
45	C₂×C₈×Dic₃	—	5	1	no	—
46	Dic₃×C₄⋊C₄	—	5	2	no	—
47	C₂×#807	—	5	2	no	—
48	C₃×#340	—	5	1	no	—
49	#4551	—	5	1	no	—
50	C₂×C₄⋊AGL(1,𝔽₅)	—	3	2	no	—
51	C₂×C₄⋊AGL(1,𝔽₅)	C₂×D₃	1	2	yes	—
52	C₂×C₄×AGL(1,𝔽₅)	—	6	2	no	—
53	C₂×C₈×D₅	—	3	1	no	—
54	C₂×C₈×D₅	C₂×D₃	1	1	yes	characteristic
55	C₄²×D₅	—	3	1	no	—
56	(C₂×AGL(1,𝔽₅))⋊C₄	—	6	2	no	—
57	#3994	—	3	8	no	—
58	C₄×C₅×C₈	—	3	1	no	—
59	C₈×Dic₅	—	3	1	no	—
60	C₈×Dic₅	—	3	1	no	—
61	C₄×#168	—	3	1	no	—
62	C₄×#107	—	15	1	no	Sylow
63	C₂²×C₃×D₅	C₂×D₄	1	1	yes	characteristic
64	C₂×ΓL(1,𝔽₁₆)	—	2	4	no	—
65	C₂×ΓL(1,𝔽₁₆)	—	4	2	no	—
66	C₂×ΓL(1,𝔽₁₆)	—	4	2	no	—
67	C₂×C₃×AGL(1,𝔽₅)	—	2	4	no	—
68	C₂×C₃×AGL(1,𝔽₅)	—	4	2	no	—
69	C₂×C₃×AGL(1,𝔽₅)	—	4	2	no	—
70	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₂×Dic₁₅	C₄⋊C₄	1	1	yes	characteristic
72	C₂×Dic₁₅	C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₅×Dic₃	C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₅×Dic₃	C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₃×Dic₅	C₂×Q₈	1	1	yes	characteristic
76	C₃×C₄×D₅	C₂²×C₄	1	2	yes	—
77	C₃×C₄×D₅	#41	1	2	yes	—
78	D₅×Dic₃	—	2	2	no	—
79	D₅×Dic₃	D₈	1	2	yes	—
80	D₅×Dic₃	Q₁₆	1	2	yes	—
81	C₃×C₅×C₈	—	2	1	no	—
82	C₃⋊#168	C₄²	1	2	yes	complemented
83	C₃×#168	—	2	1	no	—
84	C₅×C₃⋊C₈	C₄²	1	2	yes	complemented
85	C₂²×C₃×C₈	—	5	1	no	—
86	C₂×C₃×C₄⋊C₄	—	5	2	no	—
87	C₂×C₄⋊Dic₃	—	5	2	no	—
88	C₂×C₄×Dic₃	—	5	1	no	—
89	C₂×C₄×Dic₃	—	10	2	no	—
90	C₂²×C₃⋊C₈	—	5	1	no	—
91	C₃⋊#95	—	10	2	no	—
92	C₈×Dic₃	—	5	1	no	—
93	C₈×Dic₃	—	5	1	no	—
94	C₈×Dic₃	—	5	1	no	—
95	C₈×Dic₃	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
96	#807	—	5	8	no	—
97	C₂²×AGL(1,𝔽₅)	—	2	2	no	—
98	C₂²×AGL(1,𝔽₅)	—	6	2	no	—
99	C₂×C₄×D₅	—	3	1	no	—
100	C₂×C₄×D₅	—	3	1	no	—
101	C₂×C₄×D₅	C₂²×D₃	1	1	yes	characteristic
102	C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
103	C₄⋊AGL(1,𝔽₅)	—	3	4	no	—
104	C₄⋊AGL(1,𝔽₅)	—	6	2	no	—
105	C₄⋊AGL(1,𝔽₅)	C₄×D₃	1	4	yes	—
106	C₄×AGL(1,𝔽₅)	—	12	4	no	—
107	C₂×C₅×C₈	—	3	1	no	—
108	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
109	C₄²×C₅	—	3	1	no	—
110	C₄×Dic₅	—	3	1	no	—
111	C₄×Dic₅	—	3	1	no	—
112	C₄×Dic₅	—	3	1	no	—
113	C₂×#168	—	3	1	no	—
114	C₂×#168	C₄×D₃	1	1	yes	characteristic
115	C₈×D₅	—	2	2	no	—
116	C₈×D₅	—	3	4	no	—
117	C₂×#107	—	15	2	no	—
118	C₂×C₄×C₈	—	15	1	no	—
119	C₄×C₄⋊C₄	—	15	2	no	—
120	#340	—	5	1	no	—
121	#340	—	15	1	no	—
122	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₃×D₅	C₄×D₄	1	2	yes	—
124	C₂×C₃×D₅	C₂×D₈	1	1	yes	characteristic
125	C₂×C₃×D₅	C₂×Q₁₆	1	1	yes	characteristic
126	C₂×C₃×D₅	#135	1	2	yes	—
127	ΓL(1,𝔽₁₆)	—	4	4	no	—
128	C₃×AGL(1,𝔽₅)	—	4	4	no	—
129	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
130	C₃×C₄×C₅	#117	1	1	yes	characteristic
131	Dic₁₅	—	2	1	no	—
132	Dic₁₅	#107	1	2	yes	complemented
133	C₃×Dic₅	C₄×Q₈	1	2	yes	—
134	C₅×Dic₃	—	2	1	no	—
135	C₅×Dic₃	#107	1	2	yes	complemented
136	C₂²×C₃×C₄	—	5	1	no	—
137	C₂²×C₃×C₄	—	10	2	no	—
138	C₂²×Dic₃	—	5	1	no	—
139	C₂²×Dic₃	—	5	1	no	—
140	C₂²×Dic₃	—	10	2	no	—
141	C₂×C₃×C₈	—	5	1	no	—
142	C₂×C₃×C₈	—	10	2	no	—
143	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
144	C₃×C₄⋊C₄	—	5	4	no	—
145	C₃×C₄⋊C₄	—	10	2	no	—
146	C₄⋊Dic₃	—	5	4	no	—
147	C₄⋊Dic₃	—	10	2	no	—
148	C₄×Dic₃	—	5	1	no	—
149	C₄×Dic₃	—	5	1	no	—
150	C₄×Dic₃	—	5	1	no	—
151	C₄×Dic₃	—	20	4	no	—
152	C₄×Dic₃	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
153	C₂×C₃⋊C₈	—	5	1	no	—
154	C₂×C₃⋊C₈	—	5	4	no	—
155	C₂×C₃⋊C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
156	C₂²×D₅	D₃×D₄	1	1	yes	characteristic
157	C₂×AGL(1,𝔽₅)	—	2	4	no	—
158	C₂×AGL(1,𝔽₅)	—	4	2	no	—
159	C₂×AGL(1,𝔽₅)	—	4	2	no	—
160	C₂×AGL(1,𝔽₅)	—	6	4	no	—
161	C₂×AGL(1,𝔽₅)	—	12	2	no	—
162	C₂×AGL(1,𝔽₅)	—	12	2	no	—
163	C₂×C₄×C₅	—	3	1	no	—
164	C₂×C₄×C₅	—	3	1	no	—
165	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
166	C₂×Dic₅	—	3	1	no	—
167	C₂×Dic₅	—	3	1	no	—
168	C₂×Dic₅	D₃×Q₈	1	1	yes	characteristic
169	C₄×D₅	—	3	2	no	—
170	C₄×D₅	—	3	2	no	—
171	C₄×D₅	—	6	2	no	—
172	C₄×D₅	C₂×C₄×D₃	1	2	yes	—
173	C₄×D₅	#237	1	2	yes	—
174	C₅×C₈	—	2	1	no	—
175	C₅×C₈	—	3	2	no	—
176	#168	—	2	1	no	—
177	#168	—	3	2	no	—
178	C₂²×C₈	—	5	1	no	—
179	C₂²×C₈	—	15	1	no	—
180	C₂×C₄⋊C₄	—	5	2	no	—
181	C₂×C₄⋊C₄	—	15	2	no	—
182	C₂×C₄²	—	15	1	no	—
183	C₂×C₄²	—	30	2	no	—
184	#107	—	15	8	no	—
185	C₄×C₈	—	3	1	no	—
186	C₄×C₈	—	15	1	no	—
187	C₄×C₈	—	15	1	no	—
188	C₄×C₈	—	15	1	no	—
189	#95	—	30	2	no	—
190	C₂×C₃×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
191	C₂×C₃×C₅	C₂×#107	1	1	yes	characteristic
192	C₂×C₃×C₅	#427	1	1	yes	characteristic
193	C₃×D₅	C₄×D₈	1	2	yes	—
194	C₃×D₅	C₄×Q₁₆	1	2	yes	—
195	C₂³×C₃	—	5	1	no	—
196	C₂×C₃×C₄	—	5	1	no	—
197	C₂×C₃×C₄	—	5	2	no	—
198	C₂×C₃×C₄	—	5	2	no	—
199	C₂×C₃×C₄	—	10	4	no	—
200	C₂×C₃×C₄	—	20	2	no	—
201	C₂×C₃×C₄	—	20	2	no	—
202	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
203	C₂×Dic₃	—	5	1	no	—
204	C₂×Dic₃	—	5	2	no	—
205	C₂×Dic₃	—	5	2	no	—
206	C₂×Dic₃	—	5	1	no	—
207	C₂×Dic₃	—	10	2	no	—
208	C₂×Dic₃	—	10	4	no	—
209	C₂×Dic₃	—	20	2	no	—
210	C₂×Dic₃	—	20	2	no	—
211	C₂×Dic₃	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
212	C₂×Dic₃	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
213	C₃×C₈	—	2	1	no	—
214	C₃×C₈	—	10	1	no	—
215	C₃⋊C₈	—	5	2	no	—
216	C₃⋊C₈	C₄×AGL(1,𝔽₅)	1	2	yes	complemented
217	C₂²×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
218	C₂×D₅	#67⋊C₄	1	2	yes	—
219	C₂×D₅	D₃×D₈	1	1	yes	characteristic
220	C₂×D₅	#916	1	2	yes	—
221	C₂×D₅	D₃×Q₁₆	1	1	yes	characteristic
222	AGL(1,𝔽₅)	—	4	4	no	—
223	AGL(1,𝔽₅)	—	12	4	no	—
224	C₄×C₅	—	3	2	no	—
225	C₄×C₅	—	6	1	no	—
226	C₄×C₅	C₄²×D₃	1	1	yes	characteristic
227	C₄×C₅	#877	1	1	yes	characteristic
228	Dic₅	—	3	2	no	—
229	Dic₅	—	6	1	no	—
230	Dic₅	#887	1	2	yes	—
231	C₂²×C₄	—	5	1	no	—
232	C₂²×C₄	—	10	2	no	—
233	C₂²×C₄	—	15	1	no	—
234	C₂²×C₄	—	15	1	no	—
235	C₂²×C₄	—	30	2	no	—
236	C₂×C₈	—	3	1	no	—
237	C₂×C₈	—	5	1	no	—
238	C₂×C₈	—	10	2	no	—
239	C₂×C₈	—	15	4	no	—
240	C₂×C₈	—	15	1	no	—
241	C₂×C₈	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
242	C₄⋊C₄	—	5	4	no	—
243	C₄⋊C₄	—	10	2	no	—
244	C₄⋊C₄	—	15	4	no	—
245	C₄⋊C₄	—	30	2	no	—
246	C₄²	—	3	1	no	—
247	C₄²	—	15	1	no	—
248	C₄²	—	15	1	no	—
249	C₄²	—	15	1	no	—
250	C₄²	—	60	4	no	—
251	C₃×C₅	C₄×#107	1	1	yes	characteristic, complemented, Hall
252	C₂²×C₃	—	5	2	no	—
253	C₂²×C₃	—	5	1	no	—
254	C₂²×C₃	—	5	1	no	—
255	C₂²×C₃	—	5	2	no	—
256	C₂²×C₃	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
257	C₃×C₄	—	5	2	no	—
258	C₃×C₄	—	20	4	no	—
259	C₃×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
260	C₃×C₄	#4130	1	1	yes	characteristic
261	Dic₃	—	2	1	no	—
262	Dic₃	—	5	2	no	—
263	Dic₃	—	10	1	no	—
264	Dic₃	—	20	4	no	—
265	Dic₃	#3994	1	2	yes	complemented
266	C₂×C₅	#4649	1	1	yes	characteristic
267	C₂×C₅	#4860	1	1	yes	characteristic
268	C₂×C₅	D₃×#107	1	1	yes	characteristic
269	D₅	#4756	1	2	yes	—
270	D₅	#4790	1	2	yes	—
271	C₂³	—	5	1	no	—
272	C₂×C₄	—	3	1	no	—
273	C₂×C₄	—	3	1	no	—
274	C₂×C₄	—	5	1	no	—
275	C₂×C₄	—	5	2	no	—
276	C₂×C₄	—	5	2	no	—
277	C₂×C₄	—	10	4	no	—
278	C₂×C₄	—	15	2	no	—
279	C₂×C₄	—	15	2	no	—
280	C₂×C₄	—	15	1	no	—
281	C₂×C₄	—	15	1	no	—
282	C₂×C₄	—	20	2	no	—
283	C₂×C₄	—	20	2	no	—
284	C₂×C₄	—	30	2	no	—
285	C₂×C₄	—	30	4	no	—
286	C₂×C₄	—	60	2	no	—
287	C₂×C₄	—	60	2	no	—
288	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
289	C₈	—	2	1	no	—
290	C₈	—	3	2	no	—
291	C₈	—	10	1	no	—
292	C₈	—	15	2	no	—
293	C₂×C₃	—	5	2	no	—
294	C₂×C₃	—	5	2	no	—
295	C₂×C₃	C₄×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
296	C₂×C₃	C₂×#3994	1	1	yes	characteristic
297	C₂×C₃	#65721	1	1	yes	characteristic
298	C₅	#68785	1	1	yes	characteristic, complemented, Sylow
299	C₂²	—	5	2	no	—
300	C₂²	—	5	1	no	—
301	C₂²	—	5	1	no	—
302	C₂²	—	5	2	no	—
303	C₂²	D₃×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
304	C₄	—	3	2	no	—
305	C₄	—	5	2	no	—
306	C₄	—	6	1	no	—
307	C₄	—	15	2	no	—
308	C₄	—	20	4	no	—
309	C₄	—	30	1	no	—
310	C₄	—	60	4	no	—
311	C₄	C₄×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
312	C₄	#91923	1	1	yes	characteristic
313	C₃	C₄×#3994	1	1	yes	characteristic, complemented, Sylow
314	C₂	—	5	2	no	—
315	C₂	—	5	2	no	—
316	C₂	D₃×#3994	1	1	yes	characteristic
317	C₂	#11751657	1	1	yes	characteristic
318	C₂	#11751834	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₃×C₄⋊AGL(1,𝔽₅)
commutator	C₃×C₄×C₅	C₂×C₄²
Frattini	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)
Fitting	C₂×C₃×C₅×C₈	C₂×C₄
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂²×C₃×D₅	C₂×D₄
cosocle	C₂×C₃×C₄×D₅	C₂³
hypercenter	C₂×C₈	D₃×AGL(1,𝔽₅)
hypocenter	C₃×C₅	C₄×#107
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
14. #91160	C₄	semidirect
15. C₈×Dic₁₅	C₄	semidirect
16. C₅×C₈×Dic₃	C₄	semidirect
17. #90951	C₄	semidirect
18. Dic₃×#168	C₄	semidirect
82. C₃⋊#168	C₄²	semidirect
84. C₅×C₃⋊C₈	C₄²	semidirect
95. C₈×Dic₃	AGL(1,𝔽₅)	semidirect
132. Dic₁₅	#107	semidirect
135. C₅×Dic₃	#107	semidirect
216. C₃⋊C₈	C₄×AGL(1,𝔽₅)	semidirect
251. C₃×C₅	C₄×#107	semidirect
265. Dic₃	#3994	semidirect
298. C₅	#68785	semidirect
313. C₃	C₄×#3994	semidirect

Maximal subgroups

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

subgroup	index
Dic₃×C₄⋊AGL(1,𝔽₅)	2
C₂×#91160	2
C₈×D₅×Dic₃	2
#11746764	2
C₃×#64893	2
C₄×#3994	3
#68785	5