Subgroups — #14299094

Order 1920.

560 conjugacy classes of subgroups.

The subgroup lattice has 308 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×C₃×#3994	C₂	1	1	yes	maximal, characteristic
3	#11751860	C₂	1	2	yes	maximal
4	C₂×#91160	C₂	1	2	yes	maximal
5	C₈×D₅×Dic₃	C₂	1	1	yes	maximal, characteristic
6	#10603848	—	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	4	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	#91223	—	2	2	no	—
14	C₃×#3994	—	2	2	no	—
15	#91160	—	2	4	no	—
16	C₈×Dic₁₅	C₄	1	1	yes	characteristic, complemented
17	C₅×C₈×Dic₃	C₄	1	1	yes	characteristic, complemented
18	#90951	C₄	1	1	yes	characteristic, complemented
19	Dic₃×#168	C₄	1	1	yes	characteristic, complemented
20	#70606	—	5	1	no	maximal, Hall
21	C₂×#3994	—	3	1	no	—
22	C₂×#3994	—	3	2	no	—
23	C₂×#3994	D₃	1	1	yes	characteristic
24	#65688	—	3	2	no	—
25	C₄×C₈×D₅	—	3	1	no	—
26	C₂²×ΓL(1,𝔽₁₆)	—	2	2	no	—
27	C₂²×C₃×AGL(1,𝔽₅)	—	2	2	no	—
28	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
29	C₂×D₅×Dic₃	D₄	1	2	yes	—
30	C₄⋊ΓL(1,𝔽₁₆)	—	2	2	no	—
31	C₄⋊ΓL(1,𝔽₁₆)	—	2	2	no	—
32	C₃×C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
33	C₃×C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
34	Dic₃⋊AGL(1,𝔽₅)	—	4	8	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	D₄	1	2	yes	—
42	C₃×C₈×D₅	Q₈	1	2	yes	—
43	Dic₃×Dic₅	Q₈	1	2	yes	—
44	D₅×C₃⋊C₈	D₄	1	2	yes	—
45	D₅×C₃⋊C₈	Q₈	1	2	yes	—
46	C₂×C₃×#107	—	5	1	no	—
47	C₂×#818	—	5	1	no	—
48	C₂×C₈×Dic₃	—	5	1	no	—
49	#4975	—	5	2	no	—
50	C₂×#807	—	5	2	no	—
51	C₂×C₄⋊AGL(1,𝔽₅)	—	3	2	no	—
52	C₂×C₄⋊AGL(1,𝔽₅)	—	6	4	no	—
53	C₂×C₄⋊AGL(1,𝔽₅)	C₂×D₃	1	2	yes	—
54	C₂×C₈×D₅	—	3	1	no	—
55	C₂×C₈×D₅	C₂×D₃	1	1	yes	characteristic
56	C₄²×D₅	—	3	1	no	—
57	#3994	—	2	2	no	—
58	#3994	—	6	4	no	—
59	#3994	—	6	2	no	—
60	C₄×C₅×C₈	—	3	1	no	—
61	C₈×Dic₅	—	3	1	no	—
62	C₈×Dic₅	—	3	1	no	—
63	C₄×#168	—	3	1	no	—
64	#1845	—	15	1	no	Sylow
65	C₂²×C₃×D₅	C₂×D₄	1	1	yes	characteristic
66	C₂×ΓL(1,𝔽₁₆)	—	4	2	no	—
67	C₂×ΓL(1,𝔽₁₆)	—	4	4	no	—
68	C₂×C₃×AGL(1,𝔽₅)	—	4	2	no	—
69	C₂×C₃×AGL(1,𝔽₅)	—	4	4	no	—
70	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₂×Dic₁₅	C₄⋊C₄	1	2	yes	—
72	C₂×C₅×Dic₃	C₄⋊C₄	1	2	yes	—
73	C₂×C₃×Dic₅	C₂×Q₈	1	1	yes	characteristic
74	C₃×C₄×D₅	C₂×D₄	1	1	yes	characteristic
75	C₃×C₄×D₅	C₂×D₄	1	1	yes	characteristic
76	C₃×C₄×D₅	C₂×Q₈	1	1	yes	characteristic
77	C₃×C₄×D₅	C₂×Q₈	1	1	yes	characteristic
78	D₅×Dic₃	D₈	1	4	yes	—
79	D₅×Dic₃	Q₁₆	1	4	yes	—
80	C₃×C₅×C₈	C₄⋊C₄	1	2	yes	complemented
81	C₃⋊#168	C₄⋊C₄	1	2	yes	complemented
82	C₃×#168	C₄⋊C₄	1	2	yes	complemented
83	C₅×C₃⋊C₈	C₄⋊C₄	1	2	yes	complemented
84	C₂²×C₃×C₈	—	5	1	no	—
85	C₂×C₃×C₄⋊C₄	—	5	2	no	—
86	C₂×C₄⋊Dic₃	—	5	2	no	—
87	C₂×Dic₃⋊C₄	—	10	4	no	—
88	C₂×C₄×Dic₃	—	5	1	no	—
89	C₂²×C₃⋊C₈	—	5	1	no	—
90	C₃×#107	—	10	2	no	—
91	#818	—	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	—	10	4	no	—
97	C₂²×AGL(1,𝔽₅)	—	2	2	no	—
98	C₂²×AGL(1,𝔽₅)	—	6	2	no	—
99	C₂×C₄×D₅	—	3	2	no	—
100	C₂×C₄×D₅	C₂²×D₃	1	1	yes	characteristic
101	C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
102	C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
103	C₄⋊AGL(1,𝔽₅)	—	6	2	no	—
104	C₄⋊AGL(1,𝔽₅)	—	6	2	no	—
105	C₄⋊AGL(1,𝔽₅)	—	12	8	no	—
106	C₂×C₅×C₈	—	3	1	no	—
107	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
108	C₄²×C₅	—	3	1	no	—
109	C₄×Dic₅	—	3	1	no	—
110	C₄×Dic₅	—	3	2	no	—
111	C₂×#168	—	3	1	no	—
112	C₂×#168	C₄×D₃	1	1	yes	characteristic
113	C₈×D₅	—	3	2	no	—
114	C₈×D₅	—	3	2	no	—
115	C₈×D₅	Dic₆	1	2	yes	—
116	C₈×D₅	#67	1	2	yes	—
117	C₂×#107	—	5	1	no	—
118	C₂×#107	—	15	2	no	—
119	C₂×#107	—	15	1	no	—
120	C₂×C₄×C₈	—	15	1	no	—
121	#384	—	15	2	no	—
122	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₃×D₅	#127	1	1	yes	characteristic
124	C₂×C₃×D₅	C₄⋊Q₈	1	1	yes	characteristic
125	C₂×C₃×D₅	C₂×D₈	1	2	yes	—
126	C₂×C₃×D₅	C₂×Q₁₆	1	2	yes	—
127	ΓL(1,𝔽₁₆)	—	8	2	no	—
128	C₃×AGL(1,𝔽₅)	—	8	2	no	—
129	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
131	Dic₁₅	#107	1	4	yes	complemented
132	C₃×Dic₅	C₄⋊Q₈	1	1	yes	characteristic
133	C₃×Dic₅	C₄⋊Q₈	1	1	yes	characteristic
134	C₅×Dic₃	#107	1	4	yes	complemented
135	C₂²×C₃×C₄	—	5	1	no	—
136	C₂²×C₃×C₄	—	10	2	no	—
137	C₂²×Dic₃	—	5	2	no	—
138	C₂²×Dic₃	—	10	2	no	—
139	C₂×C₃×C₈	—	5	1	no	—
140	C₂×C₃×C₈	—	5	2	no	—
141	C₂×C₃×C₈	—	5	2	no	—
142	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
143	C₃×C₄⋊C₄	—	10	2	no	—
144	C₃×C₄⋊C₄	—	10	2	no	—
145	C₄⋊Dic₃	—	10	2	no	—
146	C₄⋊Dic₃	—	10	2	no	—
147	Dic₃⋊C₄	—	20	8	no	—
148	C₄×Dic₃	—	5	1	no	—
149	C₄×Dic₃	—	5	2	no	—
150	C₄×Dic₃	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
151	C₂×C₃⋊C₈	—	5	1	no	—
152	C₂×C₃⋊C₈	—	5	2	no	—
153	C₂×C₃⋊C₈	—	5	2	no	—
154	C₂×C₃⋊C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
155	C₂²×D₅	D₃×D₄	1	1	yes	characteristic
156	C₂×AGL(1,𝔽₅)	—	4	4	no	—
157	C₂×AGL(1,𝔽₅)	—	4	2	no	—
158	C₂×AGL(1,𝔽₅)	—	12	4	no	—
159	C₂×AGL(1,𝔽₅)	—	12	2	no	—
160	C₂×C₄×C₅	—	3	2	no	—
161	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
162	C₂×Dic₅	—	3	2	no	—
163	C₂×Dic₅	D₃×Q₈	1	1	yes	characteristic
164	C₄×D₅	—	3	4	no	—
165	C₄×D₅	—	3	4	no	—
166	C₄×D₅	C₂×Dic₆	1	1	yes	characteristic
167	C₄×D₅	D₃×D₄	1	1	yes	characteristic
168	C₄×D₅	D₃×Q₈	1	1	yes	characteristic
169	C₄×D₅	C₂×#67	1	1	yes	characteristic
170	C₅×C₈	—	3	2	no	—
171	C₅×C₈	Dic₃⋊C₄	1	2	yes	complemented
172	#168	—	3	2	no	—
173	#168	Dic₃⋊C₄	1	2	yes	complemented
174	C₂²×C₈	—	5	1	no	—
175	C₂²×C₈	—	15	1	no	—
176	C₂×C₄⋊C₄	—	5	2	no	—
177	C₂×C₄⋊C₄	—	15	2	no	—
178	C₂×C₄⋊C₄	—	30	4	no	—
179	C₂×C₄²	—	15	1	no	—
180	#107	—	10	2	no	—
181	#107	—	30	4	no	—
182	#107	—	30	2	no	—
183	C₄×C₈	—	3	1	no	—
184	C₄×C₈	—	15	1	no	—
185	C₄×C₈	—	15	1	no	—
186	C₄×C₈	—	15	1	no	—
187	C₂×C₃×C₅	#384	1	1	yes	characteristic
188	C₂×C₃×C₅	C₂×#107	1	2	yes	—
189	C₃×D₅	#493	1	1	yes	characteristic
190	C₃×D₅	#494	1	1	yes	characteristic
191	C₃×D₅	#500	1	2	yes	—
192	C₂³×C₃	—	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	—	20	4	no	—
199	C₂×C₃×C₄	—	20	2	no	—
200	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
201	C₂×Dic₃	—	5	2	no	—
202	C₂×Dic₃	—	5	4	no	—
203	C₂×Dic₃	—	5	4	no	—
204	C₂×Dic₃	—	20	4	no	—
205	C₂×Dic₃	—	20	2	no	—
206	C₂×Dic₃	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
207	C₃×C₈	—	5	2	no	—
208	C₃×C₈	C₄⋊AGL(1,𝔽₅)	1	2	yes	complemented
209	C₃⋊C₈	—	5	2	no	—
210	C₃⋊C₈	C₄⋊AGL(1,𝔽₅)	1	2	yes	complemented
211	C₂²×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
212	C₂×D₅	#888	1	1	yes	characteristic
213	C₂×D₅	D₃×D₈	1	2	yes	—
214	C₂×D₅	D₃×Q₁₆	1	2	yes	—
215	C₂×D₅	#940	1	1	yes	characteristic
216	AGL(1,𝔽₅)	—	8	2	no	—
217	AGL(1,𝔽₅)	—	24	2	no	—
218	C₄×C₅	—	3	4	no	—
219	C₄×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
220	C₄×C₅	C₂×Dic₃⋊C₄	1	1	yes	characteristic
221	Dic₅	—	3	4	no	—
222	Dic₅	#888	1	1	yes	characteristic
223	Dic₅	#944	1	1	yes	characteristic
224	C₂²×C₄	—	5	1	no	—
225	C₂²×C₄	—	10	2	no	—
226	C₂²×C₄	—	15	2	no	—
227	C₂²×C₄	—	30	2	no	—
228	C₂×C₈	—	3	1	no	—
229	C₂×C₈	—	5	1	no	—
230	C₂×C₈	—	5	2	no	—
231	C₂×C₈	—	5	2	no	—
232	C₂×C₈	—	15	2	no	—
233	C₂×C₈	—	15	2	no	—
234	C₂×C₈	—	15	1	no	—
235	C₂×C₈	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
236	C₄⋊C₄	—	10	2	no	—
237	C₄⋊C₄	—	10	2	no	—
238	C₄⋊C₄	—	30	2	no	—
239	C₄⋊C₄	—	30	2	no	—
240	C₄⋊C₄	—	60	8	no	—
241	C₄²	—	3	1	no	—
242	C₄²	—	15	2	no	—
243	C₄²	—	15	1	no	—
244	C₃×C₅	#1845	1	1	yes	characteristic, complemented, Hall
245	C₂²×C₃	—	5	1	no	—
246	C₂²×C₃	—	5	1	no	—
247	C₂²×C₃	—	5	2	no	—
248	C₂²×C₃	—	5	2	no	—
249	C₂²×C₃	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
250	C₃×C₄	—	5	1	no	—
251	C₃×C₄	—	5	1	no	—
252	C₃×C₄	—	40	2	no	—
253	C₃×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
254	C₃×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
255	Dic₃	—	5	4	no	—
256	Dic₃	—	40	2	no	—
257	Dic₃	#3994	1	4	yes	complemented
258	C₂×C₅	D₃×#107	1	2	yes	—
259	C₂×C₅	#4975	1	1	yes	characteristic
260	D₅	#4874	1	2	yes	—
261	D₅	#5151	1	1	yes	characteristic
262	D₅	#5183	1	1	yes	characteristic
263	C₂³	—	5	1	no	—
264	C₂×C₄	—	3	2	no	—
265	C₂×C₄	—	5	1	no	—
266	C₂×C₄	—	5	1	no	—
267	C₂×C₄	—	5	1	no	—
268	C₂×C₄	—	5	1	no	—
269	C₂×C₄	—	5	1	no	—
270	C₂×C₄	—	15	4	no	—
271	C₂×C₄	—	15	4	no	—
272	C₂×C₄	—	15	2	no	—
273	C₂×C₄	—	20	2	no	—
274	C₂×C₄	—	20	4	no	—
275	C₂×C₄	—	60	2	no	—
276	C₂×C₄	—	60	4	no	—
277	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
278	C₈	—	3	2	no	—
279	C₈	—	5	2	no	—
280	C₈	—	15	2	no	—
281	C₈	Dic₃⋊AGL(1,𝔽₅)	1	2	yes	complemented
282	C₂×C₃	—	5	1	no	—
283	C₂×C₃	—	5	1	no	—
284	C₂×C₃	—	5	2	no	—
285	C₂×C₃	#65688	1	1	yes	characteristic
286	C₂×C₃	C₂×#3994	1	2	yes	—
287	C₅	#70606	1	1	yes	characteristic, complemented, Sylow
288	C₂²	—	5	1	no	—
289	C₂²	—	5	1	no	—
290	C₂²	—	5	2	no	—
291	C₂²	—	5	2	no	—
292	C₂²	D₃×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
293	C₄	—	3	4	no	—
294	C₄	—	5	1	no	—
295	C₄	—	5	1	no	—
296	C₄	—	15	4	no	—
297	C₄	—	40	2	no	—
298	C₄	—	120	2	no	—
299	C₄	D₃×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
300	C₄	C₂×Dic₃⋊AGL(1,𝔽₅)	1	1	yes	characteristic
301	C₃	#10603848	1	1	yes	characteristic, complemented, Sylow
302	C₂	—	5	1	no	—
303	C₂	—	5	2	no	—
304	C₂	—	5	1	no	—
305	C₂	D₃×#3994	1	2	yes	—
306	C₂	#11751860	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₃×C₄⋊AGL(1,𝔽₅)
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
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	#1845
LCS	G ▷ C₂×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
16. C₈×Dic₁₅	C₄	semidirect
17. C₅×C₈×Dic₃	C₄	semidirect
18. #90951	C₄	semidirect
19. Dic₃×#168	C₄	semidirect
80. C₃×C₅×C₈	C₄⋊C₄	semidirect
81. C₃⋊#168	C₄⋊C₄	semidirect
82. C₃×#168	C₄⋊C₄	semidirect
83. C₅×C₃⋊C₈	C₄⋊C₄	semidirect
95. C₈×Dic₃	AGL(1,𝔽₅)	semidirect
131. Dic₁₅	#107	semidirect
134. C₅×Dic₃	#107	semidirect
171. C₅×C₈	Dic₃⋊C₄	semidirect
173. #168	Dic₃⋊C₄	semidirect
208. C₃×C₈	C₄⋊AGL(1,𝔽₅)	semidirect
210. C₃⋊C₈	C₄⋊AGL(1,𝔽₅)	semidirect
244. C₃×C₅	#1845	semidirect
257. Dic₃	#3994	semidirect
281. C₈	Dic₃⋊AGL(1,𝔽₅)	semidirect
287. C₅	#70606	semidirect
301. C₃	#10603848	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×C₃×#3994	2
#11751860	2
C₂×#91160	2
C₈×D₅×Dic₃	2
#10603848	3
#70606	5