Subgroups — #14299072

Order 1920.

560 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#91222	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3993	C₂	1	1	yes	maximal, characteristic
3	#11751860	C₂	1	2	yes	maximal
4	C₂×#91159	C₂	1	2	yes	maximal
5	C₈×D₅×Dic₃	C₂	1	1	yes	maximal, characteristic
6	#10603847	—	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	#91222	—	2	2	no	—
14	C₃×#3993	—	2	2	no	—
15	#91159	—	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	#70554	—	5	1	no	maximal, Hall
21	C₂×#3993	—	3	1	no	—
22	C₂×#3993	—	3	2	no	—
23	C₂×#3993	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×#106	—	5	1	no	—
47	C₂×#817	—	5	1	no	—
48	C₂×C₈×Dic₃	—	5	1	no	—
49	#4975	—	5	2	no	—
50	C₂×#808	—	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	#3993	—	2	2	no	—
58	#3993	—	6	4	no	—
59	#3993	—	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	#1844	—	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	SD₁₆	1	8	yes	—
79	C₃×C₅×C₈	C₄⋊C₄	1	2	yes	complemented
80	C₃⋊#168	C₄⋊C₄	1	2	yes	complemented
81	C₃×#168	C₄⋊C₄	1	2	yes	complemented
82	C₅×C₃⋊C₈	C₄⋊C₄	1	2	yes	complemented
83	C₂²×C₃×C₈	—	5	1	no	—
84	C₂×C₃×C₄⋊C₄	—	5	2	no	—
85	C₂×C₄⋊Dic₃	—	5	2	no	—
86	C₂×Dic₃⋊C₄	—	10	4	no	—
87	C₂×C₄×Dic₃	—	5	1	no	—
88	C₂²×C₃⋊C₈	—	5	1	no	—
89	C₃×#106	—	10	2	no	—
90	#817	—	10	2	no	—
91	C₈×Dic₃	—	5	1	no	—
92	C₈×Dic₃	—	5	1	no	—
93	C₈×Dic₃	—	5	1	no	—
94	C₈×Dic₃	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
95	#808	—	10	4	no	—
96	C₂²×AGL(1,𝔽₅)	—	2	2	no	—
97	C₂²×AGL(1,𝔽₅)	—	6	2	no	—
98	C₂×C₄×D₅	—	3	2	no	—
99	C₂×C₄×D₅	C₂²×D₃	1	1	yes	characteristic
100	C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
101	C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
102	C₄⋊AGL(1,𝔽₅)	—	6	2	no	—
103	C₄⋊AGL(1,𝔽₅)	—	6	2	no	—
104	C₄⋊AGL(1,𝔽₅)	—	12	8	no	—
105	C₂×C₅×C₈	—	3	1	no	—
106	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
107	C₄²×C₅	—	3	1	no	—
108	C₄×Dic₅	—	3	1	no	—
109	C₄×Dic₅	—	3	2	no	—
110	C₂×#168	—	3	1	no	—
111	C₂×#168	C₄×D₃	1	1	yes	characteristic
112	C₈×D₅	—	3	2	no	—
113	C₈×D₅	—	3	2	no	—
114	C₈×D₅	Dic₆	1	2	yes	—
115	C₈×D₅	#67	1	2	yes	—
116	C₂×#106	—	5	1	no	—
117	C₂×#106	—	15	2	no	—
118	C₂×#106	—	15	1	no	—
119	C₂×C₄×C₈	—	15	1	no	—
120	#384	—	15	2	no	—
121	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₃×D₅	#127	1	1	yes	characteristic
123	C₂×C₃×D₅	C₄⋊Q₈	1	1	yes	characteristic
124	C₂×C₃×D₅	C₂×SD₁₆	1	4	yes	—
125	ΓL(1,𝔽₁₆)	—	8	2	no	—
126	C₃×AGL(1,𝔽₅)	—	8	2	no	—
127	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
129	Dic₁₅	#106	1	4	yes	complemented
130	C₃×Dic₅	C₄⋊Q₈	1	1	yes	characteristic
131	C₃×Dic₅	C₄⋊Q₈	1	1	yes	characteristic
132	C₅×Dic₃	#106	1	4	yes	complemented
133	C₂²×C₃×C₄	—	5	1	no	—
134	C₂²×C₃×C₄	—	10	2	no	—
135	C₂²×Dic₃	—	5	2	no	—
136	C₂²×Dic₃	—	10	2	no	—
137	C₂×C₃×C₈	—	5	1	no	—
138	C₂×C₃×C₈	—	5	2	no	—
139	C₂×C₃×C₈	—	5	2	no	—
140	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
141	C₃×C₄⋊C₄	—	10	2	no	—
142	C₃×C₄⋊C₄	—	10	2	no	—
143	C₄⋊Dic₃	—	10	2	no	—
144	C₄⋊Dic₃	—	10	2	no	—
145	Dic₃⋊C₄	—	20	8	no	—
146	C₄×Dic₃	—	5	1	no	—
147	C₄×Dic₃	—	5	2	no	—
148	C₄×Dic₃	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
149	C₂×C₃⋊C₈	—	5	1	no	—
150	C₂×C₃⋊C₈	—	5	2	no	—
151	C₂×C₃⋊C₈	—	5	2	no	—
152	C₂×C₃⋊C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
153	C₂²×D₅	D₃×D₄	1	1	yes	characteristic
154	C₂×AGL(1,𝔽₅)	—	4	4	no	—
155	C₂×AGL(1,𝔽₅)	—	4	2	no	—
156	C₂×AGL(1,𝔽₅)	—	12	4	no	—
157	C₂×AGL(1,𝔽₅)	—	12	2	no	—
158	C₂×C₄×C₅	—	3	2	no	—
159	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
160	C₂×Dic₅	—	3	2	no	—
161	C₂×Dic₅	D₃×Q₈	1	1	yes	characteristic
162	C₄×D₅	—	3	8	no	—
163	C₄×D₅	C₂×Dic₆	1	1	yes	characteristic
164	C₄×D₅	D₃×D₄	1	1	yes	characteristic
165	C₄×D₅	D₃×Q₈	1	1	yes	characteristic
166	C₄×D₅	C₂×#67	1	1	yes	characteristic
167	C₅×C₈	—	3	2	no	—
168	C₅×C₈	Dic₃⋊C₄	1	2	yes	complemented
169	#168	—	3	2	no	—
170	#168	Dic₃⋊C₄	1	2	yes	complemented
171	C₂²×C₈	—	5	1	no	—
172	C₂²×C₈	—	15	1	no	—
173	C₂×C₄⋊C₄	—	5	2	no	—
174	C₂×C₄⋊C₄	—	15	2	no	—
175	C₂×C₄⋊C₄	—	30	4	no	—
176	C₂×C₄²	—	15	1	no	—
177	#106	—	10	2	no	—
178	#106	—	30	4	no	—
179	#106	—	30	2	no	—
180	C₄×C₈	—	3	1	no	—
181	C₄×C₈	—	15	1	no	—
182	C₄×C₈	—	15	1	no	—
183	C₄×C₈	—	15	1	no	—
184	C₂×C₃×C₅	#384	1	1	yes	characteristic
185	C₂×C₃×C₅	C₂×#106	1	2	yes	—
186	C₃×D₅	#492	1	2	yes	—
187	C₃×D₅	#498	1	2	yes	—
188	C₂³×C₃	—	5	1	no	—
189	C₂×C₃×C₄	—	5	1	no	—
190	C₂×C₃×C₄	—	5	1	no	—
191	C₂×C₃×C₄	—	5	1	no	—
192	C₂×C₃×C₄	—	5	1	no	—
193	C₂×C₃×C₄	—	5	1	no	—
194	C₂×C₃×C₄	—	20	4	no	—
195	C₂×C₃×C₄	—	20	2	no	—
196	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
197	C₂×Dic₃	—	5	2	no	—
198	C₂×Dic₃	—	5	8	no	—
199	C₂×Dic₃	—	20	4	no	—
200	C₂×Dic₃	—	20	2	no	—
201	C₂×Dic₃	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
202	C₃×C₈	—	5	2	no	—
203	C₃×C₈	C₄⋊AGL(1,𝔽₅)	1	2	yes	complemented
204	C₃⋊C₈	—	5	2	no	—
205	C₃⋊C₈	C₄⋊AGL(1,𝔽₅)	1	2	yes	complemented
206	C₂²×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
207	C₂×D₅	#888	1	1	yes	characteristic
208	C₂×D₅	D₃×SD₁₆	1	4	yes	—
209	C₂×D₅	#940	1	1	yes	characteristic
210	AGL(1,𝔽₅)	—	8	2	no	—
211	AGL(1,𝔽₅)	—	24	2	no	—
212	C₄×C₅	—	3	4	no	—
213	C₄×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
214	C₄×C₅	C₂×Dic₃⋊C₄	1	1	yes	characteristic
215	Dic₅	—	3	4	no	—
216	Dic₅	#888	1	1	yes	characteristic
217	Dic₅	#944	1	1	yes	characteristic
218	C₂²×C₄	—	5	1	no	—
219	C₂²×C₄	—	10	2	no	—
220	C₂²×C₄	—	15	2	no	—
221	C₂²×C₄	—	30	2	no	—
222	C₂×C₈	—	3	1	no	—
223	C₂×C₈	—	5	1	no	—
224	C₂×C₈	—	5	2	no	—
225	C₂×C₈	—	5	2	no	—
226	C₂×C₈	—	15	2	no	—
227	C₂×C₈	—	15	2	no	—
228	C₂×C₈	—	15	1	no	—
229	C₂×C₈	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
230	C₄⋊C₄	—	10	2	no	—
231	C₄⋊C₄	—	10	2	no	—
232	C₄⋊C₄	—	30	2	no	—
233	C₄⋊C₄	—	30	2	no	—
234	C₄⋊C₄	—	60	8	no	—
235	C₄²	—	3	1	no	—
236	C₄²	—	15	2	no	—
237	C₄²	—	15	1	no	—
238	C₃×C₅	#1844	1	1	yes	characteristic, complemented, Hall
239	C₂²×C₃	—	5	1	no	—
240	C₂²×C₃	—	5	1	no	—
241	C₂²×C₃	—	5	4	no	—
242	C₂²×C₃	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
243	C₃×C₄	—	5	1	no	—
244	C₃×C₄	—	5	1	no	—
245	C₃×C₄	—	40	2	no	—
246	C₃×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
247	C₃×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
248	Dic₃	—	5	4	no	—
249	Dic₃	—	40	2	no	—
250	Dic₃	#3993	1	4	yes	complemented
251	C₂×C₅	D₃×#106	1	2	yes	—
252	C₂×C₅	#4975	1	1	yes	characteristic
253	D₅	#4855	1	2	yes	—
254	D₅	#5175	1	2	yes	—
255	C₂³	—	5	1	no	—
256	C₂×C₄	—	3	2	no	—
257	C₂×C₄	—	5	1	no	—
258	C₂×C₄	—	5	1	no	—
259	C₂×C₄	—	5	1	no	—
260	C₂×C₄	—	5	1	no	—
261	C₂×C₄	—	5	1	no	—
262	C₂×C₄	—	15	8	no	—
263	C₂×C₄	—	15	2	no	—
264	C₂×C₄	—	20	2	no	—
265	C₂×C₄	—	20	4	no	—
266	C₂×C₄	—	60	2	no	—
267	C₂×C₄	—	60	4	no	—
268	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
269	C₈	—	3	2	no	—
270	C₈	—	5	2	no	—
271	C₈	—	15	2	no	—
272	C₈	Dic₃⋊AGL(1,𝔽₅)	1	2	yes	complemented
273	C₂×C₃	—	5	2	no	—
274	C₂×C₃	—	5	2	no	—
275	C₂×C₃	#65688	1	1	yes	characteristic
276	C₂×C₃	C₂×#3993	1	2	yes	—
277	C₅	#70554	1	1	yes	characteristic, complemented, Sylow
278	C₂²	—	5	1	no	—
279	C₂²	—	5	1	no	—
280	C₂²	—	5	4	no	—
281	C₂²	D₃×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
282	C₄	—	3	4	no	—
283	C₄	—	5	1	no	—
284	C₄	—	5	1	no	—
285	C₄	—	15	4	no	—
286	C₄	—	40	2	no	—
287	C₄	—	120	2	no	—
288	C₄	D₃×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
289	C₄	C₂×Dic₃⋊AGL(1,𝔽₅)	1	1	yes	characteristic
290	C₃	#10603847	1	1	yes	characteristic, complemented, Sylow
291	C₂	—	5	2	no	—
292	C₂	—	5	2	no	—
293	C₂	D₃×#3993	1	2	yes	—
294	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	#1844
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
79. C₃×C₅×C₈	C₄⋊C₄	semidirect
80. C₃⋊#168	C₄⋊C₄	semidirect
81. C₃×#168	C₄⋊C₄	semidirect
82. C₅×C₃⋊C₈	C₄⋊C₄	semidirect
94. C₈×Dic₃	AGL(1,𝔽₅)	semidirect
129. Dic₁₅	#106	semidirect
132. C₅×Dic₃	#106	semidirect
168. C₅×C₈	Dic₃⋊C₄	semidirect
170. #168	Dic₃⋊C₄	semidirect
203. C₃×C₈	C₄⋊AGL(1,𝔽₅)	semidirect
205. C₃⋊C₈	C₄⋊AGL(1,𝔽₅)	semidirect
238. C₃×C₅	#1844	semidirect
250. Dic₃	#3993	semidirect
272. C₈	Dic₃⋊AGL(1,𝔽₅)	semidirect
277. C₅	#70554	semidirect
290. C₃	#10603847	semidirect

Maximal subgroups

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

subgroup	index
C₂×#91222	2
C₂×C₃×#3993	2
#11751860	2
C₂×#91159	2
C₈×D₅×Dic₃	2
#10603847	3
#70554	5