Subgroups — #14285643

Order 1920.

360 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅⋊(C₄⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic
2	C₃×#65680	C₂	1	1	yes	maximal, characteristic
3	#11747003	C₂	1	1	yes	maximal, characteristic
4	#11746515	C₂	1	1	yes	maximal, characteristic, complemented
5	#11746514	C₂	1	1	yes	maximal, characteristic, complemented
6	#11746513	C₂	1	1	yes	maximal, characteristic, complemented
7	#11746512	C₂	1	1	yes	maximal, characteristic, complemented
8	#10603865	—	3	1	no	maximal, Hall
9	C₂×D₅⋊(C₃⋊C₈)	—	2	1	no	—
10	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
11	C₂×#6649	C₄	1	1	yes	characteristic
12	C₂×#6649	C₄	1	1	yes	characteristic
13	#91233	—	2	1	no	—
14	C₄⋊(C₃⋊#170)	C₂²	1	1	yes	characteristic
15	C₄⋊(C₃⋊#170)	C₂²	1	1	yes	characteristic
16	C₃×#4003	—	2	1	no	—
17	C₃×C₄⋊#170	C₂²	1	1	yes	characteristic
18	C₃×C₄⋊#170	C₂²	1	1	yes	characteristic
19	C₃×C₄×#170	—	2	1	no	—
20	#91087	C₄	1	1	yes	characteristic
21	C₅×#803	C₄	1	1	yes	characteristic
22	Dic₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
23	Dic₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
24	#70746	—	5	1	no	maximal, Hall
25	#65680	D₃	1	1	yes	characteristic
26	D₅⋊(C₄⋊C₈)	—	3	1	no	—
27	#64996	—	3	1	no	—
28	#64884	—	3	1	no	—
29	#64883	—	3	1	no	—
30	#64882	—	3	1	no	—
31	#64881	—	3	1	no	—
32	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
34	C₂×C₃×C₄×D₅	Q₈	1	1	yes	characteristic
35	C₂×C₃⋊#170	—	2	1	no	—
36	C₂×C₃⋊#170	—	2	1	no	—
37	D₅⋊(C₃⋊C₈)	—	4	2	no	—
38	C₂×C₃×#170	—	2	1	no	—
39	C₂×C₃×#170	—	2	1	no	—
40	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
42	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
43	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
45	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
46	C₃×C₄×Dic₅	Q₈	1	1	yes	characteristic
47	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
48	#6649	—	2	2	no	—
49	#6649	C₈	1	4	yes	complemented
50	C₃×#431	—	5	1	no	—
51	C₂×C₄⋊(C₃⋊C₈)	—	5	1	no	—
52	C₄×#208	—	5	1	no	—
53	#4480	—	5	1	no	—
54	#4480	—	5	1	no	—
55	#4479	—	5	1	no	—
56	#4479	—	5	1	no	—
57	C₂×D₅⋊C₈	—	6	1	no	—
58	C₂×#676	—	3	1	no	—
59	C₂×#676	—	3	1	no	—
60	C₄²×D₅	C₂×D₃	1	1	yes	characteristic
61	#4004	—	6	1	no	—
62	#4003	—	2	1	no	—
63	C₄⋊#170	—	3	1	no	—
64	C₄⋊#170	—	3	1	no	—
65	C₄⋊#170	C₂×D₃	1	1	yes	characteristic
66	C₄⋊#170	C₂×D₃	1	1	yes	characteristic
67	C₄×#170	—	2	1	no	—
68	C₅×#97	—	3	1	no	—
69	C₈×Dic₅	—	3	1	no	—
70	C₈×Dic₅	—	3	1	no	—
71	#3935	—	3	1	no	—
72	#1565	—	15	1	no	Sylow
73	C₂²×C₃×D₅	C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₃×Dic₅	C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
79	C₂×C₃×Dic₅	C₂×Q₈	1	1	yes	characteristic
80	C₃×C₄×D₅	—	2	1	no	—
81	C₃×C₄×D₅	—	2	1	no	—
82	C₃×C₄×D₅	—	4	1	no	—
83	C₃×C₄×D₅	C₂×C₈	1	2	yes	—
84	C₃×C₄×D₅	M₁₆	1	2	yes	—
85	C₃⋊#170	—	4	1	no	—
86	C₃⋊#170	—	4	1	no	—
87	C₃×#170	—	4	1	no	—
88	C₃×#170	—	4	1	no	—
89	C₃⋊#168	—	2	1	no	—
90	C₃⋊#168	C₂×C₈	1	2	yes	complemented
91	C₅×C₃⋊C₈	—	2	1	no	—
92	C₅×C₃⋊C₈	C₂×C₈	1	2	yes	complemented
93	C₂×C₃×C₄²	—	5	1	no	—
94	C₂×#208	—	5	1	no	—
95	C₂×#208	—	5	1	no	—
96	C₂²×C₃⋊C₈	—	10	1	no	—
97	C₃×C₄⋊C₈	—	5	1	no	—
98	C₃×C₄⋊C₈	—	5	1	no	—
99	C₃×C₂²⋊C₈	—	10	1	no	—
100	C₃×C₄×C₈	—	10	1	no	—
101	C₄⋊(C₃⋊C₈)	—	5	1	no	—
102	C₄⋊(C₃⋊C₈)	—	5	1	no	—
103	C₄⋊(C₃⋊C₈)	—	10	1	no	—
104	#803	—	5	1	no	—
105	#803	AGL(1,𝔽₅)	1	1	yes	characteristic
106	C₄×C₃⋊C₈	—	5	1	no	—
107	C₄×C₃⋊C₈	—	5	1	no	—
108	C₂×C₄×D₅	Dic₆	1	1	yes	characteristic
109	C₂×C₄×D₅	C₄×D₃	1	1	yes	characteristic
110	C₂×C₄×D₅	#67	1	1	yes	characteristic
111	C₂×#170	—	2	1	no	—
112	C₂×#170	—	2	1	no	—
113	C₂×#170	—	6	1	no	—
114	C₂×#170	—	6	1	no	—
115	D₅⋊C₈	—	12	2	no	—
116	C₂×C₅×C₈	—	3	1	no	—
117	C₂×C₅×C₈	—	3	1	no	—
118	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
119	C₄×Dic₅	Dic₆	1	1	yes	characteristic
120	C₄×Dic₅	#67	1	1	yes	characteristic
121	C₄×Dic₅	C₂²×D₃	1	1	yes	characteristic
122	C₂×#168	—	3	1	no	—
123	C₂×#168	—	3	1	no	—
124	#676	—	3	4	no	—
125	#676	—	6	2	no	—
126	#431	—	5	1	no	—
127	C₂×C₄⋊C₈	—	15	1	no	—
128	C₄×M₁₆	—	15	1	no	—
129	#335	—	15	1	no	—
130	#335	—	15	1	no	—
131	#334	—	15	1	no	—
132	#334	—	15	1	no	—
133	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₃×D₅	—	2	1	no	—
135	C₂×C₃×D₅	—	2	1	no	—
136	C₂×C₃×D₅	C₄⋊C₈	1	2	yes	—
137	C₃×C₄×C₅	—	2	1	no	—
138	C₃×C₄×C₅	—	2	1	no	—
139	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
140	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
141	C₃×Dic₅	—	2	1	no	—
142	C₃×Dic₅	C₄⋊C₈	1	2	yes	—
143	C₃×Dic₅	AGL(1,ℤ/8)	1	1	yes	characteristic
144	C₃×Dic₅	#137	1	1	yes	characteristic
145	C₂²×C₃×C₄	—	5	1	no	—
146	C₂²×C₃×C₄	—	5	1	no	—
147	C₂²×C₃×C₄	—	5	1	no	—
148	C₂×C₃×C₈	—	10	1	no	—
149	C₂×C₃×C₈	—	10	1	no	—
150	C₃×C₄²	—	5	1	no	—
151	C₃×C₄²	—	5	1	no	—
152	C₃×C₄²	—	5	1	no	—
153	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
154	#208	—	5	4	no	—
155	#208	—	10	2	no	—
156	C₂×C₃⋊C₈	—	5	1	no	—
157	C₂×C₃⋊C₈	—	5	1	no	—
158	C₂×C₃⋊C₈	—	10	1	no	—
159	C₂×C₃⋊C₈	—	10	1	no	—
160	C₂×C₃⋊C₈	—	20	2	no	—
161	C₂×C₃⋊C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
162	C₂×C₃⋊C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
163	C₂²×D₅	Dic₃⋊C₄	1	1	yes	characteristic
164	C₂×C₄×C₅	Dic₃⋊C₄	1	1	yes	characteristic
165	C₂×C₄×C₅	Dic₃⋊C₄	1	1	yes	characteristic
166	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
167	C₂×Dic₅	Dic₃⋊C₄	1	1	yes	characteristic
168	C₂×Dic₅	C₂×Dic₆	1	1	yes	characteristic
169	C₂×Dic₅	C₂×#67	1	1	yes	characteristic
170	C₄×D₅	—	2	1	no	—
171	C₄×D₅	—	2	1	no	—
172	C₄×D₅	—	4	1	no	—
173	C₄×D₅	D₃×C₈	1	2	yes	—
174	C₄×D₅	ΓL(1,𝔽₂₅)	1	2	yes	—
175	#170	—	4	1	no	—
176	#170	—	4	1	no	—
177	#170	—	12	1	no	—
178	#170	—	12	1	no	—
179	C₅×C₈	—	3	2	no	—
180	C₅×C₈	—	6	1	no	—
181	#168	—	3	2	no	—
182	#168	—	6	1	no	—
183	C₂×M₁₆	—	15	1	no	—
184	C₂×M₁₆	—	15	1	no	—
185	C₂²×C₈	—	30	1	no	—
186	C₂×C₄²	—	5	1	no	—
187	C₄⋊C₈	—	5	1	no	—
188	C₄⋊C₈	—	5	1	no	—
189	C₄⋊C₈	—	15	1	no	—
190	C₄⋊C₈	—	15	1	no	—
191	C₄⋊C₈	—	30	1	no	—
192	C₂²⋊C₈	—	10	1	no	—
193	#97	—	3	1	no	—
194	#97	—	15	1	no	—
195	C₄×C₈	—	10	1	no	—
196	C₄×C₈	—	15	1	no	—
197	C₄×C₈	—	15	1	no	—
198	C₂×C₃×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
199	C₂×C₃×C₅	#428	1	1	yes	characteristic
200	C₂×C₃×C₅	#430	1	1	yes	characteristic
201	C₃×D₅	—	2	2	no	—
202	C₂³×C₃	—	5	1	no	—
203	C₂×C₃×C₄	—	5	1	no	—
204	C₂×C₃×C₄	—	5	1	no	—
205	C₂×C₃×C₄	—	5	2	no	—
206	C₂×C₃×C₄	—	5	2	no	—
207	C₂×C₃×C₄	—	5	1	no	—
208	C₂×C₃×C₄	—	10	1	no	—
209	C₂×C₃×C₄	—	10	1	no	—
210	C₂×C₃×C₄	—	20	1	no	—
211	C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
212	C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
214	C₃×C₈	—	20	1	no	—
215	C₃×C₈	—	20	1	no	—
216	C₃⋊C₈	—	2	1	no	—
217	C₃⋊C₈	—	5	2	no	—
218	C₃⋊C₈	—	10	1	no	—
219	C₃⋊C₈	—	20	1	no	—
220	C₃⋊C₈	—	20	1	no	—
221	C₃⋊C₈	D₅⋊C₈	1	2	yes	complemented
222	C₂²×C₅	C₂×Dic₃⋊C₄	1	1	yes	characteristic
223	C₂×D₅	—	2	1	no	—
224	C₂×D₅	—	2	1	no	—
225	C₂×D₅	Dic₃⋊C₈	1	2	yes	—
226	C₄×C₅	—	2	1	no	—
227	C₄×C₅	—	2	1	no	—
228	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
229	C₄×C₅	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
230	Dic₅	—	2	1	no	—
231	Dic₅	Dic₃⋊C₈	1	2	yes	—
232	Dic₅	#949	1	1	yes	characteristic
233	Dic₅	#951	1	1	yes	characteristic
234	C₂²×C₄	—	5	1	no	—
235	C₂²×C₄	—	5	1	no	—
236	C₂²×C₄	—	5	1	no	—
237	M₁₆	—	15	4	no	—
238	M₁₆	—	30	2	no	—
239	C₂×C₈	—	3	1	no	—
240	C₂×C₈	—	3	1	no	—
241	C₂×C₈	—	10	1	no	—
242	C₂×C₈	—	10	1	no	—
243	C₂×C₈	—	15	1	no	—
244	C₂×C₈	—	15	1	no	—
245	C₂×C₈	—	30	1	no	—
246	C₂×C₈	—	30	1	no	—
247	C₂×C₈	—	60	2	no	—
248	C₄²	—	5	1	no	—
249	C₄²	—	5	1	no	—
250	C₄²	—	5	1	no	—
251	C₄²	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
252	C₃×C₅	#1565	1	1	yes	characteristic, complemented, Hall
253	C₂²×C₃	—	5	2	no	—
254	C₂²×C₃	—	10	1	no	—
255	C₂²×C₃	—	10	1	no	—
256	C₂²×C₃	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
257	C₃×C₄	—	2	1	no	—
258	C₃×C₄	—	2	1	no	—
259	C₃×C₄	—	5	2	no	—
260	C₃×C₄	—	5	1	no	—
261	C₃×C₄	—	5	1	no	—
262	C₃×C₄	—	10	1	no	—
263	C₃×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
264	C₃×C₄	D₅⋊M₁₆	1	1	yes	characteristic
265	C₂×C₅	#4995	1	1	yes	characteristic
266	C₂×C₅	C₂×Dic₃⋊C₈	1	1	yes	characteristic
267	C₂×C₅	#5124	1	1	yes	characteristic
268	D₅	—	2	2	no	—
269	C₂³	—	5	1	no	—
270	C₂×C₄	—	5	1	no	—
271	C₂×C₄	—	5	1	no	—
272	C₂×C₄	—	5	2	no	—
273	C₂×C₄	—	5	2	no	—
274	C₂×C₄	—	5	1	no	—
275	C₂×C₄	—	10	1	no	—
276	C₂×C₄	—	10	1	no	—
277	C₂×C₄	—	20	1	no	—
278	C₂×C₄	Dic₃⋊AGL(1,𝔽₅)	1	1	yes	characteristic
279	C₂×C₄	Dic₃⋊AGL(1,𝔽₅)	1	1	yes	characteristic
280	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
281	C₈	—	3	2	no	—
282	C₈	—	6	1	no	—
283	C₈	—	15	2	no	—
284	C₈	—	20	1	no	—
285	C₈	—	20	1	no	—
286	C₈	—	30	1	no	—
287	C₈	—	60	1	no	—
288	C₈	—	60	1	no	—
289	C₂×C₃	—	10	2	no	—
290	C₂×C₃	D₅⋊(C₄⋊C₈)	1	1	yes	characteristic
291	C₂×C₃	#65727	1	1	yes	characteristic
292	C₂×C₃	#65729	1	1	yes	characteristic
293	C₅	#70746	1	1	yes	characteristic, complemented, Sylow
294	C₂²	—	5	2	no	—
295	C₂²	—	10	1	no	—
296	C₂²	—	10	1	no	—
297	C₂²	C₂×Dic₃⋊AGL(1,𝔽₅)	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	5	1	no	—
301	C₄	—	5	1	no	—
302	C₄	—	5	2	no	—
303	C₄	—	10	1	no	—
304	C₄	D₃×D₅⋊C₈	1	1	yes	characteristic
305	C₄	D₅⋊ΓL(1,𝔽₂₅)	1	1	yes	characteristic
306	C₃	#10603865	1	1	yes	characteristic, complemented, Sylow
307	C₂	—	10	2	no	—
308	C₂	#11751728	1	1	yes	characteristic
309	C₂	#11751732	1	1	yes	characteristic
310	C₂	D₅⋊(Dic₃⋊C₈)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×Dic₃⋊AGL(1,𝔽₅)
commutator	C₃×C₄×C₅	C₂²×C₈
Frattini	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)
Fitting	C₃×C₄²×C₅	C₂×C₄
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂²×C₃×D₅	C₄⋊C₄
cosocle	C₃×C₄×Dic₅	C₂³
hypercenter	C₄²	D₃×AGL(1,𝔽₅)
hypocenter	C₃×C₅	#1565
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
4. #11746515	C₂	semidirect
5. #11746514	C₂	semidirect
6. #11746513	C₂	semidirect
7. #11746512	C₂	semidirect
49. #6649	C₈	semidirect
90. C₃⋊#168	C₂×C₈	semidirect
92. C₅×C₃⋊C₈	C₂×C₈	semidirect
221. C₃⋊C₈	D₅⋊C₈	semidirect
252. C₃×C₅	#1565	semidirect
293. C₅	#70746	semidirect
306. C₃	#10603865	semidirect

Maximal subgroups

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

subgroup	index
D₅⋊(C₄⋊(C₃⋊C₈))	2
C₃×#65680	2
#11747003	2
#11746515	2
#11746514	2
#11746513	2
#11746512	2
#10603865	3
#70746	5