Subgroups — #14280662

Order 1920.

380 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11751795	C₂	1	1	yes	maximal, characteristic
2	#11751795	C₂	1	1	yes	maximal, characteristic
3	Dic₅×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic
4	#11746751	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#170⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#11746485	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃⋊C₈×#170	C₂	1	1	yes	maximal, characteristic, complemented
8	#10604736	—	3	1	no	maximal, Hall
9	C₂×C₅×ΓL(1,𝔽₂₅)	C₄	1	1	yes	characteristic
10	C₄×D₃×Dic₅	C₂²	1	1	yes	characteristic
11	C₂×#6650	C₄	1	1	yes	characteristic
12	C₄×C₃⋊#170	C₂²	1	1	yes	characteristic
13	C₄×C₃⋊#170	C₂²	1	1	yes	characteristic
14	C₃×C₄×#170	C₂²	1	1	yes	characteristic
15	C₃×C₄×#170	C₂²	1	1	yes	characteristic
16	#91177	C₄	1	2	yes	—
17	#91177	C₄	1	2	yes	—
18	(C₂×D₃)⋊#170	C₄	1	2	yes	—
19	(C₂×D₃)⋊#170	C₄	1	2	yes	—
20	#91099	C₄	1	1	yes	characteristic
21	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
22	#90954	C₄	1	1	yes	characteristic
23	Dic₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
24	#68072	—	5	1	no	maximal, Hall
25	#65754	—	3	1	no	—
26	#65754	—	3	1	no	—
27	M₁₆×Dic₅	—	3	1	no	—
28	#170⋊C₈	—	3	1	no	—
29	#170⋊C₈	D₃	1	1	yes	characteristic, complemented
30	#64879	—	3	1	no	—
31	C₈×#170	—	3	1	no	—
32	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
33	C₂×D₃×Dic₅	C₂×C₄	1	1	yes	characteristic
34	C₂×D₃×Dic₅	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃⋊#170	C₂×C₄	1	2	yes	—
36	C₂×C₃⋊#170	C₂×C₄	1	2	yes	—
37	C₂×C₃×#170	C₂×C₄	1	2	yes	—
38	C₂×C₃×#170	C₂×C₄	1	2	yes	—
39	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
40	C₄×Dic₁₅	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
42	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
43	C₅×ΓL(1,𝔽₂₅)	—	2	2	no	—
44	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
45	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
46	Dic₃×Dic₅	C₂×C₄	1	1	yes	characteristic
47	Dic₃×Dic₅	C₂×C₄	1	1	yes	characteristic
48	#6650	—	2	2	no	—
49	C₄×ΓL(1,𝔽₂₅)	—	5	1	no	—
50	#4685	—	5	1	no	—
51	#4685	—	5	1	no	—
52	C₃×#322	—	5	1	no	—
53	#4454	—	5	1	no	—
54	#4454	—	5	1	no	—
55	C₈×C₃⋊C₈	—	5	1	no	—
56	C₂×C₅×M₁₆	—	3	1	no	—
57	C₂×C₄×Dic₅	—	3	1	no	—
58	C₂×#681	—	3	1	no	—
59	C₂²⋊#170	—	3	2	no	—
60	C₂²⋊#170	—	3	2	no	—
61	#4004	—	3	2	no	—
62	#4004	—	3	2	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	#3947	—	3	1	no	—
68	#3947	—	3	1	no	—
69	C₈×Dic₅	—	3	1	no	—
70	C₈×Dic₅	C₂×D₃	1	1	yes	characteristic
71	#1453	—	15	1	no	Sylow
72	C₂²×C₅×D₃	C₄²	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂×Dic₁₅	C₄²	1	1	yes	characteristic
75	C₂×Dic₁₅	C₄²	1	1	yes	characteristic
76	C₂×C₅×Dic₃	C₄²	1	1	yes	characteristic
77	C₄×C₅×D₃	—	2	1	no	—
78	C₄×C₅×D₃	—	2	1	no	—
79	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
81	D₃×Dic₅	—	2	2	no	—
82	D₃×Dic₅	—	2	2	no	—
83	C₃⋊#170	—	2	2	no	—
84	C₃⋊#170	—	2	2	no	—
85	C₃×#170	—	2	2	no	—
86	C₃×#170	—	2	2	no	—
87	C₃×C₅×C₈	—	2	1	no	—
88	C₃⋊#168	M₁₆	1	2	yes	—
89	C₃×#168	—	2	1	no	—
90	C₅×C₃⋊C₈	M₁₆	1	2	yes	—
91	C₂×ΓL(1,𝔽₂₅)	—	5	1	no	—
92	C₂×ΓL(1,𝔽₂₅)	AGL(1,𝔽₅)	1	1	yes	characteristic
93	C₄²×D₃	—	5	1	no	—
94	C₃×C₄×C₈	—	5	1	no	—
95	C₃×C₄×C₈	—	5	1	no	—
96	C₃×C₄×C₈	—	5	1	no	—
97	(C₂×D₃)⋊C₈	—	5	2	no	—
98	(C₂×D₃)⋊C₈	—	5	2	no	—
99	#815	—	5	1	no	—
100	#815	—	5	1	no	—
101	Dic₃⋊C₈	—	5	2	no	—
102	Dic₃⋊C₈	—	5	2	no	—
103	C₄×C₃⋊C₈	—	5	1	no	—
104	C₄×C₃⋊C₈	—	5	1	no	—
105	C₄×C₃⋊C₈	—	5	1	no	—
106	C₂²×C₄×C₅	—	3	1	no	—
107	C₂²×Dic₅	—	3	1	no	—
108	C₂²×Dic₅	—	3	1	no	—
109	C₂×#170	—	3	2	no	—
110	C₂×#170	—	3	2	no	—
111	C₂×#170	C₄×D₃	1	2	yes	—
112	C₂×#170	C₄×D₃	1	2	yes	—
113	C₅×M₁₆	—	6	2	no	—
114	C₂×C₅×C₈	—	3	1	no	—
115	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
116	C₄×Dic₅	—	3	1	no	—
117	C₄×Dic₅	—	3	1	no	—
118	C₄×Dic₅	—	3	1	no	—
119	C₄×Dic₅	C₂²×D₃	1	1	yes	characteristic
120	#681	—	6	2	no	—
121	C₂×#168	—	3	1	no	—
122	C₂×#168	C₄×D₃	1	1	yes	characteristic
123	#431	—	15	1	no	—
124	#431	—	15	1	no	—
125	C₄×M₁₆	—	15	1	no	—
126	#322	—	5	1	no	—
127	#322	—	15	1	no	—
128	#322	—	15	1	no	—
129	C₈²	—	15	1	no	—
130	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
131	C₂×C₅×D₃	—	2	1	no	—
132	C₂×C₅×D₃	—	2	1	no	—
133	C₂×C₅×D₃	—	2	1	no	—
134	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
135	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
136	Dic₁₅	—	2	1	no	—
137	Dic₁₅	—	2	1	no	—
138	C₃×Dic₅	#131	1	2	yes	—
139	C₃×Dic₅	#131	1	2	yes	—
140	C₅×Dic₃	—	2	1	no	—
141	C₂×C₄×D₃	—	5	1	no	—
142	C₂×C₄×D₃	—	5	1	no	—
143	C₂×C₄×D₃	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
144	C₂×C₃×C₈	—	5	2	no	—
145	C₂×C₃×C₈	—	5	2	no	—
146	C₂×C₃×C₈	—	5	1	no	—
147	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
148	C₃×C₄²	—	5	1	no	—
149	C₄×Dic₃	—	5	1	no	—
150	C₄×Dic₃	—	5	1	no	—
151	C₄×Dic₃	—	5	1	no	—
152	C₂×C₃⋊C₈	—	5	1	no	—
153	C₂×C₃⋊C₈	—	5	2	no	—
154	C₂×C₃⋊C₈	—	5	2	no	—
155	C₂×C₃⋊C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
156	ΓL(1,𝔽₂₅)	—	2	2	no	—
157	ΓL(1,𝔽₂₅)	—	10	2	no	—
158	C₂³×C₅	—	3	1	no	—
159	C₂×C₄×C₅	—	3	1	no	—
160	C₂×C₄×C₅	—	6	1	no	—
161	C₂×C₄×C₅	—	6	1	no	—
162	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
163	C₂×Dic₅	—	3	1	no	—
164	C₂×Dic₅	—	3	1	no	—
165	C₂×Dic₅	—	6	2	no	—
166	C₂×Dic₅	—	6	2	no	—
167	C₂×Dic₅	C₂×C₄×D₃	1	1	yes	characteristic
168	C₂×Dic₅	C₂×C₄×D₃	1	1	yes	characteristic
169	#170	—	2	2	no	—
170	#170	—	2	2	no	—
171	#170	—	6	2	no	—
172	#170	—	6	2	no	—
173	C₅×C₈	—	2	1	no	—
174	C₅×C₈	—	3	2	no	—
175	#168	—	2	1	no	—
176	#168	—	3	2	no	—
177	C₂×M₁₆	—	3	1	no	—
178	C₂×M₁₆	—	15	1	no	—
179	C₂×C₄²	—	15	1	no	—
180	C₄⋊C₈	—	15	2	no	—
181	C₄⋊C₈	—	15	2	no	—
182	C₂²⋊C₈	—	15	2	no	—
183	C₂²⋊C₈	—	15	2	no	—
184	#97	—	15	1	no	—
185	#97	—	15	1	no	—
186	C₄×C₈	—	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	—	15	1	no	—
192	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
193	C₂×C₃×C₅	#405	1	1	yes	characteristic
194	C₂×C₃×C₅	#405	1	1	yes	characteristic
195	C₅×D₃	—	4	1	no	—
196	C₂²×D₃	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
197	C₂×C₃×C₄	—	5	1	no	—
198	C₂×C₃×C₄	—	5	1	no	—
199	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
200	C₂×Dic₃	—	5	1	no	—
201	C₂×Dic₃	—	5	1	no	—
202	C₂×Dic₃	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
203	C₄×D₃	—	2	1	no	—
204	C₄×D₃	—	2	1	no	—
205	C₄×D₃	—	10	2	no	—
206	C₄×D₃	—	10	2	no	—
207	C₃×C₈	—	2	1	no	—
208	C₃×C₈	—	10	2	no	—
209	C₃×C₈	—	10	2	no	—
210	C₃×C₈	—	10	1	no	—
211	C₃⋊C₈	—	5	2	no	—
212	C₃⋊C₈	—	10	2	no	—
213	C₃⋊C₈	—	10	2	no	—
214	C₃⋊C₈	#704	1	2	yes	—
215	C₂²×C₅	—	6	1	no	—
216	C₂²×C₅	—	6	1	no	—
217	C₂²×C₅	—	6	1	no	—
218	C₂²×C₅	C₄²×D₃	1	1	yes	characteristic
219	C₄×C₅	—	6	1	no	—
220	C₄×C₅	D₃×M₁₆	1	1	yes	characteristic
221	C₄×C₅	D₃×M₁₆	1	1	yes	characteristic
222	Dic₅	—	6	1	no	—
223	Dic₅	—	6	1	no	—
224	Dic₅	#901	1	2	yes	—
225	Dic₅	#907	1	2	yes	—
226	C₂²×C₄	—	3	1	no	—
227	C₂²×C₄	—	15	1	no	—
228	C₂²×C₄	—	15	1	no	—
229	M₁₆	—	6	2	no	—
230	M₁₆	—	30	2	no	—
231	C₂×C₈	—	3	1	no	—
232	C₂×C₈	—	5	2	no	—
233	C₂×C₈	—	5	2	no	—
234	C₂×C₈	—	5	1	no	—
235	C₂×C₈	—	15	1	no	—
236	C₂×C₈	—	15	2	no	—
237	C₂×C₈	—	15	2	no	—
238	C₂×C₈	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
239	C₄²	—	5	1	no	—
240	C₄²	—	15	1	no	—
241	C₄²	—	15	1	no	—
242	C₄²	—	15	1	no	—
243	C₃×C₅	#1453	1	1	yes	characteristic, complemented, Hall
244	C₂²×C₃	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
245	C₂×D₃	—	2	1	no	—
246	C₂×D₃	—	2	1	no	—
247	C₂×D₃	—	2	1	no	—
248	C₃×C₄	—	5	2	no	—
249	C₃×C₄	—	5	2	no	—
250	C₃×C₄	D₅⋊M₁₆	1	1	yes	characteristic
251	C₃×C₄	C₂×#704	1	1	yes	characteristic
252	Dic₃	—	2	1	no	—
253	Dic₃	—	10	1	no	—
254	Dic₃	—	10	1	no	—
255	C₂×C₅	—	12	1	no	—
256	C₂×C₅	#4689	1	1	yes	characteristic
257	C₂×C₅	#4707	1	1	yes	characteristic
258	C₂×C₅	#4708	1	1	yes	characteristic
259	C₂³	—	3	1	no	—
260	C₂×C₄	—	3	1	no	—
261	C₂×C₄	—	5	1	no	—
262	C₂×C₄	—	5	1	no	—
263	C₂×C₄	—	6	1	no	—
264	C₂×C₄	—	6	1	no	—
265	C₂×C₄	—	15	1	no	—
266	C₂×C₄	—	15	1	no	—
267	C₂×C₄	—	30	2	no	—
268	C₂×C₄	—	30	2	no	—
269	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
270	C₈	—	2	1	no	—
271	C₈	—	3	2	no	—
272	C₈	—	10	2	no	—
273	C₈	—	10	2	no	—
274	C₈	—	10	1	no	—
275	C₈	—	15	2	no	—
276	C₈	—	30	2	no	—
277	C₈	—	30	2	no	—
278	C₂×C₃	#65728	1	1	yes	characteristic
279	C₂×C₃	#65728	1	1	yes	characteristic
280	C₂×C₃	C₄×#704	1	1	yes	characteristic
281	D₃	—	4	1	no	—
282	C₅	#68072	1	1	yes	characteristic, complemented, Sylow
283	C₂²	—	6	1	no	—
284	C₂²	—	6	1	no	—
285	C₂²	—	6	1	no	—
286	C₂²	C₄×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
287	C₄	—	5	2	no	—
288	C₄	—	5	2	no	—
289	C₄	—	6	1	no	—
290	C₄	—	30	1	no	—
291	C₄	—	30	1	no	—
292	C₄	#91915	1	1	yes	characteristic
293	C₄	D₃×#704	1	1	yes	characteristic
294	C₃	#10604736	1	1	yes	characteristic, complemented, Sylow
295	C₂	—	12	1	no	—
296	C₂	#11751671	1	1	yes	characteristic
297	C₂	#11751672	1	1	yes	characteristic
298	C₂	#11751907	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)
commutator, socle	C₂²×C₃×C₅	C₂×C₄²
Fitting	C₂×C₃×C₅×C₈	C₂×C₄
prime radical	C₂²×C₅×D₃	C₄²
cosocle	C₃×C₄×Dic₅	C₂³
hypercenter	C₂×C₈	D₃×AGL(1,𝔽₅)
hypocenter	C₃×C₅	#1453
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
4. #11746751	C₂	semidirect
5. C₃×#170⋊C₈	C₂	semidirect
6. #11746485	C₂	semidirect
7. C₃⋊C₈×#170	C₂	semidirect
29. #170⋊C₈	D₃	semidirect
243. C₃×C₅	#1453	semidirect
282. C₅	#68072	semidirect
294. C₃	#10604736	semidirect

Maximal subgroups

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

subgroup	index
#11751795	2
Dic₅×ΓL(1,𝔽₂₅)	2
#11746751	2
C₃×#170⋊C₈	2
#11746485	2
C₃⋊C₈×#170	2
#10604736	3
#68072	5