Finite Group Databasebeta

← #14295379

Subgroups — #14295379

Order 1920.

824 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4027C211yesmaximal, characteristic
2C2×#91240C212yesmaximal
3C4×#6763C212yesmaximal, complemented
4#11746788C212yesmaximal, complemented
5#10604216—31nomaximal, Hall
6C22×C3×C22⋊D5C2211yescharacteristic
7C23×C3×Dic5C411yescharacteristic, complemented
8C22×C3×C4×D5C2211yescharacteristic
9C2×#6763C2214yes—
10C2×C4×ΓL(1,𝔽16)—22no—
11C2×C3×C5×C22⋊C4C411yescharacteristic, complemented
12C2×C3×#685C2211yescharacteristic
13C2×C3×#683C411yescharacteristic, complemented
14C2×C3×C4×Dic5C411yescharacteristic, complemented
15C3×#4027—22no—
16C3×#4027—22no—
17#91240C418yescomplemented
18#91235—22no—
19C4×#834—51nomaximal, Hall
20C2×#4027D311yescharacteristic
21C2×#4011—32no—
22C4×C22⋊AGL(1,𝔽5)—32no—
23#64917—32no—
24C24×C3×C5C2×C411yescharacteristic
25C23×C3×D5C2311yescharacteristic
26C22×ΓL(1,𝔽16)—24no—
27C22×C3×C4×C5C2×C411yescharacteristic
28C22×C3×C4×C5C2×C411yescharacteristic
29C2×C3×C22⋊D5—22no—
30C2×C3×C22⋊D5—22no—
31C2×C3×C22⋊D5C2×C414yes—
32C22×C3×Dic5—21no—
33C22×C3×Dic5—21no—
34C22×C3×Dic5—42no—
35C22×C3×Dic5C2×C411yescharacteristic
36C22×C3×Dic5C2×C411yescharacteristic
37C22×C3×Dic5C2×C411yescharacteristic
38C2×C3×C4×D5—42no—
39C2×C3×C4×D5D412yes—
40C2×C3×C4×D5D412yes—
41#6763—24no—
42#6763C2×C418yescomplemented
43C4×ΓL(1,𝔽16)—44no—
44C3×C5×C22⋊C4—22no—
45C3×#685—22no—
46C3×#683—22no—
47C3×C4×Dic5—21no—
48C3×C4×Dic5—21no—
49C2×C3×C4×D4—51no—
50C2×#834—52no—
51C4×#217—52no—
52#4526—52no—
53C22×C22⋊D5C2×D311yescharacteristic
54C23×Dic5Dic311yescharacteristic, complemented
55C22×C4×D5C2×D311yescharacteristic
56C2×C22⋊AGL(1,𝔽5)—34no—
57C2×C4×AGL(1,𝔽5)—62no—
58C2×C5×C22⋊C4Dic311yescharacteristic, complemented
59C2×#685C2×D311yescharacteristic
60C2×#683Dic311yescharacteristic, complemented
61C2×C4×Dic5Dic311yescharacteristic, complemented
62#4027—22no—
63#4027—22no—
64#4011—38no—
65(C2×AGL(1,𝔽5))⋊C4—62no—
66C4×#99—151noSylow
67C23×C3×C5—21no—
68C23×C3×C5—21no—
69C23×C3×C5—41no—
70C23×C3×C5—41no—
71C23×C3×C5C4212yescomplemented
72C23×C3×C5C22×C411yescharacteristic
73C22×C3×D5—41no—
74C22×C3×D5—41no—
75C22×C3×D5C22×C412yes—
76C22×C3×D5C2×D412yes—
77C22×C3×D5#4112yes—
78C2×ΓL(1,𝔽16)—28no—
79C2×ΓL(1,𝔽16)—44no—
80C2×ΓL(1,𝔽16)—44no—
81C2×C3×C4×C5—21no—
82C2×C3×C4×C5—41no—
83C2×C3×C4×C5—41no—
84C2×C3×C4×C5C22⋊C412yes—
85C3×C22⋊D5—28no—
86C2×C3×Dic5—21no—
87C2×C3×Dic5—21no—
88C2×C3×Dic5—22no—
89C2×C3×Dic5—21no—
90C2×C3×Dic5—42no—
91C2×C3×Dic5—42no—
92C2×C3×Dic5—41no—
93C2×C3×Dic5C4212yescomplemented
94C2×C3×Dic5C22⋊C411yescharacteristic
95C2×C3×Dic5C22⋊C411yescharacteristic
96C3×C4×D5—24no—
97C3×C4×D5—42no—
98C22×C3×D4—51no—
99C23×C3×C4—51no—
100C23×C3×C4—51no—
101C3×C4×D4—102no—
102C3×C4×D4—102no—
103C2×C3×C4⋊C4—51no—
104C2×C3×C22⋊C4—51no—
105C2×C3×C22⋊C4AGL(1,𝔽5)11yescharacteristic, complemented
106C2×C3×C42—51no—
107C2×#217—54no—
108C2×C4×Dic3—102no—
109#834—58no—
110C3⋊#95—102no—
111C24×C5C2×Dic311yescharacteristic
112C23×D5C22×D311yescharacteristic
113C22×AGL(1,𝔽5)—64no—
114C22×C4×C5C2×Dic311yescharacteristic
115C22×C4×C5C2×Dic311yescharacteristic
116C2×C22⋊D5—22no—
117C2×C22⋊D5—22no—
118C2×C22⋊D5C4×D314yes—
119C22×Dic5—21no—
120C22×Dic5—21no—
121C22×Dic5—42no—
122C22×Dic5C2×Dic311yescharacteristic
123C22×Dic5C2×Dic311yescharacteristic
124C22×Dic5C2×Dic311yescharacteristic
125C2×C4×D5—42no—
126C2×C4×D5#6712yes—
127C2×C4×D5#6712yes—
128C22⋊AGL(1,𝔽5)—38no—
129C22⋊AGL(1,𝔽5)—64no—
130C4×AGL(1,𝔽5)—124no—
131C5×C22⋊C4—22no—
132#685—22no—
133#683—22no—
134C4×Dic5—21no—
135C4×Dic5—21no—
136C2×C4×D4—51no—
137C2×#99—152no—
138C4×C22⋊C4—152no—
139#342—152no—
140C22×C3×C5—21no—
141C22×C3×C5—21no—
142C22×C3×C5—22no—
143C22×C3×C5—42no—
144C22×C3×C5—42no—
145C22×C3×C5—41no—
146C22×C3×C5—41no—
147C22×C3×C5C2×C4211yescharacteristic
148C22×C3×C5C2×C22⋊C411yescharacteristic
149C22×C3×C5#11711yescharacteristic
150C2×C3×D5—22no—
151C2×C3×D5—22no—
152C2×C3×D5—42no—
153C2×C3×D5—42no—
154C2×C3×D5C4×D414yes—
155ΓL(1,𝔽16)—48no—
156C3×C4×C5—22no—
157C3×C4×C5—41no—
158C3×Dic5—22no—
159C3×Dic5—21no—
160C3×Dic5—41no—
161C3×Dic5#9912yescomplemented
162C24×C3—51no—
163C24×C3C2×AGL(1,𝔽5)11yescharacteristic
164C2×C3×D4—54no—
165C2×C3×D4—102no—
166C2×C3×D4—102no—
167C22×C3×C4—51no—
168C22×C3×C4—51no—
169C22×C3×C4—52no—
170C22×C3×C4—52no—
171C22×C3×C4—51no—
172C22×C3×C4—101no—
173C22×C3×C4—101no—
174C22×C3×C4—202no—
175C22×C3×C4—202no—
176C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
177C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
178C22×Dic3—104no—
179C3×C4⋊C4—102no—
180C3×C22⋊C4—22no—
181C3×C22⋊C4—102no—
182C3×C42—101no—
183C3×C42—101no—
184#217—58no—
185#217—104no—
186C4×Dic3—204no—
187C23×C5—21no—
188C23×C5—21no—
189C23×C5—41no—
190C23×C5—41no—
191C23×C5C4×Dic312yescomplemented
192C23×C5C22×Dic311yescharacteristic
193C22×D5—41no—
194C22×D5—41no—
195C22×D5C2×C4×D312yes—
196C22×D5#23512yes—
197C22×D5C2×#6712yes—
198C2×AGL(1,𝔽5)—68no—
199C2×AGL(1,𝔽5)—124no—
200C2×AGL(1,𝔽5)—124no—
201C2×C4×C5—21no—
202C2×C4×C5—41no—
203C2×C4×C5—41no—
204C2×C4×C5#21712yes—
205C22⋊D5—28no—
206C2×Dic5—21no—
207C2×Dic5—21no—
208C2×Dic5—22no—
209C2×Dic5—21no—
210C2×Dic5—42no—
211C2×Dic5—42no—
212C2×Dic5—41no—
213C2×Dic5C4×Dic312yescomplemented
214C2×Dic5#21711yescharacteristic
215C2×Dic5#21711yescharacteristic
216C4×D5—24no—
217C4×D5—42no—
218C22×D4—51no—
219C23×C4—51no—
220C23×C4—51no—
221C4×D4—102no—
222C4×D4—102no—
223C2×C4⋊C4—51no—
224C2×C22⋊C4—51no—
225C2×C22⋊C4—154no—
226C2×C22⋊C4ΓL(1,𝔽16)11yescharacteristic, complemented
227C2×C42—51no—
228C2×C42—302no—
229#99—158no—
230#95—302no—
231C2×C3×C5—21no—
232C2×C3×C5—21no—
233C2×C3×C5—42no—
234C2×C3×C5C4×C22⋊C411yescharacteristic
235C2×C3×C5C2×#9911yescharacteristic
236C2×C3×C5#41011yescharacteristic
237C3×D5—24no—
238C23×C3—21no—
239C23×C3—21no—
240C23×C3—41no—
241C23×C3—41no—
242C23×C3—52no—
243C23×C3—52no—
244C23×C3—52no—
245C23×C3—201no—
246C23×C3—201no—
247C23×C3C4×AGL(1,𝔽5)12yescomplemented
248C23×C3C22×AGL(1,𝔽5)11yescharacteristic
249C3×D4—108no—
250C2×C3×C4—21no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—51no—
254C2×C3×C4—52no—
255C2×C3×C4—51no—
256C2×C3×C4—104no—
257C2×C3×C4—102no—
258C2×C3×C4—101no—
259C2×C3×C4—101no—
260C2×C3×C4—101no—
261C2×C3×C4—202no—
262C2×C3×C4—202no—
263C2×C3×C4—202no—
264C2×C3×C4—201no—
265C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
266C2×Dic3—108no—
267C2×Dic3—204no—
268C2×Dic3—204no—
269C22×C5—21no—
270C22×C5—21no—
271C22×C5—22no—
272C22×C5—42no—
273C22×C5—42no—
274C22×C5—41no—
275C22×C5—41no—
276C22×C5C2×C4×Dic311yescharacteristic
277C22×C5#92611yescharacteristic
278C22×C5C2×#21711yescharacteristic
279C2×D5—22no—
280C2×D5—22no—
281C2×D5—42no—
282C2×D5—42no—
283C2×D5C4×#6714yes—
284AGL(1,𝔽5)—128no—
285C4×C5—22no—
286C4×C5—41no—
287Dic5—22no—
288Dic5—21no—
289Dic5—41no—
290Dic5#83412yescomplemented
291C24—51no—
292C24C2×ΓL(1,𝔽16)11yescharacteristic
293C2×D4—54no—
294C2×D4—102no—
295C2×D4—102no—
296C22×C4—51no—
297C22×C4—51no—
298C22×C4—52no—
299C22×C4—52no—
300C22×C4—51no—
301C22×C4—101no—
302C22×C4—101no—
303C22×C4—202no—
304C22×C4—202no—
305C22×C4—304no—
306C22×C4C2×ΓL(1,𝔽16)11yescharacteristic
307C22×C4C2×ΓL(1,𝔽16)11yescharacteristic
308C4⋊C4—102no—
309C22⋊C4—22no—
310C22⋊C4—102no—
311C22⋊C4—158no—
312C22⋊C4—304no—
313C42—101no—
314C42—101no—
315C42—604no—
316C3×C5C4×#9911yescharacteristic, complemented, Hall
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—22no—
320C22×C3—42no—
321C22×C3—42no—
322C22×C3—41no—
323C22×C3—41no—
324C22×C3—54no—
325C22×C3—102no—
326C22×C3—102no—
327C22×C3—202no—
328C22×C3—202no—
329C22×C3C2×C4×AGL(1,𝔽5)11yescharacteristic
330C22×C3#413011yescharacteristic
331C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
332C3×C4—22no—
333C3×C4—41no—
334C3×C4—52no—
335C3×C4—102no—
336C3×C4—101no—
337C3×C4—201no—
338Dic3—208no—
339C2×C5—21no—
340C2×C5—21no—
341C2×C5—42no—
342C2×C5C4×#21711yescharacteristic
343C2×C5C2×#83411yescharacteristic
344C2×C5#524011yescharacteristic
345D5—24no—
346C23—21no—
347C23—21no—
348C23—41no—
349C23—41no—
350C23—52no—
351C23—52no—
352C23—52no—
353C23—201no—
354C23—201no—
355C23C4×ΓL(1,𝔽16)12yescomplemented
356C23C22×ΓL(1,𝔽16)11yescharacteristic
357D4—108no—
358C2×C4—21no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—51no—
362C2×C4—52no—
363C2×C4—51no—
364C2×C4—104no—
365C2×C4—102no—
366C2×C4—101no—
367C2×C4—101no—
368C2×C4—101no—
369C2×C4—202no—
370C2×C4—202no—
371C2×C4—202no—
372C2×C4—201no—
373C2×C4—308no—
374C2×C4—604no—
375C2×C4—604no—
376C2×C4#676312yes—
377C2×C3—21no—
378C2×C3—21no—
379C2×C3—42no—
380C2×C3—104no—
381C2×C3C4×C22⋊AGL(1,𝔽5)11yescharacteristic
382C2×C3D5⋊#9911yescharacteristic
383C2×C3#6577111yescharacteristic
384C5C4×#83411yescharacteristic, complemented, Sylow
385C22—21no—
386C22—21no—
387C22—22no—
388C22—42no—
389C22—42no—
390C22—41no—
391C22—41no—
392C22—54no—
393C22—102no—
394C22—102no—
395C22—202no—
396C22—202no—
397C22C2×C4×ΓL(1,𝔽16)11yescharacteristic
398C22#9198911yescharacteristic
399C22C2×#676311yescharacteristic
400C4—22no—
401C4—41no—
402C4—52no—
403C4—102no—
404C4—101no—
405C4—201no—
406C4—608no—
407C3#1060421611yescharacteristic, complemented, Sylow
408C2—21no—
409C2—21no—
410C2—42no—
411C2—104no—
412C2C4×#676311yescharacteristic
413C2D5⋊#83411yescharacteristic
414C2#1175243011yescharacteristic
subgroupG/H
centerC22C2×#6763
commutatorC22×C3×C5C2×C42
FrattiniC23C22×ΓL(1,𝔽16)
FittingC2×C3×C5×C22⋊C4C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC22×C3×C22⋊D5C22
cosocleC23×C3×D5C23
hypercenterC2×C22⋊C4ΓL(1,𝔽16)
hypocenterC3×C5C4×#99
LCSG ▷ C22×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C4×#6763C2semidirect
4. #11746788C2semidirect
7. C23×C3×Dic5C4semidirect
11. C2×C3×C5×C22⋊C4C4semidirect
13. C2×C3×#683C4semidirect
14. C2×C3×C4×Dic5C4semidirect
17. #91240C4semidirect
42. #6763C2×C4semidirect
54. C23×Dic5Dic3semidirect
58. C2×C5×C22⋊C4Dic3semidirect
60. C2×#683Dic3semidirect
61. C2×C4×Dic5Dic3semidirect
71. C23×C3×C5C42semidirect
93. C2×C3×Dic5C42semidirect
105. C2×C3×C22⋊C4AGL(1,𝔽5)semidirect
161. C3×Dic5#99semidirect
191. C23×C5C4×Dic3semidirect
213. C2×Dic5C4×Dic3semidirect
226. C2×C22⋊C4ΓL(1,𝔽16)semidirect
247. C23×C3C4×AGL(1,𝔽5)semidirect
290. Dic5#834semidirect
316. C3×C5C4×#99semidirect
355. C23C4×ΓL(1,𝔽16)semidirect
384. C5C4×#834semidirect
407. C3#10604216semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#40272
C2×#912402
C4×#67632
#117467882
#106042163
C4×#8345