Finite Group Databasebeta

← #14330125

Subgroups — #14330125

Order 1920.

896 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D5×#870C211yesmaximal, characteristic
2#11752345C211yesmaximal, characteristic
3C3×#65686C211yesmaximal, characteristic
4#11751840C212yesmaximal
5#11751839C212yesmaximal
6#11751833C212yesmaximal
7AGL(1,𝔽5)×Dic3⋊C4C212yesmaximal
8#11751747C212yesmaximal
9#11751745C212yesmaximal
10#10617347—31nomaximal, Hall
11C2×C4×ΓL(1,𝔽16)C2212yes—
12C2×C3×C4×AGL(1,𝔽5)C2212yes—
13C2×Dic3⋊AGL(1,𝔽5)C2212yes—
14C2×Dic3⋊AGL(1,𝔽5)C2212yes—
15C2×Dic3×AGL(1,𝔽5)C2212yes—
16C2×Dic3×AGL(1,𝔽5)C2212yes—
17#91759C411yescharacteristic, complemented
18C5×#870C411yescharacteristic, complemented
19C3×C42×D5C2211yescharacteristic
20D5×C4⋊Dic3C2211yescharacteristic
21D5×C4⋊Dic3C2211yescharacteristic
22D5×Dic3⋊C4C2212yes—
23D5×Dic3⋊C4C2212yes—
24#91388C411yescharacteristic, complemented
25#91388C411yescharacteristic, complemented
26#91378C412yescomplemented
27#91378C412yescomplemented
28#91235C2211yescharacteristic
29#91235C2211yescharacteristic
30#91235C2212yes—
31C3×(C2×AGL(1,𝔽5))⋊C4C2211yescharacteristic
32C3×(C2×AGL(1,𝔽5))⋊C4C2211yescharacteristic
33C3×(C2×AGL(1,𝔽5))⋊C4C2212yes—
34(C2×AGL(1,𝔽5))⋊Dic3C2212yes—
35(C2×AGL(1,𝔽5))⋊Dic3C2212yes—
36(C2×AGL(1,𝔽5))⋊Dic3C2212yes—
37(C2×AGL(1,𝔽5))⋊Dic3C2212yes—
38#73394—51nomaximal, Hall
39D5×#125—31no—
40#65711—32no—
41#65711—32no—
42#65711—32no—
43#65711—32no—
44C4⋊C4×AGL(1,𝔽5)—32no—
45C4⋊C4×AGL(1,𝔽5)—32no—
46#65686—31no—
47#65686D311yescharacteristic
48C22×ΓL(1,𝔽16)C2312yes—
49C22×ΓL(1,𝔽16)C2312yes—
50C22×C3×AGL(1,𝔽5)C2312yes—
51C22×C3×AGL(1,𝔽5)C2312yes—
52C2×C3×C4×D5C2311yescharacteristic
53C2×C3×C4×D5C2311yescharacteristic
54C2×C3×C4×D5C2311yescharacteristic
55C2×D5×Dic3C2312yes—
56C2×D5×Dic3C2312yes—
57C4×ΓL(1,𝔽16)—24no—
58C3×C4×AGL(1,𝔽5)—24no—
59Dic3⋊AGL(1,𝔽5)—24no—
60Dic3⋊AGL(1,𝔽5)—24no—
61Dic3×AGL(1,𝔽5)—24no—
62Dic3×AGL(1,𝔽5)—24no—
63C3×C42×C5C2×C411yescharacteristic
64C4⋊Dic15C2×C411yescharacteristic
65C4⋊Dic15C2×C411yescharacteristic
66#6712C2×C412yes—
67#6712C2×C412yes—
68C5×C4⋊Dic3C2×C411yescharacteristic
69C5×C4⋊Dic3C2×C411yescharacteristic
70C5×Dic3⋊C4C2×C412yes—
71C5×Dic3⋊C4C2×C412yes—
72C3×C4×Dic5C2×C411yescharacteristic
73C3×C4×Dic5C2×C411yescharacteristic
74C3×C4×Dic5C2×C411yescharacteristic
75Dic3⋊Dic5C2×C412yes—
76Dic3⋊Dic5C2×C412yes—
77#6669C2×C412yes—
78#6669C2×C412yes—
79#6668C2×C412yes—
80#6668C2×C412yes—
81C2×#870—51no—
82C3×#376—51no—
83#4933—52no—
84#4932—51no—
85C4×Dic3⋊C4—52no—
86#4654—52no—
87#4653—52no—
88#4652—52no—
89#4649—52no—
90C2×C4⋊AGL(1,𝔽5)—32no—
91C2×C4⋊AGL(1,𝔽5)—32no—
92C2×C4×AGL(1,𝔽5)—32no—
93C2×C4×AGL(1,𝔽5)—32no—
94C2×C4×AGL(1,𝔽5)—32no—
95C2×C4×AGL(1,𝔽5)C2×D312yes—
96C5×#125—31no—
97D5×C4⋊C4—32no—
98D5×C4⋊C4—32no—
99D5×C4⋊C4—31no—
100D5×C4⋊C4—31no—
101#4036—31no—
102#4036—31no—
103#4035—32no—
104#4035—32no—
105C42×D5C2×D311yescharacteristic
106#4016—31no—
107(C2×AGL(1,𝔽5))⋊C4—32no—
108(C2×AGL(1,𝔽5))⋊C4—32no—
109(C2×AGL(1,𝔽5))⋊C4—32no—
110(C2×AGL(1,𝔽5))⋊C4—32no—
111(C2×AGL(1,𝔽5))⋊C4—31no—
112(C2×AGL(1,𝔽5))⋊C4—31no—
113(C2×AGL(1,𝔽5))⋊C4—32no—
114(C2×AGL(1,𝔽5))⋊C4C2×D311yescharacteristic
115(C2×AGL(1,𝔽5))⋊C4C2×D311yescharacteristic
116(C2×AGL(1,𝔽5))⋊C4C2×D312yes—
117#2371—151noSylow
118C22×C3×D5C2411yescharacteristic
119C2×ΓL(1,𝔽16)—22no—
120C2×ΓL(1,𝔽16)—22no—
121C2×ΓL(1,𝔽16)—22no—
122C2×ΓL(1,𝔽16)—22no—
123C2×ΓL(1,𝔽16)—22no—
124C2×ΓL(1,𝔽16)#4114yes—
125C2×C3×AGL(1,𝔽5)—22no—
126C2×C3×AGL(1,𝔽5)—22no—
127C2×C3×AGL(1,𝔽5)—22no—
128C2×C3×AGL(1,𝔽5)—22no—
129C2×C3×AGL(1,𝔽5)—22no—
130C2×C3×AGL(1,𝔽5)#4114yes—
131C2×C3×C4×C5C22×C411yescharacteristic
132C2×C3×C4×C5C22×C411yescharacteristic
133C2×C3×C4×C5C22×C411yescharacteristic
134C2×Dic15C22×C412yes—
135C2×Dic15C22×C412yes—
136C2×C5×Dic3C22×C412yes—
137C2×C5×Dic3C22×C412yes—
138C2×C3×Dic5C22×C411yescharacteristic
139C2×C3×Dic5C22×C411yescharacteristic
140C2×C3×Dic5C22×C411yescharacteristic
141C3×C4×D5—22no—
142C3×C4×D5—22no—
143C3×C4×D5—21no—
144C3×C4×D5—21no—
145D5×Dic3—22no—
146D5×Dic3—22no—
147D5×Dic3—22no—
148D5×Dic3—22no—
149C2×C3×C42—51no—
150C2×C3×C42—52no—
151C2×C4⋊Dic3—51no—
152C2×C4⋊Dic3—51no—
153C2×Dic3⋊C4—52no—
154C2×Dic3⋊C4—52no—
155C2×Dic3⋊C4—52no—
156C2×Dic3⋊C4—52no—
157C2×C4×Dic3—52no—
158C2×C4×Dic3—52no—
159C2×C4×Dic3—52no—
160#870—51no—
161#870—51no—
162#870—51no—
163#870—52no—
164#870—52no—
165#870AGL(1,𝔽5)11yescharacteristic, complemented
166C3×#95—51no—
167C3×#95—51no—
168C3×#95—52no—
169C3⋊#95—52no—
170C3⋊#95—52no—
171C3⋊#95—52no—
172C3⋊#95—52no—
173C3⋊#95—52no—
174C3⋊#95—51no—
175C3⋊#95—51no—
176C22×AGL(1,𝔽5)—32no—
177C22×AGL(1,𝔽5)—32no—
178C22×AGL(1,𝔽5)C22×D312yes—
179C22×AGL(1,𝔽5)C22×D312yes—
180C2×C4×D5—32no—
181C2×C4×D5—32no—
182C2×C4×D5C22×D311yescharacteristic
183C2×C4×D5C22×D311yescharacteristic
184C2×C4×D5C22×D311yescharacteristic
185C4⋊AGL(1,𝔽5)—64no—
186C4⋊AGL(1,𝔽5)—64no—
187C4×AGL(1,𝔽5)—24no—
188C4×AGL(1,𝔽5)—64no—
189C4×AGL(1,𝔽5)—64no—
190C4×AGL(1,𝔽5)—64no—
191C5×C4⋊C4—32no—
192C5×C4⋊C4—32no—
193C5×C4⋊C4—31no—
194C5×C4⋊C4—31no—
195C42×C5C4×D311yescharacteristic
196C4⋊Dic5—31no—
197C4⋊Dic5—31no—
198C4⋊Dic5—32no—
199C4⋊Dic5—32no—
200#683—32no—
201#683—32no—
202#683—32no—
203#683—32no—
204#683—32no—
205#683—32no—
206C4×Dic5C4×D311yescharacteristic
207C4×Dic5C4×D311yescharacteristic
208C4×Dic5C4×D311yescharacteristic
209C2×#125—151no—
210#387—152no—
211#387—152no—
212#387—152no—
213#387—152no—
214C4×C4⋊C4—152no—
215C4×C4⋊C4—152no—
216#376—51no—
217#376—151no—
218C22×C3×C5C23×C411yescharacteristic
219C2×C3×D5C2×#4112yes—
220C2×C3×D5C2×#4112yes—
221C2×C3×D52−1+411yescharacteristic
222C2×C3×D52−1+411yescharacteristic
223ΓL(1,𝔽16)—24no—
224ΓL(1,𝔽16)—42no—
225C3×AGL(1,𝔽5)—24no—
226C3×AGL(1,𝔽5)—42no—
227C3×C4×C5—21no—
228C3×C4×C5—21no—
229C3×C4×C5—21no—
230Dic15—22no—
231Dic15—22no—
232C3×Dic5—21no—
233C3×Dic5—21no—
234C3×Dic5—21no—
235C5×Dic3—22no—
236C5×Dic3—22no—
237C22×C3×C4—52no—
238C22×C3×C4—51no—
239C22×C3×C4—51no—
240C22×C3×C4—51no—
241C22×C3×C4—52no—
242C22×Dic3—52no—
243C22×Dic3—52no—
244C22×Dic3—52no—
245C22×Dic3—52no—
246C3×C42—51no—
247C3×C42—51no—
248C3×C42—51no—
249C3×C42—104no—
250C3×C42C2×AGL(1,𝔽5)11yescharacteristic
251C4⋊Dic3—51no—
252C4⋊Dic3—51no—
253C4⋊Dic3—52no—
254C4⋊Dic3—52no—
255C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
256C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
257Dic3⋊C4—52no—
258Dic3⋊C4—52no—
259Dic3⋊C4—52no—
260Dic3⋊C4—52no—
261Dic3⋊C4—52no—
262Dic3⋊C4—52no—
263Dic3⋊C4—104no—
264Dic3⋊C4—104no—
265Dic3⋊C4C2×AGL(1,𝔽5)12yes—
266Dic3⋊C4C2×AGL(1,𝔽5)12yes—
267C4×Dic3—104no—
268C4×Dic3—104no—
269C4×Dic3—104no—
270C22×D5C23×D311yescharacteristic
271C2×AGL(1,𝔽5)—22no—
272C2×AGL(1,𝔽5)—22no—
273C2×AGL(1,𝔽5)—22no—
274C2×AGL(1,𝔽5)—22no—
275C2×AGL(1,𝔽5)—22no—
276C2×AGL(1,𝔽5)—34no—
277C2×AGL(1,𝔽5)—62no—
278C2×AGL(1,𝔽5)—62no—
279C2×AGL(1,𝔽5)—62no—
280C2×AGL(1,𝔽5)—62no—
281C2×AGL(1,𝔽5)—62no—
282C2×AGL(1,𝔽5)#23514yes—
283C2×C4×C5—32no—
284C2×C4×C5—32no—
285C2×C4×C5C2×C4×D311yescharacteristic
286C2×C4×C5C2×C4×D311yescharacteristic
287C2×C4×C5C2×C4×D311yescharacteristic
288C2×Dic5—32no—
289C2×Dic5—32no—
290C2×Dic5C2×C4×D311yescharacteristic
291C2×Dic5C2×C4×D311yescharacteristic
292C2×Dic5C2×C4×D311yescharacteristic
293C4×D5—22no—
294C4×D5—22no—
295C4×D5—21no—
296C4×D5—21no—
297C4×D5—62no—
298C4×D5—62no—
299C4×D5—62no—
300C4×D5—62no—
301#125—31no—
302#125—152no—
303#125—152no—
304#125—151no—
305#125—151no—
306#125—151no—
307C2×C4⋊C4—152no—
308C2×C4⋊C4—152no—
309C2×C4⋊C4—151no—
310C2×C4⋊C4—151no—
311C2×C4⋊C4—152no—
312C2×C4⋊C4—152no—
313C2×C42—51no—
314C2×C42—52no—
315C2×C42—152no—
316C2×C42—152no—
317C2×C42—152no—
318#95—51no—
319#95—52no—
320#95—51no—
321#95—151no—
322#95—152no—
323#95—151no—
324#95—152no—
325#95—152no—
326#95—152no—
327#95—152no—
328C2×C3×C5C4×#4111yescharacteristic
329C2×C3×C5C4×#4111yescharacteristic
330C2×C3×C5#51911yescharacteristic
331C3×D5#55212yes—
332C3×D5#55212yes—
333C23×C3—51no—
334C2×C3×C4—54no—
335C2×C3×C4—51no—
336C2×C3×C4—51no—
337C2×C3×C4—51no—
338C2×C3×C4—102no—
339C2×C3×C4—102no—
340C2×C3×C4—101no—
341C2×C3×C4—101no—
342C2×C3×C4—102no—
343C2×C3×C4—102no—
344C2×C3×C4—102no—
345C2×C3×C4—102no—
346C2×C3×C4—102no—
347C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
348C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
349C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
350C2×Dic3—54no—
351C2×Dic3—52no—
352C2×Dic3—52no—
353C2×Dic3—102no—
354C2×Dic3—102no—
355C2×Dic3—102no—
356C2×Dic3—102no—
357C2×Dic3—102no—
358C2×Dic3—102no—
359C2×Dic3—102no—
360C2×Dic3—102no—
361C2×Dic3—102no—
362C2×Dic3C22×AGL(1,𝔽5)12yes—
363C2×Dic3C22×AGL(1,𝔽5)12yes—
364C22×C5C22×C4×D311yescharacteristic
365C2×D5C2×#23512yes—
366C2×D5D3×#4112yes—
367C2×D5#101011yescharacteristic
368C2×D5#101011yescharacteristic
369AGL(1,𝔽5)—24no—
370AGL(1,𝔽5)—42no—
371AGL(1,𝔽5)—64no—
372AGL(1,𝔽5)—122no—
373C4×C5—21no—
374C4×C5—21no—
375C4×C5—21no—
376C4×C5—62no—
377C4×C5—62no—
378Dic5—21no—
379Dic5—21no—
380Dic5—21no—
381Dic5—62no—
382Dic5—62no—
383C22×C4—52no—
384C22×C4—51no—
385C22×C4—51no—
386C22×C4—51no—
387C22×C4—52no—
388C22×C4—152no—
389C22×C4—152no—
390C22×C4—152no—
391C22×C4—152no—
392C4⋊C4—32no—
393C4⋊C4—32no—
394C4⋊C4—31no—
395C4⋊C4—31no—
396C4⋊C4—152no—
397C4⋊C4—152no—
398C4⋊C4—152no—
399C4⋊C4—152no—
400C4⋊C4—152no—
401C4⋊C4—152no—
402C4⋊C4—152no—
403C4⋊C4—152no—
404C4⋊C4—151no—
405C4⋊C4—151no—
406C4⋊C4—304no—
407C4⋊C4—304no—
408C42—51no—
409C42—51no—
410C42—51no—
411C42—104no—
412C42—304no—
413C42—304no—
414C42—304no—
415C42D3×AGL(1,𝔽5)11yescharacteristic
416C3×C5#237111yescharacteristic, complemented, Hall
417C22×C3—51no—
418C22×C3—51no—
419C22×C3—52no—
420C22×C3—52no—
421C22×C3C23×AGL(1,𝔽5)11yescharacteristic
422C3×C4—21no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—104no—
426C3×C4—101no—
427C3×C4—101no—
428C3×C4—101no—
429C3×C4—202no—
430Dic3—22no—
431Dic3—22no—
432Dic3—104no—
433Dic3—102no—
434Dic3—102no—
435Dic3—202no—
436C2×C5C4×#23511yescharacteristic
437C2×C5#551611yescharacteristic
438C2×C5#552211yescharacteristic
439D5#552912yes—
440D5#553112yes—
441C23—51no—
442C2×C4—32no—
443C2×C4—32no—
444C2×C4—54no—
445C2×C4—51no—
446C2×C4—51no—
447C2×C4—51no—
448C2×C4—102no—
449C2×C4—102no—
450C2×C4—101no—
451C2×C4—101no—
452C2×C4—102no—
453C2×C4—102no—
454C2×C4—102no—
455C2×C4—102no—
456C2×C4—102no—
457C2×C4—154no—
458C2×C4—152no—
459C2×C4—152no—
460C2×C4—302no—
461C2×C4—302no—
462C2×C4—302no—
463C2×C4—302no—
464C2×C4—302no—
465C2×C4—302no—
466C2×C4—302no—
467C2×C4—302no—
468C2×C4—302no—
469C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
470C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
471C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
472C2×C3—52no—
473C2×C3—52no—
474C2×C3#6626211yescharacteristic
475C2×C3#41×AGL(1,𝔽5)11yescharacteristic
476C2×C3#41×AGL(1,𝔽5)11yescharacteristic
477C5#7339411yescharacteristic, complemented, Sylow
478C22—51no—
479C22—51no—
480C22—52no—
481C22—52no—
482C22C22×D3×AGL(1,𝔽5)11yescharacteristic
483C4—21no—
484C4—21no—
485C4—21no—
486C4—62no—
487C4—62no—
488C4—104no—
489C4—101no—
490C4—101no—
491C4—101no—
492C4—202no—
493C4—304no—
494C4—302no—
495C4—302no—
496C4—602no—
497C3#1061734711yescharacteristic, complemented, Sylow
498C2—52no—
499C2—52no—
500C2#1175681211yescharacteristic
501C2AGL(1,𝔽5)×#23511yescharacteristic
502C2#1175683511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×AGL(1,𝔽5)
commutator, socleC22×C3×C5C23×C4
FittingC3×C42×C5C2×C4
prime radical, cosocleC22×C3×D5C24
hypercenterC42D3×AGL(1,𝔽5)
hypocenterC3×C5#2371
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
17. #91759C4semidirect
18. C5×#870C4semidirect
24. #91388C4semidirect
25. #91388C4semidirect
26. #91378C4semidirect
27. #91378C4semidirect
165. #870AGL(1,𝔽5)semidirect
416. C3×C5#2371semidirect
477. C5#73394semidirect
497. C3#10617347semidirect

Maximal subgroups

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

subgroupindex
D5×#8702
#117523452
C3×#656862
#117518402
#117518392
#117518332
AGL(1,𝔽5)×Dic3⋊C42
#117517472
#117517452
#106173473
#733945