Finite Group Databasebeta

← #14471692

Subgroups — #14471692

Order 1920.

700 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#66064C211yesmaximal, characteristic, complemented
2#11752486C211yesmaximal, characteristic, complemented
3#11751185C211yesmaximal, characteristic, complemented
4C5×#5093C211yesmaximal, characteristic, complemented
5#11747371C212yesmaximal
6#11747287C212yesmaximal, complemented
7#11747261C212yesmaximal, complemented
8#11747145C212yesmaximal, complemented
9#11747059C211yesmaximal, characteristic, complemented
10C4×#6653C211yesmaximal, characteristic
11C3⋊C8×Dic10C211yesmaximal, characteristic, complemented
12#10613352—31nomaximal, Hall
13C3×C5×C4⋊Q8C2211yescharacteristic
14C4×D60C2211yescharacteristic
15C5×#869C2211yescharacteristic
16C3×#4092—22no—
17C3×#4043—21no—
18C3×#4039C2212yes—
19C3×D5×C4⋊C4—21no—
20C3×#4036C2212yes—
21C3×C4×D20C2211yescharacteristic
22C3×C4×Dic10C2211yescharacteristic
23#91390—21no—
24#91375C2212yes—
25#91347—21no—
26#91337—22no—
27Dic5×Dic6C2212yes—
28C2×#6655C2212yes—
29C2×#6653C2211yescharacteristic
30#91093C2212yes—
31#91092C2212yes—
32C4⋊(C3⋊#168)C2211yescharacteristic
33C5×#810C2212yes—
34C5×#810C2212yes—
35C4×C5×C3⋊C8C2211yescharacteristic
36#90990C2211yescharacteristic
37#90982C2212yes—
38#90972C2212yes—
39#90968C2211yescharacteristic
40#90965C2211yescharacteristic
41Dic5×C3⋊C8C2212yes—
42#83637—51nomaximal, Hall
43#66066—31no—
44#66064D311yescharacteristic, complemented
45C5×#487—31no—
46#65373—31no—
47#65372—31no—
48#65108—32no—
49#65089—32no—
50#65085—32no—
51#65077—32no—
52C4×D40—31no—
53C8×Dic10—31no—
54C2×C3×C5×Q8—21no—
55C2×D60C2311yescharacteristic
56C2×C4×D15—21no—
57C2×C5×Dic6C2312yes—
58C2×C3×D20C2311yescharacteristic
59C2×C3×C4×D5—21no—
60C2×C3×C4×D5—22no—
61C2×C3×Dic10D411yescharacteristic
62C3×C5×C4⋊C4—21no—
63C3×C5×C4⋊C4C2312yes—
64C3×C42×C5C2311yescharacteristic
65#6714—21no—
66C4⋊Dic15C2311yescharacteristic
67C2×C3⋊#168C2312yes—
68C5×C4⋊Dic3—21no—
69C5×C4⋊Dic3—21no—
70C2×C5×C3⋊C8C2311yescharacteristic
71C2×C5×C3⋊C8C2311yescharacteristic
72C3×#685—21no—
73C3×#685—22no—
74C3×C4⋊Dic5—22no—
75C3×C4⋊Dic5D411yescharacteristic
76C3×C4⋊Dic5C2311yescharacteristic
77C3×#683—22no—
78C3×#683D412yes—
79C3×C4×Dic5C2312yes—
80Dic3⋊Dic5—22no—
81#6667—22no—
82#6667—22no—
83Dic3×Dic5—22no—
84#6655—24no—
85#6653—22no—
86C3×#554—51no—
87#5584—51no—
88#5093D511yescharacteristic, complemented
89#5084—51no—
90#5068—52no—
91#5063—52no—
92Q8×C3⋊C8—51no—
93#5019—51no—
94C4×#214—52no—
95#5016—52no—
96C4×#213—51no—
97C5×C4⋊Q8—31no—
98C5×C4⋊Q8C2×D311yescharacteristic
99#4093—32no—
100#4092—22no—
101Q8×Dic5—32no—
102C2×#688—32no—
103C2×D40—31no—
104#4044—62no—
105#4043—21no—
106#4043—61no—
107#4039C2×D312yes—
108#4038—61no—
109D5×C4⋊C4—21no—
110#4036C2×D312yes—
111C4×D20—31no—
112C4×D20C2×D311yescharacteristic
113C4×Dic10C2×D311yescharacteristic
114C5×#103—32no—
115C5×#103—32no—
116C4×C5×C8—31no—
117#3967—32no—
118#3953—31no—
119#3953—31no—
120#3950—31no—
121#3946—32no—
122C8×Dic5—32no—
123#3941—32no—
124#3940—32no—
125C4⋊#168—31no—
126#3360—151noSylow
127C22×D15—21no—
128C22×C3×D5—21no—
129C3×C5×Q8—81no—
130C2×C3×C4×C5—22no—
131C2×C3×C4×C5C2×D411yescharacteristic
132C2×C3×C4×C5C2×D411yescharacteristic
133C2×C3×C4×C5C2411yescharacteristic
134C2×Dic15—21no—
135D60—21no—
136D60—21no—
137C4×D15—42no—
138C2×C5×Dic3—22no—
139C5×Dic6—22no—
140C5×Dic6—22no—
141C2×C3×Dic5—21no—
142C2×C3×Dic5C2×D412yes—
143C2×C3×Dic5C2×D412yes—
144C3×D20—21no—
145C3×D20#4112yes—
146C3×C4×D5—42no—
147C3×C4×D5—82no—
148C3×Dic10—22no—
149C3⋊#168—22no—
150C5×C3⋊C8—21no—
151C5×C3⋊C8—21no—
152C3×C4⋊Q8C2×D511yescharacteristic
153C3×#125—52no—
154C3×C22⋊Q8—101no—
155C3×C22⋊Q8—102no—
156C3×C4×Q8—51no—
157C3×C4×D4—52no—
158C3×C4×D4—51no—
159C2×C3×C4⋊C4—101no—
160#946—101no—
161C2×#214—52no—
162C2×#213—51no—
163#898—102no—
164#892—101no—
165#875—52no—
166C4×D12—51no—
167#869C2×D511yescharacteristic
168C4×Dic6—52no—
169#832—51no—
170#810—52no—
171#810C2×D512yes—
172#810C2×D512yes—
173#809—52no—
174#809—51no—
175#808—52no—
176#807—51no—
177C4⋊(C3⋊C8)—51no—
178C4⋊(C3⋊C8)—52no—
179C4×C3⋊C8—52no—
180C4×C3⋊C8C2×D511yescharacteristic
181C2×C5×Q8—21no—
182C2×C5×Q8—32no—
183C2×D20—31no—
184C2×D20C22×D311yescharacteristic
185C2×C4×D5—21no—
186C2×C4×D5—22no—
187C2×C4×D5—61no—
188C2×Dic10#6711yescharacteristic
189C2×C5×C8—31no—
190C2×C5×C8—31no—
191C5×C4⋊C4—21no—
192C5×C4⋊C4—61no—
193C5×C4⋊C4—61no—
194C5×C4⋊C4C22×D312yes—
195C42×C5C22×D311yescharacteristic
196#688—64no—
197#685—21no—
198#685—22no—
199#685—61no—
200#685—62no—
201#685—62no—
202C4⋊Dic5—22no—
203C4⋊Dic5—31no—
204C4⋊Dic5—62no—
205C4⋊Dic5#6711yescharacteristic
206C4⋊Dic5C22×D311yescharacteristic
207#683—22no—
208#683#6712yes—
209C4×Dic5—62no—
210C4×Dic5C22×D312yes—
211C2×#168—32no—
212D40—62no—
213#556—151no—
214#554—51no—
215#488—152no—
216#487—31no—
217#479—152no—
218#476—151no—
219#463—152no—
220#461—151no—
221C8×Q8—151no—
222C4×SD16—152no—
223C4×D8—151no—
224C22×C3×C5C22×D411yescharacteristic
225C2×D15—41no—
226C2×D15—41no—
227C2×D15—41no—
228C2×C3×D5—22no—
229C2×C3×D5—41no—
230C2×C3×D5—41no—
231C3×C4×C5—21no—
232C3×C4×C5—42no—
233C3×C4×C5#13512yes—
234C3×C4×C5C2×#4111yescharacteristic
235C3×C4×C52−1+411yescharacteristic
236Dic15—41no—
237C3×Dic5—22no—
238C3×Dic5—22no—
239C3×Dic5—41no—
240C5×Dic3—42no—
241C2×C3×Q8—21no—
242C2×C3×Q8—51no—
243C2×C3×D4—51no—
244C22×C3×C4—102no—
245C22×C3×C4—101no—
246C2×D12—51no—
247C2×C4×D3—101no—
248C2×Dic6C22×D512yes—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—52no—
251C3×C4⋊C4—51no—
252C3×C4⋊C4—51no—
253C3×C4⋊C4—102no—
254C3×C4⋊C4—102no—
255C3×C4⋊C4C22×D512yes—
256C3×C22⋊C4—102no—
257C3×C22⋊C4—101no—
258C3×C42—52no—
259C3×C42C22×D511yescharacteristic
260#214—104no—
261#213—102no—
262(C2×D3)⋊C4—102no—
263(C2×D3)⋊C4—102no—
264(C2×D3)⋊C4—101no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—51no—
268Dic3⋊C4—102no—
269C4×Dic3—102no—
270C2×C3⋊C8—52no—
271C2×C3⋊C8C22×D511yescharacteristic
272C2×C3⋊C8C22×D511yescharacteristic
273C22×D5—21no—
274C22×D5—61no—
275C5×Q8—62no—
276C5×Q8—62no—
277C5×Q8—81no—
278C2×C4×C5—22no—
279C2×C4×C5—62no—
280C2×C4×C5C2×#6711yescharacteristic
281C2×C4×C5C2×#6711yescharacteristic
282C2×C4×C5C23×D311yescharacteristic
283C2×Dic5—21no—
284C2×Dic5—61no—
285C2×Dic5C2×#6712yes—
286C2×Dic5C2×#6712yes—
287D20—21no—
288D20—61no—
289D20—61no—
290D20#23912yes—
291C4×D5—42no—
292C4×D5—82no—
293C4×D5—122no—
294Dic10—22no—
295C5×C8—61no—
296C5×C8—61no—
297#168—62no—
298C2×SD16—152no—
299C2×D8—151no—
300C4⋊Q8—31no—
301C4⋊Q8D3×D511yescharacteristic
302#126—302no—
303#125—52no—
304#124—152no—
305C22⋊Q8—101no—
306C22⋊Q8—102no—
307C22⋊Q8—301no—
308C4×Q8—51no—
309C4×Q8—152no—
310C4×D4—52no—
311C4×D4—51no—
312C4×D4—151no—
313#117—301no—
314C2×C4⋊C4—101no—
315#107—151no—
316#106—152no—
317C4⋊C8—152no—
318C4⋊C8—151no—
319#103—32no—
320#103—32no—
321#103—152no—
322#102—151no—
323#102—151no—
324#102—152no—
325C4×C8—31no—
326C4×C8—152no—
327C2×C3×C5#54811yescharacteristic
328C2×C3×C5C2×#13511yescharacteristic
329C2×C3×C5#57711yescharacteristic
330D15—81no—
331C3×D5—42no—
332C23×C3—101no—
333C22×D3—101no—
334C3×Q8—81no—
335C3×Q8—102no—
336C3×D4—52no—
337C3×D4—101no—
338C2×C3×C4—22no—
339C2×C3×C4—52no—
340C2×C3×C4—52no—
341C2×C3×C4—101no—
342C2×C3×C4—202no—
343C2×C3×C4—402no—
344C2×C3×C4D4×D511yescharacteristic
345C2×C3×C4D4×D511yescharacteristic
346C2×C3×C4C23×D511yescharacteristic
347C2×Dic3—22no—
348C2×Dic3—101no—
349D12—101no—
350D12—101no—
351C4×D3—202no—
352Dic6—22no—
353Dic6—22no—
354C3⋊C8—21no—
355C3⋊C8—21no—
356C3⋊C8—102no—
357C22×C5C22×#6711yescharacteristic
358C2×D5—22no—
359C2×D5—41no—
360C2×D5—41no—
361C2×D5—121no—
362C2×D5—121no—
363C2×D5—121no—
364C4×C5—21no—
365C4×C5—42no—
366C4×C5—122no—
367C4×C5#95012yes—
368C4×C5C2×#23911yescharacteristic
369C4×C5#100711yescharacteristic
370Dic5—22no—
371Dic5—22no—
372Dic5—41no—
373Dic5—121no—
374C2×Q8—21no—
375C2×Q8—32no—
376C2×Q8—51no—
377C2×D4—51no—
378C2×D4—151no—
379C22×C4—102no—
380C22×C4—101no—
381C22×C4—301no—
382SD16—304no—
383D8—302no—
384C2×C8—31no—
385C2×C8—31no—
386C2×C8—152no—
387C4⋊C4—21no—
388C4⋊C4—51no—
389C4⋊C4—52no—
390C4⋊C4—51no—
391C4⋊C4—61no—
392C4⋊C4—61no—
393C4⋊C4—102no—
394C4⋊C4—102no—
395C4⋊C4—151no—
396C4⋊C4—302no—
397C4⋊C4C2×D3×D512yes—
398C22⋊C4—102no—
399C22⋊C4—101no—
400C22⋊C4—302no—
401C22⋊C4—302no—
402C22⋊C4—301no—
403C42—52no—
404C42—302no—
405C42C2×D3×D511yescharacteristic
406C3×C5#336011yescharacteristic, complemented, Hall
407C22×C3—102no—
408C22×C3—201no—
409C22×C3—201no—
410C22×C3C2×D4×D511yescharacteristic
411C2×D3—201no—
412C2×D3—201no—
413C2×D3—201no—
414C3×C4—21no—
415C3×C4—42no—
416C3×C4—102no—
417C3×C4—102no—
418C3×C4—201no—
419C3×C4#406512yes—
420C3×C4D5×#4111yescharacteristic
421C3×C4#415011yescharacteristic
422Dic3—42no—
423Dic3—201no—
424C2×C5#581711yescharacteristic
425C2×C5C2×#95011yescharacteristic
426C2×C5#583611yescharacteristic
427D5—42no—
428D5—241no—
429C23—101no—
430C23—301no—
431Q8—62no—
432Q8—62no—
433Q8—81no—
434Q8—102no—
435D4—52no—
436D4—101no—
437D4—301no—
438D4—301no—
439C2×C4—22no—
440C2×C4—52no—
441C2×C4—52no—
442C2×C4—62no—
443C2×C4—101no—
444C2×C4—202no—
445C2×C4—301no—
446C2×C4—402no—
447C2×C4—602no—
448C2×C4D5×#6711yescharacteristic
449C2×C4D5×#6711yescharacteristic
450C2×C4C22×D3×D511yescharacteristic
451C8—61no—
452C8—61no—
453C8—302no—
454C2×C3—202no—
455C2×C3#6601111yescharacteristic
456C2×C3#6609611yescharacteristic
457C2×C3C2×#406511yescharacteristic
458D3—401no—
459C5#8363711yescharacteristic, complemented, Sylow
460C22—102no—
461C22—201no—
462C22—201no—
463C22—601no—
464C22—601no—
465C22—601no—
466C22C2×D5×#6711yescharacteristic
467C4—21no—
468C4—42no—
469C4—102no—
470C4—102no—
471C4—122no—
472C4—201no—
473C4—601no—
474C4#9151612yes—
475C4#9203011yescharacteristic
476C4D5×#23911yescharacteristic
477C3#1061335211yescharacteristic, complemented, Sylow
478C2—202no—
479C2—1201no—
480C2#1175470711yescharacteristic
481C2C2×#9151611yescharacteristic
482C2#1175522611yescharacteristic
subgroupG/H
centerC22C2×D5×#67
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C5×C4⋊Q8C22
socleC22×C3×C5C22×D4
prime radicalC2×#6653C22
hypercenterC4⋊Q8D3×D5
hypocenterC3×C5#3360
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C3×#66064C2semidirect
2. #11752486C2semidirect
3. #11751185C2semidirect
4. C5×#5093C2semidirect
6. #11747287C2semidirect
7. #11747261C2semidirect
8. #11747145C2semidirect
9. #11747059C2semidirect
11. C3⋊C8×Dic10C2semidirect
44. #66064D3semidirect
88. #5093D5semidirect
406. C3×C5#3360semidirect
459. C5#83637semidirect
477. C3#10613352semidirect

Maximal subgroups

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

subgroupindex
C3×#660642
#117524862
#117511852
C5×#50932
#117473712
#117472872
#117472612
#117471452
#117470592
C4×#66532
C3⋊C8×Dic102
#106133523
#836375