Finite Group Databasebeta

← #14471664

Subgroups — #14471664

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
1#11756338C211yesmaximal, characteristic, complemented
2C5×#5584C211yesmaximal, characteristic, complemented
3C3×Q8×#168C211yesmaximal, characteristic, complemented
4#11747087C211yesmaximal, characteristic, complemented
5C4×#6660C212yesmaximal, complemented
6#11747046C212yesmaximal
7C4×#6654C211yesmaximal, characteristic
8#11746965C212yesmaximal, complemented
9#11746929C211yesmaximal, characteristic, complemented
10#11746907C212yesmaximal, complemented
11#11746883C211yesmaximal, characteristic, complemented
12#10615398—31nomaximal, Hall
13C3×C4×C5×Q8C2211yescharacteristic
14#91835—21no—
15#91785—22no—
16C4⋊C4×D15—21no—
17C4×D60C2211yescharacteristic
18C4×D60C2212yes—
19#91759C2212yes—
20#91758C2211yescharacteristic
21C5×#946—21no—
22C5×#898—22no—
23C5×#892—21no—
24C5×#875C2212yes—
25C4×C5×D12C2211yescharacteristic
26C5×#869C2211yescharacteristic
27C4×C5×Dic6C2212yes—
28C2×#6660C2212yes—
29C2×#6654C2211yescharacteristic
30C3×C4⋊#168C2211yescharacteristic
31C3×C4⋊#168C2212yes—
32C3×C4×#168C2212yes—
33C3×C4×#168C2211yescharacteristic
34#90988C2211yescharacteristic
35#90985C2212yes—
36#90973C2212yes—
37#90973C2212yes—
38#90973C2212yes—
39#90969C2211yescharacteristic
40#90967C2212yes—
41#90967C2211yescharacteristic
42#79926—51nomaximal, Hall
43C5×#556—31no—
44#65913—31no—
45#65372—31no—
46#65370—31no—
47#65346—32no—
48#65345—32no—
49C4×#688—32no—
50Q8×#168D311yescharacteristic, complemented
51#65311—32no—
52C4×#686—31no—
53#65297—31no—
54C2×C3×C5×Q8D411yescharacteristic
55C2×D60C2311yescharacteristic
56C2×C4×D15—21no—
57C2×C4×D15—22no—
58C2×Dic30—21no—
59C2×C5×D12C2311yescharacteristic
60C2×C4×C5×D3—21no—
61C2×C5×Dic6C2312yes—
62C3×C5×C4⋊C4D411yescharacteristic
63C3×C5×C4⋊C4D412yes—
64C3×C42×C5C2312yes—
65C3×C42×C5C2311yescharacteristic
66#6714—21no—
67#6714—22no—
68C4⋊Dic15—21no—
69C4⋊Dic15C2312yes—
70C4⋊Dic15C2311yescharacteristic
71#6712—22no—
72#6712—22no—
73C5×(C2×D3)⋊C4—21no—
74C5×(C2×D3)⋊C4—22no—
75C5×(C2×D3)⋊C4—22no—
76C5×C4⋊Dic3—21no—
77C5×C4⋊Dic3—21no—
78C5×C4⋊Dic3C2311yescharacteristic
79C5×Dic3⋊C4—22no—
80C4×C5×Dic3—22no—
81C2×C3×#168C2311yescharacteristic
82C2×C3×#168C2312yes—
83C2×C3×#168C2311yescharacteristic
84#6660—24no—
85#6654—22no—
86#5584D511yescharacteristic, complemented
87#5579—51no—
88C3×C8×Q8—51no—
89#4844—52no—
90#4842—51no—
91#4841—51no—
92#4829—52no—
93#4697—52no—
94C4×D24—51no—
95C4×#204—52no—
96#4681—51no—
97C5×C4⋊Q8—31no—
98C5×#126—62no—
99C5×#124—32no—
100C5×C22⋊Q8—61no—
101C4×C5×Q8—32no—
102C4×C5×Q8C2×D311yescharacteristic
103C4×C5×D4—31no—
104C5×#117—61no—
105#4092—61no—
106C2×#688—32no—
107C2×#686—31no—
108#4042—62no—
109D5×C4⋊C4—61no—
110C4×D20—31no—
111C4×D20—32no—
112#4016—32no—
113#4015—31no—
114#3967—32no—
115#3967—32no—
116#3967—32no—
117#3964—32no—
118#3964—31no—
119#3941—31no—
120#3940—32no—
121#3939—31no—
122C4⋊#168C2×D311yescharacteristic
123C4⋊#168C2×D312yes—
124C4×#168C2×D311yescharacteristic
125C4×#168C2×D312yes—
126#3360—151noSylow
127C22×D15—21no—
128C22×C5×D3—21no—
129C3×C5×Q8—22no—
130C2×C3×C4×C5C2×D412yes—
131C2×C3×C4×C5C2×D412yes—
132C2×C3×C4×C5C2×D411yescharacteristic
133C2×C3×C4×C5C2×D411yescharacteristic
134C2×C3×C4×C5C2411yescharacteristic
135C2×Dic15—21no—
136C2×Dic15—22no—
137D60—21no—
138D60#4112yes—
139C4×D15—42no—
140C4×D15—82no—
141Dic30—81no—
142C2×C5×Dic3—21no—
143C2×C5×Dic3—22no—
144C5×D12—21no—
145C5×D12—21no—
146C4×C5×D3—42no—
147C5×Dic6—22no—
148C5×Dic6—22no—
149C3×#168—22no—
150C3×#168—21no—
151C3×#168—21no—
152C3×C4×Q8C2×D511yescharacteristic
153#946—21no—
154#946—101no—
155C2×D24—51no—
156C2×#204—52no—
157#898—22no—
158#896—102no—
159#892—21no—
160D3×C4⋊C4—101no—
161#875C2×D512yes—
162C4×D12—52no—
163C4×D12—51no—
164C4×D12C2×D511yescharacteristic
165#870—52no—
166#869—51no—
167#869C2×D511yescharacteristic
168C4×Dic6C2×D512yes—
169C3×C4⋊C8—52no—
170C3×C4⋊C8—51no—
171C3×C4×C8—51no—
172C3×C4×C8—52no—
173#821—51no—
174#821—52no—
175#821—51no—
176#818—51no—
177#817—52no—
178#816—52no—
179#816—52no—
180#816—52no—
181C2×C5×Q8—32no—
182C2×C5×Q8D1211yescharacteristic
183C2×C5×D4—31no—
184C22×C4×C5—61no—
185C2×D20—31no—
186C2×C4×D5—61no—
187C2×C4×D5—62no—
188C2×Dic10—61no—
189C5×C4⋊C4—31no—
190C5×C4⋊C4—61no—
191C5×C4⋊C4—62no—
192C5×C4⋊C4—61no—
193C5×C4⋊C4D1211yescharacteristic
194C5×C4⋊C4D1212yes—
195C5×C22⋊C4—62no—
196C5×C22⋊C4—61no—
197C5×C22⋊C4—62no—
198C42×C5—62no—
199C42×C5C22×D311yescharacteristic
200C42×C5C22×D312yes—
201#688—64no—
202#686—62no—
203#685—61no—
204#685—62no—
205C4⋊Dic5—31no—
206C4⋊Dic5—32no—
207C4⋊Dic5—61no—
208#683—62no—
209#683—62no—
210C2×#168C22×D311yescharacteristic
211C2×#168C22×D311yescharacteristic
212C2×#168C22×D312yes—
213#556—31no—
214#554—151no—
215#488—152no—
216#487—151no—
217#479—152no—
218#476—151no—
219#463—152no—
220#461—151no—
221C8×Q8—51no—
222C4×SD16—152no—
223C4×D8—151no—
224C22×C3×C5C22×D411yescharacteristic
225C2×D15—22no—
226C2×D15—41no—
227C2×D15—41no—
228C2×C5×D3—41no—
229C2×C5×D3—41no—
230C2×C5×D3—41no—
231C3×C4×C5—22no—
232C3×C4×C5—21no—
233C3×C4×C5—22no—
234C3×C4×C5#13512yes—
235C3×C4×C5C2×#4111yescharacteristic
236C3×C4×C52−1+411yescharacteristic
237Dic15—41no—
238Dic15—42no—
239C5×Dic3—42no—
240C5×Dic3—41no—
241C2×C3×Q8C22⋊D511yescharacteristic
242C2×D12—51no—
243C2×D12C22×D511yescharacteristic
244C2×C4×D3—21no—
245C2×C4×D3—101no—
246C2×C4×D3—102no—
247C2×Dic6—101no—
248C2×Dic6C22×D512yes—
249C2×C3×C8—51no—
250C2×C3×C8—51no—
251C2×C3×C8—52no—
252C3×C4⋊C4C22⋊D511yescharacteristic
253C3×C4⋊C4C22⋊D512yes—
254C3×C42C22×D512yes—
255C3×C42C22×D511yescharacteristic
256(C2×D3)⋊C4—21no—
257(C2×D3)⋊C4—22no—
258(C2×D3)⋊C4—22no—
259(C2×D3)⋊C4—101no—
260(C2×D3)⋊C4—102no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—52no—
264C4⋊Dic3—51no—
265C4⋊Dic3—101no—
266C4⋊Dic3C22×D511yescharacteristic
267Dic3⋊C4—22no—
268Dic3⋊C4—102no—
269Dic3⋊C4—102no—
270C4×Dic3—22no—
271D24—102no—
272#204—104no—
273C23×C5—61no—
274C22×D5—61no—
275C5×Q8—22no—
276C5×Q8—62no—
277C5×Q8—62no—
278C5×D4—61no—
279C5×D4—61no—
280C2×C4×C5—61no—
281C2×C4×C5—62no—
282C2×C4×C5—122no—
283C2×C4×C5C2×D1212yes—
284C2×C4×C5C2×D1212yes—
285C2×C4×C5C2×D1211yescharacteristic
286C2×C4×C5C2×D1211yescharacteristic
287C2×C4×C5C23×D311yescharacteristic
288C2×Dic5—61no—
289C2×Dic5—62no—
290D20—32no—
291D20—61no—
292C4×D5—122no—
293C4×D5—242no—
294Dic10—241no—
295#168—21no—
296#168—21no—
297#168—22no—
298C2×SD16—152no—
299C2×D8—151no—
300C4⋊Q8—31no—
301C4⋊Q8—151no—
302#126—62no—
303#125—152no—
304#124—32no—
305C22⋊Q8—61no—
306C22⋊Q8—301no—
307C22⋊Q8—302no—
308C4×Q8—32no—
309C4×Q8D3×D511yescharacteristic
310C4×D4—31no—
311C4×D4—151no—
312C4×D4—152no—
313#117—61no—
314C2×C4⋊C4—301no—
315#107—151no—
316#106—152no—
317C4⋊C8—51no—
318C4⋊C8—52no—
319#103—152no—
320#103—152no—
321#103—152no—
322#102—151no—
323#102—151no—
324#102—152no—
325C4×C8—51no—
326C4×C8—52no—
327C2×C3×C5#54811yescharacteristic
328C2×C3×C5C2×#13511yescharacteristic
329C2×C3×C5#57711yescharacteristic
330D15—42no—
331C5×D3—81no—
332C22×D3—21no—
333C22×D3—101no—
334C3×Q8—22no—
335C2×C3×C4C2×C22⋊D512yes—
336C2×C3×C4C2×C22⋊D512yes—
337C2×C3×C4C2×C22⋊D511yescharacteristic
338C2×C3×C4C2×C22⋊D511yescharacteristic
339C2×C3×C4C23×D511yescharacteristic
340C2×Dic3—21no—
341C2×Dic3—22no—
342C2×Dic3—101no—
343C2×Dic3—102no—
344D12—21no—
345D12—21no—
346D12—52no—
347D12—101no—
348C4×D3—42no—
349C4×D3—202no—
350C4×D3—402no—
351Dic6—22no—
352Dic6—22no—
353Dic6—401no—
354C3×C8—102no—
355C3×C8—101no—
356C3×C8—101no—
357C22×C5—121no—
358C22×C5—121no—
359C22×C5—121no—
360C22×C5C22×D1211yescharacteristic
361C2×D5—62no—
362C2×D5—121no—
363C2×D5—121no—
364C4×C5—22no—
365C4×C5—21no—
366C4×C5—22no—
367C4×C5—121no—
368C4×C5—122no—
369C4×C5#90412yes—
370C4×C5#100711yescharacteristic
371C4×C5D3×#4111yescharacteristic
372Dic5—121no—
373Dic5—122no—
374C2×Q8—32no—
375C2×Q8—301no—
376C2×Q8#120311yescharacteristic
377C2×D4—31no—
378C2×D4—151no—
379C22×C4—61no—
380C22×C4—301no—
381C22×C4—302no—
382SD16—304no—
383D8—302no—
384C2×C8—51no—
385C2×C8—52no—
386C2×C8—51no—
387C4⋊C4—31no—
388C4⋊C4—61no—
389C4⋊C4—62no—
390C4⋊C4—61no—
391C4⋊C4—152no—
392C4⋊C4—151no—
393C4⋊C4—302no—
394C4⋊C4—301no—
395C4⋊C4—302no—
396C4⋊C4#120311yescharacteristic
397C4⋊C4#120312yes—
398C22⋊C4—62no—
399C22⋊C4—61no—
400C22⋊C4—62no—
401C22⋊C4—301no—
402C22⋊C4—302no—
403C42—62no—
404C42C2×D3×D511yescharacteristic
405C42C2×D3×D512yes—
406C3×C5#336011yescharacteristic, complemented, Hall
407C22×C3C22×C22⋊D511yescharacteristic
408C2×D3—41no—
409C2×D3—41no—
410C2×D3—41no—
411C2×D3—102no—
412C2×D3—201no—
413C2×D3—201no—
414C3×C4—22no—
415C3×C4—21no—
416C3×C4—22no—
417C3×C4#409612yes—
418C3×C4D5×#4111yescharacteristic
419C3×C4#415011yescharacteristic
420Dic3—41no—
421Dic3—42no—
422Dic3—201no—
423Dic3—202no—
424C2×C5—241no—
425C2×C5#557611yescharacteristic
426C2×C5C2×#90411yescharacteristic
427C2×C5#575111yescharacteristic
428D5—122no—
429C23—61no—
430C23—301no—
431Q8—22no—
432Q8—62no—
433Q8—62no—
434Q8—1201no—
435D4—61no—
436D4—61no—
437D4—152no—
438D4—301no—
439C2×C4—61no—
440C2×C4—62no—
441C2×C4—122no—
442C2×C4—301no—
443C2×C4—302no—
444C2×C4—602no—
445C2×C4—1202no—
446C2×C4C2×#120312yes—
447C2×C4C2×#120312yes—
448C2×C4C2×#120311yescharacteristic
449C2×C4C2×#120311yescharacteristic
450C2×C4C22×D3×D511yescharacteristic
451C8—101no—
452C8—101no—
453C8—102no—
454C2×C3C2×#409611yescharacteristic
455C2×C3#6616511yescharacteristic
456C2×C3#6617111yescharacteristic
457D3—81no—
458D3—202no—
459C5#7992611yescharacteristic, complemented, Sylow
460C22—121no—
461C22—121no—
462C22—121no—
463C22—302no—
464C22—601no—
465C22—601no—
466C22C22×#120311yescharacteristic
467C4—22no—
468C4—21no—
469C4—22no—
470C4—122no—
471C4—121no—
472C4—602no—
473C4—601no—
474C4#9130312yes—
475C4#9203011yescharacteristic
476C4#9203411yescharacteristic
477C3#1061539811yescharacteristic, complemented, Sylow
478C2—241no—
479C2—602no—
480C2C2×#9130311yescharacteristic
481C2#1175513411yescharacteristic
482C2#1175530811yescharacteristic
subgroupG/H
centerC22C22×#1203
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C4×C5×Q8C22
socleC22×C3×C5C22×D4
prime radicalC2×#6654C22
hypercenterC4×Q8D3×D5
hypocenterC3×C5#3360
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. #11756338C2semidirect
2. C5×#5584C2semidirect
3. C3×Q8×#168C2semidirect
4. #11747087C2semidirect
5. C4×#6660C2semidirect
8. #11746965C2semidirect
9. #11746929C2semidirect
10. #11746907C2semidirect
11. #11746883C2semidirect
50. Q8×#168D3semidirect
86. #5584D5semidirect
406. C3×C5#3360semidirect
459. C5#79926semidirect
477. C3#10615398semidirect

Maximal subgroups

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

subgroupindex
#117563382
C5×#55842
C3×Q8×#1682
#117470872
C4×#66602
#117470462
C4×#66542
#117469652
#117469292
#117469072
#117468832
#106153983
#799265