Finite Group Databasebeta

← #14472142

Subgroups — #14472142

Order 1920.

672 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756343C211yesmaximal, characteristic
2C5×#5567C211yesmaximal, characteristic, complemented
3C3×Q8×#168C211yesmaximal, characteristic, complemented
4#11747102C212yesmaximal
5#11747101C211yesmaximal, characteristic, complemented
6C4×#6663C212yesmaximal, complemented
7C4×#6660C211yesmaximal, characteristic
8#11746967C212yesmaximal, complemented
9#11746966C211yesmaximal, characteristic
10#11746930C211yesmaximal, characteristic, complemented
11#11746913C212yesmaximal
12#10615420—31nomaximal, Hall
13C3×C4×C5×Q8C2211yescharacteristic
14#91835—21no—
15#91787—22no—
16#91781—21no—
17#91764C2212yes—
18C4×D60C2211yescharacteristic
19#91758C2211yescharacteristic
20C4×Dic30C2212yes—
21C5×#944—21no—
22C5×#889—22no—
23C5×#887—21no—
24C5×#870C2212yes—
25C5×#869C2211yescharacteristic
26C4×C5×Dic6C2212yes—
27C4×C5×Dic6C2211yescharacteristic
28C2×#6663C2212yes—
29C2×#6660C2211yescharacteristic
30C3×C4⋊#168C2211yescharacteristic
31C3×C4⋊#168C2212yes—
32C3×C4×#168C2212yes—
33C3×C4×#168C2211yescharacteristic
34#90988C2212yes—
35#90985C2211yescharacteristic
36#90975C2212yes—
37#90975C2212yes—
38#90975C2212yes—
39#90973C2211yescharacteristic
40#90969C2212yes—
41#90969C2211yescharacteristic
42#79934—51nomaximal, Hall
43C5×#557—31no—
44#65918—31no—
45#65382—31no—
46#65376—31no—
47#65357—32no—
48#65352—32no—
49#65320—31no—
50C4×#689—32no—
51#65317—32no—
52C4×#688—31no—
53Q8×#168D311yescharacteristic, complemented
54C2×C3×C5×Q8D411yescharacteristic
55C2×D60C2311yescharacteristic
56C2×C4×D15—21no—
57C2×Dic30C2312yes—
58C2×C5×Dic6—21no—
59C2×C5×Dic6C2311yescharacteristic
60C3×C5×C4⋊C4D411yescharacteristic
61C3×C5×C4⋊C4D412yes—
62C3×C42×C5C2311yescharacteristic
63C3×C42×C5C2312yes—
64#6714—21no—
65#6714—22no—
66#6714—22no—
67C4⋊Dic15—21no—
68C4⋊Dic15—21no—
69C4⋊Dic15C2311yescharacteristic
70#6712—22no—
71C4×Dic15—22no—
72C5×C4⋊Dic3—21no—
73C5×C4⋊Dic3C2312yes—
74C5×C4⋊Dic3C2311yescharacteristic
75C5×Dic3⋊C4—21no—
76C5×Dic3⋊C4—22no—
77C5×Dic3⋊C4—22no—
78C5×Dic3⋊C4—22no—
79C4×C5×Dic3—22no—
80C4×C5×Dic3—21no—
81C2×C3×#168C2311yescharacteristic
82C2×C3×#168C2312yes—
83C2×C3×#168C2311yescharacteristic
84#6663—24no—
85#6660—22no—
86#5584—51no—
87#5567D511yescharacteristic, complemented
88C3×C8×Q8—51no—
89#4849—51no—
90#4846—51no—
91#4845—52no—
92#4829—52no—
93C4×Dic12—52no—
94#4697—52no—
95C4×#204—51no—
96#4681—51no—
97C5×C4⋊Q8—31no—
98C5×C4⋊Q8—61no—
99C5×#125—32no—
100C5×#125—62no—
101C4×C5×Q8—31no—
102C4×C5×Q8—32no—
103C4×C5×Q8—61no—
104C4×C5×Q8C2×D311yescharacteristic
105#4092—61no—
106C2×#689—32no—
107C2×#688—31no—
108#4044—62no—
109#4038—61no—
110#4021—32no—
111C4×D20—31no—
112#4015—31no—
113C4×Dic10—32no—
114#3967—31no—
115#3942—32no—
116#3942—32no—
117#3942—32no—
118#3941—31no—
119#3941—32no—
120#3940—31no—
121#3939—32no—
122C4⋊#168C2×D312yes—
123C4⋊#168C2×D311yescharacteristic
124C4×#168C2×D311yescharacteristic
125C4×#168C2×D312yes—
126#3364—151noSylow
127C22×D15—21no—
128C3×C5×Q8—22no—
129C2×C3×C4×C5C2×D412yes—
130C2×C3×C4×C5C2×D412yes—
131C2×C3×C4×C5C2×D411yescharacteristic
132C2×C3×C4×C5C2×D411yescharacteristic
133C2×C3×C4×C5C2411yescharacteristic
134C2×Dic15—21no—
135C2×Dic15—22no—
136D60—21no—
137D60—21no—
138C4×D15—42no—
139Dic30—22no—
140Dic30—22no—
141C2×C5×Dic3—21no—
142C2×C5×Dic3—21no—
143C2×C5×Dic3—22no—
144C5×Dic6—21no—
145C5×Dic6—42no—
146C5×Dic6#4112yes—
147C3×#168—21no—
148C3×#168—22no—
149C3×#168—21no—
150C3×C4×Q8C2×D511yescharacteristic
151#946—101no—
152#944—21no—
153C2×Dic12—52no—
154C2×#204—51no—
155#898—102no—
156#892—101no—
157#889—22no—
158#887—21no—
159#875—52no—
160C4×D12—51no—
161#870C2×D512yes—
162#869—51no—
163#869C2×D511yescharacteristic
164C4×Dic6—52no—
165C4×Dic6C2×D511yescharacteristic
166C4×Dic6C2×D512yes—
167C3×C4⋊C8—52no—
168C3×C4⋊C8—51no—
169C3×C4×C8—51no—
170C3×C4×C8—52no—
171#821—51no—
172#821—52no—
173#818—52no—
174#817—51no—
175#816—51no—
176#816—52no—
177#816—52no—
178#816—52no—
179C2×C5×Q8—31no—
180C2×C5×Q8—61no—
181C2×C5×Q8D1211yescharacteristic
182C2×D20—31no—
183C2×C4×D5—61no—
184C2×Dic10—32no—
185C5×C4⋊C4—31no—
186C5×C4⋊C4—32no—
187C5×C4⋊C4—61no—
188C5×C4⋊C4—62no—
189C5×C4⋊C4—62no—
190C5×C4⋊C4—62no—
191C5×C4⋊C4—61no—
192C5×C4⋊C4D1211yescharacteristic
193C5×C4⋊C4D1212yes—
194C42×C5—62no—
195C42×C5—61no—
196C42×C5C22×D312yes—
197C42×C5C22×D311yescharacteristic
198#689—64no—
199#688—62no—
200#685—61no—
201#685—62no—
202#685—62no—
203C4⋊Dic5—31no—
204C4⋊Dic5—61no—
205C4⋊Dic5—61no—
206#683—62no—
207C4×Dic5—62no—
208C2×#168C22×D311yescharacteristic
209C2×#168C22×D312yes—
210C2×#168C22×D311yescharacteristic
211#557—31no—
212#556—151no—
213#488—152no—
214#487—151no—
215#479—152no—
216#474—151no—
217#464—152no—
218#462—151no—
219C8×Q8—51no—
220C4×Q16—152no—
221C4×SD16—151no—
222C22×C3×C5C22×D411yescharacteristic
223C2×D15—41no—
224C2×D15—41no—
225C2×D15—41no—
226C3×C4×C5—22no—
227C3×C4×C5—22no—
228C3×C4×C5—21no—
229C3×C4×C5#13512yes—
230C3×C4×C5C2×#4111yescharacteristic
231C3×C4×C52−1+411yescharacteristic
232Dic15—42no—
233Dic15—41no—
234C5×Dic3—22no—
235C5×Dic3—42no—
236C5×Dic3—41no—
237C2×C3×Q8C22⋊D511yescharacteristic
238C2×D12—51no—
239C2×C4×D3—101no—
240C2×Dic6—21no—
241C2×Dic6—52no—
242C2×Dic6C22×D511yescharacteristic
243C2×C3×C8—51no—
244C2×C3×C8—51no—
245C2×C3×C8—52no—
246C3×C4⋊C4C22⋊D511yescharacteristic
247C3×C4⋊C4C22⋊D512yes—
248C3×C42C22×D512yes—
249C3×C42C22×D511yescharacteristic
250(C2×D3)⋊C4—101no—
251(C2×D3)⋊C4—102no—
252(C2×D3)⋊C4—102no—
253C4⋊Dic3—21no—
254C4⋊Dic3—51no—
255C4⋊Dic3—101no—
256C4⋊Dic3—101no—
257C4⋊Dic3C22×D511yescharacteristic
258C4⋊Dic3C22×D512yes—
259Dic3⋊C4—21no—
260Dic3⋊C4—22no—
261Dic3⋊C4—22no—
262Dic3⋊C4—22no—
263Dic3⋊C4—102no—
264C4×Dic3—22no—
265C4×Dic3—21no—
266C4×Dic3—102no—
267Dic12—104no—
268#204—102no—
269C22×D5—61no—
270C5×Q8—22no—
271C5×Q8—32no—
272C5×Q8—61no—
273C5×Q8—122no—
274C2×C4×C5—61no—
275C2×C4×C5—61no—
276C2×C4×C5—62no—
277C2×C4×C5C2×D1212yes—
278C2×C4×C5C2×D1212yes—
279C2×C4×C5C2×D1211yescharacteristic
280C2×C4×C5C2×D1211yescharacteristic
281C2×C4×C5C23×D311yescharacteristic
282C2×Dic5—61no—
283C2×Dic5—62no—
284D20—61no—
285D20—61no—
286C4×D5—122no—
287Dic10—62no—
288Dic10—62no—
289#168—21no—
290#168—21no—
291#168—22no—
292C2×Q16—152no—
293C2×SD16—151no—
294C4⋊Q8—31no—
295C4⋊Q8—61no—
296C4⋊Q8—151no—
297#126—302no—
298#125—32no—
299#125—62no—
300#124—152no—
301C22⋊Q8—301no—
302C4×Q8—31no—
303C4×Q8—32no—
304C4×Q8—61no—
305C4×Q8—152no—
306C4×Q8D3×D511yescharacteristic
307C4×D4—151no—
308#117—301no—
309#107—152no—
310#106—151no—
311C4⋊C8—51no—
312C4⋊C8—52no—
313#103—152no—
314#103—152no—
315#103—152no—
316#103—151no—
317#102—151no—
318#102—152no—
319C4×C8—51no—
320C4×C8—52no—
321C2×C3×C5#54811yescharacteristic
322C2×C3×C5C2×#13511yescharacteristic
323C2×C3×C5#57811yescharacteristic
324D15—81no—
325C22×D3—101no—
326C3×Q8—22no—
327C2×C3×C4C2×C22⋊D512yes—
328C2×C3×C4C2×C22⋊D512yes—
329C2×C3×C4C2×C22⋊D511yescharacteristic
330C2×C3×C4C2×C22⋊D511yescharacteristic
331C2×C3×C4C23×D511yescharacteristic
332C2×Dic3—21no—
333C2×Dic3—21no—
334C2×Dic3—22no—
335C2×Dic3—101no—
336C2×Dic3—102no—
337D12—101no—
338D12—101no—
339C4×D3—202no—
340Dic6—21no—
341Dic6—42no—
342Dic6—102no—
343Dic6—102no—
344Dic6#71312yes—
345C3×C8—101no—
346C3×C8—102no—
347C3×C8—101no—
348C22×C5C22×D1211yescharacteristic
349C2×D5—121no—
350C2×D5—121no—
351C2×D5—121no—
352C4×C5—22no—
353C4×C5—22no—
354C4×C5—21no—
355C4×C5—62no—
356C4×C5—122no—
357C4×C5—121no—
358C4×C5#90412yes—
359C4×C5#100711yescharacteristic
360C4×C5D3×#4111yescharacteristic
361Dic5—121no—
362Dic5—122no—
363C2×Q8—31no—
364C2×Q8—61no—
365C2×Q8—152no—
366C2×Q8#120311yescharacteristic
367C2×D4—151no—
368C22×C4—301no—
369Q16—304no—
370SD16—302no—
371C2×C8—51no—
372C2×C8—52no—
373C2×C8—51no—
374C4⋊C4—31no—
375C4⋊C4—32no—
376C4⋊C4—61no—
377C4⋊C4—62no—
378C4⋊C4—62no—
379C4⋊C4—62no—
380C4⋊C4—61no—
381C4⋊C4—151no—
382C4⋊C4—301no—
383C4⋊C4—302no—
384C4⋊C4—301no—
385C4⋊C4#120311yescharacteristic
386C4⋊C4#120312yes—
387C22⋊C4—302no—
388C22⋊C4—301no—
389C22⋊C4—302no—
390C42—62no—
391C42—61no—
392C42—302no—
393C42C2×D3×D511yescharacteristic
394C42C2×D3×D512yes—
395C3×C5#336411yescharacteristic, complemented, Hall
396C22×C3C22×C22⋊D511yescharacteristic
397C2×D3—201no—
398C2×D3—201no—
399C2×D3—201no—
400C3×C4—22no—
401C3×C4—22no—
402C3×C4—21no—
403C3×C4#409612yes—
404C3×C4C2×#71311yescharacteristic
405C3×C4#414711yescharacteristic
406Dic3—22no—
407Dic3—42no—
408Dic3—41no—
409Dic3—202no—
410Dic3—201no—
411C2×C5#557611yescharacteristic
412C2×C5C2×#90411yescharacteristic
413C2×C5#575311yescharacteristic
414D5—241no—
415C23—301no—
416Q8—22no—
417Q8—32no—
418Q8—61no—
419Q8—122no—
420Q8—302no—
421Q8—302no—
422D4—301no—
423D4—301no—
424C2×C4—61no—
425C2×C4—61no—
426C2×C4—62no—
427C2×C4—301no—
428C2×C4—302no—
429C2×C4—602no—
430C2×C4C2×#120312yes—
431C2×C4C2×#120311yescharacteristic
432C2×C4C2×#120312yes—
433C2×C4C2×#120311yescharacteristic
434C2×C4C22×D3×D511yescharacteristic
435C8—101no—
436C8—101no—
437C8—102no—
438C2×C3#6615111yescharacteristic
439C2×C3C2×#409611yescharacteristic
440C2×C3#6617211yescharacteristic
441D3—401no—
442C5#7993411yescharacteristic, complemented, Sylow
443C22—601no—
444C22—601no—
445C22—601no—
446C22C22×#120311yescharacteristic
447C4—22no—
448C4—21no—
449C4—22no—
450C4—62no—
451C4—121no—
452C4—122no—
453C4—602no—
454C4—601no—
455C4#9130312yes—
456C4#9202911yescharacteristic
457C4D3×#71311yescharacteristic
458C3#1061542011yescharacteristic, complemented, Sylow
459C2—1201no—
460C2C2×#9130311yescharacteristic
461C2#1175513111yescharacteristic
462C2#1175530611yescharacteristic
subgroupG/H
centerC22C22×#1203
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C4×C5×Q8C22
socleC22×C3×C5C22×D4
prime radicalC2×D60C23
hypercenterC4×Q8D3×D5
hypocenterC3×C5#3364
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
2. C5×#5567C2semidirect
3. C3×Q8×#168C2semidirect
5. #11747101C2semidirect
6. C4×#6663C2semidirect
8. #11746967C2semidirect
10. #11746930C2semidirect
53. Q8×#168D3semidirect
87. #5567D5semidirect
395. C3×C5#3364semidirect
442. C5#79934semidirect
458. C3#10615420semidirect

Maximal subgroups

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

subgroupindex
#117563432
C5×#55672
C3×Q8×#1682
#117471022
#117471012
C4×#66632
C4×#66602
#117469672
#117469662
#117469302
#117469132
#106154203
#799345