Finite Group Databasebeta

← #14472075

Subgroups — #14472075

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
1C3×C5×#557C211yesmaximal, characteristic, complemented
2#11756343C211yesmaximal, characteristic
3#11751194C211yesmaximal, characteristic, complemented
4#11751188C211yesmaximal, characteristic
5#11751169C212yesmaximal
6#11751164C212yesmaximal, complemented
7#11751132C211yesmaximal, characteristic, complemented
8C4×#6718C212yesmaximal, complemented
9#11751129C212yesmaximal
10C4×#6717C211yesmaximal, characteristic
11Q8×C3⋊#168C211yesmaximal, characteristic, complemented
12#10615420—31nomaximal, Hall
13C3×C5×C4⋊Q8—21no—
14C3×C5×C4⋊Q8C2211yescharacteristic
15C3×C5×#125—22no—
16C3×C5×#125C2212yes—
17C3×C4×C5×Q8—21no—
18C3×C4×C5×Q8C2211yescharacteristic
19C3×C4×C5×Q8C2212yes—
20C3×C4×C5×Q8C2211yescharacteristic
21#91835—21no—
22C2×#6718C2212yes—
23C2×#6717C2211yescharacteristic
24#91787—22no—
25#91781—21no—
26#91764C2212yes—
27C4×D60C2211yescharacteristic
28#91758C2211yescharacteristic
29C4×Dic30C2212yes—
30#91119C2211yescharacteristic
31#91094C2212yes—
32#91094C2212yes—
33#91094C2212yes—
34#91093C2211yescharacteristic
35#91093C2212yes—
36#91092C2211yescharacteristic
37#91091C2212yes—
38C4⋊(C3⋊#168)C2211yescharacteristic
39C4⋊(C3⋊#168)C2212yes—
40C4×C3⋊#168C2212yes—
41C4×C3⋊#168C2211yescharacteristic
42#83689—51nomaximal, Hall
43C5×#557D311yescharacteristic, complemented
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×#168—31no—
54C2×C3×C5×Q8—21no—
55C2×C3×C5×Q8D411yescharacteristic
56C2×C3×C5×Q8C2311yescharacteristic
57C2×D60C2311yescharacteristic
58C2×C4×D15—21no—
59C2×Dic30C2312yes—
60C3×C5×C4⋊C4—22no—
61C3×C5×C4⋊C4—21no—
62C3×C5×C4⋊C4—22no—
63C3×C5×C4⋊C4—21no—
64C3×C5×C4⋊C4—22no—
65C3×C5×C4⋊C4D411yescharacteristic
66C3×C5×C4⋊C4D412yes—
67C3×C5×C4⋊C4C2311yescharacteristic
68C3×C5×C4⋊C4C2312yes—
69C3×C42×C5—22no—
70C3×C42×C5—21no—
71C3×C42×C5C2312yes—
72C3×C42×C5C2311yescharacteristic
73#6718—24no—
74#6717—22no—
75#6714—21no—
76#6714—22no—
77#6714—22no—
78C4⋊Dic15—21no—
79C4⋊Dic15—21no—
80C4⋊Dic15C2311yescharacteristic
81#6712—22no—
82C4×Dic15—22no—
83C2×C3⋊#168C2311yescharacteristic
84C2×C3⋊#168C2312yes—
85C2×C3⋊#168C2311yescharacteristic
86C3×#557D511yescharacteristic, complemented
87#5584—51no—
88#5093—51no—
89#5087—51no—
90#5068—52no—
91#5063—52no—
92#5031—51no—
93C4×#216—52no—
94#5028—52no—
95C4×#215—51no—
96Q8×C3⋊C8—51no—
97C5×C4⋊Q8—21no—
98C5×C4⋊Q8C2×D311yescharacteristic
99C5×#125—22no—
100C5×#125C2×D312yes—
101C4×C5×Q8—21no—
102C4×C5×Q8C2×D311yescharacteristic
103C4×C5×Q8C2×D312yes—
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⋊#168—32no—
123C4⋊#168—31no—
124C4×#168—31no—
125C4×#168—32no—
126#3364—151noSylow
127C22×D15—21no—
128C3×C5×Q8—22no—
129C3×C5×Q8—21no—
130C3×C5×Q8—42no—
131C3×C5×Q8#4112yes—
132C2×C3×C4×C5—21no—
133C2×C3×C4×C5—21no—
134C2×C3×C4×C5—22no—
135C2×C3×C4×C5C2×D412yes—
136C2×C3×C4×C5C2×D412yes—
137C2×C3×C4×C5C2×D411yescharacteristic
138C2×C3×C4×C5C2×D411yescharacteristic
139C2×C3×C4×C5C2411yescharacteristic
140C2×Dic15—21no—
141C2×Dic15—22no—
142D60—21no—
143D60—21no—
144C4×D15—42no—
145Dic30—22no—
146Dic30—22no—
147C3⋊#168—21no—
148C3⋊#168—22no—
149C3⋊#168—21no—
150C3×C4⋊Q8—21no—
151C3×C4⋊Q8C2×D511yescharacteristic
152C3×#125—22no—
153C3×#125C2×D512yes—
154C3×C4×Q8—21no—
155C3×C4×Q8C2×D511yescharacteristic
156C3×C4×Q8C2×D512yes—
157C3×C4×Q8C2×D511yescharacteristic
158#946—101no—
159C2×#216—52no—
160C2×#215—51no—
161#898—102no—
162#892—101no—
163#875—52no—
164C4×D12—51no—
165#869—51no—
166C4×Dic6—52no—
167#835—51no—
168#810—52no—
169#810—52no—
170#810—52no—
171#809—51no—
172#809—52no—
173#808—51no—
174#807—52no—
175C4⋊(C3⋊C8)—52no—
176C4⋊(C3⋊C8)—51no—
177C4×C3⋊C8—51no—
178C4×C3⋊C8—52no—
179C2×C5×Q8—21no—
180C2×C5×Q8#6711yescharacteristic
181C2×C5×Q8C22×D311yescharacteristic
182C2×D20—31no—
183C2×C4×D5—61no—
184C2×Dic10—32no—
185C5×C4⋊C4—22no—
186C5×C4⋊C4—21no—
187C5×C4⋊C4—22no—
188C5×C4⋊C4—21no—
189C5×C4⋊C4—22no—
190C5×C4⋊C4#6711yescharacteristic
191C5×C4⋊C4#6712yes—
192C5×C4⋊C4C22×D311yescharacteristic
193C5×C4⋊C4C22×D312yes—
194C42×C5—22no—
195C42×C5—21no—
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×#168—31no—
209C2×#168—32no—
210C2×#168—31no—
211#557D1511yescharacteristic, complemented
212#556—151no—
213#488—152no—
214#487—151no—
215#479—152no—
216#474—151no—
217#464—152no—
218#462—151no—
219C8×Q8—151no—
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—22no—
229C3×C4×C5—21no—
230C3×C4×C5—42no—
231C3×C4×C5—41no—
232C3×C4×C5#13512yes—
233C3×C4×C5C2×#4111yescharacteristic
234C3×C4×C52−1+411yescharacteristic
235Dic15—42no—
236Dic15—41no—
237C2×C3×Q8—21no—
238C2×C3×Q8C22⋊D511yescharacteristic
239C2×C3×Q8C22×D511yescharacteristic
240C2×D12—51no—
241C2×C4×D3—101no—
242C2×Dic6—52no—
243C3×C4⋊C4—22no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—22no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—22no—
248C3×C4⋊C4C22⋊D511yescharacteristic
249C3×C4⋊C4C22⋊D512yes—
250C3×C4⋊C4C22×D512yes—
251C3×C4⋊C4C22×D511yescharacteristic
252C3×C42—22no—
253C3×C42—21no—
254C3×C42C22×D512yes—
255C3×C42C22×D511yescharacteristic
256#216—104no—
257#215—102no—
258(C2×D3)⋊C4—101no—
259(C2×D3)⋊C4—102no—
260(C2×D3)⋊C4—102no—
261C4⋊Dic3—51no—
262C4⋊Dic3—101no—
263C4⋊Dic3—101no—
264Dic3⋊C4—102no—
265C4×Dic3—102no—
266C2×C3⋊C8—51no—
267C2×C3⋊C8—52no—
268C2×C3⋊C8—51no—
269C22×D5—61no—
270C5×Q8—22no—
271C5×Q8—21no—
272C5×Q8—42no—
273C5×Q8#23912yes—
274C2×C4×C5—21no—
275C2×C4×C5—21no—
276C2×C4×C5—22no—
277C2×C4×C5C2×#6712yes—
278C2×C4×C5C2×#6712yes—
279C2×C4×C5C2×#6711yescharacteristic
280C2×C4×C5C2×#6711yescharacteristic
281C2×C4×C5C23×D311yescharacteristic
282C2×Dic5—61no—
283C2×Dic5—62no—
284D20—61no—
285D20—61no—
286C4×D5—122no—
287Dic10—62no—
288Dic10—62no—
289#168—61no—
290#168—62no—
291#168—61no—
292C2×Q16—152no—
293C2×SD16—151no—
294C4⋊Q8—21no—
295C4⋊Q8—151no—
296C4⋊Q8C2×D1511yescharacteristic
297#126—302no—
298#125—22no—
299#125C2×D1512yes—
300#124—152no—
301C22⋊Q8—301no—
302C4×Q8—21no—
303C4×Q8—152no—
304C4×Q8C2×D1511yescharacteristic
305C4×Q8C2×D1512yes—
306C4×Q8C2×D1511yescharacteristic
307C4×D4—151no—
308#117—301no—
309#107—152no—
310#106—151no—
311C4⋊C8—151no—
312C4⋊C8—152no—
313#103—151no—
314#103—152no—
315#103—152no—
316#103—152no—
317#102—151no—
318#102—152no—
319C4×C8—151no—
320C4×C8—152no—
321C2×C3×C5#54811yescharacteristic
322C2×C3×C5C2×#13511yescharacteristic
323C2×C3×C5#57811yescharacteristic
324D15—81no—
325C22×D3—101no—
326C3×Q8—22no—
327C3×Q8—21no—
328C3×Q8—42no—
329C3×Q8#71312yes—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—22no—
333C2×C3×C4C2×C22⋊D512yes—
334C2×C3×C4C2×C22⋊D512yes—
335C2×C3×C4C2×C22⋊D511yescharacteristic
336C2×C3×C4C2×C22⋊D511yescharacteristic
337C2×C3×C4C23×D511yescharacteristic
338C2×Dic3—101no—
339C2×Dic3—102no—
340D12—101no—
341D12—101no—
342C4×D3—202no—
343Dic6—102no—
344Dic6—102no—
345C3⋊C8—101no—
346C3⋊C8—102no—
347C3⋊C8—101no—
348C22×C5C22×#6711yescharacteristic
349C2×D5—121no—
350C2×D5—121no—
351C2×D5—121no—
352C4×C5—22no—
353C4×C5—22no—
354C4×C5—22no—
355C4×C5—21no—
356C4×C5—42no—
357C4×C5—41no—
358C4×C5#95012yes—
359C4×C5C2×#23911yescharacteristic
360C4×C5#100711yescharacteristic
361Dic5—122no—
362Dic5—121no—
363C2×Q8—21no—
364C2×Q8—152no—
365C2×Q8#122011yescharacteristic
366C2×Q8C22×D1511yescharacteristic
367C2×D4—151no—
368C22×C4—301no—
369Q16—304no—
370SD16—302no—
371C2×C8—151no—
372C2×C8—152no—
373C2×C8—151no—
374C4⋊C4—22no—
375C4⋊C4—21no—
376C4⋊C4—22no—
377C4⋊C4—21no—
378C4⋊C4—22no—
379C4⋊C4—151no—
380C4⋊C4—301no—
381C4⋊C4—302no—
382C4⋊C4—301no—
383C4⋊C4#122011yescharacteristic
384C4⋊C4#122012yes—
385C4⋊C4C22×D1511yescharacteristic
386C4⋊C4C22×D1512yes—
387C22⋊C4—302no—
388C22⋊C4—301no—
389C22⋊C4—302no—
390C42—22no—
391C42—21no—
392C42—302no—
393C42C22×D1511yescharacteristic
394C42C22×D1512yes—
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—22no—
403C3×C4—21no—
404C3×C4—42no—
405C3×C4—41no—
406C3×C4#409612yes—
407C3×C4C2×#71311yescharacteristic
408C3×C4#414711yescharacteristic
409Dic3—202no—
410Dic3—201no—
411C2×C5#581711yescharacteristic
412C2×C5C2×#95011yescharacteristic
413C2×C5#583811yescharacteristic
414D5—241no—
415C23—301no—
416Q8—22no—
417Q8—21no—
418Q8—42no—
419Q8—302no—
420Q8—302no—
421Q8#682112yes—
422D4—301no—
423D4—301no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—301no—
428C2×C4—302no—
429C2×C4—602no—
430C2×C4C2×#122012yes—
431C2×C4C2×#122011yescharacteristic
432C2×C4C2×#122012yes—
433C2×C4C2×#122011yescharacteristic
434C2×C4C23×D1511yescharacteristic
435C8—301no—
436C8—302no—
437C8—301no—
438C2×C3#6615111yescharacteristic
439C2×C3C2×#409611yescharacteristic
440C2×C3#6617211yescharacteristic
441D3—401no—
442C5#8368911yescharacteristic, complemented, Sylow
443C22—601no—
444C22—601no—
445C22—601no—
446C22C22×#122011yescharacteristic
447C4—22no—
448C4—22no—
449C4—22no—
450C4—21no—
451C4—42no—
452C4—41no—
453C4—602no—
454C4—601no—
455C4#9183912yes—
456C4C2×#682111yescharacteristic
457C4#9209811yescharacteristic
458C3#1061542011yescharacteristic, complemented, Sylow
459C2—1201no—
460C2#1175657611yescharacteristic
461C2C2×#9183911yescharacteristic
462C2#1175659711yescharacteristic
subgroupG/H
centerC22C22×#1220
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC3×C5×#557C2
socleC22×C3×C5C22×D4
prime radicalC2×D60C23
hypercenter#557D15
hypocenterC3×C5#3364
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #557

Complements

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

NH ≅ G/Ntype
1. C3×C5×#557C2semidirect
3. #11751194C2semidirect
6. #11751164C2semidirect
7. #11751132C2semidirect
8. C4×#6718C2semidirect
11. Q8×C3⋊#168C2semidirect
43. C5×#557D3semidirect
86. C3×#557D5semidirect
211. #557D15semidirect
395. C3×C5#3364semidirect
442. C5#83689semidirect
458. C3#10615420semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5572
#117563432
#117511942
#117511882
#117511692
#117511642
#117511322
C4×#67182
#117511292
C4×#67172
Q8×C3⋊#1682
#106154203
#836895