Finite Group Databasebeta

← #14472165

Subgroups — #14472165

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
1C5×#5721C211yesmaximal, characteristic, complemented
2#11752507C211yesmaximal, characteristic
3#11751194C211yesmaximal, characteristic, complemented
4C3×#65376C211yesmaximal, characteristic
5#11747374C212yesmaximal
6#11747373C212yesmaximal
7#11747290C212yesmaximal, complemented
8#11747275C212yesmaximal, complemented
9#11747101C211yesmaximal, characteristic, complemented
10C4×#6655C211yesmaximal, characteristic
11#11746857C211yesmaximal, characteristic, complemented
12#10615420—31nomaximal, Hall
13C3×C5×C4⋊Q8C2211yescharacteristic
14#91758C2211yescharacteristic
15C5×Q8×Dic3—21no—
16C5×#944—21no—
17C5×#890C2212yes—
18C5×#889—22no—
19C5×#887C2212yes—
20C4×C5×Dic6C2211yescharacteristic
21C4×C5×Dic6C2211yescharacteristic
22C3×C4×D20C2211yescharacteristic
23#91364C2212yes—
24#91362—22no—
25#91359—21no—
26#91358—21no—
27#91340C2212yes—
28C2×#6663C2212yes—
29C2×#6655C2211yescharacteristic
30#91094C2212yes—
31#91094C2212yes—
32C4×C3⋊#168C2211yescharacteristic
33C3×#3941C2212yes—
34C3×#3939C2212yes—
35C3×C4⋊#168C2211yescharacteristic
36#90986C2211yescharacteristic
37Dic3⋊#168C2212yes—
38#90975C2212yes—
39#90972C2211yescharacteristic
40#90964C2211yescharacteristic
41Dic3×#168C2212yes—
42#80994—51nomaximal, Hall
43C5×#557—31no—
44#65918—31no—
45#65382—31no—
46#65376D311yescharacteristic
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×Dic30C2312yes—
56C2×C5×Dic6D411yescharacteristic
57C2×C5×Dic6C2311yescharacteristic
58C2×C3×D20C2311yescharacteristic
59C2×C3×C4×D5—21no—
60C3×C5×C4⋊C4—21no—
61C3×C5×C4⋊C4C2312yes—
62C3×C42×C5C2311yescharacteristic
63C4⋊Dic15—21no—
64C4⋊Dic15—21no—
65C2×C3⋊#168C2311yescharacteristic
66C2×C3⋊#168C2311yescharacteristic
67C5×C4⋊Dic3—22no—
68C5×C4⋊Dic3D411yescharacteristic
69C5×C4⋊Dic3C2311yescharacteristic
70C5×Dic3⋊C4—21no—
71C5×Dic3⋊C4—22no—
72C5×Dic3⋊C4—22no—
73C5×Dic3⋊C4D412yes—
74C4×C5×Dic3—21no—
75C4×C5×Dic3—22no—
76C4×C5×Dic3C2312yes—
77C3×#685—21no—
78C3×C4⋊Dic5C2311yescharacteristic
79C2×C3×#168C2312yes—
80#6669—22no—
81#6665—22no—
82#6665—22no—
83Dic3×Dic5—22no—
84#6663—24no—
85#6655—22no—
86#5721D511yescharacteristic, complemented
87#5542—51no—
88C3×#474—51no—
89#5093—51no—
90C4×#214—51no—
91#4877—52no—
92#4873—52no—
93#4849—51no—
94#4830—51no—
95#4768—52no—
96#4767—52no—
97C5×C4⋊Q8—61no—
98C5×C4⋊Q8C2×D311yescharacteristic
99C5×#125—32no—
100C5×#125—62no—
101C4×C5×Q8—31no—
102C4×C5×Q8—32no—
103C4×C5×Q8—31no—
104C4×C5×Q8—61no—
105#4092—61no—
106C2×#689—32no—
107C2×#688—31no—
108#4044—62no—
109#4038—61no—
110#4021—32no—
111C4×D20C2×D311yescharacteristic
112#4015—31no—
113C4×Dic10—32no—
114#3967—31no—
115#3942—32no—
116#3942—32no—
117#3942—32no—
118#3941—31no—
119#3941C2×D312yes—
120#3940—31no—
121#3939C2×D312yes—
122C4⋊#168—32no—
123C4⋊#168C2×D311yescharacteristic
124C4×#168—31no—
125C4×#168—32no—
126#3364—151noSylow
127C22×C3×D5—21no—
128C3×C5×Q8—42no—
129C2×C3×C4×C5—22no—
130C2×C3×C4×C5C2×D411yescharacteristic
131C2×C3×C4×C5C2×D411yescharacteristic
132C2×C3×C4×C5C2411yescharacteristic
133C2×Dic15—22no—
134Dic30—22no—
135Dic30—22no—
136C2×C5×Dic3—21no—
137C2×C5×Dic3—21no—
138C2×C5×Dic3C2×D412yes—
139C2×C5×Dic3C2×D412yes—
140C5×Dic6—22no—
141C5×Dic6—21no—
142C5×Dic6#4112yes—
143C2×C3×Dic5—21no—
144C3×D20—21no—
145C3×D20—21no—
146C3×C4×D5—42no—
147C3⋊#168—21no—
148C3⋊#168—21no—
149C3×#168—22no—
150C3×C4⋊Q8C2×D511yescharacteristic
151C3×C4×D4—51no—
152Q8×Dic3—21no—
153#944—21no—
154#936—52no—
155C2×#214—51no—
156#926—101no—
157#924—101no—
158C2×Dic12—52no—
159#890C2×D512yes—
160#889—22no—
161#887—52no—
162#887C2×D512yes—
163#879—102no—
164#869—51no—
165C4×Dic6C2×D511yescharacteristic
166C4×Dic6C2×D511yescharacteristic
167C3×#107—52no—
168C3×C4⋊C8—51no—
169C3×#102—52no—
170#832—51no—
171#816—52no—
172Dic3⋊C8—52no—
173C8×Dic3—52no—
174#810—51no—
175#810—52no—
176#810—52no—
177#808—51no—
178C4×C3⋊C8—51no—
179C2×C5×Q8—21no—
180C2×C5×Q8—31no—
181C2×C5×Q8—31no—
182C2×D20C22×D311yescharacteristic
183C2×C4×D5—21no—
184C2×Dic10—32no—
185C5×C4⋊C4—21no—
186C5×C4⋊C4—31no—
187C5×C4⋊C4—32no—
188C5×C4⋊C4—31no—
189C5×C4⋊C4—62no—
190C5×C4⋊C4—62no—
191C5×C4⋊C4—62no—
192C5×C4⋊C4—61no—
193C5×C4⋊C4C22×D312yes—
194C42×C5—32no—
195C42×C5—62no—
196C42×C5—61no—
197C42×C5C22×D311yescharacteristic
198#689—64no—
199#688—62no—
200#685—21no—
201#685—62no—
202#685—62no—
203C4⋊Dic5—61no—
204C4⋊Dic5—61no—
205C4⋊Dic5C22×D311yescharacteristic
206#683—62no—
207C4×Dic5—62no—
208C2×#168—31no—
209C2×#168—31no—
210C2×#168C22×D312yes—
211#557—31no—
212#556—151no—
213#488—152no—
214#487—151no—
215#479—152no—
216#474—51no—
217#464—152no—
218#462—151no—
219C8×Q8—151no—
220C4×Q16—152no—
221C4×SD16—151no—
222C22×C3×C5C22×D411yescharacteristic
223C2×C3×D5—41no—
224C2×C3×D5—41no—
225C2×C3×D5—41no—
226C3×C4×C5—21no—
227C3×C4×C5—42no—
228C3×C4×C5#13512yes—
229C3×C4×C5C2×#4111yescharacteristic
230C3×C4×C52−1+411yescharacteristic
231Dic15—42no—
232C3×Dic5—41no—
233C5×Dic3—22no—
234C5×Dic3—22no—
235C5×Dic3—22no—
236C5×Dic3—41no—
237C2×C3×Q8—21no—
238C2×C3×D4—51no—
239C22×C3×C4—101no—
240C2×Dic6—52no—
241C2×Dic6C22⋊D511yescharacteristic
242C2×Dic6C22×D511yescharacteristic
243C2×C3×C8—52no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—51no—
246C3×C4⋊C4C22×D512yes—
247C3×C22⋊C4—101no—
248C3×C42C22×D511yescharacteristic
249#217—102no—
250#217—102no—
251#214—102no—
252C4⋊Dic3—22no—
253C4⋊Dic3—101no—
254C4⋊Dic3—101no—
255C4⋊Dic3C22⋊D511yescharacteristic
256C4⋊Dic3C22×D511yescharacteristic
257Dic3⋊C4—21no—
258Dic3⋊C4—22no—
259Dic3⋊C4—22no—
260Dic3⋊C4—102no—
261Dic3⋊C4C22⋊D512yes—
262C4×Dic3—21no—
263C4×Dic3—22no—
264C4×Dic3—102no—
265C4×Dic3C22×D512yes—
266C2×C3⋊C8—51no—
267C2×C3⋊C8—51no—
268Dic12—104no—
269C22×D5—21no—
270C5×Q8—32no—
271C5×Q8—42no—
272C5×Q8—62no—
273C5×Q8—61no—
274C2×C4×C5—22no—
275C2×C4×C5—32no—
276C2×C4×C5—32no—
277C2×C4×C5—61no—
278C2×C4×C5—61no—
279C2×C4×C5D3×D411yescharacteristic
280C2×C4×C5D3×D411yescharacteristic
281C2×C4×C5C23×D311yescharacteristic
282C2×Dic5—21no—
283C2×Dic5—62no—
284D20—21no—
285D20—21no—
286C4×D5—42no—
287Dic10—62no—
288Dic10—62no—
289#168—22no—
290#168—61no—
291#168—61no—
292C2×Q16—152no—
293C2×SD16—151no—
294C4⋊Q8—61no—
295C4⋊Q8—151no—
296C4⋊Q8D3×D511yescharacteristic
297#126—302no—
298#125—32no—
299#125—62no—
300#124—152no—
301C22⋊Q8—301no—
302C4×Q8—31no—
303C4×Q8—32no—
304C4×Q8—31no—
305C4×Q8—61no—
306C4×Q8—152no—
307C4×D4—51no—
308#117—301no—
309#107—52no—
310#106—151no—
311C4⋊C8—51no—
312C4⋊C8—152no—
313#103—152no—
314#103—151no—
315#103—152no—
316#103—152no—
317#102—52no—
318#102—151no—
319C4×C8—152no—
320C4×C8—151no—
321C2×C3×C5#54811yescharacteristic
322C2×C3×C5C2×#13511yescharacteristic
323C2×C3×C5#57811yescharacteristic
324C3×D5—81no—
325C23×C3—101no—
326C3×Q8—42no—
327C3×D4—101no—
328C3×D4—101no—
329C2×C3×C4—22no—
330C2×C3×C4—101no—
331C2×C3×C4—202no—
332C2×C3×C4C2×C22⋊D511yescharacteristic
333C2×C3×C4C2×C22⋊D511yescharacteristic
334C2×C3×C4C23×D511yescharacteristic
335C2×Dic3—21no—
336C2×Dic3—21no—
337C2×Dic3—102no—
338C2×Dic3C2×C22⋊D512yes—
339C2×Dic3C2×C22⋊D512yes—
340Dic6—22no—
341Dic6—21no—
342Dic6—102no—
343Dic6—102no—
344Dic6#71312yes—
345C3×C8—102no—
346C3⋊C8—101no—
347C3⋊C8—101no—
348C22×C5C2×D3×D411yescharacteristic
349C2×D5—41no—
350C2×D5—41no—
351C2×D5—41no—
352C4×C5—21no—
353C4×C5—42no—
354C4×C5—62no—
355C4×C5—62no—
356C4×C5—62no—
357C4×C5—121no—
358C4×C5#91212yes—
359C4×C5#100711yescharacteristic
360C4×C5D3×#4111yescharacteristic
361Dic5—41no—
362Dic5—122no—
363C2×Q8—21no—
364C2×Q8—31no—
365C2×Q8—31no—
366C2×Q8—152no—
367C2×D4—51no—
368C22×C4—101no—
369Q16—304no—
370SD16—302no—
371C2×C8—52no—
372C2×C8—151no—
373C2×C8—151no—
374C4⋊C4—21no—
375C4⋊C4—31no—
376C4⋊C4—32no—
377C4⋊C4—31no—
378C4⋊C4—51no—
379C4⋊C4—62no—
380C4⋊C4—62no—
381C4⋊C4—62no—
382C4⋊C4—61no—
383C4⋊C4—302no—
384C4⋊C4—301no—
385C4⋊C4—301no—
386C4⋊C4C2×D3×D512yes—
387C22⋊C4—101no—
388C22⋊C4—302no—
389C22⋊C4—302no—
390C42—32no—
391C42—62no—
392C42—61no—
393C42—302no—
394C42C2×D3×D511yescharacteristic
395C3×C5#336411yescharacteristic, complemented, Hall
396C22×C3—201no—
397C22×C3—201no—
398C22×C3—201no—
399C22×C3C22×C22⋊D511yescharacteristic
400C3×C4—21no—
401C3×C4—42no—
402C3×C4—201no—
403C3×C4#409612yes—
404C3×C4C2×#71311yescharacteristic
405C3×C4#414711yescharacteristic
406Dic3—22no—
407Dic3—22no—
408Dic3—22no—
409Dic3—41no—
410Dic3—202no—
411C2×C5#563311yescharacteristic
412C2×C5C2×#91211yescharacteristic
413C2×C5#577911yescharacteristic
414D5—81no—
415C23—101no—
416Q8—32no—
417Q8—42no—
418Q8—62no—
419Q8—61no—
420Q8—302no—
421Q8—302no—
422D4—101no—
423D4—101no—
424C2×C4—22no—
425C2×C4—32no—
426C2×C4—32no—
427C2×C4—61no—
428C2×C4—61no—
429C2×C4—101no—
430C2×C4—202no—
431C2×C4—302no—
432C2×C4D3×C22⋊D511yescharacteristic
433C2×C4D3×C22⋊D511yescharacteristic
434C2×C4C22×D3×D511yescharacteristic
435C8—102no—
436C8—301no—
437C8—301no—
438C2×C3—401no—
439C2×C3#6615111yescharacteristic
440C2×C3C2×#409611yescharacteristic
441C2×C3#6617211yescharacteristic
442C5#8099411yescharacteristic, complemented, Sylow
443C22—201no—
444C22—201no—
445C22—201no—
446C22C2×D3×C22⋊D511yescharacteristic
447C4—21no—
448C4—42no—
449C4—62no—
450C4—62no—
451C4—62no—
452C4—121no—
453C4—201no—
454C4—602no—
455C4#9149312yes—
456C4#9202911yescharacteristic
457C4D3×#71311yescharacteristic
458C3#1061542011yescharacteristic, complemented, Sylow
459C2—401no—
460C2C2×#9149311yescharacteristic
461C2#1175511411yescharacteristic
462C2#1175523411yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊D5
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C5×C4⋊Q8C22
socleC22×C3×C5C22×D4
prime radicalC2×C3×D20C23
hypercenterC4⋊Q8D3×D5
hypocenterC3×C5#3364
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. C5×#5721C2semidirect
3. #11751194C2semidirect
7. #11747290C2semidirect
8. #11747275C2semidirect
9. #11747101C2semidirect
11. #11746857C2semidirect
86. #5721D5semidirect
395. C3×C5#3364semidirect
442. C5#80994semidirect
458. C3#10615420semidirect

Maximal subgroups

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

subgroupindex
C5×#57212
#117525072
#117511942
C3×#653762
#117473742
#117473732
#117472902
#117472752
#117471012
C4×#66552
#117468572
#106154203
#809945