Finite Group Databasebeta

← #14438039

Subgroups — #14438039

Order 1920.

784 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5517C211yesmaximal, characteristic, complemented
2#11754581C211yesmaximal, characteristic
3#11751103C211yesmaximal, characteristic, complemented
4C3×#65291C211yesmaximal, characteristic, complemented
5#11748393C211yesmaximal, characteristic, complemented
6#11748364C211yesmaximal, characteristic, complemented
7C2×#90953C211yesmaximal, characteristic, complemented
8#11747423C212yesmaximal, complemented
9#11747405C212yesmaximal, complemented
10#11747380C212yesmaximal, complemented
11#11747376C212yesmaximal, complemented
12#10615066—31nomaximal, Hall
13C2×#6817C2211yescharacteristic
14C2×C3×#709C2211yescharacteristic
15C3×C5×#117C2211yescharacteristic
16C22×C3⋊#168C2211yescharacteristic
17C5×#926C2211yescharacteristic
18C2×C4×C5×Dic3C2211yescharacteristic
19C5×#890C2212yes—
20C5×#888C2212yes—
21C5×#878—22no—
22C4×C5×Dic6—22no—
23C22×C3×#168C2211yescharacteristic
24#91569—22no—
25#91458C2211yescharacteristic
26C4×#1202—22no—
27#91389C2211yescharacteristic
28#91374C2211yescharacteristic
29#91364C2211yescharacteristic
30#91094C2212yes—
31#91093C2212yes—
32C3×#3942C2212yes—
33C3×#3941C2212yes—
34#90975C2212yes—
35#90970C2212yes—
36#90969C2212yes—
37#90964C2212yes—
38#90953C2212yes—
39#90953C2212yes—
40#78826—51nomaximal, Hall
41C5×#533—31no—
42#65811—31no—
43#65378—32no—
44#65378—32no—
45#65358—32no—
46#65356—32no—
47#65291—31no—
48#65291—31no—
49#65291—31no—
50#65291D311yescharacteristic, complemented
51C2×#3935—31no—
52C22×C3×C4×C5C2311yescharacteristic
53C2×#1220—21no—
54#6817—22no—
55#6817—22no—
56C2×D60C2311yescharacteristic
57C2×C4×D15—21no—
58C2×Dic30C2311yescharacteristic
59C22×C5×Dic3D412yes—
60C2×C5×Dic6—22no—
61C2×C3×C22⋊D5—21no—
62C3×#709—22no—
63C3×#709—22no—
64C2×C3×D20C2311yescharacteristic
65C2×C3×C4×D5—21no—
66C2×C3×Dic10C2311yescharacteristic
67C2×#1202—22no—
68C3×C5×C4⋊C4C2312yes—
69C3×C5×C22⋊C4—21no—
70C3×C42×C5—21no—
71C2×C3⋊#168—22no—
72C2×C3⋊#168C2312yes—
73C5×#217—21no—
74C5×C4⋊Dic3—22no—
75C5×C4⋊Dic3C2312yes—
76C5×Dic3⋊C4—22no—
77C5×Dic3⋊C4—22no—
78C4×C5×Dic3—21no—
79C4×C5×Dic3D412yes—
80C4×C5×Dic3C2311yescharacteristic
81C4×C5×Dic3C2311yescharacteristic
82C2×C3×#168—22no—
83C2×C3×#168C2312yes—
84Dic3⋊Dic5—22no—
85#6667—22no—
86#6665—22no—
87#5832—51no—
88#5517D511yescharacteristic, complemented
89C3×#416—51no—
90#5233—51no—
91#5113—51no—
92C2×#815—51no—
93#5002—51no—
94#4797—52no—
95#4794—52no—
96#4765—52no—
97#4764—52no—
98C2×#709—31no—
99C2×#709C2×D311yescharacteristic
100C5×C4⋊Q8—32no—
101C5×#125—32no—
102C5×C22⋊Q8—62no—
103C4×C5×Q8—62no—
104C5×#117—31no—
105C5×#117C2×D311yescharacteristic
106C2×C42×C5—31no—
107#4076—62no—
108C22×#168—31no—
109C22×#168C2×D311yescharacteristic
110#4021—31no—
111#4021—31no—
112#4020—31no—
113C4×D20—62no—
114#4015—31no—
115#3942—32no—
116#3942—32no—
117#3942—32no—
118#3942C2×D312yes—
119#3941—32no—
120#3941—32no—
121#3941—32no—
122#3941C2×D312yes—
123#3935—32no—
124#3935—32no—
125#3135—151noSylow
126C23×C3×C5C2×D411yescharacteristic
127C22×D15—21no—
128C22×C3×D5—21no—
129C2×C3×C4×C5—22no—
130C2×C3×C4×C5C2×D411yescharacteristic
131C2×C3×C4×C5#4112yes—
132C2×C3×C4×C5#4112yes—
133C2×C3×C4×C5C2411yescharacteristic
134#1220—42no—
135C2×Dic15—21no—
136D60—21no—
137D60—21no—
138C4×D15—42no—
139Dic30—21no—
140Dic30—21no—
141C2×C5×Dic3—22no—
142C2×C5×Dic3—42no—
143C2×C5×Dic3C2×D412yes—
144C2×C5×Dic3C2×D412yes—
145C5×Dic6—44no—
146C3×C22⋊D5—42no—
147C2×C3×Dic5—21no—
148C3×D20—21no—
149C3×D20—21no—
150C3×C4×D5—42no—
151C3×Dic10—21no—
152C3×Dic10—21no—
153#1202—44no—
154C3⋊#168—22no—
155C3×#168—22no—
156C2×C3×#41—51no—
157C2×#235—51no—
158C22×C3×C8—51no—
159C3×#117C2×D511yescharacteristic
160#947—51no—
161#944—51no—
162#940—51no—
163#939—102no—
164#936—51no—
165C4×#67—102no—
166#926C2×D511yescharacteristic
167C2×C4×Dic3C2×D511yescharacteristic
168C22×C3⋊C8—51no—
169#890C2×D512yes—
170#888C2×D512yes—
171#878—22no—
172C4×Dic6—22no—
173C3×#103—52no—
174C3×#102—52no—
175#835—52no—
176#832—52no—
177#821—52no—
178#816—52no—
179#815—52no—
180#815—52no—
181#810—52no—
182#809—52no—
183C2×C5×Q8—62no—
184C22×C4×C5—32no—
185C22×C4×C5C22×D311yescharacteristic
186C2×C22⋊D5—21no—
187C2×C22⋊D5—61no—
188#709—22no—
189#709—22no—
190#709—62no—
191#709—62no—
192C2×D20—31no—
193C2×D20—62no—
194C2×D20C22×D311yescharacteristic
195C2×C4×D5—21no—
196C2×C4×D5—61no—
197C2×Dic10—31no—
198C2×Dic10C22×D311yescharacteristic
199C5×C4⋊C4—32no—
200C5×C4⋊C4—62no—
201C5×C4⋊C4—62no—
202C5×C4⋊C4—62no—
203C5×C4⋊C4C22×D312yes—
204C5×C22⋊C4—21no—
205C5×C22⋊C4—61no—
206C42×C5—21no—
207C42×C5—32no—
208C42×C5—31no—
209C42×C5—31no—
210C42×C5—61no—
211#685—62no—
212#685—62no—
213C4⋊Dic5—62no—
214C2×#168—22no—
215C2×#168—32no—
216C2×#168—62no—
217C2×#168C22×D312yes—
218#533—31no—
219#532—151no—
220#491—152no—
221#490—152no—
222#489—152no—
223#489—152no—
224#416—51no—
225#416—151no—
226#416—151no—
227#416—151no—
228C2×#97—151no—
229C22×C3×C5—22no—
230C22×C3×C5#12412yes—
231C22×C3×C5C22×D411yescharacteristic
232C2×D15—41no—
233C2×D15—42no—
234C2×C3×D5—41no—
235C2×C3×D5—42no—
236C3×C4×C5—42no—
237C3×C4×C5#12412yes—
238C3×C4×C5C2×#4111yescharacteristic
239C3×C4×C5C2×#4111yescharacteristic
240Dic15—41no—
241C3×Dic5—41no—
242C5×Dic3—22no—
243C5×Dic3—22no—
244C5×Dic3—42no—
245C3×#41—102no—
246C3×#41—102no—
247C2×C3×Q8—51no—
248C2×C3×D4—51no—
249C2×C3×D4—101no—
250C22×C3×C4—101no—
251C22×C3×C4C22×D511yescharacteristic
252C2×#67—101no—
253C2×#67—102no—
254C22×Dic3C22⋊D512yes—
255#235—102no—
256#235—102no—
257C2×D12—51no—
258C2×C4×D3—101no—
259C2×Dic6—22no—
260C2×Dic6—51no—
261C2×C3×C8—52no—
262C2×C3×C8—102no—
263C3×C4⋊C4C22×D512yes—
264C3×C22⋊C4—21no—
265C3×C42—21no—
266#217—21no—
267#217—102no—
268(C2×D3)⋊C4—102no—
269C4⋊Dic3—22no—
270C4⋊Dic3C22×D512yes—
271Dic3⋊C4—22no—
272Dic3⋊C4—22no—
273Dic3⋊C4—102no—
274C4×Dic3—21no—
275C4×Dic3C22⋊D512yes—
276C4×Dic3C22×D511yescharacteristic
277C4×Dic3C22×D511yescharacteristic
278C2×C3⋊C8—52no—
279C2×C3⋊C8—102no—
280C23×C5D3×D411yescharacteristic
281C22×D5—21no—
282C22×D5—61no—
283C5×Q8—124no—
284C2×C4×C5—22no—
285C2×C4×C5—32no—
286C2×C4×C5—32no—
287C2×C4×C5—62no—
288C2×C4×C5—122no—
289C2×C4×C5#23512yes—
290C2×C4×C5D3×D411yescharacteristic
291C2×C4×C5#23712yes—
292C2×C4×C5C23×D311yescharacteristic
293C22⋊D5—42no—
294C22⋊D5—122no—
295C2×Dic5—21no—
296C2×Dic5—61no—
297D20—21no—
298D20—21no—
299D20—61no—
300D20—61no—
301D20—124no—
302C4×D5—42no—
303C4×D5—122no—
304Dic10—21no—
305Dic10—21no—
306Dic10—61no—
307Dic10—61no—
308#168—22no—
309#168—62no—
310C2×#41—51no—
311C2×#41—151no—
312C22×C8—51no—
313C22×C8—151no—
314C4⋊Q8—32no—
315C4⋊Q8—151no—
316#127—151no—
317#125—32no—
318#124—151no—
319#124—151no—
320C22⋊Q8—62no—
321#121—302no—
322C4×Q8—62no—
323C4×D4—302no—
324#117—31no—
325#117D3×D511yescharacteristic
326C2×C42—31no—
327#103—52no—
328#103—152no—
329#103—152no—
330#103—152no—
331#102—52no—
332#102—152no—
333#102—152no—
334#102—152no—
335#97—152no—
336#97—152no—
337C2×C3×C5—22no—
338C2×C3×C5C2×#12411yescharacteristic
339C2×C3×C5#57512yes—
340D15—81no—
341C3×D5—81no—
342C23×C3—101no—
343C23×C3C2×C22⋊D511yescharacteristic
344C22×D3—101no—
345C3×Q8—101no—
346C3×Q8—101no—
347C3×D4—101no—
348C3×D4—101no—
349C3×D4—202no—
350C2×C3×C4—22no—
351C2×C3×C4—101no—
352C2×C3×C4—202no—
353C2×C3×C4#71312yes—
354C2×C3×C4#71312yes—
355C2×C3×C4C2×C22⋊D511yescharacteristic
356C2×C3×C4C23×D511yescharacteristic
357#67—204no—
358#67—202no—
359C2×Dic3—22no—
360C2×Dic3—42no—
361C2×Dic3—101no—
362C2×Dic3C2×C22⋊D512yes—
363C2×Dic3C2×C22⋊D512yes—
364D12—101no—
365D12—101no—
366C4×D3—202no—
367Dic6—44no—
368Dic6—101no—
369Dic6—101no—
370C3×C8—102no—
371C3⋊C8—102no—
372C22×C5—22no—
373C22×C5#88512yes—
374C22×C5C2×D3×D411yescharacteristic
375C2×D5—41no—
376C2×D5—42no—
377C2×D5—121no—
378C2×D5—122no—
379C4×C5—42no—
380C4×C5—62no—
381C4×C5—62no—
382C4×C5—122no—
383C4×C5#88512yes—
384C4×C5C2×#23511yescharacteristic
385C4×C5C2×#23711yescharacteristic
386Dic5—41no—
387Dic5—121no—
388#41—102no—
389#41—102no—
390#41—302no—
391#41—302no—
392C2×Q8—51no—
393C2×Q8—62no—
394C2×Q8—151no—
395C2×D4—51no—
396C2×D4—101no—
397C2×D4—151no—
398C2×D4—302no—
399C2×D4—301no—
400C22×C4—32no—
401C22×C4—101no—
402C22×C4—301no—
403C22×C4C2×D3×D511yescharacteristic
404C2×C8—52no—
405C2×C8—102no—
406C2×C8—152no—
407C2×C8—302no—
408C4⋊C4—32no—
409C4⋊C4—62no—
410C4⋊C4—62no—
411C4⋊C4—62no—
412C4⋊C4—302no—
413C4⋊C4C2×D3×D512yes—
414C22⋊C4—21no—
415C22⋊C4—61no—
416C22⋊C4—302no—
417C22⋊C4—302no—
418C42—21no—
419C42—32no—
420C42—31no—
421C42—31no—
422C42—61no—
423C3×C5#313511yescharacteristic, complemented, Hall
424C22×C3—22no—
425C22×C3—201no—
426C22×C3—202no—
427C22×C3#409312yes—
428C22×C3C22×C22⋊D511yescharacteristic
429C2×D3—201no—
430C2×D3—202no—
431C3×C4—42no—
432C3×C4—201no—
433C3×C4#409312yes—
434C3×C4C2×#71311yescharacteristic
435C3×C4C2×#71311yescharacteristic
436Dic3—22no—
437Dic3—22no—
438Dic3—42no—
439Dic3—201no—
440C2×C5—22no—
441C2×C5C2×#88511yescharacteristic
442C2×C5#576812yes—
443D5—81no—
444D5—241no—
445C23—101no—
446C23—301no—
447C23D3×C22⋊D511yescharacteristic
448Q8—101no—
449Q8—101no—
450Q8—124no—
451Q8—301no—
452Q8—301no—
453D4—101no—
454D4—101no—
455D4—202no—
456D4—301no—
457D4—301no—
458D4—604no—
459D4—602no—
460C2×C4—22no—
461C2×C4—32no—
462C2×C4—32no—
463C2×C4—62no—
464C2×C4—101no—
465C2×C4—122no—
466C2×C4—202no—
467C2×C4—301no—
468C2×C4—602no—
469C2×C4#676612yes—
470C2×C4#677312yes—
471C2×C4D3×C22⋊D511yescharacteristic
472C2×C4C22×D3×D511yescharacteristic
473C8—102no—
474C8—302no—
475C2×C3—22no—
476C2×C3—401no—
477C2×C3C2×#409311yescharacteristic
478C2×C3#6615612yes—
479D3—401no—
480C5#7882611yescharacteristic, complemented, Sylow
481C22—22no—
482C22—201no—
483C22—202no—
484C22—601no—
485C22—602no—
486C22#9145512yes—
487C22C2×D3×C22⋊D511yescharacteristic
488C4—42no—
489C4—62no—
490C4—62no—
491C4—122no—
492C4—201no—
493C4—601no—
494C4#9145512yes—
495C4C2×#676611yescharacteristic
496C4C2×#677311yescharacteristic
497C3#1061506611yescharacteristic, complemented, Sylow
498C2—22no—
499C2—401no—
500C2—1201no—
501C2C2×#9145511yescharacteristic
502C2#1175532212yes—
subgroupG/H
centerC2×C4D3×C22⋊D5
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C5×#117C22
socleC22×C3×C5C22×D4
prime radical#11754581C2
hypercenter#117D3×D5
hypocenterC3×C5#3135
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. C5×#5517C2semidirect
3. #11751103C2semidirect
4. C3×#65291C2semidirect
5. #11748393C2semidirect
6. #11748364C2semidirect
7. C2×#90953C2semidirect
8. #11747423C2semidirect
9. #11747405C2semidirect
10. #11747380C2semidirect
11. #11747376C2semidirect
50. #65291D3semidirect
88. #5517D5semidirect
423. C3×C5#3135semidirect
480. C5#78826semidirect
497. C3#10615066semidirect

Maximal subgroups

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

subgroupindex
C5×#55172
#117545812
#117511032
C3×#652912
#117483932
#117483642
C2×#909532
#117474232
#117474052
#117473802
#117473762
#106150663
#788265