Finite Group Databasebeta

← #14417366

Subgroups — #14417366

Order 1920.

1056 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91256C211yesmaximal, characteristic
2#11752729C212yesmaximal, complemented
3#11750790C211yesmaximal, characteristic, complemented
4C4×C5×D24C211yesmaximal, characteristic, complemented
5C3×C4×D40C211yesmaximal, characteristic, complemented
6#11748109C212yesmaximal, complemented
7#11747810C211yesmaximal, characteristic, complemented
8#11747809C211yesmaximal, characteristic, complemented
9#11747734C212yesmaximal, complemented
10#11747690C211yesmaximal, characteristic, complemented
11#11747021C212yesmaximal, complemented
12#10611319—31nomaximal, Hall
13C2×#6777C2212yes—
14C2×#6705C2211yescharacteristic
15C2×C8×D15C2211yescharacteristic
16#91761C2211yescharacteristic
17C2×C5×D24C2211yescharacteristic
18C4×C5×D12C2212yescomplemented
19C2×C3×D40C2211yescharacteristic
20C3×C4×D20C2212yescomplemented
21D3×#685—22no—
22D5×(C2×D3)⋊C4—22no—
23#91441—22no—
24#91440—22no—
25#91435C2212yescomplemented
26#91434C2212yescomplemented
27#91347C2212yes—
28C2×#6652C2212yes—
29#91256C418yescomplemented
30C3×C4×C5×C8C2211yescharacteristic, complemented
31#91099C2211yescharacteristic, complemented
32C8×Dic15C2211yescharacteristic, complemented
33#91087C2211yescharacteristic, complemented
34C5×#821C2212yescomplemented
35C5×#818C2211yescharacteristic, complemented
36C3×#3953C2212yescomplemented
37C3×#3950C2211yescharacteristic, complemented
38#90986C2212yescomplemented
39#90966C2212yescomplemented
40#90964C2212yescomplemented
41#79588—51nomaximal, Hall
42C2×D5×D8—31no—
43#65879—32no—
44C4×C5×D8—31no—
45D8×Dic5—31no—
46#65303—32no—
47#65169—31no—
48#65161—31no—
49#65074—32no—
50#65059—32no—
51C4×D40D311yescharacteristic, complemented
52#64978—31no—
53C22×D3×D5—22no—
54C2×C4×D15C2311yescharacteristic
55C2×C5×D12C2312yes—
56C2×C4×C5×D3—22no—
57C2×C3×D20C2312yes—
58C2×C3×C4×D5—22no—
59C2×#1201—22no—
60C2×D3×Dic5—22no—
61C2×D5×Dic3—22no—
62#6777—24no—
63#6777C2×C418yescomplemented
64C2×C3×C5×C8C2311yescharacteristic
65C2×C3×C5×C8C2311yescharacteristic
66C3×C42×C5C2311yescharacteristic
67#6714D411yescharacteristic
68#6714D411yescharacteristic
69#6712D411yescharacteristic
70#6712D411yescharacteristic
71C4×Dic15C2311yescharacteristic
72C2×C3⋊#168C2311yescharacteristic
73C2×C3⋊#168C2311yescharacteristic
74#6705—22no—
75C8×D15C2×C414yescomplemented
76C5×(C2×D3)⋊C4—22no—
77C5×C4⋊Dic3C2312yes—
78C5×D24C2×C414yescomplemented
79C3×#685—22no—
80C3×C4⋊Dic5C2312yes—
81C3×D40C2×C414yescomplemented
82#6667—22no—
83(C2×D3)⋊Dic5—22no—
84#6665—22no—
85Dic3×Dic5—22no—
86#6652C2×C418yescomplemented
87C2×D3×D8—51no—
88#5545—52no—
89C3×C4×D8—51no—
90Dic3×D8—51no—
91#5014—52no—
92#4880—51no—
93#4872—51no—
94#4785—52no—
95#4770—52no—
96C4×D24D511yescharacteristic, complemented
97#4689—51no—
98C2×D4×D5—32no—
99C2×C5×D8—31no—
100C4×C5×D4—32no—
101D4×Dic5—32no—
102C2×#686—32no—
103C4×C22⋊D5—62no—
104C2×#685—62no—
105D5×D8—38no—
106C2×D40C2×D311yescharacteristic
107C2×#676—31no—
108C2×C8×D5—31no—
109#4039—32no—
110#4038—32no—
111#4027—62no—
112D5×C22⋊C4—62no—
113C4×D20C2×D312yescomplemented
114#4018—31no—
115C5×#107—31no—
116C5×#102—32no—
117C4×C5×C8C2×D311yescharacteristic, complemented
118#3964—32no—
119#3953C2×D312yescomplemented
120#3950C2×D311yescharacteristic, complemented
121#3947—31no—
122C8×Dic5—31no—
123#3941—32no—
124#3940—32no—
125#3935—31no—
126#2948—151noSylow
127C22×D15C2×D411yescharacteristic
128C22×C5×D3—22no—
129C22×C3×D5—22no—
130C2×D3×D5—28no—
131C2×D3×D5—82no—
132C2×D3×D5—82no—
133C2×C3×C4×C5C2×D411yescharacteristic
134C2×C3×C4×C5C2×D411yescharacteristic
135C2×C3×C4×C5C2411yescharacteristic
136C2×Dic15C2×D411yescharacteristic
137C2×Dic15C2×D411yescharacteristic
138C2×Dic15C2×D411yescharacteristic
139C4×D15C22×C412yes—
140C4×D15#4112yes—
141C2×C5×Dic3—22no—
142C5×D12—22no—
143C5×D12C22×C414yes—
144C4×C5×D3—82no—
145C2×C3×Dic5—22no—
146C3×D20—22no—
147C3×D20C22×C414yes—
148C3×C4×D5—82no—
149#1201—28no—
150D3×Dic5—82no—
151D5×Dic3—82no—
152C3×C5×C8—21no—
153C3×C5×C8C22×C412yes—
154C3⋊#168—21no—
155C3⋊#168C22×C412yes—
156C2×D3×D4—52no—
157C2×C3×D8—51no—
158C3×C4×D4—52no—
159D4×Dic3—52no—
160C2×#213—52no—
161C4×#67—102no—
162C2×(C2×D3)⋊C4—102no—
163D3×D8—58no—
164C2×D24C2×D511yescharacteristic
165C2×ΓL(1,𝔽25)—51no—
166C2×D3×C8—51no—
167#893—52no—
168#892—52no—
169#67⋊C4—102no—
170D3×C22⋊C4—102no—
171C4×D12C2×D512yescomplemented
172#872—51no—
173C3×#107—51no—
174C3×#102—52no—
175C3×C4×C8C2×D511yescharacteristic, complemented
176#832—52no—
177#821C2×D512yescomplemented
178#818C2×D511yescharacteristic, complemented
179#815—51no—
180C8×Dic3—51no—
181#809—52no—
182#808—52no—
183#803—51no—
184C23×D5—62no—
185C2×C5×D4—32no—
186C22×C4×C5—62no—
187C2×C22⋊D5—62no—
188C22×Dic5—62no—
189D4×D5—38no—
190D4×D5—64no—
191C2×D20C22×D312yes—
192C2×C4×D5—22no—
193C2×C4×D5—31no—
194C2×C4×D5—62no—
195C5×D8—34no—
196C2×C5×C8C22×D311yescharacteristic
197C2×C5×C8C22×D311yescharacteristic
198C5×C4⋊C4—32no—
199C5×C22⋊C4—62no—
200C42×C5C22×D311yescharacteristic
201C22⋊Dic5—62no—
202#686—38no—
203#685—22no—
204#685—31no—
205#685—31no—
206#685—62no—
207#685—62no—
208C4⋊Dic5C22×D312yes—
209#683—31no—
210#683—31no—
211C4×Dic5—31no—
212C4×Dic5—62no—
213C2×#168—31no—
214C2×#168—31no—
215D40C4×D314yescomplemented
216#676—62no—
217C8×D5—34no—
218C22×D8—151no—
219#518—152no—
220#442—152no—
221#442—152no—
222C4×D8—31no—
223C4×D8—51no—
224C4×D8—151no—
225C4×D8—151no—
226#427—151no—
227#418—152no—
228#405—151no—
229C22×C3×C5C22×D411yescharacteristic
230C2×D15—21no—
231C2×D15—21no—
232C2×D15C4×D412yes—
233C2×C5×D3—24no—
234C2×C5×D3—42no—
235C2×C5×D3—42no—
236C2×C3×D5—24no—
237C2×C3×D5—42no—
238C2×C3×D5—42no—
239D3×D5—88no—
240C3×C4×C5—21no—
241C3×C4×C5—21no—
242C3×C4×C5C23×C411yescharacteristic
243C3×C4×C5C2×#4111yescharacteristic
244Dic15—21no—
245Dic15—21no—
246Dic15C4×D412yes—
247C3×Dic5—42no—
248C5×Dic3—42no—
249C23×D3—102no—
250C2×C3×D4—52no—
251C22×C3×C4—102no—
252C2×#67—102no—
253C22×Dic3—102no—
254D3×D4—58no—
255D3×D4—104no—
256C2×D12C22×D512yes—
257C2×C4×D3—22no—
258C2×C4×D3—51no—
259C2×C4×D3—102no—
260C3×D8—54no—
261C2×C3×C8C22×D511yescharacteristic
262C2×C3×C8C22×D511yescharacteristic
263C3×C4⋊C4—52no—
264C3×C22⋊C4—102no—
265C3×C42C22×D511yescharacteristic
266#217—102no—
267#213—58no—
268(C2×D3)⋊C4—22no—
269(C2×D3)⋊C4—51no—
270(C2×D3)⋊C4—51no—
271(C2×D3)⋊C4—102no—
272(C2×D3)⋊C4—102no—
273C4⋊Dic3C22×D512yes—
274Dic3⋊C4—51no—
275Dic3⋊C4—51no—
276C4×Dic3—51no—
277C4×Dic3—102no—
278C2×C3⋊C8—51no—
279C2×C3⋊C8—51no—
280D24C4×D514yescomplemented
281ΓL(1,𝔽25)—102no—
282D3×C8—54no—
283C23×C5—62no—
284C22×D5—22no—
285C22×D5—31no—
286C22×D5—68no—
287C22×D5—242no—
288C22×D5—242no—
289C5×D4—34no—
290C5×D4—62no—
291C2×C4×C5—62no—
292C2×C4×C5—242no—
293C2×C4×C5D3×D411yescharacteristic
294C2×C4×C5D3×D411yescharacteristic
295C2×C4×C5C23×D311yescharacteristic
296C22⋊D5—68no—
297C2×Dic5—22no—
298C2×Dic5—31no—
299C2×Dic5—31no—
300C2×Dic5—31no—
301C2×Dic5—242no—
302D20—22no—
303D20C2×C4×D314yes—
304C4×D5—32no—
305C4×D5—32no—
306C4×D5—82no—
307C4×D5—242no—
308C5×C8—21no—
309C5×C8C2×C4×D312yes—
310#168—32no—
311#168—61no—
312C22×D4—152no—
313C2×D8—31no—
314C2×D8—51no—
315C2×D8—158no—
316C2×D8—152no—
317C2×M16—151no—
318C22×C8—151no—
319C4×D4—32no—
320C4×D4—52no—
321C4×D4—152no—
322C4×D4—152no—
323C4×D4—302no—
324C4×D4—302no—
325#117—151no—
326#117—152no—
327C2×C22⋊C4—302no—
328C2×C22⋊C4—302no—
329#107—31no—
330#107—51no—
331#106—152no—
332#102—32no—
333#102—52no—
334#102—152no—
335#102—152no—
336#97—151no—
337#97—151no—
338C4×C8—151no—
339C4×C8D3×D511yescharacteristic, complemented
340C2×C3×C5C2×C4×D411yescharacteristic
341C2×C3×C5#57611yescharacteristic
342C2×C3×C5#57611yescharacteristic
343D15—22no—
344C3×D5—44no—
345C5×D3—44no—
346C23×C3—102no—
347C22×D3—22no—
348C22×D3—51no—
349C22×D3—108no—
350C22×D3—402no—
351C22×D3—402no—
352C3×D4—54no—
353C3×D4—102no—
354C2×C3×C4—102no—
355C2×C3×C4—402no—
356C2×C3×C4D4×D511yescharacteristic
357C2×C3×C4D4×D511yescharacteristic
358C2×C3×C4C23×D511yescharacteristic
359#67—108no—
360C2×Dic3—22no—
361C2×Dic3—51no—
362C2×Dic3—51no—
363C2×Dic3—51no—
364C2×Dic3—402no—
365D12—22no—
366D12C2×C4×D514yes—
367C4×D3—52no—
368C4×D3—52no—
369C4×D3—82no—
370C4×D3—402no—
371C3×C8—21no—
372C3×C8C2×C4×D512yes—
373C3⋊C8—52no—
374C3⋊C8—101no—
375C22×C5—64no—
376C22×C5—122no—
377C22×C5—122no—
378C22×C5C2×D3×D411yescharacteristic
379C2×D5—24no—
380C2×D5—32no—
381C2×D5—42no—
382C2×D5—42no—
383C2×D5—61no—
384C2×D5—61no—
385C2×D5—248no—
386C4×C5—21no—
387C4×C5—21no—
388C4×C5—122no—
389C4×C5C22×C4×D311yescharacteristic
390C4×C5D3×#4111yescharacteristic
391Dic5—32no—
392Dic5—42no—
393Dic5—61no—
394Dic5—61no—
395C24—302no—
396C2×D4—32no—
397C2×D4—52no—
398C2×D4—158no—
399C2×D4—304no—
400C2×D4—302no—
401C22×C4—62no—
402C22×C4—102no—
403C22×C4—151no—
404C22×C4—302no—
405C22×C4—302no—
406D8—34no—
407D8—54no—
408D8—158no—
409M16—302no—
410C2×C8—151no—
411C2×C8—154no—
412C2×C8—151no—
413C2×C8C2×D3×D511yescharacteristic
414C2×C8C2×D3×D511yescharacteristic
415C4⋊C4—32no—
416C4⋊C4—52no—
417C4⋊C4—151no—
418C4⋊C4—151no—
419C22⋊C4—62no—
420C22⋊C4—102no—
421C22⋊C4—151no—
422C22⋊C4—151no—
423C22⋊C4—302no—
424C22⋊C4—302no—
425C22⋊C4—302no—
426C42—151no—
427C42—302no—
428C42C2×D3×D511yescharacteristic
429C3×C5#294811yescharacteristic, complemented, Hall
430C22×C3—104no—
431C22×C3—202no—
432C22×C3—202no—
433C22×C3C2×D4×D511yescharacteristic
434C2×D3—24no—
435C2×D3—42no—
436C2×D3—42no—
437C2×D3—52no—
438C2×D3—101no—
439C2×D3—101no—
440C2×D3—408no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—202no—
444C3×C4C22×C4×D511yescharacteristic
445C3×C4D5×#4111yescharacteristic
446Dic3—42no—
447Dic3—52no—
448Dic3—101no—
449Dic3—101no—
450C2×C5—124no—
451C2×C5C4×D3×D411yescharacteristic
452C2×C5#575711yescharacteristic
453C2×C5#576911yescharacteristic
454D5—44no—
455D5—62no—
456C23—62no—
457C23—102no—
458C23—151no—
459C23—308no—
460C23—1202no—
461C23—1202no—
462D4—34no—
463D4—54no—
464D4—62no—
465D4—102no—
466D4—308no—
467C2×C4—62no—
468C2×C4—102no—
469C2×C4—151no—
470C2×C4—152no—
471C2×C4—152no—
472C2×C4—151no—
473C2×C4—151no—
474C2×C4—242no—
475C2×C4—402no—
476C2×C4—1202no—
477C2×C4—1202no—
478C2×C4#677711yescharacteristic
479C2×C4#677711yescharacteristic
480C2×C4C22×D3×D511yescharacteristic
481C8—21no—
482C8—152no—
483C8—301no—
484C8C4×D3×D512yes—
485C2×C3—204no—
486C2×C3C4×D4×D511yescharacteristic
487C2×C3#6609111yescharacteristic
488C2×C3#6610311yescharacteristic
489D3—44no—
490D3—102no—
491C5#7958811yescharacteristic, complemented, Sylow
492C22—64no—
493C22—104no—
494C22—122no—
495C22—122no—
496C22—152no—
497C22—202no—
498C22—202no—
499C22—301no—
500C22—301no—
501C22—1208no—
502C22C2×#677711yescharacteristic
503C4—21no—
504C4—21no—
505C4—122no—
506C4—152no—
507C4—202no—
508C4—301no—
509C4—301no—
510C4C2×C4×D3×D511yescharacteristic
511C4#9201611yescharacteristic
512C3#1061131911yescharacteristic, complemented, Sylow
513C2—124no—
514C2—204no—
515C2—302no—
516C2C4×#677711yescharacteristic
517C2#1175398111yescharacteristic
518C2#1175398311yescharacteristic
subgroupG/H
centerC22C2×#6777
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC3×C4×C5×C8C22
socleC22×C3×C5C22×D4
prime radicalC2×#91256C2
cosocleC2×C3×C4×C5C24
hypercenterC4×C8D3×D5
hypocenterC3×C5#2948
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8

Complements

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

NH ≅ G/Ntype
2. #11752729C2semidirect
3. #11750790C2semidirect
4. C4×C5×D24C2semidirect
5. C3×C4×D40C2semidirect
6. #11748109C2semidirect
7. #11747810C2semidirect
8. #11747809C2semidirect
9. #11747734C2semidirect
10. #11747690C2semidirect
11. #11747021C2semidirect
18. C4×C5×D12C22semidirect
20. C3×C4×D20C22semidirect
25. #91435C22semidirect
26. #91434C22semidirect
29. #91256C4semidirect
30. C3×C4×C5×C8C22semidirect
31. #91099C22semidirect
32. C8×Dic15C22semidirect
33. #91087C22semidirect
34. C5×#821C22semidirect
35. C5×#818C22semidirect
36. C3×#3953C22semidirect
37. C3×#3950C22semidirect
38. #90986C22semidirect
39. #90966C22semidirect
40. #90964C22semidirect
51. C4×D40D3semidirect
63. #6777C2×C4semidirect
75. C8×D15C2×C4semidirect
78. C5×D24C2×C4semidirect
81. C3×D40C2×C4semidirect
86. #6652C2×C4semidirect
96. C4×D24D5semidirect
113. C4×D20C2×D3semidirect
117. C4×C5×C8C2×D3semidirect
119. #3953C2×D3semidirect
120. #3950C2×D3semidirect
171. C4×D12C2×D5semidirect
175. C3×C4×C8C2×D5semidirect
177. #821C2×D5semidirect
178. #818C2×D5semidirect
215. D40C4×D3semidirect
280. D24C4×D5semidirect
339. C4×C8D3×D5semidirect
429. C3×C5#2948semidirect
491. C5#79588semidirect
512. C3#10611319semidirect

Maximal subgroups

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

subgroupindex
C2×#912562
#117527292
#117507902
C4×C5×D242
C3×C4×D402
#117481092
#117478102
#117478092
#117477342
#117476902
#117470212
#106113193
#795885