Finite Group Databasebeta

← #14418180

Subgroups — #14418180

Order 1920.

916 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11754016C212yesmaximal, complemented
2#11753969C211yesmaximal, characteristic
3#11750996C211yesmaximal, characteristic, complemented
4C5×#4893C211yesmaximal, characteristic, complemented
5C3×#65183C211yesmaximal, characteristic, complemented
6D3×#3957C212yesmaximal, complemented
7#11748207C212yesmaximal, complemented
8#11748175C212yesmaximal, complemented
9#11748070C211yesmaximal, characteristic, complemented
10#11748065C211yesmaximal, characteristic, complemented
11#11747602C211yesmaximal, characteristic, complemented
12#10610989—31nomaximal, Hall
13D3×#709C2212yes—
14#91796C2212yescomplemented
15#91793C2211yescharacteristic
16C5×D3×M16C2212yes—
17C2×C5×ΓL(1,𝔽25)C2211yescharacteristic
18C3×#4053C2212yescomplemented
19C3×#4050C2211yescharacteristic
20C4×D3×Dic5C2211yescharacteristic
21#91311C2212yes—
22C2×#6650C2211yescharacteristic
23#91290C2212yescomplemented
24#91284C2212yescomplemented
25#91274C412yescomplemented
26#91267—22no—
27#91265—22no—
28#91263C412yescomplemented
29#91254C412yescomplemented
30#91253C412yescomplemented
31#91246—22no—
32D5×ΓL(1,𝔽25)—22no—
33C3×C5×#108C2211yescharacteristic, complemented
34#91109C2212yescomplemented
35C8×Dic15C2211yescharacteristic, complemented
36C5×#822C2212yescomplemented
37C5×#819C2211yescharacteristic, complemented
38C3×#3957C2212yescomplemented
39C3×#3947C2211yescharacteristic, complemented
40#90981C2212yescomplemented
41#90976C2212yescomplemented
42#90952C2211yescharacteristic, complemented
43#90951C2211yescharacteristic, complemented
44#79898—51nomaximal, Hall
45#66082—31no—
46#66079—32no—
47C5×#430—31no—
48C2×#3957—32no—
49M16×Dic5—31no—
50#65184—32no—
51#65184—31no—
52#65184—31no—
53#65183—32no—
54#65183—31no—
55#65183D311yescharacteristic, complemented
56#6817C2312yes—
57C2×C4×C5×D3C2311yescharacteristic
58C3×#709C2312yes—
59D3×C22⋊D5—22no—
60C2×D3×Dic5—21no—
61#6779—22no—
62D3×D20C2×C412yes—
63C4×D3×D5—22no—
64#6771—22no—
65#6769C2×C412yes—
66D3×Dic10C2×C412yes—
67#6765C2×C412yes—
68C3×C5×M16C2312yes—
69C2×C3×C5×C8C2311yescharacteristic
70C4×Dic15C2311yescharacteristic
71C2×C3⋊#168C2311yescharacteristic
72Dic60C2×C411yescharacteristic, complemented
73D120C2×C411yescharacteristic, complemented
74#6706C2×C412yescomplemented
75#6705—22no—
76C8×D15—22no—
77C5×#208C2312yes—
78C2×C5×C3⋊C8C2311yescharacteristic
79C5×ΓL(1,𝔽25)—22no—
80C5×ΓL(1,𝔽25)C2×C412yes—
81C5×ΓL(1,𝔽25)C2×C412yes—
82C5×D3×C8—22no—
83C3×C4×Dic5C2311yescharacteristic
84C2×C3×#168C2311yescharacteristic
85C3×Dic20C2×C411yescharacteristic, complemented
86C3×D40C2×C411yescharacteristic, complemented
87C3×#677C2×C412yescomplemented
88C3×#676—22no—
89C3×C8×D5—22no—
90Dic3×Dic5—21no—
91#6662C2×C412yescomplemented
92#6658C2×C412yescomplemented
93#6657C2×C412yescomplemented
94#6653C2×C412yescomplemented
95#6651—22no—
96#6650D412yes—
97#6650D412yes—
98#6649—22no—
99#6648—22no—
100D5×C3⋊C8—22no—
101#5768—51no—
102#5750—52no—
103C3×#444—51no—
104#5196—51no—
105#5144—52no—
106#4895—51no—
107#4893D511yescharacteristic, complemented
108#4823—52no—
109D3×#104—52no—
110#4717—51no—
111#4707—51no—
112C2×#709—32no—
113C2×C5×M16—31no—
114C2×C5×M16—32no—
115C2×C4×Dic5—31no—
116C2×#681—31no—
117#4055—32no—
118#4055—32no—
119#4054—32no—
120#4054—32no—
121#4053—32no—
122#4053—32no—
123#4053—32no—
124#4053C2×D312yescomplemented
125D5×M16—62no—
126D5×M16—62no—
127#4050—32no—
128#4050—31no—
129#4050C2×D311yescharacteristic
130#4047—62no—
131#4047—62no—
132C5×#108—32no—
133C5×#108—31no—
134C5×#108C2×D311yescharacteristic, complemented
135#3957—32no—
136#3957—32no—
137#3957—32no—
138#3957C2×D312yescomplemented
139#3947—31no—
140#3947C2×D311yescharacteristic, complemented
141C8×Dic5—31no—
142C8×Dic5—31no—
143#2956—151noSylow
144C22×C5×D3#4111yescharacteristic
145C2×D3×D5—22no—
146C2×C3×C4×C5C2411yescharacteristic
147#1220—22no—
148C2×Dic15—21no—
149D60C22×C412yes—
150C4×D15—22no—
151Dic30C22×C412yes—
152C2×C5×Dic3#4111yescharacteristic
153C4×C5×D3C22×C412yes—
154C4×C5×D3C2×D412yes—
155C3×C22⋊D5—22no—
156C2×C3×Dic5—21no—
157C3×D20C22×C412yes—
158C3×C4×D5—22no—
159C3×Dic10C22×C412yes—
160#1204—22no—
161#1203—22no—
162#1202—22no—
163#1201—22no—
164#1200—22no—
165D3×Dic5—22no—
166D3×Dic5—42no—
167D5×Dic3—22no—
168C3×C5×C8—22no—
169C3×C5×C8C22×C411yescharacteristic
170C3×C5×C8C22×C411yescharacteristic
171C3⋊#168C2×D411yescharacteristic
172C3⋊#168C2×D411yescharacteristic
173C3×#168C2×D412yes—
174C5×C3⋊C8—22no—
175C5×C3⋊C8C22×C412yes—
176D3×#41—52no—
177C3×#135—51no—
178C3×#131—52no—
179#950—52no—
180#948—52no—
181#918—52no—
182#915—52no—
183#914—52no—
184#911—52no—
185#907—52no—
186#907—102no—
187D3×M16—102no—
188D3×M16C2×D512yes—
189#904—51no—
190#901—52no—
191#901—102no—
192C2×ΓL(1,𝔽25)—51no—
193C2×ΓL(1,𝔽25)—102no—
194C2×ΓL(1,𝔽25)C2×D511yescharacteristic
195C42×D3—51no—
196C3×#108C2×D511yescharacteristic, complemented
197C3×#104—52no—
198C3×#97—51no—
199#837—52no—
200#825—52no—
201#822C2×D512yescomplemented
202#819C2×D511yescharacteristic, complemented
203C8×Dic3—51no—
204C8×Dic3—51no—
205#805—52no—
206#803—51no—
207C22×C4×C5—31no—
208C2×C22⋊D5—62no—
209C22×Dic5—61no—
210#709—32no—
211#709—32no—
212#709—32no—
213#709—62no—
214#709—62no—
215#709C22×D312yes—
216C2×D20—32no—
217C2×C4×D5—62no—
218C2×Dic10—32no—
219C5×M16—32no—
220C5×M16—32no—
221C5×M16—32no—
222C5×M16—62no—
223C5×M16C22×D312yes—
224C2×C5×C8—31no—
225C2×C5×C8—62no—
226C2×C5×C8C22×D311yescharacteristic
227C4×Dic5—31no—
228C4×Dic5—61no—
229C4×Dic5C22×D311yescharacteristic
230#681—32no—
231#681—32no—
232C2×#168—31no—
233C2×#168C22×D311yescharacteristic
234Dic20—32no—
235Dic20—31no—
236Dic20C4×D311yescharacteristic, complemented
237D40—32no—
238D40—31no—
239D40C4×D311yescharacteristic, complemented
240#677—32no—
241#677—32no—
242#677—32no—
243#677C4×D312yescomplemented
244#676—22no—
245#676—62no—
246#676—62no—
247#676—62no—
248C8×D5—22no—
249C8×D5—62no—
250C8×D5—62no—
251C8×D5—62no—
252#575—151no—
253#568—152no—
254#444—51no—
255#444—151no—
256#444—152no—
257#443—152no—
258#443—151no—
259#443—151no—
260#430—31no—
261C2×#104—152no—
262C4×M16—151no—
263C22×C3×C5C2×#4111yescharacteristic
264C2×D15—22no—
265C2×C5×D3—41no—
266C2×C5×D3C4×D412yes—
267C2×C3×D5—22no—
268D3×D5—82no—
269C3×C4×C5C23×C411yescharacteristic
270C3×C4×C5C22×D411yescharacteristic
271Dic15—22no—
272Dic15—22no—
273C3×Dic5—22no—
274C3×Dic5—41no—
275C5×Dic3C4×D412yes—
276C3×#41—52no—
277#239—52no—
278D3×Q8—52no—
279#237—52no—
280#237—102no—
281D3×D4—52no—
282D3×D4—102no—
283#235—52no—
284#235—102no—
285C2×C4×D3—102no—
286C2×C4×D3—101no—
287C2×C4×D3C22×D511yescharacteristic
288C3×Q16—51no—
289C3×SD16—52no—
290C3×D8—51no—
291C3×M16—102no—
292C3×M16C22×D512yes—
293C2×C3×C8—51no—
294C2×C3×C8—102no—
295C2×C3×C8C22×D511yescharacteristic
296C3×C42—51no—
297#216—52no—
298#215—52no—
299#214—52no—
300#213—52no—
301C4×Dic3—51no—
302C4×Dic3—101no—
303#208—102no—
304#208C22×D512yes—
305C2×C3⋊C8—51no—
306C2×C3⋊C8—102no—
307C2×C3⋊C8C22×D511yescharacteristic
308Dic12—51no—
309D24—51no—
310#204—52no—
311ΓL(1,𝔽25)—22no—
312ΓL(1,𝔽25)—52no—
313ΓL(1,𝔽25)—52no—
314ΓL(1,𝔽25)—102no—
315ΓL(1,𝔽25)—102no—
316ΓL(1,𝔽25)C4×D512yes—
317ΓL(1,𝔽25)C4×D512yes—
318D3×C8—22no—
319D3×C8—102no—
320D3×C8—102no—
321C23×C5—31no—
322C22×D5—62no—
323C2×C4×C5—31no—
324C2×C4×C5—32no—
325C2×C4×C5—32no—
326C2×C4×C5C23×D311yescharacteristic
327C22⋊D5—22no—
328C22⋊D5—62no—
329C22⋊D5—62no—
330C22⋊D5—62no—
331C2×Dic5—21no—
332C2×Dic5—62no—
333C2×Dic5—61no—
334C2×Dic5—122no—
335D20—32no—
336D20—62no—
337D20C2×C4×D312yes—
338C4×D5—22no—
339C4×D5—62no—
340C4×D5—62no—
341C4×D5—62no—
342Dic10—32no—
343Dic10—62no—
344Dic10C2×C4×D312yes—
345C5×C8—22no—
346C5×C8—32no—
347C5×C8—62no—
348C5×C8C2×C4×D311yescharacteristic
349C5×C8C2×C4×D311yescharacteristic
350#168—31no—
351#168—31no—
352#168D3×D412yes—
353C2×#41—152no—
354#137—152no—
355#137—152no—
356AGL(1,ℤ/8)—152no—
357AGL(1,ℤ/8)—152no—
358#135—51no—
359#135—152no—
360#135—151no—
361#131—52no—
362#131—152no—
363#131—152no—
364#131—152no—
365#131—302no—
366#131—302no—
367C2×M16—31no—
368C2×M16—32no—
369C2×M16—151no—
370C2×M16—302no—
371C2×M16—302no—
372C2×C42—151no—
373#108—32no—
374#108—31no—
375#108D3×D511yescharacteristic, complemented
376#104—52no—
377#104—152no—
378#104—152no—
379#104—152no—
380#97—51no—
381#97—151no—
382C4×C8—151no—
383C4×C8—151no—
384C2×C3×C5—21no—
385C2×C3×C5C2×C4×D411yescharacteristic
386D15—42no—
387C3×D5—42no—
388C5×D3—22no—
389C22×D3—102no—
390C22×D3#71311yescharacteristic
391C3×Q8—52no—
392C3×D4—52no—
393C3×D4—102no—
394C2×C3×C4—102no—
395C2×C3×C4—101no—
396C2×C3×C4C23×D511yescharacteristic
397#67—102no—
398#67—102no—
399#67—102no—
400C2×Dic3—101no—
401C2×Dic3—102no—
402C2×Dic3#71311yescharacteristic
403D12—52no—
404D12—102no—
405C4×D3—102no—
406C4×D3—102no—
407C4×D3—102no—
408C4×D3—202no—
409C4×D3C2×C4×D512yes—
410C4×D3D4×D512yes—
411Dic6—52no—
412Dic6—102no—
413C3×C8—22no—
414C3×C8—52no—
415C3×C8C2×C4×D511yescharacteristic
416C3×C8C2×C4×D511yescharacteristic
417C3⋊C8—22no—
418C3⋊C8—51no—
419C3⋊C8—51no—
420C3⋊C8C2×C4×D512yes—
421C22×C5—32no—
422C22×C5—121no—
423C22×C5D3×#4111yescharacteristic
424C2×D5—22no—
425C2×D5—62no—
426C2×D5—242no—
427C4×C5—32no—
428C4×C5C22×C4×D311yescharacteristic
429C4×C5C2×D3×D411yescharacteristic
430Dic5—22no—
431Dic5—41no—
432Dic5—62no—
433Dic5—62no—
434#41—52no—
435#41—152no—
436#41—152no—
437#41—152no—
438#41—302no—
439#41—302no—
440C2×Q8—152no—
441C2×D4—152no—
442C2×D4—302no—
443C22×C4—31no—
444C22×C4—302no—
445C22×C4—301no—
446Q16—51no—
447Q16—152no—
448Q16—151no—
449SD16—52no—
450SD16—152no—
451SD16—152no—
452SD16—152no—
453D8—51no—
454D8—152no—
455D8—151no—
456M16—32no—
457M16—32no—
458M16—32no—
459M16—62no—
460M16—102no—
461M16—152no—
462M16—152no—
463M16—302no—
464M16—302no—
465M16—302no—
466M16C2×D3×D512yes—
467C2×C8—31no—
468C2×C8—51no—
469C2×C8—62no—
470C2×C8—102no—
471C2×C8—151no—
472C2×C8—302no—
473C2×C8—302no—
474C2×C8—302no—
475C2×C8C2×D3×D511yescharacteristic
476C42—51no—
477C42—151no—
478C42—301no—
479C3×C5#295611yescharacteristic, complemented, Hall
480C22×C3—102no—
481C22×C3C2×#71311yescharacteristic
482C2×D3—41no—
483C2×D3—102no—
484C2×D3—402no—
485C2×D3#403912yes—
486C3×C4—102no—
487C3×C4—201no—
488C3×C4C22×C4×D511yescharacteristic
489C3×C4C2×D4×D511yescharacteristic
490Dic3—102no—
491Dic3—102no—
492Dic3#403912yes—
493C2×C5—21no—
494C2×C5—62no—
495C2×C5C4×D3×D411yescharacteristic
496D5—42no—
497D5—122no—
498C23—31no—
499C23—302no—
500Q8—52no—
501Q8—152no—
502Q8—302no—
503D4—52no—
504D4—102no—
505D4—152no—
506D4—302no—
507D4—302no—
508D4—302no—
509D4—302no—
510C2×C4—31no—
511C2×C4—32no—
512C2×C4—32no—
513C2×C4—102no—
514C2×C4—101no—
515C2×C4—302no—
516C2×C4—302no—
517C2×C4—302no—
518C2×C4—301no—
519C2×C4—302no—
520C2×C4—602no—
521C2×C4C22×D3×D511yescharacteristic
522C8—22no—
523C8—32no—
524C8—52no—
525C8—62no—
526C8—151no—
527C8—151no—
528C8C4×D3×D511yescharacteristic
529C8C4×D3×D511yescharacteristic
530C2×C3—21no—
531C2×C3—202no—
532C2×C3C2×#403911yescharacteristic
533D3—22no—
534D3—202no—
535C5#7989811yescharacteristic, complemented, Sylow
536C22—32no—
537C22—102no—
538C22—121no—
539C22—302no—
540C22—1202no—
541C22D3×#71311yescharacteristic
542C4—32no—
543C4—102no—
544C4—201no—
545C4—302no—
546C4—302no—
547C4C2×C4×D3×D511yescharacteristic
548C4D3×D4×D511yescharacteristic
549C3#1061098911yescharacteristic, complemented, Sylow
550C2—21no—
551C2—62no—
552C2—202no—
553C2—602no—
554C2D3×#403911yescharacteristic
subgroupG/H
centerC4D3×D4×D5
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC3×C5×#108C22
socleC2×C3×C5C2×C4×D4
prime radical#11753969C2
cosocleC2×C3×C4×C5C24
hypercenter#108D3×D5
hypocenterC3×C5#2956
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C4 ◁ C2×C4 ◁ #108

Complements

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

NH ≅ G/Ntype
1. #11754016C2semidirect
3. #11750996C2semidirect
4. C5×#4893C2semidirect
5. C3×#65183C2semidirect
6. D3×#3957C2semidirect
7. #11748207C2semidirect
8. #11748175C2semidirect
9. #11748070C2semidirect
10. #11748065C2semidirect
11. #11747602C2semidirect
14. #91796C22semidirect
18. C3×#4053C22semidirect
23. #91290C22semidirect
24. #91284C22semidirect
25. #91274C4semidirect
28. #91263C4semidirect
29. #91254C4semidirect
30. #91253C4semidirect
33. C3×C5×#108C22semidirect
34. #91109C22semidirect
35. C8×Dic15C22semidirect
36. C5×#822C22semidirect
37. C5×#819C22semidirect
38. C3×#3957C22semidirect
39. C3×#3947C22semidirect
40. #90981C22semidirect
41. #90976C22semidirect
42. #90952C22semidirect
43. #90951C22semidirect
55. #65183D3semidirect
72. Dic60C2×C4semidirect
73. D120C2×C4semidirect
74. #6706C2×C4semidirect
85. C3×Dic20C2×C4semidirect
86. C3×D40C2×C4semidirect
87. C3×#677C2×C4semidirect
91. #6662C2×C4semidirect
92. #6658C2×C4semidirect
93. #6657C2×C4semidirect
94. #6653C2×C4semidirect
107. #4893D5semidirect
124. #4053C2×D3semidirect
134. C5×#108C2×D3semidirect
138. #3957C2×D3semidirect
140. #3947C2×D3semidirect
196. C3×#108C2×D5semidirect
201. #822C2×D5semidirect
202. #819C2×D5semidirect
236. Dic20C4×D3semidirect
239. D40C4×D3semidirect
243. #677C4×D3semidirect
375. #108D3×D5semidirect
479. C3×C5#2956semidirect
535. C5#79898semidirect
549. C3#10610989semidirect

Maximal subgroups

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

subgroupindex
#117540162
#117539692
#117509962
C5×#48932
C3×#651832
D3×#39572
#117482072
#117481752
#117480702
#117480652
#117476022
#106109893
#798985