Finite Group Databasebeta

← #14418063

Subgroups — #14418063

Order 1920.

912 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11754145C211yesmaximal, characteristic
2C2×#91290C211yesmaximal, characteristic, complemented
3C2×#91286C211yesmaximal, characteristic, complemented
4#11751106C211yesmaximal, characteristic, complemented
5C5×#5005C211yesmaximal, characteristic, complemented
6C3×#65286C211yesmaximal, characteristic, complemented
7#11748476C211yesmaximal, characteristic, complemented
8#11747019C214yesmaximal, complemented
9#11747018C214yesmaximal, complemented
10#10610720—31nomaximal, Hall
11C2×#6817C2211yescharacteristic
12C2×C5×#235C2211yescharacteristic
13C3×C5×#117C2211yescharacteristic
14C2×#6710C2211yescharacteristic
15C2×C5×#208C2211yescharacteristic
16C2×C3×#681C2211yescharacteristic
17C22×C3×#168C2211yescharacteristic
18#91322C412yescomplemented
19#91321C412yescomplemented
20#91305C412yescomplemented
21#91304C412yescomplemented
22#91303C2214yes—
23C2×#6651—21no—
24C2×#6650—21no—
25#91290—21no—
26#91290—21no—
27#91290C2212yescomplemented
28#91290C2212yescomplemented
29C2×#6648—21no—
30#91286—21no—
31#91286—21no—
32#91286C2212yescomplemented
33#91286C2212yescomplemented
34C2×D3×#168—21no—
35#91089C2214yescomplemented
36C5×#805C2214yescomplemented
37C3×#3935C2212yescomplemented
38C3×C4×#168C2212yescomplemented
39#90991C2214yescomplemented
40#78579—51nomaximal, Hall
41#66156—31no—
42C2×#4094—31no—
43C2×#4053—31no—
44C5×#421—31no—
45#65413—31no—
46#65294—31no—
47#65286D311yescharacteristic, complemented
48#65113—34no—
49#65112—34no—
50C22×C3×C4×C5C2311yescharacteristic
51C2×#1220—21no—
52#6817—21no—
53#6817—21no—
54#6817C2×C412yes—
55#6817C2312yes—
56C2×D60C2×C411yescharacteristic
57C2×C4×D15—21no—
58C2×Dic30C2×C411yescharacteristic
59C2×C5×#67—21no—
60C5×#235—21no—
61C5×#235—21no—
62C5×#235C2×C412yes—
63C5×#235C2312yes—
64C2×C5×D12C2×C411yescharacteristic
65C2×C4×C5×D3—21no—
66C2×C5×Dic6C2×C411yescharacteristic
67C3×C5×C4⋊C4—21no—
68C3×C5×C22⋊C4—21no—
69C3×C42×C5C2312yes—
70#6710—21no—
71#6710C2312yes—
72C2×C3⋊#168—21no—
73C5×#208—21no—
74C5×#208C2312yes—
75C2×C5×C3⋊C8—21no—
76C3×#681C2×C412yes—
77C3×#681C2×C412yes—
78C2×C3×#168D411yescharacteristic
79C2×C3×#168D411yescharacteristic
80C2×C3×#168D411yescharacteristic
81C2×C3×#168D411yescharacteristic
82C2×C3×#168C2312yes—
83C2×C3×#168C2311yescharacteristic
84C2×C3×#168C2311yescharacteristic
85#6663C2×C414yes—
86#6660C2×C414yes—
87#6659C2×C414yes—
88#6654C2×C414yes—
89#6651—22no—
90#6651—22no—
91#6650—22no—
92#6650—22no—
93#6648—22no—
94#6648—22no—
95D3×#168—22no—
96D3×#168—22no—
97#5748—51no—
98C2×#901—51no—
99C2×#901—51no—
100C3×#405—51no—
101#5125—51no—
102#5005—51no—
103#5005D511yescharacteristic, complemented
104#4717—54no—
105#4700—54no—
106C2×C5×#41—31no—
107C2×#709—31no—
108C2×C5×M16—31no—
109C5×#117C2×D311yescharacteristic
110#4097—32no—
111#4096—34no—
112#4095—32no—
113#4094—32no—
114#4094—32no—
115#4094—61no—
116#4094—61no—
117#4088—32no—
118#4078—32no—
119C2×#681—31no—
120C2×#681—61no—
121C2×#681C2×D311yescharacteristic
122C22×#168—61no—
123C22×#168C2×D311yescharacteristic
124#4053—32no—
125#4053—32no—
126#4053—61no—
127#4053—61no—
128C2×#676—61no—
129C2×C8×D5—61no—
130C5×#104—34no—
131#3954—34no—
132#3937—34no—
133#3935C2×D312yescomplemented
134C4×#168C2×D312yescomplemented
135#2955—151noSylow
136C23×C3×C5#4111yescharacteristic
137C22×D15—21no—
138C22×C5×D3—21no—
139C2×C3×C4×C5—22no—
140C2×C3×C4×C5C22×C411yescharacteristic
141C2×C3×C4×C5C22×C411yescharacteristic
142C2×C3×C4×C5C2×D411yescharacteristic
143C2×C3×C4×C5C2×D411yescharacteristic
144C2×C3×C4×C5#4111yescharacteristic
145C2×C3×C4×C5C2411yescharacteristic
146#1220—22no—
147#1220—22no—
148C2×Dic15—21no—
149D60—21no—
150D60C22×C412yes—
151C4×D15—22no—
152C4×D15—22no—
153Dic30—21no—
154Dic30C22×C412yes—
155C5×#67—22no—
156C5×#67—22no—
157C2×C5×Dic3—21no—
158C5×D12—21no—
159C5×D12C22×C412yes—
160C4×C5×D3—22no—
161C4×C5×D3—22no—
162C5×Dic6—21no—
163C5×Dic6C22×C412yes—
164C3⋊#168—22no—
165C3×#168C22×C414yes—
166C3×#168C2×D411yescharacteristic
167C3×#168C2×D411yescharacteristic
168C3×#168C2×D411yescharacteristic
169C3×#168C2×D411yescharacteristic
170C5×C3⋊C8—22no—
171C2×#235—51no—
172C2×#235C2×D511yescharacteristic
173C2×C3×M16—51no—
174C22×C3×C8—51no—
175C3×#117C2×D511yescharacteristic
176C2×#208—51no—
177C2×#208C2×D511yescharacteristic
178#909—52no—
179#909—52no—
180#908—52no—
181#908—52no—
182#904—54no—
183#901—52no—
184#901—52no—
185#901—52no—
186#901—52no—
187#901—101no—
188#901—101no—
189#901—101no—
190#901—101no—
191C2×ΓL(1,𝔽25)—101no—
192C2×ΓL(1,𝔽25)—101no—
193C2×D3×C8—101no—
194C2×D3×C8—101no—
195C3×#97—52no—
196C3×C4×C8—52no—
197#819—54no—
198#805—54no—
199#805C2×D514yescomplemented
200C5×#41—32no—
201C5×#41—32no—
202C5×#41—61no—
203C5×#41—61no—
204C2×C5×Q8—31no—
205C2×C5×D4—31no—
206C2×C5×D4—61no—
207C22×C4×C5—61no—
208C22×C4×C5C22×D311yescharacteristic
209C2×C22⋊D5—61no—
210#709—32no—
211#709—32no—
212#709—61no—
213#709—61no—
214C2×D20—31no—
215C2×C4×D5—61no—
216C2×Dic10—31no—
217C5×M16—32no—
218C5×M16—61no—
219C2×C5×C8—61no—
220C5×C4⋊C4—21no—
221C5×C22⋊C4—21no—
222C42×C5C22×D312yes—
223#689—34no—
224#688—34no—
225#687—34no—
226#686—34no—
227#681—32no—
228#681—61no—
229#681—62no—
230#681—62no—
231#681C4×D312yes—
232#681C4×D312yes—
233C2×#168—61no—
234C2×#168—62no—
235C2×#168—62no—
236C2×#168D1211yescharacteristic
237C2×#168D1211yescharacteristic
238C2×#168D1211yescharacteristic
239C2×#168D1211yescharacteristic
240C2×#168C22×D312yes—
241C2×#168C22×D311yescharacteristic
242C2×#168C22×D311yescharacteristic
243#676—62no—
244#676—62no—
245C8×D5—62no—
246C8×D5—62no—
247#575—151no—
248C2×#131—151no—
249C2×#131—151no—
250#444—154no—
251#443—154no—
252#430—151no—
253#421—31no—
254#421—151no—
255#405—51no—
256C22×C3×C5—41no—
257C22×C3×C5C4×D411yescharacteristic
258C22×C3×C5C4×D411yescharacteristic
259C22×C3×C5C2×#4111yescharacteristic
260C2×D15—22no—
261C2×D15—42no—
262C2×C5×D3—22no—
263C2×C5×D3—42no—
264C3×C4×C5—42no—
265C3×C4×C5C4×D411yescharacteristic
266C3×C4×C5C4×D411yescharacteristic
267C3×C4×C5C23×C411yescharacteristic
268C3×C4×C5C22×D411yescharacteristic
269Dic15—22no—
270C5×Dic3—22no—
271C22×C3×C4C22×D511yescharacteristic
272C2×#67—21no—
273C2×#67—101no—
274#235—21no—
275#235—21no—
276#235—52no—
277#235—52no—
278#235—101no—
279#235—101no—
280#235C4×D512yes—
281#235C22×D512yes—
282C2×D12—51no—
283C2×D12C4×D511yescharacteristic
284C2×C4×D3—21no—
285C2×C4×D3—101no—
286C2×Dic6—51no—
287C2×Dic6C4×D511yescharacteristic
288C3×M16—52no—
289C3×M16—52no—
290C2×C3×C8—51no—
291C2×C3×C8—51no—
292C2×C3×C8—51no—
293C2×C3×C8—51no—
294C2×C3×C8—52no—
295C2×C3×C8—51no—
296C2×C3×C8—51no—
297C3×C4⋊C4—21no—
298C3×C22⋊C4—21no—
299C3×C42C22×D512yes—
300#208—21no—
301#208—52no—
302#208—101no—
303#208C22×D512yes—
304C2×C3⋊C8—21no—
305C2×C3⋊C8—101no—
306Dic12—54no—
307D24—54no—
308#204—54no—
309#204—54no—
310ΓL(1,𝔽25)—102no—
311ΓL(1,𝔽25)—102no—
312ΓL(1,𝔽25)—102no—
313ΓL(1,𝔽25)—102no—
314D3×C8—102no—
315D3×C8—102no—
316D3×C8—102no—
317D3×C8—102no—
318C23×C5—61no—
319C23×C5#23511yescharacteristic
320C22×D5—61no—
321C5×Q8—32no—
322C5×Q8—61no—
323C5×D4—32no—
324C5×D4—62no—
325C5×D4—62no—
326C5×D4—61no—
327C2×C4×C5—22no—
328C2×C4×C5—61no—
329C2×C4×C5—62no—
330C2×C4×C5—62no—
331C2×C4×C5C2×C4×D311yescharacteristic
332C2×C4×C5C2×C4×D311yescharacteristic
333C2×C4×C5C2×D1211yescharacteristic
334C2×C4×C5C2×D1211yescharacteristic
335C2×C4×C5#23511yescharacteristic
336C2×C4×C5C23×D311yescharacteristic
337C22⋊D5—62no—
338C22⋊D5—62no—
339C2×Dic5—61no—
340D20—32no—
341D20—61no—
342C4×D5—62no—
343C4×D5—62no—
344Dic10—32no—
345Dic10—61no—
346C5×C8—62no—
347#168—62no—
348#168C2×C4×D314yes—
349#168C2×D1211yescharacteristic
350#168C2×D1211yescharacteristic
351#168C2×D1211yescharacteristic
352#168C2×D1211yescharacteristic
353C2×#41—31no—
354C2×#41—151no—
355#137—152no—
356#137—152no—
357AGL(1,ℤ/8)—152no—
358AGL(1,ℤ/8)—152no—
359#135—154no—
360#131—152no—
361#131—152no—
362#131—152no—
363#131—152no—
364#131—301no—
365#131—301no—
366#131—301no—
367#131—301no—
368C2×M16—31no—
369C2×M16—51no—
370C2×M16—151no—
371C2×M16—301no—
372C2×M16—301no—
373C22×C8—51no—
374C22×C8—301no—
375C22×C8—301no—
376#117D3×D511yescharacteristic
377#108—154no—
378#104—34no—
379#104—154no—
380#97—52no—
381C4×C8—52no—
382C2×C3×C5—21no—
383C2×C3×C5—21no—
384C2×C3×C5—21no—
385C2×C3×C5C2×C4×D411yescharacteristic
386D15—42no—
387C5×D3—42no—
388C23×C3#71111yescharacteristic
389C22×D3—21no—
390C22×D3—101no—
391C2×C3×C4—22no—
392C2×C3×C4C2×C4×D511yescharacteristic
393C2×C3×C4C2×C4×D511yescharacteristic
394C2×C3×C4D4×D511yescharacteristic
395C2×C3×C4D4×D511yescharacteristic
396C2×C3×C4#71111yescharacteristic
397C2×C3×C4C23×D511yescharacteristic
398#67—22no—
399#67—22no—
400#67—102no—
401#67—102no—
402C2×Dic3—21no—
403C2×Dic3—101no—
404D12—21no—
405D12—52no—
406D12—101no—
407D12C2×C4×D512yes—
408C4×D3—22no—
409C4×D3—22no—
410C4×D3—102no—
411C4×D3—102no—
412Dic6—21no—
413Dic6—52no—
414Dic6—101no—
415Dic6C2×C4×D512yes—
416C3×C8—54no—
417C3×C8—51no—
418C3×C8—51no—
419C3×C8—51no—
420C3×C8—51no—
421C3⋊C8—22no—
422C3⋊C8—102no—
423C22×C5—41no—
424C22×C5—62no—
425C22×C5—122no—
426C22×C5C4×D1211yescharacteristic
427C22×C5C4×D1211yescharacteristic
428C22×C5C2×#23511yescharacteristic
429C2×D5—62no—
430C2×D5—122no—
431C4×C5—42no—
432C4×C5—62no—
433C4×C5C4×D1211yescharacteristic
434C4×C5C4×D1211yescharacteristic
435C4×C5C22×C4×D311yescharacteristic
436C4×C5C22×D1211yescharacteristic
437Dic5—62no—
438#41—32no—
439#41—32no—
440#41—61no—
441#41—61no—
442#41—152no—
443#41—152no—
444#41—301no—
445#41—301no—
446C2×Q8—31no—
447C2×Q8—151no—
448C2×D4—31no—
449C2×D4—61no—
450C2×D4—151no—
451C2×D4—301no—
452C22×C4—61no—
453C22×C4—301no—
454C22×C4C2×D3×D511yescharacteristic
455Q16—154no—
456SD16—154no—
457SD16—154no—
458D8—154no—
459M16—32no—
460M16—52no—
461M16—52no—
462M16—61no—
463M16—152no—
464M16—302no—
465M16—302no—
466M16—302no—
467M16—302no—
468M16—301no—
469C2×C8—51no—
470C2×C8—51no—
471C2×C8—51no—
472C2×C8—51no—
473C2×C8—51no—
474C2×C8—52no—
475C2×C8—51no—
476C2×C8—61no—
477C2×C8—302no—
478C2×C8—302no—
479C2×C8—302no—
480C2×C8—302no—
481C2×C8—301no—
482C4⋊C4—21no—
483C22⋊C4—21no—
484C42C2×D3×D512yes—
485C3×C5#295511yescharacteristic, complemented, Hall
486C22×C3—41no—
487C22×C3#402711yescharacteristic
488C22×C3#402711yescharacteristic
489C22×C3C2×#71111yescharacteristic
490C2×D3—22no—
491C2×D3—42no—
492C2×D3—102no—
493C2×D3—202no—
494C3×C4—42no—
495C3×C4#402711yescharacteristic
496C3×C4#402711yescharacteristic
497C3×C4C22×C4×D511yescharacteristic
498C3×C4C2×D4×D511yescharacteristic
499Dic3—22no—
500Dic3—102no—
501C2×C5—21no—
502C2×C5—21no—
503C2×C5—21no—
504C2×C5—122no—
505C2×C5C2×C4×D1211yescharacteristic
506D5—122no—
507C23—61no—
508C23—301no—
509C23#678211yescharacteristic
510Q8—32no—
511Q8—61no—
512Q8—152no—
513Q8—301no—
514D4—32no—
515D4—62no—
516D4—62no—
517D4—61no—
518D4—152no—
519D4—302no—
520D4—302no—
521D4—301no—
522C2×C4—22no—
523C2×C4—61no—
524C2×C4—62no—
525C2×C4—62no—
526C2×C4—301no—
527C2×C4—302no—
528C2×C4—302no—
529C2×C4C4×D3×D511yescharacteristic
530C2×C4C4×D3×D511yescharacteristic
531C2×C4D5×D1211yescharacteristic
532C2×C4D5×D1211yescharacteristic
533C2×C4#678211yescharacteristic
534C2×C4C22×D3×D511yescharacteristic
535C8—54no—
536C8—51no—
537C8—51no—
538C8—51no—
539C8—51no—
540C8—62no—
541C8—302no—
542C2×C3—21no—
543C2×C3—21no—
544C2×C3—21no—
545C2×C3C2×#402711yescharacteristic
546D3—42no—
547D3—202no—
548C5#7857911yescharacteristic, complemented, Sylow
549C22—41no—
550C22—62no—
551C22—122no—
552C22—302no—
553C22—602no—
554C22#9140511yescharacteristic
555C22#9140511yescharacteristic
556C22C2×#678211yescharacteristic
557C4—42no—
558C4—62no—
559C4—302no—
560C4#9140511yescharacteristic
561C4#9140511yescharacteristic
562C4C2×C4×D3×D511yescharacteristic
563C4C2×D5×D1211yescharacteristic
564C3#1061072011yescharacteristic, complemented, Sylow
565C2—21no—
566C2—21no—
567C2—21no—
568C2—122no—
569C2—602no—
570C2C2×#9140511yescharacteristic
subgroupG/H
centerC4C2×D5×D12
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC3×C5×#117C22
socleC2×C3×C5C2×C4×D4
prime radical#11754145C2
cosocleC2×C3×C4×C5C24
hypercenter#117D3×D5
hypocenterC3×C5#2955
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
2. C2×#91290C2semidirect
3. C2×#91286C2semidirect
4. #11751106C2semidirect
5. C5×#5005C2semidirect
6. C3×#65286C2semidirect
7. #11748476C2semidirect
8. #11747019C2semidirect
9. #11747018C2semidirect
18. #91322C4semidirect
19. #91321C4semidirect
20. #91305C4semidirect
21. #91304C4semidirect
27. #91290C22semidirect
28. #91290C22semidirect
32. #91286C22semidirect
33. #91286C22semidirect
35. #91089C22semidirect
36. C5×#805C22semidirect
37. C3×#3935C22semidirect
38. C3×C4×#168C22semidirect
39. #90991C22semidirect
47. #65286D3semidirect
103. #5005D5semidirect
133. #3935C2×D3semidirect
134. C4×#168C2×D3semidirect
199. #805C2×D5semidirect
485. C3×C5#2955semidirect
548. C5#78579semidirect
564. C3#10610720semidirect

Maximal subgroups

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

subgroupindex
#117541452
C2×#912902
C2×#912862
#117511062
C5×#50052
C3×#652862
#117484762
#117470192
#117470182
#106107203
#785795