Finite Group Databasebeta

← #14429252

Subgroups — #14429252

Order 1920.

944 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5510C211yesmaximal, characteristic
2#11754271C211yesmaximal, characteristic, complemented
3C2×#91322C211yesmaximal, characteristic, complemented
4C2×#91288C211yesmaximal, characteristic, complemented
5C2×#91094C211yesmaximal, characteristic, complemented
6C3×#65253C211yesmaximal, characteristic, complemented
7#11748382C211yesmaximal, characteristic, complemented
8#11747467C212yesmaximal, complemented
9#11747464C212yesmaximal, complemented
10#11747448C212yesmaximal, complemented
11#11747447C212yesmaximal, complemented
12#10615154—31nomaximal, Hall
13C22×Dic30C2211yescharacteristic
14C2×C5×#235C2211yescharacteristic
15C2×C5×#235C2211yescharacteristic
16C2×C3×C5×C4⋊C4C2211yescharacteristic
17C22×C3⋊#168C2211yescharacteristic
18C5×#930—22no—
19C5×#897—21no—
20C5×#897C2212yes—
21C5×#895—21no—
22C5×#895C2212yes—
23C3×#4072C2211yescharacteristic
24C2×C3×#681C2211yescharacteristic
25#91543—22no—
26#91376—21no—
27#91373—21no—
28#91365C2212yes—
29#91365C2212yes—
30#91322—24no—
31C2×#6663C2212yes—
32C2×#6659C2212yes—
33C2×#6650C2212yes—
34#91288—24no—
35C2×D3×#168C2212yes—
36#91094C2212yes—
37#91094C2212yes—
38C3×#3940C2212yes—
39C3×#3939C2212yes—
40#90972C2211yescharacteristic
41#90972C2211yescharacteristic
42#90966C2211yescharacteristic
43#90966C2211yescharacteristic
44#79214—51nomaximal, Hall
45C5×#537—31no—
46#66159—31no—
47C2×#4097—31no—
48C2×#4094—31no—
49#65522—31no—
50#65341—32no—
51#65340—32no—
52#65331—32no—
53#65330—32no—
54C2×#3942—31no—
55#65253D311yescharacteristic, complemented
56C22×C3×C4×C5—21no—
57C22×C3×C4×C5C2311yescharacteristic
58C22×Dic15—21no—
59C2×Dic30—22no—
60C2×Dic30—22no—
61C2×Dic30—21no—
62C2×Dic30C2312yes—
63C2×C5×#67—21no—
64C2×C5×#67D412yes—
65C5×#235—22no—
66C5×#235—22no—
67C5×#235—22no—
68C5×#235—22no—
69C2×C5×D12—22no—
70C2×C5×D12D411yescharacteristic
71C2×C5×D12C2311yescharacteristic
72C2×C4×C5×D3—21no—
73C2×C4×C5×D3C2312yes—
74C2×C5×Dic6D411yescharacteristic
75C2×C5×Dic6C2311yescharacteristic
76C3×C5×C4⋊C4—21no—
77C3×C5×C4⋊C4C2312yes—
78C2×C3⋊#168—22no—
79C2×C3⋊#168C2311yescharacteristic
80C2×C3⋊#168C2311yescharacteristic
81C5×(C2×D3)⋊C4—22no—
82C5×(C2×D3)⋊C4—22no—
83C5×C4⋊Dic3—22no—
84C3×C22⋊Dic5—21no—
85C3×C4⋊Dic5C2311yescharacteristic
86C3×C4⋊Dic5C2311yescharacteristic
87C3×C4×Dic5—21no—
88C3×#681D414yescomplemented
89C2×C3×#168C2312yes—
90#6668—22no—
91#6668—22no—
92(C2×D3)⋊Dic5—22no—
93(C2×D3)⋊Dic5—22no—
94#6663—24no—
95#6659—24no—
96#6650—24no—
97D3×#168—24no—
98C2×#909—51no—
99C2×#907—51no—
100#5526—51no—
101#5510D511yescharacteristic
102C3×#428—51no—
103#5002—51no—
104C2×#810—51no—
105#4887—52no—
106#4886—52no—
107#4867—52no—
108#4864—52no—
109C2×C5×#41—31no—
110C2×C5×#41—31no—
111C22×Dic10—31no—
112C5×C22⋊Q8—32no—
113C5×C22⋊Q8—61no—
114C5×#121—32no—
115C5×#121—61no—
116C5×#121—62no—
117C2×C5×C4⋊C4C2×D311yescharacteristic
118#4097—64no—
119#4094—64no—
120#4090—61no—
121C2×#689—32no—
122#4082—61no—
123#4081—62no—
124C2×#687—32no—
125#4072C2×D311yescharacteristic
126#4070—32no—
127#4070—32no—
128C2×#681—32no—
129C2×#681C2×D311yescharacteristic
130C22×#168—31no—
131C22×#168—32no—
132#3967—31no—
133#3967—31no—
134#3964—31no—
135#3964—31no—
136#3942—32no—
137#3942—32no—
138#3940C2×D312yes—
139#3939C2×D312yes—
140#3063—151noSylow
141C23×C3×C5C2×D411yescharacteristic
142C22×C5×D3—21no—
143C22×C5×D3C2×D412yes—
144C2×C3×C4×C5—22no—
145C2×C3×C4×C5—81no—
146C2×C3×C4×C5C2×D411yescharacteristic
147C2×C3×C4×C5C2×D412yes—
148C2×C3×C4×C5#4112yes—
149C2×C3×C4×C5C2411yescharacteristic
150C2×Dic15—22no—
151C2×Dic15—42no—
152Dic30—22no—
153Dic30—22no—
154Dic30—42no—
155C5×#67—24no—
156C5×#67—42no—
157C2×C5×Dic3—21no—
158C2×C5×Dic3C2×D412yes—
159C5×D12—22no—
160C5×D12—21no—
161C5×D12—21no—
162C5×D12—82no—
163C4×C5×D3—22no—
164C4×C5×D3—22no—
165C4×C5×D3—42no—
166C5×Dic6—22no—
167C5×Dic6—21no—
168C5×Dic6—21no—
169C2×C3×Dic5—21no—
170C2×C3×Dic5—21no—
171C3⋊#168—21no—
172C3⋊#168—21no—
173C3×#168C2×D414yes—
174C2×#235C2×D511yescharacteristic
175C2×#235C2×D511yescharacteristic
176C22×Dic6—51no—
177C2×C3×M16—51no—
178C3×#117—51no—
179C2×C3×C4⋊C4C2×D511yescharacteristic
180#930—22no—
181C22×C3⋊C8—51no—
182#909—104no—
183#907—104no—
184C2×Dic12—52no—
185C2×#204—52no—
186C2×ΓL(1,𝔽25)—52no—
187C2×D3×C8—52no—
188#897—21no—
189#897—52no—
190#897—52no—
191#897C2×D512yes—
192#895—21no—
193#895C2×D512yes—
194#886—102no—
195#875—101no—
196#869—101no—
197C3×#107—52no—
198C3×#106—52no—
199#810—51no—
200#810—51no—
201#810—52no—
202#810—52no—
203#809—51no—
204#809—51no—
205C5×#41—62no—
206C5×#41—62no—
207C5×#41—62no—
208C5×#41—62no—
209C2×C5×Q8—31no—
210C2×C5×Q8—31no—
211C2×C5×D4—32no—
212C2×C5×D4—31no—
213C2×C5×D4—31no—
214C2×C5×D4—62no—
215C2×C5×D4—61no—
216C22×C4×C5—21no—
217C22×C4×C5—32no—
218C22×C4×C5—61no—
219C22×C4×C5C22×D311yescharacteristic
220C22×Dic5—61no—
221C2×Dic10—32no—
222C2×Dic10—62no—
223C2×Dic10—62no—
224C2×Dic10—61no—
225C5×C4⋊C4—21no—
226C5×C4⋊C4—62no—
227C5×C4⋊C4C22×D312yes—
228C5×C22⋊C4—62no—
229C5×C22⋊C4—62no—
230C22⋊Dic5—21no—
231C22⋊Dic5—62no—
232C22⋊Dic5—62no—
233#689—64no—
234#687—64no—
235C4⋊Dic5C22×D311yescharacteristic
236C4⋊Dic5C22×D311yescharacteristic
237#683—62no—
238#683—62no—
239C4×Dic5—21no—
240#681—64no—
241#681D1214yescomplemented
242C2×#168—31no—
243C2×#168—31no—
244C2×#168—62no—
245C2×#168—64no—
246C2×#168C22×D312yes—
247C2×#137—151no—
248C2×#131—151no—
249#537—31no—
250#536—151no—
251#470—152no—
252#468—152no—
253#467—152no—
254#465—152no—
255#428—51no—
256#416—151no—
257C2×#103—151no—
258C22×C3×C5—21no—
259C22×C3×C5—21no—
260C22×C3×C5#12112yes—
261C22×C3×C5C22×D411yescharacteristic
262C2×C5×D3—22no—
263C2×C5×D3—22no—
264C2×C5×D3—22no—
265C2×C5×D3—41no—
266C2×C5×D3—41no—
267C2×C5×D3—41no—
268C3×C4×C5—42no—
269C3×C4×C5#12112yes—
270C3×C4×C5C22×D411yescharacteristic
271C3×C4×C5C2×#4111yescharacteristic
272Dic15—42no—
273C3×Dic5—41no—
274C3×Dic5—41no—
275C5×Dic3—22no—
276C5×Dic3—41no—
277C22×C3×C4—21no—
278C22×C3×C4C22×D511yescharacteristic
279C2×#67—21no—
280C2×#67C22⋊D512yes—
281C22×Dic3—101no—
282#235—22no—
283#235—22no—
284#235—22no—
285#235—22no—
286C2×D12—22no—
287C2×D12C22⋊D511yescharacteristic
288C2×D12C22×D511yescharacteristic
289C2×C4×D3—21no—
290C2×C4×D3C22×D512yes—
291C2×Dic6—52no—
292C2×Dic6—102no—
293C2×Dic6—102no—
294C2×Dic6—101no—
295C2×Dic6C22⋊D511yescharacteristic
296C2×Dic6C22×D511yescharacteristic
297C3×M16—54no—
298C2×C3×C8—52no—
299C3×C4⋊C4—21no—
300C3×C4⋊C4—51no—
301C3×C4⋊C4—51no—
302C3×C4⋊C4C22×D512yes—
303C3×C22⋊C4—101no—
304C3×C42—101no—
305(C2×D3)⋊C4—22no—
306(C2×D3)⋊C4—22no—
307(C2×D3)⋊C4—102no—
308(C2×D3)⋊C4—102no—
309C4⋊Dic3—22no—
310C4⋊Dic3—102no—
311C4⋊Dic3—102no—
312C2×C3⋊C8—51no—
313C2×C3⋊C8—51no—
314C2×C3⋊C8—102no—
315Dic12—104no—
316#204—104no—
317ΓL(1,𝔽25)—104no—
318D3×C8—104no—
319C23×C5—32no—
320C23×C5—61no—
321C23×C5D3×D411yescharacteristic
322C5×Q8—62no—
323C5×Q8—61no—
324C5×Q8—61no—
325C5×D4—64no—
326C5×D4—62no—
327C5×D4—61no—
328C5×D4—61no—
329C5×D4—122no—
330C5×D4—242no—
331C2×C4×C5—22no—
332C2×C4×C5—32no—
333C2×C4×C5—62no—
334C2×C4×C5—62no—
335C2×C4×C5—61no—
336C2×C4×C5—81no—
337C2×C4×C5—122no—
338C2×C4×C5C2×D1212yes—
339C2×C4×C5D3×D411yescharacteristic
340C2×C4×C5#23912yes—
341C2×C4×C5C23×D311yescharacteristic
342C2×Dic5—21no—
343C2×Dic5—21no—
344C2×Dic5—62no—
345C2×Dic5—122no—
346Dic10—62no—
347Dic10—62no—
348Dic10—122no—
349#168—61no—
350#168—61no—
351#168C2×D1214yes—
352C2×#41—31no—
353C2×#41—31no—
354C22×Q8—151no—
355#137—304no—
356C2×Q16—152no—
357C2×SD16—152no—
358#131—304no—
359C2×M16—51no—
360C2×M16—152no—
361C22×C8—152no—
362C22×C8—151no—
363C4⋊Q8—301no—
364#124—301no—
365#123—302no—
366C22⋊Q8—32no—
367C22⋊Q8—61no—
368C22⋊Q8—152no—
369C22⋊Q8—152no—
370#121—32no—
371#121—61no—
372#121—62no—
373#117—51no—
374C2×C4⋊C4D3×D511yescharacteristic
375#107—52no—
376#106—52no—
377#103—151no—
378#103—151no—
379#103—152no—
380#103—152no—
381#102—151no—
382#102—151no—
383C2×C3×C5—22no—
384C2×C3×C5C2×#12111yescharacteristic
385C2×C3×C5#57711yescharacteristic
386C2×C3×C5#57811yescharacteristic
387C5×D3—42no—
388C5×D3—81no—
389C23×C3C2×C22⋊D511yescharacteristic
390C22×D3—21no—
391C22×D3C2×C22⋊D512yes—
392C2×C3×C4—22no—
393C2×C3×C4—81no—
394C2×C3×C4—101no—
395C2×C3×C4—101no—
396C2×C3×C4D4×D512yes—
397C2×C3×C4#71112yes—
398C2×C3×C4C2×C22⋊D511yescharacteristic
399C2×C3×C4C23×D511yescharacteristic
400#67—24no—
401#67—42no—
402C2×Dic3—21no—
403C2×Dic3—102no—
404C2×Dic3—202no—
405C2×Dic3C2×C22⋊D512yes—
406D12—21no—
407D12—21no—
408D12—22no—
409D12—82no—
410C4×D3—22no—
411C4×D3—22no—
412C4×D3—42no—
413Dic6—22no—
414Dic6—21no—
415Dic6—21no—
416Dic6—102no—
417Dic6—102no—
418Dic6—202no—
419C3×C8—54no—
420C3⋊C8—101no—
421C3⋊C8—101no—
422C22×C5—21no—
423C22×C5—21no—
424C22×C5—62no—
425C22×C5—62no—
426C22×C5—62no—
427C22×C5—121no—
428C22×C5—121no—
429C22×C5—121no—
430C22×C5#89512yes—
431C22×C5C2×D3×D411yescharacteristic
432C4×C5—42no—
433C4×C5—62no—
434C4×C5—121no—
435C4×C5#89512yes—
436C4×C5C22×D1211yescharacteristic
437C4×C5C2×#23911yescharacteristic
438Dic5—41no—
439Dic5—41no—
440Dic5—122no—
441#41—62no—
442#41—62no—
443#41—62no—
444#41—62no—
445C2×Q8—31no—
446C2×Q8—31no—
447C2×Q8—152no—
448C2×Q8—301no—
449C2×Q8—302no—
450C2×Q8—302no—
451C2×D4—32no—
452C2×D4—31no—
453C2×D4—31no—
454C2×D4—62no—
455C2×D4—61no—
456C22×C4—21no—
457C22×C4—32no—
458C22×C4—61no—
459C22×C4—301no—
460C22×C4C2×D3×D511yescharacteristic
461Q16—304no—
462SD16—304no—
463M16—54no—
464M16—304no—
465C2×C8—52no—
466C2×C8—151no—
467C2×C8—151no—
468C2×C8—304no—
469C2×C8—302no—
470C4⋊C4—21no—
471C4⋊C4—51no—
472C4⋊C4—51no—
473C4⋊C4—62no—
474C4⋊C4—302no—
475C4⋊C4—302no—
476C4⋊C4C2×D3×D512yes—
477C22⋊C4—62no—
478C22⋊C4—62no—
479C22⋊C4—101no—
480C22⋊C4—302no—
481C22⋊C4—302no—
482C42—101no—
483C3×C5#306311yescharacteristic, complemented, Hall
484C22×C3—21no—
485C22×C3—21no—
486C22×C3#408512yes—
487C22×C3C22×C22⋊D511yescharacteristic
488C2×D3—22no—
489C2×D3—22no—
490C2×D3—22no—
491C2×D3—41no—
492C2×D3—41no—
493C2×D3—41no—
494C3×C4—42no—
495C3×C4—201no—
496C3×C4—201no—
497C3×C4#408512yes—
498C3×C4C2×D4×D511yescharacteristic
499C3×C4C2×#71111yescharacteristic
500Dic3—22no—
501Dic3—41no—
502Dic3—202no—
503C2×C5—22no—
504C2×C5—122no—
505C2×C5—241no—
506C2×C5C2×#89511yescharacteristic
507C2×C5#577511yescharacteristic
508C2×C5#577911yescharacteristic
509C23—32no—
510C23—61no—
511C23D3×C22⋊D511yescharacteristic
512Q8—62no—
513Q8—61no—
514Q8—61no—
515Q8—302no—
516Q8—302no—
517Q8—602no—
518D4—64no—
519D4—62no—
520D4—61no—
521D4—61no—
522D4—122no—
523D4—242no—
524C2×C4—22no—
525C2×C4—32no—
526C2×C4—62no—
527C2×C4—62no—
528C2×C4—61no—
529C2×C4—81no—
530C2×C4—101no—
531C2×C4—101no—
532C2×C4—122no—
533C2×C4—302no—
534C2×C4—602no—
535C2×C4#676812yes—
536C2×C4D5×D1212yes—
537C2×C4D3×C22⋊D511yescharacteristic
538C2×C4C22×D3×D511yescharacteristic
539C8—54no—
540C8—301no—
541C8—301no—
542C2×C3—22no—
543C2×C3C2×#408511yescharacteristic
544C2×C3#6617011yescharacteristic
545C2×C3#6617211yescharacteristic
546D3—42no—
547D3—81no—
548C5#7921411yescharacteristic, complemented, Sylow
549C22—21no—
550C22—21no—
551C22—62no—
552C22—62no—
553C22—62no—
554C22—121no—
555C22—121no—
556C22—121no—
557C22#9141612yes—
558C22C2×D3×C22⋊D511yescharacteristic
559C4—42no—
560C4—62no—
561C4—121no—
562C4—201no—
563C4—201no—
564C4—602no—
565C4#9141612yes—
566C4C2×#676811yescharacteristic
567C4C2×D5×D1211yescharacteristic
568C3#1061515411yescharacteristic, complemented, Sylow
569C2—22no—
570C2—122no—
571C2—241no—
572C2C2×#9141611yescharacteristic
573C2#1175472011yescharacteristic
574C2#1175511411yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊D5
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×C4⋊C4C22
socleC22×C3×C5C22×D4
prime radicalC5×#5510C2
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5#3063
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #11754271C2semidirect
3. C2×#91322C2semidirect
4. C2×#91288C2semidirect
5. C2×#91094C2semidirect
6. C3×#65253C2semidirect
7. #11748382C2semidirect
8. #11747467C2semidirect
9. #11747464C2semidirect
10. #11747448C2semidirect
11. #11747447C2semidirect
55. #65253D3semidirect
88. C3×#681D4semidirect
241. #681D12semidirect
483. C3×C5#3063semidirect
548. C5#79214semidirect
568. C3#10615154semidirect

Maximal subgroups

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

subgroupindex
C5×#55102
#117542712
C2×#913222
C2×#912882
C2×#910942
C3×#652532
#117483822
#117474672
#117474642
#117474482
#117474472
#106151543
#792145