Finite Group Databasebeta

← #14439700

Subgroups — #14439700

Order 1920.

1064 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×#5526C211yesmaximal, characteristic, complemented
2C3×#65860C211yesmaximal, characteristic, complemented
3C22×#6661C211yesmaximal, characteristic, complemented
4C2×#91309C211yesmaximal, characteristic, complemented
5C2×#91307C211yesmaximal, characteristic, complemented
6C2×#91297C211yesmaximal, characteristic
7#11751098C211yesmaximal, characteristic, complemented
8#11747103C212yesmaximal, complemented
9#11747082C212yesmaximal, complemented
10#11747080C212yesmaximal, complemented
11#11747040C212yesmaximal, complemented
12#10615134—31nomaximal, Hall
13C2×C5×#235C2211yescharacteristic
14C22×C5×Dic6C2211yescharacteristic
15C2×C3×#709C2211yescharacteristic
16C22×C3×Dic10C2211yescharacteristic
17C3×C5×#117C2211yescharacteristic
18C2×#6710C2211yescharacteristic
19C22×C3⋊#168C2211yescharacteristic
20C5×#897—22no—
21C5×#886—22no—
22C5×#875C2212yes—
23C5×#869C2212yes—
24C3×#4043—22no—
25C3×#4032—22no—
26C3×#4021C2212yes—
27C3×#4015C2212yes—
28C2×#6661—24no—
29C2×#6661C2212yes—
30C2×#6661C2212yes—
31C2×#6661C2212yes—
32C2×#6661C2211yescharacteristic
33C2×#6661C2212yes—
34#91309—24no—
35C2×#6656C2211yescharacteristic
36C2×#6656C2212yes—
37#91307—24no—
38C2×#6655C2211yescharacteristic
39C2×#6655C2212yes—
40#91297—24no—
41C2×#6652C2211yescharacteristic
42#91087C2212yes—
43C4×C3⋊#168C2212yes—
44#83403—51nomaximal, Hall
45C5×#536—31no—
46C2×#4097—31no—
47C2×#4096—31no—
48C22×#689—31no—
49C2×#4088—31no—
50#65860D311yescharacteristic, complemented
51#65381—32no—
52#65375—32no—
53#65349—32no—
54#65348—32no—
55#65286—31no—
56C22×C3×C4×C5C2311yescharacteristic
57C2×C5×#67—21no—
58C22×C5×Dic3—21no—
59C5×#235—22no—
60C5×#235—22no—
61C2×C5×D12C2311yescharacteristic
62C2×C4×C5×D3—21no—
63C2×C5×Dic6—22no—
64C2×C5×Dic6—22no—
65C2×C5×Dic6—21no—
66C2×C5×Dic6C2311yescharacteristic
67C2×C5×Dic6C2312yes—
68C2×C3×C22⋊D5—21no—
69C22×C3×Dic5—21no—
70C3×#709—22no—
71C3×#709—22no—
72C2×C3×D20C2311yescharacteristic
73C2×C3×C4×D5—21no—
74C2×C3×Dic10—22no—
75C2×C3×Dic10—22no—
76C2×C3×Dic10—21no—
77C2×C3×Dic10C2312yes—
78C2×C3×Dic10C2311yescharacteristic
79C3×C5×C4⋊C4D412yes—
80C3×C5×C22⋊C4D412yes—
81C3×C42×C5C2312yes—
82#6710D414yes—
83C2×C3⋊#168D414yes—
84C2×C3⋊#168C2311yescharacteristic
85C2×C3⋊#168C2311yescharacteristic
86C2×C3⋊#168C2312yes—
87C5×(C2×D3)⋊C4—24no—
88C5×C4⋊Dic3—24no—
89C3×#685—24no—
90C3×C4⋊Dic5—24no—
91#6661—22no—
92#6661—24no—
93#6661—24no—
94#6661—24no—
95#6661—24no—
96#6656—22no—
97#6656—24no—
98#6655—22no—
99#6655—24no—
100#6652—22no—
101C3×#536—51no—
102C2×#951—51no—
103C2×#950—51no—
104C22×#216—51no—
105C2×#942—51no—
106#5526D511yescharacteristic, complemented
107#5092—52no—
108#5086—52no—
109#5060—52no—
110#5059—52no—
111#4997—51no—
112C2×C5×#41—31no—
113C22×C5×Q8—31no—
114C2×#709C2×D311yescharacteristic
115C22×Dic10C2×D311yescharacteristic
116C5×C4⋊Q8—32no—
117C5×#124—32no—
118C5×#123—62no—
119C5×C22⋊Q8—62no—
120C5×#117C2×D311yescharacteristic
121#4097—64no—
122#4096—64no—
123C2×#689—32no—
124C2×#689—32no—
125C2×#689—32no—
126C2×#689—32no—
127C2×#689—31no—
128C2×#689—64no—
129#4088—64no—
130C2×#688—32no—
131C2×#688—31no—
132C2×#687—32no—
133C2×#687—31no—
134C2×#686—31no—
135C2×#681—31no—
136C22×#168—31no—
137#4043—22no—
138#4032—22no—
139#4021C2×D312yes—
140#4015C2×D312yes—
141#3935—32no—
142C4×#168—32no—
143#3156—151noSylow
144C23×C3×C5C2×D411yescharacteristic
145C22×C5×D3—21no—
146C22×C3×D5—21no—
147C2×C3×C4×C5C2×D414yes—
148C2×C3×C4×C5C2×D411yescharacteristic
149C2×C3×C4×C5C2×D412yes—
150C2×C3×C4×C5C2×D412yes—
151C2×C3×C4×C5C2411yescharacteristic
152C5×#67—42no—
153C2×C5×Dic3—21no—
154C2×C5×Dic3—22no—
155C2×C5×Dic3—42no—
156C5×D12—21no—
157C5×D12—21no—
158C4×C5×D3—42no—
159C5×Dic6—22no—
160C5×Dic6—21no—
161C5×Dic6—22no—
162C5×Dic6—21no—
163C5×Dic6—42no—
164C3×C22⋊D5—42no—
165C2×C3×Dic5—21no—
166C2×C3×Dic5—22no—
167C2×C3×Dic5—42no—
168C3×D20—21no—
169C3×D20—21no—
170C3×C4×D5—42no—
171C3×Dic10—22no—
172C3×Dic10—21no—
173C3×Dic10—22no—
174C3×Dic10—21no—
175C3×Dic10—42no—
176C3⋊#168C2×D412yes—
177C3⋊#168C2×D414yes—
178C3⋊#168C2×D412yes—
179C2×C3×#41—51no—
180C22×C3×Q8—51no—
181C2×#235C2×D511yescharacteristic
182C22×Dic6C2×D511yescharacteristic
183C3×C4⋊Q8—52no—
184C3×#124—52no—
185C3×#123—102no—
186C3×C22⋊Q8—102no—
187C3×#117C2×D511yescharacteristic
188#951—104no—
189#950—104no—
190C2×#216—52no—
191C2×#216—52no—
192C2×#216—52no—
193C2×#216—51no—
194C2×#216—52no—
195C2×#216—104no—
196#942—104no—
197C2×#215—52no—
198C2×#215—51no—
199C2×#214—51no—
200C2×#214—52no—
201C2×#213—51no—
202C2×#208—51no—
203C22×C3⋊C8—51no—
204#897—22no—
205#886—22no—
206#875C2×D512yes—
207#869C2×D512yes—
208#803—52no—
209C4×C3⋊C8—52no—
210C5×#41—62no—
211C5×#41—62no—
212C2×C5×Q8—32no—
213C2×C5×Q8—31no—
214C2×C5×Q8—61no—
215C2×C5×Q8—62no—
216C2×C5×Q8—62no—
217C2×C5×D4—31no—
218C2×C5×D4—61no—
219C22×C4×C5—61no—
220C22×C4×C5—61no—
221C22×C4×C5C22×D311yescharacteristic
222C2×C22⋊D5—21no—
223C22×Dic5—21no—
224#709—22no—
225#709—22no—
226C2×D20C22×D311yescharacteristic
227C2×C4×D5—21no—
228C2×Dic10—22no—
229C2×Dic10—22no—
230C2×Dic10—21no—
231C2×Dic10C22×D312yes—
232C2×Dic10C22×D311yescharacteristic
233C5×C4⋊C4—64no—
234C5×C4⋊C4#6712yes—
235C5×C22⋊C4—64no—
236C5×C22⋊C4#6712yes—
237C42×C5C22×D312yes—
238#689—62no—
239#689—64no—
240#689—64no—
241#689—64no—
242#689—64no—
243#688—62no—
244#688—64no—
245#687—62no—
246#687—64no—
247#686—62no—
248#685—24no—
249C4⋊Dic5—24no—
250#681—34no—
251C2×#168—34no—
252C2×#168—31no—
253C2×#168—32no—
254C2×#168—31no—
255C2×#137—151no—
256C2×#137—151no—
257C2×#135—151no—
258C22×Q16—151no—
259#536—31no—
260#536—51no—
261#497—152no—
262#497—152no—
263#495—152no—
264#494—152no—
265#405—151no—
266C22×C3×C5—22no—
267C22×C3×C5#12712yes—
268C22×C3×C5C22×D411yescharacteristic
269C2×C5×D3—41no—
270C2×C5×D3—42no—
271C2×C3×D5—41no—
272C2×C3×D5—42no—
273C3×C4×C5—24no—
274C3×C4×C5#12712yes—
275C3×C4×C5C22×D411yescharacteristic
276C3×C4×C5C22×D411yescharacteristic
277C3×Dic5—42no—
278C3×Dic5—41no—
279C5×Dic3—42no—
280C5×Dic3—41no—
281C3×#41—102no—
282C3×#41—102no—
283C2×C3×Q8—52no—
284C2×C3×Q8—51no—
285C2×C3×Q8—101no—
286C2×C3×Q8—102no—
287C2×C3×Q8—102no—
288C2×C3×D4—51no—
289C2×C3×D4—101no—
290C22×C3×C4—101no—
291C22×C3×C4—101no—
292C22×C3×C4C22×D511yescharacteristic
293C2×#67—21no—
294C22×Dic3—21no—
295#235—22no—
296#235—22no—
297C2×D12C22×D511yescharacteristic
298C2×C4×D3—21no—
299C2×Dic6—22no—
300C2×Dic6—22no—
301C2×Dic6—21no—
302C2×Dic6C22×D511yescharacteristic
303C2×Dic6C22×D512yes—
304C3×C4⋊C4—104no—
305C3×C4⋊C4C22⋊D512yes—
306C3×C22⋊C4—104no—
307C3×C22⋊C4C22⋊D512yes—
308C3×C42C22×D512yes—
309#216—104no—
310#216—104no—
311#216—104no—
312#216—102no—
313#216—104no—
314#215—104no—
315#215—102no—
316#214—102no—
317#214—104no—
318#213—102no—
319(C2×D3)⋊C4—24no—
320C4⋊Dic3—24no—
321#208—54no—
322C2×C3⋊C8—54no—
323C2×C3⋊C8—51no—
324C2×C3⋊C8—52no—
325C2×C3⋊C8—51no—
326C23×C5—61no—
327C23×C5C2×#6711yescharacteristic
328C22×D5—21no—
329C5×Q8—62no—
330C5×Q8—61no—
331C5×Q8—62no—
332C5×Q8—61no—
333C5×Q8—122no—
334C5×D4—61no—
335C5×D4—61no—
336C5×D4—122no—
337C2×C4×C5—62no—
338C2×C4×C5—61no—
339C2×C4×C5—122no—
340C2×C4×C5—122no—
341C2×C4×C5D3×D412yes—
342C2×C4×C5D3×D412yes—
343C2×C4×C5C2×#6714yes—
344C2×C4×C5C2×#6711yescharacteristic
345C2×C4×C5C23×D311yescharacteristic
346C22⋊D5—42no—
347C2×Dic5—21no—
348C2×Dic5—22no—
349C2×Dic5—42no—
350D20—21no—
351D20—21no—
352C4×D5—42no—
353Dic10—22no—
354Dic10—21no—
355Dic10—22no—
356Dic10—21no—
357Dic10—42no—
358#168—32no—
359#168—32no—
360#168—34no—
361C2×#41—31no—
362C2×#41—51no—
363C22×Q8—31no—
364C22×Q8—51no—
365#137—304no—
366#137—304no—
367#135—304no—
368C2×Q16—152no—
369C2×Q16—152no—
370C2×Q16—151no—
371C2×Q16—152no—
372C2×Q16—152no—
373C2×Q16—304no—
374C2×SD16—151no—
375C2×SD16—152no—
376C2×SD16—151no—
377C2×SD16—152no—
378C2×D8—151no—
379C2×M16—151no—
380C22×C8—151no—
381C4⋊Q8—32no—
382C4⋊Q8—52no—
383#124—32no—
384#124—52no—
385#123—62no—
386#123—102no—
387C22⋊Q8—62no—
388C22⋊Q8—102no—
389#117D3×D511yescharacteristic
390#97—152no—
391C4×C8—152no—
392C2×C3×C5—22no—
393C2×C3×C5C2×#12711yescharacteristic
394C2×C3×C5#57812yes—
395C3×D5—81no—
396C5×D3—81no—
397C23×C3—101no—
398C23×C3C2×C22⋊D511yescharacteristic
399C22×D3—21no—
400C3×Q8—102no—
401C3×Q8—101no—
402C3×Q8—102no—
403C3×Q8—101no—
404C3×Q8—202no—
405C3×D4—101no—
406C3×D4—101no—
407C3×D4—202no—
408C2×C3×C4—102no—
409C2×C3×C4—101no—
410C2×C3×C4—202no—
411C2×C3×C4—202no—
412C2×C3×C4D4×D512yes—
413C2×C3×C4D4×D512yes—
414C2×C3×C4C2×C22⋊D514yes—
415C2×C3×C4C2×C22⋊D511yescharacteristic
416C2×C3×C4C23×D511yescharacteristic
417#67—42no—
418C2×Dic3—21no—
419C2×Dic3—22no—
420C2×Dic3—42no—
421D12—21no—
422D12—21no—
423C4×D3—42no—
424Dic6—22no—
425Dic6—21no—
426Dic6—22no—
427Dic6—21no—
428Dic6—42no—
429C3⋊C8—52no—
430C3⋊C8—52no—
431C3⋊C8—54no—
432C22×C5—22no—
433C22×C5—121no—
434C22×C5—122no—
435C22×C5#94012yes—
436C22×C5C22×#6711yescharacteristic
437C2×D5—41no—
438C2×D5—42no—
439C4×C5—24no—
440C4×C5—121no—
441C4×C5—122no—
442C4×C5#94012yes—
443C4×C5C2×D3×D411yescharacteristic
444C4×C5C2×D3×D411yescharacteristic
445Dic5—42no—
446Dic5—41no—
447#41—62no—
448#41—62no—
449#41—102no—
450#41—102no—
451C2×Q8—32no—
452C2×Q8—31no—
453C2×Q8—52no—
454C2×Q8—51no—
455C2×Q8—61no—
456C2×Q8—62no—
457C2×Q8—62no—
458C2×Q8—101no—
459C2×Q8—102no—
460C2×Q8—102no—
461C2×D4—31no—
462C2×D4—51no—
463C2×D4—61no—
464C2×D4—101no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4—101no—
468C22×C4—101no—
469C22×C4C2×D3×D511yescharacteristic
470Q16—304no—
471Q16—304no—
472Q16—302no—
473Q16—304no—
474Q16—304no—
475SD16—302no—
476SD16—304no—
477SD16—302no—
478SD16—304no—
479D8—302no—
480M16—154no—
481C2×C8—154no—
482C2×C8—151no—
483C2×C8—152no—
484C2×C8—151no—
485C4⋊C4—64no—
486C4⋊C4—104no—
487C4⋊C4#120112yes—
488C22⋊C4—64no—
489C22⋊C4—104no—
490C22⋊C4#120112yes—
491C42C2×D3×D512yes—
492C3×C5#315611yescharacteristic, complemented, Hall
493C22×C3—22no—
494C22×C3—201no—
495C22×C3—202no—
496C22×C3#408612yes—
497C22×C3C22×C22⋊D511yescharacteristic
498C2×D3—41no—
499C2×D3—42no—
500C3×C4—24no—
501C3×C4—201no—
502C3×C4—202no—
503C3×C4#408612yes—
504C3×C4C2×D4×D511yescharacteristic
505C3×C4C2×D4×D511yescharacteristic
506Dic3—42no—
507Dic3—41no—
508C2×C5—22no—
509C2×C5—241no—
510C2×C5C2×#94011yescharacteristic
511C2×C5#583812yes—
512D5—81no—
513C23—61no—
514C23—101no—
515C23C2×#120111yescharacteristic
516Q8—62no—
517Q8—61no—
518Q8—62no—
519Q8—61no—
520Q8—102no—
521Q8—101no—
522Q8—102no—
523Q8—101no—
524Q8—122no—
525Q8—202no—
526D4—61no—
527D4—61no—
528D4—101no—
529D4—101no—
530D4—122no—
531D4—202no—
532C2×C4—62no—
533C2×C4—61no—
534C2×C4—102no—
535C2×C4—101no—
536C2×C4—122no—
537C2×C4—122no—
538C2×C4—202no—
539C2×C4—202no—
540C2×C4#677712yes—
541C2×C4#677712yes—
542C2×C4C2×#120114yes—
543C2×C4C2×#120111yescharacteristic
544C2×C4C22×D3×D511yescharacteristic
545C8—152no—
546C8—152no—
547C8—154no—
548C2×C3—22no—
549C2×C3—401no—
550C2×C3C2×#408611yescharacteristic
551C2×C3#6617212yes—
552D3—81no—
553C5#8340311yescharacteristic, complemented, Sylow
554C22—22no—
555C22—122no—
556C22—121no—
557C22—202no—
558C22—201no—
559C22#9146312yes—
560C22C22×#120111yescharacteristic
561C4—24no—
562C4—121no—
563C4—122no—
564C4—201no—
565C4—202no—
566C4#9146312yes—
567C4C2×#677711yescharacteristic
568C4C2×#677711yescharacteristic
569C3#1061513411yescharacteristic, complemented, Sylow
570C2—22no—
571C2—241no—
572C2—401no—
573C2C2×#9146311yescharacteristic
574C2#1175530212yes—
subgroupG/H
centerC22C22×#1201
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C5×#117C22
socleC22×C3×C5C22×D4
prime radicalC2×#91297C2
hypercenter#117D3×D5
hypocenterC3×C5#3156
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ #117

Complements

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

NH ≅ G/Ntype
1. C5×#5526C2semidirect
2. C3×#65860C2semidirect
3. C22×#6661C2semidirect
4. C2×#91309C2semidirect
5. C2×#91307C2semidirect
7. #11751098C2semidirect
8. #11747103C2semidirect
9. #11747082C2semidirect
10. #11747080C2semidirect
11. #11747040C2semidirect
50. #65860D3semidirect
106. #5526D5semidirect
492. C3×C5#3156semidirect
553. C5#83403semidirect
569. C3#10615134semidirect

Maximal subgroups

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

subgroupindex
C5×#55262
C3×#658602
C22×#66612
C2×#913092
C2×#913072
C2×#912972
#117510982
#117471032
#117470822
#117470802
#117470402
#106151343
#834035