Finite Group Databasebeta

← #14328944

Subgroups — #14328944

Order 1920.

1152 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
1C2×C5×#924C211yesmaximal, characteristic
2C2×C3×#4072C211yesmaximal, characteristic
3C2×#91344C211yesmaximal, characteristic, complemented
4C4×#6719C211yesmaximal, characteristic
5#11749479C212yesmaximal
6#11749478C212yesmaximal
7#11748868C211yesmaximal, characteristic, complemented
8#11748699C211yesmaximal, characteristic, complemented
9Dic5×Dic3⋊C4C212yesmaximal, complemented
10#11748607C211yesmaximal, characteristic, complemented
11#11748573C212yesmaximal, complemented
12#10614479—31nomaximal, Hall
13C23×C3×C4×C5C2211yescharacteristic
14C22×C5×Dic6C2211yescharacteristic
15C2×#6719C2211yescharacteristic
16C2×#6719C2211yescharacteristic
17C2×C4×Dic15C2211yescharacteristic
18C2×C4×Dic15C2211yescharacteristic
19C2×C5×#217C2212yes—
20C2×C5×C4⋊Dic3C2211yescharacteristic
21C5×#924C418yescomplemented
22C2×C5×Dic3⋊C4C2212yes—
23C2×C3×C22⋊Dic5C2212yes—
24C3×#4072—24no—
25C2×C3×C4⋊Dic5C2211yescharacteristic
26C2×C3×C4⋊Dic5C2211yescharacteristic
27C2×C3×C4×Dic5C2212yes—
28C2×Dic3⋊Dic5C2212yes—
29C2×Dic3⋊Dic5C2212yes—
30C2×Dic3×Dic5C2212yes—
31C2×Dic3×Dic5C2212yes—
32#91344—24no—
33#91115C2211yescharacteristic
34#91115C2211yescharacteristic
35#90994C2212yes—
36#90994C2212yes—
37#90994C2212yes—
38#90994C2212yes—
39#82243—51nomaximal, Hall
40C2×C5×C22⋊Q8—31no—
41C2×Q8×Dic5—31no—
42C2×#4072D311yescharacteristic
43#65519—31no—
44C4×C22⋊Dic5—31no—
45#65274—31no—
46#65270—32no—
47C4⋊C4×Dic5—31no—
48C4⋊C4×Dic5—32no—
49#65240—32no—
50#65240—32no—
51C24×C3×C5C2311yescharacteristic
52C22×C3×C4×C5—22no—
53C22×C3×C4×C5—22no—
54C22×C3×C4×C5C2×C414yes—
55C22×C3×C4×C5C2311yescharacteristic
56C22×C3×C4×C5C2311yescharacteristic
57C22×Dic15C2311yescharacteristic
58C22×Dic15C2311yescharacteristic
59C22×Dic15C2311yescharacteristic
60C22×Dic15C2311yescharacteristic
61C22×C5×Dic3C2312yes—
62C22×C5×Dic3C2312yes—
63C2×C5×Dic6—22no—
64C2×C5×Dic6—22no—
65C2×C5×Dic6C2×C414yes—
66C22×C3×Dic5C2312yes—
67C22×C3×Dic5C2312yes—
68#6719—22no—
69#6719—22no—
70C4×Dic15—22no—
71C4×Dic15—22no—
72C5×#217C2×C418yes—
73C5×C4⋊Dic3C2×C414yes—
74C5×Dic3⋊C4C2×C418yes—
75C3×C22⋊Dic5—24no—
76C3×C4⋊Dic5—22no—
77C3×C4⋊Dic5D414yes—
78C3×C4×Dic5—24no—
79Dic3⋊Dic5—24no—
80Dic3⋊Dic5—24no—
81Dic3×Dic5—24no—
82Dic3×Dic5—24no—
83C2×C3×#117—51no—
84C2×#924D511yescharacteristic
85C2×#887—51no—
86C4×#217—51no—
87Dic3×C4⋊C4—51no—
88#4973—51no—
89#4972—51no—
90#4945—52no—
91#4944—52no—
92#4933—52no—
93C4×Dic3⋊C4—52no—
94C22×C5×Q8—31no—
95C23×C4×C5C2×D311yescharacteristic
96C5×C22⋊Q8—38no—
97C2×C5×C4⋊C4—32no—
98C2×C5×C4⋊C4—31no—
99C2×C5×C22⋊C4—32no—
100C2×C22⋊Dic5—31no—
101C2×C22⋊Dic5—31no—
102C2×C22⋊Dic5C2×D312yes—
103Q8×Dic5—64no—
104#4072—24no—
105C2×C4⋊Dic5—32no—
106C2×C4⋊Dic5—32no—
107C2×C4⋊Dic5C2×D311yescharacteristic
108C2×C4⋊Dic5C2×D311yescharacteristic
109C2×C4×Dic5—31no—
110C2×C4×Dic5—31no—
111C2×C4×Dic5—32no—
112C2×C4×Dic5—32no—
113C2×C4×Dic5C2×D312yes—
114#3963—31no—
115#3963—31no—
116#3963—32no—
117#3963—32no—
118#3963—32no—
119#3963—32no—
120#2362—151noSylow
121C23×C3×C5—21no—
122C23×C3×C5—21no—
123C23×C3×C5—21no—
124C23×C3×C5—21no—
125C23×C3×C5—21no—
126C23×C3×C5—21no—
127C23×C3×C5C22×C412yes—
128C23×C3×C5C2411yescharacteristic
129C2×C3×C4×C5—21no—
130C2×C3×C4×C5—21no—
131C2×C3×C4×C5—28no—
132C2×C3×C4×C5C22×C412yes—
133C2×C3×C4×C5C22×C412yes—
134C2×C3×C4×C5C2×D412yes—
135C2×C3×C4×C5#4112yes—
136C2×Dic15—21no—
137C2×Dic15—21no—
138C2×Dic15—21no—
139C2×Dic15—21no—
140C2×Dic15—21no—
141C2×Dic15—21no—
142C2×Dic15—21no—
143C2×Dic15—21no—
144C2×Dic15—21no—
145C2×Dic15—21no—
146C2×Dic15#4112yes—
147C2×Dic15#4112yes—
148C2×C5×Dic3—22no—
149C2×C5×Dic3—22no—
150C2×C5×Dic3—22no—
151C2×C5×Dic3—22no—
152C2×C5×Dic3C22×C414yes—
153C2×C5×Dic3C22×C414yes—
154C5×Dic6—28no—
155C2×C3×Dic5—22no—
156C2×C3×Dic5—22no—
157C2×C3×Dic5—22no—
158C2×C3×Dic5—22no—
159C2×C3×Dic5—22no—
160C2×C3×Dic5C2×D414yes—
161C23×C3×C4C2×D511yescharacteristic
162C22×Dic6C2×D511yescharacteristic
163C3×#117—104no—
164C2×C3×C4⋊C4—51no—
165C2×C3×C4⋊C4—51no—
166C2×C3×C22⋊C4—52no—
167C2×C3×C42—52no—
168C2×#217—51no—
169C2×#217—51no—
170C2×#217C2×D512yes—
171C2×C4⋊Dic3C2×D511yescharacteristic
172#924Dic518yescomplemented
173C2×Dic3⋊C4—52no—
174C2×Dic3⋊C4—52no—
175C2×Dic3⋊C4C2×D512yes—
176C2×C4×Dic3—51no—
177C2×C4×Dic3—51no—
178C2×C4×Dic3—52no—
179C2×C4×Dic3—52no—
180#887—104no—
181C3⋊#95—51no—
182C3⋊#95—51no—
183C3⋊#95—52no—
184C3⋊#95—52no—
185C3⋊#95—52no—
186C3⋊#95—52no—
187C24×C5C22×D311yescharacteristic
188C2×C5×Q8—34no—
189C2×C5×Q8—62no—
190C2×C5×Q8—62no—
191C22×C4×C5—22no—
192C22×C4×C5—22no—
193C22×C4×C5—32no—
194C22×C4×C5—32no—
195C22×C4×C5C4×D314yes—
196C22×C4×C5C22×D311yescharacteristic
197C22×C4×C5C22×D311yescharacteristic
198C22×Dic5—31no—
199C22×Dic5—31no—
200C22×Dic5—31no—
201C22×Dic5—31no—
202C22×Dic5C22×D312yes—
203C22×Dic5C22×D312yes—
204C5×C4⋊C4—38no—
205C5×C4⋊C4—34no—
206C5×C22⋊C4—38no—
207C22⋊Dic5—24no—
208C22⋊Dic5—62no—
209C22⋊Dic5—62no—
210C4⋊Dic5—22no—
211C4⋊Dic5—64no—
212C4⋊Dic5—64no—
213C4⋊Dic5#6714yes—
214C4×Dic5—24no—
215C4×Dic5—64no—
216C4×Dic5—62no—
217C4×Dic5—62no—
218C4×Dic5—64no—
219C2×C22⋊Q8—31no—
220C2×C4×Q8—151no—
221C2×#117—51no—
222#391—151no—
223#389—151no—
224#388—152no—
225#388—152no—
226#387—152no—
227C4×C4⋊C4—152no—
228C4×C4⋊C4—151no—
229C4×C22⋊C4—151no—
230C22×C3×C5—22no—
231C22×C3×C5—22no—
232C22×C3×C5—22no—
233C22×C3×C5—41no—
234C22×C3×C5—41no—
235C22×C3×C5—41no—
236C22×C3×C5—41no—
237C22×C3×C5C23×C411yescharacteristic
238C22×C3×C5C22×D411yescharacteristic
239C22×C3×C5C2×#4111yescharacteristic
240C22×C3×C5C2×#4111yescharacteristic
241C22×C3×C5C2×#4111yescharacteristic
242C22×C3×C52−1+411yescharacteristic
243C22×C3×C52−1+411yescharacteristic
244C3×C4×C5—22no—
245C3×C4×C5C4×D414yes—
246Dic15—22no—
247Dic15—22no—
248Dic15—41no—
249Dic15—41no—
250C3×Dic5—24no—
251C3×Dic5—42no—
252C5×Dic3—24no—
253C5×Dic3—24no—
254C24×C3C22×D511yescharacteristic
255C22×C3×C4—22no—
256C22×C3×C4—22no—
257C22×C3×C4—52no—
258C22×C3×C4—52no—
259C22×C3×C4C2×Dic514yes—
260C22×C3×C4C22×D511yescharacteristic
261C22×C3×C4C22×D511yescharacteristic
262C22×Dic3—51no—
263C22×Dic3—51no—
264C22×Dic3—51no—
265C22×Dic3—51no—
266C22×Dic3C22×D512yes—
267C22×Dic3C22×D512yes—
268C2×Dic6—22no—
269C2×Dic6—22no—
270C2×Dic6C2×Dic514yes—
271C3×C4⋊C4—54no—
272C3×C4⋊C4—102no—
273C3×C22⋊C4—104no—
274C3×C42—104no—
275#217—102no—
276#217—102no—
277#217C2×Dic518yes—
278C4⋊Dic3C2×Dic514yes—
279Dic3⋊C4—104no—
280Dic3⋊C4—104no—
281Dic3⋊C4C2×Dic518yes—
282C4×Dic3—104no—
283C4×Dic3—104no—
284C4×Dic3—102no—
285C4×Dic3—102no—
286C23×C5—21no—
287C23×C5—21no—
288C23×C5—21no—
289C23×C5—21no—
290C23×C5—21no—
291C23×C5—21no—
292C23×C5C2×C4×D312yes—
293C23×C5C23×D311yescharacteristic
294C5×Q8—68no—
295C2×C4×C5—21no—
296C2×C4×C5—21no—
297C2×C4×C5—28no—
298C2×C4×C5—34no—
299C2×C4×C5—34no—
300C2×C4×C5—62no—
301C2×C4×C5—62no—
302C2×C4×C5—62no—
303C2×C4×C5—62no—
304C2×C4×C5C2×C4×D312yes—
305C2×C4×C5C2×C4×D312yes—
306C2×C4×C5#23512yes—
307C2×C4×C5C2×#6712yes—
308C2×Dic5—22no—
309C2×Dic5—22no—
310C2×Dic5—22no—
311C2×Dic5—22no—
312C2×Dic5—22no—
313C2×Dic5—32no—
314C2×Dic5—32no—
315C2×Dic5—61no—
316C2×Dic5—61no—
317C2×Dic5—61no—
318C2×Dic5—61no—
319C2×Dic5—61no—
320C2×Dic5—61no—
321C2×Dic5—61no—
322C2×Dic5—61no—
323C2×Dic5—61no—
324C2×Dic5—61no—
325C2×Dic5C2×#6714yes—
326C22×Q8—31no—
327C23×C4D3×D511yescharacteristic
328C22⋊Q8—38no—
329C4×Q8—304no—
330#117—104no—
331C2×C4⋊C4—32no—
332C2×C4⋊C4—31no—
333C2×C4⋊C4—51no—
334C2×C4⋊C4—51no—
335C2×C4⋊C4—152no—
336C2×C4⋊C4—152no—
337C2×C22⋊C4—32no—
338C2×C22⋊C4—52no—
339C2×C22⋊C4—151no—
340C2×C22⋊C4—151no—
341C2×C42—52no—
342C2×C42—152no—
343C2×C42—152no—
344C2×C42—151no—
345C2×C42—151no—
346#95—152no—
347#95—152no—
348#95—152no—
349#95—152no—
350#95—151no—
351#95—151no—
352C2×C3×C5—42no—
353C2×C3×C5C2×C4×D411yescharacteristic
354C2×C3×C5C4×#4111yescharacteristic
355C2×C3×C5#51911yescharacteristic
356C2×C3×C5#54711yescharacteristic
357C2×C3×C5#54811yescharacteristic
358C2×C3×C5#55211yescharacteristic
359C2×C3×C5#55611yescharacteristic
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—21no—
363C23×C3—21no—
364C23×C3—21no—
365C23×C3—21no—
366C23×C3C22×Dic512yes—
367C23×C3C23×D511yescharacteristic
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—28no—
371C2×C3×C4—54no—
372C2×C3×C4—102no—
373C2×C3×C4—102no—
374C2×C3×C4—102no—
375C2×C3×C4—102no—
376C2×C3×C4—102no—
377C2×C3×C4D4×D512yes—
378C2×C3×C4#71112yes—
379C2×C3×C4C22×Dic512yes—
380C2×C3×C4C22×Dic512yes—
381C2×Dic3—22no—
382C2×Dic3—22no—
383C2×Dic3—22no—
384C2×Dic3—22no—
385C2×Dic3—52no—
386C2×Dic3—52no—
387C2×Dic3—101no—
388C2×Dic3—101no—
389C2×Dic3—101no—
390C2×Dic3—101no—
391C2×Dic3—101no—
392C2×Dic3—101no—
393C2×Dic3—101no—
394C2×Dic3—101no—
395C2×Dic3—101no—
396C2×Dic3—101no—
397C2×Dic3C22×Dic514yes—
398C2×Dic3C22×Dic514yes—
399Dic6—28no—
400C22×C5—22no—
401C22×C5—22no—
402C22×C5—22no—
403C22×C5—41no—
404C22×C5—41no—
405C22×C5—41no—
406C22×C5—41no—
407C22×C5C22×C4×D311yescharacteristic
408C22×C5C2×#23511yescharacteristic
409C22×C5D3×#4111yescharacteristic
410C22×C5D3×#4111yescharacteristic
411C22×C5#101011yescharacteristic
412C22×C5#101011yescharacteristic
413C22×C5C22×#6711yescharacteristic
414C4×C5—22no—
415C4×C5—64no—
416C4×C5—64no—
417C4×C5C4×#6714yes—
418Dic5—24no—
419Dic5—42no—
420Dic5—62no—
421Dic5—62no—
422Dic5—121no—
423Dic5—121no—
424C24C2×D3×D511yescharacteristic
425C2×Q8—34no—
426C2×Q8—62no—
427C2×Q8—62no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—32no—
431C22×C4—32no—
432C22×C4—52no—
433C22×C4—52no—
434C22×C4—151no—
435C22×C4—151no—
436C22×C4—151no—
437C22×C4—151no—
438C22×C4D3×Dic514yes—
439C22×C4C2×D3×D511yescharacteristic
440C22×C4C2×D3×D511yescharacteristic
441C4⋊C4—38no—
442C4⋊C4—34no—
443C4⋊C4—54no—
444C4⋊C4—102no—
445C4⋊C4—304no—
446C4⋊C4—304no—
447C22⋊C4—38no—
448C22⋊C4—104no—
449C22⋊C4—302no—
450C22⋊C4—302no—
451C42—104no—
452C42—304no—
453C42—304no—
454C42—302no—
455C42—302no—
456C3×C5#236211yescharacteristic, complemented, Hall
457C22×C3—22no—
458C22×C3—22no—
459C22×C3—22no—
460C22×C3—41no—
461C22×C3—41no—
462C22×C3—41no—
463C22×C3—41no—
464C22×C3C2×D4×D511yescharacteristic
465C22×C3C2×#71111yescharacteristic
466C22×C3#414711yescharacteristic
467C22×C3#414711yescharacteristic
468C22×C3D5×#4111yescharacteristic
469C22×C3D5×#4111yescharacteristic
470C22×C3C23×Dic511yescharacteristic
471C3×C4—22no—
472C3×C4—104no—
473C3×C4—202no—
474C3×C4D4×Dic514yes—
475Dic3—24no—
476Dic3—24no—
477Dic3—102no—
478Dic3—102no—
479Dic3—201no—
480Dic3—201no—
481C2×C5—42no—
482C2×C5#551611yescharacteristic
483C2×C5#552211yescharacteristic
484C2×C5#552911yescharacteristic
485C2×C5#553311yescharacteristic
486C2×C5C2×C4×#6711yescharacteristic
487C2×C5#582411yescharacteristic
488C2×C5#583111yescharacteristic
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23C2×D3×Dic512yes—
496C23C22×D3×D511yescharacteristic
497Q8—68no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—28no—
501C2×C4—34no—
502C2×C4—34no—
503C2×C4—54no—
504C2×C4—62no—
505C2×C4—62no—
506C2×C4—62no—
507C2×C4—62no—
508C2×C4—102no—
509C2×C4—102no—
510C2×C4—102no—
511C2×C4—102no—
512C2×C4—102no—
513C2×C4—152no—
514C2×C4—152no—
515C2×C4—301no—
516C2×C4—301no—
517C2×C4—301no—
518C2×C4—301no—
519C2×C4—301no—
520C2×C4—301no—
521C2×C4—301no—
522C2×C4—301no—
523C2×C4—301no—
524C2×C4—301no—
525C2×C4C2×D3×Dic512yes—
526C2×C4C2×D3×Dic512yes—
527C2×C4#678212yes—
528C2×C4D5×#6712yes—
529C2×C3—42no—
530C2×C3#6596511yescharacteristic
531C2×C3#6596711yescharacteristic
532C2×C3#6597311yescharacteristic
533C2×C3#6597511yescharacteristic
534C2×C3C2×D4×Dic511yescharacteristic
535C2×C3#6614611yescharacteristic
536C2×C3#41×Dic511yescharacteristic
537C5#8224311yescharacteristic, complemented, Sylow
538C22—22no—
539C22—22no—
540C22—22no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22#9200011yescharacteristic
546C22#9200011yescharacteristic
547C22#9201611yescharacteristic
548C22#9201611yescharacteristic
549C22C22×D3×Dic511yescharacteristic
550C22C2×#678211yescharacteristic
551C22C2×D5×#6711yescharacteristic
552C4—22no—
553C4—64no—
554C4—64no—
555C4—104no—
556C4—202no—
557C4—302no—
558C4—302no—
559C4—601no—
560C4—601no—
561C4Dic5×#6714yes—
562C3#1061447911yescharacteristic, complemented, Sylow
563C2—42no—
564C2#1175417711yescharacteristic
565C2#1175421111yescharacteristic
566C2#1175421511yescharacteristic
567C2#1175430911yescharacteristic
568C2#1175445511yescharacteristic
569C2#1175446311yescharacteristic
570C2C2×Dic5×#6711yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC23×C3×C4×C5C22
socle, cosocleC23×C3×C5C24
prime radicalC24×C3×C5C23
hypercenterC23×C4D3×D5
hypocenterC3×C5#2362
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
3. C2×#91344C2semidirect
7. #11748868C2semidirect
8. #11748699C2semidirect
9. Dic5×Dic3⋊C4C2semidirect
10. #11748607C2semidirect
11. #11748573C2semidirect
21. C5×#924C4semidirect
172. #924Dic5semidirect
456. C3×C5#2362semidirect
537. C5#82243semidirect
562. C3#10614479semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#9242
C2×C3×#40722
C2×#913442
C4×#67192
#117494792
#117494782
#117488682
#117486992
Dic5×Dic3⋊C42
#117486072
#117485732
#106144793
#822435