Finite Group Databasebeta

← #14323692

Subgroups — #14323692

Order 1920.

1112 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
1C2×C3×C4×C5×Q8C211yesmaximal, characteristic, complemented
2C2×#91835C211yesmaximal, characteristic
3C2×#91761C211yesmaximal, characteristic
4#11751330C211yesmaximal, characteristic, complemented
5#11751091C212yesmaximal
6#11751088C211yesmaximal, characteristic
7#11751077C212yesmaximal, complemented
8#11751073C211yesmaximal, characteristic, complemented
9#11751041C212yesmaximal
10C4×C4⋊Dic15C211yesmaximal, characteristic, complemented
11#11751034C212yesmaximal, complemented
12#10614219—31nomaximal, Hall
13C22×C3×C5×Q8C2211yescharacteristic
14C22×C4×D15C2211yescharacteristic
15C3×C4×C5×Q8—24no—
16C2×C3×C5×C4⋊C4C2211yescharacteristic
17C2×C3×C5×C4⋊C4C2212yes—
18C2×C3×C42×C5C2212yes—
19C2×C3×C42×C5C2211yescharacteristic
20#91835C418yescomplemented
21C2×#6714C2211yescharacteristic
22C2×#6714C2211yescharacteristic
23C2×#6714C2212yes—
24C2×#6714C2212yes—
25C2×C4⋊Dic15C2211yescharacteristic
26C2×C4⋊Dic15C2211yescharacteristic
27C2×#6712C2211yescharacteristic
28C2×#6712C2211yescharacteristic
29C2×#6712C2212yes—
30C2×#6712C2212yes—
31C2×C4×Dic15C2211yescharacteristic
32C2×C4×Dic15C2212yes—
33#91761—24no—
34#91115C2212yes—
35#91115C2212yes—
36#91115C2212yes—
37#91115C2212yes—
38#91115C2212yes—
39#82488—51nomaximal, Hall
40C2×C4×C5×Q8D311yescharacteristic, complemented
41C2×#4092—31no—
42C2×#4018—31no—
43#65519—31no—
44#65279—32no—
45#65276—31no—
46#65262—31no—
47#65261—32no—
48#65229—32no—
49C4×C4⋊Dic5—31no—
50#65222—32no—
51C23×D15C2311yescharacteristic
52C2×C3×C5×Q8—22no—
53C2×C3×C5×Q8—22no—
54C2×C3×C5×Q8C2×C414yes—
55C22×C3×C4×C5C2312yes—
56C22×C3×C4×C5C2312yes—
57C22×C3×C4×C5C2311yescharacteristic
58C22×C3×C4×C5C2311yescharacteristic
59C22×C3×C4×C5C2311yescharacteristic
60C22×Dic15C2311yescharacteristic
61C22×Dic15C2311yescharacteristic
62C22×Dic15C2312yes—
63C22×Dic15C2312yes—
64C22×Dic15C2311yescharacteristic
65C2×C4×D15—22no—
66C2×C4×D15—22no—
67C2×C4×D15C2×C414yes—
68C3×C5×C4⋊C4—24no—
69C3×C5×C4⋊C4—22no—
70C3×C42×C5—24no—
71C3×C42×C5D414yes—
72#6714—22no—
73#6714—22no—
74#6714—42no—
75#6714C2×C418yes—
76C4⋊Dic15—22no—
77C4⋊Dic15C2×C414yes—
78#6712—22no—
79#6712—22no—
80#6712—42no—
81#6712C2×C418yes—
82C4×Dic15—22no—
83C4×Dic15—42no—
84C2×C3×C4×Q8D511yescharacteristic, complemented
85C2×#946—51no—
86C2×#872—51no—
87#5229—51no—
88#4990—52no—
89#4987—51no—
90#4976—52no—
91#4972—51no—
92#4940—52no—
93C4×C4⋊Dic3—51no—
94#4933—52no—
95C22×C5×Q8C2×D311yescharacteristic
96C22×C4×D5—31no—
97C4×C5×Q8—24no—
98C2×C5×C4⋊C4C2×D312yes—
99C2×C5×C4⋊C4C2×D311yescharacteristic
100C2×C42×C5C2×D312yes—
101C2×C42×C5C2×D311yescharacteristic
102#4092—38no—
103C2×#685—32no—
104C2×#685—31no—
105C2×#685—31no—
106C2×#685—32no—
107C2×C4⋊Dic5—31no—
108C2×C4⋊Dic5—31no—
109C2×#683—32no—
110C2×#683—31no—
111C2×#683—31no—
112C2×#683—32no—
113C2×C4×Dic5—31no—
114C2×C4×Dic5—32no—
115#4018—64no—
116#3963—32no—
117#3963—32no—
118#3963—32no—
119#3963—32no—
120#3963—32no—
121#2323—151noSylow
122C23×C3×C5C2411yescharacteristic
123C22×D15—21no—
124C22×D15—21no—
125C22×D15—21no—
126C22×D15—21no—
127C22×D15—21no—
128C22×D15—21no—
129C22×D15C22×C412yes—
130C3×C5×Q8—28no—
131C2×C3×C4×C5—22no—
132C2×C3×C4×C5—22no—
133C2×C3×C4×C5—21no—
134C2×C3×C4×C5—21no—
135C2×C3×C4×C5—21no—
136C2×C3×C4×C5—21no—
137C2×C3×C4×C5—22no—
138C2×C3×C4×C5—22no—
139C2×C3×C4×C5—22no—
140C2×C3×C4×C5C22×C414yes—
141C2×C3×C4×C5C22×C412yes—
142C2×C3×C4×C5C2×D412yes—
143C2×C3×C4×C5C2×D412yes—
144C2×C3×C4×C5C2×D412yes—
145C2×C3×C4×C5#4112yes—
146C2×Dic15—21no—
147C2×Dic15—21no—
148C2×Dic15—22no—
149C2×Dic15—22no—
150C2×Dic15—21no—
151C2×Dic15—21no—
152C2×Dic15—22no—
153C2×Dic15—22no—
154C2×Dic15—21no—
155C2×Dic15—21no—
156C2×Dic15—21no—
157C2×Dic15—21no—
158C2×Dic15—22no—
159C2×Dic15C22×C412yes—
160C2×Dic15C22×C414yes—
161C4×D15—28no—
162C22×C3×Q8C2×D511yescharacteristic
163C22×C4×D3—51no—
164C3×C4×Q8—24no—
165C2×C3×C4⋊C4C2×D511yescharacteristic
166C2×C3×C4⋊C4C2×D512yes—
167C2×C3×C42C2×D512yes—
168C2×C3×C42C2×D511yescharacteristic
169#946—58no—
170C2×(C2×D3)⋊C4—52no—
171C2×(C2×D3)⋊C4—51no—
172C2×(C2×D3)⋊C4—51no—
173C2×(C2×D3)⋊C4—52no—
174C2×C4⋊Dic3—51no—
175C2×C4⋊Dic3—51no—
176C2×Dic3⋊C4—52no—
177C2×Dic3⋊C4—51no—
178C2×Dic3⋊C4—51no—
179C2×Dic3⋊C4—52no—
180C2×C4×Dic3—51no—
181C2×C4×Dic3—52no—
182#872—104no—
183C3⋊#95—52no—
184C3⋊#95—52no—
185C3⋊#95—52no—
186C3⋊#95—52no—
187C3⋊#95—52no—
188C23×D5—31no—
189C2×C5×Q8—22no—
190C2×C5×Q8—22no—
191C2×C5×Q8C4×D314yes—
192C22×C4×C5C22×D312yes—
193C22×C4×C5C22×D312yes—
194C22×C4×C5C22×D311yescharacteristic
195C22×C4×C5C22×D311yescharacteristic
196C22×C4×C5C22×D311yescharacteristic
197C22×Dic5—31no—
198C22×Dic5—32no—
199C22×Dic5—31no—
200C22×Dic5—32no—
201C22×Dic5—31no—
202C2×C4×D5—34no—
203C2×C4×D5—62no—
204C2×C4×D5—62no—
205C5×C4⋊C4—24no—
206C5×C4⋊C4—22no—
207C42×C5—24no—
208C42×C5#6714yes—
209#685—38no—
210#685—62no—
211#685—62no—
212#685—122no—
213C4⋊Dic5—34no—
214C4⋊Dic5—62no—
215#683—38no—
216#683—62no—
217#683—62no—
218#683—122no—
219C4×Dic5—62no—
220C4×Dic5—122no—
221C2×C22⋊Q8—151no—
222C2×C4×Q8D1511yescharacteristic, complemented
223C2×#117—151no—
224#391—151no—
225#389—151no—
226#389—152no—
227#388—152no—
228#388—152no—
229#387—152no—
230#380—151no—
231C4×C4⋊C4—151no—
232C22×C3×C5C23×C411yescharacteristic
233C22×C3×C5C22×D411yescharacteristic
234C22×C3×C5C2×#4111yescharacteristic
235C22×C3×C5H(5,𝔽2)11yescharacteristic
236C22×C3×C52−1+411yescharacteristic
237C22×C3×C52−1+411yescharacteristic
238C22×C3×C52−1+411yescharacteristic
239C2×D15—22no—
240C2×D15—22no—
241C2×D15—22no—
242C2×D15—41no—
243C2×D15—41no—
244C2×D15—41no—
245C2×D15—41no—
246C3×C4×C5—24no—
247C3×C4×C5—22no—
248C3×C4×C5—22no—
249C3×C4×C5—42no—
250C3×C4×C5C4×D414yes—
251Dic15—22no—
252Dic15—24no—
253Dic15—41no—
254Dic15—41no—
255Dic15—42no—
256C23×D3—51no—
257C2×C3×Q8—22no—
258C2×C3×Q8—22no—
259C2×C3×Q8C4×D514yes—
260C22×C3×C4C22×D512yes—
261C22×C3×C4C22×D512yes—
262C22×C3×C4C22×D511yescharacteristic
263C22×C3×C4C22×D511yescharacteristic
264C22×C3×C4C22×D511yescharacteristic
265C22×Dic3—52no—
266C22×Dic3—51no—
267C22×Dic3—51no—
268C22×Dic3—52no—
269C22×Dic3—51no—
270C2×C4×D3—54no—
271C2×C4×D3—102no—
272C2×C4×D3—102no—
273C3×C4⋊C4—24no—
274C3×C4⋊C4—22no—
275C3×C42—24no—
276C3×C42C22⋊D514yes—
277(C2×D3)⋊C4—58no—
278(C2×D3)⋊C4—102no—
279(C2×D3)⋊C4—102no—
280(C2×D3)⋊C4—202no—
281C4⋊Dic3—54no—
282C4⋊Dic3—102no—
283Dic3⋊C4—58no—
284Dic3⋊C4—102no—
285Dic3⋊C4—102no—
286Dic3⋊C4—202no—
287C4×Dic3—102no—
288C4×Dic3—202no—
289C23×C5C23×D311yescharacteristic
290C22×D5—32no—
291C22×D5—61no—
292C22×D5—61no—
293C22×D5—61no—
294C22×D5—61no—
295C22×D5—61no—
296C22×D5—61no—
297C5×Q8—28no—
298C2×C4×C5—22no—
299C2×C4×C5—21no—
300C2×C4×C5—21no—
301C2×C4×C5—21no—
302C2×C4×C5—21no—
303C2×C4×C5—22no—
304C2×C4×C5—22no—
305C2×C4×C5—22no—
306C2×C4×C5—22no—
307C2×C4×C5C2×C4×D314yes—
308C2×C4×C5C2×C4×D312yes—
309C2×C4×C5#23512yes—
310C2×C4×C5C2×#6712yes—
311C2×C4×C5C2×#6712yes—
312C2×C4×C5C2×#6712yes—
313C2×Dic5—32no—
314C2×Dic5—34no—
315C2×Dic5—61no—
316C2×Dic5—62no—
317C2×Dic5—62no—
318C2×Dic5—61no—
319C2×Dic5—62no—
320C2×Dic5—61no—
321C2×Dic5—61no—
322C2×Dic5—62no—
323C2×Dic5—62no—
324C2×Dic5—61no—
325C2×Dic5—61no—
326C2×Dic5—61no—
327C2×Dic5—61no—
328C4×D5—68no—
329C22×Q8C2×D1511yescharacteristic
330C23×C4—151no—
331C22⋊Q8—158no—
332C4×Q8—24no—
333#117—304no—
334C2×C4⋊C4—151no—
335C2×C4⋊C4—152no—
336C2×C4⋊C4—151no—
337C2×C4⋊C4—151no—
338C2×C4⋊C4—151no—
339C2×C4⋊C4—152no—
340C2×C4⋊C4C2×D1511yescharacteristic
341C2×C4⋊C4C2×D1512yes—
342C2×C22⋊C4—152no—
343C2×C22⋊C4—151no—
344C2×C22⋊C4—151no—
345C2×C22⋊C4—152no—
346C2×C42—151no—
347C2×C42—152no—
348C2×C42C2×D1512yes—
349C2×C42C2×D1511yescharacteristic
350#95—152no—
351#95—152no—
352#95—152no—
353#95—152no—
354#95—152no—
355C2×C3×C5C2×C4×D411yescharacteristic
356C2×C3×C5#51911yescharacteristic
357C2×C3×C5#52011yescharacteristic
358C2×C3×C5#53611yescharacteristic
359C2×C3×C5#53711yescharacteristic
360C2×C3×C5#54111yescharacteristic
361C2×C3×C5#54211yescharacteristic
362D15—42no—
363C23×C3C23×D511yescharacteristic
364C22×D3—52no—
365C22×D3—101no—
366C22×D3—101no—
367C22×D3—101no—
368C22×D3—101no—
369C22×D3—101no—
370C22×D3—101no—
371C3×Q8—28no—
372C2×C3×C4—22no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—22no—
380C2×C3×C4—22no—
381C2×C3×C4C2×C4×D514yes—
382C2×C3×C4C2×C4×D512yes—
383C2×C3×C4#70912yes—
384C2×C3×C4C2×C22⋊D512yes—
385C2×C3×C4C2×C22⋊D512yes—
386C2×C3×C4C2×C22⋊D512yes—
387C2×Dic3—52no—
388C2×Dic3—54no—
389C2×Dic3—101no—
390C2×Dic3—102no—
391C2×Dic3—102no—
392C2×Dic3—101no—
393C2×Dic3—102no—
394C2×Dic3—101no—
395C2×Dic3—101no—
396C2×Dic3—102no—
397C2×Dic3—102no—
398C2×Dic3—101no—
399C2×Dic3—101no—
400C2×Dic3—101no—
401C2×Dic3—101no—
402C4×D3—108no—
403C22×C5C22×C4×D311yescharacteristic
404C22×C5C2×#23511yescharacteristic
405C22×C5#100711yescharacteristic
406C22×C5#100711yescharacteristic
407C22×C5#100911yescharacteristic
408C22×C5#101011yescharacteristic
409C22×C5C22×#6711yescharacteristic
410C2×D5—62no—
411C2×D5—62no—
412C2×D5—62no—
413C2×D5—121no—
414C2×D5—121no—
415C2×D5—121no—
416C2×D5—121no—
417C4×C5—24no—
418C4×C5—22no—
419C4×C5—22no—
420C4×C5—42no—
421C4×C5C4×#6714yes—
422Dic5—62no—
423Dic5—64no—
424Dic5—121no—
425Dic5—121no—
426Dic5—122no—
427C24—151no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8C4×D1514yes—
431C22×C4—152no—
432C22×C4—151no—
433C22×C4—154no—
434C22×C4—152no—
435C22×C4—151no—
436C22×C4—151no—
437C22×C4—302no—
438C22×C4—302no—
439C22×C4C22×D1512yes—
440C22×C4C22×D1512yes—
441C22×C4C22×D1511yescharacteristic
442C22×C4C22×D1511yescharacteristic
443C22×C4C22×D1511yescharacteristic
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—158no—
447C4⋊C4—154no—
448C4⋊C4—302no—
449C4⋊C4—302no—
450C4⋊C4—302no—
451C4⋊C4—602no—
452C22⋊C4—158no—
453C22⋊C4—302no—
454C22⋊C4—302no—
455C22⋊C4—602no—
456C42—24no—
457C42—302no—
458C42—602no—
459C42#122014yes—
460C3×C5#232311yescharacteristic, complemented, Hall
461C22×C3C22×C4×D511yescharacteristic
462C22×C3C2×#70911yescharacteristic
463C22×C3#414711yescharacteristic
464C22×C3#414711yescharacteristic
465C22×C3#414911yescharacteristic
466C22×C3#415011yescharacteristic
467C22×C3C22×C22⋊D511yescharacteristic
468C2×D3—102no—
469C2×D3—102no—
470C2×D3—102no—
471C2×D3—201no—
472C2×D3—201no—
473C2×D3—201no—
474C2×D3—201no—
475C3×C4—24no—
476C3×C4—22no—
477C3×C4—22no—
478C3×C4—42no—
479C3×C4C4×C22⋊D514yes—
480Dic3—102no—
481Dic3—104no—
482Dic3—201no—
483Dic3—201no—
484Dic3—202no—
485C2×C5#557211yescharacteristic
486C2×C5#557411yescharacteristic
487C2×C5#558211yescharacteristic
488C2×C5#558311yescharacteristic
489C2×C5C2×C4×#6711yescharacteristic
490C2×C5#581611yescharacteristic
491C2×C5#583311yescharacteristic
492D5—122no—
493C23—152no—
494C23—301no—
495C23—301no—
496C23—301no—
497C23—301no—
498C23—301no—
499C23—301no—
500C23C23×D1511yescharacteristic
501Q8—28no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—154no—
512C2×C4—152no—
513C2×C4—301no—
514C2×C4—302no—
515C2×C4—302no—
516C2×C4—301no—
517C2×C4—308no—
518C2×C4—302no—
519C2×C4—301no—
520C2×C4—301no—
521C2×C4—302no—
522C2×C4—302no—
523C2×C4—301no—
524C2×C4—301no—
525C2×C4—301no—
526C2×C4—301no—
527C2×C4C2×C4×D1514yes—
528C2×C4C2×C4×D1512yes—
529C2×C4#681712yes—
530C2×C4C2×#122012yes—
531C2×C4C2×#122012yes—
532C2×C4C2×#122012yes—
533C2×C3#6590611yescharacteristic
534C2×C3#6590811yescharacteristic
535C2×C3#6591611yescharacteristic
536C2×C3#6591711yescharacteristic
537C2×C3C2×C4×C22⋊D511yescharacteristic
538C2×C3#6615011yescharacteristic
539C2×C3#6616711yescharacteristic
540D3—202no—
541C5#8248811yescharacteristic, complemented, Sylow
542C22—302no—
543C22—302no—
544C22—302no—
545C22—601no—
546C22—601no—
547C22—601no—
548C22—601no—
549C22C22×C4×D1511yescharacteristic
550C22C2×#681711yescharacteristic
551C22#9209811yescharacteristic
552C22#9209811yescharacteristic
553C22#9210011yescharacteristic
554C22#9210111yescharacteristic
555C22C22×#122011yescharacteristic
556C4—24no—
557C4—22no—
558C4—22no—
559C4—42no—
560C4—302no—
561C4—304no—
562C4—601no—
563C4—601no—
564C4—602no—
565C4C4×#122014yes—
566C3#1061421911yescharacteristic, complemented, Sylow
567C2—602no—
568C2#1175633111yescharacteristic
569C2#1175633311yescharacteristic
570C2#1175634111yescharacteristic
571C2#1175634211yescharacteristic
572C2C2×C4×#122011yescharacteristic
573C2#1175657511yescharacteristic
574C2#1175659211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D15
commutatorC22×C3×C5C23×C4
FittingC2×C3×C4×C5×Q8C2
socle, cosocleC23×C3×C5C24
prime radicalC23×D15C23
hypercenterC2×C4×Q8D15
hypocenterC3×C5#2323
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×C5×Q8C2semidirect
4. #11751330C2semidirect
7. #11751077C2semidirect
8. #11751073C2semidirect
10. C4×C4⋊Dic15C2semidirect
11. #11751034C2semidirect
20. #91835C4semidirect
40. C2×C4×C5×Q8D3semidirect
84. C2×C3×C4×Q8D5semidirect
222. C2×C4×Q8D15semidirect
460. C3×C5#2323semidirect
541. C5#82488semidirect
566. C3#10614219semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×C5×Q82
C2×#918352
C2×#917612
#117513302
#117510912
#117510882
#117510772
#117510732
#117510412
C4×C4⋊Dic152
#117510342
#106142193
#824885