Finite Group Databasebeta

← #14320813

Subgroups — #14320813

Order 1920.

1304 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91782C211yesmaximal, characteristic
2C2×#91781C212yesmaximal, complemented
3C3×C5×#389C211yesmaximal, characteristic, complemented
4#11751091C212yesmaximal, complemented
5#11751089C211yesmaximal, characteristic
6#11751085C211yesmaximal, characteristic, complemented
7C4⋊C4×Dic15C211yesmaximal, characteristic, complemented
8#11751040C211yesmaximal, characteristic
9C42×Dic15C211yesmaximal, characteristic, complemented
10#11750771C212yesmaximal, complemented
11#11750767C212yesmaximal, complemented
12#10607692—31nomaximal, Hall
13C22×D60C2211yescharacteristic
14C22×C4×D15C2212yes—
15C2×C3×C5×C4⋊C4C2211yescharacteristic
16C2×C3×C5×C4⋊C4C2212yes—
17C2×C3×C5×C4⋊C4C2211yescharacteristic
18C2×C3×C42×C5C2211yescharacteristic
19C2×#6714C2212yes—
20C2×#6714C2212yes—
21C2×#6714C2212yes—
22C2×#6714C2212yes—
23C2×#6714C2212yes—
24C2×C4⋊Dic15C2212yes—
25C2×C4⋊Dic15C2211yescharacteristic
26C2×C4×Dic15C2211yescharacteristic
27C2×C4×Dic15C2212yes—
28C2×C4×Dic15C2211yescharacteristic
29C2×C4×Dic15C2211yescharacteristic
30C2×C4×Dic15C2211yescharacteristic
31C2×C4×Dic15C2211yescharacteristic
32C2×C4×Dic15C2211yescharacteristic
33#91782C418yescomplemented
34#91781—28no—
35C3×C5×#95C2212yes—
36#91115C2212yes—
37#91115C2212yes—
38#75961—51nomaximal, Hall
39C2×#4039—31no—
40C2×#4038—32no—
41C5×#389D311yescharacteristic, complemented
42#65279—32no—
43#65277—31no—
44#65274—31no—
45C4⋊C4×Dic5—31no—
46#65228—31no—
47C42×Dic5—31no—
48#64959—32no—
49#64955—32no—
50C23×D15C2312yes—
51C22×C3×C4×C5C2312yes—
52C22×C3×C4×C5C2312yes—
53C22×C3×C4×C5C2311yescharacteristic
54C22×C3×C4×C5C2311yescharacteristic
55C22×C3×C4×C5C2311yescharacteristic
56C22×Dic15C2312yes—
57C22×Dic15C2311yescharacteristic
58C22×Dic15C2311yescharacteristic
59C22×Dic15C2311yescharacteristic
60C22×Dic15C2311yescharacteristic
61C2×D60—22no—
62C2×D60—22no—
63C2×D60C2×C414yes—
64C2×C4×D15—24no—
65C2×C4×D15—24no—
66C2×C4×D15C2×C418yes—
67C3×C5×C4⋊C4—24no—
68C3×C5×C4⋊C4—22no—
69C3×C5×C4⋊C4C2×C414yes—
70C3×C42×C5—22no—
71#6714—24no—
72#6714—24no—
73#6714—24no—
74#6714—24no—
75#6714C2×C418yes—
76C4⋊Dic15—22no—
77C4⋊Dic15—24no—
78C4×Dic15—22no—
79C4×Dic15—22no—
80C4×Dic15—24no—
81C4×Dic15—22no—
82C4×Dic15—22no—
83C4×Dic15C2×C414yes—
84C4×Dic15D414yescomplemented
85C2×#893—51no—
86C2×#892—52no—
87C3×#389D511yescharacteristic, complemented
88#4990—52no—
89#4988—51no—
90#4984—51no—
91Dic3×C4⋊C4—51no—
92#4939—51no—
93C42×Dic3—51no—
94#4670—52no—
95#4666—52no—
96C22×D20—31no—
97C22×C4×D5—32no—
98C2×C5×C4⋊C4C2×D312yes—
99C2×C5×C4⋊C4C2×D311yescharacteristic
100C2×C5×C4⋊C4C2×D311yescharacteristic
101C2×C42×C5C2×D311yescharacteristic
102C2×#685—32no—
103C2×#685—32no—
104C2×#685—32no—
105C2×#685—32no—
106C2×#685—32no—
107C2×C4⋊Dic5—32no—
108C2×C4⋊Dic5—31no—
109C2×C4×Dic5—31no—
110C2×C4×Dic5—31no—
111C2×C4×Dic5—32no—
112C2×C4×Dic5—31no—
113C2×C4×Dic5—31no—
114C2×C4×Dic5—31no—
115C2×C4×Dic5—31no—
116#4039—38no—
117#4038—68no—
118C5×#95C2×D312yes—
119#3963—32no—
120#3963—32no—
121#2293—151noSylow
122C23×C3×C5C2411yescharacteristic
123C22×D15—22no—
124C22×D15—22no—
125C22×D15—22no—
126C22×D15—22no—
127C22×D15—22no—
128C22×D15—22no—
129C22×D15C22×C414yes—
130C2×C3×C4×C5—22no—
131C2×C3×C4×C5—21no—
132C2×C3×C4×C5—21no—
133C2×C3×C4×C5—21no—
134C2×C3×C4×C5—21no—
135C2×C3×C4×C5—22no—
136C2×C3×C4×C5—22no—
137C2×C3×C4×C5—22no—
138C2×C3×C4×C5—22no—
139C2×C3×C4×C5C22×C414yes—
140C2×C3×C4×C5C22×C412yes—
141C2×C3×C4×C5C2×D412yes—
142C2×C3×C4×C5#4112yes—
143C2×C3×C4×C5#4112yes—
144C2×C3×C4×C5#4112yes—
145C2×Dic15—21no—
146C2×Dic15—21no—
147C2×Dic15—21no—
148C2×Dic15—21no—
149C2×Dic15—21no—
150C2×Dic15—21no—
151C2×Dic15—22no—
152C2×Dic15—22no—
153C2×Dic15—22no—
154C2×Dic15C22×C412yes—
155C2×Dic15C22×C412yes—
156C2×Dic15C2×D412yes—
157C2×Dic15C2×D412yes—
158C2×Dic15#4112yes—
159C2×Dic15#4112yes—
160D60—28no—
161C4×D15—216no—
162C22×D12—51no—
163C22×C4×D3—52no—
164C2×C3×C4⋊C4C2×D511yescharacteristic
165C2×C3×C4⋊C4C2×D512yes—
166C2×C3×C4⋊C4C2×D511yescharacteristic
167C2×C3×C42C2×D511yescharacteristic
168C2×(C2×D3)⋊C4—52no—
169C2×(C2×D3)⋊C4—52no—
170C2×(C2×D3)⋊C4—52no—
171C2×(C2×D3)⋊C4—52no—
172C2×(C2×D3)⋊C4—52no—
173C2×C4⋊Dic3—52no—
174C2×C4⋊Dic3—51no—
175C2×C4×Dic3—51no—
176C2×C4×Dic3—51no—
177C2×C4×Dic3—51no—
178C2×C4×Dic3—52no—
179C2×C4×Dic3—51no—
180C2×C4×Dic3—51no—
181C2×C4×Dic3—51no—
182#893—58no—
183#892—108no—
184C3×#95C2×D512yes—
185C3⋊#95—52no—
186C3⋊#95—52no—
187C23×D5—32no—
188C22×C4×C5C22×D312yes—
189C22×C4×C5C22×D312yes—
190C22×C4×C5C22×D311yescharacteristic
191C22×C4×C5C22×D311yescharacteristic
192C22×C4×C5C22×D311yescharacteristic
193C22×Dic5—31no—
194C22×Dic5—31no—
195C22×Dic5—31no—
196C22×Dic5—31no—
197C22×Dic5—32no—
198C2×D20—34no—
199C2×D20—62no—
200C2×D20—62no—
201C2×C4×D5—38no—
202C2×C4×D5—64no—
203C2×C4×D5—64no—
204C5×C4⋊C4—24no—
205C5×C4⋊C4—22no—
206C5×C4⋊C4C4×D314yes—
207C42×C5—22no—
208#685—38no—
209#685—64no—
210#685—64no—
211#685—64no—
212#685—64no—
213C4⋊Dic5—64no—
214C4⋊Dic5—62no—
215C4×Dic5—34no—
216C4×Dic5—34no—
217C4×Dic5—64no—
218C4×Dic5—62no—
219C4×Dic5—62no—
220C4×Dic5—62no—
221C4×Dic5—62no—
222C2×C4×D4—151no—
223C2×#117—152no—
224#390—151no—
225#390—151no—
226#389—151no—
227#389D1511yescharacteristic, complemented
228#388—152no—
229#388—152no—
230C4×C4⋊C4—151no—
231C4×C22⋊C4—152no—
232C43—151no—
233C22×C3×C5C23×C411yescharacteristic
234C22×C3×C5C22×D411yescharacteristic
235C22×C3×C5C2×#4111yescharacteristic
236C22×C3×C5C2×#4111yescharacteristic
237C22×C3×C5C2×#4111yescharacteristic
238C22×C3×C5C2×#4111yescharacteristic
239C22×C3×C5C2×#4111yescharacteristic
240C2×D15—24no—
241C2×D15—24no—
242C2×D15—24no—
243C2×D15—42no—
244C2×D15—42no—
245C2×D15—42no—
246C2×D15—42no—
247C3×C4×C5—24no—
248C3×C4×C5—22no—
249C3×C4×C5—22no—
250C3×C4×C5—42no—
251C3×C4×C5#11714yescomplemented
252Dic15—22no—
253Dic15—22no—
254Dic15—22no—
255Dic15—42no—
256Dic15C4×D414yes—
257C23×D3—52no—
258C22×C3×C4C22×D512yes—
259C22×C3×C4C22×D512yes—
260C22×C3×C4C22×D511yescharacteristic
261C22×C3×C4C22×D511yescharacteristic
262C22×C3×C4C22×D511yescharacteristic
263C22×Dic3—51no—
264C22×Dic3—51no—
265C22×Dic3—51no—
266C22×Dic3—51no—
267C22×Dic3—52no—
268C2×D12—54no—
269C2×D12—102no—
270C2×D12—102no—
271C2×C4×D3—58no—
272C2×C4×D3—104no—
273C2×C4×D3—104no—
274C3×C4⋊C4—24no—
275C3×C4⋊C4—22no—
276C3×C4⋊C4C4×D514yes—
277C3×C42—22no—
278(C2×D3)⋊C4—58no—
279(C2×D3)⋊C4—104no—
280(C2×D3)⋊C4—104no—
281(C2×D3)⋊C4—104no—
282(C2×D3)⋊C4—104no—
283C4⋊Dic3—104no—
284C4⋊Dic3—102no—
285C4×Dic3—54no—
286C4×Dic3—54no—
287C4×Dic3—104no—
288C4×Dic3—102no—
289C4×Dic3—102no—
290C4×Dic3—102no—
291C4×Dic3—102no—
292C23×C5C23×D311yescharacteristic
293C22×D5—34no—
294C22×D5—62no—
295C22×D5—62no—
296C22×D5—62no—
297C22×D5—62no—
298C22×D5—62no—
299C22×D5—62no—
300C2×C4×C5—22no—
301C2×C4×C5—21no—
302C2×C4×C5—21no—
303C2×C4×C5—21no—
304C2×C4×C5—21no—
305C2×C4×C5—22no—
306C2×C4×C5—22no—
307C2×C4×C5—22no—
308C2×C4×C5—22no—
309C2×C4×C5C2×C4×D314yes—
310C2×C4×C5C2×C4×D312yes—
311C2×C4×C5D3×D412yes—
312C2×C4×C5#23712yes—
313C2×C4×C5#23912yes—
314C2×C4×C5#23912yes—
315C2×Dic5—32no—
316C2×Dic5—32no—
317C2×Dic5—32no—
318C2×Dic5—32no—
319C2×Dic5—32no—
320C2×Dic5—32no—
321C2×Dic5—61no—
322C2×Dic5—61no—
323C2×Dic5—61no—
324C2×Dic5—61no—
325C2×Dic5—61no—
326C2×Dic5—61no—
327C2×Dic5—62no—
328C2×Dic5—62no—
329C2×Dic5—62no—
330D20—68no—
331C4×D5—616no—
332C22×D4—151no—
333C23×C4—152no—
334C4×D4—158no—
335#117—308no—
336C2×C4⋊C4—152no—
337C2×C4⋊C4—151no—
338C2×C4⋊C4C2×D1512yes—
339C2×C4⋊C4C2×D1511yescharacteristic
340C2×C4⋊C4C2×D1511yescharacteristic
341C2×C22⋊C4—152no—
342C2×C22⋊C4—152no—
343C2×C22⋊C4—152no—
344C2×C22⋊C4—152no—
345C2×C22⋊C4—152no—
346C2×C42—151no—
347C2×C42—151no—
348C2×C42—152no—
349C2×C42—151no—
350C2×C42—151no—
351C2×C42—151no—
352C2×C42—151no—
353C2×C42C2×D1511yescharacteristic
354#95—152no—
355#95—152no—
356#95C2×D1512yes—
357C2×C3×C5C2×#11711yescharacteristic
358C2×C3×C5C2×C4×D411yescharacteristic
359C2×C3×C5C4×#4111yescharacteristic
360C2×C3×C5#52911yescharacteristic
361C2×C3×C5#52911yescharacteristic
362C2×C3×C5#53211yescharacteristic
363C2×C3×C5#53211yescharacteristic
364D15—44no—
365C23×C3C23×D511yescharacteristic
366C22×D3—54no—
367C22×D3—102no—
368C22×D3—102no—
369C22×D3—102no—
370C22×D3—102no—
371C22×D3—102no—
372C22×D3—102no—
373C2×C3×C4—22no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—22no—
379C2×C3×C4—22no—
380C2×C3×C4—22no—
381C2×C3×C4—22no—
382C2×C3×C4C2×C4×D514yes—
383C2×C3×C4C2×C4×D512yes—
384C2×C3×C4D4×D512yes—
385C2×C3×C4#71112yes—
386C2×C3×C4#71312yes—
387C2×C3×C4#71312yes—
388C2×Dic3—52no—
389C2×Dic3—52no—
390C2×Dic3—52no—
391C2×Dic3—52no—
392C2×Dic3—52no—
393C2×Dic3—52no—
394C2×Dic3—101no—
395C2×Dic3—101no—
396C2×Dic3—101no—
397C2×Dic3—101no—
398C2×Dic3—101no—
399C2×Dic3—101no—
400C2×Dic3—102no—
401C2×Dic3—102no—
402C2×Dic3—102no—
403D12—108no—
404C4×D3—1016no—
405C22×C5C22×C4×D311yescharacteristic
406C22×C5C2×D3×D411yescharacteristic
407C22×C5C2×#23711yescharacteristic
408C22×C5C2×#23911yescharacteristic
409C22×C5C2×#23911yescharacteristic
410C22×C5D3×#4111yescharacteristic
411C22×C5D3×#4111yescharacteristic
412C2×D5—64no—
413C2×D5—64no—
414C2×D5—64no—
415C2×D5—122no—
416C2×D5—122no—
417C2×D5—122no—
418C2×D5—122no—
419C4×C5—24no—
420C4×C5—22no—
421C4×C5—22no—
422C4×C5—42no—
423C4×C5#89214yescomplemented
424Dic5—34no—
425Dic5—62no—
426Dic5—62no—
427Dic5—62no—
428Dic5—122no—
429C24—152no—
430C2×D4—154no—
431C2×D4—302no—
432C2×D4—302no—
433C22×C4—151no—
434C22×C4—151no—
435C22×C4—151no—
436C22×C4—151no—
437C22×C4—158no—
438C22×C4—152no—
439C22×C4—304no—
440C22×C4—304no—
441C22×C4C22×D1512yes—
442C22×C4C22×D1512yes—
443C22×C4C22×D1511yescharacteristic
444C22×C4C22×D1511yescharacteristic
445C22×C4C22×D1511yescharacteristic
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—304no—
449C4⋊C4—302no—
450C4⋊C4C4×D1514yes—
451C22⋊C4—158no—
452C22⋊C4—304no—
453C22⋊C4—304no—
454C22⋊C4—304no—
455C22⋊C4—304no—
456C42—22no—
457C42—154no—
458C42—154no—
459C42—304no—
460C42—302no—
461C42—302no—
462C42—302no—
463C42—302no—
464C3×C5#229311yescharacteristic, complemented, Hall
465C22×C3C22×C4×D511yescharacteristic
466C22×C3C2×D4×D511yescharacteristic
467C22×C3C2×#71111yescharacteristic
468C22×C3C2×#71311yescharacteristic
469C22×C3C2×#71311yescharacteristic
470C22×C3D5×#4111yescharacteristic
471C22×C3D5×#4111yescharacteristic
472C2×D3—104no—
473C2×D3—104no—
474C2×D3—104no—
475C2×D3—202no—
476C2×D3—202no—
477C2×D3—202no—
478C2×D3—202no—
479C3×C4—24no—
480C3×C4—22no—
481C3×C4—22no—
482C3×C4—42no—
483C3×C4#403814yescomplemented
484Dic3—54no—
485Dic3—102no—
486Dic3—102no—
487Dic3—102no—
488Dic3—202no—
489C2×C5C2×#89211yescharacteristic
490C2×C5C4×D3×D411yescharacteristic
491C2×C5C4×#23911yescharacteristic
492C2×C5#559611yescharacteristic
493C2×C5#562411yescharacteristic
494C2×C5#569111yescharacteristic
495C2×C5#572511yescharacteristic
496D5—124no—
497C23—154no—
498C23—302no—
499C23—302no—
500C23—302no—
501C23—302no—
502C23—302no—
503C23—302no—
504C23C23×D1511yescharacteristic
505D4—308no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—152no—
516C2×C4—152no—
517C2×C4—152no—
518C2×C4—152no—
519C2×C4—152no—
520C2×C4—152no—
521C2×C4—3016no—
522C2×C4—301no—
523C2×C4—301no—
524C2×C4—301no—
525C2×C4—301no—
526C2×C4—301no—
527C2×C4—301no—
528C2×C4—302no—
529C2×C4—302no—
530C2×C4—302no—
531C2×C4C2×C4×D1514yes—
532C2×C4C2×C4×D1512yes—
533C2×C4D4×D1512yes—
534C2×C4#681912yes—
535C2×C4#682112yes—
536C2×C4#682112yes—
537C2×C3C2×#403811yescharacteristic
538C2×C3C4×D4×D511yescharacteristic
539C2×C3C4×#71311yescharacteristic
540C2×C3#6593011yescharacteristic
541C2×C3#6595811yescharacteristic
542C2×C3#6602511yescharacteristic
543C2×C3#6605911yescharacteristic
544D3—204no—
545C5#7596111yescharacteristic, complemented, Sylow
546C22—304no—
547C22—304no—
548C22—304no—
549C22—602no—
550C22—602no—
551C22—602no—
552C22—602no—
553C22C22×C4×D1511yescharacteristic
554C22C2×D4×D1511yescharacteristic
555C22C2×#681911yescharacteristic
556C22C2×#682111yescharacteristic
557C22C2×#682111yescharacteristic
558C22#41×D1511yescharacteristic
559C22#41×D1511yescharacteristic
560C4—24no—
561C4—22no—
562C4—22no—
563C4—42no—
564C4—154no—
565C4—302no—
566C4—302no—
567C4—302no—
568C4—602no—
569C4#9178114yescomplemented
570C3#1060769211yescharacteristic, complemented, Sylow
571C2—604no—
572C2C2×#9178111yescharacteristic
573C2C4×D4×D1511yescharacteristic
574C2C4×#682111yescharacteristic
575C2#1175635511yescharacteristic
576C2#1175638311yescharacteristic
577C2#1175645011yescharacteristic
578C2#1175648411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D15
commutatorC22×C3×C5C23×C4
FittingC3×C5×#389C2
socle, cosocleC23×C3×C5C24
prime radicalC22×D60C22
hypercenter#389D15
hypocenterC3×C5#2293
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
2. C2×#91781C2semidirect
3. C3×C5×#389C2semidirect
4. #11751091C2semidirect
6. #11751085C2semidirect
7. C4⋊C4×Dic15C2semidirect
9. C42×Dic15C2semidirect
10. #11750771C2semidirect
11. #11750767C2semidirect
33. #91782C4semidirect
41. C5×#389D3semidirect
84. C4×Dic15D4semidirect
87. C3×#389D5semidirect
227. #389D15semidirect
251. C3×C4×C5#117semidirect
423. C4×C5#892semidirect
464. C3×C5#2293semidirect
483. C3×C4#4038semidirect
545. C5#75961semidirect
569. C4#91781semidirect
570. C3#10607692semidirect

Maximal subgroups

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

subgroupindex
C2×#917822
C2×#917812
C3×C5×#3892
#117510912
#117510892
#117510852
C4⋊C4×Dic152
#117510402
C42×Dic152
#117507712
#117507672
#106076923
#759615