Finite Group Databasebeta

← #14322749

Subgroups — #14322749

Order 1920.

968 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91761C211yesmaximal, characteristic
2C4×C5×C4⋊Dic3C211yesmaximal, characteristic, complemented
3C3×C4×C4⋊Dic5C211yesmaximal, characteristic, complemented
4#11748866C211yesmaximal, characteristic, complemented
5#11748854C212yesmaximal
6#11748847C212yesmaximal, complemented
7#11748829C212yesmaximal
8#11748824C212yesmaximal, complemented
9#11748799C211yesmaximal, characteristic
10#11748799C211yesmaximal, characteristic
11#11748623C211yesmaximal, characteristic, complemented
12#10605491—31nomaximal, Hall
13C22×C4×D15C2211yescharacteristic
14C2×C3×C42×C5C2211yescharacteristic
15C2×#6714C2211yescharacteristic
16C2×#6714C2211yescharacteristic
17C2×#6712C2211yescharacteristic
18C2×#6712C2211yescharacteristic
19C2×C4×Dic15C2211yescharacteristic
20#91761C418yescomplemented
21C2×C5×C4⋊Dic3C2211yescharacteristic
22C2×C5×C4⋊Dic3C2211yescharacteristic
23C2×C4×C5×Dic3C2212yes—
24C2×C3×C4⋊Dic5C2211yescharacteristic
25C2×C3×C4⋊Dic5C2211yescharacteristic
26C2×C3×C4×Dic5C2212yes—
27C2×#6669C2212yes—
28C2×#6669C2212yes—
29C2×#6667C2212yes—
30C2×#6667C2212yes—
31C5×C3⋊#95C2212yes—
32C3×#3963C2212yes—
33#90994C2212yes—
34#90994C2212yes—
35#90994C2212yes—
36#90994C2212yes—
37#73760—51nomaximal, Hall
38C2×#4018—31no—
39C4×C5×C4⋊C4—31no—
40#65276—31no—
41#65276—31no—
42#65266—31no—
43#65265—31no—
44#65229—32no—
45C4×C4⋊Dic5D311yescharacteristic, complemented
46#65222—32no—
47#64958—32no—
48#64941—32no—
49C23×D15C2311yescharacteristic
50C22×C3×C4×C5C2311yescharacteristic
51C22×C3×C4×C5C2311yescharacteristic
52C22×C3×C4×C5C2311yescharacteristic
53C22×Dic15C2311yescharacteristic
54C22×Dic15C2311yescharacteristic
55C22×Dic15C2311yescharacteristic
56C2×C4×D15—22no—
57C2×C4×D15—22no—
58C2×C4×D15C2×C414yes—
59C22×C5×Dic3C2312yes—
60C22×C5×Dic3C2312yes—
61C22×C3×Dic5C2312yes—
62C22×C3×Dic5C2312yes—
63C3×C42×C5C2×C414yes—
64#6714C2×C414yes—
65#6714C2×C414yes—
66#6712C2×C414yes—
67#6712C2×C414yes—
68C4×Dic15C2×C414yes—
69C5×C4⋊Dic3—22no—
70C5×C4⋊Dic3—22no—
71C4×C5×Dic3—42no—
72C3×C4⋊Dic5—22no—
73C3×C4⋊Dic5—22no—
74C3×C4×Dic5—42no—
75#6669—42no—
76#6669—42no—
77#6667—42no—
78#6667—42no—
79C2×#872—51no—
80C3×C4×C4⋊C4—51no—
81#4987—51no—
82#4987—51no—
83#4977—51no—
84#4975—51no—
85#4940—52no—
86C4×C4⋊Dic3D511yescharacteristic, complemented
87#4933—52no—
88#4669—52no—
89#4652—52no—
90C22×C4×D5—31no—
91C2×C5×C4⋊C4—31no—
92C2×C5×C4⋊C4—31no—
93C2×C42×C5—32no—
94C2×C42×C5C2×D311yescharacteristic
95C2×#685—31no—
96C2×#685—31no—
97C2×#685—32no—
98C2×#685—32no—
99C2×C4⋊Dic5C2×D311yescharacteristic
100C2×C4⋊Dic5C2×D311yescharacteristic
101C2×#683—31no—
102C2×#683—31no—
103C2×#683—32no—
104C2×#683—32no—
105C2×C4×Dic5—31no—
106C2×C4×Dic5C2×D312yes—
107#4018—38no—
108C5×#95—32no—
109#3963—32no—
110#3963—32no—
111#3963—32no—
112#3963—32no—
113#3963C2×D312yes—
114#2314—151noSylow
115C23×C3×C5C2411yescharacteristic
116C22×D15—21no—
117C22×D15—21no—
118C22×D15—21no—
119C22×D15—21no—
120C22×D15—21no—
121C22×D15—21no—
122C22×D15C22×C412yes—
123C2×C3×C4×C5—21no—
124C2×C3×C4×C5—21no—
125C2×C3×C4×C5—21no—
126C2×C3×C4×C5—21no—
127C2×C3×C4×C5C22×C412yes—
128C2×C3×C4×C5C22×C412yes—
129C2×C3×C4×C5C22×C412yes—
130C2×C3×C4×C5#4112yes—
131C2×C3×C4×C5#4112yes—
132C2×Dic15—21no—
133C2×Dic15—21no—
134C2×Dic15—21no—
135C2×Dic15—21no—
136C2×Dic15—21no—
137C2×Dic15—21no—
138C2×Dic15C22×C412yes—
139C2×Dic15C22×C412yes—
140C2×Dic15C22×C412yes—
141C4×D15—28no—
142C2×C5×Dic3—22no—
143C2×C5×Dic3—22no—
144C2×C5×Dic3—22no—
145C2×C5×Dic3—22no—
146C2×C5×Dic3—22no—
147C2×C5×Dic3—22no—
148C2×C3×Dic5—22no—
149C2×C3×Dic5—22no—
150C2×C3×Dic5—22no—
151C2×C3×Dic5—22no—
152C2×C3×Dic5—22no—
153C2×C3×Dic5—22no—
154C22×C4×D3—51no—
155C2×C3×C4⋊C4—51no—
156C2×C3×C4⋊C4—51no—
157C2×C3×C42—52no—
158C2×C3×C42C2×D511yescharacteristic
159C2×(C2×D3)⋊C4—51no—
160C2×(C2×D3)⋊C4—51no—
161C2×(C2×D3)⋊C4—52no—
162C2×(C2×D3)⋊C4—52no—
163C2×C4⋊Dic3C2×D511yescharacteristic
164C2×C4⋊Dic3C2×D511yescharacteristic
165C2×Dic3⋊C4—51no—
166C2×Dic3⋊C4—51no—
167C2×Dic3⋊C4—52no—
168C2×Dic3⋊C4—52no—
169C2×C4×Dic3—51no—
170C2×C4×Dic3C2×D512yes—
171#872—58no—
172C3×#95—52no—
173C3⋊#95—52no—
174C3⋊#95—52no—
175C3⋊#95—52no—
176C3⋊#95—52no—
177C3⋊#95C2×D512yes—
178C23×D5—31no—
179C22×C4×C5—32no—
180C22×C4×C5—32no—
181C22×C4×C5C22×D311yescharacteristic
182C22×C4×C5C22×D311yescharacteristic
183C22×C4×C5C22×D311yescharacteristic
184C22×Dic5—31no—
185C22×Dic5—31no—
186C22×Dic5—31no—
187C22×Dic5C22×D312yes—
188C22×Dic5C22×D312yes—
189C2×C4×D5—34no—
190C2×C4×D5—62no—
191C2×C4×D5—62no—
192C5×C4⋊C4—62no—
193C5×C4⋊C4—62no—
194C42×C5—122no—
195C42×C5C4×D314yes—
196#685—34no—
197#685—34no—
198#685—122no—
199#685—122no—
200C4⋊Dic5—22no—
201C4⋊Dic5—22no—
202#683—34no—
203#683—34no—
204#683—122no—
205#683—122no—
206C4×Dic5—34no—
207C4×Dic5—42no—
208C2×#117—151no—
209#388—152no—
210#388—152no—
211#387—152no—
212#387—152no—
213#384—151no—
214#382—151no—
215#380—151no—
216#380—151no—
217C4×C4⋊C4—31no—
218C4×C4⋊C4—51no—
219C22×C3×C5C23×C411yescharacteristic
220C22×C3×C5C2×#4111yescharacteristic
221C22×C3×C5C2×#4111yescharacteristic
222C22×C3×C5H(5,𝔽2)11yescharacteristic
223C22×C3×C5H(5,𝔽2)11yescharacteristic
224C22×C3×C52−1+411yescharacteristic
225C22×C3×C52−1+411yescharacteristic
226C2×D15—22no—
227C2×D15—22no—
228C2×D15—22no—
229C2×D15—41no—
230C2×D15—41no—
231C2×D15—41no—
232C2×D15—41no—
233C3×C4×C5—22no—
234C3×C4×C5—22no—
235C3×C4×C5#11714yes—
236Dic15—22no—
237Dic15—22no—
238Dic15—22no—
239C3×Dic5—42no—
240C3×Dic5—42no—
241C5×Dic3—42no—
242C5×Dic3—42no—
243C23×D3—51no—
244C22×C3×C4—52no—
245C22×C3×C4—52no—
246C22×C3×C4C22×D511yescharacteristic
247C22×C3×C4C22×D511yescharacteristic
248C22×C3×C4C22×D511yescharacteristic
249C22×Dic3—51no—
250C22×Dic3—51no—
251C22×Dic3—51no—
252C22×Dic3C22×D512yes—
253C22×Dic3C22×D512yes—
254C2×C4×D3—54no—
255C2×C4×D3—102no—
256C2×C4×D3—102no—
257C3×C4⋊C4—102no—
258C3×C4⋊C4—102no—
259C3×C42—202no—
260C3×C42C4×D514yes—
261(C2×D3)⋊C4—54no—
262(C2×D3)⋊C4—54no—
263(C2×D3)⋊C4—202no—
264(C2×D3)⋊C4—202no—
265C4⋊Dic3—22no—
266C4⋊Dic3—22no—
267Dic3⋊C4—54no—
268Dic3⋊C4—54no—
269Dic3⋊C4—202no—
270Dic3⋊C4—202no—
271C4×Dic3—42no—
272C4×Dic3—54no—
273C23×C5C23×D311yescharacteristic
274C22×D5—32no—
275C22×D5—61no—
276C22×D5—61no—
277C22×D5—61no—
278C22×D5—61no—
279C22×D5—61no—
280C22×D5—61no—
281C2×C4×C5—21no—
282C2×C4×C5—21no—
283C2×C4×C5—21no—
284C2×C4×C5—21no—
285C2×C4×C5—62no—
286C2×C4×C5—62no—
287C2×C4×C5—62no—
288C2×C4×C5—62no—
289C2×C4×C5—62no—
290C2×C4×C5—62no—
291C2×C4×C5C2×C4×D312yes—
292C2×C4×C5C2×C4×D312yes—
293C2×C4×C5C2×C4×D312yes—
294C2×C4×C5#23512yes—
295C2×C4×C5#23512yes—
296C2×Dic5—22no—
297C2×Dic5—22no—
298C2×Dic5—22no—
299C2×Dic5—22no—
300C2×Dic5—22no—
301C2×Dic5—22no—
302C2×Dic5—32no—
303C2×Dic5—32no—
304C2×Dic5—32no—
305C2×Dic5—61no—
306C2×Dic5—61no—
307C2×Dic5—61no—
308C2×Dic5—61no—
309C2×Dic5—61no—
310C2×Dic5—61no—
311C4×D5—68no—
312C23×C4—151no—
313#117—158no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—51no—
317C2×C4⋊C4—51no—
318C2×C4⋊C4—151no—
319C2×C4⋊C4—151no—
320C2×C4⋊C4—152no—
321C2×C4⋊C4—152no—
322C2×C22⋊C4—151no—
323C2×C22⋊C4—151no—
324C2×C22⋊C4—152no—
325C2×C22⋊C4—152no—
326C2×C42—32no—
327C2×C42—52no—
328C2×C42—151no—
329C2×C42D3×D511yescharacteristic
330#95—32no—
331#95—52no—
332#95—152no—
333#95—152no—
334#95—152no—
335#95—152no—
336C2×C3×C5C2×#11711yescharacteristic
337C2×C3×C5#52011yescharacteristic
338C2×C3×C5#52011yescharacteristic
339C2×C3×C5#54011yescharacteristic
340C2×C3×C5#54111yescharacteristic
341C2×C3×C5#54211yescharacteristic
342C2×C3×C5#54211yescharacteristic
343D15—42no—
344C23×C3C23×D511yescharacteristic
345C22×D3—52no—
346C22×D3—101no—
347C22×D3—101no—
348C22×D3—101no—
349C22×D3—101no—
350C22×D3—101no—
351C22×D3—101no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—102no—
357C2×C3×C4—102no—
358C2×C3×C4—102no—
359C2×C3×C4—102no—
360C2×C3×C4—102no—
361C2×C3×C4—102no—
362C2×C3×C4C2×C4×D512yes—
363C2×C3×C4C2×C4×D512yes—
364C2×C3×C4C2×C4×D512yes—
365C2×C3×C4#70912yes—
366C2×C3×C4#70912yes—
367C2×Dic3—22no—
368C2×Dic3—22no—
369C2×Dic3—22no—
370C2×Dic3—22no—
371C2×Dic3—22no—
372C2×Dic3—22no—
373C2×Dic3—52no—
374C2×Dic3—52no—
375C2×Dic3—52no—
376C2×Dic3—101no—
377C2×Dic3—101no—
378C2×Dic3—101no—
379C2×Dic3—101no—
380C2×Dic3—101no—
381C2×Dic3—101no—
382C4×D3—108no—
383C22×C5C22×C4×D311yescharacteristic
384C22×C5C2×#23511yescharacteristic
385C22×C5C2×#23511yescharacteristic
386C22×C5#100411yescharacteristic
387C22×C5#100711yescharacteristic
388C22×C5#100911yescharacteristic
389C22×C5#101011yescharacteristic
390C2×D5—62no—
391C2×D5—62no—
392C2×D5—62no—
393C2×D5—121no—
394C2×D5—121no—
395C2×D5—121no—
396C2×D5—121no—
397C4×C5—22no—
398C4×C5—22no—
399C4×C5—122no—
400C4×C5—122no—
401C4×C5#87214yes—
402Dic5—42no—
403Dic5—42no—
404Dic5—62no—
405Dic5—62no—
406Dic5—62no—
407C24—151no—
408C22×C4—32no—
409C22×C4—32no—
410C22×C4—52no—
411C22×C4—52no—
412C22×C4—151no—
413C22×C4—151no—
414C22×C4—151no—
415C22×C4—154no—
416C22×C4—302no—
417C22×C4—302no—
418C22×C4C2×D3×D511yescharacteristic
419C22×C4C2×D3×D511yescharacteristic
420C22×C4C2×D3×D511yescharacteristic
421C4⋊C4—62no—
422C4⋊C4—62no—
423C4⋊C4—102no—
424C4⋊C4—102no—
425C4⋊C4—154no—
426C4⋊C4—154no—
427C4⋊C4—602no—
428C4⋊C4—602no—
429C22⋊C4—154no—
430C22⋊C4—154no—
431C22⋊C4—602no—
432C22⋊C4—602no—
433C42—122no—
434C42—154no—
435C42—202no—
436C42#120014yes—
437C3×C5#231411yescharacteristic, complemented, Hall
438C22×C3C22×C4×D511yescharacteristic
439C22×C3C2×#70911yescharacteristic
440C22×C3C2×#70911yescharacteristic
441C22×C3#414411yescharacteristic
442C22×C3#414711yescharacteristic
443C22×C3#414911yescharacteristic
444C22×C3#415011yescharacteristic
445C2×D3—102no—
446C2×D3—102no—
447C2×D3—102no—
448C2×D3—201no—
449C2×D3—201no—
450C2×D3—201no—
451C2×D3—201no—
452C3×C4—22no—
453C3×C4—22no—
454C3×C4—202no—
455C3×C4—202no—
456C3×C4#401814yes—
457Dic3—42no—
458Dic3—42no—
459Dic3—102no—
460Dic3—102no—
461Dic3—102no—
462C2×C5C2×#87211yescharacteristic
463C2×C5#550411yescharacteristic
464C2×C5#552311yescharacteristic
465C2×C5#556111yescharacteristic
466C2×C5#556211yescharacteristic
467C2×C5#558211yescharacteristic
468C2×C5#558311yescharacteristic
469D5—122no—
470C23—152no—
471C23—301no—
472C23—301no—
473C23—301no—
474C23—301no—
475C23—301no—
476C23—301no—
477C23C22×D3×D511yescharacteristic
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—62no—
483C2×C4—62no—
484C2×C4—62no—
485C2×C4—62no—
486C2×C4—62no—
487C2×C4—62no—
488C2×C4—102no—
489C2×C4—102no—
490C2×C4—102no—
491C2×C4—102no—
492C2×C4—102no—
493C2×C4—102no—
494C2×C4—152no—
495C2×C4—152no—
496C2×C4—152no—
497C2×C4—301no—
498C2×C4—301no—
499C2×C4—301no—
500C2×C4—301no—
501C2×C4—301no—
502C2×C4—301no—
503C2×C4—308no—
504C2×C4#677112yes—
505C2×C4#677112yes—
506C2×C4C2×#120012yes—
507C2×C4C2×#120012yes—
508C2×C4C2×#120012yes—
509C2×C3C2×#401811yescharacteristic
510C2×C3#6583811yescharacteristic
511C2×C3#6585711yescharacteristic
512C2×C3#6589511yescharacteristic
513C2×C3#6589611yescharacteristic
514C2×C3#6591611yescharacteristic
515C2×C3#6591711yescharacteristic
516D3—202no—
517C5#7376011yescharacteristic, complemented, Sylow
518C22—302no—
519C22—302no—
520C22—302no—
521C22—601no—
522C22—601no—
523C22—601no—
524C22—601no—
525C22#9200011yescharacteristic
526C22#9200411yescharacteristic
527C22C2×#677111yescharacteristic
528C22C2×#677111yescharacteristic
529C22#9201711yescharacteristic
530C22#9201811yescharacteristic
531C22C22×#120011yescharacteristic
532C4—22no—
533C4—22no—
534C4—122no—
535C4—122no—
536C4—202no—
537C4—202no—
538C4—302no—
539C4—302no—
540C4—302no—
541C4#9138614yes—
542C3#1060549111yescharacteristic, complemented, Sylow
543C2—602no—
544C2#1175250311yescharacteristic
545C2#1175251411yescharacteristic
546C2#1175270711yescharacteristic
547C2#1175271611yescharacteristic
548C2C2×#9138611yescharacteristic
549C2#1175449611yescharacteristic
550C2#1175450611yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C42×C5C22
socle, cosocleC23×C3×C5C24
prime radicalC23×D15C23
hypercenterC2×C42D3×D5
hypocenterC3×C5#2314
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
2. C4×C5×C4⋊Dic3C2semidirect
3. C3×C4×C4⋊Dic5C2semidirect
4. #11748866C2semidirect
6. #11748847C2semidirect
8. #11748824C2semidirect
11. #11748623C2semidirect
20. #91761C4semidirect
45. C4×C4⋊Dic5D3semidirect
86. C4×C4⋊Dic3D5semidirect
437. C3×C5#2314semidirect
517. C5#73760semidirect
542. C3#10605491semidirect

Maximal subgroups

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

subgroupindex
C2×#917612
C4×C5×C4⋊Dic32
C3×C4×C4⋊Dic52
#117488662
#117488542
#117488472
#117488292
#117488242
#117487992
#117486232
#106054913
#737605