Finite Group Databasebeta

← #14328061

Subgroups — #14328061

Order 1920.

1120 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×Dic5C211yesmaximal, characteristic
2#11751322C211yesmaximal, characteristic
3C5×#5221C211yesmaximal, characteristic
4#11749486C212yesmaximal, complemented
5#11749485C212yesmaximal, complemented
6Dic5×#217C212yesmaximal, complemented
7#11749471C212yesmaximal, complemented
8C4×#6668C211yesmaximal, characteristic, complemented
9#11748699C211yesmaximal, characteristic, complemented
10Dic3×C4⋊Dic5C211yesmaximal, characteristic, complemented
11#11748613C211yesmaximal, characteristic, complemented
12#10608034—31nomaximal, Hall
13C22×C3×C5×D4C2211yescharacteristic
14C23×C3×Dic5C2212yes—
15C2×#6719C2212yes—
16C2×#6719C2212yes—
17C2×C4⋊Dic15C2211yescharacteristic
18C2×C4×Dic15C2211yescharacteristic
19C2×C5×#217C2212yes—
20C2×C5×#217C2212yes—
21C2×C5×C4⋊Dic3C2211yescharacteristic
22C2×C4×C5×Dic3C2211yescharacteristic
23C2×C3×C22⋊Dic5C2212yes—
24C3×D4×Dic5C418yescomplemented
25C2×C3×C4⋊Dic5C2211yescharacteristic
26C2×C3×C4×Dic5C2211yescharacteristic
27C2×#6668C2211yescharacteristic
28C2×#6668C2211yescharacteristic
29C2×Dic3×Dic5C2212yes—
30C2×Dic3×Dic5C2212yes—
31#90994C2212yes—
32#90994C2212yes—
33#90994C2212yes—
34#90994C2212yes—
35#90994C2211yescharacteristic
36#90994C2211yescharacteristic
37#82338—51nomaximal, Hall
38C2×D4×Dic5D311yescharacteristic
39C5×#390—31no—
40#65510—31no—
41#65266—31no—
42C4⋊C4×Dic5—31no—
43#65234—32no—
44#65233—32no—
45#65232—32no—
46C22⋊C4×Dic5—32no—
47C4×C4⋊Dic5—31no—
48C4×#683—31no—
49C24×C3×C5C2312yes—
50C2×C3×C5×D4—22no—
51C2×C3×C5×D4—22no—
52C2×C3×C5×D4C2×C414yes—
53C22×C3×C4×C5C2311yescharacteristic
54C22×Dic15C2312yes—
55C22×Dic15C2311yescharacteristic
56C22×Dic15C2311yescharacteristic
57C22×C5×Dic3C2311yescharacteristic
58C22×C5×Dic3C2312yes—
59C22×C5×Dic3C2311yescharacteristic
60C22×C3×Dic5—42no—
61C22×C3×Dic5—42no—
62C22×C3×Dic5C2×C418yes—
63C22×C3×Dic5C2311yescharacteristic
64C22×C3×Dic5C2312yes—
65C22×C3×Dic5C2311yescharacteristic
66#6719—24no—
67#6719—42no—
68C4⋊Dic15—22no—
69C4×Dic15—22no—
70C5×#217—24no—
71C5×#217—42no—
72C5×C4⋊Dic3—22no—
73C4×C5×Dic3—22no—
74C3×C22⋊Dic5C2×C418yes—
75C3×C4⋊Dic5C2×C414yes—
76C3×C4×Dic5C2×C414yes—
77#6668—22no—
78#6668—22no—
79Dic3×Dic5—42no—
80Dic3×Dic5—42no—
81C2×C3×C4×D4—51no—
82#5221—51no—
83#5221D511yescharacteristic
84#5211—52no—
85C4×#217—52no—
86Dic3×C4⋊C4—51no—
87Dic3×C4⋊C4—51no—
88#4951—52no—
89#4951—52no—
90#4936—51no—
91C4×C4⋊Dic3—51no—
92C22×C5×D4C2×D311yescharacteristic
93C23×Dic5C2×D312yes—
94C2×C5×C4⋊C4—31no—
95C2×C5×C22⋊C4—32no—
96C2×C5×C22⋊C4—32no—
97C2×C42×C5—31no—
98C2×C22⋊Dic5—32no—
99C2×C22⋊Dic5—32no—
100C2×C22⋊Dic5C2×D312yes—
101D4×Dic5Dic318yescomplemented
102C2×C4⋊Dic5—31no—
103C2×C4⋊Dic5C2×D311yescharacteristic
104C2×#683—31no—
105C2×#683—31no—
106C2×C4×Dic5—31no—
107C2×C4×Dic5—32no—
108C2×C4×Dic5—32no—
109C2×C4×Dic5C2×D311yescharacteristic
110#3963—31no—
111#3963—31no—
112#3963—32no—
113#3963—32no—
114#3963—32no—
115#3963—32no—
116#2357—151noSylow
117C23×C3×C5—22no—
118C23×C3×C5—22no—
119C23×C3×C5—22no—
120C23×C3×C5—22no—
121C23×C3×C5—22no—
122C23×C3×C5—22no—
123C23×C3×C5C22×C414yes—
124C23×C3×C5C2411yescharacteristic
125C3×C5×D4—28no—
126C2×C3×C4×C5C22×C412yes—
127C2×C3×C4×C5#4112yes—
128C2×C3×C4×C5#4112yes—
129C2×Dic15—21no—
130C2×Dic15—21no—
131C2×Dic15—21no—
132C2×Dic15—21no—
133C2×Dic15—22no—
134C2×Dic15—22no—
135C2×Dic15—22no—
136C2×Dic15—21no—
137C2×Dic15#4112yes—
138C2×C5×Dic3—21no—
139C2×C5×Dic3—21no—
140C2×C5×Dic3—21no—
141C2×C5×Dic3—21no—
142C2×C5×Dic3—22no—
143C2×C5×Dic3—22no—
144C2×C5×Dic3—22no—
145C2×C5×Dic3—21no—
146C2×C5×Dic3#4112yes—
147C2×C3×Dic5—21no—
148C2×C3×Dic5—21no—
149C2×C3×Dic5—22no—
150C2×C3×Dic5—22no—
151C2×C3×Dic5—21no—
152C2×C3×Dic5—21no—
153C2×C3×Dic5—48no—
154C2×C3×Dic5C22×C412yes—
155C2×C3×Dic5C22×C412yes—
156C2×C3×Dic5C22×C414yes—
157C22×C3×D4C2×D511yescharacteristic
158C23×C3×C4—52no—
159C3×C4×D4—58no—
160C2×C3×C4⋊C4—51no—
161C2×C3×C22⋊C4—52no—
162C2×C3×C42—51no—
163C2×#217—52no—
164C2×#217—52no—
165C2×#217C2×D512yes—
166C2×#217C2×D512yes—
167C2×C4⋊Dic3—51no—
168C2×C4⋊Dic3—51no—
169C2×C4⋊Dic3—51no—
170C2×C4⋊Dic3C2×D511yescharacteristic
171C2×C4×Dic3—51no—
172C2×C4×Dic3—52no—
173C2×C4×Dic3—52no—
174C2×C4×Dic3C2×D511yescharacteristic
175C3⋊#95—51no—
176C3⋊#95—51no—
177C3⋊#95—52no—
178C3⋊#95—52no—
179C3⋊#95—52no—
180C3⋊#95—52no—
181C24×C5C22×D312yes—
182C2×C5×D4—22no—
183C2×C5×D4—22no—
184C2×C5×D4C2×Dic314yes—
185C22×C4×C5—31no—
186C22×C4×C5—31no—
187C22×C4×C5—32no—
188C22×C4×C5C22×D311yescharacteristic
189C22×Dic5—31no—
190C22×Dic5—31no—
191C22×Dic5—32no—
192C22×Dic5—42no—
193C22×Dic5—42no—
194C22×Dic5C2×Dic318yes—
195C22×Dic5C22×D311yescharacteristic
196C22×Dic5C22×D312yes—
197C22×Dic5C22×D311yescharacteristic
198C5×C4⋊C4—62no—
199C5×C22⋊C4—64no—
200C5×C22⋊C4—122no—
201C42×C5—62no—
202C22⋊Dic5—64no—
203C22⋊Dic5—122no—
204C22⋊Dic5C2×Dic318yes—
205C4⋊Dic5—62no—
206C4⋊Dic5C2×Dic314yes—
207#683—62no—
208#683—62no—
209C4×Dic5—62no—
210C4×Dic5—122no—
211C4×Dic5—122no—
212C4×Dic5C2×Dic314yes—
213C2×C4×D4—51no—
214#390—31no—
215#390—151no—
216#388—152no—
217#388—152no—
218#382—151no—
219#381—152no—
220C4×C4⋊C4—151no—
221C4×C4⋊C4—151no—
222C4×C4⋊C4—151no—
223C4×C22⋊C4—152no—
224C22×C3×C5—24no—
225C22×C3×C5—24no—
226C22×C3×C5—24no—
227C22×C3×C5—42no—
228C22×C3×C5—42no—
229C22×C3×C5—42no—
230C22×C3×C5—42no—
231C22×C3×C5C23×C411yescharacteristic
232C22×C3×C5C2×#4111yescharacteristic
233C22×C3×C5C2×#4111yescharacteristic
234C22×C3×C5C2×#4111yescharacteristic
235C22×C3×C5C2×#4111yescharacteristic
236C22×C3×C5H(5,𝔽2)11yescharacteristic
237C22×C3×C5H(5,𝔽2)11yescharacteristic
238C3×C4×C5#11714yes—
239Dic15—22no—
240Dic15—41no—
241Dic15—42no—
242C3×Dic5—22no—
243C3×Dic5—22no—
244C3×Dic5—24no—
245C5×Dic3—22no—
246C5×Dic3—41no—
247C5×Dic3—42no—
248C24×C3C22×D512yes—
249C2×C3×D4—22no—
250C2×C3×D4—22no—
251C2×C3×D4C4×D514yes—
252C22×C3×C4—51no—
253C22×C3×C4—52no—
254C22×C3×C4—51no—
255C22×C3×C4—58no—
256C22×C3×C4—202no—
257C22×C3×C4—202no—
258C22×C3×C4C22×D511yescharacteristic
259C22×Dic3—51no—
260C22×Dic3—51no—
261C22×Dic3—52no—
262C22×Dic3C22×D512yes—
263C22×Dic3C22×D511yescharacteristic
264C22×Dic3C22×D511yescharacteristic
265C3×C4⋊C4—54no—
266C3×C22⋊C4—58no—
267C3×C42—54no—
268#217—24no—
269#217—42no—
270#217—104no—
271#217—202no—
272C4⋊Dic3—22no—
273C4⋊Dic3—102no—
274C4⋊Dic3—102no—
275C4⋊Dic3—102no—
276C4×Dic3—22no—
277C4×Dic3—102no—
278C4×Dic3—202no—
279C4×Dic3—202no—
280C23×C5—22no—
281C23×C5—22no—
282C23×C5—22no—
283C23×C5—22no—
284C23×C5—22no—
285C23×C5—22no—
286C23×C5C22×Dic314yes—
287C23×C5C23×D311yescharacteristic
288C5×D4—28no—
289C2×C4×C5—32no—
290C2×C4×C5—61no—
291C2×C4×C5—61no—
292C2×C4×C5—61no—
293C2×C4×C5—62no—
294C2×C4×C5—62no—
295C2×C4×C5—61no—
296C2×C4×C5—61no—
297C2×C4×C5—62no—
298C2×C4×C5#23512yes—
299C2×C4×C5#23512yes—
300C2×C4×C5C22×Dic312yes—
301C2×Dic5—22no—
302C2×Dic5—22no—
303C2×Dic5—21no—
304C2×Dic5—21no—
305C2×Dic5—21no—
306C2×Dic5—21no—
307C2×Dic5—32no—
308C2×Dic5—48no—
309C2×Dic5—61no—
310C2×Dic5—61no—
311C2×Dic5—61no—
312C2×Dic5—62no—
313C2×Dic5—62no—
314C2×Dic5—61no—
315C2×Dic5—61no—
316C2×Dic5—62no—
317C2×Dic5C22×Dic312yes—
318C2×Dic5C22×Dic312yes—
319C2×Dic5C22×Dic314yes—
320C22×D4D3×D511yescharacteristic
321C23×C4—52no—
322C4×D4—58no—
323C2×C4⋊C4—31no—
324C2×C4⋊C4—51no—
325C2×C4⋊C4—151no—
326C2×C4⋊C4—151no—
327C2×C4⋊C4—151no—
328C2×C22⋊C4—32no—
329C2×C22⋊C4—32no—
330C2×C22⋊C4—52no—
331C2×C22⋊C4—152no—
332C2×C22⋊C4—152no—
333C2×C42—31no—
334C2×C42—51no—
335C2×C42—151no—
336C2×C42—152no—
337C2×C42—152no—
338#95—151no—
339#95—152no—
340#95—151no—
341#95—152no—
342#95—152no—
343#95—152no—
344C2×C3×C5—44no—
345C2×C3×C5C2×#11711yescharacteristic
346C2×C3×C5C4×#4111yescharacteristic
347C2×C3×C5#51811yescharacteristic
348C2×C3×C5#55311yescharacteristic
349C2×C3×C5#55311yescharacteristic
350C2×C3×C5#55911yescharacteristic
351C2×C3×C5#55911yescharacteristic
352C23×C3—22no—
353C23×C3—22no—
354C23×C3—22no—
355C23×C3—22no—
356C23×C3—22no—
357C23×C3—22no—
358C23×C3C2×C4×D514yes—
359C23×C3C23×D511yescharacteristic
360C3×D4—28no—
361C2×C3×C4—52no—
362C2×C3×C4—52no—
363C2×C3×C4—54no—
364C2×C3×C4—101no—
365C2×C3×C4—102no—
366C2×C3×C4—101no—
367C2×C3×C4—101no—
368C2×C3×C4—102no—
369C2×C3×C4—101no—
370C2×C3×C4—208no—
371C2×C3×C4C2×C4×D512yes—
372C2×C3×C4#71112yes—
373C2×C3×C4#71112yes—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—22no—
378C2×Dic3—22no—
379C2×Dic3—22no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—52no—
383C2×Dic3—101no—
384C2×Dic3—101no—
385C2×Dic3—101no—
386C2×Dic3—101no—
387C2×Dic3—102no—
388C2×Dic3—102no—
389C2×Dic3—102no—
390C2×Dic3—101no—
391C2×Dic3#71112yes—
392C22×C5—24no—
393C22×C5—24no—
394C22×C5—24no—
395C22×C5—42no—
396C22×C5—42no—
397C22×C5—42no—
398C22×C5—42no—
399C22×C5C2×#23511yescharacteristic
400C22×C5C2×#23511yescharacteristic
401C22×C5#100411yescharacteristic
402C22×C5#100411yescharacteristic
403C22×C5D3×#4111yescharacteristic
404C22×C5D3×#4111yescharacteristic
405C22×C5C23×Dic311yescharacteristic
406C4×C5—62no—
407C4×C5—121no—
408C4×C5—122no—
409C4×C5#92614yes—
410Dic5—22no—
411Dic5—24no—
412Dic5—22no—
413Dic5—62no—
414Dic5—121no—
415Dic5—122no—
416C24C2×D3×D512yes—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4D5×Dic314yes—
420C22×C4—31no—
421C22×C4—31no—
422C22×C4—32no—
423C22×C4—51no—
424C22×C4—51no—
425C22×C4—52no—
426C22×C4—58no—
427C22×C4—151no—
428C22×C4—151no—
429C22×C4—152no—
430C22×C4—202no—
431C22×C4—202no—
432C22×C4C2×D3×D511yescharacteristic
433C4⋊C4—54no—
434C4⋊C4—62no—
435C4⋊C4—302no—
436C4⋊C4—302no—
437C4⋊C4—302no—
438C22⋊C4—58no—
439C22⋊C4—64no—
440C22⋊C4—122no—
441C22⋊C4—304no—
442C22⋊C4—602no—
443C42—54no—
444C42—62no—
445C42—302no—
446C42—602no—
447C42—602no—
448C3×C5#235711yescharacteristic, complemented, Hall
449C22×C3—24no—
450C22×C3—24no—
451C22×C3—24no—
452C22×C3—42no—
453C22×C3—42no—
454C22×C3—42no—
455C22×C3—42no—
456C22×C3C22×C4×D511yescharacteristic
457C22×C3C2×#71111yescharacteristic
458C22×C3C2×#71111yescharacteristic
459C22×C3C2×#71111yescharacteristic
460C22×C3#414411yescharacteristic
461C22×C3D5×#4111yescharacteristic
462C22×C3#414911yescharacteristic
463C3×C4—102no—
464C3×C4—104no—
465C3×C4—102no—
466C3×C4#402314yes—
467Dic3—22no—
468Dic3—41no—
469Dic3—42no—
470Dic3—102no—
471Dic3—201no—
472Dic3—202no—
473C2×C5—44no—
474C2×C5#555811yescharacteristic
475C2×C5#555811yescharacteristic
476C2×C5#555911yescharacteristic
477C2×C5#555911yescharacteristic
478C2×C5C2×#92611yescharacteristic
479C2×C5#579811yescharacteristic
480C2×C5Dic3×#4111yescharacteristic
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23C2×D5×Dic314yes—
488C23C22×D3×D511yescharacteristic
489D4—28no—
490C2×C4—32no—
491C2×C4—52no—
492C2×C4—52no—
493C2×C4—54no—
494C2×C4—61no—
495C2×C4—61no—
496C2×C4—62no—
497C2×C4—61no—
498C2×C4—62no—
499C2×C4—61no—
500C2×C4—61no—
501C2×C4—62no—
502C2×C4—101no—
503C2×C4—101no—
504C2×C4—102no—
505C2×C4—101no—
506C2×C4—102no—
507C2×C4—101no—
508C2×C4—152no—
509C2×C4—208no—
510C2×C4—301no—
511C2×C4—301no—
512C2×C4—301no—
513C2×C4—302no—
514C2×C4—302no—
515C2×C4—301no—
516C2×C4—301no—
517C2×C4—302no—
518C2×C4C2×D5×Dic312yes—
519C2×C4#678212yes—
520C2×C4#678212yes—
521C2×C3—44no—
522C2×C3C2×#402311yescharacteristic
523C2×C3C4×#71111yescharacteristic
524C2×C3#6587911yescharacteristic
525C2×C3#6594811yescharacteristic
526C2×C3#6595211yescharacteristic
527C2×C3#6601511yescharacteristic
528C2×C3#6604711yescharacteristic
529C5#8233811yescharacteristic, complemented, Sylow
530C22—24no—
531C22—24no—
532C22—24no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22D3×#71111yescharacteristic
538C22#9202411yescharacteristic
539C22#9202511yescharacteristic
540C22#9202611yescharacteristic
541C22C22×D5×Dic311yescharacteristic
542C22C2×#678211yescharacteristic
543C22C2×#678211yescharacteristic
544C4—62no—
545C4—102no—
546C4—102no—
547C4—104no—
548C4—122no—
549C4—121no—
550C4—302no—
551C4—601no—
552C4—602no—
553C4#9152814yes—
554C3#1060803411yescharacteristic, complemented, Sylow
555C2—44no—
556C2#1175473811yescharacteristic
557C2Dic3×#71111yescharacteristic
558C2#1175488311yescharacteristic
559C2#1175488711yescharacteristic
560C2#1175490211yescharacteristic
561C2#1175491011yescharacteristic
562C2C2×#9152811yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
Fitting, prime radicalC22×C3×C5×D4C22
socle, cosocleC23×C3×C5C24
hypercenterC22×D4D3×D5
hypocenterC3×C5#2357
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
4. #11749486C2semidirect
5. #11749485C2semidirect
6. Dic5×#217C2semidirect
7. #11749471C2semidirect
8. C4×#6668C2semidirect
9. #11748699C2semidirect
10. Dic3×C4⋊Dic5C2semidirect
11. #11748613C2semidirect
24. C3×D4×Dic5C4semidirect
101. D4×Dic5Dic3semidirect
448. C3×C5#2357semidirect
529. C5#82338semidirect
554. C3#10608034semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×Dic52
#117513222
C5×#52212
#117494862
#117494852
Dic5×#2172
#117494712
C4×#66682
#117486992
Dic3×C4⋊Dic52
#117486132
#106080343
#823385