Finite Group Databasebeta

← #14328968

Subgroups — #14328968

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×C3×#4043C211yesmaximal, characteristic
2C2×#91358C211yesmaximal, characteristic
3C2×Dic3×Dic10C211yesmaximal, characteristic, complemented
4C4⋊C4×Dic15C211yesmaximal, characteristic, complemented
5C5×#4984C211yesmaximal, characteristic, complemented
6#11748791C212yesmaximal
7#11748771C212yesmaximal
8C4×#6665C211yesmaximal, characteristic
9#11748699C212yesmaximal, complemented
10#11748698C212yesmaximal, complemented
11#11748610C211yesmaximal, characteristic, complemented
12#10607715—31nomaximal, Hall
13C22×C3×C4×D5C2211yescharacteristic
14C22×C3×Dic10C2211yescharacteristic
15C2×C3×C5×C4⋊C4C2211yescharacteristic
16C2×C4⋊Dic15C2211yescharacteristic
17C2×C4×Dic15C2211yescharacteristic
18C2×C4×Dic15C2212yes—
19C2×C5×C4⋊Dic3C2211yescharacteristic
20C2×C5×C4⋊Dic3C2212yes—
21C2×C4×C5×Dic3C2211yescharacteristic
22C2×C3×#685C2212yes—
23C2×C3×C4⋊Dic5C2212yes—
24C3×#4043C418yescomplemented
25C2×#6668C2212yes—
26C2×#6665C2211yescharacteristic
27C2×#6665C2211yescharacteristic
28C2×#6665C2212yes—
29C2×Dic3×Dic5C2212yes—
30C2×Dic3×Dic5C2212yes—
31#91358—24no—
32Dic3×Dic10—24no—
33#91115C2212yes—
34C5×C3⋊#95C2212yes—
35#90994C2212yes—
36#90994C2212yes—
37#90994C2211yescharacteristic
38#90994C2211yescharacteristic
39#82748—51nomaximal, Hall
40C2×#4043D311yescharacteristic
41C2×#4038—31no—
42C2×C4×Dic10—31no—
43C5×#389—31no—
44#65279—32no—
45C4⋊C4×Dic5—32no—
46C4⋊C4×Dic5—31no—
47#65263—31no—
48C4×#685—31no—
49#64958—32no—
50#64943—32no—
51C23×C3×D5C2311yescharacteristic
52C22×C3×C4×C5C2312yes—
53C22×C3×C4×C5C2311yescharacteristic
54C22×Dic15C2311yescharacteristic
55C22×Dic15C2312yes—
56C22×Dic15C2311yescharacteristic
57C22×C5×Dic3C2311yescharacteristic
58C22×C5×Dic3C2312yes—
59C22×C5×Dic3C2311yescharacteristic
60C22×C3×Dic5C2311yescharacteristic
61C22×C3×Dic5C2312yes—
62C2×C3×C4×D5—22no—
63C2×C3×C4×D5—22no—
64C2×C3×C4×D5C2×C414yes—
65C2×C3×Dic10—22no—
66C2×C3×Dic10—22no—
67C2×C3×Dic10C2×C414yes—
68C3×C5×C4⋊C4C2×C414yes—
69C4⋊Dic15D414yes—
70C4×Dic15—22no—
71C4×Dic15—24no—
72C5×C4⋊Dic3—22no—
73C5×C4⋊Dic3—24no—
74C4×C5×Dic3—22no—
75C3×#685C2×C418yes—
76C3×C4⋊Dic5C2×C418yes—
77#6668—24no—
78#6665—24no—
79#6665—22no—
80#6665—22no—
81Dic3×Dic5—24no—
82Dic3×Dic5—24no—
83C2×C3×C22⋊Q8—51no—
84C2×Q8×Dic3—51no—
85C2×#926—51no—
86#5229—51no—
87C4×#217—51no—
88#4984D511yescharacteristic, complemented
89#4980—52no—
90Dic3×C4⋊C4—51no—
91Dic3×C4⋊C4—52no—
92#4951—52no—
93#4951—52no—
94C22×C4×D5C2×D311yescharacteristic
95C22×Dic10C2×D311yescharacteristic
96C2×C5×C4⋊C4—31no—
97C2×C5×C4⋊C4—32no—
98C2×C5×C4⋊C4C2×D311yescharacteristic
99C2×C42×C5—31no—
100C2×#685—32no—
101C2×#685—31no—
102C2×#685—31no—
103C2×#685C2×D312yes—
104C2×C4⋊Dic5—31no—
105C2×C4⋊Dic5C2×D312yes—
106C2×#683—32no—
107C2×C4×Dic5—32no—
108C2×C4×Dic5—31no—
109C2×C4×Dic5—32no—
110C2×C4×Dic5—32no—
111#4043Dic318yescomplemented
112#4038—64no—
113C4×Dic10—64no—
114C5×#95—32no—
115#3963—32no—
116#3963—31no—
117#3963—32no—
118#3963—32no—
119#3963—31no—
120#2362—151noSylow
121C23×C3×C5C2411yescharacteristic
122C22×C3×D5—21no—
123C22×C3×D5—21no—
124C22×C3×D5—21no—
125C22×C3×D5—21no—
126C22×C3×D5—21no—
127C22×C3×D5—21no—
128C22×C3×D5C22×C412yes—
129C2×C3×C4×C5—22no—
130C2×C3×C4×C5—22no—
131C2×C3×C4×C5C22×C414yes—
132C2×C3×C4×C5C22×C412yes—
133C2×C3×C4×C5C2×D412yes—
134C2×C3×C4×C5#4112yes—
135C2×Dic15—22no—
136C2×Dic15—21no—
137C2×Dic15—22no—
138C2×Dic15—21no—
139C2×Dic15—21no—
140C2×Dic15—21no—
141C2×Dic15—21no—
142C2×Dic15C2×D414yes—
143C2×Dic15#4112yes—
144C2×C5×Dic3—21no—
145C2×C5×Dic3—21no—
146C2×C5×Dic3—22no—
147C2×C5×Dic3—21no—
148C2×C5×Dic3—21no—
149C2×C5×Dic3—22no—
150C2×C5×Dic3—21no—
151C2×C5×Dic3—22no—
152C2×C5×Dic3#4112yes—
153C2×C3×Dic5—21no—
154C2×C3×Dic5—22no—
155C2×C3×Dic5—22no—
156C2×C3×Dic5—21no—
157C2×C3×Dic5C22×C412yes—
158C2×C3×Dic5C22×C414yes—
159C3×C4×D5—28no—
160C3×Dic10—28no—
161C22×C3×Q8—51no—
162C23×C3×C4—51no—
163C3×C22⋊Q8—58no—
164C2×C3×C4⋊C4—52no—
165C2×C3×C4⋊C4C2×D511yescharacteristic
166C2×C3×C22⋊C4—52no—
167C2×#217—51no—
168C2×#217—51no—
169C2×#217—52no—
170Q8×Dic3—104no—
171#926—104no—
172C2×C4⋊Dic3—51no—
173C2×C4⋊Dic3—52no—
174C2×C4⋊Dic3C2×D512yes—
175C2×C4⋊Dic3C2×D511yescharacteristic
176C2×C4×Dic3—52no—
177C2×C4×Dic3—51no—
178C2×C4×Dic3—52no—
179C2×C4×Dic3—52no—
180C2×C4×Dic3C2×D511yescharacteristic
181C3⋊#95—52no—
182C3⋊#95—52no—
183C3⋊#95—52no—
184C3⋊#95—51no—
185C3⋊#95—51no—
186C3⋊#95C2×D512yes—
187C23×D5C22×D311yescharacteristic
188C22×C4×C5—31no—
189C22×C4×C5—31no—
190C22×C4×C5—32no—
191C22×C4×C5C22×D312yes—
192C22×C4×C5C22×D311yescharacteristic
193C22×Dic5—31no—
194C22×Dic5—32no—
195C22×Dic5—31no—
196C22×Dic5C22×D312yes—
197C22×Dic5C22×D311yescharacteristic
198C2×C4×D5—22no—
199C2×C4×D5—22no—
200C2×C4×D5C2×Dic314yes—
201C2×Dic10—22no—
202C2×Dic10—22no—
203C2×Dic10C2×Dic314yes—
204C5×C4⋊C4—64no—
205C5×C4⋊C4—62no—
206C5×C4⋊C4C2×Dic314yes—
207C42×C5—62no—
208#685—64no—
209#685—62no—
210#685—62no—
211#685C2×Dic318yes—
212C4⋊Dic5—34no—
213C4⋊Dic5C2×Dic318yes—
214#683—64no—
215C4×Dic5—64no—
216C4×Dic5—64no—
217C4×Dic5—64no—
218C4×Dic5—62no—
219C2×C22⋊Q8—51no—
220C2×C4×Q8—151no—
221C2×#117—151no—
222#391—151no—
223#389—31no—
224#388—152no—
225#388—152no—
226#387—152no—
227C4×C4⋊C4—152no—
228C4×C4⋊C4—151no—
229C4×C22⋊C4—151no—
230C22×C3×C5C23×C411yescharacteristic
231C22×C3×C5C22×D411yescharacteristic
232C22×C3×C5C2×#4111yescharacteristic
233C22×C3×C5C2×#4111yescharacteristic
234C22×C3×C5C2×#4111yescharacteristic
235C22×C3×C52−1+411yescharacteristic
236C22×C3×C52−1+411yescharacteristic
237C2×C3×D5—22no—
238C2×C3×D5—22no—
239C2×C3×D5—22no—
240C2×C3×D5—41no—
241C2×C3×D5—41no—
242C2×C3×D5—41no—
243C2×C3×D5—41no—
244C3×C4×C5—24no—
245C3×C4×C5C4×D414yes—
246Dic15—24no—
247Dic15—22no—
248Dic15—41no—
249C3×Dic5—24no—
250C3×Dic5—22no—
251C5×Dic3—22no—
252C5×Dic3—41no—
253C5×Dic3—42no—
254C24×C3—51no—
255C2×C3×Q8—54no—
256C2×C3×Q8—102no—
257C2×C3×Q8—102no—
258C22×C3×C4—51no—
259C22×C3×C4—52no—
260C22×C3×C4—54no—
261C22×C3×C4—102no—
262C22×C3×C4—102no—
263C22×C3×C4C22×D512yes—
264C22×C3×C4C22×D511yescharacteristic
265C22×Dic3—51no—
266C22×Dic3—52no—
267C22×Dic3—51no—
268C22×Dic3C22×D512yes—
269C22×Dic3C22×D511yescharacteristic
270C22×Dic3C22×D511yescharacteristic
271C3×C4⋊C4—58no—
272C3×C4⋊C4C4×D514yes—
273C3×C22⋊C4—58no—
274#217—104no—
275#217—102no—
276#217—102no—
277C4⋊Dic3—22no—
278C4⋊Dic3—24no—
279C4⋊Dic3—54no—
280C4⋊Dic3—104no—
281C4×Dic3—22no—
282C4×Dic3—104no—
283C4×Dic3—104no—
284C4×Dic3—104no—
285C4×Dic3—102no—
286C23×C5C23×D311yescharacteristic
287C22×D5—21no—
288C22×D5—21no—
289C22×D5—21no—
290C22×D5—21no—
291C22×D5—21no—
292C22×D5—21no—
293C22×D5C22×Dic312yes—
294C2×C4×C5—22no—
295C2×C4×C5—22no—
296C2×C4×C5—32no—
297C2×C4×C5—61no—
298C2×C4×C5—61no—
299C2×C4×C5—62no—
300C2×C4×C5—61no—
301C2×C4×C5—61no—
302C2×C4×C5—62no—
303C2×C4×C5—61no—
304C2×C4×C5—62no—
305C2×C4×C5D3×D412yes—
306C2×C4×C5#23712yes—
307C2×C4×C5C22×Dic314yes—
308C2×C4×C5C22×Dic312yes—
309C2×Dic5—21no—
310C2×Dic5—22no—
311C2×Dic5—22no—
312C2×Dic5—21no—
313C2×Dic5—34no—
314C2×Dic5—32no—
315C2×Dic5—61no—
316C2×Dic5—62no—
317C2×Dic5—61no—
318C2×Dic5—62no—
319C2×Dic5—61no—
320C2×Dic5—61no—
321C2×Dic5—61no—
322C2×Dic5C22×Dic312yes—
323C2×Dic5C22×Dic314yes—
324C4×D5—28no—
325Dic10—28no—
326C22×Q8—51no—
327C23×C4—51no—
328C22⋊Q8—58no—
329C4×Q8—304no—
330#117—304no—
331C2×C4⋊C4—31no—
332C2×C4⋊C4—32no—
333C2×C4⋊C4—52no—
334C2×C4⋊C4—151no—
335C2×C4⋊C4—152no—
336C2×C4⋊C4D3×D511yescharacteristic
337C2×C22⋊C4—52no—
338C2×C22⋊C4—152no—
339C2×C22⋊C4—151no—
340C2×C22⋊C4—151no—
341C2×C42—31no—
342C2×C42—152no—
343C2×C42—152no—
344C2×C42—151no—
345C2×C42—152no—
346#95—32no—
347#95—151no—
348#95—151no—
349#95—152no—
350#95—152no—
351#95—152no—
352C2×C3×C5C2×C4×D411yescharacteristic
353C2×C3×C5C4×#4111yescharacteristic
354C2×C3×C5#51911yescharacteristic
355C2×C3×C5#54711yescharacteristic
356C2×C3×C5#54811yescharacteristic
357C2×C3×C5#55211yescharacteristic
358C2×C3×C5#55611yescharacteristic
359C3×D5—42no—
360C23×C3—52no—
361C23×C3—101no—
362C23×C3—101no—
363C23×C3—101no—
364C23×C3—101no—
365C23×C3—101no—
366C23×C3—101no—
367C23×C3C23×D511yescharacteristic
368C3×Q8—108no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—54no—
372C2×C3×C4—52no—
373C2×C3×C4—101no—
374C2×C3×C4—102no—
375C2×C3×C4—101no—
376C2×C3×C4—102no—
377C2×C3×C4—108no—
378C2×C3×C4C2×C4×D514yes—
379C2×C3×C4C2×C4×D512yes—
380C2×C3×C4D4×D512yes—
381C2×C3×C4#71312yes—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—22no—
386C2×Dic3—22no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—22no—
390C2×Dic3—54no—
391C2×Dic3—52no—
392C2×Dic3—101no—
393C2×Dic3—102no—
394C2×Dic3—102no—
395C2×Dic3—101no—
396C2×Dic3—101no—
397C2×Dic3—101no—
398C2×Dic3—101no—
399C2×Dic3#71112yes—
400C22×C5C2×D3×D411yescharacteristic
401C22×C5C2×#23711yescharacteristic
402C22×C5#100711yescharacteristic
403C22×C5#100711yescharacteristic
404C22×C5D3×#4111yescharacteristic
405C22×C5D3×#4111yescharacteristic
406C22×C5C23×Dic311yescharacteristic
407C2×D5—22no—
408C2×D5—22no—
409C2×D5—22no—
410C2×D5—41no—
411C2×D5—41no—
412C2×D5—41no—
413C2×D5—41no—
414C4×C5—24no—
415C4×C5—62no—
416C4×C5—122no—
417C4×C5—121no—
418C4×C5D4×Dic314yes—
419Dic5—24no—
420Dic5—22no—
421Dic5—62no—
422Dic5—64no—
423Dic5—121no—
424C24—51no—
425C2×Q8—54no—
426C2×Q8—102no—
427C2×Q8—102no—
428C22×C4—31no—
429C22×C4—31no—
430C22×C4—32no—
431C22×C4—51no—
432C22×C4—52no—
433C22×C4—54no—
434C22×C4—102no—
435C22×C4—102no—
436C22×C4—151no—
437C22×C4—152no—
438C22×C4—151no—
439C22×C4C2×D3×D512yes—
440C22×C4C2×D3×D511yescharacteristic
441C4⋊C4—58no—
442C4⋊C4—64no—
443C4⋊C4—62no—
444C4⋊C4—154no—
445C4⋊C4—304no—
446C4⋊C4D5×Dic314yes—
447C22⋊C4—58no—
448C22⋊C4—302no—
449C22⋊C4—304no—
450C22⋊C4—302no—
451C42—62no—
452C42—304no—
453C42—304no—
454C42—304no—
455C42—302no—
456C3×C5#236211yescharacteristic, complemented, Hall
457C22×C3—102no—
458C22×C3—102no—
459C22×C3—102no—
460C22×C3—201no—
461C22×C3—201no—
462C22×C3—201no—
463C22×C3—201no—
464C22×C3C22×C4×D511yescharacteristic
465C22×C3C2×D4×D511yescharacteristic
466C22×C3C2×#71111yescharacteristic
467C22×C3C2×#71311yescharacteristic
468C22×C3#414711yescharacteristic
469C22×C3D5×#4111yescharacteristic
470C22×C3#415011yescharacteristic
471C3×C4—24no—
472C3×C4—102no—
473C3×C4—104no—
474C3×C4#403914yes—
475Dic3—22no—
476Dic3—42no—
477Dic3—41no—
478Dic3—104no—
479Dic3—102no—
480Dic3—201no—
481C2×C5#563111yescharacteristic
482C2×C5#563311yescharacteristic
483C2×C5#563911yescharacteristic
484C2×C5#564111yescharacteristic
485C2×C5C2×D4×Dic311yescharacteristic
486C2×C5#581211yescharacteristic
487C2×C5Dic3×#4111yescharacteristic
488D5—42no—
489C23—52no—
490C23—101no—
491C23—101no—
492C23—101no—
493C23—101no—
494C23—101no—
495C23—101no—
496C23C22×D3×D511yescharacteristic
497Q8—108no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—32no—
501C2×C4—54no—
502C2×C4—52no—
503C2×C4—61no—
504C2×C4—61no—
505C2×C4—61no—
506C2×C4—62no—
507C2×C4—62no—
508C2×C4—61no—
509C2×C4—61no—
510C2×C4—62no—
511C2×C4—101no—
512C2×C4—102no—
513C2×C4—101no—
514C2×C4—102no—
515C2×C4—108no—
516C2×C4—154no—
517C2×C4—152no—
518C2×C4—301no—
519C2×C4—302no—
520C2×C4—302no—
521C2×C4—301no—
522C2×C4—301no—
523C2×C4—301no—
524C2×C4—301no—
525C2×C4#676612yes—
526C2×C4#677712yes—
527C2×C4C2×D5×Dic314yes—
528C2×C4C2×D5×Dic312yes—
529C2×C3—202no—
530C2×C3C2×#403911yescharacteristic
531C2×C3C4×#71111yescharacteristic
532C2×C3#6590611yescharacteristic
533C2×C3#6593211yescharacteristic
534C2×C3#6596511yescharacteristic
535C2×C3#6602811yescharacteristic
536C2×C3#6606611yescharacteristic
537C5#8274811yescharacteristic, complemented, Sylow
538C22—102no—
539C22—102no—
540C22—102no—
541C22—201no—
542C22—201no—
543C22—201no—
544C22—201no—
545C22C2×#676611yescharacteristic
546C22C2×#677711yescharacteristic
547C22D3×#71111yescharacteristic
548C22#9202411yescharacteristic
549C22#9202911yescharacteristic
550C22#9203011yescharacteristic
551C22C22×D5×Dic311yescharacteristic
552C4—24no—
553C4—62no—
554C4—102no—
555C4—104no—
556C4—122no—
557C4—121no—
558C4—302no—
559C4—304no—
560C4—601no—
561C4#9143414yes—
562C3#1060771511yescharacteristic, complemented, Sylow
563C2—202no—
564C2#1175370311yescharacteristic
565C2#1175370411yescharacteristic
566C2#1175376911yescharacteristic
567C2#1175380011yescharacteristic
568C2C2×#9143411yescharacteristic
569C2Dic3×#71111yescharacteristic
570C2#1175514011yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C5×C4⋊C4C22
socle, cosocleC23×C3×C5C24
prime radicalC23×C3×D5C23
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5#2362
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×Dic3×Dic10C2semidirect
4. C4⋊C4×Dic15C2semidirect
5. C5×#4984C2semidirect
9. #11748699C2semidirect
10. #11748698C2semidirect
11. #11748610C2semidirect
24. C3×#4043C4semidirect
88. #4984D5semidirect
111. #4043Dic3semidirect
456. C3×C5#2362semidirect
537. C5#82748semidirect
562. C3#10607715semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#40432
C2×#913582
C2×Dic3×Dic102
C4⋊C4×Dic152
C5×#49842
#117487912
#117487712
C4×#66652
#117486992
#117486982
#117486102
#106077153
#827485