Finite Group Databasebeta

← #14326955

Subgroups — #14326955

Order 1920.

1008 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91358C211yesmaximal, characteristic
2#11751073C211yesmaximal, characteristic, complemented
3C5×Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
4C3×#65276C211yesmaximal, characteristic
5#11748858C211yesmaximal, characteristic, complemented
6#11748707C212yesmaximal
7#11748705C212yesmaximal
8C4×#6665C211yesmaximal, characteristic
9#11748699C211yesmaximal, characteristic, complemented
10#11748646C212yesmaximal, complemented
11#11748645C212yesmaximal, complemented
12#10605365—31nomaximal, Hall
13C22×C3×C4×D5C2211yescharacteristic
14C2×C3×C5×C4⋊C4C2211yescharacteristic
15C2×C4⋊Dic15C2211yescharacteristic
16C2×#6712C2212yes—
17C2×C4×Dic15C2211yescharacteristic
18C2×C5×C4⋊Dic3C2211yescharacteristic
19C2×C4×C5×Dic3C2211yescharacteristic
20C2×C4×C5×Dic3C2212yes—
21C2×C3×#685C2212yes—
22C2×C3×C4⋊Dic5C2211yescharacteristic
23C2×#6669C2212yes—
24C2×#6665C2211yescharacteristic
25C2×#6665C2211yescharacteristic
26C2×#6665C2212yes—
27C2×Dic3×Dic5C2212yes—
28C2×Dic3×Dic5C2212yes—
29#91358C418yescomplemented
30#91115C2212yes—
31C5×C3⋊#95C2212yes—
32C3×#3963C2212yes—
33#90994C2211yescharacteristic
34#90994C2211yescharacteristic
35#90994C2212yes—
36#90994C2212yes—
37#73882—51nomaximal, Hall
38C2×#4038—31no—
39C4×C5×C4⋊C4—31no—
40#65276D311yescharacteristic
41C4⋊C4×Dic5—31no—
42#65262—31no—
43#65229—32no—
44C4×#685—31no—
45C4×#683—32no—
46#65217—31no—
47#64958—32no—
48#64940—32no—
49C23×C3×D5C2311yescharacteristic
50C22×C3×C4×C5C2312yes—
51C22×C3×C4×C5C2311yescharacteristic
52C22×Dic15C2311yescharacteristic
53C22×Dic15C2312yes—
54C22×Dic15C2311yescharacteristic
55C22×C5×Dic3C2311yescharacteristic
56C22×C5×Dic3C2312yes—
57C22×C5×Dic3C2311yescharacteristic
58C22×C3×Dic5C2312yes—
59C22×C3×Dic5C2311yescharacteristic
60C2×C3×C4×D5—22no—
61C2×C3×C4×D5—22no—
62C2×C3×C4×D5C2×C414yes—
63C3×C5×C4⋊C4—22no—
64C4⋊Dic15C2×C414yes—
65#6712—24no—
66C4×Dic15—22no—
67C5×C4⋊Dic3C2×C414yes—
68C4×C5×Dic3—22no—
69C4×C5×Dic3—42no—
70C3×#685—42no—
71C3×C4⋊Dic5—22no—
72#6669—24no—
73#6665—22no—
74#6665—22no—
75#6665C2×C418yes—
76Dic3×Dic5—42no—
77Dic3×Dic5C2×C418yes—
78C2×#926—51no—
79C3×#380—51no—
80C4×#217—51no—
81Dic3×C4⋊C4—51no—
82Dic3×C4⋊C4D511yescharacteristic, complemented
83#4972—51no—
84#4972—51no—
85#4944—52no—
86#4944—52no—
87#4651—52no—
88#4649—52no—
89C22×C4×D5C2×D311yescharacteristic
90C2×C5×C4⋊C4—31no—
91C2×C5×C4⋊C4C2×D311yescharacteristic
92C2×C42×C5—31no—
93C2×C42×C5—32no—
94C2×#685—32no—
95C2×#685—31no—
96C2×#685—31no—
97C2×#685C2×D312yes—
98C2×C4⋊Dic5—31no—
99C2×C4⋊Dic5C2×D311yescharacteristic
100C2×#683—32no—
101C2×#683—32no—
102C2×C4×Dic5—32no—
103C2×C4×Dic5—31no—
104C2×C4×Dic5—32no—
105#4038—38no—
106C5×#95—32no—
107#3963—32no—
108#3963—31no—
109#3963—32no—
110#3963—32no—
111#3963—31no—
112#3963C2×D312yes—
113#2347—151noSylow
114C23×C3×C5C2411yescharacteristic
115C22×C3×D5—21no—
116C22×C3×D5—21no—
117C22×C3×D5—21no—
118C22×C3×D5—21no—
119C22×C3×D5—21no—
120C22×C3×D5—21no—
121C22×C3×D5C22×C412yes—
122C2×C3×C4×C5—22no—
123C2×C3×C4×C5—22no—
124C2×C3×C4×C5—22no—
125C2×C3×C4×C5C22×C412yes—
126C2×C3×C4×C5#4112yes—
127C2×C3×C4×C5#4112yes—
128C2×Dic15—22no—
129C2×Dic15—21no—
130C2×Dic15—22no—
131C2×Dic15—21no—
132C2×Dic15—21no—
133C2×Dic15—21no—
134C2×Dic15—21no—
135C2×Dic15C22×C414yes—
136C2×Dic15#4112yes—
137C2×C5×Dic3—21no—
138C2×C5×Dic3—22no—
139C2×C5×Dic3—21no—
140C2×C5×Dic3—22no—
141C2×C5×Dic3—21no—
142C2×C5×Dic3—21no—
143C2×C5×Dic3—21no—
144C2×C5×Dic3C22×C414yes—
145C2×C5×Dic3#4112yes—
146C2×C3×Dic5—21no—
147C2×C3×Dic5—21no—
148C2×C3×Dic5—22no—
149C2×C3×Dic5—22no—
150C2×C3×Dic5—22no—
151C2×C3×Dic5C22×C412yes—
152C3×C4×D5—28no—
153C23×C3×C4—51no—
154C2×C3×C4⋊C4—51no—
155C2×C3×C4⋊C4C2×D511yescharacteristic
156C2×C3×C22⋊C4—52no—
157C2×#217—52no—
158C2×#217—51no—
159C2×#217—51no—
160#926—58no—
161C2×C4⋊Dic3—51no—
162C2×C4⋊Dic3C2×D511yescharacteristic
163C2×Dic3⋊C4—52no—
164C2×Dic3⋊C4—52no—
165C2×C4×Dic3—52no—
166C2×C4×Dic3—51no—
167C2×C4×Dic3—52no—
168C2×C4×Dic3C2×D511yescharacteristic
169C2×C4×Dic3C2×D512yes—
170C3×#95—52no—
171C3⋊#95—52no—
172C3⋊#95—51no—
173C3⋊#95—51no—
174C3⋊#95—52no—
175C3⋊#95—52no—
176C3⋊#95C2×D512yes—
177C23×D5C22×D311yescharacteristic
178C22×C4×C5—31no—
179C22×C4×C5—32no—
180C22×C4×C5—31no—
181C22×C4×C5C22×D312yes—
182C22×C4×C5C22×D311yescharacteristic
183C22×Dic5—32no—
184C22×Dic5—31no—
185C22×Dic5—31no—
186C22×Dic5C22×D312yes—
187C22×Dic5C22×D311yescharacteristic
188C2×C4×D5—22no—
189C2×C4×D5—22no—
190C2×C4×D5C4×D314yes—
191C5×C4⋊C4—22no—
192C5×C4⋊C4—34no—
193C42×C5—62no—
194C42×C5—122no—
195#685—38no—
196#685—42no—
197#685—62no—
198#685—62no—
199C4⋊Dic5—22no—
200C4⋊Dic5—34no—
201#683—64no—
202#683—64no—
203C4×Dic5—38no—
204C4×Dic5—62no—
205C4×Dic5—122no—
206C2×#117—151no—
207#389—151no—
208#389—151no—
209#388—152no—
210#388—152no—
211#383—152no—
212#380—51no—
213C4×C4⋊C4—31no—
214C4×C4⋊C4—152no—
215C4×C4⋊C4—151no—
216C4×C22⋊C4—151no—
217C22×C3×C5C23×C411yescharacteristic
218C22×C3×C5C2×#4111yescharacteristic
219C22×C3×C5C2×#4111yescharacteristic
220C22×C3×C5C2×#4111yescharacteristic
221C22×C3×C5C2×#4111yescharacteristic
222C22×C3×C5H(5,𝔽2)11yescharacteristic
223C22×C3×C52−1+411yescharacteristic
224C2×C3×D5—22no—
225C2×C3×D5—22no—
226C2×C3×D5—22no—
227C2×C3×D5—41no—
228C2×C3×D5—41no—
229C2×C3×D5—41no—
230C2×C3×D5—41no—
231C3×C4×C5—42no—
232C3×C4×C5#11714yes—
233Dic15—24no—
234Dic15—22no—
235Dic15—41no—
236C3×Dic5—22no—
237C3×Dic5—42no—
238C5×Dic3—24no—
239C5×Dic3—22no—
240C5×Dic3—41no—
241C24×C3—51no—
242C22×C3×C4—51no—
243C22×C3×C4—54no—
244C22×C3×C4—52no—
245C22×C3×C4—102no—
246C22×C3×C4—102no—
247C22×C3×C4C22×D512yes—
248C22×C3×C4C22×D511yescharacteristic
249C22×Dic3—52no—
250C22×Dic3—51no—
251C22×Dic3—51no—
252C22×Dic3C22×D511yescharacteristic
253C22×Dic3C22×D512yes—
254C22×Dic3C22×D511yescharacteristic
255C3×C4⋊C4—22no—
256C3×C4⋊C4—102no—
257C3×C22⋊C4—202no—
258#217—58no—
259#217—102no—
260#217—102no—
261C4⋊Dic3—54no—
262C4⋊Dic3C4×D514yes—
263Dic3⋊C4—104no—
264Dic3⋊C4—104no—
265C4×Dic3—22no—
266C4×Dic3—42no—
267C4×Dic3—58no—
268C4×Dic3—102no—
269C4×Dic3—202no—
270C23×C5C23×D311yescharacteristic
271C22×D5—21no—
272C22×D5—21no—
273C22×D5—21no—
274C22×D5—21no—
275C22×D5—21no—
276C22×D5—21no—
277C22×D5C2×C4×D312yes—
278C2×C4×C5—22no—
279C2×C4×C5—22no—
280C2×C4×C5—22no—
281C2×C4×C5—34no—
282C2×C4×C5—32no—
283C2×C4×C5—61no—
284C2×C4×C5—62no—
285C2×C4×C5—61no—
286C2×C4×C5—62no—
287C2×C4×C5—61no—
288C2×C4×C5—61no—
289C2×C4×C5—61no—
290C2×C4×C5C2×C4×D312yes—
291C2×C4×C5#23712yes—
292C2×C4×C5#23712yes—
293C2×Dic5—21no—
294C2×Dic5—21no—
295C2×Dic5—22no—
296C2×Dic5—22no—
297C2×Dic5—22no—
298C2×Dic5—34no—
299C2×Dic5—32no—
300C2×Dic5—61no—
301C2×Dic5—61no—
302C2×Dic5—62no—
303C2×Dic5—62no—
304C2×Dic5—61no—
305C2×Dic5—61no—
306C2×Dic5—61no—
307C2×Dic5C2×C4×D312yes—
308C4×D5—28no—
309C23×C4—51no—
310#117—158no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—51no—
313C2×C4⋊C4—151no—
314C2×C4⋊C4—152no—
315C2×C4⋊C4—152no—
316C2×C4⋊C4D3×D511yescharacteristic
317C2×C22⋊C4—52no—
318C2×C22⋊C4—152no—
319C2×C22⋊C4—151no—
320C2×C22⋊C4—151no—
321C2×C42—31no—
322C2×C42—32no—
323C2×C42—152no—
324C2×C42—151no—
325C2×C42—152no—
326#95—32no—
327#95—52no—
328#95—151no—
329#95—151no—
330#95—152no—
331#95—152no—
332#95—152no—
333C2×C3×C5C2×#11711yescharacteristic
334C2×C3×C5C4×#4111yescharacteristic
335C2×C3×C5#52011yescharacteristic
336C2×C3×C5#55311yescharacteristic
337C2×C3×C5#55311yescharacteristic
338C2×C3×C5#55611yescharacteristic
339C2×C3×C5#55611yescharacteristic
340C3×D5—42no—
341C23×C3—52no—
342C23×C3—101no—
343C23×C3—101no—
344C23×C3—101no—
345C23×C3—101no—
346C23×C3—101no—
347C23×C3—101no—
348C23×C3C23×D511yescharacteristic
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—52no—
353C2×C3×C4—101no—
354C2×C3×C4—101no—
355C2×C3×C4—108no—
356C2×C3×C4—102no—
357C2×C3×C4—102no—
358C2×C3×C4—102no—
359C2×C3×C4C2×C4×D512yes—
360C2×C3×C4#70912yes—
361C2×C3×C4#70912yes—
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—22no—
365C2×Dic3—22no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—54no—
370C2×Dic3—52no—
371C2×Dic3—101no—
372C2×Dic3—101no—
373C2×Dic3—102no—
374C2×Dic3—102no—
375C2×Dic3—101no—
376C2×Dic3—101no—
377C2×Dic3—101no—
378C2×Dic3C2×C4×D514yes—
379C2×Dic3#71312yes—
380C22×C5C22×C4×D311yescharacteristic
381C22×C5C2×#23711yescharacteristic
382C22×C5C2×#23711yescharacteristic
383C22×C5#100411yescharacteristic
384C22×C5#100711yescharacteristic
385C22×C5D3×#4111yescharacteristic
386C22×C5D3×#4111yescharacteristic
387C2×D5—22no—
388C2×D5—22no—
389C2×D5—22no—
390C2×D5—41no—
391C2×D5—41no—
392C2×D5—41no—
393C2×D5—41no—
394C4×C5—42no—
395C4×C5—64no—
396C4×C5—62no—
397C4×C5—121no—
398C4×C5#87714yes—
399Dic5—22no—
400Dic5—42no—
401Dic5—64no—
402Dic5—62no—
403Dic5—121no—
404C24—51no—
405C22×C4—32no—
406C22×C4—31no—
407C22×C4—31no—
408C22×C4—51no—
409C22×C4—54no—
410C22×C4—52no—
411C22×C4—102no—
412C22×C4—102no—
413C22×C4—152no—
414C22×C4—151no—
415C22×C4—151no—
416C22×C4C2×D3×D512yes—
417C22×C4C2×D3×D511yescharacteristic
418C4⋊C4—22no—
419C4⋊C4—34no—
420C4⋊C4—102no—
421C4⋊C4—154no—
422C4⋊C4—304no—
423C4⋊C4—304no—
424C22⋊C4—158no—
425C22⋊C4—202no—
426C22⋊C4—302no—
427C22⋊C4—302no—
428C42—62no—
429C42—122no—
430C42—158no—
431C42—302no—
432C42—602no—
433C3×C5#234711yescharacteristic, complemented, Hall
434C22×C3—102no—
435C22×C3—102no—
436C22×C3—102no—
437C22×C3—201no—
438C22×C3—201no—
439C22×C3—201no—
440C22×C3—201no—
441C22×C3C22×C4×D511yescharacteristic
442C22×C3C2×#70911yescharacteristic
443C22×C3C2×#70911yescharacteristic
444C22×C3#414411yescharacteristic
445C22×C3C2×#71311yescharacteristic
446C22×C3D5×#4111yescharacteristic
447C22×C3#415011yescharacteristic
448C3×C4—42no—
449C3×C4—102no—
450C3×C4—202no—
451C3×C4#401814yes—
452Dic3—22no—
453Dic3—24no—
454Dic3—41no—
455Dic3—102no—
456Dic3—104no—
457Dic3—201no—
458C2×C5C2×#87711yescharacteristic
459C2×C5#550411yescharacteristic
460C2×C5#552211yescharacteristic
461C2×C5#561411yescharacteristic
462C2×C5#561411yescharacteristic
463C2×C5#564111yescharacteristic
464C2×C5#564111yescharacteristic
465D5—42no—
466C23—52no—
467C23—101no—
468C23—101no—
469C23—101no—
470C23—101no—
471C23—101no—
472C23—101no—
473C23C22×D3×D511yescharacteristic
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—34no—
478C2×C4—32no—
479C2×C4—52no—
480C2×C4—61no—
481C2×C4—61no—
482C2×C4—62no—
483C2×C4—62no—
484C2×C4—61no—
485C2×C4—61no—
486C2×C4—61no—
487C2×C4—101no—
488C2×C4—101no—
489C2×C4—108no—
490C2×C4—102no—
491C2×C4—102no—
492C2×C4—102no—
493C2×C4—154no—
494C2×C4—152no—
495C2×C4—302no—
496C2×C4—301no—
497C2×C4—302no—
498C2×C4—301no—
499C2×C4—301no—
500C2×C4—301no—
501C2×C4—301no—
502C2×C4#676512yes—
503C2×C4C4×D3×D512yes—
504C2×C4#677912yes—
505C2×C3—202no—
506C2×C3C2×#401811yescharacteristic
507C2×C3#6584811yescharacteristic
508C2×C3#6586711yescharacteristic
509C2×C3#6588011yescharacteristic
510C2×C3#6589211yescharacteristic
511C2×C3C4×#71311yescharacteristic
512C2×C3#6591811yescharacteristic
513C5#7388211yescharacteristic, complemented, Sylow
514C22—102no—
515C22—102no—
516C22—102no—
517C22—201no—
518C22—201no—
519C22—201no—
520C22—201no—
521C22C2×#676511yescharacteristic
522C22C2×C4×D3×D511yescharacteristic
523C22#9202411yescharacteristic
524C22#9202511yescharacteristic
525C22#9203011yescharacteristic
526C22D3×#71311yescharacteristic
527C22C2×#677911yescharacteristic
528C4—42no—
529C4—62no—
530C4—64no—
531C4—102no—
532C4—121no—
533C4—202no—
534C4—304no—
535C4—302no—
536C4—601no—
537C4#9139314yes—
538C3#1060536511yescharacteristic, complemented, Sylow
539C2—202no—
540C2#1175340311yescharacteristic
541C2#1175367111yescharacteristic
542C2#1175377511yescharacteristic
543C2#1175379711yescharacteristic
544C2C2×#9139311yescharacteristic
545C2#1175474411yescharacteristic
546C2#1175525211yescharacteristic
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#2347
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
2. #11751073C2semidirect
3. C5×Dic3×C4⋊C4C2semidirect
5. #11748858C2semidirect
9. #11748699C2semidirect
10. #11748646C2semidirect
11. #11748645C2semidirect
29. #91358C4semidirect
82. Dic3×C4⋊C4D5semidirect
433. C3×C5#2347semidirect
513. C5#73882semidirect
538. C3#10605365semidirect

Maximal subgroups

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

subgroupindex
C2×#913582
#117510732
C5×Dic3×C4⋊C42
C3×#652762
#117488582
#117487072
#117487052
C4×#66652
#117486992
#117486462
#117486452
#106053653
#738825