Finite Group Databasebeta

← #14322848

Subgroups — #14322848

Order 1920.

1200 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×C3×#4038C211yesmaximal, characteristic
2C2×D5×C4⋊Dic3C211yesmaximal, characteristic
3C2×#91357C211yesmaximal, characteristic
4#11751073C211yesmaximal, characteristic, complemented
5C5×#4972C211yesmaximal, characteristic, complemented
6#11748885C212yesmaximal
7#11748878C212yesmaximal
8#11748868C211yesmaximal, characteristic, complemented
9#11748858C211yesmaximal, characteristic, complemented
10#11748802C212yesmaximal, complemented
11#11748801C212yesmaximal, complemented
12#10607926—31nomaximal, Hall
13C22×C3×C4×D5C2211yescharacteristic
14C22×D5×Dic3C2212yes—
15C2×C3×C5×C4⋊C4C2211yescharacteristic
16C2×C4⋊Dic15C2211yescharacteristic
17C2×#6712C2212yes—
18C2×C4×Dic15C2211yescharacteristic
19C2×C5×C4⋊Dic3C2211yescharacteristic
20C2×C5×Dic3⋊C4C2212yes—
21C2×C4×C5×Dic3C2211yescharacteristic
22C2×C3×#685C2212yes—
23C2×C3×C4⋊Dic5C2211yescharacteristic
24C2×C3×C4×Dic5C2212yes—
25C3×#4038—24no—
26C2×Dic3⋊Dic5C2212yes—
27C2×#6669C2212yes—
28C2×#6668C2211yescharacteristic
29C2×#6668C2211yescharacteristic
30C2×#6668C2212yes—
31C2×#6665C2211yescharacteristic
32C2×#6665C2211yescharacteristic
33D5×C4⋊Dic3C418yescomplemented
34#91357—24no—
35#91115C2212yes—
36C5×C3⋊#95C2212yes—
37#90994C2212yes—
38#90994C2212yes—
39#82232—51nomaximal, Hall
40C2×#4038D311yescharacteristic
41C2×D5×C4⋊C4—31no—
42C2×#4018—31no—
43C5×#389—31no—
44#65276—32no—
45#65274—31no—
46#65262—31no—
47#65261—32no—
48#65217—31no—
49#64956—32no—
50#64936—32no—
51C23×C3×D5C2311yescharacteristic
52C22×C3×C4×C5C2312yes—
53C22×C3×C4×C5C2311yescharacteristic
54C22×Dic15C2311yescharacteristic
55C22×Dic15C2311yescharacteristic
56C22×Dic15C2312yes—
57C22×C5×Dic3C2311yescharacteristic
58C22×C5×Dic3C2311yescharacteristic
59C22×C5×Dic3C2312yes—
60C22×C3×Dic5C2312yes—
61C22×C3×Dic5C2311yescharacteristic
62C2×C3×C4×D5C2×C414yes—
63C2×C3×C4×D5D414yes—
64C2×C3×C4×D5Q814yes—
65C2×D5×Dic3—42no—
66C2×D5×Dic3—42no—
67C2×D5×Dic3C2×C418yes—
68C3×C5×C4⋊C4—22no—
69C4⋊Dic15C2×C414yes—
70#6712—42no—
71C4×Dic15—22no—
72C5×C4⋊Dic3C2×C414yes—
73C5×Dic3⋊C4—42no—
74C4×C5×Dic3—22no—
75C3×#685—24no—
76C3×C4⋊Dic5—22no—
77C3×C4×Dic5—24no—
78Dic3⋊Dic5—42no—
79#6669—42no—
80#6668—22no—
81#6668—22no—
82#6668C2×C418yes—
83#6665—22no—
84#6665—22no—
85C2×C3×#117—51no—
86C2×#926—51no—
87C22×C4⋊Dic3—51no—
88#4972—51no—
89#4972—51no—
90#4972—51no—
91#4972D511yescharacteristic, complemented
92#4942—52no—
93#4942—52no—
94#4928—52no—
95#4928—52no—
96C22×C4×D5—32no—
97C22×C4×D5C2×D311yescharacteristic
98C2×C5×C4⋊C4—31no—
99C2×C5×C4⋊C4—32no—
100C2×C5×C4⋊C4C2×D311yescharacteristic
101C2×C42×C5—31no—
102C2×#685—31no—
103C2×#685—31no—
104C2×#685C2×D312yes—
105C2×C4⋊Dic5—31no—
106C2×C4⋊Dic5—32no—
107C2×C4⋊Dic5C2×D311yescharacteristic
108C2×#683—32no—
109C2×#683—31no—
110C2×#683—31no—
111C2×#683—32no—
112C2×#683—32no—
113C2×C4×Dic5—31no—
114C2×C4×Dic5C2×D312yes—
115#4038—24no—
116D5×C4⋊C4—38no—
117#4018—64no—
118C5×#95—32no—
119#3963—32no—
120#3963—32no—
121#3963—32no—
122#2316—151noSylow
123C23×C3×C5C2411yescharacteristic
124C22×C3×D5—21no—
125C22×C3×D5—21no—
126C22×C3×D5—21no—
127C22×C3×D5—21no—
128C22×C3×D5C22×C412yes—
129C22×C3×D5C2×D412yes—
130C22×C3×D5C2×Q812yes—
131C2×C3×C4×C5—22no—
132C2×C3×C4×C5—22no—
133C2×C3×C4×C5—22no—
134C2×C3×C4×C5C22×C412yes—
135C2×C3×C4×C5C2×D412yes—
136C2×C3×C4×C5C2×Q812yes—
137C2×Dic15—22no—
138C2×Dic15—22no—
139C2×Dic15—21no—
140C2×Dic15—21no—
141C2×Dic15—21no—
142C2×Dic15—21no—
143C2×Dic15—21no—
144C2×Dic15—21no—
145C2×Dic15C22×C414yes—
146C2×C5×Dic3—22no—
147C2×C5×Dic3—22no—
148C2×C5×Dic3—21no—
149C2×C5×Dic3—21no—
150C2×C5×Dic3—21no—
151C2×C5×Dic3—21no—
152C2×C5×Dic3—21no—
153C2×C5×Dic3—21no—
154C2×C5×Dic3C22×C414yes—
155C2×C3×Dic5—22no—
156C2×C3×Dic5—22no—
157C2×C3×Dic5—22no—
158C2×C3×Dic5C22×C412yes—
159C2×C3×Dic5C2×D412yes—
160C2×C3×Dic5C2×Q812yes—
161C3×C4×D5C4⋊C4116yescomplemented
162D5×Dic3—48no—
163C23×C3×C4—51no—
164C23×Dic3—52no—
165C3×#117—104no—
166C2×C3×C4⋊C4—51no—
167C2×C3×C4⋊C4C2×D511yescharacteristic
168C2×C3×C22⋊C4—52no—
169C2×C3×C42—52no—
170C2×#217—51no—
171C2×#217—51no—
172#926—104no—
173C2×C4⋊Dic3—51no—
174C2×C4⋊Dic3—52no—
175C2×C4⋊Dic3—58no—
176C2×C4⋊Dic3—51no—
177C2×C4⋊Dic3—51no—
178C2×C4⋊Dic3C2×D511yescharacteristic
179C2×Dic3⋊C4—52no—
180C2×Dic3⋊C4—52no—
181C2×Dic3⋊C4—52no—
182C2×Dic3⋊C4C2×D512yes—
183C2×C4×Dic3—51no—
184C2×C4×Dic3C2×D511yescharacteristic
185C3⋊#95—52no—
186C3⋊#95—52no—
187C3⋊#95—52no—
188C3⋊#95C2×D512yes—
189C23×D5C22×D311yescharacteristic
190C22×C4×C5—32no—
191C22×C4×C5—31no—
192C22×C4×C5—31no—
193C22×C4×C5C22×D312yes—
194C22×C4×C5C22×D311yescharacteristic
195C22×Dic5—32no—
196C22×Dic5—31no—
197C22×Dic5—31no—
198C22×Dic5C22×D312yes—
199C22×Dic5C22×D311yescharacteristic
200C2×C4×D5—38no—
201C2×C4×D5—122no—
202C2×C4×D5—122no—
203C2×C4×D5Dic614yes—
204C2×C4×D5C4×D314yes—
205C2×C4×D5#6714yes—
206C5×C4⋊C4—22no—
207C5×C4⋊C4—34no—
208C5×C4⋊C4—122no—
209C42×C5—62no—
210#685—24no—
211#685—62no—
212#685—62no—
213C4⋊Dic5—22no—
214C4⋊Dic5—34no—
215C4⋊Dic5—122no—
216#683—38no—
217#683—62no—
218#683—62no—
219#683—122no—
220#683—122no—
221C4×Dic5—24no—
222C4×Dic5—62no—
223C2×#117—51no—
224C2×#117—151no—
225C22×C4⋊C4—151no—
226#389—31no—
227#389—152no—
228#389—151no—
229#389—152no—
230#389—151no—
231#389—151no—
232#380—152no—
233#380—152no—
234C22×C3×C5C23×C411yescharacteristic
235C22×C3×C5C22×D411yescharacteristic
236C22×C3×C5C22×Q811yescharacteristic
237C22×C3×C5H(5,𝔽2)11yescharacteristic
238C22×C3×C5H(5,𝔽2)11yescharacteristic
239C22×C3×C52−1+411yescharacteristic
240C22×C3×C52−1+411yescharacteristic
241C2×C3×D5—22no—
242C2×C3×D5—22no—
243C2×C3×D5—22no—
244C2×C3×D5—22no—
245C2×C3×D5—22no—
246C2×C3×D5—22no—
247C2×C3×D5C2×C4⋊C414yes—
248C3×C4×C5—42no—
249C3×C4×C5C2×C4⋊C414yes—
250Dic15—24no—
251Dic15—41no—
252Dic15—41no—
253C3×Dic5—42no—
254C3×Dic5C2×C4⋊C414yes—
255C5×Dic3—24no—
256C5×Dic3—41no—
257C5×Dic3—41no—
258C24×C3—51no—
259C22×C3×C4—51no—
260C22×C3×C4—54no—
261C22×C3×C4—54no—
262C22×C3×C4—54no—
263C22×C3×C4—52no—
264C22×C3×C4C22×D512yes—
265C22×C3×C4C22×D511yescharacteristic
266C22×Dic3—52no—
267C22×Dic3—58no—
268C22×Dic3—51no—
269C22×Dic3—51no—
270C22×Dic3—202no—
271C22×Dic3—202no—
272C22×Dic3C22×D511yescharacteristic
273C22×Dic3C22×D511yescharacteristic
274C22×Dic3C22×D512yes—
275C3×C4⋊C4—22no—
276C3×C4⋊C4—102no—
277C3×C22⋊C4—104no—
278C3×C42—104no—
279#217—102no—
280#217—102no—
281C4⋊Dic3—54no—
282C4⋊Dic3—58no—
283C4⋊Dic3—102no—
284C4⋊Dic3—102no—
285C4⋊Dic3C4×D514yes—
286Dic3⋊C4—42no—
287Dic3⋊C4—202no—
288Dic3⋊C4—202no—
289Dic3⋊C4—202no—
290C4×Dic3—22no—
291C4×Dic3—102no—
292C23×C5C23×D311yescharacteristic
293C22×D5—21no—
294C22×D5—21no—
295C22×D5—21no—
296C22×D5—21no—
297C22×D5C2×Dic612yes—
298C22×D5C2×C4×D312yes—
299C22×D5C2×#6712yes—
300C2×C4×C5—22no—
301C2×C4×C5—22no—
302C2×C4×C5—22no—
303C2×C4×C5—34no—
304C2×C4×C5—62no—
305C2×C4×C5—62no—
306C2×C4×C5—61no—
307C2×C4×C5—61no—
308C2×C4×C5—61no—
309C2×C4×C5—61no—
310C2×C4×C5—61no—
311C2×C4×C5—61no—
312C2×C4×C5C2×Dic612yes—
313C2×C4×C5C2×C4×D312yes—
314C2×C4×C5C2×#6712yes—
315C2×Dic5—22no—
316C2×Dic5—22no—
317C2×Dic5—22no—
318C2×Dic5—34no—
319C2×Dic5—62no—
320C2×Dic5—62no—
321C2×Dic5—61no—
322C2×Dic5—61no—
323C2×Dic5—61no—
324C2×Dic5—61no—
325C2×Dic5—61no—
326C2×Dic5—61no—
327C2×Dic5C2×Dic612yes—
328C2×Dic5C2×C4×D312yes—
329C2×Dic5C2×#6712yes—
330C4×D5—128no—
331C4×D5Dic3⋊C4116yescomplemented
332C23×C4—51no—
333C23×C4—152no—
334#117—104no—
335#117—304no—
336C2×C4⋊C4—31no—
337C2×C4⋊C4—32no—
338C2×C4⋊C4—51no—
339C2×C4⋊C4—158no—
340C2×C4⋊C4—152no—
341C2×C4⋊C4—151no—
342C2×C4⋊C4—151no—
343C2×C4⋊C4—151no—
344C2×C4⋊C4—152no—
345C2×C4⋊C4—152no—
346C2×C4⋊C4—152no—
347C2×C4⋊C4D3×D511yescharacteristic
348C2×C22⋊C4—52no—
349C2×C22⋊C4—151no—
350C2×C22⋊C4—151no—
351C2×C42—31no—
352C2×C42—52no—
353C2×C42—151no—
354#95—32no—
355#95—152no—
356#95—152no—
357#95—152no—
358C2×C3×C5C22×C4⋊C411yescharacteristic
359C2×C3×C5#51811yescharacteristic
360C2×C3×C5#51911yescharacteristic
361C2×C3×C5#53711yescharacteristic
362C2×C3×C5#53711yescharacteristic
363C2×C3×C5#54411yescharacteristic
364C2×C3×C5#54411yescharacteristic
365C3×D5—24no—
366C23×C3—52no—
367C23×C3—52no—
368C23×C3—52no—
369C23×C3—101no—
370C23×C3—101no—
371C23×C3—101no—
372C23×C3—101no—
373C23×C3C23×D511yescharacteristic
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—52no—
378C2×C3×C4—52no—
379C2×C3×C4—52no—
380C2×C3×C4—516no—
381C2×C3×C4—102no—
382C2×C3×C4—102no—
383C2×C3×C4—102no—
384C2×C3×C4C2×C4×D512yes—
385C2×C3×C4D4×D512yes—
386C2×C3×C4Q8×D512yes—
387C2×Dic3—22no—
388C2×Dic3—22no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—54no—
396C2×Dic3—102no—
397C2×Dic3—102no—
398C2×Dic3—101no—
399C2×Dic3—101no—
400C2×Dic3—101no—
401C2×Dic3—101no—
402C2×Dic3—101no—
403C2×Dic3—101no—
404C2×Dic3—208no—
405C2×Dic3C2×C4×D514yes—
406C22×C5C22×Dic611yescharacteristic
407C22×C5C22×C4×D311yescharacteristic
408C22×C5#100911yescharacteristic
409C22×C5#100911yescharacteristic
410C22×C5#101011yescharacteristic
411C22×C5#101011yescharacteristic
412C22×C5C22×#6711yescharacteristic
413C2×D5—22no—
414C2×D5—22no—
415C2×D5—22no—
416C2×D5—22no—
417C2×D5—22no—
418C2×D5—22no—
419C2×D5C2×Dic3⋊C414yes—
420C4×C5—42no—
421C4×C5—64no—
422C4×C5—121no—
423C4×C5—121no—
424C4×C5C2×Dic3⋊C414yes—
425Dic5—42no—
426Dic5—64no—
427Dic5—121no—
428Dic5—121no—
429Dic5C2×Dic3⋊C414yes—
430C24—51no—
431C22×C4—32no—
432C22×C4—31no—
433C22×C4—31no—
434C22×C4—51no—
435C22×C4—54no—
436C22×C4—54no—
437C22×C4—54no—
438C22×C4—52no—
439C22×C4—158no—
440C22×C4—152no—
441C22×C4—151no—
442C22×C4—151no—
443C22×C4—602no—
444C22×C4—602no—
445C22×C4C2×D3×D512yes—
446C22×C4C2×D3×D511yescharacteristic
447C4⋊C4—22no—
448C4⋊C4—34no—
449C4⋊C4—102no—
450C4⋊C4—122no—
451C4⋊C4—158no—
452C4⋊C4—154no—
453C4⋊C4—302no—
454C4⋊C4—302no—
455C4⋊C4—602no—
456C4⋊C4—602no—
457C4⋊C4—602no—
458C22⋊C4—104no—
459C22⋊C4—302no—
460C22⋊C4—302no—
461C42—62no—
462C42—104no—
463C42—302no—
464C3×C5#231611yescharacteristic, complemented, Hall
465C22×C3—54no—
466C22×C3—102no—
467C22×C3—102no—
468C22×C3—102no—
469C22×C3—102no—
470C22×C3—102no—
471C22×C3—102no—
472C22×C3C22×C4×D511yescharacteristic
473C22×C3C2×D4×D511yescharacteristic
474C22×C3#414411yescharacteristic
475C22×C3C2×Q8×D511yescharacteristic
476C22×C3#414711yescharacteristic
477C22×C3#414911yescharacteristic
478C22×C3#415011yescharacteristic
479C3×C4—42no—
480C3×C4—54no—
481C3×C4—202no—
482C3×C4D5×C4⋊C414yes—
483Dic3—24no—
484Dic3—41no—
485Dic3—41no—
486Dic3—104no—
487Dic3—201no—
488Dic3—201no—
489C2×C5#551611yescharacteristic
490C2×C5#551911yescharacteristic
491C2×C5#551911yescharacteristic
492C2×C5#552111yescharacteristic
493C2×C5C22×Dic3⋊C411yescharacteristic
494C2×C5#583311yescharacteristic
495C2×C5#583311yescharacteristic
496D5—24no—
497C23—52no—
498C23—52no—
499C23—52no—
500C23—101no—
501C23—101no—
502C23—101no—
503C23—101no—
504C23C22×D3×D511yescharacteristic
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—34no—
509C2×C4—52no—
510C2×C4—52no—
511C2×C4—52no—
512C2×C4—516no—
513C2×C4—62no—
514C2×C4—62no—
515C2×C4—61no—
516C2×C4—61no—
517C2×C4—61no—
518C2×C4—61no—
519C2×C4—61no—
520C2×C4—61no—
521C2×C4—102no—
522C2×C4—102no—
523C2×C4—102no—
524C2×C4—154no—
525C2×C4—302no—
526C2×C4—302no—
527C2×C4—301no—
528C2×C4—301no—
529C2×C4—301no—
530C2×C4—301no—
531C2×C4—301no—
532C2×C4—301no—
533C2×C4—608no—
534C2×C4D5×Dic612yes—
535C2×C4C4×D3×D512yes—
536C2×C4D5×#6712yes—
537C2×C3—104no—
538C2×C3C2×D5×C4⋊C411yescharacteristic
539C2×C3#6587911yescharacteristic
540C2×C3#6590611yescharacteristic
541C2×C3#6594211yescharacteristic
542C2×C3#6596311yescharacteristic
543C2×C3#6602311yescharacteristic
544C2×C3#6606311yescharacteristic
545C5#8223211yescharacteristic, complemented, Sylow
546C22—54no—
547C22—102no—
548C22—102no—
549C22—102no—
550C22—102no—
551C22—102no—
552C22—102no—
553C22C2×D5×Dic611yescharacteristic
554C22C2×C4×D3×D511yescharacteristic
555C22#9202011yescharacteristic
556C22#9202711yescharacteristic
557C22#9202811yescharacteristic
558C22#9203511yescharacteristic
559C22C2×D5×#6711yescharacteristic
560C4—42no—
561C4—54no—
562C4—64no—
563C4—121no—
564C4—121no—
565C4—202no—
566C4—304no—
567C4—601no—
568C4—601no—
569C4D5×Dic3⋊C414yes—
570C3#1060792611yescharacteristic, complemented, Sylow
571C2—104no—
572C2#1175339711yescharacteristic
573C2#1175365911yescharacteristic
574C2#1175366011yescharacteristic
575C2#1175383711yescharacteristic
576C2C2×D5×Dic3⋊C411yescharacteristic
577C2#1175493411yescharacteristic
578C2#1175522311yescharacteristic
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#2316
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
4. #11751073C2semidirect
5. C5×#4972C2semidirect
8. #11748868C2semidirect
9. #11748858C2semidirect
10. #11748802C2semidirect
11. #11748801C2semidirect
33. D5×C4⋊Dic3C4semidirect
91. #4972D5semidirect
161. C3×C4×D5C4⋊C4semidirect
331. C4×D5Dic3⋊C4semidirect
464. C3×C5#2316semidirect
545. C5#82232semidirect
570. C3#10607926semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#40382
C2×D5×C4⋊Dic32
C2×#913572
#117510732
C5×#49722
#117488852
#117488782
#117488682
#117488582
#117488022
#117488012
#106079263
#822325