Finite Group Databasebeta

← #14327187

Subgroups — #14327187

Order 1920.

1408 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D60C211yesmaximal, characteristic
2C2×#91428C211yesmaximal, characteristic
3C2×#91428C211yesmaximal, characteristic
4C4×C5×Dic3⋊C4C211yesmaximal, characteristic, complemented
5C3×#65217C211yesmaximal, characteristic, complemented
6#11748894C212yesmaximal, complemented
7#11748888C212yesmaximal, complemented
8#11748876C212yesmaximal, complemented
9#11748857C211yesmaximal, characteristic, complemented
10#11748731C212yesmaximal, complemented
11#11748619C211yesmaximal, characteristic, complemented
12#10605501—31nomaximal, Hall
13C22×D60C2211yescharacteristic
14C22×C4×D15C2212yes—
15C22×#1200C2212yes—
16C22×#1200C2212yes—
17C2×C3×C42×C5C2211yescharacteristic
18C2×#6714C2212yes—
19C2×C4⋊Dic15C2211yescharacteristic
20C4×D60—24no—
21C2×C5×Dic3⋊C4C2211yescharacteristic
22C2×C5×Dic3⋊C4C2211yescharacteristic
23C2×C4×C5×Dic3C2211yescharacteristic
24C2×C4×C5×Dic3C2211yescharacteristic
25C2×C3×C4⋊Dic5C2211yescharacteristic
26C2×C3×C4⋊Dic5C2211yescharacteristic
27C2×C3×#683C2212yes—
28C2×#6668C2212yes—
29C2×#6667C2212yes—
30C2×#6667C2212yes—
31#91428—24no—
32#91428C418yescomplemented
33C5×C3⋊#95C2211yescharacteristic
34C5×C3⋊#95C2211yescharacteristic
35C3×#3963C2212yes—
36#90994C2212yes—
37#90994C2212yes—
38#90994C2212yes—
39#76203—51nomaximal, Hall
40C2×C4×D20—31no—
41C2×C4×D20—31no—
42C2×C4×D20—31no—
43C4×C5×C4⋊C4—31no—
44#65278—32no—
45#65278—32no—
46#65225—31no—
47#65217—31no—
48#65217D311yescharacteristic, complemented
49#64956—32no—
50D5×#95—32no—
51C23×D15C2312yes—
52C22×C3×C4×C5C2311yescharacteristic
53C22×C3×C4×C5C2311yescharacteristic
54C22×C3×C4×C5C2311yescharacteristic
55C22×Dic15C2312yes—
56C2×D60C2×C414yes—
57C2×D60D414yes—
58C2×D60Q814yes—
59C2×C4×D15—24no—
60C2×C4×D15—24no—
61C2×C4×D15—42no—
62C22×C5×Dic3C2311yescharacteristic
63C22×C5×Dic3C2311yescharacteristic
64C22×C5×Dic3C2311yescharacteristic
65C22×C5×Dic3C2311yescharacteristic
66C22×C3×Dic5C2312yes—
67C22×C3×Dic5C2312yes—
68C2×#1200—24no—
69C2×#1200—24no—
70C2×#1200—42no—
71C2×#1200—42no—
72C2×#1200—42no—
73C2×#1200C2×C418yescomplemented
74C3×C42×C5—22no—
75#6714—24no—
76C4⋊Dic15—22no—
77C5×Dic3⋊C4—22no—
78C5×Dic3⋊C4—22no—
79C4×C5×Dic3—22no—
80C4×C5×Dic3C2×C414yescomplemented
81C3×C4⋊Dic5—22no—
82C3×C4⋊Dic5C2×C414yescomplemented
83C3×#683—42no—
84#6668—42no—
85#6667—24no—
86#6667C2×C418yescomplemented
87C2×#893—51no—
88C2×#893—51no—
89C2×C4×D12—51no—
90C3×#389—51no—
91#4989—52no—
92#4987—52no—
93#4984—51no—
94#4977—51no—
95C4×Dic3⋊C4D511yescharacteristic, complemented
96#4668—52no—
97D3×#95—52no—
98C22×D20—31no—
99C22×C4×D5—32no—
100C22×C4×D5—32no—
101C22×C4×D5—32no—
102C2×C5×C4⋊C4—31no—
103C2×C5×C4⋊C4—31no—
104C2×C42×C5—31no—
105C2×C42×C5—31no—
106C2×C42×C5C2×D311yescharacteristic
107C2×#685—32no—
108C2×#685—32no—
109C2×#685—32no—
110C2×C4⋊Dic5—31no—
111C2×C4⋊Dic5C2×D311yescharacteristic
112C2×C4⋊Dic5C2×D311yescharacteristic
113C2×#683—32no—
114C2×#683C2×D312yes—
115C4×D20—38no—
116C4×D20—64no—
117C4×D20—64no—
118C5×#95—31no—
119C5×#95—31no—
120#3963—32no—
121#3963—32no—
122#3963—32no—
123#3963C2×D312yes—
124#2349—151noSylow
125C23×C3×C5C2411yescharacteristic
126C22×D15—22no—
127C22×D15—22no—
128C22×D15—22no—
129C22×D15—22no—
130C22×D15C22×C414yes—
131C22×D15C2×D414yes—
132C22×D15C2×Q814yes—
133C2×C3×C4×C5—21no—
134C2×C3×C4×C5—21no—
135C2×C3×C4×C5—21no—
136C2×C3×C4×C5—21no—
137C2×C3×C4×C5—21no—
138C2×C3×C4×C5C22×C412yes—
139C2×C3×C4×C5C2×D412yes—
140C2×C3×C4×C5C2×Q812yes—
141C2×C3×C4×C5#4112yes—
142C2×Dic15—22no—
143C2×Dic15—22no—
144C2×Dic15—22no—
145D60C4⋊C4116yescomplemented
146C4×D15—48no—
147C2×C5×Dic3—21no—
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×Dic3—21no—
155C2×C5×Dic3—21no—
156C2×C5×Dic3C22×C412yes—
157C2×C5×Dic3C22×C412yes—
158C2×C5×Dic3#4112yes—
159C2×C3×Dic5—22no—
160C2×C3×Dic5—22no—
161C2×C3×Dic5—22no—
162C2×C3×Dic5—22no—
163C2×C3×Dic5—22no—
164C2×C3×Dic5C22×C414yes—
165#1200—48no—
166#1200—48no—
167C22×D12—51no—
168C22×C4×D3—52no—
169C22×C4×D3—52no—
170C22×C4×D3—52no—
171C2×C3×C4⋊C4—51no—
172C2×C3×C4⋊C4—51no—
173C2×C3×C4⋊C4—52no—
174C2×C3×C42C2×D511yescharacteristic
175C2×(C2×D3)⋊C4—52no—
176C2×(C2×D3)⋊C4—52no—
177C2×(C2×D3)⋊C4—52no—
178C2×C4⋊Dic3—51no—
179C2×C4⋊Dic3—52no—
180C2×Dic3⋊C4C2×D511yescharacteristic
181C2×Dic3⋊C4C2×D511yescharacteristic
182C2×C4×Dic3C2×D511yescharacteristic
183C2×C4×Dic3C2×D511yescharacteristic
184#893—58no—
185#893—104no—
186C4×D12—104no—
187C3×#95—52no—
188C3⋊#95—52no—
189C3⋊#95—52no—
190C3⋊#95—52no—
191C3⋊#95C2×D511yescharacteristic
192C3⋊#95C2×D511yescharacteristic
193C23×D5—32no—
194C22×C4×C5—31no—
195C22×C4×C5—31no—
196C22×C4×C5—31no—
197C22×C4×C5—31no—
198C22×C4×C5C22×D311yescharacteristic
199C22×C4×C5C22×D311yescharacteristic
200C22×C4×C5C22×D311yescharacteristic
201C22×Dic5—32no—
202C22×Dic5C22×D312yes—
203C22×Dic5C22×D312yes—
204C2×D20—34no—
205C2×D20—34no—
206C2×D20—34no—
207C2×C4×D5—38no—
208C2×C4×D5—64no—
209C2×C4×D5—64no—
210C2×C4×D5—64no—
211C2×C4×D5—64no—
212C2×C4×D5—122no—
213C2×C4×D5—122no—
214C2×C4×D5—122no—
215C2×C4×D5—122no—
216C5×C4⋊C4—62no—
217C5×C4⋊C4—62no—
218C42×C5—22no—
219C42×C5—34no—
220C42×C5—62no—
221#685—38no—
222#685—64no—
223#685—64no—
224C4⋊Dic5—22no—
225C4⋊Dic5—62no—
226C4⋊Dic5C4×D314yescomplemented
227#683—42no—
228#683—122no—
229C2×C4×D4—151no—
230C2×C4×D4—151no—
231C2×C4×D4—151no—
232#389—51no—
233#389—151no—
234#385—152no—
235#385—152no—
236#382—151no—
237#380—152no—
238C4×C4⋊C4—31no—
239C2×#95—152no—
240C22×C3×C5C23×C411yescharacteristic
241C22×C3×C5C22×D411yescharacteristic
242C22×C3×C5C22×Q811yescharacteristic
243C22×C3×C5C2×#4111yescharacteristic
244C22×C3×C5C2×#4111yescharacteristic
245C22×C3×C5H(5,𝔽2)11yescharacteristic
246C22×C3×C5H(5,𝔽2)11yescharacteristic
247C2×D15—24no—
248C2×D15—24no—
249C2×D15—24no—
250C2×D15—24no—
251C2×D15—24no—
252C2×D15—24no—
253C2×D15C2×C4⋊C418yes—
254C3×C4×C5—22no—
255C3×C4×C5—41no—
256C3×C4×C5C2×C4⋊C414yes—
257Dic15—42no—
258C3×Dic5—24no—
259C3×Dic5—42no—
260C5×Dic3—22no—
261C5×Dic3—22no—
262C5×Dic3—22no—
263C5×Dic3—41no—
264C23×D3—52no—
265C22×C3×C4—52no—
266C22×C3×C4—52no—
267C22×C3×C4C22×D511yescharacteristic
268C22×C3×C4C22×D511yescharacteristic
269C22×C3×C4C22×D511yescharacteristic
270C22×Dic3—52no—
271C22×Dic3C22×D511yescharacteristic
272C22×Dic3C22×D511yescharacteristic
273C22×Dic3C22×D511yescharacteristic
274C22×Dic3C22×D511yescharacteristic
275C2×D12—54no—
276C2×D12—54no—
277C2×D12—54no—
278C2×C4×D3—58no—
279C2×C4×D3—104no—
280C2×C4×D3—104no—
281C2×C4×D3—104no—
282C2×C4×D3—104no—
283C2×C4×D3—202no—
284C2×C4×D3—202no—
285C2×C4×D3—202no—
286C2×C4×D3—202no—
287C3×C4⋊C4—54no—
288C3×C4⋊C4—102no—
289C3×C4⋊C4—202no—
290C3×C42—22no—
291(C2×D3)⋊C4—58no—
292(C2×D3)⋊C4—104no—
293(C2×D3)⋊C4—104no—
294C4⋊Dic3—102no—
295C4⋊Dic3—202no—
296Dic3⋊C4—22no—
297Dic3⋊C4—22no—
298C4×Dic3—22no—
299C4×Dic3C4×D514yescomplemented
300C23×C5C23×D311yescharacteristic
301C22×D5—34no—
302C22×D5—34no—
303C22×D5—34no—
304C22×D5—62no—
305C22×D5—62no—
306C22×D5—62no—
307C22×D5—62no—
308C2×C4×C5—21no—
309C2×C4×C5—21no—
310C2×C4×C5—21no—
311C2×C4×C5—21no—
312C2×C4×C5—21no—
313C2×C4×C5—32no—
314C2×C4×C5—32no—
315C2×C4×C5—32no—
316C2×C4×C5—61no—
317C2×C4×C5—61no—
318C2×C4×C5—61no—
319C2×C4×C5—61no—
320C2×C4×C5—61no—
321C2×C4×C5—61no—
322C2×C4×C5—61no—
323C2×C4×C5—61no—
324C2×C4×C5—61no—
325C2×C4×C5C2×C4×D312yes—
326C2×C4×C5D3×D412yes—
327C2×C4×C5D3×Q812yes—
328C2×C4×C5#23912yes—
329C2×Dic5—22no—
330C2×Dic5—22no—
331C2×Dic5—22no—
332C2×Dic5—22no—
333C2×Dic5—22no—
334C2×Dic5—62no—
335C2×Dic5—62no—
336C2×Dic5—62no—
337C2×Dic5C2×C4×D314yes—
338D20—316no—
339C4×D5—128no—
340C4×D5—128no—
341C4×D5—128no—
342C22×D4—151no—
343C23×C4—152no—
344C23×C4—152no—
345C23×C4—152no—
346C4×D4—158no—
347C4×D4—304no—
348C4×D4—304no—
349C2×C4⋊C4—31no—
350C2×C4⋊C4—31no—
351C2×C4⋊C4—51no—
352C2×C4⋊C4—51no—
353C2×C4⋊C4—52no—
354C2×C4⋊C4—151no—
355C2×C4⋊C4—152no—
356C2×C22⋊C4—152no—
357C2×C22⋊C4—152no—
358C2×C22⋊C4—152no—
359C2×C42—31no—
360C2×C42—31no—
361C2×C42D3×D511yescharacteristic
362#95—31no—
363#95—31no—
364#95—52no—
365#95—152no—
366#95—152no—
367#95—152no—
368C2×C3×C5C22×C4⋊C411yescharacteristic
369C2×C3×C5C4×#4111yescharacteristic
370C2×C3×C5#51811yescharacteristic
371C2×C3×C5#54611yescharacteristic
372C2×C3×C5#54611yescharacteristic
373C2×C3×C5#55411yescharacteristic
374C2×C3×C5#55411yescharacteristic
375D15—28no—
376C23×C3C23×D511yescharacteristic
377C22×D3—54no—
378C22×D3—54no—
379C22×D3—54no—
380C22×D3—102no—
381C22×D3—102no—
382C22×D3—102no—
383C22×D3—102no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—54no—
390C2×C3×C4—102no—
391C2×C3×C4—102no—
392C2×C3×C4—102no—
393C2×C3×C4—102no—
394C2×C3×C4—102no—
395C2×C3×C4C2×C4×D512yes—
396C2×C3×C4#70912yes—
397C2×C3×C4D4×D512yes—
398C2×C3×C4Q8×D512yes—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—102no—
409C2×Dic3—102no—
410C2×Dic3—102no—
411C2×Dic3C2×C4×D512yes—
412C2×Dic3C2×C4×D512yes—
413C2×Dic3#70912yes—
414D12—516no—
415C4×D3—208no—
416C4×D3—208no—
417C4×D3—208no—
418C22×C5C22×C4×D311yescharacteristic
419C22×C5C2×D3×D411yescharacteristic
420C22×C5#100411yescharacteristic
421C22×C5C2×D3×Q811yescharacteristic
422C22×C5C2×#23911yescharacteristic
423C22×C5D3×#4111yescharacteristic
424C22×C5#100911yescharacteristic
425C2×D5—38no—
426C2×D5—64no—
427C2×D5—64no—
428C2×D5—64no—
429C2×D5—64no—
430C2×D5—64no—
431C2×D5—64no—
432C4×C5—22no—
433C4×C5—41no—
434C4×C5—62no—
435C4×C5—62no—
436C4×C5—62no—
437C4×C5—121no—
438C4×C5D3×C4⋊C414yes—
439Dic5—24no—
440Dic5—42no—
441Dic5—122no—
442C24—152no—
443C2×D4—154no—
444C2×D4—154no—
445C2×D4—154no—
446C22×C4—31no—
447C22×C4—31no—
448C22×C4—31no—
449C22×C4—31no—
450C22×C4—52no—
451C22×C4—52no—
452C22×C4—158no—
453C22×C4—152no—
454C22×C4—304no—
455C22×C4—304no—
456C22×C4—304no—
457C22×C4—304no—
458C22×C4—602no—
459C22×C4—602no—
460C22×C4—602no—
461C22×C4—602no—
462C22×C4C2×D3×D511yescharacteristic
463C22×C4C2×D3×D511yescharacteristic
464C22×C4C2×D3×D511yescharacteristic
465C4⋊C4—54no—
466C4⋊C4—62no—
467C4⋊C4—62no—
468C4⋊C4—102no—
469C4⋊C4—202no—
470C4⋊C4—302no—
471C4⋊C4—602no—
472C22⋊C4—158no—
473C22⋊C4—304no—
474C22⋊C4—304no—
475C42—22no—
476C42—34no—
477C42—62no—
478C3×C5#234911yescharacteristic, complemented, Hall
479C22×C3C22×C4×D511yescharacteristic
480C22×C3C2×#70911yescharacteristic
481C22×C3C2×#70911yescharacteristic
482C22×C3C2×D4×D511yescharacteristic
483C22×C3C2×Q8×D511yescharacteristic
484C22×C3#414911yescharacteristic
485C22×C3#414911yescharacteristic
486C2×D3—58no—
487C2×D3—104no—
488C2×D3—104no—
489C2×D3—104no—
490C2×D3—104no—
491C2×D3—104no—
492C2×D3—104no—
493C3×C4—22no—
494C3×C4—41no—
495C3×C4—104no—
496C3×C4—202no—
497C3×C4D5×C4⋊C414yes—
498Dic3—22no—
499Dic3—22no—
500Dic3—22no—
501Dic3—41no—
502Dic3—202no—
503C2×C5C2×D3×C4⋊C411yescharacteristic
504C2×C5#554511yescharacteristic
505C2×C5C4×#23911yescharacteristic
506C2×C5#561211yescharacteristic
507C2×C5#563011yescharacteristic
508C2×C5#569011yescharacteristic
509C2×C5#573011yescharacteristic
510D5—68no—
511C23—154no—
512C23—154no—
513C23—154no—
514C23—302no—
515C23—302no—
516C23—302no—
517C23—302no—
518C23C22×D3×D511yescharacteristic
519D4—1516no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—32no—
526C2×C4—32no—
527C2×C4—32no—
528C2×C4—54no—
529C2×C4—61no—
530C2×C4—61no—
531C2×C4—61no—
532C2×C4—61no—
533C2×C4—61no—
534C2×C4—61no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—102no—
539C2×C4—102no—
540C2×C4—102no—
541C2×C4—102no—
542C2×C4—102no—
543C2×C4—302no—
544C2×C4—302no—
545C2×C4—302no—
546C2×C4—608no—
547C2×C4—608no—
548C2×C4—608no—
549C2×C4#676912yes—
550C2×C4D15⋊Q812yes—
551C2×C4C4×D3×D512yes—
552C2×C4#679012yes—
553C2×C3C4×#70911yescharacteristic
554C2×C3C2×D5×C4⋊C411yescharacteristic
555C2×C3#6585511yescharacteristic
556C2×C3#6588411yescharacteristic
557C2×C3#6588411yescharacteristic
558C2×C3#6591311yescharacteristic
559C2×C3#6591311yescharacteristic
560D3—108no—
561C5#7620311yescharacteristic, complemented, Sylow
562C22—158no—
563C22—304no—
564C22—304no—
565C22—304no—
566C22—304no—
567C22—304no—
568C22—304no—
569C22C2×#676911yescharacteristic
570C22C2×D15⋊Q811yescharacteristic
571C22C2×C4×D3×D511yescharacteristic
572C22D3×#70911yescharacteristic
573C22#9201711yescharacteristic
574C22#9201911yescharacteristic
575C22C2×#679011yescharacteristic
576C4—22no—
577C4—41no—
578C4—62no—
579C4—62no—
580C4—62no—
581C4—104no—
582C4—121no—
583C4—202no—
584C4—602no—
585C4#9140414yes—
586C3#1060550111yescharacteristic, complemented, Sylow
587C2—308no—
588C2C4×#676911yescharacteristic
589C2#1175261611yescharacteristic
590C2#1175264511yescharacteristic
591C2#1175272711yescharacteristic
592C2C2×#9140411yescharacteristic
593C2#1175438211yescharacteristic
594C2#1175466711yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C42×C5C22
socle, cosocleC23×C3×C5C24
prime radicalC22×D60C22
hypercenterC2×C42D3×D5
hypocenterC3×C5#2349
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
4. C4×C5×Dic3⋊C4C2semidirect
5. C3×#65217C2semidirect
6. #11748894C2semidirect
7. #11748888C2semidirect
8. #11748876C2semidirect
9. #11748857C2semidirect
10. #11748731C2semidirect
11. #11748619C2semidirect
32. #91428C4semidirect
48. #65217D3semidirect
73. C2×#1200C2×C4semidirect
80. C4×C5×Dic3C2×C4semidirect
82. C3×C4⋊Dic5C2×C4semidirect
86. #6667C2×C4semidirect
95. C4×Dic3⋊C4D5semidirect
145. D60C4⋊C4semidirect
226. C4⋊Dic5C4×D3semidirect
299. C4×Dic3C4×D5semidirect
478. C3×C5#2349semidirect
561. C5#76203semidirect
586. C3#10605501semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D602
C2×#914282
C4×C5×Dic3⋊C42
C3×#652172
#117488942
#117488882
#117488762
#117488572
#117487312
#117486192
#106055013
#762035