Finite Group Databasebeta

← #14327117

Subgroups — #14327117

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×#91782C211yesmaximal, characteristic
2C2×#91782C211yesmaximal, characteristic
3C2×C4×D60C211yesmaximal, characteristic
4C3×C5×#389C211yesmaximal, characteristic, complemented
5#11751090C212yesmaximal, complemented
6#11751088C212yesmaximal, complemented
7#11751085C211yesmaximal, characteristic, complemented
8#11751078C211yesmaximal, characteristic, complemented
9C4×#6712C211yesmaximal, characteristic, complemented
10#11750769C212yesmaximal, complemented
11D15×#95C212yesmaximal, complemented
12#10607934—31nomaximal, Hall
13C22×D60C2211yescharacteristic
14C22×C4×D15C2212yes—
15C22×C4×D15C2212yes—
16C22×C4×D15C2212yes—
17C2×C3×C5×C4⋊C4C2211yescharacteristic
18C2×C3×C5×C4⋊C4C2212yes—
19C2×C3×C5×C4⋊C4C2211yescharacteristic
20C2×C3×C42×C5C2211yescharacteristic
21C2×#6714C2212yes—
22C2×#6714C2212yes—
23C2×#6714C2212yes—
24C2×C4⋊Dic15C2211yescharacteristic
25C2×C4⋊Dic15C2212yes—
26C2×#6712C2211yescharacteristic
27C2×#6712C2211yescharacteristic
28C2×C4×Dic15C2211yescharacteristic
29C2×C4×Dic15C2211yescharacteristic
30#91782—24no—
31#91782C418yescomplemented
32C4×D60—24no—
33C3×C5×#95C2212yes—
34#91115C2211yescharacteristic
35#91115C2212yes—
36#91115C2212yes—
37#91115C2212yes—
38#91115C2211yescharacteristic
39#76203—51nomaximal, Hall
40C2×#4039—31no—
41C2×#4039—31no—
42C2×C4×D20—31no—
43C5×#389D311yescharacteristic, complemented
44#65278—32no—
45#65276—32no—
46#65274—31no—
47#65266—31no—
48C4×#683—31no—
49#64957—32no—
50D5×#95—32no—
51C23×D15C2312yes—
52C22×C3×C4×C5C2311yescharacteristic
53C22×C3×C4×C5C2312yes—
54C22×C3×C4×C5C2312yes—
55C22×C3×C4×C5C2311yescharacteristic
56C22×C3×C4×C5C2311yescharacteristic
57C22×Dic15C2311yescharacteristic
58C22×Dic15C2312yes—
59C22×Dic15C2311yescharacteristic
60C22×Dic15C2311yescharacteristic
61C22×Dic15C2311yescharacteristic
62C2×D60C2×C414yes—
63C2×D60D414yes—
64C2×D60Q814yes—
65C2×C4×D15—24no—
66C2×C4×D15—24no—
67C2×C4×D15—24no—
68C2×C4×D15—24no—
69C2×C4×D15—42no—
70C2×C4×D15—42no—
71C2×C4×D15—42no—
72C2×C4×D15—42no—
73C2×C4×D15C2×C418yescomplemented
74C3×C5×C4⋊C4—22no—
75C3×C5×C4⋊C4—42no—
76C3×C5×C4⋊C4C2×C414yescomplemented
77C3×C42×C5—22no—
78#6714—24no—
79#6714—24no—
80#6714C2×C418yescomplemented
81C4⋊Dic15—22no—
82C4⋊Dic15—42no—
83#6712—22no—
84#6712—22no—
85C4×Dic15—22no—
86C4×Dic15C2×C414yescomplemented
87C2×#893—51no—
88C2×#893—51no—
89C2×C4×D12—51no—
90C3×#389D511yescharacteristic, complemented
91#4989—52no—
92#4987—52no—
93#4984—51no—
94#4977—51no—
95C4×Dic3⋊C4—51no—
96#4668—52no—
97D3×#95—52no—
98C22×D20—31no—
99C22×C4×D5—32no—
100C22×C4×D5—32no—
101C22×C4×D5—32no—
102C2×C5×C4⋊C4C2×D311yescharacteristic
103C2×C5×C4⋊C4C2×D312yes—
104C2×C5×C4⋊C4C2×D311yescharacteristic
105C2×C42×C5C2×D311yescharacteristic
106C2×#685—32no—
107C2×#685—32no—
108C2×#685—32no—
109C2×C4⋊Dic5—31no—
110C2×C4⋊Dic5—32no—
111C2×#683—31no—
112C2×#683—31no—
113C2×C4×Dic5—31no—
114C2×C4×Dic5—31no—
115#4039—38no—
116#4039—64no—
117C4×D20—64no—
118C5×#95C2×D312yes—
119#3963—31no—
120#3963—32no—
121#3963—32no—
122#3963—32no—
123#3963—31no—
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—22no—
135C2×C3×C4×C5—22no—
136C2×C3×C4×C5—21no—
137C2×C3×C4×C5—21no—
138C2×C3×C4×C5—22no—
139C2×C3×C4×C5—21no—
140C2×C3×C4×C5—22no—
141C2×C3×C4×C5—21no—
142C2×C3×C4×C5—22no—
143C2×C3×C4×C5C22×C414yes—
144C2×C3×C4×C5C22×C412yes—
145C2×C3×C4×C5C2×D412yes—
146C2×C3×C4×C5C2×Q812yes—
147C2×C3×C4×C5#4112yes—
148C2×Dic15—21no—
149C2×Dic15—21no—
150C2×Dic15—21no—
151C2×Dic15—21no—
152C2×Dic15—21no—
153C2×Dic15—21no—
154C2×Dic15—21no—
155C2×Dic15—22no—
156C2×Dic15—21no—
157C2×Dic15—22no—
158C2×Dic15—21no—
159C2×Dic15—22no—
160C2×Dic15C22×C412yes—
161C2×Dic15C22×C412yes—
162C2×Dic15#4112yes—
163D60C4⋊C4116yescomplemented
164C4×D15—48no—
165C4×D15—48no—
166C4×D15—48no—
167C22×D12—51no—
168C22×C4×D3—52no—
169C22×C4×D3—52no—
170C22×C4×D3—52no—
171C2×C3×C4⋊C4C2×D511yescharacteristic
172C2×C3×C4⋊C4C2×D512yes—
173C2×C3×C4⋊C4C2×D511yescharacteristic
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⋊C4—51no—
181C2×Dic3⋊C4—51no—
182C2×C4×Dic3—51no—
183C2×C4×Dic3—51no—
184#893—58no—
185#893—104no—
186C4×D12—104no—
187C3×#95C2×D512yes—
188C3⋊#95—51no—
189C3⋊#95—52no—
190C3⋊#95—52no—
191C3⋊#95—52no—
192C3⋊#95—51no—
193C23×D5—32no—
194C22×C4×C5C22×D311yescharacteristic
195C22×C4×C5C22×D312yes—
196C22×C4×C5C22×D312yes—
197C22×C4×C5C22×D311yescharacteristic
198C22×C4×C5C22×D311yescharacteristic
199C22×Dic5—31no—
200C22×Dic5—31no—
201C22×Dic5—31no—
202C22×Dic5—32no—
203C22×Dic5—31no—
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—22no—
217C5×C4⋊C4—42no—
218C5×C4⋊C4C4×D314yescomplemented
219C42×C5—22no—
220#685—38no—
221#685—64no—
222#685—64no—
223C4⋊Dic5—62no—
224C4⋊Dic5—122no—
225#683—62no—
226#683—62no—
227C4×Dic5—34no—
228C4×Dic5—62no—
229C2×C4×D4—151no—
230C2×C4×D4—151no—
231C2×C4×D4—151no—
232#389—151no—
233#389D1511yescharacteristic, complemented
234#385—152no—
235#385—152no—
236#382—151no—
237#380—152no—
238C4×C4⋊C4—151no—
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—24no—
255C3×C4×C5—22no—
256C3×C4×C5—42no—
257C3×C4×C5—41no—
258C3×C4×C5C2×C4⋊C414yes—
259Dic15—22no—
260Dic15—22no—
261Dic15—22no—
262Dic15—42no—
263Dic15—41no—
264C23×D3—52no—
265C22×C3×C4C22×D512yes—
266C22×C3×C4C22×D511yescharacteristic
267C22×C3×C4C22×D512yes—
268C22×C3×C4C22×D511yescharacteristic
269C22×C3×C4C22×D511yescharacteristic
270C22×Dic3—51no—
271C22×Dic3—51no—
272C22×Dic3—51no—
273C22×Dic3—52no—
274C22×Dic3—51no—
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—22no—
288C3×C4⋊C4—42no—
289C3×C4⋊C4C4×D514yescomplemented
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—102no—
297Dic3⋊C4—102no—
298C4×Dic3—54no—
299C4×Dic3—102no—
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—22no—
310C2×C4×C5—22no—
311C2×C4×C5—21no—
312C2×C4×C5—21no—
313C2×C4×C5—22no—
314C2×C4×C5—21no—
315C2×C4×C5—22no—
316C2×C4×C5—21no—
317C2×C4×C5—22no—
318C2×C4×C5C2×C4×D314yes—
319C2×C4×C5C2×C4×D312yes—
320C2×C4×C5D3×D412yes—
321C2×C4×C5D3×Q812yes—
322C2×C4×C5#23912yes—
323C2×Dic5—32no—
324C2×Dic5—32no—
325C2×Dic5—32no—
326C2×Dic5—61no—
327C2×Dic5—61no—
328C2×Dic5—61no—
329C2×Dic5—61no—
330C2×Dic5—61no—
331C2×Dic5—61no—
332C2×Dic5—61no—
333C2×Dic5—62no—
334C2×Dic5—61no—
335C2×Dic5—62no—
336C2×Dic5—61no—
337C2×Dic5—62no—
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—151no—
350C2×C4⋊C4—151no—
351C2×C4⋊C4—151no—
352C2×C4⋊C4—152no—
353C2×C4⋊C4C2×D1511yescharacteristic
354C2×C4⋊C4C2×D1512yes—
355C2×C4⋊C4C2×D1511yescharacteristic
356C2×C22⋊C4—152no—
357C2×C22⋊C4—152no—
358C2×C22⋊C4—152no—
359C2×C42—151no—
360C2×C42—151no—
361C2×C42C2×D1511yescharacteristic
362#95—151no—
363#95—152no—
364#95—152no—
365#95—151no—
366#95—152no—
367#95C2×D1512yes—
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—22no—
386C2×C3×C4—22no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—22no—
390C2×C3×C4—21no—
391C2×C3×C4—22no—
392C2×C3×C4—21no—
393C2×C3×C4—22no—
394C2×C3×C4C2×C4×D514yes—
395C2×C3×C4C2×C4×D512yes—
396C2×C3×C4D4×D512yes—
397C2×C3×C4Q8×D512yes—
398C2×C3×C4#71312yes—
399C2×Dic3—52no—
400C2×Dic3—52no—
401C2×Dic3—52no—
402C2×Dic3—101no—
403C2×Dic3—101no—
404C2×Dic3—101no—
405C2×Dic3—101no—
406C2×Dic3—101no—
407C2×Dic3—101no—
408C2×Dic3—101no—
409C2×Dic3—102no—
410C2×Dic3—101no—
411C2×Dic3—102no—
412C2×Dic3—101no—
413C2×Dic3—102no—
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—24no—
433C4×C5—22no—
434C4×C5—42no—
435C4×C5—41no—
436C4×C5D3×C4⋊C414yes—
437Dic5—62no—
438Dic5—62no—
439Dic5—62no—
440Dic5—122no—
441Dic5—121no—
442C24—152no—
443C2×D4—154no—
444C2×D4—154no—
445C2×D4—154no—
446C22×C4—151no—
447C22×C4—151no—
448C22×C4—151no—
449C22×C4—158no—
450C22×C4—152no—
451C22×C4—151no—
452C22×C4—304no—
453C22×C4—304no—
454C22×C4—304no—
455C22×C4—304no—
456C22×C4—602no—
457C22×C4—602no—
458C22×C4—602no—
459C22×C4—602no—
460C22×C4C22×D1512yes—
461C22×C4C22×D1511yescharacteristic
462C22×C4C22×D1512yes—
463C22×C4C22×D1511yescharacteristic
464C22×C4C22×D1511yescharacteristic
465C4⋊C4—22no—
466C4⋊C4—42no—
467C4⋊C4—302no—
468C4⋊C4—302no—
469C4⋊C4—302no—
470C4⋊C4—602no—
471C4⋊C4C4×D1514yescomplemented
472C22⋊C4—158no—
473C22⋊C4—304no—
474C22⋊C4—304no—
475C42—22no—
476C42—154no—
477C42—302no—
478C3×C5#234911yescharacteristic, complemented, Hall
479C22×C3C22×C4×D511yescharacteristic
480C22×C3C2×D4×D511yescharacteristic
481C22×C3#414411yescharacteristic
482C22×C3C2×Q8×D511yescharacteristic
483C22×C3C2×#71311yescharacteristic
484C22×C3D5×#4111yescharacteristic
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—24no—
494C3×C4—22no—
495C3×C4—42no—
496C3×C4—41no—
497C3×C4D5×C4⋊C414yes—
498Dic3—102no—
499Dic3—102no—
500Dic3—102no—
501Dic3—202no—
502Dic3—201no—
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—
518C23C23×D1511yescharacteristic
519D4—1516no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—152no—
531C2×C4—152no—
532C2×C4—152no—
533C2×C4—301no—
534C2×C4—301no—
535C2×C4—301no—
536C2×C4—301no—
537C2×C4—301no—
538C2×C4—301no—
539C2×C4—301no—
540C2×C4—302no—
541C2×C4—301no—
542C2×C4—302no—
543C2×C4—301no—
544C2×C4—302no—
545C2×C4—608no—
546C2×C4—608no—
547C2×C4—608no—
548C2×C4C2×C4×D1514yes—
549C2×C4C2×C4×D1512yes—
550C2×C4D4×D1512yes—
551C2×C4Q8×D1512yes—
552C2×C4#682112yes—
553C2×C3C2×D5×C4⋊C411yescharacteristic
554C2×C3#6587911yescharacteristic
555C2×C3C4×#71311yescharacteristic
556C2×C3#6594611yescharacteristic
557C2×C3#6596411yescharacteristic
558C2×C3#6602411yescharacteristic
559C2×C3#6606411yescharacteristic
560D3—108no—
561C5#7620311yescharacteristic, complemented, Sylow
562C22—158no—
563C22—304no—
564C22—304no—
565C22—304no—
566C22—304no—
567C22—304no—
568C22—304no—
569C22C22×C4×D1511yescharacteristic
570C22C2×D4×D1511yescharacteristic
571C22#9209511yescharacteristic
572C22C2×Q8×D1511yescharacteristic
573C22C2×#682111yescharacteristic
574C22#41×D1511yescharacteristic
575C22#9210011yescharacteristic
576C4—24no—
577C4—22no—
578C4—42no—
579C4—41no—
580C4—302no—
581C4—302no—
582C4—302no—
583C4—602no—
584C4—601no—
585C4C4⋊C4×D1514yes—
586C3#1060793411yescharacteristic, complemented, Sylow
587C2—308no—
588C2C2×C4⋊C4×D1511yescharacteristic
589C2#1175630411yescharacteristic
590C2C4×#682111yescharacteristic
591C2#1175637111yescharacteristic
592C2#1175638911yescharacteristic
593C2#1175644911yescharacteristic
594C2#1175648911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D15
commutatorC22×C3×C5C23×C4
FittingC3×C5×#389C2
socle, cosocleC23×C3×C5C24
prime radicalC22×D60C22
hypercenter#389D15
hypocenterC3×C5#2349
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
4. C3×C5×#389C2semidirect
5. #11751090C2semidirect
6. #11751088C2semidirect
7. #11751085C2semidirect
8. #11751078C2semidirect
9. C4×#6712C2semidirect
10. #11750769C2semidirect
11. D15×#95C2semidirect
31. #91782C4semidirect
43. C5×#389D3semidirect
73. C2×C4×D15C2×C4semidirect
76. C3×C5×C4⋊C4C2×C4semidirect
80. #6714C2×C4semidirect
86. C4×Dic15C2×C4semidirect
90. C3×#389D5semidirect
163. D60C4⋊C4semidirect
218. C5×C4⋊C4C4×D3semidirect
233. #389D15semidirect
289. C3×C4⋊C4C4×D5semidirect
471. C4⋊C4C4×D15semidirect
478. C3×C5#2349semidirect
561. C5#76203semidirect
586. C3#10607934semidirect

Maximal subgroups

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

subgroupindex
C2×#917822
C2×C4×D602
C3×C5×#3892
#117510902
#117510882
#117510852
#117510782
C4×#67122
#117507692
D15×#952
#106079343
#762035