Finite Group Databasebeta

← #14319756

Subgroups — #14319756

Order 1920.

1560 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4039C211yesmaximal, characteristic
2C2×#91434C211yesmaximal, characteristic
3C2×Dic3×D20C211yesmaximal, characteristic
4C4⋊C4×Dic15C211yesmaximal, characteristic, complemented
5C5×Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
6#11748884C212yesmaximal, complemented
7Dic3×#685C212yesmaximal, complemented
8D5×C3⋊#95C212yesmaximal, complemented
9C4×#6665C212yesmaximal, complemented
10Dic5×C4⋊Dic3C211yesmaximal, characteristic, complemented
11#11748611C211yesmaximal, characteristic, complemented
12#10605482—31nomaximal, Hall
13C22×C3×D20C2211yescharacteristic
14C22×C3×C4×D5C2212yes—
15C22×D5×Dic3C2212yes—
16C22×D5×Dic3C2212yes—
17C2×C3×C5×C4⋊C4C2211yescharacteristic
18C2×C4⋊Dic15C2211yescharacteristic
19C2×C4×Dic15C2212yes—
20C2×C4×Dic15C2211yescharacteristic
21C2×C5×C4⋊Dic3C2211yescharacteristic
22C2×C4×C5×Dic3C2212yes—
23C2×C4×C5×Dic3C2211yescharacteristic
24C2×C3×#685C2212yes—
25C2×C3×C4×Dic5C2211yescharacteristic
26C3×#4039C418yescomplemented
27C2×#6665C2212yes—
28C2×#6665C2212yes—
29C2×Dic3×Dic5C2212yes—
30#91434C418yescomplemented
31Dic3×D20C418yescomplemented
32#91115C2212yes—
33C5×C3⋊#95C2212yes—
34#90994C2211yescharacteristic
35#90994C2211yescharacteristic
36#90994C2211yescharacteristic
37#90994C2211yescharacteristic
38#90994C2212yes—
39#82317—51nomaximal, Hall
40C2×#4039—31no—
41C2×#4039D311yescharacteristic
42C2×C4×D20—31no—
43C4×C5×C4⋊C4—31no—
44C4⋊C4×Dic5—31no—
45C4⋊C4×Dic5—31no—
46C4×#685—32no—
47C4×#685—32no—
48#65221—31no—
49#64955—32no—
50D5×#95—32no—
51C23×C3×D5C2312yes—
52C22×C3×C4×C5C2312yes—
53C22×C3×C4×C5C2311yescharacteristic
54C22×Dic15C2311yescharacteristic
55C22×Dic15C2311yescharacteristic
56C22×Dic15C2312yes—
57C22×C5×Dic3C2311yescharacteristic
58C22×C5×Dic3C2312yes—
59C22×C5×Dic3C2311yescharacteristic
60C22×C3×Dic5C2311yescharacteristic
61C22×C3×Dic5C2311yescharacteristic
62C2×C3×D20C2×C414yes—
63C2×C3×D20C2×C414yes—
64C2×C3×D20C2×C414yes—
65C2×C3×C4×D5—42no—
66C2×C3×C4×D5—42no—
67C2×C3×C4×D5C2×C418yescomplemented
68C2×D5×Dic3—42no—
69C2×D5×Dic3—42no—
70C2×D5×Dic3—42no—
71C2×D5×Dic3—42no—
72C2×D5×Dic3C2×C418yescomplemented
73C2×D5×Dic3C2×C418yescomplemented
74C3×C5×C4⋊C4C2×C414yescomplemented
75C4⋊Dic15C2×C414yescomplemented
76C4×Dic15—42no—
77C4×Dic15C2×C414yescomplemented
78C5×C4⋊Dic3C2×C414yescomplemented
79C4×C5×Dic3—42no—
80C4×C5×Dic3C2×C414yescomplemented
81C3×#685C2×C418yescomplemented
82C3×C4×Dic5C2×C414yescomplemented
83#6665C2×C418yescomplemented
84#6665C2×C418yescomplemented
85Dic3×Dic5—42no—
86C2×C3×C4×D4—51no—
87C2×D4×Dic3—51no—
88C2×D4×Dic3—51no—
89C4×#217—52no—
90C2×C3⋊#95—52no—
91Dic3×C4⋊C4—51no—
92Dic3×C4⋊C4D511yescharacteristic, complemented
93Dic3×C22⋊C4—52no—
94Dic3×C22⋊C4—52no—
95C4×C4⋊Dic3—51no—
96#4932—51no—
97C22×D20C2×D311yescharacteristic
98C22×C4×D5—32no—
99C22×C4×D5—32no—
100C22×C4×D5C2×D312yes—
101C2×C5×C4⋊C4—31no—
102C2×C5×C4⋊C4C2×D311yescharacteristic
103C2×C42×C5—31no—
104C2×C42×C5—32no—
105C2×#685—32no—
106C2×#685—32no—
107C2×#685C2×D312yes—
108C2×C4⋊Dic5—31no—
109C2×C4×Dic5—31no—
110C2×C4×Dic5—32no—
111C2×C4×Dic5—32no—
112C2×C4×Dic5C2×D311yescharacteristic
113#4039—38no—
114#4039Dic318yescomplemented
115C4×D20—38no—
116C5×#95—32no—
117#3963—32no—
118#3963—31no—
119#3963—32no—
120#3963—31no—
121#3963—31no—
122#3963—31no—
123#2275—151noSylow
124C23×C3×C5C2411yescharacteristic
125C22×C3×D5—22no—
126C22×C3×D5—22no—
127C22×C3×D5—22no—
128C22×C3×D5—22no—
129C22×C3×D5C22×C414yes—
130C22×C3×D5C22×C414yes—
131C22×C3×D5C22×C414yes—
132C2×C3×C4×C5—22no—
133C2×C3×C4×C5—22no—
134C2×C3×C4×C5C22×C414yes—
135C2×C3×C4×C5C22×C412yes—
136C2×C3×C4×C5C22×C412yes—
137C2×C3×C4×C5C22×C412yes—
138C2×Dic15—22no—
139C2×Dic15—21no—
140C2×Dic15—22no—
141C2×Dic15—21no—
142C2×Dic15—21no—
143C2×Dic15—21no—
144C2×Dic15C22×C412yes—
145C2×Dic15C22×C414yes—
146C2×Dic15C22×C412yes—
147C2×C5×Dic3—22no—
148C2×C5×Dic3—21no—
149C2×C5×Dic3—22no—
150C2×C5×Dic3—21no—
151C2×C5×Dic3—21no—
152C2×C5×Dic3—21no—
153C2×C5×Dic3C22×C412yes—
154C2×C5×Dic3C22×C414yes—
155C2×C5×Dic3C22×C412yes—
156C2×C3×Dic5—21no—
157C2×C3×Dic5—21no—
158C2×C3×Dic5—21no—
159C2×C3×Dic5—21no—
160C2×C3×Dic5C22×C412yes—
161C2×C3×Dic5C22×C412yes—
162C3×D20C42116yescomplemented
163C3×C4×D5—48no—
164D5×Dic3—48no—
165D5×Dic3—48no—
166C22×C3×D4—51no—
167C23×C3×C4—52no—
168C23×Dic3—52no—
169C23×Dic3—52no—
170C3×C4×D4—58no—
171C2×C3×C4⋊C4C2×D511yescharacteristic
172C2×C3×C22⋊C4—52no—
173C2×C3×C42—51no—
174C2×#217—52no—
175C2×#217—52no—
176D4×Dic3—58no—
177D4×Dic3—58no—
178C2×C4⋊Dic3—51no—
179C2×C4⋊Dic3C2×D511yescharacteristic
180C2×C4×Dic3—51no—
181C2×C4×Dic3—52no—
182C2×C4×Dic3—52no—
183C2×C4×Dic3C2×D512yes—
184C2×C4×Dic3C2×D511yescharacteristic
185C3⋊#95—52no—
186C3⋊#95—51no—
187C3⋊#95—51no—
188C3⋊#95—51no—
189C3⋊#95—51no—
190C3⋊#95—52no—
191C3⋊#95C2×D512yes—
192C23×D5C22×D312yes—
193C22×C4×C5—31no—
194C22×C4×C5—32no—
195C22×C4×C5—31no—
196C22×C4×C5C22×D312yes—
197C22×C4×C5C22×D311yescharacteristic
198C22×Dic5—31no—
199C22×Dic5—32no—
200C22×Dic5—31no—
201C22×Dic5C22×D311yescharacteristic
202C22×Dic5C22×D311yescharacteristic
203C2×D20C4×D314yes—
204C2×D20C4×D314yes—
205C2×D20C2×Dic314yes—
206C2×C4×D5—38no—
207C2×C4×D5—38no—
208C2×C4×D5—42no—
209C2×C4×D5—42no—
210C2×C4×D5—122no—
211C2×C4×D5—122no—
212C2×C4×D5—122no—
213C2×C4×D5—122no—
214C2×C4×D5C2×Dic318yescomplemented
215C5×C4⋊C4—34no—
216C5×C4⋊C4C2×Dic314yescomplemented
217C42×C5—34no—
218C42×C5—122no—
219#685—38no—
220#685—38no—
221#685C2×Dic318yescomplemented
222C4⋊Dic5—34no—
223C4×Dic5—34no—
224C4×Dic5—122no—
225C4×Dic5—122no—
226C4×Dic5C2×Dic314yescomplemented
227C2×C4×D4—51no—
228C2×C4×D4—151no—
229C2×C4×D4—151no—
230C4×C4⋊C4—31no—
231C4×C4⋊C4—151no—
232C4×C4⋊C4—151no—
233C4×C22⋊C4—152no—
234C4×C22⋊C4—152no—
235C4×C22⋊C4—152no—
236#376—151no—
237C2×#95—152no—
238C22×C3×C5C23×C411yescharacteristic
239C22×C3×C5C23×C411yescharacteristic
240C22×C3×C5C23×C411yescharacteristic
241C22×C3×C5H(5,𝔽2)11yescharacteristic
242C22×C3×C5H(5,𝔽2)11yescharacteristic
243C22×C3×C5H(5,𝔽2)11yescharacteristic
244C22×C3×C52−1+411yescharacteristic
245C2×C3×D5—24no—
246C2×C3×D5—24no—
247C2×C3×D5—24no—
248C2×C3×D5—24no—
249C2×C3×D5—24no—
250C2×C3×D5—24no—
251C2×C3×D5C2×C4218yes—
252C3×C4×C5—24no—
253C3×C4×C5C2×C4214yes—
254Dic15—22no—
255Dic15—24no—
256Dic15—22no—
257C3×Dic5—22no—
258C3×Dic5—22no—
259C5×Dic3—22no—
260C5×Dic3—24no—
261C5×Dic3—22no—
262C24×C3—52no—
263C2×C3×D4—54no—
264C2×C3×D4—54no—
265C2×C3×D4—54no—
266C22×C3×C4—51no—
267C22×C3×C4—51no—
268C22×C3×C4—58no—
269C22×C3×C4—202no—
270C22×C3×C4—202no—
271C22×C3×C4C22×D512yes—
272C22×C3×C4C22×D511yescharacteristic
273C22×Dic3—51no—
274C22×Dic3—51no—
275C22×Dic3—52no—
276C22×Dic3—58no—
277C22×Dic3—58no—
278C22×Dic3—202no—
279C22×Dic3—202no—
280C22×Dic3—202no—
281C22×Dic3—202no—
282C22×Dic3C22×D511yescharacteristic
283C22×Dic3C22×D511yescharacteristic
284C22×Dic3C22×D512yes—
285C3×C4⋊C4C4×D514yescomplemented
286C3×C22⋊C4—58no—
287C3×C42—54no—
288#217—58no—
289#217—58no—
290C4⋊Dic3—54no—
291C4⋊Dic3C4×D514yescomplemented
292C4×Dic3—42no—
293C4×Dic3—54no—
294C4×Dic3—202no—
295C4×Dic3—202no—
296C4×Dic3C4×D514yescomplemented
297C23×C5C23×D311yescharacteristic
298C22×D5—22no—
299C22×D5—22no—
300C22×D5—22no—
301C22×D5—22no—
302C22×D5C2×C4×D314yes—
303C22×D5C2×C4×D314yes—
304C22×D5C22×Dic314yes—
305C2×C4×C5—22no—
306C2×C4×C5—22no—
307C2×C4×C5—32no—
308C2×C4×C5—34no—
309C2×C4×C5—32no—
310C2×C4×C5—61no—
311C2×C4×C5—62no—
312C2×C4×C5—61no—
313C2×C4×C5—62no—
314C2×C4×C5—61no—
315C2×C4×C5—61no—
316C2×C4×C5C2×C4×D312yes—
317C2×C4×C5C2×C4×D312yes—
318C2×C4×C5C22×Dic314yes—
319C2×C4×C5C22×Dic312yes—
320C2×Dic5—21no—
321C2×Dic5—21no—
322C2×Dic5—21no—
323C2×Dic5—21no—
324C2×Dic5—32no—
325C2×Dic5—32no—
326C2×Dic5—34no—
327C2×Dic5—61no—
328C2×Dic5—62no—
329C2×Dic5—61no—
330C2×Dic5—62no—
331C2×Dic5—61no—
332C2×Dic5—61no—
333C2×Dic5C22×Dic312yes—
334C2×Dic5C22×Dic312yes—
335D20C4×Dic3116yescomplemented
336C4×D5—48no—
337C4×D5—128no—
338C4×D5—128no—
339C22×D4—51no—
340C23×C4—52no—
341C23×C4—152no—
342C23×C4—152no—
343C4×D4—58no—
344C4×D4—158no—
345C4×D4—158no—
346C2×C4⋊C4—31no—
347C2×C4⋊C4—151no—
348C2×C4⋊C4D3×D511yescharacteristic
349C2×C22⋊C4—52no—
350C2×C22⋊C4—152no—
351C2×C22⋊C4—152no—
352C2×C42—31no—
353C2×C42—32no—
354C2×C42—51no—
355C2×C42—151no—
356C2×C42—152no—
357C2×C42—152no—
358#95—32no—
359#95—151no—
360#95—152no—
361#95—151no—
362#95—151no—
363#95—151no—
364#95—152no—
365C2×C3×C5C22×C4211yescharacteristic
366C2×C3×C5#51811yescharacteristic
367C2×C3×C5#51811yescharacteristic
368C2×C3×C5#51811yescharacteristic
369C2×C3×C5#52011yescharacteristic
370C2×C3×C5#52011yescharacteristic
371C2×C3×C5#52011yescharacteristic
372C3×D5—28no—
373C23×C3—54no—
374C23×C3—54no—
375C23×C3—54no—
376C23×C3—102no—
377C23×C3—102no—
378C23×C3—102no—
379C23×C3—102no—
380C23×C3C23×D511yescharacteristic
381C3×D4—516no—
382C2×C3×C4—22no—
383C2×C3×C4—22no—
384C2×C3×C4—52no—
385C2×C3×C4—52no—
386C2×C3×C4—101no—
387C2×C3×C4—101no—
388C2×C3×C4—101no—
389C2×C3×C4—101no—
390C2×C3×C4—208no—
391C2×C3×C4C2×C4×D512yes—
392C2×C3×C4C2×C4×D512yes—
393C2×C3×C4C2×C4×D514yes—
394C2×C3×C4C2×C4×D512yes—
395C2×Dic3—21no—
396C2×Dic3—22no—
397C2×Dic3—21no—
398C2×Dic3—22no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—52no—
402C2×Dic3—52no—
403C2×Dic3—54no—
404C2×Dic3—101no—
405C2×Dic3—101no—
406C2×Dic3—101no—
407C2×Dic3—102no—
408C2×Dic3—102no—
409C2×Dic3—101no—
410C2×Dic3—208no—
411C2×Dic3—208no—
412C2×Dic3C2×C4×D512yes—
413C2×Dic3C2×C4×D512yes—
414C2×Dic3C2×C4×D514yes—
415C22×C5C22×C4×D311yescharacteristic
416C22×C5C22×C4×D311yescharacteristic
417C22×C5#100411yescharacteristic
418C22×C5#100411yescharacteristic
419C22×C5#100911yescharacteristic
420C22×C5#101011yescharacteristic
421C22×C5C23×Dic311yescharacteristic
422C2×D5—24no—
423C2×D5—24no—
424C2×D5—24no—
425C2×D5—24no—
426C2×D5—24no—
427C2×D5—24no—
428C2×D5C2×C4×Dic318yes—
429C4×C5—24no—
430C4×C5—62no—
431C4×C5—64no—
432C4×C5—62no—
433C4×C5C2×C4×Dic314yes—
434Dic5—22no—
435Dic5—22no—
436Dic5—62no—
437Dic5—64no—
438Dic5—62no—
439C24—52no—
440C2×D4—54no—
441C2×D4—54no—
442C2×D4—54no—
443C22×C4—31no—
444C22×C4—32no—
445C22×C4—31no—
446C22×C4—51no—
447C22×C4—51no—
448C22×C4—58no—
449C22×C4—158no—
450C22×C4—158no—
451C22×C4—151no—
452C22×C4—151no—
453C22×C4—152no—
454C22×C4—202no—
455C22×C4—202no—
456C22×C4—602no—
457C22×C4—602no—
458C22×C4—602no—
459C22×C4—602no—
460C22×C4C2×D3×D512yes—
461C22×C4C2×D3×D511yescharacteristic
462C4⋊C4—34no—
463C4⋊C4—154no—
464C4⋊C4D5×Dic314yescomplemented
465C22⋊C4—58no—
466C22⋊C4—158no—
467C22⋊C4—158no—
468C42—34no—
469C42—54no—
470C42—122no—
471C42—154no—
472C42—602no—
473C42—602no—
474C3×C5#227511yescharacteristic, complemented, Hall
475C22×C3—58no—
476C22×C3—104no—
477C22×C3—104no—
478C22×C3—104no—
479C22×C3—104no—
480C22×C3—104no—
481C22×C3—104no—
482C22×C3C22×C4×D511yescharacteristic
483C22×C3C22×C4×D511yescharacteristic
484C22×C3C22×C4×D511yescharacteristic
485C22×C3#414411yescharacteristic
486C22×C3#414711yescharacteristic
487C22×C3#414911yescharacteristic
488C22×C3#414911yescharacteristic
489C3×C4—24no—
490C3×C4—102no—
491C3×C4—102no—
492C3×C4C42×D514yes—
493Dic3—22no—
494Dic3—22no—
495Dic3—24no—
496Dic3—104no—
497Dic3—102no—
498Dic3—102no—
499C2×C5#554511yescharacteristic
500C2×C5#554511yescharacteristic
501C2×C5#554611yescharacteristic
502C2×C5#554611yescharacteristic
503C2×C5C22×C4×Dic311yescharacteristic
504C2×C5#579811yescharacteristic
505C2×C5#582711yescharacteristic
506D5—28no—
507C23—54no—
508C23—54no—
509C23—54no—
510C23—102no—
511C23—102no—
512C23—102no—
513C23—102no—
514C23C22×D3×D511yescharacteristic
515D4—516no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—32no—
519C2×C4—32no—
520C2×C4—34no—
521C2×C4—52no—
522C2×C4—52no—
523C2×C4—61no—
524C2×C4—62no—
525C2×C4—61no—
526C2×C4—61no—
527C2×C4—62no—
528C2×C4—61no—
529C2×C4—101no—
530C2×C4—101no—
531C2×C4—101no—
532C2×C4—101no—
533C2×C4—152no—
534C2×C4—152no—
535C2×C4—154no—
536C2×C4—208no—
537C2×C4—301no—
538C2×C4—301no—
539C2×C4—302no—
540C2×C4—301no—
541C2×C4—302no—
542C2×C4—301no—
543C2×C4—608no—
544C2×C4—608no—
545C2×C4C4×D3×D512yes—
546C2×C4C4×D3×D512yes—
547C2×C4C2×D5×Dic314yes—
548C2×C4C2×D5×Dic312yes—
549C2×C3—108no—
550C2×C3C2×C42×D511yescharacteristic
551C2×C3#6583811yescharacteristic
552C2×C3#6585511yescharacteristic
553C2×C3#6587911yescharacteristic
554C2×C3#6587911yescharacteristic
555C2×C3#6590811yescharacteristic
556C2×C3#6590811yescharacteristic
557C5#8231711yescharacteristic, complemented, Sylow
558C22—58no—
559C22—104no—
560C22—104no—
561C22—104no—
562C22—104no—
563C22—104no—
564C22—104no—
565C22C2×C4×D3×D511yescharacteristic
566C22C2×C4×D3×D511yescharacteristic
567C22#9202511yescharacteristic
568C22#9202611yescharacteristic
569C22#9202811yescharacteristic
570C22#9203511yescharacteristic
571C22C22×D5×Dic311yescharacteristic
572C4—24no—
573C4—62no—
574C4—64no—
575C4—62no—
576C4—102no—
577C4—102no—
578C4—304no—
579C4—302no—
580C4—302no—
581C4C4×D5×Dic314yes—
582C3#1060548211yescharacteristic, complemented, Sylow
583C2—108no—
584C2#1175339911yescharacteristic
585C2#1175340111yescharacteristic
586C2#1175383911yescharacteristic
587C2#1175384111yescharacteristic
588C2C2×C4×D5×Dic311yescharacteristic
589C2#1175491011yescharacteristic
590C2#1175521711yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC2×C3×C5C22×C42
FittingC2×C3×C5×C4⋊C4C22
socle, cosocleC23×C3×C5C24
prime radicalC22×C3×D20C22
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5#2275
LCSG ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. C4⋊C4×Dic15C2semidirect
5. C5×Dic3×C4⋊C4C2semidirect
6. #11748884C2semidirect
7. Dic3×#685C2semidirect
8. D5×C3⋊#95C2semidirect
9. C4×#6665C2semidirect
10. Dic5×C4⋊Dic3C2semidirect
11. #11748611C2semidirect
26. C3×#4039C4semidirect
30. #91434C4semidirect
31. Dic3×D20C4semidirect
67. C2×C3×C4×D5C2×C4semidirect
72. C2×D5×Dic3C2×C4semidirect
73. C2×D5×Dic3C2×C4semidirect
74. C3×C5×C4⋊C4C2×C4semidirect
75. C4⋊Dic15C2×C4semidirect
77. C4×Dic15C2×C4semidirect
78. C5×C4⋊Dic3C2×C4semidirect
80. C4×C5×Dic3C2×C4semidirect
81. C3×#685C2×C4semidirect
82. C3×C4×Dic5C2×C4semidirect
83. #6665C2×C4semidirect
84. #6665C2×C4semidirect
92. Dic3×C4⋊C4D5semidirect
114. #4039Dic3semidirect
162. C3×D20C42semidirect
214. C2×C4×D5C2×Dic3semidirect
216. C5×C4⋊C4C2×Dic3semidirect
221. #685C2×Dic3semidirect
226. C4×Dic5C2×Dic3semidirect
285. C3×C4⋊C4C4×D5semidirect
291. C4⋊Dic3C4×D5semidirect
296. C4×Dic3C4×D5semidirect
335. D20C4×Dic3semidirect
464. C4⋊C4D5×Dic3semidirect
474. C3×C5#2275semidirect
557. C5#82317semidirect
582. C3#10605482semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#40392
C2×#914342
C2×Dic3×D202
C4⋊C4×Dic152
C5×Dic3×C4⋊C42
#117488842
Dic3×#6852
D5×C3⋊#952
C4×#66652
Dic5×C4⋊Dic32
#117486112
#106054823
#823175