Finite Group Databasebeta

← #14453112

Subgroups — #14453112

Order 1920.

692 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5534C211yesmaximal, characteristic, complemented
2C3×#65868C211yesmaximal, characteristic, complemented
3#11751100C211yesmaximal, characteristic, complemented
4#11748563C211yesmaximal, characteristic, complemented
5#11748547C211yesmaximal, characteristic, complemented
6#11748545C211yesmaximal, characteristic, complemented
7#11748523C211yesmaximal, characteristic
8#11747069C211yesmaximal, characteristic, complemented
9#11747061C211yesmaximal, characteristic, complemented
10#11747034C211yesmaximal, characteristic, complemented
11#11747027C211yesmaximal, characteristic, complemented
12#11746927C211yesmaximal, characteristic, complemented
13#11746918C211yesmaximal, characteristic, complemented
14#11746882C211yesmaximal, characteristic, complemented
15#11746880C211yesmaximal, characteristic, complemented
16#10615200—31nomaximal, Hall
17C3×C5×#117C2211yescharacteristic
18C2×#6710C2211yescharacteristic
19C22×C3⋊#168C2211yescharacteristic
20C5×#930C2211yescharacteristic
21C5×#924C2211yescharacteristic
22C5×#898—21no—
23C5×#896—21no—
24C5×#885—21no—
25C5×#883—21no—
26C5×#875C2211yescharacteristic
27C4×C5×D12C2211yescharacteristic
28C5×#869C2211yescharacteristic
29C4×C5×Dic6C2211yescharacteristic
30C3×#4076C2211yescharacteristic
31C3×#4070C2211yescharacteristic
32C3×#4044—21no—
33C3×#4042—21no—
34C3×#4031—21no—
35C3×#4029—21no—
36C3×#4021C2211yescharacteristic
37C3×C4×D20C2211yescharacteristic
38C3×#4015C2211yescharacteristic
39C3×C4×Dic10C2211yescharacteristic
40C2×#6661C2211yescharacteristic
41C2×#6656C2211yescharacteristic
42C2×#6655C2211yescharacteristic
43C2×#6652C2211yescharacteristic
44C4⋊(C3⋊#168)C2211yescharacteristic
45C4⋊(C3⋊#168)C2211yescharacteristic
46C4⋊(C3⋊#168)C2211yescharacteristic
47C4⋊(C3⋊#168)C2211yescharacteristic
48#90986C2211yescharacteristic
49#90986C2211yescharacteristic
50#90986C2211yescharacteristic
51#90986C2211yescharacteristic
52#90972C2211yescharacteristic
53#90972C2211yescharacteristic
54#90970C2211yescharacteristic
55#90970C2211yescharacteristic
56#90966C2211yescharacteristic
57#90966C2211yescharacteristic
58#90964C2211yescharacteristic
59#90964C2211yescharacteristic
60#83469—51nomaximal, Hall
61C5×#542—31no—
62#65868D311yescharacteristic, complemented
63#65383—31no—
64#65373—31no—
65#65346—31no—
66#65343—31no—
67#65341—31no—
68#65337—31no—
69#65330—31no—
70#65328—31no—
71#65317—31no—
72#65309—31no—
73#65305—31no—
74#65297—31no—
75#65288—31no—
76C22×C3×C4×C5C2311yescharacteristic
77C2×C5×#67—21no—
78C2×C5×D12C2311yescharacteristic
79C2×C4×C5×D3—21no—
80C2×C5×Dic6—21no—
81C2×C5×Dic6C2311yescharacteristic
82C2×C3×C22⋊D5—21no—
83C2×C3×D20C2311yescharacteristic
84C2×C3×C4×D5—21no—
85C2×C3×Dic10—21no—
86C2×C3×Dic10C2311yescharacteristic
87C3×C5×C4⋊C4D411yescharacteristic
88C3×C5×C4⋊C4D411yescharacteristic
89C3×C5×C22⋊C4D411yescharacteristic
90C3×C5×C22⋊C4D411yescharacteristic
91C3×C42×C5C2311yescharacteristic
92C3×C42×C5C2311yescharacteristic
93#6710—41no—
94C2×C3⋊#168—41no—
95C2×C3⋊#168C2311yescharacteristic
96C2×C3⋊#168C2311yescharacteristic
97C2×C3⋊#168C2311yescharacteristic
98C2×C3⋊#168C2311yescharacteristic
99C5×#217—21no—
100C5×(C2×D3)⋊C4—21no—
101C5×(C2×D3)⋊C4—21no—
102C5×(C2×D3)⋊C4—21no—
103C5×(C2×D3)⋊C4—21no—
104C5×C4⋊Dic3—21no—
105C5×C4⋊Dic3C2311yescharacteristic
106C5×C4⋊Dic3C2311yescharacteristic
107C5×Dic3⋊C4—21no—
108C5×Dic3⋊C4—21no—
109C4×C5×Dic3—21no—
110C3×C22⋊Dic5—21no—
111C3×#685—21no—
112C3×#685—21no—
113C3×#685—21no—
114C3×#685—21no—
115C3×C4⋊Dic5—21no—
116C3×C4⋊Dic5C2311yescharacteristic
117C3×C4⋊Dic5C2311yescharacteristic
118C3×#683—21no—
119C3×#683—21no—
120C3×C4×Dic5—21no—
121#6661—41no—
122#6656—41no—
123#6655—41no—
124#6652—41no—
125C3×#542—51no—
126#5534D511yescharacteristic, complemented
127#5094—51no—
128#5084—51no—
129#5057—51no—
130#5054—51no—
131#5052—51no—
132#5048—51no—
133#5041—51no—
134#5039—51no—
135#5028—51no—
136#5020—51no—
137#5016—51no—
138#5008—51no—
139#4999—51no—
140C5×C4⋊Q8—31no—
141C5×#126—61no—
142C5×#124—31no—
143C5×#124—61no—
144C5×#123—61no—
145C5×C22⋊Q8—31no—
146C5×C22⋊Q8—61no—
147C5×#121—31no—
148C4×C5×Q8—31no—
149C4×C5×D4—31no—
150C5×#117C2×D311yescharacteristic
151C2×#689—31no—
152C2×#688—31no—
153C2×#687—31no—
154C2×#686—31no—
155#4076C2×D311yescharacteristic
156#4070C2×D311yescharacteristic
157C2×#681—31no—
158C22×#168—31no—
159#4044—21no—
160#4042—21no—
161#4031—21no—
162#4029—21no—
163#4021C2×D311yescharacteristic
164C4×D20C2×D311yescharacteristic
165#4015C2×D311yescharacteristic
166C4×Dic10C2×D311yescharacteristic
167#3967—31no—
168#3967—31no—
169#3964—31no—
170#3964—31no—
171#3942—31no—
172#3942—31no—
173#3941—31no—
174#3941—31no—
175#3940—31no—
176#3940—31no—
177#3940—31no—
178#3940—31no—
179C4⋊#168—31no—
180C4⋊#168—31no—
181C4⋊#168—31no—
182C4⋊#168—31no—
183#3263—151noSylow
184C23×C3×C5C2×D411yescharacteristic
185C22×C5×D3—21no—
186C22×C3×D5—21no—
187C2×C3×C4×C5—21no—
188C2×C3×C4×C5—21no—
189C2×C3×C4×C5C2×D411yescharacteristic
190C2×C3×C4×C5C2×D411yescharacteristic
191C2×C3×C4×C5C2×D411yescharacteristic
192C2×C3×C4×C5C2×D411yescharacteristic
193C2×C3×C4×C5C2×D411yescharacteristic
194C2×C3×C4×C5C2411yescharacteristic
195C5×#67—81no—
196C2×C5×Dic3—21no—
197C2×C5×Dic3—21no—
198C2×C5×Dic3—21no—
199C5×D12—21no—
200C5×D12—21no—
201C4×C5×D3—81no—
202C5×Dic6—21no—
203C5×Dic6—21no—
204C5×Dic6—81no—
205C3×C22⋊D5—81no—
206C2×C3×Dic5—21no—
207C2×C3×Dic5—21no—
208C2×C3×Dic5—21no—
209C3×D20—21no—
210C3×D20—21no—
211C3×C4×D5—81no—
212C3×Dic10—21no—
213C3×Dic10—21no—
214C3×Dic10—81no—
215C3⋊#168—21no—
216C3⋊#168—21no—
217C3⋊#168—21no—
218C3⋊#168—21no—
219C3×C4⋊Q8—51no—
220C3×#126—101no—
221C3×#124—51no—
222C3×#124—101no—
223C3×#123—101no—
224C3×C22⋊Q8—51no—
225C3×C22⋊Q8—101no—
226C3×#121—51no—
227C3×C4×Q8—51no—
228C3×C4×D4—51no—
229C3×#117C2×D511yescharacteristic
230C2×#216—51no—
231C2×#215—51no—
232C2×#214—51no—
233C2×#213—51no—
234#930C2×D511yescharacteristic
235#924C2×D511yescharacteristic
236C2×#208—51no—
237C22×C3⋊C8—51no—
238#898—21no—
239#896—21no—
240#885—21no—
241#883—21no—
242#875C2×D511yescharacteristic
243C4×D12C2×D511yescharacteristic
244#869C2×D511yescharacteristic
245C4×Dic6C2×D511yescharacteristic
246#835—51no—
247#835—51no—
248#832—51no—
249#832—51no—
250#810—51no—
251#810—51no—
252#809—51no—
253#809—51no—
254#808—51no—
255#808—51no—
256#808—51no—
257#808—51no—
258C4⋊(C3⋊C8)—51no—
259C4⋊(C3⋊C8)—51no—
260C4⋊(C3⋊C8)—51no—
261C4⋊(C3⋊C8)—51no—
262C2×C5×Q8—31no—
263C2×C5×Q8—61no—
264C2×C5×D4—31no—
265C2×C5×D4—61no—
266C22×C4×C5—61no—
267C22×C4×C5C22×D311yescharacteristic
268C2×C22⋊D5—21no—
269C2×D20C22×D311yescharacteristic
270C2×C4×D5—21no—
271C2×Dic10—21no—
272C2×Dic10C22×D311yescharacteristic
273C5×C4⋊C4—31no—
274C5×C4⋊C4—31no—
275C5×C4⋊C4—61no—
276C5×C4⋊C4—61no—
277C5×C4⋊C4—61no—
278C5×C4⋊C4#6711yescharacteristic
279C5×C4⋊C4#6711yescharacteristic
280C5×C22⋊C4—61no—
281C5×C22⋊C4—61no—
282C5×C22⋊C4—61no—
283C5×C22⋊C4—61no—
284C5×C22⋊C4—61no—
285C5×C22⋊C4#6711yescharacteristic
286C5×C22⋊C4#6711yescharacteristic
287C42×C5—61no—
288C42×C5C22×D311yescharacteristic
289C42×C5C22×D311yescharacteristic
290C22⋊Dic5—21no—
291#689—121no—
292#688—121no—
293#687—121no—
294#686—121no—
295#685—21no—
296#685—21no—
297#685—21no—
298#685—21no—
299C4⋊Dic5—21no—
300C4⋊Dic5C22×D311yescharacteristic
301C4⋊Dic5C22×D311yescharacteristic
302#683—21no—
303#683—21no—
304C4×Dic5—21no—
305#681—121no—
306C2×#168—31no—
307C2×#168—31no—
308C2×#168—31no—
309C2×#168—31no—
310C2×#168—121no—
311#542—31no—
312#542—51no—
313#476—151no—
314#476—151no—
315#475—151no—
316#475—151no—
317#470—151no—
318#469—151no—
319#465—151no—
320#465—151no—
321#464—151no—
322#464—151no—
323#463—151no—
324#463—151no—
325#424—151no—
326C22×C3×C5—21no—
327C22×C3×C5—21no—
328C22×C3×C5—21no—
329C22×C3×C5C22×D411yescharacteristic
330C2×C5×D3—41no—
331C2×C5×D3—41no—
332C2×C5×D3—41no—
333C2×C3×D5—41no—
334C2×C3×D5—41no—
335C2×C3×D5—41no—
336C3×C4×C5—21no—
337C3×C4×C5—21no—
338C3×C4×C5—21no—
339C3×C4×C5—21no—
340C3×C4×C5—21no—
341C3×C4×C5H(5,𝔽2)11yescharacteristic
342C3×C4×C52−1+411yescharacteristic
343C3×Dic5—41no—
344C3×Dic5—41no—
345C3×Dic5—41no—
346C5×Dic3—41no—
347C5×Dic3—41no—
348C5×Dic3—41no—
349C2×C3×Q8—51no—
350C2×C3×Q8—101no—
351C2×C3×D4—51no—
352C2×C3×D4—101no—
353C22×C3×C4—101no—
354C22×C3×C4C22×D511yescharacteristic
355C2×#67—21no—
356C2×D12C22×D511yescharacteristic
357C2×C4×D3—21no—
358C2×Dic6—21no—
359C2×Dic6C22×D511yescharacteristic
360C3×C4⋊C4—51no—
361C3×C4⋊C4—51no—
362C3×C4⋊C4—101no—
363C3×C4⋊C4—101no—
364C3×C4⋊C4—101no—
365C3×C4⋊C4C22⋊D511yescharacteristic
366C3×C4⋊C4C22⋊D511yescharacteristic
367C3×C22⋊C4—101no—
368C3×C22⋊C4—101no—
369C3×C22⋊C4—101no—
370C3×C22⋊C4—101no—
371C3×C22⋊C4—101no—
372C3×C22⋊C4C22⋊D511yescharacteristic
373C3×C22⋊C4C22⋊D511yescharacteristic
374C3×C42—101no—
375C3×C42C22×D511yescharacteristic
376C3×C42C22×D511yescharacteristic
377#217—21no—
378#216—201no—
379#215—201no—
380#214—201no—
381#213—201no—
382(C2×D3)⋊C4—21no—
383(C2×D3)⋊C4—21no—
384(C2×D3)⋊C4—21no—
385(C2×D3)⋊C4—21no—
386C4⋊Dic3—21no—
387C4⋊Dic3C22×D511yescharacteristic
388C4⋊Dic3C22×D511yescharacteristic
389Dic3⋊C4—21no—
390Dic3⋊C4—21no—
391C4×Dic3—21no—
392#208—201no—
393C2×C3⋊C8—51no—
394C2×C3⋊C8—51no—
395C2×C3⋊C8—51no—
396C2×C3⋊C8—51no—
397C2×C3⋊C8—201no—
398C23×C5—61no—
399C23×C5C2×#6711yescharacteristic
400C22×D5—21no—
401C5×Q8—61no—
402C5×Q8—61no—
403C5×Q8—241no—
404C5×D4—61no—
405C5×D4—61no—
406C5×D4—241no—
407C2×C4×C5—21no—
408C2×C4×C5—21no—
409C2×C4×C5—61no—
410C2×C4×C5—61no—
411C2×C4×C5—61no—
412C2×C4×C5—241no—
413C2×C4×C5C2×#6711yescharacteristic
414C2×C4×C5C2×#6711yescharacteristic
415C2×C4×C5C2×#6711yescharacteristic
416C2×C4×C5C2×#6711yescharacteristic
417C2×C4×C5C2×#6711yescharacteristic
418C2×C4×C5C23×D311yescharacteristic
419C22⋊D5—81no—
420C2×Dic5—21no—
421C2×Dic5—21no—
422C2×Dic5—21no—
423D20—21no—
424D20—21no—
425C4×D5—81no—
426Dic10—21no—
427Dic10—21no—
428Dic10—81no—
429#168—61no—
430#168—61no—
431#168—61no—
432#168—61no—
433C2×Q16—151no—
434C2×SD16—151no—
435C2×SD16—151no—
436C2×D8—151no—
437C2×M16—151no—
438C22×C8—151no—
439C4⋊Q8—31no—
440C4⋊Q8—51no—
441#126—61no—
442#126—101no—
443#124—31no—
444#124—51no—
445#124—61no—
446#124—101no—
447#123—61no—
448#123—101no—
449C22⋊Q8—31no—
450C22⋊Q8—51no—
451C22⋊Q8—61no—
452C22⋊Q8—101no—
453#121—31no—
454#121—51no—
455C4×Q8—31no—
456C4×Q8—51no—
457C4×D4—31no—
458C4×D4—51no—
459#117D3×D511yescharacteristic
460#106—151no—
461#106—151no—
462#106—151no—
463#106—151no—
464C4⋊C8—151no—
465C4⋊C8—151no—
466C4⋊C8—151no—
467C4⋊C8—151no—
468#103—151no—
469#103—151no—
470#103—151no—
471#103—151no—
472#102—151no—
473#102—151no—
474#102—151no—
475#102—151no—
476C2×C3×C5—41no—
477C2×C3×C5#53711yescharacteristic
478C2×C3×C5#57711yescharacteristic
479C2×C3×C5#57711yescharacteristic
480C3×D5—81no—
481C5×D3—81no—
482C23×C3—101no—
483C23×C3C2×C22⋊D511yescharacteristic
484C22×D3—21no—
485C3×Q8—101no—
486C3×Q8—101no—
487C3×Q8—401no—
488C3×D4—101no—
489C3×D4—101no—
490C3×D4—401no—
491C2×C3×C4—21no—
492C2×C3×C4—21no—
493C2×C3×C4—101no—
494C2×C3×C4—101no—
495C2×C3×C4—101no—
496C2×C3×C4—401no—
497C2×C3×C4C2×C22⋊D511yescharacteristic
498C2×C3×C4C2×C22⋊D511yescharacteristic
499C2×C3×C4C2×C22⋊D511yescharacteristic
500C2×C3×C4C2×C22⋊D511yescharacteristic
501C2×C3×C4C2×C22⋊D511yescharacteristic
502C2×C3×C4C23×D511yescharacteristic
503#67—81no—
504C2×Dic3—21no—
505C2×Dic3—21no—
506C2×Dic3—21no—
507D12—21no—
508D12—21no—
509C4×D3—81no—
510Dic6—21no—
511Dic6—21no—
512Dic6—81no—
513C3⋊C8—101no—
514C3⋊C8—101no—
515C3⋊C8—101no—
516C3⋊C8—101no—
517C22×C5—21no—
518C22×C5—21no—
519C22×C5—21no—
520C22×C5—121no—
521C22×C5—121no—
522C22×C5—121no—
523C22×C5C22×#6711yescharacteristic
524C2×D5—41no—
525C2×D5—41no—
526C2×D5—41no—
527C4×C5—21no—
528C4×C5—21no—
529C4×C5—21no—
530C4×C5—21no—
531C4×C5—21no—
532C4×C5—121no—
533C4×C5—121no—
534C4×C5—121no—
535C4×C5#100911yescharacteristic
536C4×C5#101011yescharacteristic
537Dic5—41no—
538Dic5—41no—
539Dic5—41no—
540C2×Q8—31no—
541C2×Q8—51no—
542C2×Q8—61no—
543C2×Q8—101no—
544C2×D4—31no—
545C2×D4—51no—
546C2×D4—61no—
547C2×D4—101no—
548C22×C4—61no—
549C22×C4—101no—
550C22×C4C2×D3×D511yescharacteristic
551Q16—601no—
552SD16—601no—
553SD16—601no—
554D8—601no—
555M16—601no—
556C2×C8—151no—
557C2×C8—151no—
558C2×C8—151no—
559C2×C8—151no—
560C2×C8—601no—
561C4⋊C4—31no—
562C4⋊C4—31no—
563C4⋊C4—51no—
564C4⋊C4—51no—
565C4⋊C4—61no—
566C4⋊C4—61no—
567C4⋊C4—61no—
568C4⋊C4—101no—
569C4⋊C4—101no—
570C4⋊C4—101no—
571C4⋊C4#120111yescharacteristic
572C4⋊C4#120111yescharacteristic
573C22⋊C4—61no—
574C22⋊C4—61no—
575C22⋊C4—61no—
576C22⋊C4—61no—
577C22⋊C4—61no—
578C22⋊C4—101no—
579C22⋊C4—101no—
580C22⋊C4—101no—
581C22⋊C4—101no—
582C22⋊C4—101no—
583C22⋊C4#120111yescharacteristic
584C22⋊C4#120111yescharacteristic
585C42—61no—
586C42—101no—
587C42C2×D3×D511yescharacteristic
588C42C2×D3×D511yescharacteristic
589C3×C5#326311yescharacteristic, complemented, Hall
590C22×C3—21no—
591C22×C3—21no—
592C22×C3—21no—
593C22×C3—201no—
594C22×C3—201no—
595C22×C3—201no—
596C22×C3C22×C22⋊D511yescharacteristic
597C2×D3—41no—
598C2×D3—41no—
599C2×D3—41no—
600C3×C4—21no—
601C3×C4—21no—
602C3×C4—21no—
603C3×C4—21no—
604C3×C4—21no—
605C3×C4—201no—
606C3×C4—201no—
607C3×C4—201no—
608C3×C4#414911yescharacteristic
609C3×C4#415011yescharacteristic
610Dic3—41no—
611Dic3—41no—
612Dic3—41no—
613C2×C5—41no—
614C2×C5—241no—
615C2×C5#583311yescharacteristic
616C2×C5#583611yescharacteristic
617C2×C5#583711yescharacteristic
618D5—81no—
619C23—61no—
620C23—101no—
621C23C2×#120111yescharacteristic
622Q8—61no—
623Q8—61no—
624Q8—101no—
625Q8—101no—
626Q8—241no—
627Q8—401no—
628D4—61no—
629D4—61no—
630D4—101no—
631D4—101no—
632D4—241no—
633D4—401no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—61no—
637C2×C4—61no—
638C2×C4—61no—
639C2×C4—101no—
640C2×C4—101no—
641C2×C4—101no—
642C2×C4—241no—
643C2×C4—401no—
644C2×C4C2×#120111yescharacteristic
645C2×C4C2×#120111yescharacteristic
646C2×C4C2×#120111yescharacteristic
647C2×C4C2×#120111yescharacteristic
648C2×C4C2×#120111yescharacteristic
649C2×C4C22×D3×D511yescharacteristic
650C8—301no—
651C8—301no—
652C8—301no—
653C8—301no—
654C2×C3—41no—
655C2×C3—401no—
656C2×C3#6616711yescharacteristic
657C2×C3#6617011yescharacteristic
658C2×C3#6617111yescharacteristic
659D3—81no—
660C5#8346911yescharacteristic, complemented, Sylow
661C22—21no—
662C22—21no—
663C22—21no—
664C22—121no—
665C22—121no—
666C22—121no—
667C22—201no—
668C22—201no—
669C22—201no—
670C22C22×#120111yescharacteristic
671C4—21no—
672C4—21no—
673C4—21no—
674C4—21no—
675C4—21no—
676C4—121no—
677C4—121no—
678C4—121no—
679C4—201no—
680C4—201no—
681C4—201no—
682C4#9200111yescharacteristic
683C4#9201911yescharacteristic
684C3#1061520011yescharacteristic, complemented, Sylow
685C2—41no—
686C2—241no—
687C2—401no—
688C2#1175460711yescharacteristic
689C2#1175530311yescharacteristic
690C2#1175530411yescharacteristic
subgroupG/H
centerC22C22×#1201
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C5×#117C22
socleC22×C3×C5C22×D4
prime radical#11748523C2
hypercenter#117D3×D5
hypocenterC3×C5#3263
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ #117

Complements

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

NH ≅ G/Ntype
1. C5×#5534C2semidirect
2. C3×#65868C2semidirect
3. #11751100C2semidirect
4. #11748563C2semidirect
5. #11748547C2semidirect
6. #11748545C2semidirect
8. #11747069C2semidirect
9. #11747061C2semidirect
10. #11747034C2semidirect
11. #11747027C2semidirect
12. #11746927C2semidirect
13. #11746918C2semidirect
14. #11746882C2semidirect
15. #11746880C2semidirect
62. #65868D3semidirect
126. #5534D5semidirect
589. C3×C5#3263semidirect
660. C5#83469semidirect
684. C3#10615200semidirect

Maximal subgroups

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

subgroupindex
C5×#55342
C3×#658682
#117511002
#117485632
#117485472
#117485452
#117485232
#117470692
#117470612
#117470342
#117470272
#117469272
#117469182
#117468822
#117468802
#106152003
#834695