Finite Group Databasebeta

← #14430746

Subgroups — #14430746

Order 1920.

768 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#4036C211yesmaximal, characteristic, complemented
2#11753398C211yesmaximal, characteristic
3#11750978C211yesmaximal, characteristic, complemented
4C5×#4881C211yesmaximal, characteristic, complemented
5C3×#65180C211yesmaximal, characteristic, complemented
6#11748138C211yesmaximal, characteristic
7#11747852C211yesmaximal, characteristic, complemented
8#11747805C211yesmaximal, characteristic, complemented
9#11747590C211yesmaximal, characteristic, complemented
10#11747503C211yesmaximal, characteristic
11#11747389C211yesmaximal, characteristic, complemented
12#11747286C211yesmaximal, characteristic, complemented
13#11747259C211yesmaximal, characteristic, complemented
14#11747191C211yesmaximal, characteristic, complemented
15#11747187C211yesmaximal, characteristic, complemented
16#10610972—31nomaximal, Hall
17C2×#6765C2211yescharacteristic
18#91779C2211yescharacteristic
19#91776C2211yescharacteristic
20C2×C5×ΓL(1,𝔽25)C2211yescharacteristic
21C5×#892C2211yescharacteristic
22C5×D3×C4⋊C4C2211yescharacteristic
23C3×#4039C2211yescharacteristic
24C3×#4036C2211yescharacteristic
25#91442—21no—
26#91440—21no—
27#91434C2211yescharacteristic
28D3×C4⋊Dic5—21no—
29D3×C4⋊Dic5C2211yescharacteristic
30D3×#683—21no—
31C4×D3×Dic5C2211yescharacteristic
32#91393—21no—
33C4×D5×Dic3—21no—
34#91388C2211yescharacteristic
35#91384C2211yescharacteristic
36#91346—21no—
37#91343—21no—
38#91333C2211yescharacteristic
39C2×#6650C2211yescharacteristic
40C3×C5×#107C2211yescharacteristic
41#91100C2211yescharacteristic
42#91098C2211yescharacteristic
43#91094C2211yescharacteristic
44#91092C2211yescharacteristic
45C5×#817C2211yescharacteristic
46C5×#808C2211yescharacteristic
47C5×#807C2211yescharacteristic
48C3×#3953C2211yescharacteristic
49C3×#3946C2211yescharacteristic
50C3×#3941C2211yescharacteristic
51C3×#3940C2211yescharacteristic
52#90989C2211yescharacteristic
53#90988C2211yescharacteristic
54#90983C2211yescharacteristic
55#90982C2211yescharacteristic
56#90975C2211yescharacteristic
57#90974C2211yescharacteristic
58#90965C2211yescharacteristic
59#90964C2211yescharacteristic
60#81167—51nomaximal, Hall
61#65856—31no—
62C2×#4036—31no—
63C5×#428—31no—
64#65422—31no—
65#65407—31no—
66#65284—31no—
67#65255—31no—
68#65180—31no—
69#65180D311yescharacteristic, complemented
70#65166—31no—
71#65166—31no—
72#65160—31no—
73#65160—31no—
74#65146—31no—
75#65146—31no—
76C2×Dic30C2311yescharacteristic
77C2×C4×C5×D3—21no—
78C2×C4×C5×D3C2311yescharacteristic
79C2×C3×D20C2311yescharacteristic
80C2×C3×C4×D5—21no—
81C2×#1201—21no—
82C2×D3×Dic5—21no—
83C2×D3×Dic5D411yescharacteristic
84C2×D3×Dic5D411yescharacteristic
85C2×D5×Dic3—21no—
86C2×D5×Dic3—21no—
87#6765—22no—
88#6765Q814yes—
89C2×C3×C5×C8C2311yescharacteristic
90C3×C5×C4⋊C4C2311yescharacteristic
91C3×C5×C4⋊C4C2311yescharacteristic
92C4⋊Dic15—21no—
93C4⋊Dic15C2311yescharacteristic
94#6712—21no—
95#6712—21no—
96C4×Dic15—21no—
97C4×Dic15C2311yescharacteristic
98C2×C3⋊#168C2311yescharacteristic
99C5×(C2×D3)⋊C4—21no—
100C5×C4⋊Dic3C2311yescharacteristic
101C5×C4⋊Dic3C2311yescharacteristic
102C5×Dic3⋊C4—21no—
103C4×C5×Dic3—21no—
104C2×C5×C3⋊C8C2311yescharacteristic
105C5×ΓL(1,𝔽25)—22no—
106C3×#685—21no—
107C3×C4⋊Dic5—21no—
108C3×C4⋊Dic5C2311yescharacteristic
109C3×#683—21no—
110C3×C4×Dic5C2311yescharacteristic
111C2×C3×#168C2311yescharacteristic
112Dic3⋊Dic5—21no—
113Dic3⋊Dic5—21no—
114Dic3⋊Dic5—21no—
115#6669—21no—
116#6669—21no—
117#6668—21no—
118#6665—21no—
119Dic3×Dic5—21no—
120Dic3×Dic5D411yescharacteristic
121Dic3×Dic5D411yescharacteristic
122#6650—22no—
123D3×#125—51no—
124C4×#237—51no—
125C3×#478—51no—
126#5063—51no—
127#5009—51no—
128#4881D511yescharacteristic, complemented
129#4877—51no—
130#4861—51no—
131#4857—51no—
132#4837—51no—
133#4829—51no—
134#4771—51no—
135#4771—51no—
136#4766—51no—
137#4763—51no—
138C2×#709—31no—
139C2×C5×M16—31no—
140C5×#117—31no—
141C2×C5×C4⋊C4—31no—
142C2×C4⋊Dic5—31no—
143C2×C4⋊Dic5—61no—
144C2×#683—61no—
145C2×C4×Dic5—31no—
146C2×#681—31no—
147#4039—31no—
148#4039C2×D311yescharacteristic
149#4036—31no—
150#4036—31no—
151#4036—31no—
152#4036—61no—
153#4036—61no—
154#4036C2×D311yescharacteristic
155#4033—31no—
156#4033—31no—
157#4027—61no—
158#4027—61no—
159#4018—61no—
160C42×D5—61no—
161C5×#107—31no—
162C5×#107C2×D311yescharacteristic
163C5×#106—31no—
164C5×#106—31no—
165#3953—31no—
166#3953C2×D311yescharacteristic
167#3948—31no—
168#3948—31no—
169#3946—31no—
170#3946—31no—
171#3946—31no—
172#3946C2×D311yescharacteristic
173#3942—31no—
174#3942—31no—
175#3941—31no—
176#3941C2×D311yescharacteristic
177#3940—31no—
178#3940C2×D311yescharacteristic
179#3939—31no—
180#3939—31no—
181#3080—151noSylow
182C22×C5×D3C2×D411yescharacteristic
183C22×C3×D5—21no—
184C2×C3×C4×C5—21no—
185C2×C3×C4×C5—21no—
186C2×C3×C4×C5C2411yescharacteristic
187C2×Dic15—21no—
188C2×Dic15—21no—
189C2×Dic15C2×D411yescharacteristic
190C2×Dic15C2×D411yescharacteristic
191Dic30—21no—
192Dic30C2×Q812yes—
193C2×C5×Dic3—21no—
194C2×C5×Dic3—21no—
195C2×C5×Dic3C2×D411yescharacteristic
196C4×C5×D3—42no—
197C4×C5×D3C2×Q812yes—
198C4×C5×D3#4112yes—
199C2×C3×Dic5—21no—
200C2×C3×Dic5C2×D411yescharacteristic
201C2×C3×Dic5C2×D411yescharacteristic
202C3×D20—21no—
203C3×D20C2×Q812yes—
204C3×C4×D5—42no—
205#1201—24no—
206D3×Dic5—41no—
207D3×Dic5—41no—
208D3×Dic5—42no—
209D5×Dic3—24no—
210D5×Dic3—42no—
211C3×C5×C8—21no—
212C3⋊#168—21no—
213C3×#168—21no—
214C5×C3⋊C8—21no—
215C2×#237—51no—
216C3×#125—51no—
217C3×C4×D4—51no—
218D4×Dic3—51no—
219C4×#67—101no—
220C2×C4×Dic3—101no—
221C2×ΓL(1,𝔽25)—51no—
222C2×ΓL(1,𝔽25)C2×D511yescharacteristic
223#892C2×D511yescharacteristic
224D3×C4⋊C4—51no—
225D3×C4⋊C4—101no—
226D3×C4⋊C4—101no—
227D3×C4⋊C4C2×D511yescharacteristic
228#890—51no—
229#890—51no—
230#889—101no—
231#889—101no—
232#887—51no—
233#67⋊C4—101no—
234#877—101no—
235C42×D3—51no—
236#870—51no—
237C4×Dic6—51no—
238C3×#107C2×D511yescharacteristic
239C3×#106—51no—
240C3×C4⋊C8—51no—
241C3×#102—51no—
242C3×#102—51no—
243#832—51no—
244#832—51no—
245#818—51no—
246#817C2×D511yescharacteristic
247#816—51no—
248#816—51no—
249Dic3⋊C8—51no—
250Dic3⋊C8—51no—
251#810—51no—
252#810—51no—
253#808—51no—
254#808C2×D511yescharacteristic
255#807—51no—
256#807C2×D511yescharacteristic
257C4⋊(C3⋊C8)—51no—
258C22×C4×C5—31no—
259C22×C4×C5—61no—
260C2×C22⋊D5—61no—
261C22×Dic5—31no—
262C22×Dic5—31no—
263C22×Dic5—61no—
264#709—34no—
265#709—62no—
266C2×D20C22×D311yescharacteristic
267C2×C4×D5—21no—
268C2×C4×D5—61no—
269C2×C4×D5—61no—
270C2×Dic10—31no—
271C5×M16—62no—
272C2×C5×C8—31no—
273C2×C5×C8C22×D311yescharacteristic
274C5×C4⋊C4—31no—
275C5×C4⋊C4—31no—
276C5×C4⋊C4—61no—
277C5×C4⋊C4C22×D311yescharacteristic
278C5×C4⋊C4C22×D311yescharacteristic
279C5×C22⋊C4—61no—
280C42×C5—61no—
281#685—21no—
282#685—61no—
283C4⋊Dic5—21no—
284C4⋊Dic5—31no—
285C4⋊Dic5—61no—
286C4⋊Dic5—61no—
287C4⋊Dic5—61no—
288C4⋊Dic5—61no—
289C4⋊Dic5C22×D311yescharacteristic
290#683—21no—
291#683—61no—
292#683—61no—
293#683—61no—
294#683—61no—
295#683—61no—
296C4×Dic5—31no—
297C4×Dic5—31no—
298C4×Dic5—31no—
299C4×Dic5—61no—
300C4×Dic5—61no—
301C4×Dic5C22×D311yescharacteristic
302#681—62no—
303C2×#168—31no—
304C2×#168C22×D311yescharacteristic
305C2×#125—151no—
306C4×#41—151no—
307#479—151no—
308#479—151no—
309#479—151no—
310#479—151no—
311#478—51no—
312#478—151no—
313#478—151no—
314#478—151no—
315#428—31no—
316#428—151no—
317#423—151no—
318#417—151no—
319#417—151no—
320C22×C3×C5C22×D411yescharacteristic
321C2×C5×D3—21no—
322C2×C5×D3—21no—
323C2×C5×D3C22⋊Q812yes—
324C2×C3×D5—22no—
325C2×C3×D5—41no—
326C2×C3×D5—41no—
327C3×C4×C5—41no—
328C3×C4×C5—41no—
329C3×C4×C5C22×Q811yescharacteristic
330C3×C4×C5C2×#4111yescharacteristic
331Dic15—22no—
332Dic15—21no—
333Dic15—21no—
334Dic15—41no—
335C3×Dic5—21no—
336C3×Dic5—21no—
337C3×Dic5—41no—
338C5×Dic3—41no—
339C5×Dic3—41no—
340C5×Dic3C22⋊Q812yes—
341C2×C3×D4—51no—
342C22×C3×C4—101no—
343C2×#67—101no—
344C22×Dic3—101no—
345C22×Dic3—101no—
346#237—54no—
347#237—102no—
348C2×C4×D3—21no—
349C2×C4×D3—51no—
350C2×C4×D3—51no—
351C2×C4×D3—101no—
352C2×C4×D3C22×D511yescharacteristic
353C2×Dic6—51no—
354C2×C3×C8—51no—
355C2×C3×C8C22×D511yescharacteristic
356C3×C4⋊C4—51no—
357C3×C4⋊C4—101no—
358C3×C4⋊C4—101no—
359C3×C4⋊C4C22×D511yescharacteristic
360C3×C4⋊C4C22×D511yescharacteristic
361C3×C22⋊C4—101no—
362C3×C42—51no—
363#217—101no—
364(C2×D3)⋊C4—21no—
365C4⋊Dic3—51no—
366C4⋊Dic3—101no—
367C4⋊Dic3—101no—
368C4⋊Dic3C22×D511yescharacteristic
369C4⋊Dic3C22×D511yescharacteristic
370Dic3⋊C4—21no—
371Dic3⋊C4—101no—
372Dic3⋊C4—101no—
373Dic3⋊C4—101no—
374Dic3⋊C4—101no—
375Dic3⋊C4—101no—
376Dic3⋊C4—101no—
377Dic3⋊C4—101no—
378C4×Dic3—21no—
379C4×Dic3—51no—
380C4×Dic3—51no—
381C4×Dic3—51no—
382C4×Dic3—101no—
383C4×Dic3—101no—
384C2×C3⋊C8—51no—
385C2×C3⋊C8C22×D511yescharacteristic
386ΓL(1,𝔽25)—22no—
387ΓL(1,𝔽25)—102no—
388C23×C5—31no—
389C22×D5—21no—
390C2×C4×C5—21no—
391C2×C4×C5—21no—
392C2×C4×C5—31no—
393C2×C4×C5—32no—
394C2×C4×C5—32no—
395C2×C4×C5—61no—
396C2×C4×C5—61no—
397C2×C4×C5—122no—
398C2×C4×C5C23×D311yescharacteristic
399C22⋊D5—64no—
400C2×Dic5—21no—
401C2×Dic5—31no—
402C2×Dic5—31no—
403C2×Dic5—61no—
404C2×Dic5—61no—
405C2×Dic5—121no—
406C2×Dic5—121no—
407C2×Dic5—122no—
408C2×Dic5D3×D411yescharacteristic
409C2×Dic5D3×D411yescharacteristic
410D20—21no—
411D20D3×Q812yes—
412C4×D5—42no—
413C4×D5—64no—
414C4×D5—122no—
415Dic10—32no—
416Dic10—61no—
417C5×C8—21no—
418C5×C8—61no—
419#168—21no—
420#168—61no—
421C2×#41—151no—
422C2×M16—31no—
423C2×M16—151no—
424#125—51no—
425#125—151no—
426#125—151no—
427#125—151no—
428#125—301no—
429#125—301no—
430C4×Q8—151no—
431C4×Q8—151no—
432C4×D4—51no—
433C4×D4—151no—
434C4×D4—301no—
435C4×D4—301no—
436#117—31no—
437#117—301no—
438C2×C4⋊C4—31no—
439C2×C4⋊C4—151no—
440C2×C4⋊C4—301no—
441C2×C4⋊C4—301no—
442C2×C42—151no—
443C2×C42—301no—
444#107—31no—
445#107—151no—
446#107—151no—
447#107D3×D511yescharacteristic
448#106—31no—
449#106—31no—
450#106—51no—
451#106—151no—
452C4⋊C8—51no—
453C4⋊C8—151no—
454C4⋊C8—151no—
455C4⋊C8—151no—
456#103—151no—
457#103—151no—
458#103—151no—
459#103—151no—
460#102—51no—
461#102—51no—
462#102—151no—
463#102—151no—
464C2×C3×C5C2×C22⋊Q811yescharacteristic
465C2×C3×C5#57511yescharacteristic
466C2×C3×C5#57511yescharacteristic
467C3×D5—42no—
468C5×D3—22no—
469C23×C3—101no—
470C22×D3D4×D511yescharacteristic
471C3×D4—52no—
472C3×D4—101no—
473C2×C3×C4—21no—
474C2×C3×C4—21no—
475C2×C3×C4—51no—
476C2×C3×C4—51no—
477C2×C3×C4—101no—
478C2×C3×C4—202no—
479C2×C3×C4C23×D511yescharacteristic
480#67—104no—
481C2×Dic3—21no—
482C2×Dic3—21no—
483C2×Dic3—51no—
484C2×Dic3—51no—
485C2×Dic3—101no—
486C2×Dic3—104no—
487C2×Dic3—101no—
488C2×Dic3—202no—
489C2×Dic3D4×D511yescharacteristic
490C4×D3—42no—
491C4×D3—201no—
492C4×D3—201no—
493C4×D3—202no—
494C4×D3#70912yes—
495C4×D3Q8×D512yes—
496Dic6—52no—
497Dic6—101no—
498C3×C8—21no—
499C3×C8—101no—
500C3⋊C8—21no—
501C3⋊C8—101no—
502C22×C5—32no—
503C22×C5—61no—
504C22×C5—61no—
505C22×C5C2×D3×D411yescharacteristic
506C2×D5—22no—
507C2×D5—41no—
508C2×D5—41no—
509C4×C5—32no—
510C4×C5—41no—
511C4×C5—41no—
512C4×C5—121no—
513C4×C5—121no—
514C4×C5C2×D3×Q811yescharacteristic
515C4×C5D3×#4111yescharacteristic
516Dic5—21no—
517Dic5—21no—
518Dic5—41no—
519Dic5—61no—
520Dic5—61no—
521Dic5—62no—
522Dic5—121no—
523#41—154no—
524#41—302no—
525C2×Q8—151no—
526C2×D4—51no—
527C2×D4—301no—
528C22×C4—31no—
529C22×C4—61no—
530C22×C4—101no—
531C22×C4—151no—
532C22×C4—151no—
533C22×C4—301no—
534C22×C4—301no—
535C22×C4—301no—
536M16—62no—
537M16—302no—
538C2×C8—31no—
539C2×C8—51no—
540C2×C8—151no—
541C2×C8C2×D3×D511yescharacteristic
542C4⋊C4—31no—
543C4⋊C4—31no—
544C4⋊C4—51no—
545C4⋊C4—61no—
546C4⋊C4—101no—
547C4⋊C4—101no—
548C4⋊C4—151no—
549C4⋊C4—301no—
550C4⋊C4—301no—
551C4⋊C4—301no—
552C4⋊C4—301no—
553C4⋊C4—301no—
554C4⋊C4—301no—
555C4⋊C4—301no—
556C4⋊C4—301no—
557C4⋊C4—301no—
558C4⋊C4C2×D3×D511yescharacteristic
559C4⋊C4C2×D3×D511yescharacteristic
560C22⋊C4—61no—
561C22⋊C4—101no—
562C22⋊C4—301no—
563C42—51no—
564C42—61no—
565C42—151no—
566C42—151no—
567C42—151no—
568C42—301no—
569C42—301no—
570C3×C5#308011yescharacteristic, complemented, Hall
571C22×C3—102no—
572C22×C3—201no—
573C22×C3—201no—
574C22×C3C2×D4×D511yescharacteristic
575C2×D3—21no—
576C2×D3—21no—
577C2×D3#404212yes—
578C3×C4—41no—
579C3×C4—41no—
580C3×C4—101no—
581C3×C4—101no—
582C3×C4—201no—
583C3×C4C2×#70911yescharacteristic
584C3×C4C2×Q8×D511yescharacteristic
585Dic3—41no—
586Dic3—41no—
587Dic3—102no—
588Dic3—101no—
589Dic3—101no—
590Dic3—201no—
591Dic3#404212yes—
592C2×C5—62no—
593C2×C5D3×C22⋊Q811yescharacteristic
594C2×C5#576811yescharacteristic
595C2×C5#577311yescharacteristic
596D5—42no—
597C23—31no—
598C23—101no—
599Q8—152no—
600Q8—301no—
601D4—52no—
602D4—101no—
603D4—304no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—31no—
607C2×C4—32no—
608C2×C4—32no—
609C2×C4—51no—
610C2×C4—51no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—101no—
614C2×C4—122no—
615C2×C4—151no—
616C2×C4—151no—
617C2×C4—202no—
618C2×C4—304no—
619C2×C4—301no—
620C2×C4—301no—
621C2×C4—601no—
622C2×C4—601no—
623C2×C4—602no—
624C2×C4—602no—
625C2×C4C22×D3×D511yescharacteristic
626C8—21no—
627C8—61no—
628C8—101no—
629C8—301no—
630C2×C3—202no—
631C2×C3C2×#404211yescharacteristic
632C2×C3#6610711yescharacteristic
633C2×C3#6611111yescharacteristic
634D3—22no—
635C5#8116711yescharacteristic, complemented, Sylow
636C22—32no—
637C22—61no—
638C22—61no—
639C22—102no—
640C22—201no—
641C22—201no—
642C22D3×D4×D511yescharacteristic
643C4—32no—
644C4—41no—
645C4—41no—
646C4—101no—
647C4—101no—
648C4—121no—
649C4—121no—
650C4—201no—
651C4—301no—
652C4—301no—
653C4—302no—
654C4—601no—
655C4D3×#70911yescharacteristic
656C4D3×Q8×D511yescharacteristic
657C3#1061097211yescharacteristic, complemented, Sylow
658C2—62no—
659C2—202no—
660C2D3×#404211yescharacteristic
661C2#1175404911yescharacteristic
662C2#1175410411yescharacteristic
subgroupG/H
centerC22D3×D4×D5
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C5×#107C22
socleC22×C3×C5C22×D4
prime radicalC2×#6765C22
hypercenter#107D3×D5
hypocenterC3×C5#3080
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
1. D3×#4036C2semidirect
3. #11750978C2semidirect
4. C5×#4881C2semidirect
5. C3×#65180C2semidirect
7. #11747852C2semidirect
8. #11747805C2semidirect
9. #11747590C2semidirect
11. #11747389C2semidirect
12. #11747286C2semidirect
13. #11747259C2semidirect
14. #11747191C2semidirect
15. #11747187C2semidirect
69. #65180D3semidirect
128. #4881D5semidirect
570. C3×C5#3080semidirect
635. C5#81167semidirect
657. C3#10610972semidirect

Maximal subgroups

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

subgroupindex
D3×#40362
#117533982
#117509782
C5×#48812
C3×#651802
#117481382
#117478522
#117478052
#117475902
#117475032
#117473892
#117472862
#117472592
#117471912
#117471872
#106109723
#811675