Finite Group Databasebeta

← #14418190

Subgroups — #14418190

Order 1920.

916 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#66256C211yesmaximal, characteristic, complemented
2#11756823C211yesmaximal, characteristic, complemented
3#11754690C211yesmaximal, characteristic
4C2×#91237C211yesmaximal, characteristic, complemented
5#11751954C212yesmaximal, complemented
6#11751953C212yesmaximal, complemented
7#11751952C212yesmaximal, complemented
8#11751951C212yesmaximal, complemented
9#11751733C211yesmaximal, characteristic, complemented
10D5⋊#825C211yesmaximal, characteristic, complemented
11#11751698C211yesmaximal, characteristic, complemented
12#10617449—31nomaximal, Hall
13C2×C3×#711C2211yescharacteristic
14D5×#235C2211yescharacteristic
15C2×C4×ΓL(1,𝔽16)C2211yescharacteristic
16C2×#6758C2211yescharacteristic
17C2×C3×#704—22no—
18C3×#4131—21no—
19C3×#4131—21no—
20C3×#4131C2212yescomplemented
21C3×#4131C2212yescomplemented
22C3×D5⋊M16C2211yescharacteristic
23C3×D5⋊M16C2211yescharacteristic
24#91929—22no—
25#91915—21no—
26#91914C2212yescomplemented
27D3×#700—21no—
28#91911C2212yescomplemented
29#91821C411yescharacteristic, complemented
30C5×#932C411yescharacteristic, complemented
31#91492C2212yes—
32#91490C2212yes—
33#91484C412yescomplemented
34#91478C412yescomplemented
35#91315C411yescharacteristic, complemented
36#91309C411yescharacteristic, complemented
37D5×#208C2211yescharacteristic
38#91237C2212yescomplemented
39#91237C2212yescomplemented
40#91162C2212yescomplemented
41#91161C2212yescomplemented
42#91157C2211yescharacteristic, complemented
43#91157C2211yescharacteristic, complemented
44#91155C2211yescharacteristic, complemented
45#91155C2211yescharacteristic, complemented
46#91150C2212yescomplemented
47ΓL(1,𝔽16)⋊C8C2212yescomplemented
48#83811—51nomaximal, Hall
49#66264—31no—
50#66256D311yescharacteristic, complemented
51#66107—31no—
52D5⋊#104—31no—
53C2×#4008—31no—
54#65737—32no—
55#65736—32no—
56#65733—32no—
57#65732—32no—
58#65729—31no—
59M16×AGL(1,𝔽5)—31no—
60C22×ΓL(1,𝔽16)—21no—
61C2×C3×C5×D4C2×C411yescharacteristic
62#6817C2×C411yescharacteristic
63C5×#235C2×C411yescharacteristic
64C2×C3×C22⋊D5—21no—
65C22×C3×Dic5—21no—
66C3×#711—21no—
67C3×#711—21no—
68C3×#711C2×C412yes—
69C3×#711C2312yes—
70C2×C3×C4×D5C2311yescharacteristic
71C2×C3×Dic10C2×C411yescharacteristic
72D5×#67—21no—
73#6782—21no—
74D5×D12C2×C411yescharacteristic
75C4×D3×D5—21no—
76#6773C2311yescharacteristic
77#6772C2311yescharacteristic
78#6771—21no—
79D5×Dic6C2×C411yescharacteristic
80C2×C3⋊#170—21no—
81C4×ΓL(1,𝔽16)—21no—
82C4×ΓL(1,𝔽16)C2311yescharacteristic
83C4×ΓL(1,𝔽16)C2311yescharacteristic
84#6758—21no—
85#6758C2312yes—
86C3×#704—21no—
87C3×#704—22no—
88C3×#704—22no—
89C3×#704D412yes—
90C2×C3×#170—22no—
91C2×C3×#170—22no—
92C3×#700D411yescharacteristic
93C3×#700D411yescharacteristic
94C3×#700C2311yescharacteristic
95C3×#700C2311yescharacteristic
96C3×D5⋊C8—21no—
97C3×D5⋊C8C2312yes—
98#6740—22no—
99#6739—22no—
100#6738—22no—
101D3×#170—22no—
102#6716C2×C412yescomplemented
103#6715C2×C412yescomplemented
104#6710C2×C411yescharacteristic
105C5×#214C2×C412yescomplemented
106C5×#213C2×C412yescomplemented
107C5×#208C2×C411yescharacteristic
108#6662C2×C412yescomplemented
109#6661C2×C412yescomplemented
110#6658C2×C412yescomplemented
111#6656C2×C412yescomplemented
112#6649C2×C412yes—
113D5×C3⋊C8C2312yes—
114C3×#568—51no—
115#5822—51no—
116#5745—51no—
117C2×#837—51no—
118#5145—52no—
119#5145—52no—
120#5144—52no—
121#5144—52no—
122C2×#825—51no—
123#5124—51no—
124Dic3×M16—51no—
125D5×#41—31no—
126C2×#711C2×D311yescharacteristic
127C2×#704—22no—
128#4133—32no—
129#4131—21no—
130#4131—21no—
131#4131—32no—
132#4131—62no—
133#4131C2×D312yescomplemented
134#4131C2×D312yescomplemented
135C2×C4×AGL(1,𝔽5)—31no—
136D5⋊M16—61no—
137D5⋊M16C2×D311yescharacteristic
138D5⋊M16C2×D311yescharacteristic
139C2×#700—31no—
140C2×#700—61no—
141C5×AGL(1,ℤ/8)—31no—
142#4097—31no—
143#4078—31no—
144#4062—32no—
145#4061—32no—
146#4058—32no—
147#4057—32no—
148#4055—31no—
149D5×M16—31no—
150#4010—31no—
151#4010—31no—
152#4008—32no—
153#4008—32no—
154#4008—31no—
155#4008—31no—
156#3996—32no—
157#3995—32no—
158#3992—32no—
159C8×AGL(1,𝔽5)—32no—
160#2956—151noSylow
161C23×C3×C5—21no—
162C22×C3×D5#4111yescharacteristic
163C2×D3×D5—21no—
164C2×ΓL(1,𝔽16)—21no—
165C2×ΓL(1,𝔽16)—21no—
166C2×ΓL(1,𝔽16)—42no—
167C3×C5×D4—21no—
168C3×C5×D4C22×C412yes—
169C2×C3×C4×C5C22×C411yescharacteristic
170#1220—21no—
171D60C22×C411yescharacteristic
172C4×D15—21no—
173Dic30C22×C411yescharacteristic
174C5×#67—21no—
175C5×D12C22×C411yescharacteristic
176C4×C5×D3—21no—
177C5×Dic6C22×C411yescharacteristic
178C3×C22⋊D5—22no—
179C3×C22⋊D5—22no—
180C2×C3×Dic5—21no—
181C2×C3×Dic5—22no—
182C2×C3×Dic5—22no—
183C2×C3×Dic5C2×D411yescharacteristic
184C3×C4×D5C22×C411yescharacteristic
185C3×C4×D5C2×D411yescharacteristic
186C3×C4×D5#4111yescharacteristic
187C3×C4×D5C2411yescharacteristic
188C3×Dic10—21no—
189C3×Dic10C22×C412yes—
190#1204—21no—
191#1203—21no—
192#1202—21no—
193#1201—21no—
194#1200—21no—
195D3×Dic5—21no—
196D5×Dic3—21no—
197C3⋊#170—22no—
198C3×#170—21no—
199C3×#170—21no—
200C3×#170C2×D412yes—
201C3×#170C2×D412yes—
202C3⋊#168C22×C412yes—
203C5×C3⋊C8C22×C412yes—
204C2×C3×#41—51no—
205C2×#235—51no—
206C3×#131—52no—
207C3×#131—52no—
208C3×#131—101no—
209C3×#131—101no—
210C2×C3×M16—51no—
211C2×C3×M16—51no—
212C2×C3×M16—102no—
213#951—52no—
214#950—52no—
215#950—52no—
216#949—52no—
217#942—51no—
218#942—51no—
219#932—51no—
220#932AGL(1,𝔽5)11yescharacteristic, complemented
221C2×C4×Dic3—51no—
222C2×#208—51no—
223C2×#208—51no—
224#907—102no—
225D3×M16—101no—
226D3×M16—101no—
227#901—52no—
228#901—52no—
229#837—52no—
230#837—52no—
231#825—51no—
232#825—51no—
233#825—51no—
234#825—51no—
235#822—52no—
236#822—52no—
237#815—52no—
238C8×Dic3—52no—
239C22×AGL(1,𝔽5)—61no—
240C5×#41—31no—
241C2×C5×D4C4×D311yescharacteristic
242C2×C22⋊D5—21no—
243C22×Dic5—21no—
244#713—31no—
245Q8×D5—31no—
246#711—21no—
247#711—21no—
248#711—31no—
249#711—61no—
250#711C4×D312yes—
251#711C22×D312yes—
252D4×D5—31no—
253D4×D5—61no—
254#709—31no—
255#709—61no—
256C2×C4×D5—61no—
257C2×C4×D5C22×D311yescharacteristic
258C2×Dic10C4×D311yescharacteristic
259#704—21no—
260#704—22no—
261#704—22no—
262#704—62no—
263#704#6712yes—
264C2×#170—22no—
265C2×#170—22no—
266C2×#170—61no—
267C2×#170—62no—
268C4×AGL(1,𝔽5)—31no—
269C4×AGL(1,𝔽5)—31no—
270C4×AGL(1,𝔽5)—61no—
271#700—32no—
272#700—61no—
273#700—62no—
274#700#6711yescharacteristic
275#700#6711yescharacteristic
276#700C22×D311yescharacteristic
277#700C22×D311yescharacteristic
278D5⋊C8—21no—
279D5⋊C8—62no—
280D5⋊C8C22×D312yes—
281C5×SD16—32no—
282C5×D8—32no—
283C5×M16—31no—
284#689—32no—
285#687—32no—
286#687—32no—
287#686—32no—
288#681—31no—
289Dic20—32no—
290#677—32no—
291#676—32no—
292C8×D5—32no—
293#575—151no—
294#568—51no—
295#568—151no—
296#444—152no—
297#444—152no—
298#443—152no—
299#443—152no—
300#430—151no—
301C2×#104—151no—
302C2×#104—151no—
303C4×M16—151no—
304C22×C3×C5—22no—
305C22×C3×C5—81no—
306C22×C3×C5C4×D411yescharacteristic
307C2×D15—21no—
308C2×C5×D3—21no—
309C2×C3×D5—41no—
310C2×C3×D5C4×D411yescharacteristic
311C2×C3×D5C2×#4111yescharacteristic
312D3×D5—81no—
313ΓL(1,𝔽16)—22no—
314ΓL(1,𝔽16)—41no—
315C3×C4×C5C4×D411yescharacteristic
316C3×C4×C5C23×C411yescharacteristic
317Dic15—21no—
318C3×Dic5—22no—
319C3×Dic5C4×D411yescharacteristic
320C3×Dic5C22×D411yescharacteristic
321C5×Dic3—21no—
322C3×#41—52no—
323C3×#41—52no—
324C3×#41—101no—
325C3×#41—101no—
326C2×C3×Q8—51no—
327C2×C3×D4—101no—
328C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
329C22×C3×C4—51no—
330C22×C3×C4—101no—
331C2×#67—101no—
332C22×Dic3—101no—
333#235—51no—
334#235—51no—
335#235—51no—
336#235—101no—
337#235—101no—
338#235C2×AGL(1,𝔽5)11yescharacteristic
339C2×D12—51no—
340C2×C4×D3—101no—
341C2×Dic6—51no—
342C3×M16—51no—
343C3×M16—51no—
344C3×M16—52no—
345C3×M16—51no—
346C3×M16—51no—
347C3×M16—102no—
348C3×M16—102no—
349C3×M16—101no—
350C2×C3×C8—52no—
351C2×C3×C8—102no—
352C2×C3×C8—102no—
353C2×C3×C8—101no—
354#216—52no—
355#216—52no—
356#215—52no—
357#215—52no—
358#214—52no—
359#214C2×AGL(1,𝔽5)12yescomplemented
360#213—52no—
361#213C2×AGL(1,𝔽5)12yescomplemented
362C4×Dic3—51no—
363C4×Dic3—51no—
364C4×Dic3—101no—
365#208—51no—
366#208—52no—
367#208—52no—
368#208—101no—
369#208C2×AGL(1,𝔽5)11yescharacteristic
370C2×C3⋊C8—52no—
371C2×C3⋊C8—101no—
372ΓL(1,𝔽25)—102no—
373ΓL(1,𝔽25)—102no—
374D3×C8—102no—
375D3×C8—102no—
376C23×C5—21no—
377C22×D5—61no—
378C22×D5#23511yescharacteristic
379C2×AGL(1,𝔽5)—61no—
380C2×AGL(1,𝔽5)—61no—
381C2×AGL(1,𝔽5)—122no—
382C5×Q8—31no—
383C5×D4—21no—
384C5×D4—31no—
385C5×D4—61no—
386C5×D4C2×C4×D312yes—
387C2×C4×C5—61no—
388C2×C4×C5C2×C4×D311yescharacteristic
389C22⋊D5—22no—
390C22⋊D5—22no—
391C22⋊D5—61no—
392C22⋊D5—61no—
393C22⋊D5—61no—
394C2×Dic5—21no—
395C2×Dic5—22no—
396C2×Dic5—22no—
397C2×Dic5—61no—
398C2×Dic5C2×#6711yescharacteristic
399D20—31no—
400D20—61no—
401C4×D5—61no—
402C4×D5—61no—
403C4×D5—61no—
404C4×D5C2×C4×D311yescharacteristic
405C4×D5#23511yescharacteristic
406C4×D5C2×#6711yescharacteristic
407C4×D5C23×D311yescharacteristic
408Dic10—21no—
409Dic10—31no—
410Dic10—61no—
411Dic10C2×C4×D312yes—
412#170—21no—
413#170—21no—
414#170—62no—
415#170C2×#6712yes—
416#170C2×#6712yes—
417C5×C8—32no—
418#168—32no—
419C2×#41—51no—
420C2×#41—151no—
421#137—151no—
422#137—152no—
423#137—151no—
424AGL(1,ℤ/8)—31no—
425AGL(1,ℤ/8)—152no—
426AGL(1,ℤ/8)—151no—
427#135—152no—
428#135—152no—
429#131—52no—
430#131—52no—
431#131—101no—
432#131—101no—
433#131—152no—
434#131—152no—
435#131—302no—
436C2×M16—51no—
437C2×M16—51no—
438C2×M16—102no—
439C2×M16—151no—
440C2×M16—151no—
441C2×M16—301no—
442C2×M16—301no—
443C2×C42—151no—
444#108—152no—
445#108—152no—
446#104—151no—
447#104—151no—
448#104—151no—
449#104—151no—
450#104—152no—
451#104—152no—
452#97—152no—
453C4×C8—152no—
454C2×C3×C5—21no—
455C2×C3×C5—42no—
456C2×C3×C5C2×C4×D411yescharacteristic
457D15—41no—
458C3×D5—21no—
459C3×D5—21no—
460C5×D3—41no—
461C23×C3—21no—
462C23×C3—51no—
463C22×D3—101no—
464C3×Q8—52no—
465C3×Q8—101no—
466C3×D4—21no—
467C3×D4—102no—
468C3×D4—102no—
469C3×D4C22×AGL(1,𝔽5)12yes—
470C2×C3×C4—51no—
471C2×C3×C4—51no—
472C2×C3×C4—51no—
473C2×C3×C4—51no—
474C2×C3×C4—51no—
475C2×C3×C4—102no—
476C2×C3×C4—102no—
477C2×C3×C4—101no—
478C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
479#67—21no—
480#67—101no—
481#67—101no—
482#67—101no—
483C2×Dic3—101no—
484C2×Dic3—101no—
485C2×Dic3—101no—
486C2×Dic3—202no—
487D12—51no—
488D12—101no—
489D12C22×AGL(1,𝔽5)11yescharacteristic
490C4×D3—21no—
491C4×D3—101no—
492C4×D3—101no—
493C4×D3—101no—
494Dic6—51no—
495Dic6—101no—
496Dic6C22×AGL(1,𝔽5)11yescharacteristic
497C3×C8—52no—
498C3×C8—52no—
499C3×C8—101no—
500C3×C8—101no—
501C3⋊C8—52no—
502C3⋊C8—102no—
503C3⋊C8C22×AGL(1,𝔽5)12yes—
504C22×C5—22no—
505C22×C5—61no—
506C22×C5—81no—
507C22×C5C4×#6711yescharacteristic
508C2×D5—41no—
509C2×D5—61no—
510C2×D5—241no—
511C2×D5C4×#6711yescharacteristic
512C2×D5C2×#23511yescharacteristic
513AGL(1,𝔽5)—62no—
514AGL(1,𝔽5)—121no—
515C4×C5—61no—
516C4×C5C4×#6711yescharacteristic
517C4×C5C22×C4×D311yescharacteristic
518Dic5—22no—
519Dic5—61no—
520Dic5C4×#6711yescharacteristic
521Dic5C22×#6711yescharacteristic
522#41—31no—
523#41—52no—
524#41—52no—
525#41—101no—
526#41—101no—
527#41—151no—
528#41—151no—
529#41—151no—
530#41—301no—
531#41—301no—
532C2×Q8—51no—
533C2×Q8—151no—
534C2×D4—101no—
535C2×D4—151no—
536C2×D4—301no—
537C2×D4D3×AGL(1,𝔽5)11yescharacteristic
538C22×C4—51no—
539C22×C4—101no—
540C22×C4—301no—
541C22×C4—301no—
542Q16—152no—
543Q16—152no—
544SD16—32no—
545SD16—152no—
546SD16—152no—
547SD16—152no—
548D8—32no—
549D8—152no—
550M16—31no—
551M16—52no—
552M16—51no—
553M16—51no—
554M16—51no—
555M16—51no—
556M16—102no—
557M16—102no—
558M16—101no—
559M16—152no—
560M16—151no—
561M16—152no—
562M16—302no—
563M16—302no—
564M16—301no—
565C2×C8—52no—
566C2×C8—102no—
567C2×C8—102no—
568C2×C8—101no—
569C2×C8—152no—
570C2×C8—302no—
571C2×C8—302no—
572C2×C8—301no—
573C42—151no—
574C42—151no—
575C42—301no—
576C3×C5#295611yescharacteristic, complemented, Hall
577C22×C3—22no—
578C22×C3—51no—
579C22×C3—51no—
580C22×C3—81no—
581C22×C3—201no—
582C22×C3D4×AGL(1,𝔽5)11yescharacteristic
583C2×D3—21no—
584C2×D3—101no—
585C2×D3—401no—
586C3×C4—51no—
587C3×C4—51no—
588C3×C4—102no—
589C3×C4D4×AGL(1,𝔽5)11yescharacteristic
590C3×C4C23×AGL(1,𝔽5)11yescharacteristic
591Dic3—21no—
592Dic3—102no—
593Dic3—101no—
594Dic3—201no—
595C2×C5—21no—
596C2×C5—42no—
597C2×C5—121no—
598C2×C5C2×C4×#6711yescharacteristic
599D5—21no—
600D5—21no—
601D5—121no—
602C23—21no—
603C23—51no—
604C23—301no—
605Q8—31no—
606Q8—52no—
607Q8—101no—
608Q8—151no—
609Q8—301no—
610D4—21no—
611D4—31no—
612D4—61no—
613D4—102no—
614D4—102no—
615D4—151no—
616D4—301no—
617D4—301no—
618D4—301no—
619D4—301no—
620D4C2×D3×AGL(1,𝔽5)12yes—
621C2×C4—51no—
622C2×C4—51no—
623C2×C4—51no—
624C2×C4—51no—
625C2×C4—51no—
626C2×C4—61no—
627C2×C4—102no—
628C2×C4—102no—
629C2×C4—101no—
630C2×C4—301no—
631C2×C4—301no—
632C2×C4—301no—
633C2×C4—301no—
634C2×C4—301no—
635C2×C4—301no—
636C2×C4—602no—
637C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
638C8—32no—
639C8—52no—
640C8—52no—
641C8—101no—
642C8—101no—
643C8—152no—
644C8—302no—
645C2×C3—21no—
646C2×C3—42no—
647C2×C3—101no—
648C2×C3—101no—
649C2×C3C2×D4×AGL(1,𝔽5)11yescharacteristic
650D3—41no—
651D3—201no—
652C5#8381111yescharacteristic, complemented, Sylow
653C22—22no—
654C22—51no—
655C22—51no—
656C22—61no—
657C22—81no—
658C22—201no—
659C22—301no—
660C22—1201no—
661C22AGL(1,𝔽5)×#6711yescharacteristic
662C4—51no—
663C4—51no—
664C4—61no—
665C4—102no—
666C4—302no—
667C4—301no—
668C4—601no—
669C4AGL(1,𝔽5)×#6711yescharacteristic
670C4C22×D3×AGL(1,𝔽5)11yescharacteristic
671C3#1061744911yescharacteristic, complemented, Sylow
672C2—21no—
673C2—42no—
674C2—101no—
675C2—101no—
676C2—121no—
677C2—601no—
678C2C2×AGL(1,𝔽5)×#6711yescharacteristic
subgroupG/H
centerC2C2×AGL(1,𝔽5)×#67
commutatorC3×C4×C5C23×C4
FrattiniC4C22×D3×AGL(1,𝔽5)
FittingC2×C3×C5×D4C2×C4
socleC2×C3×C5C2×C4×D4
prime radical#11754690C2
cosocleC3×C4×D5C24
hypercenterC2×D4D3×AGL(1,𝔽5)
hypocenterC3×C5#2956
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C3×#66256C2semidirect
2. #11756823C2semidirect
4. C2×#91237C2semidirect
5. #11751954C2semidirect
6. #11751953C2semidirect
7. #11751952C2semidirect
8. #11751951C2semidirect
9. #11751733C2semidirect
10. D5⋊#825C2semidirect
11. #11751698C2semidirect
20. C3×#4131C22semidirect
21. C3×#4131C22semidirect
26. #91914C22semidirect
28. #91911C22semidirect
29. #91821C4semidirect
30. C5×#932C4semidirect
33. #91484C4semidirect
34. #91478C4semidirect
35. #91315C4semidirect
36. #91309C4semidirect
38. #91237C22semidirect
39. #91237C22semidirect
40. #91162C22semidirect
41. #91161C22semidirect
42. #91157C22semidirect
43. #91157C22semidirect
44. #91155C22semidirect
45. #91155C22semidirect
46. #91150C22semidirect
47. ΓL(1,𝔽16)⋊C8C22semidirect
50. #66256D3semidirect
102. #6716C2×C4semidirect
103. #6715C2×C4semidirect
105. C5×#214C2×C4semidirect
106. C5×#213C2×C4semidirect
108. #6662C2×C4semidirect
109. #6661C2×C4semidirect
110. #6658C2×C4semidirect
111. #6656C2×C4semidirect
133. #4131C2×D3semidirect
134. #4131C2×D3semidirect
220. #932AGL(1,𝔽5)semidirect
359. #214C2×AGL(1,𝔽5)semidirect
361. #213C2×AGL(1,𝔽5)semidirect
576. C3×C5#2956semidirect
652. C5#83811semidirect
671. C3#10617449semidirect

Maximal subgroups

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

subgroupindex
C3×#662562
#117568232
#117546902
C2×#912372
#117519542
#117519532
#117519522
#117519512
#117517332
D5⋊#8252
#117516982
#106174493
#838115