Finite Group Databasebeta

← #14418102

Subgroups — #14418102

Order 1920.

912 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×D5⋊#131C211yesmaximal, characteristic, complemented
2C2×#91914C211yesmaximal, characteristic, complemented
3#11755344C211yesmaximal, characteristic
4#11752454C211yesmaximal, characteristic, complemented
5#11751978C212yesmaximal, complemented
6#11751977C212yesmaximal, complemented
7#11751954C212yesmaximal, complemented
8#11751953C212yesmaximal, complemented
9#11751733C211yesmaximal, characteristic, complemented
10#11751723C211yesmaximal, characteristic, complemented
11#11751694C211yesmaximal, characteristic, complemented
12#10617452—31nomaximal, Hall
13C3×D5×#41C2211yescharacteristic
14C2×#6773C2211yescharacteristic
15#91989C2211yescharacteristic
16C2×#6758C2211yescharacteristic
17C3×#4133C2211yescharacteristic, complemented
18C3×#4133C2211yescharacteristic, complemented
19C3×#4131—21no—
20C3×#4131—21no—
21C3×#4131C2211yescharacteristic, complemented
22C3×#4131C2211yescharacteristic, complemented
23C3×D5⋊M16—21no—
24C3×D5⋊M16C2211yescharacteristic
25C2×C3×D5⋊C8—21no—
26C2×C3×D5⋊C8C2211yescharacteristic
27C2×#6740—21no—
28C2×#6738—21no—
29#91914—21no—
30#91914—21no—
31#91914C2212yescomplemented
32#91914C2212yescomplemented
33#91838C411yescharacteristic, complemented
34C5×#951C411yescharacteristic, complemented
35#91516C2212yes—
36#91515C412yescomplemented
37#91492C2212yes—
38#91491C412yescomplemented
39#91307C411yescharacteristic, complemented
40#91301C411yescharacteristic, complemented
41D5×#208C2211yescharacteristic
42#91239C2212yescomplemented
43#91237C2212yescomplemented
44#91162C2212yescomplemented
45#91161C2212yescomplemented
46#91157C2212yescomplemented
47#91157C2212yescomplemented
48#91150C2212yescomplemented
49ΓL(1,𝔽16)⋊C8C2212yescomplemented
50#83792—51nomaximal, Hall
51D5⋊#131D311yescharacteristic, complemented
52C2×#4133—31no—
53#66111—31no—
54#65795—31no—
55#65784—31no—
56#65741—32no—
57#65740—32no—
58#65737—32no—
59#65736—32no—
60#65729—31no—
61#65728—31no—
62C3×C5×#41C2×C411yescharacteristic
63C2×D60C2×C411yescharacteristic
64C2×C5×Dic6C2×C411yescharacteristic
65C3×#713C2311yescharacteristic
66C3×Q8×D5C2×C411yescharacteristic
67C3×#711—21no—
68C3×#711C2311yescharacteristic
69C3×D4×D5—21no—
70C3×D4×D5C2×C411yescharacteristic
71C3×#709—21no—
72C3×#709C2×C411yescharacteristic
73C2×C3×C4×D5—21no—
74C2×C3×C4×D5C2311yescharacteristic
75C2×#1202—21no—
76C2×#1200—21no—
77#6773—21no—
78#6773—21no—
79#6773C2×C412yes—
80#6773C2312yes—
81#6763—21no—
82C2×C3⋊#170—21no—
83C4⋊ΓL(1,𝔽16)—21no—
84C4×ΓL(1,𝔽16)C2312yes—
85#6758—21no—
86#6758C2312yes—
87C3×#704—21no—
88C3×#704—21no—
89C3×#704—21no—
90C2×C3×#170—21no—
91C2×C3×#170—21no—
92C2×C3×#170D411yescharacteristic
93C2×C3×#170D411yescharacteristic
94C3×#700—21no—
95C3×#700—21no—
96C3×#700C2311yescharacteristic
97C3×#700C2311yescharacteristic
98C3×D5⋊C8—21no—
99C3×D5⋊C8—21no—
100C3×D5⋊C8—21no—
101C3×D5⋊C8D411yescharacteristic
102C3×D5⋊C8D411yescharacteristic
103C3×D5⋊C8C2311yescharacteristic
104C3×D5⋊C8C2311yescharacteristic
105#6740—22no—
106#6740—22no—
107#6738—22no—
108#6738—22no—
109#6717C2×C412yes—
110#6715C2×C412yes—
111#6710C2×C411yescharacteristic
112C5×#216C2×C412yes—
113C5×#214C2×C412yes—
114C5×#208C2×C411yescharacteristic
115#6661C2×C412yes—
116#6658C2×C412yes—
117#6655C2×C412yes—
118#6653C2×C412yes—
119#6649C2×C412yes—
120D5×C3⋊C8C2312yes—
121C2×C3×#131—51no—
122#5822—51no—
123C2×#901—51no—
124#5236—51no—
125#5145—52no—
126#5145—52no—
127#5144—52no—
128#5144—52no—
129#5139—51no—
130#5124—51no—
131#5101—51no—
132D5×#41C2×D311yescharacteristic
133C2×#713—31no—
134#4133—32no—
135#4133—32no—
136#4133—61no—
137#4133—61no—
138#4133C2×D311yescharacteristic, complemented
139#4133C2×D311yescharacteristic, complemented
140#4131—21no—
141#4131—21no—
142#4131C2×D311yescharacteristic, complemented
143#4131C2×D311yescharacteristic, complemented
144#4130—31no—
145D5⋊M16—21no—
146D5⋊M16C2×D311yescharacteristic
147C2×#700—31no—
148C2×#700—61no—
149C2×D5⋊C8—21no—
150C2×D5⋊C8—61no—
151C2×D5⋊C8C2×D311yescharacteristic
152C5×#137—31no—
153#4095—31no—
154#4088—31no—
155#4065—32no—
156#4064—32no—
157#4062—32no—
158#4060—32no—
159#4054—31no—
160D5×M16—31no—
161#4010—32no—
162#4010—32no—
163#4010—32no—
164#4008—32no—
165#3996—32no—
166#3995—32no—
167#3992—32no—
168C8×AGL(1,𝔽5)—32no—
169#2955—151noSylow
170C22×D15—21no—
171C22×C3×D5—21no—
172C22×C3×D5#4111yescharacteristic
173C2×ΓL(1,𝔽16)—22no—
174C3×C5×Q8C22×C411yescharacteristic
175C3×C5×D4—21no—
176C3×C5×D4C22×C411yescharacteristic
177C2×C3×C4×C5—21no—
178C2×C3×C4×C5C22×C411yescharacteristic
179D60—21no—
180D60C22×C412yes—
181C2×C5×Dic3—21no—
182C5×Dic6—21no—
183C5×Dic6C22×C412yes—
184C3×C22⋊D5—21no—
185C3×C22⋊D5—21no—
186C3×C22⋊D5—21no—
187C2×C3×Dic5—21no—
188C2×C3×Dic5C2×D411yescharacteristic
189C3×D20—21no—
190C3×D20C22×C411yescharacteristic
191C3×C4×D5—21no—
192C3×C4×D5—21no—
193C3×C4×D5—21no—
194C3×C4×D5C22×C411yescharacteristic
195C3×C4×D5C2×D411yescharacteristic
196C3×C4×D5#4111yescharacteristic
197C3×C4×D5C2411yescharacteristic
198C3×Dic10—21no—
199C3×Dic10C22×C411yescharacteristic
200#1202—22no—
201#1202—22no—
202#1200—22no—
203#1200—22no—
204C3⋊#170—22no—
205C3×#170—21no—
206C3×#170—21no—
207C3×#170C2×D411yescharacteristic
208C3×#170C2×D411yescharacteristic
209C3×#170C2×D411yescharacteristic
210C3×#170C2×D411yescharacteristic
211C3⋊#168C22×C412yes—
212C5×C3⋊C8C22×C412yes—
213C2×C3×#41—51no—
214C2×#235—51no—
215C3×#131—51no—
216C3×#131—51no—
217C3×#131—51no—
218C3×#131—51no—
219C3×#131—101no—
220C3×#131—101no—
221C2×C3×M16—51no—
222C2×C3×M16—101no—
223C22×C3×C8—51no—
224C22×C3×C8—101no—
225#951—51no—
226#951AGL(1,𝔽5)11yescharacteristic, complemented
227#950—52no—
228#950—52no—
229#949—51no—
230#949—51no—
231#942—52no—
232#932—52no—
233#926—51no—
234C2×#208—51no—
235C2×#208—51no—
236#901—52no—
237#901—52no—
238#901—101no—
239#901—101no—
240C2×ΓL(1,𝔽25)—101no—
241C2×D3×C8—101no—
242#837—52no—
243#837—52no—
244#825—52no—
245#825—52no—
246#822—52no—
247#822—52no—
248#815—52no—
249C8×Dic3—52no—
250C5×#41C4×D311yescharacteristic
251C2×C5×Q8—31no—
252#713—32no—
253#713—32no—
254#713—61no—
255#713—61no—
256#713C22×D311yescharacteristic
257Q8×D5C4×D311yescharacteristic
258#711—21no—
259#711C22×D311yescharacteristic
260D4×D5—21no—
261D4×D5C4×D311yescharacteristic
262#709—21no—
263#709C4×D311yescharacteristic
264C2×D20—31no—
265C2×D20—61no—
266C2×C4×D5—21no—
267C2×C4×D5—61no—
268C2×C4×D5C22×D311yescharacteristic
269C22⋊AGL(1,𝔽5)—61no—
270#704—21no—
271#704—21no—
272#704—21no—
273C2×#170—21no—
274C2×#170—21no—
275C2×#170—61no—
276C2×#170#6711yescharacteristic
277C2×#170#6711yescharacteristic
278C4⋊AGL(1,𝔽5)—61no—
279C4×AGL(1,𝔽5)—32no—
280#700—21no—
281#700—21no—
282#700—32no—
283#700—61no—
284#700—62no—
285#700—62no—
286#700C22×D311yescharacteristic
287#700C22×D311yescharacteristic
288D5⋊C8—21no—
289D5⋊C8—21no—
290D5⋊C8—21no—
291D5⋊C8—62no—
292D5⋊C8—62no—
293D5⋊C8#6711yescharacteristic
294D5⋊C8#6711yescharacteristic
295D5⋊C8C22×D311yescharacteristic
296D5⋊C8C22×D311yescharacteristic
297C5×Q16—32no—
298C5×SD16—32no—
299C5×M16—31no—
300#689—32no—
301#688—32no—
302#688—32no—
303#686—32no—
304#681—31no—
305D40—32no—
306#677—32no—
307#676—32no—
308C8×D5—32no—
309#575—151no—
310C2×#131—51no—
311C2×#131—151no—
312#444—152no—
313#444—152no—
314#443—152no—
315#443—152no—
316#430—151no—
317#421—151no—
318#421—151no—
319#405—151no—
320C22×C3×C5—21no—
321C22×C3×C5C4×D411yescharacteristic
322C2×D15—22no—
323C2×D15—42no—
324C2×C3×D5—21no—
325C2×C3×D5—41no—
326C2×C3×D5—42no—
327C2×C3×D5C4×D411yescharacteristic
328C2×C3×D5C2×#4111yescharacteristic
329ΓL(1,𝔽16)—42no—
330C3×C4×C5—21no—
331C3×C4×C5C4×D411yescharacteristic
332C3×C4×C5C23×C411yescharacteristic
333C3×Dic5—21no—
334C3×Dic5C4×D411yescharacteristic
335C3×Dic5C22×D411yescharacteristic
336C5×Dic3—22no—
337C3×#41—51no—
338C3×#41—51no—
339C3×#41—51no—
340C3×#41—101no—
341C3×#41—101no—
342C3×#41C2×AGL(1,𝔽5)11yescharacteristic
343C2×C3×Q8—51no—
344C2×C3×D4—51no—
345C2×C3×D4—101no—
346C22×C3×C4—51no—
347C22×C3×C4—101no—
348C2×#67—101no—
349#235—52no—
350#235—52no—
351#235—101no—
352#235—101no—
353C2×D12—51no—
354C2×C4×D3—101no—
355C2×Dic6C2×AGL(1,𝔽5)11yescharacteristic
356C3×M16—51no—
357C3×M16—51no—
358C3×M16—101no—
359C3×M16—101no—
360C3×M16—101no—
361C3×M16—101no—
362C3×M16—101no—
363C2×C3×C8—51no—
364C2×C3×C8—51no—
365C2×C3×C8—51no—
366C2×C3×C8—51no—
367C2×C3×C8—51no—
368C2×C3×C8—51no—
369C2×C3×C8—101no—
370C2×C3×C8—101no—
371C2×C3×C8—101no—
372C2×C3×C8—101no—
373C2×C3×C8—101no—
374#217—101no—
375#216—52no—
376#216C2×AGL(1,𝔽5)12yes—
377#215—52no—
378#215—52no—
379#214—52no—
380#214C2×AGL(1,𝔽5)12yes—
381#213—52no—
382#213—52no—
383C4⋊Dic3—101no—
384C4×Dic3—52no—
385#208—51no—
386#208—52no—
387#208—52no—
388#208—101no—
389#208C2×AGL(1,𝔽5)11yescharacteristic
390C2×C3⋊C8—52no—
391C2×C3⋊C8—101no—
392ΓL(1,𝔽25)—102no—
393ΓL(1,𝔽25)—102no—
394D3×C8—102no—
395D3×C8—102no—
396C22×D5—21no—
397C22×D5—61no—
398C22×D5#23511yescharacteristic
399C2×AGL(1,𝔽5)—62no—
400C5×Q8—32no—
401C5×Q8—61no—
402C5×Q8C2×C4×D311yescharacteristic
403C5×D4—21no—
404C5×D4C2×C4×D311yescharacteristic
405C2×C4×C5—21no—
406C2×C4×C5—61no—
407C2×C4×C5C2×C4×D311yescharacteristic
408C22⋊D5—21no—
409C22⋊D5—21no—
410C22⋊D5—21no—
411C2×Dic5—21no—
412C2×Dic5C2×#6711yescharacteristic
413D20—21no—
414D20—32no—
415D20—62no—
416D20—62no—
417D20—61no—
418D20C2×C4×D311yescharacteristic
419C4×D5—21no—
420C4×D5—21no—
421C4×D5—21no—
422C4×D5—62no—
423C4×D5—62no—
424C4×D5C2×C4×D311yescharacteristic
425C4×D5#23511yescharacteristic
426C4×D5C2×#6711yescharacteristic
427C4×D5C23×D311yescharacteristic
428Dic10—21no—
429Dic10C2×C4×D311yescharacteristic
430#170—21no—
431#170—21no—
432#170—62no—
433#170C2×#6711yescharacteristic
434#170C2×#6711yescharacteristic
435#170C2×#6711yescharacteristic
436#170C2×#6711yescharacteristic
437C5×C8—32no—
438#168—32no—
439C2×#41—51no—
440C2×#41—151no—
441#137—31no—
442#137—152no—
443#137—151no—
444AGL(1,ℤ/8)—151no—
445AGL(1,ℤ/8)—152no—
446AGL(1,ℤ/8)—151no—
447#135—152no—
448#135—152no—
449#131—51no—
450#131—51no—
451#131—51no—
452#131—51no—
453#131—101no—
454#131—101no—
455#131—152no—
456#131—152no—
457#131—301no—
458#131—301no—
459C2×M16—51no—
460C2×M16—101no—
461C2×M16—151no—
462C2×M16—151no—
463C2×M16—301no—
464C22×C8—51no—
465C22×C8—101no—
466C22×C8—301no—
467#117—151no—
468#108—152no—
469#108—152no—
470#104—152no—
471#104—152no—
472#104—152no—
473#104—152no—
474#97—152no—
475C4×C8—152no—
476C2×C3×C5—21no—
477C2×C3×C5—41no—
478C2×C3×C5C2×C4×D411yescharacteristic
479D15—42no—
480C3×D5—21no—
481C3×D5—21no—
482C3×D5—41no—
483C23×C3—51no—
484C23×C3—101no—
485C22×D3—101no—
486C3×Q8—51no—
487C3×Q8—101no—
488C3×Q8C22×AGL(1,𝔽5)11yescharacteristic
489C3×D4—21no—
490C3×D4—51no—
491C3×D4—101no—
492C3×D4—101no—
493C3×D4—101no—
494C3×D4—101no—
495C3×D4C22×AGL(1,𝔽5)11yescharacteristic
496C2×C3×C4—21no—
497C2×C3×C4—51no—
498C2×C3×C4—51no—
499C2×C3×C4—51no—
500C2×C3×C4—51no—
501C2×C3×C4—51no—
502C2×C3×C4—101no—
503C2×C3×C4—101no—
504C2×C3×C4—101no—
505C2×C3×C4—101no—
506C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
507#67—102no—
508#67—102no—
509C2×Dic3—21no—
510C2×Dic3—102no—
511D12—52no—
512D12—101no—
513C4×D3—102no—
514C4×D3—102no—
515Dic6—21no—
516Dic6C22×AGL(1,𝔽5)12yes—
517C3×C8—51no—
518C3×C8—51no—
519C3×C8—51no—
520C3×C8—51no—
521C3×C8—101no—
522C3×C8—101no—
523C3⋊C8—52no—
524C3⋊C8—102no—
525C3⋊C8C22×AGL(1,𝔽5)12yes—
526C22×C5—21no—
527C22×C5C4×#6711yescharacteristic
528C2×D5—21no—
529C2×D5—41no—
530C2×D5—42no—
531C2×D5—62no—
532C2×D5—122no—
533C2×D5C4×#6711yescharacteristic
534C2×D5C2×#23511yescharacteristic
535AGL(1,𝔽5)—122no—
536C4×C5—21no—
537C4×C5—62no—
538C4×C5C4×#6711yescharacteristic
539C4×C5C22×C4×D311yescharacteristic
540Dic5—21no—
541Dic5C4×#6711yescharacteristic
542Dic5C22×#6711yescharacteristic
543#41—51no—
544#41—51no—
545#41—51no—
546#41—101no—
547#41—101no—
548#41—152no—
549#41—152no—
550#41—301no—
551#41—301no—
552#41D3×AGL(1,𝔽5)11yescharacteristic
553C2×Q8—31no—
554C2×Q8—51no—
555C2×D4—51no—
556C2×D4—101no—
557C2×D4—151no—
558C2×D4—301no—
559C22×C4—51no—
560C22×C4—101no—
561C22×C4—301no—
562Q16—32no—
563Q16—152no—
564SD16—32no—
565SD16—152no—
566SD16—152no—
567SD16—152no—
568D8—152no—
569D8—152no—
570M16—31no—
571M16—51no—
572M16—51no—
573M16—101no—
574M16—101no—
575M16—101no—
576M16—101no—
577M16—101no—
578M16—152no—
579M16—151no—
580M16—152no—
581M16—302no—
582M16—302no—
583M16—301no—
584C2×C8—51no—
585C2×C8—51no—
586C2×C8—51no—
587C2×C8—51no—
588C2×C8—51no—
589C2×C8—51no—
590C2×C8—101no—
591C2×C8—101no—
592C2×C8—101no—
593C2×C8—101no—
594C2×C8—101no—
595C2×C8—152no—
596C2×C8—302no—
597C2×C8—302no—
598C2×C8—301no—
599C4⋊C4—301no—
600C22⋊C4—301no—
601C42—152no—
602C3×C5#295511yescharacteristic, complemented, Hall
603C22×C3—21no—
604C22×C3—51no—
605C22×C3—51no—
606C22×C3—101no—
607C22×C3—201no—
608C22×C3—202no—
609C22×C3D4×AGL(1,𝔽5)11yescharacteristic
610C2×D3—102no—
611C2×D3—202no—
612C3×C4—21no—
613C3×C4—51no—
614C3×C4—51no—
615C3×C4—101no—
616C3×C4D4×AGL(1,𝔽5)11yescharacteristic
617C3×C4C23×AGL(1,𝔽5)11yescharacteristic
618Dic3—22no—
619Dic3—202no—
620C2×C5—21no—
621C2×C5—41no—
622C2×C5C2×C4×#6711yescharacteristic
623D5—21no—
624D5—21no—
625D5—41no—
626D5—122no—
627C23—51no—
628C23—101no—
629C23—301no—
630Q8—32no—
631Q8—51no—
632Q8—61no—
633Q8—101no—
634Q8C2×D3×AGL(1,𝔽5)11yescharacteristic
635D4—21no—
636D4—51no—
637D4—101no—
638D4—101no—
639D4—101no—
640D4—101no—
641D4—152no—
642D4—302no—
643D4—302no—
644D4—301no—
645D4C2×D3×AGL(1,𝔽5)11yescharacteristic
646C2×C4—21no—
647C2×C4—51no—
648C2×C4—51no—
649C2×C4—51no—
650C2×C4—51no—
651C2×C4—51no—
652C2×C4—61no—
653C2×C4—101no—
654C2×C4—101no—
655C2×C4—101no—
656C2×C4—101no—
657C2×C4—302no—
658C2×C4—302no—
659C2×C4—302no—
660C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
661C8—32no—
662C8—51no—
663C8—51no—
664C8—51no—
665C8—51no—
666C8—101no—
667C8—101no—
668C8—152no—
669C8—302no—
670C2×C3—21no—
671C2×C3—41no—
672C2×C3—101no—
673C2×C3—101no—
674C2×C3—201no—
675C2×C3C2×D4×AGL(1,𝔽5)11yescharacteristic
676D3—202no—
677C5#8379211yescharacteristic, complemented, Sylow
678C22—21no—
679C22—51no—
680C22—51no—
681C22—101no—
682C22—201no—
683C22—202no—
684C22—302no—
685C22—602no—
686C22AGL(1,𝔽5)×#6711yescharacteristic
687C4—21no—
688C4—51no—
689C4—51no—
690C4—62no—
691C4—101no—
692C4—602no—
693C4AGL(1,𝔽5)×#6711yescharacteristic
694C4C22×D3×AGL(1,𝔽5)11yescharacteristic
695C3#1061745211yescharacteristic, complemented, Sylow
696C2—21no—
697C2—41no—
698C2—101no—
699C2—101no—
700C2—201no—
701C2—602no—
702C2C2×AGL(1,𝔽5)×#6711yescharacteristic
subgroupG/H
centerC2C2×AGL(1,𝔽5)×#67
commutatorC3×C4×C5C23×C4
FrattiniC4C22×D3×AGL(1,𝔽5)
FittingC3×C5×#41C2×C4
socleC2×C3×C5C2×C4×D4
prime radical#11755344C2
cosocleC3×C4×D5C24
hypercenter#41D3×AGL(1,𝔽5)
hypocenterC3×C5#2955
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
1. C3×D5⋊#131C2semidirect
2. C2×#91914C2semidirect
4. #11752454C2semidirect
5. #11751978C2semidirect
6. #11751977C2semidirect
7. #11751954C2semidirect
8. #11751953C2semidirect
9. #11751733C2semidirect
10. #11751723C2semidirect
11. #11751694C2semidirect
17. C3×#4133C22semidirect
18. C3×#4133C22semidirect
21. C3×#4131C22semidirect
22. C3×#4131C22semidirect
31. #91914C22semidirect
32. #91914C22semidirect
33. #91838C4semidirect
34. C5×#951C4semidirect
36. #91515C4semidirect
38. #91491C4semidirect
39. #91307C4semidirect
40. #91301C4semidirect
42. #91239C22semidirect
43. #91237C22semidirect
44. #91162C22semidirect
45. #91161C22semidirect
46. #91157C22semidirect
47. #91157C22semidirect
48. #91150C22semidirect
49. ΓL(1,𝔽16)⋊C8C22semidirect
51. D5⋊#131D3semidirect
138. #4133C2×D3semidirect
139. #4133C2×D3semidirect
142. #4131C2×D3semidirect
143. #4131C2×D3semidirect
226. #951AGL(1,𝔽5)semidirect
602. C3×C5#2955semidirect
677. C5#83792semidirect
695. C3#10617452semidirect

Maximal subgroups

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

subgroupindex
C3×D5⋊#1312
C2×#919142
#117553442
#117524542
#117519782
#117519772
#117519542
#117519532
#117517332
#117517232
#117516942
#106174523
#837925