Finite Group Databasebeta

← #14436332

Subgroups — #14436332

Order 1920.

740 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#542C211yesmaximal, characteristic
2#11756275C211yesmaximal, characteristic, complemented
3#11751195C211yesmaximal, characteristic, complemented
4#11751192C211yesmaximal, characteristic, complemented
5#11751155C211yesmaximal, characteristic, complemented
6#11751155C211yesmaximal, characteristic, complemented
7#11751151C211yesmaximal, characteristic, complemented
8#11751146C211yesmaximal, characteristic, complemented
9#11751141C211yesmaximal, characteristic
10#11751136C211yesmaximal, characteristic
11#11751131C211yesmaximal, characteristic, complemented
12C4×#6718C211yesmaximal, characteristic, complemented
13#11751119C211yesmaximal, characteristic, complemented
14C4×#6716C211yesmaximal, characteristic, complemented
15#11751101C211yesmaximal, characteristic, complemented
16#10615198—31nomaximal, Hall
17C22×Dic30C2211yescharacteristic
18C3×C5×C4⋊Q8C2211yescharacteristic
19C3×C5×#126—21no—
20C3×C5×#124—21no—
21C3×C5×#124C2211yescharacteristic
22C3×C5×#123—21no—
23C3×C5×C22⋊Q8—21no—
24C3×C5×C22⋊Q8C2211yescharacteristic
25C3×C5×#121C2211yescharacteristic
26C3×C4×C5×Q8C2211yescharacteristic
27C3×C4×C5×D4C2211yescharacteristic
28C3×C5×#117C2211yescharacteristic
29C2×#6718C2211yescharacteristic
30C2×#6718C2211yescharacteristic
31C2×#6716C2211yescharacteristic
32C2×#6716C2211yescharacteristic
33#91815C2211yescharacteristic
34#91776—21no—
35#91776—21no—
36#91766—21no—
37#91766—21no—
38C4×Dic30C2211yescharacteristic
39C4×Dic30C2211yescharacteristic
40C4×Dic30C2211yescharacteristic
41C4×Dic30C2211yescharacteristic
42#91119C2211yescharacteristic
43#91119C2211yescharacteristic
44#91116C2211yescharacteristic
45#91116C2211yescharacteristic
46#91114C2211yescharacteristic
47#91114C2211yescharacteristic
48#91094C2211yescharacteristic
49#91094C2211yescharacteristic
50#91094C2211yescharacteristic
51#91094C2211yescharacteristic
52#91092C2211yescharacteristic
53#91092C2211yescharacteristic
54#91091C2211yescharacteristic
55#91091C2211yescharacteristic
56C4⋊(C3⋊#168)C2211yescharacteristic
57C4⋊(C3⋊#168)C2211yescharacteristic
58#91087C2211yescharacteristic
59C4×C3⋊#168C2211yescharacteristic
60#83467—51nomaximal, Hall
61C5×#542D311yescharacteristic
62#65850—31no—
63#65383—31no—
64#65380—31no—
65#65343—31no—
66#65343—31no—
67#65339—31no—
68#65334—31no—
69#65329—31no—
70#65324—31no—
71#65319—31no—
72C4×#689—31no—
73#65307—31no—
74C4×#687—31no—
75#65289—31no—
76C2×C3×C5×Q8—21no—
77C2×C3×C5×Q8C2311yescharacteristic
78C2×C3×C5×D4—21no—
79C2×C3×C5×D4C2311yescharacteristic
80C22×C3×C4×C5—21no—
81C22×C3×C4×C5C2311yescharacteristic
82C22×Dic15—21no—
83C2×Dic30—21no—
84C2×Dic30—22no—
85C2×Dic30—22no—
86C2×Dic30C2311yescharacteristic
87C2×Dic30C2311yescharacteristic
88C3×C5×C4⋊C4—21no—
89C3×C5×C4⋊C4—21no—
90C3×C5×C4⋊C4—21no—
91C3×C5×C4⋊C4D411yescharacteristic
92C3×C5×C4⋊C4D411yescharacteristic
93C3×C5×C4⋊C4C2311yescharacteristic
94C3×C5×C4⋊C4C2311yescharacteristic
95C3×C5×C22⋊C4—21no—
96C3×C5×C22⋊C4—21no—
97C3×C5×C22⋊C4—21no—
98C3×C5×C22⋊C4—21no—
99C3×C5×C22⋊C4—21no—
100C3×C5×C22⋊C4D411yescharacteristic
101C3×C5×C22⋊C4D411yescharacteristic
102C3×C42×C5—21no—
103C3×C42×C5C2311yescharacteristic
104C3×C42×C5C2311yescharacteristic
105#6719—21no—
106#6718—22no—
107#6718—22no—
108#6716—22no—
109#6716—22no—
110C4⋊Dic15C2311yescharacteristic
111C4⋊Dic15C2311yescharacteristic
112#6712—21no—
113#6712—21no—
114#6712—21no—
115#6712—21no—
116C4×Dic15—21no—
117C4×Dic15—21no—
118C4×Dic15—21no—
119C4×Dic15—21no—
120C4×Dic15—21no—
121C2×C3⋊#168C2311yescharacteristic
122C2×C3⋊#168C2311yescharacteristic
123C2×C3⋊#168C2311yescharacteristic
124C2×C3⋊#168C2311yescharacteristic
125C3×#542D511yescharacteristic
126#5516—51no—
127#5094—51no—
128#5091—51no—
129#5054—51no—
130#5054—51no—
131#5050—51no—
132#5045—51no—
133#5040—51no—
134#5035—51no—
135#5030—51no—
136C4×#216—51no—
137#5018—51no—
138C4×#214—51no—
139#5000—51no—
140C22×Dic10—31no—
141C5×C4⋊Q8C2×D311yescharacteristic
142C5×#126—21no—
143C5×#124—21no—
144C5×#124C2×D311yescharacteristic
145C5×#123—21no—
146C5×C22⋊Q8—21no—
147C5×C22⋊Q8C2×D311yescharacteristic
148C5×#121C2×D311yescharacteristic
149C4×C5×Q8C2×D311yescharacteristic
150C4×C5×D4C2×D311yescharacteristic
151C5×#117C2×D311yescharacteristic
152C2×#689—31no—
153C2×#689—31no—
154C2×#687—31no—
155C2×#687—31no—
156#4072—31no—
157#4033—61no—
158#4033—61no—
159#4023—61no—
160#4023—61no—
161C4×Dic10—31no—
162C4×Dic10—31no—
163C4×Dic10—31no—
164C4×Dic10—31no—
165#3967—31no—
166#3967—31no—
167#3964—31no—
168#3964—31no—
169C22⋊#168—31no—
170C22⋊#168—31no—
171#3942—31no—
172#3942—31no—
173#3942—31no—
174#3942—31no—
175#3940—31no—
176#3940—31no—
177#3939—31no—
178#3939—31no—
179C4⋊#168—31no—
180C4⋊#168—31no—
181#3935—31no—
182C4×#168—31no—
183#3121—151noSylow
184C23×C3×C5—21no—
185C23×C3×C5C2×D411yescharacteristic
186C3×C5×Q8—21no—
187C3×C5×Q8—21no—
188C3×C5×Q8—81no—
189C3×C5×D4—21no—
190C3×C5×D4—21no—
191C3×C5×D4—81no—
192C2×C3×C4×C5—21no—
193C2×C3×C4×C5—21no—
194C2×C3×C4×C5—21no—
195C2×C3×C4×C5—21no—
196C2×C3×C4×C5—21no—
197C2×C3×C4×C5—81no—
198C2×C3×C4×C5C2×D411yescharacteristic
199C2×C3×C4×C5C2×D411yescharacteristic
200C2×C3×C4×C5C2×D411yescharacteristic
201C2×C3×C4×C5C2×D411yescharacteristic
202C2×C3×C4×C5C2×D411yescharacteristic
203C2×C3×C4×C5C2411yescharacteristic
204C2×Dic15—24no—
205C2×Dic15—21no—
206C2×Dic15—21no—
207C2×Dic15—21no—
208C2×Dic15—21no—
209Dic30—24no—
210Dic30—21no—
211Dic30—21no—
212Dic30#4112yes—
213Dic30#4112yes—
214C3⋊#168—21no—
215C3⋊#168—21no—
216C3⋊#168—21no—
217C3⋊#168—21no—
218C22×Dic6—51no—
219C3×C4⋊Q8C2×D511yescharacteristic
220C3×#126—21no—
221C3×#124—21no—
222C3×#124C2×D511yescharacteristic
223C3×#123—21no—
224C3×C22⋊Q8—21no—
225C3×C22⋊Q8C2×D511yescharacteristic
226C3×#121C2×D511yescharacteristic
227C3×C4×Q8C2×D511yescharacteristic
228C3×C4×D4C2×D511yescharacteristic
229C3×#117C2×D511yescharacteristic
230C2×#216—51no—
231C2×#216—51no—
232C2×#214—51no—
233C2×#214—51no—
234#926—51no—
235#887—101no—
236#887—101no—
237#877—101no—
238#877—101no—
239C4×Dic6—51no—
240C4×Dic6—51no—
241C4×Dic6—51no—
242C4×Dic6—51no—
243#835—51no—
244#835—51no—
245#832—51no—
246#832—51no—
247#830—51no—
248#830—51no—
249#810—51no—
250#810—51no—
251#810—51no—
252#810—51no—
253#808—51no—
254#808—51no—
255#807—51no—
256#807—51no—
257C4⋊(C3⋊C8)—51no—
258C4⋊(C3⋊C8)—51no—
259#803—51no—
260C4×C3⋊C8—51no—
261C2×C5×Q8—21no—
262C2×C5×Q8C22×D311yescharacteristic
263C2×C5×D4—21no—
264C2×C5×D4C22×D311yescharacteristic
265C22×C4×C5—21no—
266C22×C4×C5C22×D311yescharacteristic
267C22×Dic5—61no—
268C2×Dic10—31no—
269C2×Dic10—31no—
270C2×Dic10—62no—
271C2×Dic10—62no—
272C2×Dic10—61no—
273C5×C4⋊C4—21no—
274C5×C4⋊C4—21no—
275C5×C4⋊C4—21no—
276C5×C4⋊C4#6711yescharacteristic
277C5×C4⋊C4#6711yescharacteristic
278C5×C4⋊C4C22×D311yescharacteristic
279C5×C4⋊C4C22×D311yescharacteristic
280C5×C22⋊C4—21no—
281C5×C22⋊C4—21no—
282C5×C22⋊C4—21no—
283C5×C22⋊C4—21no—
284C5×C22⋊C4—21no—
285C5×C22⋊C4#6711yescharacteristic
286C5×C22⋊C4#6711yescharacteristic
287C42×C5—21no—
288C42×C5C22×D311yescharacteristic
289C42×C5C22×D311yescharacteristic
290C22⋊Dic5—61no—
291#689—62no—
292#689—62no—
293#687—62no—
294#687—62no—
295C4⋊Dic5—31no—
296C4⋊Dic5—31no—
297#683—61no—
298#683—61no—
299#683—61no—
300#683—61no—
301C4×Dic5—61no—
302C4×Dic5—61no—
303C4×Dic5—61no—
304C4×Dic5—61no—
305C4×Dic5—61no—
306C2×#168—31no—
307C2×#168—31no—
308C2×#168—31no—
309C2×#168—31no—
310#542D1511yescharacteristic
311#519—151no—
312#485—151no—
313#482—151no—
314#477—151no—
315#475—151no—
316#464—151no—
317#464—151no—
318#451—151no—
319#448—151no—
320#441—151no—
321#440—151no—
322C4×Q16—151no—
323C4×SD16—151no—
324#433—151no—
325C22×C3×C5—21no—
326C22×C3×C5—21no—
327C22×C3×C5—21no—
328C22×C3×C5—41no—
329C22×C3×C5—41no—
330C22×C3×C5—41no—
331C22×C3×C5C22×D411yescharacteristic
332C3×C4×C5—21no—
333C3×C4×C5—21no—
334C3×C4×C5—21no—
335C3×C4×C5—21no—
336C3×C4×C5—21no—
337C3×C4×C5—41no—
338C3×C4×C5—41no—
339C3×C4×C5—41no—
340C3×C4×C5C2×#4111yescharacteristic
341C3×C4×C5C2×#4111yescharacteristic
342Dic15—22no—
343Dic15—22no—
344Dic15—41no—
345Dic15—41no—
346C2×C3×Q8—21no—
347C2×C3×Q8C22×D511yescharacteristic
348C2×C3×D4—21no—
349C2×C3×D4C22×D511yescharacteristic
350C22×C3×C4—21no—
351C22×C3×C4C22×D511yescharacteristic
352C22×Dic3—101no—
353C2×Dic6—51no—
354C2×Dic6—51no—
355C2×Dic6—102no—
356C2×Dic6—102no—
357C2×Dic6—101no—
358C3×C4⋊C4—21no—
359C3×C4⋊C4—21no—
360C3×C4⋊C4—21no—
361C3×C4⋊C4C22⋊D511yescharacteristic
362C3×C4⋊C4C22⋊D511yescharacteristic
363C3×C4⋊C4C22×D511yescharacteristic
364C3×C4⋊C4C22×D511yescharacteristic
365C3×C22⋊C4—21no—
366C3×C22⋊C4—21no—
367C3×C22⋊C4—21no—
368C3×C22⋊C4—21no—
369C3×C22⋊C4—21no—
370C3×C22⋊C4C22⋊D511yescharacteristic
371C3×C22⋊C4C22⋊D511yescharacteristic
372C3×C42—21no—
373C3×C42C22×D511yescharacteristic
374C3×C42C22×D511yescharacteristic
375#217—101no—
376#216—102no—
377#216—102no—
378#214—102no—
379#214—102no—
380C4⋊Dic3—51no—
381C4⋊Dic3—51no—
382Dic3⋊C4—101no—
383Dic3⋊C4—101no—
384Dic3⋊C4—101no—
385Dic3⋊C4—101no—
386C4×Dic3—101no—
387C4×Dic3—101no—
388C4×Dic3—101no—
389C4×Dic3—101no—
390C4×Dic3—101no—
391C2×C3⋊C8—51no—
392C2×C3⋊C8—51no—
393C2×C3⋊C8—51no—
394C2×C3⋊C8—51no—
395C23×C5—21no—
396C23×C5C2×#6711yescharacteristic
397C5×Q8—21no—
398C5×Q8—21no—
399C5×Q8—81no—
400C5×D4—21no—
401C5×D4—21no—
402C5×D4—81no—
403C2×C4×C5—21no—
404C2×C4×C5—21no—
405C2×C4×C5—21no—
406C2×C4×C5—21no—
407C2×C4×C5—21no—
408C2×C4×C5—81no—
409C2×C4×C5C2×#6711yescharacteristic
410C2×C4×C5C2×#6711yescharacteristic
411C2×C4×C5C2×#6711yescharacteristic
412C2×C4×C5C2×#6711yescharacteristic
413C2×C4×C5C2×#6711yescharacteristic
414C2×C4×C5C23×D311yescharacteristic
415C2×Dic5—64no—
416C2×Dic5—61no—
417C2×Dic5—61no—
418C2×Dic5—61no—
419C2×Dic5—61no—
420Dic10—32no—
421Dic10—32no—
422Dic10—64no—
423Dic10—61no—
424Dic10—61no—
425#168—61no—
426#168—61no—
427#168—61no—
428#168—61no—
429C22×Q8—151no—
430C2×Q16—151no—
431C2×Q16—151no—
432C2×SD16—151no—
433C2×SD16—151no—
434C4⋊Q8C2×D1511yescharacteristic
435#126—21no—
436#124—21no—
437#124C2×D1511yescharacteristic
438#123—21no—
439C22⋊Q8—21no—
440C22⋊Q8C2×D1511yescharacteristic
441#121C2×D1511yescharacteristic
442C4×Q8—151no—
443C4×Q8—151no—
444C4×Q8—151no—
445C4×Q8—151no—
446C4×Q8—301no—
447C4×Q8—301no—
448C4×Q8C2×D1511yescharacteristic
449C4×D4C2×D1511yescharacteristic
450#117—151no—
451#117—301no—
452#117—301no—
453#117C2×D1511yescharacteristic
454#107—151no—
455#107—151no—
456#106—151no—
457#106—151no—
458C4⋊C8—151no—
459C4⋊C8—151no—
460#103—151no—
461#103—151no—
462#103—151no—
463#103—151no—
464#103—151no—
465#103—151no—
466#102—151no—
467#102—151no—
468C22⋊C8—151no—
469C22⋊C8—151no—
470#97—151no—
471C4×C8—151no—
472C2×C3×C5—41no—
473C2×C3×C5—81no—
474C2×C3×C5#52511yescharacteristic
475C2×C3×C5#57711yescharacteristic
476C2×C3×C5#57811yescharacteristic
477C23×C3—21no—
478C23×C3C2×C22⋊D511yescharacteristic
479C3×Q8—21no—
480C3×Q8—21no—
481C3×Q8—81no—
482C3×D4—21no—
483C3×D4—21no—
484C3×D4—81no—
485C2×C3×C4—21no—
486C2×C3×C4—21no—
487C2×C3×C4—21no—
488C2×C3×C4—21no—
489C2×C3×C4—21no—
490C2×C3×C4—81no—
491C2×C3×C4C2×C22⋊D511yescharacteristic
492C2×C3×C4C2×C22⋊D511yescharacteristic
493C2×C3×C4C2×C22⋊D511yescharacteristic
494C2×C3×C4C2×C22⋊D511yescharacteristic
495C2×C3×C4C2×C22⋊D511yescharacteristic
496C2×C3×C4C23×D511yescharacteristic
497C2×Dic3—104no—
498C2×Dic3—101no—
499C2×Dic3—101no—
500C2×Dic3—101no—
501C2×Dic3—101no—
502Dic6—52no—
503Dic6—52no—
504Dic6—104no—
505Dic6—101no—
506Dic6—101no—
507C3⋊C8—101no—
508C3⋊C8—101no—
509C3⋊C8—101no—
510C3⋊C8—101no—
511C22×C5—21no—
512C22×C5—21no—
513C22×C5—21no—
514C22×C5—41no—
515C22×C5—41no—
516C22×C5—41no—
517C22×C5C22×#6711yescharacteristic
518C4×C5—21no—
519C4×C5—21no—
520C4×C5—21no—
521C4×C5—21no—
522C4×C5—21no—
523C4×C5—41no—
524C4×C5—41no—
525C4×C5—41no—
526C4×C5D3×#4111yescharacteristic
527C4×C5D3×#4111yescharacteristic
528Dic5—62no—
529Dic5—62no—
530Dic5—121no—
531Dic5—121no—
532C2×Q8—21no—
533C2×Q8—151no—
534C2×Q8—151no—
535C2×Q8—302no—
536C2×Q8—302no—
537C2×Q8—301no—
538C2×Q8C22×D1511yescharacteristic
539C2×D4—21no—
540C2×D4C22×D1511yescharacteristic
541C22×C4—21no—
542C22×C4—301no—
543C22×C4C22×D1511yescharacteristic
544Q16—302no—
545Q16—302no—
546SD16—302no—
547SD16—302no—
548C2×C8—151no—
549C2×C8—151no—
550C2×C8—151no—
551C2×C8—151no—
552C4⋊C4—21no—
553C4⋊C4—21no—
554C4⋊C4—21no—
555C4⋊C4—151no—
556C4⋊C4—151no—
557C4⋊C4—301no—
558C4⋊C4—301no—
559C4⋊C4—301no—
560C4⋊C4—301no—
561C4⋊C4#122011yescharacteristic
562C4⋊C4#122011yescharacteristic
563C4⋊C4C22×D1511yescharacteristic
564C4⋊C4C22×D1511yescharacteristic
565C22⋊C4—21no—
566C22⋊C4—21no—
567C22⋊C4—21no—
568C22⋊C4—21no—
569C22⋊C4—21no—
570C22⋊C4—301no—
571C22⋊C4#122011yescharacteristic
572C22⋊C4#122011yescharacteristic
573C42—21no—
574C42—301no—
575C42—301no—
576C42—301no—
577C42—301no—
578C42—301no—
579C42C22×D1511yescharacteristic
580C42C22×D1511yescharacteristic
581C3×C5#312111yescharacteristic, complemented, Hall
582C22×C3—21no—
583C22×C3—21no—
584C22×C3—21no—
585C22×C3—41no—
586C22×C3—41no—
587C22×C3—41no—
588C22×C3C22×C22⋊D511yescharacteristic
589C3×C4—21no—
590C3×C4—21no—
591C3×C4—21no—
592C3×C4—21no—
593C3×C4—21no—
594C3×C4—41no—
595C3×C4—41no—
596C3×C4—41no—
597C3×C4D5×#4111yescharacteristic
598C3×C4D5×#4111yescharacteristic
599Dic3—102no—
600Dic3—102no—
601Dic3—201no—
602Dic3—201no—
603C2×C5—41no—
604C2×C5—81no—
605C2×C5#582811yescharacteristic
606C2×C5#583611yescharacteristic
607C2×C5#583811yescharacteristic
608C23—21no—
609C23C2×#122011yescharacteristic
610Q8—21no—
611Q8—21no—
612Q8—81no—
613Q8—152no—
614Q8—152no—
615Q8—304no—
616Q8—301no—
617Q8—301no—
618D4—21no—
619D4—21no—
620D4—81no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—81no—
627C2×C4—304no—
628C2×C4—301no—
629C2×C4—301no—
630C2×C4—301no—
631C2×C4—301no—
632C2×C4C2×#122011yescharacteristic
633C2×C4C2×#122011yescharacteristic
634C2×C4C2×#122011yescharacteristic
635C2×C4C2×#122011yescharacteristic
636C2×C4C2×#122011yescharacteristic
637C2×C4C23×D1511yescharacteristic
638C8—301no—
639C8—301no—
640C8—301no—
641C8—301no—
642C2×C3—41no—
643C2×C3—81no—
644C2×C3#6616211yescharacteristic
645C2×C3#6617011yescharacteristic
646C2×C3#6617211yescharacteristic
647C5#8346711yescharacteristic, complemented, Sylow
648C22—21no—
649C22—21no—
650C22—21no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22C22×#122011yescharacteristic
655C4—21no—
656C4—21no—
657C4—21no—
658C4—21no—
659C4—21no—
660C4—41no—
661C4—41no—
662C4—41no—
663C4—302no—
664C4—302no—
665C4—601no—
666C4—601no—
667C4#41×D1511yescharacteristic
668C4#41×D1511yescharacteristic
669C3#1061519811yescharacteristic, complemented, Sylow
670C2—41no—
671C2—81no—
672C2#1175658711yescharacteristic
673C2#1175659511yescharacteristic
674C2#1175659711yescharacteristic
subgroupG/H
centerC22C22×#1220
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC3×C5×#542C2
socleC22×C3×C5C22×D4
prime radicalC3×C5×#121C22
hypercenter#542D15
hypocenterC3×C5#3121
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
2. #11756275C2semidirect
3. #11751195C2semidirect
4. #11751192C2semidirect
5. #11751155C2semidirect
6. #11751155C2semidirect
7. #11751151C2semidirect
8. #11751146C2semidirect
11. #11751131C2semidirect
12. C4×#6718C2semidirect
13. #11751119C2semidirect
14. C4×#6716C2semidirect
15. #11751101C2semidirect
581. C3×C5#3121semidirect
647. C5#83467semidirect
669. C3#10615198semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5422
#117562752
#117511952
#117511922
#117511552
#117511512
#117511462
#117511412
#117511362
#117511312
C4×#67182
#117511192
C4×#67162
#117511012
#106151983
#834675