Finite Group Databasebeta

← #14415444

Subgroups — #14415444

Order 1920.

1004 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C5×M16C211yesmaximal, characteristic, complemented
2#11756550C211yesmaximal, characteristic
3C2×#91796C211yesmaximal, characteristic, complemented
4C2×#91114C211yesmaximal, characteristic, complemented
5#11751230C211yesmaximal, characteristic, complemented
6#11751229C212yesmaximal, complemented
7#11751228C212yesmaximal, complemented
8#11751223C211yesmaximal, characteristic, complemented
9#11751220C211yesmaximal, characteristic, complemented
10#11751212C212yesmaximal, complemented
11C8×#1220C212yesmaximal, complemented
12#10615488—31nomaximal, Hall
13C23×C3×C4×C5C2211yescharacteristic
14C2×#6817C2211yescharacteristic
15C2×C3×C5×M16—24no—
16C2×C3×C5×M16C2211yescharacteristic
17C2×C3×C5×M16C2212yes—
18C2×C3×C5×M16C2211yescharacteristic
19C22×C3×C5×C8C2212yes—
20#91842C412yes—
21#91819C411yescharacteristic
22#91819C411yescharacteristic
23#91818C412yes—
24C4×#1220C2212yes—
25C4×#1220C2212yes—
26#91815C2211yescharacteristic
27#91813C411yescharacteristic
28#91813C411yescharacteristic
29C22×C3⋊#168C2211yescharacteristic
30C22×C3⋊#168C2211yescharacteristic
31#91796—24no—
32C2×#6705C2212yescomplemented
33C2×C8×D15C2212yescomplemented
34#91114C2211yescharacteristic
35#91114C2211yescharacteristic
36#91114C2211yescharacteristic
37#91114C2211yescharacteristic
38#91104C2212yescomplemented
39#91104C2212yescomplemented
40#91099C2212yescomplemented
41#91098C2212yescomplemented
42#91098C2212yescomplemented
43C8×Dic15C2212yescomplemented
44#83757—51nomaximal, Hall
45C22×C5×M16D311yescharacteristic, complemented
46#66125—31no—
47C2×#4053—31no—
48C2×C22⋊#168—31no—
49#65418—31no—
50#65417—32no—
51#65416—32no—
52#65411—31no—
53#65408—31no—
54#65400—32no—
55C8×C22⋊D5—32no—
56C24×C3×C5C2×C411yescharacteristic
57C22×C3×C4×C5—22no—
58C22×C3×C4×C5—22no—
59C22×C3×C4×C5C2×C411yescharacteristic
60C22×C3×C4×C5C2×C411yescharacteristic
61C22×C3×C4×C5C2×C411yescharacteristic
62C22×C3×C4×C5C2311yescharacteristic
63C22×C3×C4×C5C2311yescharacteristic
64C22×C3×C4×C5C2311yescharacteristic
65C2×#1220C2×C412yes—
66C2×#1220C2×C412yes—
67C2×#1220C2×C412yes—
68#6817—22no—
69#6817—22no—
70C2×D60C2×C411yescharacteristic
71C2×C4×D15C2312yes—
72C2×Dic30C2×C411yescharacteristic
73C3×C5×M16—22no—
74C3×C5×M16—24no—
75C3×C5×M16D414yescomplemented
76C2×C3×C5×C8—24no—
77C2×C3×C5×C8C2312yes—
78C2×C3×C5×C8C2312yes—
79#6719C2×C412yes—
80#6719C2×C412yes—
81#6719C2×C412yes—
82#6714C2×C412yes—
83#6714C2×C412yes—
84C4⋊Dic15C2×C411yescharacteristic
85C4⋊Dic15C2×C411yescharacteristic
86#6712C2×C412yes—
87#6712C2×C412yes—
88C4×Dic15C2312yes—
89C2×C3⋊#168—22no—
90C2×C3⋊#168—22no—
91C2×C3⋊#168C2311yescharacteristic
92C2×C3⋊#168C2311yescharacteristic
93C2×C3⋊#168C2311yescharacteristic
94C2×C3⋊#168C2311yescharacteristic
95#6705—24no—
96C8×D15—24no—
97C22×C3×M16D511yescharacteristic, complemented
98#5791—51no—
99C2×#907—51no—
100C2×#830—51no—
101#5129—51no—
102#5128—52no—
103#5127—52no—
104#5122—51no—
105#5119—51no—
106#5111—52no—
107C8×#67—52no—
108C23×C4×C5C2×D311yescharacteristic
109C2×#709—31no—
110C2×C5×M16—24no—
111C2×C5×M16C2×D311yescharacteristic
112C2×C5×M16C2×D312yes—
113C2×C5×M16C2×D311yescharacteristic
114C22×C5×C8C2×D312yes—
115#4099—32no—
116#4076—31no—
117#4076—31no—
118#4075—32no—
119C4×C22⋊D5—32no—
120C4×C22⋊D5—32no—
121#4072—31no—
122#4070—31no—
123#4070—31no—
124C22×#168—31no—
125C22×#168—31no—
126#4053—64no—
127C2×#676—32no—
128C2×C8×D5—32no—
129C22⋊#168—31no—
130C22⋊#168—31no—
131C22⋊#168—31no—
132C22⋊#168—31no—
133#3952—32no—
134#3952—32no—
135#3947—32no—
136#3946—32no—
137#3946—32no—
138C8×Dic5—32no—
139#2933—151noSylow
140C23×C3×C5—22no—
141C23×C3×C5—22no—
142C23×C3×C5—41no—
143C23×C3×C5C22×C411yescharacteristic
144C23×C3×C5C22×C411yescharacteristic
145C23×C3×C5C22×C411yescharacteristic
146C22×D15C22×C412yes—
147C2×C3×C4×C5—22no—
148C2×C3×C4×C5—22no—
149C2×C3×C4×C5—22no—
150C2×C3×C4×C5—21no—
151C2×C3×C4×C5—21no—
152C2×C3×C4×C5—21no—
153C2×C3×C4×C5—21no—
154C2×C3×C4×C5C22×C411yescharacteristic
155C2×C3×C4×C5C22×C411yescharacteristic
156C2×C3×C4×C5C22×C411yescharacteristic
157C2×C3×C4×C5C2×D412yes—
158C2×C3×C4×C5#4112yes—
159C2×C3×C4×C5C2411yescharacteristic
160#1220—24no—
161#1220—24no—
162#1220—42no—
163C2×Dic15C22×C412yes—
164C2×Dic15C22×C412yes—
165C2×Dic15C22×C412yes—
166D60—22no—
167C4×D15—22no—
168C4×D15—22no—
169Dic30—22no—
170C3×C5×C8—22no—
171C3×C5×C8C2×D414yes—
172C3⋊#168—21no—
173C3⋊#168—21no—
174C3⋊#168—21no—
175C3⋊#168—21no—
176C23×C3×C4C2×D511yescharacteristic
177C2×#235—51no—
178C2×C3×M16—24no—
179C2×C3×M16C2×D512yes—
180C2×C3×M16C2×D511yescharacteristic
181C2×C3×M16C2×D511yescharacteristic
182C22×C3×C8C2×D512yes—
183#953—52no—
184#930—51no—
185#930—51no—
186#929—52no—
187C4×#67—52no—
188C4×#67—52no—
189#926—51no—
190#924—51no—
191#924—51no—
192C22×C3⋊C8—51no—
193C22×C3⋊C8—51no—
194#907—104no—
195C2×ΓL(1,𝔽25)—52no—
196C2×D3×C8—52no—
197#830—51no—
198#830—51no—
199#830—51no—
200#830—51no—
201(C2×D3)⋊C8—52no—
202(C2×D3)⋊C8—52no—
203#815—52no—
204Dic3⋊C8—52no—
205Dic3⋊C8—52no—
206C8×Dic3—52no—
207C24×C5C4×D311yescharacteristic
208C22×C4×C5—22no—
209C22×C4×C5—22no—
210C22×C4×C5C4×D311yescharacteristic
211C22×C4×C5C4×D311yescharacteristic
212C22×C4×C5C4×D311yescharacteristic
213C22×C4×C5C22×D311yescharacteristic
214C22×C4×C5C22×D311yescharacteristic
215C22×C4×C5C22×D311yescharacteristic
216C2×C22⋊D5—32no—
217C2×C22⋊D5—32no—
218C2×C22⋊D5—32no—
219#709—62no—
220#709—62no—
221C2×D20—31no—
222C2×C4×D5—32no—
223C2×Dic10—31no—
224C5×M16—22no—
225C5×M16—24no—
226C5×M16#6714yescomplemented
227C2×C5×C8—24no—
228C2×C5×C8C22×D312yes—
229C2×C5×C8C22×D312yes—
230C22⋊Dic5—32no—
231C22⋊Dic5—32no—
232C22⋊Dic5—32no—
233#685—32no—
234#685—32no—
235C4⋊Dic5—31no—
236C4⋊Dic5—31no—
237#683—32no—
238#683—32no—
239C4×Dic5—32no—
240C2×#168—31no—
241C2×#168—31no—
242C2×#168—31no—
243C2×#168—31no—
244C2×#168—62no—
245C2×#168—62no—
246#676—64no—
247C8×D5—64no—
248C2×#131—151no—
249C22×M16D1511yescharacteristic, complemented
250#525—151no—
251#436—152no—
252#435—152no—
253#435—152no—
254C8×D4—152no—
255#424—151no—
256#408—151no—
257C2×C22⋊C8—151no—
258#405—151no—
259C22×C3×C5—21no—
260C22×C3×C5—21no—
261C22×C3×C5—21no—
262C22×C3×C5—21no—
263C22×C3×C5—21no—
264C22×C3×C5—21no—
265C22×C3×C5—21no—
266C22×C3×C5—44no—
267C22×C3×C5C4×D412yes—
268C22×C3×C5#13112yescomplemented
269C22×C3×C5C23×C411yescharacteristic
270C2×D15—22no—
271C2×D15—22no—
272C2×D15—22no—
273C3×C4×C5—21no—
274C3×C4×C5—21no—
275C3×C4×C5C4×D412yes—
276C3×C4×C5C22×D411yescharacteristic
277C3×C4×C5C2×#4111yescharacteristic
278Dic15—22no—
279Dic15—22no—
280Dic15—22no—
281C24×C3C4×D511yescharacteristic
282C22×C3×C4—22no—
283C22×C3×C4—22no—
284C22×C3×C4C4×D511yescharacteristic
285C22×C3×C4C4×D511yescharacteristic
286C22×C3×C4C4×D511yescharacteristic
287C22×C3×C4C22×D511yescharacteristic
288C22×C3×C4C22×D511yescharacteristic
289C22×C3×C4C22×D511yescharacteristic
290C2×#67—52no—
291C2×#67—52no—
292C2×#67—52no—
293#235—102no—
294#235—102no—
295C2×D12—51no—
296C2×C4×D3—52no—
297C2×Dic6—51no—
298C3×M16—22no—
299C3×M16—24no—
300C3×M16C22⋊D514yescomplemented
301C2×C3×C8—24no—
302C2×C3×C8C22×D512yes—
303C2×C3×C8C22×D512yes—
304#217—52no—
305#217—52no—
306#217—52no—
307(C2×D3)⋊C4—52no—
308(C2×D3)⋊C4—52no—
309C4⋊Dic3—51no—
310C4⋊Dic3—51no—
311Dic3⋊C4—52no—
312Dic3⋊C4—52no—
313C4×Dic3—52no—
314C2×C3⋊C8—51no—
315C2×C3⋊C8—51no—
316C2×C3⋊C8—51no—
317C2×C3⋊C8—51no—
318C2×C3⋊C8—102no—
319C2×C3⋊C8—102no—
320ΓL(1,𝔽25)—104no—
321D3×C8—104no—
322C23×C5—22no—
323C23×C5—22no—
324C23×C5—41no—
325C23×C5C2×C4×D311yescharacteristic
326C23×C5C2×C4×D311yescharacteristic
327C23×C5C2×C4×D311yescharacteristic
328C22×D5—32no—
329C2×C4×C5—22no—
330C2×C4×C5—22no—
331C2×C4×C5—22no—
332C2×C4×C5—21no—
333C2×C4×C5—21no—
334C2×C4×C5—21no—
335C2×C4×C5—21no—
336C2×C4×C5C2×C4×D311yescharacteristic
337C2×C4×C5C2×C4×D311yescharacteristic
338C2×C4×C5C2×C4×D311yescharacteristic
339C2×C4×C5#23512yes—
340C2×C4×C5C2×#6712yes—
341C2×C4×C5C23×D311yescharacteristic
342C22⋊D5—64no—
343C22⋊D5—64no—
344C22⋊D5—122no—
345C2×Dic5—32no—
346C2×Dic5—32no—
347C2×Dic5—32no—
348D20—62no—
349C4×D5—62no—
350C4×D5—62no—
351Dic10—62no—
352C5×C8—22no—
353C5×C8C2×#6714yes—
354#168—61no—
355#168—61no—
356#168—61no—
357#168—61no—
358C2×#41—151no—
359C23×C4C2×D1511yescharacteristic
360#131—304no—
361C2×M16—24no—
362C2×M16—152no—
363C2×M16C2×D1511yescharacteristic
364C2×M16C2×D1512yes—
365C2×M16C2×D1511yescharacteristic
366C22×C8—151no—
367C22×C8—151no—
368C22×C8—152no—
369C22×C8C2×D1512yes—
370#123—152no—
371C22⋊Q8—151no—
372C22⋊Q8—151no—
373#121—151no—
374#121—151no—
375#120—152no—
376C4×D4—152no—
377C4×D4—152no—
378#117—151no—
379C4⋊C8—152no—
380C4⋊C8—152no—
381C22⋊C8—151no—
382C22⋊C8—151no—
383C22⋊C8—151no—
384C22⋊C8—151no—
385C22⋊C8—152no—
386C22⋊C8—152no—
387#97—152no—
388C4×C8—152no—
389C2×C3×C5—22no—
390C2×C3×C5—22no—
391C2×C3×C5—41no—
392C2×C3×C5C2×C4×D411yescharacteristic
393C2×C3×C5C2×#13111yescharacteristic
394C2×C3×C5#56811yescharacteristic
395D15—42no—
396C23×C3—22no—
397C23×C3—22no—
398C23×C3—41no—
399C23×C3C2×C4×D511yescharacteristic
400C23×C3C2×C4×D511yescharacteristic
401C23×C3C2×C4×D511yescharacteristic
402C22×D3—52no—
403C2×C3×C4—22no—
404C2×C3×C4—21no—
405C2×C3×C4—21no—
406C2×C3×C4—21no—
407C2×C3×C4—21no—
408C2×C3×C4—22no—
409C2×C3×C4—22no—
410C2×C3×C4C2×C4×D511yescharacteristic
411C2×C3×C4C2×C4×D511yescharacteristic
412C2×C3×C4C2×C4×D511yescharacteristic
413C2×C3×C4#70912yes—
414C2×C3×C4C2×C22⋊D512yes—
415C2×C3×C4C23×D511yescharacteristic
416#67—104no—
417#67—104no—
418#67—202no—
419C2×Dic3—52no—
420C2×Dic3—52no—
421C2×Dic3—52no—
422D12—102no—
423C4×D3—102no—
424C4×D3—102no—
425Dic6—102no—
426C3×C8—22no—
427C3×C8C2×C22⋊D514yes—
428C3⋊C8—101no—
429C3⋊C8—101no—
430C3⋊C8—101no—
431C3⋊C8—101no—
432C22×C5—21no—
433C22×C5—21no—
434C22×C5—21no—
435C22×C5—21no—
436C22×C5—21no—
437C22×C5—21no—
438C22×C5—21no—
439C22×C5—44no—
440C22×C5#90712yescomplemented
441C22×C5C4×#6712yes—
442C22×C5C22×C4×D311yescharacteristic
443C2×D5—62no—
444C2×D5—62no—
445C2×D5—62no—
446C4×C5—21no—
447C4×C5—21no—
448C4×C5C4×#6712yes—
449C4×C5C2×#23511yescharacteristic
450C4×C5C22×#6711yescharacteristic
451Dic5—62no—
452Dic5—62no—
453Dic5—62no—
454C24C4×D1511yescharacteristic
455#41—302no—
456#41—302no—
457C2×Q8—151no—
458C2×D4—152no—
459C2×D4—152no—
460C2×D4—152no—
461C2×D4—151no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—152no—
465C22×C4C4×D1511yescharacteristic
466C22×C4C4×D1511yescharacteristic
467C22×C4C4×D1511yescharacteristic
468C22×C4C22×D1511yescharacteristic
469C22×C4C22×D1511yescharacteristic
470C22×C4C22×D1511yescharacteristic
471M16—22no—
472M16—24no—
473M16—304no—
474M16#122014yescomplemented
475C2×C8—24no—
476C2×C8—151no—
477C2×C8—151no—
478C2×C8—151no—
479C2×C8—151no—
480C2×C8—302no—
481C2×C8—302no—
482C2×C8—304no—
483C2×C8C22×D1512yes—
484C2×C8C22×D1512yes—
485C4⋊C4—151no—
486C4⋊C4—151no—
487C4⋊C4—152no—
488C4⋊C4—152no—
489C22⋊C4—152no—
490C22⋊C4—152no—
491C22⋊C4—152no—
492C22⋊C4—152no—
493C22⋊C4—152no—
494C42—152no—
495C3×C5#293311yescharacteristic, complemented, Hall
496C22×C3—21no—
497C22×C3—21no—
498C22×C3—21no—
499C22×C3—21no—
500C22×C3—21no—
501C22×C3—21no—
502C22×C3—21no—
503C22×C3—44no—
504C22×C3#405312yescomplemented
505C22×C3C4×C22⋊D512yes—
506C22×C3C22×C4×D511yescharacteristic
507C2×D3—102no—
508C2×D3—102no—
509C2×D3—102no—
510C3×C4—21no—
511C3×C4—21no—
512C3×C4C4×C22⋊D512yes—
513C3×C4C2×#70911yescharacteristic
514C3×C4C22×C22⋊D511yescharacteristic
515Dic3—102no—
516Dic3—102no—
517Dic3—102no—
518C2×C5—22no—
519C2×C5—22no—
520C2×C5—41no—
521C2×C5C2×#90711yescharacteristic
522C2×C5#574511yescharacteristic
523C2×C5C2×C4×#6711yescharacteristic
524D5—122no—
525C23—22no—
526C23—22no—
527C23—41no—
528C23—152no—
529C23C2×C4×D1511yescharacteristic
530C23C2×C4×D1511yescharacteristic
531C23C2×C4×D1511yescharacteristic
532Q8—302no—
533D4—304no—
534D4—304no—
535D4—302no—
536D4—602no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—152no—
545C2×C4—152no—
546C2×C4—152no—
547C2×C4—302no—
548C2×C4—302no—
549C2×C4C2×C4×D1511yescharacteristic
550C2×C4C2×C4×D1511yescharacteristic
551C2×C4C2×C4×D1511yescharacteristic
552C2×C4#681712yes—
553C2×C4C2×#122012yes—
554C2×C4C23×D1511yescharacteristic
555C8—22no—
556C8—301no—
557C8—301no—
558C8—301no—
559C8—301no—
560C8C2×#122014yes—
561C2×C3—22no—
562C2×C3—22no—
563C2×C3—41no—
564C2×C3C2×#405311yescharacteristic
565C2×C3#6607911yescharacteristic
566C2×C3C2×C4×C22⋊D511yescharacteristic
567D3—202no—
568C5#8375711yescharacteristic, complemented, Sylow
569C22—21no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—44no—
577C22—302no—
578C22—302no—
579C22—302no—
580C22#9179612yescomplemented
581C22C4×#122012yes—
582C22C22×C4×D1511yescharacteristic
583C4—21no—
584C4—21no—
585C4—302no—
586C4—302no—
587C4—302no—
588C4C4×#122012yes—
589C4C2×#681711yescharacteristic
590C4C22×#122011yescharacteristic
591C3#1061548811yescharacteristic, complemented, Sylow
592C2—22no—
593C2—22no—
594C2—41no—
595C2—602no—
596C2C2×#9179611yescharacteristic
597C2#1175650411yescharacteristic
598C2C2×C4×#122011yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D15
commutator, socleC22×C3×C5C23×C4
FittingC22×C3×C5×M16C2
prime radical#11756550C2
cosocleC2×C3×C4×C5C24
hypercenterC22×M16D15
hypocenterC3×C5#2933
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
1. C22×C3×C5×M16C2semidirect
3. C2×#91796C2semidirect
4. C2×#91114C2semidirect
5. #11751230C2semidirect
6. #11751229C2semidirect
7. #11751228C2semidirect
8. #11751223C2semidirect
9. #11751220C2semidirect
10. #11751212C2semidirect
11. C8×#1220C2semidirect
32. C2×#6705C22semidirect
33. C2×C8×D15C22semidirect
38. #91104C22semidirect
39. #91104C22semidirect
40. #91099C22semidirect
41. #91098C22semidirect
42. #91098C22semidirect
43. C8×Dic15C22semidirect
45. C22×C5×M16D3semidirect
75. C3×C5×M16D4semidirect
97. C22×C3×M16D5semidirect
226. C5×M16#67semidirect
249. C22×M16D15semidirect
268. C22×C3×C5#131semidirect
300. C3×M16C22⋊D5semidirect
440. C22×C5#907semidirect
474. M16#1220semidirect
495. C3×C5#2933semidirect
504. C22×C3#4053semidirect
568. C5#83757semidirect
580. C22#91796semidirect
591. C3#10615488semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C5×M162
#117565502
C2×#917962
C2×#911142
#117512302
#117512292
#117512282
#117512232
#117512202
#117512122
C8×#12202
#106154883
#837575