Finite Group Databasebeta

← #14326045

Subgroups — #14326045

Order 1920.

1856 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6817C211yesmaximal, characteristic
2C2×#91552C212yesmaximal, complemented
3C5×#5211C211yesmaximal, characteristic, complemented
4C2×C3×#3963C211yesmaximal, characteristic, complemented
5#11749494C212yesmaximal, complemented
6#11748904C212yesmaximal, complemented
7#11748888C211yesmaximal, characteristic, complemented
8#11748863C211yesmaximal, characteristic, complemented
9#11748694C212yesmaximal, complemented
10#11748670C211yesmaximal, characteristic, complemented
11#11748606C211yesmaximal, characteristic, complemented
12#10605852—31nomaximal, Hall
13C23×C3×C4×C5C2211yescharacteristic
14C22×#1220C2211yescharacteristic
15C22×#1220C2211yescharacteristic
16C2×#6817—28no—
17C2×#6817C418yescomplemented
18C22×D60C2211yescharacteristic
19C22×C4×D15C2211yescharacteristic
20C22×C4×D15C2211yescharacteristic
21C22×Dic30C2211yescharacteristic
22C23×C3×Dic5C2212yes—
23C22×#1200C2212yes—
24C2×C5×#217C2212yes—
25#91552—28no—
26C2×#6669C2212yes—
27C2×#6667C2212yes—
28C2×#6667C2212yes—
29C2×#6667C2212yes—
30C2×Dic3×Dic5C2212yes—
31C5×C3⋊#95C2211yescharacteristic
32C5×C3⋊#95C2212yes—
33C5×C3⋊#95C2211yescharacteristic
34C3×#3963C2211yescharacteristic
35C3×#3963C2212yes—
36C3×#3963C2211yescharacteristic
37#90994C2212yes—
38#90994C2212yes—
39#73368—51nomaximal, Hall
40C22×#709—31no—
41C2×#4027—32no—
42C5×#381—31no—
43C2×#3963D311yescharacteristic, complemented
44#65247—32no—
45#64956—31no—
46#64955—31no—
47#64954—32no—
48#64953—31no—
49D5×#95—31no—
50#64935—32no—
51C24×C3×C5C2311yescharacteristic
52C23×D15C2311yescharacteristic
53C23×D15C2311yescharacteristic
54C22×C3×C4×C5—24no—
55C22×C3×C4×C5C2×C414yescomplemented
56C22×C3×C4×C5C2311yescharacteristic
57C22×C3×C4×C5C2311yescharacteristic
58C2×#1220—24no—
59C2×#1220C2×C414yes—
60C2×#1220C2×C414yescomplemented
61C2×#1220D418yes—
62C22×Dic15C2311yescharacteristic
63C22×Dic15C2311yescharacteristic
64#6817—232no—
65C2×D60—24no—
66C2×D60C2×C414yescomplemented
67C2×C4×D15—24no—
68C2×C4×D15—24no—
69C2×C4×D15C2×C414yescomplemented
70C2×C4×D15C2×C414yescomplemented
71C2×Dic30—24no—
72C2×Dic30C2×C414yescomplemented
73C22×C5×Dic3C2312yes—
74C22×C5×Dic3C2312yes—
75C22×C3×Dic5—24no—
76C22×C3×Dic5—24no—
77C22×C3×Dic5—42no—
78C22×C3×Dic5C2312yes—
79C22×C3×Dic5C2312yes—
80C2×#1200—24no—
81C2×#1200—24no—
82C2×#1200—42no—
83C5×#217—24no—
84#6669—24no—
85#6667—24no—
86#6667—24no—
87#6667—42no—
88Dic3×Dic5—24no—
89C22×#235—51no—
90C2×C4×#67—52no—
91C2×C3×#95—51no—
92#5213—52no—
93#5211D511yescharacteristic, complemented
94#4667—51no—
95#4666—51no—
96#4665—52no—
97#4664—51no—
98D3×#95—51no—
99#4646—52no—
100C23×C4×C5C2×D311yescharacteristic
101C22×C22⋊D5—31no—
102C22×C22⋊D5—31no—
103C23×Dic5C2×D312yes—
104C2×#709—38no—
105C2×#709—68no—
106C22×D20—31no—
107C22×C4×D5—31no—
108C22×C4×D5—31no—
109C22×C4×D5—32no—
110C22×Dic10—31no—
111C2×C5×C22⋊C4—32no—
112C2×#685—32no—
113C2×#685—32no—
114C2×#685—32no—
115C2×#683—32no—
116C2×C4×Dic5—32no—
117#4027—68no—
118C5×#95—32no—
119C5×#95—31no—
120C5×#95—31no—
121#3963—32no—
122#3963—32no—
123#3963C2×D312yes—
124#3963C2×D311yescharacteristic
125#3963C2×D311yescharacteristic
126#2339—151noSylow
127C23×C3×C5—22no—
128C23×C3×C5—21no—
129C23×C3×C5—21no—
130C23×C3×C5C22×C412yes—
131C23×C3×C5C2×D414yes—
132C23×C3×C5C2411yescharacteristic
133C22×D15—22no—
134C22×D15—21no—
135C22×D15—21no—
136C22×D15—22no—
137C22×D15—22no—
138C22×D15—21no—
139C22×D15—21no—
140C22×D15C22×C412yes—
141C22×D15C22×C412yes—
142C22×D15C2×D414yes—
143C2×C3×C4×C5—28no—
144C2×C3×C4×C5—22no—
145C2×C3×C4×C5—22no—
146C2×C3×C4×C5C22×C412yes—
147C2×C3×C4×C5C22×C412yes—
148#1220—28no—
149#1220C22⋊C4116yescomplemented
150C2×Dic15—22no—
151C2×Dic15C22×C412yes—
152C2×Dic15C22×C412yes—
153C2×Dic15C2×D414yes—
154D60—28no—
155C4×D15—28no—
156C4×D15—28no—
157Dic30—28no—
158C2×C5×Dic3—22no—
159C2×C5×Dic3—22no—
160C2×C5×Dic3—22no—
161C2×C5×Dic3—22no—
162C2×C5×Dic3—22no—
163C2×C5×Dic3—22no—
164C2×C3×Dic5—22no—
165C2×C3×Dic5—22no—
166C2×C3×Dic5—22no—
167C2×C3×Dic5—22no—
168C2×C3×Dic5—22no—
169C2×C3×Dic5—48no—
170C2×C3×Dic5#4114yescomplemented
171#1200—48no—
172C23×C3×C4—52no—
173C23×C3×C4C2×D511yescharacteristic
174C22×#67—51no—
175C22×#67—51no—
176C2×#235—58no—
177C2×#235—108no—
178C22×D12—51no—
179C22×C4×D3—51no—
180C22×C4×D3—51no—
181C22×C4×D3—52no—
182C22×Dic6—51no—
183C2×#217C2×D512yes—
184C4×#67—108no—
185C2×(C2×D3)⋊C4—52no—
186C2×(C2×D3)⋊C4—52no—
187C2×(C2×D3)⋊C4—52no—
188C2×Dic3⋊C4—52no—
189C2×C4×Dic3—52no—
190C3×#95—52no—
191C3×#95—51no—
192C3×#95—51no—
193C3⋊#95—52no—
194C3⋊#95—52no—
195C3⋊#95C2×D512yes—
196C3⋊#95C2×D511yescharacteristic
197C3⋊#95C2×D511yescharacteristic
198C24×C5C22×D311yescharacteristic
199C23×D5—31no—
200C23×D5—31no—
201C22×C4×C5—24no—
202C22×C4×C5—32no—
203C22×C4×C5—32no—
204C22×C4×C5C4×D314yescomplemented
205C22×C4×C5C22×D311yescharacteristic
206C22×C4×C5C22×D311yescharacteristic
207C2×C22⋊D5—34no—
208C2×C22⋊D5—38no—
209C2×C22⋊D5—34no—
210C2×C22⋊D5—64no—
211C22×Dic5—24no—
212C22×Dic5—24no—
213C22×Dic5—31no—
214C22×Dic5—31no—
215C22×Dic5—42no—
216C22×Dic5C22×D312yes—
217C22×Dic5C22×D312yes—
218#709—632no—
219C2×D20—34no—
220C2×D20—64no—
221C2×C4×D5—34no—
222C2×C4×D5—34no—
223C2×C4×D5—64no—
224C2×C4×D5—64no—
225C2×C4×D5—64no—
226C2×C4×D5—64no—
227C2×C4×D5—122no—
228C2×Dic10—34no—
229C2×Dic10—64no—
230C5×C22⋊C4—64no—
231#685—64no—
232#685—64no—
233#685—122no—
234#683—64no—
235C4×Dic5—64no—
236C22×#41—151no—
237C2×C4×D4—152no—
238#391—152no—
239#386—152no—
240#386—152no—
241#381—31no—
242#381—151no—
243#380—151no—
244C4×C22⋊C4—151no—
245C2×#95—51no—
246C2×#95—151no—
247C22×C3×C5—22no—
248C22×C3×C5—22no—
249C22×C3×C5—24no—
250C22×C3×C5—24no—
251C22×C3×C5C2×C22⋊C414yes—
252C22×C3×C5C23×C411yescharacteristic
253C22×C3×C5C22×D412yes—
254C22×C3×C5C2×#4112yes—
255C22×C3×C5H(5,𝔽2)11yescharacteristic
256C22×C3×C52−1+411yescharacteristic
257C2×D15—22no—
258C2×D15—22no—
259C2×D15—24no—
260C2×D15—24no—
261C2×D15—22no—
262C2×D15—22no—
263C2×D15—22no—
264C2×D15—42no—
265C2×D15—42no—
266C2×D15C2×C22⋊C414yes—
267C3×C4×C5—22no—
268C3×C4×C5—22no—
269Dic15—22no—
270Dic15C2×C22⋊C414yes—
271C3×Dic5—24no—
272C3×Dic5—42no—
273C5×Dic3—42no—
274C5×Dic3—42no—
275C24×C3C22×D511yescharacteristic
276C23×D3—51no—
277C23×D3—51no—
278C22×C3×C4—24no—
279C22×C3×C4—52no—
280C22×C3×C4—52no—
281C22×C3×C4—104no—
282C22×C3×C4—104no—
283C22×C3×C4—202no—
284C22×C3×C4C4×D514yescomplemented
285C22×C3×C4C22×D511yescharacteristic
286C22×C3×C4C22×D511yescharacteristic
287C2×#67—54no—
288C2×#67—58no—
289C2×#67—54no—
290C2×#67—104no—
291C22×Dic3—51no—
292C22×Dic3—51no—
293C22×Dic3C22×D512yes—
294C22×Dic3C22×D512yes—
295#235—1032no—
296C2×D12—54no—
297C2×D12—104no—
298C2×C4×D3—54no—
299C2×C4×D3—54no—
300C2×C4×D3—104no—
301C2×C4×D3—104no—
302C2×C4×D3—104no—
303C2×C4×D3—104no—
304C2×C4×D3—202no—
305C2×Dic6—54no—
306C2×Dic6—104no—
307#217—24no—
308(C2×D3)⋊C4—104no—
309(C2×D3)⋊C4—104no—
310(C2×D3)⋊C4—202no—
311Dic3⋊C4—104no—
312C4×Dic3—104no—
313C23×C5—22no—
314C23×C5—21no—
315C23×C5—21no—
316C23×C5C2×C4×D312yes—
317C23×C5D3×D414yes—
318C23×C5C23×D311yescharacteristic
319C22×D5—32no—
320C22×D5—34no—
321C22×D5—32no—
322C22×D5—62no—
323C22×D5—61no—
324C22×D5—61no—
325C22×D5—62no—
326C22×D5—62no—
327C22×D5—61no—
328C22×D5—61no—
329C2×C4×C5—28no—
330C2×C4×C5—22no—
331C2×C4×C5—22no—
332C2×C4×C5—62no—
333C2×C4×C5—62no—
334C2×C4×C5—62no—
335C2×C4×C5—62no—
336C2×C4×C5—62no—
337C2×C4×C5—62no—
338C2×C4×C5C2×C4×D312yes—
339C2×C4×C5C2×C4×D312yes—
340C22⋊D5—316no—
341C22⋊D5—68no—
342C2×Dic5—22no—
343C2×Dic5—22no—
344C2×Dic5—22no—
345C2×Dic5—22no—
346C2×Dic5—22no—
347C2×Dic5—32no—
348C2×Dic5—34no—
349C2×Dic5—32no—
350C2×Dic5—48no—
351C2×Dic5—62no—
352C2×Dic5#23514yescomplemented
353D20—68no—
354C4×D5—68no—
355C4×D5—68no—
356C4×D5—128no—
357Dic10—68no—
358C2×#41—158no—
359C2×#41—308no—
360C22×Q8—151no—
361C22×D4—151no—
362C22×D4—151no—
363C22×D4—151no—
364C23×C4—52no—
365C23×C4—152no—
366C23×C4—151no—
367C23×C4—151no—
368C23×C4D3×D511yescharacteristic
369C4×D4—308no—
370C2×C4⋊C4—152no—
371C2×C22⋊C4—32no—
372C2×C22⋊C4—152no—
373C2×C22⋊C4—152no—
374C2×C22⋊C4—152no—
375C2×C42—152no—
376#95—31no—
377#95—31no—
378#95—32no—
379#95—51no—
380#95—51no—
381#95—52no—
382#95—152no—
383#95—152no—
384C2×C3×C5—24no—
385C2×C3×C5C22×C22⋊C411yescharacteristic
386C2×C3×C5C4×#4111yescharacteristic
387C2×C3×C5#52011yescharacteristic
388C2×C3×C5#54612yes—
389C2×C3×C5#54712yes—
390D15—24no—
391D15—42no—
392C23×C3—22no—
393C23×C3—21no—
394C23×C3—21no—
395C23×C3C2×C4×D512yes—
396C23×C3D4×D514yes—
397C23×C3C23×D511yescharacteristic
398C22×D3—52no—
399C22×D3—54no—
400C22×D3—52no—
401C22×D3—102no—
402C22×D3—101no—
403C22×D3—101no—
404C22×D3—102no—
405C22×D3—102no—
406C22×D3—101no—
407C22×D3—101no—
408C2×C3×C4—22no—
409C2×C3×C4—22no—
410C2×C3×C4—28no—
411C2×C3×C4—54no—
412C2×C3×C4—102no—
413C2×C3×C4—102no—
414C2×C3×C4—102no—
415C2×C3×C4—102no—
416C2×C3×C4—102no—
417C2×C3×C4—208no—
418C2×C3×C4C2×C4×D512yes—
419C2×C3×C4C2×C4×D512yes—
420#67—516no—
421#67—108no—
422C2×Dic3—22no—
423C2×Dic3—22no—
424C2×Dic3—22no—
425C2×Dic3—22no—
426C2×Dic3—22no—
427C2×Dic3—22no—
428C2×Dic3—52no—
429C2×Dic3—54no—
430C2×Dic3—52no—
431C2×Dic3—102no—
432D12—108no—
433C4×D3—108no—
434C4×D3—108no—
435C4×D3—208no—
436Dic6—108no—
437C22×C5—22no—
438C22×C5—24no—
439C22×C5—22no—
440C22×C5—24no—
441C22×C5D3×C22⋊C414yes—
442C22×C5C22×C4×D311yescharacteristic
443C22×C5C2×#23512yes—
444C22×C5C2×D3×D412yes—
445C22×C5#100411yescharacteristic
446C22×C5#100711yescharacteristic
447C2×D5—34no—
448C2×D5—62no—
449C2×D5—64no—
450C2×D5—62no—
451C2×D5—64no—
452C2×D5—62no—
453C2×D5—62no—
454C2×D5—62no—
455C2×D5—122no—
456C2×D5—122no—
457C4×C5—22no—
458C4×C5—22no—
459C4×C5—122no—
460C4×C5—122no—
461Dic5—24no—
462Dic5—34no—
463Dic5—42no—
464Dic5—62no—
465C24—151no—
466C24—151no—
467C24C2×D3×D511yescharacteristic
468#41—3032no—
469C2×Q8—154no—
470C2×Q8—304no—
471C2×D4—154no—
472C2×D4—158no—
473C2×D4—154no—
474C2×D4—154no—
475C2×D4—304no—
476C2×D4—304no—
477C22×C4—24no—
478C22×C4—32no—
479C22×C4—32no—
480C22×C4—52no—
481C22×C4—52no—
482C22×C4—104no—
483C22×C4—104no—
484C22×C4—151no—
485C22×C4—151no—
486C22×C4—154no—
487C22×C4—154no—
488C22×C4—202no—
489C22×C4—304no—
490C22×C4—304no—
491C22×C4—304no—
492C22×C4—304no—
493C22×C4—602no—
494C22×C4#120014yescomplemented
495C22×C4C2×D3×D511yescharacteristic
496C22×C4C2×D3×D511yescharacteristic
497C4⋊C4—304no—
498C22⋊C4—64no—
499C22⋊C4—304no—
500C22⋊C4—304no—
501C22⋊C4—602no—
502C42—304no—
503C3×C5#233911yescharacteristic, complemented, Hall
504C22×C3—22no—
505C22×C3—22no—
506C22×C3—24no—
507C22×C3—24no—
508C22×C3D5×C22⋊C414yes—
509C22×C3C22×C4×D511yescharacteristic
510C22×C3C2×D4×D512yes—
511C22×C3D5×#4112yes—
512C22×C3#414911yescharacteristic
513C22×C3#415011yescharacteristic
514C2×D3—54no—
515C2×D3—102no—
516C2×D3—102no—
517C2×D3—104no—
518C2×D3—104no—
519C2×D3—102no—
520C2×D3—102no—
521C2×D3—102no—
522C2×D3—202no—
523C2×D3—202no—
524C3×C4—22no—
525C3×C4—22no—
526C3×C4—104no—
527C3×C4—202no—
528Dic3—42no—
529Dic3—42no—
530Dic3—54no—
531Dic3—102no—
532C2×C5—24no—
533C2×C5C4×#23511yescharacteristic
534C2×C5C2×D3×C22⋊C411yescharacteristic
535C2×C5#549012yes—
536C2×C5#550411yescharacteristic
537C2×C5#550712yes—
538D5—64no—
539D5—122no—
540C23—21no—
541C23—22no—
542C23—21no—
543C23—152no—
544C23—154no—
545C23—152no—
546C23—301no—
547C23—302no—
548C23—301no—
549C23—301no—
550C23—302no—
551C23—301no—
552C23—302no—
553C23C2×#120012yes—
554C23#679014yes—
555C23C22×D3×D511yescharacteristic
556Q8—308no—
557D4—1516no—
558D4—308no—
559D4—308no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—28no—
563C2×C4—54no—
564C2×C4—62no—
565C2×C4—62no—
566C2×C4—62no—
567C2×C4—62no—
568C2×C4—62no—
569C2×C4—62no—
570C2×C4—102no—
571C2×C4—102no—
572C2×C4—102no—
573C2×C4—102no—
574C2×C4—102no—
575C2×C4—152no—
576C2×C4—154no—
577C2×C4—152no—
578C2×C4—208no—
579C2×C4—308no—
580C2×C4—308no—
581C2×C4—302no—
582C2×C4—608no—
583C2×C4C2×#120012yes—
584C2×C4C2×#120012yes—
585C2×C3—24no—
586C2×C3C2×D5×C22⋊C411yescharacteristic
587C2×C3#6585611yescharacteristic
588C2×C3#6585711yescharacteristic
589C2×C3#6598812yes—
590C2×C3#6598912yes—
591D3—104no—
592D3—202no—
593C5#7336811yescharacteristic, complemented, Sylow
594C22—22no—
595C22—24no—
596C22—24no—
597C22—22no—
598C22—154no—
599C22—302no—
600C22—304no—
601C22—304no—
602C22—302no—
603C22—302no—
604C22—302no—
605C22—302no—
606C22—602no—
607C22—602no—
608C22#9156114yes—
609C22#9200411yescharacteristic
610C22D5×#23512yes—
611C22#9201711yescharacteristic
612C22C22×#120011yescharacteristic
613C22C2×#679012yes—
614C4—22no—
615C4—22no—
616C4—104no—
617C4—122no—
618C4—122no—
619C4—154no—
620C4—202no—
621C4—302no—
622C3#1060585211yescharacteristic, complemented, Sylow
623C2—24no—
624C2—304no—
625C2—602no—
626C2#1175437612yes—
627C2#1175441411yescharacteristic
628C2#1175449611yescharacteristic
629C2#1175466412yes—
630C2C2×#9156111yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC23×C3×C4×C5C22
socle, cosocleC23×C3×C5C24
prime radicalC22×#6817C2
hypercenterC23×C4D3×D5
hypocenterC3×C5#2339
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
2. C2×#91552C2semidirect
3. C5×#5211C2semidirect
4. C2×C3×#3963C2semidirect
5. #11749494C2semidirect
6. #11748904C2semidirect
7. #11748888C2semidirect
8. #11748863C2semidirect
9. #11748694C2semidirect
10. #11748670C2semidirect
11. #11748606C2semidirect
17. C2×#6817C4semidirect
43. C2×#3963D3semidirect
55. C22×C3×C4×C5C2×C4semidirect
60. C2×#1220C2×C4semidirect
66. C2×D60C2×C4semidirect
69. C2×C4×D15C2×C4semidirect
70. C2×C4×D15C2×C4semidirect
72. C2×Dic30C2×C4semidirect
93. #5211D5semidirect
149. #1220C22⋊C4semidirect
170. C2×C3×Dic5#41semidirect
204. C22×C4×C5C4×D3semidirect
284. C22×C3×C4C4×D5semidirect
352. C2×Dic5#235semidirect
494. C22×C4#1200semidirect
503. C3×C5#2339semidirect
593. C5#73368semidirect
622. C3#10605852semidirect

Maximal subgroups

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

subgroupindex
C22×#68172
C2×#915522
C5×#52112
C2×C3×#39632
#117494942
#117489042
#117488882
#117488632
#117486942
#117486702
#117486062
#106058523
#733685