Finite Group Databasebeta

← #14329108

Subgroups — #14329108

Order 1920.

1376 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91781C211yesmaximal, characteristic, complemented
2C2×C3×#4039C211yesmaximal, characteristic, complemented
3C2×#91465C211yesmaximal, characteristic
4C5×Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
5#11748912C212yesmaximal, complemented
6#11748907C212yesmaximal, complemented
7#11748842C211yesmaximal, characteristic, complemented
8#11748785C211yesmaximal, characteristic, complemented
9C4×#6667C211yesmaximal, characteristic, complemented
10#11748710C212yesmaximal, complemented
11C4×#6665C212yesmaximal, complemented
12#10605380—31nomaximal, Hall
13C22×C4×D15C2211yescharacteristic
14C22×C3×D20C2211yescharacteristic
15C22×C3×C4×D5C2212yes—
16C22×#1202C2212yes—
17C2×C3×C5×C4⋊C4C2211yescharacteristic
18C2×#6714C2212yes—
19C2×C4⋊Dic15C2211yescharacteristic
20C2×C4×Dic15C2212yes—
21#91781—24no—
22C2×C5×C4⋊Dic3C2211yescharacteristic
23C2×C4×C5×Dic3C2211yescharacteristic
24C2×C4×C5×Dic3C2212yes—
25C2×C3×#685C2212yes—
26C2×C3×C4×Dic5C2211yescharacteristic
27C3×#4039—24no—
28C2×Dic3⋊Dic5C2212yes—
29C2×#6667C2211yescharacteristic
30C2×#6667C2211yescharacteristic
31C2×#6665C2212yes—
32C2×#6665C2212yes—
33C2×#6665C2212yes—
34#91465C418yescomplemented
35C5×C3⋊#95C2212yes—
36#90994C2212yes—
37#90994C2211yescharacteristic
38#90994C2211yescharacteristic
39#82341—51nomaximal, Hall
40C2×#4041—31no—
41C2×#4039D311yescharacteristic, complemented
42C2×#4038—31no—
43C4×C5×C4⋊C4—31no—
44#65274—31no—
45#65228—32no—
46#65228—31no—
47C4×#685—31no—
48C4×#685—32no—
49#64959—32no—
50#64954—32no—
51C23×D15C2311yescharacteristic
52C23×C3×D5C2312yes—
53C22×C3×C4×C5C2312yes—
54C22×C3×C4×C5C2311yescharacteristic
55C22×Dic15C2312yes—
56C22×Dic15C2311yescharacteristic
57C2×C4×D15—22no—
58C2×C4×D15—22no—
59C2×C4×D15C2×C414yescomplemented
60C22×C5×Dic3C2311yescharacteristic
61C22×C5×Dic3C2312yes—
62C22×C5×Dic3C2311yescharacteristic
63C22×C3×Dic5C2311yescharacteristic
64C22×C3×Dic5C2311yescharacteristic
65C2×C3×D20—22no—
66C2×C3×D20—22no—
67C2×C3×D20C2×C414yes—
68C2×C3×C4×D5—24no—
69C2×C3×C4×D5—24no—
70C2×C3×C4×D5—42no—
71C2×#1202—42no—
72C2×#1202—42no—
73C2×#1202C2×C418yescomplemented
74C3×C5×C4⋊C4D414yescomplemented
75#6714—24no—
76C4⋊Dic15—22no—
77C4×Dic15—24no—
78C5×C4⋊Dic3C2×C414yescomplemented
79C4×C5×Dic3—22no—
80C4×C5×Dic3—42no—
81C3×#685—24no—
82C3×C4×Dic5—22no—
83Dic3⋊Dic5—42no—
84#6667—22no—
85#6667—22no—
86#6665—24no—
87#6665—42no—
88#6665C2×C418yescomplemented
89C2×C3×C4×D4—51no—
90C2×#938—51no—
91C2×#892—51no—
92#5221—51no—
93#5213—52no—
94C4×#217—52no—
95Dic3×C4⋊C4D511yescharacteristic, complemented
96#4957—52no—
97#4944—52no—
98C4×(C2×D3)⋊C4—51no—
99#4928—51no—
100C22×D20—32no—
101C22×D20C2×D311yescharacteristic
102C22×C4×D5—31no—
103C22×C4×D5C2×D312yes—
104C2×C5×C4⋊C4—31no—
105C2×C5×C4⋊C4C2×D311yescharacteristic
106C2×C42×C5—31no—
107C2×C42×C5—32no—
108C2×#685—32no—
109C2×#685—32no—
110C2×#685—32no—
111C2×#685—31no—
112C2×#685—31no—
113C2×#685—32no—
114C2×#685C2×D312yes—
115C2×C4⋊Dic5—31no—
116C2×C4⋊Dic5—32no—
117C2×C4×Dic5—32no—
118C2×C4×Dic5C2×D311yescharacteristic
119#4041—38no—
120#4039—24no—
121#4038—64no—
122C5×#95—32no—
123#3963—32no—
124#3963—31no—
125#3963—31no—
126#2363—151noSylow
127C23×C3×C5C2411yescharacteristic
128C22×D15—21no—
129C22×D15—21no—
130C22×D15—21no—
131C22×D15—21no—
132C22×D15—21no—
133C22×D15—21no—
134C22×D15C22×C412yes—
135C22×C3×D5—22no—
136C22×C3×D5—22no—
137C22×C3×D5—22no—
138C22×C3×D5—22no—
139C22×C3×D5—22no—
140C22×C3×D5—22no—
141C22×C3×D5C22×C414yes—
142C2×C3×C4×C5—22no—
143C2×C3×C4×C5—22no—
144C2×C3×C4×C5C22×C412yes—
145C2×C3×C4×C5C2×D414yes—
146C2×C3×C4×C5C2×D412yes—
147C2×C3×C4×C5#4112yes—
148C2×Dic15—21no—
149C2×Dic15—21no—
150C2×Dic15—22no—
151C2×Dic15—22no—
152C2×Dic15—22no—
153C2×Dic15C22×C412yes—
154C4×D15—28no—
155C2×C5×Dic3—21no—
156C2×C5×Dic3—22no—
157C2×C5×Dic3—21no—
158C2×C5×Dic3—22no—
159C2×C5×Dic3—21no—
160C2×C5×Dic3—21no—
161C2×C5×Dic3—21no—
162C2×C5×Dic3C22×C414yes—
163C2×C5×Dic3#4112yes—
164C2×C3×Dic5—21no—
165C2×C3×Dic5—21no—
166C2×C3×Dic5—21no—
167C2×C3×Dic5—21no—
168C2×C3×Dic5—21no—
169C2×C3×Dic5#4112yes—
170C3×D20—28no—
171C3×C4×D5—48no—
172#1202—48no—
173C22×C3×D4—51no—
174C23×C3×C4—52no—
175C22×#67—52no—
176C22×C4×D3—51no—
177C3×C4×D4—104no—
178C2×C3×C4⋊C4C2×D511yescharacteristic
179C2×C3×C22⋊C4—52no—
180C2×C3×C42—51no—
181C2×#217—52no—
182C2×#217—52no—
183C2×#217—52no—
184#938—58no—
185C2×(C2×D3)⋊C4—52no—
186C2×(C2×D3)⋊C4—51no—
187C2×(C2×D3)⋊C4—51no—
188C2×C4⋊Dic3—51no—
189C2×C4⋊Dic3C2×D511yescharacteristic
190C2×Dic3⋊C4—52no—
191C2×C4×Dic3—52no—
192C2×C4×Dic3C2×D511yescharacteristic
193C2×C4×Dic3C2×D512yes—
194#892—104no—
195C3⋊#95—52no—
196C3⋊#95—51no—
197C3⋊#95—51no—
198C3⋊#95C2×D512yes—
199C23×D5—31no—
200C23×D5C22×D312yes—
201C22×C4×C5—31no—
202C22×C4×C5—32no—
203C22×C4×C5—31no—
204C22×C4×C5C22×D312yes—
205C22×C4×C5C22×D311yescharacteristic
206C22×Dic5—31no—
207C22×Dic5—32no—
208C22×Dic5C22×D311yescharacteristic
209C22×Dic5C22×D311yescharacteristic
210C2×D20—22no—
211C2×D20—22no—
212C2×D20—38no—
213C2×D20—122no—
214C2×D20—122no—
215C2×D20C4×D314yes—
216C2×C4×D5—24no—
217C2×C4×D5—24no—
218C2×C4×D5—34no—
219C2×C4×D5—42no—
220C2×C4×D5—62no—
221C2×C4×D5—62no—
222C5×C4⋊C4—34no—
223C5×C4⋊C4#6714yescomplemented
224C42×C5—62no—
225C42×C5—122no—
226#685—24no—
227#685—38no—
228#685—62no—
229#685—64no—
230#685—64no—
231#685—62no—
232#685—122no—
233C4⋊Dic5—62no—
234C4⋊Dic5—122no—
235C4×Dic5—22no—
236C4×Dic5—64no—
237C2×#121—151no—
238C2×C4×D4—51no—
239C2×#117—151no—
240#390—152no—
241#390—151no—
242#389—151no—
243#388—152no—
244#386—152no—
245C4×C4⋊C4—31no—
246C4×C22⋊C4—151no—
247C4×C22⋊C4—152no—
248C22×C3×C5C23×C411yescharacteristic
249C22×C3×C5C22×D411yescharacteristic
250C22×C3×C5C2×#4111yescharacteristic
251C22×C3×C5C2×#4111yescharacteristic
252C22×C3×C5C2×#4111yescharacteristic
253C22×C3×C5H(5,𝔽2)11yescharacteristic
254C22×C3×C5H(5,𝔽2)11yescharacteristic
255C2×D15—22no—
256C2×D15—22no—
257C2×D15—22no—
258C2×D15—41no—
259C2×D15—41no—
260C2×D15—41no—
261C2×D15—41no—
262C2×C3×D5—24no—
263C2×C3×D5—24no—
264C2×C3×D5—24no—
265C2×C3×D5—42no—
266C2×C3×D5—42no—
267C2×C3×D5—42no—
268C2×C3×D5—42no—
269C3×C4×C5—24no—
270C3×C4×C5C4×D414yes—
271Dic15—22no—
272Dic15—42no—
273C3×Dic5—22no—
274C3×Dic5—41no—
275C5×Dic3—24no—
276C5×Dic3—22no—
277C5×Dic3—41no—
278C24×C3—52no—
279C23×D3—51no—
280C2×C3×D4—54no—
281C2×C3×D4—102no—
282C2×C3×D4—102no—
283C22×C3×C4—51no—
284C22×C3×C4—51no—
285C22×C3×C4—104no—
286C22×C3×C4—104no—
287C22×C3×C4—202no—
288C22×C3×C4C22×D512yes—
289C22×C3×C4C22×D511yescharacteristic
290C2×#67—58no—
291C2×#67—202no—
292C2×#67—202no—
293C22×Dic3—51no—
294C22×Dic3—52no—
295C22×Dic3C22×D511yescharacteristic
296C22×Dic3C22×D512yes—
297C22×Dic3C22×D511yescharacteristic
298C2×C4×D3—54no—
299C2×C4×D3—102no—
300C2×C4×D3—102no—
301C3×C4⋊C4D2014yescomplemented
302C3×C22⋊C4—104no—
303C3×C42—102no—
304#217—58no—
305#217—104no—
306#217—202no—
307(C2×D3)⋊C4—102no—
308(C2×D3)⋊C4—104no—
309(C2×D3)⋊C4—102no—
310C4⋊Dic3—102no—
311C4⋊Dic3C4×D514yescomplemented
312Dic3⋊C4—202no—
313C4×Dic3—22no—
314C4×Dic3—42no—
315C4×Dic3—104no—
316C23×C5C23×D311yescharacteristic
317C22×D5—22no—
318C22×D5—22no—
319C22×D5—22no—
320C22×D5—22no—
321C22×D5—22no—
322C22×D5—22no—
323C22×D5—32no—
324C22×D5—61no—
325C22×D5—61no—
326C22×D5—61no—
327C22×D5—61no—
328C22×D5—61no—
329C22×D5—61no—
330C22×D5C2×C4×D314yes—
331C2×C4×C5—22no—
332C2×C4×C5—22no—
333C2×C4×C5—34no—
334C2×C4×C5—32no—
335C2×C4×C5—61no—
336C2×C4×C5—61no—
337C2×C4×C5—62no—
338C2×C4×C5—62no—
339C2×C4×C5—61no—
340C2×C4×C5—61no—
341C2×C4×C5—61no—
342C2×C4×C5C2×C4×D312yes—
343C2×C4×C5#23512yes—
344C2×C4×C5C2×#6714yes—
345C2×C4×C5C2×#6712yes—
346C2×Dic5—21no—
347C2×Dic5—21no—
348C2×Dic5—21no—
349C2×Dic5—21no—
350C2×Dic5—21no—
351C2×Dic5—32no—
352C2×Dic5—61no—
353C2×Dic5—61no—
354C2×Dic5—62no—
355C2×Dic5—62no—
356C2×Dic5—62no—
357C2×Dic5#23712yes—
358D20—28no—
359D20—128no—
360C4×D5—48no—
361C4×D5—68no—
362C22×D4—51no—
363C22×D4—152no—
364C23×C4—52no—
365C23×C4—151no—
366#121—158no—
367C4×D4—104no—
368#117—304no—
369C2×C4⋊C4—31no—
370C2×C4⋊C4—151no—
371C2×C4⋊C4—152no—
372C2×C4⋊C4D3×D511yescharacteristic
373C2×C22⋊C4—52no—
374C2×C22⋊C4—152no—
375C2×C22⋊C4—151no—
376C2×C22⋊C4—151no—
377C2×C22⋊C4—152no—
378C2×C22⋊C4—152no—
379C2×C22⋊C4—152no—
380C2×C42—31no—
381C2×C42—32no—
382C2×C42—51no—
383C2×C42—152no—
384#95—32no—
385#95—151no—
386#95—152no—
387#95—151no—
388C2×C3×C5C2×C4×D411yescharacteristic
389C2×C3×C5C4×#4111yescharacteristic
390C2×C3×C5#51811yescharacteristic
391C2×C3×C5#54611yescharacteristic
392C2×C3×C5#55011yescharacteristic
393C2×C3×C5#55311yescharacteristic
394C2×C3×C5#55511yescharacteristic
395D15—42no—
396C3×D5—44no—
397C23×C3—54no—
398C23×C3—102no—
399C23×C3—102no—
400C23×C3—102no—
401C23×C3—102no—
402C23×C3—102no—
403C23×C3—102no—
404C23×C3C23×D511yescharacteristic
405C22×D3—52no—
406C22×D3—101no—
407C22×D3—101no—
408C22×D3—101no—
409C22×D3—101no—
410C22×D3—101no—
411C22×D3—101no—
412C3×D4—108no—
413C2×C3×C4—22no—
414C2×C3×C4—22no—
415C2×C3×C4—52no—
416C2×C3×C4—101no—
417C2×C3×C4—101no—
418C2×C3×C4—101no—
419C2×C3×C4—101no—
420C2×C3×C4—101no—
421C2×C3×C4—208no—
422C2×C3×C4C2×C4×D512yes—
423C2×C3×C4C2×D2014yes—
424C2×C3×C4C2×D2012yes—
425C2×C3×C4#70912yes—
426#67—208no—
427C2×Dic3—21no—
428C2×Dic3—21no—
429C2×Dic3—22no—
430C2×Dic3—22no—
431C2×Dic3—21no—
432C2×Dic3—21no—
433C2×Dic3—21no—
434C2×Dic3—52no—
435C2×Dic3—101no—
436C2×Dic3—101no—
437C2×Dic3—102no—
438C2×Dic3—102no—
439C2×Dic3—102no—
440C2×Dic3C2×C4×D514yes—
441C2×Dic3#71312yes—
442C4×D3—108no—
443C22×C5C22×C4×D311yescharacteristic
444C22×C5C2×#23511yescharacteristic
445C22×C5C2×#23711yescharacteristic
446C22×C5#100411yescharacteristic
447C22×C5D3×#4111yescharacteristic
448C22×C5#100911yescharacteristic
449C22×C5C22×#6711yescharacteristic
450C2×D5—24no—
451C2×D5—24no—
452C2×D5—24no—
453C2×D5—42no—
454C2×D5—42no—
455C2×D5—42no—
456C2×D5—42no—
457C2×D5—62no—
458C2×D5—62no—
459C2×D5—62no—
460C2×D5—121no—
461C2×D5—121no—
462C2×D5—121no—
463C2×D5—121no—
464C4×C5—24no—
465C4×C5—64no—
466C4×C5—62no—
467C4×C5—121no—
468C4×C5C4×#6714yes—
469Dic5—22no—
470Dic5—41no—
471Dic5—62no—
472Dic5—122no—
473C24—52no—
474C24—151no—
475C2×D4—54no—
476C2×D4—102no—
477C2×D4—102no—
478C2×D4—158no—
479C2×D4—602no—
480C2×D4—602no—
481C22×C4—32no—
482C22×C4—31no—
483C22×C4—31no—
484C22×C4—51no—
485C22×C4—51no—
486C22×C4—104no—
487C22×C4—104no—
488C22×C4—151no—
489C22×C4—154no—
490C22×C4—152no—
491C22×C4—202no—
492C22×C4—302no—
493C22×C4—302no—
494C22×C4C2×D3×D512yes—
495C22×C4C2×D3×D511yescharacteristic
496C4⋊C4—34no—
497C4⋊C4—302no—
498C4⋊C4—602no—
499C4⋊C4#120214yescomplemented
500C22⋊C4—104no—
501C22⋊C4—158no—
502C22⋊C4—302no—
503C22⋊C4—304no—
504C22⋊C4—302no—
505C22⋊C4—304no—
506C22⋊C4—602no—
507C42—62no—
508C42—102no—
509C42—122no—
510C42—304no—
511C3×C5#236311yescharacteristic, complemented, Hall
512C22×C3—104no—
513C22×C3—104no—
514C22×C3—104no—
515C22×C3—202no—
516C22×C3—202no—
517C22×C3—202no—
518C22×C3—202no—
519C22×C3C22×C4×D511yescharacteristic
520C22×C3C22×D2011yescharacteristic
521C22×C3C2×#70911yescharacteristic
522C22×C3#414411yescharacteristic
523C22×C3C2×#71311yescharacteristic
524C22×C3D5×#4111yescharacteristic
525C22×C3#414911yescharacteristic
526C2×D3—102no—
527C2×D3—102no—
528C2×D3—102no—
529C2×D3—201no—
530C2×D3—201no—
531C2×D3—201no—
532C2×D3—201no—
533C3×C4—24no—
534C3×C4—102no—
535C3×C4—201no—
536C3×C4C4×D2014yes—
537Dic3—22no—
538Dic3—24no—
539Dic3—41no—
540Dic3—102no—
541Dic3—202no—
542C2×C5C4×#23711yescharacteristic
543C2×C5#554511yescharacteristic
544C2×C5#555811yescharacteristic
545C2×C5#556311yescharacteristic
546C2×C5C2×C4×#6711yescharacteristic
547C2×C5#580611yescharacteristic
548C2×C5#583411yescharacteristic
549D5—44no—
550D5—122no—
551C23—54no—
552C23—102no—
553C23—102no—
554C23—102no—
555C23—102no—
556C23—102no—
557C23—102no—
558C23—152no—
559C23—301no—
560C23—301no—
561C23—301no—
562C23—301no—
563C23—301no—
564C23—301no—
565C23C22×D3×D511yescharacteristic
566D4—108no—
567D4—608no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—34no—
571C2×C4—32no—
572C2×C4—52no—
573C2×C4—61no—
574C2×C4—61no—
575C2×C4—62no—
576C2×C4—62no—
577C2×C4—61no—
578C2×C4—61no—
579C2×C4—61no—
580C2×C4—101no—
581C2×C4—101no—
582C2×C4—101no—
583C2×C4—101no—
584C2×C4—101no—
585C2×C4—152no—
586C2×C4—208no—
587C2×C4—301no—
588C2×C4—301no—
589C2×C4—308no—
590C2×C4—302no—
591C2×C4—302no—
592C2×C4—302no—
593C2×C4#677112yes—
594C2×C4C4×D3×D512yes—
595C2×C4C2×#120214yes—
596C2×C4C2×#120212yes—
597C2×C3—204no—
598C2×C3C2×C4×D2011yescharacteristic
599C2×C3#6584811yescharacteristic
600C2×C3#6586611yescharacteristic
601C2×C3#6587911yescharacteristic
602C2×C3#6588811yescharacteristic
603C2×C3C4×#71311yescharacteristic
604C2×C3#6591111yescharacteristic
605D3—202no—
606C5#8234111yescharacteristic, complemented, Sylow
607C22—104no—
608C22—104no—
609C22—104no—
610C22—202no—
611C22—202no—
612C22—202no—
613C22—202no—
614C22—302no—
615C22—302no—
616C22—302no—
617C22—601no—
618C22—601no—
619C22—601no—
620C22—601no—
621C22C2×#677111yescharacteristic
622C22C2×C4×D3×D511yescharacteristic
623C22D5×#23711yescharacteristic
624C22#9202511yescharacteristic
625C22D3×#71311yescharacteristic
626C22#9203511yescharacteristic
627C22C22×#120211yescharacteristic
628C4—24no—
629C4—62no—
630C4—64no—
631C4—102no—
632C4—121no—
633C4—201no—
634C4—302no—
635C4—602no—
636C4C4×#120214yes—
637C3#1060538011yescharacteristic, complemented, Sylow
638C2—204no—
639C2—602no—
640C2#1175340011yescharacteristic
641C2#1175373611yescharacteristic
642C2#1175377611yescharacteristic
643C2#1175384111yescharacteristic
644C2C2×C4×#120211yescharacteristic
645C2#1175496311yescharacteristic
646C2#1175525111yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C5×C4⋊C4C22
socle, cosocleC23×C3×C5C24
prime radicalC2×#91465C2
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5#2363
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#91781C2semidirect
2. C2×C3×#4039C2semidirect
4. C5×Dic3×C4⋊C4C2semidirect
5. #11748912C2semidirect
6. #11748907C2semidirect
7. #11748842C2semidirect
8. #11748785C2semidirect
9. C4×#6667C2semidirect
10. #11748710C2semidirect
11. C4×#6665C2semidirect
34. #91465C4semidirect
41. C2×#4039D3semidirect
59. C2×C4×D15C2×C4semidirect
73. C2×#1202C2×C4semidirect
74. C3×C5×C4⋊C4D4semidirect
78. C5×C4⋊Dic3C2×C4semidirect
88. #6665C2×C4semidirect
95. Dic3×C4⋊C4D5semidirect
223. C5×C4⋊C4#67semidirect
301. C3×C4⋊C4D20semidirect
311. C4⋊Dic3C4×D5semidirect
499. C4⋊C4#1202semidirect
511. C3×C5#2363semidirect
606. C5#82341semidirect
637. C3#10605380semidirect

Maximal subgroups

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

subgroupindex
C2×#917812
C2×C3×#40392
C2×#914652
C5×Dic3×C4⋊C42
#117489122
#117489072
#117488422
#117487852
C4×#66672
#117487102
C4×#66652
#106053803
#823415