Finite Group Databasebeta

← #14327369

Subgroups — #14327369

Order 1920.

1320 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×D15C211yesmaximal, characteristic, complemented
2C2×C4×#1202C211yesmaximal, characteristic
3C2×#91358C211yesmaximal, characteristic, complemented
4C5×#4977C211yesmaximal, characteristic, complemented
5C3×#65276C211yesmaximal, characteristic, complemented
6#11748917C212yesmaximal
7#11748799C211yesmaximal, characteristic, complemented
8#11748707C212yesmaximal, complemented
9Dic3×C4⋊Dic5C211yesmaximal, characteristic, complemented
10#11748670C212yesmaximal, complemented
11#11748660C212yesmaximal, complemented
12#10605960—31nomaximal, Hall
13C22×C4×D15C2212yes—
14C22×C4×D15C2211yescharacteristic
15C22×C3×C4×D5C2211yescharacteristic
16C22×#1202C2211yescharacteristic
17C2×C3×C5×C4⋊C4C2211yescharacteristic
18C2×C4⋊Dic15C2211yescharacteristic
19C2×#6712C2212yes—
20C4⋊C4×D15—24no—
21C2×C5×C4⋊Dic3C2211yescharacteristic
22C2×C4×C5×Dic3C2211yescharacteristic
23C2×C3×#685C2212yes—
24C2×C3×C4⋊Dic5C2211yescharacteristic
25C2×Dic3⋊Dic5C2211yescharacteristic
26C2×#6667C2212yes—
27C2×#6667C2211yescharacteristic
28C2×#6665C2212yes—
29C2×#6665C2211yescharacteristic
30C2×Dic3×Dic5C2212yes—
31C2×Dic3×Dic5C2212yes—
32C4×#1202C418yescomplemented
33#91358—24no—
34C5×C3⋊#95C2212yes—
35C5×C3⋊#95C2212yes—
36C3×#3963C2212yes—
37#90994C2212yes—
38#90994C2212yes—
39#74012—51nomaximal, Hall
40C2×#4038—31no—
41C2×D5×C4⋊C4—31no—
42C2×C4×D20—31no—
43C5×#382—31no—
44#65276—31no—
45#65276D311yescharacteristic, complemented
46C4×C4⋊Dic5—31no—
47#64958—32no—
48#64955—32no—
49#64954—32no—
50#64943—32no—
51C23×D15C2311yescharacteristic
52C23×C3×D5C2311yescharacteristic
53C22×C3×C4×C5C2312yes—
54C22×C3×C4×C5C2311yescharacteristic
55C22×Dic15C2311yescharacteristic
56C22×Dic15C2312yes—
57C2×C4×D15—22no—
58C2×C4×D15—22no—
59C2×C4×D15—24no—
60C2×C4×D15—24no—
61C2×C4×D15—42no—
62C2×C4×D15C2×C414yes—
63C22×C5×Dic3C2312yes—
64C22×C5×Dic3C2311yescharacteristic
65C22×C5×Dic3C2311yescharacteristic
66C22×C3×Dic5C2312yes—
67C22×C3×Dic5C2311yescharacteristic
68C2×C3×C4×D5—22no—
69C2×C3×C4×D5—22no—
70C2×C3×C4×D5C2×C414yescomplemented
71C2×#1202—22no—
72C2×#1202—22no—
73C2×#1202C2×C414yes—
74C3×C5×C4⋊C4—22no—
75C4⋊Dic15D414yescomplemented
76#6712—24no—
77C5×C4⋊Dic3—22no—
78C4×C5×Dic3C2×C414yescomplemented
79C3×#685—42no—
80C3×C4⋊Dic5—22no—
81Dic3⋊Dic5C2×C414yescomplemented
82#6667—24no—
83#6667C2×C414yes—
84#6665—24no—
85#6665C2×C414yescomplemented
86Dic3×Dic5—24no—
87Dic3×Dic5—42no—
88C2×C4×#67—51no—
89C2×#926—51no—
90C2×D3×C4⋊C4—51no—
91C3×#380—51no—
92#4987—51no—
93#4977D511yescharacteristic, complemented
94Dic3×C4⋊C4—51no—
95#4956—52no—
96#4944—52no—
97#4666—52no—
98#4652—52no—
99C22×D20—31no—
100C22×C4×D5—31no—
101C22×C4×D5—32no—
102C22×C4×D5C2×D311yescharacteristic
103C2×C5×C4⋊C4—31no—
104C2×C5×C4⋊C4C2×D311yescharacteristic
105C2×C42×C5—31no—
106C2×#685—31no—
107C2×#685—31no—
108C2×#685—32no—
109C2×#685—32no—
110C2×#685C2×D312yes—
111C2×C4⋊Dic5—31no—
112C2×C4⋊Dic5—31no—
113C2×C4⋊Dic5C2×D311yescharacteristic
114C2×#683—32no—
115C2×C4×Dic5—32no—
116C2×C4×Dic5—32no—
117#4038—64no—
118D5×C4⋊C4—64no—
119C4×D20—38no—
120C5×#95—32no—
121C5×#95—32no—
122#3963—32no—
123#3963—32no—
124#3963C2×D312yes—
125#2352—151noSylow
126C23×C3×C5C2411yescharacteristic
127C22×D15—21no—
128C22×D15—21no—
129C22×D15—21no—
130C22×D15—21no—
131C22×D15C22×C412yes—
132C22×D15#4112yes—
133C22×D15#4112yes—
134C22×C3×D5—21no—
135C22×C3×D5—21no—
136C22×C3×D5—21no—
137C22×C3×D5—21no—
138C22×C3×D5—21no—
139C22×C3×D5—21no—
140C22×C3×D5C22×C412yes—
141C2×C3×C4×C5—22no—
142C2×C3×C4×C5—22no—
143C2×C3×C4×C5—22no—
144C2×C3×C4×C5C22×C412yes—
145C2×C3×C4×C5C2×D412yes—
146C2×C3×C4×C5#4112yes—
147C2×Dic15—21no—
148C2×Dic15—21no—
149C2×Dic15—22no—
150C2×Dic15—22no—
151C2×Dic15C22×C412yes—
152C2×Dic15C2×D414yes—
153C4×D15—28no—
154C4×D15—48no—
155C2×C5×Dic3—21no—
156C2×C5×Dic3—21no—
157C2×C5×Dic3—21no—
158C2×C5×Dic3—21no—
159C2×C5×Dic3—22no—
160C2×C5×Dic3—22no—
161C2×C5×Dic3—22no—
162C2×C5×Dic3C22×C412yes—
163C2×C5×Dic3C22×C412yes—
164C2×C3×Dic5—21no—
165C2×C3×Dic5—21no—
166C2×C3×Dic5—22no—
167C2×C3×Dic5—22no—
168C2×C3×Dic5—22no—
169C2×C3×Dic5C22×C412yes—
170C3×C4×D5—28no—
171#1202—28no—
172C23×C3×C4—51no—
173C22×#67—51no—
174C22×C4×D3—51no—
175C22×C4×D3—52no—
176C2×C3×C4⋊C4—51no—
177C2×C3×C4⋊C4C2×D511yescharacteristic
178C2×C3×C22⋊C4—52no—
179C2×#217—51no—
180C2×#217—52no—
181C4×#67—58no—
182C2×(C2×D3)⋊C4—51no—
183C2×(C2×D3)⋊C4—52no—
184#926—104no—
185C2×C4⋊Dic3—51no—
186C2×C4⋊Dic3C2×D511yescharacteristic
187C2×Dic3⋊C4—51no—
188C2×Dic3⋊C4—52no—
189C2×C4×Dic3—52no—
190C2×C4×Dic3—52no—
191C2×C4×Dic3C2×D511yescharacteristic
192D3×C4⋊C4—104no—
193C3×#95—52no—
194C3⋊#95—52no—
195C3⋊#95—52no—
196C3⋊#95C2×D512yes—
197C3⋊#95C2×D512yes—
198C23×D5—31no—
199C23×D5C22×D311yescharacteristic
200C22×C4×C5—31no—
201C22×C4×C5—31no—
202C22×C4×C5—32no—
203C22×C4×C5C22×D312yes—
204C22×C4×C5C22×D311yescharacteristic
205C22×Dic5—31no—
206C22×Dic5—32no—
207C22×Dic5C22×D312yes—
208C22×Dic5C22×D311yescharacteristic
209C2×D20—34no—
210C2×D20—62no—
211C2×D20—62no—
212C2×C4×D5—22no—
213C2×C4×D5—22no—
214C2×C4×D5—34no—
215C2×C4×D5—62no—
216C2×C4×D5—62no—
217C2×C4×D5—64no—
218C2×C4×D5—64no—
219C2×C4×D5—122no—
220C2×C4×D5C4×D314yescomplemented
221C5×C4⋊C4—22no—
222C5×C4⋊C4—62no—
223C42×C5—34no—
224#685—34no—
225#685—34no—
226#685—42no—
227#685—64no—
228#685—64no—
229C4⋊Dic5—22no—
230C4⋊Dic5—34no—
231C4⋊Dic5—34no—
232#683—64no—
233C4×Dic5—64no—
234C4×Dic5—122no—
235C2×C4×D4—151no—
236C2×#117—151no—
237C22×C4⋊C4—151no—
238#388—152no—
239#387—152no—
240#386—152no—
241#382—31no—
242#380—51no—
243#380—151no—
244C4×C4⋊C4—151no—
245C4×C22⋊C4—152no—
246C22×C3×C5C23×C411yescharacteristic
247C22×C3×C5C22×D411yescharacteristic
248C22×C3×C5C2×#4111yescharacteristic
249C22×C3×C5C2×#4111yescharacteristic
250C22×C3×C5C2×#4111yescharacteristic
251C22×C3×C5H(5,𝔽2)11yescharacteristic
252C22×C3×C52−1+411yescharacteristic
253C2×D15—22no—
254C2×D15—22no—
255C2×D15—22no—
256C2×D15—22no—
257C2×D15—22no—
258C2×D15—22no—
259C2×D15#11714yescomplemented
260C2×C3×D5—22no—
261C2×C3×D5—22no—
262C2×C3×D5—22no—
263C2×C3×D5—41no—
264C2×C3×D5—41no—
265C2×C3×D5—41no—
266C2×C3×D5—41no—
267C3×C4×C5—42no—
268C3×C4×C5C4×D414yes—
269Dic15—22no—
270Dic15—24no—
271C3×Dic5—22no—
272C3×Dic5—42no—
273C5×Dic3—22no—
274C5×Dic3—22no—
275C5×Dic3—42no—
276C24×C3—51no—
277C23×D3—51no—
278C22×C3×C4—51no—
279C22×C3×C4—54no—
280C22×C3×C4—52no—
281C22×C3×C4—102no—
282C22×C3×C4—102no—
283C22×C3×C4C22×D512yes—
284C22×C3×C4C22×D511yescharacteristic
285C2×#67—54no—
286C2×#67—102no—
287C2×#67—102no—
288C22×Dic3—51no—
289C22×Dic3—52no—
290C22×Dic3C22×D512yes—
291C22×Dic3C22×D511yescharacteristic
292C22×Dic3C22×D511yescharacteristic
293C2×C4×D3—54no—
294C2×C4×D3—102no—
295C2×C4×D3—102no—
296C2×C4×D3—104no—
297C2×C4×D3—104no—
298C2×C4×D3—202no—
299C3×C4⋊C4—22no—
300C3×C4⋊C4—102no—
301C3×C22⋊C4—202no—
302#217—54no—
303#217—104no—
304(C2×D3)⋊C4—54no—
305(C2×D3)⋊C4—104no—
306C4⋊Dic3—22no—
307C4⋊Dic3—54no—
308Dic3⋊C4—54no—
309Dic3⋊C4—104no—
310C4×Dic3—104no—
311C4×Dic3—202no—
312C4×Dic3C4×D514yescomplemented
313C23×C5C23×D311yescharacteristic
314C22×D5—21no—
315C22×D5—21no—
316C22×D5—21no—
317C22×D5—21no—
318C22×D5—21no—
319C22×D5—21no—
320C22×D5—32no—
321C22×D5—32no—
322C22×D5—32no—
323C22×D5—61no—
324C22×D5—61no—
325C22×D5—61no—
326C22×D5—61no—
327C22×D5C2×C4×D312yes—
328C2×C4×C5—22no—
329C2×C4×C5—22no—
330C2×C4×C5—22no—
331C2×C4×C5—32no—
332C2×C4×C5—32no—
333C2×C4×C5—61no—
334C2×C4×C5—61no—
335C2×C4×C5—61no—
336C2×C4×C5—61no—
337C2×C4×C5—62no—
338C2×C4×C5—62no—
339C2×C4×C5—62no—
340C2×C4×C5C2×C4×D312yes—
341C2×C4×C5D3×D412yes—
342C2×C4×C5#23712yes—
343C2×Dic5—21no—
344C2×Dic5—21no—
345C2×Dic5—22no—
346C2×Dic5—22no—
347C2×Dic5—22no—
348C2×Dic5—32no—
349C2×Dic5—34no—
350C2×Dic5—61no—
351C2×Dic5—61no—
352C2×Dic5—62no—
353C2×Dic5—62no—
354C2×Dic5C2×C4×D312yes—
355D20—68no—
356C4×D5—28no—
357C4×D5—68no—
358C4×D5—128no—
359C22×D4—151no—
360C23×C4—51no—
361C23×C4—151no—
362C23×C4—152no—
363C4×D4—158no—
364#117—304no—
365C2×C4⋊C4—31no—
366C2×C4⋊C4—51no—
367C2×C4⋊C4—151no—
368C2×C4⋊C4—151no—
369C2×C4⋊C4—152no—
370C2×C4⋊C4—304no—
371C2×C4⋊C4D3×D511yescharacteristic
372C2×C22⋊C4—52no—
373C2×C22⋊C4—151no—
374C2×C22⋊C4—151no—
375C2×C22⋊C4—152no—
376C2×C22⋊C4—152no—
377C2×C42—31no—
378C2×C42—152no—
379C2×C42—152no—
380#95—32no—
381#95—32no—
382#95—52no—
383#95—152no—
384#95—152no—
385C2×C3×C5C2×#11711yescharacteristic
386C2×C3×C5C2×C4×D411yescharacteristic
387C2×C3×C5#52011yescharacteristic
388C2×C3×C5#54611yescharacteristic
389C2×C3×C5#54811yescharacteristic
390C2×C3×C5#55211yescharacteristic
391C2×C3×C5#55311yescharacteristic
392D15—24no—
393C3×D5—42no—
394C23×C3—52no—
395C23×C3—101no—
396C23×C3—101no—
397C23×C3—101no—
398C23×C3—101no—
399C23×C3—101no—
400C23×C3—101no—
401C23×C3C23×D511yescharacteristic
402C22×D3—52no—
403C22×D3—52no—
404C22×D3—52no—
405C22×D3—101no—
406C22×D3—101no—
407C22×D3—101no—
408C22×D3—101no—
409C2×C3×C4—22no—
410C2×C3×C4—22no—
411C2×C3×C4—22no—
412C2×C3×C4—52no—
413C2×C3×C4—101no—
414C2×C3×C4—101no—
415C2×C3×C4—108no—
416C2×C3×C4—102no—
417C2×C3×C4—102no—
418C2×C3×C4—102no—
419C2×C3×C4C2×C4×D512yes—
420C2×C3×C4D4×D512yes—
421C2×C3×C4#71312yes—
422#67—108no—
423C2×Dic3—21no—
424C2×Dic3—21no—
425C2×Dic3—21no—
426C2×Dic3—21no—
427C2×Dic3—22no—
428C2×Dic3—22no—
429C2×Dic3—22no—
430C2×Dic3—52no—
431C2×Dic3—54no—
432C2×Dic3—101no—
433C2×Dic3—101no—
434C2×Dic3—102no—
435C2×Dic3—102no—
436C2×Dic3C2×C4×D512yes—
437C2×Dic3C2×C4×D512yes—
438C4×D3—108no—
439C4×D3—208no—
440C22×C5C22×C4×D311yescharacteristic
441C22×C5C2×D3×D411yescharacteristic
442C22×C5C2×#23711yescharacteristic
443C22×C5#100411yescharacteristic
444C22×C5#100711yescharacteristic
445C22×C5D3×#4111yescharacteristic
446C22×C5D3×#4111yescharacteristic
447C2×D5—22no—
448C2×D5—22no—
449C2×D5—22no—
450C2×D5—34no—
451C2×D5—41no—
452C2×D5—41no—
453C2×D5—41no—
454C2×D5—41no—
455C2×D5—62no—
456C2×D5—62no—
457C2×D5—62no—
458C2×D5—62no—
459C2×D5—62no—
460C2×D5—62no—
461C4×C5—42no—
462C4×C5—62no—
463C4×C5—62no—
464C4×C5—122no—
465C4×C5#67⋊C414yes—
466Dic5—22no—
467Dic5—42no—
468Dic5—62no—
469Dic5—64no—
470C24—51no—
471C24—151no—
472C2×D4—154no—
473C2×D4—302no—
474C2×D4—302no—
475C22×C4—31no—
476C22×C4—31no—
477C22×C4—32no—
478C22×C4—51no—
479C22×C4—54no—
480C22×C4—52no—
481C22×C4—102no—
482C22×C4—102no—
483C22×C4—151no—
484C22×C4—152no—
485C22×C4—154no—
486C22×C4—304no—
487C22×C4—304no—
488C22×C4—302no—
489C22×C4—302no—
490C22×C4—602no—
491C22×C4C2×D3×D512yes—
492C22×C4C2×D3×D511yescharacteristic
493C4⋊C4—22no—
494C4⋊C4—62no—
495C4⋊C4—102no—
496C4⋊C4—154no—
497C4⋊C4—154no—
498C4⋊C4—304no—
499C22⋊C4—154no—
500C22⋊C4—154no—
501C22⋊C4—202no—
502C22⋊C4—304no—
503C22⋊C4—304no—
504C42—34no—
505C42—304no—
506C42—602no—
507C3×C5#235211yescharacteristic, complemented, Hall
508C22×C3—102no—
509C22×C3—102no—
510C22×C3—102no—
511C22×C3—201no—
512C22×C3—201no—
513C22×C3—201no—
514C22×C3—201no—
515C22×C3C22×C4×D511yescharacteristic
516C22×C3C2×D4×D511yescharacteristic
517C22×C3C2×#71311yescharacteristic
518C22×C3D5×#4111yescharacteristic
519C22×C3D5×#4111yescharacteristic
520C22×C3#414911yescharacteristic
521C22×C3#415011yescharacteristic
522C2×D3—54no—
523C2×D3—102no—
524C2×D3—102no—
525C2×D3—102no—
526C2×D3—102no—
527C2×D3—102no—
528C2×D3—102no—
529C3×C4—42no—
530C3×C4—102no—
531C3×C4—202no—
532C3×C4#403914yes—
533Dic3—22no—
534Dic3—22no—
535Dic3—42no—
536Dic3—102no—
537Dic3—104no—
538C2×C5C2×#67⋊C411yescharacteristic
539C2×C5#550411yescharacteristic
540C2×C5D3×#11711yescharacteristic
541C2×C5#561211yescharacteristic
542C2×C5#561411yescharacteristic
543C2×C5#563311yescharacteristic
544C2×C5#563911yescharacteristic
545D5—42no—
546D5—64no—
547C23—52no—
548C23—101no—
549C23—101no—
550C23—101no—
551C23—101no—
552C23—101no—
553C23—101no—
554C23—152no—
555C23—152no—
556C23—152no—
557C23—301no—
558C23—301no—
559C23—301no—
560C23—301no—
561C23C22×D3×D511yescharacteristic
562D4—308no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—32no—
567C2×C4—32no—
568C2×C4—52no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—62no—
574C2×C4—62no—
575C2×C4—62no—
576C2×C4—101no—
577C2×C4—101no—
578C2×C4—108no—
579C2×C4—102no—
580C2×C4—102no—
581C2×C4—102no—
582C2×C4—152no—
583C2×C4—154no—
584C2×C4—301no—
585C2×C4—302no—
586C2×C4—302no—
587C2×C4—301no—
588C2×C4—308no—
589C2×C4—608no—
590C2×C4#676612yes—
591C2×C4C4×D3×D512yes—
592C2×C4#679012yes—
593C2×C3—202no—
594C2×C3C2×#403911yescharacteristic
595C2×C3D5×#11711yescharacteristic
596C2×C3#6585711yescharacteristic
597C2×C3#6601011yescharacteristic
598C2×C3#6601111yescharacteristic
599C2×C3#6602811yescharacteristic
600C2×C3#6602911yescharacteristic
601D3—104no—
602C5#7401211yescharacteristic, complemented, Sylow
603C22—102no—
604C22—102no—
605C22—102no—
606C22—154no—
607C22—201no—
608C22—201no—
609C22—201no—
610C22—201no—
611C22—302no—
612C22—302no—
613C22—302no—
614C22—302no—
615C22—302no—
616C22—302no—
617C22C2×#676611yescharacteristic
618C22C2×C4×D3×D511yescharacteristic
619C22#9202411yescharacteristic
620C22#9202611yescharacteristic
621C22#9203011yescharacteristic
622C22#9203411yescharacteristic
623C22C2×#679011yescharacteristic
624C4—42no—
625C4—62no—
626C4—62no—
627C4—102no—
628C4—122no—
629C4—202no—
630C4—302no—
631C4—304no—
632C4#9139614yes—
633C3#1060596011yescharacteristic, complemented, Sylow
634C2—202no—
635C2—304no—
636C2#1175363111yescharacteristic
637C2#1175366811yescharacteristic
638C2#1175380011yescharacteristic
639C2#1175383511yescharacteristic
640C2C2×#9139611yescharacteristic
641C2#1175495811yescharacteristic
642C2#1175523811yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C5×C4⋊C4C22
socle, cosocleC23×C3×C5C24
prime radicalC22×#1202C22
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5#2352
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×C4⋊C4×D15C2semidirect
3. C2×#91358C2semidirect
4. C5×#4977C2semidirect
5. C3×#65276C2semidirect
7. #11748799C2semidirect
8. #11748707C2semidirect
9. Dic3×C4⋊Dic5C2semidirect
10. #11748670C2semidirect
11. #11748660C2semidirect
32. C4×#1202C4semidirect
45. #65276D3semidirect
70. C2×C3×C4×D5C2×C4semidirect
75. C4⋊Dic15D4semidirect
78. C4×C5×Dic3C2×C4semidirect
81. Dic3⋊Dic5C2×C4semidirect
85. #6665C2×C4semidirect
93. #4977D5semidirect
220. C2×C4×D5C4×D3semidirect
259. C2×D15#117semidirect
312. C4×Dic3C4×D5semidirect
507. C3×C5#2352semidirect
602. C5#74012semidirect
633. C3#10605960semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D152
C2×C4×#12022
C2×#913582
C5×#49772
C3×#652762
#117489172
#117487992
#117487072
Dic3×C4⋊Dic52
#117486702
#117486602
#106059603
#740125