Finite Group Databasebeta

← #14319757

Subgroups — #14319757

Order 1920.

1560 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91782C211yesmaximal, characteristic
2C2×#91428C211yesmaximal, characteristic
3C2×#91418C211yesmaximal, characteristic
4C5×Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
5C3×C4⋊C4×Dic5C211yesmaximal, characteristic, complemented
6#11748888C212yesmaximal, complemented
7#11748881C212yesmaximal, complemented
8#11748875C212yesmaximal, complemented
9C4×#6667C212yesmaximal, complemented
10#11748699C211yesmaximal, characteristic, complemented
11#11748611C211yesmaximal, characteristic, complemented
12#10605482—31nomaximal, Hall
13C22×D60C2211yescharacteristic
14C22×C4×D15C2212yes—
15C22×#1200C2212yes—
16C22×#1200C2212yes—
17C2×C3×C5×C4⋊C4C2211yescharacteristic
18C2×#6714C2212yes—
19C2×C4×Dic15C2211yescharacteristic
20#91782C418yescomplemented
21C2×C5×C4⋊Dic3C2211yescharacteristic
22C2×C4×C5×Dic3C2212yes—
23C2×C4×C5×Dic3C2211yescharacteristic
24C2×C3×C4⋊Dic5C2211yescharacteristic
25C2×C3×C4×Dic5C2212yes—
26C2×C3×C4×Dic5C2211yescharacteristic
27C2×#6667C2212yes—
28C2×#6667C2212yes—
29C2×Dic3×Dic5C2212yes—
30#91428C418yescomplemented
31#91418C418yescomplemented
32C5×C3⋊#95C2212yes—
33C3×#3963C2212yes—
34#90994C2211yescharacteristic
35#90994C2211yescharacteristic
36#90994C2211yescharacteristic
37#90994C2211yescharacteristic
38#90994C2212yes—
39#73751—51nomaximal, Hall
40C2×#4039—31no—
41C2×#4039—31no—
42C2×C4×D20—31no—
43C4×C5×C4⋊C4—31no—
44C4⋊C4×Dic5—31no—
45C4⋊C4×Dic5D311yescharacteristic, complemented
46C4×#685—32no—
47C4×#685—32no—
48#65221—31no—
49#64955—32no—
50D5×#95—32no—
51C23×D15C2312yes—
52C22×C3×C4×C5C2312yes—
53C22×C3×C4×C5C2311yescharacteristic
54C22×Dic15C2311yescharacteristic
55C22×Dic15C2311yescharacteristic
56C2×D60C2×C414yes—
57C2×D60C2×C414yes—
58C2×D60C2×C414yes—
59C2×C4×D15—42no—
60C2×C4×D15—42no—
61C2×C4×D15C2×C418yescomplemented
62C22×C5×Dic3C2311yescharacteristic
63C22×C5×Dic3C2312yes—
64C22×C5×Dic3C2311yescharacteristic
65C22×C3×Dic5C2311yescharacteristic
66C22×C3×Dic5C2311yescharacteristic
67C22×C3×Dic5C2312yes—
68C2×#1200—42no—
69C2×#1200—42no—
70C2×#1200—42no—
71C2×#1200—42no—
72C2×#1200C2×C418yescomplemented
73C2×#1200C2×C418yescomplemented
74C3×C5×C4⋊C4C2×C414yescomplemented
75#6714C2×C418yescomplemented
76C4×Dic15C2×C414yescomplemented
77C5×C4⋊Dic3C2×C414yescomplemented
78C4×C5×Dic3—42no—
79C4×C5×Dic3C2×C414yescomplemented
80C3×C4⋊Dic5C2×C414yescomplemented
81C3×C4×Dic5—42no—
82C3×C4×Dic5C2×C414yescomplemented
83#6667C2×C418yescomplemented
84#6667C2×C418yescomplemented
85Dic3×Dic5—42no—
86C2×#893—51no—
87C2×#893—51no—
88C2×C4×D12—51no—
89C3×C4×C4⋊C4—51no—
90Dic3×C4⋊C4—51no—
91Dic3×C4⋊C4D511yescharacteristic, complemented
92C4×(C2×D3)⋊C4—52no—
93C4×(C2×D3)⋊C4—52no—
94#4932—51no—
95#4666—52no—
96D3×#95—52no—
97C22×D20—31no—
98C22×C4×D5—32no—
99C22×C4×D5—32no—
100C22×C4×D5—32no—
101C2×C5×C4⋊C4—31no—
102C2×C5×C4⋊C4C2×D311yescharacteristic
103C2×C42×C5—31no—
104C2×C42×C5—32no—
105C2×#685—32no—
106C2×#685—32no—
107C2×#685—32no—
108C2×C4⋊Dic5C2×D311yescharacteristic
109C2×C4×Dic5—31no—
110C2×C4×Dic5—32no—
111C2×C4×Dic5C2×D312yes—
112C2×C4×Dic5C2×D311yescharacteristic
113#4039—38no—
114#4039—38no—
115C4×D20—38no—
116C5×#95—32no—
117#3963—31no—
118#3963—32no—
119#3963—31no—
120#3963—31no—
121#3963—31no—
122#3963C2×D312yes—
123#2275—151noSylow
124C23×C3×C5C2411yescharacteristic
125C22×D15—22no—
126C22×D15—22no—
127C22×D15—22no—
128C22×D15—22no—
129C22×D15C22×C414yes—
130C22×D15C22×C414yes—
131C22×D15C22×C414yes—
132C2×C3×C4×C5—22no—
133C2×C3×C4×C5—22no—
134C2×C3×C4×C5C22×C414yes—
135C2×C3×C4×C5C22×C412yes—
136C2×C3×C4×C5C22×C412yes—
137C2×C3×C4×C5C22×C412yes—
138C2×Dic15—21no—
139C2×Dic15—21no—
140C2×Dic15—21no—
141C2×Dic15—21no—
142C2×Dic15C22×C412yes—
143C2×Dic15C22×C412yes—
144D60C42116yescomplemented
145C4×D15—48no—
146C2×C5×Dic3—22no—
147C2×C5×Dic3—21no—
148C2×C5×Dic3—22no—
149C2×C5×Dic3—21no—
150C2×C5×Dic3—21no—
151C2×C5×Dic3—21no—
152C2×C5×Dic3C22×C412yes—
153C2×C5×Dic3C22×C414yes—
154C2×C5×Dic3C22×C412yes—
155C2×C3×Dic5—22no—
156C2×C3×Dic5—21no—
157C2×C3×Dic5—22no—
158C2×C3×Dic5—21no—
159C2×C3×Dic5—21no—
160C2×C3×Dic5—21no—
161C2×C3×Dic5C22×C412yes—
162C2×C3×Dic5C22×C414yes—
163C2×C3×Dic5C22×C412yes—
164#1200—48no—
165#1200—48no—
166C22×D12—51no—
167C22×C4×D3—52no—
168C22×C4×D3—52no—
169C22×C4×D3—52no—
170C2×C3×C4⋊C4—51no—
171C2×C3×C4⋊C4C2×D511yescharacteristic
172C2×C3×C42—51no—
173C2×C3×C42—52no—
174C2×(C2×D3)⋊C4—52no—
175C2×(C2×D3)⋊C4—52no—
176C2×(C2×D3)⋊C4—52no—
177C2×C4⋊Dic3C2×D511yescharacteristic
178C2×C4×Dic3—51no—
179C2×C4×Dic3—52no—
180C2×C4×Dic3C2×D512yes—
181C2×C4×Dic3C2×D511yescharacteristic
182#893—58no—
183#893—58no—
184C4×D12—58no—
185C3×#95—52no—
186C3⋊#95—51no—
187C3⋊#95—52no—
188C3⋊#95—51no—
189C3⋊#95—51no—
190C3⋊#95—51no—
191C3⋊#95C2×D512yes—
192C23×D5—32no—
193C22×C4×C5—31no—
194C22×C4×C5—32no—
195C22×C4×C5—31no—
196C22×C4×C5C22×D312yes—
197C22×C4×C5C22×D311yescharacteristic
198C22×Dic5—31no—
199C22×Dic5—31no—
200C22×Dic5C22×D311yescharacteristic
201C22×Dic5C22×D312yes—
202C22×Dic5C22×D311yescharacteristic
203C2×D20—34no—
204C2×D20—34no—
205C2×D20—34no—
206C2×C4×D5—38no—
207C2×C4×D5—38no—
208C2×C4×D5—38no—
209C2×C4×D5—122no—
210C2×C4×D5—122no—
211C2×C4×D5—122no—
212C2×C4×D5—122no—
213C2×C4×D5—122no—
214C2×C4×D5—122no—
215C5×C4⋊C4—34no—
216C5×C4⋊C4C4×D314yescomplemented
217C42×C5—34no—
218C42×C5—122no—
219#685—38no—
220#685—38no—
221#685—38no—
222C4⋊Dic5C4×D314yescomplemented
223C4×Dic5—34no—
224C4×Dic5—42no—
225C4×Dic5—122no—
226C4×Dic5C4×D314yescomplemented
227C2×C4×D4—151no—
228C2×C4×D4—151no—
229C2×C4×D4—151no—
230C4×C4⋊C4—31no—
231C4×C4⋊C4—51no—
232C4×C4⋊C4—151no—
233C4×C22⋊C4—152no—
234C4×C22⋊C4—152no—
235C4×C22⋊C4—152no—
236#376—151no—
237C2×#95—152no—
238C22×C3×C5C23×C411yescharacteristic
239C22×C3×C5C23×C411yescharacteristic
240C22×C3×C5C23×C411yescharacteristic
241C22×C3×C5H(5,𝔽2)11yescharacteristic
242C22×C3×C5H(5,𝔽2)11yescharacteristic
243C22×C3×C5H(5,𝔽2)11yescharacteristic
244C22×C3×C52−1+411yescharacteristic
245C2×D15—24no—
246C2×D15—24no—
247C2×D15—24no—
248C2×D15—24no—
249C2×D15—24no—
250C2×D15—24no—
251C2×D15C2×C4218yes—
252C3×C4×C5—24no—
253C3×C4×C5C2×C4214yes—
254Dic15—22no—
255Dic15—22no—
256C3×Dic5—22no—
257C3×Dic5—24no—
258C3×Dic5—22no—
259C5×Dic3—22no—
260C5×Dic3—24no—
261C5×Dic3—22no—
262C23×D3—52no—
263C22×C3×C4—51no—
264C22×C3×C4—51no—
265C22×C3×C4—52no—
266C22×C3×C4C22×D512yes—
267C22×C3×C4C22×D511yescharacteristic
268C22×Dic3—51no—
269C22×Dic3—51no—
270C22×Dic3C22×D511yescharacteristic
271C22×Dic3C22×D511yescharacteristic
272C22×Dic3C22×D512yes—
273C2×D12—54no—
274C2×D12—54no—
275C2×D12—54no—
276C2×C4×D3—58no—
277C2×C4×D3—58no—
278C2×C4×D3—58no—
279C2×C4×D3—202no—
280C2×C4×D3—202no—
281C2×C4×D3—202no—
282C2×C4×D3—202no—
283C2×C4×D3—202no—
284C2×C4×D3—202no—
285C3×C4⋊C4—54no—
286C3×C4⋊C4C4×D514yescomplemented
287C3×C42—54no—
288C3×C42—202no—
289(C2×D3)⋊C4—58no—
290(C2×D3)⋊C4—58no—
291(C2×D3)⋊C4—58no—
292C4⋊Dic3C4×D514yescomplemented
293C4×Dic3—42no—
294C4×Dic3—54no—
295C4×Dic3—202no—
296C4×Dic3C4×D514yescomplemented
297C23×C5C23×D311yescharacteristic
298C22×D5—34no—
299C22×D5—34no—
300C22×D5—34no—
301C22×D5—62no—
302C22×D5—62no—
303C22×D5—62no—
304C22×D5—62no—
305C2×C4×C5—22no—
306C2×C4×C5—22no—
307C2×C4×C5—32no—
308C2×C4×C5—34no—
309C2×C4×C5—32no—
310C2×C4×C5—61no—
311C2×C4×C5—61no—
312C2×C4×C5—62no—
313C2×C4×C5—61no—
314C2×C4×C5—62no—
315C2×C4×C5—61no—
316C2×C4×C5C2×C4×D312yes—
317C2×C4×C5C2×C4×D312yes—
318C2×C4×C5C2×C4×D314yes—
319C2×C4×C5C2×C4×D312yes—
320C2×Dic5—21no—
321C2×Dic5—22no—
322C2×Dic5—21no—
323C2×Dic5—22no—
324C2×Dic5—21no—
325C2×Dic5—21no—
326C2×Dic5—32no—
327C2×Dic5—32no—
328C2×Dic5—61no—
329C2×Dic5—61no—
330C2×Dic5—61no—
331C2×Dic5—61no—
332C2×Dic5C2×C4×D312yes—
333C2×Dic5C2×C4×D312yes—
334C2×Dic5C2×C4×D314yes—
335D20—316no—
336C4×D5—128no—
337C4×D5—128no—
338C4×D5—128no—
339C22×D4—151no—
340C23×C4—152no—
341C23×C4—152no—
342C23×C4—152no—
343C4×D4—158no—
344C4×D4—158no—
345C4×D4—158no—
346C2×C4⋊C4—31no—
347C2×C4⋊C4—51no—
348C2×C4⋊C4D3×D511yescharacteristic
349C2×C22⋊C4—152no—
350C2×C22⋊C4—152no—
351C2×C22⋊C4—152no—
352C2×C42—31no—
353C2×C42—32no—
354C2×C42—51no—
355C2×C42—52no—
356C2×C42—151no—
357C2×C42—152no—
358#95—32no—
359#95—52no—
360#95—151no—
361#95—152no—
362#95—151no—
363#95—151no—
364#95—151no—
365C2×C3×C5C22×C4211yescharacteristic
366C2×C3×C5#51811yescharacteristic
367C2×C3×C5#51811yescharacteristic
368C2×C3×C5#51811yescharacteristic
369C2×C3×C5#52011yescharacteristic
370C2×C3×C5#52011yescharacteristic
371C2×C3×C5#52011yescharacteristic
372D15—28no—
373C23×C3C23×D511yescharacteristic
374C22×D3—54no—
375C22×D3—54no—
376C22×D3—54no—
377C22×D3—102no—
378C22×D3—102no—
379C22×D3—102no—
380C22×D3—102no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—52no—
384C2×C3×C4—54no—
385C2×C3×C4—52no—
386C2×C3×C4—101no—
387C2×C3×C4—101no—
388C2×C3×C4—102no—
389C2×C3×C4—101no—
390C2×C3×C4—102no—
391C2×C3×C4—101no—
392C2×C3×C4C2×C4×D512yes—
393C2×C3×C4C2×C4×D512yes—
394C2×C3×C4C2×C4×D514yes—
395C2×C3×C4C2×C4×D512yes—
396C2×Dic3—21no—
397C2×Dic3—22no—
398C2×Dic3—21no—
399C2×Dic3—22no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—52no—
403C2×Dic3—52no—
404C2×Dic3—101no—
405C2×Dic3—101no—
406C2×Dic3—101no—
407C2×Dic3—101no—
408C2×Dic3C2×C4×D512yes—
409C2×Dic3C2×C4×D512yes—
410C2×Dic3C2×C4×D514yes—
411D12—516no—
412C4×D3—208no—
413C4×D3—208no—
414C4×D3—208no—
415C22×C5C22×C4×D311yescharacteristic
416C22×C5C22×C4×D311yescharacteristic
417C22×C5C22×C4×D311yescharacteristic
418C22×C5#100411yescharacteristic
419C22×C5#100711yescharacteristic
420C22×C5#100911yescharacteristic
421C22×C5#100911yescharacteristic
422C2×D5—38no—
423C2×D5—64no—
424C2×D5—64no—
425C2×D5—64no—
426C2×D5—64no—
427C2×D5—64no—
428C2×D5—64no—
429C4×C5—24no—
430C4×C5—64no—
431C4×C5—62no—
432C4×C5—62no—
433C4×C5C42×D314yes—
434Dic5—22no—
435Dic5—22no—
436Dic5—24no—
437Dic5—62no—
438Dic5—62no—
439C24—152no—
440C2×D4—154no—
441C2×D4—154no—
442C2×D4—154no—
443C22×C4—31no—
444C22×C4—32no—
445C22×C4—31no—
446C22×C4—51no—
447C22×C4—52no—
448C22×C4—51no—
449C22×C4—158no—
450C22×C4—158no—
451C22×C4—151no—
452C22×C4—151no—
453C22×C4—158no—
454C22×C4—602no—
455C22×C4—602no—
456C22×C4—602no—
457C22×C4—602no—
458C22×C4—602no—
459C22×C4—602no—
460C22×C4C2×D3×D512yes—
461C22×C4C2×D3×D511yescharacteristic
462C4⋊C4—34no—
463C4⋊C4—54no—
464C4⋊C4#120014yescomplemented
465C22⋊C4—158no—
466C22⋊C4—158no—
467C22⋊C4—158no—
468C42—34no—
469C42—54no—
470C42—122no—
471C42—154no—
472C42—202no—
473C42—602no—
474C3×C5#227511yescharacteristic, complemented, Hall
475C22×C3C22×C4×D511yescharacteristic
476C22×C3C22×C4×D511yescharacteristic
477C22×C3C22×C4×D511yescharacteristic
478C22×C3#414411yescharacteristic
479C22×C3#414711yescharacteristic
480C22×C3#414911yescharacteristic
481C22×C3#414911yescharacteristic
482C2×D3—58no—
483C2×D3—104no—
484C2×D3—104no—
485C2×D3—104no—
486C2×D3—104no—
487C2×D3—104no—
488C2×D3—104no—
489C3×C4—24no—
490C3×C4—104no—
491C3×C4—102no—
492C3×C4—102no—
493C3×C4C42×D514yes—
494Dic3—22no—
495Dic3—22no—
496Dic3—24no—
497Dic3—102no—
498Dic3—102no—
499C2×C5C2×C42×D311yescharacteristic
500C2×C5#550411yescharacteristic
501C2×C5#552111yescharacteristic
502C2×C5#554511yescharacteristic
503C2×C5#554511yescharacteristic
504C2×C5#557411yescharacteristic
505C2×C5#557411yescharacteristic
506D5—68no—
507C23—154no—
508C23—154no—
509C23—154no—
510C23—302no—
511C23—302no—
512C23—302no—
513C23—302no—
514C23C22×D3×D511yescharacteristic
515D4—1516no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—32no—
519C2×C4—32no—
520C2×C4—34no—
521C2×C4—52no—
522C2×C4—52no—
523C2×C4—54no—
524C2×C4—61no—
525C2×C4—61no—
526C2×C4—62no—
527C2×C4—61no—
528C2×C4—62no—
529C2×C4—61no—
530C2×C4—101no—
531C2×C4—102no—
532C2×C4—101no—
533C2×C4—101no—
534C2×C4—102no—
535C2×C4—101no—
536C2×C4—152no—
537C2×C4—152no—
538C2×C4—301no—
539C2×C4—301no—
540C2×C4—301no—
541C2×C4—301no—
542C2×C4—608no—
543C2×C4—608no—
544C2×C4—608no—
545C2×C4C4×D3×D512yes—
546C2×C4C4×D3×D512yes—
547C2×C4C2×#120014yes—
548C2×C4C2×#120012yes—
549C2×C3C2×C42×D511yescharacteristic
550C2×C3#6583811yescharacteristic
551C2×C3#6585511yescharacteristic
552C2×C3#6587911yescharacteristic
553C2×C3#6587911yescharacteristic
554C2×C3#6590811yescharacteristic
555C2×C3#6590811yescharacteristic
556D3—108no—
557C5#7375111yescharacteristic, complemented, Sylow
558C22—158no—
559C22—304no—
560C22—304no—
561C22—304no—
562C22—304no—
563C22—304no—
564C22—304no—
565C22C2×C4×D3×D511yescharacteristic
566C22C2×C4×D3×D511yescharacteristic
567C22#9202611yescharacteristic
568C22#9202711yescharacteristic
569C22#9202911yescharacteristic
570C22#9203511yescharacteristic
571C22C22×#120011yescharacteristic
572C4—24no—
573C4—64no—
574C4—62no—
575C4—62no—
576C4—102no—
577C4—104no—
578C4—102no—
579C4—302no—
580C4—302no—
581C4C4×#120014yes—
582C3#1060548211yescharacteristic, complemented, Sylow
583C2—308no—
584C2#1175333611yescharacteristic
585C2#1175333711yescharacteristic
586C2#1175383711yescharacteristic
587C2#1175383911yescharacteristic
588C2C2×C4×#120011yescharacteristic
589C2#1175492211yescharacteristic
590C2#1175514911yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC2×C3×C5C22×C42
FittingC2×C3×C5×C4⋊C4C22
socle, cosocleC23×C3×C5C24
prime radicalC22×D60C22
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5#2275
LCSG ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. C5×Dic3×C4⋊C4C2semidirect
5. C3×C4⋊C4×Dic5C2semidirect
6. #11748888C2semidirect
7. #11748881C2semidirect
8. #11748875C2semidirect
9. C4×#6667C2semidirect
10. #11748699C2semidirect
11. #11748611C2semidirect
20. #91782C4semidirect
30. #91428C4semidirect
31. #91418C4semidirect
45. C4⋊C4×Dic5D3semidirect
61. C2×C4×D15C2×C4semidirect
72. C2×#1200C2×C4semidirect
73. C2×#1200C2×C4semidirect
74. C3×C5×C4⋊C4C2×C4semidirect
75. #6714C2×C4semidirect
76. C4×Dic15C2×C4semidirect
77. C5×C4⋊Dic3C2×C4semidirect
79. C4×C5×Dic3C2×C4semidirect
80. C3×C4⋊Dic5C2×C4semidirect
82. C3×C4×Dic5C2×C4semidirect
83. #6667C2×C4semidirect
84. #6667C2×C4semidirect
91. Dic3×C4⋊C4D5semidirect
144. D60C42semidirect
216. C5×C4⋊C4C4×D3semidirect
222. C4⋊Dic5C4×D3semidirect
226. C4×Dic5C4×D3semidirect
286. C3×C4⋊C4C4×D5semidirect
292. C4⋊Dic3C4×D5semidirect
296. C4×Dic3C4×D5semidirect
464. C4⋊C4#1200semidirect
474. C3×C5#2275semidirect
557. C5#73751semidirect
582. C3#10605482semidirect

Maximal subgroups

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

subgroupindex
C2×#917822
C2×#914282
C2×#914182
C5×Dic3×C4⋊C42
C3×C4⋊C4×Dic52
#117488882
#117488812
#117488752
C4×#66672
#117486992
#117486112
#106054823
#737515