Finite Group Databasebeta

← #14329110

Subgroups — #14329110

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×#91819C211yesmaximal, characteristic
2C2×C3×#4072C211yesmaximal, characteristic, complemented
3C2×#91428C211yesmaximal, characteristic, complemented
4C4×C5×#217C211yesmaximal, characteristic, complemented
5#11749520C212yesmaximal, complemented
6#11749483C212yesmaximal, complemented
7#11748875C212yesmaximal, complemented
8#11748858C211yesmaximal, characteristic, complemented
9#11748829C212yesmaximal, complemented
10#11748789C211yesmaximal, characteristic, complemented
11Dic3×C4⋊Dic5C211yesmaximal, characteristic, complemented
12#10605200—31nomaximal, Hall
13C23×C3×C4×C5C2211yescharacteristic
14C22×#1220C2212yes—
15C22×D60C2211yescharacteristic
16C22×#1200C2212yes—
17#91819C418yescomplemented
18C2×#6714C2212yes—
19C2×C4⋊Dic15C2211yescharacteristic
20C2×C5×#217C2211yescharacteristic
21C2×C5×#217C2211yescharacteristic
22C2×C4×C5×Dic3C2211yescharacteristic
23C2×C4×C5×Dic3C2211yescharacteristic
24C2×C3×C22⋊Dic5C2212yes—
25C3×#4072—24no—
26C2×C3×C4⋊Dic5C2211yescharacteristic
27C2×C3×C4⋊Dic5C2211yescharacteristic
28C2×C3×C4×Dic5C2212yes—
29C2×#6669C2212yes—
30C2×#6667C2212yes—
31C2×#6667C2212yes—
32C2×#6667C2212yes—
33C2×Dic3×Dic5C2212yes—
34#91428—24no—
35C5×C3⋊#95C2211yescharacteristic
36C5×C3⋊#95C2211yescharacteristic
37#90994C2212yes—
38#90994C2212yes—
39#82244—51nomaximal, Hall
40C2×#4076—31no—
41C2×#4072D311yescharacteristic, complemented
42C2×C4×D20—31no—
43C4×C5×C22⋊C4—31no—
44#65246—32no—
45#65245—32no—
46#65228—31no—
47C4×C4⋊Dic5—31no—
48#65217—31no—
49#64958—32no—
50#64955—32no—
51C24×C3×C5C2311yescharacteristic
52C23×D15C2312yes—
53C22×C3×C4×C5—22no—
54C22×C3×C4×C5—22no—
55C22×C3×C4×C5C2×C414yescomplemented
56C22×C3×C4×C5C2311yescharacteristic
57C22×C3×C4×C5C2311yescharacteristic
58C2×#1220—42no—
59C2×#1220—42no—
60C2×#1220C2×C418yescomplemented
61C22×Dic15C2312yes—
62C2×D60—22no—
63C2×D60—22no—
64C2×D60C2×C414yes—
65C22×C5×Dic3C2311yescharacteristic
66C22×C5×Dic3C2311yescharacteristic
67C22×C5×Dic3C2311yescharacteristic
68C22×C5×Dic3C2311yescharacteristic
69C22×C3×Dic5C2312yes—
70C22×C3×Dic5C2312yes—
71C2×#1200—24no—
72C2×#1200—24no—
73C2×#1200—42no—
74#6714C2×C418yescomplemented
75C4⋊Dic15C2×C414yescomplemented
76C5×#217—22no—
77C5×#217—22no—
78C4×C5×Dic3—22no—
79C4×C5×Dic3—22no—
80C3×C22⋊Dic5—24no—
81C3×C4⋊Dic5—22no—
82C3×C4⋊Dic5D414yescomplemented
83C3×C4×Dic5—24no—
84#6669—42no—
85#6667—24no—
86#6667—24no—
87#6667—42no—
88Dic3×Dic5—42no—
89C2×C3×#117—51no—
90C2×#930—51no—
91C2×#893—51no—
92C4×#217D511yescharacteristic, complemented
93#4988—51no—
94Dic3×C4⋊C4—51no—
95#4972—51no—
96#4957—52no—
97#4956—52no—
98#4940—52no—
99C4×(C2×D3)⋊C4—52no—
100C23×C4×C5C2×D311yescharacteristic
101C22×C22⋊D5—32no—
102C22×D20—31no—
103C22×C4×D5—32no—
104C2×C5×C22⋊C4—31no—
105C2×C5×C22⋊C4—31no—
106C2×C42×C5—31no—
107C2×C42×C5—31no—
108C2×C22⋊Dic5C2×D312yes—
109#4076—38no—
110C2×#685—32no—
111C2×#685—32no—
112C2×#685—32no—
113C2×#685—32no—
114#4072—24no—
115C2×C4⋊Dic5—31no—
116C2×C4⋊Dic5C2×D311yescharacteristic
117C2×C4⋊Dic5C2×D311yescharacteristic
118C2×#683—32no—
119C2×C4×Dic5—32no—
120C2×C4×Dic5C2×D312yes—
121C4×D20—64no—
122C5×#95—31no—
123C5×#95—31no—
124#3963—32no—
125#3963—32no—
126#2363—151noSylow
127C23×C3×C5—21no—
128C23×C3×C5—21no—
129C23×C3×C5—21no—
130C23×C3×C5—21no—
131C23×C3×C5—21no—
132C23×C3×C5—21no—
133C23×C3×C5C22×C412yes—
134C23×C3×C5C2411yescharacteristic
135C22×D15—22no—
136C22×D15—22no—
137C22×D15—22no—
138C22×D15—22no—
139C22×D15—22no—
140C22×D15—22no—
141C22×D15C22×C414yes—
142C2×C3×C4×C5—21no—
143C2×C3×C4×C5—21no—
144C2×C3×C4×C5—28no—
145C2×C3×C4×C5C22×C412yes—
146C2×C3×C4×C5C22×C412yes—
147C2×C3×C4×C5C2×D412yes—
148C2×C3×C4×C5#4112yes—
149#1220—48no—
150C2×Dic15—22no—
151C2×Dic15—22no—
152C2×Dic15C22×C414yes—
153D60—28no—
154C2×C5×Dic3—21no—
155C2×C5×Dic3—21no—
156C2×C5×Dic3—21no—
157C2×C5×Dic3—21no—
158C2×C5×Dic3—21no—
159C2×C5×Dic3—21no—
160C2×C5×Dic3—21no—
161C2×C5×Dic3—21no—
162C2×C5×Dic3—21no—
163C2×C5×Dic3—21no—
164C2×C5×Dic3#4112yes—
165C2×C5×Dic3#4112yes—
166C2×C3×Dic5—22no—
167C2×C3×Dic5—22no—
168C2×C3×Dic5—22no—
169C2×C3×Dic5—22no—
170C2×C3×Dic5—22no—
171C2×C3×Dic5C2×D414yes—
172#1200—48no—
173C23×C3×C4C2×D511yescharacteristic
174C22×#67—52no—
175C22×D12—51no—
176C22×C4×D3—52no—
177C3×#117—104no—
178C2×C3×C4⋊C4—51no—
179C2×C3×C4⋊C4—51no—
180C2×C3×C22⋊C4—52no—
181C2×C3×C42—52no—
182C2×#217C2×D511yescharacteristic
183C2×#217C2×D511yescharacteristic
184#930—58no—
185C2×(C2×D3)⋊C4—52no—
186C2×(C2×D3)⋊C4—52no—
187C2×(C2×D3)⋊C4—52no—
188C2×(C2×D3)⋊C4—52no—
189C2×C4⋊Dic3—51no—
190C2×Dic3⋊C4—52no—
191C2×C4×Dic3—52no—
192C2×C4×Dic3C2×D511yescharacteristic
193C2×C4×Dic3C2×D511yescharacteristic
194#893—104no—
195C3⋊#95—52no—
196C3⋊#95—52no—
197C3⋊#95C2×D511yescharacteristic
198C3⋊#95C2×D511yescharacteristic
199C24×C5C22×D311yescharacteristic
200C23×D5—32no—
201C22×C4×C5—22no—
202C22×C4×C5—22no—
203C22×C4×C5—31no—
204C22×C4×C5—31no—
205C22×C4×C5—31no—
206C22×C4×C5—31no—
207C22×C4×C5C4×D314yescomplemented
208C22×C4×C5C22×D311yescharacteristic
209C22×C4×C5C22×D311yescharacteristic
210C2×C22⋊D5—38no—
211C2×C22⋊D5—122no—
212C2×C22⋊D5—122no—
213C22×Dic5—32no—
214C22×Dic5C22×D312yes—
215C22×Dic5C22×D312yes—
216C2×D20—34no—
217C2×D20—62no—
218C2×D20—62no—
219C2×C4×D5—64no—
220C2×C4×D5—64no—
221C2×C4×D5—122no—
222C5×C22⋊C4—62no—
223C5×C22⋊C4—62no—
224C42×C5—62no—
225C42×C5—62no—
226C22⋊Dic5—24no—
227#685—38no—
228#685—64no—
229#685—64no—
230#685—122no—
231C4⋊Dic5—22no—
232C4⋊Dic5—34no—
233C4⋊Dic5#6714yescomplemented
234#683—122no—
235C4×Dic5—24no—
236C4×Dic5—122no—
237C2×#121—151no—
238C2×C4×D4—151no—
239C2×#117—51no—
240#390—151no—
241#390—152no—
242#389—151no—
243#388—152no—
244#386—152no—
245C4×C4⋊C4—151no—
246C4×C22⋊C4—31no—
247C4×C22⋊C4—152no—
248C22×C3×C5—22no—
249C22×C3×C5—22no—
250C22×C3×C5—22no—
251C22×C3×C5—41no—
252C22×C3×C5—41no—
253C22×C3×C5—41no—
254C22×C3×C5—41no—
255C22×C3×C5C23×C411yescharacteristic
256C22×C3×C5C22×D411yescharacteristic
257C22×C3×C5C2×#4111yescharacteristic
258C22×C3×C5C2×#4111yescharacteristic
259C22×C3×C5C2×#4111yescharacteristic
260C22×C3×C5H(5,𝔽2)11yescharacteristic
261C22×C3×C5H(5,𝔽2)11yescharacteristic
262C2×D15—24no—
263C2×D15—24no—
264C2×D15—24no—
265C2×D15—42no—
266C2×D15—42no—
267C2×D15—42no—
268C2×D15—42no—
269C3×C4×C5—22no—
270C3×C4×C5C4×D414yes—
271Dic15—24no—
272C3×Dic5—24no—
273C3×Dic5—42no—
274C5×Dic3—22no—
275C5×Dic3—22no—
276C5×Dic3—41no—
277C5×Dic3—41no—
278C24×C3C22×D511yescharacteristic
279C23×D3—52no—
280C22×C3×C4—22no—
281C22×C3×C4—22no—
282C22×C3×C4—52no—
283C22×C3×C4—52no—
284C22×C3×C4C4×D514yescomplemented
285C22×C3×C4C22×D511yescharacteristic
286C22×C3×C4C22×D511yescharacteristic
287C2×#67—58no—
288C2×#67—202no—
289C2×#67—202no—
290C22×Dic3—52no—
291C22×Dic3C22×D511yescharacteristic
292C22×Dic3C22×D511yescharacteristic
293C22×Dic3C22×D511yescharacteristic
294C22×Dic3C22×D511yescharacteristic
295C2×D12—54no—
296C2×D12—102no—
297C2×D12—102no—
298C2×C4×D3—104no—
299C2×C4×D3—104no—
300C2×C4×D3—202no—
301C3×C4⋊C4—54no—
302C3×C4⋊C4—102no—
303C3×C22⋊C4—104no—
304C3×C42—104no—
305#217—22no—
306#217—22no—
307(C2×D3)⋊C4—58no—
308(C2×D3)⋊C4—104no—
309(C2×D3)⋊C4—104no—
310(C2×D3)⋊C4—202no—
311C4⋊Dic3—54no—
312Dic3⋊C4—202no—
313C4×Dic3—22no—
314C4×Dic3—22no—
315C4×Dic3—202no—
316C23×C5—21no—
317C23×C5—21no—
318C23×C5—21no—
319C23×C5—21no—
320C23×C5—21no—
321C23×C5—21no—
322C23×C5C2×C4×D312yes—
323C23×C5C23×D311yescharacteristic
324C22×D5—34no—
325C22×D5—62no—
326C22×D5—62no—
327C22×D5—62no—
328C22×D5—62no—
329C22×D5—62no—
330C22×D5—62no—
331C2×C4×C5—21no—
332C2×C4×C5—21no—
333C2×C4×C5—28no—
334C2×C4×C5—32no—
335C2×C4×C5—32no—
336C2×C4×C5—61no—
337C2×C4×C5—61no—
338C2×C4×C5—61no—
339C2×C4×C5—61no—
340C2×C4×C5—61no—
341C2×C4×C5—61no—
342C2×C4×C5—61no—
343C2×C4×C5—61no—
344C2×C4×C5—61no—
345C2×C4×C5—61no—
346C2×C4×C5C2×C4×D312yes—
347C2×C4×C5C2×C4×D312yes—
348C2×C4×C5#23512yes—
349C2×C4×C5C2×#6712yes—
350C22⋊D5—128no—
351C2×Dic5—22no—
352C2×Dic5—22no—
353C2×Dic5—22no—
354C2×Dic5—22no—
355C2×Dic5—22no—
356C2×Dic5—34no—
357C2×Dic5—62no—
358C2×Dic5—62no—
359C2×Dic5C2×#6714yes—
360D20—68no—
361C4×D5—128no—
362C22×D4—152no—
363C22×D4—151no—
364C23×C4—152no—
365C23×C4D3×D511yescharacteristic
366#121—158no—
367C4×D4—304no—
368#117—104no—
369C2×C4⋊C4—51no—
370C2×C4⋊C4—51no—
371C2×C4⋊C4—151no—
372C2×C4⋊C4—152no—
373C2×C22⋊C4—31no—
374C2×C22⋊C4—31no—
375C2×C22⋊C4—52no—
376C2×C22⋊C4—152no—
377C2×C22⋊C4—152no—
378C2×C22⋊C4—152no—
379C2×C22⋊C4—152no—
380C2×C42—31no—
381C2×C42—31no—
382C2×C42—52no—
383C2×C42—152no—
384#95—31no—
385#95—31no—
386#95—152no—
387#95—152no—
388C2×C3×C5—42no—
389C2×C3×C5C2×C4×D411yescharacteristic
390C2×C3×C5C4×#4111yescharacteristic
391C2×C3×C5#51811yescharacteristic
392C2×C3×C5#54611yescharacteristic
393C2×C3×C5#55011yescharacteristic
394C2×C3×C5#55311yescharacteristic
395C2×C3×C5#55511yescharacteristic
396D15—44no—
397C23×C3—21no—
398C23×C3—21no—
399C23×C3—21no—
400C23×C3—21no—
401C23×C3—21no—
402C23×C3—21no—
403C23×C3C2×C4×D512yes—
404C23×C3C23×D511yescharacteristic
405C22×D3—54no—
406C22×D3—102no—
407C22×D3—102no—
408C22×D3—102no—
409C22×D3—102no—
410C22×D3—102no—
411C22×D3—102no—
412C2×C3×C4—21no—
413C2×C3×C4—21no—
414C2×C3×C4—28no—
415C2×C3×C4—54no—
416C2×C3×C4—102no—
417C2×C3×C4—102no—
418C2×C3×C4—102no—
419C2×C3×C4—102no—
420C2×C3×C4—102no—
421C2×C3×C4C2×C4×D512yes—
422C2×C3×C4C2×C4×D512yes—
423C2×C3×C4D4×D512yes—
424C2×C3×C4#71112yes—
425#67—208no—
426C2×Dic3—21no—
427C2×Dic3—21no—
428C2×Dic3—21no—
429C2×Dic3—21no—
430C2×Dic3—21no—
431C2×Dic3—21no—
432C2×Dic3—21no—
433C2×Dic3—21no—
434C2×Dic3—21no—
435C2×Dic3—21no—
436C2×Dic3—54no—
437C2×Dic3—102no—
438C2×Dic3—102no—
439C2×Dic3#70912yes—
440C2×Dic3#70912yes—
441D12—108no—
442C4×D3—208no—
443C22×C5—22no—
444C22×C5—22no—
445C22×C5—22no—
446C22×C5—41no—
447C22×C5—41no—
448C22×C5—41no—
449C22×C5—41no—
450C22×C5C22×C4×D311yescharacteristic
451C22×C5C2×#23511yescharacteristic
452C22×C5D3×#4111yescharacteristic
453C22×C5D3×#4111yescharacteristic
454C22×C5#100911yescharacteristic
455C22×C5#100911yescharacteristic
456C22×C5C22×#6711yescharacteristic
457C2×D5—64no—
458C2×D5—64no—
459C2×D5—64no—
460C2×D5—122no—
461C2×D5—122no—
462C2×D5—122no—
463C2×D5—122no—
464C4×C5—22no—
465C4×C5—62no—
466C4×C5—62no—
467C4×C5—121no—
468C4×C5—121no—
469C4×C5C4×#6714yes—
470Dic5—24no—
471Dic5—42no—
472Dic5—64no—
473C24—152no—
474C24C2×D3×D511yescharacteristic
475C2×D4—158no—
476C2×D4—154no—
477C2×D4—302no—
478C2×D4—302no—
479C2×D4—602no—
480C2×D4—602no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—31no—
484C22×C4—31no—
485C22×C4—31no—
486C22×C4—31no—
487C22×C4—52no—
488C22×C4—52no—
489C22×C4—152no—
490C22×C4—304no—
491C22×C4—304no—
492C22×C4—602no—
493C22×C4#120014yescomplemented
494C22×C4C2×D3×D511yescharacteristic
495C22×C4C2×D3×D511yescharacteristic
496C4⋊C4—54no—
497C4⋊C4—102no—
498C4⋊C4—154no—
499C4⋊C4—602no—
500C22⋊C4—62no—
501C22⋊C4—62no—
502C22⋊C4—104no—
503C22⋊C4—158no—
504C22⋊C4—304no—
505C22⋊C4—304no—
506C22⋊C4—602no—
507C42—62no—
508C42—62no—
509C42—104no—
510C42—602no—
511C3×C5#236311yescharacteristic, complemented, Hall
512C22×C3—22no—
513C22×C3—22no—
514C22×C3—22no—
515C22×C3—41no—
516C22×C3—41no—
517C22×C3—41no—
518C22×C3—41no—
519C22×C3C22×C4×D511yescharacteristic
520C22×C3C2×#70911yescharacteristic
521C22×C3C2×#70911yescharacteristic
522C22×C3C2×D4×D511yescharacteristic
523C22×C3C2×#71111yescharacteristic
524C22×C3#414911yescharacteristic
525C22×C3#414911yescharacteristic
526C2×D3—104no—
527C2×D3—104no—
528C2×D3—104no—
529C2×D3—202no—
530C2×D3—202no—
531C2×D3—202no—
532C2×D3—202no—
533C3×C4—22no—
534C3×C4—104no—
535C3×C4—202no—
536C3×C4#402714yes—
537Dic3—22no—
538Dic3—22no—
539Dic3—41no—
540Dic3—41no—
541Dic3—104no—
542C2×C5—42no—
543C2×C5#552111yescharacteristic
544C2×C5#552211yescharacteristic
545C2×C5#553011yescharacteristic
546C2×C5#553211yescharacteristic
547C2×C5C2×C4×#6711yescharacteristic
548C2×C5#582911yescharacteristic
549C2×C5#583411yescharacteristic
550D5—124no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—154no—
558C23—302no—
559C23—302no—
560C23—302no—
561C23—302no—
562C23—302no—
563C23—302no—
564C23C2×#120012yes—
565C23C22×D3×D511yescharacteristic
566D4—308no—
567D4—608no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—28no—
571C2×C4—32no—
572C2×C4—32no—
573C2×C4—54no—
574C2×C4—61no—
575C2×C4—61no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—61no—
579C2×C4—61no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—102no—
585C2×C4—102no—
586C2×C4—102no—
587C2×C4—102no—
588C2×C4—102no—
589C2×C4—154no—
590C2×C4—302no—
591C2×C4—302no—
592C2×C4—608no—
593C2×C4#678212yes—
594C2×C4C2×#120012yes—
595C2×C4C2×#120012yes—
596C2×C4D5×#6712yes—
597C2×C3—42no—
598C2×C3C4×#70911yescharacteristic
599C2×C3C2×#402711yescharacteristic
600C2×C3#6585511yescharacteristic
601C2×C3#6588411yescharacteristic
602C2×C3#6588711yescharacteristic
603C2×C3#6589711yescharacteristic
604C2×C3#6589911yescharacteristic
605D3—204no—
606C5#8224411yescharacteristic, complemented, Sylow
607C22—22no—
608C22—22no—
609C22—22no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—304no—
615C22—304no—
616C22—304no—
617C22—602no—
618C22—602no—
619C22—602no—
620C22—602no—
621C22D3×#70911yescharacteristic
622C22D3×#70911yescharacteristic
623C22#9201911yescharacteristic
624C22#9201911yescharacteristic
625C22C2×#678211yescharacteristic
626C22C22×#120011yescharacteristic
627C22C2×D5×#6711yescharacteristic
628C4—22no—
629C4—62no—
630C4—62no—
631C4—104no—
632C4—121no—
633C4—121no—
634C4—202no—
635C4—304no—
636C4#9155214yes—
637C3#1060520011yescharacteristic, complemented, Sylow
638C2—42no—
639C2—604no—
640C2#1175443611yescharacteristic
641C2#1175443911yescharacteristic
642C2#1175444311yescharacteristic
643C2#1175451511yescharacteristic
644C2#1175453011yescharacteristic
645C2#1175463811yescharacteristic
646C2C2×#9155211yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC23×C3×C4×C5C22
socle, cosocleC23×C3×C5C24
prime radicalC2×#91819C2
hypercenterC23×C4D3×D5
hypocenterC3×C5#2363
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×C3×#4072C2semidirect
3. C2×#91428C2semidirect
4. C4×C5×#217C2semidirect
5. #11749520C2semidirect
6. #11749483C2semidirect
7. #11748875C2semidirect
8. #11748858C2semidirect
9. #11748829C2semidirect
10. #11748789C2semidirect
11. Dic3×C4⋊Dic5C2semidirect
17. #91819C4semidirect
41. C2×#4072D3semidirect
55. C22×C3×C4×C5C2×C4semidirect
60. C2×#1220C2×C4semidirect
74. #6714C2×C4semidirect
75. C4⋊Dic15C2×C4semidirect
82. C3×C4⋊Dic5D4semidirect
92. C4×#217D5semidirect
207. C22×C4×C5C4×D3semidirect
233. C4⋊Dic5#67semidirect
284. C22×C3×C4C4×D5semidirect
493. C22×C4#1200semidirect
511. C3×C5#2363semidirect
606. C5#82244semidirect
637. C3#10605200semidirect

Maximal subgroups

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

subgroupindex
C2×#918192
C2×C3×#40722
C2×#914282
C4×C5×#2172
#117495202
#117494832
#117488752
#117488582
#117488292
#117487892
Dic3×C4⋊Dic52
#106052003
#822445