Finite Group Databasebeta

← #14323362

Subgroups — #14323362

Order 1920.

1616 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C5×C22⋊C4C211yesmaximal, characteristic, complemented
2C2×#91842C211yesmaximal, characteristic
3C2×C4×#1220C211yesmaximal, characteristic, complemented
4#11751348C211yesmaximal, characteristic, complemented
5#11751314C211yesmaximal, characteristic, complemented
6#11751311C211yesmaximal, characteristic, complemented
7C4×#6719C211yesmaximal, characteristic, complemented
8#11751058C212yesmaximal, complemented
9#11751057C212yesmaximal, complemented
10#11751046C212yesmaximal, complemented
11#11751045C212yesmaximal, complemented
12#10613827—31nomaximal, Hall
13C25×C3×C5C2211yescharacteristic
14C23×C3×C4×C5C2211yescharacteristic
15C23×C3×C4×C5C2211yescharacteristic
16C22×#1220C2211yescharacteristic
17C22×#1220C2212yes—
18C22×C4×D15C2211yescharacteristic, complemented
19C2×C3×C5×C22⋊C4—24no—
20C2×C3×C5×C22⋊C4C2212yes—
21C2×C3×C5×C22⋊C4C2212yes—
22#91842C418yescomplemented
23C2×#6719C2211yescharacteristic
24C2×#6719C2211yescharacteristic
25C2×#6719C2212yes—
26C2×#6719C2212yes—
27C2×#6719C2212yes—
28C2×#6719C2211yescharacteristic
29C4×#1220—24no—
30C2×#6714C2211yescharacteristic, complemented
31C2×#6714C2212yescomplemented
32C2×#6712C2211yescharacteristic, complemented
33C2×#6712C2212yescomplemented
34C2×C4×Dic15C2211yescharacteristic, complemented
35C2×C4×Dic15C2212yescomplemented
36#91115C2212yescomplemented
37#91115C2212yescomplemented
38#91115C2212yescomplemented
39#82096—51nomaximal, Hall
40C22×C5×C22⋊C4D311yescharacteristic, complemented
41C2×#4099—31no—
42C2×C4×C22⋊D5—31no—
43#65536—31no—
44#65502—31no—
45#65499—31no—
46C4×C22⋊Dic5—31no—
47#65246—32no—
48#65245—32no—
49#65234—32no—
50#65233—32no—
51C24×C3×C5—22no—
52C24×C3×C5—22no—
53C24×C3×C5—42no—
54C24×C3×C5—42no—
55C24×C3×C5C2×C414yescomplemented
56C24×C3×C5C2312yes—
57C24×C3×C5C2311yescharacteristic
58C23×D15C2311yescharacteristic
59C22×C3×C4×C5—22no—
60C22×C3×C4×C5—42no—
61C22×C3×C4×C5—42no—
62C22×C3×C4×C5D414yescomplemented
63C22×C3×C4×C5C2312yes—
64C22×C3×C4×C5C2311yescharacteristic
65C22×C3×C4×C5C2311yescharacteristic
66C2×#1220—22no—
67C2×#1220—22no—
68C2×#1220—42no—
69C2×#1220—42no—
70C2×#1220C2×C414yes—
71C2×#1220C2×C418yescomplemented
72C22×Dic15C2311yescharacteristic
73C22×Dic15C2312yes—
74C22×Dic15C2311yescharacteristic
75C22×Dic15C2311yescharacteristic
76C22×Dic15C2312yes—
77C2×C4×D15—22no—
78C2×C4×D15—42no—
79C3×C5×C22⋊C4—24no—
80C3×C5×C22⋊C4—24no—
81#6719—22no—
82#6719—22no—
83#6719—42no—
84#6719—42no—
85#6719C2×C418yescomplemented
86#6719C2×C414yescomplemented
87#6714—22no—
88#6714—42no—
89#6712—22no—
90#6712—42no—
91C4×Dic15—22no—
92C4×Dic15—42no—
93C22×C3×C22⋊C4D511yescharacteristic, complemented
94C2×#953—51no—
95C2×C4×#67—51no—
96#5247—51no—
97#5213—51no—
98#5210—51no—
99C4×#217—51no—
100#4957—52no—
101#4956—52no—
102#4945—52no—
103#4944—52no—
104C25×C5C2×D311yescharacteristic
105C23×C4×C5C2×D311yescharacteristic
106C23×C4×C5C2×D311yescharacteristic
107C22×C22⋊D5—31no—
108C22×C22⋊D5—32no—
109C22×C4×D5—31no—
110C2×C5×C22⋊C4—24no—
111C2×C5×C22⋊C4C2×D312yes—
112C2×C5×C22⋊C4C2×D312yes—
113#4099—38no—
114C2×C22⋊Dic5—32no—
115C2×C22⋊Dic5—31no—
116C2×C22⋊Dic5—31no—
117C2×C22⋊Dic5—31no—
118C2×C22⋊Dic5—32no—
119C2×C22⋊Dic5—32no—
120C4×C22⋊D5—64no—
121C2×#685—31no—
122C2×#685—32no—
123C2×#683—31no—
124C2×#683—32no—
125C2×C4×Dic5—31no—
126C2×C4×Dic5—32no—
127#3963—32no—
128#3963—32no—
129#3963—32no—
130#2321—151noSylow
131C23×C3×C5—22no—
132C23×C3×C5—22no—
133C23×C3×C5—28no—
134C23×C3×C5—22no—
135C23×C3×C5—22no—
136C23×C3×C5—22no—
137C23×C3×C5—22no—
138C23×C3×C5—22no—
139C23×C3×C5—22no—
140C23×C3×C5—48no—
141C23×C3×C5—44no—
142C23×C3×C5—44no—
143C23×C3×C5—48no—
144C23×C3×C5C22×C414yes—
145C23×C3×C5C22×C412yes—
146C23×C3×C5C2×D412yes—
147C23×C3×C5#4112yes—
148C23×C3×C5C2411yescharacteristic
149C22×D15—21no—
150C22×D15—21no—
151C22×D15—22no—
152C22×D15—22no—
153C22×D15C22×C412yes—
154C2×C3×C4×C5—22no—
155C2×C3×C4×C5—22no—
156C2×C3×C4×C5—22no—
157C2×C3×C4×C5—22no—
158C2×C3×C4×C5—22no—
159C2×C3×C4×C5—44no—
160C2×C3×C4×C5—44no—
161C2×C3×C4×C5C2×D412yes—
162C2×C3×C4×C5C2×D412yes—
163#1220—28no—
164#1220—48no—
165C2×Dic15—21no—
166C2×Dic15—21no—
167C2×Dic15—22no—
168C2×Dic15—22no—
169C2×Dic15—21no—
170C2×Dic15—21no—
171C2×Dic15—22no—
172C2×Dic15—22no—
173C2×Dic15—22no—
174C2×Dic15—22no—
175C2×Dic15—22no—
176C2×Dic15C22×C412yes—
177C2×Dic15C22×C414yes—
178C4×D15—44no—
179C25×C3C2×D511yescharacteristic
180C23×C3×C4C2×D511yescharacteristic
181C23×C3×C4C2×D511yescharacteristic
182C22×#67—51no—
183C22×#67—52no—
184C22×C4×D3—51no—
185C2×C3×C22⋊C4—24no—
186C2×C3×C22⋊C4C2×D512yes—
187C2×C3×C22⋊C4C2×D512yes—
188#953—58no—
189C2×#217—52no—
190C2×#217—51no—
191C2×#217—51no—
192C2×#217—51no—
193C2×#217—52no—
194C2×#217—52no—
195C4×#67—104no—
196C2×(C2×D3)⋊C4—51no—
197C2×(C2×D3)⋊C4—52no—
198C2×Dic3⋊C4—51no—
199C2×Dic3⋊C4—52no—
200C2×C4×Dic3—51no—
201C2×C4×Dic3—52no—
202C3⋊#95—52no—
203C3⋊#95—52no—
204C3⋊#95—52no—
205C24×C5—22no—
206C24×C5—22no—
207C24×C5—42no—
208C24×C5—42no—
209C24×C5C4×D314yescomplemented
210C24×C5C22×D312yes—
211C24×C5C22×D311yescharacteristic
212C23×D5—31no—
213C22×C4×C5—22no—
214C22×C4×C5—42no—
215C22×C4×C5—42no—
216C22×C4×C5#6714yescomplemented
217C22×C4×C5C22×D312yes—
218C22×C4×C5C22×D311yescharacteristic
219C22×C4×C5C22×D311yescharacteristic
220C2×C22⋊D5—34no—
221C2×C22⋊D5—38no—
222C2×C22⋊D5—62no—
223C2×C22⋊D5—62no—
224C2×C22⋊D5—122no—
225C2×C22⋊D5—122no—
226C22×Dic5—31no—
227C22×Dic5—32no—
228C22×Dic5—31no—
229C22×Dic5—32no—
230C22×Dic5—31no—
231C2×C4×D5—62no—
232C2×C4×D5—122no—
233C5×C22⋊C4—24no—
234C5×C22⋊C4—24no—
235C22⋊Dic5—38no—
236C22⋊Dic5—34no—
237C22⋊Dic5—62no—
238C22⋊Dic5—62no—
239C22⋊Dic5—122no—
240C22⋊Dic5—122no—
241#685—62no—
242#685—122no—
243#683—62no—
244#683—122no—
245C4×Dic5—62no—
246C4×Dic5—122no—
247C2×#120—151no—
248C2×C4×D4—151no—
249C22×C22⋊C4D1511yescharacteristic, complemented
250#390—152no—
251#388—152no—
252#388—152no—
253#386—151no—
254#386—152no—
255#385—151no—
256#379—151no—
257C4×C22⋊C4—151no—
258C22×C3×C5—24no—
259C22×C3×C5—22no—
260C22×C3×C5—22no—
261C22×C3×C5—24no—
262C22×C3×C5—24no—
263C22×C3×C5—24no—
264C22×C3×C5—24no—
265C22×C3×C5—416no—
266C22×C3×C5—42no—
267C22×C3×C5—42no—
268C22×C3×C5—42no—
269C22×C3×C5—42no—
270C22×C3×C5C4×D414yes—
271C22×C3×C5C23×C411yescharacteristic
272C22×C3×C5C22×D411yescharacteristic
273C22×C3×C5C2×#4111yescharacteristic
274C22×C3×C5H(5,𝔽2)12yes—
275C22×C3×C5H(5,𝔽2)12yes—
276C2×D15—22no—
277C2×D15—24no—
278C2×D15—41no—
279C2×D15—41no—
280C2×D15—42no—
281C3×C4×C5—22no—
282C3×C4×C5—22no—
283C3×C4×C5—42no—
284Dic15—22no—
285Dic15—24no—
286Dic15—41no—
287Dic15—41no—
288Dic15—42no—
289C24×C3—22no—
290C24×C3—22no—
291C24×C3—42no—
292C24×C3—42no—
293C24×C3C4×D514yescomplemented
294C24×C3C22×D512yes—
295C24×C3C22×D511yescharacteristic
296C23×D3—51no—
297C22×C3×C4—22no—
298C22×C3×C4—42no—
299C22×C3×C4—42no—
300C22×C3×C4C22⋊D514yescomplemented
301C22×C3×C4C22×D512yes—
302C22×C3×C4C22×D511yescharacteristic
303C22×C3×C4C22×D511yescharacteristic
304C2×#67—54no—
305C2×#67—58no—
306C2×#67—102no—
307C2×#67—102no—
308C2×#67—202no—
309C2×#67—202no—
310C22×Dic3—51no—
311C22×Dic3—52no—
312C22×Dic3—52no—
313C22×Dic3—51no—
314C22×Dic3—51no—
315C2×C4×D3—102no—
316C2×C4×D3—202no—
317C3×C22⋊C4—24no—
318C3×C22⋊C4—24no—
319#217—58no—
320#217—54no—
321#217—102no—
322#217—102no—
323#217—202no—
324#217—202no—
325(C2×D3)⋊C4—102no—
326(C2×D3)⋊C4—202no—
327Dic3⋊C4—102no—
328Dic3⋊C4—202no—
329C4×Dic3—102no—
330C4×Dic3—202no—
331C23×C5—22no—
332C23×C5—22no—
333C23×C5—28no—
334C23×C5—22no—
335C23×C5—22no—
336C23×C5—22no—
337C23×C5—22no—
338C23×C5—22no—
339C23×C5—22no—
340C23×C5—48no—
341C23×C5—48no—
342C23×C5—44no—
343C23×C5—44no—
344C23×C5C2×C4×D314yes—
345C23×C5C2×C4×D312yes—
346C23×C5#23512yes—
347C23×C5C2×#6712yes—
348C23×C5C23×D311yescharacteristic
349C22×D5—32no—
350C22×D5—61no—
351C22×D5—62no—
352C22×D5—62no—
353C22×D5—61no—
354C2×C4×C5—22no—
355C2×C4×C5—22no—
356C2×C4×C5—22no—
357C2×C4×C5—22no—
358C2×C4×C5—22no—
359C2×C4×C5—44no—
360C2×C4×C5—44no—
361C2×C4×C5C2×#6712yes—
362C2×C4×C5C2×#6712yes—
363C22⋊D5—68no—
364C22⋊D5—128no—
365C2×Dic5—34no—
366C2×Dic5—32no—
367C2×Dic5—61no—
368C2×Dic5—62no—
369C2×Dic5—62no—
370C2×Dic5—61no—
371C2×Dic5—62no—
372C2×Dic5—62no—
373C2×Dic5—62no—
374C2×Dic5—62no—
375C2×Dic5—62no—
376C2×Dic5—61no—
377C2×Dic5—61no—
378C4×D5—124no—
379C25C2×D1511yescharacteristic
380C22×D4—151no—
381C22×D4—152no—
382C23×C4—151no—
383C23×C4C2×D1511yescharacteristic
384C23×C4C2×D1511yescharacteristic
385#120—158no—
386C4×D4—304no—
387C2×C4⋊C4—151no—
388C2×C4⋊C4—152no—
389C2×C22⋊C4—24no—
390C2×C22⋊C4—151no—
391C2×C22⋊C4—152no—
392C2×C22⋊C4—151no—
393C2×C22⋊C4—151no—
394C2×C22⋊C4—151no—
395C2×C22⋊C4—152no—
396C2×C22⋊C4—152no—
397C2×C22⋊C4—152no—
398C2×C22⋊C4C2×D1512yes—
399C2×C22⋊C4C2×D1512yes—
400C2×C42—151no—
401C2×C42—152no—
402#95—152no—
403#95—152no—
404#95—152no—
405C2×C3×C5—24no—
406C2×C3×C5—44no—
407C2×C3×C5C2×C4×D411yescharacteristic
408C2×C3×C5#51812yes—
409C2×C3×C5#53411yescharacteristic
410C2×C3×C5#53511yescharacteristic
411C2×C3×C5#53812yes—
412D15—42no—
413C23×C3—22no—
414C23×C3—22no—
415C23×C3—22no—
416C23×C3—22no—
417C23×C3—22no—
418C23×C3—22no—
419C23×C3—22no—
420C23×C3—22no—
421C23×C3—28no—
422C23×C3—48no—
423C23×C3—48no—
424C23×C3—44no—
425C23×C3—44no—
426C23×C3C2×C4×D514yes—
427C23×C3C2×C4×D512yes—
428C23×C3#70912yes—
429C23×C3C2×C22⋊D512yes—
430C23×C3C23×D511yescharacteristic
431C22×D3—52no—
432C22×D3—101no—
433C22×D3—102no—
434C22×D3—102no—
435C22×D3—101no—
436C2×C3×C4—22no—
437C2×C3×C4—22no—
438C2×C3×C4—22no—
439C2×C3×C4—22no—
440C2×C3×C4—22no—
441C2×C3×C4—44no—
442C2×C3×C4—44no—
443C2×C3×C4C2×C22⋊D512yes—
444C2×C3×C4C2×C22⋊D512yes—
445#67—108no—
446#67—208no—
447C2×Dic3—54no—
448C2×Dic3—52no—
449C2×Dic3—101no—
450C2×Dic3—102no—
451C2×Dic3—102no—
452C2×Dic3—101no—
453C2×Dic3—102no—
454C2×Dic3—102no—
455C2×Dic3—102no—
456C2×Dic3—102no—
457C2×Dic3—102no—
458C2×Dic3—101no—
459C2×Dic3—101no—
460C4×D3—204no—
461C22×C5—24no—
462C22×C5—24no—
463C22×C5—22no—
464C22×C5—22no—
465C22×C5—24no—
466C22×C5—24no—
467C22×C5—24no—
468C22×C5—416no—
469C22×C5—42no—
470C22×C5—42no—
471C22×C5—42no—
472C22×C5—42no—
473C22×C5C4×#6714yes—
474C22×C5C22×C4×D311yescharacteristic
475C22×C5C2×#23511yescharacteristic
476C22×C5#100412yes—
477C22×C5#100412yes—
478C22×C5C22×#6711yescharacteristic
479C2×D5—64no—
480C2×D5—62no—
481C2×D5—122no—
482C2×D5—121no—
483C2×D5—121no—
484C4×C5—22no—
485C4×C5—22no—
486C4×C5—42no—
487Dic5—64no—
488Dic5—62no—
489Dic5—122no—
490Dic5—121no—
491Dic5—121no—
492C24—22no—
493C24—22no—
494C24—42no—
495C24—42no—
496C24—151no—
497C24C4×D1514yescomplemented
498C24C22×D1512yes—
499C24C22×D1511yescharacteristic
500C2×D4—154no—
501C2×D4—158no—
502C2×D4—302no—
503C2×D4—302no—
504C2×D4—602no—
505C2×D4—602no—
506C22×C4—22no—
507C22×C4—42no—
508C22×C4—42no—
509C22×C4—151no—
510C22×C4—152no—
511C22×C4—152no—
512C22×C4—151no—
513C22×C4—151no—
514C22×C4—302no—
515C22×C4—602no—
516C22×C4#122014yescomplemented
517C22×C4C22×D1512yes—
518C22×C4C22×D1511yescharacteristic
519C22×C4C22×D1511yescharacteristic
520C4⋊C4—302no—
521C4⋊C4—602no—
522C22⋊C4—24no—
523C22⋊C4—24no—
524C22⋊C4—158no—
525C22⋊C4—154no—
526C22⋊C4—302no—
527C22⋊C4—302no—
528C22⋊C4—302no—
529C22⋊C4—602no—
530C22⋊C4—602no—
531C22⋊C4—602no—
532C42—302no—
533C42—602no—
534C3×C5#232111yescharacteristic, complemented, Hall
535C22×C3—24no—
536C22×C3—24no—
537C22×C3—22no—
538C22×C3—22no—
539C22×C3—24no—
540C22×C3—24no—
541C22×C3—24no—
542C22×C3—416no—
543C22×C3—42no—
544C22×C3—42no—
545C22×C3—42no—
546C22×C3—42no—
547C22×C3C4×C22⋊D514yes—
548C22×C3C22×C4×D511yescharacteristic
549C22×C3C2×#70911yescharacteristic
550C22×C3#414412yes—
551C22×C3#414412yes—
552C22×C3C22×C22⋊D511yescharacteristic
553C2×D3—104no—
554C2×D3—102no—
555C2×D3—202no—
556C2×D3—201no—
557C2×D3—201no—
558C3×C4—22no—
559C3×C4—22no—
560C3×C4—42no—
561Dic3—104no—
562Dic3—102no—
563Dic3—202no—
564Dic3—201no—
565Dic3—201no—
566C2×C5—24no—
567C2×C5—44no—
568C2×C5#548612yes—
569C2×C5#549412yes—
570C2×C5C2×C4×#6711yescharacteristic
571C2×C5#580411yescharacteristic
572C2×C5#580511yescharacteristic
573D5—122no—
574C23—22no—
575C23—22no—
576C23—28no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23—22no—
583C23—48no—
584C23—44no—
585C23—44no—
586C23—48no—
587C23—152no—
588C23—301no—
589C23—302no—
590C23—302no—
591C23—301no—
592C23C2×C4×D1514yes—
593C23C2×C4×D1512yes—
594C23#681712yes—
595C23C2×#122012yes—
596C23C23×D1511yescharacteristic
597D4—308no—
598D4—608no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—152no—
607C2×C4—154no—
608C2×C4—301no—
609C2×C4—302no—
610C2×C4—302no—
611C2×C4—301no—
612C2×C4—302no—
613C2×C4—302no—
614C2×C4—302no—
615C2×C4—302no—
616C2×C4—302no—
617C2×C4—301no—
618C2×C4—301no—
619C2×C4—604no—
620C2×C4C2×#122012yes—
621C2×C4C2×#122012yes—
622C2×C3—24no—
623C2×C3—44no—
624C2×C3#6582012yes—
625C2×C3#6582812yes—
626C2×C3C2×C4×C22⋊D511yescharacteristic
627C2×C3#6613811yescharacteristic
628C2×C3#6613911yescharacteristic
629D3—202no—
630C5#8209611yescharacteristic, complemented, Sylow
631C22—22no—
632C22—22no—
633C22—24no—
634C22—24no—
635C22—24no—
636C22—24no—
637C22—24no—
638C22—416no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—302no—
644C22—304no—
645C22—601no—
646C22—601no—
647C22—602no—
648C22C4×#122014yes—
649C22C22×C4×D1511yescharacteristic
650C22C2×#681711yescharacteristic
651C22#9209512yes—
652C22#9209512yes—
653C22C22×#122011yescharacteristic
654C4—22no—
655C4—22no—
656C4—42no—
657C4—302no—
658C4—304no—
659C4—601no—
660C4—601no—
661C4—602no—
662C3#1061382711yescharacteristic, complemented, Sylow
663C2—24no—
664C2—44no—
665C2—602no—
666C2#1175624512yes—
667C2#1175625312yes—
668C2C2×C4×#122011yescharacteristic
669C2#1175656311yescharacteristic
670C2#1175656411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D15
commutatorC22×C3×C5C23×C4
FittingC22×C3×C5×C22⋊C4C2
socle, cosocleC23×C3×C5C24
prime radicalC2×#91842C2
hypercenterC22×C22⋊C4D15
hypocenterC3×C5#2321
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×C5×C22⋊C4C2semidirect
3. C2×C4×#1220C2semidirect
4. #11751348C2semidirect
5. #11751314C2semidirect
6. #11751311C2semidirect
7. C4×#6719C2semidirect
8. #11751058C2semidirect
9. #11751057C2semidirect
10. #11751046C2semidirect
11. #11751045C2semidirect
18. C22×C4×D15C22semidirect
22. #91842C4semidirect
30. C2×#6714C22semidirect
31. C2×#6714C22semidirect
32. C2×#6712C22semidirect
33. C2×#6712C22semidirect
34. C2×C4×Dic15C22semidirect
35. C2×C4×Dic15C22semidirect
36. #91115C22semidirect
37. #91115C22semidirect
38. #91115C22semidirect
40. C22×C5×C22⋊C4D3semidirect
55. C24×C3×C5C2×C4semidirect
62. C22×C3×C4×C5D4semidirect
71. C2×#1220C2×C4semidirect
85. #6719C2×C4semidirect
86. #6719C2×C4semidirect
93. C22×C3×C22⋊C4D5semidirect
209. C24×C5C4×D3semidirect
216. C22×C4×C5#67semidirect
249. C22×C22⋊C4D15semidirect
293. C24×C3C4×D5semidirect
300. C22×C3×C4C22⋊D5semidirect
497. C24C4×D15semidirect
516. C22×C4#1220semidirect
534. C3×C5#2321semidirect
630. C5#82096semidirect
662. C3#10613827semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C5×C22⋊C42
C2×#918422
C2×C4×#12202
#117513482
#117513142
#117513112
C4×#67192
#117510582
#117510572
#117510462
#117510452
#106138273
#820965