Finite Group Databasebeta

← #14323948

Subgroups — #14323948

Order 1920.

1336 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91465C211yesmaximal, characteristic
2C2×Dic3×D20C211yesmaximal, characteristic, complemented
3C2×#91386C211yesmaximal, characteristic, complemented
4#11751088C211yesmaximal, characteristic, complemented
5C5×Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
6C3×#65277C211yesmaximal, characteristic, complemented
7#11748912C212yesmaximal, complemented
8#11748890C212yesmaximal, complemented
9#11748872C212yesmaximal, complemented
10#11748842C211yesmaximal, characteristic, complemented
11#11748705C212yesmaximal, complemented
12#10605494—31nomaximal, Hall
13C22×C4×D15C2211yescharacteristic
14C22×C3×D20C2211yescharacteristic
15C22×#1202C2212yes—
16C22×D5×Dic3C2212yes—
17C2×C3×C5×C4⋊C4C2211yescharacteristic
18C2×#6714C2212yes—
19C2×C4⋊Dic15C2211yescharacteristic
20C2×C5×C4⋊Dic3C2211yescharacteristic
21C2×C4×C5×Dic3C2212yes—
22C2×C4×C5×Dic3C2211yescharacteristic
23C2×C3×#685C2212yes—
24C2×C3×#685C2212yes—
25C2×C3×C4×Dic5C2211yescharacteristic
26C2×Dic3⋊Dic5C2212yes—
27C2×#6669C2211yescharacteristic
28C2×#6669C2211yescharacteristic
29C2×#6667C2211yescharacteristic
30C2×#6667C2211yescharacteristic
31C2×#6665C2212yes—
32C2×#6665C2212yes—
33#91465C418yescomplemented
34Dic3×D20—24no—
35#91386—24no—
36#91115C2212yes—
37C5×C3⋊#95C2212yes—
38#90994C2212yes—
39#76430—51nomaximal, Hall
40C2×#4041—31no—
41C2×C4×D20—31no—
42C2×#4018—31no—
43C4×C5×C4⋊C4—31no—
44#65277D311yescharacteristic, complemented
45#65276—31no—
46#65274—31no—
47#65229—32no—
48#65228—32no—
49#64957—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×Dic3C2312yes—
61C22×C5×Dic3C2311yescharacteristic
62C22×C5×Dic3C2311yescharacteristic
63C22×C3×Dic5C2311yescharacteristic
64C22×C3×Dic5C2311yescharacteristic
65C2×C3×D20—22no—
66C2×C3×D20—22no—
67C2×C3×D20C2×C414yes—
68C2×#1202—42no—
69C2×#1202—42no—
70C2×#1202C2×C418yescomplemented
71C2×D5×Dic3—24no—
72C2×D5×Dic3—42no—
73C2×D5×Dic3—42no—
74C3×C5×C4⋊C4—22no—
75#6714—42no—
76C4⋊Dic15—22no—
77C5×C4⋊Dic3C2×C414yescomplemented
78C4×C5×Dic3—42no—
79C4×C5×Dic3D414yescomplemented
80C3×#685—42no—
81C3×#685—42no—
82C3×C4×Dic5—22no—
83Dic3⋊Dic5—42no—
84#6669—22no—
85#6669—22no—
86#6667—22no—
87#6667—22no—
88#6665—24no—
89#6665C2×C418yescomplemented
90C2×#938—51no—
91C2×D4×Dic3—51no—
92C2×#872—51no—
93C3×#390—51no—
94#4987—51no—
95Dic3×C4⋊C4D511yescharacteristic, complemented
96#4958—52no—
97#4957—52no—
98#4946—52no—
99#4944—52no—
100#4928—51no—
101C22×D20—32no—
102C22×D20C2×D311yescharacteristic
103C22×C4×D5—31no—
104C22×C4×D5—32no—
105C2×C5×C4⋊C4—31no—
106C2×C5×C4⋊C4C2×D311yescharacteristic
107C2×C42×C5—31no—
108C2×C42×C5—32no—
109C2×#685—32no—
110C2×#685—32no—
111C2×#685—31no—
112C2×#685—31no—
113C2×#685—32no—
114C2×#685C2×D312yes—
115C2×#685C2×D312yes—
116C2×C4⋊Dic5—31no—
117C2×C4⋊Dic5—32no—
118C2×#683—31no—
119C2×#683—31no—
120C2×C4×Dic5C2×D311yescharacteristic
121#4041—38no—
122C4×D20—64no—
123#4018—64no—
124C5×#95—32no—
125#3963—32no—
126#3963—32no—
127#2324—151noSylow
128C23×C3×C5C2411yescharacteristic
129C22×D15—21no—
130C22×D15—21no—
131C22×D15—21no—
132C22×D15—21no—
133C22×D15—21no—
134C22×D15—21no—
135C22×D15C22×C412yes—
136C22×C3×D5—22no—
137C22×C3×D5—22no—
138C22×C3×D5—22no—
139C22×C3×D5—22no—
140C22×C3×D5—22no—
141C22×C3×D5—22no—
142C22×C3×D5C22×C414yes—
143C2×C3×C4×C5—22no—
144C2×C3×C4×C5—22no—
145C2×C3×C4×C5—22no—
146C2×C3×C4×C5C22×C412yes—
147C2×C3×C4×C5C2×D412yes—
148C2×C3×C4×C5#4112yes—
149C2×Dic15—21no—
150C2×Dic15—21no—
151C2×Dic15—22no—
152C2×Dic15—22no—
153C2×Dic15—22no—
154C2×Dic15C22×C412yes—
155C4×D15—28no—
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×Dic3C2×D412yes—
164C2×C5×Dic3C2×D412yes—
165C2×C3×Dic5—21no—
166C2×C3×Dic5—21no—
167C2×C3×Dic5—21no—
168C2×C3×Dic5—21no—
169C2×C3×Dic5—21no—
170C2×C3×Dic5—21no—
171C3×D20—28no—
172#1202—48no—
173D5×Dic3—48no—
174C22×C3×D4—51no—
175C22×#67—52no—
176C23×Dic3—52no—
177C22×C4×D3—51no—
178C2×C3×C4⋊C4C2×D511yescharacteristic
179C2×C3×C22⋊C4—52no—
180C2×C3×C22⋊C4—52no—
181C2×C3×C42—51no—
182C2×#217—52no—
183C2×#217—52no—
184#938—58no—
185D4×Dic3—104no—
186C2×(C2×D3)⋊C4—51no—
187C2×(C2×D3)⋊C4—51no—
188C2×(C2×D3)⋊C4—52no—
189C2×C4⋊Dic3—51no—
190C2×C4⋊Dic3C2×D511yescharacteristic
191C2×Dic3⋊C4—51no—
192C2×Dic3⋊C4—51no—
193C2×Dic3⋊C4—52no—
194C2×C4×Dic3C2×D512yes—
195C2×C4×Dic3C2×D511yescharacteristic
196#872—104no—
197C3⋊#95—52no—
198C3⋊#95—52no—
199C3⋊#95C2×D512yes—
200C23×D5—31no—
201C23×D5C22×D312yes—
202C22×C4×C5—31no—
203C22×C4×C5—32no—
204C22×C4×C5—31no—
205C22×C4×C5C22×D312yes—
206C22×C4×C5C22×D311yescharacteristic
207C22×Dic5—31no—
208C22×Dic5—32no—
209C22×Dic5C22×D311yescharacteristic
210C22×Dic5C22×D311yescharacteristic
211C2×D20—22no—
212C2×D20—22no—
213C2×D20—38no—
214C2×D20—122no—
215C2×D20—122no—
216C2×D20C4×D314yes—
217C2×C4×D5—34no—
218C2×C4×D5—62no—
219C2×C4×D5—62no—
220C2×C4×D5—64no—
221C2×C4×D5—122no—
222C2×C4×D5—122no—
223C5×C4⋊C4—22no—
224C5×C4⋊C4—34no—
225C42×C5—34no—
226C42×C5—122no—
227#685—38no—
228#685—42no—
229#685—42no—
230#685—62no—
231#685—62no—
232#685—64no—
233#685—122no—
234C4⋊Dic5—62no—
235C4⋊Dic5—122no—
236#683—62no—
237#683—62no—
238C4×Dic5—22no—
239C2×#121—151no—
240C2×C4×D4—151no—
241C2×#117—151no—
242#390—51no—
243#390—152no—
244#389—151no—
245#388—152no—
246#386—152no—
247#385—152no—
248#380—151no—
249C4×C4⋊C4—31no—
250C22×C3×C5C23×C411yescharacteristic
251C22×C3×C5C22×D411yescharacteristic
252C22×C3×C5C2×#4111yescharacteristic
253C22×C3×C5H(5,𝔽2)11yescharacteristic
254C22×C3×C5H(5,𝔽2)11yescharacteristic
255C22×C3×C5H(5,𝔽2)11yescharacteristic
256C22×C3×C52−1+411yescharacteristic
257C2×D15—22no—
258C2×D15—22no—
259C2×D15—22no—
260C2×D15—41no—
261C2×D15—41no—
262C2×D15—41no—
263C2×D15—41no—
264C2×C3×D5—24no—
265C2×C3×D5—24no—
266C2×C3×D5—24no—
267C2×C3×D5—42no—
268C2×C3×D5—42no—
269C2×C3×D5—42no—
270C2×C3×D5—42no—
271C3×C4×C5—42no—
272C3×C4×C5C4×D414yes—
273Dic15—22no—
274Dic15—42no—
275C3×Dic5—41no—
276C3×Dic5—41no—
277C5×Dic3—24no—
278C5×Dic3—22no—
279C5×Dic3—22no—
280C24×C3—52no—
281C23×D3—51no—
282C2×C3×D4—54no—
283C2×C3×D4—102no—
284C2×C3×D4—102no—
285C22×C3×C4—51no—
286C22×C3×C4—51no—
287C22×C3×C4C22×D512yes—
288C22×C3×C4C22×D511yescharacteristic
289C2×#67—58no—
290C2×#67—202no—
291C2×#67—202no—
292C22×Dic3—51no—
293C22×Dic3—52no—
294C22×Dic3—104no—
295C22×Dic3—202no—
296C22×Dic3—202no—
297C22×Dic3C22×D512yes—
298C22×Dic3C22×D511yescharacteristic
299C22×Dic3C22×D511yescharacteristic
300C2×C4×D3—54no—
301C2×C4×D3—102no—
302C2×C4×D3—102no—
303C3×C4⋊C4—22no—
304C3×C22⋊C4—202no—
305C3×C22⋊C4—202no—
306C3×C42—102no—
307#217—58no—
308#217—104no—
309(C2×D3)⋊C4—102no—
310(C2×D3)⋊C4—102no—
311(C2×D3)⋊C4—202no—
312C4⋊Dic3—102no—
313C4⋊Dic3C4×D514yescomplemented
314Dic3⋊C4—102no—
315Dic3⋊C4—102no—
316Dic3⋊C4—202no—
317C4×Dic3—42no—
318C4×Dic3D2014yescomplemented
319C23×C5C23×D311yescharacteristic
320C22×D5—22no—
321C22×D5—22no—
322C22×D5—22no—
323C22×D5—22no—
324C22×D5—22no—
325C22×D5—22no—
326C22×D5—32no—
327C22×D5—61no—
328C22×D5—61no—
329C22×D5—61no—
330C22×D5—61no—
331C22×D5—61no—
332C22×D5—61no—
333C22×D5C2×C4×D314yes—
334C2×C4×C5—22no—
335C2×C4×C5—22no—
336C2×C4×C5—22no—
337C2×C4×C5—34no—
338C2×C4×C5—32no—
339C2×C4×C5—32no—
340C2×C4×C5—61no—
341C2×C4×C5—62no—
342C2×C4×C5—61no—
343C2×C4×C5—62no—
344C2×C4×C5—61no—
345C2×C4×C5—61no—
346C2×C4×C5C2×C4×D312yes—
347C2×C4×C5D3×D412yes—
348C2×C4×C5#23712yes—
349C2×Dic5—21no—
350C2×Dic5—21no—
351C2×Dic5—21no—
352C2×Dic5—21no—
353C2×Dic5—21no—
354C2×Dic5—21no—
355C2×Dic5—32no—
356C2×Dic5—61no—
357C2×Dic5—61no—
358C2×Dic5—62no—
359C2×Dic5—62no—
360C2×Dic5—62no—
361D20—28no—
362D20—128no—
363C4×D5—68no—
364C4×D5—128no—
365C22×D4—51no—
366C22×D4—152no—
367C23×C4—152no—
368C23×C4—151no—
369#121—158no—
370C4×D4—304no—
371#117—304no—
372C2×C4⋊C4—31no—
373C2×C4⋊C4—151no—
374C2×C4⋊C4—151no—
375C2×C4⋊C4—151no—
376C2×C4⋊C4—152no—
377C2×C4⋊C4D3×D511yescharacteristic
378C2×C22⋊C4—52no—
379C2×C22⋊C4—52no—
380C2×C22⋊C4—152no—
381C2×C22⋊C4—151no—
382C2×C22⋊C4—151no—
383C2×C22⋊C4—152no—
384C2×C22⋊C4—152no—
385C2×C42—31no—
386C2×C42—32no—
387C2×C42—51no—
388#95—32no—
389#95—152no—
390#95—152no—
391C2×C3×C5C2×C4×D411yescharacteristic
392C2×C3×C5#51811yescharacteristic
393C2×C3×C5#52011yescharacteristic
394C2×C3×C5#53511yescharacteristic
395C2×C3×C5#53711yescharacteristic
396C2×C3×C5#54011yescharacteristic
397C2×C3×C5#54211yescharacteristic
398D15—42no—
399C3×D5—44no—
400C23×C3—54no—
401C23×C3—102no—
402C23×C3—102no—
403C23×C3—102no—
404C23×C3—102no—
405C23×C3—102no—
406C23×C3—102no—
407C23×C3C23×D511yescharacteristic
408C22×D3—52no—
409C22×D3—101no—
410C22×D3—101no—
411C22×D3—101no—
412C22×D3—101no—
413C22×D3—101no—
414C22×D3—101no—
415C3×D4—108no—
416C2×C3×C4—22no—
417C2×C3×C4—22no—
418C2×C3×C4—22no—
419C2×C3×C4—101no—
420C2×C3×C4—101no—
421C2×C3×C4—101no—
422C2×C3×C4—101no—
423C2×C3×C4—101no—
424C2×C3×C4—101no—
425C2×C3×C4C2×C4×D512yes—
426C2×C3×C4C2×D2012yes—
427C2×C3×C4#70912yes—
428#67—208no—
429C2×Dic3—21no—
430C2×Dic3—22no—
431C2×Dic3—21no—
432C2×Dic3—22no—
433C2×Dic3—21no—
434C2×Dic3—21no—
435C2×Dic3—52no—
436C2×Dic3—101no—
437C2×Dic3—101no—
438C2×Dic3—102no—
439C2×Dic3—102no—
440C2×Dic3—102no—
441C2×Dic3—208no—
442C2×Dic3C2×C4×D514yes—
443C2×Dic3C2×D2012yes—
444C2×Dic3C2×D2012yes—
445C4×D3—108no—
446C22×C5C22×C4×D311yescharacteristic
447C22×C5C2×D3×D411yescharacteristic
448C22×C5C2×#23711yescharacteristic
449C22×C5#100411yescharacteristic
450C22×C5#100411yescharacteristic
451C22×C5#100911yescharacteristic
452C22×C5#101011yescharacteristic
453C2×D5—24no—
454C2×D5—24no—
455C2×D5—24no—
456C2×D5—42no—
457C2×D5—42no—
458C2×D5—42no—
459C2×D5—42no—
460C2×D5—62no—
461C2×D5—62no—
462C2×D5—62no—
463C2×D5—121no—
464C2×D5—121no—
465C2×D5—121no—
466C2×D5—121no—
467C4×C5—42no—
468C4×C5—64no—
469C4×C5—62no—
470C4×C5—62no—
471C4×C5#67⋊C414yes—
472Dic5—41no—
473Dic5—41no—
474Dic5—62no—
475Dic5—122no—
476C24—52no—
477C24—151no—
478C2×D4—54no—
479C2×D4—102no—
480C2×D4—102no—
481C2×D4—158no—
482C2×D4—602no—
483C2×D4—602no—
484C22×C4—31no—
485C22×C4—32no—
486C22×C4—31no—
487C22×C4—51no—
488C22×C4—51no—
489C22×C4—151no—
490C22×C4—154no—
491C22×C4—152no—
492C22×C4—304no—
493C22×C4—302no—
494C22×C4—302no—
495C22×C4—602no—
496C22×C4—602no—
497C22×C4C2×D3×D512yes—
498C22×C4C2×D3×D511yescharacteristic
499C4⋊C4—22no—
500C4⋊C4—34no—
501C4⋊C4—302no—
502C4⋊C4—302no—
503C4⋊C4—302no—
504C4⋊C4—602no—
505C22⋊C4—158no—
506C22⋊C4—202no—
507C22⋊C4—202no—
508C22⋊C4—302no—
509C22⋊C4—302no—
510C22⋊C4—304no—
511C22⋊C4—602no—
512C42—34no—
513C42—102no—
514C42—122no—
515C3×C5#232411yescharacteristic, complemented, Hall
516C22×C3—104no—
517C22×C3—104no—
518C22×C3—104no—
519C22×C3—202no—
520C22×C3—202no—
521C22×C3—202no—
522C22×C3—202no—
523C22×C3C22×C4×D511yescharacteristic
524C22×C3C22×D2011yescharacteristic
525C22×C3C2×#70911yescharacteristic
526C22×C3#414411yescharacteristic
527C22×C3#414711yescharacteristic
528C22×C3#414911yescharacteristic
529C22×C3#414911yescharacteristic
530C2×D3—102no—
531C2×D3—102no—
532C2×D3—102no—
533C2×D3—201no—
534C2×D3—201no—
535C2×D3—201no—
536C2×D3—201no—
537C3×C4—42no—
538C3×C4—201no—
539C3×C4—201no—
540C3×C4C4×D2014yes—
541Dic3—22no—
542Dic3—22no—
543Dic3—24no—
544Dic3—102no—
545Dic3—202no—
546C2×C5C2×#67⋊C411yescharacteristic
547C2×C5#554511yescharacteristic
548C2×C5#554611yescharacteristic
549C2×C5#560711yescharacteristic
550C2×C5#560811yescharacteristic
551C2×C5#568211yescharacteristic
552C2×C5#571911yescharacteristic
553D5—44no—
554D5—122no—
555C23—54no—
556C23—102no—
557C23—102no—
558C23—102no—
559C23—102no—
560C23—102no—
561C23—102no—
562C23—152no—
563C23—301no—
564C23—301no—
565C23—301no—
566C23—301no—
567C23—301no—
568C23—301no—
569C23C22×D3×D511yescharacteristic
570D4—108no—
571D4—608no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—34no—
576C2×C4—32no—
577C2×C4—32no—
578C2×C4—61no—
579C2×C4—62no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—62no—
583C2×C4—61no—
584C2×C4—101no—
585C2×C4—101no—
586C2×C4—101no—
587C2×C4—101no—
588C2×C4—101no—
589C2×C4—101no—
590C2×C4—152no—
591C2×C4—301no—
592C2×C4—301no—
593C2×C4—308no—
594C2×C4—302no—
595C2×C4—302no—
596C2×C4—302no—
597C2×C4—608no—
598C2×C4C4×D3×D512yes—
599C2×C4D3×D2012yes—
600C2×C4#677912yes—
601C2×C3—204no—
602C2×C3C2×C4×D2011yescharacteristic
603C2×C3#6584411yescharacteristic
604C2×C3#6585911yescharacteristic
605C2×C3#6587911yescharacteristic
606C2×C3#6589611yescharacteristic
607C2×C3#6590811yescharacteristic
608C2×C3#6591611yescharacteristic
609D3—202no—
610C5#7643011yescharacteristic, complemented, Sylow
611C22—104no—
612C22—104no—
613C22—104no—
614C22—202no—
615C22—202no—
616C22—202no—
617C22—202no—
618C22—302no—
619C22—302no—
620C22—302no—
621C22—601no—
622C22—601no—
623C22—601no—
624C22—601no—
625C22C2×C4×D3×D511yescharacteristic
626C22C2×D3×D2011yescharacteristic
627C22#9202511yescharacteristic
628C22#9202611yescharacteristic
629C22#9202811yescharacteristic
630C22#9203511yescharacteristic
631C22C2×#677911yescharacteristic
632C4—42no—
633C4—64no—
634C4—62no—
635C4—62no—
636C4—201no—
637C4—201no—
638C4—302no—
639C4—602no—
640C4#9139514yes—
641C3#1060549411yescharacteristic, complemented, Sylow
642C2—204no—
643C2—602no—
644C2#1175340111yescharacteristic
645C2#1175344111yescharacteristic
646C2#1175384111yescharacteristic
647C2#1175388211yescharacteristic
648C2C2×#9139511yescharacteristic
649C2#1175491311yescharacteristic
650C2#1175521811yescharacteristic
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#2324
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
2. C2×Dic3×D20C2semidirect
3. C2×#91386C2semidirect
4. #11751088C2semidirect
5. C5×Dic3×C4⋊C4C2semidirect
6. C3×#65277C2semidirect
7. #11748912C2semidirect
8. #11748890C2semidirect
9. #11748872C2semidirect
10. #11748842C2semidirect
11. #11748705C2semidirect
33. #91465C4semidirect
44. #65277D3semidirect
59. C2×C4×D15C2×C4semidirect
70. C2×#1202C2×C4semidirect
77. C5×C4⋊Dic3C2×C4semidirect
79. C4×C5×Dic3D4semidirect
89. #6665C2×C4semidirect
95. Dic3×C4⋊C4D5semidirect
313. C4⋊Dic3C4×D5semidirect
318. C4×Dic3D20semidirect
515. C3×C5#2324semidirect
610. C5#76430semidirect
641. C3#10605494semidirect

Maximal subgroups

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

subgroupindex
C2×#914652
C2×Dic3×D202
C2×#913862
#117510882
C5×Dic3×C4⋊C42
C3×#652772
#117489122
#117488902
#117488722
#117488422
#117487052
#106054943
#764305