Finite Group Databasebeta

← #14327085

Subgroups — #14327085

Order 1920.

1552 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91782C211yesmaximal, characteristic
2C2×#91437C212yesmaximal, complemented
3C2×#91428C211yesmaximal, characteristic
4C2×#91418C211yesmaximal, characteristic
5C5×#4984C211yesmaximal, characteristic, complemented
6C3×#65274C211yesmaximal, characteristic, complemented
7#11748902C212yesmaximal, complemented
8#11748898C212yesmaximal, complemented
9#11748866C211yesmaximal, characteristic, complemented
10#11748863C212yesmaximal, complemented
11C4×#6669C211yesmaximal, characteristic, complemented
12#10607933—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×C5×C4⋊Dic3C2212yes—
23C2×C4×C5×Dic3C2211yescharacteristic
24C2×C3×C4⋊Dic5C2211yescharacteristic
25C2×C3×C4⋊Dic5C2212yes—
26C2×C3×C4×Dic5C2211yescharacteristic
27C2×#6669C2212yes—
28C2×#6669C2211yescharacteristic
29C2×#6669C2212yes—
30C2×#6669C2211yescharacteristic
31C2×#6667C2212yes—
32C2×#6667C2212yes—
33#91437—28no—
34#91428—24no—
35#91418—24no—
36C5×C3⋊#95C2212yes—
37C3×#3963C2212yes—
38#90994C2211yescharacteristic
39#90994C2211yescharacteristic
40#76202—51nomaximal, Hall
41C2×#4039—31no—
42C2×#4039—31no—
43C2×D5×C4⋊C4—32no—
44C2×C4×D20—31no—
45C5×#389—31no—
46#65278—32no—
47#65274D311yescharacteristic, complemented
48#65265—31no—
49C4×#683—31no—
50#64957—32no—
51#64956—32no—
52C23×D15C2312yes—
53C22×C3×C4×C5C2312yes—
54C22×C3×C4×C5C2311yescharacteristic
55C22×Dic15C2311yescharacteristic
56C22×Dic15C2311yescharacteristic
57C2×D60C2×C414yes—
58C2×D60D414yes—
59C2×D60Q814yes—
60C2×C4×D15—24no—
61C2×C4×D15—24no—
62C2×C4×D15C2×C418yescomplemented
63C22×C5×Dic3C2311yescharacteristic
64C22×C5×Dic3C2312yes—
65C22×C5×Dic3C2311yescharacteristic
66C22×C3×Dic5C2311yescharacteristic
67C22×C3×Dic5C2312yes—
68C22×C3×Dic5C2311yescharacteristic
69C2×#1200—24no—
70C2×#1200—24no—
71C2×#1200—24no—
72C2×#1200—24no—
73C2×#1200—42no—
74C2×#1200—42no—
75C3×C5×C4⋊C4C2×C414yescomplemented
76#6714C2×C418yescomplemented
77C4×Dic15C2×C414yescomplemented
78C5×C4⋊Dic3—22no—
79C5×C4⋊Dic3—24no—
80C4×C5×Dic3—22no—
81C3×C4⋊Dic5—22no—
82C3×C4⋊Dic5—24no—
83C3×C4×Dic5—22no—
84#6669—24no—
85#6669—22no—
86#6669—42no—
87#6669D414yescomplemented
88#6667—24no—
89#6667—24no—
90C2×#893—51no—
91C2×#893—51no—
92C2×D3×C4⋊C4—52no—
93C2×C4×D12—51no—
94C3×#389—51no—
95#4989—52no—
96#4984D511yescharacteristic, complemented
97#4975—51no—
98C4×Dic3⋊C4—51no—
99#4668—52no—
100#4667—52no—
101C22×D20—31no—
102C22×C4×D5—32no—
103C22×C4×D5—32no—
104C22×C4×D5—32no—
105C2×C5×C4⋊C4—32no—
106C2×C5×C4⋊C4—31no—
107C2×C5×C4⋊C4C2×D311yescharacteristic
108C2×C42×C5—31no—
109C2×#685—32no—
110C2×#685—32no—
111C2×#685—32no—
112C2×C4⋊Dic5C2×D311yescharacteristic
113C2×C4⋊Dic5C2×D312yes—
114C2×#683—31no—
115C2×#683—32no—
116C2×#683—31no—
117C2×#683—32no—
118C2×C4×Dic5—31no—
119C2×C4×Dic5C2×D311yescharacteristic
120#4039—38no—
121#4039—64no—
122D5×C4⋊C4—68no—
123C4×D20—64no—
124C5×#95—32no—
125#3963—31no—
126#3963—31no—
127#3963C2×D312yes—
128D4×C4⋊C4—151noSylow
129C23×C3×C5C2411yescharacteristic
130C22×D15—22no—
131C22×D15—22no—
132C22×D15—22no—
133C22×D15—22no—
134C22×D15C22×C414yes—
135C22×D15C2×D414yes—
136C22×D15C2×Q814yes—
137C2×C3×C4×C5—22no—
138C2×C3×C4×C5—22no—
139C2×C3×C4×C5C22×C414yes—
140C2×C3×C4×C5C22×C412yes—
141C2×C3×C4×C5C2×D412yes—
142C2×C3×C4×C5C2×Q812yes—
143C2×Dic15—21no—
144C2×Dic15—21no—
145C2×Dic15C22×C412yes—
146C2×Dic15C22×C412yes—
147C2×Dic15C2×D412yes—
148C2×Dic15#4112yes—
149D60C4⋊C4116yescomplemented
150C4×D15—216no—
151C2×C5×Dic3—21no—
152C2×C5×Dic3—21no—
153C2×C5×Dic3—21no—
154C2×C5×Dic3—22no—
155C2×C5×Dic3—22no—
156C2×C5×Dic3—21no—
157C2×C5×Dic3—22no—
158C2×C5×Dic3—21no—
159C2×C5×Dic3C2×D412yes—
160C2×C3×Dic5—21no—
161C2×C3×Dic5—21no—
162C2×C3×Dic5—21no—
163C2×C3×Dic5—22no—
164C2×C3×Dic5—22no—
165C2×C3×Dic5—21no—
166C2×C3×Dic5—22no—
167C2×C3×Dic5—21no—
168C2×C3×Dic5C2×D412yes—
169#1200—48no—
170#1200—48no—
171C22×D12—51no—
172C22×C4×D3—52no—
173C22×C4×D3—52no—
174C22×C4×D3—52no—
175C2×C3×C4⋊C4—51no—
176C2×C3×C4⋊C4—52no—
177C2×C3×C4⋊C4C2×D511yescharacteristic
178C2×C3×C42—51no—
179C2×(C2×D3)⋊C4—52no—
180C2×(C2×D3)⋊C4—52no—
181C2×(C2×D3)⋊C4—52no—
182C2×C4⋊Dic3C2×D512yes—
183C2×C4⋊Dic3C2×D511yescharacteristic
184C2×Dic3⋊C4—51no—
185C2×Dic3⋊C4—51no—
186C2×Dic3⋊C4—52no—
187C2×Dic3⋊C4—52no—
188C2×C4×Dic3—51no—
189C2×C4×Dic3C2×D511yescharacteristic
190#893—58no—
191#893—104no—
192D3×C4⋊C4—108no—
193C4×D12—104no—
194C3×#95—52no—
195C3⋊#95—51no—
196C3⋊#95—51no—
197C3⋊#95C2×D512yes—
198C23×D5—32no—
199C22×C4×C5—31no—
200C22×C4×C5—31no—
201C22×C4×C5—32no—
202C22×C4×C5C22×D312yes—
203C22×C4×C5C22×D311yescharacteristic
204C22×Dic5—31no—
205C22×Dic5—31no—
206C22×Dic5C22×D312yes—
207C22×Dic5C22×D311yescharacteristic
208C22×Dic5C22×D311yescharacteristic
209C2×D20—34no—
210C2×D20—34no—
211C2×D20—34no—
212C2×C4×D5—38no—
213C2×C4×D5—64no—
214C2×C4×D5—64no—
215C2×C4×D5—64no—
216C2×C4×D5—64no—
217C2×C4×D5—64no—
218C2×C4×D5—64no—
219C2×C4×D5—122no—
220C2×C4×D5—122no—
221C5×C4⋊C4—64no—
222C5×C4⋊C4—62no—
223C5×C4⋊C4C4×D314yescomplemented
224C42×C5—62no—
225#685—38no—
226#685—64no—
227#685—64no—
228C4⋊Dic5—22no—
229C4⋊Dic5—24no—
230#683—34no—
231#683—64no—
232#683—62no—
233#683—122no—
234C4×Dic5—22no—
235C4×Dic5—34no—
236C2×C4×D4—151no—
237C2×C4×D4—151no—
238C2×C4×D4—151no—
239C22×C4⋊C4—152no—
240#389—31no—
241#389—51no—
242#385—152no—
243#385—152no—
244#384—151no—
245#380—152no—
246C4×C4⋊C4—151no—
247C22×C3×C5C23×C411yescharacteristic
248C22×C3×C5C22×D411yescharacteristic
249C22×C3×C5C22×D411yescharacteristic
250C22×C3×C5C22×Q811yescharacteristic
251C22×C3×C5C2×#4111yescharacteristic
252C22×C3×C5H(5,𝔽2)11yescharacteristic
253C22×C3×C52−1+411yescharacteristic
254C2×D15—24no—
255C2×D15—24no—
256C2×D15—24no—
257C2×D15—24no—
258C2×D15—24no—
259C2×D15—24no—
260C2×D15C2×C4⋊C418yescomplemented
261C3×C4×C5—24no—
262C3×C4×C5C2×C4⋊C414yescomplemented
263Dic15—22no—
264Dic15C4×D414yescomplemented
265C3×Dic5—22no—
266C3×Dic5—42no—
267C3×Dic5—41no—
268C5×Dic3—22no—
269C5×Dic3—42no—
270C5×Dic3—41no—
271C23×D3—52no—
272C22×C3×C4—51no—
273C22×C3×C4—51no—
274C22×C3×C4—52no—
275C22×C3×C4C22×D512yes—
276C22×C3×C4C22×D511yescharacteristic
277C22×Dic3—51no—
278C22×Dic3—51no—
279C22×Dic3C22×D511yescharacteristic
280C22×Dic3C22×D512yes—
281C22×Dic3C22×D511yescharacteristic
282C2×D12—54no—
283C2×D12—54no—
284C2×D12—54no—
285C2×C4×D3—58no—
286C2×C4×D3—104no—
287C2×C4×D3—104no—
288C2×C4×D3—104no—
289C2×C4×D3—104no—
290C2×C4×D3—104no—
291C2×C4×D3—104no—
292C2×C4×D3—202no—
293C2×C4×D3—202no—
294C3×C4⋊C4—104no—
295C3×C4⋊C4—102no—
296C3×C4⋊C4C4×D514yescomplemented
297C3×C42—102no—
298(C2×D3)⋊C4—58no—
299(C2×D3)⋊C4—104no—
300(C2×D3)⋊C4—104no—
301C4⋊Dic3—22no—
302C4⋊Dic3—24no—
303Dic3⋊C4—54no—
304Dic3⋊C4—104no—
305Dic3⋊C4—102no—
306Dic3⋊C4—202no—
307C4×Dic3—22no—
308C4×Dic3—54no—
309C23×C5C23×D311yescharacteristic
310C22×D5—34no—
311C22×D5—34no—
312C22×D5—34no—
313C22×D5—62no—
314C22×D5—62no—
315C22×D5—62no—
316C22×D5—62no—
317C2×C4×C5—22no—
318C2×C4×C5—22no—
319C2×C4×C5—32no—
320C2×C4×C5—61no—
321C2×C4×C5—61no—
322C2×C4×C5—62no—
323C2×C4×C5—61no—
324C2×C4×C5—62no—
325C2×C4×C5—61no—
326C2×C4×C5—61no—
327C2×C4×C5—62no—
328C2×C4×C5C2×C4×D314yes—
329C2×C4×C5C2×C4×D312yes—
330C2×C4×C5D3×D412yes—
331C2×C4×C5D3×Q812yes—
332C2×Dic5—21no—
333C2×Dic5—21no—
334C2×Dic5—21no—
335C2×Dic5—22no—
336C2×Dic5—22no—
337C2×Dic5—21no—
338C2×Dic5—21no—
339C2×Dic5—22no—
340C2×Dic5—32no—
341C2×Dic5—32no—
342C2×Dic5—32no—
343C2×Dic5—32no—
344C2×Dic5—61no—
345C2×Dic5—61no—
346C2×Dic5D3×D412yes—
347D20—316no—
348C4×D5—616no—
349C4×D5—128no—
350C4×D5—128no—
351C22×D4—151no—
352C23×C4—152no—
353C23×C4—152no—
354C23×C4—152no—
355C4×D4—158no—
356C4×D4—304no—
357C4×D4—304no—
358C2×C4⋊C4—32no—
359C2×C4⋊C4—31no—
360C2×C4⋊C4—52no—
361C2×C4⋊C4—51no—
362C2×C4⋊C4—151no—
363C2×C4⋊C4—152no—
364C2×C4⋊C4—151no—
365C2×C4⋊C4—152no—
366C2×C4⋊C4—308no—
367C2×C4⋊C4D3×D511yescharacteristic
368C2×C22⋊C4—152no—
369C2×C22⋊C4—152no—
370C2×C22⋊C4—152no—
371C2×C42—31no—
372C2×C42—51no—
373C2×C42—151no—
374#95—32no—
375#95—52no—
376#95—151no—
377#95—151no—
378C2×C3×C5C22×C4⋊C411yescharacteristic
379C2×C3×C5C2×C4×D411yescharacteristic
380C2×C3×C5#52011yescharacteristic
381C2×C3×C5D4211yescharacteristic
382C2×C3×C5#54711yescharacteristic
383C2×C3×C5D4×Q811yescharacteristic
384C2×C3×C5#55411yescharacteristic
385D15—28no—
386C23×C3C23×D511yescharacteristic
387C22×D3—54no—
388C22×D3—54no—
389C22×D3—54no—
390C22×D3—102no—
391C22×D3—102no—
392C22×D3—102no—
393C22×D3—102no—
394C2×C3×C4—22no—
395C2×C3×C4—22no—
396C2×C3×C4—52no—
397C2×C3×C4—101no—
398C2×C3×C4—101no—
399C2×C3×C4—102no—
400C2×C3×C4—101no—
401C2×C3×C4—102no—
402C2×C3×C4—101no—
403C2×C3×C4—101no—
404C2×C3×C4—102no—
405C2×C3×C4C2×C4×D514yes—
406C2×C3×C4C2×C4×D512yes—
407C2×C3×C4D4×D512yes—
408C2×C3×C4Q8×D512yes—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—22no—
413C2×Dic3—22no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—22no—
417C2×Dic3—52no—
418C2×Dic3—52no—
419C2×Dic3—52no—
420C2×Dic3—52no—
421C2×Dic3—101no—
422C2×Dic3—101no—
423C2×Dic3D4×D512yes—
424D12—516no—
425C4×D3—1016no—
426C4×D3—208no—
427C4×D3—208no—
428C22×C5C22×C4×D311yescharacteristic
429C22×C5C2×D3×D411yescharacteristic
430C22×C5C2×D3×D411yescharacteristic
431C22×C5C2×D3×Q811yescharacteristic
432C22×C5#100711yescharacteristic
433C22×C5D3×#4111yescharacteristic
434C22×C5#100911yescharacteristic
435C2×D5—38no—
436C2×D5—64no—
437C2×D5—64no—
438C2×D5—64no—
439C2×D5—64no—
440C2×D5—64no—
441C2×D5—64no—
442C4×C5—24no—
443C4×C5—62no—
444C4×C5—122no—
445C4×C5—121no—
446C4×C5D3×C4⋊C414yescomplemented
447Dic5—22no—
448Dic5—34no—
449Dic5—42no—
450Dic5—41no—
451Dic5—62no—
452C24—152no—
453C2×D4—154no—
454C2×D4—154no—
455C2×D4—154no—
456C22×C4—31no—
457C22×C4—31no—
458C22×C4—32no—
459C22×C4—51no—
460C22×C4—51no—
461C22×C4—52no—
462C22×C4—151no—
463C22×C4—151no—
464C22×C4—158no—
465C22×C4—304no—
466C22×C4—304no—
467C22×C4—304no—
468C22×C4—304no—
469C22×C4—304no—
470C22×C4—304no—
471C22×C4—602no—
472C22×C4—602no—
473C22×C4C2×D3×D512yes—
474C22×C4C2×D3×D511yescharacteristic
475C4⋊C4—64no—
476C4⋊C4—62no—
477C4⋊C4—104no—
478C4⋊C4—102no—
479C4⋊C4—154no—
480C4⋊C4—304no—
481C4⋊C4—302no—
482C4⋊C4—602no—
483C4⋊C4#120014yescomplemented
484C22⋊C4—158no—
485C22⋊C4—304no—
486C22⋊C4—304no—
487C42—62no—
488C42—102no—
489C42—154no—
490C3×C5D4×C4⋊C411yescharacteristic, complemented, Hall
491C22×C3C22×C4×D511yescharacteristic
492C22×C3C2×D4×D511yescharacteristic
493C22×C3C2×D4×D511yescharacteristic
494C22×C3C2×Q8×D511yescharacteristic
495C22×C3#414711yescharacteristic
496C22×C3D5×#4111yescharacteristic
497C22×C3#414911yescharacteristic
498C2×D3—58no—
499C2×D3—104no—
500C2×D3—104no—
501C2×D3—104no—
502C2×D3—104no—
503C2×D3—104no—
504C2×D3—104no—
505C3×C4—24no—
506C3×C4—102no—
507C3×C4—202no—
508C3×C4—201no—
509C3×C4D5×C4⋊C414yescomplemented
510Dic3—22no—
511Dic3—42no—
512Dic3—41no—
513Dic3—54no—
514Dic3—102no—
515C2×C5C2×D3×C4⋊C411yescharacteristic
516C2×C5C4×D3×D411yescharacteristic
517C2×C5#557411yescharacteristic
518C2×C5#560911yescharacteristic
519C2×C5#563111yescharacteristic
520C2×C5#569011yescharacteristic
521C2×C5#572711yescharacteristic
522D5—68no—
523C23—154no—
524C23—154no—
525C23—154no—
526C23—302no—
527C23—302no—
528C23—302no—
529C23—302no—
530C23C22×D3×D511yescharacteristic
531D4—1516no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—32no—
535C2×C4—52no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—62no—
540C2×C4—62no—
541C2×C4—61no—
542C2×C4—61no—
543C2×C4—62no—
544C2×C4—101no—
545C2×C4—101no—
546C2×C4—101no—
547C2×C4—102no—
548C2×C4—102no—
549C2×C4—101no—
550C2×C4—101no—
551C2×C4—102no—
552C2×C4—152no—
553C2×C4—152no—
554C2×C4—152no—
555C2×C4—152no—
556C2×C4—3016no—
557C2×C4—301no—
558C2×C4—301no—
559C2×C4—608no—
560C2×C4—608no—
561C2×C4D15⋊Q812yes—
562C2×C4#677712yes—
563C2×C4C2×#120014yes—
564C2×C4C2×#120012yes—
565C2×C3C2×D5×C4⋊C411yescharacteristic
566C2×C3C4×D4×D511yescharacteristic
567C2×C3#6590811yescharacteristic
568C2×C3#6594311yescharacteristic
569C2×C3#6596511yescharacteristic
570C2×C3#6602411yescharacteristic
571C2×C3#6606111yescharacteristic
572D3—108no—
573C5#7620211yescharacteristic, complemented, Sylow
574C22—158no—
575C22—304no—
576C22—304no—
577C22—304no—
578C22—304no—
579C22—304no—
580C22—304no—
581C22C2×D15⋊Q811yescharacteristic
582C22C2×#677711yescharacteristic
583C22D3×D4×D511yescharacteristic
584C22#9202411yescharacteristic
585C22#9202911yescharacteristic
586C22#9203511yescharacteristic
587C22C22×#120011yescharacteristic
588C4—24no—
589C4—62no—
590C4—102no—
591C4—122no—
592C4—121no—
593C4—154no—
594C4—202no—
595C4—201no—
596C4—302no—
597C4#9143714yescomplemented
598C3#1060793311yescharacteristic, complemented, Sylow
599C2—308no—
600C2#1175343411yescharacteristic
601C2#1175362511yescharacteristic
602C2#1175370411yescharacteristic
603C2#1175385711yescharacteristic
604C2C2×#9143711yescharacteristic
605C2D4×#120011yescharacteristic
606C2#1175514911yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C5×C4⋊C4C22
socle, cosocleC23×C3×C5C24
prime radicalC22×D60C22
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5D4×C4⋊C4
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×#91437C2semidirect
5. C5×#4984C2semidirect
6. C3×#65274C2semidirect
7. #11748902C2semidirect
8. #11748898C2semidirect
9. #11748866C2semidirect
10. #11748863C2semidirect
11. C4×#6669C2semidirect
20. #91782C4semidirect
47. #65274D3semidirect
62. C2×C4×D15C2×C4semidirect
75. C3×C5×C4⋊C4C2×C4semidirect
76. #6714C2×C4semidirect
77. C4×Dic15C2×C4semidirect
87. #6669D4semidirect
96. #4984D5semidirect
149. D60C4⋊C4semidirect
223. C5×C4⋊C4C4×D3semidirect
260. C2×D15C2×C4⋊C4semidirect
262. C3×C4×C5C2×C4⋊C4semidirect
264. Dic15C4×D4semidirect
296. C3×C4⋊C4C4×D5semidirect
446. C4×C5D3×C4⋊C4semidirect
483. C4⋊C4#1200semidirect
490. C3×C5D4×C4⋊C4semidirect
509. C3×C4D5×C4⋊C4semidirect
573. C5#76202semidirect
597. C4#91437semidirect
598. C3#10607933semidirect

Maximal subgroups

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

subgroupindex
C2×#917822
C2×#914372
C2×#914282
C2×#914182
C5×#49842
C3×#652742
#117489022
#117488982
#117488662
#117488632
C4×#66692
#106079333
#762025