Finite Group Databasebeta

← #14425692

Subgroups — #14425692

Order 1920.

1532 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
1C5×#5964C211yesmaximal, characteristic, complemented
2#11756587C211yesmaximal, characteristic, complemented
3#11754592C212yesmaximal, complemented
4C3×#65529C211yesmaximal, characteristic, complemented
5#11749184C214yesmaximal, complemented
6#11749178C214yesmaximal, complemented
7#11748625C212yesmaximal, complemented
8#10615856—31nomaximal, Hall
9C2×C3×C5×#41C2211yescharacteristic
10C22×C4×D15C2211yescharacteristic, complemented
11C5×#1010—22no—
12C5×#1009—22no—
13C5×D3×#41—24no—
14C5×#1007—21no—
15C5×#1004—21no—
16C5×#1004C2214yes—
17C2×C5×#235—22no—
18C2×C5×#235C2212yes—
19#91835—21no—
20#91827—21no—
21#91826C2212yescomplemented
22#91824C2212yescomplemented
23C4×#1220—22no—
24#91815C2211yescharacteristic, complemented
25C3×#4072C2212yescomplemented
26#91573C2214yescomplemented
27#91543C2214yescomplemented
28#91466—22no—
29C4×#1203—24no—
30#91383—22no—
31C3×#3966C2214yescomplemented
32#90996C2218yescomplemented
33#78690—51nomaximal, Hall
34C5×#585—31no—
35#66162—32no—
36#66162—31no—
37#65533—34no—
38#65532—34no—
39#65529—32no—
40#65529D311yescharacteristic, complemented
41C23×D15C2311yescharacteristic
42C3×C5×#41—22no—
43C3×C5×#41—22no—
44C2×C3×C5×Q8D411yescharacteristic, complemented
45C2×C3×C5×D4D411yescharacteristic, complemented
46C2×C3×C5×D4C2312yes—
47C22×C3×C4×C5D412yescomplemented
48C22×C3×C4×C5C2311yescharacteristic
49C2×#1220—22no—
50C22×Dic15C2311yescharacteristic
51C2×C4×D15—22no—
52C2×C4×D15—22no—
53C2×C4×D15—21no—
54C2×C4×D15—21no—
55C2×C5×#67—24no—
56C2×C5×#67C2314yes—
57C5×#239—24no—
58C5×D3×Q8—24no—
59C5×#237—24no—
60C5×#237—28no—
61C5×D3×D4—24no—
62C5×D3×D4—28no—
63C5×#235—24no—
64C5×#235—24no—
65C5×#235—22no—
66C5×#235—24no—
67C5×#235—22no—
68C5×#235—22no—
69C5×#235—22no—
70C2×C5×D12—22no—
71C2×C5×D12D412yescomplemented
72C2×C4×C5×D3—24no—
73C2×C4×C5×D3D414yescomplemented
74C2×C5×Dic6—22no—
75C2×C5×Dic6D412yescomplemented
76C2×#1203—24no—
77#6719—22no—
78#6719C2312yes—
79#6714—22no—
80C4⋊Dic15—21no—
81#6712—22no—
82C4×Dic15—21no—
83C3×C22⋊Dic5C2314yes—
84C3×C4⋊Dic5—22no—
85C3×C4×Dic5—22no—
86#6668—24no—
87#6667—24no—
88(C2×D3)⋊Dic5—24no—
89#5964D511yescharacteristic, complemented
90#5828—51no—
91#5524—52no—
92C3×#410—51no—
93#5006—52no—
94#4742—54no—
95#4741—54no—
96C5×2−1+4—62no—
97C5×2−1+4—61no—
98C5×H(5,𝔽2)—34no—
99C5×H(5,𝔽2)—62no—
100C5×H(5,𝔽2)—61no—
101C2×C5×#41—32no—
102C2×C5×#41—64no—
103C2×C5×#41—62no—
104C2×C5×#41C2×D311yescharacteristic
105C22×C4×D5—31no—
106#4092—62no—
107#4092—61no—
108#4084—62no—
109#4084—61no—
110#4083—32no—
111#4083—34no—
112#4081—32no—
113#4081—34no—
114C4×C22⋊D5—64no—
115C4×C22⋊D5—62no—
116#4072—31no—
117#4072C2×D312yescomplemented
118#3966—38no—
119#3966C2×D314yescomplemented
120#3025—151noSylow
121C23×C3×C5C2×D412yes—
122C23×C3×C5C2411yescharacteristic
123C22×D15—22no—
124C22×D15—21no—
125C22×D15—42no—
126C22×C5×D3—24no—
127C22×C5×D3C2×D414yes—
128C3×C5×Q8—22no—
129C3×C5×D4—24no—
130C3×C5×D4—22no—
131C2×C3×C4×C5—22no—
132C2×C3×C4×C5—22no—
133C2×C3×C4×C5C2×D412yes—
134C2×C3×C4×C5C2×D411yescharacteristic
135C2×C3×C4×C5C2×D411yescharacteristic
136C2×C3×C4×C5C2×D412yes—
137C2×C3×C4×C5C2×D412yes—
138#1220—44no—
139C2×Dic15—22no—
140C2×Dic15—22no—
141C2×Dic15—21no—
142C4×D15—41no—
143C4×D15—42no—
144C4×D15—41no—
145C5×#67—28no—
146C5×#67—24no—
147C5×#67—24no—
148C2×C5×Dic3—24no—
149C2×C5×Dic3C2×D414yes—
150C5×D12—24no—
151C5×D12—22no—
152C5×D12—22no—
153C4×C5×D3—28no—
154C4×C5×D3—24no—
155C4×C5×D3—24no—
156C5×Dic6—24no—
157C5×Dic6—22no—
158C5×Dic6—22no—
159C2×C3×Dic5—24no—
160#1203—48no—
161C2×C3×#41C2×D511yescharacteristic
162#1010—22no—
163#1009—22no—
164D3×#41—24no—
165#1007—21no—
166#1004—21no—
167#1004C2×D514yes—
168C2×#235—22no—
169C2×#235C2×D512yes—
170C22×C4×D3—51no—
171C3×#117—52no—
172#946—101no—
173#938—101no—
174#937—52no—
175#935—52no—
176C4×#67—102no—
177#926—51no—
178#897—102no—
179#895—102no—
180#886—54no—
181#882—54no—
182C4×D12—104no—
183C3×#99—54no—
184#806—58no—
185C23×D5—31no—
186C5×#41—22no—
187C5×#41—22no—
188C5×#41—64no—
189C5×#41—64no—
190C5×#41—68no—
191C5×#41—64no—
192C5×#41—64no—
193C5×#41—62no—
194C5×#41—62no—
195C5×#41—64no—
196C5×#41—62no—
197C5×#41—62no—
198C2×C5×Q8—32no—
199C2×C5×Q8—62no—
200C2×C5×Q8—64no—
201C2×C5×Q8D1211yescharacteristic, complemented
202C2×C5×D4—34no—
203C2×C5×D4—32no—
204C2×C5×D4—62no—
205C2×C5×D4—68no—
206C2×C5×D4—64no—
207C2×C5×D4—64no—
208C2×C5×D4D1211yescharacteristic, complemented
209C2×C5×D4C22×D312yes—
210C22×C4×C5—34no—
211C22×C4×C5—64no—
212C22×C4×C5D1212yescomplemented
213C22×C4×C5C22×D311yescharacteristic
214C2×C22⋊D5—64no—
215C2×C22⋊D5—62no—
216C22×Dic5—31no—
217C2×C4×D5—61no—
218C2×C4×D5—61no—
219C2×C4×D5—62no—
220C2×C4×D5—62no—
221C22⋊Dic5—32no—
222C22⋊Dic5—64no—
223C22⋊Dic5—62no—
224C22⋊Dic5C22×D314yes—
225#685—64no—
226#685—62no—
227C4⋊Dic5—22no—
228C4⋊Dic5—61no—
229#683—64no—
230#683—62no—
231C4×Dic5—22no—
232C4×Dic5—61no—
233#585—31no—
234#525—152no—
235#525—151no—
236#458—154no—
237B(4,𝔽2)—154no—
238#410—51no—
239#410—152no—
240C22×C3×C5—22no—
241C22×C3×C5—41no—
242C22×C3×C5—42no—
243C22×C3×C5C22×D411yescharacteristic
244C22×C3×C5C22×D412yes—
245C2×D15—24no—
246C2×D15—22no—
247C2×D15—42no—
248C2×D15—41no—
249C2×D15—41no—
250C2×C5×D3—24no—
251C2×C5×D3—44no—
252C2×C5×D3—84no—
253C3×C4×C5—22no—
254C3×C4×C5#12011yescharacteristic
255C3×C4×C5#12012yes—
256C3×C4×C5#12011yescharacteristic
257Dic15—42no—
258Dic15—41no—
259C3×Dic5—44no—
260C5×Dic3—24no—
261C23×D3—51no—
262C3×#41—22no—
263C3×#41—22no—
264C2×C3×Q8C22⋊D511yescharacteristic, complemented
265C2×C3×D4C22⋊D511yescharacteristic, complemented
266C2×C3×D4C22×D512yes—
267C22×C3×C4C22⋊D512yescomplemented
268C22×C3×C4C22×D511yescharacteristic
269C2×#67—24no—
270C2×#67—102no—
271C2×#67C22×D514yes—
272C22×Dic3—51no—
273#239—24no—
274D3×Q8—24no—
275#237—24no—
276#237—28no—
277D3×D4—24no—
278D3×D4—28no—
279#235—24no—
280#235—24no—
281#235—22no—
282#235—24no—
283#235—22no—
284#235—22no—
285#235—22no—
286C2×D12—22no—
287C2×D12—104no—
288C2×D12C22⋊D512yescomplemented
289C2×C4×D3—24no—
290C2×C4×D3—102no—
291C2×C4×D3—102no—
292C2×C4×D3—101no—
293C2×C4×D3—101no—
294C2×C4×D3C22⋊D514yescomplemented
295C2×Dic6—22no—
296C2×Dic6C22⋊D512yescomplemented
297C3×C4⋊C4—102no—
298C3×C22⋊C4—54no—
299C3×C42—102no—
300#217—52no—
301#217—102no—
302(C2×D3)⋊C4—104no—
303(C2×D3)⋊C4—102no—
304(C2×D3)⋊C4—104no—
305C4⋊Dic3—101no—
306C4⋊Dic3—104no—
307Dic3⋊C4—102no—
308C4×Dic3—101no—
309C23×C5—34no—
310C23×C5—64no—
311C23×C5C2×D1212yes—
312C23×C5C23×D311yescharacteristic
313C22×D5—61no—
314C22×D5—62no—
315C22×D5—122no—
316C5×Q8—22no—
317C5×Q8—64no—
318C5×Q8—62no—
319C5×Q8—62no—
320C5×D4—24no—
321C5×D4—22no—
322C5×D4—64no—
323C5×D4—68no—
324C5×D4—64no—
325C5×D4—64no—
326C5×D4—62no—
327C5×D4—62no—
328C2×C4×C5—22no—
329C2×C4×C5—22no—
330C2×C4×C5—34no—
331C2×C4×C5—68no—
332C2×C4×C5—64no—
333C2×C4×C5—64no—
334C2×C4×C5—64no—
335C2×C4×C5C2×D1212yes—
336C2×C4×C5C2×D1211yescharacteristic
337C2×C4×C5C2×D1211yescharacteristic
338C2×C4×C5D3×D412yes—
339C2×C4×C5D3×D412yes—
340C22⋊D5—128no—
341C22⋊D5—124no—
342C2×Dic5—24no—
343C2×Dic5—62no—
344C2×Dic5—61no—
345C2×Dic5—62no—
346C4×D5—121no—
347C4×D5—122no—
348C4×D5—121no—
3492−1+4—62no—
3502−1+4—61no—
351H(5,𝔽2)—34no—
352H(5,𝔽2)—62no—
353H(5,𝔽2)—61no—
354C2×#41—32no—
355C2×#41—64no—
356C2×#41—62no—
357C2×#41D3×D511yescharacteristic
358C23×C4—151no—
359#123—154no—
360#123—152no—
361C22⋊Q8—302no—
362C22⋊Q8—301no—
363#121—302no—
364#121—301no—
365#120—154no—
366#120—152no—
367C4×D4—304no—
368C4×D4—302no—
369#117—52no—
370#117—151no—
371#99—54no—
372#99—158no—
373C2×C3×C5—22no—
374C2×C3×C5—21no—
375C2×C3×C5—42no—
376C2×C3×C5C2×#12011yescharacteristic
377D15—42no—
378C5×D3—44no—
379C23×C3C2×C22⋊D512yes—
380C23×C3C23×D511yescharacteristic
381C22×D3—24no—
382C22×D3—102no—
383C22×D3—101no—
384C22×D3—202no—
385C22×D3C2×C22⋊D514yes—
386C3×Q8—22no—
387C3×D4—24no—
388C3×D4—22no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—104no—
392C2×C3×C4C2×C22⋊D512yes—
393C2×C3×C4C2×C22⋊D511yescharacteristic
394C2×C3×C4C2×C22⋊D511yescharacteristic
395C2×C3×C4C2×C22⋊D512yes—
396C2×C3×C4C2×C22⋊D512yes—
397#67—28no—
398#67—24no—
399#67—24no—
400#67—204no—
401C2×Dic3—24no—
402C2×Dic3—102no—
403C2×Dic3—101no—
404C2×Dic3—102no—
405C2×Dic3C2×C22⋊D514yes—
406D12—24no—
407D12—22no—
408D12—22no—
409D12—208no—
410C4×D3—28no—
411C4×D3—24no—
412C4×D3—24no—
413C4×D3—201no—
414C4×D3—202no—
415C4×D3—201no—
416Dic6—24no—
417Dic6—22no—
418Dic6—22no—
419C22×C5—22no—
420C22×C5—41no—
421C22×C5—42no—
422C22×C5—64no—
423C22×C5—124no—
424C22×C5—244no—
425C22×C5C22×D1211yescharacteristic
426C22×C5C2×D3×D412yes—
427C2×D5—64no—
428C2×D5—62no—
429C2×D5—122no—
430C2×D5—121no—
431C2×D5—121no—
432C4×C5—22no—
433C4×C5—64no—
434C4×C5#88211yescharacteristic
435C4×C5#88212yes—
436C4×C5#88211yescharacteristic
437Dic5—44no—
438Dic5—122no—
439Dic5—121no—
440C24—151no—
441#41—22no—
442#41—22no—
443#41—64no—
444#41—64no—
445#41—68no—
446#41—64no—
447#41—64no—
448#41—62no—
449#41—62no—
450#41—64no—
451#41—62no—
452#41—62no—
453C2×Q8—32no—
454C2×Q8—62no—
455C2×Q8—64no—
456C2×Q8#120311yescharacteristic, complemented
457C2×D4—34no—
458C2×D4—32no—
459C2×D4—62no—
460C2×D4—68no—
461C2×D4—64no—
462C2×D4—64no—
463C2×D4—304no—
464C2×D4—302no—
465C2×D4#120311yescharacteristic, complemented
466C2×D4C2×D3×D512yes—
467C22×C4—34no—
468C22×C4—64no—
469C22×C4—151no—
470C22×C4—301no—
471C22×C4—301no—
472C22×C4—302no—
473C22×C4—302no—
474C22×C4#120312yescomplemented
475C22×C4C2×D3×D511yescharacteristic
476C4⋊C4—102no—
477C4⋊C4—304no—
478C4⋊C4—301no—
479C4⋊C4—302no—
480C22⋊C4—54no—
481C22⋊C4—152no—
482C22⋊C4—304no—
483C22⋊C4—304no—
484C22⋊C4—302no—
485C22⋊C4—302no—
486C42—102no—
487C42—301no—
488C3×C5#302511yescharacteristic, complemented, Hall
489C22×C3—22no—
490C22×C3—41no—
491C22×C3—42no—
492C22×C3C22×C22⋊D511yescharacteristic
493C22×C3C22×C22⋊D512yes—
494C2×D3—24no—
495C2×D3—44no—
496C2×D3—84no—
497C2×D3—104no—
498C2×D3—102no—
499C2×D3—201no—
500C2×D3—202no—
501C2×D3—201no—
502C3×C4—22no—
503C3×C4—204no—
504C3×C4#409911yescharacteristic
505C3×C4#409912yes—
506C3×C4#409911yescharacteristic
507Dic3—24no—
508Dic3—202no—
509Dic3—201no—
510C2×C5—22no—
511C2×C5—21no—
512C2×C5—42no—
513C2×C5—124no—
514C2×C5C2×#88211yescharacteristic
515D5—122no—
516C23—34no—
517C23—64no—
518C23—301no—
519C23—302no—
520C23—602no—
521C23C2×#120312yes—
522C23C22×D3×D511yescharacteristic
523Q8—22no—
524Q8—64no—
525Q8—62no—
526Q8—62no—
527D4—24no—
528D4—22no—
529D4—64no—
530D4—68no—
531D4—64no—
532D4—64no—
533D4—62no—
534D4—62no—
535D4—608no—
536D4—604no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—34no—
540C2×C4—68no—
541C2×C4—64no—
542C2×C4—64no—
543C2×C4—64no—
544C2×C4—104no—
545C2×C4—302no—
546C2×C4—301no—
547C2×C4—302no—
548C2×C4—601no—
549C2×C4—602no—
550C2×C4—601no—
551C2×C4C2×#120312yes—
552C2×C4C2×#120311yescharacteristic
553C2×C4C2×#120311yescharacteristic
554C2×C4D3×C22⋊D512yes—
555C2×C4D3×C22⋊D512yes—
556C2×C3—21no—
557C2×C3—22no—
558C2×C3—42no—
559C2×C3C2×#409911yescharacteristic
560D3—44no—
561D3—202no—
562C5#7869011yescharacteristic, complemented, Sylow
563C22—22no—
564C22—41no—
565C22—42no—
566C22—64no—
567C22—124no—
568C22—244no—
569C22—304no—
570C22—302no—
571C22—602no—
572C22—601no—
573C22—601no—
574C22C22×#120311yescharacteristic
575C22C2×D3×C22⋊D512yes—
576C4—22no—
577C4—64no—
578C4—204no—
579C4—602no—
580C4—601no—
581C4#9157011yescharacteristic
582C4#9157012yes—
583C4#9157011yescharacteristic
584C3#1061585611yescharacteristic, complemented, Sylow
585C2—22no—
586C2—21no—
587C2—42no—
588C2—124no—
589C2—602no—
590C2C2×#9157011yescharacteristic
subgroupG/H
centerC4#91570
commutator, cosocleC23×C3×C5C24
FrattiniC23C22×D3×D5
FittingC2×C3×C5×#41C22
socleC2×C3×C5C2×#120
hypercenterC2×#41D3×D5
hypocenterC3×C5#3025
LCSG ▷ C23×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C5×#5964C2semidirect
2. #11756587C2semidirect
3. #11754592C2semidirect
4. C3×#65529C2semidirect
5. #11749184C2semidirect
6. #11749178C2semidirect
7. #11748625C2semidirect
10. C22×C4×D15C22semidirect
21. #91826C22semidirect
22. #91824C22semidirect
24. #91815C22semidirect
25. C3×#4072C22semidirect
26. #91573C22semidirect
27. #91543C22semidirect
31. C3×#3966C22semidirect
32. #90996C22semidirect
40. #65529D3semidirect
44. C2×C3×C5×Q8D4semidirect
45. C2×C3×C5×D4D4semidirect
47. C22×C3×C4×C5D4semidirect
71. C2×C5×D12D4semidirect
73. C2×C4×C5×D3D4semidirect
75. C2×C5×Dic6D4semidirect
89. #5964D5semidirect
117. #4072C2×D3semidirect
119. #3966C2×D3semidirect
201. C2×C5×Q8D12semidirect
208. C2×C5×D4D12semidirect
212. C22×C4×C5D12semidirect
264. C2×C3×Q8C22⋊D5semidirect
265. C2×C3×D4C22⋊D5semidirect
267. C22×C3×C4C22⋊D5semidirect
288. C2×D12C22⋊D5semidirect
294. C2×C4×D3C22⋊D5semidirect
296. C2×Dic6C22⋊D5semidirect
456. C2×Q8#1203semidirect
465. C2×D4#1203semidirect
474. C22×C4#1203semidirect
488. C3×C5#3025semidirect
562. C5#78690semidirect
584. C3#10615856semidirect

Maximal subgroups

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

subgroupindex
C5×#59642
#117565872
#117545922
C3×#655292
#117491842
#117491782
#117486252
#106158563
#786905