Finite Group Databasebeta

← #14313275

Subgroups — #14313275

Order 1920.

532 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#65926C211yesmaximal, characteristic
2#11749172C211yesmaximal, characteristic, complemented
3#11749123C211yesmaximal, characteristic, complemented
4#11746357C211yesmaximal, characteristic, complemented
5C5×#4536C211yesmaximal, characteristic, complemented
6#11745995C211yesmaximal, characteristic, complemented
7#11745994C211yesmaximal, characteristic, complemented
8#10599930—31nomaximal, Hall
9C22×C3×C22⋊D5C411yescharacteristic, complemented
10C2×C3×#711C2211yescharacteristic
11C3×C5×#120C2211yescharacteristic
12C3×#4099C411yescharacteristic, complemented
13C3×#4086—21no—
14C3×#4085—21no—
15C3×#4083C2211yescharacteristic
16C3×#4031—21no—
17C3×#4030—21no—
18C3×#4028C411yescharacteristic, complemented
19C3×#4023C411yescharacteristic, complemented
20#91528—21no—
21D5×#208—21no—
22#91118C2211yescharacteristic
23#91117C2211yescharacteristic
24C5×#834C2211yescharacteristic
25C5×#833C2211yescharacteristic
26#90995—21no—
27#90961—21no—
28#71859—51nomaximal, Hall
29#65926D311yescharacteristic
30#65034—31no—
31#65029—31no—
32C5×AΣL(1,𝔽16)—31no—
33#64756—31no—
34#64695—31no—
35#64691—31no—
36C24×C3×C5C2×C411yescharacteristic
37C23×C3×D5C2×C411yescharacteristic
38C2×C3×C5×D4—21no—
39C2×C3×C5×D4C2311yescharacteristic
40C2×C3×C22⋊D5—21no—
41C2×C3×C22⋊D5—21no—
42C2×C3×C22⋊D5—21no—
43C2×C3×C22⋊D5—21no—
44C2×C3×C22⋊D5—41no—
45C2×C3×C22⋊D5—41no—
46C2×C3×C22⋊D5C2×C411yescharacteristic
47C2×C3×C22⋊D5D411yescharacteristic
48C22×C3×Dic5C2×C411yescharacteristic
49C22×C3×Dic5D411yescharacteristic
50C3×#711—41no—
51C3×#711—41no—
52C2×C3×D20—21no—
53C2×C3×C4×D5D411yescharacteristic
54C2×C3×Dic10D411yescharacteristic
55C3×C5×C22⋊C4—21no—
56C3×C5×C22⋊C4C2×C411yescharacteristic
57#6719—21no—
58#6710—21no—
59C5×#217—21no—
60C5×#208—21no—
61C3×C22⋊Dic5—21no—
62C3×C22⋊Dic5C2×C411yescharacteristic
63C3×#685—21no—
64C3×#685—21no—
65C3×#683—21no—
66C3×C4×Dic5—21no—
67#6668—41no—
68Dic3×Dic5—41no—
69#6649—41no—
70D5×C3⋊C8—41no—
71C3×#535—51no—
72#5240—51no—
73#5237—51no—
74#4536—51no—
75#4536—51no—
76#4536—51no—
77#4536D511yescharacteristic, complemented
78C22×C22⋊D5Dic311yescharacteristic, complemented
79C2×#711C2×D311yescharacteristic
80C5×#120C2×D311yescharacteristic
81#4099Dic311yescharacteristic, complemented
82#4086—21no—
83#4085—21no—
84#4083C2×D311yescharacteristic
85D5×M16—61no—
86#4031—21no—
87#4030—21no—
88#4028Dic311yescharacteristic, complemented
89#4023—61no—
90#4023Dic311yescharacteristic, complemented
91C5×#100—31no—
92C5×#99—31no—
93#3966—31no—
94#3965—31no—
95#3956—61no—
96#3938—61no—
97#2122—151noSylow
98C23×C3×C5—21no—
99C23×C3×C5—41no—
100C23×C3×C5—81no—
101C23×C3×C5C22×C411yescharacteristic
102C23×C3×C5C2×D411yescharacteristic
103C22×C3×D5—21no—
104C22×C3×D5—41no—
105C22×C3×D5—81no—
106C22×C3×D5C22⋊C411yescharacteristic
107C3×C5×D4—41no—
108C3×C5×D4—81no—
109C2×C3×C4×C5—21no—
110C2×C3×C4×C5C2×D411yescharacteristic
111C2×Dic15—41no—
112C2×C5×Dic3—41no—
113C3×C22⋊D5—41no—
114C3×C22⋊D5—41no—
115C3×C22⋊D5—42no—
116C3×C22⋊D5—41no—
117C3×C22⋊D5—41no—
118C3×C22⋊D5—81no—
119C3×C22⋊D5—81no—
120C2×C3×Dic5—21no—
121C2×C3×Dic5—21no—
122C2×C3×Dic5—21no—
123C2×C3×Dic5—21no—
124C2×C3×Dic5—21no—
125C2×C3×Dic5C22⋊C411yescharacteristic
126C2×C3×Dic5C22⋊C411yescharacteristic
127C2×C3×Dic5C22⋊C411yescharacteristic
128C3×D20—81no—
129C3×C4×D5—21no—
130C3×C4×D5—21no—
131C3×Dic10—41no—
132C3⋊#168—41no—
133C5×C3⋊C8—41no—
134C2×C3×#41—51no—
135C22×C3×D4—51no—
136C3×#127—101no—
137C3×#124—101no—
138C3×#121—101no—
139C3×#121—101no—
140C3×#120—51no—
141C3×#120—51no—
142C3×#120—51no—
143C3×#120C2×D511yescharacteristic
144C3×#117—51no—
145#926—101no—
146C2×#208—101no—
147#836—101no—
148#834—51no—
149#834—101no—
150#834C2×D511yescharacteristic
151#833—51no—
152#833C2×D511yescharacteristic
153C24×C5C2×Dic311yescharacteristic
154C23×D5C2×Dic311yescharacteristic
155C2×C5×D4—21no—
156C2×C5×D4C22×D311yescharacteristic
157C2×C22⋊D5—21no—
158C2×C22⋊D5—21no—
159C2×C22⋊D5—21no—
160C2×C22⋊D5—21no—
161C2×C22⋊D5—41no—
162C2×C22⋊D5—41no—
163C2×C22⋊D5C2×Dic311yescharacteristic
164C2×C22⋊D5#6711yescharacteristic
165C22×Dic5C2×Dic311yescharacteristic
166C22×Dic5#6711yescharacteristic
167#711—41no—
168#711—41no—
169C2×D20—21no—
170C2×C4×D5#6711yescharacteristic
171C2×Dic10#6711yescharacteristic
172C5×M16—61no—
173C5×C22⋊C4—21no—
174C5×C22⋊C4—61no—
175C5×C22⋊C4C2×Dic311yescharacteristic
176C22⋊Dic5—21no—
177C22⋊Dic5—61no—
178C22⋊Dic5C2×Dic311yescharacteristic
179#685—21no—
180#685—21no—
181#683—21no—
182#683—121no—
183C4×Dic5—21no—
184C4×Dic5—121no—
185#681—61no—
186#676—121no—
187C8×D5—121no—
188#535—51no—
189#413—151no—
190#410—151no—
191AΣL(1,𝔽16)—31no—
192AΣL(1,𝔽16)—151no—
193AΣL(1,𝔽16)—151no—
194AΣL(1,𝔽16)—151no—
195C22×C3×C5—21no—
196C22×C3×C5—21no—
197C22×C3×C5—41no—
198C22×C3×C5—41no—
199C22×C3×C5—41no—
200C22×C3×C5—41no—
201C22×C3×C5—41no—
202C22×C3×C5—82no—
203C22×C3×C5C2×C22⋊C411yescharacteristic
204C2×C3×D5—21no—
205C2×C3×D5—41no—
206C2×C3×D5—41no—
207C2×C3×D5—41no—
208C2×C3×D5—41no—
209C2×C3×D5—82no—
210C3×C4×C5—21no—
211C3×C4×C5—41no—
212Dic15—81no—
213C3×Dic5—21no—
214C3×Dic5—21no—
215C3×Dic5—41no—
216C3×Dic5#9911yescharacteristic
217C3×Dic5#9911yescharacteristic
218C5×Dic3—81no—
219C24×C3—51no—
220C24×C3C4×D511yescharacteristic
221C3×#41—201no—
222C3×#41—201no—
223C2×C3×Q8—51no—
224C2×C3×D4—21no—
225C2×C3×D4—51no—
226C2×C3×D4—51no—
227C2×C3×D4—101no—
228C2×C3×D4—101no—
229C2×C3×D4—101no—
230C2×C3×D4—101no—
231C2×C3×D4—101no—
232C2×C3×D4—201no—
233C2×C3×D4—201no—
234C2×C3×D4C22×D511yescharacteristic
235C22×C3×C4—51no—
236C22×C3×C4—51no—
237C22×C3×C4—51no—
238C3×C4⋊C4—101no—
239C3×C22⋊C4—21no—
240C3×C22⋊C4—51no—
241C3×C22⋊C4—101no—
242C3×C22⋊C4—101no—
243C3×C22⋊C4—101no—
244C3×C22⋊C4C4×D511yescharacteristic
245C3×C42—101no—
246#217—21no—
247#217—101no—
248C4⋊Dic3—201no—
249C4×Dic3—201no—
250#208—21no—
251#208—101no—
252#208—201no—
253C2×C3⋊C8—201no—
254C23×C5—21no—
255C23×C5—41no—
256C23×C5—81no—
257C23×C5C22×Dic311yescharacteristic
258C23×C5C2×#6711yescharacteristic
259C22×D5—21no—
260C22×D5—41no—
261C22×D5—81no—
262C22×D5#21711yescharacteristic
263C5×D4—41no—
264C5×D4—81no—
265C2×C4×C5—21no—
266C2×C4×C5—121no—
267C2×C4×C5C2×#6711yescharacteristic
268C22⋊D5—41no—
269C22⋊D5—41no—
270C22⋊D5—42no—
271C22⋊D5—41no—
272C22⋊D5—41no—
273C22⋊D5—81no—
274C22⋊D5—81no—
275C2×Dic5—21no—
276C2×Dic5—21no—
277C2×Dic5—21no—
278C2×Dic5—21no—
279C2×Dic5—21no—
280C2×Dic5—121no—
281C2×Dic5#21711yescharacteristic
282C2×Dic5#21711yescharacteristic
283C2×Dic5#21711yescharacteristic
284D20—81no—
285C4×D5—21no—
286C4×D5—21no—
287Dic10—41no—
288C5×C8—121no—
289#168—121no—
290C2×#41—51no—
291C22×D4—51no—
292C2×M16—301no—
293#127—101no—
294#124—101no—
295#121—101no—
296#121—101no—
297#120—51no—
298#120—51no—
299#120—51no—
300#120D3×D511yescharacteristic
301#117—51no—
302#117—301no—
303#101—301no—
304#100—31no—
305#100—151no—
306#99—31no—
307#99—151no—
308#99—301no—
309C2×C3×C5—21no—
310C2×C3×C5—41no—
311C2×C3×C5—41no—
312C2×C3×C5—81no—
313C2×C3×C5C2×#9911yescharacteristic
314C3×D5—41no—
315C3×D5—81no—
316C23×C3—21no—
317C23×C3—41no—
318C23×C3—51no—
319C23×C3—81no—
320C23×C3—101no—
321C23×C3—201no—
322C23×C3—401no—
323C23×C3C2×C4×D511yescharacteristic
324C23×C3D4×D511yescharacteristic
325C3×Q8—201no—
326C3×D4—41no—
327C3×D4—81no—
328C3×D4—201no—
329C3×D4—201no—
330C3×D4—201no—
331C3×D4—201no—
332C3×D4—202no—
333C3×D4—401no—
334C3×D4—401no—
335C3×D4—401no—
336C2×C3×C4—21no—
337C2×C3×C4—51no—
338C2×C3×C4—51no—
339C2×C3×C4—51no—
340C2×C3×C4—101no—
341C2×C3×C4—101no—
342C2×C3×C4—101no—
343C2×C3×C4—101no—
344C2×C3×C4—101no—
345C2×C3×C4—101no—
346C2×C3×C4—101no—
347C2×C3×C4D4×D511yescharacteristic
348C2×Dic3—41no—
349C2×Dic3—201no—
350C3⋊C8—41no—
351C3⋊C8—201no—
352C22×C5—21no—
353C22×C5—21no—
354C22×C5—41no—
355C22×C5—41no—
356C22×C5—41no—
357C22×C5—41no—
358C22×C5—41no—
359C22×C5—82no—
360C22×C5C2×#21711yescharacteristic
361C2×D5—21no—
362C2×D5—41no—
363C2×D5—41no—
364C2×D5—41no—
365C2×D5—41no—
366C2×D5—82no—
367C4×C5—21no—
368C4×C5—41no—
369C4×C5—241no—
370Dic5—21no—
371Dic5—21no—
372Dic5—41no—
373Dic5—241no—
374Dic5#83411yescharacteristic
375Dic5#83411yescharacteristic
376C24—51no—
377C24D5×Dic311yescharacteristic
378#41—201no—
379#41—201no—
380C2×Q8—51no—
381C2×D4—21no—
382C2×D4—51no—
383C2×D4—51no—
384C2×D4—101no—
385C2×D4—101no—
386C2×D4—101no—
387C2×D4—101no—
388C2×D4—101no—
389C2×D4—201no—
390C2×D4—201no—
391C2×D4C2×D3×D511yescharacteristic
392C22×C4—51no—
393C22×C4—51no—
394C22×C4—51no—
395M16—61no—
396M16—301no—
397M16—601no—
398C2×C8—601no—
399C4⋊C4—101no—
400C4⋊C4—601no—
401C22⋊C4—21no—
402C22⋊C4—51no—
403C22⋊C4—61no—
404C22⋊C4—101no—
405C22⋊C4—101no—
406C22⋊C4—101no—
407C22⋊C4—301no—
408C22⋊C4D5×Dic311yescharacteristic
409C42—101no—
410C42—601no—
411C3×C5#212211yescharacteristic, complemented, Hall
412C22×C3—21no—
413C22×C3—21no—
414C22×C3—41no—
415C22×C3—41no—
416C22×C3—41no—
417C22×C3—41no—
418C22×C3—41no—
419C22×C3—82no—
420C22×C3—101no—
421C22×C3—201no—
422C22×C3—201no—
423C22×C3—201no—
424C22×C3—201no—
425C22×C3—402no—
426C22×C3D5×C22⋊C411yescharacteristic
427C3×C4—21no—
428C3×C4—41no—
429C3×C4—51no—
430C3×C4—51no—
431C3×C4—101no—
432C3×C4—101no—
433C3×C4—201no—
434Dic3—81no—
435Dic3—401no—
436C2×C5—21no—
437C2×C5—41no—
438C2×C5—41no—
439C2×C5—81no—
440C2×C5C2×#83411yescharacteristic
441D5—41no—
442D5—81no—
443C23—21no—
444C23—41no—
445C23—51no—
446C23—81no—
447C23—101no—
448C23—201no—
449C23—401no—
450C23C2×D5×Dic311yescharacteristic
451C23D5×#6711yescharacteristic
452Q8—201no—
453D4—41no—
454D4—81no—
455D4—201no—
456D4—201no—
457D4—201no—
458D4—201no—
459D4—202no—
460D4—401no—
461D4—401no—
462D4—401no—
463C2×C4—21no—
464C2×C4—51no—
465C2×C4—51no—
466C2×C4—51no—
467C2×C4—101no—
468C2×C4—101no—
469C2×C4—101no—
470C2×C4—101no—
471C2×C4—101no—
472C2×C4—101no—
473C2×C4—101no—
474C2×C4—121no—
475C2×C4—601no—
476C2×C4D5×#6711yescharacteristic
477C8—121no—
478C8—601no—
479C2×C3—21no—
480C2×C3—41no—
481C2×C3—41no—
482C2×C3—81no—
483C2×C3—201no—
484C2×C3—401no—
485C2×C3D5×#9911yescharacteristic
486C5#7185911yescharacteristic, complemented, Sylow
487C22—21no—
488C22—21no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—82no—
495C22—101no—
496C22—201no—
497C22—201no—
498C22—201no—
499C22—201no—
500C22—402no—
501C22D5×#21711yescharacteristic
502C4—21no—
503C4—41no—
504C4—51no—
505C4—51no—
506C4—101no—
507C4—101no—
508C4—201no—
509C4—241no—
510C4—1201no—
511C3#1059993011yescharacteristic, complemented, Sylow
512C2—21no—
513C2—41no—
514C2—41no—
515C2—81no—
516C2—201no—
517C2—401no—
518C2D5×#83411yescharacteristic
subgroupG/H
centerC2D5×#834
commutatorC23×C3×C5C22×C4
FrattiniC2×D4C2×D3×D5
FittingC3×C5×#120C22
socleC2×C3×C5C2×#99
prime radicalC3×#65926C2
cosocleC2×C3×C5×D4C23
hypercenter#120D3×D5
hypocenterC3×C5#2122
LCSG ▷ C23×C3×C5 ▷ C22×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #120

Complements

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

NH ≅ G/Ntype
2. #11749172C2semidirect
3. #11749123C2semidirect
4. #11746357C2semidirect
5. C5×#4536C2semidirect
6. #11745995C2semidirect
7. #11745994C2semidirect
9. C22×C3×C22⋊D5C4semidirect
12. C3×#4099C4semidirect
18. C3×#4028C4semidirect
19. C3×#4023C4semidirect
77. #4536D5semidirect
78. C22×C22⋊D5Dic3semidirect
81. #4099Dic3semidirect
88. #4028Dic3semidirect
90. #4023Dic3semidirect
411. C3×C5#2122semidirect
486. C5#71859semidirect
511. C3#10599930semidirect

Maximal subgroups

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

subgroupindex
C3×#659262
#117491722
#117491232
#117463572
C5×#45362
#117459952
#117459942
#105999303
#718595