Finite Group Databasebeta

← #14301394

Subgroups — #14301394

Order 1920.

664 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11754581C211yesmaximal, characteristic
2#11754120C211yesmaximal, characteristic, complemented
3#11751106C211yesmaximal, characteristic, complemented
4C3×#65295C211yesmaximal, characteristic, complemented
5#11748429C211yesmaximal, characteristic, complemented
6C5×#4532C211yesmaximal, characteristic, complemented
7#11745974C211yesmaximal, characteristic, complemented
8#10599570—31nomaximal, Hall
9C2×#6817C2211yescharacteristic
10C2×C3×#709C2211yescharacteristic
11C3×C5×#117C2211yescharacteristic
12C2×#6710C2211yescharacteristic
13C2×C4×C5×Dic3C2211yescharacteristic
14C2×C5×#208C2211yescharacteristic
15C3×#4072C2211yescharacteristic
16#91569—22no—
17#91458C411yescharacteristic, complemented
18C4×#1202—22no—
19#91389C411yescharacteristic, complemented
20#91374C411yescharacteristic, complemented
21#91364C411yescharacteristic, complemented
22#91284—22no—
23#91283—22no—
24D5×#208—22no—
25#91089—22no—
26C3×#3938—22no—
27#90981—22no—
28#70045—51nomaximal, Hall
29#66079—31no—
30#65811—31no—
31#65428—31no—
32#65295D311yescharacteristic, complemented
33#65294—31no—
34C5×#339—31no—
35#64743—31no—
36C22×C3×C4×C5C2311yescharacteristic
37C2×#1220—21no—
38#6817—21no—
39#6817—21no—
40#6817—21no—
41#6817—21no—
42C2×D60C2×C411yescharacteristic
43C2×C4×D15—21no—
44C2×Dic30C2×C411yescharacteristic
45C22×C5×Dic3—21no—
46C2×C3×C22⋊D5D412yes—
47C3×#709—21no—
48C3×#709—21no—
49C3×#709—21no—
50C3×#709—21no—
51C2×C3×D20C2×C411yescharacteristic
52C2×C3×C4×D5D412yes—
53C2×C3×Dic10C2×C411yescharacteristic
54C2×#1202—22no—
55C3×C5×C4⋊C4—21no—
56C3×C5×C22⋊C4—21no—
57C3×C42×C5—21no—
58#6710—21no—
59#6710D412yescomplemented
60C2×C3⋊#168—21no—
61C4×C5×Dic3—21no—
62C4×C5×Dic3C2×C411yescharacteristic
63C4×C5×Dic3C2×C411yescharacteristic
64C5×#208—21no—
65C5×#208D412yescomplemented
66C2×C5×C3⋊C8—21no—
67C3×C22⋊Dic5—21no—
68C3×C4⋊Dic5—21no—
69C3×C4×Dic5—21no—
70Dic3⋊Dic5—22no—
71#6667—22no—
72#6665—22no—
73#6649—24no—
74D5×C3⋊C8—24no—
75#5832—51no—
76#5819—51no—
77C3×#410—51no—
78#5005—51no—
79#5005—51no—
80#4532—51no—
81#4532D511yescharacteristic, complemented
82C2×#709—31no—
83C2×#709C2×D311yescharacteristic
84C2×C5×M16—31no—
85C5×#117C2×D311yescharacteristic
86C2×C42×C5—31no—
87#4076—62no—
88#4072C2×D311yescharacteristic
89C2×#681—31no—
90#4053—62no—
91D5×M16—62no—
92#4047—62no—
93#4021—31no—
94#4021—31no—
95#4020—31no—
96C4×D20—62no—
97#4015—31no—
98#3957—62no—
99#3938—22no—
100#3937—62no—
101#1884—151noSylow
102C23×C3×C5C2×D411yescharacteristic
103C22×D15—21no—
104C22×C3×D5C22⋊C412yes—
105C2×C3×C4×C5—41no—
106C2×C3×C4×C5C22×C411yescharacteristic
107C2×C3×C4×C5C2×D411yescharacteristic
108C2×C3×C4×C5C2×D411yescharacteristic
109C2×C3×C4×C5C2×D411yescharacteristic
110C2×C3×C4×C5#4111yescharacteristic
111C2×C3×C4×C5#4111yescharacteristic
112#1220—41no—
113#1220—41no—
114C2×Dic15—21no—
115D60—21no—
116D60—21no—
117C4×D15—41no—
118C4×D15—41no—
119Dic30—21no—
120Dic30—21no—
121C2×C5×Dic3—21no—
122C2×C5×Dic3—21no—
123C2×C5×Dic3—42no—
124C3×C22⋊D5—22no—
125C3×C22⋊D5—22no—
126C2×C3×Dic5—41no—
127C2×C3×Dic5C22⋊C412yes—
128C3×D20—21no—
129C3×D20—21no—
130C3×C4×D5—22no—
131C3×C4×D5—22no—
132C3×Dic10—21no—
133C3×Dic10—21no—
134#1202—44no—
135C3⋊#168—22no—
136C5×C3⋊C8—22no—
137C2×C3×#41—51no—
138C2×#235—51no—
139C3×#117—51no—
140C3×#117C2×D511yescharacteristic
141#948—102no—
142#948—102no—
143#947—51no—
144#944—51no—
145#940—51no—
146#939—102no—
147#936—51no—
148C4×#67—102no—
149C2×C4×Dic3C2×D511yescharacteristic
150C2×#208—51no—
151C2×#208—102no—
152C2×#208C2×D511yescharacteristic
153C3×#99—102no—
154#805—102no—
155#805—102no—
156C22×C4×C5—61no—
157C22×C4×C5C22×D311yescharacteristic
158C2×C22⋊D5—61no—
159C2×C22⋊D5#6712yes—
160#709—21no—
161#709—21no—
162#709—21no—
163#709—21no—
164#709—61no—
165#709—61no—
166#709—61no—
167#709—61no—
168C2×D20—31no—
169C2×D20—62no—
170C2×D20C4×D311yescharacteristic
171C2×C4×D5—61no—
172C2×C4×D5D1212yes—
173C2×Dic10—31no—
174C2×Dic10C4×D311yescharacteristic
175C5×M16—32no—
176C5×M16—61no—
177C2×C5×C8—61no—
178C5×C4⋊C4—21no—
179C5×C22⋊C4—21no—
180C42×C5—21no—
181C42×C5—31no—
182C42×C5—31no—
183C42×C5—61no—
184C22⋊Dic5—21no—
185#685—62no—
186#685—62no—
187C4⋊Dic5—21no—
188C4⋊Dic5—62no—
189C4×Dic5—21no—
190#681—32no—
191#681—61no—
192C2×#168—61no—
193#676—64no—
194C8×D5—64no—
195#568—151no—
196#532—151no—
197#421—151no—
198#421—151no—
199#410—51no—
200#339—31no—
201#339—151no—
202C22×C3×C5—41no—
203C22×C3×C5C2×C22⋊C411yescharacteristic
204C22×C3×C5#12111yescharacteristic
205C22×C3×C5#12111yescharacteristic
206C2×D15—42no—
207C2×D15—41no—
208C2×C3×D5—22no—
209C2×C3×D5—42no—
210C3×C4×C5—81no—
211C3×C4×C5C4×D411yescharacteristic
212C3×C4×C5C4×D411yescharacteristic
213C3×C4×C5#12011yescharacteristic
214C3×C4×C5#12311yescharacteristic
215Dic15—41no—
216C3×Dic5—22no—
217C3×Dic5—81no—
218C5×Dic3—22no—
219C5×Dic3—41no—
220C3×#41—101no—
221C3×#41—101no—
222C3×#41—101no—
223C3×#41—101no—
224C2×C3×Q8—51no—
225C2×C3×D4—52no—
226C2×C3×D4—51no—
227C22×C3×C4—52no—
228C22×C3×C4C22×D511yescharacteristic
229C2×#67—102no—
230C2×#67—101no—
231C22×Dic3—21no—
232#235—101no—
233#235—101no—
234#235—101no—
235#235—101no—
236C2×D12—51no—
237C2×C4×D3—101no—
238C2×Dic6—51no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—101no—
241C3×C22⋊C4—21no—
242C3×C22⋊C4—101no—
243C3×C42—21no—
244C3×C42—101no—
245#217—102no—
246(C2×D3)⋊C4—102no—
247Dic3⋊C4—102no—
248C4×Dic3—21no—
249C4×Dic3C4×D511yescharacteristic
250C4×Dic3C4×D511yescharacteristic
251#208—21no—
252#208—52no—
253#208—104no—
254#208—101no—
255#208D2012yescomplemented
256C2×C3⋊C8—21no—
257C2×C3⋊C8—104no—
258C2×C3⋊C8—101no—
259C23×C5C2×#6711yescharacteristic
260C22×D5—61no—
261C22×D5(C2×D3)⋊C412yes—
262C2×C4×C5—41no—
263C2×C4×C5—61no—
264C2×C4×C5—61no—
265C2×C4×C5—122no—
266C2×C4×C5C2×C4×D311yescharacteristic
267C2×C4×C5C2×D1211yescharacteristic
268C2×C4×C5D3×D411yescharacteristic
269C2×C4×C5D3×D411yescharacteristic
270C2×C4×C5#23711yescharacteristic
271C2×C4×C5#23711yescharacteristic
272C22⋊D5—22no—
273C22⋊D5—22no—
274C22⋊D5—121no—
275C22⋊D5—121no—
276C2×Dic5—41no—
277C2×Dic5—61no—
278C2×Dic5(C2×D3)⋊C412yes—
279D20—21no—
280D20—21no—
281D20—61no—
282D20—61no—
283D20—124no—
284C4×D5—22no—
285C4×D5—22no—
286C4×D5—121no—
287C4×D5—121no—
288Dic10—21no—
289Dic10—21no—
290Dic10—61no—
291Dic10—61no—
292C5×C8—62no—
293#168—62no—
294C2×#41—51no—
295C2×#41—151no—
296#131—302no—
297#131—302no—
298C2×M16—31no—
299C2×M16—151no—
300C2×M16—302no—
301C4⋊Q8—151no—
302#127—151no—
303#124—151no—
304#124—151no—
305#121—302no—
306C4×D4—302no—
307#117—51no—
308#117D3×D511yescharacteristic
309C2×C42—31no—
310#104—302no—
311#104—302no—
312#99—102no—
313C2×C3×C5—21no—
314C2×C3×C5—21no—
315C2×C3×C5—21no—
316C2×C3×C5#38611yescharacteristic
317D15—81no—
318C3×D5—42no—
319C23×C3—52no—
320C23×C3D4×D511yescharacteristic
321C22×D3—101no—
322C3×Q8—101no—
323C3×Q8—101no—
324C3×D4—102no—
325C3×D4—102no—
326C3×D4—101no—
327C3×D4—101no—
328C2×C3×C4—41no—
329C2×C3×C4—52no—
330C2×C3×C4—102no—
331C2×C3×C4—102no—
332C2×C3×C4—201no—
333C2×C3×C4C2×C4×D511yescharacteristic
334C2×C3×C4C2×D2011yescharacteristic
335C2×C3×C4#70911yescharacteristic
336C2×C3×C4D4×D511yescharacteristic
337C2×C3×C4D4×D511yescharacteristic
338C2×C3×C4#71311yescharacteristic
339#67—204no—
340#67—201no—
341#67—201no—
342C2×Dic3—21no—
343C2×Dic3—21no—
344C2×Dic3—42no—
345C2×Dic3—101no—
346D12—101no—
347D12—101no—
348C4×D3—201no—
349C4×D3—201no—
350Dic6—101no—
351Dic6—101no—
352C3⋊C8—22no—
353C3⋊C8—102no—
354C22×C5—41no—
355C22×C5C2×(C2×D3)⋊C411yescharacteristic
356C22×C5#93911yescharacteristic
357C22×C5#93911yescharacteristic
358C2×D5—22no—
359C2×D5—42no—
360C2×D5—122no—
361C2×D5—121no—
362C4×C5—62no—
363C4×C5—81no—
364C4×C5—121no—
365C4×C5#67⋊C411yescharacteristic
366C4×C5#67⋊C411yescharacteristic
367C4×C5#88211yescharacteristic
368C4×C5#88611yescharacteristic
369Dic5—22no—
370Dic5—81no—
371Dic5—121no—
372#41—101no—
373#41—101no—
374#41—101no—
375#41—101no—
376#41—301no—
377#41—301no—
378#41—301no—
379#41—301no—
380C2×Q8—51no—
381C2×Q8—151no—
382C2×D4—52no—
383C2×D4—51no—
384C2×D4—151no—
385C2×D4—302no—
386C2×D4—301no—
387C22×C4—52no—
388C22×C4—61no—
389C22×C4—301no—
390C22×C4C2×D3×D511yescharacteristic
391M16—32no—
392M16—61no—
393M16—152no—
394M16—304no—
395M16—301no—
396C2×C8—61no—
397C2×C8—304no—
398C2×C8—301no—
399C4⋊C4—21no—
400C4⋊C4—101no—
401C4⋊C4—302no—
402C22⋊C4—21no—
403C22⋊C4—101no—
404C22⋊C4—302no—
405C22⋊C4—302no—
406C42—21no—
407C42—31no—
408C42—31no—
409C42—61no—
410C42—101no—
411C3×C5#188411yescharacteristic, complemented, Hall
412C22×C3—41no—
413C22×C3—102no—
414C22×C3—202no—
415C22×C3D5×C22⋊C411yescharacteristic
416C22×C3#403011yescharacteristic
417C22×C3#404111yescharacteristic
418C2×D3—202no—
419C2×D3—201no—
420C3×C4—81no—
421C3×C4—102no—
422C3×C4—401no—
423C3×C4C4×D2011yescharacteristic
424C3×C4#402811yescharacteristic
425C3×C4#403911yescharacteristic
426C3×C4#404011yescharacteristic
427Dic3—22no—
428Dic3—41no—
429Dic3—201no—
430C2×C5—21no—
431C2×C5—21no—
432C2×C5—21no—
433C2×C5#495811yescharacteristic
434D5—42no—
435D5—241no—
436C23—52no—
437C23—301no—
438C23D5×#6711yescharacteristic
439Q8—101no—
440Q8—101no—
441Q8—301no—
442Q8—301no—
443D4—102no—
444D4—102no—
445D4—101no—
446D4—101no—
447D4—301no—
448D4—301no—
449D4—604no—
450D4—601no—
451D4—601no—
452C2×C4—41no—
453C2×C4—52no—
454C2×C4—61no—
455C2×C4—61no—
456C2×C4—102no—
457C2×C4—102no—
458C2×C4—122no—
459C2×C4—201no—
460C2×C4—301no—
461C2×C4—601no—
462C2×C4—601no—
463C2×C4#676611yescharacteristic
464C2×C4C4×D3×D511yescharacteristic
465C2×C4D5×D1211yescharacteristic
466C2×C4D3×D2011yescharacteristic
467C2×C4#677911yescharacteristic
468C2×C4#679011yescharacteristic
469C8—62no—
470C8—302no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—202no—
475C2×C3#6495411yescharacteristic
476D3—401no—
477C5#7004511yescharacteristic, complemented, Sylow
478C22—41no—
479C22—102no—
480C22—202no—
481C22—602no—
482C22—601no—
483C22#9140911yescharacteristic
484C22D5×(C2×D3)⋊C411yescharacteristic
485C22#9155011yescharacteristic
486C4—62no—
487C4—81no—
488C4—102no—
489C4—121no—
490C4—401no—
491C4—601no—
492C4#9139511yescharacteristic
493C4#9139611yescharacteristic
494C4#9141311yescharacteristic
495C4#9147611yescharacteristic
496C3#1059957011yescharacteristic, complemented, Sylow
497C2—21no—
498C2—21no—
499C2—21no—
500C2—202no—
501C2—1201no—
502C2#1174887211yescharacteristic
subgroupG/H
centerC4#91476
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC3×C5×#117C22
socleC2×C3×C5#386
prime radical#11754581C2
cosocleC22×C3×C4×C5C23
hypercenter#117D3×D5
hypocenterC3×C5#1884
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
2. #11754120C2semidirect
3. #11751106C2semidirect
4. C3×#65295C2semidirect
5. #11748429C2semidirect
6. C5×#4532C2semidirect
7. #11745974C2semidirect
17. #91458C4semidirect
19. #91389C4semidirect
20. #91374C4semidirect
21. #91364C4semidirect
32. #65295D3semidirect
59. #6710D4semidirect
65. C5×#208D4semidirect
81. #4532D5semidirect
255. #208D20semidirect
411. C3×C5#1884semidirect
477. C5#70045semidirect
496. C3#10599570semidirect

Maximal subgroups

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

subgroupindex
#117545812
#117541202
#117511062
C3×#652952
#117484292
C5×#45322
#117459742
#105995703
#700455