Finite Group Databasebeta

← #14302845

Subgroups — #14302845

Order 1920.

688 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11754600C211yesmaximal, characteristic
2#11754125C211yesmaximal, characteristic, complemented
3C5×#5006C211yesmaximal, characteristic, complemented
4C3×#65290C211yesmaximal, characteristic, complemented
5#11748366C211yesmaximal, characteristic, complemented
6#11746352C211yesmaximal, characteristic, complemented
7#11745971C211yesmaximal, characteristic, complemented
8#10600840—31nomaximal, Hall
9C2×C5×#235C2211yescharacteristic
10C22×C3×D20C2211yescharacteristic
11C3×C5×#117C2211yescharacteristic
12#91815C2211yescharacteristic
13C2×#6710C2211yescharacteristic
14C5×#926C2211yescharacteristic
15C2×C3×#681C2211yescharacteristic
16#91563—22no—
17#91463C412yescomplemented
18#91456—22no—
19#91368C412yescomplemented
20#91288—22no—
21D3×#681—22no—
22#91286—22no—
23C5×#806—22no—
24C3×#3941—22no—
25#90964—22no—
26#70330—51nomaximal, Hall
27#66164—31no—
28#66153—31no—
29C5×#410—31no—
30#65290—31no—
31#65290D311yescharacteristic, complemented
32#64751—31no—
33#64751—31no—
34C23×C3×D5—21no—
35C22×C3×C4×C5C2311yescharacteristic
36C2×C5×#67D412yes—
37C5×#235—21no—
38C5×#235—21no—
39C5×#235—21no—
40C5×#235—21no—
41C2×C5×D12C2×C411yescharacteristic
42C2×C4×C5×D3D412yes—
43C2×C5×Dic6C2×C411yescharacteristic
44C2×C3×D20—21no—
45C2×C3×D20—21no—
46C2×C3×D20—21no—
47C2×C3×D20—21no—
48C2×C3×D20—21no—
49C2×C3×D20C2×C412yes—
50C2×#1201—24no—
51C3×C5×C4⋊C4—21no—
52C3×C5×C22⋊C4—21no—
53C3×C42×C5—21no—
54#6719—21no—
55C4⋊Dic15—21no—
56C4×Dic15C2×C412yes—
57#6710—21no—
58#6710—21no—
59C2×C3⋊#168D412yescomplemented
60C5×#217—21no—
61C5×C4⋊Dic3—21no—
62C4×C5×Dic3—21no—
63C3×#681—21no—
64C3×#681—21no—
65C2×C3×#168D412yescomplemented
66#6665—24no—
67#6650—24no—
68D3×#168—24no—
69#5805—51no—
70#5745—51no—
71C3×#418—51no—
72#5215—51no—
73#5006D511yescharacteristic, complemented
74#4557—51no—
75#4531—51no—
76C2×C5×#41—31no—
77C22×D20C2×D311yescharacteristic
78C5×#117—31no—
79C5×#117C2×D311yescharacteristic
80#4094—62no—
81#4094—62no—
82#4093—32no—
83#4086—32no—
84#4083—62no—
85#4076—62no—
86#4072—31no—
87C2×#681—31no—
88C2×#681—62no—
89C2×#681C2×D311yescharacteristic
90C5×#99—62no—
91#3941—22no—
92#3941—62no—
93#1910—151noSylow
94C23×C3×C5C2×D411yescharacteristic
95C22×C5×D3C22⋊C412yes—
96C22×C3×D5—22no—
97C22×C3×D5—41no—
98C22×C3×D5—41no—
99C22×C3×D5—82no—
100C2×C3×C4×C5—41no—
101C2×C3×C4×C5C22×C411yescharacteristic
102C2×C3×C4×C5C2×D411yescharacteristic
103C2×C3×C4×C5C2×D411yescharacteristic
104C2×C3×C4×C5C2×D411yescharacteristic
105C2×C3×C4×C5#4111yescharacteristic
106C2×C3×C4×C5#4111yescharacteristic
107C2×Dic15—22no—
108C5×#67—22no—
109C5×#67—22no—
110C2×C5×Dic3—41no—
111C2×C5×Dic3C22⋊C412yes—
112C5×D12—21no—
113C5×D12—21no—
114C4×C5×D3—22no—
115C4×C5×D3—22no—
116C5×Dic6—21no—
117C5×Dic6—21no—
118C3×D20—22no—
119C3×D20—22no—
120C3×D20—41no—
121C3×D20—41no—
122#1201—84no—
123C3⋊#168—22no—
124C3×#168—22no—
125C22×C3×D4—51no—
126C2×#235C2×D511yescharacteristic
127C2×C3×M16—51no—
128C3×#117C2×D511yescharacteristic
129#953—102no—
130#940—52no—
131#938—102no—
132#936—52no—
133#926—51no—
134#926C2×D511yescharacteristic
135C2×#208—51no—
136#907—102no—
137D3×M16—102no—
138#901—102no—
139C3×#102—102no—
140#832—102no—
141#806—22no—
142C23×D5—21no—
143C5×#41—61no—
144C5×#41—61no—
145C5×#41—61no—
146C5×#41—61no—
147C2×C5×Q8—31no—
148C2×C5×D4—32no—
149C2×C5×D4—31no—
150C22×C4×C5—32no—
151C22×C4×C5C22×D311yescharacteristic
152C2×C22⋊D5—64no—
153C2×D20—21no—
154C2×D20—21no—
155C2×D20—21no—
156C2×D20—21no—
157C2×D20—21no—
158C2×D20C4×D312yes—
159C5×C4⋊C4—21no—
160C5×C4⋊C4—61no—
161C5×C22⋊C4—21no—
162C5×C22⋊C4—61no—
163C42×C5—21no—
164C42×C5—61no—
165C22⋊Dic5—61no—
166#685—64no—
167C4⋊Dic5—61no—
168C4×Dic5—32no—
169#681—21no—
170#681—21no—
171#681—64no—
172#681—61no—
173#681—61no—
174C2×#168—32no—
175C2×#168—64no—
176C2×#168#6712yescomplemented
177#568—151no—
178#535—151no—
179#418—51no—
180#418—151no—
181#410—31no—
182H(3,ℤ/4)—151no—
183H(3,ℤ/4)—151no—
184C22×C3×C5—41no—
185C22×C3×C5C2×C22⋊C411yescharacteristic
186C22×C3×C5#12111yescharacteristic
187C22×C3×C5#12111yescharacteristic
188C2×C5×D3—22no—
189C2×C5×D3—42no—
190C2×C3×D5—44no—
191C2×C3×D5—42no—
192C2×C3×D5—82no—
193C2×C3×D5—82no—
194C3×C4×C5—81no—
195C3×C4×C5C4×D411yescharacteristic
196C3×C4×C5C4×D411yescharacteristic
197C3×C4×C5#12011yescharacteristic
198C3×C4×C5#12311yescharacteristic
199Dic15—42no—
200C5×Dic3—22no—
201C5×Dic3—81no—
202C24×C3—101no—
203C2×C3×D4—52no—
204C2×C3×D4—101no—
205C2×C3×D4—101no—
206C2×C3×D4—101no—
207C2×C3×D4—101no—
208C2×C3×D4—101no—
209C22×C3×C4C22×D511yescharacteristic
210C2×#67—104no—
211C2×#67C22⋊D512yes—
212#235—21no—
213#235—21no—
214#235—21no—
215#235—21no—
216C2×D12C4×D511yescharacteristic
217C2×C4×D3D2012yes—
218C2×Dic6C4×D511yescharacteristic
219C3×M16—101no—
220C3×M16—101no—
221C2×C3×C8—52no—
222C3×C4⋊C4—21no—
223C3×C22⋊C4—21no—
224C3×C42—21no—
225#217—21no—
226#217—101no—
227#217—104no—
228C4⋊Dic3—21no—
229C4⋊Dic3—101no—
230C4×Dic3—21no—
231C4×Dic3—52no—
232#208—101no—
233#208—101no—
234C2×C3⋊C8—52no—
235ΓL(1,𝔽25)—104no—
236D3×C8—104no—
237C23×C5—32no—
238C23×C5D3×D411yescharacteristic
239C22×D5—22no—
240C22×D5—41no—
241C22×D5—41no—
242C22×D5—82no—
243C5×Q8—61no—
244C5×Q8—61no—
245C5×D4—62no—
246C5×D4—62no—
247C5×D4—61no—
248C5×D4—61no—
249C2×C4×C5—32no—
250C2×C4×C5—41no—
251C2×C4×C5—62no—
252C2×C4×C5—62no—
253C2×C4×C5—121no—
254C2×C4×C5C2×C4×D311yescharacteristic
255C2×C4×C5#23511yescharacteristic
256C2×C4×C5D3×D411yescharacteristic
257C2×C4×C5D3×D411yescharacteristic
258C2×C4×C5#23711yescharacteristic
259C2×C4×C5C2×#6711yescharacteristic
260C22⋊D5—244no—
261C2×Dic5—62no—
262D20—22no—
263D20—22no—
264D20—41no—
265D20—41no—
266#168—22no—
267#168—62no—
268C2×#41—31no—
269C22×D4—51no—
270#131—302no—
271#131—302no—
272C2×M16—51no—
273C2×M16—151no—
274C2×M16—302no—
275#127—152no—
276#124—152no—
277#121—302no—
278#120—302no—
279#117—31no—
280#117—151no—
281#117D3×D511yescharacteristic
282#102—102no—
283#102—302no—
284#99—62no—
285C2×C3×C5—21no—
286C2×C3×C5—21no—
287C2×C3×C5—21no—
288C2×C3×C5#38611yescharacteristic
289C3×D5—82no—
290C5×D3—42no—
291C23×C3—102no—
292C23×C3—201no—
293C23×C3—201no—
294C23×C3—402no—
295C23×C3C2×C22⋊D511yescharacteristic
296C22×D3#68512yes—
297C3×D4—102no—
298C3×D4—102no—
299C3×D4—201no—
300C3×D4—201no—
301C2×C3×C4—41no—
302C2×C3×C4C2×C4×D511yescharacteristic
303C2×C3×C4C2×D2011yescharacteristic
304C2×C3×C4D4×D511yescharacteristic
305C2×C3×C4D4×D511yescharacteristic
306C2×C3×C4#71111yescharacteristic
307C2×C3×C4#71111yescharacteristic
308#67—22no—
309#67—22no—
310#67—404no—
311C2×Dic3—41no—
312C2×Dic3—102no—
313C2×Dic3#68512yes—
314D12—21no—
315D12—21no—
316C4×D3—22no—
317C4×D3—22no—
318Dic6—21no—
319Dic6—21no—
320C3×C8—102no—
321C3⋊C8—102no—
322C22×C5—41no—
323C22×C5—62no—
324C22×C5—122no—
325C22×C5D3×C22⋊C411yescharacteristic
326C22×C5#88411yescharacteristic
327C22×C5#93811yescharacteristic
328C2×D5—44no—
329C2×D5—42no—
330C2×D5—82no—
331C2×D5—82no—
332C4×C5—62no—
333C4×C5—81no—
334C4×C5—241no—
335C4×C5#67⋊C411yescharacteristic
336C4×C5#88311yescharacteristic
337C4×C5C4×#6711yescharacteristic
338C4×C5#93711yescharacteristic
339Dic5—122no—
340C24—101no—
341#41—61no—
342#41—61no—
343#41—61no—
344#41—61no—
345C2×Q8—31no—
346C2×D4—32no—
347C2×D4—31no—
348C2×D4—52no—
349C2×D4—101no—
350C2×D4—101no—
351C2×D4—101no—
352C2×D4—101no—
353C2×D4—101no—
354C2×D4—304no—
355C22×C4—32no—
356C22×C4C2×D3×D511yescharacteristic
357M16—101no—
358M16—101no—
359M16—304no—
360M16—301no—
361M16—301no—
362C2×C8—52no—
363C2×C8—152no—
364C2×C8—304no—
365C4⋊C4—21no—
366C4⋊C4—61no—
367C4⋊C4—301no—
368C22⋊C4—21no—
369C22⋊C4—61no—
370C22⋊C4—304no—
371C22⋊C4—301no—
372C42—21no—
373C42—61no—
374C42—152no—
375C3×C5#191011yescharacteristic, complemented, Hall
376C22×C3—41no—
377C22×C3—204no—
378C22×C3—202no—
379C22×C3—402no—
380C22×C3—402no—
381C22×C3C2×#68511yescharacteristic
382C22×C3#408511yescharacteristic
383C22×C3#408511yescharacteristic
384C2×D3—22no—
385C2×D3—42no—
386C3×C4—81no—
387C3×C4#402711yescharacteristic
388C3×C4#402711yescharacteristic
389C3×C4#402811yescharacteristic
390C3×C4#403211yescharacteristic
391Dic3—22no—
392Dic3—81no—
393Dic3—202no—
394C2×C5—21no—
395C2×C5—21no—
396C2×C5—21no—
397C2×C5—122no—
398C2×C5#495611yescharacteristic
399D5—82no—
400C23—32no—
401C23—102no—
402C23—201no—
403C23—201no—
404C23—402no—
405C23D3×C22⋊D511yescharacteristic
406Q8—61no—
407Q8—61no—
408D4—62no—
409D4—62no—
410D4—61no—
411D4—61no—
412D4—102no—
413D4—102no—
414D4—201no—
415D4—201no—
416D4—1204no—
417C2×C4—32no—
418C2×C4—41no—
419C2×C4—62no—
420C2×C4—62no—
421C2×C4—121no—
422C2×C4—302no—
423C2×C4#677211yescharacteristic
424C2×C4C4×D3×D511yescharacteristic
425C2×C4D3×D2011yescharacteristic
426C2×C4#677711yescharacteristic
427C2×C4#678011yescharacteristic
428C2×C4D5×#6711yescharacteristic
429C8—102no—
430C8—302no—
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—402no—
435C2×C3#6524711yescharacteristic
436D3—42no—
437C5#7033011yescharacteristic, complemented, Sylow
438C22—41no—
439C22—62no—
440C22—122no—
441C22—204no—
442C22—202no—
443C22—402no—
444C22—402no—
445C22#9145311yescharacteristic
446C22D3×#68511yescharacteristic
447C22#9156511yescharacteristic
448C4—62no—
449C4—81no—
450C4—241no—
451C4—602no—
452C4#9144011yescharacteristic
453C4#9144211yescharacteristic
454C4#9145711yescharacteristic
455C4#9147411yescharacteristic
456C3#1060084011yescharacteristic, complemented, Sylow
457C2—21no—
458C2—21no—
459C2—21no—
460C2—122no—
461C2—402no—
462C2#1174891111yescharacteristic
subgroupG/H
centerC2#11748911
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC3×C5×#117C22
socleC2×C3×C5#386
prime radical#11754600C2
cosocleC22×C3×C4×C5C23
hypercenter#117D3×D5
hypocenterC3×C5#1910
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
2. #11754125C2semidirect
3. C5×#5006C2semidirect
4. C3×#65290C2semidirect
5. #11748366C2semidirect
6. #11746352C2semidirect
7. #11745971C2semidirect
17. #91463C4semidirect
19. #91368C4semidirect
31. #65290D3semidirect
59. C2×C3⋊#168D4semidirect
65. C2×C3×#168D4semidirect
73. #5006D5semidirect
176. C2×#168#67semidirect
375. C3×C5#1910semidirect
437. C5#70330semidirect
456. C3#10600840semidirect

Maximal subgroups

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

subgroupindex
#117546002
#117541252
C5×#50062
C3×#652902
#117483662
#117463522
#117459712
#106008403
#703305