Finite Group Databasebeta

← #14417249

Subgroups — #14417249

Order 1920.

940 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756263C211yesmaximal, characteristic
2#11754211C211yesmaximal, characteristic, complemented
3C2×#91321C211yesmaximal, characteristic
4C2×C5×#810C211yesmaximal, characteristic, complemented
5C3×#65253C211yesmaximal, characteristic, complemented
6#11748411C211yesmaximal, characteristic
7#11748326C211yesmaximal, characteristic, complemented
8#11747281C212yesmaximal, complemented
9#11747280C212yesmaximal, complemented
10#11747279C212yesmaximal, complemented
11#11747278C212yesmaximal, complemented
12#10610505—31nomaximal, Hall
13C2×#6817C2211yescharacteristic
14C22×C5×Dic6C2211yescharacteristic
15C2×C3×C5×C4⋊C4C2211yescharacteristic
16C4×#1220—22no—
17#91815C2211yescharacteristic
18#91782C2212yes—
19#91781—21no—
20C4⋊C4×D15—21no—
21#91776C2212yes—
22C22×C5×C3⋊C8C2211yescharacteristic
23C3×#4072C2211yescharacteristic
24C2×C3×#681C2211yescharacteristic
25#91529—22no—
26#91344C2212yes—
27#91344C2212yes—
28Dic5×Dic6—21no—
29Dic5×Dic6—21no—
30#91321C418yescomplemented
31C2×#6663C2212yes—
32C2×#6660C2212yes—
33C5×#810C2212yes—
34C5×#810C2212yes—
35C3×#3940C2212yes—
36C3×#3939C2212yes—
37#90974C2211yescharacteristic
38#90974C2211yescharacteristic
39#90968C2211yescharacteristic
40#90968C2211yescharacteristic
41#90954C2212yes—
42#90951C2212yes—
43#79202—51nomaximal, Hall
44#66146—31no—
45C2×#4088—31no—
46#65838—31no—
47C2×C5×#103—31no—
48#65402—31no—
49#65390—31no—
50#65253D311yescharacteristic, complemented
51#65106—32no—
52#65079—32no—
53#65078—32no—
54#65077—32no—
55C22×C3×C4×C5—21no—
56C22×C3×C4×C5C2311yescharacteristic
57C2×#1220—21no—
58#6817—22no—
59#6817C2×C414yes—
60C2×D60C2311yescharacteristic
61C2×C4×D15—21no—
62C2×C4×D15—22no—
63C2×Dic30C2311yescharacteristic
64C22×C5×Dic3—21no—
65C2×C5×Dic6—21no—
66C2×C5×Dic6—22no—
67C2×C5×Dic6C2×C414yescomplemented
68C2×C5×Dic6C2312yes—
69C3×C5×C4⋊C4—21no—
70C3×C5×C4⋊C4C2312yes—
71#6719D412yes—
72#6714—22no—
73C4⋊Dic15D411yescharacteristic
74C4⋊Dic15D411yescharacteristic
75#6712—22no—
76C4×Dic15—22no—
77C4×Dic15C2312yes—
78C2×C5×C3⋊C8—22no—
79C2×C5×C3⋊C8C2311yescharacteristic
80C2×C5×C3⋊C8C2311yescharacteristic
81C3×C22⋊Dic5—21no—
82C3×C4⋊Dic5C2311yescharacteristic
83C3×C4⋊Dic5C2311yescharacteristic
84C3×C4×Dic5—21no—
85C3×#681C2×C414yescomplemented
86C2×C3×#168C2312yes—
87Dic3⋊Dic5—22no—
88Dic3⋊Dic5—22no—
89Dic3×Dic5—22no—
90Dic3×Dic5—22no—
91#6663C2×C418yescomplemented
92#6660C2×C418yescomplemented
93C2×#909—51no—
94#5516—51no—
95#5504—51no—
96C3×#428—51no—
97#5122—51no—
98#5002—51no—
99C2×#810D511yescharacteristic, complemented
100#4889—52no—
101#4873—52no—
102#4853—52no—
103#4852—52no—
104C22×C5×Q8—31no—
105C2×#709—31no—
106C22×C5×C8—31no—
107C2×C5×C4⋊C4C2×D311yescharacteristic
108Q8×Dic5—32no—
109Q8×Dic5—32no—
110Q8×Dic5—61no—
111Q8×Dic5—61no—
112C2×#689—32no—
113#4088—38no—
114C2×#688—32no—
115C4×C22⋊D5—62no—
116#4072—31no—
117#4072—62no—
118#4072C2×D311yescharacteristic
119C2×#681C2×D311yescharacteristic
120#4039—32no—
121#4038—61no—
122D5×C4⋊C4—61no—
123#4033—32no—
124C5×#103—32no—
125C5×#103—32no—
126#3953—31no—
127#3953—31no—
128#3948—31no—
129#3948—31no—
130#3947—32no—
131C8×Dic5—32no—
132#3940C2×D312yes—
133#3939C2×D312yes—
134#2947—151noSylow
135C23×C3×C5C2×D411yescharacteristic
136C22×D15—21no—
137C2×C3×C4×C5—22no—
138C2×C3×C4×C5—81no—
139C2×C3×C4×C5C22×C412yes—
140C2×C3×C4×C5C2×D411yescharacteristic
141C2×C3×C4×C5#4112yes—
142C2×C3×C4×C5C2411yescharacteristic
143#1220—24no—
144C2×Dic15—21no—
145C2×Dic15C2×D412yes—
146C2×Dic15C2×D412yes—
147D60—21no—
148D60C22×C412yes—
149C4×D15—24no—
150C4×D15—82no—
151Dic30—21no—
152Dic30C22×C412yes—
153C2×C5×Dic3—24no—
154C2×C5×Dic3—22no—
155C5×Dic6—24no—
156C5×Dic6—22no—
157C5×Dic6C22×C414yes—
158C2×C3×Dic5—21no—
159C2×C3×Dic5—21no—
160C3×#168C22×C414yes—
161C5×C3⋊C8—21no—
162C5×C3⋊C8—21no—
163C2×#235—51no—
164C22×Dic6C2×D511yescharacteristic
165C2×C3×M16—51no—
166C3×#117—51no—
167C2×C3×C4⋊C4C2×D511yescharacteristic
168C4×#67—102no—
169#926—51no—
170C22×C3⋊C8C2×D511yescharacteristic
171#909—58no—
172C2×Dic12—52no—
173C2×#204—52no—
174#893—52no—
175#892—101no—
176D3×C4⋊C4—101no—
177#887—52no—
178#887—52no—
179#887—52no—
180#877—102no—
181C4×Dic6—101no—
182C4×Dic6—101no—
183C3×#107—52no—
184C3×#106—52no—
185#815—52no—
186C8×Dic3—52no—
187#810—51no—
188#810—51no—
189#810C2×D512yes—
190#810C2×D512yes—
191#809—51no—
192#809—51no—
193C2×C5×Q8—34no—
194C2×C5×Q8—32no—
195C2×C5×Q8—61no—
196C2×C5×Q8—62no—
197C22×C4×C5—21no—
198C22×C4×C5—61no—
199C22×C4×C5C22×D311yescharacteristic
200C2×C22⋊D5—61no—
201#709—34no—
202#709—62no—
203C2×D20—31no—
204C2×C4×D5—61no—
205C2×C4×D5—62no—
206C2×Dic10—31no—
207C2×C5×C8—31no—
208C2×C5×C8—31no—
209C2×C5×C8—62no—
210C5×C4⋊C4—21no—
211C5×C4⋊C4C22×D312yes—
212C22⋊Dic5—21no—
213C22⋊Dic5—32no—
214#689—38no—
215#688—38no—
216#685—62no—
217C4⋊Dic5—31no—
218C4⋊Dic5—31no—
219C4⋊Dic5—62no—
220C4⋊Dic5—62no—
221C4⋊Dic5C22×D311yescharacteristic
222C4⋊Dic5C22×D311yescharacteristic
223#683—62no—
224C4×Dic5—21no—
225C4×Dic5—32no—
226C4×Dic5—62no—
227C4×Dic5—62no—
228C4×Dic5—62no—
229#681C4×D314yescomplemented
230C2×#168C22×D312yes—
231C2×#137—151no—
232#520—151no—
233#519—151no—
234#441—152no—
235#440—152no—
236C4×Q16—152no—
237C4×SD16—152no—
238#428—51no—
239#416—151no—
240C2×#103—31no—
241#405—151no—
242C22×C3×C5—21no—
243C22×C3×C5—21no—
244C22×C3×C5C4×D412yes—
245C22×C3×C5C22×D411yescharacteristic
246C2×D15—22no—
247C2×D15—41no—
248C2×D15—41no—
249C3×C4×C5—42no—
250C3×C4×C5C4×D412yes—
251C3×C4×C5C23×C411yescharacteristic
252C3×C4×C5C2×#4111yescharacteristic
253Dic15—22no—
254Dic15—22no—
255Dic15—22no—
256C3×Dic5—41no—
257C3×Dic5—41no—
258C5×Dic3—24no—
259C22×C3×C4—21no—
260C22×C3×C4C22×D511yescharacteristic
261C2×#67—101no—
262C22×Dic3—21no—
263#235—54no—
264#235—102no—
265C2×D12—51no—
266C2×C4×D3—101no—
267C2×C4×D3—102no—
268C2×Dic6—21no—
269C2×Dic6—22no—
270C2×Dic6—51no—
271C2×Dic6C4×D514yescomplemented
272C2×Dic6C22×D512yes—
273C3×M16—54no—
274C2×C3×C8—52no—
275C3×C4⋊C4—21no—
276C3×C4⋊C4—51no—
277C3×C4⋊C4—51no—
278C3×C4⋊C4C22×D512yes—
279C3×C22⋊C4—101no—
280C3×C42—101no—
281#217—52no—
282(C2×D3)⋊C4—102no—
283C4⋊Dic3—51no—
284C4⋊Dic3—51no—
285Dic3⋊C4—102no—
286Dic3⋊C4—102no—
287Dic3⋊C4—102no—
288C4×Dic3—52no—
289C4×Dic3—102no—
290C4×Dic3—102no—
291C4×Dic3—102no—
292C2×C3⋊C8—22no—
293C2×C3⋊C8C22×D511yescharacteristic
294C2×C3⋊C8C22×D511yescharacteristic
295Dic12—58no—
296#204—58no—
297C23×C5D3×D411yescharacteristic
298C22×D5—61no—
299C5×Q8—34no—
300C5×Q8—64no—
301C5×Q8—62no—
302C2×C4×C5—22no—
303C2×C4×C5—64no—
304C2×C4×C5—62no—
305C2×C4×C5—81no—
306C2×C4×C5C2×C4×D312yes—
307C2×C4×C5D3×D411yescharacteristic
308C2×C4×C5#23912yes—
309C2×C4×C5C23×D311yescharacteristic
310C22⋊D5—64no—
311C2×Dic5—21no—
312C2×Dic5—21no—
313C2×Dic5—32no—
314C2×Dic5—32no—
315C2×Dic5—61no—
316D20—32no—
317D20—61no—
318C4×D5—64no—
319C4×D5—242no—
320Dic10—32no—
321Dic10—61no—
322C5×C8—61no—
323C5×C8—61no—
324#168C2×C4×D314yes—
325C2×#41—151no—
326C22×Q8—31no—
327#137—158no—
328C2×Q16—152no—
329C2×SD16—152no—
330C2×M16—51no—
331C22×C8—31no—
332C4×Q8—152no—
333C4×Q8—152no—
334C4×Q8—152no—
335C4×Q8—301no—
336C4×Q8—301no—
337C4×D4—152no—
338C4×D4—302no—
339#117—51no—
340#117—151no—
341#117—302no—
342#117—301no—
343C2×C4⋊C4—301no—
344C2×C4⋊C4D3×D511yescharacteristic
345#107—52no—
346#106—52no—
347#103—32no—
348#103—32no—
349#103—151no—
350#103—151no—
351#102—151no—
352#102—151no—
353#97—152no—
354C4×C8—152no—
355C2×C3×C5—22no—
356C2×C3×C5C2×C4×D411yescharacteristic
357C2×C3×C5#57711yescharacteristic
358C2×C3×C5#57811yescharacteristic
359D15—42no—
360C23×C3D4×D511yescharacteristic
361C22×D3—101no—
362C2×C3×C4—22no—
363C2×C3×C4—81no—
364C2×C3×C4—101no—
365C2×C3×C4—101no—
366C2×C3×C4C2×C4×D512yes—
367C2×C3×C4D4×D511yescharacteristic
368C2×C3×C4#71112yes—
369C2×C3×C4C23×D511yescharacteristic
370#67—104no—
371C2×Dic3—22no—
372C2×Dic3—24no—
373C2×Dic3—52no—
374C2×Dic3—52no—
375C2×Dic3—101no—
376D12—52no—
377D12—101no—
378C4×D3—104no—
379C4×D3—402no—
380Dic6—22no—
381Dic6—24no—
382Dic6—52no—
383Dic6—101no—
384Dic6C2×C4×D514yes—
385C3×C8—54no—
386C3⋊C8—21no—
387C3⋊C8—21no—
388C22×C5—21no—
389C22×C5—21no—
390C22×C5#89312yes—
391C22×C5C2×D3×D411yescharacteristic
392C2×D5—62no—
393C2×D5—121no—
394C2×D5—121no—
395C4×C5—42no—
396C4×C5—64no—
397C4×C5#89312yes—
398C4×C5C22×C4×D311yescharacteristic
399C4×C5C2×#23911yescharacteristic
400Dic5—41no—
401Dic5—41no—
402Dic5—62no—
403Dic5—62no—
404Dic5—62no—
405#41—154no—
406#41—302no—
407C2×Q8—34no—
408C2×Q8—32no—
409C2×Q8—62no—
410C2×Q8—61no—
411C2×Q8—151no—
412C2×D4—151no—
413C2×D4—301no—
414C22×C4—21no—
415C22×C4—61no—
416C22×C4—302no—
417C22×C4—301no—
418C22×C4C2×D3×D511yescharacteristic
419Q16—158no—
420SD16—158no—
421M16—54no—
422C2×C8—31no—
423C2×C8—31no—
424C2×C8—52no—
425C2×C8—62no—
426C4⋊C4—21no—
427C4⋊C4—51no—
428C4⋊C4—51no—
429C4⋊C4—151no—
430C4⋊C4—151no—
431C4⋊C4—302no—
432C4⋊C4—302no—
433C4⋊C4—302no—
434C4⋊C4C2×D3×D512yes—
435C22⋊C4—101no—
436C22⋊C4—152no—
437C22⋊C4—302no—
438C42—101no—
439C42—152no—
440C42—302no—
441C42—302no—
442C42—302no—
443C3×C5#294711yescharacteristic, complemented, Hall
444C22×C3—21no—
445C22×C3—21no—
446C22×C3#402712yes—
447C22×C3C2×D4×D511yescharacteristic
448C2×D3—102no—
449C2×D3—201no—
450C2×D3—201no—
451C3×C4—42no—
452C3×C4—201no—
453C3×C4—201no—
454C3×C4#402712yes—
455C3×C4C22×C4×D511yescharacteristic
456C3×C4C2×#71111yescharacteristic
457Dic3—24no—
458Dic3—102no—
459Dic3—102no—
460Dic3—102no—
461C2×C5—22no—
462C2×C5C2×#89311yescharacteristic
463C2×C5#577511yescharacteristic
464C2×C5#577911yescharacteristic
465D5—122no—
466C23—301no—
467C23#679011yescharacteristic
468Q8—34no—
469Q8—64no—
470Q8—62no—
471Q8—152no—
472Q8—301no—
473D4—152no—
474D4—304no—
475D4—301no—
476C2×C4—22no—
477C2×C4—64no—
478C2×C4—62no—
479C2×C4—81no—
480C2×C4—101no—
481C2×C4—101no—
482C2×C4—152no—
483C2×C4—152no—
484C2×C4—301no—
485C2×C4—304no—
486C2×C4—1202no—
487C2×C4#676812yes—
488C2×C4C4×D3×D512yes—
489C2×C4#679011yescharacteristic
490C2×C4C22×D3×D511yescharacteristic
491C8—54no—
492C8—61no—
493C8—61no—
494C2×C3—22no—
495C2×C3C2×#402711yescharacteristic
496C2×C3#6609611yescharacteristic
497C2×C3#6610011yescharacteristic
498D3—202no—
499C5#7920211yescharacteristic, complemented, Sylow
500C22—21no—
501C22—21no—
502C22—302no—
503C22—601no—
504C22—601no—
505C22#9140612yes—
506C22C2×#679011yescharacteristic
507C4—42no—
508C4—64no—
509C4—201no—
510C4—201no—
511C4—302no—
512C4—302no—
513C4—302no—
514C4#9140612yes—
515C4C2×#676811yescharacteristic
516C4C2×C4×D3×D511yescharacteristic
517C3#1061050511yescharacteristic, complemented, Sylow
518C2—22no—
519C2—602no—
520C2C2×#9140611yescharacteristic
521C2#1175472511yescharacteristic
522C2#1175511911yescharacteristic
subgroupG/H
centerC22C2×#6790
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×C4⋊C4C22
socleC22×C3×C5C22×D4
prime radicalC2×#6817C22
cosocleC2×C3×C4×C5C24
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5#2947
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #11754211C2semidirect
4. C2×C5×#810C2semidirect
5. C3×#65253C2semidirect
7. #11748326C2semidirect
8. #11747281C2semidirect
9. #11747280C2semidirect
10. #11747279C2semidirect
11. #11747278C2semidirect
30. #91321C4semidirect
50. #65253D3semidirect
67. C2×C5×Dic6C2×C4semidirect
85. C3×#681C2×C4semidirect
91. #6663C2×C4semidirect
92. #6660C2×C4semidirect
99. C2×#810D5semidirect
229. #681C4×D3semidirect
271. C2×Dic6C4×D5semidirect
443. C3×C5#2947semidirect
499. C5#79202semidirect
517. C3#10610505semidirect

Maximal subgroups

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

subgroupindex
#117562632
#117542112
C2×#913212
C2×C5×#8102
C3×#652532
#117484112
#117483262
#117472812
#117472802
#117472792
#117472782
#106105053
#792025