Finite Group Databasebeta

← #14413530

Subgroups — #14413530

Order 1920.

952 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
1C5×#5523C211yesmaximal, characteristic
2#11754492C211yesmaximal, characteristic
3C2×#91288C211yesmaximal, characteristic
4#11751099C211yesmaximal, characteristic, complemented
5C3×#65284C211yesmaximal, characteristic, complemented
6C2×#90989C211yesmaximal, characteristic, complemented
7#11748459C211yesmaximal, characteristic, complemented
8#11747454C212yesmaximal, complemented
9#11747437C212yesmaximal, complemented
10D3×#3939C212yesmaximal, complemented
11#11747191C212yesmaximal, complemented
12#10614293—31nomaximal, Hall
13C2×C5×#235C2211yescharacteristic
14C3×C5×#117C2211yescharacteristic
15#91815C2211yescharacteristic
16C22×C3⋊#168C2211yescharacteristic
17C2×C5×C4⋊Dic3C2211yescharacteristic
18C5×#892C2212yes—
19C5×D3×C4⋊C4C2212yes—
20C5×#67⋊C4—22no—
21C4×C5×D12—21no—
22C4×C5×Dic6—21no—
23C2×C3×C4⋊Dic5C2211yescharacteristic
24C2×C3×#681C2211yescharacteristic
25Dic5×#67—22no—
26#91435—21no—
27D3×C4⋊Dic5C2212yes—
28#91344—21no—
29#91338C2212yes—
30C2×#6650C2212yes—
31#91288C418yescomplemented
32C2×D3×#168C2212yes—
33#91092C2211yescharacteristic
34#91092C2211yescharacteristic
35#91091C2211yescharacteristic
36#91091C2211yescharacteristic
37C3×#3940C2212yes—
38C3×#3939C2212yes—
39#90989C2212yes—
40#90989C2212yes—
41#90988C2212yes—
42#90985C2212yes—
43#79210—51nomaximal, Hall
44C5×#520—31no—
45#66161—31no—
46C2×#4094—31no—
47#65287—32no—
48#65287—31no—
49#65284—32no—
50#65284D311yescharacteristic, complemented
51#65253—31no—
52#65253—32no—
53C2×#3939—31no—
54C2×#3939—32no—
55C22×C3×C4×C5C2311yescharacteristic
56C2×C5×#67D412yes—
57C22×C5×Dic3—21no—
58C5×#235C2×C414yes—
59C5×#235Q814yes—
60C2×C5×D12D411yescharacteristic
61C2×C4×C5×D3—22no—
62C2×C4×C5×D3C2312yes—
63C2×C5×Dic6D411yescharacteristic
64C22×C3×Dic5—21no—
65C2×D3×Dic5—22no—
66C3×C5×C4⋊C4C2311yescharacteristic
67C3×C5×C4⋊C4C2311yescharacteristic
68C3×C5×C22⋊C4—21no—
69C3×C42×C5—21no—
70#6719—21no—
71C4⋊Dic15C2311yescharacteristic
72C4⋊Dic15C2311yescharacteristic
73C4×Dic15—21no—
74C2×C3⋊#168C2×C414yescomplemented
75C2×C3⋊#168C2311yescharacteristic
76C2×C3⋊#168C2311yescharacteristic
77C5×(C2×D3)⋊C4—22no—
78C5×C4⋊Dic3—21no—
79C5×C4⋊Dic3C2312yes—
80C5×Dic3⋊C4—22no—
81C4×C5×Dic3—22no—
82C3×C4⋊Dic5—21no—
83C3×C4⋊Dic5C2312yes—
84C3×#681C2×C414yescomplemented
85C2×C3×#168C2312yes—
86Dic3⋊Dic5—22no—
87(C2×D3)⋊Dic5—22no—
88Dic3×Dic5—22no—
89#6650C2×C418yescomplemented
90D3×#168C2×C418yescomplemented
91C2×#907—51no—
92#5523D511yescharacteristic
93#5504—51no—
94C3×#428—51no—
95#5121—51no—
96#4998—51no—
97C2×#807—51no—
98#4881—52no—
99D3×#107—52no—
100#4861—52no—
101#4860—52no—
102C2×C5×#41—31no—
103C4×C5×Q8—61no—
104C4×C5×D4—61no—
105C4×C5×D4—62no—
106C5×#117—32no—
107C5×#117C2×D311yescharacteristic
108C2×C5×C4⋊C4—31no—
109C2×C5×C4⋊C4—32no—
110#4094—38no—
111Q8×Dic5—61no—
112D4×Dic5—61no—
113D4×Dic5—62no—
114#4072—31no—
115#4072—32no—
116C2×C4⋊Dic5—32no—
117C2×C4⋊Dic5C2×D311yescharacteristic
118C2×#681—32no—
119C2×#681C2×D311yescharacteristic
120C22×#168—31no—
121C22×#168—32no—
122#3940—31no—
123#3940—31no—
124#3940—32no—
125#3940C2×D312yes—
126#3939—31no—
127#3939—31no—
128#3939—32no—
129#3939—32no—
130#3939—32no—
131#3939C2×D312yes—
132#2913—151noSylow
133C23×C3×C5C2×D411yescharacteristic
134C22×C5×D3C2×D412yes—
135C2×C3×C4×C5—21no—
136C2×C3×C4×C5—21no—
137C2×C3×C4×C5C22×C412yes—
138C2×C3×C4×C5C2×D411yescharacteristic
139C2×C3×C4×C5C2×Q812yes—
140C2×C3×C4×C5C2411yescharacteristic
141C2×Dic15—21no—
142C2×Dic15—21no—
143C5×#67C4⋊C418yes—
144C2×C5×Dic3—22no—
145C2×C5×Dic3—81no—
146C2×C5×Dic3C2×D412yes—
147C5×D12C4⋊C414yes—
148C4×C5×D3—82no—
149C4×C5×D3C22×C414yes—
150C4×C5×D3C2×Q814yes—
151C5×Dic6C4⋊C414yes—
152C2×C3×Dic5—22no—
153C2×C3×Dic5—81no—
154D3×Dic5—82no—
155C3⋊#168C22×C412yes—
156C3⋊#168C22×C412yes—
157C3×#168C22×C414yes—
158C2×#235C2×D511yescharacteristic
159C2×C3×M16—51no—
160C3×#117C2×D511yescharacteristic
161C2×C3×C4⋊C4—51no—
162C4×#67—102no—
163#926—51no—
164C2×C4⋊Dic3C2×D511yescharacteristic
165C22×C3⋊C8—51no—
166#907—58no—
167C2×ΓL(1,𝔽25)—52no—
168C2×D3×C8—52no—
169#893—101no—
170#892—52no—
171#892C2×D512yes—
172D3×C4⋊C4—52no—
173D3×C4⋊C4C2×D512yes—
174#887—101no—
175#67⋊C4—22no—
176C4×D12—21no—
177C4×Dic6—21no—
178C3×#107—52no—
179C3×#106—52no—
180#818—52no—
181#817—52no—
182#808—51no—
183#808—51no—
184#807—51no—
185#807—51no—
186#807—52no—
187#807—52no—
188C5×#41—34no—
189C5×#41—34no—
190C2×C5×Q8—31no—
191C2×C5×D4—32no—
192C2×C5×D4—31no—
193C22×C4×C5—32no—
194C22×C4×C5—62no—
195C22×C4×C5—61no—
196C22×C4×C5C22×D311yescharacteristic
197C22×Dic5—21no—
198C22×Dic5—62no—
199C5×C4⋊C4—32no—
200C5×C4⋊C4—62no—
201C5×C4⋊C4—61no—
202C5×C4⋊C4C22×D311yescharacteristic
203C5×C4⋊C4C22×D311yescharacteristic
204C5×C22⋊C4—21no—
205C5×C22⋊C4—62no—
206C42×C5—21no—
207C42×C5—62no—
208C22⋊Dic5—62no—
209C22⋊Dic5—61no—
210C4⋊Dic5—21no—
211C4⋊Dic5—31no—
212C4⋊Dic5—31no—
213C4⋊Dic5—62no—
214C4⋊Dic5C22×D312yes—
215C4×Dic5—62no—
216C4×Dic5—61no—
217#681—38no—
218#681C4×D314yescomplemented
219C2×#168—34no—
220C2×#168—31no—
221C2×#168—31no—
222C2×#168—38no—
223C2×#168C22×D312yes—
224C2×#131—151no—
225#520—31no—
226#520—151no—
227#428—51no—
228#428—152no—
229#428—152no—
230#428—151no—
231#427—152no—
232#427—151no—
233C2×#107—152no—
234C2×#107—151no—
235C22×C3×C5—21no—
236C22×C3×C5—21no—
237C22×C3×C5C2×C4⋊C412yes—
238C22×C3×C5C22×D411yescharacteristic
239C2×C5×D3—22no—
240C2×C5×D3—22no—
241C2×C5×D3C2×C4⋊C414yes—
242C3×C4×C5—41no—
243C3×C4×C5—41no—
244C3×C4×C5C2×C4⋊C412yes—
245C3×C4×C5C23×C411yescharacteristic
246C3×C4×C5C22×Q811yescharacteristic
247Dic15—41no—
248Dic15—41no—
249C3×Dic5—42no—
250C5×Dic3—42no—
251C5×Dic3C2×C4⋊C414yes—
252C22×C3×C4—101no—
253C22×C3×C4C22×D511yescharacteristic
254C2×#67C22⋊D512yes—
255C22×Dic3—21no—
256#235Dic1014yes—
257#235C4×D514yes—
258C2×D12C22⋊D511yescharacteristic
259C2×C4×D3—22no—
260C2×C4×D3—102no—
261C2×C4×D3C22×D512yes—
262C2×Dic6C22⋊D511yescharacteristic
263C3×M16—54no—
264C2×C3×C8—52no—
265C3×C4⋊C4—52no—
266C3×C4⋊C4—101no—
267C3×C4⋊C4C22×D511yescharacteristic
268C3×C4⋊C4C22×D511yescharacteristic
269C3×C22⋊C4—21no—
270C3×C42—21no—
271#217—101no—
272(C2×D3)⋊C4—22no—
273(C2×D3)⋊C4—102no—
274C4⋊Dic3—21no—
275C4⋊Dic3—51no—
276C4⋊Dic3—51no—
277C4⋊Dic3C22×D512yes—
278Dic3⋊C4—22no—
279Dic3⋊C4—102no—
280C4×Dic3—22no—
281C4×Dic3—101no—
282C4×Dic3—102no—
283C2×C3⋊C8—54no—
284C2×C3⋊C8—51no—
285C2×C3⋊C8—51no—
286ΓL(1,𝔽25)—58no—
287D3×C8—58no—
288C23×C5—32no—
289C23×C5D3×D411yescharacteristic
290C5×Q8—34no—
291C5×D4—38no—
292C5×D4—34no—
293C2×C4×C5—21no—
294C2×C4×C5—21no—
295C2×C4×C5—32no—
296C2×C4×C5—34no—
297C2×C4×C5—34no—
298C2×C4×C5—62no—
299C2×C4×C5—242no—
300C2×C4×C5—241no—
301C2×C4×C5C2×C4×D312yes—
302C2×C4×C5D3×D411yescharacteristic
303C2×C4×C5D3×Q812yes—
304C2×C4×C5C23×D311yescharacteristic
305C2×Dic5—22no—
306C2×Dic5—61no—
307C2×Dic5—61no—
308C2×Dic5—81no—
309C2×Dic5—242no—
310#168—32no—
311#168—32no—
312#168C2×C4×D314yes—
313C2×#41—31no—
314#131—158no—
315C2×M16—51no—
316C2×M16—152no—
317C22×C8—152no—
318C22×C8—151no—
319C4×Q8—61no—
320C4×Q8—301no—
321C4×D4—61no—
322C4×D4—62no—
323C4×D4—301no—
324C4×D4—302no—
325#117—32no—
326#117—152no—
327#117—151no—
328#117D3×D511yescharacteristic
329C2×C4⋊C4—31no—
330C2×C4⋊C4—32no—
331C2×C4⋊C4—51no—
332C2×C4⋊C4—152no—
333#107—52no—
334#107—152no—
335#107—152no—
336#107—152no—
337#107—151no—
338#107—151no—
339#106—52no—
340#106—152no—
341#106—151no—
342#106—151no—
343C2×C3×C5—22no—
344C2×C3×C5C22×C4⋊C411yescharacteristic
345C2×C3×C5#57611yescharacteristic
346C2×C3×C5#57811yescharacteristic
347C5×D3—24no—
348C23×C3C2×C22⋊D511yescharacteristic
349C22×D3C2×C22⋊D512yes—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—102no—
353C2×C3×C4—401no—
354C2×C3×C4C2×Dic1012yes—
355C2×C3×C4C2×C4×D512yes—
356C2×C3×C4C2×C22⋊D511yescharacteristic
357C2×C3×C4C23×D511yescharacteristic
358#67#68318yes—
359C2×Dic3—22no—
360C2×Dic3—81no—
361C2×Dic3—101no—
362C2×Dic3—101no—
363C2×Dic3C2×C22⋊D512yes—
364D12#68314yes—
365C4×D3—82no—
366C4×D3—402no—
367C4×D3C2×Dic1014yes—
368C4×D3C2×C4×D514yes—
369Dic6#68314yes—
370C3×C8—54no—
371C3⋊C8—52no—
372C3⋊C8—52no—
373C22×C5—21no—
374C22×C5—21no—
375C22×C5—34no—
376C22×C5—62no—
377C22×C5—62no—
378C22×C5D3×C4⋊C412yes—
379C22×C5C2×D3×D411yescharacteristic
380C4×C5—34no—
381C4×C5—41no—
382C4×C5—41no—
383C4×C5—122no—
384C4×C5D3×C4⋊C412yes—
385C4×C5C22×C4×D311yescharacteristic
386C4×C5C2×D3×Q811yescharacteristic
387Dic5—42no—
388Dic5—121no—
389Dic5—121no—
390#41—34no—
391#41—34no—
392C2×Q8—31no—
393C2×D4—32no—
394C2×D4—31no—
395C22×C4—32no—
396C22×C4—62no—
397C22×C4—61no—
398C22×C4—101no—
399C22×C4—302no—
400C22×C4C2×D3×D511yescharacteristic
401M16—54no—
402M16—158no—
403C2×C8—52no—
404C2×C8—158no—
405C2×C8—154no—
406C2×C8—151no—
407C2×C8—151no—
408C4⋊C4—32no—
409C4⋊C4—52no—
410C4⋊C4—62no—
411C4⋊C4—61no—
412C4⋊C4—101no—
413C4⋊C4—151no—
414C4⋊C4—151no—
415C4⋊C4—302no—
416C4⋊C4C2×D3×D511yescharacteristic
417C4⋊C4C2×D3×D511yescharacteristic
418C22⋊C4—21no—
419C22⋊C4—62no—
420C22⋊C4—302no—
421C22⋊C4—301no—
422C42—21no—
423C42—62no—
424C42—302no—
425C42—301no—
426C3×C5#291311yescharacteristic, complemented, Hall
427C22×C3—21no—
428C22×C3—21no—
429C22×C3C2×#68312yes—
430C22×C3C22×C22⋊D511yescharacteristic
431C2×D3—22no—
432C2×D3—22no—
433C2×D3C2×#68314yes—
434C3×C4—41no—
435C3×C4—41no—
436C3×C4—202no—
437C3×C4C2×#68312yes—
438C3×C4C22×Dic1011yescharacteristic
439C3×C4C22×C4×D511yescharacteristic
440Dic3—42no—
441Dic3—201no—
442Dic3—201no—
443Dic3C2×#68314yes—
444C2×C5—22no—
445C2×C5—64no—
446C2×C5C2×D3×C4⋊C411yescharacteristic
447C2×C5#577411yescharacteristic
448C2×C5#577911yescharacteristic
449C23—32no—
450C23D3×C22⋊D511yescharacteristic
451Q8—34no—
452D4—38no—
453D4—34no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—32no—
457C2×C4—34no—
458C2×C4—34no—
459C2×C4—62no—
460C2×C4—102no—
461C2×C4—242no—
462C2×C4—241no—
463C2×C4—301no—
464C2×C4—301no—
465C2×C4—401no—
466C2×C4—1202no—
467C2×C4D3×Dic1012yes—
468C2×C4C4×D3×D512yes—
469C2×C4D3×C22⋊D511yescharacteristic
470C2×C4C22×D3×D511yescharacteristic
471C8—54no—
472C8—152no—
473C8—152no—
474C2×C3—22no—
475C2×C3C22×#68311yescharacteristic
476C2×C3#6616911yescharacteristic
477C2×C3#6617211yescharacteristic
478D3—24no—
479C5#7921011yescharacteristic, complemented, Sylow
480C22—21no—
481C22—21no—
482C22—34no—
483C22—62no—
484C22—62no—
485C22D3×#68312yes—
486C22C2×D3×C22⋊D511yescharacteristic
487C4—34no—
488C4—41no—
489C4—41no—
490C4—122no—
491C4—202no—
492C4—601no—
493C4—601no—
494C4D3×#68312yes—
495C4C2×D3×Dic1011yescharacteristic
496C4C2×C4×D3×D511yescharacteristic
497C3#1061429311yescharacteristic, complemented, Sylow
498C2—22no—
499C2—64no—
500C2C2×D3×#68311yescharacteristic
501C2#1175532311yescharacteristic
502C2#1175534111yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊D5
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC3×C5×#117C22
socleC22×C3×C5C22×D4
prime radicalC2×C5×#235C22
cosocleC2×C3×C4×C5C24
hypercenter#117D3×D5
hypocenterC3×C5#2913
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
4. #11751099C2semidirect
5. C3×#65284C2semidirect
6. C2×#90989C2semidirect
7. #11748459C2semidirect
8. #11747454C2semidirect
9. #11747437C2semidirect
10. D3×#3939C2semidirect
11. #11747191C2semidirect
31. #91288C4semidirect
50. #65284D3semidirect
74. C2×C3⋊#168C2×C4semidirect
84. C3×#681C2×C4semidirect
89. #6650C2×C4semidirect
90. D3×#168C2×C4semidirect
218. #681C4×D3semidirect
426. C3×C5#2913semidirect
479. C5#79210semidirect
497. C3#10614293semidirect

Maximal subgroups

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

subgroupindex
C5×#55232
#117544922
C2×#912882
#117510992
C3×#652842
C2×#909892
#117484592
#117474542
#117474372
D3×#39392
#117471912
#106142933
#792105