Finite Group Databasebeta

← #14460045

Subgroups — #14460045

Order 1920.

772 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
1C3×#66066C211yesmaximal, characteristic, complemented
2#11752676C211yesmaximal, characteristic
3#11751194C211yesmaximal, characteristic, complemented
4C5×#5087C211yesmaximal, characteristic, complemented
5#11747511C212yesmaximal, complemented
6#11747499C212yesmaximal, complemented
7#11747445C212yesmaximal, complemented
8#11747170C212yesmaximal, complemented
9#11747095C211yesmaximal, characteristic, complemented
10C4×#6656C211yesmaximal, characteristic, complemented
11#11746996C211yesmaximal, characteristic, complemented
12#10612712—31nomaximal, Hall
13C2×#6772C2212yes—
14C3×C5×C4⋊Q8C2211yescharacteristic
15#91758C2211yescharacteristic
16C4×C5×D12C2211yescharacteristic
17C3×#4093—21no—
18C3×Q8×Dic5—21no—
19C3×#4044—22no—
20C3×#4043C2212yes—
21C3×#4038C2212yes—
22C3×C4×D20C2211yescharacteristic
23C3×C4×Dic10C2211yescharacteristic
24#91457—22no—
25#91456—21no—
26D3×C4⋊Dic5—21no—
27#91360C2212yes—
28C2×#6662C2212yes—
29C2×#6656C2211yescharacteristic
30C2×D5×C3⋊C8C2212yes—
31#91094C2212yes—
32#91094C2212yes—
33C4×C3⋊#168C2211yescharacteristic
34C5×#809C2212yes—
35C5×#807C2212yes—
36C5×C4⋊(C3⋊C8)C2211yescharacteristic
37#90986C2211yescharacteristic
38#90974C2212yes—
39#90970C2211yescharacteristic
40#90966C2211yescharacteristic
41#90955C2212yes—
42#83664—51nomaximal, Hall
43#66066D311yescharacteristic, complemented
44#65889—31no—
45C5×#474—31no—
46#65382—31no—
47C4×#687—31no—
48#65177—32no—
49#65176—32no—
50#65138—31no—
51#65123—31no—
52#65072—32no—
53#65061—32no—
54C2×C3×C5×Q8—21no—
55C2×Dic30C2312yes—
56C2×C5×D12C2311yescharacteristic
57C2×C4×C5×D3—21no—
58C2×C3×D20D411yescharacteristic
59C2×C3×C4×D5C2312yes—
60C2×C3×Dic10C2311yescharacteristic
61C2×#1201—22no—
62C2×D3×Dic5—22no—
63#6772—24no—
64#6772—42no—
65C3×C5×C4⋊C4—21no—
66C3×C5×C4⋊C4C2312yes—
67C3×C42×C5C2311yescharacteristic
68C4⋊Dic15—21no—
69C4⋊Dic15—21no—
70C2×C3⋊#168C2311yescharacteristic
71C2×C3⋊#168C2311yescharacteristic
72C5×(C2×D3)⋊C4—21no—
73C5×C4⋊Dic3C2311yescharacteristic
74C2×C5×C3⋊C8C2312yes—
75C3×#685—22no—
76C3×#685—22no—
77C3×#685D412yes—
78C3×C4⋊Dic5—22no—
79C3×C4⋊Dic5D411yescharacteristic
80C3×C4⋊Dic5C2311yescharacteristic
81C3×#683—21no—
82C3×C4×Dic5—21no—
83C3×C4×Dic5—22no—
84Dic3⋊Dic5—22no—
85#6665—22no—
86#6662—42no—
87#6656—22no—
88D5×C3⋊C8—42no—
89C3×#556—51no—
90#5551—51no—
91#5093—51no—
92#5087D511yescharacteristic, complemented
93#5051—52no—
94#5047—52no—
95#5031—51no—
96C4×#215—51no—
97D4×C3⋊C8—51no—
98#5002—52no—
99#4996—52no—
100C2×#711—32no—
101C5×C4⋊Q8C2×D311yescharacteristic
102C4×C5×D4—31no—
103#4093—21no—
104Q8×Dic5—21no—
105C2×#687—31no—
106#4076—61no—
107C2×C4⋊Dic5—61no—
108C2×Dic20—32no—
109C2×C8×D5—32no—
110#4044—22no—
111#4043—32no—
112#4043C2×D312yes—
113#4038C2×D312yes—
114#4032—62no—
115C4×D20C2×D311yescharacteristic
116#4015—31no—
117C4×Dic10C2×D311yescharacteristic
118C5×#107—32no—
119C5×C4⋊C8—31no—
120C5×#102—32no—
121#3964—31no—
122#3952—32no—
123#3948—32no—
124#3942—31no—
125#3942—32no—
126#3942—32no—
127#3940—31no—
128C4×#168—31no—
129#3302—151noSylow
130C22×C5×D3—21no—
131C22×C3×D5C2×D412yes—
132C3×C5×Q8—42no—
133C2×C3×C4×C5—22no—
134C2×C3×C4×C5C2×D411yescharacteristic
135C2×C3×C4×C5C2×D411yescharacteristic
136C2×C3×C4×C5C2411yescharacteristic
137C2×Dic15—22no—
138Dic30—22no—
139Dic30—22no—
140C2×C5×Dic3—21no—
141C5×D12—21no—
142C5×D12#4112yes—
143C4×C5×D3—42no—
144C2×C3×Dic5—21no—
145C2×C3×Dic5—21no—
146C2×C3×Dic5C2×D412yes—
147C3×D20—22no—
148C3×C4×D5—22no—
149C3×C4×D5—22no—
150C3×Dic10—21no—
151C3×Dic10—21no—
152#1201—44no—
153D3×Dic5—44no—
154C3⋊#168—21no—
155C3⋊#168—21no—
156C5×C3⋊C8—22no—
157C2×#235—52no—
158C3×C4⋊Q8C2×D511yescharacteristic
159C3×#126—102no—
160C3×#124—101no—
161C3×C22⋊Q8—52no—
162C3×C4×Q8—51no—
163C3×C4×Q8—101no—
164C3×C4×D4—51no—
165C3×#117—52no—
166C2×#216—52no—
167C2×#215—51no—
168#938—101no—
169#924—52no—
170C22×C3⋊C8—52no—
171D3×C4⋊C4—101no—
172#883—102no—
173C4×D12C2×D511yescharacteristic
174#869—51no—
175#835—51no—
176#830—52no—
177#810—52no—
178#810—52no—
179#810—52no—
180#809—51no—
181#809C2×D512yes—
182#808—51no—
183#807C2×D512yes—
184C4⋊(C3⋊C8)C2×D511yescharacteristic
185C4×C3⋊C8—51no—
186C2×C5×Q8—21no—
187C2×C5×D4—31no—
188C22×C4×C5—61no—
189C2×C22⋊D5—62no—
190C22×Dic5—62no—
191#711—64no—
192#711—122no—
193C2×D20#6711yescharacteristic
194C2×C4×D5C22×D312yes—
195C2×Dic10—32no—
196C2×Dic10C22×D311yescharacteristic
197C2×C5×C8—32no—
198C5×C4⋊C4—21no—
199C5×C4⋊C4—31no—
200C5×C4⋊C4C22×D312yes—
201C5×C22⋊C4—61no—
202C42×C5C22×D311yescharacteristic
203#687—62no—
204#685—22no—
205#685—22no—
206#685—62no—
207#685#6712yes—
208C4⋊Dic5—22no—
209C4⋊Dic5—61no—
210C4⋊Dic5—61no—
211C4⋊Dic5—62no—
212C4⋊Dic5#6711yescharacteristic
213C4⋊Dic5C22×D311yescharacteristic
214#683—21no—
215C4×Dic5—21no—
216C4×Dic5—22no—
217C2×#168—31no—
218C2×#168—31no—
219Dic20—122no—
220C8×D5—122no—
221#556—51no—
222#547—151no—
223#487—151no—
224#485—152no—
225#474—31no—
226#467—152no—
227#462—151no—
228#452—152no—
229C4×SD16—151no—
230C8×D4—151no—
231#416—152no—
232C22×C3×C5C22×D411yescharacteristic
233C2×C5×D3—22no—
234C2×C5×D3—41no—
235C2×C5×D3—41no—
236C2×C3×D5—22no—
237C2×C3×D5—22no—
238C2×C3×D5—22no—
239C3×C4×C5—21no—
240C3×C4×C5—42no—
241C3×C4×C5#13512yes—
242C3×C4×C5C2×#4111yescharacteristic
243C3×C4×C5H(5,𝔽2)11yescharacteristic
244Dic15—42no—
245C3×Dic5—22no—
246C3×Dic5—41no—
247C3×Dic5—41no—
248C5×Dic3—41no—
249C2×C3×Q8—21no—
250C2×C3×Q8—51no—
251C2×C3×D4—51no—
252C22×C3×C4—52no—
253C2×#67—102no—
254#235—104no—
255#235—202no—
256C2×D12C22×D511yescharacteristic
257C2×C4×D3—21no—
258C2×C4×D3—102no—
259C2×Dic6—52no—
260C3×C4⋊C4—21no—
261C3×C4⋊C4—51no—
262C3×C4⋊C4—51no—
263C3×C4⋊C4—102no—
264C3×C4⋊C4—101no—
265C3×C4⋊C4C22×D512yes—
266C3×C22⋊C4—52no—
267C3×C22⋊C4—102no—
268C3×C22⋊C4—102no—
269C3×C42—102no—
270C3×C42—101no—
271C3×C42C22×D511yescharacteristic
272#217—102no—
273#216—202no—
274#215—102no—
275(C2×D3)⋊C4—21no—
276C4⋊Dic3—101no—
277C4⋊Dic3—101no—
278C4⋊Dic3C22×D511yescharacteristic
279Dic3⋊C4—102no—
280C2×C3⋊C8—51no—
281C2×C3⋊C8—51no—
282C2×C3⋊C8—202no—
283C2×C3⋊C8C22×D512yes—
284C23×C5—61no—
285C22×D5C2×#6712yes—
286C5×Q8—42no—
287C5×D4—32no—
288C5×D4—61no—
289C2×C4×C5—22no—
290C2×C4×C5—61no—
291C2×C4×C5—122no—
292C2×C4×C5C2×#6711yescharacteristic
293C2×C4×C5C2×#6711yescharacteristic
294C2×C4×C5C23×D311yescharacteristic
295C22⋊D5—124no—
296C2×Dic5—21no—
297C2×Dic5—21no—
298C2×Dic5—62no—
299C2×Dic5—124no—
300C2×Dic5C2×#6712yes—
301D20—22no—
302C4×D5—22no—
303C4×D5—22no—
304Dic10—21no—
305Dic10—21no—
306Dic10—62no—
307Dic10—62no—
308C5×C8—62no—
309#168—61no—
310#168—61no—
311C2×#41—152no—
312C2×Q16—152no—
313C2×SD16—151no—
314C22×C8—152no—
315C4⋊Q8—151no—
316C4⋊Q8D3×D511yescharacteristic
317#126—102no—
318#124—101no—
319#123—302no—
320C22⋊Q8—52no—
321C22⋊Q8—152no—
322#121—301no—
323C4×Q8—51no—
324C4×Q8—101no—
325C4×D4—31no—
326C4×D4—51no—
327#117—52no—
328C2×C4⋊C4—301no—
329#107—32no—
330#106—151no—
331C4⋊C8—31no—
332#103—151no—
333#103—152no—
334#103—152no—
335#103—152no—
336#102—32no—
337#102—151no—
338C22⋊C8—152no—
339C4×C8—151no—
340C2×C3×C5#54611yescharacteristic
341C2×C3×C5C2×#13511yescharacteristic
342C2×C3×C5#57811yescharacteristic
343C3×D5—42no—
344C5×D3—42no—
345C23×C3—52no—
346C22×D3—21no—
347C3×Q8—42no—
348C3×Q8—101no—
349C3×Q8—101no—
350C3×D4—102no—
351C2×C3×C4—22no—
352C2×C3×C4—52no—
353C2×C3×C4—102no—
354C2×C3×C4—102no—
355C2×C3×C4—101no—
356C2×C3×C4—101no—
357C2×C3×C4D4×D511yescharacteristic
358C2×C3×C4D4×D511yescharacteristic
359C2×C3×C4C23×D511yescharacteristic
360#67—204no—
361C2×Dic3—21no—
362C2×Dic3—102no—
363D12—21no—
364D12#71312yes—
365C4×D3—42no—
366C4×D3—204no—
367Dic6—102no—
368Dic6—102no—
369C3⋊C8—22no—
370C3⋊C8—101no—
371C3⋊C8—101no—
372C22×C5—62no—
373C22×C5—121no—
374C22×C5—121no—
375C22×C5C22×#6711yescharacteristic
376C2×D5—22no—
377C2×D5—22no—
378C2×D5—22no—
379C4×C5—21no—
380C4×C5—42no—
381C4×C5—121no—
382C4×C5#95012yes—
383C4×C5D3×#4111yescharacteristic
384C4×C5#100911yescharacteristic
385Dic5—22no—
386Dic5—41no—
387Dic5—41no—
388Dic5—122no—
389#41—304no—
390#41—602no—
391C2×Q8—21no—
392C2×Q8—51no—
393C2×Q8—152no—
394C2×D4—31no—
395C2×D4—51no—
396C2×D4—302no—
397C22×C4—52no—
398C22×C4—61no—
399C22×C4—302no—
400Q16—602no—
401SD16—302no—
402C2×C8—32no—
403C2×C8—151no—
404C2×C8—151no—
405C2×C8—602no—
406C4⋊C4—21no—
407C4⋊C4—31no—
408C4⋊C4—51no—
409C4⋊C4—51no—
410C4⋊C4—102no—
411C4⋊C4—101no—
412C4⋊C4—302no—
413C4⋊C4—301no—
414C4⋊C4—301no—
415C4⋊C4C2×D3×D512yes—
416C22⋊C4—52no—
417C22⋊C4—61no—
418C22⋊C4—102no—
419C22⋊C4—102no—
420C22⋊C4—302no—
421C42—102no—
422C42—101no—
423C42C2×D3×D511yescharacteristic
424C3×C5#330211yescharacteristic, complemented, Hall
425C22×C3—102no—
426C22×C3—102no—
427C22×C3—102no—
428C22×C3C2×D4×D511yescharacteristic
429C2×D3—22no—
430C2×D3—41no—
431C2×D3—41no—
432C3×C4—21no—
433C3×C4—42no—
434C3×C4—102no—
435C3×C4—201no—
436C3×C4—201no—
437C3×C4#405812yes—
438C3×C4C2×#71311yescharacteristic
439C3×C4#414911yescharacteristic
440Dic3—41no—
441Dic3—202no—
442C2×C5—122no—
443C2×C5C2×#95011yescharacteristic
444C2×C5#582911yescharacteristic
445C2×C5#583811yescharacteristic
446D5—42no—
447C23—52no—
448C23—61no—
449Q8—42no—
450Q8—101no—
451Q8—101no—
452Q8—302no—
453Q8—302no—
454D4—32no—
455D4—61no—
456D4—102no—
457D4—604no—
458C2×C4—22no—
459C2×C4—52no—
460C2×C4—61no—
461C2×C4—102no—
462C2×C4—102no—
463C2×C4—101no—
464C2×C4—101no—
465C2×C4—122no—
466C2×C4—302no—
467C2×C4—604no—
468C2×C4D5×#6711yescharacteristic
469C2×C4D5×#6711yescharacteristic
470C2×C4C22×D3×D511yescharacteristic
471C8—62no—
472C8—301no—
473C8—301no—
474C2×C3—202no—
475C2×C3#6596411yescharacteristic
476C2×C3C2×#405811yescharacteristic
477C2×C3#6611311yescharacteristic
478D3—42no—
479C5#8366411yescharacteristic, complemented, Sylow
480C22—62no—
481C22—102no—
482C22—102no—
483C22—102no—
484C22—121no—
485C22—121no—
486C22C2×D5×#6711yescharacteristic
487C4—21no—
488C4—42no—
489C4—102no—
490C4—121no—
491C4—201no—
492C4—201no—
493C4—602no—
494C4#9149012yes—
495C4D3×#71311yescharacteristic
496C4#9203511yescharacteristic
497C3#1061271211yescharacteristic, complemented, Sylow
498C2—122no—
499C2—202no—
500C2C2×#9149011yescharacteristic
501C2#1175510211yescharacteristic
502C2#1175527311yescharacteristic
subgroupG/H
centerC22C2×D5×#67
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C5×C4⋊Q8C22
socleC22×C3×C5C22×D4
prime radical#11752676C2
hypercenterC4⋊Q8D3×D5
hypocenterC3×C5#3302
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C3×#66066C2semidirect
3. #11751194C2semidirect
4. C5×#5087C2semidirect
5. #11747511C2semidirect
6. #11747499C2semidirect
7. #11747445C2semidirect
8. #11747170C2semidirect
9. #11747095C2semidirect
10. C4×#6656C2semidirect
11. #11746996C2semidirect
43. #66066D3semidirect
92. #5087D5semidirect
424. C3×C5#3302semidirect
479. C5#83664semidirect
497. C3#10612712semidirect

Maximal subgroups

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

subgroupindex
C3×#660662
#117526762
#117511942
C5×#50872
#117475112
#117474992
#117474452
#117471702
#117470952
C4×#66562
#117469962
#106127123
#836645