Finite Group Databasebeta

← #14301270

Subgroups — #14301270

Order 1920.

668 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4088C211yesmaximal, characteristic
2#11754499C211yesmaximal, characteristic
3#11751326C211yesmaximal, characteristic, complemented
4C5×#5229C211yesmaximal, characteristic, complemented
5C2×#90977C211yesmaximal, characteristic
6#11745986C211yesmaximal, characteristic, complemented
7#11745976C211yesmaximal, characteristic, complemented
8#10600486—31nomaximal, Hall
9C22×C3×C5×Q8C2211yescharacteristic
10C2×C3×#709C2211yescharacteristic
11#91815C2211yescharacteristic
12C2×#6710C2211yescharacteristic
13C2×C5×C4⋊Dic3C2211yescharacteristic
14C2×C4×C5×Dic3C2211yescharacteristic
15C2×C3×#689—21no—
16C3×#4088—22no—
17C3×#4088C414yescomplemented
18C2×C3×#688—21no—
19C2×C3×#681C2211yescharacteristic
20Dic3×C22⋊D5—22no—
21#91434—21no—
22D5×C4⋊Dic3—21no—
23#91358—21no—
24#91331—21no—
25#91119—21no—
26#91119—21no—
27C5×C3⋊#95—22no—
28#90977—22no—
29#72100—51nomaximal, Hall
30C2×#4088D311yescharacteristic
31#65838—31no—
32C5×#391—31no—
33#65514—31no—
34C2×#3937—31no—
35#64750—31no—
36#64743—31no—
37C2×C3×C5×Q8—21no—
38C2×C3×C5×Q8—21no—
39C2×C3×C5×Q8—21no—
40C2×C3×C5×Q8—21no—
41C2×C3×C5×Q8—21no—
42C2×C3×C5×Q8C2×C412yescomplemented
43C22×C3×C4×C5—21no—
44C22×C3×C4×C5C2311yescharacteristic
45C22×C5×Dic3—21no—
46C22×C5×Dic3D412yes—
47C2×C3×C22⋊D5—21no—
48C3×#709—21no—
49C3×#709—21no—
50C3×#709C2×C412yes—
51C3×#709D412yes—
52C2×C3×D20D411yescharacteristic
53C2×C3×C4×D5—21no—
54C2×C3×Dic10D411yescharacteristic
55C2×D5×Dic3—22no—
56#6719—21no—
57C4⋊Dic15D411yescharacteristic
58C4⋊Dic15D411yescharacteristic
59C4×Dic15—21no—
60#6710—21no—
61#6710—21no—
62C2×C3⋊#168—21no—
63C5×C4⋊Dic3—21no—
64C5×C4⋊Dic3—21no—
65C4×C5×Dic3—21no—
66C4×C5×Dic3—21no—
67C3×#689—24no—
68C3×#688—24no—
69C3×#681—21no—
70C3×#681C2×C412yescomplemented
71C2×C3×#168—21no—
72#6668—22no—
73#6665—22no—
74Dic3×Dic5—22no—
75C2×C3×#137—51no—
76#5827—51no—
77C2×#837—51no—
78#5229D511yescharacteristic, complemented
79#5225—51no—
80#4555—51no—
81#4554—51no—
82C22×C5×Q8C2×D311yescharacteristic
83C2×#709C2×D311yescharacteristic
84C2×C5×C4⋊C4—31no—
85C2×C42×C5—31no—
86C2×#689—21no—
87#4088—22no—
88#4088Dic314yescomplemented
89C2×#688—21no—
90C4×C22⋊D5—62no—
91#4072—31no—
92C2×#681—31no—
93C2×#681C2×D311yescharacteristic
94#4039—61no—
95#4038—61no—
96D5×C4⋊C4—61no—
97#4033—61no—
98C5×#95—62no—
99#3967—61no—
100#3967—61no—
101#3937—62no—
102#1882—151noSylow
103C23×C3×C5C2×D411yescharacteristic
104C22×C3×D5—21no—
105C3×C5×Q8—22no—
106C3×C5×Q8—22no—
107C3×C5×Q8—41no—
108C3×C5×Q8—41no—
109C2×C3×C4×C5—22no—
110C2×C3×C4×C5—41no—
111C2×C3×C4×C5—41no—
112C2×C3×C4×C5C22×C411yescharacteristic
113C2×C3×C4×C5C2×D411yescharacteristic
114C2×C3×C4×C5C2×D411yescharacteristic
115C2×C3×C4×C5C2×D411yescharacteristic
116C2×C3×C4×C5#4111yescharacteristic
117C2×C3×C4×C5#4111yescharacteristic
118C2×Dic15—21no—
119C2×Dic15—21no—
120C2×C5×Dic3—22no—
121C2×C5×Dic3—22no—
122C2×C5×Dic3—41no—
123C2×C5×Dic3—41no—
124C2×C5×Dic3—42no—
125C3×C22⋊D5—22no—
126C3×C22⋊D5—22no—
127C2×C3×Dic5—21no—
128C3×D20—21no—
129C3×D20C22⋊C412yes—
130C3×C4×D5—22no—
131C3×C4×D5—22no—
132C3×Dic10—21no—
133C3×Dic10C22⋊C412yes—
134D5×Dic3—82no—
135C3⋊#168—41no—
136C3×#168—22no—
137C2×C3×#41—51no—
138C22×C3×Q8C2×D511yescharacteristic
139C3×#137—54no—
140C3×#137—102no—
141C2×C3×Q16—101no—
142C2×C3×SD16—101no—
143C2×C3×M16—51no—
144Q8×Dic3—101no—
145D4×Dic3—101no—
146D4×Dic3—102no—
147#926—51no—
148#926—101no—
149C2×C4⋊Dic3—101no—
150C2×C4⋊Dic3C2×D511yescharacteristic
151C2×C4×Dic3C2×D511yescharacteristic
152C2×#208—51no—
153#837—102no—
154#835—101no—
155#835—101no—
156C3⋊#95—22no—
157C2×C5×Q8—21no—
158C2×C5×Q8—21no—
159C2×C5×Q8—21no—
160C2×C5×Q8—21no—
161C2×C5×Q8—21no—
162C2×C5×Q8C2×Dic312yescomplemented
163C22×C4×C5—21no—
164C22×C4×C5—32no—
165C22×C4×C5—61no—
166C22×C4×C5C22×D311yescharacteristic
167C2×C22⋊D5—21no—
168#709—21no—
169#709—21no—
170#709C2×Dic312yes—
171#709#6712yes—
172C2×D20#6711yescharacteristic
173C2×C4×D5—21no—
174C2×C4×D5—62no—
175C2×Dic10#6711yescharacteristic
176C5×C4⋊C4—61no—
177C5×C4⋊C4—61no—
178C42×C5—61no—
179C42×C5—61no—
180C22⋊Dic5—61no—
181#689—24no—
182#688—24no—
183#685—62no—
184C4⋊Dic5—31no—
185C4⋊Dic5—31no—
186#683—62no—
187C4×Dic5—62no—
188C4×Dic5—61no—
189#681—21no—
190#681—61no—
191#681—61no—
192#681C2×Dic312yescomplemented
193C2×#168—21no—
194C2×#168—61no—
195C2×#137—51no—
196#520—151no—
197C2×#104—151no—
198#419—151no—
199#391—31no—
200#343—151no—
201#339—151no—
202C22×C3×C5—21no—
203C22×C3×C5—21no—
204C22×C3×C5C4×D411yescharacteristic
205C22×C3×C5#12011yescharacteristic
206C22×C3×C5#12111yescharacteristic
207C2×C3×D5—22no—
208C2×C3×D5—41no—
209C2×C3×D5—41no—
210C3×C4×C5—42no—
211C3×C4×C5C2×C22⋊C411yescharacteristic
212C3×C4×C5C4×D411yescharacteristic
213C3×C4×C5#12111yescharacteristic
214C3×C4×C5#12311yescharacteristic
215Dic15—41no—
216Dic15—41no—
217C3×Dic5—22no—
218C5×Dic3—42no—
219C5×Dic3—42no—
220C3×#41—52no—
221C3×#41—52no—
222C3×#41—101no—
223C3×#41—101no—
224C2×C3×Q8—21no—
225C2×C3×Q8—21no—
226C2×C3×Q8—21no—
227C2×C3×Q8—21no—
228C2×C3×Q8—21no—
229C2×C3×Q8—51no—
230C2×C3×Q8C4×D512yescomplemented
231C2×C3×D4—51no—
232C2×C3×D4—101no—
233C22×C3×C4—21no—
234C22×C3×C4—101no—
235C22×C3×C4C22×D511yescharacteristic
236C22×Dic3—21no—
237C22×Dic3—102no—
238C22×Dic3C22⋊D512yes—
239C3×Q16—104no—
240C3×SD16—104no—
241C3×M16—52no—
242C3×M16—101no—
243C2×C3×C8—101no—
244#217—102no—
245#217—101no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—51no—
249C4⋊Dic3—51no—
250C4⋊Dic3—102no—
251C4×Dic3—21no—
252C4×Dic3—21no—
253C4×Dic3—102no—
254C4×Dic3—101no—
255#208—101no—
256#208—101no—
257C2×C3⋊C8—101no—
258C23×C5D3×D411yescharacteristic
259C22×D5—21no—
260C5×Q8—22no—
261C5×Q8—22no—
262C5×Q8—41no—
263C5×Q8—41no—
264C2×C4×C5—22no—
265C2×C4×C5—41no—
266C2×C4×C5—41no—
267C2×C4×C5—62no—
268C2×C4×C5—62no—
269C2×C4×C5—121no—
270C2×C4×C5—121no—
271C2×C4×C5—122no—
272C2×C4×C5D3×D411yescharacteristic
273C2×C4×C5#23711yescharacteristic
274C2×C4×C5#23711yescharacteristic
275C2×C4×C5C22×Dic311yescharacteristic
276C2×C4×C5C2×#6711yescharacteristic
277C2×C4×C5C2×#6711yescharacteristic
278C22⋊D5—22no—
279C22⋊D5—22no—
280C2×Dic5—21no—
281C2×Dic5—61no—
282C2×Dic5—61no—
283D20—21no—
284D20#21712yes—
285C4×D5—22no—
286C4×D5—22no—
287C4×D5—242no—
288Dic10—21no—
289Dic10#21712yes—
290#168—22no—
291#168—121no—
292C2×#41—51no—
293C22×Q8D3×D511yescharacteristic
294#137—54no—
295#137—102no—
296C2×Q16—101no—
297C2×SD16—101no—
298C2×M16—51no—
299C2×M16—151no—
300C4×Q8—301no—
301C4×D4—301no—
302C4×D4—302no—
303#117—151no—
304#117—301no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—301no—
307C2×C42—31no—
308#104—302no—
309#103—301no—
310#103—301no—
311#95—62no—
312C2×C3×C5—21no—
313C2×C3×C5—21no—
314C2×C3×C5#38611yescharacteristic
315C2×C3×C5#45511yescharacteristic
316C2×C3×C5#45611yescharacteristic
317C3×D5—42no—
318C23×C3—101no—
319C23×C3C2×C22⋊D511yescharacteristic
320C3×Q8—22no—
321C3×Q8—22no—
322C3×Q8—41no—
323C3×Q8—41no—
324C3×Q8—52no—
325C3×Q8—101no—
326C3×D4—52no—
327C3×D4—102no—
328C3×D4—102no—
329C3×D4—101no—
330C2×C3×C4—22no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—101no—
334C2×C3×C4—102no—
335C2×C3×C4—102no—
336C2×C3×C4C2×C4×D511yescharacteristic
337C2×C3×C4#70911yescharacteristic
338C2×C3×C4D4×D511yescharacteristic
339C2×C3×C4D4×D511yescharacteristic
340C2×C3×C4D4×D511yescharacteristic
341C2×C3×C4#71111yescharacteristic
342C2×Dic3—22no—
343C2×Dic3—22no—
344C2×Dic3—41no—
345C2×Dic3—41no—
346C2×Dic3—42no—
347C2×Dic3—101no—
348C2×Dic3—101no—
349C2×Dic3—402no—
350C3×C8—102no—
351C3⋊C8—201no—
352C22×C5—21no—
353C22×C5—21no—
354C22×C5D4×Dic311yescharacteristic
355C22×C5#93711yescharacteristic
356C22×C5#93911yescharacteristic
357C2×D5—22no—
358C2×D5—41no—
359C2×D5—41no—
360C4×C5—42no—
361C4×C5—122no—
362C4×C5—122no—
363C4×C5D4×Dic311yescharacteristic
364C4×C5#93511yescharacteristic
365C4×C5#93911yescharacteristic
366C4×C5C2×#21711yescharacteristic
367Dic5—22no—
368Dic5—121no—
369Dic5—121no—
370#41—52no—
371#41—52no—
372#41—101no—
373#41—101no—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—51no—
380C2×Q8D5×Dic312yescomplemented
381C2×D4—51no—
382C2×D4—101no—
383C22×C4—21no—
384C22×C4—32no—
385C22×C4—61no—
386C22×C4—101no—
387C22×C4—302no—
388C22×C4C2×D3×D511yescharacteristic
389Q16—104no—
390SD16—104no—
391M16—52no—
392M16—101no—
393M16—301no—
394M16—301no—
395C2×C8—101no—
396C2×C8—301no—
397C4⋊C4—61no—
398C4⋊C4—61no—
399C4⋊C4—151no—
400C4⋊C4—151no—
401C4⋊C4—302no—
402C22⋊C4—302no—
403C22⋊C4—301no—
404C42—61no—
405C42—61no—
406C42—302no—
407C42—301no—
408C3×C5#188211yescharacteristic, complemented, Hall
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—102no—
412C22×C3—201no—
413C22×C3—201no—
414C22×C3C4×C22⋊D511yescharacteristic
415C22×C3#408311yescharacteristic
416C22×C3#408411yescharacteristic
417C3×C4—42no—
418C3×C4—102no—
419C3×C4D5×C22⋊C411yescharacteristic
420C3×C4#402711yescharacteristic
421C3×C4#402911yescharacteristic
422C3×C4#403011yescharacteristic
423Dic3—42no—
424Dic3—42no—
425Dic3—201no—
426Dic3—201no—
427C2×C5—21no—
428C2×C5—21no—
429C2×C5#520211yescharacteristic
430C2×C5#520511yescharacteristic
431C2×C5#522011yescharacteristic
432D5—42no—
433C23—101no—
434C23D3×C22⋊D511yescharacteristic
435Q8—22no—
436Q8—22no—
437Q8—41no—
438Q8—41no—
439Q8—52no—
440Q8—101no—
441D4—52no—
442D4—102no—
443D4—102no—
444D4—101no—
445C2×C4—22no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—62no—
449C2×C4—62no—
450C2×C4—101no—
451C2×C4—102no—
452C2×C4—102no—
453C2×C4—121no—
454C2×C4—121no—
455C2×C4—122no—
456C2×C4—301no—
457C2×C4—301no—
458C2×C4—1202no—
459C2×C4C2×D5×Dic311yescharacteristic
460C2×C4#677911yescharacteristic
461C2×C4#678011yescharacteristic
462C2×C4D5×#6711yescharacteristic
463C2×C4D5×#6711yescharacteristic
464C2×C4#679011yescharacteristic
465C8—102no—
466C8—601no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—202no—
470C2×C3#6524511yescharacteristic
471C2×C3#6535011yescharacteristic
472C2×C3#6538411yescharacteristic
473C5#7210011yescharacteristic, complemented, Sylow
474C22—21no—
475C22—21no—
476C22—102no—
477C22—201no—
478C22—201no—
479C22#9154911yescharacteristic
480C22Dic3×C22⋊D511yescharacteristic
481C22#9157711yescharacteristic
482C4—42no—
483C4—102no—
484C4—122no—
485C4—122no—
486C4—601no—
487C4—601no—
488C4D5×#21711yescharacteristic
489C4#9154811yescharacteristic
490C4#9155011yescharacteristic
491C4#9155911yescharacteristic
492C3#1060048611yescharacteristic, complemented, Sylow
493C2—21no—
494C2—21no—
495C2—202no—
496C2#1174941111yescharacteristic
497C2#1174941311yescharacteristic
498C2#1174951811yescharacteristic
subgroupG/H
centerC22#91577
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC22×C3×C5×Q8C22
socleC22×C3×C5#120
prime radicalC2×C3×#709C22
cosocleC22×C3×C4×C5C23
hypercenterC22×Q8D3×D5
hypocenterC3×C5#1882
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
3. #11751326C2semidirect
4. C5×#5229C2semidirect
6. #11745986C2semidirect
7. #11745976C2semidirect
17. C3×#4088C4semidirect
42. C2×C3×C5×Q8C2×C4semidirect
70. C3×#681C2×C4semidirect
78. #5229D5semidirect
88. #4088Dic3semidirect
162. C2×C5×Q8C2×Dic3semidirect
192. #681C2×Dic3semidirect
230. C2×C3×Q8C4×D5semidirect
380. C2×Q8D5×Dic3semidirect
408. C3×C5#1882semidirect
473. C5#72100semidirect
492. C3#10600486semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#40882
#117544992
#117513262
C5×#52292
C2×#909772
#117459862
#117459762
#106004863
#721005