Finite Group Databasebeta

← #14459123

Subgroups — #14459123

Order 1920.

840 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×D4×Q8C211yesmaximal, characteristic, complemented
2#11756314C211yesmaximal, characteristic
3#11751328C212yesmaximal, complemented
4C2×#91119C212yesmaximal, complemented
5#11751184C211yesmaximal, characteristic, complemented
6#11751180C211yesmaximal, characteristic, complemented
7#11751149C212yesmaximal, complemented
8#11751144C212yesmaximal, complemented
9C4×#6717C211yesmaximal, characteristic, complemented
10#11751121C211yesmaximal, characteristic, complemented
11D4×C3⋊#168C211yesmaximal, characteristic, complemented
12#10615281—31nomaximal, Hall
13C22×C3×C5×Q8C2212yes—
14C2×#6819—21no—
15C3×C5×C4⋊Q8—21no—
16C3×C5×C4⋊Q8C2211yescharacteristic
17C3×C5×C22⋊Q8—22no—
18C3×C5×C22⋊Q8C2212yes—
19C3×C4×C5×Q8C2211yescharacteristic
20C3×C4×C5×D4—21no—
21C3×C4×C5×D4C2211yescharacteristic
22C2×#6717C2211yescharacteristic
23C2×#6717C2212yes—
24#91819C2212yes—
25C2×C4⋊Dic15C2212yes—
26C22×C3⋊#168C2212yes—
27#91786—21no—
28#91775—22no—
29C4×D60C2211yescharacteristic
30#91758C2211yescharacteristic
31#91119C2212yes—
32#91119C2212yes—
33#91119C2212yes—
34#91119C2211yescharacteristic
35#91119C2212yes—
36#91114C2212yes—
37#91093C2211yescharacteristic
38#91092C2212yes—
39#91092C2211yescharacteristic
40C4⋊(C3⋊#168)C2211yescharacteristic
41C4×C3⋊#168C2211yescharacteristic
42#83550—51nomaximal, Hall
43C5×D4×Q8D311yescharacteristic, complemented
44#65889—31no—
45#65516—32no—
46C2×#3967—32no—
47#65372—31no—
48#65370—31no—
49#65337—32no—
50#65332—32no—
51C4×#688—31no—
52#65309—31no—
53D4×#168—31no—
54C2×C3×C5×Q8—22no—
55C2×C3×C5×Q8—24no—
56C2×C3×C5×Q8—42no—
57C2×C3×C5×Q8C2312yes—
58C2×C3×C5×Q8C2311yescharacteristic
59C2×C3×C5×D4D411yescharacteristic
60C22×C3×C4×C5—22no—
61C22×C3×C4×C5C2312yes—
62C2×#1220—22no—
63C22×Dic15—22no—
64#6819—44no—
65C2×D60C2311yescharacteristic
66C2×C4×D15—21no—
67C2×Dic30—21no—
68C3×C5×C4⋊C4—21no—
69C3×C5×C4⋊C4—21no—
70C3×C5×C4⋊C4—21no—
71C3×C5×C4⋊C4—22no—
72C3×C5×C4⋊C4D411yescharacteristic
73C3×C5×C4⋊C4C2311yescharacteristic
74C3×C5×C22⋊C4—22no—
75C3×C5×C22⋊C4D412yes—
76C3×C42×C5—21no—
77C3×C42×C5C2311yescharacteristic
78#6717—22no—
79#6717—42no—
80#6714—21no—
81#6714—22no—
82C4⋊Dic15—21no—
83C4⋊Dic15—22no—
84C4⋊Dic15C2312yes—
85C4⋊Dic15C2311yescharacteristic
86C2×C3⋊#168—42no—
87C2×C3⋊#168C2311yescharacteristic
88C2×C3⋊#168C2312yes—
89C2×C3⋊#168C2311yescharacteristic
90C3×D4×Q8D511yescharacteristic, complemented
91#5555—51no—
92#5227—52no—
93C2×#835—52no—
94#5083—51no—
95#5079—51no—
96#5048—52no—
97#5043—52no—
98C4×#215—51no—
99#5020—51no—
100D4×C3⋊C8—51no—
101C22×C5×Q8C2×D312yes—
102C2×#711—61no—
103C5×C4⋊Q8—21no—
104C5×C4⋊Q8C2×D311yescharacteristic
105C5×C22⋊Q8—22no—
106C5×C22⋊Q8C2×D312yes—
107C4×C5×Q8C2×D311yescharacteristic
108C4×C5×D4—21no—
109C4×C5×D4C2×D311yescharacteristic
110C2×#688—31no—
111C2×#688—32no—
112#4076—32no—
113C2×C4⋊Dic5—32no—
114C22×#168—32no—
115#4043—61no—
116#4032—62no—
117C4×D20—31no—
118#4015—31no—
119#3967—32no—
120#3967—31no—
121#3967—32no—
122#3967—32no—
123#3967—32no—
124C22⋊#168—32no—
125#3941—31no—
126#3940—32no—
127#3940—31no—
128C4⋊#168—31no—
129C4×#168—31no—
130#3298—151noSylow
131C23×C3×C5C2×D412yes—
132C22×D15—21no—
133C3×C5×Q8—21no—
134C3×C5×Q8—22no—
135C3×C5×Q8—22no—
136C3×C5×Q8—44no—
137C3×C5×Q8#4112yescomplemented
138C3×C5×D4SD1614yescomplemented
139C2×C3×C4×C5—21no—
140C2×C3×C4×C5—22no—
141C2×C3×C4×C5—22no—
142C2×C3×C4×C5—21no—
143C2×C3×C4×C5—22no—
144C2×C3×C4×C5—44no—
145C2×C3×C4×C5C2×D411yescharacteristic
146C2×C3×C4×C5C2×D412yes—
147C2×C3×C4×C5C2×D411yescharacteristic
148C2×C3×C4×C5C2411yescharacteristic
149#1220—44no—
150C2×Dic15—22no—
151C2×Dic15—21no—
152C2×Dic15—44no—
153D60—21no—
154D60—21no—
155C4×D15—42no—
156Dic30—42no—
157C3⋊#168—21no—
158C3⋊#168—21no—
159C3⋊#168—22no—
160C22×C3×Q8C2×D512yes—
161C2×#237—101no—
162C3×C4⋊Q8—21no—
163C3×C4⋊Q8C2×D511yescharacteristic
164C3×C22⋊Q8—22no—
165C3×C22⋊Q8C2×D512yes—
166C3×C4×Q8C2×D511yescharacteristic
167C3×C4×D4—21no—
168C3×C4×D4C2×D511yescharacteristic
169C2×#215—51no—
170C2×#215—52no—
171#930—52no—
172C2×C4⋊Dic3—52no—
173C22×C3⋊C8—52no—
174#897—101no—
175#886—102no—
176C4×D12—51no—
177#869—51no—
178#835—51no—
179#835—52no—
180#835—52no—
181#835—52no—
182#835—52no—
183#830—52no—
184#809—51no—
185#808—52no—
186#808—51no—
187C4⋊(C3⋊C8)—51no—
188C4×C3⋊C8—51no—
189C2×C5×Q8—22no—
190C2×C5×Q8—24no—
191C2×C5×Q8—42no—
192C2×C5×Q8C22×D312yes—
193C2×C5×Q8C22×D311yescharacteristic
194C2×C5×D4#6711yescharacteristic
195C22×C4×C5—22no—
196C22×C4×C5C22×D312yes—
197C2×C22⋊D5—62no—
198C22×Dic5—62no—
199#711—124no—
200C2×D20—31no—
201C2×C4×D5—61no—
202C2×Dic10—61no—
203C5×C4⋊C4—21no—
204C5×C4⋊C4—21no—
205C5×C4⋊C4—21no—
206C5×C4⋊C4—22no—
207C5×C4⋊C4#6711yescharacteristic
208C5×C4⋊C4C22×D311yescharacteristic
209C5×C22⋊C4—22no—
210C5×C22⋊C4#6712yes—
211C42×C5—21no—
212C42×C5C22×D311yescharacteristic
213#688—62no—
214#688—122no—
215#685—61no—
216#685—62no—
217C4⋊Dic5—32no—
218C4⋊Dic5—31no—
219C4⋊Dic5—61no—
220C4⋊Dic5—62no—
221C2×#168—31no—
222C2×#168—31no—
223C2×#168—32no—
224C2×#168—122no—
225D4×Q8D1511yescharacteristic, complemented
226#547—151no—
227#487—151no—
228#483—152no—
229#475—151no—
230#465—152no—
231#461—151no—
232#448—152no—
233C4×SD16—151no—
234C8×D4—151no—
235C2×#103—152no—
236C22×C3×C5—22no—
237C22×C3×C5—22no—
238C22×C3×C5C2×SD1614yes—
239C22×C3×C5C22×D411yescharacteristic
240C2×D15—41no—
241C2×D15—41no—
242C2×D15—41no—
243C3×C4×C5—22no—
244C3×C4×C5—22no—
245C3×C4×C5—21no—
246C3×C4×C5—42no—
247C3×C4×C5—41no—
248C3×C4×C5C2×SD1612yes—
249C3×C4×C5C2×#4111yescharacteristic
250C3×C4×C5H(5,𝔽2)11yescharacteristic
251Dic15—42no—
252Dic15—41no—
253C2×C3×Q8—22no—
254C2×C3×Q8—24no—
255C2×C3×Q8—42no—
256C2×C3×Q8C22×D512yes—
257C2×C3×Q8C22×D511yescharacteristic
258C2×C3×D4C22⋊D511yescharacteristic
259C22×C3×C4—22no—
260C22×C3×C4C22×D512yes—
261C2×#67—102no—
262C22×Dic3—102no—
263#237—204no—
264C2×D12—51no—
265C2×C4×D3—101no—
266C2×Dic6—101no—
267C3×C4⋊C4—21no—
268C3×C4⋊C4—21no—
269C3×C4⋊C4—21no—
270C3×C4⋊C4—22no—
271C3×C4⋊C4C22⋊D511yescharacteristic
272C3×C4⋊C4C22×D511yescharacteristic
273C3×C22⋊C4—22no—
274C3×C22⋊C4C22⋊D512yes—
275C3×C42—21no—
276C3×C42C22×D511yescharacteristic
277#215—102no—
278#215—202no—
279(C2×D3)⋊C4—101no—
280(C2×D3)⋊C4—102no—
281C4⋊Dic3—52no—
282C4⋊Dic3—51no—
283C4⋊Dic3—101no—
284C4⋊Dic3—102no—
285C2×C3⋊C8—51no—
286C2×C3⋊C8—51no—
287C2×C3⋊C8—52no—
288C2×C3⋊C8—202no—
289C23×C5C2×#6712yes—
290C22×D5—61no—
291C5×Q8—21no—
292C5×Q8—22no—
293C5×Q8—22no—
294C5×Q8—44no—
295C5×Q8#23712yescomplemented
296C5×D4#21514yescomplemented
297C2×C4×C5—21no—
298C2×C4×C5—22no—
299C2×C4×C5—22no—
300C2×C4×C5—21no—
301C2×C4×C5—22no—
302C2×C4×C5—44no—
303C2×C4×C5C2×#6711yescharacteristic
304C2×C4×C5C2×#6712yes—
305C2×C4×C5C2×#6711yescharacteristic
306C2×C4×C5C23×D311yescharacteristic
307C22⋊D5—124no—
308C2×Dic5—62no—
309C2×Dic5—61no—
310C2×Dic5—124no—
311D20—61no—
312D20—61no—
313C4×D5—122no—
314Dic10—122no—
315#168—61no—
316#168—61no—
317#168—62no—
318C2×#41—301no—
319C22×Q8C2×D1512yes—
320C2×SD16—151no—
321C2×SD16—152no—
322C22×C8—152no—
323C4⋊Q8—21no—
324C4⋊Q8—151no—
325C4⋊Q8C2×D1511yescharacteristic
326#123—302no—
327C22⋊Q8—22no—
328C22⋊Q8—301no—
329C22⋊Q8C2×D1512yes—
330#121—152no—
331C4×Q8C2×D1511yescharacteristic
332C4×D4—21no—
333C4×D4—151no—
334C4×D4C2×D1511yescharacteristic
335C2×C4⋊C4—152no—
336#106—151no—
337#106—152no—
338C4⋊C8—151no—
339#103—151no—
340#103—152no—
341#103—152no—
342#103—152no—
343#103—152no—
344#102—151no—
345C22⋊C8—152no—
346C4×C8—151no—
347C2×C3×C5—24no—
348C2×C3×C5#54611yescharacteristic
349C2×C3×C5C22×SD1611yescharacteristic
350C2×C3×C5#57811yescharacteristic
351D15—81no—
352C23×C3C2×C22⋊D512yes—
353C22×D3—101no—
354C3×Q8—21no—
355C3×Q8—22no—
356C3×Q8—22no—
357C3×Q8—44no—
358C3×Q8#71112yescomplemented
359C3×D4#68814yescomplemented
360C2×C3×C4—21no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—21no—
364C2×C3×C4—22no—
365C2×C3×C4—44no—
366C2×C3×C4C2×C22⋊D511yescharacteristic
367C2×C3×C4C2×C22⋊D512yes—
368C2×C3×C4C2×C22⋊D511yescharacteristic
369C2×C3×C4C23×D511yescharacteristic
370#67—204no—
371C2×Dic3—102no—
372C2×Dic3—101no—
373C2×Dic3—204no—
374D12—101no—
375D12—101no—
376C4×D3—202no—
377Dic6—202no—
378C3⋊C8—101no—
379C3⋊C8—101no—
380C3⋊C8—102no—
381C22×C5—22no—
382C22×C5—22no—
383C22×C5C2×#21514yes—
384C22×C5C22×#6711yescharacteristic
385C2×D5—121no—
386C2×D5—121no—
387C2×D5—121no—
388C4×C5—22no—
389C4×C5—22no—
390C4×C5—21no—
391C4×C5—42no—
392C4×C5—41no—
393C4×C5C2×#21512yes—
394C4×C5C2×#23711yescharacteristic
395C4×C5#100411yescharacteristic
396Dic5—122no—
397Dic5—121no—
398#41—604no—
399C2×Q8—22no—
400C2×Q8—24no—
401C2×Q8—42no—
402C2×Q8—301no—
403C2×Q8C22×D1512yes—
404C2×Q8C22×D1511yescharacteristic
405C2×D4—151no—
406C2×D4—302no—
407C2×D4#122011yescharacteristic
408C22×C4—22no—
409C22×C4—302no—
410C22×C4—301no—
411C22×C4C22×D1512yes—
412SD16—302no—
413SD16—602no—
414C2×C8—151no—
415C2×C8—151no—
416C2×C8—152no—
417C2×C8—602no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—22no—
422C4⋊C4—151no—
423C4⋊C4—152no—
424C4⋊C4—302no—
425C4⋊C4—301no—
426C4⋊C4#122011yescharacteristic
427C4⋊C4C22×D1511yescharacteristic
428C22⋊C4—22no—
429C22⋊C4—301no—
430C22⋊C4—302no—
431C22⋊C4#122012yes—
432C42—21no—
433C42C22×D1511yescharacteristic
434C3×C5#329811yescharacteristic, complemented, Hall
435C22×C3—22no—
436C22×C3—22no—
437C22×C3C2×#68814yes—
438C22×C3C22×C22⋊D511yescharacteristic
439C2×D3—201no—
440C2×D3—201no—
441C2×D3—201no—
442C3×C4—22no—
443C3×C4—22no—
444C3×C4—21no—
445C3×C4—42no—
446C3×C4—41no—
447C3×C4C2×#68812yes—
448C3×C4C2×#71111yescharacteristic
449C3×C4#414411yescharacteristic
450Dic3—202no—
451Dic3—201no—
452C2×C5—24no—
453C2×C5#580611yescharacteristic
454C2×C5C22×#21511yescharacteristic
455C2×C5#583811yescharacteristic
456D5—241no—
457C23—301no—
458C23C2×#122012yes—
459Q8—21no—
460Q8—22no—
461Q8—22no—
462Q8—44no—
463Q8—602no—
464Q8#681912yescomplemented
465D4—301no—
466D4—301no—
467D4—604no—
468D4#671714yescomplemented
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—44no—
475C2×C4—301no—
476C2×C4—302no—
477C2×C4—604no—
478C2×C4—602no—
479C2×C4C2×#122011yescharacteristic
480C2×C4C2×#122012yes—
481C2×C4C2×#122011yescharacteristic
482C2×C4C23×D1511yescharacteristic
483C8—301no—
484C8—301no—
485C8—302no—
486C2×C3—24no—
487C2×C3#6614011yescharacteristic
488C2×C3C22×#68811yescharacteristic
489C2×C3#6617211yescharacteristic
490D3—401no—
491C5#8355011yescharacteristic, complemented, Sylow
492C22—22no—
493C22—22no—
494C22—601no—
495C22—601no—
496C22—601no—
497C22C2×#671714yes—
498C22C22×#122011yescharacteristic
499C4—22no—
500C4—22no—
501C4—21no—
502C4—42no—
503C4—41no—
504C4—601no—
505C4—602no—
506C4C2×#671712yes—
507C4C2×#681911yescharacteristic
508C4#9209511yescharacteristic
509C3#1061528111yescharacteristic, complemented, Sylow
510C2—24no—
511C2—1201no—
512C2#1175656511yescharacteristic
513C2C22×#671711yescharacteristic
514C2#1175659711yescharacteristic
subgroupG/H
centerC22C22×#1220
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC3×C5×D4×Q8C2
socleC22×C3×C5C22×D4
prime radical#11756314C2
hypercenterD4×Q8D15
hypocenterC3×C5#3298
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C3×C5×D4×Q8C2semidirect
3. #11751328C2semidirect
4. C2×#91119C2semidirect
5. #11751184C2semidirect
6. #11751180C2semidirect
7. #11751149C2semidirect
8. #11751144C2semidirect
9. C4×#6717C2semidirect
10. #11751121C2semidirect
11. D4×C3⋊#168C2semidirect
43. C5×D4×Q8D3semidirect
90. C3×D4×Q8D5semidirect
137. C3×C5×Q8#41semidirect
138. C3×C5×D4SD16semidirect
225. D4×Q8D15semidirect
295. C5×Q8#237semidirect
296. C5×D4#215semidirect
358. C3×Q8#711semidirect
359. C3×D4#688semidirect
434. C3×C5#3298semidirect
464. Q8#6819semidirect
468. D4#6717semidirect
491. C5#83550semidirect
509. C3#10615281semidirect

Maximal subgroups

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

subgroupindex
C3×C5×D4×Q82
#117563142
#117513282
C2×#911192
#117511842
#117511802
#117511492
#117511442
C4×#67172
#117511212
D4×C3⋊#1682
#106152813
#835505