Finite Group Databasebeta

← #14459129

Subgroups — #14459129

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
1Q8×D60C211yesmaximal, characteristic, complemented
2#11756310C211yesmaximal, characteristic
3C3×C4×C5×SD16C211yesmaximal, characteristic, complemented
4#11751271C212yesmaximal, complemented
5#11751121C211yesmaximal, characteristic, complemented
6#11751120C211yesmaximal, characteristic, complemented
7#11750963C212yesmaximal, complemented
8D15×#103C212yesmaximal, complemented
9#11750879C212yesmaximal, complemented
10C8×D60C211yesmaximal, characteristic, complemented
11#11750782C211yesmaximal, characteristic, complemented
12#10611409—31nomaximal, Hall
13C2×Q8×D15C2212yes—
14C2×#6817—21no—
15C2×C3×C5×SD16C2211yescharacteristic
16C3×C4×C5×Q8C2211yescharacteristic
17C3×C4×C5×D4C2211yescharacteristic
18#91827C2212yes—
19C2×#6716C2212yes—
20#91813—21no—
21C2×C8×D15C2212yes—
22#91786—22no—
23#91786C2212yes—
24C4⋊C4×D15C2212yes—
25#91772—22no—
26C4×D60—21no—
27C4×D60C2211yescharacteristic
28#91758—21no—
29#91758C2211yescharacteristic
30#91758C2211yescharacteristic
31C3×C5×#106C2211yescharacteristic
32C3×C5×#103C2211yescharacteristic
33C3×C5×#102C2211yescharacteristic
34C3×C4×C5×C8C2211yescharacteristic
35#91119C2212yes—
36#91104C2212yes—
37#91100C2212yes—
38#91100C2212yes—
39#91094C2212yes—
40#91092C2212yes—
41C4⋊(C3⋊#168)C2211yescharacteristic
42#79678—51nomaximal, Hall
43Q8×D20—31no—
44#65885—31no—
45C4×C5×SD16D311yescharacteristic, complemented
46#65459—32no—
47#65309—31no—
48#65308—31no—
49#65151—32no—
50D5×#103—32no—
51#65067—32no—
52C8×D20—31no—
53#64970—31no—
54C2×C3×C5×Q8C2311yescharacteristic
55C2×C3×C5×D4C2311yescharacteristic
56C22×C3×C4×C5—21no—
57C2×#1220—22no—
58Q8×D15—24no—
59Q8×D15—42no—
60#6817—44no—
61C2×D60D411yescharacteristic
62C2×C4×D15—22no—
63C2×C4×D15—22no—
64C2×C4×D15C2312yes—
65C2×Dic30—21no—
66C2×Dic30—22no—
67C2×Dic30C2312yes—
68C3×C5×SD16—22no—
69C2×C3×C5×C8C2311yescharacteristic
70C2×C3×C5×C8C2311yescharacteristic
71C3×C5×C4⋊C4—21no—
72C3×C5×C4⋊C4C2311yescharacteristic
73C3×C5×C4⋊C4C2311yescharacteristic
74C3×C5×C22⋊C4—21no—
75C3×C42×C5—21no—
76C3×C42×C5C2311yescharacteristic
77#6719—22no—
78#6716—42no—
79#6714—22no—
80#6714D412yes—
81C4⋊Dic15—21no—
82C4⋊Dic15—21no—
83C4⋊Dic15—21no—
84C4⋊Dic15—22no—
85C4⋊Dic15D411yescharacteristic
86C4⋊Dic15C2312yes—
87#6712—22no—
88C2×C3⋊#168C2312yes—
89C8×D15—42no—
90Q8×D12—51no—
91#5551—51no—
92C3×C4×SD16D511yescharacteristic, complemented
93#5170—52no—
94#5020—51no—
95#5019—51no—
96#4862—52no—
97D3×#103—52no—
98#4778—52no—
99C8×D12—51no—
100#4681—51no—
101C2×Q8×D5—32no—
102C2×#709—61no—
103C2×C5×SD16C2×D311yescharacteristic
104C4×C5×Q8C2×D311yescharacteristic
105C4×C5×D4C2×D311yescharacteristic
106#4084—32no—
107C2×#687—32no—
108#4070—61no—
109C2×C8×D5—32no—
110#4043—32no—
111#4043—62no—
112D5×C4⋊C4—32no—
113#4029—62no—
114C4×D20—31no—
115C4×D20—61no—
116#4015—31no—
117#4015—31no—
118#4015—61no—
119C5×#106C2×D311yescharacteristic
120C5×#103C2×D311yescharacteristic
121C5×#102C2×D311yescharacteristic
122C4×C5×C8C2×D311yescharacteristic
123#3967—32no—
124#3952—32no—
125#3948—32no—
126#3948—32no—
127#3942—32no—
128#3940—32no—
129C4⋊#168—31no—
130#3298—151noSylow
131C23×C3×C5—21no—
132C22×D15C2×D412yes—
133C3×C5×Q8—21no—
134C3×C5×Q8#4112yescomplemented
135C3×C5×D4—21no—
136C3×C5×D4—21no—
137C2×C3×C4×C5—21no—
138C2×C3×C4×C5—21no—
139C2×C3×C4×C5—21no—
140C2×C3×C4×C5—42no—
141C2×C3×C4×C5C2×D411yescharacteristic
142C2×C3×C4×C5C2×D411yescharacteristic
143C2×C3×C4×C5C2411yescharacteristic
144#1220—44no—
145C2×Dic15—22no—
146C2×Dic15—22no—
147C2×Dic15C2×D412yes—
148D60SD1614yescomplemented
149C4×D15—22no—
150C4×D15—22no—
151C4×D15—44no—
152C4×D15—44no—
153Dic30—22no—
154Dic30—22no—
155Dic30—44no—
156Dic30—42no—
157C3×C5×C8—21no—
158C3×C5×C8—21no—
159C3⋊#168—22no—
160C2×D3×Q8—52no—
161C2×#235—101no—
162C2×C3×SD16C2×D511yescharacteristic
163C3×C4×Q8C2×D511yescharacteristic
164C3×C4×D4C2×D511yescharacteristic
165#938—52no—
166C2×#214—52no—
167#924—101no—
168C2×D3×C8—52no—
169#897—52no—
170#897—102no—
171D3×C4⋊C4—52no—
172#883—102no—
173C4×D12—51no—
174C4×D12—101no—
175#869—51no—
176#869—51no—
177#869—101no—
178C3×#106C2×D511yescharacteristic
179C3×#103C2×D511yescharacteristic
180C3×#102C2×D511yescharacteristic
181C3×C4×C8C2×D511yescharacteristic
182#835—52no—
183(C2×D3)⋊C8—52no—
184#816—52no—
185#816—52no—
186#810—52no—
187#808—52no—
188C4⋊(C3⋊C8)—51no—
189C2×C5×Q8C22×D311yescharacteristic
190C2×C5×D4C22×D311yescharacteristic
191C22×C4×C5—21no—
192C2×C22⋊D5—62no—
193Q8×D5—64no—
194Q8×D5—122no—
195#709—124no—
196C2×D20—31no—
197C2×C4×D5—32no—
198C2×C4×D5—62no—
199C2×C4×D5—62no—
200C2×Dic10—32no—
201C2×Dic10—62no—
202C2×Dic10—61no—
203C5×SD16—22no—
204C2×C5×C8C22×D311yescharacteristic
205C2×C5×C8C22×D311yescharacteristic
206C5×C4⋊C4—21no—
207C5×C4⋊C4C22×D311yescharacteristic
208C5×C4⋊C4C22×D311yescharacteristic
209C5×C22⋊C4—21no—
210C42×C5—21no—
211C42×C5C22×D311yescharacteristic
212C22⋊Dic5—62no—
213#687—122no—
214#685—32no—
215#685—62no—
216C4⋊Dic5—31no—
217C4⋊Dic5—32no—
218C4⋊Dic5—62no—
219C4⋊Dic5—61no—
220C4⋊Dic5—61no—
221C4⋊Dic5—61no—
222#683—62no—
223C2×#168—32no—
224C8×D5—122no—
225D4×Q8—151no—
226#547—151no—
227#487—151no—
228#483—152no—
229#475—151no—
230#465—152no—
231#461—151no—
232#448—152no—
233C4×SD16D1511yescharacteristic, complemented
234C8×D4—151no—
235C2×#103—152no—
236C22×C3×C5—41no—
237C22×C3×C5—41no—
238C22×C3×C5—41no—
239C22×C3×C5C22×D411yescharacteristic
240C2×D15—22no—
241C2×D15—22no—
242C2×D15C2×SD1614yes—
243C3×C4×C5—22no—
244C3×C4×C5—21no—
245C3×C4×C5—41no—
246C3×C4×C5—41no—
247C3×C4×C5C2×SD1612yes—
248C3×C4×C5C2×#4111yescharacteristic
249C3×C4×C5H(5,𝔽2)11yescharacteristic
250Dic15—22no—
251Dic15—42no—
252Dic15—42no—
253C2×C3×Q8C22×D511yescharacteristic
254C2×C3×D4C22×D511yescharacteristic
255C22×C3×C4—21no—
256C2×#67—102no—
257D3×Q8—104no—
258D3×Q8—202no—
259#235—204no—
260C2×D12—51no—
261C2×C4×D3—52no—
262C2×C4×D3—102no—
263C2×C4×D3—102no—
264C2×Dic6—52no—
265C2×Dic6—102no—
266C2×Dic6—101no—
267C3×SD16—22no—
268C2×C3×C8C22×D511yescharacteristic
269C2×C3×C8C22×D511yescharacteristic
270C3×C4⋊C4—21no—
271C3×C4⋊C4C22×D511yescharacteristic
272C3×C4⋊C4C22×D511yescharacteristic
273C3×C22⋊C4—21no—
274C3×C42—21no—
275C3×C42C22×D511yescharacteristic
276#217—102no—
277#214—202no—
278(C2×D3)⋊C4—52no—
279(C2×D3)⋊C4—102no—
280C4⋊Dic3—51no—
281C4⋊Dic3—52no—
282C4⋊Dic3—102no—
283C4⋊Dic3—101no—
284C4⋊Dic3—101no—
285C4⋊Dic3—101no—
286Dic3⋊C4—102no—
287C2×C3⋊C8—52no—
288D3×C8—202no—
289C23×C5—21no—
290C22×D5—32no—
291C5×Q8—21no—
292C5×Q8#23512yescomplemented
293C5×D4—21no—
294C5×D4—21no—
295C2×C4×C5—21no—
296C2×C4×C5—21no—
297C2×C4×C5—21no—
298C2×C4×C5—42no—
299C2×C4×C5D3×D411yescharacteristic
300C2×C4×C5D3×D411yescharacteristic
301C2×C4×C5C23×D311yescharacteristic
302C22⋊D5—124no—
303C2×Dic5—32no—
304C2×Dic5—62no—
305C2×Dic5—62no—
306D20—34no—
307C4×D5—62no—
308C4×D5—62no—
309C4×D5—124no—
310C4×D5—124no—
311Dic10—62no—
312Dic10—62no—
313Dic10—124no—
314Dic10—122no—
315C5×C8—21no—
316C5×C8—21no—
317#168—62no—
318C2×#41—301no—
319C22×Q8—152no—
320C2×SD16—152no—
321C2×SD16C2×D1511yescharacteristic
322C22×C8—152no—
323C4⋊Q8—151no—
324C4⋊Q8—151no—
325C4⋊Q8—301no—
326#123—302no—
327C22⋊Q8—152no—
328C22⋊Q8—302no—
329C22⋊Q8—301no—
330#121—152no—
331C4×Q8C2×D1511yescharacteristic
332C4×D4—151no—
333C4×D4—301no—
334C4×D4C2×D1511yescharacteristic
335C2×C4⋊C4—152no—
336#106—152no—
337#106C2×D1511yescharacteristic
338C4⋊C8—151no—
339#103—152no—
340#103—152no—
341#103—152no—
342#103—152no—
343#103C2×D1511yescharacteristic
344#102C2×D1511yescharacteristic
345C22⋊C8—152no—
346C4×C8C2×D1511yescharacteristic
347C2×C3×C5—81no—
348C2×C3×C5#54611yescharacteristic
349C2×C3×C5C22×SD1611yescharacteristic
350C2×C3×C5#57811yescharacteristic
351D15—24no—
352C23×C3—21no—
353C22×D3—52no—
354C3×Q8—21no—
355C3×Q8#70912yescomplemented
356C3×D4—21no—
357C3×D4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—42no—
362C2×C3×C4D4×D511yescharacteristic
363C2×C3×C4D4×D511yescharacteristic
364C2×C3×C4C23×D511yescharacteristic
365#67—204no—
366C2×Dic3—52no—
367C2×Dic3—102no—
368C2×Dic3—102no—
369D12—54no—
370C4×D3—102no—
371C4×D3—102no—
372C4×D3—204no—
373C4×D3—204no—
374Dic6—102no—
375Dic6—102no—
376Dic6—204no—
377Dic6—202no—
378C3×C8—21no—
379C3×C8—21no—
380C3⋊C8—102no—
381C22×C5—41no—
382C22×C5—41no—
383C22×C5—41no—
384C22×C5C2×D3×D411yescharacteristic
385C2×D5—34no—
386C2×D5—62no—
387C2×D5—62no—
388C4×C5—21no—
389C4×C5—22no—
390C4×C5—41no—
391C4×C5—41no—
392C4×C5D3×SD1612yes—
393C4×C5C2×#23511yescharacteristic
394C4×C5#100911yescharacteristic
395Dic5—62no—
396Dic5—122no—
397Dic5—122no—
398#41—604no—
399C2×Q8—152no—
400C2×Q8—302no—
401C2×Q8—301no—
402C2×Q8—304no—
403C2×Q8—602no—
404C2×Q8C22×D1511yescharacteristic
405C2×D4—151no—
406C2×D4—302no—
407C2×D4C22×D1511yescharacteristic
408C22×C4—21no—
409C22×C4—152no—
410C22×C4—302no—
411C22×C4—302no—
412SD16—22no—
413SD16—602no—
414C2×C8—152no—
415C2×C8—602no—
416C2×C8C22×D1511yescharacteristic
417C2×C8C22×D1511yescharacteristic
418C4⋊C4—21no—
419C4⋊C4—151no—
420C4⋊C4—152no—
421C4⋊C4—302no—
422C4⋊C4—302no—
423C4⋊C4—301no—
424C4⋊C4—301no—
425C4⋊C4—301no—
426C4⋊C4C22×D1511yescharacteristic
427C4⋊C4C22×D1511yescharacteristic
428C22⋊C4—21no—
429C22⋊C4—152no—
430C22⋊C4—302no—
431C22⋊C4—302no—
432C42—21no—
433C42C22×D1511yescharacteristic
434C3×C5#329811yescharacteristic, complemented, Hall
435C22×C3—41no—
436C22×C3—41no—
437C22×C3—41no—
438C22×C3C2×D4×D511yescharacteristic
439C2×D3—54no—
440C2×D3—102no—
441C2×D3—102no—
442C3×C4—21no—
443C3×C4—22no—
444C3×C4—41no—
445C3×C4—41no—
446C3×C4D5×SD1612yes—
447C3×C4C2×#70911yescharacteristic
448C3×C4#414911yescharacteristic
449Dic3—102no—
450Dic3—202no—
451Dic3—202no—
452C2×C5—81no—
453C2×C5#555011yescharacteristic
454C2×C5C2×D3×SD1611yescharacteristic
455C2×C5#577111yescharacteristic
456D5—64no—
457C23—21no—
458C23—152no—
459Q8—21no—
460Q8—302no—
461Q8—302no—
462Q8—604no—
463Q8—602no—
464Q8#681712yescomplemented
465D4—21no—
466D4—21no—
467D4—154no—
468D4—604no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—42no—
473C2×C4—152no—
474C2×C4—302no—
475C2×C4—302no—
476C2×C4—302no—
477C2×C4—302no—
478C2×C4—604no—
479C2×C4—604no—
480C2×C4D4×D1511yescharacteristic
481C2×C4D4×D1511yescharacteristic
482C2×C4C23×D1511yescharacteristic
483C8—21no—
484C8—21no—
485C8—302no—
486C2×C3—81no—
487C2×C3#6588411yescharacteristic
488C2×C3C2×D5×SD1611yescharacteristic
489C2×C3#6610511yescharacteristic
490D3—104no—
491C5#7967811yescharacteristic, complemented, Sylow
492C22—41no—
493C22—41no—
494C22—41no—
495C22—154no—
496C22—302no—
497C22—302no—
498C22C2×D4×D1511yescharacteristic
499C4—22no—
500C4—21no—
501C4—41no—
502C4—41no—
503C4—302no—
504C4—602no—
505C4—602no—
506C4SD16×D1512yes—
507C4C2×#681711yescharacteristic
508C4#9210011yescharacteristic
509C3#1061140911yescharacteristic, complemented, Sylow
510C2—81no—
511C2—304no—
512C2#1175630911yescharacteristic
513C2C2×SD16×D1511yescharacteristic
514C2#1175653011yescharacteristic
subgroupG/H
centerC22C2×D4×D15
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC3×C4×C5×SD16C2
socleC22×C3×C5C22×D4
prime radical#11756310C2
hypercenterC4×SD16D15
hypocenterC3×C5#3298
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. Q8×D60C2semidirect
3. C3×C4×C5×SD16C2semidirect
4. #11751271C2semidirect
5. #11751121C2semidirect
6. #11751120C2semidirect
7. #11750963C2semidirect
8. D15×#103C2semidirect
9. #11750879C2semidirect
10. C8×D60C2semidirect
11. #11750782C2semidirect
45. C4×C5×SD16D3semidirect
92. C3×C4×SD16D5semidirect
134. C3×C5×Q8#41semidirect
148. D60SD16semidirect
233. C4×SD16D15semidirect
292. C5×Q8#235semidirect
355. C3×Q8#709semidirect
434. C3×C5#3298semidirect
464. Q8#6817semidirect
491. C5#79678semidirect
509. C3#10611409semidirect

Maximal subgroups

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

subgroupindex
Q8×D602
#117563102
C3×C4×C5×SD162
#117512712
#117511212
#117511202
#117509632
D15×#1032
#117508792
C8×D602
#117507822
#106114093
#796785