Finite Group Databasebeta

← #14302003

Subgroups — #14302003

Order 1920.

716 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11754584C211yesmaximal, characteristic
2C2×#91371C211yesmaximal, characteristic, complemented
3C2×#91109C211yesmaximal, characteristic, complemented
4C5×#5126C211yesmaximal, characteristic, complemented
5C2×#90981C211yesmaximal, characteristic, complemented
6C3×#64775C211yesmaximal, characteristic, complemented
7#11745973C211yesmaximal, characteristic, complemented
8#10600759—31nomaximal, Hall
9C2×#6817C2211yescharacteristic
10C22×C4×C5×D3C2211yescharacteristic
11C2×C3×C5×M16C2211yescharacteristic
12C2×C4×Dic15C2211yescharacteristic
13C2×C5×#208C2211yescharacteristic
14C3×#4072C2211yescharacteristic
15C2×C3×C4×Dic5C2211yescharacteristic
16#91570—21no—
17C2×Dic3⋊Dic5—21no—
18#91542—21no—
19C2×(C2×D3)⋊Dic5—21no—
20#91460C412yescomplemented
21C4×#1203—22no—
22#91371—22no—
23#91371—22no—
24#91365C412yescomplemented
25#91109—22no—
26C5×(C2×D3)⋊C8—22no—
27#90981—22no—
28#68757—51nomaximal, Hall
29#66125—31no—
30C2×#4072—31no—
31C5×#407—31no—
32C2×#3957—31no—
33C2×#3957—31no—
34#64775—31no—
35#64775D311yescharacteristic, complemented
36C23×C5×D3D411yescharacteristic
37C22×C3×C4×C5C2311yescharacteristic
38C2×#1220—21no—
39C22×Dic15—21no—
40#6817—21no—
41#6817—21no—
42#6817—21no—
43#6817—21no—
44C2×D60C2×C411yescharacteristic
45C2×C4×D15—21no—
46C2×Dic30C2×C411yescharacteristic
47C22×C5×Dic3D411yescharacteristic
48C2×C4×C5×D3—21no—
49C2×C4×C5×D3—21no—
50C2×C4×C5×D3—21no—
51C2×C4×C5×D3—21no—
52C2×C4×C5×D3C2×C412yes—
53C2×C4×C5×D3D412yes—
54C22×C3×Dic5—21no—
55C2×#1203—22no—
56C3×C5×M16—21no—
57C3×C5×M16—21no—
58C2×C3×C5×C8—21no—
59C4×Dic15—21no—
60C4×Dic15D412yescomplemented
61C5×#208—21no—
62C5×#208—21no—
63C2×C5×C3⋊C8—21no—
64C3×C22⋊Dic5—21no—
65C3×C4⋊Dic5C2×C412yes—
66C3×C4×Dic5—21no—
67C3×C4×Dic5—21no—
68C3×C4×Dic5D412yescomplemented
69Dic3⋊Dic5—24no—
70#6668—22no—
71#6667—22no—
72(C2×D3)⋊Dic5—24no—
73(C2×D3)⋊Dic5—22no—
74#5524—51no—
75C2×#872—51no—
76C2×#825—51no—
77#5126D511yescharacteristic, complemented
78C2×#805—51no—
79C3×#339—51no—
80#4532—51no—
81C23×C4×C5—31no—
82C2×#709—31no—
83C2×C5×M16—31no—
84C2×C5×M16C2×D311yescharacteristic
85#4099—61no—
86C2×C22⋊Dic5—61no—
87#4076—32no—
88#4075—61no—
89C4×C22⋊D5—62no—
90#4072—62no—
91#4072—62no—
92#4072C2×D311yescharacteristic
93C2×C4⋊Dic5—61no—
94#4070—32no—
95C2×C4×Dic5—31no—
96C2×C4×Dic5C2×D311yescharacteristic
97C5×C22⋊C8—62no—
98#3957—62no—
99#3957—62no—
100#1897—151noSylow
101C23×C3×C5C2×D411yescharacteristic
102C22×D15—21no—
103C22×C5×D3—21no—
104C22×C5×D3—21no—
105C22×C5×D3—42no—
106C22×C5×D3C22⋊C412yes—
107C2×C3×C4×C5C22×C411yescharacteristic
108C2×C3×C4×C5C2×D411yescharacteristic
109C2×C3×C4×C5C2×D411yescharacteristic
110C2×C3×C4×C5C2×D411yescharacteristic
111C2×C3×C4×C5#4111yescharacteristic
112C2×C3×C4×C5#4111yescharacteristic
113#1220—41no—
114#1220—41no—
115C2×Dic15—21no—
116C2×Dic15—22no—
117C2×Dic15—42no—
118D60—21no—
119D60—21no—
120C4×D15—41no—
121C4×D15—41no—
122Dic30—21no—
123Dic30—21no—
124C2×C5×Dic3—21no—
125C2×C5×Dic3—21no—
126C2×C5×Dic3C22⋊C412yes—
127C4×C5×D3—22no—
128C4×C5×D3—22no—
129C4×C5×D3—22no—
130C4×C5×D3—22no—
131C2×C3×Dic5—22no—
132C2×C3×Dic5—22no—
133C2×C3×Dic5—42no—
134#1203—44no—
135C3×C5×C8—41no—
136C5×C3⋊C8—41no—
137C2×#235—51no—
138C22×C4×D3C2×D511yescharacteristic
139C2×C3×M16C2×D511yescharacteristic
140C3×#117—51no—
141C2×C3×C42—51no—
142C2×(C2×D3)⋊C4—101no—
143C2×Dic3⋊C4—101no—
144C2×C4×Dic3—51no—
145C2×#208C2×D511yescharacteristic
146#897—52no—
147#895—52no—
148#886—101no—
149#882—101no—
150C4×D12—102no—
151#872—102no—
152#872—102no—
153#825—102no—
154(C2×D3)⋊C8—22no—
155#805—102no—
156C24×C5—31no—
157C22×C4×C5—31no—
158C22×C4×C5—32no—
159C22×C4×C5—32no—
160C22×C4×C5—61no—
161C22×C4×C5—61no—
162C22×C4×C5—61no—
163C22×C4×C5—61no—
164C22×C4×C5C22×D311yescharacteristic
165C2×C22⋊D5—62no—
166C2×C22⋊D5—61no—
167C22×Dic5—21no—
168C22×Dic5—61no—
169#709—61no—
170#709—61no—
171#709—61no—
172#709—61no—
173C2×D20—31no—
174C2×C4×D5—61no—
175C2×Dic10—31no—
176C5×M16—21no—
177C5×M16—21no—
178C5×M16—61no—
179C5×M16—61no—
180C2×C5×C8—21no—
181C2×C5×C8—61no—
182C22⋊Dic5—21no—
183C22⋊Dic5—62no—
184C22⋊Dic5—64no—
185#685—62no—
186C4⋊Dic5—64no—
187C4⋊Dic5C4×D312yes—
188#683—62no—
189C4×Dic5—21no—
190C4×Dic5—21no—
191C4×Dic5—32no—
192C4×Dic5—61no—
193C4×Dic5D1212yescomplemented
194#525—151no—
195C2×#117—151no—
196C2×#104—151no—
197C2×#104—151no—
198#407—31no—
199#339—51no—
200#339—151no—
201C22×C3×C5—22no—
202C22×C3×C5C2×C22⋊C411yescharacteristic
203C22×C3×C5#12011yescharacteristic
204C22×C3×C5#12311yescharacteristic
205C2×D15—41no—
206C2×D15—42no—
207C2×C5×D3—22no—
208C2×C5×D3—22no—
209C2×C5×D3—22no—
210C2×C5×D3—42no—
211C2×C5×D3—42no—
212C3×C4×C5C4×D411yescharacteristic
213C3×C4×C5C4×D411yescharacteristic
214C3×C4×C5#12111yescharacteristic
215C3×C4×C5#12111yescharacteristic
216Dic15—42no—
217Dic15—41no—
218C3×Dic5—42no—
219C3×Dic5—42no—
220C5×Dic3—22no—
221C23×D3D2011yescharacteristic
222C22×C3×C4—101no—
223C22×C3×C4C22×D511yescharacteristic
224C2×#67—101no—
225C22×Dic3—101no—
226C22×Dic3D2011yescharacteristic
227#235—101no—
228#235—101no—
229#235—101no—
230#235—101no—
231C2×D12—51no—
232C2×D12—102no—
233C2×C4×D3—21no—
234C2×C4×D3—21no—
235C2×C4×D3—21no—
236C2×C4×D3—21no—
237C2×C4×D3—101no—
238C2×C4×D3C4×D512yes—
239C2×C4×D3C22⋊D512yes—
240C2×Dic6—51no—
241C3×M16—21no—
242C3×M16—21no—
243C2×C3×C8—21no—
244C3×C4⋊C4—52no—
245C3×C22⋊C4—101no—
246C3×C42—52no—
247C3×C42—101no—
248C3×C42—101no—
249(C2×D3)⋊C4—104no—
250(C2×D3)⋊C4—102no—
251(C2×D3)⋊C4—102no—
252C4⋊Dic3—102no—
253Dic3⋊C4—104no—
254C4×Dic3—52no—
255C4×Dic3—101no—
256#208—21no—
257#208—21no—
258C2×C3⋊C8—21no—
259C23×C5—32no—
260C23×C5—61no—
261C23×C5—61no—
262C23×C5—122no—
263C23×C5D3×D411yescharacteristic
264C22×D5—61no—
265C2×C4×C5—32no—
266C2×C4×C5—61no—
267C2×C4×C5—61no—
268C2×C4×C5—62no—
269C2×C4×C5—62no—
270C2×C4×C5—62no—
271C2×C4×C5—62no—
272C2×C4×C5C2×C4×D311yescharacteristic
273C2×C4×C5C2×D1211yescharacteristic
274C2×C4×C5#23511yescharacteristic
275C2×C4×C5D3×D411yescharacteristic
276C2×C4×C5D3×D411yescharacteristic
277C2×C4×C5#23911yescharacteristic
278C22⋊D5—124no—
279C22⋊D5—121no—
280C22⋊D5—121no—
281C2×Dic5—22no—
282C2×Dic5—22no—
283C2×Dic5—42no—
284C2×Dic5—61no—
285C2×Dic5—62no—
286C2×Dic5—122no—
287D20—61no—
288D20—61no—
289C4×D5—121no—
290C4×D5—121no—
291Dic10—61no—
292Dic10—61no—
293C5×C8—41no—
294C5×C8—121no—
295C2×#41—151no—
296C23×C4—31no—
297C2×M16—31no—
298C2×M16D3×D511yescharacteristic
299#123—301no—
300C22⋊Q8—152no—
301#121—152no—
302#120—301no—
303C4×D4—302no—
304#117—51no—
305#117—302no—
306#117—302no—
307C2×C4⋊C4—301no—
308C2×C22⋊C4—301no—
309C2×C42—51no—
310C2×C42—151no—
311#104—302no—
312#104—302no—
313C22⋊C8—62no—
314C2×C3×C5—21no—
315C2×C3×C5—21no—
316C2×C3×C5#38611yescharacteristic
317C2×C3×C5#42112yes—
318D15—81no—
319C5×D3—42no—
320C23×C3C2×D2011yescharacteristic
321C22×D3—21no—
322C22×D3—21no—
323C22×D3—42no—
324C22×D3—101no—
325C22×D3#68512yes—
326C2×C3×C4—102no—
327C2×C3×C4—102no—
328C2×C3×C4—202no—
329C2×C3×C4C2×C4×D511yescharacteristic
330C2×C3×C4D4×D511yescharacteristic
331C2×C3×C4D4×D511yescharacteristic
332C2×C3×C4#71111yescharacteristic
333C2×C3×C4#71111yescharacteristic
334C2×C3×C4C2×C22⋊D511yescharacteristic
335#67—201no—
336#67—201no—
337C2×Dic3—21no—
338C2×Dic3—21no—
339C2×Dic3—102no—
340C2×Dic3—101no—
341C2×Dic3—202no—
342C2×Dic3#68512yes—
343D12—101no—
344D12—101no—
345D12—204no—
346C4×D3—22no—
347C4×D3—22no—
348C4×D3—22no—
349C4×D3—22no—
350C4×D3—201no—
351C4×D3—201no—
352Dic6—101no—
353Dic6—101no—
354C3×C8—41no—
355C3⋊C8—41no—
356C22×C5—22no—
357C22×C5—62no—
358C22×C5—62no—
359C22×C5—62no—
360C22×C5—122no—
361C22×C5—122no—
362C22×C5D3×C22⋊C411yescharacteristic
363C22×C5#88211yescharacteristic
364C22×C5#89411yescharacteristic
365C2×D5—121no—
366C2×D5—122no—
367C4×C5—62no—
368C4×C5C4×D1211yescharacteristic
369C4×C5#88411yescharacteristic
370C4×C5#89311yescharacteristic
371C4×C5#89511yescharacteristic
372Dic5—42no—
373Dic5—42no—
374Dic5—122no—
375Dic5—121no—
376C24—31no—
377#41—301no—
378#41—301no—
379#41—301no—
380#41—301no—
381C2×Q8—151no—
382C2×D4—151no—
383C2×D4—302no—
384C2×D4—301no—
385C22×C4—31no—
386C22×C4—32no—
387C22×C4—32no—
388C22×C4—61no—
389C22×C4—61no—
390C22×C4—61no—
391C22×C4—61no—
392C22×C4—101no—
393C22×C4—301no—
394C22×C4—301no—
395C22×C4C2×D3×D511yescharacteristic
396M16—21no—
397M16—21no—
398M16—61no—
399M16—61no—
400C2×C8—21no—
401C2×C8—61no—
402C4⋊C4—52no—
403C4⋊C4—302no—
404C4⋊C4—304no—
405C22⋊C4—101no—
406C22⋊C4—302no—
407C22⋊C4—302no—
408C22⋊C4—304no—
409C42—52no—
410C42—101no—
411C42—101no—
412C42—152no—
413C42—301no—
414C3×C5#189711yescharacteristic, complemented, Hall
415C22×C3—22no—
416C22×C3#402811yescharacteristic
417C22×C3#403211yescharacteristic
418C22×C3C2×#68511yescharacteristic
419C2×D3—22no—
420C2×D3—22no—
421C2×D3—22no—
422C2×D3—42no—
423C2×D3—42no—
424C2×D3—202no—
425C2×D3—201no—
426C3×C4—202no—
427C3×C4—202no—
428C3×C4#402711yescharacteristic
429C3×C4#402711yescharacteristic
430C3×C4#408511yescharacteristic
431C3×C4#408511yescharacteristic
432Dic3—22no—
433Dic3—202no—
434Dic3—201no—
435C2×C5—21no—
436C2×C5—21no—
437C2×C5—122no—
438C2×C5#466511yescharacteristic
439C2×C5#482112yes—
440D5—241no—
441C23—32no—
442C23—61no—
443C23—61no—
444C23—122no—
445C23—301no—
446C23D3×D2011yescharacteristic
447Q8—301no—
448Q8—301no—
449D4—301no—
450D4—301no—
451D4—604no—
452D4—601no—
453D4—601no—
454C2×C4—32no—
455C2×C4—61no—
456C2×C4—61no—
457C2×C4—62no—
458C2×C4—62no—
459C2×C4—62no—
460C2×C4—62no—
461C2×C4—102no—
462C2×C4—102no—
463C2×C4—202no—
464C2×C4—301no—
465C2×C4—302no—
466C2×C4—602no—
467C2×C4—601no—
468C2×C4—601no—
469C2×C4#676811yescharacteristic
470C2×C4C4×D3×D511yescharacteristic
471C2×C4D5×D1211yescharacteristic
472C2×C4#678211yescharacteristic
473C2×C4D3×C22⋊D511yescharacteristic
474C2×C4#679011yescharacteristic
475C8—41no—
476C8—121no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3#6524711yescharacteristic
480C2×C3#6542812yes—
481D3—42no—
482D3—401no—
483C5#6875711yescharacteristic, complemented, Sylow
484C22—22no—
485C22—62no—
486C22—62no—
487C22—62no—
488C22—122no—
489C22—122no—
490C22—601no—
491C22—602no—
492C22#9142711yescharacteristic
493C22D3×#68511yescharacteristic
494C22#9147611yescharacteristic
495C4—62no—
496C4—202no—
497C4—202no—
498C4—602no—
499C4—601no—
500C4#9140511yescharacteristic
501C4#9140611yescharacteristic
502C4#9141611yescharacteristic
503C4#9155811yescharacteristic
504C3#1060075911yescharacteristic, complemented, Sylow
505C2—21no—
506C2—21no—
507C2—122no—
508C2—1201no—
509C2#1174823312yes—
510C2#1174888311yescharacteristic
subgroupG/H
centerC2×C4D3×C22⋊D5
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC2×C3×C5×M16C22
socleC22×C3×C5C2×C22⋊C4
prime radical#11754584C2
cosocleC22×C3×C4×C5C23
hypercenterC2×M16D3×D5
hypocenterC3×C5#1897
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
2. C2×#91371C2semidirect
3. C2×#91109C2semidirect
4. C5×#5126C2semidirect
5. C2×#90981C2semidirect
6. C3×#64775C2semidirect
7. #11745973C2semidirect
20. #91460C4semidirect
24. #91365C4semidirect
35. #64775D3semidirect
60. C4×Dic15D4semidirect
68. C3×C4×Dic5D4semidirect
77. #5126D5semidirect
193. C4×Dic5D12semidirect
414. C3×C5#1897semidirect
483. C5#68757semidirect
504. C3#10600759semidirect

Maximal subgroups

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

subgroupindex
#117545842
C2×#913712
C2×#911092
C5×#51262
C2×#909812
C3×#647752
#117459732
#106007593
#687575