Finite Group Databasebeta

← #14301456

Subgroups — #14301456

Order 1920.

720 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756280C211yesmaximal, characteristic
2C2×#91374C211yesmaximal, characteristic
3C3×#65293C211yesmaximal, characteristic, complemented
4#11748400C211yesmaximal, characteristic, complemented
5C2×#90960C211yesmaximal, characteristic
6C5×#4532C211yesmaximal, characteristic, complemented
7#11745976C211yesmaximal, characteristic, complemented
8#10599552—31nomaximal, Hall
9C22×D60C2211yescharacteristic
10C22×C3×Dic10C2211yescharacteristic
11C3×C5×#117C2211yescharacteristic
12#91815C2211yescharacteristic
13#91782—22no—
14C22⋊C4×D15—22no—
15C4×D60—22no—
16C2×C4×C5×Dic3C2211yescharacteristic
17C2×C5×#208C2211yescharacteristic
18C2×C3×#681C2211yescharacteristic
19C2×#6667—22no—
20#91374—22no—
21#91374C414yescomplemented
22C3×#3942—22no—
23#90971—22no—
24#90960—22no—
25#70343—51nomaximal, Hall
26#65855—31no—
27C2×#4021—31no—
28C2×#3955—31no—
29#65409—31no—
30#65293D311yescharacteristic, complemented
31C5×#339—31no—
32#64743—31no—
33C23×D15D412yes—
34C22×C3×C4×C5C2311yescharacteristic
35C2×D60—21no—
36C2×D60—21no—
37C2×D60—21no—
38C2×D60—21no—
39C2×D60C2×C412yes—
40C2×D60D412yes—
41C2×C4×D15—24no—
42C22×C5×Dic3—21no—
43C22×C3×Dic5—21no—
44C2×C3×Dic10—21no—
45C2×C3×Dic10—21no—
46C2×C3×Dic10—21no—
47C2×C3×Dic10—21no—
48C2×C3×Dic10—21no—
49C2×C3×Dic10C2×C412yescomplemented
50C3×C5×C4⋊C4D412yes—
51C3×C5×C22⋊C4—21no—
52C3×C42×C5—21no—
53#6719—21no—
54#6714—24no—
55C4⋊Dic15D412yes—
56C4×Dic15—21no—
57C4×C5×Dic3—21no—
58C4×C5×Dic3C2×C412yescomplemented
59C5×#208—21no—
60C5×#208—21no—
61C2×C5×C3⋊C8—21no—
62C3×#681—21no—
63C3×#681—21no—
64C2×C3×#168—21no—
65#6667—28no—
66C2×#947—51no—
67#5521—51no—
68C3×#419—51no—
69#5225—51no—
70C2×#823—51no—
71#4555—51no—
72#4532D511yescharacteristic, complemented
73C22×D20—31no—
74C22×Dic10C2×D311yescharacteristic
75C2×C5×M16—31no—
76C5×#117C2×D311yescharacteristic
77C2×C42×C5—31no—
78C2×#685—62no—
79#4072—31no—
80C2×#681C2×D311yescharacteristic
81#4039—62no—
82D5×C22⋊C4—62no—
83#4021—34no—
84#4021—62no—
85C4×D20—62no—
86#3955—62no—
87#3948—62no—
88#3942—22no—
89#1885—151noSylow
90C23×C3×C5C2×D411yescharacteristic
91C22×D15—22no—
92C22×D15—22no—
93C22×D15—44no—
94C22×D15C22⋊C414yes—
95C2×C3×C4×C5—22no—
96C2×C3×C4×C5C22×C411yescharacteristic
97C2×C3×C4×C5C2×D411yescharacteristic
98C2×C3×C4×C5C2×D411yescharacteristic
99C2×C3×C4×C5C2×D411yescharacteristic
100C2×C3×C4×C5#4111yescharacteristic
101C2×C3×C4×C5#4111yescharacteristic
102C2×Dic15—22no—
103D60—22no—
104D60—22no—
105D60—22no—
106D60—22no—
107C4×D15—84no—
108C2×C5×Dic3—22no—
109C2×C5×Dic3—42no—
110C2×C3×Dic5—22no—
111C2×C3×Dic5—41no—
112C2×C3×Dic5—41no—
113C3×Dic10—22no—
114C3×Dic10—22no—
115C3×Dic10—41no—
116C3×Dic10—41no—
117C3×#168—41no—
118C5×C3⋊C8—41no—
119C22×C3×Q8—51no—
120C22×D12—51no—
121C2×C3×M16—51no—
122C3×#117C2×D511yescharacteristic
123#947—54no—
124#947—102no—
125C2×(C2×D3)⋊C4—102no—
126#926—51no—
127C2×C4×Dic3C2×D511yescharacteristic
128C2×#208C2×D511yescharacteristic
129#893—102no—
130D3×C22⋊C4—102no—
131C4×D12—102no—
132C3×#103—102no—
133#835—102no—
134#823—102no—
135C23×D5—32no—
136C22×C4×C5—61no—
137C22×C4×C5C22×D311yescharacteristic
138C22×Dic5—21no—
139C2×D20—32no—
140C2×D20—32no—
141C2×D20—61no—
142C2×D20—61no—
143C2×D20—61no—
144C2×D20—61no—
145C2×C4×D5—64no—
146C2×Dic10—21no—
147C2×Dic10—21no—
148C2×Dic10—21no—
149C2×Dic10—21no—
150C2×Dic10—21no—
151C2×Dic10C4×D312yescomplemented
152C5×M16—61no—
153C5×M16—61no—
154C2×C5×C8—61no—
155C5×C4⋊C4#6712yes—
156C5×C22⋊C4—21no—
157C42×C5—21no—
158C42×C5—32no—
159C42×C5—61no—
160C22⋊Dic5—61no—
161#685—68no—
162#685—64no—
163C4⋊Dic5—32no—
164C4×Dic5—61no—
165#681—21no—
166#681—21no—
167C2×#168—21no—
168C2×#124—151no—
169#518—151no—
170#419—51no—
171#419—151no—
172C2×#100—151no—
173#339—31no—
174#339—151no—
175C22×C3×C5—22no—
176C22×C3×C5C2×C22⋊C411yescharacteristic
177C22×C3×C5#12011yescharacteristic
178C22×C3×C5#12311yescharacteristic
179C2×D15—24no—
180C2×D15—44no—
181C2×D15—44no—
182C2×D15—82no—
183C3×C4×C5—42no—
184C3×C4×C5C4×D411yescharacteristic
185C3×C4×C5C4×D411yescharacteristic
186C3×C4×C5#12111yescharacteristic
187C3×C4×C5#12111yescharacteristic
188Dic15—42no—
189C3×Dic5—42no—
190C5×Dic3—42no—
191C23×D3—52no—
192C2×C3×Q8—52no—
193C2×C3×Q8—101no—
194C2×C3×Q8—101no—
195C2×C3×Q8—101no—
196C2×C3×Q8—101no—
197C2×C3×Q8—101no—
198C22×C3×C4—101no—
199C22×C3×C4C22×D511yescharacteristic
200C22×Dic3—21no—
201C2×D12—52no—
202C2×D12—52no—
203C2×D12—101no—
204C2×D12—101no—
205C2×D12—101no—
206C2×D12—101no—
207C2×C4×D3—104no—
208C3×M16—101no—
209C3×M16—101no—
210C2×C3×C8—101no—
211C3×C4⋊C4D2012yes—
212C3×C22⋊C4—21no—
213C3×C42—21no—
214#217—101no—
215(C2×D3)⋊C4—108no—
216(C2×D3)⋊C4—104no—
217C4⋊Dic3—52no—
218C4×Dic3—21no—
219C4×Dic3—101no—
220C4×Dic3C4×D512yescomplemented
221#208—21no—
222#208—21no—
223C2×C3⋊C8—21no—
224C23×C5D3×D411yescharacteristic
225C22×D5—34no—
226C22×D5—62no—
227C22×D5—62no—
228C22×D5—124no—
229C2×C4×C5—22no—
230C2×C4×C5—62no—
231C2×C4×C5—122no—
232C2×C4×C5C2×C4×D311yescharacteristic
233C2×C4×C5#23511yescharacteristic
234C2×C4×C5D3×D411yescharacteristic
235C2×C4×C5D3×D411yescharacteristic
236C2×C4×C5#23711yescharacteristic
237C2×C4×C5C2×#6711yescharacteristic
238C2×Dic5—22no—
239C2×Dic5—41no—
240C2×Dic5—41no—
241C2×Dic5—62no—
242D20—62no—
243D20—62no—
244D20—62no—
245D20—62no—
246C4×D5—244no—
247Dic10—22no—
248Dic10—22no—
249Dic10—41no—
250Dic10—41no—
251C5×C8—121no—
252#168—41no—
253C22×Q8—51no—
254C22×D4—151no—
255C2×M16—31no—
256C2×M16—51no—
257#124—154no—
258#124—302no—
259C4×D4—302no—
260C4×D4—302no—
261#117—151no—
262#117D3×D511yescharacteristic
263C2×C22⋊C4—302no—
264C2×C22⋊C4—302no—
265C2×C42—31no—
266#103—102no—
267#103—302no—
268#100—302no—
269C2×C3×C5—21no—
270C2×C3×C5—21no—
271C2×C3×C5#38611yescharacteristic
272C2×C3×C5#45512yes—
273D15—44no—
274C23×C3D4×D511yescharacteristic
275C22×D3—54no—
276C22×D3—102no—
277C22×D3—102no—
278C22×D3—204no—
279C3×Q8—102no—
280C3×Q8—102no—
281C3×Q8—201no—
282C3×Q8—201no—
283C2×C3×C4—22no—
284C2×C3×C4—102no—
285C2×C3×C4—201no—
286C2×C3×C4—201no—
287C2×C3×C4C2×C4×D511yescharacteristic
288C2×C3×C4C2×D2011yescharacteristic
289C2×C3×C4#70911yescharacteristic
290C2×C3×C4D4×D511yescharacteristic
291C2×C3×C4D4×D511yescharacteristic
292C2×C3×C4#71311yescharacteristic
293C2×Dic3—22no—
294C2×Dic3—42no—
295C2×Dic3—102no—
296D12—102no—
297D12—102no—
298D12—102no—
299D12—102no—
300C4×D3—404no—
301C3×C8—201no—
302C3⋊C8—41no—
303C22×C5—22no—
304C22×C5D3×C22⋊C411yescharacteristic
305C22×C5#88311yescharacteristic
306C22×C5#93711yescharacteristic
307C2×D5—64no—
308C2×D5—124no—
309C2×D5—124no—
310C2×D5—242no—
311C4×C5—42no—
312C4×C5—122no—
313C4×C5#67⋊C411yescharacteristic
314C4×C5#88411yescharacteristic
315C4×C5C4×#6711yescharacteristic
316C4×C5#93811yescharacteristic
317Dic5—42no—
318Dic5—122no—
319C24—152no—
320C2×Q8—52no—
321C2×Q8—101no—
322C2×Q8—101no—
323C2×Q8—101no—
324C2×Q8—101no—
325C2×Q8—101no—
326C2×D4—152no—
327C2×D4—152no—
328C2×D4—301no—
329C2×D4—301no—
330C2×D4—301no—
331C2×D4—301no—
332C22×C4—61no—
333C22×C4—101no—
334C22×C4—304no—
335C22×C4C2×D3×D511yescharacteristic
336M16—61no—
337M16—61no—
338M16—101no—
339M16—101no—
340C2×C8—61no—
341C2×C8—101no—
342C4⋊C4—152no—
343C4⋊C4#120212yes—
344C22⋊C4—21no—
345C22⋊C4—304no—
346C22⋊C4—308no—
347C22⋊C4—301no—
348C42—21no—
349C42—32no—
350C42—61no—
351C42—301no—
352C3×C5#188511yescharacteristic, complemented, Hall
353C22×C3—22no—
354C22×C3D5×C22⋊C411yescharacteristic
355C22×C3#402811yescharacteristic
356C22×C3#404011yescharacteristic
357C2×D3—104no—
358C2×D3—204no—
359C2×D3—204no—
360C2×D3—402no—
361C3×C4—42no—
362C3×C4—202no—
363C3×C4C4×D2011yescharacteristic
364C3×C4#403011yescharacteristic
365C3×C4#403911yescharacteristic
366C3×C4#404111yescharacteristic
367Dic3—42no—
368Dic3—202no—
369C2×C5—21no—
370C2×C5—21no—
371C2×C5#495611yescharacteristic
372C2×C5#520212yes—
373D5—124no—
374C23—154no—
375C23—302no—
376C23—302no—
377C23—604no—
378C23#679011yescharacteristic
379Q8—102no—
380Q8—102no—
381Q8—201no—
382Q8—201no—
383D4—302no—
384D4—302no—
385D4—302no—
386D4—302no—
387C2×C4—22no—
388C2×C4—62no—
389C2×C4—102no—
390C2×C4—122no—
391C2×C4—201no—
392C2×C4—201no—
393C2×C4—302no—
394C2×C4—1204no—
395C2×C4#676611yescharacteristic
396C2×C4#677111yescharacteristic
397C2×C4C4×D3×D511yescharacteristic
398C2×C4#677711yescharacteristic
399C2×C4C2×#120211yescharacteristic
400C2×C4#679011yescharacteristic
401C8—121no—
402C8—201no—
403C2×C3—21no—
404C2×C3—21no—
405C2×C3#6495411yescharacteristic
406C2×C3#6511412yes—
407D3—204no—
408C5#7034311yescharacteristic, complemented, Sylow
409C22—22no—
410C22—304no—
411C22—604no—
412C22—604no—
413C22—1202no—
414C22#9135411yescharacteristic
415C22#9147311yescharacteristic
416C22#9157211yescharacteristic
417C4—42no—
418C4—122no—
419C4—202no—
420C4—602no—
421C4#9139611yescharacteristic
422C4C4×#120211yescharacteristic
423C4#9146111yescharacteristic
424C4#9146511yescharacteristic
425C3#1059955211yescharacteristic, complemented, Sylow
426C2—21no—
427C2—21no—
428C2—604no—
429C2#1174891711yescharacteristic
430C2#1174946712yes—
subgroupG/H
centerC22#91572
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC3×C5×#117C22
socleC22×C3×C5#120
prime radicalC22×D60C22
cosocleC22×C3×C4×C5C23
hypercenter#117D3×D5
hypocenterC3×C5#1885
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
3. C3×#65293C2semidirect
4. #11748400C2semidirect
6. C5×#4532C2semidirect
7. #11745976C2semidirect
21. #91374C4semidirect
30. #65293D3semidirect
49. C2×C3×Dic10C2×C4semidirect
58. C4×C5×Dic3C2×C4semidirect
72. #4532D5semidirect
151. C2×Dic10C4×D3semidirect
220. C4×Dic3C4×D5semidirect
352. C3×C5#1885semidirect
408. C5#70343semidirect
425. C3#10599552semidirect

Maximal subgroups

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

subgroupindex
#117562802
C2×#913742
C3×#652932
#117484002
C2×#909602
C5×#45322
#117459762
#105995523
#703435