Finite Group Databasebeta

← #14301464

Subgroups — #14301464

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
1C2×C3×#4093C211yesmaximal, characteristic
2#11754488C211yesmaximal, characteristic
3#11751326C211yesmaximal, characteristic, complemented
4C5×#5225C211yesmaximal, characteristic, complemented
5C2×#90958C211yesmaximal, characteristic
6#11745978C211yesmaximal, characteristic, complemented
7#11745977C211yesmaximal, characteristic, complemented
8#10601136—31nomaximal, Hall
9C22×C3×C5×Q8C2211yescharacteristic
10C22×C3×D20C2211yescharacteristic
11#91815C2211yescharacteristic
12C2×#6710C2211yescharacteristic
13C5×#926C2211yescharacteristic
14C2×C5×#208C2211yescharacteristic
15C3×#4093—22no—
16C3×#4093C414yescomplemented
17C2×C3×#685—22no—
18C2×C3×C4×Dic5C2211yescharacteristic
19D5×#217—22no—
20#91434—22no—
21Dic3×D20—22no—
22#91119—22no—
23C5×#835—22no—
24#90958—22no—
25#71723—51nomaximal, Hall
26C2×#4093D311yescharacteristic
27#65855—31no—
28C5×#419—31no—
29#65514—31no—
30C2×#3955—31no—
31#64775—31no—
32#64752—31no—
33C23×C3×D5D412yes—
34C2×C3×C5×Q8—21no—
35C2×C3×C5×Q8—21no—
36C2×C3×C5×Q8—21no—
37C2×C3×C5×Q8—21no—
38C2×C3×C5×Q8—21no—
39C2×C3×C5×Q8C2×C412yescomplemented
40C22×C3×C4×C5—21no—
41C22×C3×C4×C5C2311yescharacteristic
42C22×C3×Dic5—21no—
43C2×C3×D20—21no—
44C2×C3×D20—21no—
45C2×C3×D20—21no—
46C2×C3×D20—21no—
47C2×C3×D20C2×C412yes—
48C2×C3×D20D412yes—
49C2×D5×Dic3—24no—
50#6719—21no—
51C4⋊Dic15D412yes—
52C4×Dic15—21no—
53#6710—21no—
54#6710—21no—
55C2×C3⋊#168—21no—
56C5×#217—21no—
57C5×C4⋊Dic3D412yes—
58C4×C5×Dic3—21no—
59C5×#208—21no—
60C5×#208—21no—
61C2×C5×C3⋊C8—21no—
62C3×#685—28no—
63C3×C4×Dic5—21no—
64C3×C4×Dic5C2×C412yescomplemented
65#6665—24no—
66C2×C3×#124—51no—
67#5798—51no—
68#5225—51no—
69#5225D511yescharacteristic, complemented
70C2×#833—51no—
71#4523—51no—
72#4523—51no—
73C22×C5×Q8C2×D311yescharacteristic
74C22×D20C2×D311yescharacteristic
75C2×C5×M16—31no—
76C5×#117—31no—
77#4093—22no—
78#4093Dic314yescomplemented
79C2×#685—22no—
80#4072—31no—
81C2×C4×Dic5C2×D311yescharacteristic
82C2×#681—31no—
83#4039—62no—
84D5×C22⋊C4—62no—
85C4×D20—62no—
86C5×#103—62no—
87#3967—62no—
88#3955—62no—
89#1885—151noSylow
90C23×C3×C5C2×D411yescharacteristic
91C22×C3×D5—22no—
92C22×C3×D5—22no—
93C22×C3×D5—44no—
94C22×C3×D5C22⋊C414yes—
95C3×C5×Q8—22no—
96C3×C5×Q8—22no—
97C3×C5×Q8—41no—
98C3×C5×Q8—41no—
99C2×C3×C4×C5—22no—
100C2×C3×C4×C5—41no—
101C2×C3×C4×C5—41no—
102C2×C3×C4×C5C22×C411yescharacteristic
103C2×C3×C4×C5C2×D411yescharacteristic
104C2×C3×C4×C5C2×D411yescharacteristic
105C2×C3×C4×C5C2×D411yescharacteristic
106C2×C3×C4×C5#4111yescharacteristic
107C2×C3×C4×C5#4111yescharacteristic
108C2×Dic15—22no—
109C2×C5×Dic3—22no—
110C2×C3×Dic5—22no—
111C2×C3×Dic5—42no—
112C3×D20—22no—
113C3×D20—22no—
114C3×D20—22no—
115C3×D20—22no—
116D5×Dic3—84no—
117C3⋊#168—41no—
118C5×C3⋊C8—41no—
119C22×C3×Q8C2×D511yescharacteristic
120C22×C3×D4—51no—
121C3×#124—54no—
122C3×#124—102no—
123C2×C3×C22⋊C4—102no—
124C2×C3×C42—51no—
125C2×#217—102no—
126D4×Dic3—102no—
127D4×Dic3—102no—
128#926—51no—
129#926C2×D511yescharacteristic
130C2×#208—51no—
131C2×#208C2×D511yescharacteristic
132#835—22no—
133#835—102no—
134#833—102no—
135C23×D5#6712yes—
136C2×C5×Q8—21no—
137C2×C5×Q8—21no—
138C2×C5×Q8—21no—
139C2×C5×Q8—21no—
140C2×C5×Q8—21no—
141C2×C5×Q8C2×Dic312yescomplemented
142C22×C4×C5—21no—
143C22×C4×C5C22×D311yescharacteristic
144C22×Dic5—21no—
145C2×D20—21no—
146C2×D20—21no—
147C2×D20—21no—
148C2×D20—21no—
149C2×D20C2×Dic312yes—
150C2×D20#6712yes—
151C2×C4×D5—64no—
152C5×M16—61no—
153C5×M16—61no—
154C2×C5×C8—61no—
155C5×C4⋊C4—32no—
156C5×C22⋊C4—61no—
157C42×C5—61no—
158C22⋊Dic5—61no—
159#685—28no—
160#685—64no—
161C4⋊Dic5—32no—
162C4×Dic5—21no—
163C4×Dic5—61no—
164C4×Dic5C2×Dic312yescomplemented
165#681—61no—
166#681—61no—
167C2×#168—61no—
168C2×#124—51no—
169#518—151no—
170#419—31no—
171#419—151no—
172C2×#100—151no—
173#339—151no—
174#339—151no—
175C22×C3×C5—22no—
176C22×C3×C5C2×C22⋊C411yescharacteristic
177C22×C3×C5#12011yescharacteristic
178C22×C3×C5#12311yescharacteristic
179C2×C3×D5—24no—
180C2×C3×D5—44no—
181C2×C3×D5—44no—
182C2×C3×D5—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—
191C24×C3—52no—
192C2×C3×Q8—21no—
193C2×C3×Q8—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8C4×D512yescomplemented
198C2×C3×D4—52no—
199C2×C3×D4—52no—
200C2×C3×D4—101no—
201C2×C3×D4—101no—
202C2×C3×D4—101no—
203C2×C3×D4—101no—
204C22×C3×C4—21no—
205C22×C3×C4—101no—
206C22×C3×C4C22×D511yescharacteristic
207C22×Dic3—104no—
208C3×C22⋊C4—108no—
209C3×C42—52no—
210C3×C42—101no—
211#217—21no—
212#217—104no—
213#217—101no—
214C4⋊Dic3—52no—
215C4⋊Dic3C22⋊D512yes—
216C4×Dic3—21no—
217C4×Dic3—101no—
218#208—21no—
219#208—21no—
220#208—101no—
221#208—101no—
222C2×C3⋊C8—21no—
223C2×C3⋊C8—101no—
224C23×C5C2×#6711yescharacteristic
225C22×D5—22no—
226C22×D5—22no—
227C22×D5—44no—
228C22×D5#21714yes—
229C5×Q8—22no—
230C5×Q8—22no—
231C5×Q8—41no—
232C5×Q8—41no—
233C2×C4×C5—22no—
234C2×C4×C5—41no—
235C2×C4×C5—41no—
236C2×C4×C5—62no—
237C2×C4×C5D3×D411yescharacteristic
238C2×C4×C5D3×D411yescharacteristic
239C2×C4×C5#23711yescharacteristic
240C2×C4×C5#23711yescharacteristic
241C2×C4×C5C22×Dic311yescharacteristic
242C2×C4×C5C2×#6711yescharacteristic
243C2×Dic5—22no—
244C2×Dic5—42no—
245C2×Dic5—62no—
246D20—22no—
247D20—22no—
248D20—22no—
249D20—22no—
250C4×D5—244no—
251C5×C8—121no—
252#168—121no—
253C22×Q8D3×D511yescharacteristic
254C22×D4—51no—
255C2×M16—31no—
256C2×M16—151no—
257#124—54no—
258#124—102no—
259C4×D4—302no—
260C4×D4—302no—
261#117—31no—
262#117—151no—
263C2×C22⋊C4—102no—
264C2×C22⋊C4—302no—
265C2×C42—51no—
266#103—62no—
267#103—302no—
268#100—302no—
269C2×C3×C5—21no—
270C2×C3×C5—21no—
271C2×C3×C5#38611yescharacteristic
272C2×C3×C5#45512yes—
273C3×D5—44no—
274C23×C3—54no—
275C23×C3—102no—
276C23×C3—102no—
277C23×C3—204no—
278C23×C3D4×D511yescharacteristic
279C3×Q8—22no—
280C3×Q8—22no—
281C3×Q8—41no—
282C3×Q8—41no—
283C3×D4—102no—
284C3×D4—102no—
285C3×D4—102no—
286C3×D4—102no—
287C2×C3×C4—22no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—102no—
291C2×C3×C4—202no—
292C2×C3×C4C2×C4×D511yescharacteristic
293C2×C3×C4#70911yescharacteristic
294C2×C3×C4D4×D511yescharacteristic
295C2×C3×C4D4×D511yescharacteristic
296C2×C3×C4#71111yescharacteristic
297C2×C3×C4C2×C22⋊D511yescharacteristic
298C2×Dic3—22no—
299C2×Dic3—102no—
300C2×Dic3—404no—
301C3⋊C8—41no—
302C3⋊C8—201no—
303C22×C5—22no—
304C22×C5#93511yescharacteristic
305C22×C5#93711yescharacteristic
306C22×C5C2×#21711yescharacteristic
307C2×D5—24no—
308C2×D5—44no—
309C2×D5—44no—
310C2×D5—82no—
311C4×C5—42no—
312C4×C5—122no—
313C4×C5D4×Dic311yescharacteristic
314C4×C5D4×Dic311yescharacteristic
315C4×C5#93911yescharacteristic
316C4×C5#93911yescharacteristic
317Dic5—42no—
318Dic5—122no—
319C24—52no—
320C2×Q8—21no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8—21no—
324C2×Q8—21no—
325C2×Q8D5×Dic312yescomplemented
326C2×D4—52no—
327C2×D4—52no—
328C2×D4—101no—
329C2×D4—101no—
330C2×D4—101no—
331C2×D4—101no—
332C22×C4—21no—
333C22×C4—101no—
334C22×C4—304no—
335C22×C4C2×D3×D511yescharacteristic
336M16—61no—
337M16—61no—
338M16—301no—
339M16—301no—
340C2×C8—61no—
341C2×C8—301no—
342C4⋊C4—32no—
343C4⋊C4—152no—
344C22⋊C4—61no—
345C22⋊C4—108no—
346C22⋊C4—304no—
347C22⋊C4—301no—
348C42—52no—
349C42—61no—
350C42—101no—
351C42—301no—
352C3×C5#188511yescharacteristic, complemented, Hall
353C22×C3—22no—
354C22×C3—104no—
355C22×C3—204no—
356C22×C3—204no—
357C22×C3—402no—
358C22×C3D5×C22⋊C411yescharacteristic
359C22×C3#402911yescharacteristic
360C22×C3#408311yescharacteristic
361C3×C4—42no—
362C3×C4—202no—
363C3×C4#402711yescharacteristic
364C3×C4#403011yescharacteristic
365C3×C4C4×C22⋊D511yescharacteristic
366C3×C4#408411yescharacteristic
367Dic3—42no—
368Dic3—202no—
369C2×C5—21no—
370C2×C5—21no—
371C2×C5#506112yes—
372C2×C5#522011yescharacteristic
373D5—44no—
374C23—54no—
375C23—102no—
376C23—102no—
377C23—204no—
378C23D5×#6711yescharacteristic
379Q8—22no—
380Q8—22no—
381Q8—41no—
382Q8—41no—
383D4—102no—
384D4—102no—
385D4—102no—
386D4—102no—
387C2×C4—22no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—62no—
391C2×C4—102no—
392C2×C4—202no—
393C2×C4—302no—
394C2×C4—1204no—
395C2×C4C2×D5×Dic311yescharacteristic
396C2×C4#677911yescharacteristic
397C2×C4#678011yescharacteristic
398C2×C4D5×#6711yescharacteristic
399C2×C4D3×C22⋊D511yescharacteristic
400C2×C4#679011yescharacteristic
401C8—121no—
402C8—601no—
403C2×C3—21no—
404C2×C3—21no—
405C2×C3—204no—
406C2×C3#6524511yescharacteristic
407C2×C3#6549112yes—
408C5#7172311yescharacteristic, complemented, Sylow
409C22—22no—
410C22—104no—
411C22—204no—
412C22—204no—
413C22—402no—
414C22D5×#21711yescharacteristic
415C22#9154811yescharacteristic
416C22#9157711yescharacteristic
417C4—42no—
418C4—122no—
419C4—202no—
420C4—602no—
421C4#9154911yescharacteristic
422C4#9155011yescharacteristic
423C4Dic3×C22⋊D511yescharacteristic
424C4#9155911yescharacteristic
425C3#1060113611yescharacteristic, complemented, Sylow
426C2—21no—
427C2—21no—
428C2—204no—
429C2#1174940912yes—
430C2#1174951811yescharacteristic
subgroupG/H
centerC22#91577
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC22×C3×C5×Q8C22
socleC22×C3×C5#120
prime radicalC22×C3×D20C22
cosocleC22×C3×C4×C5C23
hypercenterC22×Q8D3×D5
hypocenterC3×C5#1885
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×#5225C2semidirect
6. #11745978C2semidirect
7. #11745977C2semidirect
16. C3×#4093C4semidirect
39. C2×C3×C5×Q8C2×C4semidirect
64. C3×C4×Dic5C2×C4semidirect
69. #5225D5semidirect
78. #4093Dic3semidirect
141. C2×C5×Q8C2×Dic3semidirect
164. C4×Dic5C2×Dic3semidirect
197. C2×C3×Q8C4×D5semidirect
325. C2×Q8D5×Dic3semidirect
352. C3×C5#1885semidirect
408. C5#71723semidirect
425. C3#10601136semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#40932
#117544882
#117513262
C5×#52252
C2×#909582
#117459782
#117459772
#106011363
#717235