Finite Group Databasebeta

← #14472141

Subgroups — #14472141

Order 1920.

672 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5567C211yesmaximal, characteristic, complemented
2C3×#65918C211yesmaximal, characteristic
3Q8×C3⋊#168C211yesmaximal, characteristic, complemented
4#11747099C211yesmaximal, characteristic, complemented
5C4×#6661C212yesmaximal, complemented
6#11747069C212yesmaximal
7C4×#6655C211yesmaximal, characteristic
8#11746924C211yesmaximal, characteristic, complemented
9#11746919C212yesmaximal, complemented
10#11746896C212yesmaximal
11#11746886C211yesmaximal, characteristic
12#10615420—31nomaximal, Hall
13C3×C4×C5×Q8C2211yescharacteristic
14C5×#944—21no—
15C5×#889—22no—
16C5×#887—21no—
17C5×#870C2212yes—
18C5×#869C2211yescharacteristic
19C4×C5×Dic6C2212yes—
20C4×C5×Dic6C2211yescharacteristic
21C3×#4092—21no—
22C3×#4044—22no—
23C3×#4038—21no—
24C3×#4021C2212yes—
25C3×C4×D20C2211yescharacteristic
26C3×#4015C2211yescharacteristic
27C3×C4×Dic10C2212yes—
28C2×#6661C2212yes—
29C2×#6655C2211yescharacteristic
30C4⋊(C3⋊#168)C2211yescharacteristic
31C4⋊(C3⋊#168)C2212yes—
32C4×C3⋊#168C2212yes—
33C4×C3⋊#168C2211yescharacteristic
34#90989C2212yes—
35#90986C2211yescharacteristic
36#90972C2211yescharacteristic
37#90970C2212yes—
38#90970C2212yes—
39#90970C2212yes—
40#90964C2212yes—
41#90964C2211yescharacteristic
42#83676—51nomaximal, Hall
43C5×#557—31no—
44#65918D311yescharacteristic
45#65382—31no—
46#65376—31no—
47#65357—32no—
48#65352—32no—
49#65320—31no—
50C4×#689—32no—
51#65317—32no—
52C4×#688—31no—
53Q8×#168—31no—
54C2×C3×C5×Q8D411yescharacteristic
55C2×C5×Dic6—21no—
56C2×C5×Dic6C2311yescharacteristic
57C2×C3×D20C2311yescharacteristic
58C2×C3×C4×D5—21no—
59C2×C3×Dic10C2312yes—
60C3×C5×C4⋊C4D411yescharacteristic
61C3×C5×C4⋊C4D412yes—
62C3×C42×C5C2311yescharacteristic
63C3×C42×C5C2312yes—
64C2×C3⋊#168C2311yescharacteristic
65C2×C3⋊#168C2312yes—
66C2×C3⋊#168C2311yescharacteristic
67C5×C4⋊Dic3—21no—
68C5×C4⋊Dic3C2312yes—
69C5×C4⋊Dic3C2311yescharacteristic
70C5×Dic3⋊C4—21no—
71C5×Dic3⋊C4—22no—
72C5×Dic3⋊C4—22no—
73C5×Dic3⋊C4—22no—
74C4×C5×Dic3—22no—
75C4×C5×Dic3—21no—
76C3×#685—22no—
77C3×#685—21no—
78C3×#685—22no—
79C3×C4⋊Dic5—21no—
80C3×C4⋊Dic5—21no—
81C3×C4⋊Dic5C2311yescharacteristic
82C3×#683—22no—
83C3×C4×Dic5—22no—
84#6661—24no—
85#6655—22no—
86C3×#556—51no—
87#5567D511yescharacteristic, complemented
88#5090—51no—
89#5079—51no—
90#5056—52no—
91#5054—52no—
92C4×#216—52no—
93Q8×C3⋊C8—51no—
94#5022—52no—
95C4×#214—51no—
96#5007—51no—
97C5×C4⋊Q8—31no—
98C5×C4⋊Q8—61no—
99C5×#125—32no—
100C5×#125—62no—
101C4×C5×Q8—31no—
102C4×C5×Q8—32no—
103C4×C5×Q8—61no—
104C4×C5×Q8C2×D311yescharacteristic
105#4092—21no—
106C2×#689—32no—
107C2×#688—31no—
108#4044—22no—
109#4038—21no—
110#4021C2×D312yes—
111C4×D20C2×D311yescharacteristic
112#4015C2×D311yescharacteristic
113C4×Dic10C2×D312yes—
114#3967—31no—
115#3942—32no—
116#3942—32no—
117#3942—32no—
118#3941—31no—
119#3941—32no—
120#3940—31no—
121#3939—32no—
122C4⋊#168—32no—
123C4⋊#168—31no—
124C4×#168—31no—
125C4×#168—32no—
126#3364—151noSylow
127C22×C3×D5—21no—
128C3×C5×Q8—22no—
129C2×C3×C4×C5C2×D412yes—
130C2×C3×C4×C5C2×D412yes—
131C2×C3×C4×C5C2×D411yescharacteristic
132C2×C3×C4×C5C2×D411yescharacteristic
133C2×C3×C4×C5C2411yescharacteristic
134C2×C5×Dic3—21no—
135C2×C5×Dic3—21no—
136C2×C5×Dic3—22no—
137C5×Dic6—21no—
138C5×Dic6—42no—
139C5×Dic6#4112yes—
140C2×C3×Dic5—21no—
141C2×C3×Dic5—22no—
142C3×D20—21no—
143C3×D20—21no—
144C3×C4×D5—42no—
145C3×Dic10—22no—
146C3×Dic10—22no—
147C3⋊#168—22no—
148C3⋊#168—21no—
149C3⋊#168—21no—
150C3×C4⋊Q8—51no—
151C3×#126—102no—
152C3×#124—52no—
153C3×C22⋊Q8—101no—
154C3×C4×Q8—52no—
155C3×C4×Q8C2×D511yescharacteristic
156C3×C4×D4—51no—
157C3×#117—101no—
158#944—21no—
159C2×#216—52no—
160C2×#214—51no—
161#889—22no—
162#887—21no—
163#870C2×D512yes—
164#869C2×D511yescharacteristic
165C4×Dic6C2×D511yescharacteristic
166C4×Dic6C2×D512yes—
167#835—52no—
168#835—52no—
169#835—52no—
170#832—52no—
171#832—51no—
172#810—51no—
173#808—51no—
174#807—52no—
175C4⋊(C3⋊C8)—51no—
176C4⋊(C3⋊C8)—52no—
177C4×C3⋊C8—51no—
178C4×C3⋊C8—52no—
179C2×C5×Q8—31no—
180C2×C5×Q8—61no—
181C2×C5×Q8#6711yescharacteristic
182C2×D20C22×D311yescharacteristic
183C2×C4×D5—21no—
184C2×Dic10C22×D312yes—
185C5×C4⋊C4—31no—
186C5×C4⋊C4—32no—
187C5×C4⋊C4—61no—
188C5×C4⋊C4—62no—
189C5×C4⋊C4—62no—
190C5×C4⋊C4—62no—
191C5×C4⋊C4—61no—
192C5×C4⋊C4#6711yescharacteristic
193C5×C4⋊C4#6712yes—
194C42×C5—62no—
195C42×C5—61no—
196C42×C5C22×D312yes—
197C42×C5C22×D311yescharacteristic
198#689—64no—
199#688—62no—
200#685—21no—
201#685—22no—
202#685—22no—
203C4⋊Dic5—21no—
204C4⋊Dic5—21no—
205C4⋊Dic5C22×D311yescharacteristic
206#683—22no—
207C4×Dic5—22no—
208C2×#168—31no—
209C2×#168—32no—
210C2×#168—31no—
211#557—31no—
212#556—51no—
213#488—152no—
214#487—151no—
215#479—152no—
216#474—151no—
217#464—152no—
218#462—151no—
219C8×Q8—151no—
220C4×Q16—152no—
221C4×SD16—151no—
222C22×C3×C5C22×D411yescharacteristic
223C2×C3×D5—41no—
224C2×C3×D5—41no—
225C2×C3×D5—41no—
226C3×C4×C5—22no—
227C3×C4×C5—22no—
228C3×C4×C5—21no—
229C3×C4×C5#13512yes—
230C3×C4×C5C2×#4111yescharacteristic
231C3×C4×C52−1+411yescharacteristic
232C3×Dic5—42no—
233C3×Dic5—41no—
234C5×Dic3—22no—
235C5×Dic3—42no—
236C5×Dic3—41no—
237C2×C3×Q8—52no—
238C2×C3×Q8C22⋊D511yescharacteristic
239C2×C3×D4—51no—
240C22×C3×C4—101no—
241C2×Dic6—21no—
242C2×Dic6C22×D511yescharacteristic
243C3×C4⋊C4—51no—
244C3×C4⋊C4—102no—
245C3×C4⋊C4—101no—
246C3×C4⋊C4—101no—
247C3×C4⋊C4C22⋊D511yescharacteristic
248C3×C4⋊C4C22⋊D512yes—
249C3×C22⋊C4—102no—
250C3×C22⋊C4—101no—
251C3×C22⋊C4—102no—
252C3×C42—102no—
253C3×C42C22×D511yescharacteristic
254C3×C42C22×D512yes—
255#216—104no—
256#214—102no—
257C4⋊Dic3—21no—
258C4⋊Dic3C22×D512yes—
259C4⋊Dic3C22×D511yescharacteristic
260Dic3⋊C4—21no—
261Dic3⋊C4—22no—
262Dic3⋊C4—22no—
263Dic3⋊C4—22no—
264C4×Dic3—22no—
265C4×Dic3—21no—
266C2×C3⋊C8—51no—
267C2×C3⋊C8—52no—
268C2×C3⋊C8—51no—
269C22×D5—21no—
270C5×Q8—22no—
271C5×Q8—32no—
272C5×Q8—61no—
273C5×Q8—122no—
274C2×C4×C5—61no—
275C2×C4×C5—61no—
276C2×C4×C5—62no—
277C2×C4×C5C2×#6712yes—
278C2×C4×C5C2×#6712yes—
279C2×C4×C5C2×#6711yescharacteristic
280C2×C4×C5C2×#6711yescharacteristic
281C2×C4×C5C23×D311yescharacteristic
282C2×Dic5—21no—
283C2×Dic5—22no—
284D20—21no—
285D20—21no—
286C4×D5—42no—
287Dic10—22no—
288Dic10—22no—
289#168—61no—
290#168—62no—
291#168—61no—
292C2×Q16—152no—
293C2×SD16—151no—
294C4⋊Q8—31no—
295C4⋊Q8—51no—
296C4⋊Q8—61no—
297#126—102no—
298#125—32no—
299#125—62no—
300#124—52no—
301C22⋊Q8—101no—
302C4×Q8—31no—
303C4×Q8—32no—
304C4×Q8—52no—
305C4×Q8—61no—
306C4×Q8D3×D511yescharacteristic
307C4×D4—51no—
308#117—101no—
309#107—152no—
310#106—151no—
311C4⋊C8—151no—
312C4⋊C8—152no—
313#103—152no—
314#103—152no—
315#103—152no—
316#103—151no—
317#102—151no—
318#102—152no—
319C4×C8—151no—
320C4×C8—152no—
321C2×C3×C5#54811yescharacteristic
322C2×C3×C5C2×#13511yescharacteristic
323C2×C3×C5#57811yescharacteristic
324C3×D5—81no—
325C23×C3—101no—
326C3×Q8—22no—
327C3×Q8—102no—
328C3×Q8—102no—
329C3×D4—101no—
330C3×D4—101no—
331C2×C3×C4—101no—
332C2×C3×C4—102no—
333C2×C3×C4—202no—
334C2×C3×C4C2×C22⋊D512yes—
335C2×C3×C4C2×C22⋊D512yes—
336C2×C3×C4C2×C22⋊D511yescharacteristic
337C2×C3×C4C2×C22⋊D511yescharacteristic
338C2×C3×C4C23×D511yescharacteristic
339C2×Dic3—21no—
340C2×Dic3—21no—
341C2×Dic3—22no—
342Dic6—21no—
343Dic6—42no—
344Dic6#71312yes—
345C3⋊C8—101no—
346C3⋊C8—102no—
347C3⋊C8—101no—
348C22×C5C22×#6711yescharacteristic
349C2×D5—41no—
350C2×D5—41no—
351C2×D5—41no—
352C4×C5—22no—
353C4×C5—22no—
354C4×C5—21no—
355C4×C5—62no—
356C4×C5—122no—
357C4×C5—121no—
358C4×C5#95012yes—
359C4×C5D3×#4111yescharacteristic
360C4×C5#101011yescharacteristic
361Dic5—41no—
362Dic5—42no—
363C2×Q8—31no—
364C2×Q8—52no—
365C2×Q8—61no—
366C2×Q8#120111yescharacteristic
367C2×D4—51no—
368C22×C4—101no—
369Q16—304no—
370SD16—302no—
371C2×C8—151no—
372C2×C8—152no—
373C2×C8—151no—
374C4⋊C4—31no—
375C4⋊C4—32no—
376C4⋊C4—51no—
377C4⋊C4—61no—
378C4⋊C4—62no—
379C4⋊C4—62no—
380C4⋊C4—62no—
381C4⋊C4—61no—
382C4⋊C4—101no—
383C4⋊C4—102no—
384C4⋊C4—101no—
385C4⋊C4#120111yescharacteristic
386C4⋊C4#120112yes—
387C22⋊C4—102no—
388C22⋊C4—101no—
389C22⋊C4—102no—
390C42—62no—
391C42—61no—
392C42—102no—
393C42C2×D3×D511yescharacteristic
394C42C2×D3×D512yes—
395C3×C5#336411yescharacteristic, complemented, Hall
396C22×C3—201no—
397C22×C3—201no—
398C22×C3—201no—
399C22×C3C22×C22⋊D511yescharacteristic
400C3×C4—22no—
401C3×C4—22no—
402C3×C4—21no—
403C3×C4—202no—
404C3×C4—201no—
405C3×C4#409612yes—
406C3×C4C2×#71311yescharacteristic
407C3×C4#414711yescharacteristic
408Dic3—22no—
409Dic3—42no—
410Dic3—41no—
411C2×C5C2×#95011yescharacteristic
412C2×C5#583111yescharacteristic
413C2×C5#583811yescharacteristic
414D5—81no—
415C23—101no—
416Q8—22no—
417Q8—32no—
418Q8—61no—
419Q8—102no—
420Q8—102no—
421Q8—122no—
422D4—101no—
423D4—101no—
424C2×C4—61no—
425C2×C4—61no—
426C2×C4—62no—
427C2×C4—101no—
428C2×C4—102no—
429C2×C4—202no—
430C2×C4C2×#120112yes—
431C2×C4C2×#120111yescharacteristic
432C2×C4C2×#120112yes—
433C2×C4C2×#120111yescharacteristic
434C2×C4C22×D3×D511yescharacteristic
435C8—301no—
436C8—301no—
437C8—302no—
438C2×C3—401no—
439C2×C3#6615111yescharacteristic
440C2×C3C2×#409611yescharacteristic
441C2×C3#6617211yescharacteristic
442C5#8367611yescharacteristic, complemented, Sylow
443C22—201no—
444C22—201no—
445C22—201no—
446C22C22×#120111yescharacteristic
447C4—22no—
448C4—21no—
449C4—22no—
450C4—62no—
451C4—121no—
452C4—122no—
453C4—201no—
454C4—202no—
455C4#9129712yes—
456C4#9202811yescharacteristic
457C4D3×#71311yescharacteristic
458C3#1061542011yescharacteristic, complemented, Sylow
459C2—401no—
460C2C2×#9129711yescharacteristic
461C2#1175524711yescharacteristic
462C2#1175530211yescharacteristic
subgroupG/H
centerC22C22×#1201
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C4×C5×Q8C22
socleC22×C3×C5C22×D4
prime radicalC2×C3×D20C23
hypercenterC4×Q8D3×D5
hypocenterC3×C5#3364
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. C5×#5567C2semidirect
3. Q8×C3⋊#168C2semidirect
4. #11747099C2semidirect
5. C4×#6661C2semidirect
8. #11746924C2semidirect
9. #11746919C2semidirect
87. #5567D5semidirect
395. C3×C5#3364semidirect
442. C5#83676semidirect
458. C3#10615420semidirect

Maximal subgroups

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

subgroupindex
C5×#55672
C3×#659182
Q8×C3⋊#1682
#117470992
C4×#66612
#117470692
C4×#66552
#117469242
#117469192
#117468962
#117468862
#106154203
#836765