Finite Group Databasebeta

← #14472140

Subgroups — #14472140

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
1#11756326C211yesmaximal, characteristic, complemented
2C5×#5584C211yesmaximal, characteristic
3C3×Q8×#168C211yesmaximal, characteristic, complemented
4#11747105C211yesmaximal, characteristic, complemented
5C4×#6663C212yesmaximal, complemented
6C4×#6659C211yesmaximal, characteristic
7#11746965C212yesmaximal, complemented
8#11746964C211yesmaximal, characteristic
9#11746929C211yesmaximal, characteristic, complemented
10#11746907C212yesmaximal
11#11746884C212yesmaximal
12#10615407—31nomaximal, Hall
13C3×C4×C5×Q8C2211yescharacteristic
14#91833—21no—
15#91778—22no—
16#91776—21no—
17#91759C2212yes—
18#91758C2211yescharacteristic
19C4×Dic30C2212yes—
20C4×Dic30C2211yescharacteristic
21C5×#946—21no—
22C5×#898—22no—
23C5×#892—21no—
24C5×#875C2212yes—
25C4×C5×D12C2211yescharacteristic
26C5×#869C2211yescharacteristic
27C4×C5×Dic6C2212yes—
28C2×#6663C2212yes—
29C2×#6659C2211yescharacteristic
30C3×C4⋊#168C2211yescharacteristic
31C3×C4⋊#168C2212yes—
32C3×C4×#168C2212yes—
33C3×C4×#168C2211yescharacteristic
34#90988C2212yes—
35#90985C2211yescharacteristic
36#90975C2211yescharacteristic
37#90973C2212yes—
38#90973C2212yes—
39#90973C2212yes—
40#90967C2212yes—
41#90967C2211yescharacteristic
42#79934—51nomaximal, Hall
43C5×#556—31no—
44#65901—31no—
45#65379—31no—
46#65370—31no—
47#65345—32no—
48#65343—32no—
49C4×#689—32no—
50Q8×#168D311yescharacteristic, complemented
51#65311—32no—
52C4×#687—31no—
53#65296—31no—
54C2×C3×C5×Q8D411yescharacteristic
55C2×Dic30—21no—
56C2×Dic30C2311yescharacteristic
57C2×C5×D12C2311yescharacteristic
58C2×C4×C5×D3—21no—
59C2×C5×Dic6C2312yes—
60C3×C5×C4⋊C4D411yescharacteristic
61C3×C5×C4⋊C4D412yes—
62C3×C42×C5C2312yes—
63C3×C42×C5C2311yescharacteristic
64C4⋊Dic15—21no—
65C4⋊Dic15C2311yescharacteristic
66C4⋊Dic15C2312yes—
67#6712—21no—
68#6712—22no—
69#6712—22no—
70#6712—22no—
71C4×Dic15—22no—
72C4×Dic15—21no—
73C5×(C2×D3)⋊C4—21no—
74C5×(C2×D3)⋊C4—22no—
75C5×(C2×D3)⋊C4—22no—
76C5×C4⋊Dic3—21no—
77C5×C4⋊Dic3—21no—
78C5×C4⋊Dic3C2311yescharacteristic
79C5×Dic3⋊C4—22no—
80C4×C5×Dic3—22no—
81C2×C3×#168C2311yescharacteristic
82C2×C3×#168C2312yes—
83C2×C3×#168C2311yescharacteristic
84#6663—24no—
85#6659—22no—
86#5584D511yescharacteristic
87#5567—51no—
88C3×C8×Q8—51no—
89#4849—51no—
90#4846—51no—
91#4845—52no—
92#4829—52no—
93C4×Dic12—52no—
94#4697—52no—
95C4×#204—51no—
96#4681—51no—
97C5×C4⋊Q8—31no—
98C5×#126—62no—
99C5×#124—32no—
100C5×C22⋊Q8—61no—
101C4×C5×Q8—32no—
102C4×C5×Q8C2×D311yescharacteristic
103C4×C5×D4—31no—
104C5×#117—61no—
105#4090—61no—
106C2×#689—32no—
107C2×#687—31no—
108#4035—62no—
109#4033—61no—
110#4016—32no—
111#4015—31no—
112C4×Dic10—31no—
113C4×Dic10—32no—
114#3967—32no—
115#3967—32no—
116#3967—32no—
117#3964—32no—
118#3964—31no—
119#3942—31no—
120#3940—31no—
121#3939—32no—
122C4⋊#168C2×D311yescharacteristic
123C4⋊#168C2×D312yes—
124C4×#168C2×D311yescharacteristic
125C4×#168C2×D312yes—
126#3364—151noSylow
127C22×C5×D3—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×Dic15—21no—
135C2×Dic15—21no—
136C2×Dic15—22no—
137Dic30—21no—
138Dic30—42no—
139Dic30#4112yes—
140C2×C5×Dic3—21no—
141C2×C5×Dic3—22no—
142C5×D12—21no—
143C5×D12—21no—
144C4×C5×D3—42no—
145C5×Dic6—22no—
146C5×Dic6—22no—
147C3×#168—21no—
148C3×#168—22no—
149C3×#168—21no—
150C3×C4×Q8C2×D511yescharacteristic
151#946—21no—
152#944—101no—
153C2×Dic12—52no—
154C2×#204—51no—
155#898—22no—
156#892—21no—
157#889—102no—
158#887—101no—
159#875C2×D512yes—
160C4×D12C2×D511yescharacteristic
161#870—52no—
162#869—51no—
163#869C2×D511yescharacteristic
164C4×Dic6—51no—
165C4×Dic6—52no—
166C4×Dic6C2×D512yes—
167C3×C4⋊C8—52no—
168C3×C4⋊C8—51no—
169C3×C4×C8—51no—
170C3×C4×C8—52no—
171#821—51no—
172#821—52no—
173#818—52no—
174#817—51no—
175#816—52no—
176#816—52no—
177#816—52no—
178#816—51no—
179C2×C5×Q8—32no—
180C2×C5×Q8D1211yescharacteristic
181C2×C5×D4—31no—
182C22×C4×C5—61no—
183C2×Dic10—31no—
184C2×Dic10—61no—
185C5×C4⋊C4—31no—
186C5×C4⋊C4—61no—
187C5×C4⋊C4—62no—
188C5×C4⋊C4—61no—
189C5×C4⋊C4D1211yescharacteristic
190C5×C4⋊C4D1212yes—
191C5×C22⋊C4—62no—
192C5×C22⋊C4—61no—
193C5×C22⋊C4—62no—
194C42×C5—62no—
195C42×C5C22×D311yescharacteristic
196C42×C5C22×D312yes—
197#689—64no—
198#687—62no—
199C4⋊Dic5—32no—
200C4⋊Dic5—31no—
201C4⋊Dic5—61no—
202#683—61no—
203#683—62no—
204#683—62no—
205#683—62no—
206C4×Dic5—62no—
207C4×Dic5—61no—
208C2×#168C22×D311yescharacteristic
209C2×#168C22×D312yes—
210C2×#168C22×D311yescharacteristic
211#557—151no—
212#556—31no—
213#488—152no—
214#487—151no—
215#479—152no—
216#474—151no—
217#464—152no—
218#462—151no—
219C8×Q8—51no—
220C4×Q16—152no—
221C4×SD16—151no—
222C22×C3×C5C22×D411yescharacteristic
223C2×C5×D3—41no—
224C2×C5×D3—41no—
225C2×C5×D3—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
232Dic15—22no—
233Dic15—42no—
234Dic15—41no—
235C5×Dic3—42no—
236C5×Dic3—41no—
237C2×C3×Q8C22⋊D511yescharacteristic
238C2×D12C22×D511yescharacteristic
239C2×C4×D3—21no—
240C2×Dic6—51no—
241C2×Dic6—101no—
242C2×Dic6C22×D512yes—
243C2×C3×C8—51no—
244C2×C3×C8—51no—
245C2×C3×C8—52no—
246C3×C4⋊C4C22⋊D511yescharacteristic
247C3×C4⋊C4C22⋊D512yes—
248C3×C42C22×D512yes—
249C3×C42C22×D511yescharacteristic
250(C2×D3)⋊C4—21no—
251(C2×D3)⋊C4—22no—
252(C2×D3)⋊C4—22no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—51no—
256C4⋊Dic3—52no—
257C4⋊Dic3—101no—
258C4⋊Dic3C22×D511yescharacteristic
259Dic3⋊C4—22no—
260Dic3⋊C4—101no—
261Dic3⋊C4—102no—
262Dic3⋊C4—102no—
263Dic3⋊C4—102no—
264C4×Dic3—22no—
265C4×Dic3—102no—
266C4×Dic3—101no—
267Dic12—104no—
268#204—102no—
269C23×C5—61no—
270C5×Q8—22no—
271C5×Q8—62no—
272C5×Q8—62no—
273C5×D4—61no—
274C5×D4—61no—
275C2×C4×C5—61no—
276C2×C4×C5—62no—
277C2×C4×C5—122no—
278C2×C4×C5C2×D1212yes—
279C2×C4×C5C2×D1212yes—
280C2×C4×C5C2×D1211yescharacteristic
281C2×C4×C5C2×D1211yescharacteristic
282C2×C4×C5C23×D311yescharacteristic
283C2×Dic5—61no—
284C2×Dic5—61no—
285C2×Dic5—62no—
286Dic10—32no—
287Dic10—61no—
288Dic10—122no—
289#168—21no—
290#168—21no—
291#168—22no—
292C2×Q16—152no—
293C2×SD16—151no—
294C4⋊Q8—31no—
295C4⋊Q8—151no—
296C4⋊Q8—301no—
297#126—62no—
298#125—152no—
299#125—302no—
300#124—32no—
301C22⋊Q8—61no—
302C4×Q8—32no—
303C4×Q8—151no—
304C4×Q8—152no—
305C4×Q8—301no—
306C4×Q8D3×D511yescharacteristic
307C4×D4—31no—
308#117—61no—
309#107—152no—
310#106—151no—
311C4⋊C8—51no—
312C4⋊C8—52no—
313#103—151no—
314#103—152no—
315#103—152no—
316#103—152no—
317#102—151no—
318#102—152no—
319C4×C8—51no—
320C4×C8—52no—
321C2×C3×C5#54811yescharacteristic
322C2×C3×C5C2×#13511yescharacteristic
323C2×C3×C5#57811yescharacteristic
324C5×D3—81no—
325C22×D3—21no—
326C3×Q8—22no—
327C2×C3×C4C2×C22⋊D512yes—
328C2×C3×C4C2×C22⋊D512yes—
329C2×C3×C4C2×C22⋊D511yescharacteristic
330C2×C3×C4C2×C22⋊D511yescharacteristic
331C2×C3×C4C23×D511yescharacteristic
332C2×Dic3—21no—
333C2×Dic3—22no—
334C2×Dic3—101no—
335C2×Dic3—101no—
336C2×Dic3—102no—
337D12—21no—
338D12—21no—
339C4×D3—42no—
340Dic6—22no—
341Dic6—22no—
342Dic6—52no—
343Dic6—101no—
344Dic6—202no—
345C3×C8—101no—
346C3×C8—102no—
347C3×C8—101no—
348C22×C5—121no—
349C22×C5—121no—
350C22×C5—121no—
351C22×C5C22×D1211yescharacteristic
352C4×C5—22no—
353C4×C5—22no—
354C4×C5—21no—
355C4×C5—121no—
356C4×C5—122no—
357C4×C5#90412yes—
358C4×C5#100711yescharacteristic
359C4×C5D3×#4111yescharacteristic
360Dic5—62no—
361Dic5—122no—
362Dic5—121no—
363C2×Q8—32no—
364C2×Q8—151no—
365C2×Q8—301no—
366C2×Q8#120311yescharacteristic
367C2×D4—31no—
368C22×C4—61no—
369Q16—304no—
370SD16—302no—
371C2×C8—51no—
372C2×C8—52no—
373C2×C8—51no—
374C4⋊C4—31no—
375C4⋊C4—61no—
376C4⋊C4—62no—
377C4⋊C4—61no—
378C4⋊C4—151no—
379C4⋊C4—152no—
380C4⋊C4—301no—
381C4⋊C4—302no—
382C4⋊C4—302no—
383C4⋊C4—302no—
384C4⋊C4—301no—
385C4⋊C4#120311yescharacteristic
386C4⋊C4#120312yes—
387C22⋊C4—62no—
388C22⋊C4—61no—
389C22⋊C4—62no—
390C42—62no—
391C42—302no—
392C42—301no—
393C42C2×D3×D511yescharacteristic
394C42C2×D3×D512yes—
395C3×C5#336411yescharacteristic, complemented, Hall
396C22×C3C22×C22⋊D511yescharacteristic
397C2×D3—41no—
398C2×D3—41no—
399C2×D3—41no—
400C3×C4—22no—
401C3×C4—22no—
402C3×C4—21no—
403C3×C4#409612yes—
404C3×C4D5×#4111yescharacteristic
405C3×C4#415011yescharacteristic
406Dic3—41no—
407Dic3—42no—
408Dic3—102no—
409Dic3—202no—
410Dic3—201no—
411C2×C5—241no—
412C2×C5#557611yescharacteristic
413C2×C5C2×#90411yescharacteristic
414C2×C5#575311yescharacteristic
415C23—61no—
416Q8—22no—
417Q8—62no—
418Q8—62no—
419Q8—152no—
420Q8—301no—
421Q8—602no—
422D4—61no—
423D4—61no—
424C2×C4—61no—
425C2×C4—62no—
426C2×C4—122no—
427C2×C4—301no—
428C2×C4—301no—
429C2×C4—302no—
430C2×C4C2×#120312yes—
431C2×C4C2×#120311yescharacteristic
432C2×C4C2×#120312yes—
433C2×C4C2×#120311yescharacteristic
434C2×C4C22×D3×D511yescharacteristic
435C8—101no—
436C8—101no—
437C8—102no—
438C2×C3C2×#409611yescharacteristic
439C2×C3#6616511yescharacteristic
440C2×C3#6617211yescharacteristic
441D3—81no—
442C5#7993411yescharacteristic, complemented, Sylow
443C22—121no—
444C22—121no—
445C22—121no—
446C22C22×#120311yescharacteristic
447C4—22no—
448C4—21no—
449C4—22no—
450C4—122no—
451C4—121no—
452C4—302no—
453C4—601no—
454C4—602no—
455C4#9130312yes—
456C4#9203011yescharacteristic
457C4#9203411yescharacteristic
458C3#1061540711yescharacteristic, complemented, Sylow
459C2—241no—
460C2C2×#9130311yescharacteristic
461C2#1175513411yescharacteristic
462C2#1175530611yescharacteristic
subgroupG/H
centerC22C22×#1203
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C4×C5×Q8C22
socleC22×C3×C5C22×D4
prime radicalC2×C5×D12C23
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. #11756326C2semidirect
3. C3×Q8×#168C2semidirect
4. #11747105C2semidirect
5. C4×#6663C2semidirect
7. #11746965C2semidirect
9. #11746929C2semidirect
50. Q8×#168D3semidirect
395. C3×C5#3364semidirect
442. C5#79934semidirect
458. C3#10615407semidirect

Maximal subgroups

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

subgroupindex
#117563262
C5×#55842
C3×Q8×#1682
#117471052
C4×#66632
C4×#66592
#117469652
#117469642
#117469292
#117469072
#117468842
#106154073
#799345