Finite Group Databasebeta

← #14471663

Subgroups — #14471663

Order 1920.

700 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5584C211yesmaximal, characteristic, complemented
2C3×#65913C211yesmaximal, characteristic, complemented
3Q8×C3⋊#168C211yesmaximal, characteristic, complemented
4#11747059C211yesmaximal, characteristic, complemented
5C4×#6655C212yesmaximal, complemented
6#11747027C212yesmaximal
7C4×#6652C211yesmaximal, characteristic
8#11746926C212yesmaximal, complemented
9#11746923C211yesmaximal, characteristic, complemented
10#11746903C212yesmaximal, complemented
11#11746882C211yesmaximal, characteristic, complemented
12#10615398—31nomaximal, Hall
13C3×C4×C5×Q8C2211yescharacteristic
14C5×#946—21no—
15C5×#898—22no—
16C5×#892—21no—
17C5×#875C2212yes—
18C4×C5×D12C2211yescharacteristic
19C5×#869C2211yescharacteristic
20C4×C5×Dic6C2212yes—
21C3×#4092—21no—
22C3×#4042—22no—
23C3×D5×C4⋊C4—21no—
24C3×C4×D20C2211yescharacteristic
25C3×C4×D20C2212yes—
26C3×#4016C2212yes—
27C3×#4015C2211yescharacteristic
28C2×#6655C2212yes—
29C2×#6652C2211yescharacteristic
30C4⋊(C3⋊#168)C2211yescharacteristic
31C4⋊(C3⋊#168)C2212yes—
32C4×C3⋊#168C2212yes—
33C4×C3⋊#168C2211yescharacteristic
34#90989C2211yescharacteristic
35#90986C2212yes—
36#90972C2212yes—
37#90972C2212yes—
38#90972C2212yes—
39#90966C2212yes—
40#90966C2211yescharacteristic
41#90964C2211yescharacteristic
42#83637—51nomaximal, Hall
43C5×#556—31no—
44#65913D311yescharacteristic, complemented
45#65372—31no—
46#65370—31no—
47#65346—32no—
48#65345—32no—
49C4×#688—32no—
50Q8×#168—31no—
51#65311—32no—
52C4×#686—31no—
53#65297—31no—
54C2×C3×C5×Q8D411yescharacteristic
55C2×C5×D12C2311yescharacteristic
56C2×C4×C5×D3—21no—
57C2×C5×Dic6C2312yes—
58C2×C3×D20C2311yescharacteristic
59C2×C3×C4×D5—21no—
60C2×C3×C4×D5—22no—
61C2×C3×Dic10—21no—
62C3×C5×C4⋊C4D411yescharacteristic
63C3×C5×C4⋊C4D412yes—
64C3×C42×C5C2312yes—
65C3×C42×C5C2311yescharacteristic
66C2×C3⋊#168C2311yescharacteristic
67C2×C3⋊#168C2312yes—
68C2×C3⋊#168C2311yescharacteristic
69C5×(C2×D3)⋊C4—21no—
70C5×(C2×D3)⋊C4—22no—
71C5×(C2×D3)⋊C4—22no—
72C5×C4⋊Dic3—21no—
73C5×C4⋊Dic3—21no—
74C5×C4⋊Dic3C2311yescharacteristic
75C5×Dic3⋊C4—22no—
76C4×C5×Dic3—22no—
77C3×#685—22no—
78C3×#685—21no—
79C3×C4⋊Dic5—21no—
80C3×C4⋊Dic5C2312yes—
81C3×C4⋊Dic5C2311yescharacteristic
82C3×#683—22no—
83C3×#683—22no—
84#6655—24no—
85#6652—22no—
86C3×#554—51no—
87#5584D511yescharacteristic, complemented
88#5093—51no—
89#5084—51no—
90#5068—52no—
91#5063—52no—
92Q8×C3⋊C8—51no—
93#5019—51no—
94C4×#214—52no—
95#5016—52no—
96C4×#213—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#4092—21no—
106C2×#688—32no—
107C2×#686—31no—
108#4042—22no—
109D5×C4⋊C4—21no—
110C4×D20C2×D311yescharacteristic
111C4×D20C2×D312yes—
112#4016C2×D312yes—
113#4015C2×D311yescharacteristic
114#3967—32no—
115#3967—32no—
116#3967—32no—
117#3964—32no—
118#3964—31no—
119#3941—31no—
120#3940—32no—
121#3939—31no—
122C4⋊#168—31no—
123C4⋊#168—32no—
124C4×#168—31no—
125C4×#168—32no—
126#3360—151noSylow
127C22×C5×D3—21no—
128C22×C3×D5—21no—
129C3×C5×Q8—22no—
130C2×C3×C4×C5C2×D412yes—
131C2×C3×C4×C5C2×D412yes—
132C2×C3×C4×C5C2×D411yescharacteristic
133C2×C3×C4×C5C2×D411yescharacteristic
134C2×C3×C4×C5C2411yescharacteristic
135C2×C5×Dic3—21no—
136C2×C5×Dic3—22no—
137C5×D12—21no—
138C5×D12—21no—
139C4×C5×D3—42no—
140C5×Dic6—22no—
141C5×Dic6—22no—
142C2×C3×Dic5—21no—
143C2×C3×Dic5—22no—
144C3×D20—21no—
145C3×D20#4112yes—
146C3×C4×D5—42no—
147C3×C4×D5—82no—
148C3×Dic10—81no—
149C3⋊#168—21no—
150C3⋊#168—22no—
151C3⋊#168—21no—
152C3×C4⋊Q8—51no—
153C3×#125—52no—
154C3×C22⋊Q8—101no—
155C3×C22⋊Q8—102no—
156C3×C4×Q8C2×D511yescharacteristic
157C3×C4×D4—51no—
158C3×C4×D4—52no—
159C2×C3×C4⋊C4—101no—
160#946—21no—
161C2×#214—52no—
162C2×#213—51no—
163#898—22no—
164#892—21no—
165#875C2×D512yes—
166C4×D12C2×D511yescharacteristic
167#869C2×D511yescharacteristic
168C4×Dic6C2×D512yes—
169#832—51no—
170#810—52no—
171#810—52no—
172#810—52no—
173#809—51no—
174#809—52no—
175#808—52no—
176#807—51no—
177C4⋊(C3⋊C8)—52no—
178C4⋊(C3⋊C8)—51no—
179C4×C3⋊C8—51no—
180C4×C3⋊C8—52no—
181C2×C5×Q8—32no—
182C2×C5×Q8#6711yescharacteristic
183C2×C5×D4—31no—
184C22×C4×C5—61no—
185C2×D20C22×D311yescharacteristic
186C2×C4×D5—21no—
187C2×C4×D5—22no—
188C2×Dic10—21no—
189C5×C4⋊C4—31no—
190C5×C4⋊C4—61no—
191C5×C4⋊C4—62no—
192C5×C4⋊C4—61no—
193C5×C4⋊C4#6711yescharacteristic
194C5×C4⋊C4#6712yes—
195C5×C22⋊C4—62no—
196C5×C22⋊C4—61no—
197C5×C22⋊C4—62no—
198C42×C5—62no—
199C42×C5C22×D311yescharacteristic
200C42×C5C22×D312yes—
201#688—64no—
202#686—62no—
203#685—21no—
204#685—22no—
205C4⋊Dic5—21no—
206C4⋊Dic5C22×D311yescharacteristic
207C4⋊Dic5C22×D312yes—
208#683—22no—
209#683—22no—
210C2×#168—31no—
211C2×#168—31no—
212C2×#168—32no—
213#556—31no—
214#554—51no—
215#488—152no—
216#487—151no—
217#479—152no—
218#476—151no—
219#463—152no—
220#461—151no—
221C8×Q8—151no—
222C4×SD16—152no—
223C4×D8—151no—
224C22×C3×C5C22×D411yescharacteristic
225C2×C5×D3—41no—
226C2×C5×D3—41no—
227C2×C5×D3—41no—
228C2×C3×D5—22no—
229C2×C3×D5—41no—
230C2×C3×D5—41no—
231C3×C4×C5—22no—
232C3×C4×C5—21no—
233C3×C4×C5—22no—
234C3×C4×C5#13512yes—
235C3×C4×C5C2×#4111yescharacteristic
236C3×C4×C52−1+411yescharacteristic
237C3×Dic5—41no—
238C3×Dic5—42no—
239C5×Dic3—42no—
240C5×Dic3—41no—
241C2×C3×Q8—101no—
242C2×C3×Q8C22⋊D511yescharacteristic
243C2×C3×D4—51no—
244C22×C3×C4—101no—
245C22×C3×C4—102no—
246C2×D12C22×D511yescharacteristic
247C2×C4×D3—21no—
248C2×Dic6C22×D512yes—
249C3×C4⋊C4—51no—
250C3×C4⋊C4—52no—
251C3×C4⋊C4—102no—
252C3×C4⋊C4—102no—
253C3×C4⋊C4—101no—
254C3×C4⋊C4C22⋊D511yescharacteristic
255C3×C4⋊C4C22⋊D512yes—
256C3×C22⋊C4—101no—
257C3×C22⋊C4—102no—
258C3×C42C22×D512yes—
259C3×C42C22×D511yescharacteristic
260#214—104no—
261#213—102no—
262(C2×D3)⋊C4—21no—
263(C2×D3)⋊C4—22no—
264(C2×D3)⋊C4—22no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3C22×D511yescharacteristic
268Dic3⋊C4—22no—
269C4×Dic3—22no—
270C2×C3⋊C8—51no—
271C2×C3⋊C8—51no—
272C2×C3⋊C8—52no—
273C23×C5—61no—
274C22×D5—21no—
275C5×Q8—22no—
276C5×Q8—62no—
277C5×Q8—62no—
278C5×D4—61no—
279C5×D4—61no—
280C2×C4×C5—61no—
281C2×C4×C5—62no—
282C2×C4×C5—122no—
283C2×C4×C5C2×#6712yes—
284C2×C4×C5C2×#6712yes—
285C2×C4×C5C2×#6711yescharacteristic
286C2×C4×C5C2×#6711yescharacteristic
287C2×C4×C5C23×D311yescharacteristic
288C2×Dic5—21no—
289C2×Dic5—22no—
290D20—21no—
291D20#23912yes—
292C4×D5—42no—
293C4×D5—82no—
294Dic10—81no—
295#168—61no—
296#168—62no—
297#168—61no—
298C2×SD16—152no—
299C2×D8—151no—
300C4⋊Q8—31no—
301C4⋊Q8—51no—
302#126—62no—
303#125—52no—
304#124—32no—
305C22⋊Q8—61no—
306C22⋊Q8—101no—
307C22⋊Q8—102no—
308C4×Q8—32no—
309C4×Q8D3×D511yescharacteristic
310C4×D4—31no—
311C4×D4—51no—
312C4×D4—52no—
313#117—61no—
314C2×C4⋊C4—101no—
315#107—151no—
316#106—152no—
317C4⋊C8—151no—
318C4⋊C8—152no—
319#103—152no—
320#103—152no—
321#103—152no—
322#102—151no—
323#102—151no—
324#102—152no—
325C4×C8—151no—
326C4×C8—152no—
327C2×C3×C5#54811yescharacteristic
328C2×C3×C5C2×#13511yescharacteristic
329C2×C3×C5#57711yescharacteristic
330C3×D5—42no—
331C5×D3—81no—
332C23×C3—101no—
333C22×D3—21no—
334C3×Q8—22no—
335C3×Q8—401no—
336C3×D4—52no—
337C3×D4—101no—
338C2×C3×C4—101no—
339C2×C3×C4—102no—
340C2×C3×C4—202no—
341C2×C3×C4—402no—
342C2×C3×C4C2×C22⋊D512yes—
343C2×C3×C4C2×C22⋊D512yes—
344C2×C3×C4C2×C22⋊D511yescharacteristic
345C2×C3×C4C2×C22⋊D511yescharacteristic
346C2×C3×C4C23×D511yescharacteristic
347C2×Dic3—21no—
348C2×Dic3—22no—
349D12—21no—
350D12—21no—
351C4×D3—42no—
352Dic6—22no—
353Dic6—22no—
354C3⋊C8—101no—
355C3⋊C8—102no—
356C3⋊C8—101no—
357C22×C5—121no—
358C22×C5—121no—
359C22×C5—121no—
360C22×C5C22×#6711yescharacteristic
361C2×D5—22no—
362C2×D5—41no—
363C2×D5—41no—
364C4×C5—22no—
365C4×C5—21no—
366C4×C5—22no—
367C4×C5—122no—
368C4×C5—121no—
369C4×C5#95012yes—
370C4×C5C2×#23911yescharacteristic
371C4×C5#100711yescharacteristic
372Dic5—41no—
373Dic5—42no—
374C2×Q8—32no—
375C2×Q8—101no—
376C2×Q8#120111yescharacteristic
377C2×D4—31no—
378C2×D4—51no—
379C22×C4—61no—
380C22×C4—101no—
381C22×C4—102no—
382SD16—304no—
383D8—302no—
384C2×C8—151no—
385C2×C8—152no—
386C2×C8—151no—
387C4⋊C4—31no—
388C4⋊C4—52no—
389C4⋊C4—51no—
390C4⋊C4—61no—
391C4⋊C4—62no—
392C4⋊C4—61no—
393C4⋊C4—102no—
394C4⋊C4—101no—
395C4⋊C4—102no—
396C4⋊C4#120111yescharacteristic
397C4⋊C4#120112yes—
398C22⋊C4—62no—
399C22⋊C4—61no—
400C22⋊C4—62no—
401C22⋊C4—101no—
402C22⋊C4—102no—
403C42—62no—
404C42C2×D3×D511yescharacteristic
405C42C2×D3×D512yes—
406C3×C5#336011yescharacteristic, complemented, Hall
407C22×C3—102no—
408C22×C3—201no—
409C22×C3—201no—
410C22×C3C22×C22⋊D511yescharacteristic
411C2×D3—41no—
412C2×D3—41no—
413C2×D3—41no—
414C3×C4—22no—
415C3×C4—22no—
416C3×C4—21no—
417C3×C4—201no—
418C3×C4—202no—
419C3×C4#409612yes—
420C3×C4D5×#4111yescharacteristic
421C3×C4#415011yescharacteristic
422Dic3—41no—
423Dic3—42no—
424C2×C5—241no—
425C2×C5#581711yescharacteristic
426C2×C5C2×#95011yescharacteristic
427C2×C5#583611yescharacteristic
428D5—42no—
429C23—61no—
430C23—101no—
431Q8—22no—
432Q8—62no—
433Q8—62no—
434Q8—401no—
435D4—52no—
436D4—61no—
437D4—61no—
438D4—101no—
439C2×C4—61no—
440C2×C4—62no—
441C2×C4—101no—
442C2×C4—102no—
443C2×C4—122no—
444C2×C4—202no—
445C2×C4—402no—
446C2×C4C2×#120112yes—
447C2×C4C2×#120112yes—
448C2×C4C2×#120111yescharacteristic
449C2×C4C2×#120111yescharacteristic
450C2×C4C22×D3×D511yescharacteristic
451C8—301no—
452C8—301no—
453C8—302no—
454C2×C3—202no—
455C2×C3C2×#409611yescharacteristic
456C2×C3#6616511yescharacteristic
457C2×C3#6617111yescharacteristic
458D3—81no—
459C5#8363711yescharacteristic, complemented, Sylow
460C22—102no—
461C22—121no—
462C22—121no—
463C22—121no—
464C22—201no—
465C22—201no—
466C22C22×#120111yescharacteristic
467C4—22no—
468C4—21no—
469C4—22no—
470C4—121no—
471C4—122no—
472C4—202no—
473C4—201no—
474C4#9129712yes—
475C4#9203011yescharacteristic
476C4D5×#23911yescharacteristic
477C3#1061539811yescharacteristic, complemented, Sylow
478C2—202no—
479C2—241no—
480C2C2×#9129711yescharacteristic
481C2#1175524411yescharacteristic
482C2#1175530311yescharacteristic
subgroupG/H
centerC22C22×#1201
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C4×C5×Q8C22
socleC22×C3×C5C22×D4
prime radicalC2×#6652C22
hypercenterC4×Q8D3×D5
hypocenterC3×C5#3360
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×#5584C2semidirect
2. C3×#65913C2semidirect
3. Q8×C3⋊#168C2semidirect
4. #11747059C2semidirect
5. C4×#6655C2semidirect
8. #11746926C2semidirect
9. #11746923C2semidirect
10. #11746903C2semidirect
11. #11746882C2semidirect
44. #65913D3semidirect
87. #5584D5semidirect
406. C3×C5#3360semidirect
459. C5#83637semidirect
477. C3#10615398semidirect

Maximal subgroups

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

subgroupindex
C5×#55842
C3×#659132
Q8×C3⋊#1682
#117470592
C4×#66552
#117470272
C4×#66522
#117469262
#117469232
#117469032
#117468822
#106153983
#836375