Finite Group Databasebeta

← #14330202

Subgroups — #14330202

Order 1920.

1120 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91836C211yesmaximal, characteristic
2C5×#5229C211yesmaximal, characteristic, complemented
3C3×#65519C211yesmaximal, characteristic, complemented
4#11748904C212yesmaximal
5#11748881C212yesmaximal, complemented
6#11748875C212yesmaximal, complemented
7#11748685C212yesmaximal, complemented
8#11748678C212yesmaximal, complemented
9#11748653C212yesmaximal, complemented
10#10608226—31nomaximal, Hall
11C22×C3×C5×Q8C2211yescharacteristic
12C22×D60C2211yescharacteristic
13C22×#1200C2212yes—
14#91836C418yescomplemented
15C2×#6714C2212yes—
16C2×#6714C2212yes—
17C2×C4×Dic15C2211yescharacteristic
18C2×C5×C4⋊Dic3C2211yescharacteristic
19C2×C4×C5×Dic3C2211yescharacteristic
20C2×C3×C4⋊Dic5C2211yescharacteristic
21C2×C3×C4×Dic5C2211yescharacteristic
22C2×#6667C2212yes—
23C2×#6667C2212yes—
24C2×#6667C2212yes—
25C2×Dic3×Dic5C2212yes—
26C2×Dic3×Dic5C2212yes—
27C5×C3⋊#95C2212yes—
28C5×C3⋊#95C2212yes—
29C3×#3963C2212yes—
30C3×#3963C2212yes—
31#90994C2212yes—
32#90994C2212yes—
33#76495—51nomaximal, Hall
34C2×#4093—31no—
35C5×#391—31no—
36#65519D311yescharacteristic, complemented
37#65279—32no—
38C4×#685—32no—
39#64959—32no—
40#64955—32no—
41#64954—32no—
42#64937—32no—
43C23×D15C2312yes—
44C2×C3×C5×Q8—41no—
45C2×C3×C5×Q8—41no—
46C2×C3×C5×Q8C2×C414yescomplemented
47C22×C3×C4×C5C2311yescharacteristic
48C22×C3×C4×C5C2311yescharacteristic
49C22×C3×C4×C5C2311yescharacteristic
50C22×Dic15C2312yes—
51C2×D60—41no—
52C2×D60—41no—
53C2×D60C2×C414yes—
54C22×C5×Dic3C2312yes—
55C22×C5×Dic3C2312yes—
56C22×C3×Dic5C2312yes—
57C22×C3×Dic5C2312yes—
58C2×#1200—24no—
59C2×#1200—24no—
60C2×#1200—42no—
61#6714C2×C418yescomplemented
62#6714C2×C418yescomplemented
63C4×Dic15C2×C414yescomplemented
64C5×C4⋊Dic3—41no—
65C4×C5×Dic3—41no—
66C3×C4⋊Dic5—41no—
67C3×C4×Dic5—41no—
68#6667—24no—
69#6667—24no—
70#6667—42no—
71Dic3×Dic5—24no—
72Dic3×Dic5—24no—
73C2×#947—51no—
74C3×#391—51no—
75#5229D511yescharacteristic, complemented
76#4990—52no—
77C4×(C2×D3)⋊C4—52no—
78#4670—52no—
79#4666—52no—
80#4665—52no—
81#4648—52no—
82C22×C5×Q8C2×D311yescharacteristic
83C22×D20—31no—
84C22×C4×D5—32no—
85C2×C5×C4⋊C4—31no—
86C2×C42×C5—31no—
87#4093—38no—
88C2×#685—32no—
89C2×#685—32no—
90C2×#685—32no—
91C2×#685—32no—
92C2×#685—32no—
93C2×C4⋊Dic5C2×D311yescharacteristic
94C2×C4×Dic5—31no—
95C2×C4×Dic5—32no—
96C2×C4×Dic5—32no—
97C2×C4×Dic5C2×D311yescharacteristic
98C5×#95—32no—
99C5×#95—32no—
100#3963—32no—
101#3963—32no—
102#3963C2×D312yes—
103#3963C2×D312yes—
104#2372—151noSylow
105C23×C3×C5C2411yescharacteristic
106C22×D15—22no—
107C22×D15—22no—
108C22×D15—22no—
109C22×D15—22no—
110C22×D15—22no—
111C22×D15—22no—
112C22×D15C22×C414yes—
113C3×C5×Q8—44no—
114C2×C3×C4×C5—21no—
115C2×C3×C4×C5—21no—
116C2×C3×C4×C5—21no—
117C2×C3×C4×C5—21no—
118C2×C3×C4×C5—21no—
119C2×C3×C4×C5—21no—
120C2×C3×C4×C5C22×C412yes—
121C2×C3×C4×C5C22×C412yes—
122C2×C3×C4×C5C22×C412yes—
123C2×Dic15—22no—
124C2×Dic15—22no—
125C2×Dic15C22×C414yes—
126D60—44no—
127C2×C5×Dic3—22no—
128C2×C5×Dic3—22no—
129C2×C5×Dic3—22no—
130C2×C5×Dic3—22no—
131C2×C5×Dic3—22no—
132C2×C5×Dic3#4114yes—
133C2×C3×Dic5—22no—
134C2×C3×Dic5—22no—
135C2×C3×Dic5—22no—
136C2×C3×Dic5—22no—
137C2×C3×Dic5—22no—
138C2×C3×Dic5#4114yes—
139#1200—48no—
140C22×C3×Q8C2×D511yescharacteristic
141C22×D12—51no—
142C22×C4×D3—52no—
143C2×C3×C4⋊C4—51no—
144C2×C3×C42—51no—
145#947—58no—
146C2×(C2×D3)⋊C4—52no—
147C2×(C2×D3)⋊C4—52no—
148C2×(C2×D3)⋊C4—52no—
149C2×(C2×D3)⋊C4—52no—
150C2×(C2×D3)⋊C4—52no—
151C2×C4⋊Dic3C2×D511yescharacteristic
152C2×C4×Dic3—51no—
153C2×C4×Dic3—52no—
154C2×C4×Dic3—52no—
155C2×C4×Dic3C2×D511yescharacteristic
156C3×#95—52no—
157C3×#95—52no—
158C3⋊#95—52no—
159C3⋊#95—52no—
160C3⋊#95C2×D512yes—
161C3⋊#95C2×D512yes—
162C23×D5—32no—
163C2×C5×Q8—41no—
164C2×C5×Q8—41no—
165C2×C5×Q8C4×D314yescomplemented
166C22×C4×C5—32no—
167C22×C4×C5—32no—
168C22×C4×C5C22×D311yescharacteristic
169C22×C4×C5C22×D311yescharacteristic
170C22×C4×C5C22×D311yescharacteristic
171C22×Dic5—32no—
172C22×Dic5C22×D312yes—
173C22×Dic5C22×D312yes—
174C2×D20—34no—
175C2×D20—121no—
176C2×D20—121no—
177C2×C4×D5—64no—
178C2×C4×D5—64no—
179C2×C4×D5—122no—
180C5×C4⋊C4—121no—
181C42×C5—121no—
182#685—38no—
183#685—38no—
184#685—64no—
185#685—64no—
186#685—122no—
187C4⋊Dic5—41no—
188C4×Dic5—34no—
189C4×Dic5—41no—
190C4×Dic5—64no—
191C4×Dic5—64no—
192C2×#124—151no—
193#391—31no—
194#391—51no—
195#388—152no—
196#388—152no—
197#386—152no—
198C4×C22⋊C4—152no—
199C4×C22⋊C4—152no—
200#376—152no—
201C22×C3×C5C23×C411yescharacteristic
202C22×C3×C5C2×#4112yes—
203C22×C3×C5C2×#4112yes—
204C22×C3×C5H(5,𝔽2)11yescharacteristic
205C22×C3×C5H(5,𝔽2)11yescharacteristic
206C2×D15—24no—
207C2×D15—24no—
208C2×D15—24no—
209C2×D15—42no—
210C2×D15—42no—
211C2×D15—42no—
212C2×D15—42no—
213C3×C4×C5—22no—
214C3×C4×C5—22no—
215C3×C4×C5—22no—
216Dic15—24no—
217C3×Dic5—24no—
218C3×Dic5—42no—
219C5×Dic3—24no—
220C5×Dic3—42no—
221C23×D3—52no—
222C2×C3×Q8—41no—
223C2×C3×Q8—41no—
224C2×C3×Q8C4×D514yescomplemented
225C22×C3×C4—52no—
226C22×C3×C4—52no—
227C22×C3×C4C22×D511yescharacteristic
228C22×C3×C4C22×D511yescharacteristic
229C22×C3×C4C22×D511yescharacteristic
230C22×Dic3—52no—
231C22×Dic3C22×D512yes—
232C22×Dic3C22×D512yes—
233C2×D12—54no—
234C2×D12—201no—
235C2×D12—201no—
236C2×C4×D3—104no—
237C2×C4×D3—104no—
238C2×C4×D3—202no—
239C3×C4⋊C4—201no—
240C3×C42—201no—
241(C2×D3)⋊C4—58no—
242(C2×D3)⋊C4—58no—
243(C2×D3)⋊C4—104no—
244(C2×D3)⋊C4—104no—
245(C2×D3)⋊C4—202no—
246C4⋊Dic3—41no—
247C4×Dic3—41no—
248C4×Dic3—54no—
249C4×Dic3—104no—
250C4×Dic3—104no—
251C23×C5C23×D311yescharacteristic
252C22×D5—34no—
253C22×D5—62no—
254C22×D5—62no—
255C22×D5—62no—
256C22×D5—62no—
257C22×D5—62no—
258C22×D5—62no—
259C5×Q8—44no—
260C2×C4×C5—21no—
261C2×C4×C5—21no—
262C2×C4×C5—21no—
263C2×C4×C5—21no—
264C2×C4×C5—21no—
265C2×C4×C5—21no—
266C2×C4×C5—34no—
267C2×C4×C5—62no—
268C2×C4×C5—62no—
269C2×C4×C5—62no—
270C2×C4×C5—62no—
271C2×C4×C5—62no—
272C2×C4×C5C2×C4×D312yes—
273C2×C4×C5C2×C4×D312yes—
274C2×C4×C5C2×C4×D312yes—
275C2×Dic5—22no—
276C2×Dic5—22no—
277C2×Dic5—22no—
278C2×Dic5—22no—
279C2×Dic5—22no—
280C2×Dic5—34no—
281C2×Dic5—62no—
282C2×Dic5—62no—
283C2×Dic5#23914yes—
284D20—124no—
285C4×D5—128no—
286C22×Q8D3×D511yescharacteristic
287C22×D4—151no—
288C23×C4—152no—
289#124—158no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—51no—
292C2×C22⋊C4—152no—
293C2×C22⋊C4—152no—
294C2×C22⋊C4—152no—
295C2×C22⋊C4—152no—
296C2×C22⋊C4—152no—
297C2×C42—31no—
298C2×C42—51no—
299C2×C42—151no—
300C2×C42—152no—
301C2×C42—152no—
302#95—32no—
303#95—32no—
304#95—52no—
305#95—52no—
306#95—152no—
307#95—152no—
308C2×C3×C5C4×#4112yes—
309C2×C3×C5#51811yescharacteristic
310C2×C3×C5#55112yes—
311C2×C3×C5#55511yescharacteristic
312C2×C3×C5#55511yescharacteristic
313D15—44no—
314C23×C3C23×D511yescharacteristic
315C22×D3—54no—
316C22×D3—102no—
317C22×D3—102no—
318C22×D3—102no—
319C22×D3—102no—
320C22×D3—102no—
321C22×D3—102no—
322C3×Q8—44no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—54no—
330C2×C3×C4—102no—
331C2×C3×C4—102no—
332C2×C3×C4—102no—
333C2×C3×C4—102no—
334C2×C3×C4—102no—
335C2×C3×C4C2×C4×D512yes—
336C2×C3×C4C2×C4×D512yes—
337C2×C3×C4C2×C4×D512yes—
338C2×Dic3—22no—
339C2×Dic3—22no—
340C2×Dic3—22no—
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—54no—
344C2×Dic3—102no—
345C2×Dic3—102no—
346C2×Dic3#71314yes—
347D12—204no—
348C4×D3—208no—
349C22×C5C22×C4×D311yescharacteristic
350C22×C5#100411yescharacteristic
351C22×C5#100411yescharacteristic
352C22×C5C2×#23912yes—
353C22×C5D3×#4112yes—
354C2×D5—64no—
355C2×D5—64no—
356C2×D5—64no—
357C2×D5—122no—
358C2×D5—122no—
359C2×D5—122no—
360C2×D5—122no—
361C4×C5—22no—
362C4×C5—22no—
363C4×C5—22no—
364C4×C5—64no—
365C4×C5—122no—
366Dic5—24no—
367Dic5—42no—
368Dic5—64no—
369C24—152no—
370C2×Q8—41no—
371C2×Q8—41no—
372C2×Q8#120014yescomplemented
373C2×D4—154no—
374C2×D4—601no—
375C2×D4—601no—
376C22×C4—32no—
377C22×C4—32no—
378C22×C4—52no—
379C22×C4—52no—
380C22×C4—152no—
381C22×C4—304no—
382C22×C4—304no—
383C22×C4—602no—
384C22×C4C2×D3×D511yescharacteristic
385C22×C4C2×D3×D511yescharacteristic
386C22×C4C2×D3×D511yescharacteristic
387C4⋊C4—121no—
388C4⋊C4—201no—
389C22⋊C4—158no—
390C22⋊C4—158no—
391C22⋊C4—304no—
392C22⋊C4—304no—
393C22⋊C4—602no—
394C42—121no—
395C42—154no—
396C42—201no—
397C42—304no—
398C42—304no—
399C3×C5#237211yescharacteristic, complemented, Hall
400C22×C3C22×C4×D511yescharacteristic
401C22×C3#414411yescharacteristic
402C22×C3#414411yescharacteristic
403C22×C3C2×#71312yes—
404C22×C3D5×#4112yes—
405C2×D3—104no—
406C2×D3—104no—
407C2×D3—104no—
408C2×D3—202no—
409C2×D3—202no—
410C2×D3—202no—
411C2×D3—202no—
412C3×C4—22no—
413C3×C4—22no—
414C3×C4—22no—
415C3×C4—104no—
416C3×C4—202no—
417Dic3—24no—
418Dic3—42no—
419Dic3—104no—
420C2×C5#548611yescharacteristic
421C2×C5C4×#23912yes—
422C2×C5#563712yes—
423C2×C5#567011yescharacteristic
424C2×C5#573411yescharacteristic
425D5—124no—
426C23—154no—
427C23—302no—
428C23—302no—
429C23—302no—
430C23—302no—
431C23—302no—
432C23—302no—
433C23C22×D3×D511yescharacteristic
434Q8—44no—
435D4—604no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—34no—
443C2×C4—54no—
444C2×C4—62no—
445C2×C4—62no—
446C2×C4—62no—
447C2×C4—62no—
448C2×C4—62no—
449C2×C4—102no—
450C2×C4—102no—
451C2×C4—102no—
452C2×C4—102no—
453C2×C4—102no—
454C2×C4—154no—
455C2×C4—302no—
456C2×C4—302no—
457C2×C4—608no—
458C2×C4C2×#120012yes—
459C2×C4C2×#120012yes—
460C2×C4C2×#120012yes—
461C2×C3#6582011yescharacteristic
462C2×C3C4×#71312yes—
463C2×C3#6597112yes—
464C2×C3#6600411yescharacteristic
465C2×C3#6606811yescharacteristic
466D3—204no—
467C5#7649511yescharacteristic, complemented, Sylow
468C22—304no—
469C22—304no—
470C22—304no—
471C22—602no—
472C22—602no—
473C22—602no—
474C22—602no—
475C22D5×#23912yes—
476C22D3×#71312yes—
477C22C22×#120011yescharacteristic
478C22#9204911yescharacteristic
479C22#9204911yescharacteristic
480C4—22no—
481C4—22no—
482C4—22no—
483C4—64no—
484C4—104no—
485C4—122no—
486C4—202no—
487C4—304no—
488C3#1060822611yescharacteristic, complemented, Sylow
489C2—604no—
490C2#1175519911yescharacteristic
491C2#1175520012yes—
492C2#1175520711yescharacteristic
493C2#1175528912yes—
494C2#1175541311yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC22×C3×C5×Q8C22
socle, cosocleC23×C3×C5C24
prime radicalC22×D60C22
hypercenterC22×Q8D3×D5
hypocenterC3×C5#2372
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
2. C5×#5229C2semidirect
3. C3×#65519C2semidirect
5. #11748881C2semidirect
6. #11748875C2semidirect
7. #11748685C2semidirect
8. #11748678C2semidirect
9. #11748653C2semidirect
14. #91836C4semidirect
36. #65519D3semidirect
46. C2×C3×C5×Q8C2×C4semidirect
61. #6714C2×C4semidirect
62. #6714C2×C4semidirect
63. C4×Dic15C2×C4semidirect
75. #5229D5semidirect
165. C2×C5×Q8C4×D3semidirect
224. C2×C3×Q8C4×D5semidirect
372. C2×Q8#1200semidirect
399. C3×C5#2372semidirect
467. C5#76495semidirect
488. C3#10608226semidirect

Maximal subgroups

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

subgroupindex
C2×#918362
C5×#52292
C3×#655192
#117489042
#117488812
#117488752
#117486852
#117486782
#117486532
#106082263
#764955