Finite Group Databasebeta

← #14458392

Subgroups — #14458392

Order 1920.

988 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5726C211yesmaximal, characteristic, complemented
2#11752745C211yesmaximal, characteristic, complemented
3#11751188C211yesmaximal, characteristic, complemented
4C3×#65377C211yesmaximal, characteristic, complemented
5#11747482C212yesmaximal, complemented
6D3×#3941C212yesmaximal, complemented
7#11747319C212yesmaximal, complemented
8#11747318C212yesmaximal, complemented
9#11747033C211yesmaximal, characteristic, complemented
10C4×#6654C211yesmaximal, characteristic, complemented
11D12×#168C211yesmaximal, characteristic, complemented
12#10615256—31nomaximal, Hall
13C2×C5×#239—21no—
14C2×#6777—21no—
15C2×D3×D20C2212yes—
16C3×C5×C4⋊Q8C2211yescharacteristic, complemented
17C4×D60C2211yescharacteristic
18C5×#946—21no—
19C5×#895C2212yescomplemented
20C5×#894—22no—
21C5×D3×C4⋊C4C2212yescomplemented
22C4×C5×D12C2211yescharacteristic
23C4×C5×D12C2211yescharacteristic, complemented
24C3×#4020C2211yescharacteristic
25#91474—22no—
26#91465—21no—
27#91456C2212yes—
28C2×#6654C2211yescharacteristic
29C2×#6652C2212yescomplemented
30C2×D3×#168C2212yescomplemented
31#91093C2212yes—
32#91091C2212yescomplemented
33C4⋊(C3⋊#168)C2211yescharacteristic, complemented
34C3×#3941C2212yescomplemented
35C3×#3941C2212yescomplemented
36C3×C4×#168C2211yescharacteristic, complemented
37#90988C2211yescharacteristic, complemented
38#90969C2211yescharacteristic
39#90967C2211yescharacteristic, complemented
40#90964C2212yescomplemented
41(C2×D3)⋊#168C2212yescomplemented
42#81576—51nomaximal, Hall
43C5×#547—31no—
44D4×D20—31no—
45#65377D311yescharacteristic, complemented
46#65376—31no—
47#65327—32no—
48#65325—32no—
49#65304—31no—
50C4×#686—31no—
51D4×#168—31no—
52#65256—32no—
53C2×#3941—32no—
54C22×D3×D5—22no—
55C2×C3×C5×Q8—21no—
56C2×D60C2311yescharacteristic
57C2×C4×D15—21no—
58C5×#239—44no—
59C2×C5×D12—22no—
60C2×C5×D12D411yescharacteristic
61C2×C5×D12C2311yescharacteristic
62C2×C4×C5×D3—22no—
63C2×C4×C5×D3—21no—
64C2×C4×C5×D3C2312yes—
65C2×C3×D20—21no—
66C2×C3×D20—21no—
67C2×C3×D20C2312yes—
68C2×#1202—22no—
69C2×#1201—22no—
70#6777—44no—
71D3×D20—24no—
72D3×D20—42no—
73C3×C5×C4⋊C4—21no—
74C3×C5×C4⋊C4C2312yes—
75C3×C42×C5C2311yescharacteristic
76#6714—21no—
77C4⋊Dic15C2311yescharacteristic
78C2×C3⋊#168C2312yes—
79C5×(C2×D3)⋊C4—21no—
80C5×(C2×D3)⋊C4—22no—
81C5×(C2×D3)⋊C4D412yes—
82C5×C4⋊Dic3D411yescharacteristic
83C5×C4⋊Dic3C2311yescharacteristic
84C5×Dic3⋊C4—22no—
85C2×C3×#168C2311yescharacteristic
86C2×C3×#168C2311yescharacteristic
87#6665—22no—
88#6654—22no—
89#6652—42no—
90D3×#168—42no—
91#5726D511yescharacteristic, complemented
92#5717—51no—
93C3×#486—51no—
94#5087—51no—
95#5072—51no—
96#4775—52no—
97#4774—52no—
98#4773—52no—
99D3×#102—52no—
100C4×D24—51no—
101C8×D12—51no—
102C2×C5×#41—61no—
103C2×D4×D5—61no—
104C22×D20—32no—
105C5×C4⋊Q8C2×D311yescharacteristic, complemented
106C5×#123—62no—
107C5×C22⋊Q8—61no—
108C5×#121—32no—
109C4×C5×D4—31no—
110C4×C5×D4—31no—
111C2×C5×C4⋊C4—32no—
112C2×#686—32no—
113C2×#686—31no—
114#4076—32no—
115C22×#168—32no—
116#4041—61no—
117#4028—62no—
118#4020C2×D311yescharacteristic
119C4×D20—31no—
120#3964—31no—
121C22⋊#168—32no—
122#3941—32no—
123#3941—32no—
124#3941—31no—
125#3941C2×D312yescomplemented
126#3941C2×D312yescomplemented
127#3939—32no—
128#3939—31no—
129C4⋊#168—31no—
130C4×#168C2×D311yescharacteristic, complemented
131#3294—151noSylow
132C22×D15—21no—
133C22×C5×D3—21no—
134C22×C5×D3C2×D412yes—
135C22×C3×D5—22no—
136C2×D3×D5—44no—
137C2×D3×D5—44no—
138C2×D3×D5—82no—
139C3×C5×Q8—42no—
140C2×C3×C4×C5—22no—
141C2×C3×C4×C5C2×D411yescharacteristic
142C2×C3×C4×C5C2×D411yescharacteristic
143C2×C3×C4×C5C2411yescharacteristic
144C2×Dic15—21no—
145D60—21no—
146D60#4112yescomplemented
147C4×D15—42no—
148C2×C5×Dic3—21no—
149C2×C5×Dic3C2×D412yes—
150C5×D12—21no—
151C5×D12—21no—
152C5×D12—44no—
153C5×D12D814yescomplemented
154C4×C5×D3—22no—
155C4×C5×D3—22no—
156C4×C5×D3—44no—
157C4×C5×D3—42no—
158C3×D20—22no—
159C3×D20—22no—
160C3×D20—42no—
161C3×D20—81no—
162#1202—44no—
163#1201—44no—
164C3⋊#168—22no—
165C3×#168—21no—
166C3×#168—21no—
167C2×#239—21no—
168C2×D3×D4—52no—
169C2×D3×D4—101no—
170C3×C4⋊Q8C2×D511yescharacteristic, complemented
171C3×#127—51no—
172#946—21no—
173#938—52no—
174#938—101no—
175#937—102no—
176C2×#213—52no—
177C2×D24—51no—
178C2×D3×C8—52no—
179#895C2×D512yescomplemented
180#894—22no—
181D3×C4⋊C4C2×D512yescomplemented
182C4×D12—51no—
183C4×D12C2×D511yescharacteristic
184C4×D12C2×D511yescharacteristic, complemented
185C3×#102—52no—
186C3×#102—52no—
187C3×C4×C8—51no—
188#832—52no—
189#821—51no—
190#821—51no—
191(C2×D3)⋊C8—52no—
192#818—51no—
193#809—52no—
194#807—52no—
195C4⋊(C3⋊C8)—51no—
196C23×D5—62no—
197C5×#41—124no—
198C2×C5×Q8—21no—
199C2×C5×D4—31no—
200C2×C5×D4—31no—
201C2×C5×D4—62no—
202C22×C4×C5—32no—
203C22×C4×C5—62no—
204C22×C4×C5—61no—
205C2×C22⋊D5—62no—
206D4×D5—124no—
207C2×D20—21no—
208C2×D20—21no—
209C2×D20—31no—
210C2×D20—64no—
211C2×D20—62no—
212C2×D20—122no—
213C2×D20C22×D312yes—
214C2×C4×D5—61no—
215C5×C4⋊C4—21no—
216C5×C4⋊C4—31no—
217C5×C4⋊C4—31no—
218C5×C4⋊C4—62no—
219C5×C4⋊C4C22×D312yes—
220C5×C22⋊C4—32no—
221C5×C22⋊C4—62no—
222C5×C22⋊C4—61no—
223C42×C5C22×D311yescharacteristic
224#686—62no—
225#686—122no—
226#685—62no—
227#685—61no—
228C4⋊Dic5—31no—
229C2×#168—32no—
230C2×#168—122no—
231C2×#168C22×D311yescharacteristic
232C2×#168C22×D311yescharacteristic
233#547—31no—
234D42—151no—
235#486—51no—
236#480—152no—
237#474—151no—
238#466—152no—
239#459—151no—
240#447—152no—
241C4×D8—151no—
242C8×D4—151no—
243C2×#102—152no—
244C22×C3×C5C22×D411yescharacteristic
245C2×D15—22no—
246C2×D15—41no—
247C2×D15—41no—
248C2×C5×D3—22no—
249C2×C5×D3—22no—
250C2×C5×D3—41no—
251C2×C5×D3—41no—
252C2×C5×D3—41no—
253C2×C5×D3C2×D814yes—
254C2×C3×D5—42no—
255C2×C3×D5—42no—
256C2×C3×D5—42no—
257D3×D5—88no—
258C3×C4×C5—21no—
259C3×C4×C5—42no—
260C3×C4×C5C2×D812yes—
261C3×C4×C5C2×#4111yescharacteristic
262C3×C4×C5H(5,𝔽2)11yescharacteristic
263Dic15—41no—
264C5×Dic3—22no—
265C5×Dic3—41no—
266C23×D3—102no—
267C2×C3×Q8—21no—
268C2×C3×D4—52no—
269C2×C3×D4—101no—
270C2×C3×D4—101no—
271C2×#67—102no—
272C2×#67—102no—
273#239—44no—
274D3×D4—104no—
275D3×D4—202no—
276D3×D4—204no—
277C2×D12—22no—
278C2×D12—51no—
279C2×D12C22⋊D511yescharacteristic
280C2×D12C22×D511yescharacteristic
281C2×C4×D3—22no—
282C2×C4×D3—21no—
283C2×C4×D3—101no—
284C2×C4×D3C22×D512yes—
285C2×C3×C8—51no—
286C2×C3×C8—51no—
287C3×C4⋊C4—21no—
288C3×C4⋊C4C22×D512yes—
289C3×C42C22×D511yescharacteristic
290#217—102no—
291#213—202no—
292(C2×D3)⋊C4—21no—
293(C2×D3)⋊C4—22no—
294(C2×D3)⋊C4—101no—
295(C2×D3)⋊C4C22⋊D512yes—
296C4⋊Dic3—51no—
297C4⋊Dic3C22⋊D511yescharacteristic
298C4⋊Dic3C22×D511yescharacteristic
299Dic3⋊C4—22no—
300C2×C3⋊C8—52no—
301D24—102no—
302D3×C8—202no—
303C23×C5—32no—
304C23×C5—61no—
305C22×D5—22no—
306C22×D5—61no—
307C22×D5—124no—
308C22×D5—124no—
309C22×D5—242no—
310C5×Q8—42no—
311C5×D4—34no—
312C5×D4—61no—
313C5×D4—61no—
314C5×D4—124no—
315C2×C4×C5—22no—
316C2×C4×C5—32no—
317C2×C4×C5—62no—
318C2×C4×C5—62no—
319C2×C4×C5—61no—
320C2×C4×C5—124no—
321C2×C4×C5—122no—
322C2×C4×C5D3×D411yescharacteristic
323C2×C4×C5D3×D411yescharacteristic
324C2×C4×C5C23×D311yescharacteristic
325C22⋊D5—124no—
326C2×Dic5—61no—
327D20—22no—
328D20—22no—
329D20—32no—
330D20—42no—
331D20—61no—
332D20—81no—
333D20—124no—
334C4×D5—122no—
335#168—21no—
336#168—21no—
337#168—62no—
338C2×#41—61no—
339C22×D4—152no—
340C22×D4—301no—
341C2×D8—151no—
342C2×D8—152no—
343C22×C8—152no—
344C4⋊Q8D3×D511yescharacteristic, complemented
345#127—51no—
346#123—62no—
347C22⋊Q8—61no—
348#121—32no—
349#121—152no—
350#121—301no—
351#120—302no—
352C4×D4—31no—
353C4×D4—31no—
354C4×D4—151no—
355C2×C4⋊C4—32no—
356#107—151no—
357#107—152no—
358C4⋊C8—151no—
359#102—52no—
360#102—52no—
361#102—151no—
362#102—152no—
363#102—151no—
364#102—152no—
365C22⋊C8—152no—
366C4×C8—51no—
367C2×C3×C5#54611yescharacteristic
368C2×C3×C5C22×D811yescharacteristic
369C2×C3×C5#57711yescharacteristic
370D15—42no—
371C3×D5—82no—
372C5×D3—24no—
373C5×D3—81no—
374C23×C3—102no—
375C22×D3—21no—
376C22×D3—101no—
377C22×D3—204no—
378C22×D3—204no—
379C22×D3—402no—
380C22×D3C2×C22⋊D512yes—
381C3×Q8—42no—
382C3×D4—102no—
383C3×D4—102no—
384C3×D4—202no—
385C3×D4—401no—
386C2×C3×C4—22no—
387C2×C3×C4C2×C22⋊D511yescharacteristic
388C2×C3×C4C2×C22⋊D511yescharacteristic
389C2×C3×C4C23×D511yescharacteristic
390#67—204no—
391#67—204no—
392C2×Dic3—21no—
393C2×Dic3—101no—
394C2×Dic3C2×C22⋊D512yes—
395D12—21no—
396D12—21no—
397D12—44no—
398D12—52no—
399D12—101no—
400D12#68614yescomplemented
401C4×D3—22no—
402C4×D3—22no—
403C4×D3—44no—
404C4×D3—42no—
405C4×D3—202no—
406C3×C8—101no—
407C3×C8—101no—
408C3⋊C8—102no—
409C22×C5—34no—
410C22×C5—62no—
411C22×C5—62no—
412C22×C5—121no—
413C22×C5—121no—
414C22×C5—121no—
415C22×C5C2×D3×D411yescharacteristic
416C2×D5—42no—
417C2×D5—42no—
418C2×D5—42no—
419C2×D5—62no—
420C2×D5—121no—
421C2×D5—121no—
422C2×D5—248no—
423C4×C5—21no—
424C4×C5—42no—
425C4×C5—62no—
426C4×C5—121no—
427C4×C5D3×D812yes—
428C4×C5D3×#4111yescharacteristic
429C4×C5#100911yescharacteristic
430Dic5—121no—
431C24—302no—
432#41—124no—
433C2×Q8—21no—
434C2×D4—31no—
435C2×D4—31no—
436C2×D4—52no—
437C2×D4—62no—
438C2×D4—101no—
439C2×D4—101no—
440C2×D4—151no—
441C2×D4—302no—
442C2×D4—304no—
443C2×D4—302no—
444C2×D4—602no—
445C2×D4—604no—
446C22×C4—32no—
447C22×C4—62no—
448C22×C4—61no—
449C22×C4—301no—
450D8—302no—
451D8—602no—
452C2×C8—51no—
453C2×C8—51no—
454C2×C8—152no—
455C2×C8—602no—
456C4⋊C4—21no—
457C4⋊C4—31no—
458C4⋊C4—31no—
459C4⋊C4—62no—
460C4⋊C4—151no—
461C4⋊C4C2×D3×D512yes—
462C22⋊C4—32no—
463C22⋊C4—62no—
464C22⋊C4—61no—
465C22⋊C4—302no—
466C22⋊C4—301no—
467C42C2×D3×D511yescharacteristic
468C3×C5#329411yescharacteristic, complemented, Hall
469C22×C3—202no—
470C22×C3—202no—
471C22×C3—202no—
472C22×C3C22×C22⋊D511yescharacteristic
473C2×D3—22no—
474C2×D3—22no—
475C2×D3—41no—
476C2×D3—41no—
477C2×D3—41no—
478C2×D3—102no—
479C2×D3—201no—
480C2×D3—201no—
481C2×D3—408no—
482C2×D3C2×#68614yes—
483C3×C4—21no—
484C3×C4—42no—
485C3×C4C2×#68612yes—
486C3×C4D5×#4111yescharacteristic
487C3×C4#414911yescharacteristic
488Dic3—22no—
489Dic3—41no—
490Dic3—201no—
491C2×C5—64no—
492C2×C5—241no—
493C2×C5#567611yescharacteristic
494C2×C5C2×D3×D811yescharacteristic
495C2×C5#576211yescharacteristic
496D5—82no—
497D5—122no—
498C23—32no—
499C23—61no—
500C23—102no—
501C23—301no—
502C23—604no—
503C23—604no—
504C23—1202no—
505Q8—42no—
506D4—34no—
507D4—61no—
508D4—61no—
509D4—102no—
510D4—102no—
511D4—124no—
512D4—152no—
513D4—202no—
514D4—301no—
515D4—401no—
516D4—604no—
517D4—604no—
518C2×C4—22no—
519C2×C4—32no—
520C2×C4—62no—
521C2×C4—62no—
522C2×C4—61no—
523C2×C4—124no—
524C2×C4—122no—
525C2×C4—301no—
526C2×C4—602no—
527C2×C4D3×C22⋊D511yescharacteristic
528C2×C4D3×C22⋊D511yescharacteristic
529C2×C4C22×D3×D511yescharacteristic
530C8—101no—
531C8—101no—
532C8—302no—
533C2×C3—402no—
534C2×C3C22×#68611yescharacteristic
535C2×C3#6616311yescharacteristic
536C2×C3#6617111yescharacteristic
537D3—24no—
538D3—81no—
539D3—202no—
540C5#8157611yescharacteristic, complemented, Sylow
541C22—34no—
542C22—62no—
543C22—62no—
544C22—121no—
545C22—121no—
546C22—121no—
547C22—202no—
548C22—202no—
549C22—202no—
550C22—302no—
551C22—601no—
552C22—601no—
553C22—1208no—
554C22C2×D3×C22⋊D511yescharacteristic
555C4—21no—
556C4—42no—
557C4—62no—
558C4—121no—
559C4—601no—
560C4D3×#68612yes—
561C4#9203411yescharacteristic
562C4#9203511yescharacteristic
563C3#1061525611yescharacteristic, complemented, Sylow
564C2—64no—
565C2—241no—
566C2—402no—
567C2—602no—
568C2C2×D3×#68611yescharacteristic
569C2#1175509111yescharacteristic
570C2#1175528011yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊D5
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C5×C4⋊Q8C22
socleC22×C3×C5C22×D4
hypercenterC4⋊Q8D3×D5
hypocenterC3×C5#3294
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C5×#5726C2semidirect
2. #11752745C2semidirect
3. #11751188C2semidirect
4. C3×#65377C2semidirect
5. #11747482C2semidirect
6. D3×#3941C2semidirect
7. #11747319C2semidirect
8. #11747318C2semidirect
9. #11747033C2semidirect
10. C4×#6654C2semidirect
11. D12×#168C2semidirect
16. C3×C5×C4⋊Q8C22semidirect
19. C5×#895C22semidirect
21. C5×D3×C4⋊C4C22semidirect
23. C4×C5×D12C22semidirect
29. C2×#6652C22semidirect
30. C2×D3×#168C22semidirect
32. #91091C22semidirect
33. C4⋊(C3⋊#168)C22semidirect
34. C3×#3941C22semidirect
35. C3×#3941C22semidirect
36. C3×C4×#168C22semidirect
37. #90988C22semidirect
39. #90967C22semidirect
40. #90964C22semidirect
41. (C2×D3)⋊#168C22semidirect
45. #65377D3semidirect
91. #5726D5semidirect
105. C5×C4⋊Q8C2×D3semidirect
125. #3941C2×D3semidirect
126. #3941C2×D3semidirect
130. C4×#168C2×D3semidirect
146. D60#41semidirect
153. C5×D12D8semidirect
170. C3×C4⋊Q8C2×D5semidirect
179. #895C2×D5semidirect
181. D3×C4⋊C4C2×D5semidirect
184. C4×D12C2×D5semidirect
344. C4⋊Q8D3×D5semidirect
400. D12#686semidirect
468. C3×C5#3294semidirect
540. C5#81576semidirect
563. C3#10615256semidirect

Maximal subgroups

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

subgroupindex
C5×#57262
#117527452
#117511882
C3×#653772
#117474822
D3×#39412
#117473192
#117473182
#117470332
C4×#66542
D12×#1682
#106152563
#815765