Finite Group Databasebeta

← #14458403

Subgroups — #14458403

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×#5551C211yesmaximal, characteristic, complemented
2D12×D20C211yesmaximal, characteristic, complemented
3#11751116C211yesmaximal, characteristic, complemented
4C3×C4×#686C211yesmaximal, characteristic, complemented
5#11749152C212yesmaximal, complemented
6#11749034C212yesmaximal, complemented
7D3×#3941C212yesmaximal, complemented
8#11747172C212yesmaximal, complemented
9D12×#168C211yesmaximal, characteristic, complemented
10#11746912C211yesmaximal, characteristic, complemented
11#11746886C211yesmaximal, characteristic, complemented
12#10615256—31nomaximal, Hall
13C2×C5×#235—21no—
14C2×D3×D20C2212yes—
15C2×D5×D12—21no—
16C3×C4×C5×D4C2211yescharacteristic, complemented
17C2×#6715C2212yescomplemented
18#91763C2211yescharacteristic
19C5×#938C2212yescomplemented
20C5×#924—21no—
21C5×D3×C4⋊C4C2212yescomplemented
22C5×#883—22no—
23C4×C5×D12C2211yescharacteristic
24C5×#869C2211yescharacteristic, complemented
25C2×C3×#686C2211yescharacteristic
26C3×C4×D20C2211yescharacteristic
27#91476—22no—
28#91460C2212yes—
29#91450—21no—
30C2×D3×#168C2212yescomplemented
31#91093C2212yescomplemented
32C4⋊(C3⋊#168)C2211yescharacteristic, complemented
33C3×#3964C2211yescharacteristic, complemented
34C3×#3941C2211yescharacteristic
35C3×#3939C2211yescharacteristic, complemented
36C3×C4×#168C2211yescharacteristic, complemented
37#90989C2212yescomplemented
38#90969C2212yescomplemented
39#90969C2212yescomplemented
40#90964C2212yes—
41(C2×D3)⋊#168C2212yescomplemented
42#79608—51nomaximal, Hall
43C5×#547—31no—
44D4×D20—31no—
45#65377—31no—
46#65376—31no—
47#65327—32no—
48#65325—32no—
49#65304—31no—
50C4×#686D311yescharacteristic, complemented
51D4×#168—31no—
52#65256—32no—
53C2×#3941—32no—
54C22×D3×D5—22no—
55C2×C3×C5×D4C2311yescharacteristic
56C22×C3×C4×C5—21no—
57C2×D60—21no—
58C2×D60—21no—
59C2×D60C2312yes—
60C2×C5×#67—22no—
61C5×#235—44no—
62C2×C5×D12D411yescharacteristic
63C2×C4×C5×D3—22no—
64C2×C4×C5×D3C2312yes—
65C2×C5×Dic6—21no—
66C2×C3×D20C2311yescharacteristic
67C2×C3×C4×D5—21no—
68C2×#1203—22no—
69C2×#1202—22no—
70D3×D20—24no—
71D3×D20—42no—
72D5×D12—44no—
73C3×C5×C4⋊C4C2311yescharacteristic
74C3×C5×C22⋊C4—21no—
75C3×C42×C5C2311yescharacteristic
76#6715—42no—
77C2×C3⋊#168C2312yes—
78C5×#217—22no—
79C5×(C2×D3)⋊C4D412yes—
80C5×C4⋊Dic3—21no—
81C5×C4⋊Dic3D411yescharacteristic
82C5×C4⋊Dic3C2312yes—
83C5×Dic3⋊C4—22no—
84C3×#686—22no—
85C3×#685—21no—
86C3×C4⋊Dic5C2311yescharacteristic
87C2×C3×#168C2311yescharacteristic
88C2×C3×#168C2311yescharacteristic
89#6667—22no—
90D3×#168—42no—
91#5551D511yescharacteristic, complemented
92D4×D12—51no—
93C3×C4×D8—51no—
94#5156—52no—
95#5015—51no—
96#5007—51no—
97#4882—52no—
98#4775—52no—
99D3×#102—52no—
100#4694—51no—
101C8×D12—51no—
102C2×C5×#41—61no—
103C2×D4×D5—61no—
104C22×D20—32no—
105C5×C4⋊Q8—31no—
106C5×#123—62no—
107C5×C22⋊Q8—61no—
108C5×#121—32no—
109C4×C5×D4—31no—
110C4×C5×D4C2×D311yescharacteristic, complemented
111C2×C5×C4⋊C4—32no—
112C2×#686—32no—
113C2×#686C2×D311yescharacteristic
114#4076—32no—
115C22×#168—32no—
116#4041—61no—
117#4028—62no—
118#4020—31no—
119C4×D20C2×D311yescharacteristic
120#3964C2×D311yescharacteristic, complemented
121C22⋊#168—32no—
122#3941—32no—
123#3941—32no—
124#3941—32no—
125#3941—32no—
126#3941C2×D311yescharacteristic
127#3939—32no—
128#3939C2×D311yescharacteristic, complemented
129C4⋊#168—31no—
130C4×#168C2×D311yescharacteristic, complemented
131#3294—151noSylow
132C23×C3×C5—21no—
133C22×D15—22no—
134C22×C5×D3C2×D412yes—
135C22×C3×D5—21no—
136C2×D3×D5—44no—
137C2×D3×D5—44no—
138C2×D3×D5—82no—
139C3×C5×D4—21no—
140C3×C5×D4—21no—
141C2×C3×C4×C5—21no—
142C2×C3×C4×C5—42no—
143C2×C3×C4×C5C2×D411yescharacteristic
144C2×C3×C4×C5C2×D411yescharacteristic
145C2×C3×C4×C5C2411yescharacteristic
146D60—22no—
147D60—22no—
148D60—42no—
149D60—81no—
150C5×#67—44no—
151C2×C5×Dic3—22no—
152C2×C5×Dic3C2×D412yes—
153C5×D12D814yescomplemented
154C4×C5×D3—22no—
155C4×C5×D3—22no—
156C4×C5×D3—44no—
157C5×Dic6—42no—
158C2×C3×Dic5—21no—
159C3×D20—21no—
160C3×D20#4112yescomplemented
161C3×C4×D5—42no—
162#1203—44no—
163#1202—44no—
164C3⋊#168—22no—
165C3×#168—21no—
166C3×#168—21no—
167C2×D3×D4—52no—
168C2×#235—21no—
169C22×D12—101no—
170C2×C3×D8—51no—
171C3×C4×D4—51no—
172C3×C4×D4C2×D511yescharacteristic, complemented
173#938C2×D512yescomplemented
174C2×#213—52no—
175#930—101no—
176#924—21no—
177C2×D3×C8—52no—
178#895—52no—
179D3×C4⋊C4C2×D512yescomplemented
180#883—22no—
181#882—102no—
182#874—51no—
183C4×D12C2×D511yescharacteristic
184#869C2×D511yescharacteristic, complemented
185C3×#107—51no—
186C3×#102—51no—
187C3×#102—51no—
188C3×C4×C8—51no—
189#832—52no—
190#821—52no—
191#821—52no—
192(C2×D3)⋊C8—52no—
193#809—52no—
194#807—52no—
195C4⋊(C3⋊C8)—51no—
196C23×D5—62no—
197C5×#41—124no—
198C2×C5×Q8—61no—
199C2×C5×D4—31no—
200C2×C5×D4—62no—
201C2×C5×D4C22×D311yescharacteristic
202C22×C4×C5—21no—
203C22×C4×C5—32no—
204C22×C4×C5—62no—
205C2×C22⋊D5—62no—
206D4×D5—124no—
207C2×D20—32no—
208C2×D20—64no—
209C2×D20—62no—
210C2×D20—61no—
211C2×D20—61no—
212C2×D20—122no—
213C2×D20C22×D311yescharacteristic
214C2×C4×D5—21no—
215C5×C4⋊C4—31no—
216C5×C4⋊C4—32no—
217C5×C4⋊C4—62no—
218C5×C4⋊C4—61no—
219C5×C4⋊C4C22×D311yescharacteristic
220C5×C22⋊C4—21no—
221C5×C22⋊C4—32no—
222C5×C22⋊C4—62no—
223C42×C5C22×D311yescharacteristic
224#686—22no—
225#686—122no—
226#685—21no—
227#685—62no—
228C4⋊Dic5C22×D311yescharacteristic
229C2×#168—32no—
230C2×#168—122no—
231C2×#168C22×D311yescharacteristic
232C2×#168C22×D311yescharacteristic
233#547—31no—
234D42—151no—
235#486—151no—
236#480—152no—
237#474—151no—
238#466—152no—
239#459—151no—
240#447—152no—
241C4×D8—51no—
242C8×D4—151no—
243C2×#102—152no—
244C22×C3×C5—41no—
245C22×C3×C5—41no—
246C22×C3×C5—41no—
247C22×C3×C5C22×D411yescharacteristic
248C2×D15—42no—
249C2×D15—42no—
250C2×D15—42no—
251C2×C5×D3—22no—
252C2×C5×D3—22no—
253C2×C5×D3C2×D814yes—
254C2×C3×D5—22no—
255C2×C3×D5—41no—
256C2×C3×D5—41no—
257D3×D5—88no—
258C3×C4×C5—21no—
259C3×C4×C5—41no—
260C3×C4×C5C2×D812yes—
261C3×C4×C5C2×#4111yescharacteristic
262C3×C4×C5H(5,𝔽2)11yescharacteristic
263C3×Dic5—41no—
264C5×Dic3—22no—
265C5×Dic3—42no—
266C23×D3—102no—
267C2×C3×D4—51no—
268C2×C3×D4C22×D511yescharacteristic
269C22×C3×C4—21no—
270C22×C3×C4—101no—
271C2×#67—22no—
272C2×#67—102no—
273D3×D4—104no—
274D3×D4—202no—
275#235—44no—
276C2×D12—52no—
277C2×D12—102no—
278C2×D12—101no—
279C2×D12—101no—
280C2×D12—204no—
281C2×D12C22⋊D511yescharacteristic
282C2×C4×D3—22no—
283C2×C4×D3C22×D512yes—
284C2×Dic6—21no—
285C3×D8—102no—
286C2×C3×C8—51no—
287C2×C3×C8—51no—
288C3×C4⋊C4—51no—
289C3×C4⋊C4C22×D511yescharacteristic
290C3×C22⋊C4—21no—
291C3×C22⋊C4—101no—
292C3×C42C22×D511yescharacteristic
293#217—22no—
294#213—202no—
295(C2×D3)⋊C4—102no—
296(C2×D3)⋊C4C22⋊D512yes—
297C4⋊Dic3—21no—
298C4⋊Dic3C22⋊D511yescharacteristic
299C4⋊Dic3C22×D512yes—
300Dic3⋊C4—22no—
301C2×C3⋊C8—52no—
302D3×C8—202no—
303C23×C5—21no—
304C23×C5—32no—
305C22×D5—21no—
306C22×D5—62no—
307C22×D5—124no—
308C22×D5—124no—
309C22×D5—242no—
310C5×Q8—122no—
311C5×D4—21no—
312C5×D4—21no—
313C5×D4—34no—
314C5×D4—124no—
315C2×C4×C5—21no—
316C2×C4×C5—32no—
317C2×C4×C5—42no—
318C2×C4×C5—62no—
319C2×C4×C5—62no—
320C2×C4×C5—62no—
321C2×C4×C5—124no—
322C2×C4×C5D3×D411yescharacteristic
323C2×C4×C5D3×D411yescharacteristic
324C2×C4×C5C23×D311yescharacteristic
325C22⋊D5—124no—
326C2×Dic5—21no—
327D20—21no—
328D20—62no—
329D20—62no—
330D20—124no—
331D20—122no—
332D20—241no—
333D20#23512yescomplemented
334C4×D5—42no—
335#168—21no—
336#168—21no—
337#168—62no—
338C2×#41—61no—
339C22×D4—152no—
340C22×D4—301no—
341C2×D8—51no—
342C2×D8—152no—
343C22×C8—152no—
344C4⋊Q8—31no—
345#127—151no—
346#123—62no—
347C22⋊Q8—61no—
348#121—32no—
349#121—152no—
350#121—301no—
351#120—302no—
352C4×D4—31no—
353C4×D4—51no—
354C4×D4D3×D511yescharacteristic, complemented
355C2×C4⋊C4—32no—
356#107—51no—
357#107—152no—
358C4⋊C8—151no—
359#102—51no—
360#102—51no—
361#102—152no—
362#102—152no—
363#102—152no—
364#102—152no—
365C22⋊C8—152no—
366C4×C8—51no—
367C2×C3×C5—81no—
368C2×C3×C5#54611yescharacteristic
369C2×C3×C5C22×D811yescharacteristic
370C2×C3×C5#57711yescharacteristic
371D15—82no—
372C3×D5—42no—
373C5×D3—24no—
374C23×C3—21no—
375C23×C3—101no—
376C22×D3—102no—
377C22×D3—204no—
378C22×D3—204no—
379C22×D3—402no—
380C22×D3C2×C22⋊D512yes—
381C3×D4—21no—
382C3×D4—21no—
383C3×D4—52no—
384C3×D4—101no—
385C2×C3×C4—21no—
386C2×C3×C4—42no—
387C2×C3×C4—101no—
388C2×C3×C4—202no—
389C2×C3×C4C2×C22⋊D511yescharacteristic
390C2×C3×C4C2×C22⋊D511yescharacteristic
391C2×C3×C4C23×D511yescharacteristic
392#67—44no—
393#67—204no—
394C2×Dic3—22no—
395C2×Dic3C2×C22⋊D512yes—
396D12—102no—
397D12—102no—
398D12—204no—
399D12—202no—
400D12—401no—
401D12#68614yescomplemented
402C4×D3—22no—
403C4×D3—22no—
404C4×D3—44no—
405Dic6—42no—
406C3×C8—101no—
407C3×C8—101no—
408C3⋊C8—102no—
409C22×C5—34no—
410C22×C5—41no—
411C22×C5—41no—
412C22×C5—41no—
413C22×C5—62no—
414C22×C5—62no—
415C22×C5C2×D3×D411yescharacteristic
416C2×D5—22no—
417C2×D5—41no—
418C2×D5—41no—
419C2×D5—122no—
420C2×D5—122no—
421C2×D5—122no—
422C2×D5—248no—
423C4×C5—21no—
424C4×C5—41no—
425C4×C5—62no—
426C4×C5—122no—
427C4×C5D3×D812yes—
428C4×C5C2×#23511yescharacteristic
429C4×C5#100911yescharacteristic
430Dic5—41no—
431C24—302no—
432#41—124no—
433C2×Q8—61no—
434C2×D4—31no—
435C2×D4—51no—
436C2×D4—62no—
437C2×D4—152no—
438C2×D4—302no—
439C2×D4—304no—
440C2×D4—302no—
441C2×D4—301no—
442C2×D4—301no—
443C2×D4—602no—
444C2×D4—604no—
445C2×D4C2×D3×D511yescharacteristic
446C22×C4—21no—
447C22×C4—32no—
448C22×C4—62no—
449C22×C4—101no—
450D8—102no—
451D8—602no—
452C2×C8—51no—
453C2×C8—51no—
454C2×C8—152no—
455C2×C8—602no—
456C4⋊C4—31no—
457C4⋊C4—32no—
458C4⋊C4—51no—
459C4⋊C4—62no—
460C4⋊C4—61no—
461C4⋊C4C2×D3×D511yescharacteristic
462C22⋊C4—21no—
463C22⋊C4—32no—
464C22⋊C4—62no—
465C22⋊C4—101no—
466C22⋊C4—302no—
467C42C2×D3×D511yescharacteristic
468C3×C5#329411yescharacteristic, complemented, Hall
469C22×C3—41no—
470C22×C3—41no—
471C22×C3—41no—
472C22×C3—102no—
473C22×C3—201no—
474C22×C3—201no—
475C22×C3C22×C22⋊D511yescharacteristic
476C2×D3—22no—
477C2×D3—22no—
478C2×D3—202no—
479C2×D3—202no—
480C2×D3—202no—
481C2×D3—408no—
482C2×D3C2×#68614yes—
483C3×C4—21no—
484C3×C4—41no—
485C3×C4—201no—
486C3×C4C2×#68612yes—
487C3×C4D5×#4111yescharacteristic
488C3×C4#414911yescharacteristic
489Dic3—22no—
490Dic3—42no—
491C2×C5—64no—
492C2×C5—81no—
493C2×C5#555011yescharacteristic
494C2×C5C2×D3×D811yescharacteristic
495C2×C5#577011yescharacteristic
496D5—42no—
497D5—242no—
498C23—21no—
499C23—32no—
500C23—101no—
501C23—302no—
502C23—604no—
503C23—604no—
504C23—1202no—
505Q8—122no—
506D4—21no—
507D4—21no—
508D4—34no—
509D4—52no—
510D4—101no—
511D4—124no—
512D4—302no—
513D4—302no—
514D4—604no—
515D4—604no—
516D4—602no—
517D4—1201no—
518C2×C4—21no—
519C2×C4—32no—
520C2×C4—42no—
521C2×C4—62no—
522C2×C4—62no—
523C2×C4—62no—
524C2×C4—101no—
525C2×C4—124no—
526C2×C4—202no—
527C2×C4D3×C22⋊D511yescharacteristic
528C2×C4D3×C22⋊D511yescharacteristic
529C2×C4C22×D3×D511yescharacteristic
530C8—101no—
531C8—101no—
532C8—302no—
533C2×C3—81no—
534C2×C3—202no—
535C2×C3C22×#68611yescharacteristic
536C2×C3#6616311yescharacteristic
537C2×C3#6617111yescharacteristic
538D3—24no—
539D3—402no—
540C5#7960811yescharacteristic, complemented, Sylow
541C22—34no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—62no—
546C22—62no—
547C22—102no—
548C22—201no—
549C22—201no—
550C22—602no—
551C22—602no—
552C22—602no—
553C22—1208no—
554C22C2×D3×C22⋊D511yescharacteristic
555C4—21no—
556C4—41no—
557C4—62no—
558C4—122no—
559C4—201no—
560C4D3×#68612yes—
561C4D5×#23511yescharacteristic
562C4#9201911yescharacteristic
563C3#1061525611yescharacteristic, complemented, Sylow
564C2—64no—
565C2—81no—
566C2—202no—
567C2—1202no—
568C2#1175465111yescharacteristic
569C2C2×D3×#68611yescharacteristic
570C2#1175532611yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊D5
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C4×C5×D4C22
socleC22×C3×C5C22×D4
hypercenterC4×D4D3×D5
hypocenterC3×C5#3294
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C5×#5551C2semidirect
2. D12×D20C2semidirect
3. #11751116C2semidirect
4. C3×C4×#686C2semidirect
5. #11749152C2semidirect
6. #11749034C2semidirect
7. D3×#3941C2semidirect
8. #11747172C2semidirect
9. D12×#168C2semidirect
10. #11746912C2semidirect
11. #11746886C2semidirect
16. C3×C4×C5×D4C22semidirect
17. C2×#6715C22semidirect
19. C5×#938C22semidirect
21. C5×D3×C4⋊C4C22semidirect
24. C5×#869C22semidirect
30. C2×D3×#168C22semidirect
31. #91093C22semidirect
32. C4⋊(C3⋊#168)C22semidirect
33. C3×#3964C22semidirect
35. C3×#3939C22semidirect
36. C3×C4×#168C22semidirect
37. #90989C22semidirect
38. #90969C22semidirect
39. #90969C22semidirect
41. (C2×D3)⋊#168C22semidirect
50. C4×#686D3semidirect
91. #5551D5semidirect
110. C4×C5×D4C2×D3semidirect
120. #3964C2×D3semidirect
128. #3939C2×D3semidirect
130. C4×#168C2×D3semidirect
153. C5×D12D8semidirect
160. C3×D20#41semidirect
172. C3×C4×D4C2×D5semidirect
173. #938C2×D5semidirect
179. D3×C4⋊C4C2×D5semidirect
184. #869C2×D5semidirect
333. D20#235semidirect
354. C4×D4D3×D5semidirect
401. D12#686semidirect
468. C3×C5#3294semidirect
540. C5#79608semidirect
563. C3#10615256semidirect

Maximal subgroups

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

subgroupindex
C5×#55512
D12×D202
#117511162
C3×C4×#6862
#117491522
#117490342
D3×#39412
#117471722
D12×#1682
#117469122
#117468862
#106152563
#796085