Finite Group Databasebeta

← #14458348

Subgroups — #14458348

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
1D4×D60C211yesmaximal, characteristic, complemented
2#11752493C211yesmaximal, characteristic, complemented
3C5×#5007C211yesmaximal, characteristic, complemented
4C3×C4×#686C211yesmaximal, characteristic, complemented
5#11749040C212yesmaximal, complemented
6#11748957C212yesmaximal, complemented
7#11747329C212yesmaximal, complemented
8#11747200C212yesmaximal, complemented
9#11747028C211yesmaximal, characteristic, complemented
10#11746988C211yesmaximal, characteristic, complemented
11#11746912C211yesmaximal, characteristic, complemented
12#10612711—31nomaximal, Hall
13C2×D4×D15C2212yes—
14C22×D60—21no—
15C2×#6773—21no—
16C3×C4×C5×D4C2211yescharacteristic
17#91819—21no—
18#91784C2212yes—
19#91771—22no—
20#91763C2211yescharacteristic
21C4×D60C2211yescharacteristic
22C5×#869C2211yescharacteristic, complemented
23C2×C3×#686C2211yescharacteristic
24C3×C4×D20C2211yescharacteristic, complemented
25#91465C2212yescomplemented
26#91437C2212yescomplemented
27#91359—21no—
28#91356—22no—
29C2×#6653C2212yescomplemented
30C2×#6648C2212yescomplemented
31C5×#832C2212yes—
32C5×#807C2212yescomplemented
33C5×C4⋊(C3⋊C8)C2211yescharacteristic, complemented
34C3×#3964C2211yescharacteristic
35C3×#3941C2211yescharacteristic, complemented
36C3×#3939C2211yescharacteristic, complemented
37C3×C4×#168C2211yescharacteristic, complemented
38#90969C2212yescomplemented
39#90969C2212yescomplemented
40#90968C2212yescomplemented
41#90957C2212yescomplemented
42#79608—51nomaximal, Hall
43#66060—31no—
44D4×D20—31no—
45C5×#474—31no—
46#65377—31no—
47C4×#686D311yescharacteristic, complemented
48#65172—32no—
49#65171—32no—
50#65132—31no—
51#65123—31no—
52#65062—32no—
53D5×#102—32no—
54C23×D15—22no—
55C2×C3×C5×D4C2311yescharacteristic
56C22×C3×C4×C5—21no—
57C2×#1220—22no—
58D4×D15—24no—
59D4×D15—42no—
60C2×D60—22no—
61C2×D60—21no—
62C2×D60—21no—
63C2×D60—44no—
64C2×D60D411yescharacteristic
65C2×D60C2312yes—
66C2×C4×D15C2312yes—
67C2×C5×Dic6—21no—
68C2×C3×D20C2311yescharacteristic
69C2×C3×C4×D5—21no—
70C2×#1202—22no—
71C2×#1200—22no—
72#6773—44no—
73C3×C5×C4⋊C4C2311yescharacteristic
74C3×C5×C22⋊C4—21no—
75C3×C42×C5C2311yescharacteristic
76#6714—22no—
77#6714D412yes—
78C4⋊Dic15D411yescharacteristic
79C5×C4⋊Dic3—21no—
80C5×C4⋊Dic3C2312yes—
81C2×C5×C3⋊C8C2312yes—
82C3×#686—22no—
83C3×#685—21no—
84C3×C4⋊Dic5C2311yescharacteristic
85C2×C3×#168C2311yescharacteristic
86C2×C3×#168C2311yescharacteristic
87#6669—22no—
88#6665—22no—
89#6653—42no—
90#6648—42no—
91#5551—51no—
92D4×D12—51no—
93C3×C4×D8—51no—
94#5156—52no—
95#5015—51no—
96#5007D511yescharacteristic, complemented
97#4882—52no—
98#4775—52no—
99D3×#102—52no—
100#4694—51no—
101C8×D12—51no—
102C2×#713—61no—
103C2×D4×D5—32no—
104C22×D20—61no—
105C5×C4⋊Q8—31no—
106C4×C5×D4C2×D311yescharacteristic
107#4092—61no—
108C2×#686C2×D311yescharacteristic
109#4076—61no—
110C2×D40—32no—
111C2×C8×D5—32no—
112#4041—32no—
113#4041—32no—
114#4040—62no—
115D5×C4⋊C4—32no—
116#4028—62no—
117#4020—31no—
118C4×D20—31no—
119C4×D20C2×D311yescharacteristic, complemented
120C5×#107—32no—
121C5×C4⋊C8—31no—
122C5×#102—32no—
123#3964C2×D311yescharacteristic
124#3953—32no—
125#3952—32no—
126#3941—32no—
127#3941—32no—
128#3941C2×D311yescharacteristic, complemented
129#3939C2×D311yescharacteristic, complemented
130C4×#168C2×D311yescharacteristic, complemented
131#3294—151noSylow
132C23×C3×C5—21no—
133C22×D15—22no—
134C22×D15—44no—
135C22×D15—44no—
136C22×D15—82no—
137C22×D15C2×D412yes—
138C22×C3×D5—21no—
139C3×C5×D4—21no—
140C3×C5×D4#4112yescomplemented
141C2×C3×C4×C5—21no—
142C2×C3×C4×C5—42no—
143C2×C3×C4×C5C2×D411yescharacteristic
144C2×C3×C4×C5C2×D411yescharacteristic
145C2×C3×C4×C5C2411yescharacteristic
146#1220—44no—
147C2×Dic15C2×D412yes—
148D60—22no—
149D60—22no—
150D60—44no—
151D60—42no—
152D60—81no—
153D60D814yescomplemented
154C4×D15—22no—
155C4×D15—22no—
156C2×C5×Dic3—22no—
157C5×Dic6—42no—
158C2×C3×Dic5—21no—
159C3×D20—21no—
160C3×D20—21no—
161C3×C4×D5—42no—
162#1202—44no—
163#1200—44no—
164C3×#168—21no—
165C3×#168—21no—
166C5×C3⋊C8—22no—
167C2×D3×D4—52no—
168C2×#235—101no—
169C22×D12—101no—
170C2×C3×D8—51no—
171C3×C4×D4—51no—
172C3×C4×D4C2×D511yescharacteristic
173#938—52no—
174C2×#213—52no—
175#930—101no—
176#924—101no—
177C2×D3×C8—52no—
178#895—52no—
179D3×C4⋊C4—52no—
180#883—102no—
181#882—102no—
182#874—51no—
183C4×D12—51no—
184#869C2×D511yescharacteristic, complemented
185C3×#107—51no—
186C3×#102—51no—
187C3×#102—51no—
188C3×C4×C8—51no—
189#832C2×D512yes—
190#821—52no—
191#821—52no—
192(C2×D3)⋊C8—52no—
193#809—52no—
194#807C2×D512yescomplemented
195C4⋊(C3⋊C8)C2×D511yescharacteristic, complemented
196C23×D5—62no—
197C2×C5×Q8—61no—
198C2×C5×D4C22×D311yescharacteristic
199C22×C4×C5—21no—
200C2×C22⋊D5—62no—
201#713—124no—
202D4×D5—64no—
203D4×D5—122no—
204C2×D20—31no—
205C2×D20—32no—
206C2×D20—62no—
207C2×D20—62no—
208C2×D20—61no—
209C2×D20—61no—
210C2×D20—124no—
211C2×D20C22×D311yescharacteristic
212C2×C4×D5—21no—
213C2×C4×D5—32no—
214C2×C4×D5—62no—
215C2×C5×C8—32no—
216C5×C4⋊C4—32no—
217C5×C4⋊C4—61no—
218C5×C4⋊C4C22×D311yescharacteristic
219C5×C22⋊C4—21no—
220C42×C5C22×D311yescharacteristic
221#686—22no—
222#685—21no—
223#685—32no—
224#685—62no—
225#685—62no—
226C4⋊Dic5—31no—
227C4⋊Dic5C22×D311yescharacteristic
228#683—62no—
229C2×#168C22×D311yescharacteristic
230C2×#168C22×D311yescharacteristic
231D40—122no—
232C8×D5—122no—
233#547—151no—
234D42—151no—
235#486—151no—
236#480—152no—
237#474—31no—
238#466—152no—
239#459—151no—
240#447—152no—
241C4×D8—51no—
242C8×D4—151no—
243C2×#102—152no—
244C22×C3×C5—22no—
245C22×C3×C5—41no—
246C22×C3×C5—41no—
247C22×C3×C5C22×D411yescharacteristic
248C2×D15—22no—
249C2×D15—22no—
250C2×D15—42no—
251C2×D15—42no—
252C2×D15—42no—
253C2×D15—88no—
254C2×D15C2×D814yes—
255C2×C3×D5—41no—
256C2×C3×D5—41no—
257C2×C3×D5—41no—
258C3×C4×C5—21no—
259C3×C4×C5—41no—
260C3×C4×C5C2×D812yes—
261C3×C4×C5C2×#4111yescharacteristic
262C3×C4×C5H(5,𝔽2)11yescharacteristic
263Dic15—22no—
264C3×Dic5—41no—
265C5×Dic3—42no—
266C23×D3—102no—
267C2×C3×D4—51no—
268C2×C3×D4C22×D511yescharacteristic
269C22×C3×C4—21no—
270C22×C3×C4—101no—
271C2×#67—102no—
272C2×#67—102no—
273D3×D4—104no—
274D3×D4—202no—
275#235—204no—
276C2×D12—51no—
277C2×D12—52no—
278C2×D12—102no—
279C2×D12—101no—
280C2×D12—101no—
281C2×D12—204no—
282C2×C4×D3—52no—
283C2×C4×D3—102no—
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—102no—
294#213—202no—
295(C2×D3)⋊C4—52no—
296(C2×D3)⋊C4—102no—
297C4⋊Dic3—21no—
298C4⋊Dic3—51no—
299C4⋊Dic3C22×D512yes—
300Dic3⋊C4—102no—
301C2×C3⋊C8C22×D512yes—
302D3×C8—202no—
303C23×C5—21no—
304C22×D5—21no—
305C22×D5—32no—
306C22×D5—62no—
307C22×D5—124no—
308C22×D5—124no—
309C22×D5—242no—
310C5×Q8—122no—
311C5×D4—21no—
312C5×D4#23512yescomplemented
313C2×C4×C5—21no—
314C2×C4×C5—42no—
315C2×C4×C5—62no—
316C2×C4×C5D3×D411yescharacteristic
317C2×C4×C5D3×D411yescharacteristic
318C2×C4×C5C23×D311yescharacteristic
319C22⋊D5—124no—
320C2×Dic5—21no—
321C2×Dic5—32no—
322D20—21no—
323D20—21no—
324D20—34no—
325D20—62no—
326D20—62no—
327D20—124no—
328D20—124no—
329D20—122no—
330D20—241no—
331C4×D5—42no—
332C4×D5—62no—
333C4×D5—62no—
334C4×D5—124no—
335C5×C8—62no—
336#168—21no—
337#168—21no—
338C2×#41—301no—
339C22×D4—152no—
340C22×D4—301no—
341C2×D8—51no—
342C2×D8—152no—
343C22×C8—152no—
344C4⋊Q8—31no—
345#127—151no—
346#123—302no—
347C22⋊Q8—301no—
348#121—152no—
349#121—152no—
350#121—301no—
351#120—302no—
352C4×D4—51no—
353C4×D4—151no—
354C4×D4D3×D511yescharacteristic
355C2×C4⋊C4—152no—
356#107—32no—
357#107—51no—
358C4⋊C8—31no—
359#102—32no—
360#102—51no—
361#102—51no—
362#102—152no—
363#102—152no—
364#102—152no—
365C22⋊C8—152no—
366C4×C8—51no—
367C2×C3×C5—42no—
368C2×C3×C5#54611yescharacteristic
369C2×C3×C5C22×D811yescharacteristic
370C2×C3×C5#57711yescharacteristic
371D15—24no—
372D15—82no—
373C3×D5—81no—
374C23×C3—21no—
375C23×C3—101no—
376C22×D3—52no—
377C22×D3—102no—
378C22×D3—204no—
379C22×D3—204no—
380C22×D3—402no—
381C3×D4—21no—
382C3×D4—101no—
383C3×D4—101no—
384C3×D4#71312yescomplemented
385C2×C3×C4—21no—
386C2×C3×C4—42no—
387C2×C3×C4—101no—
388C2×C3×C4—202no—
389C2×C3×C4D4×D511yescharacteristic
390C2×C3×C4D4×D511yescharacteristic
391C2×C3×C4C23×D511yescharacteristic
392#67—204no—
393#67—204no—
394C2×Dic3—22no—
395C2×Dic3—52no—
396D12—54no—
397D12—102no—
398D12—102no—
399D12—204no—
400D12—202no—
401D12—401no—
402C4×D3—102no—
403C4×D3—102no—
404C4×D3—204no—
405Dic6—42no—
406C3×C8—101no—
407C3×C8—101no—
408C3⋊C8—22no—
409C22×C5—22no—
410C22×C5—41no—
411C22×C5—41no—
412C22×C5C2×D3×D411yescharacteristic
413C2×D5—34no—
414C2×D5—41no—
415C2×D5—41no—
416C2×D5—41no—
417C2×D5—62no—
418C2×D5—62no—
419C2×D5—122no—
420C2×D5—122no—
421C2×D5—122no—
422C2×D5—248no—
423C4×C5—21no—
424C4×C5—41no—
425C4×C5—122no—
426C4×C5D3×D812yes—
427C4×C5C2×#23511yescharacteristic
428C4×C5#100911yescharacteristic
429Dic5—41no—
430Dic5—62no—
431C24—302no—
432#41—604no—
433C2×Q8—61no—
434C2×D4—51no—
435C2×D4—151no—
436C2×D4—152no—
437C2×D4—302no—
438C2×D4—302no—
439C2×D4—302no—
440C2×D4—304no—
441C2×D4—301no—
442C2×D4—301no—
443C2×D4—604no—
444C2×D4—602no—
445C2×D4C2×D3×D511yescharacteristic
446C22×C4—21no—
447C22×C4—101no—
448C22×C4—152no—
449C22×C4—302no—
450D8—102no—
451D8—602no—
452C2×C8—32no—
453C2×C8—51no—
454C2×C8—51no—
455C2×C8—602no—
456C4⋊C4—32no—
457C4⋊C4—51no—
458C4⋊C4—61no—
459C4⋊C4—151no—
460C4⋊C4—302no—
461C4⋊C4C2×D3×D511yescharacteristic
462C22⋊C4—21no—
463C22⋊C4—101no—
464C22⋊C4—152no—
465C22⋊C4—302no—
466C22⋊C4—302no—
467C42C2×D3×D511yescharacteristic
468C3×C5#329411yescharacteristic, complemented, Hall
469C22×C3—22no—
470C22×C3—41no—
471C22×C3—41no—
472C22×C3—201no—
473C22×C3—201no—
474C22×C3—201no—
475C22×C3C2×D4×D511yescharacteristic
476C2×D3—54no—
477C2×D3—102no—
478C2×D3—102no—
479C2×D3—202no—
480C2×D3—202no—
481C2×D3—202no—
482C2×D3—408no—
483C3×C4—21no—
484C3×C4—41no—
485C3×C4—201no—
486C3×C4D5×D812yes—
487C3×C4C2×#71311yescharacteristic
488C3×C4#414911yescharacteristic
489Dic3—42no—
490Dic3—102no—
491C2×C5—42no—
492C2×C5#555011yescharacteristic
493C2×C5C2×D3×D811yescharacteristic
494C2×C5#577011yescharacteristic
495D5—64no—
496D5—81no—
497D5—242no—
498C23—21no—
499C23—101no—
500C23—152no—
501C23—302no—
502C23—604no—
503C23—604no—
504C23—1202no—
505Q8—122no—
506D4—21no—
507D4—101no—
508D4—101no—
509D4—154no—
510D4—302no—
511D4—302no—
512D4—604no—
513D4—604no—
514D4—604no—
515D4—602no—
516D4—1201no—
517D4#677312yescomplemented
518C2×C4—21no—
519C2×C4—42no—
520C2×C4—62no—
521C2×C4—101no—
522C2×C4—152no—
523C2×C4—202no—
524C2×C4—302no—
525C2×C4—302no—
526C2×C4—604no—
527C2×C4#679011yescharacteristic
528C2×C4#679011yescharacteristic
529C2×C4C22×D3×D511yescharacteristic
530C8—62no—
531C8—101no—
532C8—101no—
533C2×C3—42no—
534C2×C3—401no—
535C2×C3#6596411yescharacteristic
536C2×C3C2×D5×D811yescharacteristic
537C2×C3#6611211yescharacteristic
538D3—104no—
539D3—402no—
540C5#7960811yescharacteristic, complemented, Sylow
541C22—22no—
542C22—41no—
543C22—41no—
544C22—154no—
545C22—201no—
546C22—201no—
547C22—201no—
548C22—302no—
549C22—302no—
550C22—602no—
551C22—602no—
552C22—602no—
553C22—1208no—
554C22C2×#679011yescharacteristic
555C4—21no—
556C4—41no—
557C4—122no—
558C4—201no—
559C4—302no—
560C4#9148112yes—
561C4C2×#677311yescharacteristic
562C4#9201911yescharacteristic
563C3#1061271111yescharacteristic, complemented, Sylow
564C2—42no—
565C2—304no—
566C2—401no—
567C2—1202no—
568C2#1175455811yescharacteristic
569C2C2×#9148111yescharacteristic
570C2#1175535111yescharacteristic
subgroupG/H
centerC22C2×#6790
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. D4×D60C2semidirect
2. #11752493C2semidirect
3. C5×#5007C2semidirect
4. C3×C4×#686C2semidirect
5. #11749040C2semidirect
6. #11748957C2semidirect
7. #11747329C2semidirect
8. #11747200C2semidirect
9. #11747028C2semidirect
10. #11746988C2semidirect
11. #11746912C2semidirect
22. C5×#869C22semidirect
24. C3×C4×D20C22semidirect
25. #91465C22semidirect
26. #91437C22semidirect
29. C2×#6653C22semidirect
30. C2×#6648C22semidirect
32. C5×#807C22semidirect
33. C5×C4⋊(C3⋊C8)C22semidirect
35. C3×#3941C22semidirect
36. C3×#3939C22semidirect
37. C3×C4×#168C22semidirect
38. #90969C22semidirect
39. #90969C22semidirect
40. #90968C22semidirect
41. #90957C22semidirect
47. C4×#686D3semidirect
96. #5007D5semidirect
119. C4×D20C2×D3semidirect
128. #3941C2×D3semidirect
129. #3939C2×D3semidirect
130. C4×#168C2×D3semidirect
140. C3×C5×D4#41semidirect
153. D60D8semidirect
184. #869C2×D5semidirect
194. #807C2×D5semidirect
195. C4⋊(C3⋊C8)C2×D5semidirect
312. C5×D4#235semidirect
384. C3×D4#713semidirect
468. C3×C5#3294semidirect
517. D4#6773semidirect
540. C5#79608semidirect
563. C3#10612711semidirect

Maximal subgroups

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

subgroupindex
D4×D602
#117524932
C5×#50072
C3×C4×#6862
#117490402
#117489572
#117473292
#117472002
#117470282
#117469882
#117469122
#106127113
#796085