Finite Group Databasebeta

← #14458357

Subgroups — #14458357

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
2C3×#65889C211yesmaximal, characteristic, complemented
3C5×D4×C3⋊C8C211yesmaximal, characteristic, complemented
4#11748524C212yesmaximal, complemented
5#11748518C212yesmaximal, complemented
6#11748474C212yesmaximal, complemented
7C2×#90968C212yesmaximal, complemented
8#11747028C211yesmaximal, characteristic, complemented
9C4×#6653C211yesmaximal, characteristic, complemented
10#11746971C211yesmaximal, characteristic, complemented
11#11746910C211yesmaximal, characteristic, complemented
12#10611169—31nomaximal, Hall
13C2×D4×D15—21no—
14C22×D60C2212yes—
15C2×C3×#711—21no—
16C3×C4×C5×D4C2211yescharacteristic
17#91819C2212yes—
18#91784—21no—
19#91771—22no—
20#91763C2211yescharacteristic
21C4×D60C2211yescharacteristic
22C22×C5×C3⋊C8C2212yescomplemented
23C3×#4076C2212yescomplemented
24C2×C3×C4⋊Dic5C2212yescomplemented
25C3×#4043—21no—
26C3×#4032—22no—
27C3×C4×D20C2211yescharacteristic, complemented
28C3×#4015C2211yescharacteristic, complemented
29C2×#6653C2211yescharacteristic
30C2×#6653C2212yescomplemented
31C5×#830C2212yescomplemented
32C5×C4⋊(C3⋊C8)C2211yescharacteristic, complemented
33C4×C5×C3⋊C8C2211yescharacteristic, complemented
34#90990C2211yescharacteristic, complemented
35#90990C2212yescomplemented
36#90968C2211yescharacteristic
37#90968C2212yes—
38#90968C2212yescomplemented
39#90968C2212yescomplemented
40#90968C2212yescomplemented
41#90965C2211yescharacteristic, complemented
42#83525—51nomaximal, Hall
43#65889D311yescharacteristic, complemented
44D4×D20—31no—
45C5×C8×D4—31no—
46#65404—32no—
47C2×#3953—32no—
48#65135—31no—
49#65132—31no—
50#65025—32no—
51#65021—32no—
52#64983—31no—
53C4×D40—31no—
54C23×D15—22no—
55C2×C3×C5×D4D411yescharacteristic
56C22×C3×C4×C5C2312yes—
57C2×#1220—22no—
58D4×D15—44no—
59C2×D60—24no—
60C2×D60—21no—
61C2×D60—22no—
62C2×D60—21no—
63C2×D60—42no—
64C2×D60C2312yes—
65C2×D60C2311yescharacteristic
66C2×C4×D15—21no—
67C2×C3×C22⋊D5—22no—
68C22×C3×Dic5—22no—
69C3×#711—44no—
70C2×C3×D20C2311yescharacteristic
71C2×C3×C4×D5—21no—
72C2×C3×Dic10—21no—
73C3×C5×C4⋊C4D411yescharacteristic
74C3×C5×C22⋊C4D412yes—
75C3×C42×C5C2311yescharacteristic
76#6714—21no—
77#6714—22no—
78C4⋊Dic15C2311yescharacteristic
79C2×C5×C3⋊C8—42no—
80C2×C5×C3⋊C8C2311yescharacteristic
81C2×C5×C3⋊C8C2312yes—
82C2×C5×C3⋊C8C2311yescharacteristic
83C3×#685—21no—
84C3×#685—22no—
85C3×C4⋊Dic5—21no—
86C3×C4⋊Dic5—22no—
87C3×C4⋊Dic5C2312yes—
88C3×C4⋊Dic5C2311yescharacteristic
89#6653—22no—
90#6653—42no—
91C3×#547—51no—
92D4×D12—51no—
93#5088—51no—
94#5087—51no—
95#5038—52no—
96#5036—52no—
97#5015—51no—
98C4×#213—51no—
99D4×C3⋊C8D511yescharacteristic, complemented
100#4967—52no—
101C2×#809—52no—
102C2×#711—21no—
103C2×D4×D5—61no—
104C22×D20—32no—
105C22×C5×C8—32no—
106C4×C5×D4C2×D311yescharacteristic
107#4076—32no—
108#4076C2×D312yescomplemented
109C2×C4⋊Dic5C2×D312yescomplemented
110C2×D40—31no—
111C2×D40—32no—
112#4043—21no—
113#4041—61no—
114#4032—22no—
115#4028—62no—
116#4020—31no—
117C4×D20—31no—
118C4×D20C2×D311yescharacteristic, complemented
119#4015C2×D311yescharacteristic, complemented
120C5×C4⋊C8—31no—
121C5×C22⋊C8—32no—
122C4×C5×C8—31no—
123#3953—31no—
124#3953—32no—
125#3953—32no—
126#3953—32no—
127#3953—31no—
128#3953—32no—
129#3950—31no—
130#3950—32no—
131#3294—151noSylow
132C23×C3×C5C2×D412yes—
133C22×D15—21no—
134C22×D15—22no—
135C22×D15—44no—
136C22×D15—44no—
137C22×D15—82no—
138C22×C3×D5—21no—
139C3×C5×D4D814yescomplemented
140C2×C3×C4×C5—22no—
141C2×C3×C4×C5—22no—
142C2×C3×C4×C5C2×D411yescharacteristic
143C2×C3×C4×C5C2×D412yes—
144C2×C3×C4×C5C2×D411yescharacteristic
145C2×C3×C4×C5C2411yescharacteristic
146#1220—44no—
147C2×Dic15—21no—
148D60—22no—
149D60—21no—
150D60—22no—
151D60—44no—
152D60—42no—
153D60—81no—
154D60#4112yescomplemented
155C4×D15—42no—
156C3×C22⋊D5—44no—
157C2×C3×Dic5—22no—
158C2×C3×Dic5—21no—
159C2×C3×Dic5—44no—
160C3×D20—21no—
161C3×D20—21no—
162C3×C4×D5—42no—
163C3×Dic10—42no—
164C5×C3⋊C8—21no—
165C5×C3⋊C8—21no—
166C5×C3⋊C8—22no—
167C2×C3×#41—101no—
168C2×D3×D4—101no—
169C22×D12—52no—
170C3×C4⋊Q8—51no—
171C3×#123—102no—
172C3×C22⋊Q8—101no—
173C3×#121—52no—
174C3×C4×D4—51no—
175C3×C4×D4C2×D511yescharacteristic
176C2×C3×C4⋊C4—52no—
177C2×#213—51no—
178C2×#213—52no—
179#930—52no—
180C22×C3⋊C8C2×D512yescomplemented
181#895—101no—
182#882—102no—
183#874—51no—
184C4×D12—51no—
185#832—51no—
186#830C2×D512yescomplemented
187#809—51no—
188#809—52no—
189#809—52no—
190#809—52no—
191#809—52no—
192#807—51no—
193#807—52no—
194C4⋊(C3⋊C8)C2×D511yescharacteristic, complemented
195C4×C3⋊C8C2×D511yescharacteristic, complemented
196C23×D5—62no—
197C2×C5×D4#6711yescharacteristic
198C22×C4×C5C22×D312yes—
199C2×C22⋊D5—22no—
200C2×C22⋊D5—62no—
201C22×Dic5—22no—
202#711—44no—
203D4×D5—124no—
204C2×D20—31no—
205C2×D20—32no—
206C2×D20—64no—
207C2×D20—62no—
208C2×D20—61no—
209C2×D20—61no—
210C2×D20—122no—
211C2×D20C22×D311yescharacteristic
212C2×C4×D5—21no—
213C2×C4×D5—61no—
214C2×Dic10—21no—
215C2×C5×C8—31no—
216C2×C5×C8—32no—
217C2×C5×C8—31no—
218C2×C5×C8—122no—
219C5×C4⋊C4#6711yescharacteristic
220C5×C22⋊C4#6712yes—
221C42×C5C22×D311yescharacteristic
222#685—21no—
223#685—22no—
224#685—61no—
225#685—62no—
226C4⋊Dic5—21no—
227C4⋊Dic5—22no—
228C4⋊Dic5—31no—
229C4⋊Dic5C22×D312yes—
230C4⋊Dic5C22×D311yescharacteristic
231D40—62no—
232D40—122no—
233#547—51no—
234D42—151no—
235#486—151no—
236#480—152no—
237#474—151no—
238#466—152no—
239#459—151no—
240#447—152no—
241C4×D8—151no—
242C8×D4—31no—
243C2×#102—152no—
244C22×C3×C5—22no—
245C22×C3×C5—22no—
246C22×C3×C5C2×D814yes—
247C22×C3×C5C22×D411yescharacteristic
248C2×D15—22no—
249C2×D15—41no—
250C2×D15—41no—
251C2×D15—42no—
252C2×D15—42no—
253C2×D15—42no—
254C2×D15—88no—
255C2×C3×D5—41no—
256C2×C3×D5—41no—
257C2×C3×D5—41no—
258C3×C4×C5—22no—
259C3×C4×C5—21no—
260C3×C4×C5C2×D812yes—
261C3×C4×C5C2×#4111yescharacteristic
262C3×C4×C5H(5,𝔽2)11yescharacteristic
263Dic15—41no—
264C3×Dic5—42no—
265C3×Dic5—41no—
266C23×D3—102no—
267C3×#41—204no—
268C2×C3×Q8—101no—
269C2×C3×D4—51no—
270C2×C3×D4—102no—
271C2×C3×D4D2011yescharacteristic
272C22×C3×C4—102no—
273C22×C3×C4—101no—
274C22×C3×C4C22×D512yes—
275C2×#67—102no—
276D3×D4—204no—
277C2×D12—51no—
278C2×D12—52no—
279C2×D12—104no—
280C2×D12—101no—
281C2×D12—102no—
282C2×D12—101no—
283C2×D12—202no—
284C2×C4×D3—101no—
285C3×C4⋊C4—51no—
286C3×C4⋊C4—52no—
287C3×C4⋊C4—102no—
288C3×C4⋊C4—101no—
289C3×C4⋊C4D2011yescharacteristic
290C3×C22⋊C4—101no—
291C3×C22⋊C4—102no—
292C3×C22⋊C4D2012yes—
293C3×C42C22×D511yescharacteristic
294#213—102no—
295#213—202no—
296(C2×D3)⋊C4—101no—
297(C2×D3)⋊C4—102no—
298C4⋊Dic3—51no—
299C2×C3⋊C8—42no—
300C2×C3⋊C8C22×D511yescharacteristic
301C2×C3⋊C8C22×D511yescharacteristic
302C2×C3⋊C8C22×D512yes—
303C23×C5C2×#6712yes—
304C22×D5—21no—
305C22×D5—61no—
306C22×D5—62no—
307C22×D5—124no—
308C22×D5—124no—
309C22×D5—242no—
310C5×D4#21314yescomplemented
311C2×C4×C5—22no—
312C2×C4×C5—22no—
313C2×C4×C5C2×#6711yescharacteristic
314C2×C4×C5C2×#6712yes—
315C2×C4×C5C2×#6711yescharacteristic
316C2×C4×C5C23×D311yescharacteristic
317C22⋊D5—44no—
318C22⋊D5—124no—
319C2×Dic5—22no—
320C2×Dic5—21no—
321C2×Dic5—44no—
322C2×Dic5—61no—
323D20—21no—
324D20—21no—
325D20—32no—
326D20—62no—
327D20—61no—
328D20—62no—
329D20—124no—
330D20—122no—
331D20—241no—
332C4×D5—42no—
333C4×D5—122no—
334Dic10—42no—
335C5×C8—61no—
336C5×C8—62no—
337C5×C8—61no—
338C2×#41—101no—
339C22×D4—152no—
340C22×D4—301no—
341C2×D8—151no—
342C2×D8—152no—
343C22×C8—32no—
344C4⋊Q8—51no—
345#127—151no—
346#123—102no—
347C22⋊Q8—101no—
348#121—52no—
349#121—152no—
350#121—301no—
351#120—302no—
352C4×D4—51no—
353C4×D4—151no—
354C4×D4D3×D511yescharacteristic
355C2×C4⋊C4—52no—
356#107—151no—
357#107—152no—
358C4⋊C8—31no—
359#102—151no—
360#102—152no—
361#102—152no—
362#102—152no—
363#102—152no—
364#102—151no—
365C22⋊C8—32no—
366C4×C8—31no—
367C2×C3×C5—24no—
368C2×C3×C5#54611yescharacteristic
369C2×C3×C5C22×D811yescharacteristic
370C2×C3×C5#57711yescharacteristic
371D15—42no—
372D15—82no—
373C3×D5—81no—
374C23×C3—101no—
375C23×C3C2×D2012yes—
376C22×D3—101no—
377C22×D3—102no—
378C22×D3—204no—
379C22×D3—204no—
380C22×D3—402no—
381C3×Q8—202no—
382C3×D4—101no—
383C3×D4—101no—
384C3×D4—204no—
385C3×D4D4014yescomplemented
386C2×C3×C4—22no—
387C2×C3×C4—22no—
388C2×C3×C4—101no—
389C2×C3×C4—102no—
390C2×C3×C4—204no—
391C2×C3×C4—202no—
392C2×C3×C4C2×D2011yescharacteristic
393C2×C3×C4C2×D2012yes—
394C2×C3×C4C2×D2011yescharacteristic
395C2×C3×C4C23×D511yescharacteristic
396#67—204no—
397C2×Dic3—101no—
398D12—52no—
399D12—101no—
400D12—102no—
401D12—102no—
402D12—204no—
403D12—202no—
404D12—401no—
405C4×D3—202no—
406C3⋊C8—21no—
407C3⋊C8—22no—
408C3⋊C8—21no—
409C22×C5—22no—
410C22×C5—22no—
411C22×C5C2×#21314yes—
412C22×C5C22×#6711yescharacteristic
413C2×D5—41no—
414C2×D5—41no—
415C2×D5—41no—
416C2×D5—62no—
417C2×D5—121no—
418C2×D5—121no—
419C2×D5—122no—
420C2×D5—122no—
421C2×D5—122no—
422C2×D5—248no—
423C4×C5—22no—
424C4×C5—21no—
425C4×C5C2×#21312yes—
426C4×C5D3×#4111yescharacteristic
427C4×C5#100911yescharacteristic
428Dic5—42no—
429Dic5—41no—
430Dic5—121no—
431C24—302no—
432#41—204no—
433C2×Q8—101no—
434C2×D4—51no—
435C2×D4—102no—
436C2×D4—151no—
437C2×D4—152no—
438C2×D4—304no—
439C2×D4—302no—
440C2×D4—302no—
441C2×D4—301no—
442C2×D4—301no—
443C2×D4—602no—
444C2×D4—604no—
445C2×D4#120211yescharacteristic
446C22×C4—102no—
447C22×C4—101no—
448C22×C4—301no—
449C22×C4C2×D3×D512yes—
450D8—302no—
451D8—602no—
452C2×C8—31no—
453C2×C8—31no—
454C2×C8—32no—
455C2×C8—122no—
456C4⋊C4—51no—
457C4⋊C4—52no—
458C4⋊C4—102no—
459C4⋊C4—101no—
460C4⋊C4—151no—
461C4⋊C4#120211yescharacteristic
462C22⋊C4—101no—
463C22⋊C4—102no—
464C22⋊C4—301no—
465C22⋊C4—302no—
466C22⋊C4#120212yes—
467C42C2×D3×D511yescharacteristic
468C3×C5#329411yescharacteristic, complemented, Hall
469C22×C3—22no—
470C22×C3—22no—
471C22×C3—201no—
472C22×C3—201no—
473C22×C3—201no—
474C22×C3C2×D4014yes—
475C22×C3C22×D2011yescharacteristic
476C2×D3—102no—
477C2×D3—201no—
478C2×D3—201no—
479C2×D3—202no—
480C2×D3—202no—
481C2×D3—202no—
482C2×D3—408no—
483C3×C4—21no—
484C3×C4—22no—
485C3×C4—201no—
486C3×C4—202no—
487C3×C4C2×D4012yes—
488C3×C4#414411yescharacteristic
489C3×C4D5×#4111yescharacteristic
490Dic3—201no—
491C2×C5—24no—
492C2×C5C22×#21311yescharacteristic
493C2×C5#582911yescharacteristic
494C2×C5#583711yescharacteristic
495D5—81no—
496D5—122no—
497D5—242no—
498C23—101no—
499C23—301no—
500C23—302no—
501C23—604no—
502C23—604no—
503C23—1202no—
504C23C2×#120212yes—
505Q8—202no—
506D4—101no—
507D4—101no—
508D4—152no—
509D4—204no—
510D4—302no—
511D4—301no—
512D4—302no—
513D4—604no—
514D4—604no—
515D4—602no—
516D4—1201no—
517D4#665314yescomplemented
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—101no—
521C2×C4—102no—
522C2×C4—204no—
523C2×C4—202no—
524C2×C4—301no—
525C2×C4—602no—
526C2×C4C2×#120211yescharacteristic
527C2×C4C2×#120212yes—
528C2×C4C2×#120211yescharacteristic
529C2×C4C22×D3×D511yescharacteristic
530C8—61no—
531C8—61no—
532C8—62no—
533C2×C3—24no—
534C2×C3—401no—
535C2×C3#6588811yescharacteristic
536C2×C3C22×D4011yescharacteristic
537C2×C3#6608511yescharacteristic
538D3—202no—
539D3—402no—
540C5#8352511yescharacteristic, complemented, Sylow
541C22—22no—
542C22—22no—
543C22—201no—
544C22—201no—
545C22—201no—
546C22—302no—
547C22—601no—
548C22—601no—
549C22—602no—
550C22—602no—
551C22—602no—
552C22—1208no—
553C22C2×#665314yes—
554C22C22×#120211yescharacteristic
555C4—21no—
556C4—22no—
557C4—201no—
558C4—202no—
559C4—601no—
560C4C2×#665312yes—
561C4#9202411yescharacteristic
562C4#9202711yescharacteristic
563C3#1061116911yescharacteristic, complemented, Sylow
564C2—24no—
565C2—401no—
566C2—602no—
567C2—1202no—
568C2C22×#665311yescharacteristic
569C2#1175498011yescharacteristic
570C2#1175531011yescharacteristic
subgroupG/H
centerC22C22×#1202
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 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. D4×D60C2semidirect
2. C3×#65889C2semidirect
3. C5×D4×C3⋊C8C2semidirect
4. #11748524C2semidirect
5. #11748518C2semidirect
6. #11748474C2semidirect
7. C2×#90968C2semidirect
8. #11747028C2semidirect
9. C4×#6653C2semidirect
10. #11746971C2semidirect
11. #11746910C2semidirect
22. C22×C5×C3⋊C8C22semidirect
23. C3×#4076C22semidirect
24. C2×C3×C4⋊Dic5C22semidirect
27. C3×C4×D20C22semidirect
28. C3×#4015C22semidirect
30. C2×#6653C22semidirect
31. C5×#830C22semidirect
32. C5×C4⋊(C3⋊C8)C22semidirect
33. C4×C5×C3⋊C8C22semidirect
34. #90990C22semidirect
35. #90990C22semidirect
38. #90968C22semidirect
39. #90968C22semidirect
40. #90968C22semidirect
41. #90965C22semidirect
43. #65889D3semidirect
99. D4×C3⋊C8D5semidirect
108. #4076C2×D3semidirect
109. C2×C4⋊Dic5C2×D3semidirect
118. C4×D20C2×D3semidirect
119. #4015C2×D3semidirect
139. C3×C5×D4D8semidirect
154. D60#41semidirect
180. C22×C3⋊C8C2×D5semidirect
186. #830C2×D5semidirect
194. C4⋊(C3⋊C8)C2×D5semidirect
195. C4×C3⋊C8C2×D5semidirect
310. C5×D4#213semidirect
385. C3×D4D40semidirect
468. C3×C5#3294semidirect
517. D4#6653semidirect
540. C5#83525semidirect
563. C3#10611169semidirect

Maximal subgroups

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

subgroupindex
D4×D602
C3×#658892
C5×D4×C3⋊C82
#117485242
#117485182
#117484742
C2×#909682
#117470282
C4×#66532
#117469712
#117469102
#106111693
#835255