Finite Group Databasebeta

← #14458353

Subgroups — #14458353

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×#5555C211yesmaximal, characteristic, complemented
2C3×D4×D20C211yesmaximal, characteristic, complemented
3D4×C3⋊#168C211yesmaximal, characteristic, complemented
4#11748522C212yesmaximal, complemented
5#11748510C212yesmaximal, complemented
6#11748370C212yesmaximal, complemented
7C2×#90964C212yesmaximal, complemented
8#11747033C211yesmaximal, characteristic, complemented
9C4×#6652C211yesmaximal, characteristic, complemented
10#11746895C211yesmaximal, characteristic, complemented
11#11746886C211yesmaximal, characteristic, complemented
12#10615256—31nomaximal, Hall
13C2×C5×#237—21no—
14C2×C3×D4×D5—21no—
15C22×C3×D20C2212yes—
16C3×C4×C5×D4C2211yescharacteristic
17C22×C3⋊#168C2212yescomplemented
18C5×#930C2212yescomplemented
19C2×C5×C4⋊Dic3C2212yescomplemented
20C5×#897—21no—
21C5×#886—22no—
22C4×C5×D12C2211yescharacteristic, complemented
23C5×#869C2211yescharacteristic, complemented
24C3×#4076C2212yes—
25C3×#4041—21no—
26C3×#4028—22no—
27C3×#4020C2211yescharacteristic
28C3×C4×D20C2211yescharacteristic
29C2×#6652C2211yescharacteristic
30C2×#6652C2212yescomplemented
31#91114C2212yescomplemented
32C4⋊(C3⋊#168)C2211yescharacteristic, complemented
33C4×C3⋊#168C2211yescharacteristic, complemented
34#90989C2211yescharacteristic, complemented
35#90989C2212yescomplemented
36#90966C2211yescharacteristic, complemented
37#90964C2211yescharacteristic
38#90964C2212yes—
39#90964C2212yescomplemented
40#90964C2212yescomplemented
41#90964C2212yescomplemented
42#83482—51nomaximal, Hall
43C5×#547—31no—
44D4×D20D311yescharacteristic, complemented
45#65377—31no—
46#65376—31no—
47#65327—32no—
48#65325—32no—
49#65304—31no—
50C4×#686—31no—
51D4×#168—31no—
52#65256—32no—
53C2×#3941—32no—
54C23×C3×D5—22no—
55C2×C3×C5×D4D411yescharacteristic
56C22×C3×C4×C5C2312yes—
57C2×C5×#67—22no—
58C22×C5×Dic3—22no—
59C5×#237—44no—
60C2×C5×D12C2311yescharacteristic
61C2×C4×C5×D3—21no—
62C2×C5×Dic6—21no—
63C2×C3×C22⋊D5—22no—
64C3×D4×D5—44no—
65C2×C3×D20—24no—
66C2×C3×D20—22no—
67C2×C3×D20—21no—
68C2×C3×D20—21no—
69C2×C3×D20—42no—
70C2×C3×D20C2312yes—
71C2×C3×D20C2311yescharacteristic
72C2×C3×C4×D5—21no—
73C3×C5×C4⋊C4D411yescharacteristic
74C3×C5×C22⋊C4D412yes—
75C3×C42×C5C2311yescharacteristic
76C2×C3⋊#168—42no—
77C2×C3⋊#168C2311yescharacteristic
78C2×C3⋊#168C2312yes—
79C2×C3⋊#168C2311yescharacteristic
80C5×(C2×D3)⋊C4—21no—
81C5×(C2×D3)⋊C4—22no—
82C5×C4⋊Dic3—21no—
83C5×C4⋊Dic3—22no—
84C5×C4⋊Dic3C2312yes—
85C5×C4⋊Dic3C2311yescharacteristic
86C3×#685—21no—
87C3×#685—22no—
88C3×C4⋊Dic5C2311yescharacteristic
89#6652—22no—
90#6652—42no—
91C3×D42—51no—
92#5555D511yescharacteristic, complemented
93#5217—52no—
94C2×#832—52no—
95#5072—51no—
96#5070—51no—
97#5038—52no—
98#5033—52no—
99C4×#213—51no—
100D4×C3⋊C8—51no—
101#5007—51no—
102C2×C5×#41—61no—
103C2×D4×D5—21no—
104C22×D20C2×D312yes—
105C5×C4⋊Q8—31no—
106C5×#123—62no—
107C5×C22⋊Q8—61no—
108C5×#121—32no—
109C4×C5×D4—31no—
110C4×C5×D4C2×D311yescharacteristic
111C2×C5×C4⋊C4—32no—
112C2×#686—31no—
113C2×#686—32no—
114#4076C2×D312yes—
115C22×#168—32no—
116#4041—21no—
117#4028—22no—
118#4020C2×D311yescharacteristic
119C4×D20C2×D311yescharacteristic
120#3964—31no—
121C22⋊#168—32no—
122#3941—31no—
123#3941—32no—
124#3941—32no—
125#3941—32no—
126#3941—32no—
127#3939—31no—
128#3939—32no—
129C4⋊#168—31no—
130C4×#168—31no—
131#3294—151noSylow
132C23×C3×C5C2×D412yes—
133C22×C5×D3—21no—
134C22×C3×D5—21no—
135C22×C3×D5—22no—
136C22×C3×D5—44no—
137C22×C3×D5—44no—
138C22×C3×D5—82no—
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
146C5×#67—44no—
147C2×C5×Dic3—22no—
148C2×C5×Dic3—21no—
149C2×C5×Dic3—44no—
150C5×D12—21no—
151C5×D12—21no—
152C4×C5×D3—42no—
153C5×Dic6—42no—
154C3×C22⋊D5—44no—
155C2×C3×Dic5—21no—
156C3×D20—22no—
157C3×D20—21no—
158C3×D20—22no—
159C3×D20—44no—
160C3×D20—42no—
161C3×D20—81no—
162C3×D20#4112yescomplemented
163C3×C4×D5—42no—
164C3⋊#168—21no—
165C3⋊#168—21no—
166C3⋊#168—22no—
167C22×C3×D4—52no—
168C22×C3×D4—101no—
169C2×#237—21no—
170C3×#127—51no—
171C3×#121—52no—
172C3×#121—101no—
173C3×#120—102no—
174C3×C4×D4—51no—
175C3×C4×D4C2×D511yescharacteristic
176C2×#213—51no—
177C2×#213—52no—
178#930C2×D512yescomplemented
179C2×C4⋊Dic3C2×D512yescomplemented
180C22×C3⋊C8—52no—
181#897—21no—
182#886—22no—
183C4×D12C2×D511yescharacteristic, complemented
184#869C2×D511yescharacteristic, complemented
185#832—51no—
186#832—52no—
187#832—52no—
188#832—52no—
189#832—52no—
190#830—52no—
191#809—51no—
192#807—52no—
193#807—51no—
194C4⋊(C3⋊C8)—51no—
195C4×C3⋊C8—51no—
196C23×D5—22no—
197C5×#41—124no—
198C2×C5×Q8—61no—
199C2×C5×D4—31no—
200C2×C5×D4—62no—
201C2×C5×D4#6711yescharacteristic
202C22×C4×C5—62no—
203C22×C4×C5—61no—
204C22×C4×C5C22×D312yes—
205C2×C22⋊D5—22no—
206D4×D5—44no—
207C2×D20—24no—
208C2×D20—22no—
209C2×D20—21no—
210C2×D20—21no—
211C2×D20—42no—
212C2×D20C22×D312yes—
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⋊C4#6711yescharacteristic
220C5×C22⋊C4—61no—
221C5×C22⋊C4—62no—
222C5×C22⋊C4#6712yes—
223C42×C5C22×D311yescharacteristic
224#686—62no—
225#686—122no—
226#685—21no—
227#685—22no—
228C4⋊Dic5C22×D311yescharacteristic
229C2×#168—31no—
230C2×#168—31no—
231C2×#168—32no—
232C2×#168—122no—
233#547—31no—
234D42—51no—
235#486—151no—
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×C5—22no—
245C22×C3×C5—22no—
246C22×C3×C5C2×D814yes—
247C22×C3×C5C22×D411yescharacteristic
248C2×C5×D3—41no—
249C2×C5×D3—41no—
250C2×C5×D3—41no—
251C2×C3×D5—22no—
252C2×C3×D5—41no—
253C2×C3×D5—41no—
254C2×C3×D5—42no—
255C2×C3×D5—42no—
256C2×C3×D5—42no—
257C2×C3×D5—88no—
258C3×C4×C5—22no—
259C3×C4×C5—21no—
260C3×C4×C5C2×D812yes—
261C3×C4×C5C2×#4111yescharacteristic
262C3×C4×C5H(5,𝔽2)11yescharacteristic
263C3×Dic5—41no—
264C5×Dic3—42no—
265C5×Dic3—41no—
266C24×C3—102no—
267C2×C3×D4—51no—
268C2×C3×D4—52no—
269C2×C3×D4—104no—
270C2×C3×D4—102no—
271C2×C3×D4—102no—
272C2×C3×D4—101no—
273C2×C3×D4—101no—
274C2×C3×D4—202no—
275C2×C3×D4—204no—
276C2×C3×D4C22⋊D511yescharacteristic
277C22×C3×C4—101no—
278C22×C3×C4C22×D512yes—
279C2×#67—22no—
280C22×Dic3—22no—
281#237—44no—
282C2×D12C22×D511yescharacteristic
283C2×C4×D3—21no—
284C2×Dic6—21no—
285C3×C4⋊C4—51no—
286C3×C4⋊C4C22⋊D511yescharacteristic
287C3×C22⋊C4—101no—
288C3×C22⋊C4—102no—
289C3×C22⋊C4C22⋊D512yes—
290C3×C42C22×D511yescharacteristic
291#213—102no—
292#213—202no—
293(C2×D3)⋊C4—21no—
294(C2×D3)⋊C4—22no—
295C4⋊Dic3—21no—
296C4⋊Dic3—22no—
297C4⋊Dic3C22×D512yes—
298C4⋊Dic3C22×D511yescharacteristic
299C2×C3⋊C8—51no—
300C2×C3⋊C8—51no—
301C2×C3⋊C8—52no—
302C2×C3⋊C8—202no—
303C23×C5—61no—
304C23×C5C2×#6712yes—
305C22×D5—21no—
306C22×D5—22no—
307C22×D5—44no—
308C22×D5—44no—
309C22×D5—82no—
310C5×Q8—122no—
311C5×D4—61no—
312C5×D4—61no—
313C5×D4—124no—
314C5×D4#21314yescomplemented
315C2×C4×C5—22no—
316C2×C4×C5—22no—
317C2×C4×C5—61no—
318C2×C4×C5—62no—
319C2×C4×C5—124no—
320C2×C4×C5—122no—
321C2×C4×C5C2×#6711yescharacteristic
322C2×C4×C5C2×#6712yes—
323C2×C4×C5C2×#6711yescharacteristic
324C2×C4×C5C23×D311yescharacteristic
325C22⋊D5—44no—
326C2×Dic5—21no—
327D20—22no—
328D20—21no—
329D20—22no—
330D20—44no—
331D20—42no—
332D20—81no—
333D20#23712yescomplemented
334C4×D5—42no—
335#168—61no—
336#168—61no—
337#168—62no—
338C2×#41—61no—
339C22×D4—52no—
340C22×D4—101no—
341C2×D8—151no—
342C2×D8—152no—
343C22×C8—152no—
344C4⋊Q8—31no—
345#127—51no—
346#123—62no—
347C22⋊Q8—61no—
348#121—32no—
349#121—52no—
350#121—101no—
351#120—102no—
352C4×D4—31no—
353C4×D4—51no—
354C4×D4D3×D511yescharacteristic
355C2×C4⋊C4—32no—
356#107—151no—
357#107—152no—
358C4⋊C8—151no—
359#102—151no—
360#102—152no—
361#102—152no—
362#102—152no—
363#102—152no—
364#102—151no—
365C22⋊C8—152no—
366C4×C8—151no—
367C2×C3×C5—24no—
368C2×C3×C5#54611yescharacteristic
369C2×C3×C5C22×D811yescharacteristic
370C2×C3×C5#57711yescharacteristic
371C3×D5—42no—
372C3×D5—82no—
373C5×D3—81no—
374C23×C3—101no—
375C23×C3—102no—
376C23×C3—204no—
377C23×C3—204no—
378C23×C3—402no—
379C23×C3C2×C22⋊D512yes—
380C22×D3—21no—
381C3×D4—52no—
382C3×D4—102no—
383C3×D4—101no—
384C3×D4—102no—
385C3×D4—204no—
386C3×D4—204no—
387C3×D4—202no—
388C3×D4—401no—
389C3×D4#68614yescomplemented
390C2×C3×C4—22no—
391C2×C3×C4—22no—
392C2×C3×C4—101no—
393C2×C3×C4—202no—
394C2×C3×C4C2×C22⋊D511yescharacteristic
395C2×C3×C4C2×C22⋊D512yes—
396C2×C3×C4C2×C22⋊D511yescharacteristic
397C2×C3×C4C23×D511yescharacteristic
398#67—44no—
399C2×Dic3—22no—
400C2×Dic3—21no—
401C2×Dic3—44no—
402D12—21no—
403D12—21no—
404C4×D3—42no—
405Dic6—42no—
406C3⋊C8—101no—
407C3⋊C8—101no—
408C3⋊C8—102no—
409C22×C5—22no—
410C22×C5—22no—
411C22×C5—121no—
412C22×C5—121no—
413C22×C5—121no—
414C22×C5C2×#21314yes—
415C22×C5C22×#6711yescharacteristic
416C2×D5—22no—
417C2×D5—41no—
418C2×D5—41no—
419C2×D5—42no—
420C2×D5—42no—
421C2×D5—42no—
422C2×D5—88no—
423C4×C5—22no—
424C4×C5—21no—
425C4×C5—121no—
426C4×C5—122no—
427C4×C5C2×#21312yes—
428C4×C5C2×#23711yescharacteristic
429C4×C5#100411yescharacteristic
430Dic5—41no—
431C24—102no—
432#41—124no—
433C2×Q8—61no—
434C2×D4—31no—
435C2×D4—51no—
436C2×D4—52no—
437C2×D4—62no—
438C2×D4—104no—
439C2×D4—102no—
440C2×D4—102no—
441C2×D4—101no—
442C2×D4—101no—
443C2×D4—202no—
444C2×D4—204no—
445C2×D4#120111yescharacteristic
446C22×C4—62no—
447C22×C4—61no—
448C22×C4—101no—
449C22×C4C2×D3×D512yes—
450D8—302no—
451D8—602no—
452C2×C8—151no—
453C2×C8—151no—
454C2×C8—152no—
455C2×C8—602no—
456C4⋊C4—31no—
457C4⋊C4—32no—
458C4⋊C4—51no—
459C4⋊C4—62no—
460C4⋊C4—61no—
461C4⋊C4#120111yescharacteristic
462C22⋊C4—61no—
463C22⋊C4—62no—
464C22⋊C4—101no—
465C22⋊C4—102no—
466C22⋊C4#120112yes—
467C42C2×D3×D511yescharacteristic
468C3×C5#329411yescharacteristic, complemented, Hall
469C22×C3—22no—
470C22×C3—22no—
471C22×C3—102no—
472C22×C3—201no—
473C22×C3—201no—
474C22×C3—202no—
475C22×C3—202no—
476C22×C3—202no—
477C22×C3—408no—
478C22×C3C2×#68614yes—
479C22×C3C22×C22⋊D511yescharacteristic
480C2×D3—41no—
481C2×D3—41no—
482C2×D3—41no—
483C3×C4—22no—
484C3×C4—21no—
485C3×C4—201no—
486C3×C4C2×#68612yes—
487C3×C4D5×#4111yescharacteristic
488C3×C4#414911yescharacteristic
489Dic3—42no—
490Dic3—41no—
491C2×C5—24no—
492C2×C5—241no—
493C2×C5C22×#21311yescharacteristic
494C2×C5#580611yescharacteristic
495C2×C5#583611yescharacteristic
496D5—42no—
497D5—82no—
498C23—61no—
499C23—101no—
500C23—102no—
501C23—204no—
502C23—204no—
503C23—402no—
504C23C2×#120112yes—
505Q8—122no—
506D4—52no—
507D4—61no—
508D4—61no—
509D4—102no—
510D4—101no—
511D4—102no—
512D4—124no—
513D4—204no—
514D4—204no—
515D4—202no—
516D4—401no—
517D4#665214yescomplemented
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—61no—
521C2×C4—62no—
522C2×C4—101no—
523C2×C4—124no—
524C2×C4—122no—
525C2×C4—202no—
526C2×C4C2×#120111yescharacteristic
527C2×C4C2×#120112yes—
528C2×C4C2×#120111yescharacteristic
529C2×C4C22×D3×D511yescharacteristic
530C8—301no—
531C8—301no—
532C8—302no—
533C2×C3—24no—
534C2×C3—202no—
535C2×C3—402no—
536C2×C3C22×#68611yescharacteristic
537C2×C3#6616311yescharacteristic
538C2×C3#6617111yescharacteristic
539D3—81no—
540C5#8348211yescharacteristic, complemented, Sylow
541C22—22no—
542C22—22no—
543C22—102no—
544C22—121no—
545C22—121no—
546C22—121no—
547C22—201no—
548C22—201no—
549C22—202no—
550C22—202no—
551C22—202no—
552C22—408no—
553C22C2×#665214yes—
554C22C22×#120111yescharacteristic
555C4—21no—
556C4—22no—
557C4—121no—
558C4—122no—
559C4—201no—
560C4C2×#665212yes—
561C4D5×#23711yescharacteristic
562C4#9202611yescharacteristic
563C3#1061525611yescharacteristic, complemented, Sylow
564C2—24no—
565C2—202no—
566C2—241no—
567C2—402no—
568C2C22×#665211yescharacteristic
569C2#1175496111yescharacteristic
570C2#1175530311yescharacteristic
subgroupG/H
centerC22C22×#1201
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. C5×#5555C2semidirect
2. C3×D4×D20C2semidirect
3. D4×C3⋊#168C2semidirect
4. #11748522C2semidirect
5. #11748510C2semidirect
6. #11748370C2semidirect
7. C2×#90964C2semidirect
8. #11747033C2semidirect
9. C4×#6652C2semidirect
10. #11746895C2semidirect
11. #11746886C2semidirect
17. C22×C3⋊#168C22semidirect
18. C5×#930C22semidirect
19. C2×C5×C4⋊Dic3C22semidirect
22. C4×C5×D12C22semidirect
23. C5×#869C22semidirect
30. C2×#6652C22semidirect
31. #91114C22semidirect
32. C4⋊(C3⋊#168)C22semidirect
33. C4×C3⋊#168C22semidirect
34. #90989C22semidirect
35. #90989C22semidirect
36. #90966C22semidirect
39. #90964C22semidirect
40. #90964C22semidirect
41. #90964C22semidirect
44. D4×D20D3semidirect
92. #5555D5semidirect
139. C3×C5×D4D8semidirect
162. C3×D20#41semidirect
178. #930C2×D5semidirect
179. C2×C4⋊Dic3C2×D5semidirect
183. C4×D12C2×D5semidirect
184. #869C2×D5semidirect
314. C5×D4#213semidirect
333. D20#237semidirect
389. C3×D4#686semidirect
468. C3×C5#3294semidirect
517. D4#6652semidirect
540. C5#83482semidirect
563. C3#10615256semidirect

Maximal subgroups

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

subgroupindex
C5×#55552
C3×D4×D202
D4×C3⋊#1682
#117485222
#117485102
#117483702
C2×#909642
#117470332
C4×#66522
#117468952
#117468862
#106152563
#834825