Finite Group Databasebeta

← #14425262

Subgroups — #14425262

Order 1920.

1448 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5964C211yesmaximal, characteristic, complemented
2C3×#66166C211yesmaximal, characteristic, complemented
3#11754138C212yesmaximal, complemented
4#11751338C211yesmaximal, characteristic, complemented
5#11749393C212yesmaximal, complemented
6#11749386C212yesmaximal, complemented
7#11749147C212yesmaximal, complemented
8#11749140C212yesmaximal, complemented
9C2×#90979C212yesmaximal, complemented
10#10615852—31nomaximal, Hall
11C2×C3×C5×#41C2211yescharacteristic
12C5×#1010—22no—
13C5×#1009—22no—
14C5×D3×#41—24no—
15C5×#1007C2212yes—
16C5×#1004—22no—
17C5×#1004C2212yes—
18C2×C5×#235—22no—
19C2×C5×#235C2212yes—
20C2×C3×#709C2211yescharacteristic, complemented
21C2×#6710C2212yescomplemented
22C3×#4093C2211yescharacteristic, complemented
23C3×#4090C2211yescharacteristic, complemented
24C3×#4086C2211yescharacteristic, complemented
25C3×#4085—22no—
26C3×#4082C2211yescharacteristic, complemented
27C3×C4×C22⋊D5—22no—
28C2×C3×C4×Dic5C2211yescharacteristic, complemented
29#91309C2214yescomplemented
30#91307—22no—
31#91299—22no—
32#91298C2214yescomplemented
33#91297—24no—
34#91120C2212yescomplemented
35#91117C2212yescomplemented
36#90979C2214yescomplemented
37#90979C2214yescomplemented
38#83322—51nomaximal, Hall
39C5×#585—31no—
40#66166D311yescharacteristic, complemented
41#66156—32no—
42#65535—32no—
43#65534—32no—
44#65531—32no—
45#65530—32no—
46#65526—31no—
47C2×#3969—32no—
48C3×C5×#41—22no—
49C3×C5×#41—22no—
50C2×C3×C5×Q8C2311yescharacteristic
51C2×C3×C5×D4D412yes—
52C2×C3×C5×D4C2311yescharacteristic
53C22×C3×C4×C5D412yes—
54C22×C3×C4×C5C2311yescharacteristic
55C2×C5×#67—24no—
56C2×C5×#67—22no—
57C5×#239—24no—
58C5×D3×Q8—24no—
59C5×#237—28no—
60C5×#237—24no—
61C5×D3×D4—28no—
62C5×D3×D4—24no—
63C5×#235—24no—
64C5×#235—24no—
65C5×#235—22no—
66C5×#235—24no—
67C5×#235—22no—
68C5×#235D414yescomplemented
69C5×#235C2314yes—
70C2×C5×D12—22no—
71C2×C5×D12D412yescomplemented
72C2×C4×C5×D3—24no—
73C2×C4×C5×D3—22no—
74C2×C5×Dic6—22no—
75C2×C5×Dic6D412yescomplemented
76C2×C3×C22⋊D5—21no—
77C2×C3×C22⋊D5—22no—
78C22×C3×Dic5—21no—
79C3×#709—22no—
80C3×#709—22no—
81C2×C3×D20C2311yescharacteristic
82C2×C3×C4×D5—21no—
83C2×C3×Dic10C2311yescharacteristic
84#6710—22no—
85#6710C2314yes—
86C2×C3⋊#168—22no—
87C3×C22⋊Dic5—22no—
88C3×#685—22no—
89C3×#683—22no—
90C3×C4×Dic5—21no—
91C3×C4×Dic5C2311yescharacteristic
92C3×C4×Dic5C2311yescharacteristic
93#6661—24no—
94#6656—24no—
95#6655—24no—
96#6652—24no—
97#5964D511yescharacteristic, complemented
98C3×#532—51no—
99#5822—52no—
100#5237—51no—
101#5095—52no—
102#5077—52no—
103#5062—52no—
104#5061—52no—
105C2×#805—52no—
106C5×2−1+4—32no—
107C5×2−1+4—62no—
108C5×H(5,𝔽2)—32no—
109C5×H(5,𝔽2)—62no—
110C5×H(5,𝔽2)—62no—
111C2×C5×#41—32no—
112C2×C5×#41—64no—
113C2×C5×#41—62no—
114C2×C5×#41C2×D311yescharacteristic
115C2×#709C2×D311yescharacteristic, complemented
116#4097—34no—
117#4096—64no—
118#4095—34no—
119#4093C2×D311yescharacteristic, complemented
120#4090C2×D311yescharacteristic, complemented
121#4088—62no—
122#4086C2×D311yescharacteristic, complemented
123#4085—22no—
124#4082C2×D311yescharacteristic, complemented
125#4078—62no—
126C4×C22⋊D5—22no—
127C2×C4×Dic5C2×D311yescharacteristic, complemented
128C2×#681—32no—
129#3969—34no—
130#3969—34no—
131#3968—32no—
132#3965—32no—
133#3018—151noSylow
134C23×C3×C5C2×D412yes—
135C23×C3×C5C2×D411yescharacteristic
136C22×C5×D3—24no—
137C22×C5×D3—22no—
138C22×C3×D5—21no—
139C3×C5×Q8—22no—
140C3×C5×D4—24no—
141C3×C5×D4—22no—
142C2×C3×C4×C5—22no—
143C2×C3×C4×C5—22no—
144C2×C3×C4×C5C2×D412yes—
145C2×C3×C4×C5C2×D411yescharacteristic
146C2×C3×C4×C5C2×D412yes—
147C2×C3×C4×C5C2×D412yes—
148C2×C3×C4×C5C2411yescharacteristic
149C5×#67—28no—
150C5×#67—24no—
151C5×#67—24no—
152C2×C5×Dic3—24no—
153C2×C5×Dic3—22no—
154C5×D12—24no—
155C5×D12—22no—
156C5×D12C2×D414yes—
157C4×C5×D3—28no—
158C4×C5×D3—24no—
159C4×C5×D3—24no—
160C5×Dic6—24no—
161C5×Dic6—22no—
162C5×Dic6C2×D414yes—
163C3×C22⋊D5—44no—
164C3×C22⋊D5—42no—
165C2×C3×Dic5—21no—
166C2×C3×Dic5—21no—
167C2×C3×Dic5—21no—
168C2×C3×Dic5—42no—
169C3×D20—22no—
170C3×C4×D5—41no—
171C3×C4×D5—41no—
172C3×Dic10—22no—
173C3⋊#168—24no—
174C2×C3×#41—51no—
175C2×C3×#41C2×D511yescharacteristic
176#1010—22no—
177#1009—22no—
178D3×#41—24no—
179#1007C2×D512yes—
180#1004—22no—
181#1004C2×D512yes—
182C2×#235—22no—
183C2×#235C2×D512yes—
184C3×C4⋊Q8—51no—
185C3×#127—51no—
186C3×#124—51no—
187C3×#124—51no—
188C3×#121—102no—
189C3×C4×D4—102no—
190C2×C3×C42—51no—
191#951—102no—
192#950—104no—
193#949—102no—
194#942—54no—
195#932—54no—
196C2×#208—52no—
197#836—52no—
198#833—52no—
199#805—54no—
200#805—54no—
201C5×#41—22no—
202C5×#41—22no—
203C5×#41—34no—
204C5×#41—34no—
205C5×#41—64no—
206C5×#41—64no—
207C5×#41—64no—
208C5×#41—68no—
209C5×#41—64no—
210C5×#41—62no—
211C5×#41—62no—
212C5×#41—64no—
213C2×C5×Q8—32no—
214C2×C5×Q8—62no—
215C2×C5×Q8—64no—
216C2×C5×Q8C22×D311yescharacteristic
217C2×C5×D4—32no—
218C2×C5×D4—62no—
219C2×C5×D4—64no—
220C2×C5×D4—68no—
221C2×C5×D4—62no—
222C2×C5×D4—64no—
223C2×C5×D4#6712yes—
224C2×C5×D4C22×D311yescharacteristic
225C22×C4×C5—64no—
226C22×C4×C5—62no—
227C22×C4×C5#6712yes—
228C22×C4×C5C22×D311yescharacteristic
229C2×C22⋊D5—21no—
230C2×C22⋊D5—22no—
231C22×Dic5—21no—
232#709—22no—
233#709—22no—
234C2×D20C22×D311yescharacteristic
235C2×C4×D5—21no—
236C2×Dic10C22×D311yescharacteristic
237C22⋊Dic5—22no—
238#689—64no—
239#688—64no—
240#687—64no—
241#686—64no—
242#685—22no—
243#683—22no—
244C4×Dic5—21no—
245C4×Dic5C22×D311yescharacteristic
246C4×Dic5C22×D311yescharacteristic
247#681—34no—
248#681—62no—
249C2×#168—62no—
250#585—31no—
251#575—152no—
252#532—51no—
253#456—152no—
254#455—152no—
255#454—152no—
256#453—152no—
257C2×#104—152no—
258#413—151no—
259C22×C3×C5—22no—
260C22×C3×C5—41no—
261C22×C3×C5—42no—
262C22×C3×C5#12012yes—
263C22×C3×C5C22×D411yescharacteristic
264C2×C5×D3—24no—
265C2×C5×D3—82no—
266C2×C5×D3—84no—
267C2×C3×D5—42no—
268C2×C3×D5—41no—
269C3×C4×C5—22no—
270C3×C4×C5#12011yescharacteristic
271C3×C4×C5#12011yescharacteristic
272C3×C4×C5C22×D412yes—
273C3×Dic5—22no—
274C3×Dic5—41no—
275C3×Dic5—41no—
276C5×Dic3—24no—
277C3×#41—22no—
278C3×#41—22no—
279C3×#41—102no—
280C3×#41—102no—
281C2×C3×Q8—51no—
282C2×C3×Q8C22×D511yescharacteristic
283C2×C3×D4—51no—
284C2×C3×D4—102no—
285C2×C3×D4—101no—
286C2×C3×D4C22⋊D512yes—
287C2×C3×D4C22×D511yescharacteristic
288C22×C3×C4—101no—
289C22×C3×C4—101no—
290C22×C3×C4C22⋊D512yes—
291C22×C3×C4C22×D511yescharacteristic
292C2×#67—24no—
293C2×#67—22no—
294#239—24no—
295D3×Q8—24no—
296#237—28no—
297#237—24no—
298D3×D4—28no—
299D3×D4—24no—
300#235—24no—
301#235—24no—
302#235—22no—
303#235—24no—
304#235—22no—
305#235C22⋊D514yescomplemented
306#235C22×D514yes—
307C2×D12—22no—
308C2×D12C22⋊D512yescomplemented
309C2×C4×D3—24no—
310C2×C4×D3—22no—
311C2×Dic6—22no—
312C2×Dic6C22⋊D512yescomplemented
313C3×C4⋊C4—102no—
314C3×C22⋊C4—102no—
315C3×C22⋊C4—102no—
316C3×C42—51no—
317C3×C42—51no—
318C3×C42—101no—
319#216—104no—
320#215—104no—
321#214—104no—
322#213—104no—
323#208—54no—
324#208—102no—
325C2×C3⋊C8—102no—
326C23×C5—64no—
327C23×C5—62no—
328C23×C5C2×#6712yes—
329C23×C5C2×#6711yescharacteristic
330C22×D5—21no—
331C5×Q8—22no—
332C5×Q8—34no—
333C5×Q8—64no—
334C5×Q8—62no—
335C5×D4—24no—
336C5×D4—22no—
337C5×D4—34no—
338C5×D4—64no—
339C5×D4—68no—
340C5×D4—64no—
341C5×D4—64no—
342C5×D4—62no—
343C2×C4×C5—22no—
344C2×C4×C5—22no—
345C2×C4×C5—68no—
346C2×C4×C5—64no—
347C2×C4×C5—62no—
348C2×C4×C5—64no—
349C2×C4×C5—64no—
350C2×C4×C5D3×D412yes—
351C2×C4×C5D3×D412yes—
352C2×C4×C5C2×#6712yes—
353C2×C4×C5C2×#6711yescharacteristic
354C2×C4×C5C23×D311yescharacteristic
355C22⋊D5—44no—
356C22⋊D5—42no—
357C2×Dic5—21no—
358C2×Dic5—21no—
359C2×Dic5—21no—
360C2×Dic5—42no—
361D20—22no—
362C4×D5—41no—
363C4×D5—41no—
364Dic10—22no—
365#168—64no—
3662−1+4—32no—
3672−1+4—62no—
368H(5,𝔽2)—32no—
369H(5,𝔽2)—62no—
370H(5,𝔽2)—62no—
371C2×#41—32no—
372C2×#41—51no—
373C2×#41—64no—
374C2×#41—62no—
375C2×#41D3×D511yescharacteristic
376#137—154no—
377#137—302no—
378AGL(1,ℤ/8)—154no—
379AGL(1,ℤ/8)—302no—
380#135—304no—
381C2×M16—152no—
382C4⋊Q8—51no—
383#127—51no—
384#124—51no—
385#124—51no—
386#121—102no—
387C4×D4—102no—
388C2×C42—51no—
389#104—154no—
390#104—154no—
391#101—152no—
392#100—152no—
393C2×C3×C5—21no—
394C2×C3×C5—22no—
395C2×C3×C5—42no—
396C2×C3×C5C2×#12011yescharacteristic
397C3×D5—81no—
398C5×D3—44no—
399C23×C3—101no—
400C23×C3C2×C22⋊D512yes—
401C23×C3C2×C22⋊D511yescharacteristic
402C22×D3—24no—
403C22×D3—22no—
404C3×Q8—22no—
405C3×Q8—102no—
406C3×D4—24no—
407C3×D4—22no—
408C3×D4—102no—
409C3×D4—204no—
410C3×D4—202no—
411C2×C3×C4—22no—
412C2×C3×C4—22no—
413C2×C3×C4—101no—
414C2×C3×C4—101no—
415C2×C3×C4—101no—
416C2×C3×C4—202no—
417C2×C3×C4—201no—
418C2×C3×C4—201no—
419C2×C3×C4C2×C22⋊D512yes—
420C2×C3×C4C2×C22⋊D511yescharacteristic
421C2×C3×C4C2×C22⋊D512yes—
422C2×C3×C4C2×C22⋊D512yes—
423C2×C3×C4C23×D511yescharacteristic
424#67—28no—
425#67—24no—
426#67—24no—
427C2×Dic3—24no—
428C2×Dic3—22no—
429D12—24no—
430D12—22no—
431D12C2×C22⋊D514yes—
432C4×D3—28no—
433C4×D3—24no—
434C4×D3—24no—
435Dic6—24no—
436Dic6—22no—
437Dic6C2×C22⋊D514yes—
438C3⋊C8—104no—
439C22×C5—22no—
440C22×C5—41no—
441C22×C5—42no—
442C22×C5—64no—
443C22×C5—242no—
444C22×C5—244no—
445C22×C5#93712yes—
446C22×C5C22×#6711yescharacteristic
447C2×D5—42no—
448C2×D5—41no—
449C4×C5—22no—
450C4×C5—64no—
451C4×C5#93711yescharacteristic
452C4×C5#93711yescharacteristic
453C4×C5C2×D3×D412yes—
454Dic5—22no—
455Dic5—41no—
456Dic5—41no—
457#41—22no—
458#41—22no—
459#41—34no—
460#41—34no—
461#41—64no—
462#41—64no—
463#41—64no—
464#41—68no—
465#41—64no—
466#41—62no—
467#41—64no—
468#41—62no—
469#41—102no—
470#41—102no—
471C2×Q8—32no—
472C2×Q8—51no—
473C2×Q8—62no—
474C2×Q8—64no—
475C2×Q8C2×D3×D511yescharacteristic
476C2×D4—32no—
477C2×D4—51no—
478C2×D4—62no—
479C2×D4—64no—
480C2×D4—68no—
481C2×D4—62no—
482C2×D4—64no—
483C2×D4—102no—
484C2×D4—101no—
485C2×D4#120112yes—
486C2×D4C2×D3×D511yescharacteristic
487C22×C4—64no—
488C22×C4—62no—
489C22×C4—101no—
490C22×C4—101no—
491C22×C4#120112yes—
492C22×C4C2×D3×D511yescharacteristic
493Q16—304no—
494SD16—304no—
495SD16—304no—
496D8—304no—
497M16—154no—
498M16—302no—
499C2×C8—302no—
500C4⋊C4—102no—
501C22⋊C4—102no—
502C22⋊C4—102no—
503C42—51no—
504C42—51no—
505C42—101no—
506C3×C5#301811yescharacteristic, complemented, Hall
507C22×C3—22no—
508C22×C3—41no—
509C22×C3—42no—
510C22×C3—202no—
511C22×C3—201no—
512C22×C3#409912yes—
513C22×C3C22×C22⋊D511yescharacteristic
514C2×D3—24no—
515C2×D3—82no—
516C2×D3—84no—
517C3×C4—22no—
518C3×C4—102no—
519C3×C4—201no—
520C3×C4—201no—
521C3×C4#409911yescharacteristic
522C3×C4#409911yescharacteristic
523C3×C4C22×C22⋊D512yes—
524Dic3—24no—
525C2×C5—21no—
526C2×C5—22no—
527C2×C5—42no—
528C2×C5—124no—
529C2×C5C2×#93711yescharacteristic
530D5—81no—
531C23—64no—
532C23—62no—
533C23—101no—
534C23C2×#120112yes—
535C23C2×#120111yescharacteristic
536Q8—22no—
537Q8—34no—
538Q8—64no—
539Q8—62no—
540Q8—102no—
541D4—24no—
542D4—22no—
543D4—34no—
544D4—64no—
545D4—68no—
546D4—64no—
547D4—64no—
548D4—62no—
549D4—102no—
550D4—204no—
551D4—202no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—68no—
555C2×C4—64no—
556C2×C4—62no—
557C2×C4—64no—
558C2×C4—64no—
559C2×C4—101no—
560C2×C4—101no—
561C2×C4—101no—
562C2×C4—202no—
563C2×C4—201no—
564C2×C4—201no—
565C2×C4C2×#120112yes—
566C2×C4C2×#120111yescharacteristic
567C2×C4D3×C22⋊D512yes—
568C2×C4D3×C22⋊D512yes—
569C2×C4C22×D3×D511yescharacteristic
570C8—304no—
571C2×C3—21no—
572C2×C3—22no—
573C2×C3—42no—
574C2×C3—401no—
575C2×C3C2×#409911yescharacteristic
576D3—44no—
577C5#8332211yescharacteristic, complemented, Sylow
578C22—22no—
579C22—41no—
580C22—42no—
581C22—64no—
582C22—202no—
583C22—201no—
584C22—242no—
585C22—244no—
586C22#9156812yes—
587C22C22×#120111yescharacteristic
588C4—22no—
589C4—64no—
590C4—102no—
591C4—201no—
592C4—201no—
593C4#9156811yescharacteristic
594C4#9156811yescharacteristic
595C4C2×D3×C22⋊D512yes—
596C3#1061585211yescharacteristic, complemented, Sylow
597C2—21no—
598C2—22no—
599C2—42no—
600C2—124no—
601C2—401no—
602C2C2×#9156811yescharacteristic
subgroupG/H
centerC4#91568
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×#41C22
socleC2×C3×C5C2×#120
hypercenterC2×#41D3×D5
hypocenterC3×C5#3018
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C5×#5964C2semidirect
2. C3×#66166C2semidirect
3. #11754138C2semidirect
4. #11751338C2semidirect
5. #11749393C2semidirect
6. #11749386C2semidirect
7. #11749147C2semidirect
8. #11749140C2semidirect
9. C2×#90979C2semidirect
20. C2×C3×#709C22semidirect
21. C2×#6710C22semidirect
22. C3×#4093C22semidirect
23. C3×#4090C22semidirect
24. C3×#4086C22semidirect
26. C3×#4082C22semidirect
28. C2×C3×C4×Dic5C22semidirect
29. #91309C22semidirect
32. #91298C22semidirect
34. #91120C22semidirect
35. #91117C22semidirect
36. #90979C22semidirect
37. #90979C22semidirect
40. #66166D3semidirect
68. C5×#235D4semidirect
71. C2×C5×D12D4semidirect
75. C2×C5×Dic6D4semidirect
97. #5964D5semidirect
115. C2×#709C2×D3semidirect
119. #4093C2×D3semidirect
120. #4090C2×D3semidirect
122. #4086C2×D3semidirect
124. #4082C2×D3semidirect
127. C2×C4×Dic5C2×D3semidirect
305. #235C22⋊D5semidirect
308. C2×D12C22⋊D5semidirect
312. C2×Dic6C22⋊D5semidirect
506. C3×C5#3018semidirect
577. C5#83322semidirect
596. C3#10615852semidirect

Maximal subgroups

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

subgroupindex
C5×#59642
C3×#661662
#117541382
#117513382
#117493932
#117493862
#117491472
#117491402
C2×#909792
#106158523
#833225