Finite Group Databasebeta

← #14460218

Subgroups — #14460218

Order 1920.

796 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5563C211yesmaximal, characteristic, complemented
2#11752532C211yesmaximal, characteristic, complemented
3#11751120C211yesmaximal, characteristic, complemented
4C3×C4×#686C211yesmaximal, characteristic, complemented
5#11749157C212yesmaximal, complemented
6#11749055C212yesmaximal, complemented
7#11747502C212yesmaximal, complemented
8#11747176C212yesmaximal, complemented
9D12×#168C211yesmaximal, characteristic, complemented
10#11746930C211yesmaximal, characteristic, complemented
11#11746880C211yesmaximal, characteristic, complemented
12#10615383—31nomaximal, Hall
13C2×#6765C2212yes—
14C3×C4×C5×D4C2211yescharacteristic
15C2×#6716C2212yes—
16#91758C2211yescharacteristic
17C5×#938C2212yes—
18C5×#930—21no—
19C5×#926—21no—
20C5×#892C2212yes—
21C5×#885—22no—
22C4×C5×D12C2211yescharacteristic
23C5×#869C2211yescharacteristic
24C2×C3×#686C2211yescharacteristic
25C3×C4×D20C2211yescharacteristic
26#91449—21no—
27#91414—22no—
28D5×C4⋊Dic3—21no—
29#91365C2212yes—
30C2×D3×#168C2212yes—
31#91094C2212yes—
32C4⋊(C3⋊#168)C2211yescharacteristic
33C3×#3964C2211yescharacteristic
34C3×#3941C2211yescharacteristic
35C3×#3939C2211yescharacteristic
36C3×C4×#168C2211yescharacteristic
37#90986C2212yes—
38#90975C2212yes—
39#90975C2212yes—
40#90964C2212yes—
41(C2×D3)⋊#168C2212yes—
42#79610—51nomaximal, Hall
43C5×#555—31no—
44#65885—31no—
45#65382—31no—
46#65373—31no—
47#65330—32no—
48#65326—32no—
49#65308—31no—
50C4×#686D311yescharacteristic, complemented
51D4×#168—31no—
52#65291—32no—
53#65285—32no—
54C2×C3×C5×D4C2311yescharacteristic
55C22×C3×C4×C5—21no—
56C2×Dic30C2312yes—
57C2×C5×#67—22no—
58C2×C5×D12D411yescharacteristic
59C2×C4×C5×D3C2312yes—
60C2×C5×Dic6—21no—
61C2×C3×D20C2311yescharacteristic
62C2×C3×C4×D5—21no—
63C2×#1201—22no—
64C2×D5×Dic3—22no—
65#6765—24no—
66#6765—42no—
67C3×C5×C4⋊C4C2311yescharacteristic
68C3×C5×C22⋊C4—21no—
69C3×C42×C5C2311yescharacteristic
70#6716—42no—
71C4⋊Dic15—21no—
72C4⋊Dic15—21no—
73C2×C3⋊#168C2312yes—
74C5×#217—22no—
75C5×(C2×D3)⋊C4—22no—
76C5×(C2×D3)⋊C4D412yes—
77C5×C4⋊Dic3—21no—
78C5×C4⋊Dic3D411yescharacteristic
79C5×C4⋊Dic3C2312yes—
80C4×C5×Dic3—22no—
81C3×#686—22no—
82C3×#685—21no—
83C3×C4⋊Dic5C2311yescharacteristic
84C2×C3×#168C2311yescharacteristic
85C2×C3×#168C2311yescharacteristic
86#6668—22no—
87(C2×D3)⋊Dic5—22no—
88D3×#168—42no—
89#5563D511yescharacteristic, complemented
90#5555—51no—
91C3×C4×D8—51no—
92#5158—52no—
93#5019—51no—
94#5008—51no—
95#4888—52no—
96#4778—52no—
97#4772—52no—
98C8×D12—51no—
99#4681—51no—
100C2×#709—32no—
101C5×C4⋊Q8—31no—
102C5×#124—62no—
103C5×#121—32no—
104C5×#121—61no—
105C4×C5×D4—31no—
106C4×C5×D4C2×D311yescharacteristic
107C5×#117—32no—
108C5×#117—61no—
109#4084—61no—
110C2×#687—32no—
111C2×#686C2×D311yescharacteristic
112#4070—32no—
113C22×#168—32no—
114D5×C4⋊C4—61no—
115#4029—62no—
116C4×D20C2×D311yescharacteristic
117#4015—31no—
118#3964C2×D311yescharacteristic
119C22⋊#168—32no—
120#3942—32no—
121#3942—32no—
122#3942—32no—
123#3941—32no—
124#3941C2×D311yescharacteristic
125#3940—32no—
126#3939C2×D311yescharacteristic
127C4⋊#168—31no—
128C4×#168C2×D311yescharacteristic
129#3303—151noSylow
130C23×C3×C5—21no—
131C22×C5×D3C2×D412yes—
132C22×C3×D5—21no—
133C3×C5×D4—21no—
134C3×C5×D4—21no—
135C2×C3×C4×C5—21no—
136C2×C3×C4×C5—42no—
137C2×C3×C4×C5C2×D411yescharacteristic
138C2×C3×C4×C5C2×D411yescharacteristic
139C2×C3×C4×C5C2411yescharacteristic
140C2×Dic15—22no—
141Dic30—22no—
142Dic30—22no—
143C5×#67—82no—
144C2×C5×Dic3—22no—
145C2×C5×Dic3C2×D412yes—
146C5×D12—22no—
147C4×C5×D3—22no—
148C4×C5×D3—22no—
149C5×Dic6—81no—
150C2×C3×Dic5—21no—
151C3×D20—21no—
152C3×D20#4112yes—
153C3×C4×D5—42no—
154#1201—44no—
155D5×Dic3—44no—
156C3⋊#168—22no—
157C3×#168—21no—
158C3×#168—21no—
159C2×#237—52no—
160C2×C3×D8—51no—
161C3×C4×D4—51no—
162C3×C4×D4C2×D511yescharacteristic
163#938C2×D512yes—
164C2×#214—52no—
165#930—21no—
166#930—101no—
167#926—21no—
168C2×C4⋊Dic3—101no—
169C2×D3×C8—52no—
170#897—52no—
171#892C2×D512yes—
172#886—102no—
173#885—22no—
174C4×D12C2×D511yescharacteristic
175#869—51no—
176#869C2×D511yescharacteristic
177C3×#107—51no—
178C3×#102—51no—
179C3×#102—51no—
180C3×C4×C8—51no—
181#832—52no—
182(C2×D3)⋊C8—52no—
183#816—52no—
184#816—52no—
185#810—52no—
186#808—52no—
187C4⋊(C3⋊C8)—51no—
188C2×C5×Q8—61no—
189C2×C5×D4—31no—
190C2×C5×D4—62no—
191C2×C5×D4C22×D311yescharacteristic
192C22×C4×C5—21no—
193C22×C4×C5—32no—
194C2×C22⋊D5—62no—
195#709—64no—
196#709—122no—
197C2×D20C22×D311yescharacteristic
198C2×C4×D5—21no—
199C2×C4×D5—62no—
200C2×Dic10—32no—
201C5×C4⋊C4—31no—
202C5×C4⋊C4—32no—
203C5×C4⋊C4—61no—
204C5×C4⋊C4C22×D311yescharacteristic
205C5×C22⋊C4—21no—
206C5×C22⋊C4—32no—
207C5×C22⋊C4—62no—
208C5×C22⋊C4—62no—
209C42×C5—62no—
210C42×C5C22×D311yescharacteristic
211C22⋊Dic5—62no—
212#687—122no—
213#686—22no—
214#685—21no—
215C4⋊Dic5—61no—
216C4⋊Dic5—61no—
217C4⋊Dic5C22×D311yescharacteristic
218#683—62no—
219C2×#168—32no—
220C2×#168—122no—
221C2×#168C22×D311yescharacteristic
222C2×#168C22×D311yescharacteristic
223#555—31no—
224#547—151no—
225#487—151no—
226#485—152no—
227#476—151no—
228#465—152no—
229#461—151no—
230#449—152no—
231C4×D8—51no—
232C8×D4—151no—
233#416—152no—
234C22×C3×C5—41no—
235C22×C3×C5—41no—
236C22×C3×C5—41no—
237C22×C3×C5C22×D411yescharacteristic
238C2×C5×D3—22no—
239C2×C5×D3—22no—
240C2×C5×D3—22no—
241C2×C3×D5—22no—
242C2×C3×D5—41no—
243C2×C3×D5—41no—
244C3×C4×C5—21no—
245C3×C4×C5—41no—
246C3×C4×C5#13512yes—
247C3×C4×C5C2×#4111yescharacteristic
248C3×C4×C5H(5,𝔽2)11yescharacteristic
249Dic15—42no—
250C3×Dic5—41no—
251C5×Dic3—22no—
252C5×Dic3—42no—
253C2×C3×D4—51no—
254C2×C3×D4C22×D511yescharacteristic
255C22×C3×C4—21no—
256C22×C3×C4—101no—
257C2×#67—22no—
258C2×#67—102no—
259C22×Dic3—102no—
260#237—104no—
261#237—202no—
262C2×D12C22⋊D511yescharacteristic
263C2×C4×D3C22×D512yes—
264C2×Dic6—21no—
265C2×Dic6—52no—
266C3×D8—102no—
267C2×C3×C8—51no—
268C2×C3×C8—51no—
269C3×C4⋊C4—51no—
270C3×C4⋊C4C22×D511yescharacteristic
271C3×C22⋊C4—21no—
272C3×C22⋊C4—101no—
273C3×C42C22×D511yescharacteristic
274#217—22no—
275#214—202no—
276(C2×D3)⋊C4—22no—
277(C2×D3)⋊C4—102no—
278(C2×D3)⋊C4C22⋊D512yes—
279C4⋊Dic3—21no—
280C4⋊Dic3—101no—
281C4⋊Dic3—101no—
282C4⋊Dic3—102no—
283C4⋊Dic3C22⋊D511yescharacteristic
284C4⋊Dic3C22×D512yes—
285C4×Dic3—22no—
286C2×C3⋊C8—52no—
287D3×C8—202no—
288C23×C5—21no—
289C23×C5—32no—
290C22×D5—21no—
291C5×Q8—241no—
292C5×D4—21no—
293C5×D4—21no—
294C5×D4—62no—
295C5×D4—242no—
296C2×C4×C5—21no—
297C2×C4×C5—32no—
298C2×C4×C5—42no—
299C2×C4×C5—62no—
300C2×C4×C5—62no—
301C2×C4×C5—62no—
302C2×C4×C5D3×D411yescharacteristic
303C2×C4×C5D3×D411yescharacteristic
304C2×C4×C5C23×D311yescharacteristic
305C22⋊D5—124no—
306C2×Dic5—21no—
307C2×Dic5—62no—
308D20—21no—
309D20#23512yes—
310C4×D5—42no—
311C4×D5—124no—
312Dic10—62no—
313Dic10—62no—
314#168—21no—
315#168—21no—
316#168—62no—
317C2×#41—152no—
318C2×SD16—152no—
319C2×D8—51no—
320C22×C8—152no—
321C4⋊Q8—31no—
322C4⋊Q8—151no—
323#124—62no—
324#123—302no—
325C22⋊Q8—152no—
326#121—32no—
327#121—61no—
328#121—301no—
329C4×D4—31no—
330C4×D4—51no—
331C4×D4D3×D511yescharacteristic
332#117—32no—
333#117—61no—
334C2×C4⋊C4—301no—
335#107—51no—
336#106—152no—
337C4⋊C8—151no—
338#103—152no—
339#103—152no—
340#103—152no—
341#102—51no—
342#102—51no—
343#102—152no—
344C22⋊C8—152no—
345C4×C8—51no—
346C2×C3×C5—81no—
347C2×C3×C5#54611yescharacteristic
348C2×C3×C5C2×#13511yescharacteristic
349C2×C3×C5#57711yescharacteristic
350C3×D5—42no—
351C5×D3—42no—
352C23×C3—21no—
353C23×C3—101no—
354C22×D3C2×C22⋊D512yes—
355C3×D4—21no—
356C3×D4—21no—
357C3×D4—52no—
358C3×D4—101no—
359C2×C3×C4—21no—
360C2×C3×C4—42no—
361C2×C3×C4—101no—
362C2×C3×C4—202no—
363C2×C3×C4C2×C22⋊D511yescharacteristic
364C2×C3×C4C2×C22⋊D511yescharacteristic
365C2×C3×C4C23×D511yescharacteristic
366#67—82no—
367#67—204no—
368C2×Dic3—22no—
369C2×Dic3—102no—
370C2×Dic3—204no—
371C2×Dic3C2×C22⋊D512yes—
372D12—22no—
373C4×D3—22no—
374C4×D3—22no—
375Dic6—81no—
376Dic6—102no—
377Dic6—102no—
378C3×C8—101no—
379C3×C8—101no—
380C3⋊C8—102no—
381C22×C5—41no—
382C22×C5—41no—
383C22×C5—41no—
384C22×C5—62no—
385C22×C5—62no—
386C22×C5—62no—
387C22×C5C2×D3×D411yescharacteristic
388C2×D5—22no—
389C2×D5—41no—
390C2×D5—41no—
391C4×C5—21no—
392C4×C5—41no—
393C4×C5—62no—
394C4×C5—122no—
395C4×C5#91212yes—
396C4×C5C2×#23511yescharacteristic
397C4×C5#100911yescharacteristic
398Dic5—41no—
399Dic5—122no—
400#41—304no—
401#41—602no—
402C2×Q8—61no—
403C2×Q8—152no—
404C2×D4—31no—
405C2×D4—51no—
406C2×D4—62no—
407C2×D4—302no—
408C2×D4C2×D3×D511yescharacteristic
409C22×C4—21no—
410C22×C4—32no—
411C22×C4—101no—
412C22×C4—302no—
413SD16—602no—
414D8—102no—
415C2×C8—51no—
416C2×C8—51no—
417C2×C8—152no—
418C2×C8—602no—
419C4⋊C4—31no—
420C4⋊C4—32no—
421C4⋊C4—51no—
422C4⋊C4—61no—
423C4⋊C4—302no—
424C4⋊C4—301no—
425C4⋊C4—301no—
426C4⋊C4C2×D3×D511yescharacteristic
427C22⋊C4—21no—
428C22⋊C4—32no—
429C22⋊C4—62no—
430C22⋊C4—62no—
431C22⋊C4—101no—
432C22⋊C4—302no—
433C42—62no—
434C42C2×D3×D511yescharacteristic
435C3×C5#330311yescharacteristic, complemented, Hall
436C22×C3—41no—
437C22×C3—41no—
438C22×C3—41no—
439C22×C3—102no—
440C22×C3—201no—
441C22×C3—201no—
442C22×C3C22×C22⋊D511yescharacteristic
443C2×D3—22no—
444C2×D3—22no—
445C2×D3—22no—
446C3×C4—21no—
447C3×C4—41no—
448C3×C4—201no—
449C3×C4#409612yes—
450C3×C4D5×#4111yescharacteristic
451C3×C4#414911yescharacteristic
452Dic3—22no—
453Dic3—42no—
454Dic3—202no—
455C2×C5—81no—
456C2×C5—122no—
457C2×C5#555011yescharacteristic
458C2×C5C2×#91211yescharacteristic
459C2×C5#577011yescharacteristic
460D5—42no—
461C23—21no—
462C23—32no—
463C23—101no—
464Q8—241no—
465Q8—302no—
466Q8—302no—
467D4—21no—
468D4—21no—
469D4—52no—
470D4—62no—
471D4—101no—
472D4—242no—
473D4—604no—
474C2×C4—21no—
475C2×C4—32no—
476C2×C4—42no—
477C2×C4—62no—
478C2×C4—62no—
479C2×C4—62no—
480C2×C4—101no—
481C2×C4—202no—
482C2×C4—302no—
483C2×C4—604no—
484C2×C4D3×C22⋊D511yescharacteristic
485C2×C4D3×C22⋊D511yescharacteristic
486C2×C4C22×D3×D511yescharacteristic
487C8—101no—
488C8—101no—
489C8—302no—
490C2×C3—81no—
491C2×C3—202no—
492C2×C3C2×#409611yescharacteristic
493C2×C3#6616311yescharacteristic
494C2×C3#6617011yescharacteristic
495D3—42no—
496C5#7961011yescharacteristic, complemented, Sylow
497C22—41no—
498C22—41no—
499C22—41no—
500C22—62no—
501C22—62no—
502C22—62no—
503C22—102no—
504C22—201no—
505C22—201no—
506C22C2×D3×C22⋊D511yescharacteristic
507C4—21no—
508C4—41no—
509C4—62no—
510C4—122no—
511C4—201no—
512C4—602no—
513C4#9149312yes—
514C4D5×#23511yescharacteristic
515C4#9201911yescharacteristic
516C3#1061538311yescharacteristic, complemented, Sylow
517C2—81no—
518C2—122no—
519C2—202no—
520C2#1175465111yescharacteristic
521C2C2×#9149311yescharacteristic
522C2#1175534711yescharacteristic
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#3303
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×#5563C2semidirect
2. #11752532C2semidirect
3. #11751120C2semidirect
4. C3×C4×#686C2semidirect
5. #11749157C2semidirect
6. #11749055C2semidirect
7. #11747502C2semidirect
8. #11747176C2semidirect
9. D12×#168C2semidirect
10. #11746930C2semidirect
11. #11746880C2semidirect
50. C4×#686D3semidirect
89. #5563D5semidirect
435. C3×C5#3303semidirect
496. C5#79610semidirect
516. C3#10615383semidirect

Maximal subgroups

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

subgroupindex
C5×#55632
#117525322
#117511202
C3×C4×#6862
#117491572
#117490552
#117475022
#117471762
D12×#1682
#117469302
#117468802
#106153833
#796105