Finite Group Databasebeta

← #14460219

Subgroups — #14460219

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#11752561C211yesmaximal, characteristic, complemented
3C4×#6715C211yesmaximal, characteristic, complemented
4C3×#65308C211yesmaximal, characteristic, complemented
5#11749162C212yesmaximal, complemented
6#11749050C212yesmaximal, complemented
7#11747504C212yesmaximal, complemented
8#11747351C212yesmaximal, complemented
9#11746998C211yesmaximal, characteristic, complemented
10#11746924C211yesmaximal, characteristic, complemented
11#11746885C211yesmaximal, characteristic, complemented
12#10615383—31nomaximal, Hall
13C2×#6769C2212yes—
14C3×C4×C5×D4C2211yescharacteristic
15C2×#6715C2211yescharacteristic
16C4×D60C2211yescharacteristic
17C5×#938C2212yes—
18C5×#930—21no—
19C5×#926—21no—
20C5×#892C2212yes—
21C5×#885—22no—
22C4×C5×D12C2211yescharacteristic
23C5×#869C2211yescharacteristic
24C2×C3×#687C2212yes—
25C3×#4015C2211yescharacteristic
26#91466—21no—
27#91438—22no—
28#91437—21no—
29#91369C2212yes—
30C2×D3×#168C2212yes—
31#91116C2211yescharacteristic
32#91093C2211yescharacteristic
33#91091C2211yescharacteristic
34C4×C3⋊#168C2211yescharacteristic
35C3×#3942C2212yes—
36C3×C4⋊#168C2211yescharacteristic
37#90985C2212yes—
38#90970C2212yes—
39#90970C2212yes—
40#90969C2212yes—
41(C2×D3)⋊#168C2212yes—
42#80451—51nomaximal, Hall
43C5×#555—31no—
44#65885—31no—
45#65382—31no—
46#65373—31no—
47#65330—32no—
48#65326—32no—
49#65308D311yescharacteristic, complemented
50C4×#686—31no—
51D4×#168—31no—
52#65291—32no—
53#65285—32no—
54C2×C3×C5×D4C2311yescharacteristic
55C22×C3×C4×C5—21no—
56C2×D60C2311yescharacteristic
57C2×C4×D15—21no—
58C2×C5×#67—22no—
59C2×C5×D12D411yescharacteristic
60C2×C4×C5×D3C2312yes—
61C2×C5×Dic6—21no—
62C2×C3×Dic10C2312yes—
63C2×#1203—22no—
64C2×#1200—22no—
65#6769—24no—
66#6769—42no—
67C3×C5×C4⋊C4C2311yescharacteristic
68C3×C5×C22⋊C4—21no—
69C3×C42×C5C2311yescharacteristic
70#6715—22no—
71#6714—21no—
72C4⋊Dic15C2311yescharacteristic
73C2×C3⋊#168C2311yescharacteristic
74C2×C3⋊#168C2311yescharacteristic
75C5×#217—22no—
76C5×(C2×D3)⋊C4—22no—
77C5×(C2×D3)⋊C4D412yes—
78C5×C4⋊Dic3—21no—
79C5×C4⋊Dic3D411yescharacteristic
80C5×C4⋊Dic3C2312yes—
81C4×C5×Dic3—22no—
82C3×#687—42no—
83C3×C4⋊Dic5—21no—
84C3×C4⋊Dic5—21no—
85C2×C3×#168C2312yes—
86#6669—22no—
87(C2×D3)⋊Dic5—22no—
88D3×#168—42no—
89#5726—51no—
90#5563D511yescharacteristic, complemented
91C3×#461—51no—
92#5172—52no—
93#5171—52no—
94#5079—51no—
95C4×#213—51no—
96#4842—51no—
97#4834—51no—
98#4815—52no—
99#4804—52no—
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×#687C2×D312yes—
111C2×#686—31no—
112#4070—32no—
113C22×#168—32no—
114D5×C4⋊C4—61no—
115#4029—62no—
116C4×D20—31no—
117#4015C2×D311yescharacteristic
118#3964—31no—
119C22⋊#168—32no—
120#3942—32no—
121#3942—32no—
122#3942C2×D312yes—
123#3941—31no—
124#3941—32no—
125#3940—32no—
126#3939—31no—
127C4⋊#168C2×D311yescharacteristic
128C4×#168—31no—
129#3303—151noSylow
130C23×C3×C5—21no—
131C22×D15—21no—
132C22×C5×D3C2×D412yes—
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—21no—
141D60—21no—
142D60#4112yes—
143C4×D15—42no—
144C5×#67—82no—
145C2×C5×Dic3—22no—
146C2×C5×Dic3C2×D412yes—
147C5×D12—22no—
148C4×C5×D3—22no—
149C4×C5×D3—22no—
150C5×Dic6—81no—
151C2×C3×Dic5—22no—
152C3×Dic10—22no—
153C3×Dic10—22no—
154#1203—44no—
155#1200—44no—
156C3⋊#168—21no—
157C3⋊#168—21no—
158C3×#168—22no—
159C2×#239—52no—
160C2×C3×SD16—52no—
161C3×C4⋊Q8—51no—
162C3×C4×D4C2×D511yescharacteristic
163#946—52no—
164#938C2×D512yes—
165C2×#213—51no—
166#930—21no—
167#926—21no—
168C2×D3×C8—52no—
169#895—101no—
170#894—102no—
171#892C2×D512yes—
172D3×C4⋊C4—101no—
173#885—22no—
174C4×D12—51no—
175C4×D12C2×D511yescharacteristic
176#869C2×D511yescharacteristic
177C3×C4⋊C8—51no—
178C3×#103—52no—
179#835—52no—
180#835—52no—
181#832—51no—
182#821—52no—
183(C2×D3)⋊C8—52no—
184#817—52no—
185#809—51no—
186#807—51no—
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×D20—31no—
198C2×C4×D5—62no—
199C2×C4×D5—61no—
200C2×Dic10C22×D312yes—
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—42no—
213#686—62no—
214#685—61no—
215C4⋊Dic5—21no—
216C4⋊Dic5—21no—
217C4⋊Dic5—31no—
218#683—62no—
219C2×#168—31no—
220C2×#168—31no—
221C2×#168—122no—
222C2×#168C22×D312yes—
223#555—31no—
224#547—151no—
225#487—151no—
226#485—152no—
227#476—151no—
228#465—152no—
229#461—51no—
230#449—152no—
231C4×D8—151no—
232C8×D4—151no—
233#416—152no—
234C22×C3×C5—41no—
235C22×C3×C5—41no—
236C22×C3×C5—41no—
237C22×C3×C5C22×D411yescharacteristic
238C2×D15—22no—
239C2×D15—41no—
240C2×D15—41no—
241C2×C5×D3—22no—
242C2×C5×D3—22no—
243C2×C5×D3—22no—
244C3×C4×C5—21no—
245C3×C4×C5—41no—
246C3×C4×C5#13512yes—
247C3×C4×C5C2×#4111yescharacteristic
248C3×C4×C5H(5,𝔽2)11yescharacteristic
249Dic15—41no—
250C3×Dic5—42no—
251C5×Dic3—22no—
252C5×Dic3—42no—
253C2×C3×Q8—52no—
254C2×C3×D4C22×D511yescharacteristic
255C22×C3×C4—21no—
256C2×#67—22no—
257#239—104no—
258#239—202no—
259C2×D12—51no—
260C2×D12—102no—
261C2×D12C22⋊D511yescharacteristic
262C2×C4×D3—101no—
263C2×C4×D3—102no—
264C2×C4×D3C22×D512yes—
265C2×Dic6—21no—
266C3×SD16—202no—
267C2×C3×C8—52no—
268C3×C4⋊C4—101no—
269C3×C4⋊C4—101no—
270C3×C4⋊C4C22×D511yescharacteristic
271C3×C22⋊C4—21no—
272C3×C42C22×D511yescharacteristic
273#217—22no—
274#213—102no—
275(C2×D3)⋊C4—22no—
276(C2×D3)⋊C4—102no—
277(C2×D3)⋊C4—101no—
278(C2×D3)⋊C4C22⋊D512yes—
279C4⋊Dic3—21no—
280C4⋊Dic3—51no—
281C4⋊Dic3C22⋊D511yescharacteristic
282C4⋊Dic3C22×D512yes—
283Dic3⋊C4—102no—
284C4×Dic3—22no—
285C2×C3⋊C8—51no—
286C2×C3⋊C8—51no—
287D3×C8—202no—
288C23×C5—21no—
289C23×C5—32no—
290C22×D5—61no—
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—22no—
307C2×Dic5—61no—
308D20—32no—
309D20—61no—
310C4×D5—124no—
311C4×D5—122no—
312Dic10—22no—
313Dic10—22no—
314#168—22no—
315#168—61no—
316#168—61no—
317C2×#41—152no—
318C2×SD16—52no—
319C2×D8—151no—
320C22×C8—152no—
321C4⋊Q8—31no—
322C4⋊Q8—51no—
323#124—62no—
324#123—302no—
325C22⋊Q8—152no—
326#121—32no—
327#121—61no—
328#121—301no—
329C4×D4—31no—
330C4×D4—151no—
331C4×D4D3×D511yescharacteristic
332#117—32no—
333#117—61no—
334C2×C4⋊C4—301no—
335#107—151no—
336#106—152no—
337C4⋊C8—51no—
338#103—52no—
339#103—152no—
340#103—152no—
341#102—152no—
342#102—151no—
343#102—151no—
344C22⋊C8—152no—
345C4×C8—151no—
346C2×C3×C5—81no—
347C2×C3×C5#54611yescharacteristic
348C2×C3×C5C2×#13511yescharacteristic
349C2×C3×C5#57711yescharacteristic
350D15—42no—
351C5×D3—42no—
352C23×C3—21no—
353C22×D3—101no—
354C22×D3C2×C22⋊D512yes—
355C3×Q8—102no—
356C3×Q8—102no—
357C3×D4—21no—
358C3×D4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—42no—
361C2×C3×C4—102no—
362C2×C3×C4C2×C22⋊D511yescharacteristic
363C2×C3×C4C2×C22⋊D511yescharacteristic
364C2×C3×C4C23×D511yescharacteristic
365#67—82no—
366C2×Dic3—22no—
367C2×Dic3—101no—
368C2×Dic3C2×C22⋊D512yes—
369D12—22no—
370D12—52no—
371D12—101no—
372D12—204no—
373C4×D3—22no—
374C4×D3—22no—
375C4×D3—204no—
376C4×D3—202no—
377Dic6—81no—
378C3×C8—102no—
379C3⋊C8—101no—
380C3⋊C8—101no—
381C22×C5—41no—
382C22×C5—41no—
383C22×C5—41no—
384C22×C5—62no—
385C22×C5—62no—
386C22×C5—62no—
387C22×C5C2×D3×D411yescharacteristic
388C2×D5—62no—
389C2×D5—121no—
390C2×D5—121no—
391C4×C5—21no—
392C4×C5—41no—
393C4×C5—62no—
394C4×C5—122no—
395C4×C5#91612yes—
396C4×C5#100411yescharacteristic
397C4×C5D3×#4111yescharacteristic
398Dic5—42no—
399Dic5—121no—
400#41—304no—
401#41—602no—
402C2×Q8—52no—
403C2×Q8—61no—
404C2×D4—31no—
405C2×D4—62no—
406C2×D4—151no—
407C2×D4—302no—
408C2×D4C2×D3×D511yescharacteristic
409C22×C4—21no—
410C22×C4—32no—
411C22×C4—302no—
412C22×C4—301no—
413SD16—202no—
414D8—302no—
415C2×C8—52no—
416C2×C8—151no—
417C2×C8—151no—
418C2×C8—602no—
419C4⋊C4—31no—
420C4⋊C4—32no—
421C4⋊C4—61no—
422C4⋊C4—101no—
423C4⋊C4—101no—
424C4⋊C4—151no—
425C4⋊C4—302no—
426C4⋊C4C2×D3×D511yescharacteristic
427C22⋊C4—21no—
428C22⋊C4—32no—
429C22⋊C4—62no—
430C22⋊C4—62no—
431C22⋊C4—302no—
432C22⋊C4—301no—
433C42—62no—
434C42C2×D3×D511yescharacteristic
435C3×C5#330311yescharacteristic, complemented, Hall
436C22×C3—41no—
437C22×C3—41no—
438C22×C3—41no—
439C22×C3C22×C22⋊D511yescharacteristic
440C2×D3—22no—
441C2×D3—22no—
442C2×D3—22no—
443C2×D3—102no—
444C2×D3—201no—
445C2×D3—201no—
446C3×C4—21no—
447C3×C4—41no—
448C3×C4—202no—
449C3×C4#409612yes—
450C3×C4D5×#4111yescharacteristic
451C3×C4#414911yescharacteristic
452Dic3—22no—
453Dic3—42no—
454Dic3—201no—
455C2×C5—81no—
456C2×C5—122no—
457C2×C5#561211yescharacteristic
458C2×C5C2×#91611yescharacteristic
459C2×C5#577811yescharacteristic
460D5—122no—
461C23—21no—
462C23—32no—
463C23—301no—
464Q8—102no—
465Q8—102no—
466Q8—241no—
467D4—21no—
468D4—21no—
469D4—62no—
470D4—152no—
471D4—242no—
472D4—301no—
473D4—604no—
474C2×C4—21no—
475C2×C4—32no—
476C2×C4—42no—
477C2×C4—62no—
478C2×C4—62no—
479C2×C4—62no—
480C2×C4—102no—
481C2×C4—301no—
482C2×C4—604no—
483C2×C4—602no—
484C2×C4D3×C22⋊D511yescharacteristic
485C2×C4D3×C22⋊D511yescharacteristic
486C2×C4C22×D3×D511yescharacteristic
487C8—102no—
488C8—301no—
489C8—301no—
490C2×C3—81no—
491C2×C3C2×#409611yescharacteristic
492C2×C3#6616311yescharacteristic
493C2×C3#6617011yescharacteristic
494D3—42no—
495D3—202no—
496C5#8045111yescharacteristic, complemented, Sylow
497C22—41no—
498C22—41no—
499C22—41no—
500C22—62no—
501C22—62no—
502C22—62no—
503C22—302no—
504C22—601no—
505C22—601no—
506C22C2×D3×C22⋊D511yescharacteristic
507C4—21no—
508C4—41no—
509C4—62no—
510C4—122no—
511C4—202no—
512C4—601no—
513C4#9149512yes—
514C4#9201611yescharacteristic
515C4#9201711yescharacteristic
516C3#1061538311yescharacteristic, complemented, Sylow
517C2—81no—
518C2—122no—
519C2—602no—
520C2#1175465811yescharacteristic
521C2C2×#9149511yescharacteristic
522C2#1175534011yescharacteristic
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. #11752561C2semidirect
3. C4×#6715C2semidirect
4. C3×#65308C2semidirect
5. #11749162C2semidirect
6. #11749050C2semidirect
7. #11747504C2semidirect
8. #11747351C2semidirect
9. #11746998C2semidirect
10. #11746924C2semidirect
11. #11746885C2semidirect
49. #65308D3semidirect
90. #5563D5semidirect
435. C3×C5#3303semidirect
496. C5#80451semidirect
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
#117525612
C4×#67152
C3×#653082
#117491622
#117490502
#117475042
#117473512
#117469982
#117469242
#117468852
#106153833
#804515