Finite Group Databasebeta

← #14460164

Subgroups — #14460164

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×#5555C211yesmaximal, characteristic, complemented
2#11752710C211yesmaximal, characteristic, complemented
3C4×#6715C211yesmaximal, characteristic, complemented
4C3×#65297C211yesmaximal, characteristic, complemented
5#11749036C212yesmaximal, complemented
6#11748965C212yesmaximal, complemented
7#11747449C212yesmaximal, complemented
8#11747211C212yesmaximal, complemented
9#11747087C211yesmaximal, characteristic, complemented
10#11746998C211yesmaximal, characteristic, complemented
11#11746923C211yesmaximal, characteristic, complemented
12#10615242—31nomaximal, Hall
13C2×C5×#237C2212yes—
14C3×C4×C5×D4C2211yescharacteristic
15C2×#6715C2211yescharacteristic
16C4×D60C2211yescharacteristic
17C5×#930—21no—
18C2×C5×C4⋊Dic3—21no—
19C5×#897C2212yes—
20C5×#886—22no—
21C4×C5×D12C2211yescharacteristic
22C5×#869C2211yescharacteristic
23C3×#4015C2211yescharacteristic
24#91466—21no—
25#91460C2212yes—
26#91347—21no—
27#91338C2212yes—
28#91326—22no—
29C2×#6660C2212yes—
30C2×D3×#168C2212yes—
31#91116C2211yescharacteristic
32#91093C2211yescharacteristic
33#91091C2211yescharacteristic
34C4×C3⋊#168C2211yescharacteristic
35C3×#3964C2212yes—
36C3×#3940C2212yes—
37C3×C4⋊#168C2211yescharacteristic
38#90973C2212yes—
39#90972C2212yes—
40#90972C2212yes—
41(C2×D3)⋊#168C2212yes—
42#81076—51nomaximal, Hall
43C5×#547—31no—
44#65897—31no—
45#65527—32no—
46#65522—32no—
47#65372—31no—
48#65370—31no—
49#65337—32no—
50#65334—32no—
51C4×#686—31no—
52D4×#168—31no—
53#65297D311yescharacteristic, complemented
54C2×C3×C5×D4C2311yescharacteristic
55C22×C3×C4×C5—21no—
56C2×D60C2311yescharacteristic
57C2×C4×D15—21no—
58C2×C5×#67—22no—
59C22×C5×Dic3—22no—
60C5×#237—24no—
61C5×#237—42no—
62C2×C5×D12D411yescharacteristic
63C2×C4×C5×D3C2312yes—
64C2×C5×Dic6C2312yes—
65C2×C3×Dic10—21no—
66C2×#1203—22no—
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×(C2×D3)⋊C4—22no—
76C5×(C2×D3)⋊C4D412yes—
77C5×C4⋊Dic3—21no—
78C5×C4⋊Dic3—21no—
79C5×C4⋊Dic3—22no—
80C5×C4⋊Dic3D411yescharacteristic
81C3×C4⋊Dic5—21no—
82C3×C4⋊Dic5C2312yes—
83C2×C3×#168C2312yes—
84#6667—22no—
85(C2×D3)⋊Dic5—22no—
86Dic3×Dic5—22no—
87#6660—42no—
88D3×#168—42no—
89#5734—51no—
90#5555D511yescharacteristic, complemented
91C3×#476—51no—
92#5093—51no—
93C4×#213—51no—
94#4868—52no—
95#4864—52no—
96#4841—51no—
97#4834—51no—
98#4781—52no—
99#4772—52no—
100C2×C5×#41—32no—
101C5×C4⋊Q8—31no—
102C5×#123—62no—
103C5×C22⋊Q8—32no—
104C5×#121—61no—
105C4×C5×D4—31no—
106C4×C5×D4C2×D311yescharacteristic
107C2×C5×C4⋊C4—61no—
108C2×#688—32no—
109#4084—61no—
110C2×#686—31no—
111#4076—32no—
112#4072—32no—
113C22×#168—32no—
114#4038—61no—
115#4031—62no—
116C4×D20—31no—
117#4015C2×D311yescharacteristic
118#3967—32no—
119#3967—32no—
120#3967—32no—
121#3964—31no—
122#3964C2×D312yes—
123C22⋊#168—32no—
124#3941—31no—
125#3940C2×D312yes—
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#4112yes—
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—21no—
143C4×D15—42no—
144C5×#67—44no—
145C2×C5×Dic3—22no—
146C2×C5×Dic3—44no—
147C2×C5×Dic3C2×D412yes—
148C5×D12—22no—
149C4×C5×D3—22no—
150C4×C5×D3—22no—
151C5×Dic6—22no—
152C5×Dic6—22no—
153C2×C3×Dic5—22no—
154C3×Dic10—81no—
155#1203—82no—
156C3⋊#168—21no—
157C3⋊#168—21no—
158C3×#168—22no—
159C2×#237C2×D512yes—
160C3×C4⋊Q8—51no—
161C3×C4×D4C2×D511yescharacteristic
162#947—102no—
163C2×#213—51no—
164#930—21no—
165C2×C4⋊Dic3—21no—
166C2×#204—52no—
167C2×D3×C8—52no—
168#897C2×D512yes—
169#895—52no—
170#895—101no—
171#892—52no—
172#892—101no—
173#886—22no—
174C4×D12—51no—
175C4×D12C2×D511yescharacteristic
176#869C2×D511yescharacteristic
177C3×#106—52no—
178C3×C4⋊C8—51no—
179C3×#102—52no—
180#832—51no—
181(C2×D3)⋊C8—52no—
182#816—52no—
183#810—52no—
184#810—52no—
185#809—51no—
186#807—51no—
187C4×C3⋊C8—51no—
188C5×#41—64no—
189C5×#41—122no—
190C2×C5×Q8—32no—
191C2×C5×D4—31no—
192C2×C5×D4—62no—
193C2×C5×D4C22×D311yescharacteristic
194C22×C4×C5—21no—
195C22×C4×C5—32no—
196C22×C4×C5—62no—
197C2×C22⋊D5—62no—
198C2×D20—31no—
199C2×C4×D5—61no—
200C2×Dic10—21no—
201C5×C4⋊C4—31no—
202C5×C4⋊C4—62no—
203C5×C4⋊C4—61no—
204C5×C4⋊C4—61no—
205C5×C4⋊C4C22×D311yescharacteristic
206C5×C22⋊C4—21no—
207C5×C22⋊C4—32no—
208C5×C22⋊C4—62no—
209C42×C5C22×D311yescharacteristic
210C22⋊Dic5—62no—
211#688—122no—
212#686—62no—
213#685—62no—
214#685—61no—
215C4⋊Dic5—21no—
216C4⋊Dic5—31no—
217C4⋊Dic5C22×D312yes—
218C4×Dic5—62no—
219C2×#168—31no—
220C2×#168—31no—
221C2×#168—122no—
222C2×#168C22×D312yes—
223#555—151no—
224#547—31no—
225#487—151no—
226#485—152no—
227#476—51no—
228#465—152no—
229#461—151no—
230#449—152no—
231C4×D8—151no—
232C8×D4—151no—
233#416—152no—
234C22×C3×C5—22no—
235C22×C3×C5—41no—
236C22×C3×C5—41no—
237C22×C3×C5C22×D411yescharacteristic
238C2×D15—41no—
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—101no—
254C2×C3×D4C22×D511yescharacteristic
255C22×C3×C4—21no—
256C2×#67—22no—
257C22×Dic3—22no—
258#237—24no—
259#237—42no—
260C2×D12—51no—
261C2×D12—102no—
262C2×D12C22⋊D511yescharacteristic
263C2×C4×D3—101no—
264C2×C4×D3C22×D512yes—
265C2×Dic6C22×D512yes—
266C2×C3×C8—52no—
267C3×C4⋊C4—52no—
268C3×C4⋊C4—101no—
269C3×C4⋊C4C22×D511yescharacteristic
270C3×C22⋊C4—21no—
271C3×C42C22×D511yescharacteristic
272#213—102no—
273(C2×D3)⋊C4—22no—
274(C2×D3)⋊C4—102no—
275(C2×D3)⋊C4—101no—
276(C2×D3)⋊C4—102no—
277(C2×D3)⋊C4C22⋊D512yes—
278C4⋊Dic3—21no—
279C4⋊Dic3—21no—
280C4⋊Dic3—22no—
281C4⋊Dic3—51no—
282C4⋊Dic3C22⋊D511yescharacteristic
283C4×Dic3—102no—
284C2×C3⋊C8—51no—
285C2×C3⋊C8—51no—
286#204—202no—
287D3×C8—202no—
288C23×C5—21no—
289C23×C5—32no—
290C22×D5—61no—
291C5×Q8—62no—
292C5×Q8—62no—
293C5×D4—21no—
294C5×D4—62no—
295C5×D4—124no—
296C5×D4#23912yes—
297C2×C4×C5—21no—
298C2×C4×C5—32no—
299C2×C4×C5—42no—
300C2×C4×C5—62no—
301C2×C4×C5—62no—
302C2×C4×C5—62no—
303C2×C4×C5—124no—
304C2×C4×C5D3×D411yescharacteristic
305C2×C4×C5D3×D411yescharacteristic
306C2×C4×C5C23×D311yescharacteristic
307C22⋊D5—242no—
308C2×Dic5—22no—
309C2×Dic5—61no—
310D20—61no—
311D20—61no—
312C4×D5—122no—
313Dic10—81no—
314#168—22no—
315#168—61no—
316#168—61no—
317C2×#41—32no—
318C2×SD16—152no—
319C2×D8—151no—
320C22×C8—152no—
321C4⋊Q8—31no—
322C4⋊Q8—51no—
323#124—302no—
324#123—62no—
325C22⋊Q8—32no—
326#121—61no—
327#121—152no—
328#121—301no—
329C4×D4—31no—
330C4×D4—151no—
331C4×D4D3×D511yescharacteristic
332#117—152no—
333#117—301no—
334C2×C4⋊C4—61no—
335#107—151no—
336#106—52no—
337C4⋊C8—51no—
338#103—152no—
339#103—152no—
340#103—152no—
341#102—52no—
342#102—151no—
343#102—151no—
344C22⋊C8—152no—
345C4×C8—151no—
346C2×C3×C5—42no—
347C2×C3×C5#54611yescharacteristic
348C2×C3×C5C2×#13511yescharacteristic
349C2×C3×C5#57711yescharacteristic
350D15—81no—
351C5×D3—42no—
352C23×C3—21no—
353C22×D3—101no—
354C22×D3C2×C22⋊D512yes—
355C3×Q8—401no—
356C3×D4—21no—
357C3×D4#71112yes—
358C2×C3×C4—21no—
359C2×C3×C4—42no—
360C2×C3×C4—102no—
361C2×C3×C4C2×C22⋊D511yescharacteristic
362C2×C3×C4C2×C22⋊D511yescharacteristic
363C2×C3×C4C23×D511yescharacteristic
364#67—44no—
365C2×Dic3—22no—
366C2×Dic3—44no—
367C2×Dic3—101no—
368C2×Dic3C2×C22⋊D512yes—
369D12—22no—
370D12—101no—
371D12—101no—
372D12—402no—
373C4×D3—22no—
374C4×D3—22no—
375C4×D3—202no—
376Dic6—22no—
377Dic6—22no—
378C3×C8—102no—
379C3⋊C8—101no—
380C3⋊C8—101no—
381C22×C5—22no—
382C22×C5—41no—
383C22×C5—41no—
384C22×C5—62no—
385C22×C5—62no—
386C22×C5—62no—
387C22×C5C2×D3×D411yescharacteristic
388C2×D5—121no—
389C2×D5—121no—
390C2×D5—121no—
391C4×C5—21no—
392C4×C5—41no—
393C4×C5—62no—
394C4×C5—122no—
395C4×C5#91612yes—
396C4×C5C2×#23911yescharacteristic
397C4×C5#100911yescharacteristic
398Dic5—42no—
399Dic5—121no—
400#41—64no—
401#41—122no—
402C2×Q8—32no—
403C2×Q8—101no—
404C2×D4—31no—
405C2×D4—62no—
406C2×D4—151no—
407C2×D4—302no—
408C2×D4C2×D3×D511yescharacteristic
409C22×C4—21no—
410C22×C4—32no—
411C22×C4—62no—
412C22×C4—301no—
413SD16—602no—
414D8—302no—
415C2×C8—52no—
416C2×C8—151no—
417C2×C8—151no—
418C2×C8—602no—
419C4⋊C4—31no—
420C4⋊C4—52no—
421C4⋊C4—62no—
422C4⋊C4—61no—
423C4⋊C4—61no—
424C4⋊C4—101no—
425C4⋊C4—151no—
426C4⋊C4C2×D3×D511yescharacteristic
427C22⋊C4—21no—
428C22⋊C4—32no—
429C22⋊C4—62no—
430C22⋊C4—302no—
431C22⋊C4—302no—
432C22⋊C4—301no—
433C42—302no—
434C42C2×D3×D511yescharacteristic
435C3×C5#330311yescharacteristic, complemented, Hall
436C22×C3—22no—
437C22×C3—41no—
438C22×C3—41no—
439C22×C3C22×C22⋊D511yescharacteristic
440C2×D3—22no—
441C2×D3—22no—
442C2×D3—22no—
443C2×D3—201no—
444C2×D3—201no—
445C2×D3—201no—
446C3×C4—21no—
447C3×C4—41no—
448C3×C4—202no—
449C3×C4#409612yes—
450C3×C4C2×#71111yescharacteristic
451C3×C4#414411yescharacteristic
452Dic3—22no—
453Dic3—42no—
454Dic3—201no—
455C2×C5—42no—
456C2×C5—122no—
457C2×C5#563011yescharacteristic
458C2×C5C2×#91611yescharacteristic
459C2×C5#577511yescharacteristic
460D5—241no—
461C23—21no—
462C23—32no—
463C23—301no—
464Q8—62no—
465Q8—62no—
466Q8—401no—
467D4—21no—
468D4—62no—
469D4—124no—
470D4—301no—
471D4—301no—
472D4—1202no—
473D4#676812yes—
474C2×C4—21no—
475C2×C4—32no—
476C2×C4—42no—
477C2×C4—62no—
478C2×C4—62no—
479C2×C4—62no—
480C2×C4—102no—
481C2×C4—124no—
482C2×C4—301no—
483C2×C4—602no—
484C2×C4D3×C22⋊D511yescharacteristic
485C2×C4D3×C22⋊D511yescharacteristic
486C2×C4C22×D3×D511yescharacteristic
487C8—102no—
488C8—301no—
489C8—301no—
490C2×C3—42no—
491C2×C3#6614011yescharacteristic
492C2×C3C2×#409611yescharacteristic
493C2×C3#6617111yescharacteristic
494D3—42no—
495D3—401no—
496C5#8107611yescharacteristic, complemented, Sylow
497C22—22no—
498C22—41no—
499C22—41no—
500C22—62no—
501C22—62no—
502C22—62no—
503C22—601no—
504C22—601no—
505C22—601no—
506C22C2×D3×C22⋊D511yescharacteristic
507C4—21no—
508C4—41no—
509C4—62no—
510C4—122no—
511C4—202no—
512C4—601no—
513C4#9149512yes—
514C4C2×#676811yescharacteristic
515C4#9201811yescharacteristic
516C3#1061524211yescharacteristic, complemented, Sylow
517C2—42no—
518C2—122no—
519C2—1201no—
520C2#1175434911yescharacteristic
521C2C2×#9149511yescharacteristic
522C2#1175532411yescharacteristic
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×#5555C2semidirect
2. #11752710C2semidirect
3. C4×#6715C2semidirect
4. C3×#65297C2semidirect
5. #11749036C2semidirect
6. #11748965C2semidirect
7. #11747449C2semidirect
8. #11747211C2semidirect
9. #11747087C2semidirect
10. #11746998C2semidirect
11. #11746923C2semidirect
53. #65297D3semidirect
90. #5555D5semidirect
435. C3×C5#3303semidirect
496. C5#81076semidirect
516. C3#10615242semidirect

Maximal subgroups

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

subgroupindex
C5×#55552
#117527102
C4×#67152
C3×#652972
#117490362
#117489652
#117474492
#117472112
#117470872
#117469982
#117469232
#106152423
#810765