Finite Group Databasebeta

← #14460206

Subgroups — #14460206

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×#5734C211yesmaximal, characteristic, complemented
2#11752532C211yesmaximal, characteristic, complemented
3#11751194C211yesmaximal, characteristic, complemented
4C3×#65373C211yesmaximal, characteristic, complemented
5#11747515C212yesmaximal, complemented
6#11747502C212yesmaximal, complemented
7#11747448C212yesmaximal, complemented
8#11747177C212yesmaximal, complemented
9#11747077C211yesmaximal, characteristic, complemented
10C4×#6652C211yesmaximal, characteristic, complemented
11#11746998C211yesmaximal, characteristic, complemented
12#10615383—31nomaximal, Hall
13C2×#6765C2212yes—
14C3×C5×C4⋊Q8C2211yescharacteristic
15#91758C2211yescharacteristic
16C5×#947—22no—
17C5×#895—21no—
18C5×#895C2212yes—
19C5×#892—21no—
20C5×#892C2212yes—
21C4×C5×D12C2211yescharacteristic
22C4×C5×D12C2211yescharacteristic
23C3×C4×D20C2211yescharacteristic
24#91449—21no—
25#91414—22no—
26D5×C4⋊Dic3—21no—
27#91365C2212yes—
28C2×#6659C2212yes—
29C2×#6652C2211yescharacteristic
30C2×D3×#168C2212yes—
31#91094C2212yes—
32#91094C2212yes—
33C4×C3⋊#168C2211yescharacteristic
34C3×#3941C2212yes—
35C3×#3940C2212yes—
36C3×C4⋊#168C2211yescharacteristic
37#90989C2211yescharacteristic
38#90975C2212yes—
39#90966C2211yescharacteristic
40#90964C2211yescharacteristic
41(C2×D3)⋊#168C2212yes—
42#81076—51nomaximal, Hall
43C5×#555—31no—
44#65885—31no—
45#65382—31no—
46#65373D311yescharacteristic, complemented
47#65330—32no—
48#65326—32no—
49#65308—31no—
50C4×#686—31no—
51D4×#168—31no—
52#65291—32no—
53#65285—32no—
54C2×C3×C5×Q8—21no—
55C2×Dic30C2312yes—
56C2×C5×D12—22no—
57C2×C5×D12D411yescharacteristic
58C2×C5×D12C2311yescharacteristic
59C2×C4×C5×D3—21no—
60C2×C4×C5×D3C2312yes—
61C2×C3×D20C2311yescharacteristic
62C2×C3×C4×D5—21no—
63C2×#1201—22no—
64C2×D5×Dic3—22no—
65#6765—24no—
66#6765—42no—
67C3×C5×C4⋊C4—21no—
68C3×C5×C4⋊C4C2312yes—
69C3×C42×C5C2311yescharacteristic
70C4⋊Dic15—21no—
71C4⋊Dic15—21no—
72C2×C3⋊#168C2311yescharacteristic
73C2×C3⋊#168C2311yescharacteristic
74C5×(C2×D3)⋊C4—22no—
75C5×(C2×D3)⋊C4—21no—
76C5×(C2×D3)⋊C4—22no—
77C5×(C2×D3)⋊C4D412yes—
78C5×C4⋊Dic3D411yescharacteristic
79C5×C4⋊Dic3C2311yescharacteristic
80C4×C5×Dic3—22no—
81C3×#685—21no—
82C3×C4⋊Dic5C2311yescharacteristic
83C2×C3×#168C2312yes—
84#6668—22no—
85(C2×D3)⋊Dic5—22no—
86#6659—42no—
87#6652—22no—
88D3×#168—42no—
89#5734D511yescharacteristic, complemented
90#5555—51no—
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×#709—32no—
101C5×C4⋊Q8C2×D311yescharacteristic
102C5×#124—62no—
103C5×#121—32no—
104C5×#121—61no—
105C4×C5×D4—31no—
106C4×C5×D4—31no—
107C5×#117—32no—
108C5×#117—61no—
109#4084—61no—
110C2×#687—32no—
111C2×#686—31no—
112#4070—32no—
113C22×#168—32no—
114D5×C4⋊C4—61no—
115#4029—62no—
116C4×D20C2×D311yescharacteristic
117#4015—31no—
118#3964—31no—
119C22⋊#168—32no—
120#3942—32no—
121#3942—32no—
122#3942—32no—
123#3941—31no—
124#3941C2×D312yes—
125#3940C2×D312yes—
126#3939—31no—
127C4⋊#168C2×D311yescharacteristic
128C4×#168—31no—
129#3303—151noSylow
130C22×C5×D3—21no—
131C22×C5×D3C2×D412yes—
132C22×C3×D5—21no—
133C3×C5×Q8—81no—
134C2×C3×C4×C5—22no—
135C2×C3×C4×C5C2×D411yescharacteristic
136C2×C3×C4×C5C2×D411yescharacteristic
137C2×C3×C4×C5C2411yescharacteristic
138C2×Dic15—22no—
139Dic30—22no—
140Dic30—22no—
141C2×C5×Dic3—21no—
142C2×C5×Dic3C2×D412yes—
143C5×D12—22no—
144C5×D12—21no—
145C5×D12—21no—
146C5×D12—82no—
147C4×C5×D3—22no—
148C4×C5×D3—22no—
149C4×C5×D3—42no—
150C2×C3×Dic5—21no—
151C3×D20—21no—
152C3×D20#4112yes—
153C3×C4×D5—42no—
154#1201—44no—
155D5×Dic3—44no—
156C3⋊#168—21no—
157C3⋊#168—21no—
158C3×#168—22no—
159C2×#237—52no—
160C3×C4⋊Q8C2×D511yescharacteristic
161C3×C4×D4—51no—
162#947—22no—
163C2×#213—51no—
164#930—101no—
165C2×C4⋊Dic3—101no—
166C2×#204—52no—
167C2×D3×C8—52no—
168#897—52no—
169#895—21no—
170#895C2×D512yes—
171#892—21no—
172#892C2×D512yes—
173#886—102no—
174C4×D12C2×D511yescharacteristic
175C4×D12C2×D511yescharacteristic
176#869—51no—
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—
188C2×C5×Q8—21no—
189C2×C5×D4—31no—
190C2×C5×D4—31no—
191C2×C5×D4—62no—
192C22×C4×C5—32no—
193C22×C4×C5—61no—
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—21no—
202C5×C4⋊C4—31no—
203C5×C4⋊C4—31no—
204C5×C4⋊C4C22×D312yes—
205C5×C22⋊C4—32no—
206C5×C22⋊C4—62no—
207C5×C22⋊C4—62no—
208C5×C22⋊C4—61no—
209C42×C5—62no—
210C42×C5C22×D311yescharacteristic
211C22⋊Dic5—62no—
212#687—122no—
213#686—62no—
214#685—21no—
215C4⋊Dic5—61no—
216C4⋊Dic5—61no—
217C4⋊Dic5C22×D311yescharacteristic
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—51no—
228#465—152no—
229#461—151no—
230#449—152no—
231C4×D8—151no—
232C8×D4—151no—
233#416—152no—
234C22×C3×C5C22×D411yescharacteristic
235C2×C5×D3—22no—
236C2×C5×D3—22no—
237C2×C5×D3—22no—
238C2×C5×D3—41no—
239C2×C5×D3—41no—
240C2×C5×D3—41no—
241C2×C3×D5—22no—
242C2×C3×D5—41no—
243C2×C3×D5—41no—
244C3×C4×C5—21no—
245C3×C4×C5—42no—
246C3×C4×C5#13512yes—
247C3×C4×C5C2×#4111yescharacteristic
248C3×C4×C5H(5,𝔽2)11yescharacteristic
249Dic15—42no—
250C3×Dic5—41no—
251C5×Dic3—22no—
252C5×Dic3—41no—
253C2×C3×Q8—21no—
254C2×C3×D4—51no—
255C22×C3×C4—101no—
256C2×#67—102no—
257C22×Dic3—102no—
258#237—104no—
259#237—202no—
260C2×D12—22no—
261C2×D12C22⋊D511yescharacteristic
262C2×D12C22×D511yescharacteristic
263C2×C4×D3—21no—
264C2×C4×D3C22×D512yes—
265C2×Dic6—52no—
266C2×C3×C8—52no—
267C3×C4⋊C4—21no—
268C3×C4⋊C4—51no—
269C3×C4⋊C4C22×D512yes—
270C3×C22⋊C4—101no—
271C3×C42C22×D511yescharacteristic
272#213—102no—
273(C2×D3)⋊C4—22no—
274(C2×D3)⋊C4—21no—
275(C2×D3)⋊C4—22no—
276(C2×D3)⋊C4—102no—
277(C2×D3)⋊C4C22⋊D512yes—
278C4⋊Dic3—101no—
279C4⋊Dic3—101no—
280C4⋊Dic3—102no—
281C4⋊Dic3C22⋊D511yescharacteristic
282C4⋊Dic3C22×D511yescharacteristic
283C4×Dic3—22no—
284C2×C3⋊C8—51no—
285C2×C3⋊C8—51no—
286#204—202no—
287D3×C8—202no—
288C23×C5—32no—
289C23×C5—61no—
290C22×D5—21no—
291C5×Q8—81no—
292C5×D4—62no—
293C5×D4—61no—
294C5×D4—61no—
295C5×D4—242no—
296C2×C4×C5—22no—
297C2×C4×C5—32no—
298C2×C4×C5—62no—
299C2×C4×C5—62no—
300C2×C4×C5—61no—
301C2×C4×C5—122no—
302C2×C4×C5D3×D411yescharacteristic
303C2×C4×C5D3×D411yescharacteristic
304C2×C4×C5C23×D311yescharacteristic
305C22⋊D5—124no—
306C2×Dic5—21no—
307C2×Dic5—62no—
308D20—21no—
309D20#23912yes—
310C4×D5—42no—
311C4×D5—124no—
312Dic10—62no—
313Dic10—62no—
314#168—22no—
315#168—61no—
316#168—61no—
317C2×#41—152no—
318C2×SD16—152no—
319C2×D8—151no—
320C22×C8—152no—
321C4⋊Q8—151no—
322C4⋊Q8D3×D511yescharacteristic
323#124—62no—
324#123—302no—
325C22⋊Q8—152no—
326#121—32no—
327#121—61no—
328#121—301no—
329C4×D4—31no—
330C4×D4—31no—
331C4×D4—51no—
332#117—32no—
333#117—61no—
334C2×C4⋊C4—301no—
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#54611yescharacteristic
347C2×C3×C5C2×#13511yescharacteristic
348C2×C3×C5#57711yescharacteristic
349C3×D5—42no—
350C5×D3—42no—
351C5×D3—81no—
352C23×C3—101no—
353C22×D3—21no—
354C22×D3C2×C22⋊D512yes—
355C3×Q8—81no—
356C3×D4—52no—
357C3×D4—101no—
358C2×C3×C4—22no—
359C2×C3×C4—101no—
360C2×C3×C4—202no—
361C2×C3×C4C2×C22⋊D511yescharacteristic
362C2×C3×C4C2×C22⋊D511yescharacteristic
363C2×C3×C4C23×D511yescharacteristic
364#67—204no—
365C2×Dic3—21no—
366C2×Dic3—102no—
367C2×Dic3—204no—
368C2×Dic3C2×C22⋊D512yes—
369D12—22no—
370D12—21no—
371D12—21no—
372D12—82no—
373C4×D3—22no—
374C4×D3—22no—
375C4×D3—42no—
376Dic6—102no—
377Dic6—102no—
378C3×C8—102no—
379C3⋊C8—101no—
380C3⋊C8—101no—
381C22×C5—62no—
382C22×C5—62no—
383C22×C5—62no—
384C22×C5—121no—
385C22×C5—121no—
386C22×C5—121no—
387C22×C5C2×D3×D411yescharacteristic
388C2×D5—22no—
389C2×D5—41no—
390C2×D5—41no—
391C4×C5—21no—
392C4×C5—42no—
393C4×C5—62no—
394C4×C5—121no—
395C4×C5#91612yes—
396C4×C5C2×#23911yescharacteristic
397C4×C5#100911yescharacteristic
398Dic5—41no—
399Dic5—122no—
400#41—304no—
401#41—602no—
402C2×Q8—21no—
403C2×Q8—152no—
404C2×D4—31no—
405C2×D4—31no—
406C2×D4—51no—
407C2×D4—62no—
408C2×D4—302no—
409C22×C4—32no—
410C22×C4—61no—
411C22×C4—101no—
412C22×C4—302no—
413SD16—602no—
414D8—302no—
415C2×C8—52no—
416C2×C8—151no—
417C2×C8—151no—
418C2×C8—602no—
419C4⋊C4—21no—
420C4⋊C4—31no—
421C4⋊C4—31no—
422C4⋊C4—51no—
423C4⋊C4—302no—
424C4⋊C4—301no—
425C4⋊C4—301no—
426C4⋊C4C2×D3×D512yes—
427C22⋊C4—32no—
428C22⋊C4—62no—
429C22⋊C4—62no—
430C22⋊C4—61no—
431C22⋊C4—101no—
432C22⋊C4—302no—
433C42—62no—
434C42C2×D3×D511yescharacteristic
435C3×C5#330311yescharacteristic, complemented, Hall
436C22×C3—102no—
437C22×C3—201no—
438C22×C3—201no—
439C22×C3C22×C22⋊D511yescharacteristic
440C2×D3—22no—
441C2×D3—22no—
442C2×D3—22no—
443C2×D3—41no—
444C2×D3—41no—
445C2×D3—41no—
446C3×C4—21no—
447C3×C4—42no—
448C3×C4—201no—
449C3×C4#409612yes—
450C3×C4D5×#4111yescharacteristic
451C3×C4#414911yescharacteristic
452Dic3—22no—
453Dic3—41no—
454Dic3—202no—
455C2×C5—122no—
456C2×C5—241no—
457C2×C5#563011yescharacteristic
458C2×C5C2×#91611yescharacteristic
459C2×C5#577511yescharacteristic
460D5—42no—
461C23—32no—
462C23—61no—
463C23—101no—
464Q8—81no—
465Q8—302no—
466Q8—302no—
467D4—52no—
468D4—62no—
469D4—61no—
470D4—61no—
471D4—101no—
472D4—242no—
473D4—604no—
474C2×C4—22no—
475C2×C4—32no—
476C2×C4—62no—
477C2×C4—62no—
478C2×C4—61no—
479C2×C4—101no—
480C2×C4—122no—
481C2×C4—202no—
482C2×C4—302no—
483C2×C4—604no—
484C2×C4D3×C22⋊D511yescharacteristic
485C2×C4D3×C22⋊D511yescharacteristic
486C2×C4C22×D3×D511yescharacteristic
487C8—102no—
488C8—301no—
489C8—301no—
490C2×C3—202no—
491C2×C3C2×#409611yescharacteristic
492C2×C3#6616311yescharacteristic
493C2×C3#6617011yescharacteristic
494D3—42no—
495D3—81no—
496C5#8107611yescharacteristic, complemented, Sylow
497C22—62no—
498C22—62no—
499C22—62no—
500C22—102no—
501C22—121no—
502C22—121no—
503C22—121no—
504C22—201no—
505C22—201no—
506C22C2×D3×C22⋊D511yescharacteristic
507C4—21no—
508C4—42no—
509C4—62no—
510C4—121no—
511C4—201no—
512C4—602no—
513C4#9151712yes—
514C4D5×#23911yescharacteristic
515C4#9203511yescharacteristic
516C3#1061538311yescharacteristic, complemented, Sylow
517C2—122no—
518C2—202no—
519C2—241no—
520C2#1175472011yescharacteristic
521C2C2×#9151711yescharacteristic
522C2#1175527711yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊D5
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C5×C4⋊Q8C22
socleC22×C3×C5C22×D4
hypercenterC4⋊Q8D3×D5
hypocenterC3×C5#3303
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C5×#5734C2semidirect
2. #11752532C2semidirect
3. #11751194C2semidirect
4. C3×#65373C2semidirect
5. #11747515C2semidirect
6. #11747502C2semidirect
7. #11747448C2semidirect
8. #11747177C2semidirect
9. #11747077C2semidirect
10. C4×#6652C2semidirect
11. #11746998C2semidirect
46. #65373D3semidirect
89. #5734D5semidirect
435. C3×C5#3303semidirect
496. C5#81076semidirect
516. C3#10615383semidirect

Maximal subgroups

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

subgroupindex
C5×#57342
#117525322
#117511942
C3×#653732
#117475152
#117475022
#117474482
#117471772
#117470772
C4×#66522
#117469982
#106153833
#810765