Finite Group Databasebeta

← #14417716

Subgroups — #14417716

Order 1920.

968 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756838C212yesmaximal, complemented
2D5×#904C211yesmaximal, characteristic
3C2×#91223C211yesmaximal, characteristic, complemented
4C3×#65718C211yesmaximal, characteristic, complemented
5#11751708C212yesmaximal, complemented
6AGL(1,𝔽5)×Dic12C212yesmaximal, complemented
7#11751661C214yesmaximal, complemented
8AGL(1,𝔽5)×D24C212yesmaximal, complemented
9#10617440—31nomaximal, Hall
10D5×#235C2212yes—
11C2×C4⋊ΓL(1,𝔽16)C2212yes—
12C3×#4130C2211yescharacteristic
13C3×D5⋊M16C2211yescharacteristic
14#67⋊AGL(1,𝔽5)—24no—
15D3×C4⋊AGL(1,𝔽5)—22no—
16AGL(1,𝔽5)×D12C2214yes—
17#91907—22no—
18AGL(1,𝔽5)×Dic6C2214yes—
19#91793C411yescharacteristic, complemented
20C5×#904C411yescharacteristic, complemented
21C2×C3×C8×D5C2211yescharacteristic
22#91303C412yescomplemented
23#91271C411yescharacteristic, complemented
24#91270C411yescharacteristic, complemented
25#91269C412yescomplemented
26D5×Dic12C2211yescharacteristic
27D5×D24C2211yescharacteristic
28D5×#204C2212yes—
29#91223C2212yes—
30#91223C2212yes—
31C3×#3992C2212yes—
32C3×C8×AGL(1,𝔽5)C2212yes—
33#91154C2214yes—
34#91152C2214yes—
35#78562—51nomaximal, Hall
36#66266—32no—
37D5×#135—31no—
38#65793—32no—
39Q16×AGL(1,𝔽5)—32no—
40#65735—34no—
41D8×AGL(1,𝔽5)—32no—
42C2×#3994—31no—
43#65718D311yescharacteristic, complemented
44C22×ΓL(1,𝔽16)—22no—
45C2×D3×AGL(1,𝔽5)—24no—
46#6817C2×C412yescomplemented
47C5×#235C2×C412yescomplemented
48C2×C3×C4×D5C2311yescharacteristic
49D5×#67—22no—
50#6782—22no—
51D5×D12C2312yes—
52C4×D3×D5—22no—
53#6773C2×C412yescomplemented
54#6772C2×C412yescomplemented
55#6771—22no—
56D5×Dic6C2312yes—
57C4⋊ΓL(1,𝔽16)—22no—
58C4⋊ΓL(1,𝔽16)C2314yes—
59C3×C22⋊AGL(1,𝔽5)D412yes—
60C3×#704—21no—
61C3×C4⋊AGL(1,𝔽5)D411yescharacteristic
62C3×C4⋊AGL(1,𝔽5)D411yescharacteristic
63C3×C4×AGL(1,𝔽5)C2312yes—
64C3×#700—21no—
65C3×D5⋊C8C2312yes—
66Dic3⋊AGL(1,𝔽5)—24no—
67(C2×D3)⋊AGL(1,𝔽5)—24no—
68Dic3×AGL(1,𝔽5)—24no—
69C2×C3×C5×C8C2×C411yescharacteristic, complemented
70Dic60C2×C411yescharacteristic, complemented
71D120C2×C411yescharacteristic, complemented
72#6706C2×C412yescomplemented
73C5×Dic12C2×C411yescharacteristic, complemented
74C5×D24C2×C411yescharacteristic, complemented
75C5×#204C2×C412yescomplemented
76C2×C3×#168C2×C411yescharacteristic, complemented
77C3×C8×D5C2×C411yescharacteristic, complemented
78C3×C8×D5C2×C411yescharacteristic, complemented
79C3×C8×D5C2311yescharacteristic
80C3×C8×D5C2311yescharacteristic
81#6663C2×C412yescomplemented
82#6660C2×C412yescomplemented
83#6659C2×C412yescomplemented
84#6654C2×C412yescomplemented
85C2×#904—51no—
86#5523—52no—
87C3×#405—51no—
88#5133—52no—
89C2×#818—51no—
90#4710—54no—
91C4×Dic12—52no—
92C4×D24—52no—
93D5×#41—32no—
94Q8×AGL(1,𝔽5)—34no—
95D4×AGL(1,𝔽5)—34no—
96D4×AGL(1,𝔽5)—64no—
97#4130—62no—
98#4130C2×D311yescharacteristic
99C2×C4⋊AGL(1,𝔽5)—32no—
100C2×C4⋊AGL(1,𝔽5)—62no—
101D5⋊M16C2×D311yescharacteristic
102C5×#135—31no—
103#4096—32no—
104#4065—31no—
105D5×Q16—31no—
106#4062—32no—
107D5×SD16—32no—
108#4058—31no—
109D5×D8—31no—
110#4050—31no—
111C2×C8×D5C2×D311yescharacteristic
112#4009—34no—
113#4007—34no—
114#3994—32no—
115#3994—32no—
116#3992C2×D312yes—
117C8×AGL(1,𝔽5)C2×D312yes—
118#2951—151noSylow
119C22×C3×D5C2×D411yescharacteristic
120C2×D3×D5—22no—
121C2×ΓL(1,𝔽16)—24no—
122C2×ΓL(1,𝔽16)—82no—
123C2×C3×AGL(1,𝔽5)C2×D412yes—
124C2×C3×AGL(1,𝔽5)C2×D412yes—
125D3×AGL(1,𝔽5)—84no—
126C2×C3×C4×C5C22×C411yescharacteristic
127#1220—22no—
128D60C22×C412yes—
129C4×D15—22no—
130Dic30C22×C412yes—
131C5×#67—22no—
132C5×D12C22×C412yes—
133C4×C5×D3—22no—
134C5×Dic6C22×C412yes—
135C2×C3×Dic5#4111yescharacteristic
136C3×C4×D5C22×C411yescharacteristic
137C3×C4×D5C2×D411yescharacteristic
138C3×C4×D5#4111yescharacteristic
139C3×C4×D5C2411yescharacteristic
140#1204—22no—
141#1203—22no—
142#1202—22no—
143#1201—22no—
144#1200—22no—
145D3×Dic5—22no—
146D5×Dic3—22no—
147C3×#170—22no—
148C3×C5×C8C22×C411yescharacteristic
149C3×C5×C8C22×C411yescharacteristic
150C3×#168C22×C411yescharacteristic
151C3×#168C22×C411yescharacteristic
152C2×#235—52no—
153C2×C3×M16—51no—
154C22×C3×C8—51no—
155C3×#117—51no—
156C2×C4⋊Dic3—52no—
157C2×Dic12—51no—
158#904—52no—
159#904—51no—
160#904—51no—
161#904—51no—
162#904—52no—
163#904AGL(1,𝔽5)11yescharacteristic, complemented
164C2×D24—51no—
165C2×#204—52no—
166#892—102no—
167D3×C4⋊C4—102no—
168#67⋊C4—104no—
169C4×D12—54no—
170C4×Dic6—54no—
171C3×#97—52no—
172C3×C4×C8—52no—
173#821—54no—
174#818—52no—
175#818—52no—
176#816—54no—
177C22×AGL(1,𝔽5)—62no—
178C22×AGL(1,𝔽5)—64no—
179C5×#41—32no—
180#713—32no—
181Q8×D5—32no—
182#711—32no—
183#711—62no—
184D4×D5—32no—
185D4×D5—62no—
186#709—32no—
187#709—62no—
188C2×C4×D5—62no—
189C2×C4×D5C22×D311yescharacteristic
190C22⋊AGL(1,𝔽5)—64no—
191C22⋊AGL(1,𝔽5)D1212yes—
192#704—21no—
193C4⋊AGL(1,𝔽5)—34no—
194C4⋊AGL(1,𝔽5)—64no—
195C4⋊AGL(1,𝔽5)—62no—
196C4⋊AGL(1,𝔽5)D1211yescharacteristic
197C4⋊AGL(1,𝔽5)D1211yescharacteristic
198C4×AGL(1,𝔽5)—64no—
199C4×AGL(1,𝔽5)C22×D312yes—
200#700—21no—
201D5⋊C8C22×D312yes—
202C5×Q16—31no—
203C5×SD16—32no—
204C5×D8—31no—
205C2×C5×C8C4×D311yescharacteristic, complemented
206#689—32no—
207#688—32no—
208#687—32no—
209#686—32no—
210C2×#168C4×D311yescharacteristic, complemented
211Dic20—31no—
212D40—31no—
213#677—32no—
214C8×D5C4×D311yescharacteristic, complemented
215C8×D5C4×D311yescharacteristic, complemented
216C8×D5C22×D311yescharacteristic
217C8×D5C22×D311yescharacteristic
218C2×#135—151no—
219#520—152no—
220#440—154no—
221C4×Q16—152no—
222C4×D8—152no—
223C2×#107—151no—
224#417—152no—
225#405—51no—
226C22×C3×C5C4×D411yescharacteristic
227C2×D15—22no—
228C2×C5×D3—22no—
229C2×C3×D5—21no—
230C2×C3×D5—21no—
231C2×C3×D5C4×D411yescharacteristic
232C2×C3×D5C22×D411yescharacteristic
233D3×D5—42no—
234D3×D5—42no—
235ΓL(1,𝔽16)—44no—
236C3×AGL(1,𝔽5)—22no—
237C3×AGL(1,𝔽5)—22no—
238C3×C4×C5C4×D411yescharacteristic
239C3×C4×C5C23×C411yescharacteristic
240Dic15—22no—
241C3×Dic5C4×D411yescharacteristic
242C3×Dic5C2×#4111yescharacteristic
243C5×Dic3—22no—
244C22×C3×C4—51no—
245C2×#67—102no—
246C22×Dic3—102no—
247#235—52no—
248#235—52no—
249#235—52no—
250#235—102no—
251#235—102no—
252#235C2×AGL(1,𝔽5)12yescomplemented
253C2×D12—52no—
254C2×C4×D3—102no—
255C2×C4×D3—104no—
256C2×Dic6—52no—
257C3×M16—101no—
258C3×M16—101no—
259C2×C3×C8—51no—
260C2×C3×C8—51no—
261C2×C3×C8—51no—
262C2×C3×C8—51no—
263C2×C3×C8—51no—
264C2×C3×C8—52no—
265C2×C3×C8C2×AGL(1,𝔽5)11yescharacteristic, complemented
266C3×C4⋊C4—51no—
267C3×C4⋊C4—51no—
268C3×C22⋊C4—52no—
269C3×C42—52no—
270(C2×D3)⋊C4—104no—
271C4⋊Dic3—54no—
272C4⋊Dic3—102no—
273Dic3⋊C4—104no—
274C4×Dic3—104no—
275Dic12—52no—
276Dic12—51no—
277Dic12C2×AGL(1,𝔽5)11yescharacteristic, complemented
278D24—52no—
279D24—51no—
280D24C2×AGL(1,𝔽5)11yescharacteristic, complemented
281#204—52no—
282#204—52no—
283#204—52no—
284#204C2×AGL(1,𝔽5)12yescomplemented
285C22×D5—62no—
286C22×D5C2×D1211yescharacteristic
287C2×AGL(1,𝔽5)—64no—
288C2×AGL(1,𝔽5)—244no—
289C2×AGL(1,𝔽5)—242no—
290C2×AGL(1,𝔽5)C2×D1212yes—
291C2×AGL(1,𝔽5)C2×D1212yes—
292C5×Q8—32no—
293C5×D4—32no—
294C5×D4—62no—
295C2×C4×C5—62no—
296C2×C4×C5C2×C4×D311yescharacteristic
297C22⋊D5—62no—
298C22⋊D5—62no—
299C22⋊D5—62no—
300C2×Dic5—62no—
301C2×Dic5#23511yescharacteristic
302D20—32no—
303D20—62no—
304C4×D5—62no—
305C4×D5—62no—
306C4×D5—62no—
307C4×D5C2×C4×D311yescharacteristic
308C4×D5C2×D1211yescharacteristic
309C4×D5#23511yescharacteristic
310C4×D5C23×D311yescharacteristic
311Dic10—32no—
312Dic10—62no—
313#170—22no—
314C5×C8C2×C4×D311yescharacteristic
315C5×C8C2×C4×D311yescharacteristic
316#168C2×C4×D311yescharacteristic
317#168C2×C4×D311yescharacteristic
318C2×#41—152no—
319#135—31no—
320#135—151no—
321#135—152no—
322#135—151no—
323#135—152no—
324#135—151no—
325C2×Q16—151no—
326C2×SD16—152no—
327C2×D8—151no—
328C2×M16—51no—
329C22×C8—51no—
330C4×Q8—154no—
331C4×D4—154no—
332C4×D4—304no—
333#117—51no—
334#117—302no—
335C2×C4⋊C4—152no—
336C2×C4⋊C4—302no—
337#107—152no—
338#107—152no—
339#103—154no—
340#102—154no—
341#97—52no—
342C4×C8—52no—
343C2×C3×C5—21no—
344C2×C3×C5C2×C4×D411yescharacteristic
345D15—42no—
346C3×D5—21no—
347C3×D5#57611yescharacteristic
348C3×D5#57811yescharacteristic
349C5×D3—42no—
350C23×C3—51no—
351C22×D3—102no—
352C2×C3×C4—51no—
353C2×C3×C4—51no—
354C2×C3×C4—51no—
355C2×C3×C4—51no—
356C2×C3×C4—51no—
357C2×C3×C4—52no—
358C2×C3×C4—52no—
359C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
360#67—22no—
361#67—102no—
362#67—102no—
363#67—102no—
364C2×Dic3—102no—
365C2×Dic3—104no—
366C2×Dic3—402no—
367D12—52no—
368D12—102no—
369D12C22×AGL(1,𝔽5)12yes—
370C4×D3—22no—
371C4×D3—102no—
372C4×D3—102no—
373C4×D3—102no—
374C4×D3—404no—
375Dic6—52no—
376Dic6—102no—
377Dic6C22×AGL(1,𝔽5)12yes—
378C3×C8—51no—
379C3×C8—51no—
380C3×C8—102no—
381C3×C8C22×AGL(1,𝔽5)11yescharacteristic
382C3×C8C22×AGL(1,𝔽5)11yescharacteristic
383C22×C5—62no—
384C22×C5C4×D1211yescharacteristic
385C2×D5—21no—
386C2×D5—21no—
387C2×D5—62no—
388C2×D5—122no—
389C2×D5—122no—
390C2×D5C4×D1211yescharacteristic
391C2×D5C22×D1211yescharacteristic
392AGL(1,𝔽5)—22no—
393AGL(1,𝔽5)—22no—
394AGL(1,𝔽5)—124no—
395C4×C5—62no—
396C4×C5C4×D1211yescharacteristic
397C4×C5C22×C4×D311yescharacteristic
398Dic5—62no—
399Dic5C4×D1211yescharacteristic
400Dic5C2×#23511yescharacteristic
401#41—32no—
402#41—152no—
403#41—152no—
404#41—152no—
405#41—302no—
406#41—302no—
407C2×Q8—152no—
408C2×D4—152no—
409C2×D4—302no—
410C22×C4—51no—
411C22×C4—302no—
412C22×C4—304no—
413C22×C4—302no—
414Q16—31no—
415Q16—152no—
416Q16—151no—
417SD16—32no—
418SD16—152no—
419SD16—152no—
420SD16—152no—
421D8—31no—
422D8—152no—
423D8—151no—
424M16—101no—
425M16—101no—
426C2×C8—51no—
427C2×C8—51no—
428C2×C8—51no—
429C2×C8—51no—
430C2×C8—51no—
431C2×C8—52no—
432C2×C8D3×AGL(1,𝔽5)11yescharacteristic, complemented
433C4⋊C4—51no—
434C4⋊C4—51no—
435C4⋊C4—154no—
436C4⋊C4—302no—
437C4⋊C4—304no—
438C22⋊C4—52no—
439C22⋊C4—304no—
440C42—52no—
441C42—304no—
442C3×C5#295111yescharacteristic, complemented, Hall
443C22×C3—51no—
444C22×C3—51no—
445C22×C3—101no—
446C22×C3—101no—
447C22×C3D4×AGL(1,𝔽5)11yescharacteristic
448C2×D3—22no—
449C2×D3—102no—
450C2×D3—202no—
451C2×D3—202no—
452C3×C4—51no—
453C3×C4—51no—
454C3×C4—102no—
455C3×C4—102no—
456C3×C4D4×AGL(1,𝔽5)11yescharacteristic
457C3×C4C23×AGL(1,𝔽5)11yescharacteristic
458Dic3—22no—
459Dic3—102no—
460Dic3—204no—
461C2×C5—21no—
462C2×C5—122no—
463C2×C5C2×C4×D1211yescharacteristic
464D5—21no—
465D5—122no—
466D5#575211yescharacteristic
467D5#575311yescharacteristic
468C23—51no—
469C23—302no—
470Q8—32no—
471Q8—152no—
472Q8—302no—
473D4—32no—
474D4—62no—
475D4—152no—
476D4—302no—
477D4—302no—
478D4—302no—
479D4—302no—
480C2×C4—51no—
481C2×C4—51no—
482C2×C4—51no—
483C2×C4—51no—
484C2×C4—51no—
485C2×C4—52no—
486C2×C4—52no—
487C2×C4—62no—
488C2×C4—302no—
489C2×C4—302no—
490C2×C4—302no—
491C2×C4—302no—
492C2×C4—304no—
493C2×C4—1204no—
494C2×C4—1202no—
495C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
496C8—51no—
497C8—51no—
498C8—102no—
499C8C2×D3×AGL(1,𝔽5)11yescharacteristic
500C8C2×D3×AGL(1,𝔽5)11yescharacteristic
501C2×C3—21no—
502C2×C3—51no—
503C2×C3—51no—
504C2×C3—101no—
505C2×C3C2×D4×AGL(1,𝔽5)11yescharacteristic
506D3—42no—
507D3—202no—
508C5#7856211yescharacteristic, complemented, Sylow
509C22—51no—
510C22—51no—
511C22—62no—
512C22—101no—
513C22—101no—
514C22—302no—
515C22—602no—
516C22—602no—
517C22AGL(1,𝔽5)×D1211yescharacteristic
518C4—51no—
519C4—51no—
520C4—62no—
521C4—102no—
522C4—102no—
523C4—302no—
524C4—604no—
525C4AGL(1,𝔽5)×D1211yescharacteristic
526C4C22×D3×AGL(1,𝔽5)11yescharacteristic
527C3#1061744011yescharacteristic, complemented, Sylow
528C2—21no—
529C2—51no—
530C2—51no—
531C2—101no—
532C2—122no—
533C2—602no—
534C2C2×AGL(1,𝔽5)×D1211yescharacteristic
subgroupG/H
centerC2C2×AGL(1,𝔽5)×D12
commutatorC3×C4×C5C23×C4
FrattiniC4C22×D3×AGL(1,𝔽5)
FittingC2×C3×C5×C8C2×C4
socleC2×C3×C5C2×C4×D4
prime radicalD5×#904C2
cosocleC3×C4×D5C24
hypercenterC2×C8D3×AGL(1,𝔽5)
hypocenterC3×C5#2951
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. #11756838C2semidirect
3. C2×#91223C2semidirect
4. C3×#65718C2semidirect
5. #11751708C2semidirect
6. AGL(1,𝔽5)×Dic12C2semidirect
7. #11751661C2semidirect
8. AGL(1,𝔽5)×D24C2semidirect
19. #91793C4semidirect
20. C5×#904C4semidirect
22. #91303C4semidirect
23. #91271C4semidirect
24. #91270C4semidirect
25. #91269C4semidirect
43. #65718D3semidirect
46. #6817C2×C4semidirect
47. C5×#235C2×C4semidirect
53. #6773C2×C4semidirect
54. #6772C2×C4semidirect
69. C2×C3×C5×C8C2×C4semidirect
70. Dic60C2×C4semidirect
71. D120C2×C4semidirect
72. #6706C2×C4semidirect
73. C5×Dic12C2×C4semidirect
74. C5×D24C2×C4semidirect
75. C5×#204C2×C4semidirect
76. C2×C3×#168C2×C4semidirect
77. C3×C8×D5C2×C4semidirect
78. C3×C8×D5C2×C4semidirect
81. #6663C2×C4semidirect
82. #6660C2×C4semidirect
83. #6659C2×C4semidirect
84. #6654C2×C4semidirect
163. #904AGL(1,𝔽5)semidirect
205. C2×C5×C8C4×D3semidirect
210. C2×#168C4×D3semidirect
214. C8×D5C4×D3semidirect
215. C8×D5C4×D3semidirect
252. #235C2×AGL(1,𝔽5)semidirect
265. C2×C3×C8C2×AGL(1,𝔽5)semidirect
277. Dic12C2×AGL(1,𝔽5)semidirect
280. D24C2×AGL(1,𝔽5)semidirect
284. #204C2×AGL(1,𝔽5)semidirect
432. C2×C8D3×AGL(1,𝔽5)semidirect
442. C3×C5#2951semidirect
508. C5#78562semidirect
527. C3#10617440semidirect

Maximal subgroups

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

subgroupindex
#117568382
D5×#9042
C2×#912232
C3×#657182
#117517082
AGL(1,𝔽5)×Dic122
#117516612
AGL(1,𝔽5)×D242
#106174403
#785625