Finite Group Databasebeta

← #14417824

Subgroups — #14417824

Order 1920.

968 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756838C212yesmaximal, complemented
2D5×#904C211yesmaximal, characteristic
3C2×#91222C211yesmaximal, characteristic, complemented
4C3×#65718C211yesmaximal, characteristic, complemented
5#11751708C212yesmaximal, complemented
6#11751668C212yesmaximal, complemented
7#11751662C212yesmaximal, complemented
8AGL(1,𝔽5)×#204C214yesmaximal, complemented
9#10617441—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#91222C2212yes—
30#91222C2212yes—
31C3×#3992C2212yes—
32C3×C8×AGL(1,𝔽5)C2212yes—
33#91154C2214yes—
34#91152C2214yes—
35#78574—51nomaximal, Hall
36#66266—32no—
37D5×#135—31no—
38#65793—32no—
39#65739—32no—
40SD16×AGL(1,𝔽5)—34no—
41#65731—32no—
42C2×#3993—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)D412yes—
62C3×C4×AGL(1,𝔽5)C2312yes—
63C3×#700—21no—
64C3×D5⋊C8C2312yes—
65Dic3⋊AGL(1,𝔽5)—24no—
66(C2×D3)⋊AGL(1,𝔽5)—24no—
67Dic3×AGL(1,𝔽5)—24no—
68C2×C3×C5×C8C2×C411yescharacteristic, complemented
69Dic60C2×C411yescharacteristic, complemented
70D120C2×C411yescharacteristic, complemented
71#6706C2×C412yescomplemented
72C5×Dic12C2×C411yescharacteristic, complemented
73C5×D24C2×C411yescharacteristic, complemented
74C5×#204C2×C412yescomplemented
75C2×C3×#168C2×C411yescharacteristic, complemented
76C3×C8×D5C2×C411yescharacteristic, complemented
77C3×C8×D5C2×C411yescharacteristic, complemented
78C3×C8×D5C2311yescharacteristic
79C3×C8×D5C2311yescharacteristic
80#6663C2×C412yescomplemented
81#6660C2×C412yescomplemented
82#6659C2×C412yescomplemented
83#6654C2×C412yescomplemented
84C2×#904—51no—
85#5523—52no—
86C3×#405—51no—
87#5133—52no—
88C2×#817—51no—
89#4716—52no—
90#4711—52no—
91C4×#204—54no—
92D5×#41—32no—
93Q8×AGL(1,𝔽5)—34no—
94D4×AGL(1,𝔽5)—34no—
95D4×AGL(1,𝔽5)—64no—
96#4130—62no—
97#4130C2×D311yescharacteristic
98C2×C4⋊AGL(1,𝔽5)—32no—
99C2×C4⋊AGL(1,𝔽5)—62no—
100D5⋊M16C2×D311yescharacteristic
101C5×#135—31no—
102#4096—32no—
103#4065—31no—
104D5×Q16—31no—
105#4062—32no—
106D5×SD16—32no—
107#4058—31no—
108D5×D8—31no—
109#4050—31no—
110C2×C8×D5C2×D311yescharacteristic
111#4009—34no—
112#4007—34no—
113#3993—32no—
114#3993—32no—
115#3992C2×D312yes—
116C8×AGL(1,𝔽5)C2×D312yes—
117#2952—151noSylow
118C22×C3×D5C2×D411yescharacteristic
119C2×D3×D5—22no—
120C2×ΓL(1,𝔽16)—24no—
121C2×ΓL(1,𝔽16)—82no—
122C2×C3×AGL(1,𝔽5)C2×D414yes—
123D3×AGL(1,𝔽5)—84no—
124C2×C3×C4×C5C22×C411yescharacteristic
125#1220—22no—
126D60C22×C412yes—
127C4×D15—22no—
128Dic30C22×C412yes—
129C5×#67—22no—
130C5×D12C22×C412yes—
131C4×C5×D3—22no—
132C5×Dic6C22×C412yes—
133C2×C3×Dic5#4111yescharacteristic
134C3×C4×D5C22×C411yescharacteristic
135C3×C4×D5C2×D411yescharacteristic
136C3×C4×D5#4111yescharacteristic
137C3×C4×D5C2411yescharacteristic
138#1204—22no—
139#1203—22no—
140#1202—22no—
141#1201—22no—
142#1200—22no—
143D3×Dic5—22no—
144D5×Dic3—22no—
145C3×#170—22no—
146C3×C5×C8C22×C411yescharacteristic
147C3×C5×C8C22×C411yescharacteristic
148C3×#168C22×C411yescharacteristic
149C3×#168C22×C411yescharacteristic
150C2×#235—52no—
151C2×C3×M16—51no—
152C22×C3×C8—51no—
153C3×#117—51no—
154C2×C4⋊Dic3—52no—
155C2×Dic12—51no—
156#904—52no—
157#904—51no—
158#904—51no—
159#904—51no—
160#904—52no—
161#904AGL(1,𝔽5)11yescharacteristic, complemented
162C2×D24—51no—
163C2×#204—52no—
164#892—102no—
165D3×C4⋊C4—102no—
166#67⋊C4—104no—
167C4×D12—54no—
168C4×Dic6—54no—
169C3×#97—52no—
170C3×C4×C8—52no—
171#821—54no—
172#817—52no—
173#817—52no—
174#816—54no—
175C22×AGL(1,𝔽5)—62no—
176C22×AGL(1,𝔽5)—64no—
177C5×#41—32no—
178#713—32no—
179Q8×D5—32no—
180#711—32no—
181#711—62no—
182D4×D5—32no—
183D4×D5—62no—
184#709—32no—
185#709—62no—
186C2×C4×D5—62no—
187C2×C4×D5C22×D311yescharacteristic
188C22⋊AGL(1,𝔽5)—64no—
189C22⋊AGL(1,𝔽5)D1212yes—
190#704—21no—
191C4⋊AGL(1,𝔽5)—34no—
192C4⋊AGL(1,𝔽5)—64no—
193C4⋊AGL(1,𝔽5)—62no—
194C4⋊AGL(1,𝔽5)D1212yes—
195C4×AGL(1,𝔽5)—64no—
196C4×AGL(1,𝔽5)C22×D312yes—
197#700—21no—
198D5⋊C8C22×D312yes—
199C5×Q16—31no—
200C5×SD16—32no—
201C5×D8—31no—
202C2×C5×C8C4×D311yescharacteristic, complemented
203#689—32no—
204#688—32no—
205#687—32no—
206#686—32no—
207C2×#168C4×D311yescharacteristic, complemented
208Dic20—31no—
209D40—31no—
210#677—32no—
211C8×D5C4×D311yescharacteristic, complemented
212C8×D5C4×D311yescharacteristic, complemented
213C8×D5C22×D311yescharacteristic
214C8×D5C22×D311yescharacteristic
215C2×#135—151no—
216#520—152no—
217#442—152no—
218#441—152no—
219C4×SD16—154no—
220C2×#106—151no—
221#417—152no—
222#405—51no—
223C22×C3×C5C4×D411yescharacteristic
224C2×D15—22no—
225C2×C5×D3—22no—
226C2×C3×D5—22no—
227C2×C3×D5C4×D411yescharacteristic
228C2×C3×D5C22×D411yescharacteristic
229D3×D5—42no—
230D3×D5—42no—
231ΓL(1,𝔽16)—44no—
232C3×AGL(1,𝔽5)—24no—
233C3×C4×C5C4×D411yescharacteristic
234C3×C4×C5C23×C411yescharacteristic
235Dic15—22no—
236C3×Dic5C4×D411yescharacteristic
237C3×Dic5C2×#4111yescharacteristic
238C5×Dic3—22no—
239C22×C3×C4—51no—
240C2×#67—102no—
241C22×Dic3—102no—
242#235—52no—
243#235—52no—
244#235—52no—
245#235—102no—
246#235—102no—
247#235C2×AGL(1,𝔽5)12yescomplemented
248C2×D12—52no—
249C2×C4×D3—102no—
250C2×C4×D3—104no—
251C2×Dic6—52no—
252C3×M16—101no—
253C3×M16—101no—
254C2×C3×C8—51no—
255C2×C3×C8—51no—
256C2×C3×C8—51no—
257C2×C3×C8—51no—
258C2×C3×C8—51no—
259C2×C3×C8—52no—
260C2×C3×C8C2×AGL(1,𝔽5)11yescharacteristic, complemented
261C3×C4⋊C4—52no—
262C3×C22⋊C4—52no—
263C3×C42—52no—
264(C2×D3)⋊C4—104no—
265C4⋊Dic3—54no—
266C4⋊Dic3—102no—
267Dic3⋊C4—104no—
268C4×Dic3—104no—
269Dic12—52no—
270Dic12—51no—
271Dic12C2×AGL(1,𝔽5)11yescharacteristic, complemented
272D24—52no—
273D24—51no—
274D24C2×AGL(1,𝔽5)11yescharacteristic, complemented
275#204—52no—
276#204—52no—
277#204—52no—
278#204C2×AGL(1,𝔽5)12yescomplemented
279C22×D5—62no—
280C22×D5C2×D1211yescharacteristic
281C2×AGL(1,𝔽5)—64no—
282C2×AGL(1,𝔽5)—244no—
283C2×AGL(1,𝔽5)—242no—
284C2×AGL(1,𝔽5)C2×D1214yes—
285C5×Q8—32no—
286C5×D4—32no—
287C5×D4—62no—
288C2×C4×C5—62no—
289C2×C4×C5C2×C4×D311yescharacteristic
290C22⋊D5—62no—
291C22⋊D5—62no—
292C22⋊D5—62no—
293C2×Dic5—62no—
294C2×Dic5#23511yescharacteristic
295D20—32no—
296D20—62no—
297C4×D5—62no—
298C4×D5—62no—
299C4×D5—62no—
300C4×D5C2×C4×D311yescharacteristic
301C4×D5C2×D1211yescharacteristic
302C4×D5#23511yescharacteristic
303C4×D5C23×D311yescharacteristic
304Dic10—32no—
305Dic10—62no—
306#170—22no—
307C5×C8C2×C4×D311yescharacteristic
308C5×C8C2×C4×D311yescharacteristic
309#168C2×C4×D311yescharacteristic
310#168C2×C4×D311yescharacteristic
311C2×#41—152no—
312#135—31no—
313#135—151no—
314#135—152no—
315#135—151no—
316#135—152no—
317#135—151no—
318C2×Q16—151no—
319C2×SD16—152no—
320C2×D8—151no—
321C2×M16—51no—
322C22×C8—51no—
323C4×Q8—154no—
324C4×D4—154no—
325C4×D4—304no—
326#117—51no—
327#117—302no—
328C2×C4⋊C4—152no—
329C2×C4⋊C4—302no—
330#106—152no—
331#106—152no—
332#103—154no—
333#102—154no—
334#97—52no—
335C4×C8—52no—
336C2×C3×C5—21no—
337C2×C3×C5C2×C4×D411yescharacteristic
338D15—42no—
339C3×D5—21no—
340C3×D5#57712yes—
341C5×D3—42no—
342C23×C3—51no—
343C22×D3—102no—
344C2×C3×C4—51no—
345C2×C3×C4—51no—
346C2×C3×C4—51no—
347C2×C3×C4—51no—
348C2×C3×C4—51no—
349C2×C3×C4—54no—
350C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
351#67—22no—
352#67—102no—
353#67—102no—
354#67—102no—
355C2×Dic3—102no—
356C2×Dic3—104no—
357C2×Dic3—402no—
358D12—52no—
359D12—102no—
360D12C22×AGL(1,𝔽5)12yes—
361C4×D3—22no—
362C4×D3—102no—
363C4×D3—102no—
364C4×D3—102no—
365C4×D3—404no—
366Dic6—52no—
367Dic6—102no—
368Dic6C22×AGL(1,𝔽5)12yes—
369C3×C8—51no—
370C3×C8—51no—
371C3×C8—102no—
372C3×C8C22×AGL(1,𝔽5)11yescharacteristic
373C3×C8C22×AGL(1,𝔽5)11yescharacteristic
374C22×C5—62no—
375C22×C5C4×D1211yescharacteristic
376C2×D5—22no—
377C2×D5—62no—
378C2×D5—122no—
379C2×D5—122no—
380C2×D5C4×D1211yescharacteristic
381C2×D5C22×D1211yescharacteristic
382AGL(1,𝔽5)—24no—
383AGL(1,𝔽5)—124no—
384C4×C5—62no—
385C4×C5C4×D1211yescharacteristic
386C4×C5C22×C4×D311yescharacteristic
387Dic5—62no—
388Dic5C4×D1211yescharacteristic
389Dic5C2×#23511yescharacteristic
390#41—32no—
391#41—152no—
392#41—152no—
393#41—152no—
394#41—302no—
395#41—302no—
396C2×Q8—152no—
397C2×D4—152no—
398C2×D4—302no—
399C22×C4—51no—
400C22×C4—302no—
401C22×C4—304no—
402C22×C4—302no—
403Q16—31no—
404Q16—152no—
405Q16—151no—
406SD16—32no—
407SD16—152no—
408SD16—152no—
409SD16—152no—
410D8—31no—
411D8—152no—
412D8—151no—
413M16—101no—
414M16—101no—
415C2×C8—51no—
416C2×C8—51no—
417C2×C8—51no—
418C2×C8—51no—
419C2×C8—51no—
420C2×C8—52no—
421C2×C8D3×AGL(1,𝔽5)11yescharacteristic, complemented
422C4⋊C4—52no—
423C4⋊C4—154no—
424C4⋊C4—302no—
425C4⋊C4—304no—
426C22⋊C4—52no—
427C22⋊C4—304no—
428C42—52no—
429C42—304no—
430C3×C5#295211yescharacteristic, complemented, Hall
431C22×C3—51no—
432C22×C3—51no—
433C22×C3—102no—
434C22×C3D4×AGL(1,𝔽5)11yescharacteristic
435C2×D3—22no—
436C2×D3—102no—
437C2×D3—202no—
438C2×D3—202no—
439C3×C4—51no—
440C3×C4—51no—
441C3×C4—104no—
442C3×C4D4×AGL(1,𝔽5)11yescharacteristic
443C3×C4C23×AGL(1,𝔽5)11yescharacteristic
444Dic3—22no—
445Dic3—102no—
446Dic3—204no—
447C2×C5—21no—
448C2×C5—122no—
449C2×C5C2×C4×D1211yescharacteristic
450D5—21no—
451D5—122no—
452D5#575112yes—
453C23—51no—
454C23—302no—
455Q8—32no—
456Q8—152no—
457Q8—302no—
458D4—32no—
459D4—62no—
460D4—152no—
461D4—302no—
462D4—302no—
463D4—302no—
464D4—302no—
465C2×C4—51no—
466C2×C4—51no—
467C2×C4—51no—
468C2×C4—51no—
469C2×C4—51no—
470C2×C4—54no—
471C2×C4—62no—
472C2×C4—302no—
473C2×C4—302no—
474C2×C4—302no—
475C2×C4—302no—
476C2×C4—304no—
477C2×C4—1204no—
478C2×C4—1202no—
479C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
480C8—51no—
481C8—51no—
482C8—102no—
483C8C2×D3×AGL(1,𝔽5)11yescharacteristic
484C8C2×D3×AGL(1,𝔽5)11yescharacteristic
485C2×C3—21no—
486C2×C3—52no—
487C2×C3—101no—
488C2×C3C2×D4×AGL(1,𝔽5)11yescharacteristic
489D3—42no—
490D3—202no—
491C5#7857411yescharacteristic, complemented, Sylow
492C22—51no—
493C22—51no—
494C22—62no—
495C22—102no—
496C22—302no—
497C22—602no—
498C22—602no—
499C22AGL(1,𝔽5)×D1211yescharacteristic
500C4—51no—
501C4—51no—
502C4—62no—
503C4—104no—
504C4—302no—
505C4—604no—
506C4AGL(1,𝔽5)×D1211yescharacteristic
507C4C22×D3×AGL(1,𝔽5)11yescharacteristic
508C3#1061744111yescharacteristic, complemented, Sylow
509C2—21no—
510C2—52no—
511C2—101no—
512C2—122no—
513C2—602no—
514C2C2×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#2952
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×#91222C2semidirect
4. C3×#65718C2semidirect
5. #11751708C2semidirect
6. #11751668C2semidirect
7. #11751662C2semidirect
8. AGL(1,𝔽5)×#204C2semidirect
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
68. C2×C3×C5×C8C2×C4semidirect
69. Dic60C2×C4semidirect
70. D120C2×C4semidirect
71. #6706C2×C4semidirect
72. C5×Dic12C2×C4semidirect
73. C5×D24C2×C4semidirect
74. C5×#204C2×C4semidirect
75. C2×C3×#168C2×C4semidirect
76. C3×C8×D5C2×C4semidirect
77. C3×C8×D5C2×C4semidirect
80. #6663C2×C4semidirect
81. #6660C2×C4semidirect
82. #6659C2×C4semidirect
83. #6654C2×C4semidirect
161. #904AGL(1,𝔽5)semidirect
202. C2×C5×C8C4×D3semidirect
207. C2×#168C4×D3semidirect
211. C8×D5C4×D3semidirect
212. C8×D5C4×D3semidirect
247. #235C2×AGL(1,𝔽5)semidirect
260. C2×C3×C8C2×AGL(1,𝔽5)semidirect
271. Dic12C2×AGL(1,𝔽5)semidirect
274. D24C2×AGL(1,𝔽5)semidirect
278. #204C2×AGL(1,𝔽5)semidirect
421. C2×C8D3×AGL(1,𝔽5)semidirect
430. C3×C5#2952semidirect
491. C5#78574semidirect
508. C3#10617441semidirect

Maximal subgroups

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

subgroupindex
#117568382
D5×#9042
C2×#912222
C3×#657182
#117517082
#117516682
#117516622
AGL(1,𝔽5)×#2042
#106174413
#785745