Finite Group Databasebeta

← #14297803

Subgroups — #14297803

Order 1920.

584 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
1D5×#887C211yesmaximal, characteristic
2D5⋊(Dic3⋊C8)C211yesmaximal, characteristic
3#11751751C211yesmaximal, characteristic
4C2×#91154C211yesmaximal, characteristic
5C2×#91153C211yesmaximal, characteristic
6#11746745C211yesmaximal, characteristic
7C3×#64874C211yesmaximal, characteristic
8#10604452—31nomaximal, Hall
9C2×D5×Dic6C2211yescharacteristic
10C2×C4⋊ΓL(1,𝔽16)C2211yescharacteristic
11C2×D5⋊(C3⋊C8)C2211yescharacteristic
12C2×C3×C4⋊AGL(1,𝔽5)C2211yescharacteristic
13C2×C3×D5⋊C8C2211yescharacteristic
14#91776C411yescharacteristic, complemented
15C5×#887C411yescharacteristic, complemented
16C3×D5×C4⋊C4C2211yescharacteristic
17D5×Dic3⋊C4—21no—
18C4×D5×Dic3—21no—
19C4×D5×Dic3C2211yescharacteristic
20#91344C411yescharacteristic, complemented
21#91340C411yescharacteristic, complemented
22#91325—21no—
23#91324—21no—
24#91177—21no—
25Dic3⋊#170—21no—
26(C2×AGL(1,𝔽5))⋊Dic3—21no—
27(C2×AGL(1,𝔽5))⋊Dic3—21no—
28#91154—22no—
29#91153—22no—
30#68905—51nomaximal, Hall
31C4×Q8×D5—31no—
32C2×#4009—31no—
33C2×#4009—31no—
34#65689—31no—
35D5⋊(C4⋊C8)—31no—
36#64874—31no—
37#64874D311yescharacteristic
38C22×ΓL(1,𝔽16)—21no—
39C22×C3×AGL(1,𝔽5)—21no—
40C2×Dic30C2×C411yescharacteristic
41C2×C5×Dic6C2×C411yescharacteristic
42C2×C3×C4×D5—21no—
43C2×C3×C4×D5C2311yescharacteristic
44C2×#1204—21no—
45C2×D5×Dic3—21no—
46C2×D5×Dic3D411yescharacteristic
47C2×D5×Dic3D411yescharacteristic
48D5×Dic6—21no—
49D5×Dic6—21no—
50D5×Dic6D412yes—
51D5×Dic6Q812yes—
52C2×C3⋊#170—21no—
53C4⋊ΓL(1,𝔽16)—21no—
54C4⋊ΓL(1,𝔽16)—21no—
55D5⋊(C3⋊C8)—21no—
56D5⋊(C3⋊C8)—21no—
57C2×C3×#170—21no—
58C3×C4⋊AGL(1,𝔽5)—21no—
59C3×C4⋊AGL(1,𝔽5)—21no—
60C3×D5⋊C8—21no—
61C3×D5⋊C8—21no—
62C3×C5×C4⋊C4C2×C411yescharacteristic
63#6712—21no—
64C4×Dic15—21no—
65C4×Dic15C2×C411yescharacteristic
66C5×Dic3⋊C4—21no—
67C4×C5×Dic3—21no—
68C4×C5×Dic3C2×C411yescharacteristic
69C3×C4⋊Dic5C2×C411yescharacteristic
70C3×#683—21no—
71Dic3⋊Dic5—21no—
72#6669—21no—
73Dic3×Dic5—21no—
74Dic3×Dic5—21no—
75Dic3×Dic5D411yescharacteristic
76Dic3×Dic5D411yescharacteristic
77C2×#887—51no—
78C2×#816—51no—
79C2×Dic3⋊C8—51no—
80#4977—51no—
81C2×#810—51no—
82C3×#340—51no—
83#4529—51no—
84C2×Q8×D5—31no—
85C2×C4⋊AGL(1,𝔽5)—31no—
86C2×C4⋊AGL(1,𝔽5)C2×D311yescharacteristic
87C2×D5⋊C8—31no—
88C2×D5⋊C8C2×D311yescharacteristic
89C4×C5×Q8—31no—
90Q8×Dic5—31no—
91D5×C4⋊C4—61no—
92D5×C4⋊C4C2×D311yescharacteristic
93#4033—31no—
94#4033—61no—
95C42×D5—31no—
96C42×D5—61no—
97C4×Dic10—31no—
98C4×Dic10—61no—
99#4009—62no—
100#4009—62no—
101(C2×AGL(1,𝔽5))⋊C4—61no—
102(C2×AGL(1,𝔽5))⋊C4—61no—
103#4004—61no—
104C4⋊#170—61no—
105#1805—151noSylow
106C22×C3×D5C2×D411yescharacteristic
107C2×ΓL(1,𝔽16)—41no—
108C2×ΓL(1,𝔽16)—41no—
109C2×ΓL(1,𝔽16)—41no—
110C2×C3×AGL(1,𝔽5)—41no—
111C2×C3×AGL(1,𝔽5)—41no—
112C2×C3×AGL(1,𝔽5)—41no—
113C2×C3×C4×C5—21no—
114C2×C3×C4×C5C22×C411yescharacteristic
115C2×Dic15—21no—
116C2×Dic15C22⋊C411yescharacteristic
117C2×Dic15C22⋊C411yescharacteristic
118Dic30—21no—
119Dic30C4⋊C412yes—
120C2×C5×Dic3—21no—
121C2×C5×Dic3C22⋊C411yescharacteristic
122C2×C5×Dic3C22⋊C411yescharacteristic
123C5×Dic6—21no—
124C5×Dic6C4⋊C412yes—
125C2×C3×Dic5—21no—
126C2×C3×Dic5C2×D411yescharacteristic
127C3×C4×D5—41no—
128C3×C4×D5—41no—
129C3×C4×D5C2×D411yescharacteristic
130C3×C4×D5C2×Q811yescharacteristic
131C3×C4×D5#4111yescharacteristic
132C3×C4×D5#4111yescharacteristic
133#1204—22no—
134#1204—22no—
135D5×Dic3—22no—
136D5×Dic3—22no—
137D5×Dic3—21no—
138D5×Dic3—21no—
139D5×Dic3—21no—
140D5×Dic3—21no—
141C3⋊#170—41no—
142C3×#170—41no—
143C22×Dic6—51no—
144C22×C3×C8—51no—
145C2×C3×C4⋊C4—51no—
146C2×C3×C4⋊C4—51no—
147C2×C4⋊Dic3—51no—
148C2×Dic3⋊C4—101no—
149C2×C4×Dic3—51no—
150C2×C4×Dic3—101no—
151C22×C3⋊C8—51no—
152#887—51no—
153#887—51no—
154#887—51no—
155#887—101no—
156#887—101no—
157#887AGL(1,𝔽5)11yescharacteristic, complemented
158C3⋊#95—101no—
159C3⋊#95—101no—
160#816—102no—
161Dic3⋊C8—101no—
162Dic3⋊C8—101no—
163#810—102no—
164C22×AGL(1,𝔽5)—21no—
165C22×AGL(1,𝔽5)—61no—
166C2×C5×Q8—31no—
167Q8×D5—32no—
168Q8×D5—32no—
169Q8×D5—61no—
170Q8×D5—61no—
171C2×C4×D5—21no—
172C2×C4×D5—31no—
173C2×C4×D5—31no—
174C2×C4×D5—61no—
175C2×C4×D5C22×D311yescharacteristic
176C2×Dic10—31no—
177C2×Dic10—61no—
178C2×#170—21no—
179C2×#170—61no—
180C4⋊AGL(1,𝔽5)—21no—
181C4⋊AGL(1,𝔽5)—21no—
182C4⋊AGL(1,𝔽5)—61no—
183C4⋊AGL(1,𝔽5)—61no—
184D5⋊C8—21no—
185D5⋊C8—21no—
186D5⋊C8—61no—
187D5⋊C8—61no—
188C5×C4⋊C4—61no—
189C5×C4⋊C4C4×D311yescharacteristic
190C42×C5—31no—
191C42×C5—61no—
192C4⋊Dic5—61no—
193C4⋊Dic5C4×D311yescharacteristic
194#683—21no—
195#683—61no—
196#683—61no—
197C4×Dic5—31no—
198C4×Dic5—31no—
199C4×Dic5—31no—
200C4×Dic5—61no—
201C4×Dic5—61no—
202C4×Dic5—61no—
203C2×C4×Q8—151no—
204C2×C4⋊C8—151no—
205C2×#103—151no—
206C2×#103—151no—
207#382—151no—
208#340—51no—
209#340—151no—
210C22×C3×C5C2×C22⋊C411yescharacteristic
211C2×C3×D5#12111yescharacteristic
212C2×C3×D5C22⋊Q811yescharacteristic
213C2×C3×D5#13511yescharacteristic
214C2×C3×D5#13511yescharacteristic
215C2×C3×D5#13711yescharacteristic
216C2×C3×D5#13711yescharacteristic
217ΓL(1,𝔽16)—81no—
218C3×AGL(1,𝔽5)—81no—
219C3×C4×C5—41no—
220C3×C4×C5C2×C4⋊C411yescharacteristic
221C3×C4×C5#11711yescharacteristic
222Dic15—22no—
223Dic15—21no—
224Dic15—21no—
225C3×Dic5—41no—
226C3×Dic5#12111yescharacteristic
227C3×Dic5C22⋊Q811yescharacteristic
228C5×Dic3—22no—
229C5×Dic3—21no—
230C5×Dic3—21no—
231C22×C3×C4—51no—
232C22×C3×C4—101no—
233C22×C3×C4—101no—
234C22×Dic3—51no—
235C22×Dic3—51no—
236C22×Dic3—101no—
237C22×Dic3—101no—
238C2×Dic6—51no—
239C2×Dic6—52no—
240C2×Dic6—52no—
241C2×Dic6—101no—
242C2×Dic6—101no—
243C2×Dic6—101no—
244C2×Dic6C2×AGL(1,𝔽5)11yescharacteristic
245C2×C3×C8—101no—
246C2×C3×C8—101no—
247C2×C3×C8—101no—
248C3×C4⋊C4—51no—
249C3×C4⋊C4—101no—
250C3×C4⋊C4—101no—
251C3×C4⋊C4—101no—
252C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
253C4⋊Dic3—101no—
254C4⋊Dic3—101no—
255Dic3⋊C4—21no—
256Dic3⋊C4—101no—
257Dic3⋊C4—101no—
258Dic3⋊C4—101no—
259C4×Dic3—21no—
260C4×Dic3—51no—
261C4×Dic3—51no—
262C4×Dic3—51no—
263C4×Dic3—101no—
264C4×Dic3—101no—
265C4×Dic3—101no—
266C4×Dic3C2×AGL(1,𝔽5)11yescharacteristic
267C2×C3⋊C8—101no—
268C2×C3⋊C8—101no—
269C2×C3⋊C8—101no—
270C22×D5D3×D411yescharacteristic
271C2×AGL(1,𝔽5)—41no—
272C2×AGL(1,𝔽5)—41no—
273C2×AGL(1,𝔽5)—41no—
274C2×AGL(1,𝔽5)—121no—
275C2×AGL(1,𝔽5)—121no—
276C2×AGL(1,𝔽5)—121no—
277C5×Q8—32no—
278C5×Q8—61no—
279C2×C4×C5—21no—
280C2×C4×C5—31no—
281C2×C4×C5—31no—
282C2×C4×C5—61no—
283C2×C4×C5C2×C4×D311yescharacteristic
284C2×Dic5—21no—
285C2×Dic5—31no—
286C2×Dic5—31no—
287C2×Dic5—61no—
288C2×Dic5D3×D411yescharacteristic
289C4×D5—41no—
290C4×D5—41no—
291C4×D5—62no—
292C4×D5—62no—
293C4×D5—61no—
294C4×D5—61no—
295C4×D5—61no—
296C4×D5—61no—
297C4×D5#23511yescharacteristic
298C4×D5#23511yescharacteristic
299C4×D5D3×D411yescharacteristic
300C4×D5D3×Q811yescharacteristic
301Dic10—32no—
302Dic10—61no—
303Dic10—62no—
304Dic10—62no—
305#170—41no—
306#170—121no—
307C22×Q8—151no—
308C22×C8—51no—
309C22×C8—151no—
310C4×Q8—31no—
311C4×Q8—151no—
312C4×Q8—151no—
313C4×Q8—151no—
314C4×Q8—301no—
315C4×Q8—301no—
316C2×C4⋊C4—51no—
317C2×C4⋊C4—51no—
318C2×C4⋊C4—151no—
319C2×C4⋊C4—301no—
320C2×C42—151no—
321C2×C42—301no—
322C4⋊C8—301no—
323C4⋊C8—301no—
324#103—302no—
325#103—302no—
326#95—301no—
327#95—301no—
328C2×C3×C5#38011yescharacteristic
329C2×C3×C5#41611yescharacteristic
330C2×C3×C5#41911yescharacteristic
331C3×D5#46411yescharacteristic
332C3×D5#46411yescharacteristic
333C3×D5#47911yescharacteristic
334C3×D5#47911yescharacteristic
335C23×C3—51no—
336C2×C3×C4—21no—
337C2×C3×C4—51no—
338C2×C3×C4—51no—
339C2×C3×C4—51no—
340C2×C3×C4—51no—
341C2×C3×C4—51no—
342C2×C3×C4—101no—
343C2×C3×C4—201no—
344C2×C3×C4—201no—
345C2×C3×C4—201no—
346C2×C3×C4—201no—
347C2×C3×C4—201no—
348C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
349C2×Dic3—21no—
350C2×Dic3—51no—
351C2×Dic3—51no—
352C2×Dic3—102no—
353C2×Dic3—102no—
354C2×Dic3—101no—
355C2×Dic3—101no—
356C2×Dic3—101no—
357C2×Dic3—101no—
358C2×Dic3—101no—
359C2×Dic3—201no—
360C2×Dic3—201no—
361C2×Dic3—201no—
362C2×Dic3C22⋊AGL(1,𝔽5)11yescharacteristic
363C2×Dic3C22⋊AGL(1,𝔽5)11yescharacteristic
364Dic6—21no—
365Dic6—52no—
366Dic6—102no—
367Dic6—102no—
368Dic6—101no—
369Dic6C4⋊AGL(1,𝔽5)12yes—
370C3×C8—201no—
371C3⋊C8—201no—
372C22×C5D3×C22⋊C411yescharacteristic
373C2×D5#88411yescharacteristic
374C2×D5#89611yescharacteristic
375C2×D5#91211yescharacteristic
376C2×D5#91511yescharacteristic
377C2×D5#91611yescharacteristic
378C2×D5#91811yescharacteristic
379AGL(1,𝔽5)—81no—
380AGL(1,𝔽5)—241no—
381C4×C5—41no—
382C4×C5—62no—
383C4×C5—61no—
384C4×C5—61no—
385C4×C5#87211yescharacteristic
386C4×C5D3×C4⋊C411yescharacteristic
387Dic5—41no—
388Dic5—62no—
389Dic5—61no—
390Dic5—61no—
391Dic5#88411yescharacteristic
392Dic5#89611yescharacteristic
393C2×Q8—31no—
394C2×Q8—152no—
395C2×Q8—152no—
396C2×Q8—151no—
397C2×Q8—301no—
398C2×Q8—301no—
399C2×Q8—301no—
400C22×C4—51no—
401C22×C4—101no—
402C22×C4—101no—
403C22×C4—151no—
404C22×C4—151no—
405C22×C4—301no—
406C22×C4—301no—
407C2×C8—101no—
408C2×C8—101no—
409C2×C8—101no—
410C2×C8—301no—
411C2×C8—301no—
412C2×C8—301no—
413C4⋊C4—51no—
414C4⋊C4—61no—
415C4⋊C4—101no—
416C4⋊C4—101no—
417C4⋊C4—101no—
418C4⋊C4—301no—
419C4⋊C4—301no—
420C4⋊C4—301no—
421C4⋊C4—301no—
422C4⋊C4—301no—
423C4⋊C4D3×AGL(1,𝔽5)11yescharacteristic
424C42—31no—
425C42—61no—
426C42—151no—
427C42—151no—
428C42—151no—
429C42—301no—
430C42—301no—
431C42—301no—
432C3×C5#180511yescharacteristic, complemented, Hall
433C22×C3—51no—
434C22×C3—51no—
435C22×C3—51no—
436C22×C3—51no—
437C22×C3—51no—
438C22×C3—51no—
439C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
440C3×C4—41no—
441C3×C4—51no—
442C3×C4—51no—
443C3×C4—201no—
444C3×C4—401no—
445C3×C4C2×C4⋊AGL(1,𝔽5)11yescharacteristic
446C3×C4#413011yescharacteristic
447Dic3—21no—
448Dic3—22no—
449Dic3—21no—
450Dic3—101no—
451Dic3—102no—
452Dic3—101no—
453Dic3—401no—
454C2×C5#466711yescharacteristic
455C2×C5#477211yescharacteristic
456C2×C5#480211yescharacteristic
457D5#476711yescharacteristic
458D5#479811yescharacteristic
459D5#485711yescharacteristic
460D5#487711yescharacteristic
461C23—51no—
462Q8—32no—
463Q8—61no—
464Q8—152no—
465Q8—302no—
466Q8—302no—
467Q8—301no—
468C2×C4—21no—
469C2×C4—31no—
470C2×C4—31no—
471C2×C4—51no—
472C2×C4—51no—
473C2×C4—51no—
474C2×C4—51no—
475C2×C4—51no—
476C2×C4—61no—
477C2×C4—101no—
478C2×C4—151no—
479C2×C4—151no—
480C2×C4—201no—
481C2×C4—201no—
482C2×C4—201no—
483C2×C4—201no—
484C2×C4—201no—
485C2×C4—301no—
486C2×C4—301no—
487C2×C4—301no—
488C2×C4—301no—
489C2×C4—302no—
490C2×C4—302no—
491C2×C4—301no—
492C2×C4—601no—
493C2×C4—601no—
494C2×C4—601no—
495C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
496C8—201no—
497C8—601no—
498C2×C3—51no—
499C2×C3—51no—
500C2×C3—51no—
501C2×C3—51no—
502C2×C3#6576511yescharacteristic
503C2×C3#6578211yescharacteristic
504C2×C3#6579411yescharacteristic
505C5#6890511yescharacteristic, complemented, Sylow
506C22—51no—
507C22—51no—
508C22—51no—
509C22—51no—
510C22—51no—
511C22—51no—
512C22D3×C22⋊AGL(1,𝔽5)11yescharacteristic
513C4—41no—
514C4—51no—
515C4—51no—
516C4—61no—
517C4—61no—
518C4—62no—
519C4—201no—
520C4—301no—
521C4—301no—
522C4—302no—
523C4—401no—
524C4—1201no—
525C4#9190911yescharacteristic
526C4D3×C4⋊AGL(1,𝔽5)11yescharacteristic
527C3#1060445211yescharacteristic, complemented, Sylow
528C2—51no—
529C2—51no—
530C2—51no—
531C2—51no—
532C2#1175185411yescharacteristic
533C2#1175194711yescharacteristic
534C2#1175196211yescharacteristic
subgroupG/H
centerC22D3×C22⋊AGL(1,𝔽5)
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC3×C5×C4⋊C4C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC22×C3×D5C2×D4
cosocleC2×C3×C4×D5C23
hypercenterC4⋊C4D3×AGL(1,𝔽5)
hypocenterC3×C5#1805
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
14. #91776C4semidirect
15. C5×#887C4semidirect
20. #91344C4semidirect
21. #91340C4semidirect
157. #887AGL(1,𝔽5)semidirect
432. C3×C5#1805semidirect
505. C5#68905semidirect
527. C3#10604452semidirect

Maximal subgroups

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

subgroupindex
D5×#8872
D5⋊(Dic3⋊C8)2
#117517512
C2×#911542
C2×#911532
#117467452
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