Finite Group Databasebeta

← #14297865

Subgroups — #14297865

Order 1920.

624 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11753402C211yesmaximal, characteristic
2C4×Dic3⋊AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
3#11751829C211yesmaximal, characteristic, complemented
4C2×#91156C211yesmaximal, characteristic
5C2×#91155C211yesmaximal, characteristic
6#11746746C211yesmaximal, characteristic, complemented
7C3×#64875C211yesmaximal, characteristic, complemented
8#10604422—31nomaximal, Hall
9C2×#6772C2211yescharacteristic
10C2×C4×ΓL(1,𝔽16)C2211yescharacteristic
11C2×#6758C2211yescharacteristic
12C2×C3×C4×AGL(1,𝔽5)C2211yescharacteristic
13C2×C3×#700C2211yescharacteristic
14C2×Dic3⋊AGL(1,𝔽5)—21no—
15#91776C411yescharacteristic, complemented
16C5×#893C411yescharacteristic, complemented
17C3×#4038C2211yescharacteristic
18#91442—21no—
19#91441—21no—
20#91435C411yescharacteristic, complemented
21#91398—21no—
22C4×D3×Dic5—21no—
23C4×D5×Dic3C2211yescharacteristic
24#91340C411yescharacteristic, complemented
25#91177—21no—
26#91177—21no—
27(C2×AGL(1,𝔽5))⋊Dic3—21no—
28#91156—22no—
29#91155—22no—
30#68770—51nomaximal, Hall
31C4×#711—31no—
32C2×#4008—31no—
33C2×#4008—31no—
34C4×C4⋊AGL(1,𝔽5)—31no—
35#65682—31no—
36#64875—31no—
37#64875D311yescharacteristic, complemented
38C22×ΓL(1,𝔽16)—21no—
39C22×C3×AGL(1,𝔽5)—21no—
40C2×Dic30C2×C411yescharacteristic
41C2×C5×D12C2×C411yescharacteristic
42C2×C4×C5×D3—21no—
43C2×C3×C4×D5C2311yescharacteristic
44C2×#1201—21no—
45C2×D3×Dic5—21no—
46C2×D3×Dic5—21no—
47C2×D5×Dic3D411yescharacteristic
48C2×D5×Dic3D411yescharacteristic
49#6772—21no—
50#6772—21no—
51#6772D412yes—
52#6772Q812yes—
53C2×C3⋊#170—21no—
54C4×ΓL(1,𝔽16)—21no—
55C4×ΓL(1,𝔽16)—21no—
56#6758—21no—
57#6758—21no—
58C2×C3×#170—21no—
59C3×C4×AGL(1,𝔽5)—21no—
60C3×C4×AGL(1,𝔽5)—21no—
61C3×#700—21no—
62C3×#700—21no—
63Dic3⋊AGL(1,𝔽5)—24no—
64C3×C5×C4⋊C4C2×C411yescharacteristic
65#6712—21no—
66C4×Dic15—21no—
67C4×Dic15C2×C411yescharacteristic
68C5×(C2×D3)⋊C4—21no—
69C4×C5×Dic3C2×C411yescharacteristic
70C3×#685—21no—
71C3×C4⋊Dic5C2×C411yescharacteristic
72C3×C4×Dic5—21no—
73#6669—21no—
74(C2×D3)⋊Dic5—21no—
75Dic3×Dic5—21no—
76Dic3×Dic5D411yescharacteristic
77Dic3×Dic5D411yescharacteristic
78#5522—51no—
79C2×#825—51no—
80#5118—51no—
81C4×Dic3⋊C4—51no—
82C2×#805—51no—
83C3×#339—51no—
84#4532—51no—
85C2×#711—31no—
86C2×C4⋊AGL(1,𝔽5)—61no—
87C2×C4×AGL(1,𝔽5)—31no—
88C2×C4×AGL(1,𝔽5)C2×D311yescharacteristic
89C2×#700—31no—
90C2×#700C2×D311yescharacteristic
91C4×C5×D4—31no—
92D4×Dic5—31no—
93C4×C22⋊D5—61no—
94C2×C4×Dic5—61no—
95#4038C2×D311yescharacteristic
96#4033—31no—
97#4027—61no—
98#4023—61no—
99C42×D5—31no—
100C4×Dic10—31no—
101#4008—62no—
102#4008—62no—
103(C2×AGL(1,𝔽5))⋊C4—61no—
104#4004—61no—
105#4004—61no—
106#1806—151noSylow
107C22×C5×D3—21no—
108C22×C3×D5C2×D411yescharacteristic
109C2×ΓL(1,𝔽16)—22no—
110C2×ΓL(1,𝔽16)—41no—
111C2×ΓL(1,𝔽16)—41no—
112C2×C3×AGL(1,𝔽5)—22no—
113C2×C3×AGL(1,𝔽5)—41no—
114C2×C3×AGL(1,𝔽5)—41no—
115C2×C3×C4×C5—21no—
116C2×C3×C4×C5C22×C411yescharacteristic
117C2×Dic15—21no—
118C2×Dic15C22⋊C411yescharacteristic
119C2×Dic15C22⋊C411yescharacteristic
120Dic30—21no—
121Dic30C4⋊C412yescomplemented
122C2×C5×Dic3C22⋊C411yescharacteristic
123C2×C5×Dic3C22⋊C411yescharacteristic
124C5×D12—21no—
125C5×D12C4⋊C412yescomplemented
126C4×C5×D3—42no—
127C2×C3×Dic5—21no—
128C2×C3×Dic5C2×D411yescharacteristic
129C3×C4×D5C2×D411yescharacteristic
130C3×C4×D5C2×Q811yescharacteristic
131C3×C4×D5#4111yescharacteristic
132C3×C4×D5#4111yescharacteristic
133#1201—22no—
134#1201—22no—
135D3×Dic5—22no—
136D3×Dic5—22no—
137D3×Dic5—42no—
138D5×Dic3—22no—
139D5×Dic3—41no—
140C3⋊#170—41no—
141C3×#170—41no—
142C2×#235—51no—
143C2×C3×M16—51no—
144C3×#117—51no—
145C2×C3×C42—51no—
146C2×Dic3⋊C4—101no—
147C2×C4×Dic3—51no—
148C2×C4×Dic3—51no—
149C2×#208—51no—
150#893—51no—
151#893AGL(1,𝔽5)11yescharacteristic, complemented
152#887—51no—
153#887—51no—
154#67⋊C4—101no—
155#67⋊C4—101no—
156#872—101no—
157C42×D3—101no—
158C3⋊#95—101no—
159#825—102no—
160Dic3⋊C8—101no—
161Dic3⋊C8—101no—
162#805—102no—
163C22×AGL(1,𝔽5)—21no—
164C22×AGL(1,𝔽5)—61no—
165C2×C5×D4—31no—
166C22×C4×C5—61no—
167C2×C22⋊D5—61no—
168C22×Dic5—61no—
169C22×Dic5—61no—
170#711—32no—
171#711—32no—
172#711—61no—
173#711—61no—
174C2×C4×D5—31no—
175C2×C4×D5—31no—
176C2×C4×D5C22×D311yescharacteristic
177C2×Dic10—31no—
178C2×#170—21no—
179C2×#170—61no—
180C4⋊AGL(1,𝔽5)—64no—
181C4×AGL(1,𝔽5)—21no—
182C4×AGL(1,𝔽5)—21no—
183C4×AGL(1,𝔽5)—61no—
184C4×AGL(1,𝔽5)—61no—
185#700—21no—
186#700—21no—
187#700—61no—
188#700—61no—
189C5×C4⋊C4C4×D311yescharacteristic
190C5×C22⋊C4—61no—
191C42×C5—31no—
192C22⋊Dic5—61no—
193#685—21no—
194C4⋊Dic5C4×D311yescharacteristic
195#683—61no—
196#683—61no—
197C4×Dic5—21no—
198C4×Dic5—31no—
199C4×Dic5—31no—
200C4×Dic5—31no—
201C4×Dic5—61no—
202C4×Dic5—61no—
203C4×#41—151no—
204#423—151no—
205C2×#104—151no—
206C2×#104—151no—
207C4×C4⋊C4—151no—
208#339—51no—
209#339—151no—
210C22×C3×C5C2×C22⋊C411yescharacteristic
211C2×C5×D3—22no—
212C2×C5×D3—41no—
213C2×C5×D3—41no—
214C2×C3×D5—21no—
215C2×C3×D5—21no—
216C2×C3×D5#12111yescharacteristic
217C2×C3×D5C22⋊Q811yescharacteristic
218ΓL(1,𝔽16)—42no—
219C3×AGL(1,𝔽5)—42no—
220C3×C4×C5—41no—
221C3×C4×C5C2×C4⋊C411yescharacteristic
222C3×C4×C5#11711yescharacteristic
223Dic15—22no—
224Dic15—21no—
225Dic15#10412yescomplemented
226C3×Dic5—41no—
227C3×Dic5#12111yescharacteristic
228C3×Dic5C22⋊Q811yescharacteristic
229C5×Dic3—21no—
230C5×Dic3—21no—
231C22×C3×C4—51no—
232C22×C3×C4—101no—
233C2×#67—101no—
234C22×Dic3—51no—
235C22×Dic3—51no—
236C22×Dic3—101no—
237#235—52no—
238#235—52no—
239#235—101no—
240#235—101no—
241C2×D12C2×AGL(1,𝔽5)11yescharacteristic
242C2×C4×D3—21no—
243C2×C4×D3—101no—
244C2×C4×D3—101no—
245C2×Dic6—51no—
246C3×M16—101no—
247C3×M16—101no—
248C2×C3×C8—101no—
249C3×C4⋊C4—51no—
250C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
251C3×C22⋊C4—101no—
252C3×C42—101no—
253C3×C42—101no—
254C3×C42—101no—
255(C2×D3)⋊C4—21no—
256(C2×D3)⋊C4—101no—
257Dic3⋊C4—101no—
258Dic3⋊C4—104no—
259Dic3⋊C4—101no—
260C4×Dic3—51no—
261C4×Dic3—51no—
262C4×Dic3—51no—
263C4×Dic3—101no—
264C4×Dic3—101no—
265C4×Dic3—101no—
266C4×Dic3—101no—
267C4×Dic3C2×AGL(1,𝔽5)11yescharacteristic
268#208—101no—
269#208—101no—
270C2×C3⋊C8—101no—
271C23×C5—61no—
272C22×D5D3×D411yescharacteristic
273C2×AGL(1,𝔽5)—22no—
274C2×AGL(1,𝔽5)—41no—
275C2×AGL(1,𝔽5)—41no—
276C2×AGL(1,𝔽5)—62no—
277C2×AGL(1,𝔽5)—121no—
278C2×AGL(1,𝔽5)—121no—
279C5×D4—32no—
280C5×D4—61no—
281C2×C4×C5—21no—
282C2×C4×C5—31no—
283C2×C4×C5—31no—
284C2×C4×C5—122no—
285C2×C4×C5C2×C4×D311yescharacteristic
286C22⋊D5—62no—
287C22⋊D5—62no—
288C2×Dic5—21no—
289C2×Dic5—31no—
290C2×Dic5—31no—
291C2×Dic5—61no—
292C2×Dic5—62no—
293C2×Dic5—62no—
294C2×Dic5—122no—
295C2×Dic5D3×D411yescharacteristic
296C4×D5—62no—
297C4×D5—121no—
298C4×D5#23511yescharacteristic
299C4×D5#23511yescharacteristic
300C4×D5D3×D411yescharacteristic
301C4×D5D3×Q811yescharacteristic
302Dic10—32no—
303Dic10—61no—
304#170—41no—
305#170—121no—
306C2×#41—151no—
307C2×M16—51no—
308C2×M16—151no—
309C4×Q8—151no—
310C4×Q8—151no—
311C4×D4—31no—
312C4×D4—151no—
313C4×D4—301no—
314C4×D4—301no—
315#117—51no—
316#117—301no—
317C2×C4⋊C4—301no—
318C2×C42—51no—
319C2×C42—151no—
320C2×C42—151no—
321C2×C42—301no—
322C4⋊C8—301no—
323C4⋊C8—301no—
324#104—302no—
325#104—302no—
326#95—301no—
327C2×C3×C5#38011yescharacteristic
328C2×C3×C5C2×#10411yescharacteristic
329C2×C3×C5#42111yescharacteristic
330C3×D5—21no—
331C3×D5—21no—
332C5×D3—42no—
333C23×C3—51no—
334C22×D3—21no—
335C2×C3×C4—21no—
336C2×C3×C4—51no—
337C2×C3×C4—51no—
338C2×C3×C4—51no—
339C2×C3×C4—51no—
340C2×C3×C4—51no—
341C2×C3×C4—102no—
342C2×C3×C4—101no—
343C2×C3×C4—201no—
344C2×C3×C4—201no—
345C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
346#67—102no—
347#67—102no—
348C2×Dic3—51no—
349C2×Dic3—51no—
350C2×Dic3—102no—
351C2×Dic3—101no—
352C2×Dic3—102no—
353C2×Dic3—201no—
354C2×Dic3—201no—
355C2×Dic3—201no—
356C2×Dic3C22⋊AGL(1,𝔽5)11yescharacteristic
357C2×Dic3C22⋊AGL(1,𝔽5)11yescharacteristic
358D12—21no—
359D12C4⋊AGL(1,𝔽5)12yescomplemented
360C4×D3—42no—
361C4×D3—102no—
362C4×D3—102no—
363C4×D3—202no—
364Dic6—52no—
365Dic6—101no—
366C3×C8—201no—
367C3⋊C8—201no—
368C22×C5—62no—
369C22×C5—121no—
370C22×C5—121no—
371C22×C5D3×C22⋊C411yescharacteristic
372C2×D5—21no—
373C2×D5—21no—
374C2×D5#88411yescharacteristic
375C2×D5#89611yescharacteristic
376AGL(1,𝔽5)—42no—
377AGL(1,𝔽5)—122no—
378C4×C5—41no—
379C4×C5—61no—
380C4×C5—61no—
381C4×C5#87211yescharacteristic
382C4×C5D3×C4⋊C411yescharacteristic
383Dic5—32no—
384Dic5—41no—
385Dic5—62no—
386Dic5—61no—
387Dic5#88411yescharacteristic
388Dic5#89611yescharacteristic
389#41—152no—
390#41—152no—
391#41—301no—
392#41—301no—
393C2×Q8—151no—
394C2×D4—31no—
395C2×D4—301no—
396C22×C4—51no—
397C22×C4—61no—
398C22×C4—101no—
399C22×C4—151no—
400C22×C4—151no—
401C22×C4—301no—
402C22×C4—301no—
403C22×C4—301no—
404M16—101no—
405M16—101no—
406M16—301no—
407M16—301no—
408C2×C8—101no—
409C2×C8—301no—
410C4⋊C4—51no—
411C4⋊C4—301no—
412C4⋊C4—301no—
413C4⋊C4—304no—
414C4⋊C4D3×AGL(1,𝔽5)11yescharacteristic
415C22⋊C4—61no—
416C22⋊C4—101no—
417C22⋊C4—301no—
418C42—31no—
419C42—101no—
420C42—101no—
421C42—101no—
422C42—151no—
423C42—151no—
424C42—151no—
425C42—301no—
426C42—301no—
427C42—301no—
428C42—301no—
429C3×C5#180611yescharacteristic, complemented, Hall
430C22×C3—51no—
431C22×C3—51no—
432C22×C3—101no—
433C22×C3—101no—
434C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
435C2×D3—22no—
436C2×D3—41no—
437C2×D3—41no—
438C3×C4—41no—
439C3×C4—51no—
440C3×C4—51no—
441C3×C4—202no—
442C3×C4—201no—
443C3×C4C2×C4⋊AGL(1,𝔽5)11yescharacteristic
444C3×C4#413011yescharacteristic
445Dic3—21no—
446Dic3—21no—
447Dic3—52no—
448Dic3—102no—
449Dic3—101no—
450Dic3—202no—
451C2×C5—122no—
452C2×C5#466711yescharacteristic
453C2×C5D3×#10411yescharacteristic
454C2×C5#482111yescharacteristic
455D5—21no—
456D5—21no—
457C23—51no—
458C23—61no—
459Q8—152no—
460Q8—301no—
461D4—32no—
462D4—61no—
463D4—302no—
464D4—302no—
465C2×C4—21no—
466C2×C4—31no—
467C2×C4—31no—
468C2×C4—51no—
469C2×C4—51no—
470C2×C4—51no—
471C2×C4—51no—
472C2×C4—51no—
473C2×C4—102no—
474C2×C4—101no—
475C2×C4—122no—
476C2×C4—151no—
477C2×C4—151no—
478C2×C4—201no—
479C2×C4—201no—
480C2×C4—302no—
481C2×C4—302no—
482C2×C4—302no—
483C2×C4—301no—
484C2×C4—302no—
485C2×C4—601no—
486C2×C4—602no—
487C2×C4—601no—
488C2×C4—601no—
489C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
490C8—201no—
491C8—601no—
492C2×C3—101no—
493C2×C3—101no—
494C2×C3#6576511yescharacteristic
495C2×C3#6576911yescharacteristic
496C2×C3D5⋊#10411yescharacteristic
497D3—42no—
498C5#6877011yescharacteristic, complemented, Sylow
499C22—51no—
500C22—51no—
501C22—62no—
502C22—101no—
503C22—101no—
504C22—121no—
505C22—121no—
506C22D3×C22⋊AGL(1,𝔽5)11yescharacteristic
507C4—41no—
508C4—51no—
509C4—51no—
510C4—61no—
511C4—61no—
512C4—152no—
513C4—202no—
514C4—201no—
515C4—301no—
516C4—302no—
517C4—602no—
518C4#9190911yescharacteristic
519C4D3×C4⋊AGL(1,𝔽5)11yescharacteristic
520C3#1060442211yescharacteristic, complemented, Sylow
521C2—101no—
522C2—101no—
523C2—122no—
524C2#1175185411yescharacteristic
525C2#1175195011yescharacteristic
526C2#1175195911yescharacteristic
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 radicalC2×#6772C22
cosocleC2×C3×C4×D5C23
hypercenterC4⋊C4D3×AGL(1,𝔽5)
hypocenterC3×C5#1806
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
2. C4×Dic3⋊AGL(1,𝔽5)C2semidirect
3. #11751829C2semidirect
6. #11746746C2semidirect
7. C3×#64875C2semidirect
15. #91776C4semidirect
16. C5×#893C4semidirect
20. #91435C4semidirect
24. #91340C4semidirect
37. #64875D3semidirect
121. Dic30C4⋊C4semidirect
125. C5×D12C4⋊C4semidirect
151. #893AGL(1,𝔽5)semidirect
225. Dic15#104semidirect
359. D12C4⋊AGL(1,𝔽5)semidirect
429. C3×C5#1806semidirect
498. C5#68770semidirect
520. C3#10604422semidirect

Maximal subgroups

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

subgroupindex
#117534022
C4×Dic3⋊AGL(1,𝔽5)2
#117518292
C2×#911562
C2×#911552
#117467462
C3×#648752
#106044223
#687705