Finite Group Databasebeta

← #14304544

Subgroups — #14304544

Order 1920.

512 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11752449C211yesmaximal, characteristic
2C2×C3×#4009C211yesmaximal, characteristic
3#11751861C211yesmaximal, characteristic
4C2×ΓL(1,𝔽16)⋊C8C211yesmaximal, characteristic
5D5×#835C211yesmaximal, characteristic
6#11746530C211yesmaximal, characteristic
7#11746529C211yesmaximal, characteristic
8#10603995—31nomaximal, Hall
9C2×C3×Q8×D5C2211yescharacteristic
10C2×C4⋊ΓL(1,𝔽16)C2211yescharacteristic
11C2×C4×ΓL(1,𝔽16)C2211yescharacteristic
12C2×C3×C4⋊AGL(1,𝔽5)C2211yescharacteristic
13C2×C3×D5⋊C8C2211yescharacteristic
14C2×Dic3⋊AGL(1,𝔽5)—21no—
15D5×C4⋊Dic3C2211yescharacteristic
16C2×D5×C3⋊C8C2211yescharacteristic
17#91235—21no—
18#91235—21no—
19C3×#4009—22no—
20(C2×AGL(1,𝔽5))⋊Dic3—21no—
21ΓL(1,𝔽16)⋊C8—22no—
22#91119C411yescharacteristic, complemented
23C5×#835C411yescharacteristic, complemented
24#90971C411yescharacteristic, complemented
25#90970C411yescharacteristic, complemented
26#70254—51nomaximal, Hall
27(C2×AGL(1,𝔽5))⋊Q8—31no—
28C2×#4009D311yescharacteristic
29C2×C8×AGL(1,𝔽5)—31no—
30#65712—31no—
31D5×#103—31no—
32#64893—31no—
33#64874—31no—
34C22×ΓL(1,𝔽16)—21no—
35C22×ΓL(1,𝔽16)D411yescharacteristic
36C22×ΓL(1,𝔽16)D411yescharacteristic
37C22×C3×AGL(1,𝔽5)—21no—
38C2×C3×C5×Q8C2×C411yescharacteristic
39C3×Q8×D5—21no—
40C3×Q8×D5—21no—
41C3×Q8×D5—21no—
42C3×Q8×D5—21no—
43C2×C3×C4×D5—21no—
44C2×C3×C4×D5C2311yescharacteristic
45C2×C3×Dic10—21no—
46C2×C3×Dic10C2×C411yescharacteristic
47C2×D5×Dic3—21no—
48C4⋊ΓL(1,𝔽16)—21no—
49C4⋊ΓL(1,𝔽16)—21no—
50C4×ΓL(1,𝔽16)—21no—
51C4×ΓL(1,𝔽16)—21no—
52C2×C3×#170—21no—
53C3×C4⋊AGL(1,𝔽5)—21no—
54C3×C4⋊AGL(1,𝔽5)—21no—
55C3×D5⋊C8—21no—
56C3×D5⋊C8D412yes—
57Dic3⋊AGL(1,𝔽5)—42no—
58C4⋊Dic15C2×C411yescharacteristic
59C2×C3⋊#168C2×C411yescharacteristic
60C5×C4⋊Dic3C2×C411yescharacteristic
61C2×C5×C3⋊C8C2×C411yescharacteristic
62#6668—21no—
63D5×C3⋊C8—21no—
64D5×C3⋊C8—21no—
65C2×C3×#103—51no—
66#5229—51no—
67C2×#835—51no—
68C2×C8×Dic3—51no—
69#4972—51no—
70#4551—51no—
71#4529—51no—
72C2×Q8×D5C2×D311yescharacteristic
73C2×C4⋊AGL(1,𝔽5)—31no—
74C2×C4⋊AGL(1,𝔽5)—61no—
75C2×C4⋊AGL(1,𝔽5)C2×D311yescharacteristic
76C2×C4×AGL(1,𝔽5)—31no—
77C2×D5⋊C8C2×D311yescharacteristic
78C2×C8×D5—31no—
79D5×C4⋊C4—31no—
80#4009—22no—
81(C2×AGL(1,𝔽5))⋊C4—61no—
82(C2×AGL(1,𝔽5))⋊C4—61no—
83(C2×AGL(1,𝔽5))⋊C4—61no—
84C8×AGL(1,𝔽5)—62no—
85C5×#103—31no—
86#3967—31no—
87#3948—31no—
88#3942—31no—
89#1933—151noSylow
90C22×C3×D5C2×D411yescharacteristic
91C2×ΓL(1,𝔽16)—21no—
92C2×ΓL(1,𝔽16)—21no—
93C2×ΓL(1,𝔽16)—21no—
94C2×ΓL(1,𝔽16)—21no—
95C2×ΓL(1,𝔽16)—41no—
96C2×ΓL(1,𝔽16)—41no—
97C2×ΓL(1,𝔽16)—41no—
98C2×ΓL(1,𝔽16)SD1612yes—
99C2×ΓL(1,𝔽16)Q1612yes—
100C2×C3×AGL(1,𝔽5)—41no—
101C2×C3×AGL(1,𝔽5)—41no—
102C2×C3×AGL(1,𝔽5)—41no—
103C3×C5×Q8—21no—
104C3×C5×Q8—21no—
105C2×C3×C4×C5—21no—
106C2×C3×C4×C5C22×C411yescharacteristic
107C2×Dic15—21no—
108C2×C5×Dic3—21no—
109C2×C3×Dic5—21no—
110C2×C3×Dic5#4111yescharacteristic
111C3×C4×D5—41no—
112C3×C4×D5—41no—
113C3×C4×D5C2×D411yescharacteristic
114C3×C4×D5#4111yescharacteristic
115C3×C4×D5#4111yescharacteristic
116C3×C4×D5#4111yescharacteristic
117C3×Dic10—21no—
118C3×Dic10—21no—
119C3×Dic10—41no—
120C3×Dic10—41no—
121D5×Dic3—41no—
122D5×Dic3—41no—
123C3×#170—22no—
124C3⋊#168—21no—
125C5×C3⋊C8—21no—
126C22×C3×Q8—51no—
127C22×C3×C8—51no—
128C2×C3×C4⋊C4—51no—
129C2×C4⋊Dic3—51no—
130C2×C4⋊Dic3—51no—
131C2×Dic3⋊C4—101no—
132C2×C4×Dic3—51no—
133C22×C3⋊C8—51no—
134C3×#103—102no—
135#835—51no—
136#835—51no—
137#835—51no—
138#835AGL(1,𝔽5)11yescharacteristic, complemented
139C3⋊#95—101no—
140C3⋊#95—101no—
141C3⋊#95—101no—
142C8×Dic3—102no—
143C22×AGL(1,𝔽5)—21no—
144C22×AGL(1,𝔽5)—31no—
145C22×AGL(1,𝔽5)—31no—
146C22×AGL(1,𝔽5)—61no—
147C2×C5×Q8C4×D311yescharacteristic
148Q8×D5—21no—
149Q8×D5—21no—
150Q8×D5—21no—
151Q8×D5—21no—
152C2×C4×D5—21no—
153C2×C4×D5—61no—
154C2×C4×D5C22×D311yescharacteristic
155C2×Dic10—21no—
156C2×Dic10C4×D311yescharacteristic
157C2×#170—21no—
158C4⋊AGL(1,𝔽5)—21no—
159C4⋊AGL(1,𝔽5)—21no—
160C4⋊AGL(1,𝔽5)—61no—
161C4⋊AGL(1,𝔽5)—61no—
162C4⋊AGL(1,𝔽5)—122no—
163C4×AGL(1,𝔽5)—61no—
164C4×AGL(1,𝔽5)—61no—
165D5⋊C8—21no—
166D5⋊C8#6712yes—
167C2×C5×C8—31no—
168C5×C4⋊C4—31no—
169C4⋊Dic5—31no—
170#683—61no—
171C2×#168—31no—
172C8×D5—61no—
173C8×D5—61no—
174C2×#103—51no—
175C2×#103—151no—
176C2×C4×C8—151no—
177#391—151no—
178#389—151no—
179#340—151no—
180#340—151no—
181C22×C3×C5C4×D411yescharacteristic
182C2×C3×D5#12111yescharacteristic
183C2×C3×D5#12411yescharacteristic
184C2×C3×D5C2×SD1611yescharacteristic
185C2×C3×D5C2×Q1611yescharacteristic
186C2×C3×D5#13511yescharacteristic
187C2×C3×D5#13511yescharacteristic
188ΓL(1,𝔽16)—22no—
189ΓL(1,𝔽16)—22no—
190ΓL(1,𝔽16)—81no—
191C3×AGL(1,𝔽5)—81no—
192C3×C4×C5—41no—
193C3×C4×C5#11711yescharacteristic
194C3×C4×C5C4×D411yescharacteristic
195Dic15—41no—
196C3×Dic5—41no—
197C3×Dic5#12311yescharacteristic
198C3×Dic5#12611yescharacteristic
199C5×Dic3—41no—
200C2×C3×Q8—51no—
201C2×C3×Q8—101no—
202C2×C3×Q8—101no—
203C2×C3×Q8—101no—
204C2×C3×Q8—101no—
205C2×C3×Q8—101no—
206C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
207C22×C3×C4—51no—
208C22×C3×C4—101no—
209C22×C3×C4—101no—
210C22×Dic3—51no—
211C22×Dic3—51no—
212C22×Dic3—101no—
213C22×Dic3—101no—
214C2×C3×C8—52no—
215C2×C3×C8—101no—
216C2×C3×C8—101no—
217C3×C4⋊C4—101no—
218C3×C4⋊C4—101no—
219C4⋊Dic3—51no—
220C4⋊Dic3—101no—
221C4⋊Dic3—101no—
222C4⋊Dic3—101no—
223C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
224Dic3⋊C4—202no—
225C4×Dic3—101no—
226C4×Dic3—101no—
227C2×C3⋊C8—51no—
228C2×C3⋊C8—101no—
229C2×C3⋊C8—101no—
230C2×C3⋊C8C2×AGL(1,𝔽5)11yescharacteristic
231C22×D5D3×D411yescharacteristic
232C2×AGL(1,𝔽5)—32no—
233C2×AGL(1,𝔽5)—32no—
234C2×AGL(1,𝔽5)—41no—
235C2×AGL(1,𝔽5)—41no—
236C2×AGL(1,𝔽5)—41no—
237C2×AGL(1,𝔽5)—61no—
238C2×AGL(1,𝔽5)—61no—
239C2×AGL(1,𝔽5)—61no—
240C2×AGL(1,𝔽5)—61no—
241C2×AGL(1,𝔽5)—121no—
242C2×AGL(1,𝔽5)—121no—
243C2×AGL(1,𝔽5)—121no—
244C5×Q8—21no—
245C5×Q8—21no—
246C2×C4×C5—21no—
247C2×C4×C5—61no—
248C2×C4×C5C2×C4×D311yescharacteristic
249C2×Dic5—21no—
250C2×Dic5—61no—
251C2×Dic5#23711yescharacteristic
252C4×D5—41no—
253C4×D5—41no—
254C4×D5—121no—
255C4×D5—121no—
256C4×D5#23511yescharacteristic
257C4×D5#23711yescharacteristic
258C4×D5#23711yescharacteristic
259C4×D5C2×#6711yescharacteristic
260Dic10—21no—
261Dic10—21no—
262Dic10—41no—
263Dic10—41no—
264#170—22no—
265C5×C8—61no—
266#168—61no—
267C22×Q8—51no—
268C22×C8—51no—
269C22×C8—151no—
270C2×C4⋊C4—51no—
271C2×C4⋊C4—151no—
272C2×C4⋊C4—151no—
273C2×C4⋊C4—301no—
274C2×C42—151no—
275#103—31no—
276#103—102no—
277#103—151no—
278#103—151no—
279#103—151no—
280C4×C8—302no—
281#95—301no—
282#95—301no—
283#95—301no—
284C2×C3×C5#38811yescharacteristic
285C2×C3×C5C4×SD1611yescharacteristic
286C2×C3×C5C4×Q1611yescharacteristic
287C3×D5#46511yescharacteristic
288C3×D5#46711yescharacteristic
289C3×D5#48711yescharacteristic
290C3×D5#48811yescharacteristic
291C23×C3—51no—
292C3×Q8—21no—
293C3×Q8—21no—
294C3×Q8—101no—
295C3×Q8—101no—
296C3×Q8—201no—
297C3×Q8—201no—
298C2×C3×C4—21no—
299C2×C3×C4—51no—
300C2×C3×C4—51no—
301C2×C3×C4—51no—
302C2×C3×C4—51no—
303C2×C3×C4—51no—
304C2×C3×C4—101no—
305C2×C3×C4—201no—
306C2×C3×C4—201no—
307C2×C3×C4—201no—
308C2×C3×C4—201no—
309C2×C3×C4—201no—
310C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
311C2×Dic3—21no—
312C2×Dic3—52no—
313C2×Dic3—52no—
314C2×Dic3—101no—
315C2×Dic3—101no—
316C2×Dic3—101no—
317C2×Dic3—101no—
318C2×Dic3—101no—
319C2×Dic3—201no—
320C2×Dic3—201no—
321C2×Dic3—201no—
322C2×Dic3—201no—
323C2×Dic3—201no—
324C3×C8—102no—
325C3⋊C8—21no—
326C3⋊C8—101no—
327C22×C5#67⋊C411yescharacteristic
328C2×D5#88511yescharacteristic
329C2×D5D3×SD1611yescharacteristic
330C2×D5#91611yescharacteristic
331C2×D5D3×Q1611yescharacteristic
332C2×D5#91911yescharacteristic
333C2×D5#93811yescharacteristic
334AGL(1,𝔽5)—62no—
335AGL(1,𝔽5)—62no—
336AGL(1,𝔽5)—81no—
337AGL(1,𝔽5)—241no—
338C4×C5—41no—
339C4×C5—121no—
340C4×C5#87711yescharacteristic
341C4×C5C4×#6711yescharacteristic
342Dic5—41no—
343Dic5—121no—
344Dic5#87911yescharacteristic
345Dic5#93511yescharacteristic
346C2×Q8—51no—
347C2×Q8—101no—
348C2×Q8—101no—
349C2×Q8—101no—
350C2×Q8—101no—
351C2×Q8—101no—
352C2×Q8D3×AGL(1,𝔽5)11yescharacteristic
353C22×C4—51no—
354C22×C4—101no—
355C22×C4—101no—
356C22×C4—151no—
357C22×C4—151no—
358C22×C4—301no—
359C22×C4—301no—
360C2×C8—31no—
361C2×C8—52no—
362C2×C8—101no—
363C2×C8—101no—
364C2×C8—151no—
365C2×C8—301no—
366C2×C8—301no—
367C4⋊C4—31no—
368C4⋊C4—101no—
369C4⋊C4—101no—
370C4⋊C4—151no—
371C4⋊C4—301no—
372C4⋊C4—301no—
373C4⋊C4—301no—
374C4⋊C4—602no—
375C42—301no—
376C42—301no—
377C3×C5#193311yescharacteristic, complemented, Hall
378C22×C3—51no—
379C22×C3—51no—
380C22×C3—51no—
381C22×C3—51no—
382C22×C3—51no—
383C22×C3—51no—
384C22×C3D4×AGL(1,𝔽5)11yescharacteristic
385C3×C4—41no—
386C3×C4—51no—
387C3×C4—51no—
388C3×C4—201no—
389C3×C4—401no—
390C3×C4#413011yescharacteristic
391C3×C4D4×AGL(1,𝔽5)11yescharacteristic
392Dic3—41no—
393Dic3—102no—
394Dic3—102no—
395Dic3—201no—
396Dic3—401no—
397C2×C5#478811yescharacteristic
398C2×C5#479011yescharacteristic
399C2×C5#494411yescharacteristic
400D5#479211yescharacteristic
401D5#480011yescharacteristic
402D5#517111yescharacteristic
403D5#518811yescharacteristic
404C23—51no—
405Q8—21no—
406Q8—21no—
407Q8—101no—
408Q8—101no—
409Q8—201no—
410Q8—201no—
411C2×C4—21no—
412C2×C4—51no—
413C2×C4—51no—
414C2×C4—51no—
415C2×C4—51no—
416C2×C4—51no—
417C2×C4—61no—
418C2×C4—101no—
419C2×C4—152no—
420C2×C4—152no—
421C2×C4—201no—
422C2×C4—201no—
423C2×C4—201no—
424C2×C4—201no—
425C2×C4—201no—
426C2×C4—301no—
427C2×C4—301no—
428C2×C4—301no—
429C2×C4—301no—
430C2×C4—301no—
431C2×C4—601no—
432C2×C4—601no—
433C2×C4—601no—
434C2×C4—601no—
435C2×C4—601no—
436C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
437C8—61no—
438C8—102no—
439C8—301no—
440C2×C3—51no—
441C2×C3—51no—
442C2×C3—51no—
443C2×C3—51no—
444C2×C3#6570011yescharacteristic
445C2×C3SD16×AGL(1,𝔽5)11yescharacteristic
446C2×C3Q16×AGL(1,𝔽5)11yescharacteristic
447C5#7025411yescharacteristic, complemented, Sylow
448C22—51no—
449C22—51no—
450C22—51no—
451C22—51no—
452C22—51no—
453C22—51no—
454C22#67⋊AGL(1,𝔽5)11yescharacteristic
455C4—41no—
456C4—51no—
457C4—51no—
458C4—121no—
459C4—201no—
460C4—302no—
461C4—302no—
462C4—401no—
463C4—601no—
464C4—1201no—
465C4#9192311yescharacteristic
466C4AGL(1,𝔽5)×#6711yescharacteristic
467C3#1060399511yescharacteristic, complemented, Sylow
468C2—51no—
469C2—51no—
470C2—51no—
471C2—51no—
472C2#1175196511yescharacteristic
473C2#1175196911yescharacteristic
474C2#1175201711yescharacteristic
subgroupG/H
centerC22#67⋊AGL(1,𝔽5)
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC2×C3×C5×Q8C2×C4
socleC22×C3×C5C4×D4
prime radicalC22×C3×D5C2×D4
cosocleC2×C3×C4×D5C23
hypercenterC2×Q8D3×AGL(1,𝔽5)
hypocenterC3×C5#1933
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
22. #91119C4semidirect
23. C5×#835C4semidirect
24. #90971C4semidirect
25. #90970C4semidirect
138. #835AGL(1,𝔽5)semidirect
377. C3×C5#1933semidirect
447. C5#70254semidirect
467. C3#10603995semidirect

Maximal subgroups

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

subgroupindex
#117524492
C2×C3×#40092
#117518612
C2×ΓL(1,𝔽16)⋊C82
D5×#8352
#117465302
#117465292
#106039953
#702545