Finite Group Databasebeta

← #14300998

Subgroups — #14300998

Order 1920.

676 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×Q8×AGL(1,𝔽5)C211yesmaximal, characteristic
2C2×D5×#216C211yesmaximal, characteristic
3C2×#91238C211yesmaximal, characteristic
4#11751749C211yesmaximal, characteristic
5C2×#91153C211yesmaximal, characteristic
6#11746529C211yesmaximal, characteristic
7(C2×AGL(1,𝔽5))⋊(C3⋊C8)C211yesmaximal, characteristic
8#10604845—31nomaximal, Hall
9C2×C3×Q8×D5C2211yescharacteristic
10C2×D5×Dic6C2211yescharacteristic
11C2×C4⋊ΓL(1,𝔽16)C2211yescharacteristic
12C2×D5⋊(C3⋊C8)C2211yescharacteristic
13C3×Q8×AGL(1,𝔽5)—22no—
14C3×Q8×AGL(1,𝔽5)—22no—
15C2×C3×C4⋊AGL(1,𝔽5)—21no—
16C2×C3×C4⋊AGL(1,𝔽5)C2211yescharacteristic
17C2×C3×C4×AGL(1,𝔽5)—21no—
18C2×C3×C4×AGL(1,𝔽5)C2211yescharacteristic
19C2×#6718C411yescharacteristic, complemented
20C2×C5×#216C411yescharacteristic, complemented
21D5×#216—22no—
22D5×#216—22no—
23C2×#6662C411yescharacteristic, complemented
24C2×#6661C411yescharacteristic, complemented
25C2×D5×C3⋊C8C2211yescharacteristic
26#91238—22no—
27(C2×AGL(1,𝔽5))⋊Dic3—21no—
28(C2×AGL(1,𝔽5))⋊Dic3—21no—
29#91153—22no—
30#71632—51nomaximal, Hall
31C2×Q8×AGL(1,𝔽5)D311yescharacteristic
32C2×D5×Q16—31no—
33(C2×AGL(1,𝔽5))⋊Q8—31no—
34C2×#4009—31no—
35C2×#4009—31no—
36#64893—31no—
37(C2×AGL(1,𝔽5))⋊C8—31no—
38C22×ΓL(1,𝔽16)—21no—
39C22×C3×AGL(1,𝔽5)—21no—
40C22×C3×AGL(1,𝔽5)D411yescharacteristic
41C22×C3×AGL(1,𝔽5)D411yescharacteristic
42C2×C3×C5×Q8C2×C411yescharacteristic
43C2×Dic30C2×C411yescharacteristic
44C2×C5×Dic6C2×C411yescharacteristic
45C3×Q8×D5—21no—
46C3×Q8×D5—21no—
47C3×Q8×D5D412yes—
48C3×Q8×D5D412yes—
49C2×C3×C4×D5—21no—
50C2×C3×C4×D5C2311yescharacteristic
51C2×C3×Dic10—21no—
52C2×C3×Dic10C2×C411yescharacteristic
53C2×#1204—21no—
54C2×D5×Dic3—21no—
55D5×Dic6—21no—
56D5×Dic6—21no—
57D5×Dic6—21no—
58D5×Dic6—21no—
59C2×C3⋊#170—21no—
60C4⋊ΓL(1,𝔽16)—21no—
61C4⋊ΓL(1,𝔽16)—21no—
62D5⋊(C3⋊C8)—21no—
63D5⋊(C3⋊C8)—21no—
64C3×C4⋊AGL(1,𝔽5)—24no—
65C3×C4⋊AGL(1,𝔽5)—21no—
66C3×C4⋊AGL(1,𝔽5)D412yes—
67C3×C4×AGL(1,𝔽5)—24no—
68C3×C4×AGL(1,𝔽5)—21no—
69C3×C4×AGL(1,𝔽5)—21no—
70#6718—22no—
71C2×C3⋊#168C2×C411yescharacteristic
72C5×#216—22no—
73C2×C5×C3⋊C8C2×C411yescharacteristic
74#6662—22no—
75#6661—22no—
76D5×C3⋊C8—21no—
77D5×C3⋊C8—21no—
78C2×C3×C4×Q8—51no—
79C22×#216—51no—
80C2×#835—51no—
81C2×#810—51no—
82#4929—51no—
83#4529—51no—
84#4524—51no—
85C2×Q8×D5—31no—
86C2×Q8×D5C2×D311yescharacteristic
87Q8×AGL(1,𝔽5)—22no—
88Q8×AGL(1,𝔽5)—22no—
89C2×C4⋊AGL(1,𝔽5)—21no—
90C2×C4⋊AGL(1,𝔽5)—31no—
91C2×C4⋊AGL(1,𝔽5)C2×D311yescharacteristic
92C2×C4×AGL(1,𝔽5)—21no—
93C2×C4×AGL(1,𝔽5)C2×D311yescharacteristic
94C2×D5⋊C8—31no—
95C2×C5×Q16—31no—
96C2×#689—31no—
97C2×#689—31no—
98D5×Q16—62no—
99D5×Q16—62no—
100C2×Dic20—31no—
101C2×C8×D5—31no—
102#4009—62no—
103#4009—62no—
104(C2×AGL(1,𝔽5))⋊C4—61no—
105(C2×AGL(1,𝔽5))⋊C4—61no—
106#1878—151noSylow
107C22×C3×D5C2×D411yescharacteristic
108C2×ΓL(1,𝔽16)—41no—
109C2×ΓL(1,𝔽16)—41no—
110C2×ΓL(1,𝔽16)—41no—
111C2×C3×AGL(1,𝔽5)—21no—
112C2×C3×AGL(1,𝔽5)—21no—
113C2×C3×AGL(1,𝔽5)—21no—
114C2×C3×AGL(1,𝔽5)—22no—
115C2×C3×AGL(1,𝔽5)—21no—
116C2×C3×AGL(1,𝔽5)—21no—
117C2×C3×AGL(1,𝔽5)—41no—
118C2×C3×AGL(1,𝔽5)—41no—
119C2×C3×AGL(1,𝔽5)Q1612yescomplemented
120C3×C5×Q8—21no—
121C3×C5×Q8C22⋊C412yes—
122C2×C3×C4×C5—21no—
123C2×C3×C4×C5C22×C411yescharacteristic
124C2×Dic15—21no—
125Dic30—21no—
126Dic30—21no—
127C2×C5×Dic3—21no—
128C5×Dic6—21no—
129C5×Dic6—21no—
130C2×C3×Dic5—21no—
131C2×C3×Dic5#4111yescharacteristic
132C3×C4×D5—22no—
133C3×C4×D5—22no—
134C3×C4×D5C2×D411yescharacteristic
135C3×C4×D5C2×D411yescharacteristic
136C3×C4×D5C2×D411yescharacteristic
137C3×C4×D5#4111yescharacteristic
138C3×Dic10—21no—
139C3×Dic10—22no—
140C3×Dic10—22no—
141C3×Dic10C22⋊C412yes—
142#1204—41no—
143#1204—41no—
144D5×Dic3—41no—
145D5×Dic3—41no—
146C3⋊#170—41no—
147C3⋊#168—21no—
148C5×C3⋊C8—21no—
149C22×C3×Q8—51no—
150C22×Dic6—51no—
151C3×C4×Q8—102no—
152C3×C4×Q8—102no—
153C2×C3×C4⋊C4—51no—
154C2×C3×C4⋊C4—101no—
155C2×C3×C42—51no—
156C2×C3×C42—101no—
157C2×#216—51no—
158C2×#216—51no—
159C2×#216—51no—
160C2×#216—102no—
161C2×#216—102no—
162C2×#216AGL(1,𝔽5)11yescharacteristic, complemented
163C2×C4⋊Dic3—51no—
164C22×C3⋊C8—51no—
165C22×C3⋊C8—51no—
166#835—102no—
167C3⋊#95—101no—
168C3⋊#95—101no—
169#810—102no—
170C22×AGL(1,𝔽5)—21no—
171C22×AGL(1,𝔽5)—61no—
172C22×AGL(1,𝔽5)#6711yescharacteristic
173C22×AGL(1,𝔽5)#6711yescharacteristic
174C2×C5×Q8—31no—
175C2×C5×Q8C4×D311yescharacteristic
176Q8×D5—21no—
177Q8×D5—21no—
178Q8×D5—61no—
179Q8×D5—61no—
180Q8×D5—61no—
181Q8×D5—61no—
182Q8×D5D1212yes—
183Q8×D5#6712yes—
184C2×C4×D5—21no—
185C2×C4×D5—61no—
186C2×C4×D5C22×D311yescharacteristic
187C2×Dic10—21no—
188C2×Dic10—31no—
189C2×Dic10—61no—
190C2×Dic10C4×D311yescharacteristic
191C2×#170—61no—
192C4⋊AGL(1,𝔽5)—24no—
193C4⋊AGL(1,𝔽5)—21no—
194C4⋊AGL(1,𝔽5)—61no—
195C4⋊AGL(1,𝔽5)—61no—
196C4⋊AGL(1,𝔽5)#6712yes—
197C4×AGL(1,𝔽5)—24no—
198C4×AGL(1,𝔽5)—21no—
199C4×AGL(1,𝔽5)—21no—
200D5⋊C8—61no—
201D5⋊C8—61no—
202C5×Q16—62no—
203C2×C5×C8—31no—
204#689—62no—
205#689—62no—
206C2×#168—31no—
207Dic20—62no—
208C8×D5—61no—
209C8×D5—61no—
210C22×Q16—151no—
211C2×C4×Q8—51no—
212C2×#103—151no—
213C2×#103—151no—
214#391—151no—
215#340—151no—
216#336—151no—
217C22×C3×C5C4×D411yescharacteristic
218C2×C3×D5#12011yescharacteristic
219C2×C3×D5#12111yescharacteristic
220C2×C3×D5C2×Q1611yescharacteristic
221C2×C3×D5#13511yescharacteristic
222C2×C3×D5#13711yescharacteristic
223C2×C3×D5#13711yescharacteristic
224ΓL(1,𝔽16)—81no—
225C3×AGL(1,𝔽5)—22no—
226C3×AGL(1,𝔽5)—41no—
227C3×AGL(1,𝔽5)—42no—
228C3×C4×C5—22no—
229C3×C4×C5C2×C22⋊C411yescharacteristic
230C3×C4×C5C4×D411yescharacteristic
231Dic15—41no—
232C3×Dic5—22no—
233C3×Dic5#12111yescharacteristic
234C3×Dic5#12311yescharacteristic
235C5×Dic3—41no—
236C2×C3×Q8—52no—
237C2×C3×Q8—52no—
238C2×C3×Q8—51no—
239C2×C3×Q8—101no—
240C2×C3×Q8—101no—
241C2×C3×Q8—101no—
242C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
243C22×C3×C4—51no—
244C22×C3×C4—51no—
245C22×C3×C4—51no—
246C22×C3×C4—101no—
247C22×C3×C4—101no—
248C22×Dic3—101no—
249C22×Dic3—101no—
250C2×Dic6—51no—
251C2×Dic6—101no—
252C2×Dic6—101no—
253C2×Dic6—101no—
254C2×Dic6—101no—
255C2×Dic6—101no—
256C2×Dic6C2×AGL(1,𝔽5)11yescharacteristic
257C3×C4⋊C4—52no—
258C3×C4⋊C4—101no—
259C3×C4⋊C4—104no—
260C3×C42—101no—
261C3×C42—101no—
262C3×C42—104no—
263#216—22no—
264#216—102no—
265#216—102no—
266#216—102no—
267C4⋊Dic3—101no—
268C4⋊Dic3—101no—
269C2×C3⋊C8—51no—
270C2×C3⋊C8—101no—
271C2×C3⋊C8—101no—
272C2×C3⋊C8—101no—
273C2×C3⋊C8—101no—
274C2×C3⋊C8—101no—
275C2×C3⋊C8C2×AGL(1,𝔽5)11yescharacteristic
276C22×D5C2×#6711yescharacteristic
277C2×AGL(1,𝔽5)—21no—
278C2×AGL(1,𝔽5)—21no—
279C2×AGL(1,𝔽5)—21no—
280C2×AGL(1,𝔽5)—21no—
281C2×AGL(1,𝔽5)—21no—
282C2×AGL(1,𝔽5)—22no—
283C2×AGL(1,𝔽5)—41no—
284C2×AGL(1,𝔽5)—41no—
285C2×AGL(1,𝔽5)—121no—
286C2×AGL(1,𝔽5)—121no—
287C2×AGL(1,𝔽5)—121no—
288C2×AGL(1,𝔽5)#21612yescomplemented
289C5×Q8—21no—
290C5×Q8—61no—
291C5×Q8—61no—
292C5×Q8(C2×D3)⋊C412yes—
293C2×C4×C5—21no—
294C2×C4×C5—61no—
295C2×C4×C5C2×C4×D311yescharacteristic
296C2×Dic5—21no—
297C2×Dic5—61no—
298C2×Dic5#23511yescharacteristic
299C4×D5—22no—
300C4×D5—22no—
301C4×D5—121no—
302C4×D5—121no—
303C4×D5C2×D1211yescharacteristic
304C4×D5#23511yescharacteristic
305C4×D5C2×#6711yescharacteristic
306C4×D5C2×#6711yescharacteristic
307Dic10—21no—
308Dic10—22no—
309Dic10—22no—
310Dic10—61no—
311Dic10—61no—
312Dic10—121no—
313Dic10—121no—
314Dic10(C2×D3)⋊C412yes—
315#170—121no—
316C5×C8—61no—
317#168—61no—
318C22×Q8—51no—
319C22×Q8—151no—
320C2×Q16—31no—
321C2×Q16—151no—
322C2×Q16—151no—
323C2×Q16—151no—
324C2×Q16—302no—
325C2×Q16—302no—
326C22×C8—151no—
327C22×C8—151no—
328C4×Q8—102no—
329C4×Q8—102no—
330C2×C4⋊C4—51no—
331C2×C4⋊C4—101no—
332C2×C4⋊C4—151no—
333C2×C42—51no—
334C2×C42—101no—
335#103—302no—
336#103—302no—
337#95—301no—
338#95—301no—
339C2×C3×C5#38611yescharacteristic
340C2×C3×C5C4×Q1611yescharacteristic
341C2×C3×C5#44111yescharacteristic
342C3×D5#45111yescharacteristic
343C3×D5#45211yescharacteristic
344C3×D5#46211yescharacteristic
345C3×D5#46411yescharacteristic
346C23×C3—51no—
347C3×Q8—21no—
348C3×Q8—52no—
349C3×Q8—102no—
350C3×Q8—102no—
351C3×Q8—101no—
352C3×Q8C22⋊AGL(1,𝔽5)12yes—
353C2×C3×C4—21no—
354C2×C3×C4—51no—
355C2×C3×C4—51no—
356C2×C3×C4—51no—
357C2×C3×C4—51no—
358C2×C3×C4—51no—
359C2×C3×C4—52no—
360C2×C3×C4—102no—
361C2×C3×C4—102no—
362C2×C3×C4—101no—
363C2×C3×C4—101no—
364C2×C3×C4—101no—
365C2×C3×C4—102no—
366C2×C3×C4—101no—
367C2×C3×C4—101no—
368C2×C3×C4—101no—
369C2×C3×C4—201no—
370C2×C3×C4—201no—
371C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
372C2×Dic3—21no—
373C2×Dic3—101no—
374C2×Dic3—201no—
375C2×Dic3—201no—
376C2×Dic3—201no—
377C2×Dic3—201no—
378C2×Dic3—201no—
379Dic6—21no—
380Dic6—21no—
381Dic6—101no—
382Dic6—101no—
383Dic6—201no—
384Dic6—201no—
385C3⋊C8—21no—
386C3⋊C8—101no—
387C3⋊C8—201no—
388C22×C5C4×#6711yescharacteristic
389C2×D5#93011yescharacteristic
390C2×D5#94211yescharacteristic
391C2×D5C2×#21611yescharacteristic
392C2×D5#95011yescharacteristic
393C2×D5#95111yescharacteristic
394C2×D5#95311yescharacteristic
395AGL(1,𝔽5)—22no—
396AGL(1,𝔽5)—41no—
397AGL(1,𝔽5)—42no—
398AGL(1,𝔽5)—241no—
399C4×C5—22no—
400C4×C5—121no—
401C4×C5C2×(C2×D3)⋊C411yescharacteristic
402C4×C5C4×#6711yescharacteristic
403Dic5—22no—
404Dic5—121no—
405Dic5#92911yescharacteristic
406Dic5#93011yescharacteristic
407C2×Q8—31no—
408C2×Q8—52no—
409C2×Q8—52no—
410C2×Q8—51no—
411C2×Q8—101no—
412C2×Q8—101no—
413C2×Q8—101no—
414C2×Q8—151no—
415C2×Q8—301no—
416C2×Q8—301no—
417C2×Q8—301no—
418C2×Q8—301no—
419C2×Q8—301no—
420C2×Q8D3×AGL(1,𝔽5)11yescharacteristic
421C22×C4—51no—
422C22×C4—51no—
423C22×C4—51no—
424C22×C4—101no—
425C22×C4—101no—
426C22×C4—301no—
427C22×C4—301no—
428Q16—62no—
429Q16—302no—
430Q16—302no—
431Q16—302no—
432C2×C8—31no—
433C2×C8—151no—
434C2×C8—301no—
435C2×C8—301no—
436C2×C8—301no—
437C2×C8—301no—
438C2×C8—301no—
439C4⋊C4—52no—
440C4⋊C4—101no—
441C4⋊C4—104no—
442C4⋊C4—301no—
443C4⋊C4—301no—
444C42—101no—
445C42—101no—
446C42—104no—
447C3×C5#187811yescharacteristic, complemented, Hall
448C22×C3—51no—
449C22×C3—51no—
450C22×C3—51no—
451C22×C3—51no—
452C22×C3—51no—
453C22×C3—51no—
454C22×C3D4×AGL(1,𝔽5)11yescharacteristic
455C3×C4—22no—
456C3×C4—51no—
457C3×C4—51no—
458C3×C4—102no—
459C3×C4—102no—
460C3×C4—201no—
461C3×C4—202no—
462C3×C4D4×AGL(1,𝔽5)11yescharacteristic
463C3×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
464Dic3—41no—
465Dic3—201no—
466Dic3—401no—
467C2×C5C4×#21611yescharacteristic
468C2×C5#503011yescharacteristic
469C2×C5#521311yescharacteristic
470D5#502811yescharacteristic
471D5#503111yescharacteristic
472D5#522811yescharacteristic
473D5#523911yescharacteristic
474C23—51no—
475Q8—21no—
476Q8—52no—
477Q8—61no—
478Q8—61no—
479Q8—102no—
480Q8—102no—
481Q8—101no—
482Q8—301no—
483Q8—301no—
484Q8—601no—
485Q8—601no—
486Q8(C2×D3)⋊AGL(1,𝔽5)12yes—
487C2×C4—21no—
488C2×C4—51no—
489C2×C4—51no—
490C2×C4—51no—
491C2×C4—51no—
492C2×C4—51no—
493C2×C4—52no—
494C2×C4—61no—
495C2×C4—102no—
496C2×C4—102no—
497C2×C4—101no—
498C2×C4—101no—
499C2×C4—101no—
500C2×C4—102no—
501C2×C4—101no—
502C2×C4—101no—
503C2×C4—101no—
504C2×C4—201no—
505C2×C4—201no—
506C2×C4—301no—
507C2×C4—601no—
508C2×C4—601no—
509C2×C4—601no—
510C2×C4—601no—
511C2×C4—601no—
512C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
513C8—61no—
514C8—301no—
515C8—601no—
516C2×C3—51no—
517C2×C3—51no—
518C2×C3—51no—
519C2×C3—51no—
520C2×C3Q16×AGL(1,𝔽5)11yescharacteristic
521C2×C3#6573911yescharacteristic
522C2×C3#6577911yescharacteristic
523C5#7163211yescharacteristic, complemented, Sylow
524C22—51no—
525C22—51no—
526C22—51no—
527C22—51no—
528C22—51no—
529C22—51no—
530C22AGL(1,𝔽5)×#6711yescharacteristic
531C4—22no—
532C4—51no—
533C4—51no—
534C4—102no—
535C4—102no—
536C4—121no—
537C4—201no—
538C4—202no—
539C4—601no—
540C4—1201no—
541C4C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
542C4AGL(1,𝔽5)×#6711yescharacteristic
543C3#1060484511yescharacteristic, complemented, Sylow
544C2—51no—
545C2—51no—
546C2—51no—
547C2—51no—
548C2AGL(1,𝔽5)×#21611yescharacteristic
549C2#1175196811yescharacteristic
550C2#1175202211yescharacteristic
subgroupG/H
centerC22AGL(1,𝔽5)×#67
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#1878
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
19. C2×#6718C4semidirect
20. C2×C5×#216C4semidirect
23. C2×#6662C4semidirect
24. C2×#6661C4semidirect
119. C2×C3×AGL(1,𝔽5)Q16semidirect
162. C2×#216AGL(1,𝔽5)semidirect
288. C2×AGL(1,𝔽5)#216semidirect
447. C3×C5#1878semidirect
523. C5#71632semidirect
543. C3#10604845semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Q8×AGL(1,𝔽5)2
C2×D5×#2162
C2×#912382
#117517492
C2×#911532
#117465292
(C2×AGL(1,𝔽5))⋊(C3⋊C8)2
#106048453
#716325