Finite Group Databasebeta

← #14331116

Subgroups — #14331116

Order 1920.

928 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D5×#924C211yesmaximal, characteristic
2#11752423C211yesmaximal, characteristic
3C3×#65765C211yesmaximal, characteristic
4#11752018C212yesmaximal
5#11752017C212yesmaximal
6#11751842C212yesmaximal, complemented
7#11751840C212yesmaximal, complemented
8#11751837C212yesmaximal, complemented
9#11751835C212yesmaximal, complemented
10#10617288—31nomaximal, Hall
11C22×C3×C4×D5C2211yescharacteristic
12C2×D5×Dic6C2211yescharacteristic
13C2×#6763C2212yes—
14C2×C3×C22⋊AGL(1,𝔽5)C2211yescharacteristic
15C2×C3×C22⋊AGL(1,𝔽5)C2211yescharacteristic
16C2×C3×C4⋊AGL(1,𝔽5)C2212yes—
17C2×Dic3⋊AGL(1,𝔽5)C2212yes—
18C2×Dic3⋊AGL(1,𝔽5)C2212yes—
19C2×Dic3×AGL(1,𝔽5)C2212yes—
20C2×Dic3×AGL(1,𝔽5)C2212yes—
21#91813C411yescharacteristic, complemented
22C5×#924C411yescharacteristic, complemented
23#91574C412yescomplemented
24D5×#217C2212yes—
25D5×C4⋊Dic3C2211yescharacteristic
26D5×Dic3⋊C4C2212yes—
27#91360C411yescharacteristic, complemented
28#91359C411yescharacteristic, complemented
29#91349C412yescomplemented
30#91235C2211yescharacteristic
31#91235C2211yescharacteristic
32#91235C2212yes—
33C3×(C2×AGL(1,𝔽5))⋊C4C2212yes—
34(C2×AGL(1,𝔽5))⋊Dic3C2212yes—
35(C2×AGL(1,𝔽5))⋊Dic3C2212yes—
36(C2×AGL(1,𝔽5))⋊Dic3C2212yes—
37(C2×AGL(1,𝔽5))⋊Dic3C2212yes—
38#73886—51nomaximal, Hall
39D5×C22⋊Q8—31no—
40(C2×AGL(1,𝔽5))⋊Q8—32no—
41#65765D311yescharacteristic
42#65764—31no—
43#65712—32no—
44#65711—32no—
45#65711—32no—
46#65700—32no—
47#65700—32no—
48C23×C3×D5C2311yescharacteristic
49C22×ΓL(1,𝔽16)C2312yes—
50C22×ΓL(1,𝔽16)C2312yes—
51C22×C3×AGL(1,𝔽5)C2312yes—
52C22×C3×AGL(1,𝔽5)C2312yes—
53C22×C3×C4×C5C2×C411yescharacteristic
54C2×Dic30C2×C411yescharacteristic
55C2×C5×Dic6C2×C411yescharacteristic
56C22×C3×Dic5C2×C411yescharacteristic
57C2×C3×C4×D5—42no—
58C2×C3×C4×D5C2×C411yescharacteristic
59C2×C3×C4×D5C2×C411yescharacteristic
60C2×C3×C4×D5C2311yescharacteristic
61C2×C3×C4×D5C2311yescharacteristic
62C2×#1204C2×C412yes—
63C2×D5×Dic3C2312yes—
64C2×D5×Dic3C2312yes—
65D5×Dic6—42no—
66#6763—42no—
67C3×C22⋊AGL(1,𝔽5)—41no—
68C3×C22⋊AGL(1,𝔽5)—41no—
69C3×C4⋊AGL(1,𝔽5)—42no—
70Dic3⋊AGL(1,𝔽5)—42no—
71Dic3⋊AGL(1,𝔽5)—42no—
72Dic3×AGL(1,𝔽5)—42no—
73Dic3×AGL(1,𝔽5)—42no—
74#6719C2×C412yes—
75C4⋊Dic15C2×C411yescharacteristic
76#6712C2×C412yes—
77C5×#217C2×C412yes—
78C5×C4⋊Dic3C2×C411yescharacteristic
79C5×Dic3⋊C4C2×C412yes—
80Dic3⋊Dic5C2×C412yes—
81#6669C2×C412yes—
82#6668C2×C412yes—
83#6665C2×C412yes—
84#6665C2×C412yes—
85C2×#924—51no—
86C3×#380—51no—
87#5211—51no—
88#4978—52no—
89#4976—52no—
90#4945—52no—
91#4944—52no—
92#4654—52no—
93#4646—52no—
94C2×Q8×D5—31no—
95C22×C4×D5C2×D311yescharacteristic
96C2×C22⋊AGL(1,𝔽5)—32no—
97C2×C22⋊AGL(1,𝔽5)C2×D311yescharacteristic
98C2×C22⋊AGL(1,𝔽5)C2×D311yescharacteristic
99C2×C4⋊AGL(1,𝔽5)—32no—
100C2×C4⋊AGL(1,𝔽5)—32no—
101C2×C4⋊AGL(1,𝔽5)C2×D312yes—
102C2×C4×AGL(1,𝔽5)—32no—
103C2×C4×AGL(1,𝔽5)—32no—
104C5×C22⋊Q8—31no—
105#4092—31no—
106#4070—31no—
107#4043—31no—
108#4042—32no—
109D5×C4⋊C4—32no—
110D5×C4⋊C4—31no—
111D5×C22⋊C4—32no—
112#4024—32no—
113(C2×AGL(1,𝔽5))⋊C4—32no—
114(C2×AGL(1,𝔽5))⋊C4—32no—
115(C2×AGL(1,𝔽5))⋊C4—32no—
116(C2×AGL(1,𝔽5))⋊C4—32no—
117(C2×AGL(1,𝔽5))⋊C4—31no—
118(C2×AGL(1,𝔽5))⋊C4—32no—
119(C2×AGL(1,𝔽5))⋊C4—31no—
120(C2×AGL(1,𝔽5))⋊C4C2×D312yes—
121#2378—151noSylow
122C23×C3×C5C22×C411yescharacteristic
123C22×C3×D5—21no—
124C22×C3×D5—22no—
125C22×C3×D5—22no—
126C22×C3×D5—21no—
127C22×C3×D5C22×C411yescharacteristic
128C22×C3×D5C2411yescharacteristic
129C2×ΓL(1,𝔽16)—22no—
130C2×ΓL(1,𝔽16)—22no—
131C2×ΓL(1,𝔽16)—22no—
132C2×ΓL(1,𝔽16)—22no—
133C2×ΓL(1,𝔽16)—22no—
134C2×ΓL(1,𝔽16)—22no—
135C2×C3×AGL(1,𝔽5)—22no—
136C2×C3×AGL(1,𝔽5)—22no—
137C2×C3×AGL(1,𝔽5)—22no—
138C2×C3×AGL(1,𝔽5)—22no—
139C2×C3×AGL(1,𝔽5)—22no—
140C2×C3×AGL(1,𝔽5)—22no—
141C2×C3×C4×C5—41no—
142C2×C3×C4×C5C22×C411yescharacteristic
143C2×C3×C4×C5C22×C411yescharacteristic
144C2×Dic15C22×C412yes—
145C2×Dic15C22×C412yes—
146Dic30—41no—
147C2×C5×Dic3C22×C412yes—
148C2×C5×Dic3C22×C412yes—
149C5×Dic6—41no—
150C2×C3×Dic5—41no—
151C2×C3×Dic5C22×C411yescharacteristic
152C2×C3×Dic5C22×C411yescharacteristic
153C3×C4×D5—22no—
154C3×C4×D5—22no—
155C3×C4×D5—41no—
156C3×C4×D5—41no—
157#1204—42no—
158D5×Dic3—22no—
159D5×Dic3—22no—
160D5×Dic3—22no—
161D5×Dic3—22no—
162C23×C3×C4—51no—
163C22×Dic6—51no—
164C2×C3×C4⋊C4—52no—
165C2×C3×C22⋊C4—51no—
166C2×C3×C22⋊C4—51no—
167C2×#217—52no—
168C2×#217—52no—
169C2×C4⋊Dic3—51no—
170#924—51no—
171#924—51no—
172#924—52no—
173#924—51no—
174#924—52no—
175#924AGL(1,𝔽5)11yescharacteristic, complemented
176C2×Dic3⋊C4—52no—
177C2×Dic3⋊C4—52no—
178C2×Dic3⋊C4—52no—
179C2×C4×Dic3—52no—
180C2×C4×Dic3—52no—
181C3×#95—52no—
182C3⋊#95—52no—
183C3⋊#95—52no—
184C3⋊#95—52no—
185C3⋊#95—52no—
186C3⋊#95—52no—
187C3⋊#95—51no—
188C3⋊#95—51no—
189C23×D5C22×D311yescharacteristic
190C22×AGL(1,𝔽5)—32no—
191C22×AGL(1,𝔽5)—32no—
192C22×AGL(1,𝔽5)C22×D312yes—
193C22×AGL(1,𝔽5)C22×D312yes—
194C2×C5×Q8—31no—
195C22×C4×C5C4×D311yescharacteristic
196C22×Dic5C4×D311yescharacteristic
197Q8×D5—122no—
198C2×C4×D5—32no—
199C2×C4×D5—32no—
200C2×C4×D5—42no—
201C2×C4×D5C4×D311yescharacteristic
202C2×C4×D5C4×D311yescharacteristic
203C2×C4×D5C22×D311yescharacteristic
204C2×C4×D5C22×D311yescharacteristic
205C2×Dic10—31no—
206C2×Dic10—32no—
207C22⋊AGL(1,𝔽5)—41no—
208C22⋊AGL(1,𝔽5)—41no—
209C22⋊AGL(1,𝔽5)—122no—
210C4⋊AGL(1,𝔽5)—42no—
211C4⋊AGL(1,𝔽5)—122no—
212C4⋊AGL(1,𝔽5)—122no—
213C4×AGL(1,𝔽5)—122no—
214C4×AGL(1,𝔽5)—122no—
215C5×C4⋊C4—32no—
216C5×C4⋊C4—31no—
217C5×C22⋊C4—32no—
218C22⋊Dic5—32no—
219#685—32no—
220#685—32no—
221C4⋊Dic5—31no—
222C4⋊Dic5—32no—
223#683—32no—
224#683—32no—
225#683—32no—
226C2×C22⋊Q8—151no—
227#391—152no—
228#389—152no—
229#388—152no—
230#388—152no—
231#387—152no—
232#387—152no—
233#381—151no—
234#380—51no—
235C22×C3×C5—21no—
236C22×C3×C5—21no—
237C22×C3×C5—21no—
238C22×C3×C5C23×C411yescharacteristic
239C2×C3×D5—21no—
240C2×C3×D5—21no—
241C2×C3×D5—21no—
242C2×C3×D5—42no—
243C2×C3×D5—42no—
244C2×C3×D5H(5,𝔽2)11yescharacteristic
245C2×C3×D5H(5,𝔽2)12yes—
246C2×C3×D52−1+411yescharacteristic
247C2×C3×D52−1+412yes—
248ΓL(1,𝔽16)—42no—
249ΓL(1,𝔽16)—42no—
250C3×AGL(1,𝔽5)—42no—
251C3×AGL(1,𝔽5)—42no—
252C3×C4×C5—21no—
253C3×C4×C5—21no—
254Dic15—22no—
255Dic15—22no—
256C3×Dic5—21no—
257C3×Dic5—21no—
258C5×Dic3—22no—
259C5×Dic3—22no—
260C24×C3—51no—
261C22×C3×C4—51no—
262C22×C3×C4—51no—
263C22×C3×C4—51no—
264C22×C3×C4—51no—
265C22×C3×C4—51no—
266C22×C3×C4—52no—
267C22×C3×C4—52no—
268C22×C3×C4—202no—
269C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
270C22×Dic3—52no—
271C22×Dic3—52no—
272C22×Dic3—52no—
273C22×Dic3—52no—
274C2×Dic6—51no—
275C2×Dic6—52no—
276C2×Dic6—202no—
277C2×Dic6C2×AGL(1,𝔽5)11yescharacteristic
278C3×C4⋊C4—202no—
279C3×C22⋊C4—201no—
280C3×C22⋊C4—201no—
281#217—52no—
282#217—52no—
283#217—52no—
284#217—202no—
285#217C2×AGL(1,𝔽5)12yes—
286C4⋊Dic3—51no—
287C4⋊Dic3—52no—
288C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
289Dic3⋊C4—52no—
290Dic3⋊C4—52no—
291Dic3⋊C4—52no—
292Dic3⋊C4—202no—
293Dic3⋊C4—202no—
294Dic3⋊C4C2×AGL(1,𝔽5)12yes—
295C4×Dic3—202no—
296C4×Dic3—202no—
297C23×C5C2×C4×D311yescharacteristic
298C22×D5—21no—
299C22×D5—22no—
300C22×D5—21no—
301C22×D5—22no—
302C22×D5C2×C4×D311yescharacteristic
303C22×D5C23×D311yescharacteristic
304C2×AGL(1,𝔽5)—22no—
305C2×AGL(1,𝔽5)—22no—
306C2×AGL(1,𝔽5)—22no—
307C2×AGL(1,𝔽5)—22no—
308C2×AGL(1,𝔽5)—22no—
309C2×AGL(1,𝔽5)—22no—
310C2×AGL(1,𝔽5)—62no—
311C2×AGL(1,𝔽5)—62no—
312C2×AGL(1,𝔽5)—62no—
313C2×AGL(1,𝔽5)—62no—
314C2×AGL(1,𝔽5)—62no—
315C2×AGL(1,𝔽5)—62no—
316C5×Q8—121no—
317C2×C4×C5—32no—
318C2×C4×C5—32no—
319C2×C4×C5—41no—
320C2×C4×C5C2×C4×D311yescharacteristic
321C2×C4×C5C2×C4×D311yescharacteristic
322C2×Dic5—32no—
323C2×Dic5—32no—
324C2×Dic5—41no—
325C2×Dic5C2×C4×D311yescharacteristic
326C2×Dic5C2×C4×D311yescharacteristic
327C4×D5—22no—
328C4×D5—22no—
329C4×D5—41no—
330C4×D5—41no—
331C4×D5—62no—
332C4×D5—62no—
333C4×D5—62no—
334C4×D5—62no—
335Dic10—122no—
336Dic10—121no—
337C22×Q8—151no—
338C23×C4—51no—
339C22⋊Q8—31no—
340C22⋊Q8—152no—
341C22⋊Q8—152no—
342C22⋊Q8—151no—
343C22⋊Q8—151no—
344C22⋊Q8—151no—
345C2×C4⋊C4—52no—
346C2×C4⋊C4—151no—
347C2×C4⋊C4—152no—
348C2×C4⋊C4—152no—
349C2×C4⋊C4—152no—
350C2×C22⋊C4—51no—
351C2×C22⋊C4—51no—
352C2×C22⋊C4—152no—
353C2×C22⋊C4—152no—
354C2×C42—152no—
355C2×C42—152no—
356#95—52no—
357#95—151no—
358#95—152no—
359#95—151no—
360#95—152no—
361#95—152no—
362#95—152no—
363#95—152no—
364C2×C3×C5—41no—
365C2×C3×C5#51811yescharacteristic
366C2×C3×C5#51911yescharacteristic
367C2×C3×C5#52011yescharacteristic
368C3×D5—41no—
369C3×D5#56212yes—
370C3×D5#56312yes—
371C23×C3—51no—
372C23×C3—51no—
373C23×C3—102no—
374C23×C3—101no—
375C23×C3—102no—
376C23×C3—101no—
377C23×C3C22×AGL(1,𝔽5)11yescharacteristic
378C2×C3×C4—41no—
379C2×C3×C4—51no—
380C2×C3×C4—51no—
381C2×C3×C4—102no—
382C2×C3×C4—102no—
383C2×C3×C4—102no—
384C2×C3×C4—102no—
385C2×C3×C4—102no—
386C2×C3×C4—102no—
387C2×C3×C4—102no—
388C2×C3×C4—102no—
389C2×C3×C4—201no—
390C2×C3×C4—201no—
391C2×C3×C4—201no—
392C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
393C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
394C2×Dic3—52no—
395C2×Dic3—52no—
396C2×Dic3—102no—
397C2×Dic3—102no—
398C2×Dic3—102no—
399C2×Dic3—102no—
400C2×Dic3—102no—
401C2×Dic3—102no—
402C2×Dic3—102no—
403C2×Dic3—102no—
404C2×Dic3—102no—
405C2×Dic3—102no—
406C2×Dic3C22×AGL(1,𝔽5)12yes—
407C2×Dic3C22×AGL(1,𝔽5)12yes—
408Dic6—41no—
409Dic6—202no—
410Dic6—201no—
411C22×C5—21no—
412C22×C5—21no—
413C22×C5—21no—
414C22×C5C22×C4×D311yescharacteristic
415C2×D5—21no—
416C2×D5—21no—
417C2×D5—21no—
418C2×D5—42no—
419C2×D5—42no—
420C2×D5#100412yes—
421C2×D5#100411yescharacteristic
422C2×D5#100711yescharacteristic
423C2×D5#101012yes—
424AGL(1,𝔽5)—42no—
425AGL(1,𝔽5)—42no—
426AGL(1,𝔽5)—122no—
427AGL(1,𝔽5)—122no—
428C4×C5—21no—
429C4×C5—21no—
430C4×C5—62no—
431C4×C5—62no—
432Dic5—21no—
433Dic5—21no—
434Dic5—62no—
435Dic5—62no—
436C24—51no—
437C2×Q8—31no—
438C2×Q8—152no—
439C2×Q8—151no—
440C2×Q8—602no—
441C22×C4—51no—
442C22×C4—51no—
443C22×C4—51no—
444C22×C4—51no—
445C22×C4—51no—
446C22×C4—52no—
447C22×C4—52no—
448C22×C4—152no—
449C22×C4—152no—
450C22×C4—152no—
451C22×C4—152no—
452C22×C4—202no—
453C22×C4D3×AGL(1,𝔽5)11yescharacteristic
454C4⋊C4—32no—
455C4⋊C4—31no—
456C4⋊C4—152no—
457C4⋊C4—152no—
458C4⋊C4—152no—
459C4⋊C4—152no—
460C4⋊C4—151no—
461C4⋊C4—202no—
462C4⋊C4—602no—
463C4⋊C4—602no—
464C22⋊C4—32no—
465C22⋊C4—152no—
466C22⋊C4—152no—
467C22⋊C4—152no—
468C22⋊C4—201no—
469C22⋊C4—201no—
470C22⋊C4—602no—
471C42—602no—
472C42—602no—
473C3×C5#237811yescharacteristic, complemented, Hall
474C22×C3—21no—
475C22×C3—21no—
476C22×C3—21no—
477C22×C3—52no—
478C22×C3—52no—
479C22×C3—51no—
480C22×C3—51no—
481C22×C3—101no—
482C22×C3—101no—
483C22×C3—101no—
484C22×C3—202no—
485C22×C3—202no—
486C22×C3C23×AGL(1,𝔽5)11yescharacteristic
487C3×C4—21no—
488C3×C4—21no—
489C3×C4—101no—
490C3×C4—101no—
491C3×C4—202no—
492C3×C4—202no—
493Dic3—22no—
494Dic3—22no—
495Dic3—102no—
496Dic3—102no—
497Dic3—202no—
498Dic3—202no—
499C2×C5—41no—
500C2×C5#548611yescharacteristic
501C2×C5#550411yescharacteristic
502C2×C5#551611yescharacteristic
503D5—41no—
504D5#562112yes—
505D5#564612yes—
506C23—51no—
507C23—51no—
508C23—101no—
509C23—102no—
510C23—102no—
511C23—101no—
512C23C2×D3×AGL(1,𝔽5)11yescharacteristic
513Q8—121no—
514Q8—601no—
515Q8—602no—
516C2×C4—32no—
517C2×C4—32no—
518C2×C4—41no—
519C2×C4—51no—
520C2×C4—51no—
521C2×C4—102no—
522C2×C4—102no—
523C2×C4—102no—
524C2×C4—102no—
525C2×C4—102no—
526C2×C4—102no—
527C2×C4—102no—
528C2×C4—102no—
529C2×C4—152no—
530C2×C4—152no—
531C2×C4—201no—
532C2×C4—201no—
533C2×C4—201no—
534C2×C4—302no—
535C2×C4—302no—
536C2×C4—302no—
537C2×C4—302no—
538C2×C4—302no—
539C2×C4—302no—
540C2×C4—302no—
541C2×C4—302no—
542C2×C4—302no—
543C2×C4—302no—
544C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
545C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
546C2×C3—41no—
547C2×C3—52no—
548C2×C3—52no—
549C2×C3—201no—
550C2×C3#6625811yescharacteristic
551C2×C3#6626211yescharacteristic
552C2×C3#6626611yescharacteristic
553C5#7388611yescharacteristic, complemented, Sylow
554C22—21no—
555C22—21no—
556C22—21no—
557C22—52no—
558C22—51no—
559C22—51no—
560C22—52no—
561C22—101no—
562C22—101no—
563C22—101no—
564C22—202no—
565C22—202no—
566C22C22×D3×AGL(1,𝔽5)11yescharacteristic
567C4—21no—
568C4—21no—
569C4—62no—
570C4—62no—
571C4—101no—
572C4—101no—
573C4—202no—
574C4—202no—
575C4—302no—
576C4—302no—
577C4—602no—
578C4—602no—
579C3#1061728811yescharacteristic, complemented, Sylow
580C2—41no—
581C2—52no—
582C2—52no—
583C2—201no—
584C2#1175681211yescharacteristic
585C2#1175683611yescharacteristic
586C2#1175687111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×AGL(1,𝔽5)
commutator, socleC22×C3×C5C23×C4
FittingC22×C3×C4×C5C2×C4
prime radicalC23×C3×D5C23
cosocleC22×C3×D5C24
hypercenterC22×C4D3×AGL(1,𝔽5)
hypocenterC3×C5#2378
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
6. #11751842C2semidirect
7. #11751840C2semidirect
8. #11751837C2semidirect
9. #11751835C2semidirect
21. #91813C4semidirect
22. C5×#924C4semidirect
23. #91574C4semidirect
27. #91360C4semidirect
28. #91359C4semidirect
29. #91349C4semidirect
175. #924AGL(1,𝔽5)semidirect
473. C3×C5#2378semidirect
553. C5#73886semidirect
579. C3#10617288semidirect

Maximal subgroups

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

subgroupindex
D5×#9242
#117524232
C3×#657652
#117520182
#117520172
#117518422
#117518402
#117518372
#117518352
#106172883
#738865