Finite Group Databasebeta

← #14331350

Subgroups — #14331350

Order 1920.

816 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×#870C211yesmaximal, characteristic
2#11752348C211yesmaximal, characteristic
3C3×#65689C211yesmaximal, characteristic
4#11751845C212yesmaximal
5#11751844C212yesmaximal
6#11751840C212yesmaximal
7#11751839C212yesmaximal
8#11751837C212yesmaximal
9#11751835C212yesmaximal
10#10617348—31nomaximal, Hall
11C2×C4⋊ΓL(1,𝔽16)C2211yescharacteristic
12C2×C4⋊ΓL(1,𝔽16)C2211yescharacteristic
13C2×C3×C4⋊AGL(1,𝔽5)C2211yescharacteristic
14C2×C3×C4⋊AGL(1,𝔽5)C2211yescharacteristic
15C2×Dic3⋊AGL(1,𝔽5)C2212yes—
16C2×Dic3⋊AGL(1,𝔽5)C2212yes—
17C2×Dic3×AGL(1,𝔽5)C2212yes—
18C2×Dic3×AGL(1,𝔽5)C2212yes—
19#91759C411yescharacteristic, complemented
20C5×#870C411yescharacteristic, complemented
21C3×C42×D5C2211yescharacteristic
22D5×C4⋊Dic3C2211yescharacteristic
23D5×C4⋊Dic3C2211yescharacteristic
24D5×Dic3⋊C4C2212yes—
25D5×Dic3⋊C4C2212yes—
26#91388C411yescharacteristic, complemented
27#91388C411yescharacteristic, complemented
28#91378C412yescomplemented
29#91378C412yescomplemented
30#91235C2212yes—
31#91235C2212yes—
32C3×(C2×AGL(1,𝔽5))⋊C4C2212yes—
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#74161—51nomaximal, Hall
39D5×#125—31no—
40#65712—32no—
41#65712—32no—
42#65711—32no—
43#65711—32no—
44#65711—32no—
45#65711—32no—
46#65689—31no—
47#65689D311yescharacteristic
48C22×ΓL(1,𝔽16)C2312yes—
49C22×ΓL(1,𝔽16)C2312yes—
50C22×C3×AGL(1,𝔽5)C2312yes—
51C22×C3×AGL(1,𝔽5)C2312yes—
52C2×C3×C4×D5C2311yescharacteristic
53C2×C3×C4×D5C2311yescharacteristic
54C2×C3×C4×D5C2311yescharacteristic
55C2×D5×Dic3C2312yes—
56C2×D5×Dic3C2312yes—
57C4⋊ΓL(1,𝔽16)—41no—
58C4⋊ΓL(1,𝔽16)—41no—
59C3×C4⋊AGL(1,𝔽5)—41no—
60C3×C4⋊AGL(1,𝔽5)—41no—
61Dic3⋊AGL(1,𝔽5)—42no—
62Dic3⋊AGL(1,𝔽5)—42no—
63Dic3×AGL(1,𝔽5)—42no—
64Dic3×AGL(1,𝔽5)—42no—
65C3×C42×C5C2×C411yescharacteristic
66C4⋊Dic15C2×C411yescharacteristic
67C4⋊Dic15C2×C411yescharacteristic
68#6712C2×C412yes—
69#6712C2×C412yes—
70C5×C4⋊Dic3C2×C411yescharacteristic
71C5×C4⋊Dic3C2×C411yescharacteristic
72C5×Dic3⋊C4C2×C412yes—
73C5×Dic3⋊C4C2×C412yes—
74C3×C4×Dic5C2×C411yescharacteristic
75C3×C4×Dic5C2×C411yescharacteristic
76C3×C4×Dic5C2×C411yescharacteristic
77Dic3⋊Dic5C2×C412yes—
78Dic3⋊Dic5C2×C412yes—
79#6669C2×C412yes—
80#6669C2×C412yes—
81#6668C2×C412yes—
82#6668C2×C412yes—
83C2×#870—51no—
84C3×#382—51no—
85#4978—52no—
86#4976—52no—
87#4936—51no—
88#4655—52no—
89#4654—52no—
90#4653—52no—
91#4647—52no—
92C2×C4⋊AGL(1,𝔽5)—32no—
93C2×C4⋊AGL(1,𝔽5)—32no—
94C2×C4⋊AGL(1,𝔽5)—31no—
95C2×C4⋊AGL(1,𝔽5)—31no—
96C2×C4⋊AGL(1,𝔽5)C2×D311yescharacteristic
97C2×C4⋊AGL(1,𝔽5)C2×D311yescharacteristic
98C2×C4×AGL(1,𝔽5)—32no—
99C2×C4×AGL(1,𝔽5)—32no—
100C5×#125—31no—
101D5×C4⋊C4—32no—
102D5×C4⋊C4—32no—
103D5×C4⋊C4—31no—
104D5×C4⋊C4—31no—
105#4036—31no—
106#4036—31no—
107#4035—32no—
108#4035—32no—
109C42×D5C2×D311yescharacteristic
110#4016—31no—
111(C2×AGL(1,𝔽5))⋊C4—32no—
112(C2×AGL(1,𝔽5))⋊C4—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))⋊C4C2×D312yes—
118(C2×AGL(1,𝔽5))⋊C4C2×D312yes—
119#2382—151noSylow
120C22×C3×D5C2411yescharacteristic
121C2×ΓL(1,𝔽16)—22no—
122C2×ΓL(1,𝔽16)—22no—
123C2×ΓL(1,𝔽16)—22no—
124C2×ΓL(1,𝔽16)—22no—
125C2×ΓL(1,𝔽16)—22no—
126C2×ΓL(1,𝔽16)—22no—
127C2×C3×AGL(1,𝔽5)—22no—
128C2×C3×AGL(1,𝔽5)—22no—
129C2×C3×AGL(1,𝔽5)—22no—
130C2×C3×AGL(1,𝔽5)—22no—
131C2×C3×AGL(1,𝔽5)—22no—
132C2×C3×AGL(1,𝔽5)—22no—
133C2×C3×C4×C5C22×C411yescharacteristic
134C2×C3×C4×C5C22×C411yescharacteristic
135C2×C3×C4×C5C22×C411yescharacteristic
136C2×Dic15C22×C412yes—
137C2×Dic15C22×C412yes—
138C2×C5×Dic3C22×C412yes—
139C2×C5×Dic3C22×C412yes—
140C2×C3×Dic5C22×C411yescharacteristic
141C2×C3×Dic5C22×C411yescharacteristic
142C2×C3×Dic5C22×C411yescharacteristic
143C3×C4×D5—21no—
144C3×C4×D5—21no—
145C3×C4×D5—21no—
146C3×C4×D5—21no—
147C3×C4×D5—21no—
148C3×C4×D5—21no—
149D5×Dic3—22no—
150D5×Dic3—22no—
151D5×Dic3—22no—
152D5×Dic3—22no—
153C2×C3×C4⋊C4—51no—
154C2×C3×C4⋊C4—51no—
155C2×C3×C42—51no—
156C2×C4⋊Dic3—51no—
157C2×C4⋊Dic3—51no—
158C2×C4⋊Dic3—51no—
159C2×C4⋊Dic3—51no—
160C2×Dic3⋊C4—52no—
161C2×Dic3⋊C4—52no—
162C2×Dic3⋊C4—52no—
163C2×Dic3⋊C4—52no—
164C2×C4×Dic3—52no—
165C2×C4×Dic3—52no—
166#870—51no—
167#870—51no—
168#870—51no—
169#870—52no—
170#870—52no—
171#870AGL(1,𝔽5)11yescharacteristic, complemented
172C3×#95—52no—
173C3×#95—52no—
174C3⋊#95—52no—
175C3⋊#95—52no—
176C3⋊#95—52no—
177C3⋊#95—52no—
178C3⋊#95—52no—
179C3⋊#95—52no—
180C22×AGL(1,𝔽5)—32no—
181C22×AGL(1,𝔽5)—32no—
182C22×AGL(1,𝔽5)C22×D312yes—
183C22×AGL(1,𝔽5)C22×D312yes—
184C2×C4×D5—32no—
185C2×C4×D5—32no—
186C2×C4×D5C22×D311yescharacteristic
187C2×C4×D5C22×D311yescharacteristic
188C2×C4×D5C22×D311yescharacteristic
189C4⋊AGL(1,𝔽5)—41no—
190C4⋊AGL(1,𝔽5)—41no—
191C4⋊AGL(1,𝔽5)—121no—
192C4⋊AGL(1,𝔽5)—121no—
193C4⋊AGL(1,𝔽5)—122no—
194C4⋊AGL(1,𝔽5)—122no—
195C4×AGL(1,𝔽5)—122no—
196C4×AGL(1,𝔽5)—122no—
197C5×C4⋊C4—32no—
198C5×C4⋊C4—32no—
199C5×C4⋊C4—31no—
200C5×C4⋊C4—31no—
201C42×C5C4×D311yescharacteristic
202C4⋊Dic5—31no—
203C4⋊Dic5—31no—
204C4⋊Dic5—32no—
205C4⋊Dic5—32no—
206#683—32no—
207#683—32no—
208#683—32no—
209#683—32no—
210#683—32no—
211#683—32no—
212C4×Dic5C4×D311yescharacteristic
213C4×Dic5C4×D311yescharacteristic
214C4×Dic5C4×D311yescharacteristic
215C2×#125—151no—
216#389—152no—
217#389—152no—
218#387—152no—
219#387—152no—
220#387—152no—
221#387—152no—
222#382—51no—
223#382—151no—
224C22×C3×C5C23×C411yescharacteristic
225C2×C3×D5H(5,𝔽2)11yescharacteristic
226C2×C3×D5H(5,𝔽2)11yescharacteristic
227C2×C3×D52−1+411yescharacteristic
228C2×C3×D52−1+411yescharacteristic
229C2×C3×D52−1+411yescharacteristic
230C2×C3×D52−1+411yescharacteristic
231ΓL(1,𝔽16)—42no—
232ΓL(1,𝔽16)—42no—
233C3×AGL(1,𝔽5)—42no—
234C3×AGL(1,𝔽5)—42no—
235C3×C4×C5—21no—
236C3×C4×C5—21no—
237C3×C4×C5—21no—
238Dic15—22no—
239Dic15—22no—
240C3×Dic5—21no—
241C3×Dic5—21no—
242C3×Dic5—21no—
243C5×Dic3—22no—
244C5×Dic3—22no—
245C22×C3×C4—51no—
246C22×C3×C4—51no—
247C22×C3×C4—51no—
248C22×C3×C4—52no—
249C22×C3×C4—52no—
250C22×Dic3—52no—
251C22×Dic3—52no—
252C22×Dic3—52no—
253C22×Dic3—52no—
254C3×C4⋊C4—201no—
255C3×C4⋊C4—201no—
256C3×C42—51no—
257C3×C42—51no—
258C3×C42—51no—
259C3×C42C2×AGL(1,𝔽5)11yescharacteristic
260C4⋊Dic3—51no—
261C4⋊Dic3—51no—
262C4⋊Dic3—52no—
263C4⋊Dic3—52no—
264C4⋊Dic3—201no—
265C4⋊Dic3—201no—
266C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
267C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
268Dic3⋊C4—52no—
269Dic3⋊C4—52no—
270Dic3⋊C4—52no—
271Dic3⋊C4—52no—
272Dic3⋊C4—52no—
273Dic3⋊C4—52no—
274Dic3⋊C4—202no—
275Dic3⋊C4—202no—
276Dic3⋊C4C2×AGL(1,𝔽5)12yes—
277Dic3⋊C4C2×AGL(1,𝔽5)12yes—
278C4×Dic3—202no—
279C4×Dic3—202no—
280C22×D5C23×D311yescharacteristic
281C2×AGL(1,𝔽5)—22no—
282C2×AGL(1,𝔽5)—22no—
283C2×AGL(1,𝔽5)—22no—
284C2×AGL(1,𝔽5)—22no—
285C2×AGL(1,𝔽5)—22no—
286C2×AGL(1,𝔽5)—22no—
287C2×AGL(1,𝔽5)—62no—
288C2×AGL(1,𝔽5)—62no—
289C2×AGL(1,𝔽5)—62no—
290C2×AGL(1,𝔽5)—62no—
291C2×AGL(1,𝔽5)—62no—
292C2×AGL(1,𝔽5)—62no—
293C2×C4×C5—32no—
294C2×C4×C5—32no—
295C2×C4×C5C2×C4×D311yescharacteristic
296C2×C4×C5C2×C4×D311yescharacteristic
297C2×C4×C5C2×C4×D311yescharacteristic
298C2×Dic5—32no—
299C2×Dic5—32no—
300C2×Dic5C2×C4×D311yescharacteristic
301C2×Dic5C2×C4×D311yescharacteristic
302C2×Dic5C2×C4×D311yescharacteristic
303C4×D5—21no—
304C4×D5—21no—
305C4×D5—21no—
306C4×D5—21no—
307C4×D5—21no—
308C4×D5—21no—
309C4×D5—62no—
310C4×D5—62no—
311C4×D5—62no—
312C4×D5—62no—
313#125—31no—
314#125—152no—
315#125—152no—
316#125—151no—
317#125—151no—
318#125—151no—
319C2×C4⋊C4—51no—
320C2×C4⋊C4—51no—
321C2×C4⋊C4—151no—
322C2×C4⋊C4—151no—
323C2×C4⋊C4—152no—
324C2×C4⋊C4—152no—
325C2×C4⋊C4—151no—
326C2×C4⋊C4—151no—
327C2×C4⋊C4—152no—
328C2×C4⋊C4—152no—
329C2×C42—51no—
330C2×C42—152no—
331C2×C42—152no—
332#95—52no—
333#95—52no—
334#95—152no—
335#95—152no—
336#95—152no—
337#95—152no—
338#95—152no—
339#95—152no—
340C2×C3×C5#51911yescharacteristic
341C2×C3×C5#52011yescharacteristic
342C2×C3×C5#52011yescharacteristic
343C3×D5#56211yescharacteristic
344C3×D5#56311yescharacteristic
345C3×D5#56311yescharacteristic
346C3×D5#56411yescharacteristic
347C23×C3—51no—
348C2×C3×C4—51no—
349C2×C3×C4—51no—
350C2×C3×C4—51no—
351C2×C3×C4—101no—
352C2×C3×C4—101no—
353C2×C3×C4—101no—
354C2×C3×C4—101no—
355C2×C3×C4—101no—
356C2×C3×C4—101no—
357C2×C3×C4—102no—
358C2×C3×C4—102no—
359C2×C3×C4—102no—
360C2×C3×C4—102no—
361C2×C3×C4—102no—
362C2×C3×C4—102no—
363C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
364C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
365C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
366C2×Dic3—52no—
367C2×Dic3—52no—
368C2×Dic3—102no—
369C2×Dic3—102no—
370C2×Dic3—102no—
371C2×Dic3—102no—
372C2×Dic3—102no—
373C2×Dic3—102no—
374C2×Dic3—102no—
375C2×Dic3—102no—
376C2×Dic3—102no—
377C2×Dic3—102no—
378C2×Dic3C22×AGL(1,𝔽5)12yes—
379C2×Dic3C22×AGL(1,𝔽5)12yes—
380C22×C5C22×C4×D311yescharacteristic
381C2×D5#100411yescharacteristic
382C2×D5#100711yescharacteristic
383C2×D5#100911yescharacteristic
384C2×D5#101011yescharacteristic
385C2×D5#101011yescharacteristic
386C2×D5#101011yescharacteristic
387AGL(1,𝔽5)—42no—
388AGL(1,𝔽5)—42no—
389AGL(1,𝔽5)—122no—
390AGL(1,𝔽5)—122no—
391C4×C5—21no—
392C4×C5—21no—
393C4×C5—21no—
394C4×C5—62no—
395C4×C5—62no—
396Dic5—21no—
397Dic5—21no—
398Dic5—21no—
399Dic5—62no—
400Dic5—62no—
401C22×C4—51no—
402C22×C4—51no—
403C22×C4—51no—
404C22×C4—52no—
405C22×C4—52no—
406C22×C4—152no—
407C22×C4—152no—
408C22×C4—152no—
409C22×C4—152no—
410C4⋊C4—32no—
411C4⋊C4—32no—
412C4⋊C4—31no—
413C4⋊C4—31no—
414C4⋊C4—152no—
415C4⋊C4—152no—
416C4⋊C4—152no—
417C4⋊C4—152no—
418C4⋊C4—152no—
419C4⋊C4—152no—
420C4⋊C4—152no—
421C4⋊C4—152no—
422C4⋊C4—151no—
423C4⋊C4—151no—
424C4⋊C4—201no—
425C4⋊C4—201no—
426C4⋊C4—602no—
427C4⋊C4—602no—
428C4⋊C4—601no—
429C4⋊C4—601no—
430C42—51no—
431C42—51no—
432C42—51no—
433C42—602no—
434C42—602no—
435C42D3×AGL(1,𝔽5)11yescharacteristic
436C3×C5#238211yescharacteristic, complemented, Hall
437C22×C3—51no—
438C22×C3—51no—
439C22×C3—51no—
440C22×C3—51no—
441C22×C3—51no—
442C22×C3—51no—
443C22×C3C23×AGL(1,𝔽5)11yescharacteristic
444C3×C4—21no—
445C3×C4—21no—
446C3×C4—21no—
447C3×C4—101no—
448C3×C4—101no—
449C3×C4—101no—
450C3×C4—202no—
451C3×C4—202no—
452Dic3—22no—
453Dic3—22no—
454Dic3—102no—
455Dic3—102no—
456Dic3—202no—
457Dic3—202no—
458C2×C5#550411yescharacteristic
459C2×C5#551611yescharacteristic
460C2×C5#552311yescharacteristic
461D5#567211yescharacteristic
462D5#568411yescharacteristic
463D5C3⋊#56411yescharacteristic
464D5#569911yescharacteristic
465C23—51no—
466C2×C4—32no—
467C2×C4—32no—
468C2×C4—51no—
469C2×C4—51no—
470C2×C4—51no—
471C2×C4—101no—
472C2×C4—101no—
473C2×C4—101no—
474C2×C4—101no—
475C2×C4—101no—
476C2×C4—101no—
477C2×C4—102no—
478C2×C4—102no—
479C2×C4—102no—
480C2×C4—102no—
481C2×C4—102no—
482C2×C4—102no—
483C2×C4—152no—
484C2×C4—152no—
485C2×C4—302no—
486C2×C4—302no—
487C2×C4—302no—
488C2×C4—302no—
489C2×C4—302no—
490C2×C4—302no—
491C2×C4—302no—
492C2×C4—302no—
493C2×C4—302no—
494C2×C4—302no—
495C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
496C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
497C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
498C2×C3—51no—
499C2×C3—51no—
500C2×C3—51no—
501C2×C3—51no—
502C2×C3#6626211yescharacteristic
503C2×C3#6626611yescharacteristic
504C2×C3#6626611yescharacteristic
505C5#7416111yescharacteristic, complemented, Sylow
506C22—51no—
507C22—51no—
508C22—51no—
509C22—51no—
510C22—51no—
511C22—51no—
512C22C22×D3×AGL(1,𝔽5)11yescharacteristic
513C4—21no—
514C4—21no—
515C4—21no—
516C4—62no—
517C4—62no—
518C4—101no—
519C4—101no—
520C4—101no—
521C4—202no—
522C4—202no—
523C4—302no—
524C4—302no—
525C4—602no—
526C4—602no—
527C3#1061734811yescharacteristic, complemented, Sylow
528C2—51no—
529C2—51no—
530C2—51no—
531C2—51no—
532C2#1175681211yescharacteristic
533C2#1175683611yescharacteristic
534C2#1175683811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×AGL(1,𝔽5)
commutator, socleC22×C3×C5C23×C4
FittingC3×C42×C5C2×C4
prime radical, cosocleC22×C3×D5C24
hypercenterC42D3×AGL(1,𝔽5)
hypocenterC3×C5#2382
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
19. #91759C4semidirect
20. C5×#870C4semidirect
26. #91388C4semidirect
27. #91388C4semidirect
28. #91378C4semidirect
29. #91378C4semidirect
171. #870AGL(1,𝔽5)semidirect
436. C3×C5#2382semidirect
505. C5#74161semidirect
527. C3#10617348semidirect

Maximal subgroups

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

subgroupindex
D5×#8702
#117523482
C3×#656892
#117518452
#117518442
#117518402
#117518392
#117518372
#117518352
#106173483
#741615