Finite Group Databasebeta

← #14313213

Subgroups — #14313213

Order 1920.

560 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 93 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#65987C211yesmaximal, characteristic
2D5⋊#833C211yesmaximal, characteristic, complemented
3D5⋊#834C211yesmaximal, characteristic, complemented
4#11746787C212yesmaximal, complemented
5#11746735C212yesmaximal, complemented
6#10604531—31nomaximal, Hall
7C2×C3×D4×D5C2211yescharacteristic
8C22×C3×D20C411yescharacteristic, complemented
9C2×#6763—21no—
10D5⋊#208—21no—
11C3×C5×#123C411yescharacteristic, complemented
12C3×#4083C411yescharacteristic, complemented
13C3×#4075C2211yescharacteristic
14C3×#4041—21no—
15C3×#4040—21no—
16C3×#4030—21no—
17C3×#4028—21no—
18C3×#4028C2211yescharacteristic
19C3×D5×C22⋊C4C411yescharacteristic, complemented
20#91242—21no—
21#91242C2212yes—
22#91240C2212yes—
23#91228—21no—
24#71847—51nomaximal, Hall
25#65987D311yescharacteristic
26D5⋊#100—31no—
27D5⋊#99—31no—
28#64916—32no—
29#64864—32no—
30C23×C3×D5C2×C411yescharacteristic
31C23×C3×D5C2×C411yescharacteristic, complemented
32C23×C3×D5D411yescharacteristic
33C22×ΓL(1,𝔽16)—41no—
34C2×C3×C5×D4C2×C411yescharacteristic
35C22×C3×C4×C5C2×C411yescharacteristic, complemented
36C2×C3×C22⋊D5—21no—
37C2×C3×C22⋊D5D411yescharacteristic
38C2×C3×C22⋊D5C2311yescharacteristic
39C3×D4×D5—41no—
40C3×D4×D5—41no—
41C2×C3×D20—21no—
42C2×C3×D20—21no—
43C2×C3×D20—21no—
44C2×C3×D20—41no—
45C2×C3×D20—41no—
46C2×C3×D20D411yescharacteristic
47C2×C3×C4×D5—21no—
48C2×C3×C4×D5D411yescharacteristic
49#6763—22no—
50#6763—41no—
51#6762—22no—
52#6758—41no—
53D5⋊(C3⋊C8)—41no—
54C3×C5×C4⋊C4—21no—
55C3×C5×C22⋊C4—21no—
56C3×C5×C22⋊C4C2×C411yescharacteristic, complemented
57C3×C22⋊Dic5C2×C411yescharacteristic, complemented
58C3×#685—21no—
59C3×#685—21no—
60C3×#685—21no—
61C3×#685—21no—
62C3×#683—21no—
63C3×#534—51no—
64C2×#834—51no—
65C2×#833—51no—
66#4539—52no—
67#4536—52no—
68C2×D4×D5C2×D311yescharacteristic
69C22×D20Dic311yescharacteristic, complemented
70C2×C22⋊AGL(1,𝔽5)—61no—
71D5⋊M16—61no—
72C5×#123Dic311yescharacteristic, complemented
73#4083Dic311yescharacteristic, complemented
74#4075C2×D311yescharacteristic
75#4041—21no—
76#4040—21no—
77#4030—21no—
78#4028—21no—
79#4028C2×D311yescharacteristic
80D5×C22⋊C4Dic311yescharacteristic, complemented
81#4013—32no—
82#4013—61no—
83#4011—32no—
84#3999—61no—
85#2121—151noSylow
86C23×C3×C5C22⋊C411yescharacteristic
87C23×C3×C5C22×C411yescharacteristic
88C22×C3×D5—21no—
89C22×C3×D5—21no—
90C22×C3×D5—21no—
91C22×C3×D5—21no—
92C22×C3×D5—21no—
93C22×C3×D5—41no—
94C22×C3×D5—81no—
95C22×C3×D5—81no—
96C22×C3×D5—81no—
97C22×C3×D5C22⋊C411yescharacteristic
98C22×C3×D5C22⋊C411yescharacteristic
99C22×C3×D5C2×D411yescharacteristic
100C2×ΓL(1,𝔽16)—42no—
101C2×ΓL(1,𝔽16)—82no—
102C3×C5×D4—41no—
103C2×C3×C4×C5—21no—
104C2×C3×C4×C5—21no—
105C2×C3×C4×C5—41no—
106C2×C3×C4×C5C22⋊C411yescharacteristic
107C3×C22⋊D5—41no—
108C3×C22⋊D5—41no—
109C3×C22⋊D5—81no—
110C2×C3×Dic5—21no—
111C2×C3×Dic5C2×D411yescharacteristic
112C3×D20—41no—
113C3×D20—42no—
114C3×D20—41no—
115C3×D20—41no—
116C3×D20—81no—
117C3×C4×D5—21no—
118C3×C4×D5—21no—
119C3×C4×D5—81no—
120C3⋊#170—42no—
121C22×C3×D4—51no—
122C22×C3×D4—51no—
123C3×#123—51no—
124C3×#123—101no—
125C3×#123AGL(1,𝔽5)11yescharacteristic, complemented
126C3×#121—101no—
127C3×#121—101no—
128C3×#120—51no—
129C3×#120—51no—
130C3×#120—101no—
131C2×C3×C22⋊C4—51no—
132C2×#217—101no—
133C2×#208—101no—
134#834—52no—
135#834—101no—
136#833—52no—
137#833—101no—
138C23×D5C2×Dic311yescharacteristic
139C23×D5C2×Dic311yescharacteristic, complemented
140C23×D5#6711yescharacteristic
141C22×AGL(1,𝔽5)—121no—
142C2×C5×D4C2×Dic311yescharacteristic
143C22×C4×C5C2×Dic311yescharacteristic, complemented
144C2×C22⋊D5—21no—
145C2×C22⋊D5#6711yescharacteristic
146C2×C22⋊D5C22×D311yescharacteristic
147D4×D5—41no—
148D4×D5—41no—
149C2×D20—21no—
150C2×D20—21no—
151C2×D20—21no—
152C2×D20—41no—
153C2×D20—41no—
154C2×D20#6711yescharacteristic
155C2×C4×D5—21no—
156C2×C4×D5#6711yescharacteristic
157C22⋊AGL(1,𝔽5)—62no—
158C22⋊AGL(1,𝔽5)—121no—
159#704—62no—
160#700—121no—
161D5⋊C8—121no—
162C5×C4⋊C4—21no—
163C5×C22⋊C4—21no—
164C5×C22⋊C4C2×Dic311yescharacteristic, complemented
165C22⋊Dic5C2×Dic311yescharacteristic, complemented
166#685—21no—
167#685—21no—
168#685—21no—
169#685—21no—
170#683—21no—
171#534—51no—
172C2×#100—151no—
173C2×#99—151no—
174#352—152no—
175AΣL(1,𝔽16)—152no—
176C22×C3×C5—21no—
177C22×C3×C5—21no—
178C22×C3×C5—41no—
179C22×C3×C5—41no—
180C22×C3×C5C2×C22⋊C411yescharacteristic
181C2×C3×D5—21no—
182C2×C3×D5—21no—
183C2×C3×D5—41no—
184C2×C3×D5—42no—
185C2×C3×D5—41no—
186C2×C3×D5—41no—
187C2×C3×D5—41no—
188C2×C3×D5—41no—
189C2×C3×D5—41no—
190C2×C3×D5—81no—
191C2×C3×D5—81no—
192C2×C3×D5—81no—
193C2×C3×D5—82no—
194C2×C3×D5#9911yescharacteristic
195C2×C3×D5#9911yescharacteristic
196ΓL(1,𝔽16)—82no—
197C3×C4×C5—21no—
198C3×C4×C5—41no—
199C3×C4×C5—41no—
200C3×Dic5—21no—
201C3×Dic5—41no—
202C24×C3—51no—
203C24×C3—51no—
204C24×C3—51no—
205C2×C3×D4—51no—
206C2×C3×D4—51no—
207C2×C3×D4—51no—
208C2×C3×D4—101no—
209C2×C3×D4—101no—
210C2×C3×D4—101no—
211C2×C3×D4—101no—
212C2×C3×D4—201no—
213C2×C3×D4—201no—
214C2×C3×D4—201no—
215C2×C3×D4—201no—
216C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
217C22×C3×C4—51no—
218C22×C3×C4—101no—
219C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic, complemented
220C22×Dic3—201no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—101no—
223C3×C22⋊C4—21no—
224C3×C22⋊C4—51no—
225C3×C22⋊C4—101no—
226C3×C22⋊C4—101no—
227C3×C22⋊C4—101no—
228C3×C22⋊C4—101no—
229C3×C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic, complemented
230#217—102no—
231#217—201no—
232#208—102no—
233#208—201no—
234C2×C3⋊C8—201no—
235C23×C5#21711yescharacteristic
236C23×C5C22×Dic311yescharacteristic
237C22×D5—21no—
238C22×D5—21no—
239C22×D5—21no—
240C22×D5—21no—
241C22×D5—21no—
242C22×D5—41no—
243C22×D5—81no—
244C22×D5—81no—
245C22×D5—81no—
246C22×D5#21711yescharacteristic
247C22×D5#21711yescharacteristic
248C22×D5C2×#6711yescharacteristic
249C2×AGL(1,𝔽5)—122no—
250C2×AGL(1,𝔽5)—242no—
251C5×D4—41no—
252C2×C4×C5—21no—
253C2×C4×C5—21no—
254C2×C4×C5—41no—
255C2×C4×C5#21711yescharacteristic
256C22⋊D5—41no—
257C22⋊D5—41no—
258C22⋊D5—81no—
259C2×Dic5—21no—
260C2×Dic5C2×#6711yescharacteristic
261D20—41no—
262D20—42no—
263D20—41no—
264D20—41no—
265D20—81no—
266C4×D5—21no—
267C4×D5—21no—
268C4×D5—81no—
269#170—122no—
270C22×D4—51no—
271C22×D4—51no—
272C2×M16—301no—
273#123—51no—
274#123—101no—
275#123ΓL(1,𝔽16)11yescharacteristic, complemented
276#121—101no—
277#121—101no—
278#120—51no—
279#120—51no—
280#120—101no—
281C2×C22⋊C4—51no—
282C2×C22⋊C4—301no—
283#100—152no—
284#100—301no—
285#99—152no—
286#99—301no—
287C2×C3×C5—21no—
288C2×C3×C5—41no—
289C2×C3×C5—41no—
290C2×C3×C5C2×#9911yescharacteristic
291C3×D5—21no—
292C3×D5—21no—
293C3×D5—41no—
294C3×D5—41no—
295C3×D5—81no—
296C23×C3—51no—
297C23×C3—51no—
298C23×C3—51no—
299C23×C3—101no—
300C23×C3—101no—
301C23×C3—101no—
302C23×C3—101no—
303C23×C3—101no—
304C23×C3—201no—
305C23×C3—401no—
306C23×C3—401no—
307C23×C3—401no—
308C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
309C23×C3C22×AGL(1,𝔽5)11yescharacteristic
310C3×D4—41no—
311C3×D4—201no—
312C3×D4—201no—
313C3×D4—201no—
314C3×D4—202no—
315C3×D4—201no—
316C3×D4—201no—
317C3×D4—401no—
318C3×D4—401no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—41no—
322C2×C3×C4—51no—
323C2×C3×C4—101no—
324C2×C3×C4—101no—
325C2×C3×C4—101no—
326C2×C3×C4—401no—
327C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
328C2×Dic3—202no—
329C2×Dic3—402no—
330C3⋊C8—202no—
331C22×C5—21no—
332C22×C5—21no—
333C22×C5—41no—
334C22×C5—41no—
335C22×C5C2×#21711yescharacteristic
336C2×D5—21no—
337C2×D5—21no—
338C2×D5—41no—
339C2×D5—42no—
340C2×D5—41no—
341C2×D5—41no—
342C2×D5—41no—
343C2×D5—41no—
344C2×D5—41no—
345C2×D5—81no—
346C2×D5—81no—
347C2×D5—81no—
348C2×D5—82no—
349C2×D5#83411yescharacteristic
350C2×D5#83411yescharacteristic
351AGL(1,𝔽5)—242no—
352C4×C5—21no—
353C4×C5—41no—
354C4×C5—41no—
355Dic5—21no—
356Dic5—41no—
357C24—51no—
358C24—51no—
359C24—51no—
360C2×D4—51no—
361C2×D4—51no—
362C2×D4—51no—
363C2×D4—101no—
364C2×D4—101no—
365C2×D4—101no—
366C2×D4—101no—
367C2×D4—201no—
368C2×D4—201no—
369C2×D4—201no—
370C2×D4—201no—
371C2×D4C2×ΓL(1,𝔽16)11yescharacteristic
372C22×C4—51no—
373C22×C4—101no—
374C22×C4—601no—
375C22×C4C2×ΓL(1,𝔽16)11yescharacteristic, complemented
376M16—302no—
377M16—601no—
378C2×C8—601no—
379C4⋊C4—21no—
380C4⋊C4—101no—
381C22⋊C4—21no—
382C22⋊C4—51no—
383C22⋊C4—101no—
384C22⋊C4—101no—
385C22⋊C4—101no—
386C22⋊C4—101no—
387C22⋊C4—302no—
388C22⋊C4—601no—
389C22⋊C4C2×ΓL(1,𝔽16)11yescharacteristic, complemented
390C3×C5#212111yescharacteristic, complemented, Hall
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—41no—
394C22×C3—41no—
395C22×C3—51no—
396C22×C3—51no—
397C22×C3—101no—
398C22×C3—101no—
399C22×C3—201no—
400C22×C3—201no—
401C22×C3—201no—
402C22×C3—201no—
403C22×C3—201no—
404C22×C3—202no—
405C22×C3—201no—
406C22×C3—401no—
407C22×C3—401no—
408C22×C3—401no—
409C22×C3—402no—
410C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
411C3×C4—21no—
412C3×C4—41no—
413C3×C4—41no—
414C3×C4—101no—
415C3×C4—201no—
416Dic3—402no—
417C2×C5—21no—
418C2×C5—41no—
419C2×C5—41no—
420C2×C5C2×#83411yescharacteristic
421D5—21no—
422D5—21no—
423D5—41no—
424D5—41no—
425D5—81no—
426C23—51no—
427C23—51no—
428C23—51no—
429C23—101no—
430C23—101no—
431C23—101no—
432C23—101no—
433C23—101no—
434C23—201no—
435C23—401no—
436C23—401no—
437C23—401no—
438C23#676311yescharacteristic
439C23C22×ΓL(1,𝔽16)11yescharacteristic
440D4—41no—
441D4—201no—
442D4—201no—
443D4—201no—
444D4—202no—
445D4—201no—
446D4—201no—
447D4—401no—
448D4—401no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—41no—
452C2×C4—51no—
453C2×C4—101no—
454C2×C4—101no—
455C2×C4—101no—
456C2×C4—401no—
457C2×C4—602no—
458C2×C4—1202no—
459C2×C4#676311yescharacteristic
460C8—602no—
461C2×C3—21no—
462C2×C3—41no—
463C2×C3—41no—
464C2×C3—101no—
465C2×C3—101no—
466C2×C3—201no—
467C2×C3—201no—
468C2×C3—401no—
469C2×C3D5⋊#9911yescharacteristic
470C5#7184711yescharacteristic, complemented, Sylow
471C22—21no—
472C22—21no—
473C22—41no—
474C22—41no—
475C22—51no—
476C22—51no—
477C22—101no—
478C22—101no—
479C22—201no—
480C22—201no—
481C22—201no—
482C22—201no—
483C22—201no—
484C22—202no—
485C22—201no—
486C22—401no—
487C22—401no—
488C22—401no—
489C22—402no—
490C22C2×#676311yescharacteristic
491C4—21no—
492C4—41no—
493C4—41no—
494C4—101no—
495C4—201no—
496C4—1202no—
497C3#1060453111yescharacteristic, complemented, Sylow
498C2—21no—
499C2—41no—
500C2—41no—
501C2—101no—
502C2—101no—
503C2—201no—
504C2—201no—
505C2—401no—
506C2D5⋊#83411yescharacteristic
subgroupG/H
centerC2D5⋊#834
commutatorC23×C3×C5C22×C4
FrattiniC23C22×ΓL(1,𝔽16)
FittingC3×C5×#123C4
socleC2×C3×C5C2×#99
prime radicalC3×#65987C2
cosocleC2×C3×C22⋊D5C23
hypercenter#123ΓL(1,𝔽16)
hypocenterC3×C5#2121
LCSG ▷ C23×C3×C5 ▷ C22×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #123

Complements

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

NH ≅ G/Ntype
2. D5⋊#833C2semidirect
3. D5⋊#834C2semidirect
4. #11746787C2semidirect
5. #11746735C2semidirect
8. C22×C3×D20C4semidirect
11. C3×C5×#123C4semidirect
12. C3×#4083C4semidirect
19. C3×D5×C22⋊C4C4semidirect
31. C23×C3×D5C2×C4semidirect
35. C22×C3×C4×C5C2×C4semidirect
56. C3×C5×C22⋊C4C2×C4semidirect
57. C3×C22⋊Dic5C2×C4semidirect
69. C22×D20Dic3semidirect
72. C5×#123Dic3semidirect
73. #4083Dic3semidirect
80. D5×C22⋊C4Dic3semidirect
125. C3×#123AGL(1,𝔽5)semidirect
139. C23×D5C2×Dic3semidirect
143. C22×C4×C5C2×Dic3semidirect
164. C5×C22⋊C4C2×Dic3semidirect
165. C22⋊Dic5C2×Dic3semidirect
219. C22×C3×C4C2×AGL(1,𝔽5)semidirect
229. C3×C22⋊C4C2×AGL(1,𝔽5)semidirect
275. #123ΓL(1,𝔽16)semidirect
375. C22×C4C2×ΓL(1,𝔽16)semidirect
389. C22⋊C4C2×ΓL(1,𝔽16)semidirect
390. C3×C5#2121semidirect
470. C5#71847semidirect
497. C3#10604531semidirect

Maximal subgroups

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

subgroupindex
C3×#659872
D5⋊#8332
D5⋊#8342
#117467872
#117467352
#106045313
#718475