Finite Group Databasebeta

← #14299610

Subgroups — #14299610

Order 1920.

648 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 102 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
1#11756837C211yesmaximal, characteristic
2#11752386C211yesmaximal, characteristic, complemented
3#11751850C211yesmaximal, characteristic
4#11751728C211yesmaximal, characteristic, complemented
5D5×#823C211yesmaximal, characteristic
6C3×#64900C211yesmaximal, characteristic, complemented
7#11746527C211yesmaximal, characteristic, complemented
8#10603925—31nomaximal, Hall
9C2×D5×D12C2211yescharacteristic
10#91989C2211yescharacteristic
11C2×C4⋊ΓL(1,𝔽16)C2211yescharacteristic
12C3×#4130C2211yescharacteristic
13C2×C3×C4×AGL(1,𝔽5)C2211yescharacteristic
14D3×C22⋊AGL(1,𝔽5)—22no—
15C2×(C2×D3)⋊AGL(1,𝔽5)—22no—
16#91921—21no—
17#91921—21no—
18AGL(1,𝔽5)×D12—21no—
19AGL(1,𝔽5)×D12—21no—
20C3×D5×M16C2211yescharacteristic
21D5×#208C2211yescharacteristic
22#91223—21no—
23#91222—21no—
24#91160—21no—
25#91159—21no—
26#91107C411yescharacteristic, complemented
27C5×#823C411yescharacteristic, complemented
28#90960C411yescharacteristic, complemented
29#90959C411yescharacteristic, complemented
30#68758—51nomaximal, Hall
31#66258—31no—
32#65780—31no—
33#65727—31no—
34#65727—31no—
35D5×#100—31no—
36#64900—31no—
37#64900D311yescharacteristic, complemented
38C22×D3×D5D412yes—
39C22×ΓL(1,𝔽16)—21no—
40C22×C3×AGL(1,𝔽5)—21no—
41C2×D3×AGL(1,𝔽5)—22no—
42C2×D3×AGL(1,𝔽5)—22no—
43C2×D60C2×C411yescharacteristic
44C2×C5×D12C2×C411yescharacteristic
45C2×C3×C4×D5C2311yescharacteristic
46C2×#1203Q812yes—
47D5×D12—21no—
48D5×D12—21no—
49D5×D12—21no—
50D5×D12—21no—
51#6763—21no—
52C4⋊ΓL(1,𝔽16)—21no—
53C4⋊ΓL(1,𝔽16)—21no—
54C4⋊ΓL(1,𝔽16)D411yescharacteristic
55C4⋊ΓL(1,𝔽16)D411yescharacteristic
56C4×ΓL(1,𝔽16)—21no—
57C3×C22⋊AGL(1,𝔽5)—21no—
58C3×C4⋊AGL(1,𝔽5)D411yescharacteristic
59C3×C4⋊AGL(1,𝔽5)D411yescharacteristic
60C3×C4×AGL(1,𝔽5)—21no—
61C3×C4×AGL(1,𝔽5)—21no—
62C3×C4×AGL(1,𝔽5)—21no—
63(C2×D3)⋊AGL(1,𝔽5)—22no—
64(C2×D3)⋊AGL(1,𝔽5)—22no—
65(C2×D3)⋊AGL(1,𝔽5)—44no—
66C3×C5×M16C2×C411yescharacteristic, complemented
67#6710C2×C411yescharacteristic, complemented
68C5×#208C2×C411yescharacteristic, complemented
69C3×#681C2×C411yescharacteristic, complemented
70C3×#676—21no—
71C3×C8×D5—21no—
72#6649—21no—
73D5×C3⋊C8—21no—
74#5521—51no—
75C2×#823—51no—
76#5121—51no—
77#4995—51no—
78#4939—51no—
79C3×#339—51no—
80#4523—51no—
81C2×D4×D5—31no—
82C2×C22⋊AGL(1,𝔽5)—62no—
83C2×C22⋊AGL(1,𝔽5)—62no—
84D4×AGL(1,𝔽5)—61no—
85D4×AGL(1,𝔽5)—61no—
86D4×AGL(1,𝔽5)—61no—
87D4×AGL(1,𝔽5)—61no—
88#4130—31no—
89#4130C2×D311yescharacteristic
90C2×C4⋊AGL(1,𝔽5)—31no—
91C2×C4×AGL(1,𝔽5)C2×D311yescharacteristic
92D5×M16—31no—
93D5×M16C2×D311yescharacteristic
94#3994—61no—
95#3994—61no—
96#3993—61no—
97#3993—61no—
98C5×#100—31no—
99#3965—31no—
100#3955—31no—
101#3955—31no—
102#1856—151noSylow
103C22×D15C4⋊C412yes—
104C22×C5×D3C4⋊C412yes—
105C22×C3×D5C2×D411yescharacteristic
106C2×D3×D5—22no—
107C2×D3×D5—22no—
108C2×D3×D5—42no—
109C2×D3×D5—42no—
110C2×ΓL(1,𝔽16)—21no—
111C2×ΓL(1,𝔽16)—21no—
112C2×ΓL(1,𝔽16)—42no—
113C2×ΓL(1,𝔽16)—41no—
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—
119D3×AGL(1,𝔽5)—82no—
120D3×AGL(1,𝔽5)—82no—
121C2×C3×C4×C5C22×C411yescharacteristic
122D60—21no—
123D60—21no—
124C5×D12—21no—
125C5×D12—21no—
126C2×C3×Dic5C2×Q811yescharacteristic
127C3×C4×D5C2×D411yescharacteristic
128C3×C4×D5C2×D411yescharacteristic
129C3×C4×D5#4111yescharacteristic
130C3×C4×D5#4111yescharacteristic
131#1203—22no—
132#1203—22no—
133C3×C5×C8—21no—
134C3⋊#168—21no—
135C3×#168—21no—
136C5×C3⋊C8—21no—
137C22×D12—51no—
138C2×C3×M16—51no—
139C3×#117—51no—
140C2×C3×C42—51no—
141C2×(C2×D3)⋊C4—102no—
142#926—51no—
143C2×C4⋊Dic3—51no—
144C2×#208—51no—
145#893—101no—
146#893—101no—
147D3×C22⋊C4—102no—
148C4×D12—101no—
149C4×D12—101no—
150#823—51no—
151#823—51no—
152#823—51no—
153#823AGL(1,𝔽5)11yescharacteristic, complemented
154#818—101no—
155#817—101no—
156#808—101no—
157#807—101no—
158C23×D5—32no—
159C22×AGL(1,𝔽5)—21no—
160C22×AGL(1,𝔽5)—62no—
161C22×AGL(1,𝔽5)—62no—
162C22×AGL(1,𝔽5)—61no—
163C2×C5×D4—31no—
164C2×C22⋊D5—32no—
165D4×D5—61no—
166D4×D5—61no—
167D4×D5—61no—
168D4×D5—61no—
169C2×D20—31no—
170C2×C4×D5C22×D311yescharacteristic
171C22⋊AGL(1,𝔽5)—21no—
172C22⋊AGL(1,𝔽5)—62no—
173C22⋊AGL(1,𝔽5)—62no—
174C22⋊AGL(1,𝔽5)—61no—
175C22⋊AGL(1,𝔽5)—124no—
176C4⋊AGL(1,𝔽5)—31no—
177C4⋊AGL(1,𝔽5)—31no—
178C4⋊AGL(1,𝔽5)—61no—
179C4⋊AGL(1,𝔽5)—61no—
180C4⋊AGL(1,𝔽5)D1211yescharacteristic
181C4⋊AGL(1,𝔽5)D1211yescharacteristic
182C4×AGL(1,𝔽5)—21no—
183C4×AGL(1,𝔽5)—21no—
184C4×AGL(1,𝔽5)—21no—
185C4×AGL(1,𝔽5)—61no—
186C5×M16—31no—
187C5×M16C4×D311yescharacteristic, complemented
188#681—31no—
189#681C4×D311yescharacteristic, complemented
190#676—21no—
191#676—61no—
192C8×D5—21no—
193C8×D5—61no—
194#518—151no—
195#428—151no—
196#428—151no—
197C2×#100—151no—
198#390—151no—
199#339—51no—
200#339—151no—
201C22×C3×C5C2×C4⋊C411yescharacteristic
202C2×D15—22no—
203C2×D15—42no—
204C2×C5×D3—22no—
205C2×C5×D3—42no—
206C2×C3×D5—21no—
207C2×C3×D5—21no—
208C2×C3×D5#12011yescharacteristic
209C2×C3×D5#12311yescharacteristic
210D3×D5—42no—
211D3×D5—42no—
212D3×D5—82no—
213ΓL(1,𝔽16)—41no—
214ΓL(1,𝔽16)—41no—
215ΓL(1,𝔽16)—81no—
216C3×AGL(1,𝔽5)—42no—
217C3×AGL(1,𝔽5)—41no—
218C3×AGL(1,𝔽5)—41no—
219C3×C4×C5C4×D411yescharacteristic
220C3×C4×C5C4×D411yescharacteristic
221C3×Dic5C22⋊Q811yescharacteristic
222C3×Dic5C22⋊Q811yescharacteristic
223C23×D3—52no—
224C22×C3×C4—51no—
225C22×C3×C4—101no—
226C22×Dic3—101no—
227C2×D12—52no—
228C2×D12—51no—
229C2×D12—101no—
230C2×D12—101no—
231C2×D12—101no—
232C2×D12—101no—
233C2×D12C2×AGL(1,𝔽5)11yescharacteristic
234C2×C4×D3—102no—
235C2×C4×D3—102no—
236C3×M16—51no—
237C3×M16—101no—
238C3×M16C2×AGL(1,𝔽5)11yescharacteristic, complemented
239C2×C3×C8—101no—
240C3×C4⋊C4—51no—
241C3×C4⋊C4—51no—
242C3×C22⋊C4—101no—
243C3×C42—101no—
244C3×C42—101no—
245C3×C42—101no—
246#217—101no—
247(C2×D3)⋊C4—102no—
248(C2×D3)⋊C4—102no—
249(C2×D3)⋊C4—204no—
250C4⋊Dic3—51no—
251C4⋊Dic3—51no—
252C4⋊Dic3—101no—
253C4⋊Dic3—101no—
254C4×Dic3—101no—
255#208—51no—
256#208—101no—
257#208C2×AGL(1,𝔽5)11yescharacteristic, complemented
258C2×C3⋊C8—101no—
259C23×C5—32no—
260C22×D5—32no—
261C22×D5—62no—
262C22×D5—62no—
263C22×D5—122no—
264C22×D5—122no—
265C22×D5D3×D411yescharacteristic
266C2×AGL(1,𝔽5)—22no—
267C2×AGL(1,𝔽5)—21no—
268C2×AGL(1,𝔽5)—21no—
269C2×AGL(1,𝔽5)—41no—
270C2×AGL(1,𝔽5)—41no—
271C2×AGL(1,𝔽5)—61no—
272C2×AGL(1,𝔽5)—61no—
273C2×AGL(1,𝔽5)—122no—
274C2×AGL(1,𝔽5)—121no—
275C2×AGL(1,𝔽5)—242no—
276C2×AGL(1,𝔽5)—242no—
277C5×D4—61no—
278C5×D4—61no—
279C2×C4×C5C2×C4×D311yescharacteristic
280C22⋊D5—62no—
281C22⋊D5—62no—
282C2×Dic5D3×Q811yescharacteristic
283D20—61no—
284D20—61no—
285C4×D5C2×D1211yescharacteristic
286C4×D5#23511yescharacteristic
287C4×D5D3×D411yescharacteristic
288C4×D5#23711yescharacteristic
289C5×C8—21no—
290C5×C8—61no—
291#168—21no—
292#168—61no—
293C22×D4—151no—
294C2×M16—51no—
295C2×M16—151no—
296C4×D4—301no—
297C4×D4—301no—
298C4×D4—301no—
299C4×D4—301no—
300#117—51no—
301#117—151no—
302C2×C4⋊C4—151no—
303C2×C22⋊C4—302no—
304C2×C22⋊C4—302no—
305C2×C42—51no—
306#107—301no—
307#107—301no—
308#106—301no—
309#106—301no—
310#100—31no—
311#100—151no—
312#100—151no—
313#100—151no—
314C2×C3×C5—21no—
315C2×C3×C5#38511yescharacteristic
316D15—42no—
317C3×D5—21no—
318C3×D5#45311yescharacteristic
319C3×D5#45511yescharacteristic
320C5×D3—42no—
321C23×C3—51no—
322C22×D3—52no—
323C22×D3—102no—
324C22×D3—102no—
325C22×D3—202no—
326C22×D3—202no—
327C22×D3C4⋊AGL(1,𝔽5)12yes—
328C2×C3×C4—51no—
329C2×C3×C4—51no—
330C2×C3×C4—51no—
331C2×C3×C4—51no—
332C2×C3×C4—51no—
333C2×C3×C4—101no—
334C2×C3×C4—101no—
335C2×C3×C4—102no—
336C2×C3×C4—201no—
337C2×C3×C4—201no—
338C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
339C2×Dic3—101no—
340C2×Dic3—101no—
341C2×Dic3—201no—
342C2×Dic3—202no—
343D12—21no—
344D12—21no—
345D12—102no—
346D12—102no—
347D12—101no—
348D12—101no—
349C4×D3—402no—
350C4×D3—402no—
351C3×C8—21no—
352C3×C8—101no—
353C3⋊C8—21no—
354C3⋊C8—101no—
355C22×C5—62no—
356C22×C5—122no—
357C22×C5D3×C4⋊C411yescharacteristic
358C2×D5—21no—
359C2×D5—21no—
360C2×D5—62no—
361C2×D5—122no—
362C2×D5—122no—
363C2×D5—122no—
364C2×D5—242no—
365C2×D5#88211yescharacteristic
366C2×D5#88311yescharacteristic
367AGL(1,𝔽5)—42no—
368AGL(1,𝔽5)—41no—
369AGL(1,𝔽5)—41no—
370AGL(1,𝔽5)—121no—
371AGL(1,𝔽5)—121no—
372AGL(1,𝔽5)—241no—
373C4×C5C4×D1211yescharacteristic
374C4×C5#67⋊C411yescharacteristic
375Dic5#89611yescharacteristic
376Dic5#89711yescharacteristic
377C24—152no—
378C2×D4—31no—
379C2×D4—152no—
380C2×D4—151no—
381C2×D4—301no—
382C2×D4—301no—
383C2×D4—301no—
384C2×D4—301no—
385C22×C4—51no—
386C22×C4—101no—
387C22×C4—302no—
388C22×C4—302no—
389C22×C4—301no—
390M16—31no—
391M16—51no—
392M16—101no—
393M16—151no—
394M16—301no—
395M16D3×AGL(1,𝔽5)11yescharacteristic, complemented
396C2×C8—101no—
397C2×C8—301no—
398C4⋊C4—51no—
399C4⋊C4—51no—
400C4⋊C4—151no—
401C4⋊C4—151no—
402C4⋊C4—301no—
403C4⋊C4—301no—
404C22⋊C4—101no—
405C22⋊C4—302no—
406C22⋊C4—302no—
407C22⋊C4—301no—
408C22⋊C4—604no—
409C42—101no—
410C42—101no—
411C42—101no—
412C42—301no—
413C3×C5#185611yescharacteristic, complemented, Hall
414C22×C3—51no—
415C22×C3—51no—
416C22×C3—101no—
417C22×C3—101no—
418C22×C3C2×C4⋊AGL(1,𝔽5)11yescharacteristic
419C2×D3—22no—
420C2×D3—42no—
421C2×D3—102no—
422C2×D3—202no—
423C2×D3—202no—
424C2×D3—202no—
425C2×D3—402no—
426C3×C4—51no—
427C3×C4—51no—
428C3×C4—201no—
429C3×C4—201no—
430C3×C4—202no—
431C3×C4D4×AGL(1,𝔽5)11yescharacteristic
432C3×C4D4×AGL(1,𝔽5)11yescharacteristic
433Dic3—201no—
434Dic3—201no—
435Dic3—401no—
436C2×C5—21no—
437C2×C5—122no—
438C2×C5#466811yescharacteristic
439D5—21no—
440D5—122no—
441D5#482211yescharacteristic
442D5#482511yescharacteristic
443C23—32no—
444C23—51no—
445C23—152no—
446C23—302no—
447C23—302no—
448C23—602no—
449C23—602no—
450D4—61no—
451D4—61no—
452D4—302no—
453D4—302no—
454D4—301no—
455D4—301no—
456C2×C4—51no—
457C2×C4—51no—
458C2×C4—51no—
459C2×C4—51no—
460C2×C4—51no—
461C2×C4—101no—
462C2×C4—101no—
463C2×C4—102no—
464C2×C4—201no—
465C2×C4—201no—
466C2×C4—301no—
467C2×C4—301no—
468C2×C4—601no—
469C2×C4—602no—
470C2×C4—1202no—
471C2×C4—1202no—
472C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
473C8—21no—
474C8—61no—
475C8—101no—
476C8—301no—
477C2×C3—21no—
478C2×C3—51no—
479C2×C3—51no—
480C2×C3—101no—
481C2×C3#6569911yescharacteristic
482D3—42no—
483D3—202no—
484C5#6875811yescharacteristic, complemented, Sylow
485C22—51no—
486C22—51no—
487C22—62no—
488C22—101no—
489C22—101no—
490C22—122no—
491C22—302no—
492C22—602no—
493C22—602no—
494C22—602no—
495C22—1202no—
496C22D3×C4⋊AGL(1,𝔽5)11yescharacteristic
497C4—51no—
498C4—51no—
499C4—201no—
500C4—201no—
501C4—202no—
502C4—601no—
503C4—601no—
504C4—1201no—
505C4AGL(1,𝔽5)×D1211yescharacteristic
506C4#67⋊AGL(1,𝔽5)11yescharacteristic
507C3#1060392511yescharacteristic, complemented, Sylow
508C2—21no—
509C2—51no—
510C2—51no—
511C2—101no—
512C2—122no—
513C2—602no—
514C2#1175193411yescharacteristic
subgroupG/H
centerC2#11751934
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC3×C5×M16C2×C4
socleC2×C3×C5#385
prime radicalC2×D5×D12C22
cosocleC2×C3×C4×D5C23
hypercenterM16D3×AGL(1,𝔽5)
hypocenterC3×C5#1856
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
2. #11752386C2semidirect
4. #11751728C2semidirect
6. C3×#64900C2semidirect
7. #11746527C2semidirect
26. #91107C4semidirect
27. C5×#823C4semidirect
28. #90960C4semidirect
29. #90959C4semidirect
37. #64900D3semidirect
66. C3×C5×M16C2×C4semidirect
67. #6710C2×C4semidirect
68. C5×#208C2×C4semidirect
69. C3×#681C2×C4semidirect
153. #823AGL(1,𝔽5)semidirect
187. C5×M16C4×D3semidirect
189. #681C4×D3semidirect
238. C3×M16C2×AGL(1,𝔽5)semidirect
257. #208C2×AGL(1,𝔽5)semidirect
395. M16D3×AGL(1,𝔽5)semidirect
413. C3×C5#1856semidirect
484. C5#68758semidirect
507. C3#10603925semidirect

Maximal subgroups

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

subgroupindex
#117568372
#117523862
#117518502
#117517282
D5×#8232
C3×#649002
#117465272
#106039253
#687585