Finite Group Databasebeta

← #14326216

Subgroups — #14326216

Order 1920.

1744 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
1C2×D5×#239C211yesmaximal, characteristic
2C2×Q8×ΓL(1,𝔽16)C211yesmaximal, characteristic, complemented
3C2×C3×Q8×AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
4#11751857C213yesmaximal, complemented
5#11751855C213yesmaximal, complemented
6#11751850C213yesmaximal, complemented
7C4×(C2×D3)⋊AGL(1,𝔽5)C213yesmaximal, complemented
8#10617558—31nomaximal, Hall
9C2×#6821C411yescharacteristic, complemented
10C2×C5×#239C411yescharacteristic, complemented
11C2×C3×Q8×D5C2211yescharacteristic
12D5×#239—24no—
13D5×#239—24no—
14C2×D5×D12C2213yes—
15C2×C4×D3×D5C2213yes—
16C2×#6769C413yescomplemented
17C2×#6768C413yescomplemented
18Q8×ΓL(1,𝔽16)—24no—
19C2×C4⋊ΓL(1,𝔽16)C2213yes—
20C2×C4×ΓL(1,𝔽16)C2213yes—
21C3×Q8×AGL(1,𝔽5)—24no—
22C2×C3×C4⋊AGL(1,𝔽5)C2213yes—
23C2×C3×C4×AGL(1,𝔽5)C2213yes—
24C2×(C2×D3)⋊AGL(1,𝔽5)C2213yes—
25C2×(C2×D3)⋊AGL(1,𝔽5)C2213yes—
26C2×(C2×D3)⋊AGL(1,𝔽5)C2216yes—
27(C2×AGL(1,𝔽5))⋊Dic3C2211yescharacteristic
28(C2×AGL(1,𝔽5))⋊Dic3C2213yes—
29#82457—51nomaximal, Hall
30C2×D5×#41—31no—
31C2×Q8×AGL(1,𝔽5)—31no—
32C2×Q8×AGL(1,𝔽5)D311yescharacteristic, complemented
33#65780—33no—
34#65780—33no—
35#65765—33no—
36C4×C22⋊AGL(1,𝔽5)—33no—
37C22×D3×D5C2313yes—
38C22×ΓL(1,𝔽16)C2313yes—
39C22×ΓL(1,𝔽16)C2311yescharacteristic
40C22×C3×AGL(1,𝔽5)C2313yes—
41C22×C3×AGL(1,𝔽5)C2311yescharacteristic
42C2×C3×C5×Q8C2×C411yescharacteristic
43#6821—24no—
44C2×D60C2×C413yes—
45C2×C4×D15C2×C413yes—
46C5×#239—24no—
47C2×C5×D12C2×C413yes—
48C2×C4×C5×D3C2×C413yes—
49C3×Q8×D5D414yes—
50C3×Q8×D5D414yes—
51C2×C3×C4×D5C2313yes—
52C2×C3×Dic10C2×C413yes—
53C2×#1203C2×C416yes—
54C2×#1200C2×C413yes—
55C2×D3×Dic5C2×C413yes—
56C2×D5×Dic3C2311yescharacteristic
57D5×D12—26no—
58D5×D12—26no—
59C4×D3×D5—26no—
60C4×D3×D5—26no—
61#6769—212no—
62#6768—212no—
63C4⋊ΓL(1,𝔽16)—26no—
64C4×ΓL(1,𝔽16)—26no—
65C3×C4⋊AGL(1,𝔽5)—26no—
66C3×C4×AGL(1,𝔽5)—26no—
67(C2×D3)⋊AGL(1,𝔽5)—26no—
68(C2×D3)⋊AGL(1,𝔽5)—26no—
69(C2×D3)⋊AGL(1,𝔽5)—46no—
70C22×#239—51no—
71C2×C3×C4×Q8—51no—
72C2×Q8×Dic3—51no—
73#4988—53no—
74#4987—53no—
75#4939—53no—
76C4×(C2×D3)⋊C4—53no—
77C2×C5×#41—31no—
78D5×#41—64no—
79D5×#41—64no—
80C2×#713—31no—
81C2×Q8×D5C2×D311yescharacteristic
82C2×#711—33no—
83C2×D4×D5—33no—
84C2×#709—33no—
85C22×C4×D5—33no—
86C2×C22⋊AGL(1,𝔽5)—33no—
87C2×C22⋊AGL(1,𝔽5)—33no—
88C2×C22⋊AGL(1,𝔽5)—36no—
89Q8×AGL(1,𝔽5)—24no—
90Q8×AGL(1,𝔽5)—64no—
91C2×C4⋊AGL(1,𝔽5)—33no—
92C2×C4⋊AGL(1,𝔽5)C2×D313yes—
93C2×C4×AGL(1,𝔽5)—33no—
94C2×C4×AGL(1,𝔽5)C2×D313yes—
95(C2×AGL(1,𝔽5))⋊C4—31no—
96(C2×AGL(1,𝔽5))⋊C4—33no—
97#2341—151noSylow
98C22×D15C22×C413yes—
99C22×C5×D3C22×C413yes—
100C22×C3×D5C2411yescharacteristic
101C2×D3×D5—23no—
102C2×D3×D5—23no—
103C2×D3×D5—23no—
104C2×D3×D5—23no—
105C2×D3×D5—23no—
106C2×D3×D5—23no—
107C2×ΓL(1,𝔽16)—21no—
108C2×ΓL(1,𝔽16)—21no—
109C2×ΓL(1,𝔽16)—23no—
110C2×ΓL(1,𝔽16)—23no—
111C2×ΓL(1,𝔽16)—23no—
112C2×ΓL(1,𝔽16)#4112yescomplemented
113C2×C3×AGL(1,𝔽5)—21no—
114C2×C3×AGL(1,𝔽5)—21no—
115C2×C3×AGL(1,𝔽5)—23no—
116C2×C3×AGL(1,𝔽5)—23no—
117C2×C3×AGL(1,𝔽5)—23no—
118C2×C3×AGL(1,𝔽5)#4112yescomplemented
119C3×C5×Q8C22⋊C414yescomplemented
120C2×C3×C4×C5C22×C413yes—
121C2×Dic15C22×C411yescharacteristic
122D60—26no—
123C4×D15—26no—
124C2×C5×Dic3C22×C411yescharacteristic
125C5×D12—26no—
126C4×C5×D3—26no—
127C2×C3×Dic5C22×C413yes—
128C3×C4×D5C2×D416yes—
129C3×C4×D5C2×D416yes—
130C3×Dic10C22⋊C4112yescomplemented
131#1203—212no—
132#1200—26no—
133D3×Dic5—26no—
134D5×Dic3—21no—
135D5×Dic3—21no—
136C22×C3×Q8—51no—
137C2×#239—51no—
138C2×#239—53no—
139C2×#239—53no—
140C2×#239—104no—
141C2×#239—104no—
142C2×#239AGL(1,𝔽5)11yescharacteristic, complemented
143C22×D12—53no—
144C22×C4×D3—53no—
145C3×C4×Q8—104no—
146C2×C3×C4⋊C4—53no—
147C2×C3×C42—53no—
148Q8×Dic3—104no—
149C2×(C2×D3)⋊C4—53no—
150C2×(C2×D3)⋊C4—53no—
151C2×(C2×D3)⋊C4—56no—
152C2×C4⋊Dic3—53no—
153C2×C4×Dic3—53no—
154C3⋊#95—51no—
155C3⋊#95—53no—
156C23×D5—33no—
157C22×AGL(1,𝔽5)—31no—
158C22×AGL(1,𝔽5)—33no—
159C22×AGL(1,𝔽5)C22×D313yes—
160C22×AGL(1,𝔽5)C22×D311yescharacteristic
161C5×#41—64no—
162C2×C5×Q8C4×D311yescharacteristic
163C2×C5×D4—33no—
164C22×C4×C5—33no—
165C2×C22⋊D5—36no—
166C22×Dic5—33no—
167#713—64no—
168Q8×D5D1214yes—
169Q8×D5#6714yes—
170#711—612no—
171D4×D5—66no—
172D4×D5—66no—
173#709—612no—
174C2×D20—33no—
175C2×C4×D5—31no—
176C2×C4×D5—33no—
177C2×C4×D5—33no—
178C2×C4×D5—66no—
179C2×C4×D5—66no—
180C2×C4×D5C22×D313yes—
181C2×Dic10C4×D313yes—
182C22⋊AGL(1,𝔽5)—66no—
183C22⋊AGL(1,𝔽5)—66no—
184C22⋊AGL(1,𝔽5)—126no—
185C4⋊AGL(1,𝔽5)—26no—
186C4⋊AGL(1,𝔽5)—66no—
187C4×AGL(1,𝔽5)—26no—
188C4×AGL(1,𝔽5)—66no—
189C22×#41—151no—
190C2×C4×Q8—51no—
191C2×C4×Q8—151no—
192#390—153no—
193#390—153no—
194#380—153no—
195C4×C22⋊C4—153no—
196C22×C3×C5C23×C411yescharacteristic
197C2×D15—23no—
198C2×D15—23no—
199C2×D15—23no—
200C2×C5×D3—23no—
201C2×C5×D3—23no—
202C2×C5×D3—23no—
203C2×C3×D5C22×D411yescharacteristic
204C2×C3×D5C22×D411yescharacteristic
205C2×C3×D5C2×#4111yescharacteristic
206C2×C3×D5C2×#4111yescharacteristic
207C2×C3×D5H(5,𝔽2)11yescharacteristic
208C2×C3×D52−1+411yescharacteristic
209D3×D5—43no—
210D3×D5—43no—
211D3×D5—43no—
212D3×D5—43no—
213ΓL(1,𝔽16)—22no—
214ΓL(1,𝔽16)—43no—
215C3×AGL(1,𝔽5)—22no—
216C3×AGL(1,𝔽5)—43no—
217C3×C4×C5C2×C22⋊C416yes—
218Dic15—21no—
219C3×Dic5C2×C22⋊C416yes—
220C5×Dic3—21no—
221C23×D3—53no—
222C2×C3×Q8—54no—
223C2×C3×Q8—54no—
224C2×C3×Q8—53no—
225C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
226C22×C3×C4—53no—
227C22×C3×C4—51no—
228C22×C3×C4—53no—
229C22×Dic3—51no—
230C22×Dic3—51no—
231C22×Dic3—53no—
232#239—24no—
233#239—1012no—
234#239—1012no—
235#239—104no—
236C2×D12—56no—
237C2×D12—53no—
238C2×D12—106no—
239C2×D12—106no—
240C2×D12C2×AGL(1,𝔽5)13yes—
241C2×C4×D3—53no—
242C2×C4×D3—53no—
243C2×C4×D3—53no—
244C2×C4×D3—106no—
245C2×C4×D3—106no—
246C2×C4×D3C2×AGL(1,𝔽5)13yes—
247C3×C4⋊C4—106no—
248C3×C42—106no—
249(C2×D3)⋊C4—106no—
250(C2×D3)⋊C4—106no—
251(C2×D3)⋊C4—206no—
252C4⋊Dic3—106no—
253C4×Dic3—106no—
254C23×C5—33no—
255C22×D5—33no—
256C22×D5—63no—
257C22×D5—63no—
258C22×D5—63no—
259C22×D5—63no—
260C22×D5—63no—
261C22×D5—63no—
262C22×D5C23×D311yescharacteristic
263C2×AGL(1,𝔽5)—21no—
264C2×AGL(1,𝔽5)—21no—
265C2×AGL(1,𝔽5)—23no—
266C2×AGL(1,𝔽5)—23no—
267C2×AGL(1,𝔽5)—23no—
268C2×AGL(1,𝔽5)—32no—
269C2×AGL(1,𝔽5)—61no—
270C2×AGL(1,𝔽5)—61no—
271C2×AGL(1,𝔽5)—63no—
272C2×AGL(1,𝔽5)—63no—
273C2×AGL(1,𝔽5)—63no—
274C2×AGL(1,𝔽5)#23912yescomplemented
275C5×Q8(C2×D3)⋊C414yescomplemented
276C5×D4—66no—
277C2×C4×C5—31no—
278C2×C4×C5—66no—
279C2×C4×C5C2×C4×D313yes—
280C22⋊D5—612no—
281C2×Dic5—31no—
282C2×Dic5—66no—
283C2×Dic5C2×C4×D313yes—
284D20—66no—
285C4×D5—61no—
286C4×D5—61no—
287C4×D5—66no—
288C4×D5—66no—
289C4×D5C2×D1216yes—
290C4×D5C2×#6716yes—
291Dic10(C2×D3)⋊C4112yescomplemented
292C2×#41—31no—
293C2×#41—153no—
294C2×#41—153no—
295C2×#41—151no—
296C2×#41—304no—
297C2×#41—304no—
298C22×Q8—51no—
299C22×D4—153no—
300C23×C4—153no—
301C4×Q8—104no—
302C4×Q8—304no—
303C2×C4⋊C4—53no—
304C2×C4⋊C4—153no—
305C2×C22⋊C4—153no—
306C2×C22⋊C4—153no—
307C2×C22⋊C4—156no—
308C2×C42—53no—
309C2×C42—153no—
310#95—151no—
311#95—153no—
312C2×C3×C5C22×C22⋊C411yescharacteristic
313C2×C3×C5C4×#4111yescharacteristic
314C2×C3×C5#52011yescharacteristic
315D15—43no—
316C3×D5#54811yescharacteristic
317C3×D5#54811yescharacteristic
318C3×D5#55011yescharacteristic
319C3×D5#55011yescharacteristic
320C5×D3—43no—
321C23×C3—51no—
322C22×D3—53no—
323C22×D3—103no—
324C22×D3—103no—
325C22×D3—103no—
326C22×D3—103no—
327C22×D3—103no—
328C22×D3—103no—
329C22×D3C22×AGL(1,𝔽5)13yes—
330C3×Q8—512no—
331C3×Q8C22⋊AGL(1,𝔽5)14yescomplemented
332C2×C3×C4—53no—
333C2×C3×C4—56no—
334C2×C3×C4—56no—
335C2×C3×C4—52no—
336C2×C3×C4—101no—
337C2×C3×C4—101no—
338C2×C3×C4—103no—
339C2×C3×C4—103no—
340C2×C3×C4—103no—
341C2×C3×C4C22×AGL(1,𝔽5)13yes—
342C2×Dic3—52no—
343C2×Dic3—51no—
344C2×Dic3—101no—
345C2×Dic3—101no—
346C2×Dic3—101no—
347C2×Dic3—101no—
348C2×Dic3—103no—
349C2×Dic3—103no—
350C2×Dic3—103no—
351C2×Dic3C22×AGL(1,𝔽5)11yescharacteristic
352D12—26no—
353D12—1012no—
354D12—106no—
355C4×D3—26no—
356C4×D3—106no—
357C4×D3—106no—
358C4×D3—106no—
359C22×C5—63no—
360C22×C5—63no—
361C22×C5—63no—
362C22×C5C22×C4×D311yescharacteristic
363C2×D5—63no—
364C2×D5—63no—
365C2×D5—63no—
366C2×D5—123no—
367C2×D5—123no—
368C2×D5—123no—
369C2×D5—123no—
370C2×D5C22×D1211yescharacteristic
371C2×D5C2×#23911yescharacteristic
372C2×D5#100711yescharacteristic
373C2×D5D3×#4111yescharacteristic
374C2×D5#100911yescharacteristic
375C2×D5C22×#6711yescharacteristic
376AGL(1,𝔽5)—22no—
377AGL(1,𝔽5)—43no—
378AGL(1,𝔽5)—62no—
379AGL(1,𝔽5)—123no—
380C4×C5—61no—
381C4×C5C2×(C2×D3)⋊C416yes—
382Dic5—61no—
383Dic5C2×(C2×D3)⋊C416yes—
384C24—153no—
385#41—64no—
386#41—3012no—
387#41—3012no—
388#41—304no—
389C2×Q8—54no—
390C2×Q8—54no—
391C2×Q8—53no—
392C2×Q8D3×AGL(1,𝔽5)11yescharacteristic
393C2×D4—33no—
394C2×D4—156no—
395C2×D4—153no—
396C2×D4—306no—
397C2×D4—306no—
398C22×C4—33no—
399C22×C4—53no—
400C22×C4—51no—
401C22×C4—53no—
402C22×C4—153no—
403C22×C4—151no—
404C22×C4—153no—
405C22×C4—151no—
406C22×C4—153no—
407C22×C4—153no—
408C22×C4—306no—
409C22×C4—306no—
410C4⋊C4—106no—
411C4⋊C4—306no—
412C22⋊C4—306no—
413C22⋊C4—306no—
414C22⋊C4—606no—
415C42—106no—
416C42—306no—
417C3×C5#234111yescharacteristic, complemented, Hall
418C22×C3—51no—
419C22×C3—51no—
420C22×C3—51no—
421C22×C3—51no—
422C22×C3—51no—
423C22×C3—51no—
424C22×C3C23×AGL(1,𝔽5)11yescharacteristic
425C2×D3—23no—
426C2×D3—23no—
427C2×D3—23no—
428C2×D3—103no—
429C2×D3—103no—
430C2×D3—103no—
431C2×D3—203no—
432C2×D3—203no—
433C2×D3—203no—
434C2×D3—203no—
435C3×C4—56no—
436C3×C4—102no—
437C3×C4—203no—
438C3×C4C2×C22⋊AGL(1,𝔽5)16yes—
439Dic3—21no—
440Dic3—102no—
441Dic3—101no—
442Dic3—203no—
443C2×C5—123no—
444C2×C5C4×#23911yescharacteristic
445C2×C5#557411yescharacteristic
446C2×C5C22×(C2×D3)⋊C411yescharacteristic
447D5—123no—
448D5#557611yescharacteristic
449D5#557711yescharacteristic
450D5#581711yescharacteristic
451D5#583411yescharacteristic
452C23—33no—
453C23—51no—
454C23—153no—
455C23—303no—
456C23—303no—
457C23—303no—
458C23—303no—
459C23—303no—
460C23—303no—
461Q8—512no—
462Q8(C2×D3)⋊AGL(1,𝔽5)14yescomplemented
463D4—66no—
464D4—3012no—
465D4—306no—
466C2×C4—31no—
467C2×C4—53no—
468C2×C4—56no—
469C2×C4—56no—
470C2×C4—52no—
471C2×C4—66no—
472C2×C4—101no—
473C2×C4—101no—
474C2×C4—103no—
475C2×C4—103no—
476C2×C4—103no—
477C2×C4—152no—
478C2×C4—151no—
479C2×C4—306no—
480C2×C4—306no—
481C2×C4—301no—
482C2×C4—301no—
483C2×C4—301no—
484C2×C4—301no—
485C2×C4—306no—
486C2×C4—303no—
487C2×C4—303no—
488C2×C4—303no—
489C2×C4C2×D3×AGL(1,𝔽5)13yes—
490C2×C3—51no—
491C2×C3—51no—
492C2×C3—51no—
493C2×C3—51no—
494C2×C3#41×AGL(1,𝔽5)11yescharacteristic
495C2×C3#6626611yescharacteristic
496C2×C3C22×C22⋊AGL(1,𝔽5)11yescharacteristic
497D3—43no—
498D3—203no—
499C5#8245711yescharacteristic, complemented, Sylow
500C22—51no—
501C22—51no—
502C22—51no—
503C22—51no—
504C22—51no—
505C22—51no—
506C22—63no—
507C22—63no—
508C22—63no—
509C22—303no—
510C22—303no—
511C22—303no—
512C22—603no—
513C22—603no—
514C22—603no—
515C22—603no—
516C22C22×D3×AGL(1,𝔽5)11yescharacteristic
517C4—56no—
518C4—61no—
519C4—102no—
520C4—203no—
521C4—302no—
522C4—301no—
523C4—603no—
524C4C2×(C2×D3)⋊AGL(1,𝔽5)16yes—
525C3#1061755811yescharacteristic, complemented, Sylow
526C2—51no—
527C2—51no—
528C2—51no—
529C2—51no—
530C2—123no—
531C2—603no—
532C2AGL(1,𝔽5)×#23911yescharacteristic
533C2#1175685411yescharacteristic
534C2C22×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×AGL(1,𝔽5)
commutator, socleC22×C3×C5C23×C4
FittingC2×C3×C5×Q8C2×C4
prime radicalC2×D5×#239C2
cosocleC22×C3×D5C24
hypercenterC2×Q8D3×AGL(1,𝔽5)
hypocenterC3×C5#2341
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C2×Q8×ΓL(1,𝔽16)C2semidirect
3. C2×C3×Q8×AGL(1,𝔽5)C2semidirect
4. #11751857C2semidirect
5. #11751855C2semidirect
6. #11751850C2semidirect
7. C4×(C2×D3)⋊AGL(1,𝔽5)C2semidirect
9. C2×#6821C4semidirect
10. C2×C5×#239C4semidirect
16. C2×#6769C4semidirect
17. C2×#6768C4semidirect
32. C2×Q8×AGL(1,𝔽5)D3semidirect
112. C2×ΓL(1,𝔽16)#41semidirect
118. C2×C3×AGL(1,𝔽5)#41semidirect
119. C3×C5×Q8C22⋊C4semidirect
130. C3×Dic10C22⋊C4semidirect
142. C2×#239AGL(1,𝔽5)semidirect
274. C2×AGL(1,𝔽5)#239semidirect
275. C5×Q8(C2×D3)⋊C4semidirect
291. Dic10(C2×D3)⋊C4semidirect
331. C3×Q8C22⋊AGL(1,𝔽5)semidirect
417. C3×C5#2341semidirect
462. Q8(C2×D3)⋊AGL(1,𝔽5)semidirect
499. C5#82457semidirect
525. C3#10617558semidirect

Maximal subgroups

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

subgroupindex
C2×D5×#2392
C2×Q8×ΓL(1,𝔽16)2
C2×C3×Q8×AGL(1,𝔽5)2
#117518572
#117518552
#117518502
C4×(C2×D3)⋊AGL(1,𝔽5)2
#106175583
#824575