Finite Group Databasebeta

← #72231445

Subgroups — #72231445

Order 4800.

2288 conjugacy classes of subgroups; only the 220 normal subgroups are shown.

normal subgroups only

This group has 2288 conjugacy classes of subgroups — too many to draw. Only its 220 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#72231415C212yesmaximal, complemented
2C2×(C2×D3)⋊#1043C212yesmaximal, complemented
3C2×#11925533C211yesmaximal, characteristic, complemented
4C2×#67×C5⋊D5C211yesmaximal, characteristic
5C23×C3×#1043C211yesmaximal, characteristic, complemented
6#13104791—31nomaximal, Hall
7C23×C3×C5⋊D5C2211yescharacteristic
8C22×D3×C5⋊D5C2211yescharacteristic
9C22×C3⋊#1043C2212yes—
10C22×C3×#1043—22no—
11C22×C3×#1043—24no—
12C22×C3×#1043C2212yes—
13C2×#10597159C411yescharacteristic, complemented
14C2×C52×#67C411yescharacteristic, complemented
15#67×C5⋊D5—22no—
16#67×C5⋊D5—22no—
17C2×#10597115C411yescharacteristic, complemented
18C2×#10597114C411yescharacteristic, complemented
19C2×Dic3×C5⋊D5C2211yescharacteristic
20#11925533—22no—
21(C2×D3)⋊#1043—24no—
22#11752022—56nomaximal
23C23×#1043D311yescharacteristic, complemented
24C2×D4×C5⋊D5—31no—
25C2×C22⋊#1043—31no—
26C2×C22⋊#1043—32no—
27#11716874—32no—
28C23×C3×C52C2×C411yescharacteristic
29C22×C3⋊(C5⋊D5)C2×C411yescharacteristic, complemented
30C22×C52×D3C2×C411yescharacteristic, complemented
31C22×C3×C5⋊D5—22no—
32C22×C3×C5⋊D5—22no—
33C22×C3×C5⋊D5C2×C411yescharacteristic
34C22×C3×C5⋊D5D412yes—
35C22×C3×C5⋊D5D412yes—
36C22×C3×C5⋊D5C2311yescharacteristic
37C2×D3×C5⋊D5—22no—
38C2×D3×C5⋊D5—21no—
39C2×D3×C5⋊D5—21no—
40C2×D3×C5⋊D5—22no—
41C2×C3⋊#1043—22no—
42C2×C3⋊#1043—22no—
43C2×C3⋊#1043—22no—
44C2×C3×#1043—28no—
45C2×C3×#1043—22no—
46C2×C3×#1043—22no—
47C2×C3×#1043D414yescomplemented
48#10597159—22no—
49C2×C3⋊#1039C2×C411yescharacteristic, complemented
50C52×#67—22no—
51C2×C52×Dic3C2×C411yescharacteristic, complemented
52#10597115—22no—
53#10597114—22no—
54Dic3×C5⋊D5—21no—
55Dic3×C5⋊D5—21no—
56C23×C3×AGL(1,𝔽5)—56no—
57C2×D5×#67—56no—
58C2×#6763—56no—
59C2×(C2×D3)⋊AGL(1,𝔽5)—512no—
60(C2×AGL(1,𝔽5))⋊Dic3—512no—
61C23×C5⋊D5—31no—
62C23×C5⋊D5C2×D311yescharacteristic
63C22×#1043—22no—
64C22×#1043—24no—
65C22×#1043—32no—
66C22×#1043C2×D312yes—
67C2×C52×D4—31no—
68C2×C22⋊(C5⋊D5)—31no—
69C2×C22⋊(C5⋊D5)—31no—
70D4×C5⋊D5—62no—
71D4×C5⋊D5—62no—
72C2×C4⋊(C5⋊D5)—31no—
73C2×C4×C5⋊D5—31no—
74C22⋊#1043—62no—
75C22⋊#1043—64no—
76#65779—156no—
77C22×C3×C52—22no—
78C22×C3×C52C22⋊C412yescomplemented
79C22×C3×C52C22×C411yescharacteristic
80C2×C3⋊(C5⋊D5)—21no—
81C2×C3⋊(C5⋊D5)—22no—
82C2×C52×D3—21no—
83C2×C52×D3—22no—
84C2×C3×C5⋊D5—22no—
85C2×C3×C5⋊D5—22no—
86C2×C3×C5⋊D5—24no—
87C2×C3×C5⋊D5—22no—
88C2×C3×C5⋊D5C22⋊C412yescomplemented
89C2×C3×C5⋊D5C2×D411yescharacteristic
90C2×C3×C5⋊D5C2×D412yes—
91C2×C3×C5⋊D5C2×D411yescharacteristic
92C2×C3×C5⋊D5#4112yes—
93D3×C5⋊D5—41no—
94D3×C5⋊D5—42no—
95D3×C5⋊D5—41no—
96C3⋊#1043—42no—
97C3×#1043—24no—
98C3⋊#1039—21no—
99C52×Dic3—21no—
100C23×C3×D5—56no—
101C22×D3×D5—56no—
102C22×ΓL(1,𝔽16)—512no—
103C22×C3×AGL(1,𝔽5)—512no—
104C22×C3×AGL(1,𝔽5)—1024no—
105C22×C3×AGL(1,𝔽5)—1012no—
106C2×#1220—56no—
107C2×C5×#67AGL(1,𝔽5)16yescomplemented
108D5×#67—1012no—
109D5×#67—1012no—
110C2×#1202—56no—
111C2×#1201—56no—
112C2×D5×Dic3—56no—
113#6763—1012no—
114(C2×D3)⋊AGL(1,𝔽5)—1024no—
115C23×C52—31no—
116C23×C52C4×D311yescharacteristic
117C22×C5⋊D5—22no—
118C22×C5⋊D5—22no—
119C22×C5⋊D5—31no—
120C22×C5⋊D5—61no—
121C22×C5⋊D5—62no—
122C22×C5⋊D5—62no—
123C22×C5⋊D5—61no—
124C22×C5⋊D5C4×D311yescharacteristic
125C22×C5⋊D5D1212yes—
126C22×C5⋊D5#6712yes—
127C22×C5⋊D5C22×D311yescharacteristic
128C2×#1043—28no—
129C2×#1043—22no—
130C2×#1043—22no—
131C2×#1043—62no—
132C2×#1043—62no—
133C2×#1043—62no—
134C2×#1043#6714yescomplemented
135C52×D4—62no—
136C2×C4×C52—31no—
137C22⋊(C5⋊D5)—62no—
138C22⋊(C5⋊D5)—62no—
139C2×#1039—31no—
140C4⋊(C5⋊D5)—62no—
141C4×C5⋊D5—61no—
142C4×C5⋊D5—61no—
143#5213—251noHall
144C23×AGL(1,𝔽5)—56no—
145C2×D4×D5—156no—
146C2×C22⋊AGL(1,𝔽5)—1512no—
147C2×C22⋊AGL(1,𝔽5)—156no—
148(C2×AGL(1,𝔽5))⋊C4—1512no—
149C2×C3×C52—22no—
150C2×C3×C52C2×C22⋊C411yescharacteristic
151C2×C3×C52C4×D412yes—
152C3⋊(C5⋊D5)—41no—
153C52×D3—41no—
154C3×C5⋊D5—22no—
155C3×C5⋊D5#12011yescharacteristic
156C3×C5⋊D5#12112yes—
157C3×C5⋊D5#12311yescharacteristic
158C23×C3×C5C2×AGL(1,𝔽5)16yes—
159C22×D15—56no—
160C22×C5×D3C2×AGL(1,𝔽5)16yescomplemented
161C22×C3×D5—56no—
162C22×C3×D5—512no—
163C22×C3×D5—512no—
164C22×C3×D5—56no—
165C22×C3×D5—1012no—
166C22×C3×D5—1012no—
167C2×D3×D5—1012no—
168C2×D3×D5—106no—
169C2×D3×D5—106no—
170C2×D3×D5—1012no—
171C2×ΓL(1,𝔽16)—1012no—
172C2×ΓL(1,𝔽16)—1012no—
173C2×ΓL(1,𝔽16)—1012no—
174C2×C3×AGL(1,𝔽5)—524no—
175C2×C3×AGL(1,𝔽5)—1012no—
176C2×C3×AGL(1,𝔽5)—1012no—
177C2×C3×AGL(1,𝔽5)—1048no—
178#1220—1012no—
179C2×Dic15—56no—
180C5×#67—212no—
181C2×C5×Dic3C2×AGL(1,𝔽5)16yescomplemented
182#1202—1012no—
183#1201—1012no—
184D5×Dic3—106no—
185D5×Dic3—106no—
186C22×C52—22no—
187C22×C52—61no—
188C22×C52—62no—
189C22×C52(C2×D3)⋊C412yescomplemented
190C22×C52C2×C4×D311yescharacteristic
191C2×C5⋊D5—22no—
192C2×C5⋊D5—22no—
193C2×C5⋊D5—24no—
194C2×C5⋊D5—22no—
195C2×C5⋊D5—61no—
196C2×C5⋊D5—62no—
197C2×C5⋊D5—121no—
198C2×C5⋊D5—122no—
199C2×C5⋊D5—121no—
200C2×C5⋊D5(C2×D3)⋊C412yescomplemented
201C2×C5⋊D5C2×D1211yescharacteristic
202C2×C5⋊D5#23512yes—
203C2×C5⋊D5C2×#6711yescharacteristic
204C2×C5⋊D5C2×#6712yes—
205#1043—24no—
206#1043—122no—
207C4×C52—61no—
208#1039—61no—
209C23×C3×C4—251no—
210C22×#67—251no—
211C2×#217—251no—
212C2×(C2×D3)⋊C4—252no—
213C3⋊#95—252no—
214C23×D5—56no—
215C23×D5—156no—
216C22×AGL(1,𝔽5)—512no—
217C22×AGL(1,𝔽5)—1024no—
218C22×AGL(1,𝔽5)—1012no—
219C22×AGL(1,𝔽5)—1512no—
220C2×C5×D4—36no—
221C2×C22⋊D5—156no—
222C2×C22⋊D5—156no—
223D4×D5—3012no—
224D4×D5—3012no—
225C2×D20—156no—
226C2×C4×D5—156no—
227C22⋊AGL(1,𝔽5)—3024no—
228C22⋊AGL(1,𝔽5)—3012no—
229C3×C52#38611yescharacteristic, complemented, Hall
230#386—751noSylow
231C22×C3×C5—212no—
232C22×C3×C5C22⋊AGL(1,𝔽5)112yescomplemented
233C22×C3×C5C22×AGL(1,𝔽5)16yes—
234C2×D15—1012no—
235C2×D15—106no—
236C2×C5×D3—212no—
237C2×C5×D3—26no—
238C2×C3×D5—512no—
239C2×C3×D5—512no—
240C2×C3×D5—56no—
241C2×C3×D5—56no—
242C2×C3×D5—512no—
243C2×C3×D5—1012no—
244C2×C3×D5—1024no—
245C2×C3×D5—1012no—
246C2×C3×D5—1012no—
247D3×D5—206no—
248D3×D5—2012no—
249D3×D5—206no—
250ΓL(1,𝔽16)—2012no—
251C3×AGL(1,𝔽5)—1024no—
252Dic15—106no—
253C5×Dic3—26no—
254C2×C52—22no—
255C2×C52—121no—
256C2×C52C2×(C2×D3)⋊C411yescharacteristic
257C2×C52C4×#6712yes—
258C5⋊D5—22no—
259C5⋊D5—121no—
260C5⋊D5#92911yescharacteristic
261C5⋊D5#93012yes—
262C5⋊D5#95311yescharacteristic
263C24×C3—251no—
264C23×D3—251no—
265C22×C3×C4—252no—
266C22×C3×C4—504no—
267C22×C3×C4—502no—
268C2×#67—251no—
269C2×#67—251no—
270C2×#67—251no—
271C2×#67—502no—
272C2×#67—502no—
273C2×#67#104311yescharacteristic, complemented
274C22×Dic3—251no—
275C22×Dic3—252no—
276#217—502no—
277(C2×D3)⋊C4—504no—
278C23×C5—36no—
279C23×C5D3×AGL(1,𝔽5)16yes—
280C22×D5—56no—
281C22×D5—512no—
282C22×D5—512no—
283C22×D5—56no—
284C22×D5—1012no—
285C22×D5—1012no—
286C22×D5—156no—
287C22×D5—3012no—
288C22×D5—306no—
289C22×D5—306no—
290C22×D5—3012no—
291C2×AGL(1,𝔽5)—524no—
292C2×AGL(1,𝔽5)—1012no—
293C2×AGL(1,𝔽5)—1012no—
294C2×AGL(1,𝔽5)—1048no—
295C2×AGL(1,𝔽5)—3012no—
296C2×AGL(1,𝔽5)—3012no—
297C2×AGL(1,𝔽5)—3012no—
298C5×D4—612no—
299C2×C4×C5—36no—
300C22⋊D5—3012no—
301C22⋊D5—3012no—
302C2×Dic5—156no—
303D20—3012no—
304C4×D5—306no—
305C4×D5—306no—
306C22×D4—751no—
307C23×C4—251no—
308C2×C22⋊C4—752no—
309C2×C22⋊C4—751no—
310#95—752no—
311C2×C3×C5—212no—
312C2×C3×C5D4×AGL(1,𝔽5)112yes—
313C2×C3×C5C2×C22⋊AGL(1,𝔽5)16yes—
314D15—206no—
315C3×D5—56no—
316C3×D5—512no—
317C3×D5—56no—
318C3×D5—1012no—
319C5×D3—46no—
320C52#521311yescharacteristic, complemented, Sylow
321C23×C3—251no—
322C23×C3—252no—
323C23×C3—252no—
324C23×C3—251no—
325C23×C3—502no—
326C23×C3—502no—
327C23×C3C2×#104311yescharacteristic
328C22×D3—251no—
329C22×D3—502no—
330C22×D3—501no—
331C22×D3—501no—
332C22×D3—502no—
333C22×D3C2×#104311yescharacteristic, complemented
334C2×C3×C4—254no—
335C2×C3×C4—502no—
336C2×C3×C4—502no—
337C2×C3×C4—508no—
338#67—22no—
339#67—502no—
340#67—502no—
341#67—502no—
342C2×Dic3—251no—
343C2×Dic3—501no—
344C2×Dic3—501no—
345C2×Dic3—502no—
346C2×Dic3—502no—
347C2×Dic3—502no—
348C2×Dic3C2×#104311yescharacteristic, complemented
349C22×C5—212no—
350C22×C5—66no—
351C22×C5—612no—
352C22×C5(C2×D3)⋊AGL(1,𝔽5)112yescomplemented
353C22×C5C2×D3×AGL(1,𝔽5)16yes—
354C2×D5—56no—
355C2×D5—56no—
356C2×D5—512no—
357C2×D5—512no—
358C2×D5—512no—
359C2×D5—1012no—
360C2×D5—1012no—
361C2×D5—1024no—
362C2×D5—1012no—
363C2×D5—306no—
364C2×D5—3012no—
365C2×D5—606no—
366C2×D5—6012no—
367C2×D5—606no—
368AGL(1,𝔽5)—1024no—
369AGL(1,𝔽5)—6012no—
370C4×C5—66no—
371Dic5—306no—
372C24—251no—
373C24—751no—
374C2×D4—31no—
375C2×D4—751no—
376C2×D4—751no—
377C2×D4—751no—
378C2×D4—1502no—
379C2×D4—1502no—
380C22×C4—252no—
381C22×C4—504no—
382C22×C4—502no—
383C22×C4—751no—
384C22×C4—752no—
385C22⋊C4—1504no—
386C22⋊C4—1502no—
387C3×C5#6577916yescomplemented
388C22×C3—22no—
389C22×C3—252no—
390C22×C3—252no—
391C22×C3—251no—
392C22×C3—251no—
393C22×C3—252no—
394C22×C3—502no—
395C22×C3—504no—
396C22×C3—502no—
397C22×C3—502no—
398C22×C3C22⋊#104312yescomplemented
399C22×C3C22×#104311yescharacteristic
400C2×D3—22no—
401C2×D3—21no—
402C2×D3—502no—
403C2×D3—501no—
404C2×D3—1002no—
405C2×D3—1001no—
406C2×D3—1001no—
407C3×C4—504no—
408Dic3—21no—
409Dic3—501no—
410Dic3—1002no—
411C2×C5—212no—
412C2×C5—126no—
413C2×C5C2×(C2×D3)⋊AGL(1,𝔽5)16yes—
414C2×C5AGL(1,𝔽5)×#67112yes—
415D5—56no—
416D5—512no—
417D5—56no—
418D5—1012no—
419D5—606no—
420C23—31no—
421C23—251no—
422C23—251no—
423C23—252no—
424C23—252no—
425C23—502no—
426C23—502no—
427C23—751no—
428C23—1502no—
429C23—1501no—
430C23—1501no—
431C23—1502no—
432C23D3×#104311yescharacteristic
433D4—62no—
434D4—1502no—
435D4—1502no—
436D4—1502no—
437C2×C4—31no—
438C2×C4—254no—
439C2×C4—502no—
440C2×C4—502no—
441C2×C4—508no—
442C2×C4—751no—
443C2×C4—1501no—
444C2×C4—1501no—
445C2×C4—1502no—
446C2×C4—1502no—
447C2×C4—1502no—
448C2×C3—22no—
449C2×C3—252no—
450C2×C3—251no—
451C2×C3—251no—
452C2×C3—502no—
453C2×C3D4×#104312yes—
454C2×C3C2×C22⋊#104311yescharacteristic
455D3—41no—
456D3—1001no—
457C5#1175202216yescomplemented
458C22—22no—
459C22—62no—
460C22—61no—
461C22—252no—
462C22—252no—
463C22—251no—
464C22—251no—
465C22—252no—
466C22—3002no—
467C22—3001no—
468C22—3001no—
469C22—502no—
470C22—504no—
471C22—502no—
472C22—502no—
473C22—1502no—
474C22—1501no—
475C22(C2×D3)⋊#104312yescomplemented
476C22C2×D3×#104311yescharacteristic
477C4—61no—
478C4—3002no—
479C4—504no—
480C4—1501no—
481C3#1310479111yescharacteristic, complemented, Sylow
482C2—22no—
483C2—121no—
484C2—252no—
485C2—251no—
486C2—251no—
487C2—3001no—
488C2—502no—
489C2C2×(C2×D3)⋊#104311yescharacteristic
490C2#67×#104312yes—
subgroupG/H
center, FrattiniC22C2×D3×#1043
commutator, socleC22×C3×C52C22×C4
FittingC23×C3×C52C2×C4
prime radicalC2×#67×C5⋊D5C2
cosocleC22×C3×C5⋊D5C23
hypercenterC23D3×#1043
hypocenterC3×C52#386
LCSG ▷ C22×C3×C52 ▷ C3×C52
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. #72231415C2semidirect
2. C2×(C2×D3)⋊#1043C2semidirect
3. C2×#11925533C2semidirect
5. C23×C3×#1043C2semidirect
13. C2×#10597159C4semidirect
14. C2×C52×#67C4semidirect
17. C2×#10597115C4semidirect
18. C2×#10597114C4semidirect
23. C23×#1043D3semidirect
29. C22×C3⋊(C5⋊D5)C2×C4semidirect
30. C22×C52×D3C2×C4semidirect
47. C2×C3×#1043D4semidirect
49. C2×C3⋊#1039C2×C4semidirect
51. C2×C52×Dic3C2×C4semidirect
78. C22×C3×C52C22⋊C4semidirect
88. C2×C3×C5⋊D5C22⋊C4semidirect
107. C2×C5×#67AGL(1,𝔽5)semidirect
134. C2×#1043#67semidirect
160. C22×C5×D3C2×AGL(1,𝔽5)semidirect
181. C2×C5×Dic3C2×AGL(1,𝔽5)semidirect
189. C22×C52(C2×D3)⋊C4semidirect
200. C2×C5⋊D5(C2×D3)⋊C4semidirect
229. C3×C52#386semidirect
232. C22×C3×C5C22⋊AGL(1,𝔽5)semidirect
273. C2×#67#1043semidirect
320. C52#5213semidirect
333. C22×D3C2×#1043semidirect
348. C2×Dic3C2×#1043semidirect
352. C22×C5(C2×D3)⋊AGL(1,𝔽5)semidirect
387. C3×C5#65779semidirect
398. C22×C3C22⋊#1043semidirect
457. C5#11752022semidirect
475. C22(C2×D3)⋊#1043semidirect
481. C3#13104791semidirect

Maximal subgroups

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

subgroupindex
#722314152
C2×(C2×D3)⋊#10432
C2×#119255332
C2×#67×C5⋊D52
C23×C3×#10432
#131047913
#117520225