Finite Group Databasebeta

← #72260285

Subgroups — #72260285

Order 4800.

1360 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#72231004C212yesmaximal, complemented
2#72230992C212yesmaximal, complemented
3#72230973C212yesmaximal, complemented
4#72230959C212yesmaximal, complemented
5#72229444C211yesmaximal, characteristic, complemented
6#72229438C211yesmaximal, characteristic
7C2×C3×C4⋊#1043C211yesmaximal, characteristic, complemented
8#72224585C214yesmaximal, complemented
9#13107562—31nomaximal, Hall
10C22×C3×#1043C2212yes—
11C2×#10597182C2214yes—
12C5×#6817C412yescomplemented
13C2×C3×C4×C5⋊D5C2211yescharacteristic
14#11925664C2212yes—
15#11925657C412yescomplemented
16#11925655C2211yescharacteristic
17C4×D15⋊D5C2212yes—
18#11925651C412yescomplemented
19#11925650C2211yescharacteristic
20#11925648C412yescomplemented
21#11925540C2212yes—
22C4⋊(C3⋊#1044)C2211yescharacteristic
23C4⋊(C3⋊#1044)C2211yescharacteristic
24C4×C3⋊#1044C2212yes—
25C3×C4⋊#1043C2212yes—
26C3×C4⋊#1043C2212yes—
27#11925356C2214yes—
28#11925355C2214yes—
29#11925354C2214yes—
30#11756836—52nomaximal
31#11717471—31no—
32#11717451—31no—
33C2×C4⋊#1043D311yescharacteristic, complemented
34#11717375—34no—
35#11717365—32no—
36D5⋊(C4⋊AGL(1,𝔽5))—32no—
37#11717355—32no—
38#11717354—32no—
39C22×C3×C5⋊D5C2311yescharacteristic
40C2×D15⋊D5C2312yes—
41C2×C3⋊#1044C2312yes—
42C2×C3⋊#1044C2312yes—
43C2×C3×#1043—42no—
44C2×C3×#1043C2314yes—
45#10597182—44no—
46C2×C3×C4×C52C2×C411yescharacteristic, complemented
47C5×#1220C2×C414yescomplemented
48C5×D60C2×C412yescomplemented
49C4×C5×D15C2×C414yescomplemented
50C5×Dic30C2×C412yescomplemented
51C2×C3×#1039C2×C411yescharacteristic, complemented
52C3×C4×C5⋊D5C2×C411yescharacteristic, complemented
53C3×C4×C5⋊D5C2×C411yescharacteristic, complemented
54C3×C4×C5⋊D5C2311yescharacteristic
55C3×C4×C5⋊D5C2311yescharacteristic
56#10597129C2×C412yescomplemented
57#10597128C2×C412yescomplemented
58#10597127C2×C414yescomplemented
59#10597126C2312yes—
60#10597125C2×C414yescomplemented
61D5×#235—52no—
62#91989—52no—
63C2×C3×C4⋊AGL(1,𝔽5)—52no—
64C2×C3×C4⋊AGL(1,𝔽5)—102no—
65AGL(1,𝔽5)×#67—58no—
66#91921—54no—
67D3×C4⋊AGL(1,𝔽5)—54no—
68ΓL(1,𝔽16)⋊Q8—54no—
69#91907—54no—
70C22×#1043C2×D312yes—
71C2×D5⋊AGL(1,𝔽5)—34no—
72C2×C4×C5⋊D5C2×D311yescharacteristic
73C5×#709—32no—
74#87886—32no—
75#87880—32no—
76#87877—31no—
77C4×D52—32no—
78#87873—32no—
79#87872—31no—
80#87871—32no—
81C22⋊#1044—32no—
82C4⋊#1044—31no—
83C4⋊#1044—31no—
84C4×#1044—32no—
85C4⋊#1043C2×D312yes—
86C4⋊#1043C2×D312yes—
87#87832—34no—
88#87831—34no—
89#87830—34no—
90#66266—152no—
91C22×C3×C52C22×C411yescharacteristic
92C2×C5×D15C22×C414yes—
93C2×C3×C5⋊D5—21no—
94C2×C3×C5⋊D5—21no—
95C2×C3×C5⋊D5C22×C411yescharacteristic
96C2×C3×C5⋊D5C2411yescharacteristic
97D15⋊D5—22no—
98D15⋊D5—22no—
99C3⋊#1044—22no—
100C3⋊#1044—22no—
101C3×#1043—24no—
102C3×C4×C52C22×C411yescharacteristic
103C3×C4×C52C22×C411yescharacteristic
104C5×Dic15C22×C414yes—
105C3×#1039C22×C411yescharacteristic
106C3×#1039C22×C411yescharacteristic
107C22×C3×AGL(1,𝔽5)—54no—
108C22×C3×AGL(1,𝔽5)—104no—
109C2×D3×AGL(1,𝔽5)—58no—
110#6817AGL(1,𝔽5)12yescomplemented
111C5×#235—52no—
112C2×C3×C4×D5—52no—
113C2×C3×C4×D5—102no—
114D5×#67—54no—
115#6782—54no—
116D5×D12—52no—
117C4×D3×D5—54no—
118#6773—52no—
119#6772—52no—
120#6771—54no—
121D5×Dic6—52no—
122#6763—54no—
123C4⋊ΓL(1,𝔽16)—52no—
124C4⋊ΓL(1,𝔽16)—52no—
125C4×ΓL(1,𝔽16)—54no—
126C3×C4⋊AGL(1,𝔽5)—54no—
127C3×C4⋊AGL(1,𝔽5)—54no—
128C3×C4⋊AGL(1,𝔽5)—104no—
129C3×C4⋊AGL(1,𝔽5)—104no—
130Dic3⋊AGL(1,𝔽5)—58no—
131(C2×D3)⋊AGL(1,𝔽5)—58no—
132Dic3×AGL(1,𝔽5)—58no—
133C22×C5⋊D5C22×D311yescharacteristic
134C2×D52—32no—
135C2×#1044—32no—
136C2×#1044—32no—
137C2×#1043—42no—
138C2×#1043C22×D314yes—
139D5⋊AGL(1,𝔽5)—124no—
140C2×C4×C52C4×D311yescharacteristic, complemented
141C2×#1039C4×D311yescharacteristic, complemented
142C4×C5⋊D5C4×D311yescharacteristic, complemented
143C4×C5⋊D5C4×D311yescharacteristic, complemented
144C4×C5⋊D5C22×D311yescharacteristic
145C4×C5⋊D5C22×D311yescharacteristic
146C5×C22⋊D5—34no—
147C5×D20—32no—
148C4×C5×D5—34no—
149C5×Dic10—32no—
150#6039—32no—
151#6038—34no—
152#6037—32no—
153#6036—32no—
154D5×Dic5—34no—
155#5504—251noHall
156D5×#41—152no—
157Q8×AGL(1,𝔽5)—154no—
158D4×AGL(1,𝔽5)—154no—
159D4×AGL(1,𝔽5)—158no—
160#4130—154no—
161#4130—152no—
162C2×C4⋊AGL(1,𝔽5)—52no—
163C2×C4⋊AGL(1,𝔽5)—102no—
164C2×C4⋊AGL(1,𝔽5)—154no—
165C2×C3×C52—21no—
166C2×C3×C52C23×C411yescharacteristic
167C5×D15—24no—
168C3×C5⋊D5—21no—
169C3×C5⋊D5H(5,𝔽2)11yescharacteristic
170C3×C5⋊D52−1+411yescharacteristic
171C22×C3×D5—52no—
172C22×C3×D5—102no—
173C2×D3×D5—54no—
174C2×ΓL(1,𝔽16)—54no—
175C2×ΓL(1,𝔽16)—54no—
176C2×C3×AGL(1,𝔽5)—58no—
177C2×C3×AGL(1,𝔽5)—108no—
178C2×C3×AGL(1,𝔽5)—204no—
179C2×C3×AGL(1,𝔽5)—208no—
180D3×AGL(1,𝔽5)—208no—
181C2×C3×C4×C5—22no—
182C2×C3×C4×C5C2×AGL(1,𝔽5)12yescomplemented
183#1220C2×AGL(1,𝔽5)14yescomplemented
184D60C2×AGL(1,𝔽5)12yescomplemented
185C4×D15C2×AGL(1,𝔽5)14yescomplemented
186Dic30C2×AGL(1,𝔽5)12yescomplemented
187C5×#67—54no—
188C5×D12—52no—
189C4×C5×D3—54no—
190C5×Dic6—52no—
191C2×C3×Dic5—52no—
192C2×C3×Dic5—102no—
193C3×C4×D5—52no—
194C3×C4×D5—52no—
195C3×C4×D5—52no—
196C3×C4×D5—52no—
197C3×C4×D5—102no—
198C3×C4×D5—102no—
199C3×C4×D5—102no—
200C3×C4×D5—102no—
201#1204—54no—
202#1203—54no—
203#1202—54no—
204#1201—54no—
205#1200—54no—
206D3×Dic5—54no—
207D5×Dic3—54no—
208C22×C52C2×C4×D311yescharacteristic
209C2×C5⋊D5—21no—
210C2×C5⋊D5—21no—
211C2×C5⋊D5C2×C4×D311yescharacteristic
212C2×C5⋊D5C23×D311yescharacteristic
213C2×C5×D5—34no—
214D52—62no—
215D52—62no—
216#1044—62no—
217#1044—62no—
218#1043—24no—
219C4×C52C2×C4×D311yescharacteristic
220C4×C52C2×C4×D311yescharacteristic
221#1039C2×C4×D311yescharacteristic
222#1039C2×C4×D311yescharacteristic
223C5×Dic5—34no—
224C2×#235—251no—
225C2×C3×C4⋊C4—251no—
226C4×#67—254no—
227#926—251no—
228#893—252no—
229#892—252no—
230D3×C4⋊C4—252no—
231#887—252no—
232C22×AGL(1,𝔽5)—54no—
233C22×AGL(1,𝔽5)—104no—
234C22×AGL(1,𝔽5)—158no—
235C5×#41—152no—
236#713—152no—
237Q8×D5—152no—
238#711—152no—
239#711—154no—
240D4×D5—152no—
241D4×D5—154no—
242#709—32no—
243#709—154no—
244C2×C4×D5—52no—
245C2×C4×D5—102no—
246C2×C4×D5—154no—
247C22⋊AGL(1,𝔽5)—158no—
248C22⋊AGL(1,𝔽5)—154no—
249C4⋊AGL(1,𝔽5)—54no—
250C4⋊AGL(1,𝔽5)—54no—
251C4⋊AGL(1,𝔽5)—104no—
252C4⋊AGL(1,𝔽5)—104no—
253C4⋊AGL(1,𝔽5)—152no—
254C4⋊AGL(1,𝔽5)—152no—
255C4⋊AGL(1,𝔽5)—158no—
256C4×AGL(1,𝔽5)—154no—
257C4×AGL(1,𝔽5)—158no—
258C3×C52#52011yescharacteristic, complemented, Hall
259#520—751noSylow
260C22×C3×C5—22no—
261C22×C3×C5C22×AGL(1,𝔽5)12yes—
262C2×D15C22×AGL(1,𝔽5)14yes—
263C2×C5×D3—54no—
264C2×C3×D5—52no—
265C2×C3×D5—52no—
266C2×C3×D5—102no—
267C2×C3×D5—102no—
268C2×C3×D5—102no—
269C2×C3×D5—102no—
270C2×C3×D5—104no—
271C2×C3×D5—104no—
272D3×D5—104no—
273D3×D5—104no—
274ΓL(1,𝔽16)—104no—
275ΓL(1,𝔽16)—104no—
276C3×AGL(1,𝔽5)—108no—
277C3×AGL(1,𝔽5)—208no—
278C3×C4×C5—22no—
279C3×C4×C5—22no—
280C3×C4×C5C22×AGL(1,𝔽5)12yes—
281C3×C4×C5C22×AGL(1,𝔽5)12yes—
282Dic15C22×AGL(1,𝔽5)14yes—
283C3×Dic5—52no—
284C3×Dic5—52no—
285C3×Dic5—102no—
286C3×Dic5—102no—
287C5×Dic3—54no—
288C2×C52—21no—
289C2×C52C22×C4×D311yescharacteristic
290C5⋊D5—21no—
291C5⋊D5#100411yescharacteristic
292C5⋊D5#100711yescharacteristic
293C5×D5—64no—
294C22×C3×C4—251no—
295C22×C3×C4—252no—
296C2×#67—252no—
297#235—52no—
298#235—252no—
299#235—252no—
300#235—252no—
301C2×D12—251no—
302C2×C4×D3—252no—
303C2×C4×D3—254no—
304C2×Dic6—251no—
305C3×C4⋊C4—252no—
306C3×C4⋊C4—252no—
307#217—252no—
308(C2×D3)⋊C4—254no—
309C4⋊Dic3—251no—
310C4⋊Dic3—251no—
311Dic3⋊C4—254no—
312C4×Dic3—254no—
313C4×Dic3—252no—
314C22×D5—52no—
315C22×D5—102no—
316C22×D5—154no—
317C2×AGL(1,𝔽5)—58no—
318C2×AGL(1,𝔽5)—108no—
319C2×AGL(1,𝔽5)—154no—
320C2×AGL(1,𝔽5)—154no—
321C2×AGL(1,𝔽5)—204no—
322C2×AGL(1,𝔽5)—208no—
323C2×AGL(1,𝔽5)—608no—
324C5×Q8—152no—
325C5×D4—154no—
326C5×D4—152no—
327C2×C4×C5—22no—
328C2×C4×C5—154no—
329C2×C4×C5D3×AGL(1,𝔽5)12yescomplemented
330C22⋊D5—34no—
331C22⋊D5—154no—
332C22⋊D5—154no—
333C2×Dic5—52no—
334C2×Dic5—102no—
335C2×Dic5—154no—
336D20—32no—
337D20—154no—
338C4×D5—34no—
339C4×D5—52no—
340C4×D5—52no—
341C4×D5—52no—
342C4×D5—52no—
343C4×D5—102no—
344C4×D5—102no—
345C4×D5—102no—
346C4×D5—102no—
347C4×D5—154no—
348C4×D5—154no—
349Dic10—32no—
350Dic10—154no—
351C2×#41—751no—
352C4×Q8—752no—
353C4×D4—752no—
354C4×D4—754no—
355#117—752no—
356#117—751no—
357C2×C4⋊C4—251no—
358C2×C4⋊C4—752no—
359C2×C3×C5—24no—
360C2×C3×C5—22no—
361C2×C3×C5—22no—
362C2×C3×C5C23×AGL(1,𝔽5)12yes—
363D15—24no—
364C3×D5—52no—
365C3×D5—52no—
366C3×D5—102no—
367C3×D5—102no—
368C3×D5—104no—
369C3×D5—102no—
370C5×D3—104no—
371C52#550411yescharacteristic, complemented, Sylow
372C23×C3—251no—
373C22×D3—252no—
374C2×C3×C4—251no—
375C2×C3×C4—251no—
376C2×C3×C4—251no—
377C2×C3×C4—251no—
378C2×C3×C4—251no—
379C2×C3×C4—254no—
380C2×C3×C4—1002no—
381C2×C3×C4D5⋊AGL(1,𝔽5)11yescharacteristic, complemented
382#67—54no—
383#67—254no—
384C2×Dic3—252no—
385C2×Dic3—252no—
386C2×Dic3—252no—
387D12—52no—
388D12—252no—
389C4×D3—54no—
390C4×D3—254no—
391C4×D3—1004no—
392Dic6—52no—
393Dic6—252no—
394C22×C5—22no—
395C22×C5—154no—
396C22×C5C2×D3×AGL(1,𝔽5)12yes—
397C2×D5—34no—
398C2×D5—52no—
399C2×D5—52no—
400C2×D5—102no—
401C2×D5—102no—
402C2×D5—102no—
403C2×D5—102no—
404C2×D5—104no—
405C2×D5—104no—
406C2×D5—304no—
407C2×D5—304no—
408AGL(1,𝔽5)—108no—
409AGL(1,𝔽5)—208no—
410AGL(1,𝔽5)—304no—
411AGL(1,𝔽5)—304no—
412C4×C5—22no—
413C4×C5—22no—
414C4×C5—154no—
415C4×C5C2×D3×AGL(1,𝔽5)12yes—
416C4×C5C2×D3×AGL(1,𝔽5)12yes—
417Dic5—34no—
418Dic5—52no—
419Dic5—52no—
420Dic5—102no—
421Dic5—102no—
422#41—152no—
423#41—752no—
424#41—752no—
425#41—752no—
426C2×Q8—751no—
427C2×D4—751no—
428C2×D4—752no—
429C22×C4—251no—
430C22×C4—252no—
431C22×C4—752no—
432C22×C4—754no—
433C4⋊C4—252no—
434C4⋊C4—252no—
435C4⋊C4—751no—
436C4⋊C4—754no—
437C4⋊C4—751no—
438C22⋊C4—754no—
439C22⋊C4—752no—
440C42—752no—
441C42—754no—
442C3×C5—22no—
443C3×C5#6626612yescomplemented
444C22×C3—251no—
445C22×C3—251no—
446C22×C3—501no—
447C22×C3—501no—
448C22×C3C2×D5⋊AGL(1,𝔽5)11yescharacteristic
449C2×D3—54no—
450C2×D3—502no—
451C2×D3—502no—
452C3×C4—251no—
453C3×C4—251no—
454C3×C4—504no—
455C3×C4C2×D5⋊AGL(1,𝔽5)11yescharacteristic
456C3×C4C2×D5⋊AGL(1,𝔽5)11yescharacteristic
457Dic3—54no—
458Dic3—502no—
459Dic3—502no—
460C2×C5—24no—
461C2×C5—22no—
462C2×C5—22no—
463C2×C5—304no—
464C2×C5C22×D3×AGL(1,𝔽5)12yes—
465D5—52no—
466D5—52no—
467D5—64no—
468D5—102no—
469D5—102no—
470D5—104no—
471D5—102no—
472C23—251no—
473C23—752no—
474Q8—152no—
475Q8—752no—
476D4—154no—
477D4—152no—
478D4—754no—
479D4—752no—
480C2×C4—154no—
481C2×C4—251no—
482C2×C4—251no—
483C2×C4—251no—
484C2×C4—251no—
485C2×C4—251no—
486C2×C4—254no—
487C2×C4—3004no—
488C2×C4—752no—
489C2×C4—754no—
490C2×C4—752no—
491C2×C4—752no—
492C2×C4—1002no—
493C2×C4#1059718211yescharacteristic, complemented
494C2×C3—21no—
495C2×C3—251no—
496C2×C3—251no—
497C2×C3—501no—
498C2×C3C22×D5⋊AGL(1,𝔽5)11yescharacteristic
499D3—104no—
500C5—22no—
501C5#1175683612yescomplemented
502C22—154no—
503C22—251no—
504C22—251no—
505C22—501no—
506C22—501no—
507C22—1502no—
508C22—1502no—
509C22C2×#1059718211yescharacteristic
510C4—154no—
511C4—251no—
512C4—251no—
513C4—504no—
514C4—1502no—
515C4—1502no—
516C4C2×#1059718211yescharacteristic
517C4C2×#1059718211yescharacteristic
518C3#1310756211yescharacteristic, complemented, Sylow
519C2—21no—
520C2—251no—
521C2—251no—
522C2—304no—
523C2—501no—
524C2C22×#1059718211yescharacteristic
subgroupG/H
center, FrattiniC2C22×#10597182
commutator, socleC2×C3×C52C23×C4
FittingC2×C3×C4×C52C2×C4
prime radical#72229438C2
cosocleC2×C3×C5⋊D5C24
hypercenterC2×C4#10597182
hypocenterC3×C52#520
LCSG ▷ C2×C3×C52 ▷ C3×C52
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #72231004C2semidirect
2. #72230992C2semidirect
3. #72230973C2semidirect
4. #72230959C2semidirect
5. #72229444C2semidirect
7. C2×C3×C4⋊#1043C2semidirect
8. #72224585C2semidirect
12. C5×#6817C4semidirect
15. #11925657C4semidirect
18. #11925651C4semidirect
20. #11925648C4semidirect
33. C2×C4⋊#1043D3semidirect
46. C2×C3×C4×C52C2×C4semidirect
47. C5×#1220C2×C4semidirect
48. C5×D60C2×C4semidirect
49. C4×C5×D15C2×C4semidirect
50. C5×Dic30C2×C4semidirect
51. C2×C3×#1039C2×C4semidirect
52. C3×C4×C5⋊D5C2×C4semidirect
53. C3×C4×C5⋊D5C2×C4semidirect
56. #10597129C2×C4semidirect
57. #10597128C2×C4semidirect
58. #10597127C2×C4semidirect
60. #10597125C2×C4semidirect
110. #6817AGL(1,𝔽5)semidirect
140. C2×C4×C52C4×D3semidirect
141. C2×#1039C4×D3semidirect
142. C4×C5⋊D5C4×D3semidirect
143. C4×C5⋊D5C4×D3semidirect
182. C2×C3×C4×C5C2×AGL(1,𝔽5)semidirect
183. #1220C2×AGL(1,𝔽5)semidirect
184. D60C2×AGL(1,𝔽5)semidirect
185. C4×D15C2×AGL(1,𝔽5)semidirect
186. Dic30C2×AGL(1,𝔽5)semidirect
258. C3×C52#520semidirect
329. C2×C4×C5D3×AGL(1,𝔽5)semidirect
371. C52#5504semidirect
381. C2×C3×C4D5⋊AGL(1,𝔽5)semidirect
443. C3×C5#66266semidirect
493. C2×C4#10597182semidirect
501. C5#11756836semidirect
518. C3#13107562semidirect

Maximal subgroups

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

subgroupindex
#722310042
#722309922
#722309732
#722309592
#722294442
#722294382
C2×C3×C4⋊#10432
#722245852
#131075623
#117568365