Finite Group Databasebeta

← #13106592

Subgroups — #13106592

Order 1600.

630 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11717273C211yesmaximal, characteristic, complemented
2C5×#4099C211yesmaximal, characteristic, complemented
3C2×C5×C22⋊Dic5C211yesmaximal, characteristic, complemented
4#11717096C211yesmaximal, characteristic, complemented
5#11717093C211yesmaximal, characteristic, complemented
6Dic5×C22⋊D5C211yesmaximal, characteristic, complemented
7#11717089C211yesmaximal, characteristic, complemented
8#11717087C211yesmaximal, characteristic, complemented
9#11717082C211yesmaximal, characteristic, complemented
10#11717077C211yesmaximal, characteristic, complemented
11#11717076C211yesmaximal, characteristic, complemented
12#11717074C211yesmaximal, characteristic, complemented
13#11717074C211yesmaximal, characteristic, complemented
14#11717071C211yesmaximal, characteristic, complemented
15#11717070C211yesmaximal, characteristic, complemented
16C24×C52C2211yescharacteristic
17C2×C22⋊(C5⋊D5)C2211yescharacteristic
18C2×C22⋊(C5⋊D5)C2211yescharacteristic
19C2×C22⋊(C5⋊D5)C2211yescharacteristic
20C2×C5×C22⋊D5C2211yescharacteristic
21C2×C5×C22⋊D5C2211yescharacteristic
22C2×C5×C22⋊D5C2211yescharacteristic
23C22×C5×Dic5C2211yescharacteristic
24C22×C5×Dic5C2211yescharacteristic
25C2×#6039C2211yescharacteristic, complemented
26C2×#6038C2211yescharacteristic, complemented
27C2×#6036C2211yescharacteristic, complemented
28C2×D5×Dic5C2211yescharacteristic, complemented
29C22⋊#1039C2211yescharacteristic
30C22⋊#1039C2211yescharacteristic
31C22⋊#1039C2211yescharacteristic
32C5×C22⋊Dic5C2211yescharacteristic
33C5×C22⋊Dic5C2211yescharacteristic
34C5×C22⋊Dic5C2211yescharacteristic
35C5×C22⋊Dic5C2211yescharacteristic
36C5×C22⋊Dic5C2211yescharacteristic
37C5×C22⋊Dic5C2211yescharacteristic
38C5×C22⋊Dic5C2211yescharacteristic
39#87781C2211yescharacteristic, complemented
40#87781C2211yescharacteristic, complemented
41#87780C2211yescharacteristic, complemented
42#87780C2211yescharacteristic, complemented
43#87780C2211yescharacteristic, complemented
44#87780C2211yescharacteristic, complemented
45#87779C2211yescharacteristic, complemented
46#87779C2211yescharacteristic, complemented
47#87778C2211yescharacteristic, complemented
48#87778C2211yescharacteristic, complemented
49Dic52C2211yescharacteristic, complemented
50Dic52C2211yescharacteristic, complemented
51#65927—51nomaximal
52#65828—51nomaximal
53C23×C52—22no—
54C23×C52—41no—
55C23×C52—41no—
56C23×C52C2311yescharacteristic
57C23×C52C2311yescharacteristic
58C23×C52C2311yescharacteristic
59C22×C5⋊D5C2311yescharacteristic
60C22×C5×D5C2311yescharacteristic
61C22⋊(C5⋊D5)—22no—
62C22⋊(C5⋊D5)—41no—
63C22⋊(C5⋊D5)—41no—
64C2×#1039C2311yescharacteristic
65C2×#1039C2311yescharacteristic
66C2×#1039C2311yescharacteristic
67C5×C22⋊D5—22no—
68C5×C22⋊D5—41no—
69C5×C22⋊D5—41no—
70C2×C5×Dic5—41no—
71C2×C5×Dic5—41no—
72C2×C5×Dic5C2311yescharacteristic
73C2×C5×Dic5C2311yescharacteristic
74C2×C5×Dic5C2311yescharacteristic
75C2×C5×Dic5C2311yescharacteristic
76C2×C5×Dic5C2311yescharacteristic
77C2×C5×Dic5C2311yescharacteristic
78C2×C5×Dic5C2311yescharacteristic
79#6039—41no—
80#6038—41no—
81#6036—41no—
82D5×Dic5—41no—
83C5×#120—51no—
84C2×C5×C22⋊C4—51no—
85#4099—51no—
86#4099—51no—
87#4099—102no—
88#4099D511yescharacteristic, complemented
89C2×C22⋊Dic5D511yescharacteristic, complemented
90#4085—51no—
91#4085—51no—
92#4084—51no—
93#4082—51no—
94#4081—51no—
95D4×Dic5—51no—
96#4076—51no—
97C4×C22⋊D5—51no—
98C4×C22⋊D5—51no—
99#4070—51no—
100#4031—51no—
101#4031—51no—
102#4031—51no—
103#4030—51no—
104#4030—51no—
105#4029—51no—
106#4029—51no—
107#4029—51no—
108#4027—51no—
109#4025—51no—
110#4025—51no—
111#4025—51no—
112#4025—51no—
113#4024—51no—
114C22×C52—21no—
115C22×C52—21no—
116C22×C52—21no—
117C22×C52—21no—
118C22×C52—21no—
119C22×C52—21no—
120C22×C52—21no—
121C22×C52—21no—
122C22×C52—44no—
123C22×C52#4112yes—
124C22×C52C2411yescharacteristic
125C2×C5⋊D5—21no—
126C2×C5⋊D5—21no—
127C2×C5⋊D5—21no—
128C2×C5×D5—21no—
129C2×C5×D5—21no—
130C2×C5×D5—21no—
131#1039—21no—
132#1039—21no—
133#1039—21no—
134C5×Dic5—21no—
135C5×Dic5—21no—
136C5×Dic5—21no—
137C5×Dic5—21no—
138C5×Dic5—21no—
139C5×Dic5—21no—
140C5×Dic5—21no—
141C24×C5—22no—
142C24×C5C2×D511yescharacteristic
143C24×C5C2×D511yescharacteristic
144C2×C5×D4—51no—
145C2×C5×D4—51no—
146C2×C5×D4—51no—
147C22×C4×C5—51no—
148C22×C4×C5—51no—
149C2×C22⋊D5—51no—
150C2×C22⋊D5—51no—
151C2×C22⋊D5—51no—
152C2×C22⋊D5—51no—
153C2×C22⋊D5—51no—
154C2×C22⋊D5—51no—
155C2×C22⋊D5—51no—
156C2×C22⋊D5—102no—
157C2×C22⋊D5—102no—
158C2×C22⋊D5—102no—
159C2×C22⋊D5C2×D511yescharacteristic
160C2×C22⋊D5C2×D511yescharacteristic
161C2×C22⋊D5C2×D511yescharacteristic
162C22×Dic5—51no—
163C22×Dic5C2×D511yescharacteristic
164C22×Dic5C2×D511yescharacteristic
165C2×D20—51no—
166C2×C4×D5—51no—
167C2×C4×D5—51no—
168C2×C4×D5—51no—
169C2×Dic10—51no—
170C2×Dic10—51no—
171C5×C22⋊C4—51no—
172C5×C22⋊C4—51no—
173C5×C22⋊C4—51no—
174C5×C22⋊C4—51no—
175C5×C22⋊C4—51no—
176C5×C22⋊C4—51no—
177C5×C22⋊C4—51no—
178C22⋊Dic5—51no—
179C22⋊Dic5—51no—
180C22⋊Dic5—51no—
181C22⋊Dic5—51no—
182C22⋊Dic5—51no—
183C22⋊Dic5—51no—
184C22⋊Dic5—51no—
185C22⋊Dic5—51no—
186C22⋊Dic5—102no—
187C22⋊Dic5—102no—
188C22⋊Dic5—102no—
189C22⋊Dic5C2×D511yescharacteristic
190C22⋊Dic5C2×D511yescharacteristic
191C22⋊Dic5C2×D511yescharacteristic
192C22⋊Dic5C2×D511yescharacteristic
193C22⋊Dic5C2×D511yescharacteristic
194C22⋊Dic5C2×D511yescharacteristic
195C22⋊Dic5C2×D511yescharacteristic
196#685—51no—
197#685—51no—
198#685—51no—
199#685—51no—
200#685—51no—
201#685—51no—
202C4⋊Dic5—51no—
203C4⋊Dic5—51no—
204C4⋊Dic5—51no—
205C4⋊Dic5—51no—
206#683—51no—
207#683—51no—
208#683—51no—
209#683—51no—
210#683—51no—
211#683—51no—
212#683—51no—
213#683—51no—
214C4×Dic5—51no—
215C4×Dic5—51no—
216C4×Dic5—51no—
217C4×Dic5—51no—
218#538—251noSylow
219C2×C52—22no—
220C2×C52—41no—
221C2×C52—41no—
222C2×C52C2×#4111yescharacteristic
223C2×C52H(5,𝔽2)11yescharacteristic
224C2×C52H(5,𝔽2)11yescharacteristic
225C5⋊D5—41no—
226C5×D5—41no—
227C23×C5—22no—
228C23×C5—22no—
229C23×C5—22no—
230C23×C5—22no—
231C23×C5—22no—
232C23×C5—44no—
233C23×C5—41no—
234C23×C5—41no—
235C23×C5—41no—
236C23×C5—41no—
237C23×C5—44no—
238C23×C5—44no—
239C23×C5—51no—
240C23×C5C22×D511yescharacteristic
241C23×C5C22×D511yescharacteristic
242C23×C5C22×D511yescharacteristic
243C23×C5C22×D511yescharacteristic
244C23×C5C22×D511yescharacteristic
245C23×C5C22×D511yescharacteristic
246C22×D5—51no—
247C22×D5—51no—
248C22×D5—102no—
249C22×D5C22×D511yescharacteristic
250C5×D4—102no—
251C5×D4—201no—
252C5×D4—201no—
253C2×C4×C5—51no—
254C2×C4×C5—51no—
255C2×C4×C5—51no—
256C2×C4×C5—51no—
257C2×C4×C5—51no—
258C2×C4×C5—51no—
259C2×C4×C5—51no—
260C2×C4×C5—201no—
261C2×C4×C5—201no—
262C22⋊D5—22no—
263C22⋊D5—41no—
264C22⋊D5—41no—
265C22⋊D5—102no—
266C22⋊D5—102no—
267C22⋊D5—204no—
268C22⋊D5—201no—
269C22⋊D5—201no—
270C22⋊D5—201no—
271C22⋊D5—201no—
272C22⋊D5—201no—
273C22⋊D5—204no—
274C22⋊D5—204no—
275C2×Dic5—41no—
276C2×Dic5—41no—
277C2×Dic5—51no—
278C2×Dic5—51no—
279C2×Dic5—51no—
280C2×Dic5—51no—
281C2×Dic5—51no—
282C2×Dic5—51no—
283C2×Dic5—102no—
284C2×Dic5—102no—
285C2×Dic5—102no—
286C2×Dic5—201no—
287C2×Dic5C22×D511yescharacteristic
288C2×Dic5C22×D511yescharacteristic
289C2×Dic5C22×D511yescharacteristic
290C2×Dic5C22×D511yescharacteristic
291C2×Dic5C22×D511yescharacteristic
292C2×Dic5C22×D511yescharacteristic
293C2×Dic5C22×D511yescharacteristic
294D20—201no—
295C4×D5—201no—
296C4×D5—201no—
297C4×D5—201no—
298Dic10—201no—
299Dic10—201no—
300#126—251no—
301#126—251no—
302#124—251no—
303#124—251no—
304#123—251no—
305#123—251no—
306C22⋊Q8—251no—
307#121—251no—
308#121—251no—
309#121—251no—
310#120—51no—
311#120—251no—
312C4×D4—251no—
313C4×D4—251no—
314C2×C22⋊C4—51no—
315C52#53811yescharacteristic, complemented, Sylow
316C22×C5—21no—
317C22×C5—21no—
318C22×C5—21no—
319C22×C5—21no—
320C22×C5—24no—
321C22×C5—24no—
322C22×C5—21no—
323C22×C5—21no—
324C22×C5—21no—
325C22×C5—21no—
326C22×C5—21no—
327C22×C5—21no—
328C22×C5—21no—
329C22×C5—21no—
330C22×C5—21no—
331C22×C5—21no—
332C22×C5—21no—
333C22×C5—21no—
334C22×C5—24no—
335C22×C5—22no—
336C22×C5—42no—
337C22×C5—42no—
338C22×C5—42no—
339C22×C5—42no—
340C22×C5—42no—
341C22×C5—42no—
342C22×C5—44no—
343C22×C5—44no—
344C22×C5—416no—
345C22×C5—101no—
346C22×C5—101no—
347C22×C5—101no—
348C22×C5#70912yes—
349C22×C5#71112yes—
350C22×C5C23×D511yescharacteristic
351C22×C5C23×D511yescharacteristic
352C2×D5—21no—
353C2×D5—21no—
354C2×D5—21no—
355C2×D5—101no—
356C2×D5—101no—
357C2×D5—101no—
358C2×D5—101no—
359C2×D5—101no—
360C2×D5—101no—
361C2×D5—202no—
362C2×D5—202no—
363C2×D5—202no—
364C4×C5—101no—
365C4×C5—101no—
366C4×C5—101no—
367C4×C5—101no—
368C4×C5—101no—
369C4×C5—101no—
370C4×C5—101no—
371Dic5—21no—
372Dic5—21no—
373Dic5—21no—
374Dic5—21no—
375Dic5—21no—
376Dic5—21no—
377Dic5—21no—
378Dic5—101no—
379Dic5—101no—
380Dic5—101no—
381Dic5—101no—
382Dic5—101no—
383Dic5—101no—
384Dic5—202no—
385Dic5—202no—
386Dic5—202no—
387C24D5211yescharacteristic
388C2×Q8—251no—
389C2×D4—51no—
390C2×D4—51no—
391C2×D4—51no—
392C2×D4—251no—
393C2×D4—251no—
394C2×D4—251no—
395C2×D4—251no—
396C22×C4—51no—
397C22×C4—51no—
398C22×C4—251no—
399C22×C4—251no—
400C4⋊C4—251no—
401C4⋊C4—251no—
402C4⋊C4—251no—
403C4⋊C4—251no—
404C4⋊C4—251no—
405C4⋊C4—251no—
406C22⋊C4—51no—
407C22⋊C4—51no—
408C22⋊C4—51no—
409C22⋊C4—51no—
410C22⋊C4—51no—
411C22⋊C4—51no—
412C22⋊C4—51no—
413C22⋊C4—251no—
414C22⋊C4—251no—
415C22⋊C4—251no—
416C22⋊C4—251no—
417C22⋊C4—251no—
418C22⋊C4—251no—
419C22⋊C4—251no—
420C42—251no—
421C42—251no—
422C2×C5—22no—
423C2×C5—22no—
424C2×C5—22no—
425C2×C5—22no—
426C2×C5—22no—
427C2×C5—44no—
428C2×C5—44no—
429C2×C5—44no—
430C2×C5—41no—
431C2×C5—41no—
432C2×C5—41no—
433C2×C5—41no—
434C2×C5—201no—
435C2×C5C2×#70911yescharacteristic
436C2×C5C2×#71111yescharacteristic
437C2×C5#414411yescharacteristic
438C2×C5#414411yescharacteristic
439C2×C5#414411yescharacteristic
440C2×C5#414411yescharacteristic
441D5—41no—
442D5—201no—
443D5—201no—
444D5—402no—
445C23—22no—
446C23—41no—
447C23—41no—
448C23—51no—
449C23—251no—
450C23C2×D5211yescharacteristic
451C23C2×D5211yescharacteristic
452C23C2×D5211yescharacteristic
453Q8—1001no—
454D4—102no—
455D4—201no—
456D4—201no—
457D4—502no—
458D4—1001no—
459D4—1001no—
460D4—1001no—
461C2×C4—51no—
462C2×C4—51no—
463C2×C4—51no—
464C2×C4—51no—
465C2×C4—51no—
466C2×C4—51no—
467C2×C4—51no—
468C2×C4—201no—
469C2×C4—201no—
470C2×C4—251no—
471C2×C4—251no—
472C2×C4—251no—
473C2×C4—1001no—
474C2×C4—1001no—
475C5—22no—
476C5#6582811yescharacteristic, complemented
477C5#6592711yescharacteristic, complemented
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—44no—
487C22—101no—
488C22—101no—
489C22—101no—
490C22—501no—
491C22—501no—
492C22—501no—
493C22#8787912yes—
494C22C22×D5211yescharacteristic
495C4—101no—
496C4—101no—
497C4—101no—
498C4—101no—
499C4—101no—
500C4—101no—
501C4—101no—
502C4—501no—
503C4—501no—
504C4—501no—
505C2—22no—
506C2—41no—
507C2—41no—
508C2—201no—
509C2—1001no—
510C2C2×#8787911yescharacteristic
511C2#1171749511yescharacteristic
512C2#1171749511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC24×C52C22
hypercenterC24D52
hypocenterC52#538
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. #11717273C2semidirect
2. C5×#4099C2semidirect
3. C2×C5×C22⋊Dic5C2semidirect
4. #11717096C2semidirect
5. #11717093C2semidirect
6. Dic5×C22⋊D5C2semidirect
7. #11717089C2semidirect
8. #11717087C2semidirect
9. #11717082C2semidirect
10. #11717077C2semidirect
11. #11717076C2semidirect
12. #11717074C2semidirect
13. #11717074C2semidirect
14. #11717071C2semidirect
15. #11717070C2semidirect
25. C2×#6039C22semidirect
26. C2×#6038C22semidirect
27. C2×#6036C22semidirect
28. C2×D5×Dic5C22semidirect
39. #87781C22semidirect
40. #87781C22semidirect
41. #87780C22semidirect
42. #87780C22semidirect
43. #87780C22semidirect
44. #87780C22semidirect
45. #87779C22semidirect
46. #87779C22semidirect
47. #87778C22semidirect
48. #87778C22semidirect
49. Dic52C22semidirect
50. Dic52C22semidirect
88. #4099D5semidirect
89. C2×C22⋊Dic5D5semidirect
315. C52#538semidirect
476. C5#65828semidirect
477. C5#65927semidirect

Maximal subgroups

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

subgroupindex
#117172732
C5×#40992
C2×C5×C22⋊Dic52
#117170962
#117170932
Dic5×C22⋊D52
#117170892
#117170872
#117170822
#117170772
#117170762
#117170742
#117170712
#117170702
#659275
#658285