Finite Group Databasebeta

← #13106203

Subgroups — #13106203

Order 1600.

762 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×D4×D5C211yesmaximal, characteristic, complemented
2C2×#87879C211yesmaximal, characteristic, complemented
3#11717259C211yesmaximal, characteristic, complemented
4C5×#4081C211yesmaximal, characteristic, complemented
5#11717095C211yesmaximal, characteristic, complemented
6#11717095C211yesmaximal, characteristic, complemented
7#11717091C211yesmaximal, characteristic, complemented
8D5×C22⋊Dic5C211yesmaximal, characteristic, complemented
9C2×#87778C211yesmaximal, characteristic, complemented
10#11717077C211yesmaximal, characteristic, complemented
11#11717076C211yesmaximal, characteristic, complemented
12#11717022C211yesmaximal, characteristic, complemented
13#11717005C211yesmaximal, characteristic, complemented
14#11717002C211yesmaximal, characteristic, complemented
15#11716969C211yesmaximal, characteristic, complemented
16C23×C5×D5C2211yescharacteristic
17C23×C5×D5C2211yescharacteristic
18C2×C52×D4C2211yescharacteristic
19C2×C22⋊(C5⋊D5)C2211yescharacteristic, complemented
20C2×C22⋊(C5⋊D5)C2211yescharacteristic, complemented
21C22×#1039C2211yescharacteristic, complemented
22C2×C5×C22⋊D5C2211yescharacteristic
23C2×C5×C22⋊D5C2211yescharacteristic
24C22×C5×Dic5C2211yescharacteristic, complemented
25C5×D4×D5—24no—
26C2×C5×D20C2211yescharacteristic
27C2×C4×C5×D5C2211yescharacteristic
28C2×#6039C2211yescharacteristic, complemented
29C2×#6038C2211yescharacteristic
30C2×#6038C2211yescharacteristic, complemented
31C2×#6038C2211yescharacteristic, complemented
32C2×#6036C2211yescharacteristic, complemented
33#87879—24no—
34C2×D5×Dic5C2211yescharacteristic, complemented
35C2×D5×Dic5C2211yescharacteristic, complemented
36C22⋊#1039C2211yescharacteristic, complemented
37#87808C2211yescharacteristic, complemented
38#87806C2211yescharacteristic, complemented
39C5×C22⋊Dic5C2211yescharacteristic, complemented
40C5×C22⋊Dic5C2211yescharacteristic, complemented
41C5×C22⋊Dic5C2211yescharacteristic, complemented
42C5×#683C2211yescharacteristic, complemented
43C5×#683C2211yescharacteristic, complemented
44#87780C2211yescharacteristic, complemented
45#87778C2211yescharacteristic
46#87778C2211yescharacteristic
47#87778C2211yescharacteristic
48#87778C2211yescharacteristic, complemented
49#87778C2211yescharacteristic, complemented
50#87778C2211yescharacteristic, complemented
51#87778C2211yescharacteristic, complemented
52Dic52C2211yescharacteristic, complemented
53#66140—51nomaximal
54#65988—51nomaximal
55C23×C52C2311yescharacteristic
56C23×C52C2311yescharacteristic
57C22×C5⋊D5C2311yescharacteristic
58C22×C5×D5—22no—
59C22×C5×D5—22no—
60C22×C5×D5—22no—
61C22×C5×D5—22no—
62C22×C5×D5—41no—
63C22×C5×D5—41no—
64C22×C5×D5C2311yescharacteristic
65C22×C5×D5C2311yescharacteristic
66C22×C5×D5C2311yescharacteristic
67C52×D4—22no—
68C2×C4×C52C2311yescharacteristic
69C22⋊(C5⋊D5)—22no—
70C22⋊(C5⋊D5)—41no—
71C2×#1039—41no—
72C2×#1039C2311yescharacteristic
73C2×#1039C2311yescharacteristic
74C2×#1039C2311yescharacteristic
75C5×C22⋊D5—22no—
76C5×C22⋊D5D414yescomplemented
77C2×C5×Dic5—22no—
78C2×C5×Dic5C2311yescharacteristic
79C2×C5×Dic5C2311yescharacteristic
80C2×C5×Dic5C2311yescharacteristic
81C2×C5×Dic5C2311yescharacteristic
82C2×C5×Dic5C2311yescharacteristic
83C5×D20—22no—
84C4×C5×D5—22no—
85#6039—22no—
86#6038—22no—
87#6038—22no—
88#6038—41no—
89#6036—22no—
90D5×Dic5—22no—
91D5×Dic5—41no—
92C22×C5×D4—51no—
93C2×#711—51no—
94C2×D4×D5D511yescharacteristic, complemented
95C2×#709—51no—
96C5×#123—51no—
97#4099—51no—
98#4099—51no—
99C2×C22⋊Dic5—51no—
100C2×C22⋊Dic5—51no—
101#4085—51no—
102#4085—51no—
103#4085—51no—
104#4085—102no—
105#4084—51no—
106#4082—51no—
107#4081—51no—
108#4081D511yescharacteristic, complemented
109D4×Dic5—51no—
110C4×C22⋊D5—51no—
111C2×#685—51no—
112#4070—51no—
113#4042—51no—
114#4041—51no—
115#4040—51no—
116#4039—51no—
117#4031—51no—
118#4030—51no—
119#4028—51no—
120#4028—51no—
121#4027—51no—
122D5×C22⋊C4—51no—
123C22×C52—21no—
124C22×C52—21no—
125C22×C52—21no—
126C22×C52—21no—
127C22×C52—21no—
128C22×C52C2×D412yes—
129C22×C52C2411yescharacteristic
130C2×C5⋊D5—21no—
131C2×C5⋊D5—21no—
132C2×C5⋊D5—21no—
133C2×C5×D5—21no—
134C2×C5×D5—21no—
135C2×C5×D5—21no—
136C2×C5×D5—21no—
137C2×C5×D5—21no—
138C2×C5×D5—21no—
139C2×C5×D5—21no—
140C2×C5×D5—44no—
141C2×C5×D5—44no—
142C2×C5×D5C2×D412yes—
143C2×C5×D5#4112yescomplemented
144C4×C52—21no—
145#1039—21no—
146#1039—21no—
147#1039—21no—
148C5×Dic5—21no—
149C5×Dic5—21no—
150C5×Dic5—21no—
151C5×Dic5—21no—
152C5×Dic5C2×D412yes—
153C24×C5—51no—
154C24×C5—51no—
155C23×D5C2×D511yescharacteristic
156C23×D5C2×D511yescharacteristic
157C2×C5×D4—22no—
158C2×C5×D4—51no—
159C2×C5×D4—51no—
160C2×C5×D4—51no—
161C2×C5×D4—104no—
162C2×C5×D4C2×D511yescharacteristic
163C2×C5×D4C2×D511yescharacteristic
164C22×C4×C5—51no—
165C22×C4×C5—51no—
166C2×C22⋊D5—51no—
167C2×C22⋊D5—51no—
168C2×C22⋊D5—51no—
169C2×C22⋊D5—51no—
170C2×C22⋊D5—51no—
171C2×C22⋊D5—51no—
172C2×C22⋊D5—51no—
173C2×C22⋊D5—102no—
174C2×C22⋊D5—102no—
175C2×C22⋊D5C2×D511yescharacteristic
176C2×C22⋊D5C2×D511yescharacteristic
177C22×Dic5—51no—
178C22×Dic5—51no—
179C22×Dic5—51no—
180C22×Dic5—51no—
181C22×Dic5—102no—
182C22×Dic5C2×D511yescharacteristic, complemented
183#711—104no—
184D4×D5—24no—
185#709—104no—
186C2×D20—51no—
187C2×D20—51no—
188C2×D20—51no—
189C2×D20C2×D511yescharacteristic
190C2×C4×D5—51no—
191C2×C4×D5—51no—
192C2×C4×D5—51no—
193C2×C4×D5—51no—
194C2×C4×D5C2×D511yescharacteristic
195C2×Dic10—51no—
196C2×Dic10—51no—
197C5×C4⋊C4—51no—
198C5×C4⋊C4—51no—
199C5×C22⋊C4—51no—
200C5×C22⋊C4—51no—
201C5×C22⋊C4—51no—
202C22⋊Dic5—51no—
203C22⋊Dic5—51no—
204C22⋊Dic5—51no—
205C22⋊Dic5—51no—
206C22⋊Dic5—51no—
207C22⋊Dic5—51no—
208C22⋊Dic5—51no—
209C22⋊Dic5—51no—
210C22⋊Dic5—51no—
211C22⋊Dic5—102no—
212C22⋊Dic5C2×D511yescharacteristic, complemented
213C22⋊Dic5C2×D511yescharacteristic, complemented
214C22⋊Dic5C2×D511yescharacteristic, complemented
215#685—51no—
216#685—51no—
217#685—51no—
218#685—51no—
219#685—51no—
220#685—51no—
221#685—51no—
222#685—51no—
223#685—51no—
224#685—102no—
225C4⋊Dic5—51no—
226#683—51no—
227#683—51no—
228#683—51no—
229#683—102no—
230#683C2×D511yescharacteristic, complemented
231#683C2×D511yescharacteristic, complemented
232C4×Dic5—51no—
233C4×Dic5—51no—
234#546—251noSylow
235C2×C52—22no—
236C2×C52—41no—
237C2×C52C22×D411yescharacteristic
238C2×C52C2×#4111yescharacteristic
239C2×C52H(5,𝔽2)11yescharacteristic
240C5⋊D5—41no—
241C5×D5—22no—
242C5×D5—22no—
243C5×D5—41no—
244C23×C5—22no—
245C23×C5—22no—
246C23×C5—51no—
247C23×C5—51no—
248C23×C5—51no—
249C23×C5—102no—
250C23×C5—102no—
251C23×C5—102no—
252C23×C5—102no—
253C23×C5—201no—
254C23×C5—201no—
255C23×C5C22×D511yescharacteristic
256C23×C5C22×D511yescharacteristic
257C23×C5C22×D511yescharacteristic
258C23×C5C22×D511yescharacteristic
259C22×D5—22no—
260C22×D5—22no—
261C22×D5—22no—
262C22×D5—22no—
263C22×D5—41no—
264C22×D5—41no—
265C22×D5—51no—
266C22×D5—51no—
267C22×D5—102no—
268C22×D5C22×D511yescharacteristic
269C22×D5C22×D511yescharacteristic
270C22×D5C22×D511yescharacteristic
271C5×D4—22no—
272C5×D4—22no—
273C5×D4—44no—
274C5×D4—54no—
275C5×D4—102no—
276C5×D4—102no—
277C2×C4×C5—22no—
278C2×C4×C5—51no—
279C2×C4×C5—51no—
280C2×C4×C5—51no—
281C2×C4×C5—51no—
282C2×C4×C5—51no—
283C2×C4×C5—102no—
284C2×C4×C5—102no—
285C2×C4×C5C22×D511yescharacteristic
286C2×C4×C5C22×D511yescharacteristic
287C22⋊D5—22no—
288C22⋊D5—102no—
289C22⋊D5—102no—
290C22⋊D5—102no—
291C22⋊D5—102no—
292C22⋊D5—204no—
293C22⋊D5—201no—
294C22⋊D5—201no—
295C22⋊D5—201no—
296C22⋊D5—204no—
297C22⋊D5C22⋊D514yescomplemented
298C2×Dic5—22no—
299C2×Dic5—51no—
300C2×Dic5—51no—
301C2×Dic5—51no—
302C2×Dic5—51no—
303C2×Dic5—51no—
304C2×Dic5—51no—
305C2×Dic5—102no—
306C2×Dic5—102no—
307C2×Dic5—102no—
308C2×Dic5—102no—
309C2×Dic5—201no—
310C2×Dic5—201no—
311C2×Dic5—201no—
312C2×Dic5—204no—
313C2×Dic5C22×D511yescharacteristic
314C2×Dic5C22×D511yescharacteristic
315C2×Dic5C22×D511yescharacteristic
316C2×Dic5C22×D511yescharacteristic
317C2×Dic5C22×D511yescharacteristic
318D20—22no—
319D20—102no—
320D20—102no—
321D20—201no—
322C4×D5—22no—
323C4×D5—102no—
324C4×D5—102no—
325C4×D5—102no—
326C4×D5—201no—
327Dic10—102no—
328Dic10—102no—
329C2×#41—251no—
330C22×D4—51no—
331#124—251no—
332#123—51no—
333#123—251no—
334C22⋊Q8—251no—
335#121—251no—
336#121—251no—
337#121—251no—
338#120—251no—
339#120—251no—
340C4×D4—251no—
341C4×D4—251no—
342C2×C22⋊C4—251no—
343C2×C22⋊C4—251no—
344C52#54611yescharacteristic, complemented, Sylow
345C22×C5—21no—
346C22×C5—21no—
347C22×C5—21no—
348C22×C5—21no—
349C22×C5—21no—
350C22×C5—21no—
351C22×C5—21no—
352C22×C5—21no—
353C22×C5—21no—
354C22×C5—21no—
355C22×C5—24no—
356C22×C5—22no—
357C22×C5—42no—
358C22×C5—42no—
359C22×C5—42no—
360C22×C5—42no—
361C22×C5—42no—
362C22×C5—52no—
363C22×C5—52no—
364C22×C5—101no—
365C22×C5—101no—
366C22×C5—101no—
367C22×C5—101no—
368C22×C5—101no—
369C22×C5—101no—
370C22×C5—101no—
371C22×C5—204no—
372C22×C5—204no—
373C22×C5D4×D512yes—
374C22×C5C2×C22⋊D512yes—
375C22×C5C23×D511yescharacteristic
376C22×C5C23×D511yescharacteristic
377C2×D5—21no—
378C2×D5—21no—
379C2×D5—21no—
380C2×D5—21no—
381C2×D5—21no—
382C2×D5—21no—
383C2×D5—21no—
384C2×D5—44no—
385C2×D5—44no—
386C2×D5—101no—
387C2×D5—101no—
388C2×D5—101no—
389C2×D5—101no—
390C2×D5—101no—
391C2×D5—101no—
392C2×D5—202no—
393C2×D5—202no—
394C2×D5—202no—
395C2×D5#71112yescomplemented
396C2×D5C2×C22⋊D512yes—
397C4×C5—21no—
398C4×C5—21no—
399C4×C5—42no—
400C4×C5—52no—
401C4×C5—101no—
402C4×C5—101no—
403C4×C5—101no—
404C4×C5—101no—
405Dic5—21no—
406Dic5—21no—
407Dic5—21no—
408Dic5—21no—
409Dic5—104no—
410Dic5—101no—
411Dic5—101no—
412Dic5—101no—
413Dic5—101no—
414Dic5—101no—
415Dic5—101no—
416Dic5—202no—
417Dic5—202no—
418Dic5C2×C22⋊D512yes—
419C24—51no—
420C24—51no—
421#41—504no—
422C2×Q8—251no—
423C2×D4—51no—
424C2×D4—51no—
425C2×D4—51no—
426C2×D4—104no—
427C2×D4—251no—
428C2×D4—251no—
429C2×D4—251no—
430C2×D4—251no—
431C2×D4—251no—
432C2×D4D5211yescharacteristic
433C22×C4—51no—
434C22×C4—51no—
435C22×C4—251no—
436C22×C4—251no—
437C22×C4—251no—
438C22×C4—251no—
439C4⋊C4—51no—
440C4⋊C4—51no—
441C4⋊C4—251no—
442C4⋊C4—251no—
443C22⋊C4—51no—
444C22⋊C4—51no—
445C22⋊C4—51no—
446C22⋊C4—251no—
447C22⋊C4—251no—
448C22⋊C4—251no—
449C22⋊C4—251no—
450C22⋊C4—251no—
451C22⋊C4—251no—
452C22⋊C4—251no—
453C22⋊C4—251no—
454C22⋊C4—251no—
455C42—251no—
456C2×C5—22no—
457C2×C5—22no—
458C2×C5—22no—
459C2×C5—22no—
460C2×C5—22no—
461C2×C5—44no—
462C2×C5—41no—
463C2×C5—41no—
464C2×C5—82no—
465C2×C5—102no—
466C2×C5—102no—
467C2×C5—201no—
468C2×C5C2×D4×D511yescharacteristic
469C2×C5C2×#71111yescharacteristic
470C2×C5#414411yescharacteristic
471C2×C5D5×#4111yescharacteristic
472C2×C5#414911yescharacteristic
473C2×C5C22×C22⋊D511yescharacteristic
474D5—22no—
475D5—22no—
476D5—41no—
477D5—201no—
478D5—201no—
479D5—402no—
480C23—51no—
481C23—51no—
482C23—51no—
483C23—102no—
484C23—102no—
485C23—102no—
486C23—102no—
487C23—201no—
488C23—201no—
489C23—251no—
490C23C2×D5211yescharacteristic
491C23C2×D5211yescharacteristic
492Q8—502no—
493D4—22no—
494D4—54no—
495D4—102no—
496D4—102no—
497D4—502no—
498D4—502no—
499D4—502no—
500D4—1001no—
501D4—1001no—
502C2×C4—51no—
503C2×C4—51no—
504C2×C4—51no—
505C2×C4—51no—
506C2×C4—51no—
507C2×C4—102no—
508C2×C4—102no—
509C2×C4—251no—
510C2×C4—251no—
511C2×C4—251no—
512C2×C4—502no—
513C2×C4—502no—
514C2×C4—1001no—
515C2×C4—1001no—
516C2×C4C2×D5211yescharacteristic
517C5—22no—
518C5#6598811yescharacteristic, complemented
519C5#6614011yescharacteristic, complemented
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—52no—
526C22—52no—
527C22—101no—
528C22—101no—
529C22—101no—
530C22—101no—
531C22—101no—
532C22—101no—
533C22—101no—
534C22—204no—
535C22—204no—
536C22—501no—
537C22—501no—
538C22—501no—
539C22D5×C22⋊D512yes—
540C22C22×D5211yescharacteristic
541C4—21no—
542C4—52no—
543C4—101no—
544C4—101no—
545C4—101no—
546C4—101no—
547C4—501no—
548C4—501no—
549C4—501no—
550C2—22no—
551C2—41no—
552C2—102no—
553C2—102no—
554C2—201no—
555C2—1001no—
556C2D5×#71111yescharacteristic
557C2#1171747911yescharacteristic
558C2C2×D5×C22⋊D511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC2×C52×D4C22
hypercenterC2×D4D52
hypocenterC52#546
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C5×D4×D5C2semidirect
2. C2×#87879C2semidirect
3. #11717259C2semidirect
4. C5×#4081C2semidirect
5. #11717095C2semidirect
6. #11717095C2semidirect
7. #11717091C2semidirect
8. D5×C22⋊Dic5C2semidirect
9. C2×#87778C2semidirect
10. #11717077C2semidirect
11. #11717076C2semidirect
12. #11717022C2semidirect
13. #11717005C2semidirect
14. #11717002C2semidirect
15. #11716969C2semidirect
19. C2×C22⋊(C5⋊D5)C22semidirect
20. C2×C22⋊(C5⋊D5)C22semidirect
21. C22×#1039C22semidirect
24. C22×C5×Dic5C22semidirect
28. C2×#6039C22semidirect
30. C2×#6038C22semidirect
31. C2×#6038C22semidirect
32. C2×#6036C22semidirect
34. C2×D5×Dic5C22semidirect
35. C2×D5×Dic5C22semidirect
36. C22⋊#1039C22semidirect
37. #87808C22semidirect
38. #87806C22semidirect
39. C5×C22⋊Dic5C22semidirect
40. C5×C22⋊Dic5C22semidirect
41. C5×C22⋊Dic5C22semidirect
42. C5×#683C22semidirect
43. C5×#683C22semidirect
44. #87780C22semidirect
48. #87778C22semidirect
49. #87778C22semidirect
50. #87778C22semidirect
51. #87778C22semidirect
52. Dic52C22semidirect
76. C5×C22⋊D5D4semidirect
94. C2×D4×D5D5semidirect
108. #4081D5semidirect
143. C2×C5×D5#41semidirect
182. C22×Dic5C2×D5semidirect
212. C22⋊Dic5C2×D5semidirect
213. C22⋊Dic5C2×D5semidirect
214. C22⋊Dic5C2×D5semidirect
230. #683C2×D5semidirect
231. #683C2×D5semidirect
297. C22⋊D5C22⋊D5semidirect
344. C52#546semidirect
395. C2×D5#711semidirect
518. C5#65988semidirect
519. C5#66140semidirect

Maximal subgroups

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

subgroupindex
C2×C5×D4×D52
C2×#878792
#117172592
C5×#40812
#117170952
#117170912
D5×C22⋊Dic52
C2×#877782
#117170772
#117170762
#117170222
#117170052
#117170022
#117169692
#661405
#659885