Finite Group Databasebeta

← #13106594

Subgroups — #13106594

Order 1600.

628 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊#1039C211yesmaximal, characteristic, complemented
2C5×#4099C211yesmaximal, characteristic
3C2×C5×C22⋊Dic5C211yesmaximal, characteristic, complemented
4#11717100C211yesmaximal, characteristic, complemented
5#11717091C211yesmaximal, characteristic, complemented
6Dic5×C22⋊D5C211yesmaximal, characteristic, complemented
7#11717087C211yesmaximal, characteristic, complemented
8#11717082C211yesmaximal, characteristic, complemented
9#11717080C211yesmaximal, characteristic, complemented
10#11717079C211yesmaximal, characteristic, complemented
11#11717077C211yesmaximal, characteristic, complemented
12#11717075C211yesmaximal, characteristic, complemented
13#11717074C211yesmaximal, characteristic, complemented
14#11717074C211yesmaximal, characteristic, complemented
15#11717073C211yesmaximal, characteristic, complemented
16C24×C52C2211yescharacteristic
17C22×#1039C2211yescharacteristic
18C22×#1039C2211yescharacteristic
19C2×C5×C22⋊D5C2211yescharacteristic
20C2×C5×C22⋊D5C2211yescharacteristic
21C2×C5×C22⋊D5C2211yescharacteristic
22C22×C5×Dic5C2211yescharacteristic
23C22×C5×Dic5C2211yescharacteristic
24C2×#6039C2211yescharacteristic, complemented
25C2×D5×Dic5C2211yescharacteristic, complemented
26C22⋊#1039C2211yescharacteristic
27C22⋊#1039C2211yescharacteristic
28C22⋊#1039C2211yescharacteristic
29C22⋊#1039C2211yescharacteristic
30C5×C22⋊Dic5C2211yescharacteristic
31C5×C22⋊Dic5C2211yescharacteristic
32C5×C22⋊Dic5C2211yescharacteristic
33C5×C22⋊Dic5C2211yescharacteristic
34C5×C22⋊Dic5C2211yescharacteristic
35C5×C22⋊Dic5C2211yescharacteristic
36C5×C22⋊Dic5C2211yescharacteristic
37#87781C2211yescharacteristic, complemented
38#87781C2211yescharacteristic, complemented
39#87781C2211yescharacteristic, complemented
40#87780C2211yescharacteristic, complemented
41#87780C2211yescharacteristic, complemented
42#87780C2211yescharacteristic, complemented
43#87780C2211yescharacteristic, complemented
44#87780C2211yescharacteristic, complemented
45#87778C2211yescharacteristic, complemented
46#87778C2211yescharacteristic, complemented
47#87778C2211yescharacteristic, complemented
48Dic52C2211yescharacteristic, complemented
49Dic52C2211yescharacteristic, complemented
50Dic52C2211yescharacteristic, complemented
51#65921—51nomaximal
52#65829—51nomaximal
53C23×C52—22no—
54C23×C52—22no—
55C23×C52—41no—
56C23×C52C2311yescharacteristic
57C23×C52C2311yescharacteristic
58C23×C52C2311yescharacteristic
59C22×C5×D5C2311yescharacteristic
60C2×#1039—22no—
61C2×#1039—41no—
62C2×#1039C2311yescharacteristic
63C2×#1039C2311yescharacteristic
64C2×#1039C2311yescharacteristic
65C2×#1039C2311yescharacteristic
66C5×C22⋊D5—22no—
67C5×C22⋊D5—22no—
68C5×C22⋊D5—41no—
69C2×C5×Dic5—22no—
70C2×C5×Dic5—41no—
71C2×C5×Dic5C2311yescharacteristic
72C2×C5×Dic5C2311yescharacteristic
73C2×C5×Dic5C2311yescharacteristic
74C2×C5×Dic5C2311yescharacteristic
75C2×C5×Dic5C2311yescharacteristic
76C2×C5×Dic5C2311yescharacteristic
77C2×C5×Dic5C2311yescharacteristic
78#6039—41no—
79D5×Dic5—41no—
80C5×#120—51no—
81C2×C5×C22⋊C4—51no—
82#4099D511yescharacteristic
83C2×C22⋊Dic5—51no—
84C2×C22⋊Dic5—51no—
85C2×C22⋊Dic5—102no—
86C2×C22⋊Dic5D511yescharacteristic, complemented
87#4082—51no—
88#4081—51no—
89#4081—51no—
90#4081—51no—
91D4×Dic5—51no—
92D4×Dic5—51no—
93#4075—51no—
94C4×C22⋊D5—51no—
95#4072—51no—
96#4070—51no—
97#4032—51no—
98#4031—51no—
99#4029—51no—
100#4027—51no—
101#4025—51no—
102#4025—51no—
103#4025—51no—
104#4025—51no—
105#4024—51no—
106#4024—51no—
107#4024—51no—
108#4023—51no—
109#4023—51no—
110#4023—51no—
111C22×C52—21no—
112C22×C52—21no—
113C22×C52—21no—
114C22×C52—21no—
115C22×C52—21no—
116C22×C52—21no—
117C22×C52—21no—
118C22×C52—44no—
119C22×C52#4112yes—
120C22×C52#4112yes—
121C22×C52C2411yescharacteristic
122C2×C5×D5—21no—
123C2×C5×D5—21no—
124C2×C5×D5—21no—
125#1039—21no—
126#1039—21no—
127#1039—21no—
128#1039—21no—
129C5×Dic5—21no—
130C5×Dic5—21no—
131C5×Dic5—21no—
132C5×Dic5—21no—
133C5×Dic5—21no—
134C5×Dic5—21no—
135C5×Dic5—21no—
136C24×C5—22no—
137C24×C5C2×D511yescharacteristic
138C24×C5C2×D511yescharacteristic
139C2×C5×D4—51no—
140C2×C5×D4—51no—
141C2×C5×D4—51no—
142C22×C4×C5—51no—
143C22×C4×C5—51no—
144C2×C22⋊D5C2×D511yescharacteristic
145C2×C22⋊D5C2×D511yescharacteristic
146C2×C22⋊D5C2×D511yescharacteristic
147C22×Dic5—51no—
148C22×Dic5—51no—
149C22×Dic5—51no—
150C22×Dic5—51no—
151C22×Dic5—51no—
152C22×Dic5—102no—
153C22×Dic5—102no—
154C22×Dic5C2×D511yescharacteristic
155C22×Dic5C2×D511yescharacteristic
156C2×C4×D5—51no—
157C2×Dic10—51no—
158C2×Dic10—51no—
159C5×C22⋊C4—51no—
160C5×C22⋊C4—51no—
161C5×C22⋊C4—51no—
162C5×C22⋊C4—51no—
163C5×C22⋊C4—51no—
164C5×C22⋊C4—51no—
165C5×C22⋊C4—51no—
166C22⋊Dic5—51no—
167C22⋊Dic5—51no—
168C22⋊Dic5—51no—
169C22⋊Dic5—51no—
170C22⋊Dic5—51no—
171C22⋊Dic5—51no—
172C22⋊Dic5—51no—
173C22⋊Dic5—51no—
174C22⋊Dic5—51no—
175C22⋊Dic5—51no—
176C22⋊Dic5—51no—
177C22⋊Dic5—102no—
178C22⋊Dic5—102no—
179C22⋊Dic5—102no—
180C22⋊Dic5—102no—
181C22⋊Dic5C2×D511yescharacteristic
182C22⋊Dic5C2×D511yescharacteristic
183C22⋊Dic5C2×D511yescharacteristic
184C22⋊Dic5C2×D511yescharacteristic
185C22⋊Dic5C2×D511yescharacteristic
186C22⋊Dic5C2×D511yescharacteristic
187C22⋊Dic5C2×D511yescharacteristic
188#685—51no—
189#685—51no—
190#685—51no—
191C4⋊Dic5—51no—
192C4⋊Dic5—51no—
193C4⋊Dic5—51no—
194C4⋊Dic5—51no—
195C4⋊Dic5—51no—
196#683—51no—
197#683—51no—
198#683—51no—
199#683—51no—
200#683—51no—
201#683—51no—
202#683—51no—
203#683—51no—
204#683—51no—
205#683—51no—
206#683—51no—
207C4×Dic5—51no—
208C4×Dic5—51no—
209C4×Dic5—51no—
210C4×Dic5—51no—
211C4×Dic5—51no—
212C4×Dic5—51no—
213#551—251noSylow
214C2×C52—22no—
215C2×C52—22no—
216C2×C52—41no—
217C2×C52C2×#4111yescharacteristic
218C2×C52C2×#4111yescharacteristic
219C2×C52H(5,𝔽2)11yescharacteristic
220C5×D5—41no—
221C23×C5—28no—
222C23×C5—22no—
223C23×C5—22no—
224C23×C5—22no—
225C23×C5—22no—
226C23×C5—22no—
227C23×C5—22no—
228C23×C5—22no—
229C23×C5—44no—
230C23×C5—41no—
231C23×C5—41no—
232C23×C5—44no—
233C23×C5—51no—
234C23×C5C22×D511yescharacteristic
235C23×C5C22×D511yescharacteristic
236C23×C5C22×D511yescharacteristic
237C23×C5C22×D511yescharacteristic
238C23×C5C22×D511yescharacteristic
239C23×C5C22×D511yescharacteristic
240C22×D5C22×D511yescharacteristic
241C5×D4—102no—
242C5×D4—102no—
243C5×D4—201no—
244C2×C4×C5—51no—
245C2×C4×C5—51no—
246C2×C4×C5—51no—
247C2×C4×C5—51no—
248C2×C4×C5—51no—
249C2×C4×C5—51no—
250C2×C4×C5—51no—
251C2×C4×C5—102no—
252C2×C4×C5—201no—
253C22⋊D5—22no—
254C22⋊D5—22no—
255C22⋊D5—41no—
256C2×Dic5—22no—
257C2×Dic5—41no—
258C2×Dic5—51no—
259C2×Dic5—51no—
260C2×Dic5—51no—
261C2×Dic5—51no—
262C2×Dic5—51no—
263C2×Dic5—51no—
264C2×Dic5—51no—
265C2×Dic5—51no—
266C2×Dic5—102no—
267C2×Dic5—102no—
268C2×Dic5—102no—
269C2×Dic5—102no—
270C2×Dic5—102no—
271C2×Dic5—102no—
272C2×Dic5—204no—
273C2×Dic5—201no—
274C2×Dic5—201no—
275C2×Dic5—201no—
276C2×Dic5—204no—
277C2×Dic5C22×D511yescharacteristic
278C2×Dic5C22×D511yescharacteristic
279C2×Dic5C22×D511yescharacteristic
280C2×Dic5C22×D511yescharacteristic
281C2×Dic5C22×D511yescharacteristic
282C2×Dic5C22×D511yescharacteristic
283C2×Dic5C22×D511yescharacteristic
284C4×D5—201no—
285Dic10—201no—
286Dic10—201no—
287#126—251no—
288#126—251no—
289#124—251no—
290#123—251no—
291#123—251no—
292#123—251no—
293C22⋊Q8—251no—
294C22⋊Q8—251no—
295#120—51no—
296C4×D4—251no—
297C4×D4—251no—
298#117—251no—
299#117—251no—
300C2×C22⋊C4—51no—
301C2×C22⋊C4—251no—
302C52#55111yescharacteristic, complemented, Sylow
303C22×C5—24no—
304C22×C5—24no—
305C22×C5—21no—
306C22×C5—21no—
307C22×C5—21no—
308C22×C5—21no—
309C22×C5—21no—
310C22×C5—21no—
311C22×C5—21no—
312C22×C5—21no—
313C22×C5—24no—
314C22×C5—21no—
315C22×C5—21no—
316C22×C5—21no—
317C22×C5—21no—
318C22×C5—21no—
319C22×C5—21no—
320C22×C5—24no—
321C22×C5—24no—
322C22×C5—22no—
323C22×C5—42no—
324C22×C5—42no—
325C22×C5—42no—
326C22×C5—42no—
327C22×C5—44no—
328C22×C5—44no—
329C22×C5—416no—
330C22×C5—101no—
331C22×C5—101no—
332C22×C5—101no—
333C22×C5#70912yes—
334C22×C5#71112yes—
335C22×C5#71112yes—
336C22×C5#71112yes—
337C22×C5C23×D511yescharacteristic
338C22×C5C23×D511yescharacteristic
339C2×D5—21no—
340C2×D5—21no—
341C2×D5—21no—
342C4×C5—101no—
343C4×C5—101no—
344C4×C5—101no—
345C4×C5—101no—
346C4×C5—101no—
347C4×C5—101no—
348C4×C5—101no—
349Dic5—21no—
350Dic5—21no—
351Dic5—21no—
352Dic5—21no—
353Dic5—21no—
354Dic5—21no—
355Dic5—21no—
356Dic5—101no—
357Dic5—101no—
358Dic5—101no—
359Dic5—101no—
360Dic5—101no—
361Dic5—101no—
362Dic5—101no—
363Dic5—101no—
364Dic5—202no—
365Dic5—202no—
366Dic5—202no—
367Dic5—202no—
368C24D5211yescharacteristic
369C2×Q8—251no—
370C2×D4—51no—
371C2×D4—51no—
372C2×D4—51no—
373C22×C4—51no—
374C22×C4—51no—
375C22×C4—251no—
376C22×C4—251no—
377C22×C4—251no—
378C4⋊C4—251no—
379C4⋊C4—251no—
380C4⋊C4—251no—
381C4⋊C4—251no—
382C4⋊C4—251no—
383C4⋊C4—251no—
384C4⋊C4—251no—
385C4⋊C4—251no—
386C22⋊C4—51no—
387C22⋊C4—51no—
388C22⋊C4—51no—
389C22⋊C4—51no—
390C22⋊C4—51no—
391C22⋊C4—51no—
392C22⋊C4—51no—
393C22⋊C4—251no—
394C22⋊C4—251no—
395C22⋊C4—251no—
396C22⋊C4—251no—
397C22⋊C4—251no—
398C22⋊C4—251no—
399C22⋊C4—251no—
400C42—251no—
401C42—251no—
402C42—251no—
403C2×C5—28no—
404C2×C5—22no—
405C2×C5—22no—
406C2×C5—22no—
407C2×C5—22no—
408C2×C5—22no—
409C2×C5—22no—
410C2×C5—22no—
411C2×C5—44no—
412C2×C5—44no—
413C2×C5—41no—
414C2×C5—41no—
415C2×C5—201no—
416C2×C5C2×#70911yescharacteristic
417C2×C5C2×#71111yescharacteristic
418C2×C5C2×#71111yescharacteristic
419C2×C5C2×#71111yescharacteristic
420C2×C5#414411yescharacteristic
421C2×C5#414411yescharacteristic
422D5—41no—
423C23—22no—
424C23—22no—
425C23—41no—
426C23—51no—
427C23C2×D5211yescharacteristic
428C23C2×D5211yescharacteristic
429C23C2×D5211yescharacteristic
430Q8—1001no—
431D4—102no—
432D4—102no—
433D4—201no—
434C2×C4—51no—
435C2×C4—51no—
436C2×C4—51no—
437C2×C4—51no—
438C2×C4—51no—
439C2×C4—51no—
440C2×C4—51no—
441C2×C4—102no—
442C2×C4—201no—
443C2×C4—251no—
444C2×C4—251no—
445C2×C4—251no—
446C2×C4—251no—
447C2×C4—502no—
448C2×C4—1001no—
449C2×C4—1001no—
450C5—22no—
451C5#6582911yescharacteristic, complemented
452C5#6592111yescharacteristic, complemented
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—44no—
461C22—101no—
462C22—101no—
463C22—101no—
464C22#8787912yes—
465C22#8788012yes—
466C22C22×D5211yescharacteristic
467C4—101no—
468C4—101no—
469C4—101no—
470C4—101no—
471C4—101no—
472C4—101no—
473C4—101no—
474C4—501no—
475C4—501no—
476C4—501no—
477C4—501no—
478C2—22no—
479C2—22no—
480C2—41no—
481C2—201no—
482C2C2×#8787911yescharacteristic
483C2C2×#8788011yescharacteristic
484C2#1171749511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC24×C52C22
prime radicalC5×#4099C2
hypercenterC24D52
hypocenterC52#551
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C2×C22⋊#1039C2semidirect
3. C2×C5×C22⋊Dic5C2semidirect
4. #11717100C2semidirect
5. #11717091C2semidirect
6. Dic5×C22⋊D5C2semidirect
7. #11717087C2semidirect
8. #11717082C2semidirect
9. #11717080C2semidirect
10. #11717079C2semidirect
11. #11717077C2semidirect
12. #11717075C2semidirect
13. #11717074C2semidirect
14. #11717074C2semidirect
15. #11717073C2semidirect
24. C2×#6039C22semidirect
25. C2×D5×Dic5C22semidirect
37. #87781C22semidirect
38. #87781C22semidirect
39. #87781C22semidirect
40. #87780C22semidirect
41. #87780C22semidirect
42. #87780C22semidirect
43. #87780C22semidirect
44. #87780C22semidirect
45. #87778C22semidirect
46. #87778C22semidirect
47. #87778C22semidirect
48. Dic52C22semidirect
49. Dic52C22semidirect
50. Dic52C22semidirect
86. C2×C22⋊Dic5D5semidirect
302. C52#551semidirect
451. C5#65829semidirect
452. C5#65921semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊#10392
C5×#40992
C2×C5×C22⋊Dic52
#117171002
#117170912
Dic5×C22⋊D52
#117170872
#117170822
#117170802
#117170792
#117170772
#117170752
#117170742
#117170732
#659215
#658295