Finite Group Databasebeta

← #13106204

Subgroups — #13106204

Order 1600.

678 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#711C211yesmaximal, characteristic, complemented
2C2×#87879C211yesmaximal, characteristic, complemented
3#11717259C211yesmaximal, characteristic, complemented
4C5×#4081C211yesmaximal, characteristic, complemented
5#11717091C211yesmaximal, characteristic, complemented
6#11717089C211yesmaximal, characteristic, complemented
7#11717087C211yesmaximal, characteristic, complemented
8C2×#87781C211yesmaximal, characteristic, complemented
9#11717084C211yesmaximal, characteristic, complemented
10#11717082C211yesmaximal, characteristic, complemented
11#11717070C211yesmaximal, characteristic, complemented
12#11717008C211yesmaximal, characteristic, complemented
13D5×#683C211yesmaximal, characteristic, complemented
14#11716968C211yesmaximal, characteristic, complemented
15#11716957C211yesmaximal, characteristic, complemented
16C2×C52×D4C2211yescharacteristic
17C2×C22⋊(C5⋊D5)C2211yescharacteristic
18C2×C22⋊(C5⋊D5)C2211yescharacteristic
19C22×#1039C2211yescharacteristic
20C2×C5×C22⋊D5C2211yescharacteristic
21C2×C5×C22⋊D5C2211yescharacteristic
22C22×C5×Dic5C2211yescharacteristic
23C22×C5×Dic5C2211yescharacteristic
24C22×C5×Dic5C2211yescharacteristic
25C5×#711—24no—
26C2×C4×C5×D5C2211yescharacteristic
27C2×C5×Dic10C2211yescharacteristic
28C2×#6039C2211yescharacteristic
29C2×#6038C2211yescharacteristic
30C2×#6036C2211yescharacteristic
31#87879—24no—
32C2×D5×Dic5C2211yescharacteristic
33C2×D5×Dic5C2211yescharacteristic
34C22⋊#1039C2211yescharacteristic
35#87808C2211yescharacteristic
36#87806C2211yescharacteristic
37C5×C22⋊Dic5C2211yescharacteristic
38C5×C22⋊Dic5C2211yescharacteristic
39C5×C22⋊Dic5C2211yescharacteristic
40C5×#683C2211yescharacteristic
41C5×#683C2211yescharacteristic
42#87781C2211yescharacteristic
43#87781C2211yescharacteristic
44#87781C2211yescharacteristic
45#87781C2211yescharacteristic
46#87780C2211yescharacteristic
47#87780C2211yescharacteristic
48#87780C2211yescharacteristic
49#87779C2211yescharacteristic
50#87779C2211yescharacteristic
51#87778C2211yescharacteristic
52Dic52C2211yescharacteristic
53#66158—51nomaximal
54#65989—51nomaximal
55C23×C52C2311yescharacteristic
56C23×C52C2311yescharacteristic
57C22×C5⋊D5C2311yescharacteristic
58C22×C5×D5C2311yescharacteristic
59C52×D4—22no—
60C2×C4×C52C2311yescharacteristic
61C22⋊(C5⋊D5)—22no—
62C22⋊(C5⋊D5)—22no—
63C2×#1039—22no—
64C2×#1039C2311yescharacteristic
65C2×#1039C2311yescharacteristic
66C2×#1039C2311yescharacteristic
67C5×C22⋊D5—22no—
68C5×C22⋊D5D414yescomplemented
69C2×C5×Dic5—22no—
70C2×C5×Dic5—22no—
71C2×C5×Dic5—22no—
72C2×C5×Dic5C2311yescharacteristic
73C2×C5×Dic5C2311yescharacteristic
74C2×C5×Dic5C2311yescharacteristic
75C2×C5×Dic5C2311yescharacteristic
76C2×C5×Dic5C2311yescharacteristic
77C2×C5×Dic5C2311yescharacteristic
78C2×C5×Dic5C2311yescharacteristic
79C4×C5×D5—22no—
80C5×Dic10—22no—
81#6039—22no—
82#6038—22no—
83#6036—22no—
84D5×Dic5—22no—
85D5×Dic5—22no—
86C2×C5×#41—51no—
87C2×#711—51no—
88C2×#711D511yescharacteristic, complemented
89C2×#709—51no—
90C5×#123—51no—
91#4092—51no—
92#4090—51no—
93#4085—51no—
94#4085—51no—
95#4085—51no—
96#4085—102no—
97#4081—51no—
98#4081D511yescharacteristic, complemented
99D4×Dic5—51no—
100#4075—51no—
101#4075—51no—
102C4×C22⋊D5—51no—
103#4070—51no—
104C2×#683—51no—
105C2×#683—51no—
106C2×#683—51no—
107#4042—51no—
108#4040—51no—
109D5×C4⋊C4—51no—
110#4034—51no—
111#4032—51no—
112#4030—51no—
113#4029—51no—
114#4027—51no—
115#4027—51no—
116#4024—51no—
117C22×C52—21no—
118C22×C52—21no—
119C22×C52—21no—
120C22×C52—21no—
121C22×C52—21no—
122C22×C52C2×D412yes—
123C22×C52C2411yescharacteristic
124C2×C5⋊D5—21no—
125C2×C5⋊D5—21no—
126C2×C5⋊D5—21no—
127C2×C5×D5—21no—
128C2×C5×D5—21no—
129C2×C5×D5C2×D412yes—
130C4×C52—21no—
131#1039—21no—
132#1039—21no—
133#1039#4112yescomplemented
134C5×Dic5—21no—
135C5×Dic5—21no—
136C5×Dic5—21no—
137C5×Dic5—21no—
138C5×Dic5—21no—
139C5×Dic5—21no—
140C5×Dic5C2×D412yes—
141C5×#41—104no—
142C2×C5×Q8—51no—
143C2×C5×D4—22no—
144C2×C5×D4—51no—
145C2×C5×D4—51no—
146C2×C5×D4C2×D511yescharacteristic
147C2×C5×D4C2×D511yescharacteristic
148C22×C4×C5—51no—
149C22×C4×C5—51no—
150C22×C4×C5—51no—
151C22×C4×C5—51no—
152C2×C22⋊D5—51no—
153C2×C22⋊D5—51no—
154C2×C22⋊D5—51no—
155C2×C22⋊D5—51no—
156C2×C22⋊D5—51no—
157C2×C22⋊D5—102no—
158C2×C22⋊D5—102no—
159C2×C22⋊D5C2×D511yescharacteristic
160C2×C22⋊D5C2×D511yescharacteristic
161C22×Dic5—51no—
162C22×Dic5—51no—
163C22×Dic5—51no—
164C22×Dic5—51no—
165C22×Dic5—102no—
166C22×Dic5C2×D511yescharacteristic
167C22×Dic5C2×D511yescharacteristic
168C22×Dic5C2×D511yescharacteristic
169#711—24no—
170#711—104no—
171#709—104no—
172C2×D20—51no—
173C2×C4×D5—51no—
174C2×C4×D5—51no—
175C2×C4×D5—51no—
176C2×C4×D5—51no—
177C2×C4×D5C2×D511yescharacteristic
178C2×Dic10—51no—
179C2×Dic10—51no—
180C2×Dic10C2×D511yescharacteristic
181C5×C4⋊C4—51no—
182C5×C4⋊C4—51no—
183C5×C22⋊C4—51no—
184C5×C22⋊C4—51no—
185C5×C22⋊C4—51no—
186C22⋊Dic5—51no—
187C22⋊Dic5—51no—
188C22⋊Dic5—51no—
189C22⋊Dic5—102no—
190C22⋊Dic5C2×D511yescharacteristic
191C22⋊Dic5C2×D511yescharacteristic
192C22⋊Dic5C2×D511yescharacteristic
193#685—51no—
194#685—51no—
195#685—51no—
196#685—51no—
197#685—51no—
198#685—51no—
199#685—51no—
200#685—102no—
201C4⋊Dic5—51no—
202C4⋊Dic5—51no—
203C4⋊Dic5—51no—
204#683—51no—
205#683—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—
214#683—51no—
215#683—51no—
216#683—51no—
217#683—102no—
218#683C2×D511yescharacteristic
219#683C2×D511yescharacteristic
220C4×Dic5—51no—
221C4×Dic5—51no—
222#547—251noSylow
223C2×C52—22no—
224C2×C52—41no—
225C2×C52C22×D411yescharacteristic
226C2×C52C2×#4111yescharacteristic
227C2×C522−1+411yescharacteristic
228C5⋊D5—41no—
229C5×D5—22no—
230C23×C5—22no—
231C23×C5—22no—
232C23×C5—51no—
233C23×C5C22×D511yescharacteristic
234C23×C5C22×D511yescharacteristic
235C23×C5C22×D511yescharacteristic
236C23×C5C22×D511yescharacteristic
237C22×D5—51no—
238C22×D5—51no—
239C22×D5—102no—
240C22×D5C22×D511yescharacteristic
241C5×Q8—102no—
242C5×D4—22no—
243C5×D4—22no—
244C5×D4—44no—
245C5×D4—54no—
246C5×D4—102no—
247C2×C4×C5—22no—
248C2×C4×C5—51no—
249C2×C4×C5—51no—
250C2×C4×C5—51no—
251C2×C4×C5—51no—
252C2×C4×C5—51no—
253C2×C4×C5—51no—
254C2×C4×C5—51no—
255C2×C4×C5—102no—
256C2×C4×C5—102no—
257C2×C4×C5—102no—
258C2×C4×C5—102no—
259C2×C4×C5C22×D511yescharacteristic
260C2×C4×C5C22×D511yescharacteristic
261C22⋊D5—22no—
262C22⋊D5—102no—
263C22⋊D5—102no—
264C22⋊D5—102no—
265C22⋊D5—102no—
266C22⋊D5—102no—
267C22⋊D5—204no—
268C22⋊D5—204no—
269C22⋊D5C22⋊D514yescomplemented
270C2×Dic5—22no—
271C2×Dic5—22no—
272C2×Dic5—22no—
273C2×Dic5—51no—
274C2×Dic5—51no—
275C2×Dic5—51no—
276C2×Dic5—51no—
277C2×Dic5—51no—
278C2×Dic5—51no—
279C2×Dic5—102no—
280C2×Dic5—102no—
281C2×Dic5—102no—
282C2×Dic5—102no—
283C2×Dic5—102no—
284C2×Dic5—102no—
285C2×Dic5—102no—
286C2×Dic5—204no—
287C2×Dic5C22×D511yescharacteristic
288C2×Dic5C22×D511yescharacteristic
289C2×Dic5C22×D511yescharacteristic
290C2×Dic5C22×D511yescharacteristic
291C2×Dic5C22×D511yescharacteristic
292C2×Dic5C22×D511yescharacteristic
293C2×Dic5C22×D511yescharacteristic
294D20—102no—
295C4×D5—22no—
296C4×D5—102no—
297C4×D5—102no—
298C4×D5—102no—
299C4×D5—102no—
300Dic10—22no—
301Dic10—102no—
302Dic10—102no—
303C2×#41—51no—
304C2×#41—251no—
305C4⋊Q8—251no—
306#123—51no—
307#123—251no—
308#123—251no—
309#123—251no—
310C22⋊Q8—251no—
311C22⋊Q8—251no—
312#121—251no—
313#121—251no—
314C4×D4—251no—
315C4×D4—251no—
316C2×C4⋊C4—251no—
317C2×C4⋊C4—251no—
318C52#54711yescharacteristic, complemented, Sylow
319C22×C5—21no—
320C22×C5—21no—
321C22×C5—21no—
322C22×C5—21no—
323C22×C5—21no—
324C22×C5—21no—
325C22×C5—21no—
326C22×C5—21no—
327C22×C5—21no—
328C22×C5—21no—
329C22×C5—24no—
330C22×C5—22no—
331C22×C5—42no—
332C22×C5—42no—
333C22×C5—42no—
334C22×C5—42no—
335C22×C5—42no—
336C22×C5—52no—
337C22×C5—101no—
338C22×C5—101no—
339C22×C5D4×D512yes—
340C22×C5C2×C22⋊D512yes—
341C22×C5C23×D511yescharacteristic
342C22×C5C23×D511yescharacteristic
343C2×D5—21no—
344C2×D5—21no—
345C2×D5—101no—
346C2×D5—101no—
347C2×D5—101no—
348C2×D5—101no—
349C2×D5—101no—
350C2×D5—101no—
351C2×D5—202no—
352C2×D5—202no—
353C2×D5—202no—
354C2×D5C2×C22⋊D512yes—
355C4×C5—21no—
356C4×C5—21no—
357C4×C5—42no—
358C4×C5—52no—
359C4×C5—101no—
360C4×C5—101no—
361C4×C5—101no—
362C4×C5—101no—
363C4×C5—101no—
364C4×C5—101no—
365Dic5—21no—
366Dic5—21no—
367Dic5—21no—
368Dic5—21no—
369Dic5—21no—
370Dic5—21no—
371Dic5—52no—
372Dic5—52no—
373Dic5—104no—
374Dic5—101no—
375Dic5—101no—
376Dic5—101no—
377Dic5—101no—
378Dic5—202no—
379Dic5—202no—
380Dic5C2×C22⋊D512yes—
381#41—104no—
382#41—504no—
383C2×Q8—51no—
384C2×Q8—251no—
385C2×D4—51no—
386C2×D4—51no—
387C2×D4—251no—
388C2×D4—251no—
389C2×D4—251no—
390C2×D4D5211yescharacteristic
391C22×C4—51no—
392C22×C4—51no—
393C22×C4—51no—
394C22×C4—51no—
395C22×C4—251no—
396C22×C4—251no—
397C22×C4—251no—
398C22×C4—251no—
399C4⋊C4—51no—
400C4⋊C4—51no—
401C4⋊C4—251no—
402C4⋊C4—251no—
403C4⋊C4—251no—
404C4⋊C4—251no—
405C4⋊C4—251no—
406C4⋊C4—251no—
407C4⋊C4—251no—
408C4⋊C4—251no—
409C22⋊C4—51no—
410C22⋊C4—51no—
411C22⋊C4—51no—
412C22⋊C4—251no—
413C22⋊C4—251no—
414C22⋊C4—251no—
415C22⋊C4—251no—
416C22⋊C4—251no—
417C42—251no—
418C2×C5—22no—
419C2×C5—22no—
420C2×C5—22no—
421C2×C5—22no—
422C2×C5—22no—
423C2×C5—44no—
424C2×C5—41no—
425C2×C5—41no—
426C2×C5—82no—
427C2×C5—102no—
428C2×C5C2×D4×D511yescharacteristic
429C2×C5D5×#4111yescharacteristic
430C2×C5D5×#4111yescharacteristic
431C2×C5#415011yescharacteristic
432C2×C5#415011yescharacteristic
433C2×C5C22×C22⋊D511yescharacteristic
434D5—22no—
435D5—201no—
436D5—201no—
437D5—402no—
438C23—51no—
439C23—251no—
440C23C2×D5211yescharacteristic
441C23C2×D5211yescharacteristic
442Q8—102no—
443Q8—502no—
444D4—22no—
445D4—54no—
446D4—102no—
447D4—502no—
448D4—502no—
449D4—502no—
450C2×C4—51no—
451C2×C4—51no—
452C2×C4—51no—
453C2×C4—51no—
454C2×C4—51no—
455C2×C4—51no—
456C2×C4—51no—
457C2×C4—102no—
458C2×C4—102no—
459C2×C4—102no—
460C2×C4—102no—
461C2×C4—251no—
462C2×C4—251no—
463C2×C4—251no—
464C2×C4—502no—
465C2×C4—502no—
466C2×C4—502no—
467C2×C4—502no—
468C2×C4C2×D5211yescharacteristic
469C5—22no—
470C5#6598911yescharacteristic, complemented
471C5#6615811yescharacteristic, complemented
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—52no—
478C22—101no—
479C22—101no—
480C22—501no—
481C22—501no—
482C22—501no—
483C22D5×C22⋊D512yes—
484C22C22×D5211yescharacteristic
485C4—21no—
486C4—52no—
487C4—101no—
488C4—101no—
489C4—101no—
490C4—101no—
491C4—101no—
492C4—101no—
493C4—252no—
494C4—501no—
495C4—501no—
496C2—22no—
497C2—41no—
498C2—102no—
499C2—1001no—
500C2#1171747411yescharacteristic
501C2#1171747711yescharacteristic
502C2C2×D5×C22⋊D511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC2×C52×D4C22
hypercenterC2×D4D52
hypocenterC52#547
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×#711C2semidirect
2. C2×#87879C2semidirect
3. #11717259C2semidirect
4. C5×#4081C2semidirect
5. #11717091C2semidirect
6. #11717089C2semidirect
7. #11717087C2semidirect
8. C2×#87781C2semidirect
9. #11717084C2semidirect
10. #11717082C2semidirect
11. #11717070C2semidirect
12. #11717008C2semidirect
13. D5×#683C2semidirect
14. #11716968C2semidirect
15. #11716957C2semidirect
68. C5×C22⋊D5D4semidirect
88. C2×#711D5semidirect
98. #4081D5semidirect
133. #1039#41semidirect
269. C22⋊D5C22⋊D5semidirect
318. C52#547semidirect
470. C5#65989semidirect
471. C5#66158semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#7112
C2×#878792
#117172592
C5×#40812
#117170912
#117170892
#117170872
C2×#877812
#117170842
#117170822
#117170702
#117170082
D5×#6832
#117169682
#117169572
#661585
#659895