Finite Group Databasebeta

← C2×#10617733 #13007570

Subgroups — C2×#10617733 #13007570

Order 1280.

7118 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×C4×C5×#41C211yesmaximal, characteristic, complemented
2C2×#66165C212yesmaximal, complemented
3#10617733C2132yesmaximal, direct product factor
4C2×Q8×D20C211yesmaximal, characteristic, complemented
5C2×#65906C212yesmaximal, complemented
6C2×Q8×Dic10C211yesmaximal, characteristic, complemented
7C2×#65895C216yesmaximal, complemented
8C2×#65876C213yesmaximal, complemented
9C2×#65897C213yesmaximal, complemented
10C2×#65867C216yesmaximal, complemented
11C2×#65813C213yesmaximal, complemented
12C2×#65804C213yesmaximal, complemented
13C22×C5×#41C2211yescharacteristic
14C22×Q8×D5C2212yes—
15C4×C5×#41C22116yescomplemented
16C2×C4×C5×Q8C2211yescharacteristic
17C2×C4×C5×Q8C2211yescharacteristic
18C2×C4×C5×D4C2213yes—
19C2×C4×C5×D4C2213yes—
20C2×C5×#117C2213yes—
21C22×C42×C5C2213yes—
22#66165C22132yescomplemented
23C2×Q8×Dic5C2212yes—
24C2×#4084C2216yes—
25C2×#4082C2216yes—
26C2×#4076C2213yes—
27C2×#4075C2216yes—
28C2×C4×C22⋊D5C2216yes—
29C2×#4072C2216yes—
30C2×#4070C2216yes—
31C2×#4070C2213yes—
32C2×#4070C2216yes—
33Q8×D20C22116yescomplemented
34#65906C22132yescomplemented
35Q8×Dic10C22116yescomplemented
36#65897C22148yescomplemented
37#65895C22196yescomplemented
38#65876C22148yescomplemented
39#65867C22196yescomplemented
40C2×#4043C2216yes—
41C2×#4038C2216yes—
42C2×#4034C2216yes—
43C2×#4033C2216yes—
44C2×#4031C22112yes—
45C2×#4025C22112yes—
46#65813C22148yescomplemented
47C2×#4022C2216yes—
48C2×#4021C2211yescharacteristic
49C2×C4×D20C2213yes—
50C2×#4018C2216yes—
51#65804C22148yescomplemented
52C2×#4016C2213yes—
53C2×#4015C2213yes—
54C2×#4015C2213yes—
55C2×C4×Dic10C2216yes—
56C2×C4×Dic10C2213yes—
57C2×C4×Dic10C2216yes—
58C2×#3518—51nomaximal, Sylow
59C2×C5×#41—28no—
60C2×C5×#41C2318yes—
61C22×C5×Q8C2311yescharacteristic
62C22×C5×D4C2313yes—
63C23×C4×C5C2313yes—
64C23×C4×C5C2313yes—
65C23×C4×C5C2313yes—
66C22×C22⋊D5C2316yes—
67C2×Q8×D5—216no—
68C2×Q8×D5C23116yes—
69C22×D20C2311yescharacteristic
70C22×C4×D5C2316yes—
71C22×Dic10C2311yescharacteristic
72C22×Dic10C2316yes—
73C22×Dic10C2316yes—
74C4×C5×Q8C2318yes—
75C4×C5×Q8C2318yes—
76C4×C5×D4C23124yes—
77C4×C5×D4C23124yes—
78C5×#117C23124yes—
79C2×C5×C4⋊C4C2313yes—
80C2×C5×C4⋊C4C2313yes—
81C2×C5×C22⋊C4C2316yes—
82C2×C42×C5C2313yes—
83C2×C42×C5C2313yes—
84C2×C42×C5C2313yes—
85C2×C42×C5C2311yescharacteristic
86C2×C42×C5C23124yes—
87C2×C22⋊Dic5C2316yes—
88C2×C22⋊Dic5C23112yes—
89Q8×Dic5C23116yes—
90#4084C23148yes—
91#4082C23148yes—
92#4076C23124yes—
93#4075C23148yes—
94C4×C22⋊D5C23148yes—
95C2×#685C2312yes—
96C2×#685C2316yes—
97C2×#685C2316yes—
98C2×#685C2312yes—
99#4072C23148yes—
100C2×C4⋊Dic5C2313yes—
101C2×C4⋊Dic5C2316yes—
102C2×C4⋊Dic5C2316yes—
103C2×C4⋊Dic5C2313yes—
104#4070C23148yes—
105#4070C23148yes—
106#4070C23124yes—
107C2×#683C2316yes—
108C2×#683C2316yes—
109C2×#683C2316yes—
110C2×#683C2316yes—
111C2×C4×Dic5C23112yes—
112C2×C4×Dic5C2316yes—
113#4043C23148yes—
114#4038C23148yes—
115#4034C23148yes—
116#4033C23148yes—
117#4031C23196yes—
118#4025C23196yes—
119#4022C23148yes—
120#4021C2318yes—
121C4×D20C23124yes—
122#4018C23148yes—
123#4016C23124yes—
124#4015C23124yes—
125#4015C23124yes—
126C4×Dic10C23148yes—
127C4×Dic10C23148yes—
128C4×Dic10C23124yes—
129#3518—532no—
130C2×Q82—51no—
131C2×#556—53no—
132C2×#556—56no—
133C2×#555—53no—
134C2×D4×Q8—51no—
135C2×#548—52no—
136C2×#542—56no—
137C2×#533—53no—
138C2×#529—53no—
139C2×#519—52no—
140C2×C4×#41D511yescharacteristic, complemented
141C24×C5C2413yes—
142C23×D5C2412yes—
143C5×#41—232no—
144C2×C5×Q8#4118yes—
145C2×C5×Q8C2414yes—
146C2×C5×D4—212no—
147C2×C5×D4C24112yes—
148C22×C4×C5—212no—
149C22×C4×C5—212no—
150C22×C4×C5—46no—
151C22×C4×C5C2411yescharacteristic
152C22×C4×C5C2411yescharacteristic
153C22×C4×C5C2413yes—
154C22×C4×C5C2413yes—
155C22×C4×C5C24112yes—
156C22×C4×C5C24112yes—
157C22×C4×C5C24112yes—
158C22×C4×C5C2413yes—
159C22×C4×C5C2411yescharacteristic
160C2×C22⋊D5—412no—
161C2×C22⋊D5C24124yes—
162C22×Dic5C2416yes—
163C22×Dic5C2412yes—
164C22×Dic5C2416yes—
165Q8×D5—264no—
166C2×D20—24no—
167C2×D20C2414yes—
168C2×C4×D5—224no—
169C2×C4×D5C24124yes—
170C2×Dic10—224no—
171C2×Dic10—24no—
172C2×Dic10—412no—
173C2×Dic10C24124yes—
174C2×Dic10C2414yes—
175C2×Dic10C24124yes—
176C5×C4⋊C4C24112yes—
177C5×C4⋊C4C24112yes—
178C5×C22⋊C4C24124yes—
179C42×C5C2414yes—
180C42×C5C24112yes—
181C42×C5C24112yes—
182C42×C5C24112yes—
183C22⋊Dic5C24124yes—
184C22⋊Dic5C24148yes—
185#685C2418yes—
186#685C24124yes—
187#685C24124yes—
188#685C2418yes—
189C4⋊Dic5C24112yes—
190C4⋊Dic5C24124yes—
191C4⋊Dic5C24124yes—
192C4⋊Dic5C24112yes—
193#683C24124yes—
194#683C24124yes—
195#683C24124yes—
196#683C24124yes—
197C4×Dic5C24148yes—
198C4×Dic5C24124yes—
199C22×#41C2×D511yescharacteristic
200C23×Q8—52no—
201Q82—516no—
202#556—548no—
203#556—596no—
204#555—548no—
205D4×Q8—516no—
206#548—532no—
207#542—596no—
208#533—548no—
209C2×C4⋊Q8—56no—
210C2×C4⋊Q8—53no—
211C2×C4⋊Q8—53no—
212#529—548no—
213C2×#126—512no—
214C2×#126—56no—
215C2×#125—53no—
216C2×#124—56no—
217C2×#124—512no—
218C2×#124—51no—
219C2×#123—56no—
220C2×C22⋊Q8—53no—
221C2×C22⋊Q8—56no—
222C2×C22⋊Q8—56no—
223C2×C22⋊Q8—56no—
224C2×#121—56no—
225C2×#121—53no—
226#519—532no—
227C4×#41C2×D5116yescomplemented
228C2×C4×Q8—52no—
229C2×C4×Q8—56no—
230C2×C4×Q8—56no—
231C2×C4×Q8—53no—
232C2×C4×Q8—56no—
233C2×C4×Q8C2×D511yescharacteristic
234C2×C4×Q8C2×D511yescharacteristic
235C2×C4×D4—56no—
236C2×C4×D4—53no—
237C2×C4×D4C2×D513yes—
238C2×C4×D4C2×D513yes—
239C2×#117—56no—
240C2×#117—56no—
241C2×#117—56no—
242C2×#117C2×D513yes—
243C22×C42C2×D513yes—
244C23×C5—26no—
245C23×C5—26no—
246C23×C5—26no—
247C23×C5C2516yes—
248C23×C5C2511yescharacteristic
249C22×D5—24no—
250C22×D5—24no—
251C22×D5—24no—
252C22×D5C2514yes—
253C5×Q8C2×#41116yes—
254C5×D4—224no—
255C2×C4×C5—22no—
256C2×C4×C5—26no—
257C2×C4×C5—26no—
258C2×C4×C5—22no—
259C2×C4×C5—224no—
260C2×C4×C5—224no—
261C2×C4×C5—412no—
262C2×C4×C5C2×#41112yes—
263C2×C4×C52−1+414yes—
264C2×C4×C5C2516yes—
265C2×C4×C5C2512yes—
266C2×C4×C5C2512yes—
267C2×C4×C5C2516yes—
268C2×C4×C5C2516yes—
269C2×C4×C5C2512yes—
270C22⋊D5—424no—
271C2×Dic5—212no—
272C2×Dic5—24no—
273C2×Dic5—212no—
274C2×Dic5C2514yes—
275C2×Dic5C25112yes—
276C2×Dic5C25112yes—
277D20—28no—
278C4×D5—248no—
279Dic10—248no—
280Dic10—28no—
281Dic10—424no—
282C2×#41—28no—
283C2×#41C22×D518yes—
284C22×Q8—516no—
285C22×Q8—51no—
286C22×Q8—56no—
287C22×Q8—56no—
288C22×Q8—1016no—
289C22×Q8C22×D511yescharacteristic
290C22×D4—56no—
291C22×D4—51no—
292C22×D4C22×D513yes—
293C23×C4—56no—
294C23×C4C22×D513yes—
295C23×C4C22×D513yes—
296C23×C4C22×D513yes—
297C4⋊Q8—548no—
298C4⋊Q8—524no—
299C4⋊Q8—524no—
300#126—596no—
301#126—548no—
302#125—524no—
303#124—548no—
304#124—596no—
305#124—58no—
306#123—548no—
307C22⋊Q8—524no—
308C22⋊Q8—548no—
309C22⋊Q8—548no—
310C22⋊Q8—548no—
311#121—548no—
312#121—524no—
313C4×Q8—516no—
314C4×Q8—548no—
315C4×Q8—548no—
316C4×Q8—524no—
317C4×Q8—548no—
318C4×Q8C22×D518yes—
319C4×Q8C22×D518yes—
320C4×D4—548no—
321C4×D4—524no—
322C4×D4C22×D5124yes—
323C4×D4C22×D5124yes—
324#117—548no—
325#117—548no—
326#117—548no—
327#117C22×D5124yes—
328C2×C4⋊C4—53no—
329C2×C4⋊C4—53no—
330C2×C4⋊C4—56no—
331C2×C4⋊C4—56no—
332C2×C4⋊C4—56no—
333C2×C4⋊C4—56no—
334C2×C4⋊C4—56no—
335C2×C4⋊C4—56no—
336C2×C4⋊C4C22×D513yes—
337C2×C4⋊C4C22×D513yes—
338C2×C22⋊C4—512no—
339C2×C22⋊C4—56no—
340C2×C22⋊C4—56no—
341C2×C22⋊C4—52no—
342C2×C22⋊C4—52no—
343C2×C22⋊C4—56no—
344C2×C22⋊C4C22×D516yes—
345C2×C42—512no—
346C2×C42—56no—
347C2×C42C22×D513yes—
348C2×C42C22×D513yes—
349C2×C42C22×D513yes—
350C2×C42C22×D5124yes—
351C2×C42C22×D511yescharacteristic
352C22×C5—26no—
353C22×C5—26no—
354C22×C5—26no—
355C22×C5—412no—
356C22×C5C22×#4112yes—
357C22×C5C2×2−1+412yes—
358C22×C5#58512yes—
359C22×C5C2611yescharacteristic
360C2×D5—24no—
361C2×D5—24no—
362C2×D5—24no—
363C2×D5—48no—
364C4×C5—22no—
365C4×C5—26no—
366C4×C5—26no—
367C4×C5—22no—
368C4×C5C22×#41112yes—
369C4×C5C2×2−1+414yes—
370Dic5—212no—
371Dic5—24no—
372Dic5—212no—
373C24—52no—
374C24C23×D513yes—
375#41—232no—
376C2×Q8—54no—
377C2×Q8—524no—
378C2×Q8—524no—
379C2×Q8—1064no—
380C2×Q8—1024no—
381C2×Q8—104no—
382C2×Q8—2012no—
383C2×Q8#70918yes—
384C2×Q8C23×D514yes—
385C2×D4—212no—
386C2×D4—524no—
387C2×D4—54no—
388C2×D4—104no—
389C2×D4—2012no—
390C2×D4C23×D5112yes—
391C22×C4—212no—
392C22×C4—212no—
393C22×C4—46no—
394C22×C4—52no—
395C22×C4—56no—
396C22×C4—56no—
397C22×C4—524no—
398C22×C4—1024no—
399C22×C4C23×D5112yes—
400C22×C4C23×D5112yes—
401C22×C4C23×D513yes—
402C22×C4C23×D511yescharacteristic
403C22×C4C23×D511yescharacteristic
404C22×C4C23×D513yes—
405C22×C4C23×D5112yes—
406C22×C4C23×D513yes—
407C22×C4C23×D511yescharacteristic
408C4⋊C4—512no—
409C4⋊C4—512no—
410C4⋊C4—524no—
411C4⋊C4—524no—
412C4⋊C4—524no—
413C4⋊C4—524no—
414C4⋊C4—524no—
415C4⋊C4—524no—
416C4⋊C4C23×D5112yes—
417C4⋊C4C23×D5112yes—
418C22⋊C4—548no—
419C22⋊C4—524no—
420C22⋊C4—524no—
421C22⋊C4—58no—
422C22⋊C4—58no—
423C22⋊C4—524no—
424C22⋊C4C23×D5124yes—
425C42—548no—
426C42—524no—
427C42C23×D514yes—
428C42C23×D5112yes—
429C42C23×D5112yes—
430C42C23×D5112yes—
431C2×C5—46no—
432C2×C5#351814yescomplemented
433C2×C5C23×#4111yescharacteristic
434C2×C5C22×2−1+411yescharacteristic
435C2×C5C2×#58511yescharacteristic
436D5—44no—
437C23—26no—
438C23—26no—
439C23—26no—
440C23—54no—
441C23—104no—
442C23—104no—
443C23—104no—
444C23C24×D516yes—
445C23C24×D511yescharacteristic
446Q8—1048no—
447Q8—108no—
448Q8—2024no—
449Q8C2×#709116yes—
450D4—224no—
451D4—108no—
452D4—2024no—
453C2×C4—22no—
454C2×C4—26no—
455C2×C4—22no—
456C2×C4—26no—
457C2×C4—224no—
458C2×C4—224no—
459C2×C4—412no—
460C2×C4—54no—
461C2×C4—512no—
462C2×C4—512no—
463C2×C4—1012no—
464C2×C4—104no—
465C2×C4—1012no—
466C2×C4—1048no—
467C2×C4C2×#709112yes—
468C2×C4#415014yes—
469C2×C4C24×D516yes—
470C2×C4C24×D512yes—
471C2×C4C24×D512yes—
472C2×C4C24×D516yes—
473C2×C4C24×D516yes—
474C2×C4C24×D512yes—
475C5C2×#351811yescharacteristic, complemented, Sylow
476C22—26no—
477C22—26no—
478C22—26no—
479C22—412no—
480C22—104no—
481C22—104no—
482C22—104no—
483C22—208no—
484C22C22×#70912yes—
485C22C2×#415012yes—
486C22#6628312yes—
487C22C25×D511yescharacteristic
488C4—22no—
489C4—26no—
490C4—22no—
491C4—26no—
492C4—1012no—
493C4—104no—
494C4—1012no—
495C4C22×#709112yes—
496C4C2×#415014yes—
497C2—46no—
498C2—204no—
499C2#1061773314yesdirect product factor
500C2C23×#70911yescharacteristic
501C2C22×#415011yescharacteristic
502C2C2×#6628311yescharacteristic
subgroupG/H
centerC23C24×D5
commutator, cosocleC22×C5C26
FrattiniC22C25×D5
FittingC2×C4×C5×#41C2
socleC23×C5C25
hypercenterC2×C4×#41D5
hypocenterC5C2×#3518
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C4×#41

Complements

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

NH ≅ G/Ntype
1. C2×C4×C5×#41C2semidirect
2. C2×#66165C2semidirect
3. #10617733C2direct (N × H)
4. C2×Q8×D20C2semidirect
5. C2×#65906C2semidirect
6. C2×Q8×Dic10C2semidirect
7. C2×#65895C2semidirect
8. C2×#65876C2semidirect
9. C2×#65897C2semidirect
10. C2×#65867C2semidirect
11. C2×#65813C2semidirect
12. C2×#65804C2semidirect
15. C4×C5×#41C22semidirect
22. #66165C22semidirect
33. Q8×D20C22semidirect
34. #65906C22semidirect
35. Q8×Dic10C22semidirect
36. #65897C22semidirect
37. #65895C22semidirect
38. #65876C22semidirect
39. #65867C22semidirect
46. #65813C22semidirect
51. #65804C22semidirect
140. C2×C4×#41D5semidirect
227. C4×#41C2×D5semidirect
432. C2×C5#3518semidirect
475. C5C2×#3518semidirect
499. C2#10617733direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×C5×#412
C2×#661652
#106177332
C2×Q8×D202
C2×#659062
C2×Q8×Dic102
C2×#658952
C2×#658762
C2×#658972
C2×#658672
C2×#658132
C2×#658042
C2×#35185