Finite Group Databasebeta

← C2×#10619193 #13007713

Subgroups — C2×#10619193 #13007713

Order 1280.

11126 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C5×2−1+4C211yesmaximal, characteristic, complemented
2C22×#4150C211yesmaximal, characteristic, complemented
3#10619193C2132yesmaximal, direct product factor
4C2×#66158C2112yesmaximal, complemented
5C2×#66159C213yesmaximal, complemented
6C2×#66166C213yesmaximal, complemented
7C2×#41×Dic5C211yesmaximal, characteristic, complemented
8C2×#66150C213yesmaximal, complemented
9C2×Q8×C22⋊D5C214yesmaximal, complemented
10C22×#4090C213yesmaximal, complemented
11C2×C5×2−1+4—216no—
12C2×C5×2−1+4C22116yescomplemented
13C22×C5×#41C2213yes—
14C22×C5×#41C2216yes—
15C22×C5×#41C2211yescharacteristic
16C23×C5×Q8C2213yes—
17C23×C5×Q8C2212yes—
18C2×#4150—216no—
19C2×#4150C22116yescomplemented
20C22×Q8×D5C2212yes—
21C22×#711C2216yes—
22C22×#709C2213yes—
23C23×Dic10C2213yes—
24#66166C22148yescomplemented
25#41×Dic5C22116yescomplemented
26#66159C22148yescomplemented
27#66158C221192yescomplemented
28#66150C22148yescomplemented
29Q8×C22⋊D5C22164yescomplemented
30C2×#4093C2213yes—
31C2×#4092C22112yes—
32C2×Q8×Dic5C2212yes—
33C2×#4090C2213yes—
34C2×#4090C22112yes—
35C2×#4090C2216yes—
36C2×#4090C22148yescomplemented
37C2×#4086C2211yescharacteristic
38C2×#4085C22112yes—
39C2×#4082C2213yes—
40C2×#4081C22112yes—
41C2×D4×Dic5C2216yes—
42C2×#4075C22112yes—
43C2×C4×C22⋊D5C22112yes—
44C2×#4072C2213yes—
45C2×#4070C22112yes—
46C22×#683C22112yes—
47C22×C4×Dic5C2213yes—
48C2×#3499—51nomaximal, Sylow
49C5×2−1+4—2128no—
50C2×C5×#41—248no—
51C2×C5×#41—224no—
52C2×C5×#41D4116yescomplemented
53C2×C5×#41C23148yes—
54C2×C5×#41C23124yes—
55C2×C5×#41C2318yes—
56C22×C5×Q8—216no—
57C22×C5×Q8—224no—
58C22×C5×Q8C2313yes—
59C22×C5×Q8C2316yes—
60C22×C5×Q8C23124yes—
61C22×C5×Q8C23116yes—
62C22×C5×Q8C2311yescharacteristic
63C22×C5×D4C2316yes—
64C22×C5×D4C2311yescharacteristic
65C22×C5×D4C2313yes—
66C23×C4×C5C2316yes—
67C23×C4×C5C2316yes—
68C23×C4×C5C2313yes—
69C22×C22⋊D5C2314yes—
70C22×C22⋊D5C2316yes—
71C23×Dic5C2316yes—
72C23×Dic5C2316yes—
73#4150—2128no—
74C2×Q8×D5—216no—
75C2×Q8×D5C23116yes—
76C2×#711—248no—
77C2×#711C23148yes—
78C2×#709—224no—
79C2×#709C23124yes—
80C22×D20C2311yescharacteristic
81C22×C4×D5C2316yes—
82C22×Dic10—224no—
83C22×Dic10C2316yes—
84C22×Dic10C2313yes—
85C22×Dic10C23124yes—
86C2×C22⋊Dic5C23112yes—
87C2×C22⋊Dic5C2316yes—
88#4093C23124yes—
89#4092C23196yes—
90Q8×Dic5C23116yes—
91#4090C23124yes—
92#4090C23196yes—
93#4090C23148yes—
94#4086C2318yes—
95#4085C23196yes—
96#4082C23124yes—
97#4081C23196yes—
98D4×Dic5C23148yes—
99#4075C23196yes—
100C4×C22⋊D5C23196yes—
101C2×#685C23112yes—
102#4072C23124yes—
103C2×C4⋊Dic5C2316yes—
104#4070C23196yes—
105C2×#683C23112yes—
106C2×#683C23124yes—
107C2×#683C23196yes—
108C2×C4×Dic5C2311yescharacteristic
109C2×C4×Dic5C2313yes—
110C2×C4×Dic5C2316yes—
111C2×C4×Dic5C23124yes—
112C22×2−1+4—51no—
113C22×2−1+4D511yescharacteristic, complemented
114#3499—532no—
115C2×D4×Q8—54no—
116C2×#547—512no—
117C2×#536—53no—
118C2×#536—53no—
119C2×#532—53no—
120C22×C4⋊Q8—53no—
121C2×C4×#41—51no—
122C24×C5C2412yes—
123C24×C5C2413yes—
124C23×D5C2412yes—
125C5×#41—2192no—
126C5×#41—296no—
127C5×#41C2×D4164yescomplemented
128C2×C5×Q8—264no—
129C2×C5×Q8—212no—
130C2×C5×Q8—224no—
131C2×C5×Q8—296no—
132C2×C5×Q8C2×D418yes—
133C2×C5×Q8C24112yes—
134C2×C5×Q8C24124yes—
135C2×C5×Q8C2414yes—
136C2×C5×D4—224no—
137C2×C5×D4—24no—
138C2×C5×D4C2×D4124yes—
139C2×C5×D4C24124yes—
140C2×C5×D4C2414yes—
141C2×C5×D4C24112yes—
142C22×C4×C5—224no—
143C22×C4×C5—224no—
144C22×C4×C5C2×D4124yes—
145C22×C4×C5C2416yes—
146C22×C4×C5C24124yes—
147C22×C4×C5C24124yes—
148C22×C4×C5C2411yescharacteristic
149C22×C4×C5C2413yes—
150C22×C4×C5C24112yes—
151C2×C22⋊D5—224no—
152C2×C22⋊D5—216no—
153C2×C22⋊D5C24124yes—
154C2×C22⋊D5C24116yes—
155C22×Dic5—224no—
156C22×Dic5—224no—
157C22×Dic5C2416yes—
158C22×Dic5C2416yes—
159C22×Dic5C24124yes—
160C22×Dic5C24124yes—
161C22×Dic5C2412yes—
162Q8×D5—264no—
163#711—2192no—
164#709—296no—
165C2×D20—24no—
166C2×D20C2414yes—
167C2×C4×D5—224no—
168C2×C4×D5C24124yes—
169C2×Dic10—212no—
170C2×Dic10—224no—
171C2×Dic10—296no—
172C2×Dic10C24112yes—
173C2×Dic10C24124yes—
174C22⋊Dic5C24148yes—
175C22⋊Dic5C24124yes—
176#685C24148yes—
177C4⋊Dic5C24124yes—
178#683C24148yes—
179#683C24196yes—
180C4×Dic5C2414yes—
181C4×Dic5C24112yes—
182C4×Dic5C24124yes—
183C2×2−1+4—216no—
184C2×2−1+4—516no—
185C2×2−1+4—1016no—
186C2×2−1+4C2×D5116yescomplemented
187C22×#41—56no—
188C22×#41—53no—
189C22×#41C2×D513yes—
190C22×#41C2×D516yes—
191C22×#41C2×D511yescharacteristic
192C23×Q8—52no—
193C23×Q8—53no—
194C23×Q8C2×D513yes—
195C23×Q8C2×D512yes—
196D4×Q8—564no—
197#547—5192no—
198#536—548no—
199#536—548no—
200#532—548no—
201C2×C4⋊Q8—53no—
202C2×C4⋊Q8—56no—
203C2×C4⋊Q8—512no—
204C2×C4⋊Q8—548no—
205C2×#127—51no—
206C2×#124—53no—
207C2×#124—53no—
208C2×#123—512no—
209C2×#123—512no—
210C2×C22⋊Q8—512no—
211C2×C22⋊Q8—512no—
212C2×#121—512no—
213C4×#41—516no—
214C2×C4×Q8—52no—
215C2×C4×D4—56no—
216C2×C4×D4—512no—
217C2×#117—53no—
218C22×C4⋊C4—512no—
219C22×C42—53no—
220C23×C5—212no—
221C23×C5—24no—
222C23×C5—28no—
223C23×C5C22×D4112yes—
224C23×C5C2514yes—
225C23×C5C2516yes—
226C23×C5C2511yescharacteristic
227C22×D5—24no—
228C22×D5—28no—
229C22×D5C2514yes—
230C5×Q8—224no—
231C5×Q8—248no—
232C5×Q8C22×D4116yes—
233C5×D4—248no—
234C5×D4—28no—
235C5×D4C22×D4148yes—
236C2×C4×C5—212no—
237C2×C4×C5—248no—
238C2×C4×C5—248no—
239C2×C4×C5C22×D414yes—
240C2×C4×C5C22×D4112yes—
241C2×C4×C5C22×D4148yes—
242C2×C4×C5C25112yes—
243C2×C4×C5C2512yes—
244C2×C4×C5C2516yes—
245C22⋊D5—248no—
246C22⋊D5—232no—
247C2×Dic5—212no—
248C2×Dic5—248no—
249C2×Dic5—212no—
250C2×Dic5—248no—
251C2×Dic52−1+418yescomplemented
252C2×Dic5C25112yes—
253C2×Dic5C25112yes—
254C2×Dic5C2514yes—
255D20—28no—
256C4×D5—248no—
257Dic10—224no—
258Dic10—248no—
2592−1+4—2128no—
2602−1+4—10128no—
261C2×#41—248no—
262C2×#41—224no—
263C2×#41—548no—
264C2×#41—524no—
265C2×#41—1048no—
266C2×#41—1024no—
267C2×#41C22⋊D5116yescomplemented
268C2×#41C22×D5124yes—
269C2×#41C22×D5148yes—
270C2×#41C22×D518yes—
271C22×Q8—216no—
272C22×Q8—224no—
273C22×Q8—516no—
274C22×Q8—56no—
275C22×Q8—53no—
276C22×Q8—524no—
277C22×Q8—1016no—
278C22×Q8—1024no—
279C22×Q8C22×D516yes—
280C22×Q8C22×D513yes—
281C22×Q8C22×D5124yes—
282C22×Q8C22×D5116yes—
283C22×Q8C22×D511yescharacteristic
284C22×D4—54no—
285C22×D4—56no—
286C22×D4—51no—
287C22×D4C22×D516yes—
288C22×D4C22×D511yescharacteristic
289C22×D4C22×D513yes—
290C23×C4—56no—
291C23×C4—56no—
292C23×C4—56no—
293C23×C4C22×D516yes—
294C23×C4C22×D516yes—
295C23×C4C22×D513yes—
296C4⋊Q8—524no—
297C4⋊Q8—548no—
298C4⋊Q8—596no—
299#127—58no—
300#124—524no—
301#124—524no—
302#123—596no—
303#123—596no—
304C22⋊Q8—596no—
305C22⋊Q8—596no—
306#121—596no—
307C4×Q8—516no—
308C4×D4—548no—
309C4×D4—596no—
310#117—524no—
311C2×C4⋊C4—512no—
312C2×C4⋊C4—524no—
313C2×C4⋊C4—596no—
314C2×C4⋊C4—56no—
315C2×C22⋊C4—56no—
316C2×C22⋊C4—512no—
317C2×C22⋊C4—512no—
318C2×C42—53no—
319C2×C42—56no—
320C2×C42—524no—
321C2×C42—51no—
322C22×C5—212no—
323C22×C5—224no—
324C22×C5—24no—
325C22×C5—28no—
326C22×C5—48no—
327C22×C5C23×D4112yes—
328C22×C5C23×D412yes—
329C22×C5C2×2−1+414yes—
330C22×C5C2611yescharacteristic
331C2×D5—24no—
332C2×D5—28no—
333C2×D5—48no—
334C4×C5—212no—
335C4×C5C23×D414yes—
336C4×C5C23×D4112yes—
337Dic5—212no—
338Dic5—212no—
339Dic5C2×2−1+418yescomplemented
340C24—52no—
341C24C23×D512yes—
342C24C23×D513yes—
343#41—2192no—
344#41—296no—
345#41—10192no—
346#41—1096no—
347#41C2×C22⋊D5164yescomplemented
348C2×Q8—264no—
349C2×Q8—224no—
350C2×Q8—212no—
351C2×Q8—296no—
352C2×Q8—524no—
353C2×Q8—512no—
354C2×Q8—1064no—
355C2×Q8—1024no—
356C2×Q8—1012no—
357C2×Q8—1096no—
358C2×Q8C2×C22⋊D518yes—
359C2×Q8C23×D5124yes—
360C2×Q8C23×D5112yes—
361C2×Q8C23×D514yes—
362C2×D4—224no—
363C2×D4—24no—
364C2×D4—516no—
365C2×D4—524no—
366C2×D4—54no—
367C2×D4—1016no—
368C2×D4—1024no—
369C2×D4—104no—
370C2×D4C2×C22⋊D5124yes—
371C2×D4C23×D5124yes—
372C2×D4C23×D514yes—
373C2×D4C23×D5112yes—
374C22×C4—224no—
375C22×C4—224no—
376C22×C4—52no—
377C22×C4—524no—
378C22×C4—524no—
379C22×C4—56no—
380C22×C4—56no—
381C22×C4—524no—
382C22×C4—1024no—
383C22×C4—1024no—
384C22×C4—1024no—
385C22×C4C2×C22⋊D5124yes—
386C22×C4C23×D5124yes—
387C22×C4C23×D5124yes—
388C22×C4C23×D516yes—
389C22×C4C23×D5112yes—
390C22×C4C23×D511yescharacteristic
391C22×C4C23×D513yes—
392C4⋊C4—548no—
393C4⋊C4—596no—
394C4⋊C4—524no—
395C22⋊C4—524no—
396C22⋊C4—548no—
397C22⋊C4—548no—
398C42—54no—
399C42—512no—
400C42—524no—
401C2×C5—212no—
402C2×C5—44no—
403C2×C5#349914yescomplemented
404C2×C5C24×D411yescharacteristic
405C2×C5C22×2−1+412yes—
406D5—44no—
407C23—212no—
408C23—28no—
409C23—24no—
410C23—54no—
411C23—108no—
412C23—104no—
413C23C22×C22⋊D5112yes—
414C23C24×D514yes—
415C23C24×D516yes—
416C23C24×D511yescharacteristic
417Q8—248no—
418Q8—224no—
419Q8—1048no—
420Q8—1024no—
421Q8C22×C22⋊D5116yes—
422D4—248no—
423D4—28no—
424D4—1032no—
425D4—1048no—
426D4—108no—
427D4C22×C22⋊D5148yes—
428C2×C4—212no—
429C2×C4—248no—
430C2×C4—248no—
431C2×C4—58no—
432C2×C4—54no—
433C2×C4—512no—
434C2×C4—512no—
435C2×C4—1048no—
436C2×C4—1012no—
437C2×C4—1012no—
438C2×C4—1048no—
439C2×C4—1048no—
440C2×C4C22×C22⋊D5112yes—
441C2×C4C22×C22⋊D514yes—
442C2×C4C22×C22⋊D5148yes—
443C2×C4C24×D5112yes—
444C2×C4C24×D512yes—
445C2×C4C24×D516yes—
446C5C2×#349911yescharacteristic, complemented, Sylow
447C22—212no—
448C22—224no—
449C22—28no—
450C22—24no—
451C22—48no—
452C22—108no—
453C22—104no—
454C22—208no—
455C22D5×2−1+414yes—
456C22C23×C22⋊D5112yes—
457C22C23×C22⋊D512yes—
458C22C25×D511yescharacteristic
459C4—212no—
460C4—58no—
461C4—1012no—
462C4—1012no—
463C4C23×C22⋊D5112yes—
464C4C23×C22⋊D514yes—
465C2—212no—
466C2—44no—
467C2—204no—
468C2#1061919314yesdirect product factor
469C2C2×D5×2−1+412yes—
470C2C24×C22⋊D511yescharacteristic
subgroupG/H
centerC23C24×D5
commutator, cosocleC22×C5C26
FrattiniC22C25×D5
FittingC22×C5×2−1+4C2
socleC23×C5C25
hypercenterC22×2−1+4D5
hypocenterC5C2×#3499
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C22×2−1+4

Complements

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

NH ≅ G/Ntype
1. C22×C5×2−1+4C2semidirect
2. C22×#4150C2semidirect
3. #10619193C2direct (N × H)
4. C2×#66158C2semidirect
5. C2×#66159C2semidirect
6. C2×#66166C2semidirect
7. C2×#41×Dic5C2semidirect
8. C2×#66150C2semidirect
9. C2×Q8×C22⋊D5C2semidirect
10. C22×#4090C2semidirect
12. C2×C5×2−1+4C22semidirect
19. C2×#4150C22semidirect
24. #66166C22semidirect
25. #41×Dic5C22semidirect
26. #66159C22semidirect
27. #66158C22semidirect
28. #66150C22semidirect
29. Q8×C22⋊D5C22semidirect
36. C2×#4090C22semidirect
52. C2×C5×#41D4semidirect
113. C22×2−1+4D5semidirect
127. C5×#41C2×D4semidirect
186. C2×2−1+4C2×D5semidirect
251. C2×Dic52−1+4semidirect
267. C2×#41C22⋊D5semidirect
339. Dic5C2×2−1+4semidirect
347. #41C2×C22⋊D5semidirect
403. C2×C5#3499semidirect
446. C5C2×#3499semidirect
468. C2#10619193direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C5×2−1+42
C22×#41502
#106191932
C2×#661582
C2×#661592
C2×#661662
C2×#41×Dic52
C2×#661502
C2×Q8×C22⋊D52
C22×#40902
C2×#34995