Finite Group Databasebeta

← C2×#10615207 #12966889

Subgroups — C2×#10615207 #12966889

Order 1280.

2094 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#544C211yesmaximal, characteristic, complemented
2C2×#65857C211yesmaximal, characteristic
3#10615207C2116yesmaximal, direct product factor
4C2×#65383C212yesmaximal, complemented
5C2×#65373C212yesmaximal, complemented
6C2×#65352C212yesmaximal, complemented
7C2×#65355C212yesmaximal, complemented
8C2×#65291C211yesmaximal, characteristic
9C2×#65284C211yesmaximal, characteristic, complemented
10C2×#65288C211yesmaximal, characteristic, complemented
11C2×#65255C211yesmaximal, characteristic
12C22×#3940C211yesmaximal, characteristic, complemented
13C22×#709C2211yescharacteristic
14C5×#544—24no—
15C5×#544C2218yescomplemented
16C2×C5×C4⋊Q8—21no—
17C2×C5×C4⋊Q8C2212yes—
18C2×C5×#125—21no—
19C2×C5×#125C2212yes—
20C2×C5×C22⋊Q8—22no—
21C2×C5×#117C2211yescharacteristic
22C2×C5×#117C2211yescharacteristic
23C22×C5×C4⋊C4C2211yescharacteristic
24C22×C4⋊Dic5C2211yescharacteristic
25C22×#681C2211yescharacteristic
26C23×#168C2211yescharacteristic
27#65857—24no—
28#65857C2218yes—
29C2×#4038—21no—
30C2×D5×C4⋊C4—21no—
31C2×#4027—22no—
32C2×C4×D20C2212yes—
33C2×C4×Dic10C2212yes—
34#65383C22116yescomplemented
35#65373C22116yescomplemented
36#65355C22116yescomplemented
37#65352C22116yescomplemented
38#65291C2218yes—
39#65288C2218yescomplemented
40#65284C2218yescomplemented
41C2×#3942C2212yes—
42C2×#3942C2211yescharacteristic
43C2×#3942C2211yescharacteristic
44#65255C2218yes—
45C2×#3941C2212yes—
46C2×#3941C2211yescharacteristic
47C2×#3941C2211yescharacteristic
48C2×#3940C2212yes—
49C2×#3940C2212yes—
50C2×#3940C2212yes—
51C2×#3940C2218yescomplemented
52C2×#3939C2212yes—
53C2×C4⋊#168C2212yes—
54C2×C4⋊#168C2212yes—
55C2×#3084—51nomaximal, Sylow
56C22×C5×Q8—22no—
57C23×C4×C5—21no—
58C23×C4×C5C2311yescharacteristic
59C22×C22⋊D5—21no—
60C23×Dic5—21no—
61C2×#709—22no—
62C2×#709—24no—
63C2×#709Q818yescomplemented
64C2×#709C2314yes—
65C22×D20C2311yescharacteristic
66C22×C4×D5—21no—
67C22×C4×D5—22no—
68C22×Dic10C2311yescharacteristic
69C5×C4⋊Q8—28no—
70C5×C4⋊Q8—24no—
71C5×C4⋊Q8C2318yes—
72C5×#125—28no—
73C5×#125—24no—
74C5×#125C2318yes—
75C5×C22⋊Q8—216no—
76C5×#117—22no—
77C5×#117D414yes—
78C5×#117C2314yes—
79C5×#117C2314yes—
80C2×C5×C4⋊C4—21no—
81C2×C5×C4⋊C4—22no—
82C2×C5×C4⋊C4—22no—
83C2×C5×C4⋊C4—22no—
84C2×C5×C4⋊C4—22no—
85C2×C5×C4⋊C4D411yescharacteristic
86C2×C5×C4⋊C4D411yescharacteristic
87C2×C5×C4⋊C4C2312yes—
88C2×C5×C4⋊C4C2314yes—
89C2×C5×C4⋊C4C2311yescharacteristic
90C2×C5×C4⋊C4C2311yescharacteristic
91C2×C5×C22⋊C4—21no—
92C2×C5×C22⋊C4D412yes—
93C2×C42×C5—21no—
94C2×C42×C5C2312yes—
95C2×#685—22no—
96C2×C4⋊Dic5—21no—
97C2×C4⋊Dic5—22no—
98C2×C4⋊Dic5C2314yes—
99C2×C4⋊Dic5C2312yes—
100C2×#683—22no—
101C2×C4×Dic5—22no—
102C2×#681—24no—
103C2×#681C2314yes—
104C22×#168—24no—
105C22×#168C2314yes—
106C22×#168C2311yescharacteristic
107C22×#168C2311yescharacteristic
108C22×#168C2312yes—
109#4038—28no—
110D5×C4⋊C4—28no—
111#4027—216no—
112C4×D20—24no—
113C4×D20C2318yes—
114C4×Dic10—24no—
115C4×Dic10C2318yes—
116#3942C2314yes—
117#3942C2314yes—
118#3942C2318yes—
119#3941C2314yes—
120#3941C2314yes—
121#3941C2318yes—
122#3940C2318yes—
123#3940C2318yes—
124#3940C2318yes—
125#3939C2318yes—
126C4⋊#168C2318yes—
127C4⋊#168C2318yes—
128C2×#544D511yescharacteristic, complemented
129C2×#520—51no—
130#3084—516no—
131C2×#479—52no—
132C2×#475—52no—
133C2×#478—52no—
134C2×#476—52no—
135C2×#428—51no—
136C22×#106—51no—
137C2×#424—51no—
138C2×#417—51no—
139C2×#416—51no—
140C24×C5C2×D411yescharacteristic
141C23×D5—21no—
142C2×C5×Q8—28no—
143C2×C5×Q8—84no—
144C22×C4×C5—21no—
145C22×C4×C5—21no—
146C22×C4×C5—22no—
147C22×C4×C5—24no—
148C22×C4×C5—82no—
149C22×C4×C5C2×D412yes—
150C22×C4×C5C2×D412yes—
151C22×C4×C5C2×D411yescharacteristic
152C22×C4×C5C2×D412yes—
153C22×C4×C5C2×Q814yes—
154C22×C4×C5#4114yes—
155C22×C4×C5C2411yescharacteristic
156C22×C4×C5C2412yes—
157C2×C22⋊D5—28no—
158C2×C22⋊D5—24no—
159C22×Dic5—21no—
160C22×Dic5—22no—
161C22×Dic5—24no—
162C22×Dic5—82no—
163#709—216no—
164#709—28no—
165#709C2×Q8116yescomplemented
166C2×D20—21no—
167C2×D20—22no—
168C2×D20C2×Q814yes—
169C2×D20C2412yes—
170C2×C4×D5—28no—
171C2×C4×D5—24no—
172C2×C4×D5—28no—
173C2×C4×D5—84no—
174C2×Dic10—21no—
175C2×Dic10—22no—
176C2×Dic10C2×Q814yes—
177C2×Dic10C2412yes—
178C5×C4⋊C4—24no—
179C5×C4⋊C4—22no—
180C5×C4⋊C4—28no—
181C5×C4⋊C4—28no—
182C5×C4⋊C4—28no—
183C5×C4⋊C4—21no—
184C5×C4⋊C4—21no—
185C5×C4⋊C4C2×D414yes—
186C5×C4⋊C4C2×D414yes—
187C5×C4⋊C4C2414yes—
188C5×C4⋊C4C2412yes—
189C5×C4⋊C4C2412yes—
190C5×C22⋊C4—24no—
191C5×C22⋊C4C2×D418yes—
192C42×C5—24no—
193C42×C5C2×D414yes—
194C42×C5C2414yes—
195#685—28no—
196C4⋊Dic5—22no—
197C4⋊Dic5—24no—
198C4⋊Dic5C2414yes—
199#683—28no—
200C4×Dic5—28no—
201#681—28no—
202C2×#168—22no—
203C2×#168—22no—
204C2×#168—28no—
205C2×#168—24no—
206C2×#168C2414yes—
207C2×#168C2412yes—
208C2×#168C2412yes—
209C22×#41—51no—
210C22×M16—51no—
211C23×C8—51no—
212#544—24no—
213#544C2×D518yescomplemented
214C2×C4⋊Q8—21no—
215C2×C4⋊Q8C2×D512yes—
216C2×#125—21no—
217C2×#125C2×D512yes—
218C2×C22⋊Q8—22no—
219#520—58no—
220#520—104no—
221C2×C4×Q8—52no—
222C2×C4×D4—52no—
223C2×C4×D4—102no—
224C2×#117—101no—
225C2×#117C2×D511yescharacteristic
226C2×#117C2×D511yescharacteristic
227C22×C4⋊C4—51no—
228C22×C4⋊C4—101no—
229C22×C4⋊C4C2×D511yescharacteristic
230#479—516no—
231#478—516no—
232#476—516no—
233#475—516no—
234#428—58no—
235C2×#107—52no—
236C2×#106—52no—
237C2×#106—52no—
238C2×#106—52no—
239C2×#106—58no—
240#424—58no—
241C2×C4⋊C8—52no—
242C2×C4⋊C8—52no—
243#417—58no—
244#416—58no—
245C2×#103—51no—
246C2×#103—52no—
247C2×#103—51no—
248C2×#102—51no—
249C2×#102—52no—
250C2×#102—51no—
251C23×C5—22no—
252C23×C5—22no—
253C23×C5C22⋊Q814yes—
254C23×C5C22×D412yes—
255C23×C5C22×D411yescharacteristic
256C22×D5—24no—
257C22×D5—22no—
258C22×D5—42no—
259C22×D5—42no—
260C5×Q8—88no—
261C2×C4×C5—24no—
262C2×C4×C5—24no—
263C2×C4×C5—22no—
264C2×C4×C5—22no—
265C2×C4×C5—24no—
266C2×C4×C5—42no—
267C2×C4×C5—44no—
268C2×C4×C5—42no—
269C2×C4×C5—84no—
270C2×C4×C5C22⋊Q814yes—
271C2×C4×C5C22⋊Q818yes—
272C2×C4×C5C22×D414yes—
273C2×C4×C5C22×D414yes—
274C2×C4×C5C22×D412yes—
275C2×C4×C5C22×D411yescharacteristic
276C2×C4×C5C22×Q812yes—
277C2×C4×C5C22×Q814yes—
278C2×C4×C5C2×#4112yes—
279C2×C4×C5C2×#4114yes—
280C2×C4×C5C2511yescharacteristic
281C22⋊D5—216no—
282C2×Dic5—24no—
283C2×Dic5—22no—
284C2×Dic5—24no—
285C2×Dic5—44no—
286C2×Dic5—84no—
287D20—24no—
288D20—22no—
289D20C22×Q814yes—
290C4×D5—216no—
291C4×D5—88no—
292Dic10—24no—
293Dic10—22no—
294Dic10C22×Q814yes—
295#168—22no—
296#168—22no—
297#168—24no—
298C2×#41—58no—
299C2×#41—54no—
300C2×#41—104no—
301C2×#41—102no—
302C22×Q8—22no—
303C22×Q8—51no—
304C22×D4—51no—
305C22×D4—101no—
306C23×C4—21no—
307C23×C4—101no—
308C23×C4—101no—
309C23×C4—102no—
310C23×C4C22×D511yescharacteristic
311C2×M16—54no—
312C2×M16—104no—
313C22×C8—51no—
314C22×C8—51no—
315C22×C8—54no—
316C22×C8—52no—
317C22×C8—104no—
318C4⋊Q8—28no—
319C4⋊Q8—24no—
320C4⋊Q8C22×D518yes—
321#125—28no—
322#125—24no—
323#125C22×D518yes—
324C22⋊Q8—216no—
325C4×Q8—58no—
326C4×Q8—104no—
327C4×D4—58no—
328C4×D4—1016no—
329C4×D4—104no—
330#117—22no—
331#117—108no—
332#117C22⋊D514yes—
333#117C22×D514yes—
334#117C22×D514yes—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—52no—
341C2×C4⋊C4—54no—
342C2×C4⋊C4—101no—
343C2×C4⋊C4—102no—
344C2×C4⋊C4—102no—
345C2×C4⋊C4—108no—
346C2×C4⋊C4C22⋊D511yescharacteristic
347C2×C4⋊C4C22⋊D511yescharacteristic
348C2×C4⋊C4C22×D512yes—
349C2×C4⋊C4C22×D514yes—
350C2×C4⋊C4C22×D511yescharacteristic
351C2×C4⋊C4C22×D511yescharacteristic
352C2×C22⋊C4—21no—
353C2×C22⋊C4—102no—
354C2×C22⋊C4C22⋊D512yes—
355C2×C42—21no—
356C2×C42—102no—
357C2×C42C22×D512yes—
358#107—58no—
359#106—58no—
360#106—58no—
361#106—58no—
362C4⋊C8—58no—
363C4⋊C8—58no—
364#103—54no—
365#103—54no—
366#103—58no—
367#102—54no—
368#102—54no—
369#102—58no—
370C22×C5—28no—
371C22×C5—22no—
372C22×C5—22no—
373C22×C5C2×C22⋊Q814yes—
374C22×C5C2×C22⋊Q812yes—
375C22×C5#57712yes—
376C22×C5#57712yes—
377C22×C5C23×D411yescharacteristic
378C2×D5—24no—
379C2×D5—48no—
380C2×D5—42no—
381C2×D5—42no—
382C4×C5—24no—
383C4×C5—24no—
384C4×C5—42no—
385C4×C5—44no—
386C4×C5—42no—
387C4×C5C2×C22⋊Q812yes—
388C4×C5C2×C22⋊Q814yes—
389C4×C5C23×Q811yescharacteristic
390C4×C5C22×#4111yescharacteristic
391Dic5—24no—
392Dic5—44no—
393C24—101no—
394C24C2×C22⋊D511yescharacteristic
395#41—516no—
396#41—108no—
397#41—1016no—
398C2×Q8—28no—
399C2×Q8—54no—
400C2×Q8—52no—
401C2×Q8—84no—
402C2×Q8—102no—
403C2×Q8—101no—
404C2×D4—54no—
405C2×D4—52no—
406C2×D4—104no—
407C2×D4—108no—
408C2×D4—102no—
409C2×D4—101no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—82no—
415C22×C4—101no—
416C22×C4—104no—
417C22×C4—108no—
418C22×C4—102no—
419C22×C4—104no—
420C22×C4—108no—
421C22×C4—404no—
422C22×C4—402no—
423C22×C4Q8×D514yes—
424C22×C4#71314yes—
425C22×C4C2×C22⋊D512yes—
426C22×C4C2×C22⋊D512yes—
427C22×C4C2×C22⋊D512yes—
428C22×C4C2×C22⋊D511yescharacteristic
429C22×C4C23×D512yes—
430C22×C4C23×D511yescharacteristic
431M16—108no—
432C2×C8—52no—
433C2×C8—52no—
434C2×C8—54no—
435C2×C8—102no—
436C2×C8—102no—
437C2×C8—108no—
438C2×C8—104no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—28no—
442C4⋊C4—28no—
443C4⋊C4—28no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—54no—
447C4⋊C4—104no—
448C4⋊C4—102no—
449C4⋊C4—108no—
450C4⋊C4C2×C22⋊D514yes—
451C4⋊C4C2×C22⋊D514yes—
452C4⋊C4C23×D514yes—
453C4⋊C4C23×D512yes—
454C4⋊C4C23×D512yes—
455C22⋊C4—24no—
456C22⋊C4—108no—
457C22⋊C4C2×C22⋊D518yes—
458C42—24no—
459C42—108no—
460C42C2×C22⋊D514yes—
461C42C23×D514yes—
462C2×C5—24no—
463C2×C5#308414yescomplemented
464C2×C5C22×C22⋊Q811yescharacteristic
465C2×C5C2×#57711yescharacteristic
466C2×C5C2×#57711yescharacteristic
467D5—44no—
468C23—22no—
469C23—22no—
470C23—104no—
471C23—102no—
472C23—202no—
473C23—202no—
474C23#409214yes—
475C23C22×C22⋊D512yes—
476C23C22×C22⋊D511yescharacteristic
477Q8—54no—
478Q8—88no—
479Q8—104no—
480Q8—102no—
481D4—54no—
482D4—1016no—
483D4—104no—
484D4—102no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—84no—
494C2×C4—104no—
495C2×C4—102no—
496C2×C4—1016no—
497C2×C4—104no—
498C2×C4—204no—
499C2×C4—408no—
500C2×C4—404no—
501C2×C4#409214yes—
502C2×C4#409218yes—
503C2×C4C2×Q8×D512yes—
504C2×C4C2×Q8×D514yes—
505C2×C4C2×#71312yes—
506C2×C4C2×#71314yes—
507C2×C4C22×C22⋊D514yes—
508C2×C4C22×C22⋊D514yes—
509C2×C4C22×C22⋊D512yes—
510C2×C4C22×C22⋊D511yescharacteristic
511C2×C4C24×D511yescharacteristic
512C8—102no—
513C8—102no—
514C8—104no—
515C5C2×#308411yescharacteristic, complemented, Sylow
516C22—22no—
517C22—22no—
518C22—28no—
519C22—104no—
520C22—202no—
521C22—202no—
522C22—208no—
523C22C2×#409214yes—
524C22C2×#409212yes—
525C22#6617012yes—
526C22#6617112yes—
527C22C23×C22⋊D511yescharacteristic
528C4—24no—
529C4—24no—
530C4—44no—
531C4—42no—
532C4—42no—
533C4—104no—
534C4—204no—
535C4C2×#409214yes—
536C4C2×#409212yes—
537C4C22×Q8×D511yescharacteristic
538C4C22×#71311yescharacteristic
539C2—24no—
540C2—204no—
541C2#1061520714yesdirect product factor
542C2C22×#409211yescharacteristic
543C2C2×#6617011yescharacteristic
544C2C2×#6617111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC2×C5×#544C2
socleC23×C5C22×D4
prime radicalC22×#709C22
hypercenterC2×#544D5
hypocenterC5C2×#3084
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
1. C2×C5×#544C2semidirect
3. #10615207C2direct (N × H)
4. C2×#65383C2semidirect
5. C2×#65373C2semidirect
6. C2×#65352C2semidirect
7. C2×#65355C2semidirect
9. C2×#65284C2semidirect
10. C2×#65288C2semidirect
12. C22×#3940C2semidirect
15. C5×#544C22semidirect
34. #65383C22semidirect
35. #65373C22semidirect
36. #65355C22semidirect
37. #65352C22semidirect
39. #65288C22semidirect
40. #65284C22semidirect
51. C2×#3940C22semidirect
63. C2×#709Q8semidirect
128. C2×#544D5semidirect
165. #709C2×Q8semidirect
213. #544C2×D5semidirect
463. C2×C5#3084semidirect
515. C5C2×#3084semidirect
541. C2#10615207direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#5442
C2×#658572
#106152072
C2×#653832
C2×#653732
C2×#653522
C2×#653552
C2×#652912
C2×#652842
C2×#652882
C2×#652552
C22×#39402
C2×#30845