Finite Group Databasebeta

← C2×#10618948 #13006985

Subgroups — C2×#10618948 #13006985

Order 1280.

7086 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×Q82C211yesmaximal, characteristic, complemented
2#10618948C2132yesmaximal, direct product factor
3C2×#66069C214yesmaximal, complemented
4C2×#66060C214yesmaximal, complemented
5C2×#66063C214yesmaximal, complemented
6C2×#66058C214yesmaximal, complemented
7C2×D5×C4⋊Q8C211yesmaximal, characteristic, complemented
8C2×#66068C211yesmaximal, characteristic, complemented
9C2×#65913C212yesmaximal, complemented
10C2×#65915C214yesmaximal, complemented
11C2×#65916C214yesmaximal, complemented
12C2×#65914C212yesmaximal, complemented
13C22×#713C2214yes—
14C22×Q8×D5C2212yes—
15C5×Q82C22116yescomplemented
16C2×C5×C4⋊Q8C2214yes—
17C2×C5×C4⋊Q8C2214yes—
18C2×C5×C4⋊Q8C2211yescharacteristic
19C2×C4×C5×Q8C2212yes—
20C2×C4×C5×Q8C2214yes—
21C2×#4093C2214yes—
22C2×#4093C2212yes—
23C2×#4092C2218yes—
24C2×#4092C2212yes—
25C2×#4092C2214yes—
26C2×#4090C2212yes—
27C2×#4090C2214yes—
28#66069C22164yescomplemented
29#66068C22116yescomplemented
30#66063C22164yescomplemented
31#66060C22164yescomplemented
32#66058C22164yescomplemented
33D5×C4⋊Q8C22116yescomplemented
34#65916C22164yescomplemented
35#65915C22164yescomplemented
36#65914C22132yescomplemented
37#65913C22132yescomplemented
38C2×#4044C2214yes—
39C2×#4042C2218yes—
40C2×#4042C2218yes—
41C2×#4041C2212yes—
42C2×#4041C2214yes—
43C2×#4040C2218yes—
44C2×#4040C2218yes—
45C2×#4040C2214yes—
46C2×#4038C2214yes—
47C2×D5×C4⋊C4C2214yes—
48C2×D5×C4⋊C4C2214yes—
49C2×#4036C2212yes—
50C2×#4035C2218yes—
51C2×#4034C2214yes—
52C2×#4022C2218yes—
53C2×#4021C2214yes—
54C2×C4×D20C2214yes—
55C2×C4×D20C2214yes—
56C2×C4×D20C2212yes—
57C2×#4018C2214yes—
58C2×#4018C2214yes—
59C2×C42×D5C2211yescharacteristic
60C2×#4016C2214yes—
61C2×#4015C2211yescharacteristic
62C2×#3544—51nomaximal, Sylow
63C22×C5×Q8C2312yes—
64C22×C5×Q8C2314yes—
65C22×C5×Q8C2312yes—
66C2×#713—416no—
67C2×#713C23132yes—
68C2×Q8×D5—48no—
69C2×Q8×D5C23116yes—
70C22×D20C2312yes—
71C22×D20C2314yes—
72C22×D20C2314yes—
73C22×C4×D5C2312yes—
74C22×C4×D5C2311yescharacteristic
75C22×C4×D5C2314yes—
76C22×C4×D5C2314yes—
77C22×C4×D5C2318yes—
78C22×Dic10C2312yes—
79C22×Dic10C2314yes—
80C5×C4⋊Q8C2318yes—
81C5×C4⋊Q8C23132yes—
82C5×C4⋊Q8C23132yes—
83C4×C5×Q8C23116yes—
84C4×C5×Q8C23132yes—
85C2×C5×C4⋊C4C2314yes—
86C2×C5×C4⋊C4C2318yes—
87C2×C5×C4⋊C4C2312yes—
88C2×C5×C4⋊C4C2314yes—
89C2×C42×C5C2311yescharacteristic
90C2×C42×C5C2314yes—
91C2×C42×C5C2314yes—
92#4093C23132yes—
93#4093C23116yes—
94#4092C23164yes—
95#4092C23132yes—
96#4092C23116yes—
97#4090C23116yes—
98#4090C23132yes—
99C2×#685C2314yes—
100C2×#685C2314yes—
101C2×#685C2314yes—
102C2×#685C2318yes—
103C2×#685C2314yes—
104C2×#685C2314yes—
105C2×#685C2318yes—
106C2×C4⋊Dic5C2314yes—
107C2×C4⋊Dic5C2312yes—
108C2×C4⋊Dic5C2314yes—
109C2×#683C2318yes—
110C2×#683C2318yes—
111C2×#683C2314yes—
112C2×#683C2314yes—
113C2×#683C2318yes—
114C2×C4×Dic5C2312yes—
115C2×C4×Dic5C2311yescharacteristic
116C2×C4×Dic5C2314yes—
117#4044C23132yes—
118#4042C23164yes—
119#4042C23164yes—
120#4041C23116yes—
121#4041C23132yes—
122#4040C23132yes—
123#4040C23164yes—
124#4040C23164yes—
125#4038C23132yes—
126D5×C4⋊C4C23132yes—
127D5×C4⋊C4C23132yes—
128#4036C23116yes—
129#4035C23164yes—
130#4034C23132yes—
131#4022C23164yes—
132#4021C23132yes—
133C4×D20C23132yes—
134C4×D20C23132yes—
135C4×D20C23116yes—
136#4018C23132yes—
137#4018C23132yes—
138C42×D5C2318yes—
139#4016C23132yes—
140#4015C2318yes—
141#3544—532no—
142C2×#563—54no—
143C2×Q82D511yescharacteristic, complemented
144C2×#555—51no—
145C2×#554—52no—
146C2×#552—54no—
147C2×#547—54no—
148C2×#544—54no—
149C2×#542—54no—
150C2×#536—54no—
151C22×C4⋊Q8—51no—
152C2×#529—52no—
153C23×D5C2414yes—
154C23×D5C2411yescharacteristic
155C2×C5×Q8—28no—
156C2×C5×Q8—44no—
157C2×C5×Q8—48no—
158C2×C5×Q8C2418yes—
159C2×C5×Q8C2418yes—
160C2×C5×Q8C24116yes—
161C22×C4×C5C2414yes—
162C22×C4×C5C2411yescharacteristic
163C22×C4×C5C2414yes—
164C22×C4×C5C2414yes—
165C22×C4×C5C2412yes—
166C22×Dic5C2411yescharacteristic
167C22×Dic5C2414yes—
168C22×Dic5C2412yes—
169C22×Dic5C2414yes—
170#713—464no—
171Q8×D5—432no—
172C2×D20—216no—
173C2×D20—28no—
174C2×D20—48no—
175C2×D20C24116yes—
176C2×D20C2418yes—
177C2×D20C24116yes—
178C2×C4×D5—24no—
179C2×C4×D5—216no—
180C2×C4×D5—28no—
181C2×C4×D5—216no—
182C2×C4×D5—416no—
183C2×C4×D5C2414yes—
184C2×C4×D5C24116yes—
185C2×C4×D5C2418yes—
186C2×C4×D5C24116yes—
187C2×C4×D5C24132yes—
188C2×Dic10—48no—
189C2×Dic10—44no—
190C2×Dic10C2418yes—
191C2×Dic10C24116yes—
192C5×C4⋊C4C24116yes—
193C5×C4⋊C4C24132yes—
194C5×C4⋊C4C2418yes—
195C5×C4⋊C4C24116yes—
196C42×C5C2414yes—
197C42×C5C24116yes—
198C42×C5C24116yes—
199#685C24116yes—
200#685C24116yes—
201#685C24116yes—
202#685C24132yes—
203#685C24116yes—
204#685C24116yes—
205#685C24132yes—
206C4⋊Dic5C24116yes—
207C4⋊Dic5C2418yes—
208C4⋊Dic5C24116yes—
209#683C24132yes—
210#683C24132yes—
211#683C24116yes—
212#683C24116yes—
213#683C24132yes—
214C4×Dic5C2418yes—
215C4×Dic5C2414yes—
216C4×Dic5C24116yes—
217C22×#41—54no—
218C23×Q8—52no—
219#563—564no—
220Q82C2×D5116yescomplemented
221#555—516no—
222#554—532no—
223#552—564no—
224#547—564no—
225#544—564no—
226#542—564no—
227#536—564no—
228C2×C4⋊Q8—52no—
229C2×C4⋊Q8—54no—
230C2×C4⋊Q8—516no—
231C2×C4⋊Q8—54no—
232C2×C4⋊Q8—51no—
233C2×C4⋊Q8C2×D511yescharacteristic
234C2×C4⋊Q8C2×D514yes—
235C2×C4⋊Q8C2×D514yes—
236#529—532no—
237C2×#126—54no—
238C2×#126—58no—
239C2×#125—58no—
240C2×#125—52no—
241C2×#125—54no—
242C2×#124—52no—
243C2×#124—54no—
244C2×#124—54no—
245C2×#123—54no—
246C2×#123—58no—
247C2×#123—58no—
248C2×C22⋊Q8—52no—
249C2×C22⋊Q8—54no—
250C2×C22⋊Q8—58no—
251C2×C22⋊Q8—58no—
252C2×C22⋊Q8—58no—
253C2×#121—52no—
254C2×#121—54no—
255C2×C4×Q8C2×D512yes—
256C2×C4×Q8C2×D514yes—
257C2×C4×D4—54no—
258C2×C4×D4—52no—
259C2×C4×D4—54no—
260C2×#117—54no—
261C2×#117—54no—
262C2×#117—54no—
263C22×C4⋊C4—54no—
264C22×C4⋊C4—54no—
265C22×C42—51no—
266C23×C5C2511yescharacteristic
267C22×D5—24no—
268C22×D5—28no—
269C22×D5—28no—
270C22×D5—28no—
271C22×D5H(5,𝔽2)14yes—
272C22×D5C2518yes—
273C22×D5C2512yes—
274C5×Q8—216no—
275C5×Q8—48no—
276C5×Q8—416no—
277C2×C4×C5—22no—
278C2×C4×C5—28no—
279C2×C4×C5—28no—
280C2×C4×C5—28no—
281C2×C4×C52−1+418yes—
282C2×C4×C5C2518yes—
283C2×C4×C5C2512yes—
284C2×C4×C5C2518yes—
285C2×C4×C5C2518yes—
286C2×C4×C5C2514yes—
287C2×Dic5—22no—
288C2×Dic5—28no—
289C2×Dic5—24no—
290C2×Dic5—28no—
291C2×Dic5C2514yes—
292C2×Dic5C2518yes—
293C2×Dic5C2512yes—
294C2×Dic5C2518yes—
295D20—232no—
296D20—216no—
297D20—416no—
298C4×D5—28no—
299C4×D5—232no—
300C4×D5—216no—
301C4×D5—232no—
302C4×D5—432no—
303Dic10—416no—
304Dic10—48no—
305C2×#41—532no—
306C2×#41—2016no—
307C22×Q8—516no—
308C22×Q8—52no—
309C22×Q8—54no—
310C22×Q8—208no—
311C22×Q8C22×D512yes—
312C22×Q8C22×D514yes—
313C22×Q8C22×D512yes—
314C22×D4—52no—
315C22×D4—54no—
316C22×D4—54no—
317C23×C4—54no—
318C23×C4—52no—
319C23×C4—54no—
320C23×C4—51no—
321C23×C4—58no—
322C4⋊Q8—516no—
323C4⋊Q8—532no—
324C4⋊Q8—532no—
325C4⋊Q8—58no—
326C4⋊Q8C22×D518yes—
327C4⋊Q8C22×D5132yes—
328C4⋊Q8C22×D5132yes—
329#126—532no—
330#126—564no—
331#125—516no—
332#125—564no—
333#125—532no—
334#124—516no—
335#124—532no—
336#124—532no—
337#123—532no—
338#123—564no—
339#123—564no—
340C22⋊Q8—516no—
341C22⋊Q8—532no—
342C22⋊Q8—564no—
343C22⋊Q8—564no—
344C22⋊Q8—564no—
345#121—516no—
346#121—532no—
347C4×Q8C22×D5116yes—
348C4×Q8C22×D5132yes—
349C4×D4—532no—
350C4×D4—516no—
351C4×D4—532no—
352#117—532no—
353#117—532no—
354#117—532no—
355C2×C4⋊C4—54no—
356C2×C4⋊C4—58no—
357C2×C4⋊C4—58no—
358C2×C4⋊C4—52no—
359C2×C4⋊C4—58no—
360C2×C4⋊C4—54no—
361C2×C4⋊C4—54no—
362C2×C4⋊C4—54no—
363C2×C4⋊C4—532no—
364C2×C4⋊C4—532no—
365C2×C4⋊C4C22×D512yes—
366C2×C4⋊C4C22×D514yes—
367C2×C4⋊C4C22×D514yes—
368C2×C4⋊C4C22×D518yes—
369C2×C22⋊C4—54no—
370C2×C22⋊C4—54no—
371C2×C22⋊C4—58no—
372C2×C22⋊C4—54no—
373C2×C22⋊C4—58no—
374C2×C22⋊C4—54no—
375C2×C22⋊C4—54no—
376C2×C42—54no—
377C2×C42—52no—
378C2×C42—51no—
379C2×C42—58no—
380C2×C42C22×D514yes—
381C2×C42C22×D514yes—
382C2×C42C22×D511yescharacteristic
383C22×C5C2×H(5,𝔽2)12yes—
384C22×C5C2×2−1+414yes—
385C22×C5C2611yescharacteristic
386C2×D5—24no—
387C2×D5—28no—
388C2×D5—28no—
389C2×D5—28no—
390C2×D5—28no—
391C2×D5—416no—
392C2×D5C2×H(5,𝔽2)14yes—
393C4×C5—22no—
394C4×C5—28no—
395C4×C5—28no—
396C4×C5—28no—
397C4×C5C2×2−1+418yes—
398Dic5—22no—
399Dic5—28no—
400Dic5—24no—
401Dic5—28no—
402C24—51no—
403C24—54no—
404#41—2064no—
405C2×Q8—28no—
406C2×Q8—44no—
407C2×Q8—48no—
408C2×Q8—58no—
409C2×Q8—516no—
410C2×Q8—208no—
411C2×Q8—204no—
412C2×Q8—2032no—
413C2×Q8C23×D518yes—
414C2×Q8C23×D5116yes—
415C2×Q8C23×D518yes—
416C2×D4—58no—
417C2×D4—516no—
418C2×D4—516no—
419C2×D4—1016no—
420C2×D4—108no—
421C2×D4—208no—
422C22×C4—54no—
423C22×C4—52no—
424C22×C4—51no—
425C22×C4—54no—
426C22×C4—58no—
427C22×C4—54no—
428C22×C4—516no—
429C22×C4—516no—
430C22×C4—532no—
431C22×C4—104no—
432C22×C4—1016no—
433C22×C4—108no—
434C22×C4—1016no—
435C22×C4—2016no—
436C22×C4C23×D511yescharacteristic
437C22×C4C23×D514yes—
438C22×C4C23×D514yes—
439C22×C4C23×D514yes—
440C22×C4C23×D512yes—
441C4⋊C4—516no—
442C4⋊C4—532no—
443C4⋊C4—532no—
444C4⋊C4—58no—
445C4⋊C4—532no—
446C4⋊C4—516no—
447C4⋊C4—516no—
448C4⋊C4—516no—
449C4⋊C4C23×D518yes—
450C4⋊C4C23×D5116yes—
451C4⋊C4C23×D5116yes—
452C4⋊C4C23×D5132yes—
453C22⋊C4—516no—
454C22⋊C4—516no—
455C22⋊C4—532no—
456C22⋊C4—516no—
457C22⋊C4—532no—
458C22⋊C4—516no—
459C22⋊C4—516no—
460C42—58no—
461C42—54no—
462C42—516no—
463C42C23×D514yes—
464C42C23×D5116yes—
465C42C23×D5116yes—
466C2×C5#354414yescomplemented
467C2×C5C22×H(5,𝔽2)11yescharacteristic
468C2×C5C22×2−1+412yes—
469D5—24no—
470D5—48no—
471C23—54no—
472C23—52no—
473C23—58no—
474C23—104no—
475C23—108no—
476C23—108no—
477C23—108no—
478C23C24×D511yescharacteristic
479Q8—216no—
480Q8—48no—
481Q8—416no—
482Q8—2016no—
483Q8—208no—
484D4—1032no—
485D4—1016no—
486D4—2016no—
487C2×C4—28no—
488C2×C4—28no—
489C2×C4—22no—
490C2×C4—28no—
491C2×C4—54no—
492C2×C4—58no—
493C2×C4—52no—
494C2×C4—58no—
495C2×C4—108no—
496C2×C4—1032no—
497C2×C4—108no—
498C2×C4—104no—
499C2×C4—108no—
500C2×C4—102no—
501C2×C4—1016no—
502C2×C4—1032no—
503C2×C4—2032no—
504C2×C4#414718yes—
505C2×C4C24×D512yes—
506C2×C4C24×D518yes—
507C2×C4C24×D518yes—
508C2×C4C24×D518yes—
509C2×C4C24×D514yes—
510C5C2×#354411yescharacteristic, complemented, Sylow
511C22—54no—
512C22—104no—
513C22—108no—
514C22—108no—
515C22—108no—
516C22—108no—
517C22—2016no—
518C22C2×#414714yes—
519C22D5×H(5,𝔽2)12yes—
520C22C25×D511yescharacteristic
521C4—28no—
522C4—28no—
523C4—22no—
524C4—28no—
525C4—104no—
526C4—108no—
527C4—108no—
528C4—102no—
529C4C2×#414718yes—
530C2—104no—
531C2—208no—
532C2#1061894814yesdirect product factor
533C2C22×#414712yes—
534C2C2×D5×H(5,𝔽2)11yescharacteristic
subgroupG/H
centerC23C24×D5
commutator, cosocleC22×C5C26
FrattiniC22C25×D5
FittingC2×C5×Q82C2
socleC23×C5C25
hypercenterC2×Q82D5
hypocenterC5C2×#3544
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×Q82

Complements

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

NH ≅ G/Ntype
1. C2×C5×Q82C2semidirect
2. #10618948C2direct (N × H)
3. C2×#66069C2semidirect
4. C2×#66060C2semidirect
5. C2×#66063C2semidirect
6. C2×#66058C2semidirect
7. C2×D5×C4⋊Q8C2semidirect
8. C2×#66068C2semidirect
9. C2×#65913C2semidirect
10. C2×#65915C2semidirect
11. C2×#65916C2semidirect
12. C2×#65914C2semidirect
15. C5×Q82C22semidirect
28. #66069C22semidirect
29. #66068C22semidirect
30. #66063C22semidirect
31. #66060C22semidirect
32. #66058C22semidirect
33. D5×C4⋊Q8C22semidirect
34. #65916C22semidirect
35. #65915C22semidirect
36. #65914C22semidirect
37. #65913C22semidirect
143. C2×Q82D5semidirect
220. Q82C2×D5semidirect
466. C2×C5#3544semidirect
510. C5C2×#3544semidirect
532. C2#10618948direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×Q822
#106189482
C2×#660692
C2×#660602
C2×#660632
C2×#660582
C2×D5×C4⋊Q82
C2×#660682
C2×#659132
C2×#659152
C2×#659162
C2×#659142
C2×#35445