Finite Group Databasebeta

← C2×#10618118 #13006984

Subgroups — C2×#10618118 #13006984

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
1C22×C5×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×#66150C214yesmaximal, complemented
3#10618118C2132yesmaximal, direct product factor
4C2×#66069C214yesmaximal, complemented
5C2×#66064C212yesmaximal, complemented
6C2×#66063C214yesmaximal, complemented
7C2×#66067C214yesmaximal, complemented
8C2×#66056C211yesmaximal, characteristic, complemented
9C2×#66068C211yesmaximal, characteristic, complemented
10C2×#65841C214yesmaximal, complemented
11C2×#65847C214yesmaximal, complemented
12C2×#65813C212yesmaximal, complemented
13C23×C5×Q8C2212yes—
14C22×#709C2214yes—
15C2×C5×C4⋊Q8C2211yescharacteristic
16C2×C5×C4⋊Q8C2212yes—
17C2×C5×C4⋊Q8C2211yescharacteristic
18C2×C5×C4⋊Q8C2214yes—
19C2×C5×C4⋊Q8C22116yescomplemented
20C22×C5×C4⋊C4C2214yes—
21C22×C42×C5C2211yescharacteristic
22#66150C22164yescomplemented
23C2×#4093C2214yes—
24C2×#4093C2214yes—
25C2×#4092C2218yes—
26C2×#4092C2218yes—
27C2×#4090C2214yes—
28C2×#4090C2214yes—
29C2×#4076C2212yes—
30C2×#4075C2218yes—
31C2×#4075C2214yes—
32C2×C4×C22⋊D5C2214yes—
33C2×#4072C2214yes—
34C2×#4070C2212yes—
35#66069C22164yescomplemented
36#66068C22116yescomplemented
37#66067C22164yescomplemented
38#66064C22132yescomplemented
39#66063C22164yescomplemented
40#66056C22116yescomplemented
41#65847C22164yescomplemented
42C2×#4044C2218yes—
43C2×#4043C2214yes—
44C2×#4042C2218yes—
45#65841C22164yescomplemented
46C2×#4041C2214yes—
47C2×#4040C2218yes—
48C2×#4039C2214yes—
49C2×#4038C2214yes—
50C2×D5×C4⋊C4C2214yes—
51C2×#4036C2214yes—
52C2×#4035C2218yes—
53C2×#4034C2214yes—
54C2×#4033C2214yes—
55#65813C22132yescomplemented
56C2×#4022C2214yes—
57C2×#4021C2212yes—
58C2×C4×D20C2212yes—
59C2×#4018C2214yes—
60C2×#4016C2212yes—
61C2×C4×Dic10C2212yes—
62C2×#3544—51nomaximal, Sylow
63C22×C5×Q8—48no—
64C22×C5×Q8C2312yes—
65C22×C5×Q8C2314yes—
66C22×C5×Q8C2312yes—
67C22×C5×Q8C23116yes—
68C23×C4×C5C2312yes—
69C23×C4×C5C2314yes—
70C23×C4×C5C2311yescharacteristic
71C22×C22⋊D5C2314yes—
72C2×#709—416no—
73C2×#709C23132yes—
74C22×D20C2312yes—
75C22×D20C2314yes—
76C22×C4×D5C2314yes—
77C22×C4×D5C2318yes—
78C22×Dic10C2312yes—
79C22×Dic10C2314yes—
80C5×C4⋊Q8C2318yes—
81C5×C4⋊Q8C23132yes—
82C5×C4⋊Q8C2318yes—
83C5×C4⋊Q8C23116yes—
84C2×C5×C4⋊C4C2314yes—
85C2×C5×C4⋊C4C2318yes—
86C2×C5×C4⋊C4C2314yes—
87C2×C5×C4⋊C4C23132yes—
88C2×C42×C5C2311yescharacteristic
89C2×C42×C5C2312yes—
90C2×C42×C5C2311yescharacteristic
91C2×C42×C5C2318yes—
92C2×C22⋊Dic5C2314yes—
93#4093C23132yes—
94#4093C23132yes—
95#4092C23164yes—
96#4092C23164yes—
97#4090C23132yes—
98#4090C23132yes—
99#4076C23116yes—
100#4075C23132yes—
101#4075C23164yes—
102C4×C22⋊D5C23132yes—
103C2×#685C2314yes—
104C2×#685C2314yes—
105C2×#685C2314yes—
106C2×#685C2314yes—
107C2×#685C2318yes—
108C2×#685C2318yes—
109#4072C23132yes—
110C2×C4⋊Dic5C2312yes—
111C2×C4⋊Dic5C2312yes—
112C2×C4⋊Dic5C2318yes—
113#4070C23116yes—
114C2×#683C2314yes—
115C2×#683C2314yes—
116C2×#683C2314yes—
117C2×#683C2314yes—
118C2×#683C2318yes—
119C2×#683C2318yes—
120C2×C4×Dic5C2314yes—
121C2×C4×Dic5C2314yes—
122C2×C4×Dic5C2314yes—
123#4044C23164yes—
124#4043C23132yes—
125#4042C23164yes—
126#4041C23132yes—
127#4040C23164yes—
128#4039C23132yes—
129#4038C23132yes—
130D5×C4⋊C4C23132yes—
131#4036C23132yes—
132#4035C23164yes—
133#4034C23132yes—
134#4033C23132yes—
135#4022C23132yes—
136#4021C23116yes—
137C4×D20C23116yes—
138#4018C23132yes—
139#4016C23116yes—
140C4×Dic10C23116yes—
141#3544—532no—
142C2×#563—54no—
143C2×Q82—51no—
144C2×#555—51no—
145C2×#554—52no—
146C2×#552—54no—
147C2×#547—54no—
148C2×#544—54no—
149C2×#542—54no—
150C2×#536—54no—
151C22×C4⋊Q8D511yescharacteristic, complemented
152C2×#529—52no—
153C24×C5C2411yescharacteristic
154C23×D5C2414yes—
155C2×C5×Q8—44no—
156C2×C5×Q8—48no—
157C2×C5×Q8—44no—
158C2×C5×Q8—432no—
159C2×C5×Q8C2418yes—
160C2×C5×Q8C24116yes—
161C2×C5×Q8C2418yes—
162C22×C4×C5—216no—
163C22×C4×C5—24no—
164C22×C4×C5—28no—
165C22×C4×C5C2412yes—
166C22×C4×C5C2418yes—
167C22×C4×C5C2411yescharacteristic
168C22×C4×C5C2411yescharacteristic
169C22×C4×C5C2414yes—
170C22×C4×C5C2414yes—
171C22×C4×C5C2414yes—
172C22×C4×C5C24116yes—
173C22×C4×C5C2412yes—
174C2×C22⋊D5—216no—
175C2×C22⋊D5C24116yes—
176C22×Dic5C2414yes—
177C22×Dic5C2414yes—
178C22×Dic5C2414yes—
179#709—464no—
180C2×D20—28no—
181C2×D20—48no—
182C2×D20C2418yes—
183C2×D20C24116yes—
184C2×C4×D5—216no—
185C2×C4×D5—416no—
186C2×C4×D5C24116yes—
187C2×C4×D5C24132yes—
188C2×Dic10—28no—
189C2×Dic10—48no—
190C2×Dic10C2418yes—
191C2×Dic10C24116yes—
192C5×C4⋊C4C24116yes—
193C5×C4⋊C4C24132yes—
194C5×C4⋊C4C24116yes—
195C42×C5C2414yes—
196C42×C5C2418yes—
197C42×C5C2414yes—
198C22⋊Dic5C24116yes—
199#685C24116yes—
200#685C24116yes—
201#685C24116yes—
202#685C24116yes—
203#685C24132yes—
204#685C24132yes—
205C4⋊Dic5C2418yes—
206C4⋊Dic5C2418yes—
207C4⋊Dic5C24132yes—
208#683C24116yes—
209#683C24116yes—
210#683C24116yes—
211#683C24116yes—
212#683C24132yes—
213#683C24132yes—
214C4×Dic5C24116yes—
215C4×Dic5C24116yes—
216C4×Dic5C24116yes—
217C22×#41—54no—
218C23×Q8C2×D512yes—
219#563—564no—
220Q82—516no—
221#555—516no—
222#554—532no—
223#552—564no—
224#547—564no—
225#544—564no—
226#542—564no—
227#536—564no—
228C2×C4⋊Q8—54no—
229C2×C4⋊Q8—54no—
230C2×C4⋊Q8—54no—
231C2×C4⋊Q8C2×D514yes—
232C2×C4⋊Q8C2×D511yescharacteristic
233C2×C4⋊Q8C2×D511yescharacteristic
234C2×C4⋊Q8C2×D512yes—
235C2×C4⋊Q8C2×D5116yescomplemented
236#529—532no—
237C2×#126—58no—
238C2×#126—54no—
239C2×#125—58no—
240C2×#125—54no—
241C2×#125—52no—
242C2×#124—54no—
243C2×#124—54no—
244C2×#124—52no—
245C2×#123—54no—
246C2×#123—58no—
247C2×#123—58no—
248C2×C22⋊Q8—58no—
249C2×C22⋊Q8—58no—
250C2×C22⋊Q8—52no—
251C2×C22⋊Q8—58no—
252C2×C22⋊Q8—54no—
253C2×#121—52no—
254C2×#121—54no—
255C2×C4×Q8—54no—
256C2×C4×Q8—52no—
257C2×C4×D4—54no—
258C2×C4×D4—54no—
259C2×C4×D4—52no—
260C2×#117—54no—
261C2×#117—54no—
262C2×#117—54no—
263C22×C4⋊C4—54no—
264C22×C4⋊C4C2×D514yes—
265C22×C42C2×D511yescharacteristic
266C23×C5—24no—
267C23×C5H(5,𝔽2)14yes—
268C23×C5C2512yes—
269C23×C5C2511yescharacteristic
270C22×D5—28no—
271C22×D5—28no—
272C22×D5—28no—
273C22×D5C2518yes—
274C5×Q8—48no—
275C5×Q8—416no—
276C5×Q8—48no—
277C2×C4×C5—28no—
278C2×C4×C5—28no—
279C2×C4×C5—232no—
280C2×C4×C5—28no—
281C2×C4×C5—22no—
282C2×C4×C5—22no—
283C2×C4×C5—24no—
284C2×C4×C5—216no—
285C2×C4×C52−1+418yes—
286C2×C4×C5C2518yes—
287C2×C4×C5C2514yes—
288C2×C4×C5C2512yes—
289C2×C4×C5C2512yes—
290C2×C4×C5C2518yes—
291C2×C4×C5C2514yes—
292C22⋊D5—232no—
293C2×Dic5—28no—
294C2×Dic5—28no—
295C2×Dic5—28no—
296C2×Dic5C2518yes—
297C2×Dic5C2518yes—
298C2×Dic5C2518yes—
299D20—216no—
300D20—416no—
301C4×D5—232no—
302C4×D5—432no—
303Dic10—216no—
304Dic10—416no—
305C2×#41—532no—
306C2×#41—2016no—
307C22×Q8—48no—
308C22×Q8—52no—
309C22×Q8—54no—
310C22×Q8C22×D512yes—
311C22×Q8C22×D514yes—
312C22×Q8C22×D512yes—
313C22×Q8C22×D5116yes—
314C22×D4—54no—
315C22×D4—52no—
316C22×D4—54no—
317C23×C4—54no—
318C23×C4—58no—
319C23×C4C22×D514yes—
320C23×C4C22×D512yes—
321C23×C4C22×D511yescharacteristic
322C4⋊Q8—532no—
323C4⋊Q8—532no—
324C4⋊Q8—532no—
325C4⋊Q8C22×D5132yes—
326C4⋊Q8C22×D518yes—
327C4⋊Q8C22×D518yes—
328C4⋊Q8C22×D5116yes—
329#126—564no—
330#126—532no—
331#125—532no—
332#125—564no—
333#125—516no—
334#124—532no—
335#124—532no—
336#124—516no—
337#123—532no—
338#123—564no—
339#123—564no—
340C22⋊Q8—564no—
341C22⋊Q8—564no—
342C22⋊Q8—516no—
343C22⋊Q8—532no—
344C22⋊Q8—564no—
345#121—516no—
346#121—532no—
347C4×Q8—532no—
348C4×Q8—516no—
349C4×D4—532no—
350C4×D4—532no—
351C4×D4—516no—
352#117—532no—
353#117—532no—
354#117—532no—
355C2×C4⋊C4—52no—
356C2×C4⋊C4—58no—
357C2×C4⋊C4—52no—
358C2×C4⋊C4—58no—
359C2×C4⋊C4—54no—
360C2×C4⋊C4—54no—
361C2×C4⋊C4—54no—
362C2×C4⋊C4—54no—
363C2×C4⋊C4—58no—
364C2×C4⋊C4—532no—
365C2×C4⋊C4C22×D514yes—
366C2×C4⋊C4C22×D514yes—
367C2×C4⋊C4C22×D518yes—
368C2×C4⋊C4C22×D5132yes—
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—54no—
378C2×C42—54no—
379C2×C42C22×D512yes—
380C2×C42C22×D511yescharacteristic
381C2×C42C22×D511yescharacteristic
382C2×C42C22×D518yes—
383C22×C5—24no—
384C22×C5—28no—
385C22×C5C2×H(5,𝔽2)14yes—
386C22×C5C2×H(5,𝔽2)12yes—
387C22×C5C2×2−1+414yes—
388C22×C5C2611yescharacteristic
389C2×D5—28no—
390C2×D5—28no—
391C2×D5—28no—
392C2×D5—416no—
393C4×C5—28no—
394C4×C5—28no—
395C4×C5—22no—
396C4×C5—22no—
397C4×C5—24no—
398C4×C5C2×2−1+418yes—
399Dic5—28no—
400Dic5—28no—
401Dic5—28no—
402C24—54no—
403C24C23×D511yescharacteristic
404#41—2064no—
405C2×Q8—44no—
406C2×Q8—48no—
407C2×Q8—44no—
408C2×Q8—432no—
409C2×Q8—58no—
410C2×Q8—516no—
411C2×Q8—108no—
412C2×Q8—208no—
413C2×Q8C23×D518yes—
414C2×Q8C23×D5116yes—
415C2×Q8C23×D518yes—
416C2×D4—516no—
417C2×D4—58no—
418C2×D4—516no—
419C2×D4—1016no—
420C2×D4—108no—
421C2×D4—208no—
422C22×C4—24no—
423C22×C4—216no—
424C22×C4—28no—
425C22×C4—54no—
426C22×C4—54no—
427C22×C4—54no—
428C22×C4—516no—
429C22×C4—532no—
430C22×C4—1016no—
431C22×C4—2016no—
432C22×C4C23×D518yes—
433C22×C4C23×D514yes—
434C22×C4C23×D5116yes—
435C22×C4C23×D514yes—
436C22×C4C23×D512yes—
437C22×C4C23×D511yescharacteristic
438C22×C4C23×D511yescharacteristic
439C22×C4C23×D514yes—
440C22×C4C23×D512yes—
441C4⋊C4—58no—
442C4⋊C4—532no—
443C4⋊C4—58no—
444C4⋊C4—532no—
445C4⋊C4—516no—
446C4⋊C4—516no—
447C4⋊C4—516no—
448C4⋊C4—516no—
449C4⋊C4—532no—
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—516no—
461C42—516no—
462C42—516no—
463C42C23×D518yes—
464C42C23×D514yes—
465C42C23×D514yes—
466C2×C5—24no—
467C2×C5#354414yescomplemented
468C2×C5C22×H(5,𝔽2)11yescharacteristic
469C2×C5C22×2−1+412yes—
470D5—48no—
471C23—24no—
472C23—58no—
473C23—108no—
474C23—108no—
475C23—108no—
476C23#414414yes—
477C23C24×D512yes—
478C23C24×D511yescharacteristic
479Q8—48no—
480Q8—416no—
481Q8—48no—
482Q8—1016no—
483Q8—2016no—
484D4—1032no—
485D4—1016no—
486D4—2016no—
487C2×C4—28no—
488C2×C4—28no—
489C2×C4—28no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—232no—
493C2×C4—24no—
494C2×C4—216no—
495C2×C4—58no—
496C2×C4—58no—
497C2×C4—58no—
498C2×C4—108no—
499C2×C4—108no—
500C2×C4—108no—
501C2×C4—1032no—
502C2×C4—2032no—
503C2×C4#414718yes—
504C2×C4C24×D518yes—
505C2×C4C24×D514yes—
506C2×C4C24×D512yes—
507C2×C4C24×D512yes—
508C2×C4C24×D518yes—
509C2×C4C24×D514yes—
510C5C2×#354411yescharacteristic, complemented, Sylow
511C22—28no—
512C22—24no—
513C22—108no—
514C22—108no—
515C22—108no—
516C22—2016no—
517C22C2×#414414yes—
518C22C2×#414412yes—
519C22C2×#414714yes—
520C22C25×D511yescharacteristic
521C4—28no—
522C4—28no—
523C4—22no—
524C4—22no—
525C4—24no—
526C4—108no—
527C4—108no—
528C4—108no—
529C4C2×#414718yes—
530C2—24no—
531C2—208no—
532C2#1061811814yesdirect product factor
533C2C22×#414411yescharacteristic
534C2C22×#414712yes—
subgroupG/H
centerC23C24×D5
commutator, cosocleC22×C5C26
FrattiniC22C25×D5
FittingC22×C5×C4⋊Q8C2
socleC23×C5C25
hypercenterC22×C4⋊Q8D5
hypocenterC5C2×#3544
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C22×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×C5×C4⋊Q8C2semidirect
2. C2×#66150C2semidirect
3. #10618118C2direct (N × H)
4. C2×#66069C2semidirect
5. C2×#66064C2semidirect
6. C2×#66063C2semidirect
7. C2×#66067C2semidirect
8. C2×#66056C2semidirect
9. C2×#66068C2semidirect
10. C2×#65841C2semidirect
11. C2×#65847C2semidirect
12. C2×#65813C2semidirect
19. C2×C5×C4⋊Q8C22semidirect
22. #66150C22semidirect
35. #66069C22semidirect
36. #66068C22semidirect
37. #66067C22semidirect
38. #66064C22semidirect
39. #66063C22semidirect
40. #66056C22semidirect
41. #65847C22semidirect
45. #65841C22semidirect
55. #65813C22semidirect
151. C22×C4⋊Q8D5semidirect
235. C2×C4⋊Q8C2×D5semidirect
467. C2×C5#3544semidirect
510. C5C2×#3544semidirect
532. C2#10618118direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C5×C4⋊Q82
C2×#661502
#106181182
C2×#660692
C2×#660642
C2×#660632
C2×#660672
C2×#660562
C2×#660682
C2×#658412
C2×#658472
C2×#658132
C2×#35445