Finite Group Databasebeta

← C2×#10617261 #13044792

Subgroups — C2×#10617261 #13044792

Order 1280.

1982 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65956C211yesmaximal, characteristic
2C2×#65788C211yesmaximal, characteristic, complemented
3C2×Q8×#170C211yesmaximal, characteristic, complemented
4C2×#65758C211yesmaximal, characteristic
5C22×C4×#170C211yesmaximal, characteristic
6#10617261C2116yesmaximal, direct product factor
7C2×#65707C212yesmaximal, complemented
8C2×#65709C211yesmaximal, characteristic, complemented
9C2×#65703C211yesmaximal, characteristic, complemented
10C2×#65703C212yesmaximal, complemented
11C2×#65696C212yesmaximal
12C2×#65691C212yesmaximal
13C22×#704C2212yes—
14C23×#170C2211yescharacteristic
15C23×#170C2211yescharacteristic
16C2×C5×C22⋊Q8C411yescharacteristic
17C2×Q8×Dic5C2211yescharacteristic
18C2×#4090C411yescharacteristic
19C2×#4070C411yescharacteristic
20C22×C4×Dic5C2211yescharacteristic
21#65956C418yes—
22#65956C2218yes—
23C2×#4035C412yes—
24C2×#4034C411yescharacteristic
25C2×#4033C2211yescharacteristic
26C2×#4033C2212yes—
27C2×#4024C412yes—
28C2×#4023C2212yes—
29#65788C2218yescomplemented
30Q8×#170C2218yescomplemented
31#65758C2218yes—
32C2×#4004C2212yes—
33C2×#4004C2212yes—
34C2×#4004C2212yes—
35C2×#4002C2212yes—
36C2×#4002C2212yes—
37C2×C4⋊#170C2211yescharacteristic
38C2×C4⋊#170C2211yescharacteristic
39C2×C4⋊#170C2212yes—
40C2×C4⋊#170C2212yes—
41C2×C4×#170C2218yes—
42C2×C4×#170C2211yescharacteristic
43C2×C4×#170C2211yescharacteristic
44C2×C4×#170C2211yescharacteristic
45C2×C4×#170C2211yescharacteristic
46C2×C4×#170C2212yes—
47C2×C4×#170C2212yes—
48#65709C2218yescomplemented
49#65707C22116yescomplemented
50#65703C2218yescomplemented
51#65703C22116yescomplemented
52#65696C22116yes—
53#65691C22116yes—
54C2×#2960—51nomaximal, Sylow
55C22×C5×Q8C2×C411yescharacteristic
56C23×C4×C5C2×C411yescharacteristic
57C23×Dic5C2×C411yescharacteristic
58C23×Dic5C2311yescharacteristic
59C22×Dic10C2×C412yes—
60C22×Dic10C2×C411yescharacteristic
61C2×#704—28no—
62C2×#704C2318yes—
63C22×#170—24no—
64C22×#170Q818yescomplemented
65C22×#170C2314yes—
66C22×#170C2311yescharacteristic
67C22×#170C2311yescharacteristic
68C22×#170C2312yes—
69C22×#170C2312yes—
70C22×#170C2311yescharacteristic
71C22×#170C2311yescharacteristic
72C22×#170C2314yes—
73C5×C22⋊Q8C2×C418yes—
74C2×C5×C4⋊C4C2×C411yescharacteristic
75C2×C5×C4⋊C4C2×C412yes—
76C2×C5×C22⋊C4C2×C412yes—
77C2×C22⋊Dic5C2×C412yes—
78Q8×Dic5C2×C414yes—
79Q8×Dic5C2314yes—
80#4090C2×C418yes—
81C2×C4⋊Dic5C2×C412yes—
82C2×C4⋊Dic5C2×C411yescharacteristic
83#4070C2×C418yes—
84C2×#683C2×C412yes—
85C2×#683C2×C412yes—
86C2×#683C2×C412yes—
87C2×#683C2×C412yes—
88C2×#683C2×C412yes—
89C2×C4×Dic5C2×C411yescharacteristic
90C2×C4×Dic5C2×C411yescharacteristic
91C2×C4×Dic5C2×C414yes—
92C2×C4×Dic5C2312yes—
93C2×C4×Dic5C2312yes—
94C2×C4×Dic5C2311yescharacteristic
95C2×C4×Dic5C2311yescharacteristic
96C2×C4×Dic5C2314yes—
97#4035C2×C4116yes—
98#4034C2×C418yes—
99#4033C2×C414yes—
100#4033C2×C418yes—
101#4033C2318yes—
102#4033C2314yes—
103#4024C2×C4116yes—
104#4023C2×C418yes—
105#4023C2318yes—
106#4004C2318yes—
107#4004C2318yes—
108#4004C2318yes—
109#4002C2318yes—
110#4002C2318yes—
111C4⋊#170C2318yes—
112C4⋊#170C2318yes—
113C4⋊#170C2314yes—
114C4⋊#170C2314yes—
115C4×#170C2318yes—
116C4×#170C2318yes—
117C4×#170C2314yes—
118C4×#170C2314yes—
119C4×#170C2314yes—
120C4×#170C2314yes—
121C2×#533—51no—
122#2960—516no—
123C2×#446—51no—
124C2×#446—51no—
125C2×#446—52no—
126C2×C8×Q8—51no—
127C2×C8×Q8—51no—
128C2×C8×Q8—52no—
129C2×#424—52no—
130C2×#423—51no—
131C2×#405—52no—
132C22×C4×C8—51no—
133C24×C5C22×C411yescharacteristic
134C2×C5×Q8—24no—
135C2×C5×Q8C22×C414yes—
136C22×C4×C5—24no—
137C22×C4×C5C22×C412yes—
138C22×C4×C5C22×C412yes—
139C22×C4×C5C22×C411yescharacteristic
140C22×C4×C5C22×C414yes—
141C22×C4×C5C22×C411yescharacteristic
142C22×Dic5—24no—
143C22×Dic5C22×C412yes—
144C22×Dic5C22×C412yes—
145C22×Dic5C22×C414yes—
146C22×Dic5C22×C411yescharacteristic
147C22×Dic5C22×C412yes—
148C22×Dic5C22×C411yescharacteristic
149C22×Dic5C22×C411yescharacteristic
150C22×Dic5C2×Q814yes—
151C22×Dic5#4114yes—
152C22×Dic5C2411yescharacteristic
153C22×Dic5C2412yes—
154C2×Dic10—24no—
155C2×Dic10—28no—
156C2×Dic10C22×C418yes—
157C2×Dic10C22×C414yes—
158#704—216no—
159C2×#170—22no—
160C2×#170—22no—
161C2×#170—28no—
162C2×#170—24no—
163C2×#170—24no—
164C2×#170C2×Q814yes—
165C2×#170C2×Q814yes—
166C2×#170C2×Q8116yescomplemented
167C2×#170C2412yes—
168C2×#170C2412yes—
169C2×#170C2414yes—
170C2×#170C2414yes—
171C2×#170C2412yes—
172C2×#170C2412yes—
173C5×C4⋊C4C22×C414yes—
174C5×C4⋊C4C22×C418yes—
175C5×C22⋊C4C22×C418yes—
176C22⋊Dic5C22×C418yes—
177C4⋊Dic5C22×C418yes—
178C4⋊Dic5C22×C414yes—
179#683C22×C418yes—
180#683C22×C418yes—
181#683C22×C418yes—
182#683C22×C418yes—
183#683C22×C418yes—
184C4×Dic5C22×C414yes—
185C4×Dic5C22×C414yes—
186C4×Dic5C22×C412yes—
187C4×Dic5C22×C412yes—
188C4×Dic5C22×C414yes—
189C4×Dic5C22×C414yes—
190C4×Dic5C2412yes—
191C4×Dic5C2412yes—
192C4×Dic5C2414yes—
193C4×Dic5C2414yes—
194C22×M16—52no—
195C23×C8—51no—
196C23×C8—51no—
197#533—58no—
198#533—58no—
199C2×C4⋊Q8—51no—
200C2×C4⋊Q8—51no—
201C2×#125—52no—
202C2×C22⋊Q8—51no—
203C2×C22⋊Q8—52no—
204C2×C22⋊Q8AGL(1,𝔽5)11yescharacteristic
205C2×C4×Q8—51no—
206C2×C4×Q8—51no—
207C2×C4×Q8—52no—
208C2×#117—52no—
209C22×C42—51no—
210#446—516no—
211#446—58no—
212#446—58no—
213C8×Q8—58no—
214C8×Q8—516no—
215C8×Q8—58no—
216#424—516no—
217#423—58no—
218C2×C4⋊C8—51no—
219C2×C4⋊C8—51no—
220C2×C4⋊C8—52no—
221C2×C4⋊C8—52no—
222C2×C4⋊C8—52no—
223C2×C4⋊C8—52no—
224C2×C4⋊C8—52no—
225#405—516no—
226C2×#97—52no—
227C2×#97—52no—
228C2×C4×C8—51no—
229C2×C4×C8—51no—
230C2×C4×C8—51no—
231C2×C4×C8—51no—
232C2×C4×C8—58no—
233C2×C4×C8—52no—
234C2×C4×C8—52no—
235C23×C5—22no—
236C23×C5—22no—
237C23×C5C4×Q814yes—
238C23×C5C23×C412yes—
239C23×C5C23×C411yescharacteristic
240C5×Q8—28no—
241C2×C4×C5—24no—
242C2×C4×C5—22no—
243C2×C4×C5—24no—
244C2×C4×C5—28no—
245C2×C4×C5#13114yes—
246C2×C4×C5C23×C412yes—
247C2×C4×C5C23×C414yes—
248C2×C4×C5C23×C414yes—
249C2×C4×C5C23×C412yes—
250C2×Dic5—24no—
251C2×Dic5—24no—
252C2×Dic5—22no—
253C2×Dic5—28no—
254C2×Dic5C4×Q818yes—
255C2×Dic5C4×Q814yes—
256C2×Dic5#13114yes—
257C2×Dic5C23×C412yes—
258C2×Dic5C23×C414yes—
259C2×Dic5C23×C414yes—
260C2×Dic5C23×C411yescharacteristic
261C2×Dic5C23×C412yes—
262C2×Dic5C23×C412yes—
263C2×Dic5C22×Q812yes—
264C2×Dic5C22×Q814yes—
265C2×Dic5C2×#4112yes—
266C2×Dic5C2×#4114yes—
267C2×Dic5C2511yescharacteristic
268Dic10—28no—
269Dic10—216no—
270#170—22no—
271#170—22no—
272#170—24no—
273#170—24no—
274#170C22×Q814yes—
275#170C22×Q814yes—
276C22×Q8—52no—
277C22×Q8—51no—
278C22×Q8C2×AGL(1,𝔽5)11yescharacteristic
279C23×C4—51no—
280C23×C4—51no—
281C23×C4C2×AGL(1,𝔽5)11yescharacteristic
282C2×M16—58no—
283C2×M16—108no—
284C22×C8—58no—
285C22×C8—51no—
286C22×C8—51no—
287C22×C8—54no—
288C22×C8—51no—
289C22×C8—51no—
290C22×C8—54no—
291C22×C8—52no—
292C22×C8—52no—
293C22×C8—104no—
294C4⋊Q8—58no—
295C4⋊Q8—58no—
296#125—516no—
297C22⋊Q8—58no—
298C22⋊Q8—516no—
299C22⋊Q8C2×AGL(1,𝔽5)18yes—
300C4×Q8—54no—
301C4×Q8—54no—
302C4×Q8—54no—
303C4×Q8—58no—
304C4×Q8—54no—
305C4×Q8—58no—
306#117—58no—
307#117—58no—
308C2×C4⋊C4—52no—
309C2×C4⋊C4—52no—
310C2×C4⋊C4—52no—
311C2×C4⋊C4—52no—
312C2×C4⋊C4—52no—
313C2×C4⋊C4—52no—
314C2×C4⋊C4—51no—
315C2×C4⋊C4C2×AGL(1,𝔽5)12yes—
316C2×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
317C2×C22⋊C4—52no—
318C2×C22⋊C4C2×AGL(1,𝔽5)12yes—
319C2×C42—52no—
320C2×C42—52no—
321C2×C42—51no—
322C2×C42—51no—
323C2×C42—51no—
324C2×C42—51no—
325C2×C42—54no—
326C2×C42—54no—
327C4⋊C8—54no—
328C4⋊C8—54no—
329C4⋊C8—58no—
330C4⋊C8—58no—
331C4⋊C8—58no—
332C4⋊C8—58no—
333C4⋊C8—58no—
334#97—58no—
335#97—58no—
336C4×C8—54no—
337C4×C8—54no—
338C4×C8—54no—
339C4×C8—54no—
340C4×C8—58no—
341C4×C8—58no—
342C22×C5—28no—
343C22×C5—22no—
344C22×C5—22no—
345C22×C5C2×C4×Q814yes—
346C22×C5C2×C4×Q812yes—
347C22×C5C2×#13112yes—
348C22×C5C2×#13112yes—
349C22×C5C24×C411yescharacteristic
350C4×C5—24no—
351C4×C5—22no—
352C4×C5—24no—
353C4×C5C2×#13114yes—
354Dic5—24no—
355Dic5—24no—
356Dic5—22no—
357Dic5C2×C4×Q812yes—
358Dic5C2×C4×Q814yes—
359Dic5C2×#13114yes—
360Dic5C23×Q811yescharacteristic
361Dic5C22×#4111yescharacteristic
362C24C22×AGL(1,𝔽5)11yescharacteristic
363C2×Q8—24no—
364C2×Q8—58no—
365C2×Q8—54no—
366C2×Q8—108no—
367C2×Q8—104no—
368C2×Q8C22×AGL(1,𝔽5)14yes—
369C22×C4—24no—
370C22×C4—54no—
371C22×C4—54no—
372C22×C4—52no—
373C22×C4—52no—
374C22×C4—51no—
375C22×C4—51no—
376C22×C4—51no—
377C22×C4—54no—
378C22×C4—51no—
379C22×C4—52no—
380C22×C4—52no—
381C22×C4—104no—
382C22×C4C22×AGL(1,𝔽5)14yes—
383C22×C4C22×AGL(1,𝔽5)11yescharacteristic
384C22×C4C22×AGL(1,𝔽5)12yes—
385C22×C4C22×AGL(1,𝔽5)12yes—
386C22×C4C22×AGL(1,𝔽5)11yescharacteristic
387M16—1016no—
388C2×C8—52no—
389C2×C8—52no—
390C2×C8—54no—
391C2×C8—54no—
392C2×C8—516no—
393C2×C8—52no—
394C2×C8—52no—
395C2×C8—54no—
396C2×C8—54no—
397C2×C8—102no—
398C2×C8—102no—
399C2×C8—108no—
400C2×C8—104no—
401C2×C8—104no—
402C4⋊C4—58no—
403C4⋊C4—58no—
404C4⋊C4—58no—
405C4⋊C4—58no—
406C4⋊C4—58no—
407C4⋊C4—58no—
408C4⋊C4—54no—
409C4⋊C4C22×AGL(1,𝔽5)18yes—
410C4⋊C4C22×AGL(1,𝔽5)14yes—
411C22⋊C4—58no—
412C22⋊C4C22×AGL(1,𝔽5)18yes—
413C42—52no—
414C42—52no—
415C42—54no—
416C42—54no—
417C42—52no—
418C42—52no—
419C42—54no—
420C42—54no—
421C42—54no—
422C42—54no—
423C2×C5—24no—
424C2×C5#296014yescomplemented
425C2×C5C22×C4×Q811yescharacteristic
426C2×C5C22×#13111yescharacteristic
427C2×C5C22×#13111yescharacteristic
428C23—22no—
429C23—22no—
430C23Q8×AGL(1,𝔽5)14yes—
431C23C23×AGL(1,𝔽5)12yes—
432C23C23×AGL(1,𝔽5)11yescharacteristic
433Q8—28no—
434Q8—1016no—
435Q8—108no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—28no—
440C2×C4—54no—
441C2×C4—52no—
442C2×C4—52no—
443C2×C4—54no—
444C2×C4—58no—
445C2×C4—54no—
446C2×C4—54no—
447C2×C4—51no—
448C2×C4—51no—
449C2×C4—52no—
450C2×C4—52no—
451C2×C4—52no—
452C2×C4—54no—
453C2×C4—54no—
454C2×C4—108no—
455C2×C4—104no—
456C2×C4—102no—
457C2×C4—104no—
458C2×C4#413114yes—
459C2×C4C23×AGL(1,𝔽5)12yes—
460C2×C4C23×AGL(1,𝔽5)14yes—
461C2×C4C23×AGL(1,𝔽5)14yes—
462C2×C4C23×AGL(1,𝔽5)12yes—
463C8—54no—
464C8—54no—
465C8—102no—
466C8—102no—
467C8—104no—
468C8—104no—
469C5C2×#296011yescharacteristic, complemented, Sylow
470C22—22no—
471C22—22no—
472C22—28no—
473C22C2×#413112yes—
474C22C2×Q8×AGL(1,𝔽5)14yes—
475C22C2×Q8×AGL(1,𝔽5)12yes—
476C22D5⋊#13112yes—
477C22C24×AGL(1,𝔽5)11yescharacteristic
478C4—22no—
479C4—24no—
480C4—24no—
481C4—51no—
482C4—51no—
483C4—52no—
484C4—54no—
485C4—54no—
486C4—104no—
487C4—102no—
488C4—104no—
489C4C2×#413114yes—
490C2—24no—
491C2#1061726114yesdirect product factor
492C2C22×#413111yescharacteristic
493C2C22×Q8×AGL(1,𝔽5)11yescharacteristic
494C2C2×D5⋊#13111yescharacteristic
subgroupG/H
centerC23C23×AGL(1,𝔽5)
commutatorC22×C5C24×C4
FrattiniC22C24×AGL(1,𝔽5)
FittingC2×C5×C22⋊Q8C4
socleC23×C5C23×C4
prime radicalC24×C5C22×C4
cosocleC2×Dic5C25
hypercenterC2×C22⋊Q8AGL(1,𝔽5)
hypocenterC5C2×#2960
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#65788C2semidirect
3. C2×Q8×#170C2semidirect
6. #10617261C2direct (N × H)
7. C2×#65707C2semidirect
8. C2×#65709C2semidirect
9. C2×#65703C2semidirect
10. C2×#65703C2semidirect
29. #65788C22semidirect
30. Q8×#170C22semidirect
48. #65709C22semidirect
49. #65707C22semidirect
50. #65703C22semidirect
51. #65703C22semidirect
64. C22×#170Q8semidirect
166. C2×#170C2×Q8semidirect
424. C2×C5#2960semidirect
469. C5C2×#2960semidirect
491. C2#10617261direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#659562
C2×#657882
C2×Q8×#1702
C2×#657582
C22×C4×#1702
#106172612
C2×#657072
C2×#657092
C2×#657032
C2×#656962
C2×#656912
C2×#29605