Finite Group Databasebeta

← C2×#10613704 #12968913

Subgroups — C2×#10613704 #12968913

Order 1280.

1902 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66062C211yesmaximal, characteristic
2C2×#66022C211yesmaximal, characteristic
3C2×C5×#501C211yesmaximal, characteristic, complemented
4C2×#65369C211yesmaximal, characteristic, complemented
5C2×#65354C211yesmaximal, characteristic, complemented
6#10613704C2116yesmaximal, direct product factor
7C2×#65169C212yesmaximal
8C2×#65165C212yesmaximal, complemented
9C2×#65149C212yesmaximal
10C2×#65145C212yesmaximal, complemented
11C2×D5×#97C211yesmaximal, characteristic
12C2×#64991C211yesmaximal, characteristic, complemented
13C2×C5×C4⋊Q8C2211yescharacteristic
14C2×C5×#125C2211yescharacteristic
15C2×#4092—21no—
16C2×Q8×Dic5—21no—
17C22×C8×D5C2212yes—
18#66062—24no—
19#66062C2218yes—
20#66022—24no—
21#66022C2218yes—
22C2×#4043—21no—
23C2×#4042—21no—
24C2×#4042—21no—
25C2×#4038C2211yescharacteristic
26C2×#4038C2211yescharacteristic
27C2×#4038C2212yes—
28C2×#4036C2212yes—
29C2×#4034C2211yescharacteristic
30C2×#4034C2211yescharacteristic
31C2×#4033—21no—
32C2×#4033—21no—
33C2×#4033—21no—
34C2×C42×D5C2211yescharacteristic
35C2×#4016C2211yescharacteristic
36C2×#4015C2211yescharacteristic
37C5×#501C2218yescomplemented
38C2×C5×#107C2212yes—
39C2×C5×#106C2212yes—
40C2×C5×#97C2211yescharacteristic
41C2×#3950C2212yes—
42C2×#3949C2212yes—
43C2×#3947C2212yes—
44#65369C2218yescomplemented
45#65354C2218yescomplemented
46C2×#3940C2212yes—
47C2×#3940C2212yes—
48C2×#3939C2212yes—
49C2×#3939C2212yes—
50C2×#3935C2211yescharacteristic
51#65169C22116yes—
52#65165C22116yescomplemented
53#65149C22116yes—
54#65145C22116yescomplemented
55D5×#97C2218yes—
56#64991C2218yescomplemented
57C2×#3162—51nomaximal, Sylow
58C22×C5×Q8—21no—
59C22×C4×D5D411yescharacteristic
60C22×C4×D5D411yescharacteristic
61C22×C4×D5C2311yescharacteristic
62C22×Dic10—21no—
63C22×Dic10—21no—
64C22×Dic10—21no—
65C22×C5×C8C2312yes—
66C5×C4⋊Q8—22no—
67C5×C4⋊Q8C2314yes—
68C5×#125—22no—
69C5×#125C2314yes—
70C2×C5×C4⋊C4—21no—
71C2×C5×C4⋊C4—21no—
72C2×C5×C4⋊C4—21no—
73C2×C5×C4⋊C4C2311yescharacteristic
74C2×C5×C4⋊C4C2311yescharacteristic
75C2×C5×C4⋊C4C2312yes—
76C2×C42×C5C2311yescharacteristic
77#4092—28no—
78Q8×Dic5—28no—
79C2×#685—21no—
80C2×#685—21no—
81C2×#685—22no—
82C2×C4⋊Dic5—21no—
83C2×C4⋊Dic5—22no—
84C2×C4⋊Dic5C2312yes—
85C2×C4⋊Dic5C2311yescharacteristic
86C2×C4⋊Dic5C2311yescharacteristic
87C2×#683—21no—
88C2×#683—21no—
89C2×#683—21no—
90C2×#683—21no—
91C2×#683—22no—
92C2×C4×Dic5—21no—
93C2×C4×Dic5—21no—
94C2×C4×Dic5—22no—
95C2×C4×Dic5D411yescharacteristic
96C2×C4×Dic5D411yescharacteristic
97C2×C4×Dic5C2311yescharacteristic
98C22×#168C2312yes—
99C2×C8×D5Q8116yescomplemented
100C2×C8×D5C2318yes—
101#4043—28no—
102#4042—28no—
103#4042—28no—
104#4038—24no—
105#4038—22no—
106#4038—22no—
107#4038C2318yes—
108#4038C2314yes—
109#4038C2314yes—
110#4036—24no—
111#4036C2318yes—
112#4034—22no—
113#4034—22no—
114#4034C2314yes—
115#4034C2314yes—
116#4033—28no—
117#4033—28no—
118#4033—28no—
119C42×D5D414yes—
120C42×D5C2314yes—
121#4016—22no—
122#4016C2314yes—
123#4015—22no—
124#4015C2314yes—
125C5×#107C2318yes—
126C5×#106C2318yes—
127C5×#97C2314yes—
128#3950C2318yes—
129#3949C2318yes—
130#3947C2318yes—
131#3940C2318yes—
132#3940C2318yes—
133#3939C2318yes—
134#3939C2318yes—
135#3935C2314yes—
136C2×#533—51no—
137C2×#533—51no—
138#3162—516no—
139C2×#501—52no—
140C2×#501—52no—
141C2×#501—51no—
142C2×#501—51no—
143C2×#501—51no—
144C2×#501D511yescharacteristic, complemented
145C2×#427—52no—
146C2×#427—52no—
147C22×#97—51no—
148C23×D5C2×D411yescharacteristic
149C2×C5×Q8—24no—
150C2×C5×Q8—44no—
151C22×C4×C5—21no—
152C22×C4×C5—21no—
153C22×C4×C5—22no—
154C22×C4×C5C2×D411yescharacteristic
155C22×C4×C5C2×D411yescharacteristic
156C22×C4×C5C2411yescharacteristic
157C22×Dic5—21no—
158C22×Dic5—21no—
159C22×Dic5—22no—
160C22×Dic5C2×D411yescharacteristic
161C22×Dic5C2×D411yescharacteristic
162C22×Dic5C2×D411yescharacteristic
163C2×C4×D5—42no—
164C2×C4×D5—42no—
165C2×C4×D5C2×D414yes—
166C2×C4×D5C2×D414yes—
167C2×C4×D5C2×D412yes—
168C2×C4×D5C2×Q818yes—
169C2×C4×D5C2412yes—
170C2×Dic10—24no—
171C2×Dic10—24no—
172C2×Dic10—24no—
173C2×Dic10—44no—
174C2×Dic10—44no—
175C2×Dic10—44no—
176C2×C5×C8C2×Q818yes—
177C2×C5×C8C2414yes—
178C5×C4⋊C4—24no—
179C5×C4⋊C4—24no—
180C5×C4⋊C4—24no—
181C5×C4⋊C4—21no—
182C5×C4⋊C4—21no—
183C5×C4⋊C4—22no—
184C5×C4⋊C4C2412yes—
185C5×C4⋊C4C2412yes—
186C5×C4⋊C4C2414yes—
187C42×C5C2×D412yes—
188C42×C5C2412yes—
189#685—24no—
190#685—24no—
191#685—28no—
192C4⋊Dic5—22no—
193C4⋊Dic5—24no—
194C4⋊Dic5—21no—
195C4⋊Dic5—21no—
196C4⋊Dic5—28no—
197C4⋊Dic5C2412yes—
198C4⋊Dic5C2414yes—
199C4⋊Dic5C2412yes—
200#683—24no—
201#683—24no—
202#683—24no—
203#683—24no—
204#683—28no—
205C4×Dic5—24no—
206C4×Dic5—24no—
207C4×Dic5—28no—
208C4×Dic5C2×D412yes—
209C4×Dic5C2×D414yes—
210C4×Dic5C2×D414yes—
211C4×Dic5C2412yes—
212C2×#168C2×Q818yes—
213C2×#168C2414yes—
214C8×D5C2×Q8132yescomplemented
215C23×C8—52no—
216#533—58no—
217#533—58no—
218#533—104no—
219#533—104no—
220C2×C4⋊Q8—51no—
221C2×C4⋊Q8—51no—
222C2×C4⋊Q8—51no—
223C2×C4⋊Q8C2×D511yescharacteristic
224C2×#125—52no—
225C2×#125—51no—
226C2×#125C2×D511yescharacteristic
227C2×C22⋊Q8—101no—
228C2×C22⋊Q8—101no—
229C2×C22⋊Q8—101no—
230C2×C22⋊Q8—101no—
231C2×C4×Q8—101no—
232C2×C4×Q8—101no—
233C2×C4×Q8—101no—
234C2×C4×Q8—101no—
235C2×#117—52no—
236C2×#117—51no—
237C2×#117—51no—
238C22×C42—51no—
239#501—58no—
240#501—58no—
241#501—516no—
242#501—516no—
243#501—58no—
244#501C2×D518yescomplemented
245#427—516no—
246#427—516no—
247C2×#107—52no—
248C2×#107—52no—
249C2×#107—52no—
250C2×#107C2×D512yes—
251C2×#106—52no—
252C2×#106—52no—
253C2×#106—52no—
254C2×#106C2×D512yes—
255C2×#97—52no—
256C2×#97—51no—
257C2×#97—58no—
258C2×#97C2×D511yescharacteristic
259C23×C5C22×D411yescharacteristic
260C22×D5—22no—
261C22×D5—22no—
262C22×D5C4⋊Q814yes—
263C22×D5C22×D412yes—
264C5×Q8—48no—
265C2×C4×C5—22no—
266C2×C4×C5—22no—
267C2×C4×C5—24no—
268C2×C4×C5—22no—
269C2×C4×C5—22no—
270C2×C4×C5—42no—
271C2×C4×C5—42no—
272C2×C4×C5—44no—
273C2×C4×C5C22×D412yes—
274C2×C4×C5C22×D412yes—
275C2×C4×C5C22×D411yescharacteristic
276C2×C4×C5C22×Q814yes—
277C2×C4×C5C2511yescharacteristic
278C2×Dic5—22no—
279C2×Dic5—22no—
280C2×Dic5—22no—
281C2×Dic5—22no—
282C2×Dic5—24no—
283C2×Dic5—42no—
284C2×Dic5—42no—
285C2×Dic5—44no—
286C2×Dic5C4⋊Q814yes—
287C2×Dic5C22×D412yes—
288C2×Dic5C22×D412yes—
289C2×Dic5C22×D412yes—
290C4×D5—44no—
291C4×D5—44no—
292C4×D5C4⋊Q818yes—
293C4×D5C22×Q818yes—
294Dic10—48no—
295Dic10—48no—
296Dic10—48no—
297C5×C8C22×Q818yes—
298#168C22×Q818yes—
299C22×Q8—21no—
300C22×Q8—101no—
301C22×Q8—101no—
302C22×Q8—101no—
303C23×C4—51no—
304C23×C4—51no—
305C23×C4—51no—
306C22×C8—52no—
307C22×C8—516no—
308C22×C8—58no—
309C22×C8C22×D512yes—
310C4⋊Q8—22no—
311C4⋊Q8—54no—
312C4⋊Q8—54no—
313C4⋊Q8—54no—
314C4⋊Q8—102no—
315C4⋊Q8—102no—
316C4⋊Q8—102no—
317C4⋊Q8C22×D514yes—
318#125—22no—
319#125—58no—
320#125—54no—
321#125—104no—
322#125—102no—
323#125C22×D514yes—
324C22⋊Q8—108no—
325C22⋊Q8—108no—
326C22⋊Q8—108no—
327C22⋊Q8—108no—
328C4×Q8—108no—
329C4×Q8—108no—
330C4×Q8—108no—
331C4×Q8—108no—
332#117—54no—
333#117—58no—
334#117—54no—
335#117—102no—
336#117—102no—
337#117—104no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—52no—
342C2×C4⋊C4—51no—
343C2×C4⋊C4—51no—
344C2×C4⋊C4—102no—
345C2×C4⋊C4—102no—
346C2×C4⋊C4—101no—
347C2×C4⋊C4—101no—
348C2×C4⋊C4—101no—
349C2×C4⋊C4—101no—
350C2×C4⋊C4—101no—
351C2×C4⋊C4C22×D512yes—
352C2×C4⋊C4C22×D511yescharacteristic
353C2×C4⋊C4C22×D511yescharacteristic
354C2×C22⋊C4—101no—
355C2×C22⋊C4—101no—
356C2×C22⋊C4—102no—
357C2×C42—51no—
358C2×C42—51no—
359C2×C42—51no—
360C2×C42—54no—
361C2×C42—54no—
362C2×C42—101no—
363C2×C42—101no—
364C2×C42—102no—
365C2×C42C22×D511yescharacteristic
366#107—58no—
367#107—58no—
368#107—58no—
369#107C22×D518yes—
370#106—58no—
371#106—58no—
372#106—58no—
373#106C22×D518yes—
374#97—58no—
375#97—54no—
376#97C22×D514yes—
377C22×C5C2×C4⋊Q812yes—
378C22×C5#57512yes—
379C22×C5#57512yes—
380C22×C5C23×D411yescharacteristic
381C2×D5—28no—
382C2×D5—22no—
383C2×D5—22no—
384C2×D5C2×C4⋊Q814yes—
385C4×C5—22no—
386C4×C5—22no—
387C4×C5—42no—
388C4×C5—42no—
389C4×C5—44no—
390C4×C5C2×C4⋊Q812yes—
391C4×C5C23×Q812yes—
392Dic5—22no—
393Dic5—22no—
394Dic5—42no—
395Dic5—42no—
396Dic5—44no—
397Dic5C2×C4⋊Q814yes—
398C24—51no—
399C2×Q8—24no—
400C2×Q8—44no—
401C2×Q8—104no—
402C2×Q8—104no—
403C2×Q8—104no—
404C2×Q8—204no—
405C2×Q8—204no—
406C2×Q8—204no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—51no—
411C22×C4—52no—
412C22×C4—58no—
413C22×C4—52no—
414C22×C4—51no—
415C22×C4—51no—
416C22×C4—54no—
417C22×C4—54no—
418C22×C4—101no—
419C22×C4—102no—
420C22×C4—101no—
421C22×C4—202no—
422C22×C4—202no—
423C22×C4D4×D511yescharacteristic
424C22×C4D4×D511yescharacteristic
425C22×C4C23×D511yescharacteristic
426C2×C8—58no—
427C2×C8—54no—
428C2×C8—532no—
429C2×C8Q8×D518yes—
430C2×C8C23×D514yes—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—22no—
437C4⋊C4—52no—
438C4⋊C4—54no—
439C4⋊C4—52no—
440C4⋊C4—108no—
441C4⋊C4—108no—
442C4⋊C4—104no—
443C4⋊C4—104no—
444C4⋊C4—102no—
445C4⋊C4—101no—
446C4⋊C4—101no—
447C4⋊C4—104no—
448C4⋊C4—104no—
449C4⋊C4—104no—
450C4⋊C4C23×D512yes—
451C4⋊C4C23×D514yes—
452C4⋊C4C23×D512yes—
453C22⋊C4—104no—
454C22⋊C4—104no—
455C22⋊C4—108no—
456C42—54no—
457C42—54no—
458C42—52no—
459C42—52no—
460C42—104no—
461C42—104no—
462C42—108no—
463C42D4×D512yes—
464C42C23×D512yes—
465C2×C5#316214yescomplemented
466C2×C5C22×C4⋊Q811yescharacteristic
467C2×C5C2×#57511yescharacteristic
468C2×C5C2×#57511yescharacteristic
469D5—24no—
470C23—54no—
471C23—52no—
472C23—102no—
473C23—102no—
474C23C2×D4×D511yescharacteristic
475Q8—48no—
476Q8—208no—
477Q8—208no—
478Q8—208no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—54no—
488C2×C4—52no—
489C2×C4—58no—
490C2×C4—58no—
491C2×C4—52no—
492C2×C4—52no—
493C2×C4—102no—
494C2×C4—102no—
495C2×C4—102no—
496C2×C4—104no—
497C2×C4—102no—
498C2×C4—204no—
499C2×C4—204no—
500C2×C4—202no—
501C2×C4—204no—
502C2×C4—202no—
503C2×C4C2×D4×D512yes—
504C2×C4C2×D4×D512yes—
505C2×C4C2×D4×D511yescharacteristic
506C2×C4C2×Q8×D514yes—
507C2×C4C24×D511yescharacteristic
508C8—58no—
509C8C2×Q8×D518yes—
510C5C2×#316211yescharacteristic, complemented, Sylow
511C22—54no—
512C22—108no—
513C22—102no—
514C22—102no—
515C22D5×C4⋊Q812yes—
516C22#6610712yes—
517C22#6611112yes—
518C22C22×D4×D511yescharacteristic
519C4—22no—
520C4—22no—
521C4—44no—
522C4—42no—
523C4—42no—
524C4—54no—
525C4—102no—
526C4—102no—
527C4—202no—
528C4—204no—
529C4—202no—
530C4D5×C4⋊Q812yes—
531C4C22×Q8×D512yes—
532C2—104no—
533C2#1061370414yesdirect product factor
534C2C2×D5×C4⋊Q811yescharacteristic
535C2C2×#6610711yescharacteristic
536C2C2×#6611111yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC2×C5×#501C2
socleC23×C5C22×D4
prime radicalC23×D5C2×D4
hypercenterC2×#501D5
hypocenterC5C2×#3162
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
3. C2×C5×#501C2semidirect
4. C2×#65369C2semidirect
5. C2×#65354C2semidirect
6. #10613704C2direct (N × H)
8. C2×#65165C2semidirect
10. C2×#65145C2semidirect
12. C2×#64991C2semidirect
37. C5×#501C22semidirect
44. #65369C22semidirect
45. #65354C22semidirect
52. #65165C22semidirect
54. #65145C22semidirect
56. #64991C22semidirect
99. C2×C8×D5Q8semidirect
144. C2×#501D5semidirect
214. C8×D5C2×Q8semidirect
244. #501C2×D5semidirect
465. C2×C5#3162semidirect
510. C5C2×#3162semidirect
533. C2#10613704direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#660622
C2×#660222
C2×C5×#5012
C2×#653692
C2×#653542
#106137042
C2×#651692
C2×#651652
C2×#651492
C2×#651452
C2×D5×#972
C2×#649912
C2×#31625