Finite Group Databasebeta

← C2×#10612166 #12970573

Subgroups — C2×#10612166 #12970573

Order 1280.

2146 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D5×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×#65891C211yesmaximal, characteristic
3C2×C5×#461C211yesmaximal, characteristic, complemented
4C2×#65459C212yesmaximal
5C2×#65456C212yesmaximal, complemented
6C2×#65379C211yesmaximal, characteristic, complemented
7C2×#65308C211yesmaximal, characteristic, complemented
8#10612166C2116yesmaximal, direct product factor
9C2×#65138C211yesmaximal, characteristic, complemented
10C2×#65124C211yesmaximal, characteristic, complemented
11C2×#65097C212yesmaximal, complemented
12C2×#65086C212yesmaximal, complemented
13C22×Q8×D5C2212yes—
14C22×C5×SD16C2212yes—
15C2×C5×C4⋊Q8C2211yescharacteristic
16C2×C4×C5×D4C2211yescharacteristic
17C2×#4090C2212yes—
18C22×#689C2212yes—
19C2×#4084C2211yescharacteristic
20C2×#4082C2211yescharacteristic
21C22×#687C2212yes—
22C2×C4×C22⋊D5—21no—
23C2×#4072—21no—
24C22×Dic20C2212yes—
25D5×C4⋊Q8—24no—
26D5×C4⋊Q8C2218yescomplemented
27#65891—24no—
28#65891C2218yes—
29C2×#4043C2211yescharacteristic
30C2×D5×C4⋊C4—21no—
31C2×D5×C4⋊C4—21no—
32C2×#4036C2211yescharacteristic
33C2×#4034—21no—
34C2×#4034—21no—
35C2×#4029—21no—
36C2×#4025—21no—
37C2×C42×D5C2211yescharacteristic
38C2×#4015C2211yescharacteristic
39C2×C4×Dic10C2211yescharacteristic
40C5×#461C2218yescomplemented
41C2×C5×C4⋊C8C2211yescharacteristic
42C2×C5×#103C2212yes—
43C2×#3967C2212yes—
44#65459C22116yes—
45#65456C22116yescomplemented
46C2×#3952C2212yes—
47C2×#3948C2212yes—
48C2×#3947C2212yes—
49#65379C2218yescomplemented
50#65308C2218yescomplemented
51C2×#3942C2212yes—
52C2×C4⋊#168C2211yescharacteristic
53#65138C2218yescomplemented
54#65124C2218yescomplemented
55#65097C22116yescomplemented
56#65086C22116yescomplemented
57C2×#3218—51nomaximal, Sylow
58C22×C5×Q8C2312yes—
59C22×C5×D4C2311yescharacteristic
60C23×C4×C5—21no—
61C22×C22⋊D5—21no—
62C2×Q8×D5—24no—
63C2×Q8×D5—44no—
64C2×Q8×D5—44no—
65C2×Q8×D5C2318yes—
66C22×C4×D5—22no—
67C22×C4×D5D411yescharacteristic
68C22×C4×D5D411yescharacteristic
69C22×C4×D5C2311yescharacteristic
70C22×Dic10—22no—
71C22×Dic10C2311yescharacteristic
72C22×Dic10C2312yes—
73C2×C5×SD16—44no—
74C2×C5×SD16C2318yes—
75C22×C5×C8C2312yes—
76C5×C4⋊Q8—22no—
77C5×C4⋊Q8C2314yes—
78C4×C5×D4—22no—
79C4×C5×D4C2314yes—
80C2×C5×C4⋊C4—21no—
81C2×C5×C4⋊C4—21no—
82C2×C5×C4⋊C4C2311yescharacteristic
83C2×C5×C22⋊C4—21no—
84C2×C42×C5C2311yescharacteristic
85C2×C22⋊Dic5—21no—
86C2×C22⋊Dic5—21no—
87C2×C22⋊Dic5—21no—
88#4090—24no—
89#4090C2318yes—
90C2×#689—44no—
91C2×#689C2318yes—
92#4084—22no—
93#4084C2314yes—
94#4082—22no—
95#4082C2314yes—
96C2×#687—44no—
97C2×#687C2318yes—
98C4×C22⋊D5—28no—
99C2×#685—21no—
100#4072—28no—
101C2×C4⋊Dic5—21no—
102C2×C4⋊Dic5—21no—
103C2×C4⋊Dic5—21no—
104C2×C4⋊Dic5—21no—
105C2×C4⋊Dic5C2311yescharacteristic
106C2×#683—21no—
107C2×#683—21no—
108C2×#683—22no—
109C2×#683—22no—
110C2×C4×Dic5—21no—
111C2×C4×Dic5D411yescharacteristic
112C2×C4×Dic5D411yescharacteristic
113C2×C4×Dic5C2311yescharacteristic
114C22×#168C2312yes—
115C2×Dic20—44no—
116C2×Dic20C2318yes—
117#4043—22no—
118#4043C2314yes—
119D5×C4⋊C4—28no—
120D5×C4⋊C4—28no—
121#4036—22no—
122#4036C2314yes—
123#4034—28no—
124#4034—28no—
125#4029—28no—
126#4025—28no—
127C42×D5D414yes—
128C42×D5C2314yes—
129#4015—22no—
130#4015C2314yes—
131C4×Dic10—22no—
132C4×Dic10C2314yes—
133C5×C4⋊C8C2314yes—
134C5×#103C2318yes—
135#3967C2318yes—
136#3952C2318yes—
137#3948C2318yes—
138#3947C2318yes—
139#3942C2318yes—
140C4⋊#168C2314yes—
141C22×C4⋊Q8—51no—
142C2×#529—51no—
143#3218—516no—
144C2×#497—52no—
145C2×#491—52no—
146C2×#462—51no—
147C2×#462—51no—
148C2×#461—51no—
149C2×#461D511yescharacteristic, complemented
150C2×#451—52no—
151C2×#448—52no—
152C2×#432—51no—
153C24×C5—21no—
154C23×D5C2×D411yescharacteristic
155C2×C5×Q8—22no—
156C2×C5×Q8—24no—
157C2×C5×Q8—24no—
158C2×C5×Q8C2414yes—
159C2×C5×D4—21no—
160C2×C5×D4—22no—
161C2×C5×D4—22no—
162C2×C5×D4C2412yes—
163C22×C4×C5—21no—
164C22×C4×C5—22no—
165C22×C4×C5—24no—
166C22×C4×C5—44no—
167C22×C4×C5C2×D411yescharacteristic
168C22×C4×C5C2×D411yescharacteristic
169C22×C4×C5C2411yescharacteristic
170C2×C22⋊D5—24no—
171C2×C22⋊D5—44no—
172C22×Dic5—21no—
173C22×Dic5—21no—
174C22×Dic5—22no—
175C22×Dic5C2×D411yescharacteristic
176C22×Dic5C2×D411yescharacteristic
177C22×Dic5C2×D411yescharacteristic
178Q8×D5—48no—
179Q8×D5—48no—
180Q8×D5—416no—
181C2×C4×D5—24no—
182C2×C4×D5—24no—
183C2×C4×D5—22no—
184C2×C4×D5—22no—
185C2×C4×D5—28no—
186C2×C4×D5—48no—
187C2×C4×D5C2×D414yes—
188C2×C4×D5C2×D414yes—
189C2×C4×D5C2×D412yes—
190C2×C4×D5C2412yes—
191C2×Dic10—24no—
192C2×Dic10—24no—
193C2×Dic10—22no—
194C2×Dic10—22no—
195C2×Dic10—22no—
196C2×Dic10—21no—
197C2×Dic10—28no—
198C2×Dic10—48no—
199C2×Dic10C2414yes—
200C2×Dic10C2412yes—
201C5×SD16—48no—
202C2×C5×C8—24no—
203C2×C5×C8C2414yes—
204C5×C4⋊C4—24no—
205C5×C4⋊C4—24no—
206C5×C4⋊C4—21no—
207C5×C4⋊C4C2412yes—
208C5×C22⋊C4—24no—
209C42×C5C2×D412yes—
210C42×C5C2412yes—
211C22⋊Dic5—24no—
212C22⋊Dic5—24no—
213C22⋊Dic5—24no—
214#689—48no—
215#687—48no—
216#685—24no—
217C4⋊Dic5—21no—
218C4⋊Dic5—24no—
219C4⋊Dic5—24no—
220C4⋊Dic5—24no—
221C4⋊Dic5—24no—
222C4⋊Dic5C2412yes—
223#683—24no—
224#683—24no—
225#683—28no—
226#683—28no—
227C4×Dic5—24no—
228C4×Dic5C2×D412yes—
229C4×Dic5C2×D414yes—
230C4×Dic5C2×D414yes—
231C4×Dic5C2412yes—
232C2×#168—24no—
233C2×#168C2414yes—
234Dic20—48no—
235C23×Q8—52no—
236C22×Q16—52no—
237C22×Q16—52no—
238C22×SD16—52no—
239C22×SD16C2×D512yes—
240C2×C4⋊Q8—52no—
241C2×C4⋊Q8—58no—
242C2×C4⋊Q8—51no—
243C2×C4⋊Q8—101no—
244C2×C4⋊Q8—101no—
245C2×C4⋊Q8—104no—
246C2×C4⋊Q8C2×D511yescharacteristic
247#529—58no—
248#529—104no—
249C2×#126—101no—
250C2×#125—51no—
251C2×#124—51no—
252C2×#123—101no—
253C2×C22⋊Q8—51no—
254C2×#121—51no—
255C2×C4×Q8—51no—
256C2×C4×D4—101no—
257C2×C4×D4C2×D511yescharacteristic
258C2×#117—101no—
259C22×C4⋊C4—101no—
260C22×C4⋊C4—101no—
261C22×C42—51no—
262#497—516no—
263#491—516no—
264#462—58no—
265#462—58no—
266#461—58no—
267#461C2×D518yescomplemented
268#451—516no—
269#448—516no—
270#432—58no—
271C2×C4⋊C8—51no—
272C2×C4⋊C8C2×D511yescharacteristic
273C2×#103—52no—
274C2×#103—52no—
275C2×#103—52no—
276C2×#103C2×D512yes—
277C2×C22⋊C8—52no—
278C2×#97—52no—
279C23×C5—22no—
280C23×C5—42no—
281C23×C5—42no—
282C23×C5—42no—
283C23×C5C22×D411yescharacteristic
284C22×D5—22no—
285C22×D5—22no—
286C22×D5#13714yes—
287C22×D5C22×D412yes—
288C5×Q8—24no—
289C5×Q8—24no—
290C5×Q8—44no—
291C5×D4—22no—
292C5×D4—22no—
293C5×D4—42no—
294C2×C4×C5—22no—
295C2×C4×C5—24no—
296C2×C4×C5—22no—
297C2×C4×C5—44no—
298C2×C4×C5—42no—
299C2×C4×C5—48no—
300C2×C4×C5#13714yes—
301C2×C4×C5C22×D412yes—
302C2×C4×C5C22×D411yescharacteristic
303C2×C4×C5C22×D412yes—
304C2×C4×C5H(5,𝔽2)12yes—
305C2×C4×C5H(5,𝔽2)12yes—
306C2×C4×C5C2511yescharacteristic
307C22⋊D5—48no—
308C2×Dic5—22no—
309C2×Dic5—22no—
310C2×Dic5—22no—
311C2×Dic5—22no—
312C2×Dic5—22no—
313C2×Dic5—24no—
314C2×Dic5—44no—
315C2×Dic5—42no—
316C2×Dic5—42no—
317C2×Dic5C22×D412yes—
318C2×Dic5C22×D412yes—
319C2×Dic5C22×D412yes—
320C4×D5—28no—
321C4×D5—24no—
322C4×D5—28no—
323C4×D5—22no—
324C4×D5—22no—
325C4×D5—416no—
326Dic10—22no—
327Dic10—24no—
328Dic10—24no—
329Dic10—22no—
330Dic10—416no—
331Dic10—42no—
332Dic10—44no—
333C5×C8—24no—
334#168—24no—
335C22×Q8—58no—
336C22×Q8—51no—
337C22×Q8—52no—
338C22×Q8—102no—
339C22×Q8—104no—
340C22×Q8—204no—
341C22×Q8—204no—
342C22×Q8C22×D512yes—
343C22×D4—101no—
344C22×D4C22×D511yescharacteristic
345C23×C4—21no—
346C23×C4—51no—
347C23×C4—51no—
348C23×C4—51no—
349C23×C4—102no—
350C2×Q16—58no—
351C2×Q16—58no—
352C2×Q16—204no—
353C2×Q16—204no—
354C2×SD16—44no—
355C2×SD16—58no—
356C2×SD16—204no—
357C2×SD16C22×D518yes—
358C22×C8—52no—
359C22×C8C22×D512yes—
360C4⋊Q8—22no—
361C4⋊Q8—58no—
362C4⋊Q8—54no—
363C4⋊Q8—108no—
364C4⋊Q8—108no—
365C4⋊Q8—104no—
366C4⋊Q8—102no—
367C4⋊Q8C22×D514yes—
368#126—108no—
369#125—54no—
370#125—102no—
371#124—54no—
372#124—102no—
373#123—108no—
374C22⋊Q8—54no—
375C22⋊Q8—102no—
376#121—54no—
377#121—102no—
378C4×Q8—54no—
379C4×Q8—102no—
380C4×D4—22no—
381C4×D4—108no—
382C4×D4C22×D514yes—
383#117—108no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—51no—
387C2×C4⋊C4—108no—
388C2×C4⋊C4—108no—
389C2×C4⋊C4—102no—
390C2×C4⋊C4—102no—
391C2×C4⋊C4—101no—
392C2×C4⋊C4—101no—
393C2×C4⋊C4—101no—
394C2×C4⋊C4—101no—
395C2×C4⋊C4—101no—
396C2×C4⋊C4—101no—
397C2×C4⋊C4C22×D511yescharacteristic
398C2×C22⋊C4—21no—
399C2×C22⋊C4—101no—
400C2×C22⋊C4—101no—
401C2×C22⋊C4—101no—
402C2×C22⋊C4—101no—
403C2×C42—51no—
404C2×C42—51no—
405C2×C42—51no—
406C2×C42—54no—
407C2×C42—54no—
408C2×C42—101no—
409C2×C42C22×D511yescharacteristic
410C4⋊C8—54no—
411C4⋊C8C22×D514yes—
412#103—58no—
413#103—58no—
414#103—58no—
415#103C22×D518yes—
416C22⋊C8—58no—
417#97—58no—
418C22×C5—42no—
419C22×C5—42no—
420C22×C5—42no—
421C22×C5—84no—
422C22×C5#53512yes—
423C22×C5C2×#13712yes—
424C22×C5C2×#13712yes—
425C22×C5C23×D411yescharacteristic
426C2×D5—22no—
427C2×D5—28no—
428C2×D5—22no—
429C2×D5C2×#13714yes—
430C4×C5—22no—
431C4×C5—44no—
432C4×C5—42no—
433C4×C5#53512yes—
434C4×C5C2×#13714yes—
435C4×C5C2×H(5,𝔽2)11yescharacteristic
436C4×C5C2×H(5,𝔽2)11yescharacteristic
437Dic5—22no—
438Dic5—22no—
439Dic5—22no—
440Dic5—44no—
441Dic5—42no—
442Dic5—42no—
443C24—21no—
444C24—51no—
445C2×Q8—22no—
446C2×Q8—24no—
447C2×Q8—24no—
448C2×Q8—54no—
449C2×Q8—52no—
450C2×Q8—108no—
451C2×Q8—101no—
452C2×Q8—102no—
453C2×Q8—102no—
454C2×Q8—104no—
455C2×Q8—104no—
456C2×Q8—102no—
457C2×Q8—208no—
458C2×Q8—208no—
459C2×Q8—208no—
460C2×Q8—2016no—
461C2×Q8C23×D514yes—
462C2×D4—21no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—104no—
466C2×D4—204no—
467C2×D4C23×D512yes—
468C22×C4—21no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—44no—
472C22×C4—51no—
473C22×C4—51no—
474C22×C4—51no—
475C22×C4—54no—
476C22×C4—54no—
477C22×C4—52no—
478C22×C4—52no—
479C22×C4—104no—
480C22×C4—108no—
481C22×C4—102no—
482C22×C4—102no—
483C22×C4—104no—
484C22×C4—101no—
485C22×C4—101no—
486C22×C4—102no—
487C22×C4—208no—
488C22×C4D4×D511yescharacteristic
489C22×C4D4×D511yescharacteristic
490C22×C4C23×D511yescharacteristic
491Q16—208no—
492Q16—208no—
493SD16—48no—
494SD16—208no—
495C2×C8—24no—
496C2×C8—54no—
497C2×C8—104no—
498C2×C8C23×D514yes—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—21no—
502C4⋊C4—52no—
503C4⋊C4—108no—
504C4⋊C4—108no—
505C4⋊C4—104no—
506C4⋊C4—104no—
507C4⋊C4—104no—
508C4⋊C4—101no—
509C4⋊C4—104no—
510C4⋊C4—104no—
511C4⋊C4—104no—
512C4⋊C4C23×D512yes—
513C22⋊C4—24no—
514C22⋊C4—104no—
515C22⋊C4—104no—
516C22⋊C4—104no—
517C22⋊C4—104no—
518C42—54no—
519C42—52no—
520C42—54no—
521C42—52no—
522C42—104no—
523C42D4×D512yes—
524C42C23×D512yes—
525C2×C5—82no—
526C2×C5#321814yescomplemented
527C2×C5C2×#53511yescharacteristic
528C2×C5C22×#13711yescharacteristic
529C2×C5C22×#13711yescharacteristic
530D5—24no—
531C23—22no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—54no—
536C23—52no—
537C23—102no—
538C23—102no—
539C23C2×D4×D511yescharacteristic
540Q8—24no—
541Q8—24no—
542Q8—44no—
543Q8—102no—
544Q8—104no—
545Q8—104no—
546Q8—102no—
547Q8—2016no—
548Q8—204no—
549Q8—202no—
550D4—22no—
551D4—22no—
552D4—42no—
553D4—208no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—48no—
560C2×C4—52no—
561C2×C4—52no—
562C2×C4—52no—
563C2×C4—108no—
564C2×C4—102no—
565C2×C4—102no—
566C2×C4—102no—
567C2×C4—108no—
568C2×C4—104no—
569C2×C4—102no—
570C2×C4—102no—
571C2×C4—102no—
572C2×C4—102no—
573C2×C4—104no—
574C2×C4—2016no—
575C2×C4—202no—
576C2×C4—204no—
577C2×C4—202no—
578C2×C4#406114yes—
579C2×C4C2×D4×D512yes—
580C2×C4C2×D4×D512yes—
581C2×C4C2×D4×D511yescharacteristic
582C2×C4#414412yes—
583C2×C4#414912yes—
584C2×C4C24×D511yescharacteristic
585C8—24no—
586C8—104no—
587C5C2×#321811yescharacteristic, complemented, Sylow
588C22—42no—
589C22—42no—
590C22—42no—
591C22—54no—
592C22—84no—
593C22—102no—
594C22—108no—
595C22—102no—
596C22#6594112yes—
597C22C2×#406112yes—
598C22D5×#13712yes—
599C22C22×D4×D511yescharacteristic
600C4—22no—
601C4—42no—
602C4—44no—
603C4—102no—
604C4—102no—
605C4—102no—
606C4—202no—
607C4—204no—
608C4—202no—
609C4#6594112yes—
610C4C2×#406114yes—
611C4C2×#414411yescharacteristic
612C4C2×#414911yescharacteristic
613C2—82no—
614C2—104no—
615C2#1061216614yesdirect product factor
616C2C2×#6594111yescharacteristic
617C2C22×#406111yescharacteristic
618C2C2×D5×#13711yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC2×C5×#461C2
socleC23×C5C22×D4
prime radicalC2×#4084C22
hypercenterC2×#461D5
hypocenterC5C2×#3218
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
1. C2×D5×C4⋊Q8C2semidirect
3. C2×C5×#461C2semidirect
5. C2×#65456C2semidirect
6. C2×#65379C2semidirect
7. C2×#65308C2semidirect
8. #10612166C2direct (N × H)
9. C2×#65138C2semidirect
10. C2×#65124C2semidirect
11. C2×#65097C2semidirect
12. C2×#65086C2semidirect
26. D5×C4⋊Q8C22semidirect
40. C5×#461C22semidirect
45. #65456C22semidirect
49. #65379C22semidirect
50. #65308C22semidirect
53. #65138C22semidirect
54. #65124C22semidirect
55. #65097C22semidirect
56. #65086C22semidirect
149. C2×#461D5semidirect
267. #461C2×D5semidirect
526. C2×C5#3218semidirect
587. C5C2×#3218semidirect
615. C2#10612166direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D5×C4⋊Q82
C2×#658912
C2×C5×#4612
C2×#654592
C2×#654562
C2×#653792
C2×#653082
#106121662
C2×#651382
C2×#651242
C2×#650972
C2×#650862
C2×#32185