Finite Group Databasebeta

← C2×#10610138 #12305519

Subgroups — C2×#10610138 #12305519

Order 1280.

2022 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#400C211yesmaximal, characteristic, complemented
2C2×#65282C211yesmaximal, characteristic
3C2×#65278C211yesmaximal, characteristic
4C2×#65275C211yesmaximal, characteristic, complemented
5C2×#65268C211yesmaximal, characteristic, complemented
6#10610138C2116yesmaximal, direct product factor
7C2×#64964C211yesmaximal, characteristic
8C2×#64962C211yesmaximal, characteristic
9C2×#64957C211yesmaximal, characteristic
10C2×#64957C211yesmaximal, characteristic
11C2×#64948C211yesmaximal, characteristic, complemented
12C2×#64949C211yesmaximal, characteristic, complemented
13C2×#64949C211yesmaximal, characteristic, complemented
14C2×#64947C211yesmaximal, characteristic, complemented
15C2×#64963C211yesmaximal, characteristic
16C2×#64945C211yesmaximal, characteristic, complemented
17C23×C4×D5C2211yescharacteristic
18C23×Dic10C2211yescharacteristic
19C22×C5×C4⋊C4C2211yescharacteristic
20C22×C5×C4⋊C4C2211yescharacteristic
21C22×#685C2211yescharacteristic
22C22×#685C2211yescharacteristic
23C22×#685C2211yescharacteristic
24C22×#685C2211yescharacteristic
25C22×#685C2211yescharacteristic
26C22×#685C2211yescharacteristic
27C22×C4⋊Dic5C2211yescharacteristic
28C22×C4⋊Dic5C2211yescharacteristic
29C22×C4⋊Dic5C2211yescharacteristic
30C22×#683C2211yescharacteristic
31C22×#683C2211yescharacteristic
32C22×#683C2211yescharacteristic
33C22×#683C2211yescharacteristic
34C5×#400C2218yescomplemented
35C2×C5×#95C2211yescharacteristic
36C2×C5×#95C2211yescharacteristic
37C2×C5×#95C2211yescharacteristic
38C2×C5×#95C2211yescharacteristic
39C2×C5×#95C2211yescharacteristic
40C2×#3963C2211yescharacteristic
41C2×#3963C2211yescharacteristic
42C2×#3963C2211yescharacteristic
43C2×#3963C2211yescharacteristic
44C2×#3963C2211yescharacteristic
45C2×#3963C2211yescharacteristic
46C2×#3963C2211yescharacteristic
47C2×#3963C2211yescharacteristic
48C2×#3963C2211yescharacteristic
49C2×#3963C2211yescharacteristic
50C2×#3963C2211yescharacteristic
51C2×#3963C2211yescharacteristic
52C2×#3963C2211yescharacteristic
53#65282C2218yes—
54#65278C2218yes—
55#65275C2218yescomplemented
56#65268C2218yescomplemented
57#64964C2218yes—
58#64963C2218yes—
59#64962C2218yes—
60#64957C2218yes—
61#64957C2218yes—
62#64949C2218yescomplemented
63#64949C2218yescomplemented
64#64948C2218yescomplemented
65#64947C2218yescomplemented
66#64945C2218yescomplemented
67C2×#2835—51nomaximal, Sylow
68C24×D5C2311yescharacteristic
69C23×C4×C5C2311yescharacteristic
70C23×C4×C5C2311yescharacteristic
71C23×C4×C5C2311yescharacteristic
72C23×C4×C5C2311yescharacteristic
73C23×C4×C5C2311yescharacteristic
74C23×C4×C5C2311yescharacteristic
75C23×C4×C5C2311yescharacteristic
76C23×Dic5C2311yescharacteristic
77C23×Dic5C2311yescharacteristic
78C23×Dic5C2311yescharacteristic
79C23×Dic5C2311yescharacteristic
80C23×Dic5C2311yescharacteristic
81C23×Dic5C2311yescharacteristic
82C23×Dic5C2311yescharacteristic
83C22×C4×D5—42no—
84C22×C4×D5—42no—
85C22×C4×D5—42no—
86C22×C4×D5C2314yes—
87C22×Dic10—42no—
88C22×Dic10—42no—
89C22×Dic10—42no—
90C22×Dic10C2314yes—
91C2×C5×C4⋊C4—42no—
92C2×C5×C4⋊C4—42no—
93C2×C5×C4⋊C4C2314yes—
94C2×C5×C4⋊C4C2314yes—
95C2×#685—24no—
96C2×#685—24no—
97C2×#685—24no—
98C2×#685—42no—
99C2×#685—42no—
100C2×#685—42no—
101C2×#685C2314yes—
102C2×#685C2314yes—
103C2×#685C2314yes—
104C2×#685C2314yes—
105C2×#685C2314yes—
106C2×#685C2314yes—
107C2×C4⋊Dic5—42no—
108C2×C4⋊Dic5—42no—
109C2×C4⋊Dic5—42no—
110C2×C4⋊Dic5C2314yes—
111C2×C4⋊Dic5C2314yes—
112C2×C4⋊Dic5C2314yes—
113C2×#683—42no—
114C2×#683—42no—
115C2×#683—42no—
116C2×#683—42no—
117C2×#683C2314yes—
118C2×#683C2314yes—
119C2×#683C2314yes—
120C2×#683C2314yes—
121C5×#95C2314yes—
122C5×#95C2314yes—
123C5×#95C2314yes—
124C5×#95C2314yes—
125C5×#95C2314yes—
126#3963C2314yes—
127#3963C2314yes—
128#3963C2314yes—
129#3963C2314yes—
130#3963C2314yes—
131#3963C2314yes—
132#3963C2314yes—
133#3963C2314yes—
134#3963C2314yes—
135#3963C2314yes—
136#3963C2314yes—
137#3963C2314yes—
138#3963C2314yes—
139#2835—516no—
140C2×#400—51no—
141C2×#400—51no—
142C2×#400—51no—
143C2×#400—51no—
144C2×#400—51no—
145C2×#400D511yescharacteristic, complemented
146C2×#397—51no—
147C2×#397—51no—
148C2×#397—51no—
149C2×#395—51no—
150C2×#395—51no—
151C2×#393—51no—
152C2×#385—51no—
153C2×#385—51no—
154C2×#385—51no—
155C24×C5C2411yescharacteristic
156C23×D5—22no—
157C23×D5—22no—
158C23×D5—22no—
159C23×D5—22no—
160C23×D5—22no—
161C23×D5—22no—
162C23×D5#4114yes—
163C23×D5C2412yes—
164C22×C4×C5—22no—
165C22×C4×C5—22no—
166C22×C4×C5—22no—
167C22×C4×C5—22no—
168C22×C4×C5—22no—
169C22×C4×C5—22no—
170C22×C4×C5—22no—
171C22×C4×C5—22no—
172C22×C4×C5—22no—
173C22×C4×C5—22no—
174C22×C4×C5—22no—
175C22×C4×C5—22no—
176C22×C4×C5—22no—
177C22×C4×C5—22no—
178C22×C4×C5—22no—
179C22×C4×C5—22no—
180C22×C4×C5—22no—
181C22×C4×C5—22no—
182C22×C4×C5—22no—
183C22×C4×C5—22no—
184C22×C4×C5—22no—
185C22×C4×C5C2412yes—
186C22×C4×C5C2412yes—
187C22×C4×C5C2412yes—
188C22×C4×C5C2412yes—
189C22×C4×C5C2412yes—
190C22×C4×C5C2412yes—
191C22×C4×C5C2412yes—
192C22×Dic5—22no—
193C22×Dic5—22no—
194C22×Dic5—22no—
195C22×Dic5—22no—
196C22×Dic5—22no—
197C22×Dic5—22no—
198C22×Dic5—22no—
199C22×Dic5—22no—
200C22×Dic5—22no—
201C22×Dic5—22no—
202C22×Dic5—22no—
203C22×Dic5—22no—
204C22×Dic5—22no—
205C22×Dic5—22no—
206C22×Dic5—22no—
207C22×Dic5—22no—
208C22×Dic5—22no—
209C22×Dic5—22no—
210C22×Dic5—22no—
211C22×Dic5—22no—
212C22×Dic5—22no—
213C22×Dic5C2412yes—
214C22×Dic5C2412yes—
215C22×Dic5C2412yes—
216C22×Dic5C2412yes—
217C22×Dic5C2412yes—
218C22×Dic5C2412yes—
219C22×Dic5C2412yes—
220C2×C4×D5—44no—
221C2×C4×D5—44no—
222C2×C4×D5—44no—
223C2×C4×D5—416no—
224C2×Dic10—44no—
225C2×Dic10—44no—
226C2×Dic10—44no—
227C2×Dic10—416no—
228C5×C4⋊C4—44no—
229C5×C4⋊C4—44no—
230#685—28no—
231#685—28no—
232#685—28no—
233#685—44no—
234#685—44no—
235#685—44no—
236C4⋊Dic5—44no—
237C4⋊Dic5—44no—
238C4⋊Dic5—44no—
239#683—44no—
240#683—44no—
241#683—44no—
242#683—44no—
243C23×Q8—51no—
244C24×C4—51no—
245C22×C4⋊C4—51no—
246C22×C4⋊C4—51no—
247C22×C4⋊C4—51no—
248C22×C4⋊C4—51no—
249C22×C4⋊C4—51no—
250C22×C4⋊C4—51no—
251C22×C4⋊C4—51no—
252C22×C4⋊C4C2×D511yescharacteristic
253C22×C4⋊C4C2×D511yescharacteristic
254C22×C22⋊C4—51no—
255C22×C22⋊C4—51no—
256C22×C22⋊C4—51no—
257C22×C22⋊C4—51no—
258C22×C22⋊C4—51no—
259C22×C22⋊C4—51no—
260#400—58no—
261#400—58no—
262#400—58no—
263#400—58no—
264#400—58no—
265#400C2×D518yescomplemented
266#397—58no—
267#397—58no—
268#397—58no—
269#395—58no—
270#395—58no—
271#393—58no—
272#385—58no—
273#385—58no—
274#385—58no—
275C2×#95—51no—
276C2×#95—51no—
277C2×#95—51no—
278C2×#95—51no—
279C2×#95—51no—
280C2×#95—51no—
281C2×#95—51no—
282C2×#95—51no—
283C2×#95—51no—
284C2×#95—51no—
285C2×#95—51no—
286C2×#95—51no—
287C2×#95—51no—
288C2×#95C2×D511yescharacteristic
289C2×#95C2×D511yescharacteristic
290C2×#95C2×D511yescharacteristic
291C2×#95C2×D511yescharacteristic
292C2×#95C2×D511yescharacteristic
293C23×C5C2×#4112yes—
294C23×C5H(5,𝔽2)12yes—
295C23×C5H(5,𝔽2)12yes—
296C23×C5H(5,𝔽2)12yes—
297C23×C52−1+412yes—
298C23×C52−1+412yes—
299C23×C52−1+412yes—
300C23×C5C2511yescharacteristic
301C22×D5—28no—
302C22×D5—28no—
303C22×D5—28no—
304C22×D5—22no—
305C22×D5—22no—
306C22×D5—22no—
307C22×D5—22no—
308C22×D5—22no—
309C22×D5—22no—
310C22×D5—44no—
311C22×D5—44no—
312C22×D5—44no—
313C22×D5—44no—
314C22×D5C2×#4114yes—
315C2×C4×C5—22no—
316C2×C4×C5—22no—
317C2×C4×C5—22no—
318C2×C4×C5—22no—
319C2×C4×C5—22no—
320C2×C4×C5—22no—
321C2×C4×C5—22no—
322C2×C4×C5—22no—
323C2×C4×C5—22no—
324C2×C4×C5—22no—
325C2×C4×C5—22no—
326C2×C4×C5—22no—
327C2×C4×C5—22no—
328C2×C4×C5—22no—
329C2×C4×C5—22no—
330C2×C4×C5—22no—
331C2×C4×C5—22no—
332C2×C4×C5—22no—
333C2×C4×C5—22no—
334C2×C4×C5—22no—
335C2×C4×C5—22no—
336C2×C4×C5—44no—
337C2×C4×C5—44no—
338C2×C4×C5—44no—
339C2×C4×C5—44no—
340C2×C4×C5—44no—
341C2×C4×C5—44no—
342C2×C4×C5—44no—
343C2×Dic5—22no—
344C2×Dic5—22no—
345C2×Dic5—22no—
346C2×Dic5—22no—
347C2×Dic5—22no—
348C2×Dic5—22no—
349C2×Dic5—22no—
350C2×Dic5—22no—
351C2×Dic5—22no—
352C2×Dic5—22no—
353C2×Dic5—22no—
354C2×Dic5—22no—
355C2×Dic5—22no—
356C2×Dic5—22no—
357C2×Dic5—22no—
358C2×Dic5—22no—
359C2×Dic5—22no—
360C2×Dic5—22no—
361C2×Dic5—22no—
362C2×Dic5—22no—
363C2×Dic5—22no—
364C2×Dic5—44no—
365C2×Dic5—44no—
366C2×Dic5—44no—
367C2×Dic5—44no—
368C2×Dic5—44no—
369C2×Dic5—44no—
370C2×Dic5—44no—
371C4×D5—416no—
372Dic10—416no—
373C25—51no—
374C22×Q8—54no—
375C22×Q8—202no—
376C22×Q8—202no—
377C22×Q8—202no—
378C23×C4—51no—
379C23×C4—51no—
380C23×C4—51no—
381C23×C4—51no—
382C23×C4—51no—
383C23×C4—51no—
384C23×C4—51no—
385C23×C4—54no—
386C23×C4—202no—
387C23×C4—202no—
388C23×C4—202no—
389C23×C4C22×D511yescharacteristic
390C23×C4C22×D511yescharacteristic
391C23×C4C22×D511yescharacteristic
392C23×C4C22×D511yescharacteristic
393C23×C4C22×D511yescharacteristic
394C23×C4C22×D511yescharacteristic
395C23×C4C22×D511yescharacteristic
396C2×C4⋊C4—42no—
397C2×C4⋊C4—42no—
398C2×C4⋊C4—54no—
399C2×C4⋊C4—54no—
400C2×C4⋊C4—54no—
401C2×C4⋊C4—54no—
402C2×C4⋊C4—54no—
403C2×C4⋊C4—54no—
404C2×C4⋊C4—54no—
405C2×C4⋊C4—202no—
406C2×C4⋊C4—202no—
407C2×C4⋊C4—202no—
408C2×C4⋊C4—202no—
409C2×C4⋊C4—202no—
410C2×C4⋊C4—202no—
411C2×C4⋊C4—202no—
412C2×C4⋊C4C22×D514yes—
413C2×C4⋊C4C22×D514yes—
414C2×C22⋊C4—54no—
415C2×C22⋊C4—54no—
416C2×C22⋊C4—54no—
417C2×C22⋊C4—54no—
418C2×C22⋊C4—54no—
419C2×C22⋊C4—54no—
420C2×C22⋊C4—104no—
421C2×C22⋊C4—104no—
422C2×C22⋊C4—104no—
423C2×C22⋊C4—202no—
424C2×C22⋊C4—202no—
425C2×C22⋊C4—202no—
426#95—54no—
427#95—54no—
428#95—54no—
429#95—54no—
430#95—54no—
431#95—54no—
432#95—54no—
433#95—54no—
434#95—54no—
435#95—54no—
436#95—54no—
437#95—54no—
438#95—54no—
439#95C22×D514yes—
440#95C22×D514yes—
441#95C22×D514yes—
442#95C22×D514yes—
443#95C22×D514yes—
444C22×C5#53914yes—
445C22×C5#53914yes—
446C22×C5#53914yes—
447C22×C5H(3,𝔽4)14yes—
448C22×C5#56314yes—
449C22×C5#56314yes—
450C22×C5#56314yes—
451C22×C5C22×#4111yescharacteristic
452C22×C5C2×H(5,𝔽2)11yescharacteristic
453C22×C5C2×H(5,𝔽2)11yescharacteristic
454C22×C5C2×H(5,𝔽2)11yescharacteristic
455C22×C5C2×2−1+411yescharacteristic
456C22×C5C2×2−1+411yescharacteristic
457C22×C5C2×2−1+411yescharacteristic
458C2×D5—24no—
459C2×D5—24no—
460C2×D5—24no—
461C2×D5—416no—
462C2×D5—42no—
463C2×D5—42no—
464C2×D5—42no—
465C2×D5—42no—
466C4×C5—42no—
467C4×C5—42no—
468C4×C5—42no—
469C4×C5—42no—
470C4×C5—42no—
471C4×C5—42no—
472C4×C5—42no—
473Dic5—42no—
474Dic5—42no—
475Dic5—42no—
476Dic5—42no—
477Dic5—42no—
478Dic5—42no—
479Dic5—42no—
480C24—54no—
481C24—52no—
482C24—102no—
483C24—102no—
484C24—102no—
485C24—102no—
486C24—102no—
487C24—102no—
488C24C23×D511yescharacteristic
489C2×Q8—204no—
490C2×Q8—204no—
491C2×Q8—204no—
492C2×Q8—2016no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—22no—
509C22×C4—22no—
510C22×C4—22no—
511C22×C4—22no—
512C22×C4—22no—
513C22×C4—22no—
514C22×C4—52no—
515C22×C4—52no—
516C22×C4—52no—
517C22×C4—52no—
518C22×C4—52no—
519C22×C4—52no—
520C22×C4—52no—
521C22×C4—102no—
522C22×C4—102no—
523C22×C4—102no—
524C22×C4—102no—
525C22×C4—102no—
526C22×C4—102no—
527C22×C4—102no—
528C22×C4—102no—
529C22×C4—102no—
530C22×C4—102no—
531C22×C4—102no—
532C22×C4—102no—
533C22×C4—102no—
534C22×C4—102no—
535C22×C4—102no—
536C22×C4—102no—
537C22×C4—102no—
538C22×C4—102no—
539C22×C4—102no—
540C22×C4—102no—
541C22×C4—102no—
542C22×C4—204no—
543C22×C4—204no—
544C22×C4—204no—
545C22×C4—2016no—
546C22×C4C23×D512yes—
547C22×C4C23×D512yes—
548C22×C4C23×D512yes—
549C22×C4C23×D512yes—
550C22×C4C23×D512yes—
551C22×C4C23×D512yes—
552C22×C4C23×D512yes—
553C4⋊C4—44no—
554C4⋊C4—44no—
555C4⋊C4—204no—
556C4⋊C4—204no—
557C4⋊C4—204no—
558C4⋊C4—204no—
559C4⋊C4—204no—
560C4⋊C4—204no—
561C4⋊C4—204no—
562C22⋊C4—108no—
563C22⋊C4—108no—
564C22⋊C4—108no—
565C22⋊C4—204no—
566C22⋊C4—204no—
567C22⋊C4—204no—
568C2×C5#283518yescomplemented
569C2×C5C2×#53911yescharacteristic
570C2×C5C2×#53911yescharacteristic
571C2×C5C2×#53911yescharacteristic
572C2×C5C2×H(3,𝔽4)11yescharacteristic
573C2×C5C2×#56311yescharacteristic
574C2×C5C2×#56311yescharacteristic
575C2×C5C2×#56311yescharacteristic
576D5—44no—
577C23—54no—
578C23—108no—
579C23—102no—
580C23—102no—
581C23—102no—
582C23—102no—
583C23—108no—
584C23—102no—
585C23—102no—
586C23—108no—
587C23—204no—
588C23—204no—
589C23—204no—
590C23—204no—
591C23#414412yes—
592C23#414712yes—
593C23D5×#4112yes—
594C23#414912yes—
595C23#414912yes—
596C23#415012yes—
597C23#415012yes—
598C23C24×D511yescharacteristic
599Q8—2016no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—102no—
629C2×C4—102no—
630C2×C4—102no—
631C2×C4—102no—
632C2×C4—102no—
633C2×C4—102no—
634C2×C4—102no—
635C2×C4—102no—
636C2×C4—102no—
637C2×C4—102no—
638C2×C4—102no—
639C2×C4—102no—
640C2×C4—102no—
641C2×C4—102no—
642C2×C4—102no—
643C2×C4—102no—
644C2×C4—102no—
645C2×C4—102no—
646C2×C4—102no—
647C2×C4—102no—
648C2×C4—102no—
649C2×C4—2016no—
650C2×C4—204no—
651C2×C4—204no—
652C2×C4—204no—
653C2×C4—204no—
654C2×C4—204no—
655C2×C4—204no—
656C2×C4—204no—
657C5C2×#283511yescharacteristic, complemented, Sylow
658C22—104no—
659C22—104no—
660C22—104no—
661C22—202no—
662C22—202no—
663C22—202no—
664C22—202no—
665C22—2016no—
666C22#6598014yes—
667C22#6599414yes—
668C22#6601714yes—
669C22#6602614yes—
670C22#6603314yes—
671C22#6604114yes—
672C22#6604614yes—
673C22C2×#414411yescharacteristic
674C22C2×#414711yescharacteristic
675C22C2×D5×#4111yescharacteristic
676C22C2×#414911yescharacteristic
677C22C2×#414911yescharacteristic
678C22C2×#415011yescharacteristic
679C22C2×#415011yescharacteristic
680C4—42no—
681C4—42no—
682C4—42no—
683C4—42no—
684C4—42no—
685C4—42no—
686C4—42no—
687C4—202no—
688C4—202no—
689C4—202no—
690C4—202no—
691C4—202no—
692C4—202no—
693C4—202no—
694C2—204no—
695C2#1061013818yesdirect product factor
696C2C2×#6598011yescharacteristic
697C2C2×#6599411yescharacteristic
698C2C2×#6601711yescharacteristic
699C2C2×#6602611yescharacteristic
700C2C2×#6603311yescharacteristic
701C2C2×#6604611yescharacteristic
702C2C2×#6604111yescharacteristic
subgroupG/H
centerC24C23×D5
commutator, cosocleC23×C5C25
FrattiniC23C24×D5
FittingC2×C5×#400C2
socleC24×C5C24
prime radicalC24×D5C23
hypercenterC2×#400D5
hypocenterC5C2×#2835
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ C2×#400

Complements

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

NH ≅ G/Ntype
1. C2×C5×#400C2semidirect
4. C2×#65275C2semidirect
5. C2×#65268C2semidirect
6. #10610138C2direct (N × H)
11. C2×#64948C2semidirect
12. C2×#64949C2semidirect
13. C2×#64949C2semidirect
14. C2×#64947C2semidirect
16. C2×#64945C2semidirect
34. C5×#400C22semidirect
55. #65275C22semidirect
56. #65268C22semidirect
62. #64949C22semidirect
63. #64949C22semidirect
64. #64948C22semidirect
65. #64947C22semidirect
66. #64945C22semidirect
145. C2×#400D5semidirect
265. #400C2×D5semidirect
568. C2×C5#2835semidirect
657. C5C2×#2835semidirect
695. C2#10610138direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#4002
C2×#652822
C2×#652782
C2×#652752
C2×#652682
#106101382
C2×#649642
C2×#649622
C2×#649572
C2×#649482
C2×#649492
C2×#649472
C2×#649632
C2×#649452
C2×#28355