Finite Group Databasebeta

← C2×#10617123 #13045257

Subgroups — C2×#10617123 #13045257

Order 1280.

2210 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65856C211yesmaximal, characteristic
2C2×#65757C211yesmaximal, characteristic, complemented
3C22×#4004C211yesmaximal, characteristic, complemented
4#10617123C2116yesmaximal, direct product factor
5C2×#65707C212yesmaximal, complemented
6C2×#65704C212yesmaximal, complemented
7C2×#65693C212yesmaximal, complemented
8C2×#65692C212yesmaximal, complemented
9C2×#65684C211yesmaximal, characteristic, complemented
10C2×#65678C211yesmaximal, characteristic, complemented
11C2×#65682C211yesmaximal, characteristic, complemented
12C2×D5⋊(C4⋊C8)C211yesmaximal, characteristic, complemented
13C22×#709C411yescharacteristic
14C23×#170C2212yes—
15C22×#700C2212yes—
16C22×D5⋊C8C2212yes—
17C2×C5×#117C411yescharacteristic
18C22×C4×Dic5C2211yescharacteristic
19#65856C418yes—
20#65856C2218yes—
21C2×#4039C411yescharacteristic
22C2×#4039C2211yescharacteristic
23C2×#4033C411yescharacteristic
24C2×#4033C2211yescharacteristic
25C2×#4027C411yescharacteristic
26C2×#4027C411yescharacteristic
27C2×#4027C2211yescharacteristic
28C2×#4027C2211yescharacteristic
29C2×#4018C411yescharacteristic
30C2×#4018C411yescharacteristic
31C2×C42×D5C2211yescharacteristic
32C2×C42×D5C2211yescharacteristic
33C2×C22⋊#170C2212yes—
34#65757C2218yescomplemented
35C2×#4004C2211yescharacteristic
36C2×#4004C2211yescharacteristic
37C2×#4004C2212yes—
38C2×#4004C2211yescharacteristic
39C2×#4004C2211yescharacteristic
40C2×#4004C2212yes—
41C2×#4004C2218yescomplemented
42C2×#4003C2212yes—
43C2×#4003C2212yes—
44C2×#4002C2212yes—
45C2×#4002C2212yes—
46C2×C4⋊#170C2212yes—
47C2×C4×#170C2212yes—
48#65707C22116yescomplemented
49#65704C22116yescomplemented
50#65693C22116yescomplemented
51#65692C22116yescomplemented
52#65684C2218yescomplemented
53#65682C2218yescomplemented
54D5⋊(C4⋊C8)C2218yescomplemented
55#65678C2218yescomplemented
56C2×#2990—51nomaximal, Sylow
57C23×C4×C5C2×C411yescharacteristic
58C22×C22⋊D5C2×C411yescharacteristic
59C22×C22⋊D5C2×C411yescharacteristic
60C23×Dic5C2×C411yescharacteristic
61C23×Dic5C2311yescharacteristic
62C2×#709—28no—
63C2×#709C2×C418yes—
64C22×D20C2×C411yescharacteristic
65C22×C4×D5C2×C411yescharacteristic
66C22×C4×D5C2×C411yescharacteristic
67C22×C4×D5C2×C411yescharacteristic
68C22×C4×D5C2×C411yescharacteristic
69C22×C4×D5C2311yescharacteristic
70C22×C4×D5C2311yescharacteristic
71C22×Dic10C2×C411yescharacteristic
72C22×#170—44no—
73C22×#170C2312yes—
74C22×#170C2312yes—
75C22×#170C2312yes—
76C22×#170C2312yes—
77C22×#170C2318yes—
78C2×#700—44no—
79C2×#700C2318yes—
80C2×D5⋊C8—44no—
81C2×D5⋊C8C2318yes—
82C5×#117C2×C418yes—
83C2×C5×C4⋊C4C2×C411yescharacteristic
84C2×C5×C4⋊C4C2×C411yescharacteristic
85C2×C5×C22⋊C4C2×C411yescharacteristic
86C2×C5×C22⋊C4C2×C411yescharacteristic
87C2×C42×C5C2×C411yescharacteristic
88C2×C42×C5C2×C411yescharacteristic
89C2×#685C2×C411yescharacteristic
90C2×#685C2×C411yescharacteristic
91C2×#685C2×C411yescharacteristic
92C2×#685C2×C411yescharacteristic
93C2×#683C2×C411yescharacteristic
94C2×#683C2×C411yescharacteristic
95C2×#683C2×C411yescharacteristic
96C2×#683C2×C411yescharacteristic
97C2×C4×Dic5C2×C411yescharacteristic
98C2×C4×Dic5C2×C411yescharacteristic
99C2×C4×Dic5C2×C411yescharacteristic
100C2×C4×Dic5C2×C411yescharacteristic
101C2×C4×Dic5C2×C414yes—
102C2×C4×Dic5C2311yescharacteristic
103C2×C4×Dic5C2311yescharacteristic
104C2×C4×Dic5C2311yescharacteristic
105C2×C4×Dic5C2311yescharacteristic
106C2×C4×Dic5C2314yes—
107#4039C2×C418yes—
108#4039C2×C414yes—
109#4039C2314yes—
110#4033C2×C418yes—
111#4033C2×C414yes—
112#4033C2314yes—
113#4027C2×C418yes—
114#4027C2×C418yes—
115#4027C2×C414yes—
116#4027C2×C414yes—
117#4027C2314yes—
118#4027C2314yes—
119#4018C2×C418yes—
120#4018C2×C418yes—
121C42×D5C2×C414yes—
122C42×D5C2×C414yes—
123C42×D5C2314yes—
124C42×D5C2314yes—
125C22⋊#170C2318yes—
126#4004C2314yes—
127#4004C2314yes—
128#4004C2314yes—
129#4004C2314yes—
130#4004C2318yes—
131#4004C2318yes—
132#4003C2318yes—
133#4003C2318yes—
134#4002C2318yes—
135#4002C2318yes—
136C4⋊#170C2318yes—
137C4×#170C2318yes—
138C2×C4×#41—51no—
139#2990—516no—
140C2×#446—52no—
141C2×#435—52no—
142C2×#435—52no—
143C2×C8×D4—52no—
144C2×#432—51no—
145C2×#432—51no—
146C2×#431—51no—
147C2×#423—51no—
148C22×C4⋊C8—51no—
149C22×C4⋊C8—51no—
150C24×C5C22×C411yescharacteristic
151C23×D5C22×C411yescharacteristic
152C23×D5C22×C411yescharacteristic
153C22×C4×C5—24no—
154C22×C4×C5C22×C411yescharacteristic
155C22×C4×C5C22×C411yescharacteristic
156C22×C4×C5C22×C411yescharacteristic
157C22×C4×C5C22×C411yescharacteristic
158C22×C4×C5C22×C411yescharacteristic
159C22×C4×C5C22×C411yescharacteristic
160C22×C4×C5C22×C414yes—
161C2×C22⋊D5—24no—
162C2×C22⋊D5M1618yescomplemented
163C2×C22⋊D5C22×C414yes—
164C2×C22⋊D5C22×C414yes—
165C22×Dic5—24no—
166C22×Dic5—22no—
167C22×Dic5—22no—
168C22×Dic5C22×C411yescharacteristic
169C22×Dic5C22×C414yes—
170C22×Dic5C22×C411yescharacteristic
171C22×Dic5C22×C412yes—
172C22×Dic5C22×C411yescharacteristic
173C22×Dic5C22×C411yescharacteristic
174C22×Dic5C22×C411yescharacteristic
175C22×Dic5C2412yes—
176C22×Dic5C2411yescharacteristic
177#709—232no—
178C2×D20—24no—
179C2×D20C22×C414yes—
180C2×C4×D5—24no—
181C2×C4×D5—24no—
182C2×C4×D5—24no—
183C2×C4×D5—24no—
184C2×C4×D5—22no—
185C2×C4×D5—22no—
186C2×C4×D5—22no—
187C2×C4×D5—22no—
188C2×C4×D5C22×C414yes—
189C2×C4×D5C22×C414yes—
190C2×C4×D5C22×C414yes—
191C2×C4×D5C22×C414yes—
192C2×C4×D5C22×C412yes—
193C2×C4×D5C22×C412yes—
194C2×C4×D5C2412yes—
195C2×C4×D5C2412yes—
196C2×Dic10—24no—
197C2×Dic10C22×C414yes—
198C2×#170—24no—
199C2×#170—24no—
200C2×#170—24no—
201C2×#170—24no—
202C2×#170—48no—
203C2×#170C2414yes—
204C2×#170C2414yes—
205C2×#170C2414yes—
206C2×#170C2414yes—
207#700—48no—
208D5⋊C8—48no—
209C5×C4⋊C4C22×C414yes—
210C5×C4⋊C4C22×C414yes—
211C5×C22⋊C4C22×C414yes—
212C5×C22⋊C4C22×C414yes—
213C42×C5C22×C414yes—
214C42×C5C22×C414yes—
215#685C22×C414yes—
216#685C22×C414yes—
217#685C22×C414yes—
218#685C22×C414yes—
219#683C22×C414yes—
220#683C22×C414yes—
221#683C22×C414yes—
222#683C22×C414yes—
223C4×Dic5C22×C412yes—
224C4×Dic5C22×C412yes—
225C4×Dic5C22×C412yes—
226C4×Dic5C22×C414yes—
227C4×Dic5C22×C414yes—
228C4×Dic5C22×C414yes—
229C4×Dic5C22×C414yes—
230C4×Dic5C22×C412yes—
231C4×Dic5C2412yes—
232C4×Dic5C2412yes—
233C4×Dic5C2412yes—
234C4×Dic5C2412yes—
235C22×#41—51no—
236C22×M16—52no—
237C23×C8—52no—
238C23×C8—52no—
239C4×#41—58no—
240C4×#41—58no—
241C2×C4×Q8—51no—
242C2×C4×Q8—51no—
243C2×C4×D4—51no—
244C2×C4×D4—51no—
245C2×C4×D4—51no—
246C2×C4×D4—51no—
247C2×C4×D4—51no—
248C2×C4×D4—51no—
249C2×#117—51no—
250C2×#117—51no—
251C2×#117AGL(1,𝔽5)11yescharacteristic
252C22×C42—51no—
253C22×C42—51no—
254C22×C42—51no—
255#446—516no—
256#435—516no—
257#435—516no—
258C8×D4—516no—
259#432—58no—
260#432—58no—
261#431—58no—
262#423—58no—
263C2×C4⋊C8—51no—
264C2×C4⋊C8—51no—
265C2×C4⋊C8—51no—
266C2×C4⋊C8—52no—
267C2×C4⋊C8—52no—
268C2×C4⋊C8—58no—
269C2×C4⋊C8—58no—
270C2×C4⋊C8—52no—
271C2×C4⋊C8—51no—
272C2×C22⋊C8—52no—
273C2×C22⋊C8—52no—
274C2×C22⋊C8—52no—
275C2×#97—52no—
276C2×#97—52no—
277C2×C4×C8—52no—
278C23×C5—22no—
279C23×C5—22no—
280C23×C5C2×M1614yes—
281C23×C5C23×C412yes—
282C23×C5C23×C411yescharacteristic
283C22×D5—22no—
284C22×D5—22no—
285C22×D5—22no—
286C22×D5—22no—
287C22×D5—22no—
288C22×D5C2×M1614yes—
289C22×D5C23×C412yes—
290C22×D5C23×C412yes—
291C2×C4×C5—22no—
292C2×C4×C5—22no—
293C2×C4×C5—22no—
294C2×C4×C5—22no—
295C2×C4×C5—28no—
296C2×C4×C5—22no—
297C2×C4×C5—22no—
298C2×C4×C5C23×C412yes—
299C2×C4×C5C23×C412yes—
300C2×C4×C5C23×C412yes—
301C2×C4×C5C23×C412yes—
302C2×C4×C5C23×C412yes—
303C2×C4×C5C23×C412yes—
304C22⋊D5—28no—
305C22⋊D5C2×M16116yescomplemented
306C2×Dic5—22no—
307C2×Dic5—22no—
308C2×Dic5—22no—
309C2×Dic5—24no—
310C2×Dic5—28no—
311C2×Dic5—22no—
312C2×Dic5—22no—
313C2×Dic5C2×M1614yes—
314C2×Dic5#13114yes—
315C2×Dic5C23×C412yes—
316C2×Dic5C23×C412yes—
317C2×Dic5C23×C412yes—
318C2×Dic5C23×C411yescharacteristic
319C2×Dic5C23×C412yes—
320C2×Dic5C23×C412yes—
321C2×Dic5H(5,𝔽2)12yes—
322C2×Dic52−1+412yes—
323C2×Dic5C2511yescharacteristic
324D20—28no—
325C4×D5—28no—
326C4×D5—24no—
327C4×D5—24no—
328C4×D5—28no—
329C4×D5—28no—
330C4×D5—28no—
331C4×D5—22no—
332C4×D5—22no—
333C4×D5—22no—
334C4×D5—22no—
335Dic10—28no—
336#170—24no—
337#170—24no—
338#170—24no—
339#170—24no—
340C2×#41—58no—
341C2×#41—108no—
342C22×Q8—51no—
343C22×D4—51no—
344C22×D4—51no—
345C22×D4—51no—
346C23×C4—51no—
347C23×C4—51no—
348C23×C4—51no—
349C23×C4—51no—
350C23×C4—51no—
351C23×C4—51no—
352C23×C4—51no—
353C23×C4—51no—
354C23×C4C2×AGL(1,𝔽5)11yescharacteristic
355C2×M16—58no—
356C2×M16—204no—
357C22×C8—52no—
358C22×C8—52no—
359C22×C8—52no—
360C22×C8—52no—
361C22×C8—58no—
362C22×C8—58no—
363C22×C8—204no—
364C22×C8—204no—
365C4×Q8—54no—
366C4×Q8—58no—
367C4×Q8—54no—
368C4×D4—54no—
369C4×D4—58no—
370C4×D4—54no—
371C4×D4—54no—
372C4×D4—54no—
373C4×D4—58no—
374C4×D4—54no—
375C4×D4—58no—
376C4×D4—54no—
377#117—58no—
378#117—58no—
379#117C2×AGL(1,𝔽5)18yes—
380C2×C4⋊C4—51no—
381C2×C4⋊C4—51no—
382C2×C4⋊C4—51no—
383C2×C4⋊C4—51no—
384C2×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
385C2×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
386C2×C22⋊C4—51no—
387C2×C22⋊C4—51no—
388C2×C22⋊C4—51no—
389C2×C22⋊C4—51no—
390C2×C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic
391C2×C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic
392C2×C42—51no—
393C2×C42—51no—
394C2×C42—51no—
395C2×C42—51no—
396C2×C42—51no—
397C2×C42—51no—
398C2×C42—51no—
399C2×C42—54no—
400C2×C42—54no—
401C2×C42—51no—
402C2×C42—54no—
403C2×C42—54no—
404C2×C42—54no—
405C2×C42—54no—
406C2×C42C2×AGL(1,𝔽5)11yescharacteristic
407C2×C42C2×AGL(1,𝔽5)11yescharacteristic
408C4⋊C8—58no—
409C4⋊C8—54no—
410C4⋊C8—54no—
411C4⋊C8—54no—
412C4⋊C8—58no—
413C4⋊C8—58no—
414C4⋊C8—54no—
415C22⋊C8—58no—
416C22⋊C8—58no—
417C22⋊C8—58no—
418#97—58no—
419#97—58no—
420C4×C8—58no—
421C22×C5—22no—
422C22×C5—28no—
423C22×C5—22no—
424C22×C5#52012yes—
425C22×C5C22×M1614yes—
426C22×C5C22×M1612yes—
427C22×C5C2×#13112yes—
428C22×C5C24×C411yescharacteristic
429C2×D5—22no—
430C2×D5—28no—
431C2×D5—22no—
432C2×D5—22no—
433C2×D5—22no—
434C2×D5—22no—
435C2×D5—44no—
436C2×D5C22×M1614yes—
437C4×C5—22no—
438C4×C5—22no—
439C4×C5—22no—
440C4×C5—22no—
441C4×C5—22no—
442C4×C5—22no—
443Dic5—22no—
444Dic5—22no—
445Dic5—22no—
446Dic5#52012yes—
447Dic5C22×M1614yes—
448Dic5C2×#13114yes—
449Dic5C2×H(5,𝔽2)11yescharacteristic
450Dic5C2×2−1+411yescharacteristic
451C24—51no—
452C24—51no—
453C24C22×AGL(1,𝔽5)11yescharacteristic
454#41—1032no—
455C2×Q8—54no—
456C2×Q8—104no—
457C2×D4—58no—
458C2×D4—54no—
459C2×D4—54no—
460C2×D4—54no—
461C2×D4—104no—
462C2×D4—104no—
463C22×C4—24no—
464C22×C4—51no—
465C22×C4—51no—
466C22×C4—51no—
467C22×C4—52no—
468C22×C4—54no—
469C22×C4—52no—
470C22×C4—51no—
471C22×C4—51no—
472C22×C4—51no—
473C22×C4—52no—
474C22×C4—52no—
475C22×C4—54no—
476C22×C4—54no—
477C22×C4—54no—
478C22×C4—52no—
479C22×C4—54no—
480C22×C4—52no—
481C22×C4—102no—
482C22×C4—102no—
483C22×C4—104no—
484C22×C4—104no—
485C22×C4—104no—
486C22×C4—102no—
487C22×C4—102no—
488C22×C4—104no—
489C22×C4—102no—
490C22×C4—104no—
491C22×C4—102no—
492C22×C4C22×AGL(1,𝔽5)14yes—
493C22×C4C22×AGL(1,𝔽5)11yescharacteristic
494C22×C4C22×AGL(1,𝔽5)11yescharacteristic
495C22×C4C22×AGL(1,𝔽5)11yescharacteristic
496C22×C4C22×AGL(1,𝔽5)11yescharacteristic
497C22×C4C22×AGL(1,𝔽5)11yescharacteristic
498C22×C4C22×AGL(1,𝔽5)11yescharacteristic
499M16—208no—
500C2×C8—54no—
501C2×C8—54no—
502C2×C8—54no—
503C2×C8—54no—
504C2×C8—104no—
505C2×C8—104no—
506C2×C8—104no—
507C2×C8—104no—
508C2×C8—208no—
509C2×C8—208no—
510C4⋊C4—54no—
511C4⋊C4—54no—
512C4⋊C4—54no—
513C4⋊C4—54no—
514C4⋊C4C22×AGL(1,𝔽5)14yes—
515C4⋊C4C22×AGL(1,𝔽5)14yes—
516C22⋊C4—54no—
517C22⋊C4—54no—
518C22⋊C4—54no—
519C22⋊C4—54no—
520C22⋊C4C22×AGL(1,𝔽5)14yes—
521C22⋊C4C22×AGL(1,𝔽5)14yes—
522C42—52no—
523C42—52no—
524C42—52no—
525C42—52no—
526C42—52no—
527C42—54no—
528C42—52no—
529C42—54no—
530C42—52no—
531C42—54no—
532C42—52no—
533C42—54no—
534C42C22×AGL(1,𝔽5)14yes—
535C42C22×AGL(1,𝔽5)14yes—
536C2×C5—24no—
537C2×C5#299014yescomplemented
538C2×C5C2×#52011yescharacteristic
539C2×C5C23×M1611yescharacteristic
540C2×C5C22×#13111yescharacteristic
541D5—24no—
542D5—42no—
543C23—22no—
544C23—22no—
545C23—54no—
546C23—52no—
547C23—52no—
548C23—102no—
549C23—102no—
550C23—102no—
551C23—102no—
552C23—102no—
553C23D5⋊M1614yes—
554C23C23×AGL(1,𝔽5)12yes—
555C23C23×AGL(1,𝔽5)11yescharacteristic
556Q8—108no—
557D4—516no—
558D4—108no—
559D4—108no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—28no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—54no—
568C2×C4—54no—
569C2×C4—52no—
570C2×C4—52no—
571C2×C4—51no—
572C2×C4—52no—
573C2×C4—51no—
574C2×C4—52no—
575C2×C4—52no—
576C2×C4—52no—
577C2×C4—52no—
578C2×C4—102no—
579C2×C4—102no—
580C2×C4—108no—
581C2×C4—108no—
582C2×C4—108no—
583C2×C4—102no—
584C2×C4—102no—
585C2×C4—104no—
586C2×C4—104no—
587C2×C4—108no—
588C2×C4—102no—
589C2×C4—108no—
590C2×C4—102no—
591C2×C4—104no—
592C2×C4—102no—
593C2×C4—102no—
594C2×C4—102no—
595C2×C4C23×AGL(1,𝔽5)12yes—
596C2×C4C23×AGL(1,𝔽5)12yes—
597C2×C4C23×AGL(1,𝔽5)12yes—
598C2×C4C23×AGL(1,𝔽5)12yes—
599C2×C4C23×AGL(1,𝔽5)12yes—
600C2×C4C23×AGL(1,𝔽5)12yes—
601C8—104no—
602C8—104no—
603C8—104no—
604C8—104no—
605C5C2×#299011yescharacteristic, complemented, Sylow
606C22—22no—
607C22—22no—
608C22—28no—
609C22—54no—
610C22—102no—
611C22—102no—
612C22—108no—
613C22—102no—
614C22—102no—
615C22—102no—
616C22—204no—
617C22C2×D5⋊M1614yes—
618C22C2×D5⋊M1612yes—
619C22D5⋊#13112yes—
620C22#6626612yes—
621C22C24×AGL(1,𝔽5)11yescharacteristic
622C4—22no—
623C4—22no—
624C4—22no—
625C4—22no—
626C4—22no—
627C4—22no—
628C4—54no—
629C4—54no—
630C4—51no—
631C4—51no—
632C4—52no—
633C4—102no—
634C4—102no—
635C4—102no—
636C2—24no—
637C2—104no—
638C2—202no—
639C2#1061712314yesdirect product factor
640C2C22×D5⋊M1611yescharacteristic
641C2C2×D5⋊#13111yescharacteristic
642C2C2×#6626611yescharacteristic
subgroupG/H
centerC23C23×AGL(1,𝔽5)
commutatorC22×C5C24×C4
FrattiniC22C24×AGL(1,𝔽5)
FittingC2×C5×#117C4
socleC23×C5C23×C4
prime radicalC22×#709C4
cosocleC2×Dic5C25
hypercenterC2×#117AGL(1,𝔽5)
hypocenterC5C2×#2990
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. C2×#65757C2semidirect
3. C22×#4004C2semidirect
4. #10617123C2direct (N × H)
5. C2×#65707C2semidirect
6. C2×#65704C2semidirect
7. C2×#65693C2semidirect
8. C2×#65692C2semidirect
9. C2×#65684C2semidirect
10. C2×#65678C2semidirect
11. C2×#65682C2semidirect
12. C2×D5⋊(C4⋊C8)C2semidirect
34. #65757C22semidirect
41. C2×#4004C22semidirect
48. #65707C22semidirect
49. #65704C22semidirect
50. #65693C22semidirect
51. #65692C22semidirect
52. #65684C22semidirect
53. #65682C22semidirect
54. D5⋊(C4⋊C8)C22semidirect
55. #65678C22semidirect
162. C2×C22⋊D5M16semidirect
305. C22⋊D5C2×M16semidirect
537. C2×C5#2990semidirect
605. C5C2×#2990semidirect
639. C2#10617123direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#658562
C2×#657572
C22×#40042
#106171232
C2×#657072
C2×#657042
C2×#656932
C2×#656922
C2×#656842
C2×#656782
C2×#656822
C2×D5⋊(C4⋊C8)2
C2×#29905