Finite Group Databasebeta

← C2×#22592653 #32166621

Subgroups — C2×#22592653 #32166621

Order 18176.

3374 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C22⋊D71C211yesmaximal, characteristic
2C22×C4⋊C4×D71C211yesmaximal, characteristic, complemented
3#22592653C2116yesmaximal, direct product factor
4C2×#17985940C212yesmaximal, complemented
5C2×#17985935C212yesmaximal, complemented
6C2×#17985898C211yesmaximal, characteristic, complemented
7C2×#17985895C211yesmaximal, characteristic, complemented
8C2×#17985875C212yesmaximal
9C2×#17985873C212yesmaximal, complemented
10C2×#17985859C211yesmaximal, characteristic, complemented
11C2×#17985856C211yesmaximal, characteristic, complemented
12C2×#380×C71C211yesmaximal, characteristic, complemented
13C23×C22⋊D71C2211yescharacteristic
14C23×C4×D71C2211yescharacteristic
15C23×C4×D71C2212yes—
16C2×C4×C22⋊D71—28no—
17C2×C4×C22⋊D71C2218yes—
18C22×C4⋊Dic71C2211yescharacteristic
19C22×C4⋊Dic71C2211yescharacteristic
20C22×C4⋊Dic71C2212yes—
21C22×Dic71⋊C4C2212yes—
22C22×Dic71⋊C4C2211yescharacteristic
23C22×Dic71⋊C4C2212yes—
24C2×C4⋊C4×D71—28no—
25C2×C4⋊C4×D71C2218yescomplemented
26C22×C22⋊Dic71C2211yescharacteristic
27C22×C22⋊Dic71C2212yes—
28C22×(C2×D71)⋊C4C2211yescharacteristic
29C22×(C2×D71)⋊C4C2212yes—
30C22×C4×Dic71C2211yescharacteristic
31#17985940C22116yescomplemented
32#17985935C22116yescomplemented
33#17985898C2218yescomplemented
34#17985895C2218yescomplemented
35#17985875C22116yes—
36#17985873C22116yescomplemented
37#17985859C2218yescomplemented
38#17985856C2218yescomplemented
39C2×C71⋊#95C2212yes—
40C2×C71⋊#95C2212yes—
41C2×C71⋊#95C2212yes—
42C2×C71⋊#95C2212yes—
43C24×C4×C71C2211yescharacteristic
44C22×C4⋊C4×C71C2211yescharacteristic
45C22×C4⋊C4×C71C2211yescharacteristic
46C22×C22⋊C4×C71C2212yes—
47#380×C71C2218yescomplemented
48C2×#95×C71C2212yes—
49C24×D71C2311yescharacteristic
50C22×C22⋊D71—24no—
51C22×C22⋊D71—24no—
52C22×C22⋊D71Q818yes—
53C22×C22⋊D71C2314yes—
54C23×Dic71C2312yes—
55C23×Dic71C2312yes—
56C23×Dic71C2311yescharacteristic
57C23×Dic71C2311yescharacteristic
58C23×Dic71C2311yescharacteristic
59C22×C4×D71—28no—
60C22×C4×D71—24no—
61C22×C4×D71—24no—
62C22×C4×D71—44no—
63C22×C4×D71—44no—
64C22×C4×D71D418yes—
65C22×C4×D71C2314yes—
66C22×C4×D71C2318yes—
67C4×C22⋊D71—232no—
68C2×C4⋊Dic71—24no—
69C2×C4⋊Dic71—24no—
70C2×C4⋊Dic71—44no—
71C2×C4⋊Dic71C2314yes—
72C2×C4⋊Dic71C2314yes—
73C2×C4⋊Dic71C2318yes—
74C2×Dic71⋊C4—28no—
75C2×Dic71⋊C4—24no—
76C2×Dic71⋊C4—44no—
77C2×Dic71⋊C4C2318yes—
78C2×Dic71⋊C4C2314yes—
79C2×Dic71⋊C4C2318yes—
80C4⋊C4×D71—232no—
81C2×C22⋊Dic71—24no—
82C2×C22⋊Dic71—44no—
83C2×C22⋊Dic71C2314yes—
84C2×C22⋊Dic71C2318yes—
85C2×(C2×D71)⋊C4—24no—
86C2×(C2×D71)⋊C4—28no—
87C2×(C2×D71)⋊C4C2318yes—
88C2×(C2×D71)⋊C4C2314yes—
89C2×C4×Dic71—24no—
90C2×C4×Dic71C2314yes—
91C71⋊#95C2318yes—
92C71⋊#95C2318yes—
93C71⋊#95C2318yes—
94C71⋊#95C2318yes—
95C25×C71C2311yescharacteristic
96C23×C4×C71—24no—
97C23×C4×C71—24no—
98C23×C4×C71—24no—
99C23×C4×C71C2312yes—
100C23×C4×C71C2312yes—
101C23×C4×C71C2311yescharacteristic
102C23×C4×C71C2314yes—
103C23×C4×C71C2311yescharacteristic
104C2×C4⋊C4×C71—24no—
105C2×C4⋊C4×C71—24no—
106C2×C4⋊C4×C71C2314yes—
107C2×C4⋊C4×C71C2314yes—
108C2×C22⋊C4×C71—44no—
109C2×C22⋊C4×C71C2318yes—
110#95×C71C2318yes—
111C23×D71—22no—
112C23×D71—22no—
113C23×D71—22no—
114C23×D71—22no—
115C23×D71C2×D414yes—
116C23×D71C2×Q814yes—
117C23×D71#4114yes—
118C23×D71C2412yes—
119C2×C22⋊D71—232no—
120C2×C22⋊D71—28no—
121C2×C22⋊D71—28no—
122C2×C22⋊D71C2×Q8116yes—
123C22×Dic71—22no—
124C22×Dic71—22no—
125C22×Dic71—22no—
126C22×Dic71—22no—
127C22×Dic71—24no—
128C22×Dic71—22no—
129C22×Dic71—24no—
130C22×Dic71—24no—
131C22×Dic71—24no—
132C22×Dic71—24no—
133C22×Dic71—24no—
134C22×Dic71—22no—
135C22×Dic71—22no—
136C22×Dic71C2×D414yes—
137C22×Dic71C2×Q814yes—
138C22×Dic71C2414yes—
139C22×Dic71C2414yes—
140C22×Dic71C2412yes—
141C22×Dic71C2412yes—
142C22×Dic71C2412yes—
143C2×C4×D71—232no—
144C2×C4×D71—28no—
145C2×C4×D71—28no—
146C2×C4×D71—216no—
147C2×C4×D71—48no—
148C2×C4×D71—48no—
149C2×C4×D71—432no—
150C2×C4×D71C2×D4116yes—
151C4⋊Dic71—28no—
152C4⋊Dic71—28no—
153C4⋊Dic71—48no—
154Dic71⋊C4—216no—
155Dic71⋊C4—28no—
156Dic71⋊C4—48no—
157C22⋊Dic71—28no—
158C22⋊Dic71—48no—
159(C2×D71)⋊C4—28no—
160(C2×D71)⋊C4—216no—
161C4×Dic71—28no—
162C24×C71—22no—
163C24×C71—22no—
164C24×C71—22no—
165C24×C71—22no—
166C24×C71—22no—
167C24×C71—22no—
168C24×C71C2×Q814yes—
169C24×C71C2412yes—
170C24×C71C2411yescharacteristic
171C22×C4×C71—24no—
172C22×C4×C71—22no—
173C22×C4×C71—24no—
174C22×C4×C71—22no—
175C22×C4×C71—22no—
176C22×C4×C71—24no—
177C22×C4×C71—24no—
178C22×C4×C71—24no—
179C22×C4×C71—24no—
180C22×C4×C71—28no—
181C22×C4×C71—28no—
182C22×C4×C71—28no—
183C22×C4×C71—416no—
184C22×C4×C71C2×D414yes—
185C22×C4×C71#4114yes—
186C22×C4×C71#4114yes—
187C22×C4×C71C2414yes—
188C22×C4×C71C2412yes—
189C22×C4×C71C2414yes—
190C22×C4×C71C2412yes—
191C4⋊C4×C71—28no—
192C4⋊C4×C71—28no—
193C22⋊C4×C71—48no—
194C23×C71—28no—
195C23×C71—28no—
196C23×C71—28no—
197C23×C71—22no—
198C23×C71—22no—
199C23×C71—22no—
200C23×C71—22no—
201C23×C71—22no—
202C23×C71—22no—
203C23×C71—44no—
204C23×C71—44no—
205C23×C71—44no—
206C23×C71—44no—
207C23×C71C22×D412yes—
208C23×C71C22×Q814yes—
209C23×C71C22×Q812yes—
210C23×C71C2×#4112yes—
211C23×C71C2×#4112yes—
212C23×C71C2×#4112yes—
213C23×C71H(5,𝔽2)12yes—
214C23×C71H(5,𝔽2)12yes—
215C23×C71C2511yescharacteristic
216C22×D71—28no—
217C22×D71—28no—
218C22×D71—28no—
219C22×D71—28no—
220C22×D71—28no—
221C22×D71—28no—
222C22×D71—22no—
223C22×D71—22no—
224C22×D71—22no—
225C22×D71—22no—
226C22×D71C22⋊Q8116yes—
227C22×D71C22×D414yes—
228C22×D71C22×Q814yes—
229C22×D71C2×#4114yes—
230C2×C4×C71—24no—
231C2×C4×C71—22no—
232C2×C4×C71—24no—
233C2×C4×C71—22no—
234C2×C4×C71—22no—
235C2×C4×C71—24no—
236C2×C4×C71—24no—
237C2×C4×C71—24no—
238C2×C4×C71—24no—
239C2×C4×C71—44no—
240C2×C4×C71—48no—
241C2×C4×C71—48no—
242C2×C4×C71—416no—
243C2×C4×C71#123116yes—
244C2×C4×C71C22×D414yes—
245C2×C4×C71C2×#4114yes—
246C2×C4×C71C2×#4114yes—
247C22⋊D71—232no—
248C2×Dic71—28no—
249C2×Dic71—28no—
250C2×Dic71—22no—
251C2×Dic71—22no—
252C2×Dic71—22no—
253C2×Dic71—22no—
254C2×Dic71—22no—
255C2×Dic71—24no—
256C2×Dic71—24no—
257C2×Dic71—24no—
258C2×Dic71—24no—
259C2×Dic71—24no—
260C2×Dic71—22no—
261C2×Dic71—22no—
262C2×Dic71—24no—
263C2×Dic71—48no—
264C2×Dic71—44no—
265C2×Dic71—48no—
266C2×Dic71C22×D414yes—
267C2×Dic71C22×Q814yes—
268C4×D71—232no—
269C4×D71—432no—
270C22×C71—24no—
271C22×C71—24no—
272C22×C71—24no—
273C22×C71—42no—
274C22×C71—42no—
275C22×C71—42no—
276C22×C71—42no—
277C22×C71—416no—
278C22×C71C2×C22⋊Q814yes—
279C22×C71C2×#12314yes—
280C22×C71#53514yes—
281C22×C71#55314yes—
282C22×C71#55314yes—
283C22×C71#55414yes—
284C22×C71#55414yes—
285C22×C71C23×D411yescharacteristic
286C22×C71C23×Q811yescharacteristic
287C22×C71C22×#4111yescharacteristic
288C22×C71C22×#4111yescharacteristic
289C22×C71C22×#4111yescharacteristic
290C22×C71C2×H(5,𝔽2)11yescharacteristic
291C22×C71C2×H(5,𝔽2)11yescharacteristic
292C2×D71—24no—
293C2×D71—24no—
294C2×D71—24no—
295C2×D71—24no—
296C2×D71—24no—
297C2×D71—24no—
298C2×D71—232no—
299C2×D71C2×C22⋊Q818yes—
300C4×C71—42no—
301C4×C71—44no—
302C4×C71—44no—
303C4×C71C2×#12318yes—
304Dic71—24no—
305Dic71—24no—
306Dic71—44no—
307Dic71—42no—
308Dic71—44no—
309C2×#2487—711nomaximal, Sylow
310C2×C71—44no—
311C2×C71#248718yescomplemented
312C2×C71C22×C22⋊Q811yescharacteristic
313C2×C71C22×#12311yescharacteristic
314C2×C71C2×#53511yescharacteristic
315C2×C71C2×#55311yescharacteristic
316C2×C71C2×#55311yescharacteristic
317C2×C71C2×#55411yescharacteristic
318C2×C71C2×#55411yescharacteristic
319D71—28no—
320C22×C4×D4—711no—
321C23×C4⋊C4—711no—
322#2487—7116no—
323C2×#400—712no—
324C2×#399—712no—
325C2×#389—711no—
326C2×#389—711no—
327C2×#386—712no—
328C2×#385—712no—
329C2×#380—711no—
330C2×#380—711no—
331C2×#380D7111yescharacteristic, complemented
332C71C2×#248711yescharacteristic, complemented, Sylow
333C23×D4—711no—
334C24×C4—712no—
335C24×C4—711no—
336C24×C4C2×D7111yescharacteristic
337C2×C4×D4—718no—
338C2×C4×D4—1428no—
339C22×C4⋊C4—712no—
340C22×C4⋊C4—712no—
341C22×C4⋊C4—712no—
342C22×C4⋊C4—711no—
343C22×C4⋊C4—711no—
344C22×C4⋊C4—711no—
345C22×C4⋊C4—718no—
346C22×C4⋊C4—1428no—
347C22×C4⋊C4C2×D7111yescharacteristic
348C22×C4⋊C4C2×D7111yescharacteristic
349C22×C22⋊C4—712no—
350C22×C22⋊C4—711no—
351C22×C22⋊C4—711no—
352C22×C22⋊C4—712no—
353C22×C22⋊C4C2×D7112yes—
354C22×C42—711no—
355#400—7116no—
356#399—7116no—
357#389—718no—
358#389—718no—
359#386—7116no—
360#385—7116no—
361#380—718no—
362#380—718no—
363#380C2×D7118yescomplemented
364C2×#95—712no—
365C2×#95—712no—
366C2×#95—712no—
367C2×#95—712no—
368C2×#95C2×D7112yes—
369C25—711no—
370C25C22×D7111yescharacteristic
371C22×D4—718no—
372C22×D4—714no—
373C22×D4—1424no—
374C22×D4—1424no—
375C23×C4—24no—
376C23×C4—24no—
377C23×C4—24no—
378C23×C4—2844no—
379C23×C4—2844no—
380C23×C4—718no—
381C23×C4—711no—
382C23×C4—711no—
383C23×C4—718no—
384C23×C4—714no—
385C23×C4—711no—
386C23×C4—712no—
387C23×C4—712no—
388C23×C4—1428no—
389C23×C4—1424no—
390C23×C4—1424no—
391C23×C4C22×D7114yes—
392C23×C4C22×D7112yes—
393C23×C4C22×D7111yescharacteristic
394C23×C4C22×D7112yes—
395C23×C4C22×D7111yescharacteristic
396C4×D4—14232no—
397C2×C4⋊C4—24no—
398C2×C4⋊C4—24no—
399C2×C4⋊C4—2844no—
400C2×C4⋊C4—2844no—
401C2×C4⋊C4—718no—
402C2×C4⋊C4—718no—
403C2×C4⋊C4—714no—
404C2×C4⋊C4—718no—
405C2×C4⋊C4—714no—
406C2×C4⋊C4—714no—
407C2×C4⋊C4—14232no—
408C2×C4⋊C4—1428no—
409C2×C4⋊C4—1424no—
410C2×C4⋊C4—1424no—
411C2×C4⋊C4—1424no—
412C2×C4⋊C4C22×D7114yes—
413C2×C4⋊C4C22×D7114yes—
414C2×C22⋊C4—44no—
415C2×C22⋊C4—2844no—
416C2×C22⋊C4—718no—
417C2×C22⋊C4—714no—
418C2×C22⋊C4—714no—
419C2×C22⋊C4—718no—
420C2×C22⋊C4—1424no—
421C2×C22⋊C4—1428no—
422C2×C22⋊C4—1424no—
423C2×C22⋊C4C22×D7118yes—
424C2×C42—714no—
425C2×C42—1424no—
426#95—718no—
427#95—718no—
428#95—718no—
429#95—718no—
430#95C22×D7118yes—
431C24—22no—
432C24—22no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—22no—
437C24—714no—
438C24—714no—
439C24—714no—
440C24—712no—
441C24—1422no—
442C24—1422no—
443C24—1422no—
444C24—1422no—
445C24Q8×D7114yes—
446C24C23×D7112yes—
447C24C23×D7111yescharacteristic
448C2×D4—7116no—
449C2×D4—1428no—
450C2×D4—1428no—
451C2×D4—14232no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—28no—
462C22×C4—28no—
463C22×C4—28no—
464C22×C4—416no—
465C22×C4—2848no—
466C22×C4—2848no—
467C22×C4—28432no—
468C22×C4—714no—
469C22×C4—714no—
470C22×C4—712no—
471C22×C4—712no—
472C22×C4—7116no—
473C22×C4—714no—
474C22×C4—714no—
475C22×C4—712no—
476C22×C4—14216no—
477C22×C4—1422no—
478C22×C4—1424no—
479C22×C4—1424no—
480C22×C4—1424no—
481C22×C4—1422no—
482C22×C4—1424no—
483C22×C4—1424no—
484C22×C4—1424no—
485C22×C4—1422no—
486C22×C4—1422no—
487C22×C4—1422no—
488C22×C4—1422no—
489C22×C4—1422no—
490C22×C4—1428no—
491C22×C4—1428no—
492C22×C4—14232no—
493C22×C4D4×D7114yes—
494C22×C4#1176586414yes—
495C22×C4#1176586514yes—
496C22×C4C23×D7112yes—
497C22×C4C23×D7114yes—
498C22×C4C23×D7114yes—
499C22×C4C23×D7112yes—
500C4⋊C4—28no—
501C4⋊C4—28no—
502C4⋊C4—2848no—
503C4⋊C4—2848no—
504C4⋊C4—14216no—
505C4⋊C4—1428no—
506C4⋊C4—1428no—
507C4⋊C4—1428no—
508C22⋊C4—48no—
509C22⋊C4—2848no—
510C22⋊C4—14216no—
511C22⋊C4—1428no—
512C22⋊C4—1428no—
513C42—1428no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—28no—
521C23—28no—
522C23—28no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—714no—
528C23—714no—
529C23—714no—
530C23—7116no—
531C23—1428no—
532C23—1422no—
533C23—1422no—
534C23—1422no—
535C23—1422no—
536C23—1428no—
537C23—1428no—
538C23—1428no—
539C23—1428no—
540C23—1428no—
541C23C2×D4×D7112yes—
542C23C2×Q8×D7114yes—
543C23C2×Q8×D7112yes—
544C23#41×D7112yes—
545C23C2×#1176586412yes—
546C23C2×#1176586512yes—
547C23#1455617312yes—
548C23#1455617412yes—
549C23C24×D7111yescharacteristic
550D4—14232no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—48no—
561C2×C4—48no—
562C2×C4—44no—
563C2×C4—416no—
564C2×C4—28432no—
565C2×C4—2848no—
566C2×C4—2844no—
567C2×C4—2848no—
568C2×C4—714no—
569C2×C4—714no—
570C2×C4—1422no—
571C2×C4—1424no—
572C2×C4—1424no—
573C2×C4—1424no—
574C2×C4—1422no—
575C2×C4—1424no—
576C2×C4—1424no—
577C2×C4—1424no—
578C2×C4—1422no—
579C2×C4—1422no—
580C2×C4—1422no—
581C2×C4—1422no—
582C2×C4—1422no—
583C2×C4—1428no—
584C2×C4—1428no—
585C2×C4—14232no—
586C2×C4#14556110116yes—
587C2×C4C2×D4×D7114yes—
588C2×C4C2×#1176586414yes—
589C2×C4C2×#1176586514yes—
590C22—24no—
591C22—24no—
592C22—24no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—416no—
597C22—42no—
598C22—718no—
599C22—1424no—
600C22—1424no—
601C22—1424no—
602C22—14232no—
603C22—1424no—
604C22—1424no—
605C22—1424no—
606C22C22⋊Q8×D7114yes—
607C22C2×#1455611014yes—
608C22#1798656514yes—
609C22#1798669314yes—
610C22#1798669614yes—
611C22#1798670414yes—
612C22#1798670714yes—
613C22C22×D4×D7111yescharacteristic
614C22C22×Q8×D7111yescharacteristic
615C22C2×#41×D7111yescharacteristic
616C22C22×#1176586411yescharacteristic
617C22C22×#1176586511yescharacteristic
618C22C2×#1455617311yescharacteristic
619C22C2×#1455617411yescharacteristic
620C4—42no—
621C4—44no—
622C4—44no—
623C4—2844no—
624C4—2844no—
625C4—2842no—
626C4—1424no—
627C4—1424no—
628C4C2×#1455611018yes—
629C2—44no—
630C2—1428no—
631C2#2259265318yesdirect product factor
632C2C2×C22⋊Q8×D7111yescharacteristic
633C2C22×#1455611011yescharacteristic
634C2C2×#1798656511yescharacteristic
635C2C2×#1798669311yescharacteristic
636C2C2×#1798669611yescharacteristic
637C2C2×#1798670411yescharacteristic
638C2C2×#1798670711yescharacteristic
subgroupG/H
centerC24C23×D71
commutator, cosocleC23×C71C25
FrattiniC23C24×D71
FittingC2×#380×C71C2
socleC24×C71C24
prime radicalC23×C22⋊D71C22
hypercenterC2×#380D71
hypocenterC71C2×#2487
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊C4×D71C2semidirect
3. #22592653C2direct (N × H)
4. C2×#17985940C2semidirect
5. C2×#17985935C2semidirect
6. C2×#17985898C2semidirect
7. C2×#17985895C2semidirect
9. C2×#17985873C2semidirect
10. C2×#17985859C2semidirect
11. C2×#17985856C2semidirect
12. C2×#380×C71C2semidirect
25. C2×C4⋊C4×D71C22semidirect
31. #17985940C22semidirect
32. #17985935C22semidirect
33. #17985898C22semidirect
34. #17985895C22semidirect
36. #17985873C22semidirect
37. #17985859C22semidirect
38. #17985856C22semidirect
47. #380×C71C22semidirect
311. C2×C71#2487semidirect
331. C2×#380D71semidirect
332. C71C2×#2487semidirect
363. #380C2×D71semidirect
631. C2#22592653direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×C22⋊D712
C22×C4⋊C4×D712
#225926532
C2×#179859402
C2×#179859352
C2×#179858982
C2×#179858952
C2×#179858752
C2×#179858732
C2×#179858592
C2×#179858562
C2×#380×C712
C2×#248771