Finite Group Databasebeta

← C2×#22592655 #32166625

Subgroups — C2×#22592655 #32166625

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×D284C211yesmaximal, characteristic
2C22×C4⋊C4×D71C211yesmaximal, characteristic, complemented
3#22592655C2116yesmaximal, direct product factor
4C2×#17985941C212yesmaximal, complemented
5C2×#17985934C212yesmaximal, complemented
6C2×#17985897C211yesmaximal, characteristic, complemented
7C2×#17985877C212yesmaximal
8C2×#17985870C212yesmaximal, complemented
9C2×#17985856C211yesmaximal, characteristic, complemented
10C2×#17985856C211yesmaximal, characteristic, complemented
11C2×#17985856C211yesmaximal, characteristic, complemented
12C2×#389×C71C211yesmaximal, characteristic, complemented
13C23×D284C2211yescharacteristic
14C23×C4×D71C2211yescharacteristic
15C23×C4×D71C2211yescharacteristic
16C23×C4×D71C2212yes—
17C2×C4×D284—28no—
18C2×C4×D284C2218yes—
19C22×C4⋊Dic71C2211yescharacteristic
20C22×C4⋊Dic71C2211yescharacteristic
21C22×C4⋊Dic71C2211yescharacteristic
22C22×Dic71⋊C4C2212yes—
23C22×Dic71⋊C4C2212yes—
24C2×C4⋊C4×D71—28no—
25C2×C4⋊C4×D71C2218yescomplemented
26C22×(C2×D71)⋊C4C2211yescharacteristic
27C22×(C2×D71)⋊C4C2212yes—
28C22×(C2×D71)⋊C4C2211yescharacteristic
29C22×(C2×D71)⋊C4C2212yes—
30C22×(C2×D71)⋊C4C2212yes—
31#17985941C22116yescomplemented
32#17985934C22116yescomplemented
33#17985897C2218yescomplemented
34#17985877C22116yes—
35#17985870C22116yescomplemented
36#17985856C2218yescomplemented
37#17985856C2218yescomplemented
38#17985856C2218yescomplemented
39C2×C71⋊#95C2212yes—
40C2×C71⋊#95C2212yes—
41C2×C71⋊#95C2212yes—
42C2×C71⋊#95C2212yes—
43C22×C4⋊C4×C71C2211yescharacteristic
44C22×C4⋊C4×C71C2211yescharacteristic
45C22×C4⋊C4×C71C2212yes—
46C22×C42×C71C2211yescharacteristic
47#389×C71C2218yescomplemented
48C2×#95×C71C2212yes—
49C24×D71C2311yescharacteristic
50C24×D71C2311yescharacteristic
51C22×D284—24no—
52C22×D284—24no—
53C22×D284Q818yes—
54C22×D284C2314yes—
55C23×Dic71C2311yescharacteristic
56C23×Dic71C2312yes—
57C23×Dic71C2312yes—
58C23×Dic71C2311yescharacteristic
59C22×C4×D71—28no—
60C22×C4×D71—24no—
61C22×C4×D71—24no—
62C22×C4×D71—24no—
63C22×C4×D71—24no—
64C22×C4×D71—24no—
65C22×C4×D71—44no—
66C22×C4×D71—44no—
67C22×C4×D71D418yes—
68C22×C4×D71C2314yes—
69C22×C4×D71C2314yes—
70C22×C4×D71C2318yes—
71C4×D284—232no—
72C2×C4⋊Dic71—24no—
73C2×C4⋊Dic71—24no—
74C2×C4⋊Dic71—24no—
75C2×C4⋊Dic71C2314yes—
76C2×C4⋊Dic71C2314yes—
77C2×C4⋊Dic71C2314yes—
78C2×Dic71⋊C4—28no—
79C2×Dic71⋊C4—44no—
80C2×Dic71⋊C4C2318yes—
81C2×Dic71⋊C4C2318yes—
82C4⋊C4×D71—232no—
83C2×(C2×D71)⋊C4—24no—
84C2×(C2×D71)⋊C4—28no—
85C2×(C2×D71)⋊C4—24no—
86C2×(C2×D71)⋊C4—44no—
87C2×(C2×D71)⋊C4—44no—
88C2×(C2×D71)⋊C4C2318yes—
89C2×(C2×D71)⋊C4C2314yes—
90C2×(C2×D71)⋊C4C2314yes—
91C2×(C2×D71)⋊C4C2318yes—
92C2×(C2×D71)⋊C4C2318yes—
93C71⋊#95C2318yes—
94C71⋊#95C2318yes—
95C71⋊#95C2318yes—
96C71⋊#95C2318yes—
97C23×C4×C71C2312yes—
98C23×C4×C71C2312yes—
99C23×C4×C71C2311yescharacteristic
100C23×C4×C71C2311yescharacteristic
101C23×C4×C71C2311yescharacteristic
102C2×C4⋊C4×C71—24no—
103C2×C4⋊C4×C71—24no—
104C2×C4⋊C4×C71—44no—
105C2×C4⋊C4×C71C2314yes—
106C2×C4⋊C4×C71C2314yes—
107C2×C4⋊C4×C71C2318yes—
108C2×C42×C71—24no—
109C2×C42×C71C2314yes—
110#95×C71C2318yes—
111C23×D71—22no—
112C23×D71—22no—
113C23×D71—22no—
114C23×D71—22no—
115C23×D71—22no—
116C23×D71—22no—
117C23×D71—22no—
118C23×D71—22no—
119C23×D71—22no—
120C23×D71—22no—
121C23×D71C2×D414yes—
122C23×D71C2×Q814yes—
123C23×D71C2×Q814yes—
124C23×D71#4114yes—
125C23×D71C2412yes—
126C23×D71C2412yes—
127C2×D284—232no—
128C2×D284—28no—
129C2×D284—28no—
130C2×D284C2×Q8116yes—
131C22×Dic71—22no—
132C22×Dic71—22no—
133C22×Dic71—22no—
134C22×Dic71—24no—
135C22×Dic71—22no—
136C22×Dic71—22no—
137C22×Dic71—24no—
138C22×Dic71—24no—
139C22×Dic71—24no—
140C22×Dic71—24no—
141C22×Dic71—24no—
142C22×Dic71C2×D414yes—
143C22×Dic71C2412yes—
144C22×Dic71C2414yes—
145C22×Dic71C2414yes—
146C22×Dic71C2412yes—
147C2×C4×D71—232no—
148C2×C4×D71—28no—
149C2×C4×D71—28no—
150C2×C4×D71—216no—
151C2×C4×D71—28no—
152C2×C4×D71—28no—
153C2×C4×D71—28no—
154C2×C4×D71—48no—
155C2×C4×D71—48no—
156C2×C4×D71—432no—
157C2×C4×D71—416no—
158C2×C4×D71C2×D4116yes—
159C4⋊Dic71—28no—
160C4⋊Dic71—28no—
161C4⋊Dic71—28no—
162Dic71⋊C4—216no—
163Dic71⋊C4—48no—
164(C2×D71)⋊C4—28no—
165(C2×D71)⋊C4—216no—
166(C2×D71)⋊C4—28no—
167(C2×D71)⋊C4—48no—
168(C2×D71)⋊C4—48no—
169C24×C71C2411yescharacteristic
170C22×C4×C71—22no—
171C22×C4×C71—22no—
172C22×C4×C71—24no—
173C22×C4×C71—22no—
174C22×C4×C71—24no—
175C22×C4×C71—24no—
176C22×C4×C71—24no—
177C22×C4×C71—24no—
178C22×C4×C71—22no—
179C22×C4×C71—24no—
180C22×C4×C71—22no—
181C22×C4×C71C2×D414yes—
182C22×C4×C71C2×Q814yes—
183C22×C4×C71#4114yes—
184C22×C4×C71#4114yes—
185C22×C4×C71C2414yes—
186C22×C4×C71C2414yes—
187C22×C4×C71C2412yes—
188C22×C4×C71C2412yes—
189C22×C4×C71C2412yes—
190C4⋊C4×C71—28no—
191C4⋊C4×C71—28no—
192C4⋊C4×C71—48no—
193C42×C71—28no—
194C23×C71C22×D412yes—
195C23×C71C22×Q812yes—
196C23×C71C2×#4112yes—
197C23×C71C2×#4112yes—
198C23×C71C2×#4112yes—
199C23×C71H(5,𝔽2)12yes—
200C23×C71H(5,𝔽2)12yes—
201C23×C71C2511yescharacteristic
202C22×D71—28no—
203C22×D71—28no—
204C22×D71—28no—
205C22×D71—28no—
206C22×D71—28no—
207C22×D71—28no—
208C22×D71—22no—
209C22×D71—22no—
210C22×D71—22no—
211C22×D71—22no—
212C22×D71—28no—
213C22×D71—28no—
214C22×D71—28no—
215C22×D71—22no—
216C22×D71—22no—
217C22×D71—22no—
218C22×D71—22no—
219C22×D71—22no—
220C22×D71—22no—
221C22×D71—44no—
222C22×D71—44no—
223C22×D71—44no—
224C22×D71—44no—
225C22×D71C22⋊Q8116yes—
226C22×D71C22×D414yes—
227C22×D71C22×Q814yes—
228C22×D71C22×Q814yes—
229C22×D71C2×#4114yes—
230C2×C4×C71—28no—
231C2×C4×C71—22no—
232C2×C4×C71—22no—
233C2×C4×C71—22no—
234C2×C4×C71—24no—
235C2×C4×C71—24no—
236C2×C4×C71—24no—
237C2×C4×C71—22no—
238C2×C4×C71—24no—
239C2×C4×C71—24no—
240C2×C4×C71—24no—
241C2×C4×C71—22no—
242C2×C4×C71—44no—
243C2×C4×C71—48no—
244C2×C4×C71—48no—
245C2×C4×C71#123116yes—
246C2×C4×C71C22×D414yes—
247C2×C4×C71C22×Q814yes—
248C2×C4×C71C2×#4114yes—
249C2×C4×C71C2×#4114yes—
250C2×Dic71—28no—
251C2×Dic71—22no—
252C2×Dic71—22no—
253C2×Dic71—22no—
254C2×Dic71—24no—
255C2×Dic71—24no—
256C2×Dic71—24no—
257C2×Dic71—22no—
258C2×Dic71—22no—
259C2×Dic71—24no—
260C2×Dic71—24no—
261C2×Dic71—24no—
262C2×Dic71—44no—
263C2×Dic71—48no—
264C2×Dic71—48no—
265C2×Dic71C22×D414yes—
266D284—232no—
267C4×D71—232no—
268C4×D71—432no—
269C4×D71—416no—
270C22×C71C2×C22⋊Q814yes—
271C22×C71C2×#12314yes—
272C22×C71#53514yes—
273C22×C71#55314yes—
274C22×C71#55314yes—
275C22×C71#55414yes—
276C22×C71#55414yes—
277C22×C71C23×D411yescharacteristic
278C22×C71C23×Q811yescharacteristic
279C22×C71C22×#4111yescharacteristic
280C22×C71C22×#4111yescharacteristic
281C22×C71C22×#4111yescharacteristic
282C22×C71C2×H(5,𝔽2)11yescharacteristic
283C22×C71C2×H(5,𝔽2)11yescharacteristic
284C2×D71—24no—
285C2×D71—24no—
286C2×D71—24no—
287C2×D71—24no—
288C2×D71—24no—
289C2×D71—24no—
290C2×D71—232no—
291C2×D71—24no—
292C2×D71—24no—
293C2×D71—24no—
294C2×D71—42no—
295C2×D71—42no—
296C2×D71—42no—
297C2×D71—42no—
298C2×D71—416no—
299C2×D71C2×C22⋊Q818yes—
300C4×C71—24no—
301C4×C71—44no—
302C4×C71—42no—
303C4×C71—44no—
304C4×C71C2×#12318yes—
305Dic71—24no—
306Dic71—42no—
307Dic71—44no—
308Dic71—44no—
309C2×#2487—711nomaximal, Sylow
310C2×C71#248718yescomplemented
311C2×C71C22×C22⋊Q811yescharacteristic
312C2×C71C22×#12311yescharacteristic
313C2×C71C2×#53511yescharacteristic
314C2×C71C2×#55311yescharacteristic
315C2×C71C2×#55311yescharacteristic
316C2×C71C2×#55411yescharacteristic
317C2×C71C2×#55411yescharacteristic
318D71—28no—
319D71—44no—
320C22×C4×D4—711no—
321C23×C4⋊C4—711no—
322#2487—7116no—
323C2×#400—712no—
324C2×#399—712no—
325C2×#389—711no—
326C2×#389D7111yescharacteristic, complemented
327C2×#386—712no—
328C2×#385—712no—
329C2×#380—711no—
330C2×#380—711no—
331C2×#380—711no—
332C71C2×#248711yescharacteristic, complemented, Sylow
333C23×D4—711no—
334C24×C4—712no—
335C24×C4—711no—
336C24×C4—711no—
337C2×C4×D4—718no—
338C2×C4×D4—1428no—
339C22×C4⋊C4—712no—
340C22×C4⋊C4—712no—
341C22×C4⋊C4—711no—
342C22×C4⋊C4—711no—
343C22×C4⋊C4—711no—
344C22×C4⋊C4—718no—
345C22×C4⋊C4—1428no—
346C22×C4⋊C4C2×D7111yescharacteristic
347C22×C4⋊C4C2×D7111yescharacteristic
348C22×C4⋊C4C2×D7112yes—
349C22×C22⋊C4—712no—
350C22×C22⋊C4—712no—
351C22×C22⋊C4—711no—
352C22×C22⋊C4—711no—
353C22×C22⋊C4—712no—
354C22×C42C2×D7111yescharacteristic
355#400—7116no—
356#399—7116no—
357#389—718no—
358#389C2×D7118yescomplemented
359#386—7116no—
360#385—7116no—
361#380—718no—
362#380—718no—
363#380—718no—
364C2×#95—712no—
365C2×#95—712no—
366C2×#95—712no—
367C2×#95—712no—
368C2×#95C2×D7112yes—
369C25—711no—
370C25—711no—
371C22×D4—718no—
372C22×D4—714no—
373C22×D4—1424no—
374C22×D4—1424no—
375C23×C4—2844no—
376C23×C4—2844no—
377C23×C4—718no—
378C23×C4—711no—
379C23×C4—718no—
380C23×C4—714no—
381C23×C4—712no—
382C23×C4—712no—
383C23×C4—711no—
384C23×C4—714no—
385C23×C4—1428no—
386C23×C4—1424no—
387C23×C4—1424no—
388C23×C4—1424no—
389C23×C4—1424no—
390C23×C4—1424no—
391C23×C4C22×D7111yescharacteristic
392C23×C4C22×D7112yes—
393C23×C4C22×D7112yes—
394C23×C4C22×D7111yescharacteristic
395C23×C4C22×D7111yescharacteristic
396C4×D4—14232no—
397C2×C4⋊C4—24no—
398C2×C4⋊C4—24no—
399C2×C4⋊C4—44no—
400C2×C4⋊C4—2844no—
401C2×C4⋊C4—718no—
402C2×C4⋊C4—714no—
403C2×C4⋊C4—718no—
404C2×C4⋊C4—714no—
405C2×C4⋊C4—714no—
406C2×C4⋊C4—14232no—
407C2×C4⋊C4—1428no—
408C2×C4⋊C4—1424no—
409C2×C4⋊C4—1424no—
410C2×C4⋊C4—1424no—
411C2×C4⋊C4C22×D7114yes—
412C2×C4⋊C4C22×D7114yes—
413C2×C4⋊C4C22×D7118yes—
414C2×C22⋊C4—2844no—
415C2×C22⋊C4—2844no—
416C2×C22⋊C4—718no—
417C2×C22⋊C4—718no—
418C2×C22⋊C4—714no—
419C2×C22⋊C4—714no—
420C2×C22⋊C4—718no—
421C2×C22⋊C4—1424no—
422C2×C22⋊C4—1424no—
423C2×C22⋊C4—1428no—
424C2×C42—24no—
425C2×C42C22×D7114yes—
426#95—718no—
427#95—718no—
428#95—718no—
429#95—718no—
430#95C22×D7118yes—
431C24—714no—
432C24—714no—
433C24—714no—
434C24—712no—
435C24—714no—
436C24—712no—
437C24—1422no—
438C24—1422no—
439C24—1422no—
440C24—1422no—
441C24—1422no—
442C24—1422no—
443C24—1422no—
444C24—1422no—
445C24—1422no—
446C24—1422no—
447C24C23×D7111yescharacteristic
448C2×D4—7116no—
449C2×D4—1428no—
450C2×D4—1428no—
451C2×D4—14232no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—2848no—
464C22×C4—2848no—
465C22×C4—28432no—
466C22×C4—28416no—
467C22×C4—714no—
468C22×C4—712no—
469C22×C4—7116no—
470C22×C4—712no—
471C22×C4—714no—
472C22×C4—714no—
473C22×C4—14216no—
474C22×C4—1422no—
475C22×C4—1424no—
476C22×C4—1424no—
477C22×C4—1422no—
478C22×C4—1424no—
479C22×C4—1422no—
480C22×C4—1424no—
481C22×C4—1424no—
482C22×C4—1424no—
483C22×C4—1428no—
484C22×C4—1428no—
485C22×C4—1428no—
486C22×C4—1422no—
487C22×C4—1422no—
488C22×C4—1428no—
489C22×C4—1428no—
490C22×C4—14232no—
491C22×C4D4×D7114yes—
492C22×C4Q8×D7114yes—
493C22×C4#1176586514yes—
494C22×C4#1176586614yes—
495C22×C4C23×D7114yes—
496C22×C4C23×D7114yes—
497C22×C4C23×D7112yes—
498C22×C4C23×D7112yes—
499C22×C4C23×D7112yes—
500C4⋊C4—28no—
501C4⋊C4—28no—
502C4⋊C4—48no—
503C4⋊C4—2848no—
504C4⋊C4—14216no—
505C4⋊C4—1428no—
506C4⋊C4—1428no—
507C4⋊C4—1428no—
508C22⋊C4—2848no—
509C22⋊C4—2848no—
510C22⋊C4—1428no—
511C22⋊C4—14216no—
512C22⋊C4—1428no—
513C42—28no—
514C23—2844no—
515C23—2844no—
516C23—2844no—
517C23—2844no—
518C23—714no—
519C23—714no—
520C23—714no—
521C23—7116no—
522C23—714no—
523C23—1422no—
524C23—1422no—
525C23—1422no—
526C23—1422no—
527C23—1422no—
528C23—1422no—
529C23—1428no—
530C23—1428no—
531C23—1428no—
532C23—1428no—
533C23—1422no—
534C23—1422no—
535C23—1422no—
536C23—1422no—
537C23—1428no—
538C23—1428no—
539C23—1428no—
540C23—1428no—
541C23—1428no—
542C23C2×D4×D7112yes—
543C23C2×Q8×D7112yes—
544C23#41×D7112yes—
545C23C2×#1176586512yes—
546C23C2×#1176586612yes—
547C23#1455617312yes—
548C23#1455617412yes—
549C23C24×D7111yescharacteristic
550D4—14232no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—28no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—48no—
564C2×C4—44no—
565C2×C4—48no—
566C2×C4—28432no—
567C2×C4—2848no—
568C2×C4—2848no—
569C2×C4—2844no—
570C2×C4—28416no—
571C2×C4—714no—
572C2×C4—1422no—
573C2×C4—1424no—
574C2×C4—1424no—
575C2×C4—1422no—
576C2×C4—1424no—
577C2×C4—1422no—
578C2×C4—1424no—
579C2×C4—1424no—
580C2×C4—1424no—
581C2×C4—1422no—
582C2×C4—1422no—
583C2×C4—1428no—
584C2×C4—14232no—
585C2×C4#14556113116yes—
586C2×C4C2×D4×D7114yes—
587C2×C4C2×Q8×D7114yes—
588C2×C4C2×#1176586514yes—
589C2×C4C2×#1176586614yes—
590C22—2842no—
591C22—2842no—
592C22—2842no—
593C22—28416no—
594C22—2842no—
595C22—718no—
596C22—1424no—
597C22—1424no—
598C22—1424no—
599C22—1424no—
600C22—1424no—
601C22—1424no—
602C22—14232no—
603C22—1424no—
604C22—1424no—
605C22—1424no—
606C22C22⋊Q8×D7114yes—
607C22C2×#1455611314yes—
608C22#1798656514yes—
609C22#1798669914yes—
610C22#1798670214yes—
611C22#1798670914yes—
612C22#1798671014yes—
613C22C22×D4×D7111yescharacteristic
614C22C22×Q8×D7111yescharacteristic
615C22C2×#41×D7111yescharacteristic
616C22C22×#1176586511yescharacteristic
617C22C22×#1176586611yescharacteristic
618C22C2×#1455617311yescharacteristic
619C22C2×#1455617411yescharacteristic
620C4—24no—
621C4—44no—
622C4—44no—
623C4—42no—
624C4—2842no—
625C4—2844no—
626C4—2844no—
627C4—1424no—
628C4C2×#1455611318yes—
629C2—2844no—
630C2—1428no—
631C2#2259265518yesdirect product factor
632C2C2×C22⋊Q8×D7111yescharacteristic
633C2C22×#1455611311yescharacteristic
634C2C2×#1798656511yescharacteristic
635C2C2×#1798669911yescharacteristic
636C2C2×#1798670211yescharacteristic
637C2C2×#1798670911yescharacteristic
638C2C2×#1798671011yescharacteristic
subgroupG/H
centerC24C23×D71
commutator, cosocleC23×C71C25
FrattiniC23C24×D71
FittingC2×#389×C71C2
socleC24×C71C24
prime radicalC23×D284C22
hypercenterC2×#389D71
hypocenterC71C2×#2487
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊C4×D71C2semidirect
3. #22592655C2direct (N × H)
4. C2×#17985941C2semidirect
5. C2×#17985934C2semidirect
6. C2×#17985897C2semidirect
8. C2×#17985870C2semidirect
9. C2×#17985856C2semidirect
10. C2×#17985856C2semidirect
11. C2×#17985856C2semidirect
12. C2×#389×C71C2semidirect
25. C2×C4⋊C4×D71C22semidirect
31. #17985941C22semidirect
32. #17985934C22semidirect
33. #17985897C22semidirect
35. #17985870C22semidirect
36. #17985856C22semidirect
37. #17985856C22semidirect
38. #17985856C22semidirect
47. #389×C71C22semidirect
310. C2×C71#2487semidirect
326. C2×#389D71semidirect
332. C71C2×#2487semidirect
358. #389C2×D71semidirect
631. C2#22592655direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×D2842
C22×C4⋊C4×D712
#225926552
C2×#179859412
C2×#179859342
C2×#179858972
C2×#179858772
C2×#179858702
C2×#179858562
C2×#389×C712
C2×#248771