Finite Group Databasebeta

← C2×#22592697 #32166551

Subgroups — C2×#22592697 #32166551

Order 18176.

3678 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556127C211yesmaximal, characteristic, complemented
2C22×#14556116C212yesmaximal, complemented
3C22×#14556103C211yesmaximal, characteristic
4#22592697C2116yesmaximal, direct product factor
5C2×#17985917C212yesmaximal, complemented
6C2×#17985912C212yesmaximal, complemented
7C2×#17985901C211yesmaximal, characteristic, complemented
8C2×#17985900C211yesmaximal, characteristic, complemented
9C2×#17985890C212yesmaximal, complemented
10C2×#17985856C211yesmaximal, characteristic, complemented
11C2×C4×(C2×D71)⋊C4C211yesmaximal, characteristic, complemented
12C2×C4×C4⋊C4×C71C211yesmaximal, characteristic, complemented
13C23×Dic142C2212yes—
14C23×D284C2211yescharacteristic
15C23×D284C2212yes—
16C23×C4×D71C2211yescharacteristic
17C2×#14556127—28no—
18C2×#14556127C2218yescomplemented
19C2×#14556116—216no—
20C2×#14556116C22116yescomplemented
21C2×#14556103—28no—
22C2×#14556103C2218yes—
23C22×C4⋊Dic71C2211yescharacteristic
24C22×Dic71⋊C4C2212yes—
25C22×(C2×D71)⋊C4C2212yes—
26C22×(C2×D71)⋊C4C2212yes—
27C22×(C2×D71)⋊C4C2212yes—
28C22×(C2×D71)⋊C4C2212yes—
29C22×(C2×D71)⋊C4C2211yescharacteristic
30C22×(C2×D71)⋊C4C2212yes—
31C22×(C2×D71)⋊C4C2212yes—
32C22×(C2×D71)⋊C4C2211yescharacteristic
33C22×C4×Dic71C2211yescharacteristic
34#17985917C22116yescomplemented
35#17985912C22116yescomplemented
36#17985901C2218yescomplemented
37#17985900C2218yescomplemented
38#17985890C22116yescomplemented
39#17985856C2218yescomplemented
40C4×(C2×D71)⋊C4C2218yescomplemented
41C2×C71⋊#95C2211yescharacteristic
42C2×C71⋊#95C2212yes—
43C2×C71⋊#95C2211yescharacteristic
44C22×C4⋊C4×C71C2211yescharacteristic
45C22×C4⋊C4×C71C2211yescharacteristic
46C22×C42×C71C2212yes—
47C22×C42×C71C2211yescharacteristic
48C4×C4⋊C4×C71C2218yescomplemented
49C2×#95×C71C2212yes—
50C24×D71C2312yes—
51C24×D71C2311yescharacteristic
52C22×Dic142—28no—
53C22×Dic142—28no—
54C22×Dic142—44no—
55C22×Dic142C2318yes—
56C22×D284—28no—
57C22×D284—28no—
58C22×D284—24no—
59C22×D284—24no—
60C22×D284—24no—
61C22×D284—44no—
62C22×D284C2314yes—
63C22×D284C2318yes—
64C23×Dic71C2311yescharacteristic
65C23×Dic71C2312yes—
66C23×Dic71C2311yescharacteristic
67C23×Dic71C2311yescharacteristic
68C22×C4×D71—24no—
69C22×C4×D71—24no—
70C22×C4×D71—24no—
71C22×C4×D71C2314yes—
72#14556127—232no—
73#14556116—264no—
74#14556103—232no—
75C2×C4⋊Dic71—24no—
76C2×C4⋊Dic71C2314yes—
77C2×Dic71⋊C4—28no—
78C2×Dic71⋊C4C2318yes—
79C2×(C2×D71)⋊C4—28no—
80C2×(C2×D71)⋊C4—28no—
81C2×(C2×D71)⋊C4—24no—
82C2×(C2×D71)⋊C4—28no—
83C2×(C2×D71)⋊C4—28no—
84C2×(C2×D71)⋊C4—28no—
85C2×(C2×D71)⋊C4—44no—
86C2×(C2×D71)⋊C4D418yescomplemented
87C2×(C2×D71)⋊C4C2314yes—
88C2×(C2×D71)⋊C4C2318yes—
89C2×(C2×D71)⋊C4C2318yes—
90C2×(C2×D71)⋊C4C2318yes—
91C2×(C2×D71)⋊C4C2318yes—
92C2×(C2×D71)⋊C4C2318yes—
93C2×(C2×D71)⋊C4C2318yes—
94C2×(C2×D71)⋊C4C2314yes—
95C2×C4×Dic71—24no—
96C2×C4×Dic71C2314yes—
97C71⋊#95C2314yes—
98C71⋊#95C2314yes—
99C71⋊#95C2318yes—
100C23×C4×C71C2312yes—
101C23×C4×C71C2311yescharacteristic
102C23×C4×C71C2312yes—
103C23×C4×C71C2311yescharacteristic
104C23×C4×C71C2311yescharacteristic
105C2×C4⋊C4×C71—24no—
106C2×C4⋊C4×C71D418yes—
107C2×C4⋊C4×C71C2314yes—
108C2×C4⋊C4×C71C2314yes—
109C2×C42×C71—28no—
110C2×C42×C71—24no—
111C2×C42×C71C2314yes—
112C2×C42×C71C2318yes—
113#95×C71C2318yes—
114C23×D71—22no—
115C23×D71—22no—
116C23×D71—22no—
117C23×D71—22no—
118C23×D71—22no—
119C23×D71—22no—
120C23×D71—24no—
121C23×D71—24no—
122C23×D71—24no—
123C23×D71—24no—
124C23×D71—24no—
125C23×D71—24no—
126C23×D71—24no—
127C23×D71C2×D414yes—
128C23×D71C2414yes—
129C23×D71C2412yes—
130C2×Dic142—216no—
131C2×Dic142—216no—
132C2×Dic142—48no—
133C2×Dic142—432no—
134C2×D284—216no—
135C2×D284—216no—
136C2×D284—28no—
137C2×D284—28no—
138C2×D284—28no—
139C2×D284—48no—
140C2×D284—432no—
141C2×D284—416no—
142C22×Dic71—22no—
143C22×Dic71—22no—
144C22×Dic71—22no—
145C22×Dic71—22no—
146C22×Dic71—22no—
147C22×Dic71—24no—
148C22×Dic71—24no—
149C22×Dic71—22no—
150C22×Dic71—22no—
151C22×Dic71—24no—
152C22×Dic71C2×D414yes—
153C22×Dic71#4114yes—
154C22×Dic71C2412yes—
155C22×Dic71C2414yes—
156C22×Dic71C2412yes—
157C22×Dic71C2412yes—
158C2×C4×D71—28no—
159C2×C4×D71—28no—
160C2×C4×D71—28no—
161C2×C4×D71—416no—
162C4⋊Dic71—28no—
163Dic71⋊C4—216no—
164(C2×D71)⋊C4—216no—
165(C2×D71)⋊C4—216no—
166(C2×D71)⋊C4—216no—
167(C2×D71)⋊C4—216no—
168(C2×D71)⋊C4—28no—
169(C2×D71)⋊C4—216no—
170(C2×D71)⋊C4—48no—
171(C2×D71)⋊C4C2×D4116yescomplemented
172C4×Dic71—28no—
173C24×C71C2411yescharacteristic
174C22×C4×C71—24no—
175C22×C4×C71—22no—
176C22×C4×C71—24no—
177C22×C4×C71—22no—
178C22×C4×C71—24no—
179C22×C4×C71—22no—
180C22×C4×C71—24no—
181C22×C4×C71—24no—
182C22×C4×C71—22no—
183C22×C4×C71—22no—
184C22×C4×C71C2×D414yes—
185C22×C4×C71C2×D418yes—
186C22×C4×C71C2×D414yes—
187C22×C4×C71#4114yes—
188C22×C4×C71#4114yes—
189C22×C4×C71C2412yes—
190C22×C4×C71C2414yes—
191C22×C4×C71C2414yes—
192C22×C4×C71C2412yes—
193C22×C4×C71C2412yes—
194C4⋊C4×C71—28no—
195C4⋊C4×C71C2×D4116yes—
196C42×C71—216no—
197C42×C71—28no—
198C23×C71C22×D412yes—
199C23×C71C22×D412yes—
200C23×C71C2×#4112yes—
201C23×C71C2×#4112yes—
202C23×C71C2×#4112yes—
203C23×C71H(5,𝔽2)12yes—
204C23×C712−1+412yes—
205C23×C71C2511yescharacteristic
206C22×D71—28no—
207C22×D71—28no—
208C22×D71—28no—
209C22×D71—22no—
210C22×D71—22no—
211C22×D71—22no—
212C22×D71—22no—
213C22×D71—22no—
214C22×D71—22no—
215C22×D71—24no—
216C22×D71—24no—
217C22×D71—24no—
218C22×D71—24no—
219C22×D71—24no—
220C22×D71—24no—
221C22×D71—24no—
222C22×D71—44no—
223C22×D71—44no—
224C22×D71—44no—
225C22×D71—44no—
226C22×D71—48no—
227C22×D71—48no—
228C22×D71—48no—
229C22×D71—48no—
230C22×D71—48no—
231C22×D71—48no—
232C22×D71—48no—
233C22×D71C22×D414yes—
234C2×C4×C71—216no—
235C2×C4×C71—28no—
236C2×C4×C71—24no—
237C2×C4×C71—22no—
238C2×C4×C71—24no—
239C2×C4×C71—22no—
240C2×C4×C71—24no—
241C2×C4×C71—22no—
242C2×C4×C71—24no—
243C2×C4×C71—24no—
244C2×C4×C71—22no—
245C2×C4×C71—22no—
246C2×C4×C71—48no—
247C2×C4×C71—44no—
248C2×C4×C71#124116yescomplemented
249C2×C4×C71C22×D414yes—
250C2×C4×C71C22×D418yes—
251C2×C4×C71C22×D414yes—
252C2×C4×C71C2×#4114yes—
253C2×C4×C71C2×#4114yes—
254C2×Dic71—28no—
255C2×Dic71—28no—
256C2×Dic71—22no—
257C2×Dic71—22no—
258C2×Dic71—22no—
259C2×Dic71—22no—
260C2×Dic71—22no—
261C2×Dic71—24no—
262C2×Dic71—24no—
263C2×Dic71—22no—
264C2×Dic71—22no—
265C2×Dic71—24no—
266C2×Dic71—44no—
267C2×Dic71—48no—
268C2×Dic71C22×D414yes—
269C2×Dic71C2×#4114yes—
270D284—432no—
271D284—416no—
272C4×D71—416no—
273Dic142—432no—
274C22×C71C2×#12414yes—
275C22×C71#53214yes—
276C22×C71#54214yes—
277C22×C71D4214yes—
278C22×C71#54714yes—
279C22×C71#54814yes—
280C22×C71#55514yes—
281C22×C71C23×D411yescharacteristic
282C22×C71C23×D411yescharacteristic
283C22×C71C22×#4111yescharacteristic
284C22×C71C22×#4111yescharacteristic
285C22×C71C22×#4111yescharacteristic
286C22×C71C2×H(5,𝔽2)11yescharacteristic
287C22×C71C2×2−1+411yescharacteristic
288C2×D71—24no—
289C2×D71—24no—
290C2×D71—24no—
291C2×D71—42no—
292C2×D71—42no—
293C2×D71—42no—
294C2×D71—42no—
295C2×D71—416no—
296C2×D71—44no—
297C2×D71—44no—
298C2×D71—44no—
299C2×D71—44no—
300C2×D71—44no—
301C2×D71—44no—
302C2×D71—44no—
303C2×D71—816no—
304C4×C71—28no—
305C4×C71—24no—
306C4×C71—44no—
307C4×C71—42no—
308C4×C71C2×#12418yescomplemented
309Dic71—24no—
310Dic71—24no—
311Dic71—42no—
312Dic71—44no—
313C2×#2491—711nomaximal, Sylow
314C2×C71#249118yescomplemented
315C2×C71C22×#12411yescharacteristic
316C2×C71C2×#53211yescharacteristic
317C2×C71C2×#54211yescharacteristic
318C2×C71C2×D4211yescharacteristic
319C2×C71C2×#54711yescharacteristic
320C2×C71C2×#54811yescharacteristic
321C2×C71C2×#55511yescharacteristic
322D71—44no—
323D71—84no—
324C22×C4⋊Q8—711no—
325C22×#124—712no—
326C22×#121—711no—
327#2491—7116no—
328C2×#394—712no—
329C2×#393—712no—
330C2×#390—711no—
331C2×#390—711no—
332C2×#388—712no—
333C2×#380—711no—
334C2×C4×C4⋊C4D7111yescharacteristic, complemented
335C2×C4×C22⋊C4—711no—
336C71C2×#249111yescharacteristic, complemented, Sylow
337C23×Q8—712no—
338C23×D4—712no—
339C23×D4—711no—
340C24×C4—711no—
341C2×C4⋊Q8—718no—
342C2×C4⋊Q8—1428no—
343C2×#124—7116no—
344C2×#124—14216no—
345C2×#121—718no—
346C2×#121—1428no—
347C22×C4⋊C4—712no—
348C22×C4⋊C4—711no—
349C22×C4⋊C4C2×D7111yescharacteristic
350C22×C4⋊C4C2×D7111yescharacteristic
351C22×C22⋊C4—712no—
352C22×C22⋊C4—711no—
353C22×C22⋊C4—712no—
354C22×C22⋊C4—712no—
355C22×C22⋊C4—711no—
356C22×C22⋊C4—712no—
357C22×C22⋊C4—712no—
358C22×C22⋊C4—712no—
359C22×C42—711no—
360C22×C42C2×D7112yes—
361C22×C42C2×D7111yescharacteristic
362#394—7116no—
363#393—7116no—
364#390—718no—
365#390—718no—
366#388—7116no—
367#380—718no—
368C4×C4⋊C4C2×D7118yescomplemented
369C4×C22⋊C4—718no—
370C2×#95—711no—
371C2×#95—712no—
372C2×#95—711no—
373C2×#95C2×D7112yes—
374C25—711no—
375C25—712no—
376C22×Q8—2844no—
377C22×Q8—718no—
378C22×Q8—1428no—
379C22×Q8—1428no—
380C22×D4—2844no—
381C22×D4—718no—
382C22×D4—714no—
383C22×D4—1428no—
384C22×D4—1428no—
385C22×D4—1424no—
386C22×D4—1424no—
387C22×D4—1424no—
388C23×C4—711no—
389C23×C4—711no—
390C23×C4—712no—
391C23×C4—711no—
392C23×C4—714no—
393C23×C4—1424no—
394C23×C4—1424no—
395C23×C4—1424no—
396C23×C4C22×D7112yes—
397C23×C4C22×D7111yescharacteristic
398C23×C4C22×D7111yescharacteristic
399C23×C4C22×D7112yes—
400C23×C4C22×D7111yescharacteristic
401C4⋊Q8—14232no—
402#124—14264no—
403#121—14232no—
404C2×C4⋊C4—24no—
405C2×C4⋊C4—714no—
406C2×C4⋊C4—718no—
407C2×C4⋊C4—1428no—
408C2×C4⋊C4—1424no—
409C2×C4⋊C4D28418yes—
410C2×C4⋊C4C22×D7114yes—
411C2×C4⋊C4C22×D7114yes—
412C2×C22⋊C4—2844no—
413C2×C22⋊C4—718no—
414C2×C22⋊C4—718no—
415C2×C22⋊C4—714no—
416C2×C22⋊C4—718no—
417C2×C22⋊C4—718no—
418C2×C22⋊C4—718no—
419C2×C22⋊C4—718no—
420C2×C22⋊C4—714no—
421C2×C22⋊C4—718no—
422C2×C22⋊C4—1428no—
423C2×C22⋊C4—1428no—
424C2×C22⋊C4—1424no—
425C2×C22⋊C4—1428no—
426C2×C22⋊C4—1428no—
427C2×C22⋊C4—1428no—
428C2×C42—28no—
429C2×C42—24no—
430C2×C42—714no—
431C2×C42—1424no—
432C2×C42C22×D7118yes—
433C2×C42C22×D7114yes—
434#95—714no—
435#95—714no—
436#95—718no—
437#95C22×D7118yes—
438C24—714no—
439C24—712no—
440C24—714no—
441C24—1424no—
442C24—1424no—
443C24—1424no—
444C24—1424no—
445C24—1424no—
446C24—1424no—
447C24—1424no—
448C24—1422no—
449C24—1422no—
450C24—1422no—
451C24—1422no—
452C24—1422no—
453C24—1422no—
454C24C23×D7111yescharacteristic
455C2×Q8—28432no—
456C2×Q8—2848no—
457C2×Q8—14216no—
458C2×Q8—14216no—
459C2×D4—28432no—
460C2×D4—2848no—
461C2×D4—28416no—
462C2×D4—14216no—
463C2×D4—14216no—
464C2×D4—1428no—
465C2×D4—1428no—
466C2×D4—1428no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—28416no—
478C22×C4—714no—
479C22×C4—714no—
480C22×C4—712no—
481C22×C4—712no—
482C22×C4—712no—
483C22×C4—714no—
484C22×C4—1424no—
485C22×C4—1422no—
486C22×C4—1424no—
487C22×C4—1422no—
488C22×C4—1422no—
489C22×C4—1424no—
490C22×C4—1428no—
491C22×C4—1428no—
492C22×C4—1428no—
493C22×C4—1422no—
494C22×C4—1422no—
495C22×C4—1422no—
496C22×C4—1422no—
497C22×C4D4×D7114yes—
498C22×C4C2×D28418yes—
499C22×C4C2×D28414yes—
500C22×C4#1176586514yes—
501C22×C4#1176586514yes—
502C22×C4C23×D7112yes—
503C22×C4C23×D7114yes—
504C22×C4C23×D7112yes—
505C22×C4C23×D7114yes—
506C22×C4C23×D7112yes—
507C4⋊C4—28no—
508C4⋊C4—14216no—
509C4⋊C4—1428no—
510C4⋊C4C2×D284116yes—
511C22⋊C4—2848no—
512C22⋊C4—7116no—
513C22⋊C4—14216no—
514C22⋊C4—1428no—
515C22⋊C4—14216no—
516C22⋊C4—14216no—
517C22⋊C4—14216no—
518C22⋊C4—14216no—
519C42—216no—
520C42—28no—
521C42—1428no—
522C23—2844no—
523C23—2844no—
524C23—2844no—
525C23—2844no—
526C23—2848no—
527C23—2848no—
528C23—2848no—
529C23—2848no—
530C23—2848no—
531C23—2848no—
532C23—2848no—
533C23—714no—
534C23—1424no—
535C23—1424no—
536C23—1424no—
537C23—1424no—
538C23—1424no—
539C23—1424no—
540C23—1424no—
541C23—1422no—
542C23—1422no—
543C23—1422no—
544C23—1422no—
545C23—1422no—
546C23—1422no—
547C23—1428no—
548C23—1428no—
549C23—1428no—
550C23C2×D4×D7112yes—
551C23C22×D28412yes—
552C23#41×D7112yes—
553C23C2×#1176586512yes—
554C23C2×#1176586512yes—
555C23#1455617412yes—
556C23#1455617612yes—
557C23C24×D7111yescharacteristic
558Q8—28432no—
559D4—28432no—
560D4—28416no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—216no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—44no—
574C2×C4—48no—
575C2×C4—2848no—
576C2×C4—2844no—
577C2×C4—28416no—
578C2×C4—714no—
579C2×C4—714no—
580C2×C4—1424no—
581C2×C4—1422no—
582C2×C4—1424no—
583C2×C4—1422no—
584C2×C4—1422no—
585C2×C4—1424no—
586C2×C4—1422no—
587C2×C4—1422no—
588C2×C4—1422no—
589C2×C4—1422no—
590C2×C4—1428no—
591C2×C4—1428no—
592C2×C4#14556116116yescomplemented
593C2×C4C2×D4×D7114yes—
594C2×C4C22×D28418yes—
595C2×C4C22×D28414yes—
596C2×C4C2×#1176586514yes—
597C2×C4C2×#1176586514yes—
598C22—2842no—
599C22—2842no—
600C22—2842no—
601C22—28416no—
602C22—2842no—
603C22—2844no—
604C22—2844no—
605C22—2844no—
606C22—2844no—
607C22—2844no—
608C22—2844no—
609C22—2844no—
610C22—56816no—
611C22—1424no—
612C22—1424no—
613C22—1424no—
614C22C2×#1455611614yes—
615C22#1798654614yes—
616C22#1798662114yes—
617C22D4×D28414yes—
618C22#1798665014yes—
619C22#1798666114yes—
620C22#1798671214yes—
621C22C22×D4×D7111yescharacteristic
622C22C23×D28411yescharacteristic
623C22C2×#41×D7111yescharacteristic
624C22C22×#1176586511yescharacteristic
625C22C22×#1176586511yescharacteristic
626C22C2×#1455617411yescharacteristic
627C22C2×#1455617611yescharacteristic
628C4—28no—
629C4—24no—
630C4—44no—
631C4—42no—
632C4—2844no—
633C4—2842no—
634C4—1424no—
635C4—1424no—
636C4C2×#1455611618yescomplemented
637C2—2844no—
638C2—5684no—
639C2#2259269718yesdirect product factor
640C2C22×#1455611611yescharacteristic
641C2C2×#1798654611yescharacteristic
642C2C2×#1798662111yescharacteristic
643C2C2×D4×D28411yescharacteristic
644C2C2×#1798665011yescharacteristic
645C2C2×#1798666111yescharacteristic
646C2C2×#1798671211yescharacteristic
subgroupG/H
centerC24C23×D71
commutator, cosocleC23×C71C25
FrattiniC23C24×D71
FittingC2×C4×C4⋊C4×C71C2
socleC24×C71C24
prime radicalC22×#14556103C2
hypercenterC2×C4×C4⋊C4D71
hypocenterC71C2×#2491
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#14556127C2semidirect
2. C22×#14556116C2semidirect
4. #22592697C2direct (N × H)
5. C2×#17985917C2semidirect
6. C2×#17985912C2semidirect
7. C2×#17985901C2semidirect
8. C2×#17985900C2semidirect
9. C2×#17985890C2semidirect
10. C2×#17985856C2semidirect
11. C2×C4×(C2×D71)⋊C4C2semidirect
12. C2×C4×C4⋊C4×C71C2semidirect
18. C2×#14556127C22semidirect
20. C2×#14556116C22semidirect
34. #17985917C22semidirect
35. #17985912C22semidirect
36. #17985901C22semidirect
37. #17985900C22semidirect
38. #17985890C22semidirect
39. #17985856C22semidirect
40. C4×(C2×D71)⋊C4C22semidirect
48. C4×C4⋊C4×C71C22semidirect
86. C2×(C2×D71)⋊C4D4semidirect
171. (C2×D71)⋊C4C2×D4semidirect
248. C2×C4×C71#124semidirect
308. C4×C71C2×#124semidirect
314. C2×C71#2491semidirect
334. C2×C4×C4⋊C4D71semidirect
336. C71C2×#2491semidirect
368. C4×C4⋊C4C2×D71semidirect
592. C2×C4#14556116semidirect
636. C4C2×#14556116semidirect
639. C2#22592697direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145561272
C22×#145561162
C22×#145561032
#225926972
C2×#179859172
C2×#179859122
C2×#179859012
C2×#179859002
C2×#179858902
C2×#179858562
C2×C4×(C2×D71)⋊C42
C2×C4×C4⋊C4×C712
C2×#249171