Finite Group Databasebeta

← #2853×D71 #32203333

Subgroups — #2853×D71 #32203333

Order 18176.

3354 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#123×D71C211yesmaximal, characteristic
2C22×C42×D71C211yesmaximal, characteristic, complemented
3#22596665C212yesmaximal, complemented
4#22596664C214yesmaximal, complemented
5#22596663C211yesmaximal, characteristic, complemented
6#22596662C211yesmaximal, characteristic, complemented
7#22596661C212yesmaximal, complemented
8#22596660C214yesmaximal, complemented
9#22596659C211yesmaximal, characteristic, complemented
10#388×D71C214yesmaximal, complemented
11#387×D71C214yesmaximal, complemented
12#386×D71C212yesmaximal
13#385×D71C212yesmaximal, complemented
14#381×D71C211yesmaximal, characteristic, complemented
15C71×#2853C211yesmaximal, characteristic, complemented
16C22×D4×D71C2211yescharacteristic
17C23×C4×D71C2212yes—
18C23×C4×D71C2211yescharacteristic
19C2×#14556113C2212yes—
20C2×#14556112C2211yescharacteristic
21C2×#14556111C2211yescharacteristic
22C2×#14556110C2212yes—
23C2×#14556109C2211yescharacteristic
24#123×D71—28no—
25C2×C4⋊C4×D71C2214yes—
26C2×C4⋊C4×D71C2212yes—
27C2×C22⋊C4×D71C2211yescharacteristic
28C2×C22⋊C4×D71C2212yes—
29C2×C22⋊C4×D71C2214yes—
30C2×C22⋊C4×D71C2212yes—
31C22×C4×Dic71C2211yescharacteristic, complemented
32C22×C4×Dic71C2212yescomplemented
33C2×C42×D71—28no—
34C2×C42×D71C2214yes—
35C2×C42×D71C2214yescomplemented
36#17985893C2214yescomplemented
37#17985892C2214yescomplemented
38#17985891C2214yescomplemented
39#17985890C2214yescomplemented
40#17985889C2214yescomplemented
41#17985888C2214yescomplemented
42#17985887C2214yescomplemented
43#17985886C2214yescomplemented
44#17985885C2214yescomplemented
45#17985884C2214yescomplemented
46#17985883C2214yescomplemented
47#17985882C2214yescomplemented
48#17985881C2214yescomplemented
49#17985880C2214yescomplemented
50#17985879C2212yes—
51#17985878C2212yes—
52#17985877C2214yes—
53#17985876C2212yes—
54#17985875C2214yes—
55#17985874C2212yescomplemented
56#17985873C2214yescomplemented
57#17985872C2212yescomplemented
58#17985871C2212yescomplemented
59#17985870C2214yescomplemented
60#17985862C2211yescharacteristic, complemented
61#17985861C2214yescomplemented
62#17985860C2212yescomplemented
63#95×D71C2214yes—
64#95×D71C2214yes—
65#95×D71C2214yes—
66C2×#123×C71C2211yescharacteristic
67C22×C42×C71C2211yescharacteristic, complemented
68#388×C71C2214yescomplemented
69#387×C71C2214yescomplemented
70#386×C71C2212yes—
71#385×C71C2212yescomplemented
72#381×C71C2211yescharacteristic, complemented
73C24×D71C2311yescharacteristic
74C24×D71C2311yescharacteristic
75C22×D284C2311yescharacteristic
76C22×C22⋊D71C2311yescharacteristic
77C22×C22⋊D71C2311yescharacteristic
78C2×D4×D71—24no—
79C2×D4×D71—28no—
80C23×Dic71C2312yes—
81C23×Dic71C2311yescharacteristic
82C22×C4×D71—28no—
83C22×C4×D71—28no—
84C22×C4×D71—216no—
85C22×C4×D71D418yes—
86C22×C4×D71C2314yes—
87C22×C4×D71C2311yescharacteristic
88C22×C4×D71C2314yes—
89C22×C4×D71C2312yes—
90C22×C4×D71C2312yes—
91C22×C4×D71C2312yes—
92C22×C4×D71C2314yes—
93#14556113—28no—
94#14556112—24no—
95#14556111—24no—
96#14556110—28no—
97#14556109—24no—
98C2×C4⋊Dic71C2314yes—
99C2×C4⋊Dic71C2312yes—
100C2×Dic71⋊C4C2314yes—
101C2×Dic71⋊C4C2314yes—
102C2×Dic71⋊C4C2312yes—
103C2×Dic71⋊C4C2312yes—
104C4⋊C4×D71—28no—
105C4⋊C4×D71—216no—
106C2×C22⋊Dic71C2312yes—
107C2×C22⋊Dic71C2311yescharacteristic
108C2×C22⋊Dic71C2314yes—
109C2×C22⋊Dic71C2312yes—
110C2×(C2×D71)⋊C4C2314yes—
111C2×(C2×D71)⋊C4C2312yes—
112C2×(C2×D71)⋊C4C2314yes—
113C2×(C2×D71)⋊C4C2314yes—
114C2×(C2×D71)⋊C4C2312yes—
115C2×(C2×D71)⋊C4C2312yes—
116C22⋊C4×D71—24no—
117C22⋊C4×D71—28no—
118C22⋊C4×D71—28no—
119C22⋊C4×D71—216no—
120C2×C4×Dic71—28no—
121C2×C4×Dic71—24no—
122C2×C4×Dic71C2314yes—
123C2×C4×Dic71C2314yes—
124C2×C4×Dic71C2314yes—
125C42×D71—216no—
126C42×D71—216no—
127C71⋊#95C2314yes—
128C71⋊#95C2314yes—
129C71⋊#95C2314yes—
130C71⋊#95C2314yes—
131C71⋊#95C2314yes—
132C71⋊#95C2314yes—
133C71⋊#95C2314yes—
134C71⋊#95C2314yes—
135C71⋊#95C2314yes—
136C22×D4×C71C2311yescharacteristic
137C23×C4×C71C2312yes—
138C23×C4×C71C2311yescharacteristic
139#123×C71—24no—
140C2×C4⋊C4×C71C2314yes—
141C2×C4⋊C4×C71C2312yes—
142C2×C22⋊C4×C71C2312yes—
143C2×C22⋊C4×C71C2311yescharacteristic
144C2×C22⋊C4×C71C2314yes—
145C2×C22⋊C4×C71C2312yes—
146C2×C42×C71—24no—
147C2×C42×C71C2314yes—
148#95×C71C2314yes—
149#95×C71C2314yes—
150#95×C71C2314yes—
151C23×D71—28no—
152C23×D71—22no—
153C23×D71—24no—
154C23×D71—28no—
155C23×D71C2×D414yes—
156C23×D71#4118yes—
157C23×D71C2411yescharacteristic
158C23×D71C2411yescharacteristic
159C23×D71C2411yescharacteristic
160C2×D284—22no—
161C2×D284—24no—
162C2×C22⋊D71—22no—
163C2×C22⋊D71—24no—
164C2×C22⋊D71—22no—
165C2×C22⋊D71—24no—
166D4×D71—416no—
167C22×Dic71—24no—
168C22×Dic71—24no—
169C22×Dic71—28no—
170C22×Dic71C2×D414yes—
171C22×Dic71C2411yescharacteristic
172C22×Dic71C2414yes—
173C22×Dic71C2412yes—
174C22×Dic71C2412yes—
175C22×Dic71C2414yes—
176C2×C4×D71—264no—
177C2×C4×D71—232no—
178C2×C4×D71—28no—
179C2×C4×D71—28no—
180C2×C4×D71—28no—
181C2×C4×D71—28no—
182C2×C4×D71—28no—
183C2×C4×D71—28no—
184C2×C4×D71—28no—
185C2×C4×D71—22no—
186C2×C4×D71—24no—
187C2×C4×D71—24no—
188C2×C4×D71—28no—
189C2×C4×D71—28no—
190C2×C4×D71—28no—
191C2×C4×D71—28no—
192C2×C4×D71C2×D418yes—
193C2×C4×D71C2×D418yes—
194C2×C4×D71#41116yes—
195C4⋊Dic71—24no—
196C4⋊Dic71—28no—
197Dic71⋊C4—24no—
198Dic71⋊C4—28no—
199Dic71⋊C4—24no—
200Dic71⋊C4—28no—
201C22⋊Dic71—22no—
202C22⋊Dic71—24no—
203C22⋊Dic71—24no—
204C22⋊Dic71—28no—
205(C2×D71)⋊C4—24no—
206(C2×D71)⋊C4—28no—
207(C2×D71)⋊C4—24no—
208(C2×D71)⋊C4—28no—
209(C2×D71)⋊C4—24no—
210(C2×D71)⋊C4—28no—
211C4×Dic71—28no—
212C4×Dic71—28no—
213C4×Dic71—28no—
214C24×C71C2411yescharacteristic
215C24×C71C2411yescharacteristic
216C2×D4×C71—22no—
217C2×D4×C71—24no—
218C22×C4×C71—24no—
219C22×C4×C71—24no—
220C22×C4×C71—28no—
221C22×C4×C71C2×D414yes—
222C22×C4×C71C2411yescharacteristic
223C22×C4×C71C2414yes—
224C22×C4×C71C2412yes—
225C22×C4×C71C2412yes—
226C22×C4×C71C2414yes—
227C4⋊C4×C71—24no—
228C4⋊C4×C71—28no—
229C22⋊C4×C71—22no—
230C22⋊C4×C71—24no—
231C22⋊C4×C71—24no—
232C22⋊C4×C71—28no—
233C42×C71—28no—
234C23×C71—24no—
235C23×C71—21no—
236C23×C71—24no—
237C23×C71—22no—
238C23×C71C22×D412yes—
239C23×C71C2×#4114yes—
240C23×C71C2511yescharacteristic
241C22×D71—24no—
242C22×D71—216no—
243C22×D71—232no—
244C22×D71—21no—
245C22×D71—24no—
246C22×D71—22no—
247C22×D71—44no—
248C22×D71—48no—
249C22×D71—416no—
250C22×D71#123116yes—
251C22×D71C22×D412yes—
252C22×D71C22×D412yes—
253C22×D71C2×#4114yes—
254C22×D71C2×#4114yes—
255C22×D71C2×#4118yes—
256D4×C71—44no—
257C2×C4×C71—216no—
258C2×C4×C71—28no—
259C2×C4×C71—24no—
260C2×C4×C71—24no—
261C2×C4×C71—24no—
262C2×C4×C71—22no—
263C2×C4×C71—21no—
264C2×C4×C71—24no—
265C2×C4×C71—24no—
266C2×C4×C71—24no—
267C2×C4×C71—24no—
268C2×C4×C71—22no—
269C2×C4×C71C22×D414yes—
270C2×C4×C71C2×#4118yes—
271C22⋊D71—44no—
272C22⋊D71—44no—
273C2×Dic71—216no—
274C2×Dic71—28no—
275C2×Dic71—24no—
276C2×Dic71—24no—
277C2×Dic71—24no—
278C2×Dic71—22no—
279C2×Dic71—21no—
280C2×Dic71—24no—
281C2×Dic71—24no—
282C2×Dic71—24no—
283C2×Dic71—24no—
284C2×Dic71—22no—
285C2×Dic71C22×D414yes—
286C2×Dic71C2×#4118yes—
287D284—44no—
288C4×D71—232no—
289C4×D71—216no—
290C4×D71—216no—
291C4×D71—232no—
292C4×D71—44no—
293C4×D71—48no—
294C4×D71—416no—
295C22×C71—24no—
296C22×C71—28no—
297C22×C71—41no—
298C22×C71—42no—
299C22×C71—44no—
300C22×C71C2×#12314yes—
301C22×C71C23×D411yescharacteristic
302C22×C71C22×#4112yes—
303C22×C71C22×#4114yes—
304C2×D71—24no—
305C2×D71—28no—
306C2×D71—232no—
307C2×D71—41no—
308C2×D71—42no—
309C2×D71—44no—
310C2×D71—88no—
311C2×D71C2×#12314yes—
312C2×D71C2×#12314yes—
313C2×D71#52518yes—
314C2×D71#529116yes—
315C4×C71—24no—
316C4×C71—28no—
317C4×C71—41no—
318C4×C71—42no—
319C4×C71—44no—
320Dic71—24no—
321Dic71—28no—
322Dic71—41no—
323Dic71—42no—
324Dic71—44no—
325C2×#2853—711nomaximal, Sylow
326C2×C71—24no—
327C2×C71—81no—
328C2×C71C22×#12311yescharacteristic
329C2×C71C2×#52512yes—
330C2×C71C2×#52914yes—
331D71—24no—
332D71—81no—
333D71#285318yesdirect product factor
334C22×#123—711no—
335C23×C42—711no—
336#2853—711no—
337#2853—714no—
338#2853—714no—
339#2853—712no—
340#2853—712no—
341#2853—711no—
342#2853—711no—
343#2853D7111yescharacteristic, direct product factor
344C2×#388—714no—
345C2×#387—714no—
346C2×#386—712no—
347C2×#385—712no—
348C2×#381—711no—
349C71C2×#285311yescharacteristic, complemented, Sylow
350C23×D4—711no—
351C24×C4—711no—
352C24×C4—712no—
353C2×#123—711no—
354C2×#123—711no—
355C2×#123—712no—
356C2×#123—712no—
357C2×#123—711no—
358C2×#123—1428no—
359C2×#123C2×D7111yescharacteristic
360C22×C4⋊C4—712no—
361C22×C4⋊C4—714no—
362C22×C22⋊C4—711no—
363C22×C22⋊C4—712no—
364C22×C22⋊C4—712no—
365C22×C22⋊C4—714no—
366C22×C42—712no—
367C22×C42—711no—
368C22×C42—714no—
369C22×C42—714no—
370C22×C42—1428no—
371C22×C42C2×D7111yescharacteristic, complemented
372#388—714no—
373#388—714no—
374#388—714no—
375#388—714no—
376#388—714no—
377#388—714no—
378#388—714no—
379#388C2×D7114yescomplemented
380#387—714no—
381#387—714no—
382#387—714no—
383#387—714no—
384#387—714no—
385#387—714no—
386#387—714no—
387#387C2×D7114yescomplemented
388#386—712no—
389#386—712no—
390#386—714no—
391#386—712no—
392#386—714no—
393#386C2×D7112yes—
394#385—712no—
395#385—714no—
396#385—712no—
397#385—712no—
398#385—714no—
399#385C2×D7112yescomplemented
400#381—711no—
401#381—712no—
402#381—714no—
403#381C2×D7111yescharacteristic, complemented
404C2×#95—714no—
405C2×#95—714no—
406C2×#95—714no—
407C25—711no—
408C25—711no—
409C22×D4—711no—
410C22×D4—711no—
411C22×D4—711no—
412C22×D4—1424no—
413C22×D4—1428no—
414C22×D4C22×D7111yescharacteristic
415C23×C4—711no—
416C23×C4—718no—
417C23×C4—712no—
418C23×C4—714no—
419C23×C4—712no—
420C23×C4—712no—
421C23×C4—712no—
422C23×C4—714no—
423C23×C4—711no—
424C23×C4—714no—
425C23×C4—14216no—
426C23×C4—1428no—
427C23×C4—1428no—
428C23×C4C22×D7112yes—
429C23×C4C22×D7111yescharacteristic
430#123—24no—
431#123—1428no—
432#123—1424no—
433#123—1428no—
434#123—1424no—
435#123—1424no—
436C2×C4⋊C4—712no—
437C2×C4⋊C4—712no—
438C2×C4⋊C4—712no—
439C2×C4⋊C4—714no—
440C2×C4⋊C4—714no—
441C2×C4⋊C4—714no—
442C2×C4⋊C4—1428no—
443C2×C4⋊C4—14216no—
444C2×C4⋊C4C22×D7112yes—
445C2×C4⋊C4C22×D7114yes—
446C2×C22⋊C4—711no—
447C2×C22⋊C4—712no—
448C2×C22⋊C4—712no—
449C2×C22⋊C4—714no—
450C2×C22⋊C4—712no—
451C2×C22⋊C4—712no—
452C2×C22⋊C4—712no—
453C2×C22⋊C4—714no—
454C2×C22⋊C4—714no—
455C2×C22⋊C4—714no—
456C2×C22⋊C4—1428no—
457C2×C22⋊C4—14216no—
458C2×C22⋊C4—1424no—
459C2×C22⋊C4—1428no—
460C2×C22⋊C4C22×D7111yescharacteristic
461C2×C22⋊C4C22×D7112yes—
462C2×C22⋊C4C22×D7112yes—
463C2×C22⋊C4C22×D7114yes—
464C2×C42—24no—
465C2×C42—714no—
466C2×C42—714no—
467C2×C42—714no—
468C2×C42—14216no—
469C2×C42—14216no—
470C2×C42—1428no—
471C2×C42—1424no—
472C2×C42C22×D7114yes—
473#95—714no—
474#95—714no—
475#95—714no—
476#95—714no—
477#95—714no—
478#95—714no—
479#95—714no—
480#95—714no—
481#95—714no—
482#95C22×D7114yes—
483#95C22×D7114yes—
484#95C22×D7114yes—
485C24—711no—
486C24—714no—
487C24—718no—
488C24—711no—
489C24—711no—
490C24—1422no—
491C24—1428no—
492C24—1424no—
493C24—1428no—
494C24C23×D7111yescharacteristic
495C24C23×D7111yescharacteristic
496C2×D4—22no—
497C2×D4—24no—
498C2×D4—28416no—
499C2×D4—1422no—
500C2×D4—1424no—
501C2×D4—1422no—
502C2×D4—1424no—
503C2×D4—1422no—
504C2×D4—1424no—
505C22×C4—24no—
506C22×C4—24no—
507C22×C4—28no—
508C22×C4—714no—
509C22×C4—712no—
510C22×C4—714no—
511C22×C4—718no—
512C22×C4—718no—
513C22×C4—7116no—
514C22×C4—714no—
515C22×C4—711no—
516C22×C4—712no—
517C22×C4—1424no—
518C22×C4—1428no—
519C22×C4—1428no—
520C22×C4—1422no—
521C22×C4—1428no—
522C22×C4—1428no—
523C22×C4—1424no—
524C22×C4—1428no—
525C22×C4—1428no—
526C22×C4—1428no—
527C22×C4—1428no—
528C22×C4—1428no—
529C22×C4—1428no—
530C22×C4—14232no—
531C22×C4—14264no—
532C22×C4—1428no—
533C22×C4—1424no—
534C22×C4—1424no—
535C22×C4—1428no—
536C22×C4D4×D7114yes—
537C22×C4C23×D7114yes—
538C22×C4C23×D7111yescharacteristic
539C22×C4C23×D7112yes—
540C22×C4C23×D7112yes—
541C22×C4C23×D7114yes—
542C4⋊C4—24no—
543C4⋊C4—28no—
544C4⋊C4—1428no—
545C4⋊C4—1424no—
546C4⋊C4—1428no—
547C4⋊C4—1424no—
548C4⋊C4—1424no—
549C4⋊C4—1428no—
550C22⋊C4—22no—
551C22⋊C4—24no—
552C22⋊C4—24no—
553C22⋊C4—28no—
554C22⋊C4—1428no—
555C22⋊C4—1424no—
556C22⋊C4—1428no—
557C22⋊C4—1424no—
558C22⋊C4—1428no—
559C22⋊C4—1424no—
560C22⋊C4—1422no—
561C22⋊C4—1424no—
562C22⋊C4—1424no—
563C22⋊C4—1428no—
564C42—28no—
565C42—1428no—
566C42—1428no—
567C42—1428no—
568C23—24no—
569C23—21no—
570C23—24no—
571C23—22no—
572C23—2844no—
573C23—2848no—
574C23—28416no—
575C23—712no—
576C23—718no—
577C23—714no—
578C23—712no—
579C23—714no—
580C23—7116no—
581C23—14216no—
582C23—14232no—
583C23—1421no—
584C23—1424no—
585C23—1422no—
586C23—1424no—
587C23C2×D4×D7112yes—
588C23#41×D7114yes—
589C23C24×D7111yescharacteristic
590D4—44no—
591D4—2844no—
592D4—2844no—
593D4—2844no—
594C2×C4—216no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—28no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—2844no—
607C2×C4—2848no—
608C2×C4—28416no—
609C2×C4—714no—
610C2×C4—718no—
611C2×C4—14216no—
612C2×C4—14216no—
613C2×C4—14232no—
614C2×C4—14232no—
615C2×C4—1422no—
616C2×C4—1421no—
617C2×C4—1424no—
618C2×C4—1424no—
619C2×C4—1424no—
620C2×C4—1424no—
621C2×C4—1422no—
622C2×C4—14216no—
623C2×C4—1424no—
624C2×C4—1424no—
625C2×C4—1424no—
626C2×C4—1428no—
627C2×C4C2×D4×D7114yes—
628C2×C4#41×D7118yes—
629C22—24no—
630C22—28no—
631C22—41no—
632C22—42no—
633C22—44no—
634C22—2841no—
635C22—2842no—
636C22—2844no—
637C22—5688no—
638C22—714no—
639C22—718no—
640C22—7116no—
641C22—714no—
642C22—14232no—
643C22—1424no—
644C22—1428no—
645C22#123×D7114yes—
646C22C22×D4×D7111yescharacteristic
647C22C2×#41×D7112yes—
648C22C2×#41×D7114yes—
649C4—24no—
650C4—28no—
651C4—41no—
652C4—42no—
653C4—44no—
654C4—2841no—
655C4—2842no—
656C4—2844no—
657C4—1424no—
658C4—1428no—
659C2—24no—
660C2—81no—
661C2—5681no—
662C2—718no—
663C2—1424no—
664C2C2×#123×D7111yescharacteristic
665C2#525×D7112yes—
666C2#529×D7114yes—
subgroupG/H
center, FrattiniC23C24×D71
commutator, socle, cosocleC23×C71C25
FittingC71×#2853C2
prime radicalC22×D4×D71C22
hypercenter#2853D71
hypocenterC71C2×#2853
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #2853

Complements

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

NH ≅ G/Ntype
2. C22×C42×D71C2semidirect
3. #22596665C2semidirect
4. #22596664C2semidirect
5. #22596663C2semidirect
6. #22596662C2semidirect
7. #22596661C2semidirect
8. #22596660C2semidirect
9. #22596659C2semidirect
10. #388×D71C2semidirect
11. #387×D71C2semidirect
13. #385×D71C2semidirect
14. #381×D71C2semidirect
15. C71×#2853C2semidirect
31. C22×C4×Dic71C22semidirect
32. C22×C4×Dic71C22semidirect
35. C2×C42×D71C22semidirect
36. #17985893C22semidirect
37. #17985892C22semidirect
38. #17985891C22semidirect
39. #17985890C22semidirect
40. #17985889C22semidirect
41. #17985888C22semidirect
42. #17985887C22semidirect
43. #17985886C22semidirect
44. #17985885C22semidirect
45. #17985884C22semidirect
46. #17985883C22semidirect
47. #17985882C22semidirect
48. #17985881C22semidirect
49. #17985880C22semidirect
55. #17985874C22semidirect
56. #17985873C22semidirect
57. #17985872C22semidirect
58. #17985871C22semidirect
59. #17985870C22semidirect
60. #17985862C22semidirect
61. #17985861C22semidirect
62. #17985860C22semidirect
67. C22×C42×C71C22semidirect
68. #388×C71C22semidirect
69. #387×C71C22semidirect
71. #385×C71C22semidirect
72. #381×C71C22semidirect
333. D71#2853direct (N × H)
343. #2853D71direct (N × H)
349. C71C2×#2853semidirect
371. C22×C42C2×D71semidirect
379. #388C2×D71semidirect
387. #387C2×D71semidirect
399. #385C2×D71semidirect
403. #381C2×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#123×D712
C22×C42×D712
#225966652
#225966642
#225966632
#225966622
#225966612
#225966602
#225966592
#388×D712
#387×D712
#386×D712
#385×D712
#381×D712
C71×#28532
C2×#285371