Finite Group Databasebeta

← #32157495

Subgroups — #32157495

Order 18176.

4110 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊C4×D71C211yesmaximal, characteristic
2#22591188C218yesmaximal, complemented
3#22591184C218yesmaximal, complemented
4#22590953C212yesmaximal, complemented
5#22590938C211yesmaximal, characteristic, complemented
6#22590936C212yesmaximal, complemented
7#22590722C212yesmaximal, complemented
8#22590721C212yesmaximal, complemented
9#379×D71C212yesmaximal
10#22590692C212yesmaximal, complemented
11C71×#2312C211yesmaximal, characteristic, complemented
12C25×D71C2211yescharacteristic
13C23×C4×D71C2211yescharacteristic
14C23×C4×D71C2211yescharacteristic
15C2×#14556086C2214yescomplemented
16C2×#14556085C2214yescomplemented
17C22×C22⋊Dic71C2211yescharacteristic
18C22×(C2×D71)⋊C4C2211yescharacteristic
19C22×(C2×D71)⋊C4C2211yescharacteristic
20C2×C22⋊C4×D71C4116yescomplemented
21C2×C22⋊C4×D71C2214yes—
22C2×C22⋊C4×D71C2214yes—
23C2×C22⋊C4×D71C2214yes—
24C2×C22⋊C4×D71C2214yes—
25C2×C22⋊C4×D71C2214yes—
26#17985893C2218yescomplemented
27#17985892C2218yescomplemented
28#17985887C2218yescomplemented
29#17985873C2218yescomplemented
30#17985870C2218yescomplemented
31#17985861C2214yescomplemented
32#17985861C2214yescomplemented
33#17985859C2212yescomplemented
34#17985858C2214yescomplemented
35#17985857C2214yescomplemented
36#17985856C2214yescomplemented
37#17985855C2212yes—
38#17985854C2214yes—
39#17985854C2214yes—
40#17985854C2214yes—
41C22⋊C4×Dic71C2214yescomplemented
42C4×(C2×D71)⋊C4C2214yescomplemented
43C4×(C2×D71)⋊C4C2218yescomplemented
44C4×C22⋊Dic71C2212yescomplemented
45#17985846C2214yescomplemented
46#17985846C2218yescomplemented
47C22×C22⋊C4×C71C2211yescharacteristic
48#388×C71C2218yescomplemented
49#380×C71C2212yescomplemented
50#379×C71C2212yes—
51C4×C22⋊C4×C71C2212yescomplemented
52C24×D71—28no—
53C24×D71—44no—
54C24×D71—44no—
55C24×D71C2×C418yes—
56C24×D71C2311yescharacteristic
57C24×D71C2312yes—
58C24×D71C2312yes—
59C24×D71C2311yescharacteristic
60C23×Dic71C2311yescharacteristic
61C23×Dic71C2311yescharacteristic
62C22×C4×D71—44no—
63C22×C4×D71—44no—
64C22×C4×D71C2×C418yescomplemented
65C22×C4×D71C2×C418yescomplemented
66C22×C4×D71C2312yes—
67C22×C4×D71C2312yes—
68C22×C4×D71C2318yes—
69C22×C4×D71C2312yes—
70C22×C4×D71C2312yes—
71#14556086—48no—
72#14556085—48no—
73C2×C4⋊Dic71C2314yescomplemented
74C2×Dic71⋊C4C2318yescomplemented
75C2×C22⋊Dic71C2×C418yescomplemented
76C2×C22⋊Dic71C2314yes—
77C2×C22⋊Dic71C2314yes—
78C2×C22⋊Dic71C2314yes—
79C2×C22⋊Dic71C2314yes—
80C2×(C2×D71)⋊C4C2×C418yes—
81C2×(C2×D71)⋊C4C2×C418yes—
82C2×(C2×D71)⋊C4C2318yes—
83C2×(C2×D71)⋊C4C2318yes—
84C2×(C2×D71)⋊C4C2318yes—
85C2×(C2×D71)⋊C4C2314yes—
86C2×(C2×D71)⋊C4C2314yes—
87C22⋊C4×D71—216no—
88C22⋊C4×D71—48no—
89C22⋊C4×D71—48no—
90C22⋊C4×D71C2×C4132yescomplemented
91C22⋊C4×D71C2×C4132yescomplemented
92C2×C4×Dic71C2314yescomplemented
93C2×C4×Dic71C2318yescomplemented
94C71⋊#95C2318yescomplemented
95C71⋊#95C2314yescomplemented
96C71⋊#95C2314yescomplemented
97C71⋊#95C2318yescomplemented
98C25×C71C2311yescharacteristic
99C23×C4×C71C2311yescharacteristic
100C23×C4×C71C2311yescharacteristic
101C2×C4⋊C4×C71C2314yescomplemented
102C2×C22⋊C4×C71C2×C418yescomplemented
103C2×C22⋊C4×C71C2314yes—
104C2×C22⋊C4×C71C2314yes—
105C2×C22⋊C4×C71C2314yes—
106C2×C22⋊C4×C71C2314yes—
107C2×C42×C71C2314yescomplemented
108#95×C71C2314yescomplemented
109#95×C71C2314yescomplemented
110C23×D71—24no—
111C23×D71—24no—
112C23×D71—24no—
113C23×D71—232no—
114C23×D71—28no—
115C23×D71—28no—
116C23×D71—28no—
117C23×D71—28no—
118C23×D71—416no—
119C23×D71—416no—
120C23×D71—42no—
121C23×D71—42no—
122C23×D71—44no—
123C23×D71—44no—
124C23×D71—464no—
125C23×D71C22×C414yes—
126C23×D71C22×C418yes—
127C23×D71C22×C418yes—
128C23×D71C22×C414yes—
129C23×D71#4118yes—
130C23×D71C2412yes—
131C23×D71C2411yescharacteristic
132C23×D71C2411yescharacteristic
133C22×Dic71—42no—
134C22×Dic71—42no—
135C22×Dic71C22×C414yes—
136C22×Dic71C22×C414yes—
137C22×Dic71C2418yes—
138C22×Dic71C2412yes—
139C22×Dic71C2412yes—
140C2×C4×D71—28no—
141C2×C4×D71—28no—
142C2×C4×D71—216no—
143C2×C4×D71—416no—
144C2×C4×D71—416no—
145C2×C4×D71—44no—
146C2×C4×D71—44no—
147C2×C4×D71—48no—
148C2×C4×D71—48no—
149C2×C4×D71C22×C418yes—
150C2×C4×D71C22×C418yes—
151C2×C4×D71C22×C418yes—
152C2×C4×D71C22×C418yes—
153C4⋊Dic71—44no—
154Dic71⋊C4—48no—
155C22⋊Dic71—28no—
156C22⋊Dic71—44no—
157C22⋊Dic71—44no—
158C22⋊Dic71C22×C4116yes—
159(C2×D71)⋊C4—216no—
160(C2×D71)⋊C4—216no—
161(C2×D71)⋊C4—216no—
162(C2×D71)⋊C4C22×C4116yes—
163(C2×D71)⋊C4C22×C4116yes—
164C4×Dic71—44no—
165C4×Dic71—48no—
166C24×C71—24no—
167C24×C71—42no—
168C24×C71—42no—
169C24×C71C22×C414yes—
170C24×C71C2412yes—
171C24×C71C2411yescharacteristic
172C22×C4×C71—42no—
173C22×C4×C71—42no—
174C22×C4×C71C22×C414yes—
175C22×C4×C71C22×C414yes—
176C22×C4×C71C2418yes—
177C22×C4×C71C2412yes—
178C22×C4×C71C2412yes—
179C4⋊C4×C71—44no—
180C22⋊C4×C71—28no—
181C22⋊C4×C71—44no—
182C22⋊C4×C71—44no—
183C22⋊C4×C71C22×C4116yes—
184C42×C71—44no—
185C23×C71—22no—
186C23×C71—22no—
187C23×C71—28no—
188C23×C71—24no—
189C23×C71—24no—
190C23×C71—24no—
191C23×C71—44no—
192C23×C71—44no—
193C23×C71—416no—
194C23×C71C23×C414yes—
195C23×C71C23×C412yes—
196C23×C71C2×#4114yes—
197C23×C71C2511yescharacteristic
198C22×D71—232no—
199C22×D71—28no—
200C22×D71—22no—
201C22×D71—22no—
202C22×D71—22no—
203C22×D71—22no—
204C22×D71—24no—
205C22×D71—24no—
206C22×D71—28no—
207C22×D71—28no—
208C22×D71—216no—
209C22×D71—216no—
210C22×D71—216no—
211C22×D71—216no—
212C22×D71—48no—
213C22×D71—48no—
214C22×D71—44no—
215C22×D71—44no—
216C22×D71—416no—
217C22×D71—432no—
218C22×D71—432no—
219C22×D71—4128no—
220C22×D71#117116yescomplemented
221C22×D71C23×C414yes—
222C22×D71C23×C412yes—
223C22×D71C23×C412yes—
224C22×D71C2×#4118yes—
225C22×D71H(5,𝔽2)14yes—
226C22×D71H(5,𝔽2)14yes—
227C2×C4×C71—24no—
228C2×C4×C71—24no—
229C2×C4×C71—28no—
230C2×C4×C71—28no—
231C2×C4×C71—28no—
232C2×C4×C71—44no—
233C2×C4×C71—44no—
234C2×C4×C71C23×C414yes—
235C2×C4×C71C23×C414yes—
236C2×Dic71—24no—
237C2×Dic71—24no—
238C2×Dic71—28no—
239C2×Dic71—28no—
240C2×Dic71—28no—
241C2×Dic71—44no—
242C2×Dic71—44no—
243C2×Dic71C23×C414yes—
244C2×Dic71C23×C414yes—
245C4×D71—216no—
246C4×D71—216no—
247C4×D71—48no—
248C4×D71—48no—
249C4×D71—816no—
250C22×C71—22no—
251C22×C71—22no—
252C22×C71—28no—
253C22×C71—24no—
254C22×C71—24no—
255C22×C71—24no—
256C22×C71—416no—
257C22×C71—48no—
258C22×C71C2×#11714yes—
259C22×C71C24×C411yescharacteristic
260C22×C71C22×#4112yes—
261C22×C71C2×H(5,𝔽2)12yes—
262C22×C71C2×H(5,𝔽2)12yes—
263C2×D71—232no—
264C2×D71—22no—
265C2×D71—22no—
266C2×D71—22no—
267C2×D71—22no—
268C2×D71—24no—
269C2×D71—24no—
270C2×D71—216no—
271C2×D71—28no—
272C2×D71—28no—
273C2×D71—416no—
274C2×D71—432no—
275C2×D71—432no—
276C2×D71C2×#11718yes—
277C2×D71#51814yes—
278C2×D71#51814yes—
279C4×C71—24no—
280C4×C71—24no—
281C4×C71—48no—
282Dic71—24no—
283Dic71—24no—
284Dic71—48no—
285#34176—711nomaximal, Sylow
286C2×C71—24no—
287C2×C71—44no—
288C2×C71C22×#11711yescharacteristic
289C2×C71C2×#51811yescharacteristic
290C2×C71C2×#51811yescharacteristic
291C2×C71#349014yes—
292D71—24no—
293D71—24no—
294D71—28no—
295C23×C22⋊C4—711no—
296#2342—718no—
297#2342—718no—
298#2315—712no—
299#2312—712no—
300#2312—711no—
301#2312D7111yescharacteristic, complemented
302#2280—712no—
303#2280—712no—
304C2×#379—712no—
305#2273—712no—
306C71#3417611yescharacteristic, complemented, Sylow
307C26—711no—
308C24×C4—711no—
309C24×C4—711no—
310C2×#117—714no—
311C2×#117—714no—
312C22×C22⋊C4—714no—
313C22×C22⋊C4—714no—
314C22×C22⋊C4—711no—
315C22×C22⋊C4—711no—
316C22×C22⋊C4—711no—
317C22×C22⋊C4—7116no—
318C22×C22⋊C4—714no—
319C22×C22⋊C4—714no—
320C22×C22⋊C4—714no—
321C22×C22⋊C4C2×D7111yescharacteristic
322#388—718no—
323#388—718no—
324#388—718no—
325#388C2×D7118yescomplemented
326#385—718no—
327#385—718no—
328#381—714no—
329#381—714no—
330#380—712no—
331#380—714no—
332#380—714no—
333#380—714no—
334#380C2×D7112yescomplemented
335#379—712no—
336#379—714no—
337#379—714no—
338#379—714no—
339#379C2×D7112yes—
340C4×C22⋊C4—712no—
341C4×C22⋊C4—714no—
342C4×C22⋊C4—718no—
343C4×C22⋊C4—714no—
344C4×C22⋊C4—718no—
345C4×C22⋊C4—714no—
346C4×C22⋊C4C2×D7112yescomplemented
347C25—2844no—
348C25—2844no—
349C25—711no—
350C25—718no—
351C25—712no—
352C25—712no—
353C25—711no—
354C25—1428no—
355C25C22×D7111yescharacteristic
356C23×C4—2844no—
357C23×C4—2844no—
358C23×C4—711no—
359C23×C4—711no—
360C23×C4—718no—
361C23×C4—718no—
362C23×C4—712no—
363C23×C4—712no—
364C23×C4—718no—
365C23×C4—712no—
366C23×C4—712no—
367C23×C4C22×D7111yescharacteristic
368C23×C4C22×D7111yescharacteristic
369#117—2848no—
370#117—2848no—
371C2×C4⋊C4—718no—
372C2×C4⋊C4—714no—
373C2×C4⋊C4C22×D7114yescomplemented
374C2×C22⋊C4—2848no—
375C2×C22⋊C4—2848no—
376C2×C22⋊C4—714no—
377C2×C22⋊C4—714no—
378C2×C22⋊C4—718no—
379C2×C22⋊C4—714no—
380C2×C22⋊C4—718no—
381C2×C22⋊C4—718no—
382C2×C22⋊C4—714no—
383C2×C22⋊C4—714no—
384C2×C22⋊C4—7132no—
385C2×C22⋊C4—7132no—
386C2×C22⋊C4—714no—
387C2×C22⋊C4—718no—
388C2×C22⋊C4—718no—
389C2×C22⋊C4—718no—
390C2×C22⋊C4—14216no—
391C2×C22⋊C4C4×D7118yescomplemented
392C2×C22⋊C4C22×D7114yes—
393C2×C22⋊C4C22×D7114yes—
394C2×C22⋊C4C22×D7114yes—
395C2×C22⋊C4C22×D7114yes—
396C2×C42—718no—
397C2×C42—714no—
398C2×C42C22×D7114yescomplemented
399#95—718no—
400#95—714no—
401#95—718no—
402#95—714no—
403#95C22×D7114yescomplemented
404#95C22×D7114yescomplemented
405C24—24no—
406C24—42no—
407C24—42no—
408C24—2842no—
409C24—2842no—
410C24—2844no—
411C24—2844no—
412C24—28464no—
413C24—28416no—
414C24—28416no—
415C24—714no—
416C24—718no—
417C24—718no—
418C24—718no—
419C24—712no—
420C24—711no—
421C24—711no—
422C24—714no—
423C24—1428no—
424C24—1428no—
425C24—1428no—
426C24—1428no—
427C24—14232no—
428C24—1424no—
429C24—1424no—
430C24—1424no—
431C24C2×C4×D7114yes—
432C24C23×D7112yes—
433C24C23×D7111yescharacteristic
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—2844no—
437C22×C4—2844no—
438C22×C4—2848no—
439C22×C4—2848no—
440C22×C4—28416no—
441C22×C4—28416no—
442C22×C4—2842no—
443C22×C4—2842no—
444C22×C4—714no—
445C22×C4—714no—
446C22×C4—718no—
447C22×C4—718no—
448C22×C4—712no—
449C22×C4—712no—
450C22×C4—718no—
451C22×C4—718no—
452C22×C4—718no—
453C22×C4—14216no—
454C22×C4—1428no—
455C22×C4—1428no—
456C22×C4C2×C4×D7114yes—
457C22×C4C2×C4×D7114yes—
458C22×C4C23×D7118yes—
459C22×C4C23×D7112yes—
460C22×C4C23×D7112yes—
461C4⋊C4—44no—
462C4⋊C4—2848no—
463C4⋊C4—2844no—
464C22⋊C4—28no—
465C22⋊C4—44no—
466C22⋊C4—44no—
467C22⋊C4—2844no—
468C22⋊C4—2844no—
469C22⋊C4—7116no—
470C22⋊C4—7116no—
471C22⋊C4—7116no—
472C22⋊C4—14216no—
473C22⋊C4—14216no—
474C22⋊C4—14216no—
475C22⋊C4—1428no—
476C22⋊C4C2×C4×D71116yes—
477C42—44no—
478C42—2844no—
479C42—2848no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—28no—
486C23—44no—
487C23—44no—
488C23—416no—
489C23—28416no—
490C23—2844no—
491C23—2844no—
492C23—2848no—
493C23—2848no—
494C23—284128no—
495C23—28432no—
496C23—28432no—
497C23—7116no—
498C23—714no—
499C23—712no—
500C23—712no—
501C23—718no—
502C23—714no—
503C23—714no—
504C23—14216no—
505C23—14216no—
506C23—14216no—
507C23—14216no—
508C23—1428no—
509C23—1428no—
510C23—14232no—
511C23—1422no—
512C23—1422no—
513C23—1424no—
514C23—1424no—
515C23—1422no—
516C23—1422no—
517C23—1428no—
518C23C22×C4×D7114yes—
519C23C22×C4×D7112yes—
520C23#41×D7114yes—
521C23C24×D7111yescharacteristic
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—28no—
525C2×C4—28no—
526C2×C4—28no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—2848no—
530C2×C4—2848no—
531C2×C4—2844no—
532C2×C4—2844no—
533C2×C4—56816no—
534C2×C4—714no—
535C2×C4—714no—
536C2×C4—14216no—
537C2×C4—14216no—
538C2×C4—1428no—
539C2×C4—1428no—
540C2×C4—1428no—
541C2×C4—1424no—
542C2×C4—1424no—
543C2×C4C22×C4×D7114yes—
544C2×C4C22×C4×D7114yes—
545C22—28no—
546C22—22no—
547C22—22no—
548C22—24no—
549C22—24no—
550C22—24no—
551C22—416no—
552C22—48no—
553C22—28416no—
554C22—28432no—
555C22—28432no—
556C22—718no—
557C22—714no—
558C22—714no—
559C22—14232no—
560C22—1422no—
561C22—1422no—
562C22—14216no—
563C22—1428no—
564C22—1428no—
565C22—1422no—
566C22—1422no—
567C22—1424no—
568C22—1424no—
569C22#117×D7114yes—
570C22C23×C4×D7111yescharacteristic
571C22C2×#41×D7112yes—
572C22H(5,𝔽2)×D7112yes—
573C22H(5,𝔽2)×D7112yes—
574C4—24no—
575C4—24no—
576C4—48no—
577C4—2848no—
578C4—1424no—
579C4—1424no—
580C2—24no—
581C2—44no—
582C2—1428no—
583C2—1424no—
584C2—1424no—
585C2C2×#117×D7111yescharacteristic
586C2#518×D7111yescharacteristic
587C2#518×D7111yescharacteristic
588C2#2260202614yes—
subgroupG/H
center, FrattiniC23C24×D71
commutatorC22×C71C24×C4
FittingC71×#2312C2
socle, cosocleC23×C71C25
prime radicalC25×D71C22
hypercenter#2312D71
hypocenterC71#34176
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ #2312

Complements

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

NH ≅ G/Ntype
2. #22591188C2semidirect
3. #22591184C2semidirect
4. #22590953C2semidirect
5. #22590938C2semidirect
6. #22590936C2semidirect
7. #22590722C2semidirect
8. #22590721C2semidirect
10. #22590692C2semidirect
11. C71×#2312C2semidirect
15. C2×#14556086C22semidirect
16. C2×#14556085C22semidirect
20. C2×C22⋊C4×D71C4semidirect
26. #17985893C22semidirect
27. #17985892C22semidirect
28. #17985887C22semidirect
29. #17985873C22semidirect
30. #17985870C22semidirect
31. #17985861C22semidirect
32. #17985861C22semidirect
33. #17985859C22semidirect
34. #17985858C22semidirect
35. #17985857C22semidirect
36. #17985856C22semidirect
41. C22⋊C4×Dic71C22semidirect
42. C4×(C2×D71)⋊C4C22semidirect
43. C4×(C2×D71)⋊C4C22semidirect
44. C4×C22⋊Dic71C22semidirect
45. #17985846C22semidirect
46. #17985846C22semidirect
48. #388×C71C22semidirect
49. #380×C71C22semidirect
51. C4×C22⋊C4×C71C22semidirect
64. C22×C4×D71C2×C4semidirect
65. C22×C4×D71C2×C4semidirect
73. C2×C4⋊Dic71C23semidirect
74. C2×Dic71⋊C4C23semidirect
75. C2×C22⋊Dic71C2×C4semidirect
90. C22⋊C4×D71C2×C4semidirect
91. C22⋊C4×D71C2×C4semidirect
92. C2×C4×Dic71C23semidirect
93. C2×C4×Dic71C23semidirect
94. C71⋊#95C23semidirect
95. C71⋊#95C23semidirect
96. C71⋊#95C23semidirect
97. C71⋊#95C23semidirect
101. C2×C4⋊C4×C71C23semidirect
102. C2×C22⋊C4×C71C2×C4semidirect
107. C2×C42×C71C23semidirect
108. #95×C71C23semidirect
109. #95×C71C23semidirect
220. C22×D71#117semidirect
301. #2312D71semidirect
306. C71#34176semidirect
325. #388C2×D71semidirect
334. #380C2×D71semidirect
346. C4×C22⋊C4C2×D71semidirect
373. C2×C4⋊C4C22×D71semidirect
391. C2×C22⋊C4C4×D71semidirect
398. C2×C42C22×D71semidirect
403. #95C22×D71semidirect
404. #95C22×D71semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊C4×D712
#225911882
#225911842
#225909532
#225909382
#225909362
#225907222
#225907212
#379×D712
#225906922
C71×#23122
#3417671