Finite Group Databasebeta

← #2646×D71 #32163893

Subgroups — #2646×D71 #32163893

Order 18176.

2374 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22594313C212yesmaximal, complemented
2#22594312C214yesmaximal, complemented
3#22594311C214yesmaximal, complemented
4#22594310C211yesmaximal, characteristic, complemented
5#22594309C214yesmaximal, complemented
6#400×D71C214yesmaximal, complemented
7#398×D71C214yesmaximal, complemented
8#385×D71C214yesmaximal
9#380×D71C212yesmaximal
10C2×#95×D71C211yesmaximal, characteristic
11C71×#2646C211yesmaximal, characteristic, complemented
12C23×C4×D71C2212yes—
13C23×C4×D71C2211yescharacteristic
14C2×C4⋊C4×D71C2214yes—
15C2×C4⋊C4×D71C2214yes—
16C2×C4⋊C4×D71C2212yes—
17C2×C4⋊C4×D71C2212yes—
18C2×C22⋊C4×D71C2214yes—
19#17985941C2214yescomplemented
20#17985941C2214yescomplemented
21#17985940C2214yescomplemented
22#17985940C2214yescomplemented
23#17985939C2214yescomplemented
24#17985938C2214yescomplemented
25#17985937C2214yescomplemented
26#17985933C2214yescomplemented
27#17985933C2214yescomplemented
28#17985932C2214yescomplemented
29#17985932C2214yescomplemented
30#17985931C2214yescomplemented
31#17985930C2214yescomplemented
32#17985929C2214yescomplemented
33#17985874C2214yes—
34#17985873C2214yes—
35#17985873C2214yes—
36#17985872C2214yes—
37#17985871C2214yes—
38#17985870C2214yes—
39#17985870C2214yes—
40#17985859C2212yes—
41#17985858C2214yes—
42#17985857C2214yes—
43#17985856C2212yes—
44#17985856C2212yes—
45C2×C71⋊#95C2212yes—
46C2×C71⋊#95C2211yescharacteristic
47#95×D71C2214yes—
48#95×D71C2214yes—
49#95×D71C2214yes—
50#95×D71C2214yes—
51#95×D71C2214yes—
52#400×C71C2214yescomplemented
53#398×C71C2214yescomplemented
54#385×C71C2214yes—
55#380×C71C2212yes—
56C2×#95×C71C2211yescharacteristic
57C24×D71C2311yescharacteristic
58C23×Dic71C2312yes—
59C23×Dic71C2311yescharacteristic
60C22×C4×D71—28no—
61C22×C4×D71—44no—
62C22×C4×D71—44no—
63C22×C4×D71—44no—
64C22×C4×D71Q818yes—
65C22×C4×D71C2314yes—
66C22×C4×D71C2314yes—
67C22×C4×D71C2314yes—
68C22×C4×D71C2312yes—
69C22×C4×D71C2314yes—
70C22×C4×D71C2312yes—
71C2×C4⋊Dic71C2312yes—
72C2×C4⋊Dic71C2312yes—
73C2×C4⋊Dic71C2314yes—
74C2×C4⋊Dic71C2314yes—
75C2×Dic71⋊C4C2314yes—
76C2×Dic71⋊C4C2314yes—
77C2×Dic71⋊C4C2314yes—
78C2×Dic71⋊C4C2314yes—
79C2×Dic71⋊C4C2314yes—
80C2×Dic71⋊C4C2314yes—
81C4⋊C4×D71—44no—
82C4⋊C4×D71—44no—
83C4⋊C4×D71—48no—
84C4⋊C4×D71—48no—
85C2×C22⋊Dic71C2314yes—
86C2×(C2×D71)⋊C4C2314yes—
87C2×(C2×D71)⋊C4C2314yes—
88C22⋊C4×D71—216no—
89C71⋊#95C2314yes—
90C71⋊#95C2314yes—
91C71⋊#95C2314yes—
92C71⋊#95C2314yes—
93C71⋊#95C2314yes—
94C71⋊#95C2314yes—
95C71⋊#95C2314yes—
96C71⋊#95C2314yes—
97C71⋊#95C2314yes—
98C71⋊#95C2314yes—
99C71⋊#95C2314yes—
100C71⋊#95C2314yes—
101C23×C4×C71C2312yes—
102C23×C4×C71C2311yescharacteristic
103C2×C4⋊C4×C71C2312yes—
104C2×C4⋊C4×C71C2312yes—
105C2×C4⋊C4×C71C2314yes—
106C2×C4⋊C4×C71C2314yes—
107C2×C22⋊C4×C71C2314yes—
108#95×C71C2314yes—
109#95×C71C2314yes—
110#95×C71C2314yes—
111#95×C71C2314yes—
112C23×D71—22no—
113C23×D71—24no—
114C23×D71—22no—
115C23×D71C2×Q814yes—
116C23×D71#4118yes—
117C23×D71C2411yescharacteristic
118C23×D71C2411yescharacteristic
119C22×Dic71—24no—
120C22×Dic71—42no—
121C22×Dic71—42no—
122C22×Dic71—42no—
123C22×Dic71C2×Q814yes—
124C22×Dic71C2414yes—
125C22×Dic71C2414yes—
126C22×Dic71C2414yes—
127C22×Dic71C2412yes—
128C2×C4×D71—24no—
129C2×C4×D71—24no—
130C2×C4×D71—28no—
131C2×C4×D71—28no—
132C2×C4×D71—28no—
133C2×C4×D71—28no—
134C2×C4×D71—28no—
135C2×C4×D71—28no—
136C2×C4×D71—28no—
137C2×C4×D71—28no—
138C2×C4×D71—28no—
139C2×C4×D71—28no—
140C2×C4×D71—416no—
141C2×C4×D71—42no—
142C2×C4×D71—42no—
143C2×C4×D71—44no—
144C2×C4×D71—44no—
145C2×C4×D71—432no—
146C2×C4×D71C2×Q818yes—
147C2×C4×D71C2×Q818yes—
148C4⋊Dic71—42no—
149C4⋊Dic71—44no—
150C4⋊Dic71—42no—
151C4⋊Dic71—44no—
152Dic71⋊C4—44no—
153Dic71⋊C4—44no—
154Dic71⋊C4—44no—
155Dic71⋊C4—44no—
156Dic71⋊C4—44no—
157Dic71⋊C4—44no—
158C22⋊Dic71—28no—
159(C2×D71)⋊C4—28no—
160(C2×D71)⋊C4—28no—
161C24×C71C2411yescharacteristic
162C22×C4×C71—24no—
163C22×C4×C71—42no—
164C22×C4×C71—42no—
165C22×C4×C71—42no—
166C22×C4×C71C2×Q814yes—
167C22×C4×C71C2414yes—
168C22×C4×C71C2414yes—
169C22×C4×C71C2414yes—
170C22×C4×C71C2412yes—
171C4⋊C4×C71—42no—
172C4⋊C4×C71—42no—
173C4⋊C4×C71—44no—
174C4⋊C4×C71—44no—
175C22⋊C4×C71—28no—
176C23×C71—21no—
177C23×C71—22no—
178C23×C71—21no—
179C23×C71C22×Q812yes—
180C23×C71C2×#4114yes—
181C23×C71C2511yescharacteristic
182C22×D71—21no—
183C22×D71—22no—
184C22×D71—21no—
185C22×D71—216no—
186C22×D71—216no—
187C22×D71—28no—
188C22×D71—28no—
189C22×D71#125116yes—
190C22×D71C22×Q812yes—
191C22×D71C22×Q812yes—
192C22×D71C2×#4114yes—
193C22×D71C2×#4114yes—
194C22×D71H(5,𝔽2)12yes—
195C22×D71H(5,𝔽2)14yes—
196C22×D712−1+412yes—
197C2×C4×C71—22no—
198C2×C4×C71—22no—
199C2×C4×C71—24no—
200C2×C4×C71—24no—
201C2×C4×C71—24no—
202C2×C4×C71—24no—
203C2×C4×C71—24no—
204C2×C4×C71—24no—
205C2×C4×C71—24no—
206C2×C4×C71—24no—
207C2×C4×C71—24no—
208C2×C4×C71—44no—
209C2×C4×C71—48no—
210C2×C4×C71C22×Q814yes—
211C2×Dic71—22no—
212C2×Dic71—22no—
213C2×Dic71—24no—
214C2×Dic71—24no—
215C2×Dic71—24no—
216C2×Dic71—24no—
217C2×Dic71—24no—
218C2×Dic71—24no—
219C2×Dic71—24no—
220C2×Dic71—24no—
221C2×Dic71—24no—
222C2×Dic71—44no—
223C2×Dic71—48no—
224C2×Dic71C22×Q814yes—
225C4×D71—216no—
226C4×D71—48no—
227C4×D71—416no—
228C4×D71—416no—
229C4×D71—416no—
230C4×D71—416no—
231C22×C71—24no—
232C22×C71—24no—
233C22×C71—22no—
234C22×C71—22no—
235C22×C71C2×#12514yes—
236C22×C71C23×Q811yescharacteristic
237C22×C71C22×#4112yes—
238C22×C71C2×H(5,𝔽2)11yescharacteristic
239C22×C71C2×H(5,𝔽2)12yes—
240C22×C71C2×2−1+411yescharacteristic
241C2×D71—24no—
242C2×D71—24no—
243C2×D71—22no—
244C2×D71—22no—
245C2×D71—232no—
246C2×D71C2×#12514yes—
247C2×D71C2×#12514yes—
248C2×D71#53818yes—
249C2×D71#53918yes—
250C2×D71#54314yes—
251C2×D71#54414yes—
252C4×C71—24no—
253C4×C71—44no—
254C4×C71—44no—
255C4×C71—44no—
256Dic71—24no—
257Dic71—44no—
258Dic71—44no—
259Dic71—44no—
260C2×#2646—711nomaximal, Sylow
261C2×C71—24no—
262C2×C71C22×#12511yescharacteristic
263C2×C71C2×#53812yes—
264C2×C71C2×#53912yes—
265C2×C71C2×#54311yescharacteristic
266C2×C71C2×#54411yescharacteristic
267D71—24no—
268D71#264618yesdirect product factor
269#2646—714no—
270#2646—714no—
271#2646—714no—
272#2646—712no—
273#2646—711no—
274#2646D7111yescharacteristic, direct product factor
275C2×#400—714no—
276C2×#398—714no—
277C2×#385—714no—
278C2×#380—712no—
279C22×#95—711no—
280C71C2×#264611yescharacteristic, complemented, Sylow
281C24×C4—711no—
282C24×C4—712no—
283C22×C4⋊C4—712no—
284C22×C4⋊C4—714no—
285C22×C4⋊C4—714no—
286C22×C4⋊C4—712no—
287C22×C22⋊C4—714no—
288#400—714no—
289#400—714no—
290#400—714no—
291#400—714no—
292#400—714no—
293#400—714no—
294#400—714no—
295#400C2×D7114yescomplemented
296#398—714no—
297#398—714no—
298#398—714no—
299#398—714no—
300#398—714no—
301#398—714no—
302#398—714no—
303#398C2×D7114yescomplemented
304#385—714no—
305#385—714no—
306#385—714no—
307#385—714no—
308#385—714no—
309#385—714no—
310#385—714no—
311#385C2×D7114yes—
312#380—712no—
313#380—712no—
314#380—712no—
315#380—714no—
316#380—714no—
317#380C2×D7112yes—
318C2×#95—712no—
319C2×#95—711no—
320C2×#95—714no—
321C2×#95—714no—
322C2×#95—714no—
323C2×#95—714no—
324C2×#95—714no—
325C2×#95C2×D7111yescharacteristic
326C25—711no—
327C23×C4—2844no—
328C23×C4—2844no—
329C23×C4—2844no—
330C23×C4—711no—
331C23×C4—718no—
332C23×C4—712no—
333C23×C4—712no—
334C23×C4—714no—
335C23×C4—712no—
336C23×C4—714no—
337C23×C4—714no—
338C23×C4—714no—
339C23×C4—1428no—
340C23×C4C22×D7112yes—
341C23×C4C22×D7111yescharacteristic
342C2×C4⋊C4—2848no—
343C2×C4⋊C4—2848no—
344C2×C4⋊C4—2844no—
345C2×C4⋊C4—2844no—
346C2×C4⋊C4—714no—
347C2×C4⋊C4—712no—
348C2×C4⋊C4—714no—
349C2×C4⋊C4—712no—
350C2×C4⋊C4—714no—
351C2×C4⋊C4—714no—
352C2×C4⋊C4—714no—
353C2×C4⋊C4—714no—
354C2×C4⋊C4—714no—
355C2×C4⋊C4—714no—
356C2×C4⋊C4C22×D7112yes—
357C2×C4⋊C4C22×D7112yes—
358C2×C4⋊C4C22×D7114yes—
359C2×C4⋊C4C22×D7114yes—
360C2×C22⋊C4—714no—
361C2×C22⋊C4—714no—
362C2×C22⋊C4—714no—
363C2×C22⋊C4—14216no—
364C2×C22⋊C4C22×D7114yes—
365#95—714no—
366#95—714no—
367#95—714no—
368#95—714no—
369#95—714no—
370#95—714no—
371#95—714no—
372#95—714no—
373#95—714no—
374#95—714no—
375#95—714no—
376#95—714no—
377#95C22×D7114yes—
378#95C22×D7114yes—
379#95C22×D7114yes—
380#95C22×D7114yes—
381C24—711no—
382C24—714no—
383C24—718no—
384C24—711no—
385C24—1424no—
386C24—1422no—
387C24—1422no—
388C24C23×D7111yescharacteristic
389C22×C4—24no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—2842no—
394C22×C4—2842no—
395C22×C4—2844no—
396C22×C4—2844no—
397C22×C4—28432no—
398C22×C4—28416no—
399C22×C4—2842no—
400C22×C4—2842no—
401C22×C4—2842no—
402C22×C4—714no—
403C22×C4—718no—
404C22×C4—712no—
405C22×C4—718no—
406C22×C4—714no—
407C22×C4—714no—
408C22×C4—714no—
409C22×C4—1428no—
410C22×C4—1428no—
411C22×C4—1428no—
412C22×C4—1428no—
413C22×C4—1428no—
414C22×C4—1428no—
415C22×C4—1428no—
416C22×C4—1428no—
417C22×C4—1428no—
418C22×C4—1428no—
419C22×C4—1424no—
420C22×C4—1424no—
421C22×C4—1424no—
422C22×C4Q8×D7114yes—
423C22×C4C23×D7114yes—
424C22×C4C23×D7114yes—
425C22×C4C23×D7114yes—
426C22×C4C23×D7112yes—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—2844no—
432C4⋊C4—2844no—
433C4⋊C4—2844no—
434C4⋊C4—2844no—
435C4⋊C4—2844no—
436C4⋊C4—2844no—
437C4⋊C4—2842no—
438C4⋊C4—2842no—
439C4⋊C4—2844no—
440C4⋊C4—2844no—
441C22⋊C4—28no—
442C22⋊C4—1428no—
443C22⋊C4—1428no—
444C22⋊C4—1428no—
445C23—22no—
446C23—21no—
447C23—21no—
448C23—712no—
449C23—714no—
450C23—714no—
451C23—712no—
452C23—712no—
453C23—712no—
454C23—714no—
455C23—7116no—
456C23—1428no—
457C23—14216no—
458C23—1428no—
459C23—14216no—
460C23—1422no—
461C23—1421no—
462C23—1421no—
463C23C2×Q8×D7112yes—
464C23#41×D7114yes—
465C23C24×D7111yescharacteristic
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—44no—
478C2×C4—48no—
479C2×C4—2848no—
480C2×C4—28416no—
481C2×C4—28416no—
482C2×C4—28416no—
483C2×C4—28416no—
484C2×C4—2848no—
485C2×C4—2844no—
486C2×C4—714no—
487C2×C4—14216no—
488C2×C4—1424no—
489C2×C4—1424no—
490C2×C4—1424no—
491C2×C4—1424no—
492C2×C4—1424no—
493C2×C4—1424no—
494C2×C4—1424no—
495C2×C4—1424no—
496C2×C4—1424no—
497C2×C4—1422no—
498C2×C4—1422no—
499C2×C4C2×Q8×D7114yes—
500C22—22no—
501C22—22no—
502C22—24no—
503C22—24no—
504C22—714no—
505C22—714no—
506C22—714no—
507C22—718no—
508C22—718no—
509C22—714no—
510C22—14232no—
511C22—1422no—
512C22—1422no—
513C22—1424no—
514C22—1424no—
515C22#125×D7114yes—
516C22C22×Q8×D7111yescharacteristic
517C22C2×#41×D7112yes—
518C22H(5,𝔽2)×D7112yes—
519C22H(5,𝔽2)×D7111yescharacteristic
520C222−1+4×D7111yescharacteristic
521C4—24no—
522C4—44no—
523C4—44no—
524C4—44no—
525C4—2844no—
526C4—2844no—
527C4—2844no—
528C4—1424no—
529C2—24no—
530C2—718no—
531C2—1424no—
532C2C2×#125×D7111yescharacteristic
533C2#538×D7112yes—
534C2#539×D7112yes—
535C2#543×D7111yescharacteristic
536C2#544×D7111yescharacteristic
subgroupG/H
center, FrattiniC23C24×D71
commutator, socle, cosocleC23×C71C25
FittingC71×#2646C2
prime radicalC24×D71C23
hypercenter#2646D71
hypocenterC71C2×#2646
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #2646

Complements

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

NH ≅ G/Ntype
1. #22594313C2semidirect
2. #22594312C2semidirect
3. #22594311C2semidirect
4. #22594310C2semidirect
5. #22594309C2semidirect
6. #400×D71C2semidirect
7. #398×D71C2semidirect
11. C71×#2646C2semidirect
19. #17985941C22semidirect
20. #17985941C22semidirect
21. #17985940C22semidirect
22. #17985940C22semidirect
23. #17985939C22semidirect
24. #17985938C22semidirect
25. #17985937C22semidirect
26. #17985933C22semidirect
27. #17985933C22semidirect
28. #17985932C22semidirect
29. #17985932C22semidirect
30. #17985931C22semidirect
31. #17985930C22semidirect
32. #17985929C22semidirect
52. #400×C71C22semidirect
53. #398×C71C22semidirect
268. D71#2646direct (N × H)
274. #2646D71direct (N × H)
280. C71C2×#2646semidirect
295. #400C2×D71semidirect
303. #398C2×D71semidirect

Maximal subgroups

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

subgroupindex
#225943132
#225943122
#225943112
#225943102
#225943092
#400×D712
#398×D712
#385×D712
#380×D712
C2×#95×D712
C71×#26462
C2×#264671