Finite Group Databasebeta

← #34019518

Subgroups — #34019518

Order 18688.

624 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626863C211yesmaximal, characteristic
2#22626831C211yesmaximal, characteristic, complemented
3#22625572C211yesmaximal, characteristic, complemented
4#22625568C211yesmaximal, characteristic, complemented
5#22625561C211yesmaximal, characteristic, complemented
6#22625554C211yesmaximal, characteristic, complemented
7#22625542C211yesmaximal, characteristic, complemented
8#22625541C211yesmaximal, characteristic, complemented
9#22625525C211yesmaximal, characteristic, complemented
10#22625518C211yesmaximal, characteristic, complemented
11#22625495C211yesmaximal, characteristic, complemented
12#22625262C211yesmaximal, characteristic
13#22625261C211yesmaximal, characteristic, complemented
14#22625150C211yesmaximal, characteristic, complemented
15#22618755C211yesmaximal, characteristic
16#17989639—21no—
17#17989639—21no—
18#17989637C2211yescharacteristic
19#17989637C2211yescharacteristic
20#17989636C2211yescharacteristic
21#17989636C2211yescharacteristic
22Q8×#10596833C2211yescharacteristic
23Q8×#10596833C2211yescharacteristic
24#17989622—21no—
25#17989622—21no—
26#17989622C2211yescharacteristic
27#17989622C2211yescharacteristic
28#17989608C2211yescharacteristic
29#17989607—21no—
30#17989604—21no—
31#17989604C2211yescharacteristic
32#17989604C2211yescharacteristic
33#17989603—21no—
34#17989590—21no—
35#17989590C2211yescharacteristic
36#17989451C2211yescharacteristic
37#17989450C2211yescharacteristic
38#17989449C2211yescharacteristic
39#17989448C2211yescharacteristic
40#17989437C2211yescharacteristic
41#17989437C2211yescharacteristic
42#17989437C2211yescharacteristic
43#17989437C2211yescharacteristic
44Q8⋊#10596833C2211yescharacteristic
45Q8⋊#10596833C2211yescharacteristic
46D4⋊#10596833C2211yescharacteristic
47D4⋊#10596833C2211yescharacteristic
48#17989429C2211yescharacteristic
49#17989429C2211yescharacteristic
50#17989428C2211yescharacteristic
51C8×#10596833C2211yescharacteristic
52C2×#14556621C411yescharacteristic
53#41×Dic73C2211yescharacteristic
54C2×#14556556C2211yescharacteristic
55#17988691C411yescharacteristic
56#17988691C411yescharacteristic
57#17988684C411yescharacteristic
58#17988684C411yescharacteristic
59#17988679C2211yescharacteristic
60#17988679C2211yescharacteristic
61#17988678C2211yescharacteristic
62#17988678C2211yescharacteristic
63#17988627C411yescharacteristic
64#17988597C411yescharacteristic
65M16×Dic73C2211yescharacteristic
66#417×C73C411yescharacteristic
67C2×#11924037—21no—
68C2×#11924037—21no—
69C2×#11924037—21no—
70#14556658—21no—
71#14556658—21no—
72#14556658—21no—
73#14556658—21no—
74#14556658D411yescharacteristic
75#14556658D411yescharacteristic
76C4⋊#10596833—21no—
77C4⋊#10596833—21no—
78C4⋊#10596833—21no—
79C4⋊#10596833—21no—
80C4⋊#10596833C2311yescharacteristic
81C4⋊#10596833C2311yescharacteristic
82C4⋊#10596833C2311yescharacteristic
83C4⋊#10596833C2311yescharacteristic
84C22⋊#10596833—21no—
85C22⋊#10596833—21no—
86C22⋊#10596833—21no—
87C22⋊#10596833—21no—
88C22⋊#10596833D411yescharacteristic
89C22⋊#10596833D411yescharacteristic
90#14556646—21no—
91#14556646—21no—
92#14556646—21no—
93C4×#10596833—21no—
94C4×#10596833—21no—
95C4×#10596833—21no—
96C4×#10596833—21no—
97C4×#10596833C2311yescharacteristic
98C4×#10596833C2311yescharacteristic
99C4×#10596833C2311yescharacteristic
100C4×#10596833C2311yescharacteristic
101C22×Dic146C2×C411yescharacteristic
102#14556621—24no—
103C2×#11924018C2×C411yescharacteristic
104C2×#11924018C2×C411yescharacteristic
105C2×#11924016C2×C411yescharacteristic
106C2×#11924016C2×C411yescharacteristic
107C2×#11924012C2×C411yescharacteristic
108Q8×Dic73C2×C411yescharacteristic
109Q8×Dic73C2311yescharacteristic
110#14556556—21no—
111#14556556—21no—
112#14556556C2×C411yescharacteristic
113#14556556C2×C411yescharacteristic
114#14556556C2311yescharacteristic
115#14556556C2311yescharacteristic
116D4×Dic73—21no—
117D4×Dic73—21no—
118D4×Dic73C2×C411yescharacteristic
119D4×Dic73C2311yescharacteristic
120#14556550—21no—
121#14556550C2×C411yescharacteristic
122C2×#11924007—21no—
123C2×C4×Dic73—21no—
124C2×C4×Dic73—21no—
125C2×C4×Dic73C2311yescharacteristic
126#14556524C2×C411yescharacteristic
127#14556524C2×C411yescharacteristic
128#14556522C2×C411yescharacteristic
129#14556522C2×C411yescharacteristic
130#14556516C2×C411yescharacteristic
131#14556516C2×C411yescharacteristic
132#14556516C2×C411yescharacteristic
133#14556516C2×C411yescharacteristic
134#14556501C2×C411yescharacteristic
135#14556501C2×C411yescharacteristic
136C8×Dic73C2311yescharacteristic
137C8×Dic73C2311yescharacteristic
138C2×#41×C73C2×C411yescharacteristic
139C2×M16×C73C2×C411yescharacteristic
140C2×C4⋊C4×C73C2×C411yescharacteristic
141#103×C73C2×C411yescharacteristic
142#103×C73C2×C411yescharacteristic
143#102×C73C2×C411yescharacteristic
144#102×C73C2×C411yescharacteristic
145#11924037—81no—
146#11924037—81no—
147#11924037—81no—
148C2×#10596833—21no—
149C2×#10596833—21no—
150C2×#10596833—21no—
151C2×#10596833—21no—
152C2×#10596833—21no—
153C2×#10596833—21no—
154C2×#10596833C2×D411yescharacteristic
155C2×#10596833C2×D411yescharacteristic
156C2×#10596833C2×D411yescharacteristic
157C2×#10596833C2×D411yescharacteristic
158C2×Dic146—22no—
159C2×Dic146—21no—
160C2×Dic146—41no—
161C2×Dic146C22×C411yescharacteristic
162C2×Dic146C22×C411yescharacteristic
163C22×Dic73—21no—
164C22×Dic73—21no—
165C22×Dic73—21no—
166C22×Dic73C2×D411yescharacteristic
167C22×Dic73#4111yescharacteristic
168#11924018—22no—
169#11924018M1614yescomplemented
170#11924016—22no—
171#11924016—22no—
172#11924012—22no—
173C2×#10596831C22×C411yescharacteristic
174C2×#10596831C22×C411yescharacteristic
175C4⋊Dic73—21no—
176C4⋊Dic73—21no—
177C4⋊Dic73C22×C411yescharacteristic
178C4⋊Dic73C22×C411yescharacteristic
179#11924007—21no—
180#11924007—21no—
181#11924007—21no—
182#11924007—21no—
183C22⋊Dic73—21no—
184C22⋊Dic73—21no—
185C22⋊Dic73—21no—
186C4×Dic73—21no—
187C4×Dic73—21no—
188C4×Dic73—21no—
189C4×Dic73—21no—
190C4×Dic73—21no—
191C4×Dic73—21no—
192C4×Dic73—21no—
193C4×Dic73C22×C411yescharacteristic
194C4×Dic73C2×D411yescharacteristic
195C4×Dic73#4111yescharacteristic
196C4×Dic73C2411yescharacteristic
197#41×C73—22no—
198#41×C73—41no—
199C2×Q8×C73C22×C411yescharacteristic
200C2×D4×C73—21no—
201C2×D4×C73C22×C411yescharacteristic
202C22×C4×C73—21no—
203C22×C4×C73—21no—
204C22×C4×C73C22×C411yescharacteristic
205M16×C73—41no—
206C2×C8×C73C22×C411yescharacteristic
207C2×C8×C73C22×C411yescharacteristic
208C4⋊C4×C73—21no—
209C4⋊C4×C73C22×C411yescharacteristic
210C4⋊C4×C73C22×C411yescharacteristic
211C23×C73—21no—
212C23×C73C4×D411yescharacteristic
213Q8×C73—21no—
214Q8×C73C2×M1612yes—
215D4×C73—21no—
216D4×C73—21no—
217D4×C73—42no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—42no—
224C2×C4×C73—42no—
225C2×C4×C73C4×D411yescharacteristic
226C2×C4×C73C23×C411yescharacteristic
227C2×Dic73—21no—
228C2×Dic73—21no—
229C2×Dic73—21no—
230C2×Dic73—21no—
231C2×Dic73—21no—
232C2×Dic73—21no—
233C2×Dic73—41no—
234C2×Dic73—42no—
235C2×Dic73—42no—
236C2×Dic73—41no—
237C2×Dic73—41no—
238C2×Dic73C4×D411yescharacteristic
239C2×Dic73C4×D411yescharacteristic
240C2×Dic73C22×D411yescharacteristic
241C2×Dic73C2×#4111yescharacteristic
242Dic146—21no—
243Dic146—21no—
244Dic146—21no—
245Dic146—42no—
246Dic146C2×M1612yes—
247#10596833—21no—
248#10596833—21no—
249#10596833—21no—
250#10596833—21no—
251#10596833—41no—
252#10596833—41no—
253#10596833—41no—
254#10596833—41no—
255#10596833—41no—
256#10596833—41no—
257C8×C73—21no—
258C8×C73—21no—
259#10596831—21no—
260#10596831C2×M1612yes—
261C22×C73—21no—
262C22×C73—21no—
263C22×C73—21no—
264C22×C73—41no—
265C22×C73—41no—
266C22×C73—41no—
267C22×C73C2×C4×D411yescharacteristic
268C4×C73—21no—
269C4×C73—22no—
270C4×C73—41no—
271C4×C73—41no—
272C4×C73C22×M1611yescharacteristic
273C4×C73#56811yescharacteristic
274Dic73—22no—
275Dic73—21no—
276Dic73—21no—
277Dic73—21no—
278Dic73—41no—
279Dic73—41no—
280Dic73—41no—
281Dic73#57711yescharacteristic
282Dic73#57811yescharacteristic
283#18053—731nomaximal, Sylow
284C2×C73—41no—
285C2×C73—81no—
286C2×C73D4×M1611yescharacteristic
287C2×C73#294711yescharacteristic
288C2×C73#295511yescharacteristic
289#2991—731no—
290#2984—731no—
291C4×#137—731no—
292#1592—731no—
293#1590—731no—
294#1588—731no—
295#1587—731no—
296#1584—731no—
297#1583—731no—
298#1579—731no—
299C8×Q16—731no—
300#1569—731no—
301#1488—731no—
302#1487—731no—
303#1453—731no—
304C73#1805311yescharacteristic, complemented, Sylow
305C2×#137—731no—
306C4×#41—731no—
307C2×C4×Q8—731no—
308#446—731no—
309#446—731no—
310#446—1461no—
311#446—1461no—
312C8×Q8—731no—
313C8×Q8—731no—
314C8×Q8—731no—
315C8×Q8—731no—
316#441—731no—
317#441—731no—
318#440—731no—
319#440—731no—
320C4×Q16—731no—
321C4×Q16—731no—
322C4×SD16—731no—
323C4×SD16—731no—
324#436—731no—
325#436—731no—
326#436—1461no—
327#436—1461no—
328#432—731no—
329#432—1461no—
330#431—731no—
331#431—731no—
332#431—1461no—
333#431—1461no—
334#428—731no—
335#423—731no—
336#423—1461no—
337#419—731no—
338#417#6440811yescharacteristic
339C4×M16—731no—
340#335—731no—
341#335—731no—
342#334—731no—
343#334—731no—
344Q8⋊C8—731no—
345Q8⋊C8—731no—
346Q8⋊C8—731no—
347Q8⋊C8—731no—
348Q8⋊C8—731no—
349Q8⋊C8—731no—
350D4⋊C8—731no—
351D4⋊C8—731no—
352#322—731no—
353#322—731no—
354#322—731no—
355C82—731no—
356C2×#41C2×#6440811yescharacteristic
357C22×Q8—731no—
358#137—1464no—
359C2×Q16—731no—
360C2×Q16—731no—
361C2×SD16—731no—
362C2×SD16—731no—
363C2×M16—731no—
364C2×M16—1461no—
365C2×M16—1461no—
366C2×M16—1461no—
367C2×M16C2×#6440811yescharacteristic
368C4×Q8—731no—
369C4×Q8—731no—
370C4×Q8—731no—
371C4×Q8—731no—
372C4×Q8—731no—
373C4×Q8—731no—
374C4×Q8—1461no—
375C4×Q8—1461no—
376C4×D4—731no—
377C4×D4—731no—
378C4×D4—1461no—
379C4×D4—1461no—
380#117—731no—
381#117—1461no—
382C2×C4⋊C4—1461no—
383C2×C4⋊C4C2×#6440811yescharacteristic
384C2×C42—731no—
385C2×C42—1461no—
386C2×C42—1461no—
387#107—731no—
388#107—731no—
389#106—731no—
390#106—731no—
391C4⋊C8—731no—
392C4⋊C8—731no—
393C4⋊C8—731no—
394C4⋊C8—731no—
395C4⋊C8—731no—
396C4⋊C8—731no—
397C4⋊C8—1461no—
398C4⋊C8—1461no—
399C4⋊C8—1461no—
400C4⋊C8—1461no—
401C4⋊C8—1461no—
402C4⋊C8—1461no—
403C4⋊C8—1461no—
404C4⋊C8—1461no—
405#103—731no—
406#103—731no—
407#103—731no—
408#103—731no—
409#103C2×#6440811yescharacteristic
410#103C2×#6440811yescharacteristic
411#102C2×#6440811yescharacteristic
412#102C2×#6440811yescharacteristic
413C22⋊C8—731no—
414C22⋊C8—731no—
415C22⋊C8—1461no—
416C22⋊C8—1461no—
417C22⋊C8—1461no—
418C22⋊C8—1461no—
419#97—731no—
420#97—731no—
421#97—1461no—
422#97—1461no—
423#97—1461no—
424C4×C8—731no—
425C4×C8—731no—
426C4×C8—731no—
427C4×C8—731no—
428C4×C8—731no—
429C4×C8—731no—
430C4×C8—1461no—
431C4×C8—1461no—
432C4×C8—1461no—
433C4×C8—1461no—
434#41—22no—
435#41—41no—
436C2×Q8—2921no—
437C2×Q8—731no—
438C2×Q8—731no—
439C2×Q8—1462no—
440C2×Q8—1461no—
441C2×Q8C22×#6440811yescharacteristic
442C2×D4—21no—
443C2×D4C22×#6440811yescharacteristic
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—731no—
447C22×C4—731no—
448C22×C4—1461no—
449C22×C4—1461no—
450C22×C4—1461no—
451C22×C4C22×#6440811yescharacteristic
452Q16—734no—
453Q16—1462no—
454SD16—1462no—
455SD16—1462no—
456M16—41no—
457M16—5841no—
458M16—5841no—
459M16—5841no—
460M16—1462no—
461C2×C8—731no—
462C2×C8—731no—
463C2×C8—731no—
464C2×C8—731no—
465C2×C8—731no—
466C2×C8—731no—
467C2×C8—1461no—
468C2×C8—1461no—
469C2×C8—1461no—
470C2×C8—1461no—
471C2×C8—1461no—
472C2×C8—1461no—
473C2×C8C22×#6440811yescharacteristic
474C2×C8C22×#6440811yescharacteristic
475C4⋊C4—21no—
476C4⋊C4—731no—
477C4⋊C4—731no—
478C4⋊C4—1461no—
479C4⋊C4—1461no—
480C4⋊C4—1461no—
481C4⋊C4—1461no—
482C4⋊C4—1461no—
483C4⋊C4—1461no—
484C4⋊C4C22×#6440811yescharacteristic
485C4⋊C4C22×#6440811yescharacteristic
486C22⋊C4—1461no—
487C22⋊C4—1461no—
488C22⋊C4—1461no—
489C42—731no—
490C42—731no—
491C42—731no—
492C42—731no—
493C42—1461no—
494C42—1461no—
495C42—1461no—
496C42—1461no—
497C42—1461no—
498C42—1461no—
499C42—1461no—
500C23—21no—
501C23D4×#6440811yescharacteristic
502Q8—21no—
503Q8—2922no—
504Q8—732no—
505Q8—1461no—
506Q8—1461no—
507Q8—1461no—
508Q8D73⋊M1612yes—
509D4—21no—
510D4—21no—
511D4—42no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—2921no—
520C2×C4—2921no—
521C2×C4—2922no—
522C2×C4—2922no—
523C2×C4—2921no—
524C2×C4—731no—
525C2×C4—731no—
526C2×C4—731no—
527C2×C4—731no—
528C2×C4—1461no—
529C2×C4—1461no—
530C2×C4—1461no—
531C2×C4—1461no—
532C2×C4—1461no—
533C2×C4—1461no—
534C2×C4D4×#6440811yescharacteristic
535C2×C4C23×#6440811yescharacteristic
536C8—21no—
537C8—21no—
538C8—2921no—
539C8—2921no—
540C8—2921no—
541C8—2921no—
542C8—2921no—
543C8—2921no—
544C8—732no—
545C8—1461no—
546C8—1461no—
547C8—1461no—
548C8—1461no—
549C8—1461no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22C2×D4×#6440811yescharacteristic
557C4—21no—
558C4—22no—
559C4—41no—
560C4—41no—
561C4—2921no—
562C4—2921no—
563C4—2921no—
564C4—731no—
565C4—731no—
566C4—1462no—
567C4—1461no—
568C4—1461no—
569C4—1461no—
570C4C2×D73⋊M1611yescharacteristic
571C4#1798966511yescharacteristic
572C2—41no—
573C2—81no—
574C2#2262666911yescharacteristic
575C2#2262668911yescharacteristic
576C2#2262670311yescharacteristic
subgroupG/H
centerC22C2×D4×#64408
commutatorC2×C4×C73C23×C4
FrattiniC2×C4C23×#64408
Fitting#417×C73C4
socleC22×C73C2×C4×D4
prime radicalC2×#41×C73C2×C4
cosocleC4×Dic73C24
hypercenter#417#64408
hypocenterC73#18053
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. #22626831C2semidirect
3. #22625572C2semidirect
4. #22625568C2semidirect
5. #22625561C2semidirect
6. #22625554C2semidirect
7. #22625542C2semidirect
8. #22625541C2semidirect
9. #22625525C2semidirect
10. #22625518C2semidirect
11. #22625495C2semidirect
13. #22625261C2semidirect
14. #22625150C2semidirect
169. #11924018M16semidirect
304. C73#18053semidirect

Maximal subgroups

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

subgroupindex
#226268632
#226268312
#226255722
#226255682
#226255612
#226255542
#226255422
#226255412
#226255252
#226255182
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