Finite Group Databasebeta

← #34018527

Subgroups — #34018527

Order 18688.

636 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626855C211yesmaximal, characteristic
2#22626828C211yesmaximal, characteristic, complemented
3#22625578C211yesmaximal, characteristic, complemented
4#22625576C211yesmaximal, characteristic, complemented
5#22625575C211yesmaximal, characteristic, complemented
6#22625573C211yesmaximal, characteristic, complemented
7#22625571C211yesmaximal, characteristic, complemented
8#22625569C211yesmaximal, characteristic, complemented
9#22625566C211yesmaximal, characteristic, complemented
10#22625565C211yesmaximal, characteristic, complemented
11#22625483C211yesmaximal, characteristic, complemented
12#22625225C211yesmaximal, characteristic
13#22625220C211yesmaximal, characteristic, complemented
14#22625129C211yesmaximal, characteristic, complemented
15#22618747C211yesmaximal, characteristic
16#17989639—21no—
17#17989639—21no—
18#17989638C2211yescharacteristic
19#17989638C2211yescharacteristic
20#17989638C2211yescharacteristic
21#17989638C2211yescharacteristic
22#17989637C2211yescharacteristic
23#17989637C2211yescharacteristic
24#17989621C2211yescharacteristic
25#17989621C2211yescharacteristic
26#17989617—21no—
27#17989617—21no—
28#17989610—21no—
29#17989610—21no—
30#17989609—21no—
31#17989606C2211yescharacteristic
32#17989605C2211yescharacteristic
33#17989592—21no—
34#17989588C2211yescharacteristic
35C4×#11924036C2211yescharacteristic
36#17989451C2211yescharacteristic
37#17989450C2211yescharacteristic
38#17989449C2211yescharacteristic
39#17989448C2211yescharacteristic
40#17989437C2211yescharacteristic
41#17989437C2211yescharacteristic
42Q8⋊#10596833C2211yescharacteristic
43Q8⋊#10596833C2211yescharacteristic
44Q8⋊#10596833C2211yescharacteristic
45Q8⋊#10596833C2211yescharacteristic
46#17989435C2211yescharacteristic
47#17989435C2211yescharacteristic
48#17989428C2211yescharacteristic
49#17989428C2211yescharacteristic
50C8×#10596833C2211yescharacteristic
51C8×#10596833C2211yescharacteristic
52C2×#14556623C411yescharacteristic
53C4×#11924029C2211yescharacteristic
54C4×Q8×D73C2211yescharacteristic
55#17988692C411yescharacteristic
56#17988689C411yescharacteristic
57#17988685C411yescharacteristic
58Q16×Dic73C2211yescharacteristic
59#17988679C2211yescharacteristic
60#17988674C2211yescharacteristic
61#17988672C2211yescharacteristic
62#17988626C411yescharacteristic
63#17988594C411yescharacteristic
64#17988587C411yescharacteristic
65#17988539C2211yescharacteristic
66#441×C73C411yescharacteristic
67C2×#11924038—21no—
68C2×#11924038—21no—
69C2×#11924036—21no—
70C2×#11924036—21no—
71#14556658—21no—
72#14556658—21no—
73#14556658—21no—
74#14556658—21no—
75#14556658—21no—
76#14556658D411yescharacteristic
77#14556658D411yescharacteristic
78C4⋊#10596833—21no—
79C4⋊#10596833—21no—
80C4⋊#10596833C2311yescharacteristic
81C4⋊#10596833C2311yescharacteristic
82C4⋊#10596833C2311yescharacteristic
83C4⋊#10596833C2311yescharacteristic
84#14556648—21no—
85#14556648—21no—
86#14556648—21no—
87#14556648D411yescharacteristic
88#14556648D411yescharacteristic
89#14556646—21no—
90#14556646—21no—
91#14556646—21no—
92#14556646—21no—
93#14556646—21no—
94#14556646—21no—
95C4×#10596833—21no—
96C4×#10596833—21no—
97C4×#10596833C2311yescharacteristic
98C4×#10596833C2311yescharacteristic
99C4×#10596833C2311yescharacteristic
100C4×#10596833C2311yescharacteristic
101C2×#11924029C2×C411yescharacteristic
102C2×Q8×D73C2×C411yescharacteristic
103#14556623—24no—
104C2×#11924018C2×C411yescharacteristic
105C2×#11924017C2×C411yescharacteristic
106C2×#11924015C2×C411yescharacteristic
107C2×#11924011C2×C411yescharacteristic
108C2×#11924011C2×C411yescharacteristic
109Q8×Dic73C2311yescharacteristic
110Q8×Dic73C2311yescharacteristic
111C4×Dic146—21no—
112C4×Dic146C2×C411yescharacteristic
113#14556556—21no—
114#14556556C2311yescharacteristic
115C4×D292—21no—
116C4×D292C2×C411yescharacteristic
117#14556553—21no—
118#14556553C2311yescharacteristic
119#14556549—21no—
120#14556549C2×C411yescharacteristic
121C4⋊C4×D73—21no—
122C4⋊C4×D73C2×C411yescharacteristic
123C42×D73—21no—
124C42×D73—21no—
125C42×D73C2311yescharacteristic
126#14556525C2×C411yescharacteristic
127#14556524C2×C411yescharacteristic
128#14556522C2×C411yescharacteristic
129#14556516C2×C411yescharacteristic
130#14556515C2×C411yescharacteristic
131#14556515C2×C411yescharacteristic
132#14556514C2×C411yescharacteristic
133#14556513C2×C411yescharacteristic
134#14556511C2×C411yescharacteristic
135#14556502C2×C411yescharacteristic
136C8×Dic73C2311yescharacteristic
137C8×Dic73C2311yescharacteristic
138C2×Q16×C73C2×C411yescharacteristic
139C4×Q8×C73C2×C411yescharacteristic
140C4×Q8×C73C2×C411yescharacteristic
141#106×C73C2×C411yescharacteristic
142#103×C73C2×C411yescharacteristic
143#103×C73C2×C411yescharacteristic
144#97×C73C2×C411yescharacteristic
145#11924038—81no—
146#11924038—81no—
147#11924036—42no—
148#11924036—81no—
149C2×#10596833—21no—
150C2×#10596833—21no—
151C2×#10596833—21no—
152C2×#10596833—21no—
153C2×#10596833—21no—
154C2×#10596833—21no—
155C2×#10596833C2×D411yescharacteristic
156C2×#10596833C2×D411yescharacteristic
157C2×#10596833C2×D411yescharacteristic
158C2×#10596833C2×D411yescharacteristic
159#11924029—22no—
160#11924029—22no—
161C2×Dic146—21no—
162C2×Dic146C22×C411yescharacteristic
163Q8×D73—22no—
164Q8×D73—22no—
165C2×D292—21no—
166C2×D292C22×C411yescharacteristic
167C2×C4×D73—21no—
168C2×C4×D73—21no—
169C2×C4×D73—21no—
170C2×C4×D73—21no—
171C2×C4×D73—21no—
172C2×C4×D73C22×C411yescharacteristic
173C2×C4×D73C2×D411yescharacteristic
174C2×C4×D73#4111yescharacteristic
175#11924018—22no—
176#11924017—22no—
177#11924015—22no—
178#11924011—22no—
179#11924011M1614yescomplemented
180C2×#10596831C22×C411yescharacteristic
181C2×#10596831C22×C411yescharacteristic
182C4⋊Dic73—21no—
183C4⋊Dic73—21no—
184C4⋊Dic73C22×C411yescharacteristic
185C4⋊Dic73C22×C411yescharacteristic
186#11924007—21no—
187#11924007—21no—
188#11924007—21no—
189#11924005—21no—
190#11924005—21no—
191#11924005—21no—
192C4×Dic73—21no—
193C4×Dic73—21no—
194C4×Dic73—21no—
195C4×Dic73—21no—
196C4×Dic73—21no—
197C4×Dic73C2×D411yescharacteristic
198C4×Dic73#4111yescharacteristic
199C4×Dic73C2411yescharacteristic
200C2×Q8×C73C22×C411yescharacteristic
201C2×Q8×C73C22×C411yescharacteristic
202Q16×C73—22no—
203C2×C8×C73C22×C411yescharacteristic
204C2×C8×C73C22×C411yescharacteristic
205C4⋊C4×C73—21no—
206C4⋊C4×C73—21no—
207C4⋊C4×C73C22×C411yescharacteristic
208C4⋊C4×C73C22×C411yescharacteristic
209C42×C73—21no—
210C42×C73—21no—
211C42×C73C22×C411yescharacteristic
212C22×D73—21no—
213C22×D73C4×D411yescharacteristic
214Q8×C73—21no—
215Q8×C73—21no—
216Q8×C73—21no—
217Q8×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73C4×D411yescharacteristic
223C2×C4×C73C4×D411yescharacteristic
224C2×C4×C73C23×C411yescharacteristic
225C2×Dic73—21no—
226C2×Dic73—21no—
227C2×Dic73—21no—
228C2×Dic73C4×D411yescharacteristic
229C2×Dic73C22×D411yescharacteristic
230C2×Dic73C2×#4111yescharacteristic
231D292—21no—
232D292—21no—
233D292—42no—
234C4×D73—21no—
235C4×D73—21no—
236C4×D73—41no—
237C4×D73—42no—
238C4×D73—41no—
239C4×D73—41no—
240C4×D73—42no—
241C4×D73—81no—
242C4×D73—81no—
243C4×D73C2×M1612yes—
244C4×D73#13112yes—
245Dic146—21no—
246Dic146—21no—
247Dic146—42no—
248#10596833—22no—
249#10596833—21no—
250#10596833—21no—
251#10596833—21no—
252#10596833—21no—
253#10596833—41no—
254#10596833—41no—
255#10596833—41no—
256#10596833—41no—
257#10596833—41no—
258C8×C73—21no—
259C8×C73C2×M1612yes—
260#10596831—21no—
261#10596831C2×M1612yes—
262C22×C73C2×C4×D411yescharacteristic
263C2×D73—21no—
264C2×D73—21no—
265C2×D73—41no—
266C2×D73—41no—
267C2×D73—41no—
268C2×D73#43612yes—
269C4×C73—21no—
270C4×C73—21no—
271C4×C73—41no—
272C4×C73—41no—
273C4×C73—41no—
274C4×C73—41no—
275C4×C73C22×M1611yescharacteristic
276C4×C73C2×#13111yescharacteristic
277Dic73—21no—
278Dic73—41no—
279Dic73—41no—
280Dic73—41no—
281Dic73#43612yes—
282Dic73#57711yescharacteristic
283Dic73#57811yescharacteristic
284#17894—731nomaximal, Sylow
285C2×C73C2×#43611yescharacteristic
286C2×C73#294711yescharacteristic
287C2×C73#295611yescharacteristic
288D73—22no—
289D73—81no—
290#2989—731no—
291#2983—731no—
292C4×#137—731no—
293#1593—731no—
294#1593—731no—
295#1592—731no—
296#1592—731no—
297#1591—731no—
298#1591—731no—
299#1590—731no—
300#1590—731no—
301#1565—731no—
302#1476—731no—
303C2×Q8⋊C8—731no—
304C8×M16—731no—
305C73#1789411yescharacteristic, complemented, Sylow
306C2×#137—731no—
307C4×#41—731no—
308C2×C4×Q8—731no—
309#446—731no—
310#446—731no—
311#446—731no—
312#446—731no—
313#446—731no—
314#446—731no—
315#446—1461no—
316#446—1461no—
317#441—731no—
318#441#6440811yescharacteristic
319#440—731no—
320#440—731no—
321C4×Q16—731no—
322C4×Q16—731no—
323C4×SD16—731no—
324C4×SD16—731no—
325#436—731no—
326#436—731no—
327#435—1461no—
328#435—1461no—
329#432—1461no—
330#432—1461no—
331#432—1461no—
332#431—731no—
333#431—731no—
334#428—731no—
335#423—1461no—
336C2×C4⋊C8—731no—
337#419—731no—
338#417—731no—
339C4×M16—731no—
340C2×C4×C8—731no—
341#335—731no—
342#335—731no—
343#334—731no—
344#334—731no—
345Q8⋊C8—731no—
346Q8⋊C8—731no—
347Q8⋊C8—731no—
348Q8⋊C8—731no—
349Q8⋊C8—731no—
350Q8⋊C8—731no—
351D4⋊C8—731no—
352D4⋊C8—731no—
353#322—731no—
354#322—731no—
355C82—731no—
356C82—731no—
357C2×#41—731no—
358C22×Q8—731no—
359#137—1464no—
360C2×Q16—731no—
361C2×Q16C2×#6440811yescharacteristic
362C2×SD16—731no—
363C2×SD16—731no—
364C2×M16—731no—
365C2×M16—731no—
366C2×M16—1461no—
367C2×M16—1461no—
368C22×C8—1461no—
369C22×C8—1461no—
370C4×Q8—731no—
371C4×Q8—731no—
372C4×Q8—731no—
373C4×Q8—731no—
374C4×Q8—1461no—
375C4×Q8—1461no—
376C4×Q8C2×#6440811yescharacteristic
377C4×Q8C2×#6440811yescharacteristic
378C4×D4—731no—
379C4×D4—731no—
380C4×D4—1461no—
381C4×D4—1461no—
382#117—731no—
383#117—1461no—
384C2×C4⋊C4—731no—
385C2×C4⋊C4—1461no—
386C2×C42—731no—
387C2×C42—1461no—
388C2×C42—1461no—
389#107—731no—
390#107—731no—
391#106—731no—
392#106C2×#6440811yescharacteristic
393C4⋊C8—731no—
394C4⋊C8—731no—
395C4⋊C8—731no—
396C4⋊C8—731no—
397C4⋊C8—731no—
398C4⋊C8—731no—
399C4⋊C8—1461no—
400C4⋊C8—1461no—
401C4⋊C8—1461no—
402C4⋊C8—1461no—
403C4⋊C8—1461no—
404C4⋊C8—1461no—
405C4⋊C8—1461no—
406#103—731no—
407#103—731no—
408#103—731no—
409#103—731no—
410#103C2×#6440811yescharacteristic
411#103C2×#6440811yescharacteristic
412#102—731no—
413#102—731no—
414C22⋊C8—731no—
415C22⋊C8—731no—
416C22⋊C8—1461no—
417C22⋊C8—1461no—
418C22⋊C8—1461no—
419#97—731no—
420#97—1461no—
421#97—1461no—
422#97—1461no—
423#97—1461no—
424#97—1461no—
425#97—1461no—
426#97C2×#6440811yescharacteristic
427C4×C8—731no—
428C4×C8—731no—
429C4×C8—731no—
430C4×C8—731no—
431C4×C8—731no—
432C4×C8—731no—
433C4×C8—1461no—
434C4×C8—1461no—
435#41—1462no—
436#41—1462no—
437C2×Q8—731no—
438C2×Q8—1461no—
439C2×Q8—1462no—
440C2×Q8—1462no—
441C2×Q8C22×#6440811yescharacteristic
442C2×Q8C22×#6440811yescharacteristic
443C2×D4—731no—
444C2×D4—1461no—
445C22×C4—731no—
446C22×C4—731no—
447C22×C4—731no—
448C22×C4—1461no—
449C22×C4—1461no—
450C22×C4—1461no—
451C22×C4—1461no—
452C22×C4—1461no—
453Q16—22no—
454Q16—1462no—
455SD16—1462no—
456SD16—1462no—
457M16—5841no—
458M16—5841no—
459M16—734no—
460M16—1462no—
461C2×C8—2922no—
462C2×C8—5841no—
463C2×C8—731no—
464C2×C8—731no—
465C2×C8—731no—
466C2×C8—731no—
467C2×C8—731no—
468C2×C8—731no—
469C2×C8—1461no—
470C2×C8—1461no—
471C2×C8—1461no—
472C2×C8—1461no—
473C2×C8—1461no—
474C2×C8—1461no—
475C2×C8C22×#6440811yescharacteristic
476C2×C8C22×#6440811yescharacteristic
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—731no—
480C4⋊C4—731no—
481C4⋊C4—1461no—
482C4⋊C4—1461no—
483C4⋊C4—1461no—
484C4⋊C4—1461no—
485C4⋊C4—1461no—
486C4⋊C4C22×#6440811yescharacteristic
487C4⋊C4C22×#6440811yescharacteristic
488C22⋊C4—1461no—
489C22⋊C4—1461no—
490C22⋊C4—1461no—
491C42—21no—
492C42—21no—
493C42—731no—
494C42—731no—
495C42—731no—
496C42—1461no—
497C42—1461no—
498C42—1461no—
499C42—1461no—
500C42—1461no—
501C42C22×#6440811yescharacteristic
502C23—731no—
503C23—1461no—
504Q8—21no—
505Q8—21no—
506Q8—21no—
507Q8—21no—
508Q8—2922no—
509Q8—1461no—
510Q8—1461no—
511D4—2922no—
512D4—1461no—
513D4—1461no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—2921no—
519C2×C4—2922no—
520C2×C4—2921no—
521C2×C4—2922no—
522C2×C4—2921no—
523C2×C4—5841no—
524C2×C4—5841no—
525C2×C4—731no—
526C2×C4—731no—
527C2×C4—732no—
528C2×C4—732no—
529C2×C4—731no—
530C2×C4—1461no—
531C2×C4—1461no—
532C2×C4—1461no—
533C2×C4—1461no—
534C2×C4—1461no—
535C2×C4D4×#6440811yescharacteristic
536C2×C4D4×#6440811yescharacteristic
537C2×C4C23×#6440811yescharacteristic
538C8—21no—
539C8—2921no—
540C8—2921no—
541C8—2921no—
542C8—2921no—
543C8—2921no—
544C8—732no—
545C8—1461no—
546C8—1461no—
547C8—1461no—
548C8—1461no—
549C8—1462no—
550C8—1461no—
551C8D73⋊M1612yes—
552C22—2921no—
553C22—2921no—
554C22—2921no—
555C22—732no—
556C22—1461no—
557C22—1461no—
558C22C2×D4×#6440811yescharacteristic
559C4—21no—
560C4—21no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—2921no—
566C4—2921no—
567C4—2921no—
568C4—732no—
569C4—731no—
570C4—731no—
571C4—1461no—
572C4C2×D73⋊M1611yescharacteristic
573C4D73⋊#13111yescharacteristic
574C2—5841no—
575C2—1462no—
576C2D73⋊#43611yescharacteristic
577C2#2262668911yescharacteristic
578C2#2262670611yescharacteristic
subgroupG/H
centerC22C2×D4×#64408
commutatorC2×C4×C73C23×C4
FrattiniC2×C4C23×#64408
Fitting#441×C73C4
socleC22×C73C2×C4×D4
prime radicalC2×#11924029C2×C4
cosocleC4×Dic73C24
hypercenter#441#64408
hypocenterC73#17894
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ #441

Complements

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

NH ≅ G/Ntype
2. #22626828C2semidirect
3. #22625578C2semidirect
4. #22625576C2semidirect
5. #22625575C2semidirect
6. #22625573C2semidirect
7. #22625571C2semidirect
8. #22625569C2semidirect
9. #22625566C2semidirect
10. #22625565C2semidirect
11. #22625483C2semidirect
13. #22625220C2semidirect
14. #22625129C2semidirect
179. #11924011M16semidirect
305. C73#17894semidirect

Maximal subgroups

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

subgroupindex
#226268552
#226268282
#226255782
#226255762
#226255752
#226255732
#226255712
#226255692
#226255662
#226255652
#226254832
#226252252
#226252202
#226251292
#226187472
#1789473