Finite Group Databasebeta

← #33056114

Subgroups — #33056114

Order 18688.

626 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
1#22619960C211yesmaximal, characteristic
2#22619010C211yesmaximal, characteristic, complemented
3#22619008C211yesmaximal, characteristic
4#22608809C211yesmaximal, characteristic, complemented
5#22608807C211yesmaximal, characteristic, complemented
6#22608776C211yesmaximal, characteristic, complemented
7#22608746C211yesmaximal, characteristic, complemented
8#22608740C211yesmaximal, characteristic, complemented
9SD16×#10596831C211yesmaximal, characteristic, complemented
10#22608730C211yesmaximal, characteristic, complemented
11#22608708C211yesmaximal, characteristic, complemented
12#22608376C211yesmaximal, characteristic, complemented
13#22608346C211yesmaximal, characteristic
14#22608227C211yesmaximal, characteristic, complemented
15C73×#1583C211yesmaximal, characteristic, complemented
16#17989134C2211yescharacteristic
17C4×Q8×D73C2211yescharacteristic
18#17988907C411yescharacteristic
19#17988904C411yescharacteristic
20#17988871C411yescharacteristic
21#17988870C411yescharacteristic
22#17988786C411yescharacteristic
23#17988785C411yescharacteristic
24#17988727C411yescharacteristic
25#17988724C411yescharacteristic
26Q8×#10596831C2211yescharacteristic
27C8×Dic146—21no—
28C8×Dic146C2211yescharacteristic
29#17988711—21no—
30#17988711C2211yescharacteristic
31C4×#11924017C2211yescharacteristic
32C4×#11924015C2211yescharacteristic
33C4×#11924016C2211yescharacteristic
34#17988666C2211yescharacteristic
35#17988656—21no—
36#17988655—21no—
37D4×#10596831C2211yescharacteristic
38#17988652C2211yescharacteristic
39#17988643C2211yescharacteristic
40#17988639—21no—
41#17988639C2211yescharacteristic
42#17988638—21no—
43#17988638C2211yescharacteristic
44#17988638C2211yescharacteristic
45#17988555—21no—
46#17988552—21no—
47C4×#11924011C2211yescharacteristic
48#17988339C2211yescharacteristic
49#17988339C2211yescharacteristic
50#17988338C2211yescharacteristic
51#17988319C2211yescharacteristic
52Q8⋊#10596831C2211yescharacteristic
53#17988317C2211yescharacteristic
54#17988316C2211yescharacteristic
55D4⋊#10596831C2211yescharacteristic
56#17988314C2211yescharacteristic
57#17988309C2211yescharacteristic
58#17988309C2211yescharacteristic
59C8×#10596831C2211yescharacteristic
60C8×Q8×C73C2211yescharacteristic
61C4×SD16×C73C2211yescharacteristic
62#436×C73C2211yescharacteristic
63#334×C73C2211yescharacteristic
64Q8⋊C8×C73C2211yescharacteristic
65D4⋊C8×C73C2211yescharacteristic
66#322×C73C2211yescharacteristic
67C2×#11924028—21no—
68C2×Q8×D73C2×C411yescharacteristic
69C2×#11924017C2×C411yescharacteristic
70C2×#11924016C2×C411yescharacteristic
71C2×#11924015C2×C411yescharacteristic
72C2×#11924011—21no—
73C2×#11924011—21no—
74C22×#10596831—21no—
75C2×C8×D73—21no—
76#14556590C2×C411yescharacteristic
77#14556587C2×C411yescharacteristic
78#14556577—21no—
79#14556566C2×C411yescharacteristic
80#14556564—21no—
81#14556562C2×C411yescharacteristic
82Q8×Dic73C2×C411yescharacteristic
83C4×Dic146—21no—
84C4×Dic146C2311yescharacteristic
85#14556556—21no—
86#14556556C2×C411yescharacteristic
87C4×D292C2311yescharacteristic
88C4×C22⋊D73—21no—
89C4⋊C4×D73—21no—
90C4⋊C4×D73C2×C411yescharacteristic
91C42×D73—21no—
92C42×D73C2311yescharacteristic
93#14556523C2×C411yescharacteristic
94#14556522C2×C411yescharacteristic
95#14556522C2×C411yescharacteristic
96C4⋊#10596831—21no—
97C4⋊#10596831C2311yescharacteristic
98C4⋊#10596831C2311yescharacteristic
99#14556520—21no—
100#14556520—21no—
101#14556520C2×C411yescharacteristic
102#14556520C2×C411yescharacteristic
103#14556520D411yescharacteristic
104#14556520D411yescharacteristic
105#14556516C2×C411yescharacteristic
106#14556515C2×C411yescharacteristic
107#14556514C2×C411yescharacteristic
108#14556513C2×C411yescharacteristic
109#14556512C2×C411yescharacteristic
110#14556511C2×C411yescharacteristic
111C22⋊#10596831—21no—
112#14556503—21no—
113#14556503—21no—
114#14556503—21no—
115#14556503—21no—
116#14556503C2×C411yescharacteristic
117#14556503C2×C411yescharacteristic
118#14556503D411yescharacteristic
119#14556503D411yescharacteristic
120#14556501—21no—
121#14556501—21no—
122C4×#10596831—21no—
123C4×#10596831C2311yescharacteristic
124C4×#10596831C2311yescharacteristic
125C4×#10596831C2311yescharacteristic
126C8×Dic73—21no—
127C8×Dic73—21no—
128C8×Dic73—21no—
129C2×SD16×C73C2×C411yescharacteristic
130C2×M16×C73—21no—
131C4×Q8×C73C2311yescharacteristic
132C4×D4×C73C2311yescharacteristic
133#106×C73C2×C411yescharacteristic
134C4⋊C8×C73—21no—
135C4⋊C8×C73C2311yescharacteristic
136C4⋊C8×C73C2311yescharacteristic
137#103×C73C2×C411yescharacteristic
138#102×C73C2×C411yescharacteristic
139C22⋊C8×C73—21no—
140C4×C8×C73—21no—
141C4×C8×C73C2311yescharacteristic
142C4×C8×C73C2311yescharacteristic
143C4×C8×C73C2311yescharacteristic
144#11924028—81no—
145#11924028—81no—
146C2×Dic146—21no—
147C2×Dic146—21no—
148C2×Dic146C22×C411yescharacteristic
149Q8×D73—22no—
150Q8×D73—22no—
151C2×D292—21no—
152C2×D292C22×C411yescharacteristic
153C2×C22⋊D73—21no—
154C2×C22⋊D73—21no—
155C2×C22⋊D73—21no—
156C2×C4×D73—21no—
157C2×C4×D73—21no—
158C2×C4×D73—21no—
159C2×C4×D73C22×C411yescharacteristic
160C2×C4×D73C2×D411yescharacteristic
161C2×C4×D73#4111yescharacteristic
162#11924017—22no—
163#11924016—22no—
164#11924015—22no—
165#11924011—42no—
166#11924011—81no—
167C2×#10596831—21no—
168C2×#10596831—21no—
169C2×#10596831—21no—
170C2×#10596831—81no—
171C2×#10596831C22×C411yescharacteristic
172C2×#10596831C22×C411yescharacteristic
173C2×#10596831C2×D411yescharacteristic
174C2×#10596831C2×D411yescharacteristic
175C8×D73—81no—
176C4⋊Dic73—21no—
177C4⋊Dic73C22×C411yescharacteristic
178C4⋊Dic73C22×C411yescharacteristic
179#11924007—21no—
180#11924007—21no—
181#11924007—21no—
182#11924007—21no—
183C22⋊Dic73—21no—
184#11924005—21no—
185#11924005—21no—
186C4×Dic73—21no—
187C4×Dic73—21no—
188C4×Dic73—21no—
189C4×Dic73C22×C411yescharacteristic
190C4×Dic73C2×D411yescharacteristic
191C4×Dic73#4111yescharacteristic
192C2×Q8×C73C22×C411yescharacteristic
193C2×D4×C73C22×C411yescharacteristic
194C22×C4×C73—21no—
195SD16×C73—22no—
196M16×C73—81no—
197C2×C8×C73—21no—
198C2×C8×C73—21no—
199C2×C8×C73—21no—
200C2×C8×C73C22×C411yescharacteristic
201C2×C8×C73C22×C411yescharacteristic
202C2×C8×C73C2×D411yescharacteristic
203C2×C8×C73C2×D411yescharacteristic
204C4⋊C4×C73—21no—
205C4⋊C4×C73C22×C411yescharacteristic
206C4⋊C4×C73C22×C411yescharacteristic
207C22⋊C4×C73—21no—
208C42×C73—21no—
209C42×C73C2411yescharacteristic
210C23×C73—21no—
211C22×D73—21no—
212C22×D73C4×D411yescharacteristic
213Q8×C73—21no—
214Q8×C73#13112yes—
215D4×C73—21no—
216D4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—41no—
221C2×C4×C73—41no—
222C2×C4×C73C23×C411yescharacteristic
223C2×C4×C73C22×D411yescharacteristic
224C2×C4×C73C2×#4111yescharacteristic
225C22⋊D73—81no—
226C22⋊D73—81no—
227C22⋊D73—81no—
228C2×Dic73—21no—
229C2×Dic73—21no—
230C2×Dic73—21no—
231C2×Dic73C4×D411yescharacteristic
232C2×Dic73C4×D411yescharacteristic
233C2×Dic73C4×D411yescharacteristic
234D292—21no—
235D292—21no—
236D292—81no—
237C4×D73—24no—
238C4×D73—21no—
239C4×D73—21no—
240C4×D73—21no—
241C4×D73—21no—
242C4×D73—41no—
243C4×D73—41no—
244C4×D73—41no—
245C4×D73—81no—
246Dic146—24no—
247Dic146—21no—
248Dic146—81no—
249Dic146#13112yes—
250C8×C73—21no—
251C8×C73—21no—
252C8×C73—21no—
253C8×C73—21no—
254C8×C73—41no—
255C8×C73—41no—
256C8×C73—41no—
257#10596831—22no—
258#10596831—21no—
259#10596831—21no—
260#10596831—21no—
261#10596831—21no—
262#10596831—41no—
263#10596831—41no—
264C22×C73—41no—
265C22×C73—41no—
266C22×C73—41no—
267C22×C73C2×C4×D411yescharacteristic
268C2×D73—21no—
269C2×D73—21no—
270C2×D73—21no—
271C2×D73—41no—
272C2×D73—41no—
273C2×D73—41no—
274C4×C73—22no—
275C4×C73—21no—
276C4×C73—41no—
277C4×C73—41no—
278C4×C73C2×#13111yescharacteristic
279C4×C73C2×#13111yescharacteristic
280C4×C73#57711yescharacteristic
281C4×C73#57711yescharacteristic
282Dic73—21no—
283Dic73—22no—
284Dic73—21no—
285Dic73—21no—
286Dic73—41no—
287Dic73—41no—
288#17923—731nomaximal, Sylow
289C2×C73—81no—
290C2×C73#292911yescharacteristic
291C2×C73#294911yescharacteristic
292C2×C73#295611yescharacteristic
293D73—41no—
294D73—81no—
295#3103—731no—
296#2982—731no—
297#2981—731no—
298#1588—731no—
299#1588—731no—
300#1583—731no—
301#1583D7311yescharacteristic, complemented
302SD16⋊C8—731no—
303SD16⋊C8—731no—
304C8×SD16—731no—
305C8×SD16—731no—
306#1572—731no—
307#1491—731no—
308#1484—731no—
309#1453—731no—
310C73#1792311yescharacteristic, complemented, Sylow
311#532—731no—
312C2×C4×Q8—731no—
313#481—731no—
314#481—731no—
315#475—731no—
316#475—731no—
317#460—731no—
318#460—731no—
319#448—731no—
320#448—731no—
321C8×Q8—731no—
322C8×Q8—731no—
323C8×Q8—731no—
324C8×Q8—1461no—
325C8×Q8—1461no—
326C8×Q8C2×D7311yescharacteristic
327C4×SD16—731no—
328C4×SD16—731no—
329C4×SD16—731no—
330C4×SD16C2×D7311yescharacteristic
331#436—731no—
332#436C2×D7311yescharacteristic
333#435—1461no—
334#435—1461no—
335C8×D4—731no—
336C8×D4—731no—
337#432—731no—
338#431—731no—
339#431—731no—
340#431—731no—
341#431—1461no—
342#431—1461no—
343#408—1461no—
344#408—1461no—
345C4×M16—731no—
346#335—731no—
347#335—731no—
348#334—731no—
349#334C2×D7311yescharacteristic
350Q8⋊C8—731no—
351Q8⋊C8—731no—
352Q8⋊C8—731no—
353Q8⋊C8C2×D7311yescharacteristic
354D4⋊C8—731no—
355D4⋊C8—731no—
356D4⋊C8—731no—
357D4⋊C8C2×D7311yescharacteristic
358#322—731no—
359#322—731no—
360#322C2×D7311yescharacteristic
361C82—731no—
362C2×#41—1461no—
363C22×Q8—731no—
364C2×SD16—731no—
365C2×SD16—731no—
366C2×SD16—731no—
367C2×SD16C4×D7311yescharacteristic
368C2×M16—21no—
369C2×M16—1461no—
370C2×M16—1461no—
371C22×C8—1461no—
372C22×C8—1461no—
373C4⋊Q8—731no—
374#127—731no—
375#124—1461no—
376#121—731no—
377#121—731no—
378#121—1461no—
379C4×Q8—731no—
380C4×Q8—731no—
381C4×Q8—731no—
382C4×Q8—1461no—
383C4×Q8—1461no—
384C4×Q8C22×D7311yescharacteristic
385C4×D4—731no—
386C4×D4—1461no—
387C4×D4C22×D7311yescharacteristic
388C2×C4⋊C4—731no—
389C2×C4⋊C4—1461no—
390C2×C42—731no—
391C2×C42—1461no—
392#106—731no—
393#106—731no—
394#106—731no—
395#106C4×D7311yescharacteristic
396C4⋊C8—21no—
397C4⋊C8—731no—
398C4⋊C8—731no—
399C4⋊C8—731no—
400C4⋊C8—731no—
401C4⋊C8—731no—
402C4⋊C8—731no—
403C4⋊C8—1461no—
404C4⋊C8—1461no—
405C4⋊C8—1461no—
406C4⋊C8C22×D7311yescharacteristic
407C4⋊C8C22×D7311yescharacteristic
408#103—731no—
409#103—731no—
410#103—731no—
411#103C4×D7311yescharacteristic
412#102—731no—
413#102—731no—
414#102—731no—
415#102C4×D7311yescharacteristic
416C22⋊C8—21no—
417C22⋊C8—731no—
418C22⋊C8—731no—
419C22⋊C8—731no—
420C22⋊C8—731no—
421C22⋊C8—1461no—
422C22⋊C8—1461no—
423C22⋊C8—1461no—
424C22⋊C8—1461no—
425C22⋊C8—1461no—
426#97—1461no—
427#97—1461no—
428C4×C8—21no—
429C4×C8—731no—
430C4×C8—731no—
431C4×C8—731no—
432C4×C8—1461no—
433C4×C8—1461no—
434C4×C8—1461no—
435C4×C8—1461no—
436C4×C8C22×D7311yescharacteristic
437C4×C8C22×D7311yescharacteristic
438C4×C8C22×D7311yescharacteristic
439#41—5841no—
440#41—5841no—
441C2×Q8—731no—
442C2×Q8—1461no—
443C2×Q8—1461no—
444C2×Q8—1462no—
445C2×Q8—1462no—
446C2×Q8C2×C4×D7311yescharacteristic
447C2×D4—731no—
448C2×D4—1461no—
449C2×D4—1461no—
450C2×D4—1461no—
451C2×D4—1461no—
452C2×D4C2×C4×D7311yescharacteristic
453C22×C4—21no—
454C22×C4—731no—
455C22×C4—731no—
456C22×C4—731no—
457C22×C4—1461no—
458C22×C4—1461no—
459C22×C4—1461no—
460SD16—22no—
461SD16—1462no—
462SD16—1462no—
463SD16—1462no—
464M16—81no—
465M16—2922no—
466M16—5841no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—5841no—
471C2×C8—5841no—
472C2×C8—731no—
473C2×C8—731no—
474C2×C8—731no—
475C2×C8—731no—
476C2×C8—1461no—
477C2×C8—1461no—
478C2×C8—1461no—
479C2×C8C2×C4×D7311yescharacteristic
480C2×C8C2×C4×D7311yescharacteristic
481C2×C8D4×D7311yescharacteristic
482C2×C8D4×D7311yescharacteristic
483C4⋊C4—21no—
484C4⋊C4—731no—
485C4⋊C4—731no—
486C4⋊C4—1461no—
487C4⋊C4—1461no—
488C4⋊C4—1461no—
489C4⋊C4—1461no—
490C4⋊C4—1461no—
491C4⋊C4C2×C4×D7311yescharacteristic
492C4⋊C4C2×C4×D7311yescharacteristic
493C22⋊C4—21no—
494C22⋊C4—1461no—
495C22⋊C4—1461no—
496C22⋊C4—1461no—
497C42—21no—
498C42—731no—
499C42—731no—
500C42—731no—
501C42—1461no—
502C42—1461no—
503C42—1461no—
504C42C23×D7311yescharacteristic
505C23—21no—
506C23—731no—
507C23—1461no—
508Q8—21no—
509Q8—5841no—
510Q8—732no—
511Q8—1464no—
512Q8—1461no—
513Q8#1455659912yes—
514D4—21no—
515D4—21no—
516D4—5841no—
517D4—5841no—
518D4—5841no—
519D4—5841no—
520D4—1461no—
521D4—1461no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—2921no—
528C2×C4—2921no—
529C2×C4—2921no—
530C2×C4—5841no—
531C2×C4—731no—
532C2×C4—731no—
533C2×C4—731no—
534C2×C4—1464no—
535C2×C4—1461no—
536C2×C4—1461no—
537C2×C4—1461no—
538C2×C4—1461no—
539C2×C4—1461no—
540C2×C4—1461no—
541C2×C4—1461no—
542C2×C4C22×C4×D7311yescharacteristic
543C2×C4C2×D4×D7311yescharacteristic
544C2×C4#41×D7311yescharacteristic
545C8—21no—
546C8—21no—
547C8—21no—
548C8—21no—
549C8—41no—
550C8—41no—
551C8—41no—
552C8—2921no—
553C8—2921no—
554C8—1462no—
555C8—1461no—
556C8—1461no—
557C8—1461no—
558C8—1461no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—2921no—
563C22—2921no—
564C22—2921no—
565C22—1461no—
566C22—1461no—
567C22—1461no—
568C22C4×D4×D7311yescharacteristic
569C4—21no—
570C4—22no—
571C4—41no—
572C4—41no—
573C4—2921no—
574C4—2921no—
575C4—1461no—
576C4—1461no—
577C4—1461no—
578C4—1462no—
579C4#131×D7311yescharacteristic
580C4C2×#1455659911yescharacteristic
581C4#1798939311yescharacteristic
582C4#1798939411yescharacteristic
583C2—81no—
584C2—2921no—
585C2—5841no—
586C2#2261862211yescharacteristic
587C2#2261878511yescharacteristic
588C2#2261883911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC73×#1583C2
socleC22×C73C2×C4×D4
prime radical#17989134C22
cosocleC42×C73C24
hypercenter#1583D73
hypocenterC73#17923
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1583

Complements

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

NH ≅ G/Ntype
2. #22619010C2semidirect
4. #22608809C2semidirect
5. #22608807C2semidirect
6. #22608776C2semidirect
7. #22608746C2semidirect
8. #22608740C2semidirect
9. SD16×#10596831C2semidirect
10. #22608730C2semidirect
11. #22608708C2semidirect
12. #22608376C2semidirect
14. #22608227C2semidirect
15. C73×#1583C2semidirect
301. #1583D73semidirect
310. C73#17923semidirect

Maximal subgroups

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

subgroupindex
#226199602
#226190102
#226190082
#226088092
#226088072
#226087762
#226087462
#226087402
SD16×#105968312
#226087302
#226087082
#226083762
#226083462
#226082272
C73×#15832
#1792373