Finite Group Databasebeta

← #33057832

Subgroups — #33057832

Order 18688.

656 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620520C211yesmaximal, characteristic
2#22618929C211yesmaximal, characteristic, complemented
3#22618675C211yesmaximal, characteristic, complemented
4#22608842C211yesmaximal, characteristic, complemented
5#22608824C211yesmaximal, characteristic, complemented
6#22608812C211yesmaximal, characteristic, complemented
7#22608792C211yesmaximal, characteristic, complemented
8#22608764C211yesmaximal, characteristic, complemented
9#22608763C211yesmaximal, characteristic, complemented
10#22608740C211yesmaximal, characteristic, complemented
11#22608723C211yesmaximal, characteristic, complemented
12#22608684C211yesmaximal, characteristic, complemented
13#22608345C211yesmaximal, characteristic, complemented
14#22608234C211yesmaximal, characteristic, complemented
15C73×#1484C211yesmaximal, characteristic, complemented
16C2×#14556598—21no—
17#17989121C2211yescharacteristic
18#17988993C411yescharacteristic
19#17988974C411yescharacteristic
20#17988951C411yescharacteristic
21#17988949C411yescharacteristic
22#17988838C411yescharacteristic
23#17988838C411yescharacteristic
24#17988831C411yescharacteristic
25#17988831C411yescharacteristic
26#17988714C2211yescharacteristic
27#17988711C2211yescharacteristic
28C4×#11924016C2211yescharacteristic
29C4×#11924016C2211yescharacteristic
30C4×#11924013C2211yescharacteristic
31C4×#11924013C2211yescharacteristic
32#17988665C2211yescharacteristic
33#17988665C2211yescharacteristic
34#17988664C2211yescharacteristic
35#17988662—21no—
36#17988661—21no—
37#17988659—21no—
38#17988659—21no—
39D4×#10596831C2211yescharacteristic
40D4×#10596831C2211yescharacteristic
41#17988652C2211yescharacteristic
42#17988646—21no—
43C2×C4⋊#10596831C2211yescharacteristic
44#17988554—21no—
45#17988541—21no—
46C4×#11924012C2211yescharacteristic
47C4×#11924012C2211yescharacteristic
48#17988339C2211yescharacteristic
49#17988339C2211yescharacteristic
50#17988337C2211yescharacteristic
51#17988337C2211yescharacteristic
52#17988319C2211yescharacteristic
53#17988319C2211yescharacteristic
54#17988316C2211yescharacteristic
55#17988316C2211yescharacteristic
56#17988309C2211yescharacteristic
57#17988309C2211yescharacteristic
58#17988308C2211yescharacteristic
59C8×#10596831C2211yescharacteristic
60#532×C73C2211yescharacteristic
61#431×C73C2211yescharacteristic
62C4×M16×C73C2211yescharacteristic
63D4⋊C8×C73C2211yescharacteristic
64D4⋊C8×C73C2211yescharacteristic
65D4⋊C8×C73C2211yescharacteristic
66D4⋊C8×C73C2211yescharacteristic
67C2×#11924016C2×C411yescharacteristic
68C2×#11924016C2×C411yescharacteristic
69C2×#11924013C2×C411yescharacteristic
70C2×#11924013C2×C411yescharacteristic
71#14556598—44no—
72C2×#11924012—21no—
73C2×#11924012—21no—
74C2×#11924012C2×C411yescharacteristic
75C2×#11924012C2×C411yescharacteristic
76C2×#11924012D411yescharacteristic
77C2×#11924012D411yescharacteristic
78C2×#11924011—21no—
79C22×#10596831—21no—
80C22×#10596831—21no—
81C22×#10596831—21no—
82C2×C8×D73—21no—
83#14556585—21no—
84#14556583C2×C411yescharacteristic
85#14556579C2×C411yescharacteristic
86#14556574—21no—
87#14556570C2×C411yescharacteristic
88#14556565C2×C411yescharacteristic
89C4×Dic146C2311yescharacteristic
90C4×D292C2311yescharacteristic
91C4×C22⋊D73—21no—
92#14556548—21no—
93#14556524C2×C411yescharacteristic
94#14556524C2×C411yescharacteristic
95#14556522C2×C411yescharacteristic
96#14556522C2×C411yescharacteristic
97C4⋊#10596831—21no—
98C4⋊#10596831C2311yescharacteristic
99C4⋊#10596831C2311yescharacteristic
100#14556520—21no—
101#14556520—21no—
102#14556514C2×C411yescharacteristic
103#14556514C2×C411yescharacteristic
104#14556512C2×C411yescharacteristic
105#14556512C2×C411yescharacteristic
106#14556512C2×C411yescharacteristic
107#14556512C2×C411yescharacteristic
108#14556511C2×C411yescharacteristic
109#14556511C2×C411yescharacteristic
110C22⋊#10596831—21no—
111C22⋊#10596831—21no—
112C22⋊#10596831—21no—
113C22⋊#10596831—21no—
114#14556503—21no—
115#14556503—21no—
116#14556502C2×C411yescharacteristic
117#14556502C2×C411yescharacteristic
118#14556502D411yescharacteristic
119#14556502D411yescharacteristic
120#14556501—21no—
121C4×#10596831C2311yescharacteristic
122C4×#10596831C2311yescharacteristic
123C4×#10596831C2311yescharacteristic
124C4×#10596831C2311yescharacteristic
125C8×Dic73—21no—
126C2×#41×C73—21no—
127C2×M16×C73—21no—
128C4⋊Q8×C73C2×C411yescharacteristic
129#127×C73C2×C411yescharacteristic
130#124×C73—21no—
131#121×C73—21no—
132#121×C73C2×C411yescharacteristic
133#121×C73C2×C411yescharacteristic
134C4×D4×C73—21no—
135C4×D4×C73C2311yescharacteristic
136C4×D4×C73C2311yescharacteristic
137C2×C42×C73C2311yescharacteristic
138C4⋊C8×C73C2311yescharacteristic
139C4⋊C8×C73C2311yescharacteristic
140C22⋊C8×C73—21no—
141#97×C73—21no—
142C4×C8×C73—21no—
143C4×C8×C73C2311yescharacteristic
144C4×C8×C73C2311yescharacteristic
145C2×Dic146C22×C411yescharacteristic
146C2×D292C22×C411yescharacteristic
147C2×C22⋊D73—21no—
148C2×C4×D73—21no—
149#11924016—22no—
150#11924016—41no—
151#11924013—22no—
152#11924013—41no—
153#11924012—22no—
154#11924012—22no—
155#11924012—41no—
156#11924012—41no—
157#11924012—42no—
158#11924012—42no—
159#11924011—42no—
160C2×#10596831—21no—
161C2×#10596831—21no—
162C2×#10596831—42no—
163C2×#10596831—42no—
164C2×#10596831—42no—
165C2×#10596831C22×C411yescharacteristic
166C2×#10596831C22×C411yescharacteristic
167C2×#10596831C22×C411yescharacteristic
168C2×#10596831C22×C411yescharacteristic
169C2×#10596831C2×D411yescharacteristic
170C2×#10596831C2×D411yescharacteristic
171C2×#10596831C2×D411yescharacteristic
172C2×#10596831C2×D411yescharacteristic
173C8×D73—42no—
174C4⋊Dic73C22×C411yescharacteristic
175C4⋊Dic73C22×C411yescharacteristic
176#11924007—21no—
177#11924007—21no—
178#11924007—21no—
179#11924007—21no—
180C22⋊Dic73—21no—
181#11924005—21no—
182#11924005—21no—
183#11924005—21no—
184#11924005—21no—
185C4×Dic73—21no—
186#41×C73—42no—
187#41×C73—42no—
188C2×Q8×C73—21no—
189C2×D4×C73—21no—
190C2×D4×C73—21no—
191C2×D4×C73—21no—
192C2×D4×C73—21no—
193C2×D4×C73C22×C411yescharacteristic
194C2×D4×C73C22×C411yescharacteristic
195C22×C4×C73—21no—
196C22×C4×C73—21no—
197C22×C4×C73C22×C411yescharacteristic
198C22×C4×C73C2×D411yescharacteristic
199C22×C4×C73#4111yescharacteristic
200M16×C73—42no—
201C2×C8×C73—21no—
202C2×C8×C73—21no—
203C2×C8×C73—21no—
204C2×C8×C73—21no—
205C4⋊C4×C73—21no—
206C4⋊C4×C73C22×C411yescharacteristic
207C4⋊C4×C73C22×C411yescharacteristic
208C22⋊C4×C73—21no—
209C22⋊C4×C73—21no—
210C22⋊C4×C73—21no—
211C22⋊C4×C73—21no—
212C42×C73C22×C411yescharacteristic
213C42×C73C2×D411yescharacteristic
214C42×C73#4111yescharacteristic
215C42×C73C2411yescharacteristic
216C23×C73—21no—
217C23×C73—21no—
218C23×C73C4×D411yescharacteristic
219C22×D73—21no—
220Q8×C73—42no—
221D4×C73—21no—
222D4×C73—21no—
223D4×C73—21no—
224D4×C73—21no—
225D4×C73—42no—
226D4×C73—42no—
227D4×C73—42no—
228D4×C73—81no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73—41no—
236C2×C4×C73—41no—
237C2×C4×C73—41no—
238C2×C4×C73—41no—
239C2×C4×C73—41no—
240C2×C4×C73C4×D411yescharacteristic
241C2×C4×C73C4×D411yescharacteristic
242C2×C4×C73C4×D411yescharacteristic
243C2×C4×C73C23×C411yescharacteristic
244C2×C4×C73C22×D411yescharacteristic
245C2×C4×C73C2×#4111yescharacteristic
246C22⋊D73—81no—
247C2×Dic73—21no—
248C2×Dic73—21no—
249C2×Dic73—21no—
250D292—21no—
251D292—21no—
252C4×D73—41no—
253C4×D73—41no—
254Dic146—21no—
255Dic146—21no—
256C8×C73—22no—
257C8×C73—41no—
258C8×C73—41no—
259C8×C73—41no—
260#10596831—21no—
261#10596831—21no—
262#10596831—21no—
263#10596831—21no—
264#10596831—21no—
265#10596831—21no—
266#10596831—41no—
267#10596831—41no—
268#10596831#13112yescomplemented
269#10596831AGL(1,ℤ/8)12yes—
270C22×C73—21no—
271C22×C73—21no—
272C22×C73—21no—
273C22×C73—41no—
274C22×C73—41no—
275C22×C73—41no—
276C22×C73—41no—
277C22×C73—41no—
278C22×C73—41no—
279C22×C73C2×C4×D411yescharacteristic
280C2×D73—41no—
281C2×D73—41no—
282C2×D73—41no—
283C4×C73—21no—
284C4×C73—21no—
285C4×C73—21no—
286C4×C73—21no—
287C4×C73—41no—
288C4×C73—41no—
289C4×C73C2×#13111yescharacteristic
290C4×C73#56811yescharacteristic
291C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
292C4×C73#57511yescharacteristic
293Dic73—41no—
294Dic73—41no—
295Dic73—41no—
296#18076—731nomaximal, Sylow
297C2×C73—41no—
298C2×C73—81no—
299C2×C73—81no—
300C2×C73#293511yescharacteristic
301C2×C73C4×AGL(1,ℤ/8)11yescharacteristic
302C2×C73#295611yescharacteristic
303D73—81no—
304#3171—731no—
305#2969—731no—
306#2935—731no—
307#1594—731no—
308#1590—731no—
309#1588—731no—
310#1586—731no—
311#1582—731no—
312#1581—731no—
313SD16⋊C8—731no—
314C8×D8—731no—
315#1565—731no—
316#1484D7311yescharacteristic, complemented
317#1483—731no—
318#1455—731no—
319C73#1807611yescharacteristic, complemented, Sylow
320C2×#131—1461no—
321#532C2×D7311yescharacteristic
322#529—731no—
323#501—731no—
324#496—731no—
325#490—731no—
326#489—731no—
327#469—731no—
328#469—731no—
329#468—731no—
330#468—731no—
331#446—731no—
332C8×Q8—731no—
333C4×SD16—731no—
334C4×SD16—731no—
335C4×D8—731no—
336C4×D8—731no—
337#436—731no—
338#436—731no—
339#436—731no—
340#436—1461no—
341#435—1461no—
342#435—1461no—
343#435—1461no—
344C8×D4—731no—
345C8×D4—731no—
346C8×D4—731no—
347#433—1461no—
348#431C2×D7311yescharacteristic
349C2×C4⋊C8—731no—
350#408—1461no—
351#405—1461no—
352C4×M16—731no—
353C4×M16—731no—
354C4×M16C2×D7311yescharacteristic
355#335—731no—
356#335—731no—
357#334—731no—
358#334—731no—
359Q8⋊C8—731no—
360Q8⋊C8—731no—
361D4⋊C8—731no—
362D4⋊C8—731no—
363D4⋊C8C2×D7311yescharacteristic
364D4⋊C8C2×D7311yescharacteristic
365D4⋊C8C2×D7311yescharacteristic
366D4⋊C8C2×D7311yescharacteristic
367#322—731no—
368#322—731no—
369#322—731no—
370C82—731no—
371C2×#41—21no—
372C2×SD16—731no—
373C2×SD16—731no—
374C2×D8—731no—
375C2×D8—731no—
376#131—2924no—
377C2×M16—21no—
378C2×M16—731no—
379C2×M16—731no—
380C2×M16—731no—
381C2×M16—731no—
382C2×M16—1461no—
383C2×M16—1461no—
384C2×M16—1461no—
385C22×C8—1461no—
386C22×C8—1461no—
387C22×C8—1461no—
388C22×C8—1461no—
389C4⋊Q8C4×D7311yescharacteristic
390#127C4×D7311yescharacteristic
391#126—1461no—
392#125—731no—
393#124—21no—
394#124—731no—
395#123—1461no—
396C22⋊Q8—731no—
397#121—21no—
398#121—731no—
399#121C4×D7311yescharacteristic
400#121C4×D7311yescharacteristic
401C4×Q8—731no—
402C4×D4—21no—
403C4×D4—731no—
404C4×D4—1461no—
405C4×D4C22×D7311yescharacteristic
406C4×D4C22×D7311yescharacteristic
407#117—1461no—
408C2×C42C22×D7311yescharacteristic
409#107—731no—
410#107—731no—
411#106—731no—
412#106—731no—
413C4⋊C8—731no—
414C4⋊C8—731no—
415C4⋊C8—1461no—
416C4⋊C8—1461no—
417C4⋊C8—1461no—
418C4⋊C8C22×D7311yescharacteristic
419C4⋊C8C22×D7311yescharacteristic
420#103—731no—
421#103—731no—
422#102—731no—
423#102—731no—
424#102—731no—
425#102—731no—
426#102—731no—
427#102—731no—
428C22⋊C8—21no—
429C22⋊C8—1461no—
430C22⋊C8—1461no—
431C22⋊C8—1461no—
432C22⋊C8—1461no—
433C22⋊C8—1461no—
434C22⋊C8—1461no—
435#97—21no—
436#97—731no—
437#97—731no—
438#97—731no—
439#97—731no—
440#97—1461no—
441C4×C8—21no—
442C4×C8—731no—
443C4×C8—731no—
444C4×C8—731no—
445C4×C8—731no—
446C4×C8—1461no—
447C4×C8C22×D7311yescharacteristic
448C4×C8C22×D7311yescharacteristic
449#41—42no—
450#41—42no—
451C2×Q8—21no—
452C2×Q8—731no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—731no—
458C2×D4—1461no—
459C2×D4C2×C4×D7311yescharacteristic
460C2×D4C2×C4×D7311yescharacteristic
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—1461no—
464C22×C4C2×C4×D7311yescharacteristic
465C22×C4D4×D7311yescharacteristic
466C22×C4#1192402711yescharacteristic
467SD16—2921no—
468SD16—1462no—
469D8—2921no—
470D8—1462no—
471M16—42no—
472M16—2922no—
473M16—2922no—
474M16—2922no—
475M16—2921no—
476M16—2921no—
477M16—1462no—
478M16—1462no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—2922no—
484C2×C8—2922no—
485C2×C8—2922no—
486C2×C8—2922no—
487C2×C8—731no—
488C2×C8—731no—
489C2×C8—731no—
490C2×C8—731no—
491C2×C8—731no—
492C2×C8—731no—
493C2×C8—731no—
494C2×C8—731no—
495C2×C8—1461no—
496C2×C8—1461no—
497C4⋊C4—21no—
498C4⋊C4—731no—
499C4⋊C4—731no—
500C4⋊C4—1461no—
501C4⋊C4—1461no—
502C4⋊C4—1461no—
503C4⋊C4—1461no—
504C4⋊C4C2×C4×D7311yescharacteristic
505C4⋊C4C2×C4×D7311yescharacteristic
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—1461no—
511C22⋊C4—1461no—
512C22⋊C4—1461no—
513C22⋊C4—1461no—
514C22⋊C4—1461no—
515C42—1461no—
516C42C2×C4×D7311yescharacteristic
517C42D4×D7311yescharacteristic
518C42#1192402711yescharacteristic
519C42C23×D7311yescharacteristic
520C23—21no—
521C23—21no—
522C23—1461no—
523C23#1455655111yescharacteristic
524Q8—42no—
525Q8—1461no—
526Q8—1461no—
527D4—21no—
528D4—21no—
529D4—21no—
530D4—21no—
531D4—42no—
532D4—42no—
533D4—42no—
534D4—81no—
535D4—5841no—
536D4—1461no—
537D4—1461no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—2921no—
550C2×C4—2921no—
551C2×C4—1461no—
552C2×C4—1461no—
553C2×C4—1461no—
554C2×C4#1455655111yescharacteristic
555C2×C4#1455655111yescharacteristic
556C2×C4#1455655111yescharacteristic
557C2×C4C22×C4×D7311yescharacteristic
558C2×C4C2×D4×D7311yescharacteristic
559C2×C4C2×#1192402711yescharacteristic
560C8—22no—
561C8—41no—
562C8—41no—
563C8—41no—
564C8—2921no—
565C8—2921no—
566C8—732no—
567C8—732no—
568C8—1461no—
569C8—1461no—
570C8—1461no—
571C8—1461no—
572C8—1461no—
573C8—1461no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—2921no—
584C22—2921no—
585C22—2921no—
586C22C2×#1455655111yescharacteristic
587C4—21no—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—41no—
592C4—41no—
593C4—2921no—
594C4—2921no—
595C4—2921no—
596C4#131×D7311yescharacteristic
597C4#1798935211yescharacteristic
598C4AGL(1,ℤ/8)×D7311yescharacteristic
599C4#1798938311yescharacteristic
600C2—41no—
601C2—81no—
602C2—81no—
603C2—5841no—
604C2#2261867111yescharacteristic
605C2#2261873511yescharacteristic
606C2#2261883711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC73×#1484C2
socleC22×C73C2×C4×D4
prime radical#22620520C2
cosocleC42×C73C24
hypercenter#1484D73
hypocenterC73#18076
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1484

Complements

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

NH ≅ G/Ntype
2. #22618929C2semidirect
3. #22618675C2semidirect
4. #22608842C2semidirect
5. #22608824C2semidirect
6. #22608812C2semidirect
7. #22608792C2semidirect
8. #22608764C2semidirect
9. #22608763C2semidirect
10. #22608740C2semidirect
11. #22608723C2semidirect
12. #22608684C2semidirect
13. #22608345C2semidirect
14. #22608234C2semidirect
15. C73×#1484C2semidirect
268. #10596831#131semidirect
316. #1484D73semidirect
319. C73#18076semidirect

Maximal subgroups

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

subgroupindex
#226205202
#226189292
#226186752
#226088422
#226088242
#226088122
#226087922
#226087642
#226087632
#226087402
#226087232
#226086842
#226083452
#226082342
C73×#14842
#1807673