Finite Group Databasebeta

← #33144912

Subgroups — #33144912

Order 18688.

678 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620465C211yesmaximal, characteristic, complemented
2#22619791C211yesmaximal, characteristic
3#22618524C211yesmaximal, characteristic
4#22611441C211yesmaximal, characteristic, complemented
5#22611437C211yesmaximal, characteristic, complemented
6#22611278C211yesmaximal, characteristic, complemented
7#22611274C211yesmaximal, characteristic, complemented
8#22610708C211yesmaximal, characteristic, complemented
9#22610681C211yesmaximal, characteristic, complemented
10#22610162C211yesmaximal, characteristic
11#22610157C211yesmaximal, characteristic
12#22610098C211yesmaximal, characteristic, complemented
13#22609993C211yesmaximal, characteristic
14#22609778C211yesmaximal, characteristic
15C73×#3083C211yesmaximal, characteristic
16C2×#14556598C2211yescharacteristic
17#17989215C2211yescharacteristic
18#17989070C2211yescharacteristic
19#17988997—21no—
20#17988993—21no—
21#17988990—21no—
22#17988986—21no—
23#17988892—21no—
24#17988889—21no—
25#17988875—21no—
26#17988870—21no—
27C2×#14556526C2211yescharacteristic
28#17988630—21no—
29#17988630C2211yescharacteristic
30#17988630C2211yescharacteristic
31#17988630C2211yescharacteristic
32#17988627—21no—
33#17988627C2211yescharacteristic
34#17988627C2211yescharacteristic
35#17988623—21no—
36#17988623C2211yescharacteristic
37C2×#14556524C2211yescharacteristic
38C2×#14556523C2211yescharacteristic
39C2×#14556522—21no—
40C2×#14556522C2211yescharacteristic
41#17988613C2211yescharacteristic
42C2×#14556518C2211yescharacteristic
43#17988588C2211yescharacteristic
44#17988543C2211yescharacteristic
45C2×#14556501C2211yescharacteristic
46#17988456C2211yescharacteristic
47#17988452C2211yescharacteristic
48#17988356C2211yescharacteristic
49#17988356C2211yescharacteristic
50#17988355C2211yescharacteristic
51#17988355C2211yescharacteristic
52#17988352C2211yescharacteristic
53#17988348C2211yescharacteristic
54#17988346C2211yescharacteristic
55#17988346C2211yescharacteristic
56#544×C73C2211yescharacteristic
57#520×C73C2211yescharacteristic
58#479×C73—21no—
59#478×C73—21no—
60#477×C73—21no—
61#474×C73—21no—
62#428×C73C2211yescharacteristic
63C2×#107×C73C2211yescharacteristic
64#424×C73C2211yescharacteristic
65#417×C73C2211yescharacteristic
66#416×C73C2211yescharacteristic
67#14556598—22no—
68#14556598Q814yescomplemented
69C2×#11924012—21no—
70C2×#11924012C2311yescharacteristic
71C2×#11924012C2311yescharacteristic
72C2×#11924012C2311yescharacteristic
73C22×#10596831—21no—
74C22×#10596831C2311yescharacteristic
75#14556591—21no—
76#14556590—21no—
77#14556583—21no—
78#14556582—21no—
79#14556568—41no—
80Q8×Dic73—21no—
81D4×Dic73—21no—
82D4×Dic73—41no—
83#14556550—21no—
84#14556550C2311yescharacteristic
85C2×C4⋊Dic73—21no—
86C2×C4⋊Dic73C2311yescharacteristic
87C2×C4⋊Dic73C2311yescharacteristic
88C2×#11924007—41no—
89C2×C4×Dic73C2311yescharacteristic
90#14556526—22no—
91#14556524—21no—
92#14556524—21no—
93#14556524—21no—
94#14556524—21no—
95#14556524—21no—
96#14556524—21no—
97#14556524—21no—
98#14556524D411yescharacteristic
99#14556524D411yescharacteristic
100#14556523—21no—
101#14556523—21no—
102#14556522—21no—
103#14556522—21no—
104#14556522—21no—
105#14556522—21no—
106#14556522—21no—
107#14556522—21no—
108#14556522—21no—
109#14556522D411yescharacteristic
110#14556522D411yescharacteristic
111#14556520—21no—
112#14556520—21no—
113#14556518Q814yescomplemented
114#14556515—21no—
115#14556512—21no—
116#14556502—21no—
117#14556501—22no—
118C4×#10596831—21no—
119#14556498—41no—
120C2×#41×C73C2311yescharacteristic
121C2×M16×C73C2311yescharacteristic
122C22×C8×C73C2311yescharacteristic
123C4⋊Q8×C73—21no—
124C4⋊Q8×C73—21no—
125#125×C73—21no—
126#125×C73—21no—
127C22⋊Q8×C73—41no—
128C4×Q8×C73—21no—
129C4×D4×C73—21no—
130C4×D4×C73—41no—
131#117×C73—21no—
132#117×C73C2311yescharacteristic
133#117×C73C2311yescharacteristic
134C2×C4⋊C4×C73—21no—
135C2×C4⋊C4×C73C2311yescharacteristic
136C2×C4⋊C4×C73C2311yescharacteristic
137#107×C73—21no—
138#107×C73—21no—
139#107×C73—21no—
140#106×C73—21no—
141C4⋊C8×C73—21no—
142C4⋊C8×C73—21no—
143#103×C73—21no—
144#103×C73D411yescharacteristic
145#103×C73D411yescharacteristic
146#102×C73—21no—
147#102×C73D411yescharacteristic
148#102×C73D411yescharacteristic
149C2×Dic146—41no—
150C22×Dic73—21no—
151C22×Dic73—21no—
152C22×Dic73—21no—
153C22×Dic73—41no—
154#11924012—21no—
155#11924012—21no—
156#11924012—24no—
157#11924012—21no—
158#11924012—21no—
159#11924012C2×Q812yes—
160#11924012C2×Q812yes—
161C2×#10596831—21no—
162C2×#10596831—21no—
163C2×#10596831—24no—
164C2×#10596831—21no—
165C2×#10596831C2×D411yescharacteristic
166C2×#10596831C2×D411yescharacteristic
167C2×#10596831C2×Q812yes—
168C2×#10596831#4112yes—
169C4⋊Dic73—21no—
170C4⋊Dic73—21no—
171C4⋊Dic73—21no—
172C4⋊Dic73—21no—
173C4⋊Dic73—41no—
174C4⋊Dic73—41no—
175C4⋊Dic73C2×D411yescharacteristic
176C4⋊Dic73C2×D411yescharacteristic
177#11924007—41no—
178#11924007—41no—
179#11924007—82no—
180C22⋊Dic73—21no—
181C22⋊Dic73—41no—
182C4×Dic73—21no—
183C4×Dic73—21no—
184C4×Dic73—41no—
185C4×Dic73C2×Q812yes—
186#41×C73—21no—
187#41×C73—21no—
188#41×C73C2×Q812yes—
189#41×C73C2×Q812yes—
190C2×Q8×C73—41no—
191C2×Q8×C73C2×D411yescharacteristic
192C2×D4×C73—21no—
193C2×D4×C73C2×D411yescharacteristic
194C22×C4×C73—21no—
195C22×C4×C73—21no—
196C22×C4×C73—21no—
197C22×C4×C73—41no—
198C22×C4×C73C2411yescharacteristic
199M16×C73—21no—
200M16×C73—21no—
201C2×C8×C73—21no—
202C2×C8×C73—21no—
203C2×C8×C73—21no—
204C2×C8×C73C2×D411yescharacteristic
205C2×C8×C73C2×D411yescharacteristic
206C4⋊C4×C73—21no—
207C4⋊C4×C73—21no—
208C4⋊C4×C73—21no—
209C4⋊C4×C73—21no—
210C4⋊C4×C73—41no—
211C4⋊C4×C73—41no—
212C4⋊C4×C73—41no—
213C4⋊C4×C73—41no—
214C4⋊C4×C73C2×D411yescharacteristic
215C4⋊C4×C73C2×D411yescharacteristic
216C4⋊C4×C73C2×D411yescharacteristic
217C4⋊C4×C73C2×D411yescharacteristic
218C22⋊C4×C73—21no—
219C22⋊C4×C73—21no—
220C22⋊C4×C73—41no—
221C42×C73—21no—
222C42×C73—21no—
223C42×C73—41no—
224C23×C73—21no—
225C23×C73H(5,𝔽2)11yescharacteristic
226Q8×C73—21no—
227Q8×C73—161no—
228Q8×C73C4⋊Q812yes—
229D4×C73—22no—
230D4×C73—22no—
231D4×C73—21no—
232D4×C73C4⋊Q812yes—
233C2×C4×C73—21no—
234C2×C4×C73—22no—
235C2×C4×C73—22no—
236C2×C4×C73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—41no—
241C2×C4×C73—41no—
242C2×C4×C73—81no—
243C2×C4×C73—81no—
244C2×C4×C73—161no—
245C2×C4×C73C22×D411yescharacteristic
246C2×C4×C73C22×D411yescharacteristic
247C2×C4×C73C22×Q811yescharacteristic
248C2×C4×C73C22×Q811yescharacteristic
249C2×C4×C73C2×#4111yescharacteristic
250C2×C4×C732−1+411yescharacteristic
251C2×Dic73—21no—
252C2×Dic73—21no—
253C2×Dic73—22no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—81no—
259C2×Dic73—81no—
260C2×Dic73—161no—
261Dic146—161no—
262C8×C73—21no—
263C8×C73—21no—
264C8×C73—41no—
265#10596831—22no—
266#10596831—22no—
267#10596831—41no—
268#10596831C22⋊Q812yes—
269#10596831C22⋊Q812yes—
270C22×C73—21no—
271C22×C73—21no—
272C22×C73—22no—
273C22×C73—41no—
274C22×C73—41no—
275C22×C73#54411yescharacteristic
276C22×C73D4211yescharacteristic
277C22×C73#55411yescharacteristic
278C4×C73—22no—
279C4×C73—41no—
280C4×C73—41no—
281C4×C73—41no—
282C4×C73—41no—
283C4×C73—81no—
284C4×C73—81no—
285C4×C73C2×C22⋊Q811yescharacteristic
286C4×C73C2×C4⋊Q811yescharacteristic
287C4×C73#54711yescharacteristic
288C4×C73D4×Q811yescharacteristic
289Dic73—42no—
290Dic73—41no—
291Dic73—41no—
292Dic73—81no—
293Dic73—81no—
294#26084—731nomaximal, Sylow
295C2×C73—21no—
296C2×C73—21no—
297C2×C73—42no—
298C2×C73#244011yescharacteristic
299C2×C73#328811yescharacteristic
300C2×C73#328911yescharacteristic
301#3164—731no—
302#3084—731no—
303#3083D7311yescharacteristic
304#2912—731no—
305#2090—731no—
306#2089—731no—
307#2061—731no—
308#2060—731no—
309#1951—731no—
310#1946—731no—
311#1860—731no—
312#1859—731no—
313#1847—731no—
314#1816—731no—
315#1765—731no—
316C73#2608411yescharacteristic, complemented, Sylow
317C2×#131—731no—
318#544—731no—
319#544C2×D7311yescharacteristic
320#520—731no—
321#520C2×D7311yescharacteristic
322#501—1461no—
323#501—1461no—
324#500—1461no—
325#499—1461no—
326#479—21no—
327#479—1461no—
328#478—21no—
329#478—1461no—
330#477—21no—
331#476—1461no—
332#475—1461no—
333#474—21no—
334C2×#108—731no—
335#428—731no—
336#428—731no—
337#428—731no—
338#428—731no—
339#428—731no—
340#428—1461no—
341#428—1461no—
342#428C2×D7311yescharacteristic
343#427—731no—
344#427—1461no—
345C2×#107—731no—
346C2×#107C2×D7311yescharacteristic
347C2×#106—731no—
348C2×#106—731no—
349C2×#106—1461no—
350#424—731no—
351#424C2×D7311yescharacteristic
352C2×#104—731no—
353#417—731no—
354#417C2×D7311yescharacteristic
355#416C2×D7311yescharacteristic
356#405—731no—
357C2×#97—731no—
358#384—731no—
359#382—731no—
360#343—731no—
361#343—731no—
362#343—731no—
363#343—731no—
364#341—731no—
365#339—731no—
366#339—731no—
367#339—731no—
368C2×#41C22×D7311yescharacteristic
369#131—734no—
370#131—1462no—
371C2×M16—731no—
372C2×M16—731no—
373C2×M16—731no—
374C2×M16—1461no—
375C2×M16C22×D7311yescharacteristic
376C22×C8—731no—
377C22×C8—1461no—
378C22×C8C22×D7311yescharacteristic
379C4⋊Q8—21no—
380C4⋊Q8—21no—
381C4⋊Q8—1461no—
382C4⋊Q8—1461no—
383#125—21no—
384#125—21no—
385#125—1461no—
386#125—1461no—
387C22⋊Q8—41no—
388C22⋊Q8—2921no—
389C4×Q8—21no—
390C4×Q8—1461no—
391C4×D4—21no—
392C4×D4—41no—
393C4×D4—2921no—
394C4×D4—1461no—
395#117—21no—
396#117—731no—
397#117—1461no—
398#117C22×D7311yescharacteristic
399#117C22×D7311yescharacteristic
400C2×C4⋊C4—21no—
401C2×C4⋊C4—2921no—
402C2×C4⋊C4—731no—
403C2×C4⋊C4—731no—
404C2×C4⋊C4—1461no—
405C2×C4⋊C4C22×D7311yescharacteristic
406C2×C4⋊C4C22×D7311yescharacteristic
407C2×C42—731no—
408#108—1462no—
409#107—21no—
410#107—21no—
411#107—21no—
412#107—731no—
413#107—731no—
414#107—1461no—
415#107—1461no—
416#107—1461no—
417#107—1461no—
418#107—1461no—
419#107—1461no—
420#107—1461no—
421#106—21no—
422#106—731no—
423#106—731no—
424#106—1461no—
425#106—1461no—
426#106—1461no—
427#106—1461no—
428#106—1461no—
429#106—1461no—
430#106—1461no—
431#106—1461no—
432#106—1461no—
433C4⋊C8—21no—
434C4⋊C8—21no—
435C4⋊C8—1461no—
436C4⋊C8—1461no—
437#104—734no—
438#103—21no—
439#103—1461no—
440#103C22⋊D7311yescharacteristic
441#103C22⋊D7311yescharacteristic
442#102—21no—
443#102—1461no—
444#102C22⋊D7311yescharacteristic
445#102C22⋊D7311yescharacteristic
446#97—1461no—
447#97—1462no—
448C4×C8—1461no—
449#95—2921no—
450#41—21no—
451#41—21no—
452#41Q8×D7312yes—
453#41Q8×D7312yes—
454C2×Q8—41no—
455C2×Q8—2921no—
456C2×Q8C2×C22⋊D7311yescharacteristic
457C2×D4—21no—
458C2×D4C2×C22⋊D7311yescharacteristic
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—41no—
463C22×C4—2921no—
464C22×C4—1461no—
465C22×C4—1461no—
466C22×C4—1461no—
467C22×C4C23×D7311yescharacteristic
468M16—21no—
469M16—21no—
470M16—732no—
471M16—732no—
472M16—1461no—
473M16—1461no—
474M16—1464no—
475M16—1461no—
476M16—1461no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—731no—
481C2×C8—731no—
482C2×C8—732no—
483C2×C8—732no—
484C2×C8—1461no—
485C2×C8—1464no—
486C2×C8—1461no—
487C2×C8—1461no—
488C2×C8C2×C22⋊D7311yescharacteristic
489C2×C8C2×C22⋊D7311yescharacteristic
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—2921no—
499C4⋊C4—2921no—
500C4⋊C4—2921no—
501C4⋊C4—2921no—
502C4⋊C4—5842no—
503C4⋊C4—731no—
504C4⋊C4—731no—
505C4⋊C4—1461no—
506C4⋊C4—1461no—
507C4⋊C4—1461no—
508C4⋊C4—1461no—
509C4⋊C4D4×D7311yescharacteristic
510C4⋊C4D4×D7311yescharacteristic
511C4⋊C4C2×C22⋊D7311yescharacteristic
512C4⋊C4C2×C22⋊D7311yescharacteristic
513C22⋊C4—21no—
514C22⋊C4—21no—
515C22⋊C4—41no—
516C22⋊C4—2921no—
517C22⋊C4—1461no—
518C42—21no—
519C42—21no—
520C42—41no—
521C42—2921no—
522C42—732no—
523C42—1461no—
524C42—1461no—
525C23—21no—
526C23#1455663611yescharacteristic
527Q8—21no—
528Q8—161no—
529Q8—11681no—
530Q8#1455658912yes—
531D4—22no—
532D4—22no—
533D4—21no—
534D4#1455658912yes—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—161no—
547C2×C4—2921no—
548C2×C4—2921no—
549C2×C4—2921no—
550C2×C4—2921no—
551C2×C4—5841no—
552C2×C4—5841no—
553C2×C4—11681no—
554C2×C4—1461no—
555C2×C4—1461no—
556C2×C4—1462no—
557C2×C4C2×D4×D7311yescharacteristic
558C2×C4C22×C22⋊D7311yescharacteristic
559C2×C4C2×Q8×D7311yescharacteristic
560C2×C4C2×Q8×D7311yescharacteristic
561C2×C4#41×D7311yescharacteristic
562C2×C4#1455663811yescharacteristic
563C8—21no—
564C8—21no—
565C8—41no—
566C8—2921no—
567C8—732no—
568C8—732no—
569C8—1462no—
570C8—1462no—
571C22—21no—
572C22—21no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22#1798921611yescharacteristic
577C22D4×C22⋊D7311yescharacteristic
578C22#1798929111yescharacteristic
579C4—22no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—41no—
584C4—81no—
585C4—81no—
586C4—2922no—
587C4—2921no—
588C4—2921no—
589C4—5841no—
590C4—5841no—
591C4C22⋊Q8×D7311yescharacteristic
592C4C2×#1455658911yescharacteristic
593C4#1798923611yescharacteristic
594C4#1798925011yescharacteristic
595C2—21no—
596C2—21no—
597C2—42no—
598C2#2261363011yescharacteristic
599C2#2262169511yescharacteristic
600C2#2262170211yescharacteristic
subgroupG/H
centerC22D4×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#3083C2
socleC22×C73D42
prime radicalC2×#41×C73C23
hypercenter#3083D73
hypocenterC73#26084
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ #3083

Complements

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

NH ≅ G/Ntype
1. #22620465C2semidirect
4. #22611441C2semidirect
5. #22611437C2semidirect
6. #22611278C2semidirect
7. #22611274C2semidirect
8. #22610708C2semidirect
9. #22610681C2semidirect
12. #22610098C2semidirect
68. #14556598Q8semidirect
113. #14556518Q8semidirect
316. C73#26084semidirect

Maximal subgroups

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

subgroupindex
#226204652
#226197912
#226185242
#226114412
#226114372
#226112782
#226112742
#226107082
#226106812
#226101622
#226101572
#226100982
#226099932
#226097782
C73×#30832
#2608473