Finite Group Databasebeta

← #33108065

Subgroups — #33108065

Order 18688.

708 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619717C211yesmaximal, characteristic
2#22618875C211yesmaximal, characteristic, complemented
3#22618498C211yesmaximal, characteristic
4#22610906C211yesmaximal, characteristic, complemented
5#22610705C211yesmaximal, characteristic, complemented
6#22610685C211yesmaximal, characteristic, complemented
7#22610681C211yesmaximal, characteristic, complemented
8#22610533C211yesmaximal, characteristic, complemented
9#22610209C211yesmaximal, characteristic
10#22610207C211yesmaximal, characteristic
11#22609783C211yesmaximal, characteristic, complemented
12#22609777C211yesmaximal, characteristic
13#22609774C211yesmaximal, characteristic
14#22609735C211yesmaximal, characteristic
15C73×#1922C211yesmaximal, characteristic, complemented
16C2×#14556599C2211yescharacteristic
17C2×#14556597C2211yescharacteristic
18#17989140C2211yescharacteristic
19#17989055C2211yescharacteristic
20#17988897C411yescharacteristic, complemented
21#17988893C411yescharacteristic, complemented
22#17988890C411yescharacteristic, complemented
23#17988886C411yescharacteristic, complemented
24#17988881C411yescharacteristic, complemented
25#17988876C411yescharacteristic, complemented
26#17988871C411yescharacteristic, complemented
27#17988866C411yescharacteristic, complemented
28#17988717—21no—
29#17988715—21no—
30#17988715—21no—
31C8×Dic146—21no—
32C2×#14556527C2211yescharacteristic
33C2×#14556526C2211yescharacteristic
34#17988623C2211yescharacteristic
35#17988614C2211yescharacteristic
36#17988614C2211yescharacteristic
37#17988613C2211yescharacteristic
38#17988612—21no—
39#17988612—21no—
40#17988610C2211yescharacteristic
41#17988609—21no—
42C2×#14556520C2211yescharacteristic
43C2×#14556520C2211yescharacteristic
44C4⋊C8×D73—21no—
45C2×#14556519C2211yescharacteristic
46C2×#14556517C2211yescharacteristic
47#17988584C2211yescharacteristic
48#17988581C2211yescharacteristic
49#17988545C2211yescharacteristic
50M16×Dic73C2211yescharacteristic
51C2×#14556501C2211yescharacteristic
52C2×#14556501C2211yescharacteristic
53C42×Dic73C2211yescharacteristic
54#17988351C2211yescharacteristic
55#17988348C2211yescharacteristic
56#17988347C2211yescharacteristic
57#17988346C2211yescharacteristic
58#17988346C2211yescharacteristic
59#17988342C2211yescharacteristic
60#427×C73C2211yescharacteristic
61#424×C73C2211yescharacteristic
62#405×C73C2211yescharacteristic
63#341×C73C2211yescharacteristic
64#339×C73C2211yescharacteristic
65#339×C73C2211yescharacteristic
66#336×C73C2211yescharacteristic
67C2×#11924028C2311yescharacteristic
68#14556599—22no—
69#14556599—41no—
70#14556597—22no—
71#14556597—41no—
72C2×#11924012C2311yescharacteristic
73C2×#11924012C2311yescharacteristic
74C2×#11924011—21no—
75C2×#11924011—21no—
76C22×#10596831C2311yescharacteristic
77C22×#10596831C2311yescharacteristic
78C2×C8×D73—21no—
79C2×C8×D73—21no—
80#14556590C2×C411yescharacteristic
81#14556590C2×C411yescharacteristic
82#14556582C2×C411yescharacteristic
83#14556582C2×C411yescharacteristic
84#14556568—21no—
85#14556568—21no—
86C4×Dic146—21no—
87C4×Dic146—21no—
88#14556556C2×C411yescharacteristic
89#14556556C2×C411yescharacteristic
90#14556553C2×C411yescharacteristic
91#14556553C2×C411yescharacteristic
92#14556551—21no—
93#14556551—21no—
94#14556550C2311yescharacteristic
95#14556548—21no—
96C2×C4×Dic73—21no—
97C2×C4×Dic73—21no—
98C2×C4×Dic73C2311yescharacteristic
99C2×C4×Dic73C2311yescharacteristic
100C42×D73—21no—
101#14556527—41no—
102#14556526—41no—
103#14556524C2×C411yescharacteristic
104#14556524C2×C411yescharacteristic
105#14556522C2×C411yescharacteristic
106#14556522C2×C411yescharacteristic
107C4⋊#10596831—21no—
108C4⋊#10596831—21no—
109C4⋊#10596831—21no—
110C4⋊#10596831—21no—
111#14556520—21no—
112#14556520—21no—
113#14556520—21no—
114#14556520—21no—
115#14556520C2×C411yescharacteristic
116#14556520C2×C411yescharacteristic
117#14556520C2×C411yescharacteristic
118#14556520C2×C411yescharacteristic
119#14556520D411yescharacteristic
120#14556520D411yescharacteristic
121#14556520D411yescharacteristic
122#14556520D411yescharacteristic
123#14556519—22no—
124#14556517Q814yescomplemented
125#14556516C2×C411yescharacteristic
126#14556516C2×C411yescharacteristic
127#14556514C2×C411yescharacteristic
128#14556514C2×C411yescharacteristic
129#14556513C2×C411yescharacteristic
130#14556513C2×C411yescharacteristic
131#14556511C2×C411yescharacteristic
132#14556511C2×C411yescharacteristic
133#14556501—21no—
134#14556501—21no—
135#14556501—41no—
136#14556501—41no—
137C8×Dic73—21no—
138C8×Dic73—21no—
139C2×M16×C73C2311yescharacteristic
140C2×M16×C73C2311yescharacteristic
141C22×C8×C73C2311yescharacteristic
142C22×C8×C73C2311yescharacteristic
143#117×C73C2311yescharacteristic
144#117×C73C2311yescharacteristic
145C2×C42×C73C2311yescharacteristic
146#107×C73C2×C411yescharacteristic
147#107×C73C2×C411yescharacteristic
148#106×C73C2×C411yescharacteristic
149#106×C73C2×C411yescharacteristic
150C4⋊C8×C73—21no—
151C4⋊C8×C73—21no—
152#97×C73—21no—
153C4×C8×C73—21no—
154#11924028—21no—
155#11924028—21no—
156#11924028C2×Q812yes—
157#11924028#4112yes—
158C2×Dic146—21no—
159C2×Dic146—21no—
160C2×Dic146C22×C411yescharacteristic
161C2×D292C22×C411yescharacteristic
162C2×C22⋊D73—21no—
163C22×Dic73—21no—
164C22×Dic73#4111yescharacteristic
165C22×Dic73#4111yescharacteristic
166C2×C4×D73—21no—
167C2×C4×D73—21no—
168C2×C4×D73—21no—
169#11924012—21no—
170#11924012—21no—
171#11924012—21no—
172#11924012—21no—
173#11924011—42no—
174#11924011—42no—
175C2×#10596831—21no—
176C2×#10596831—21no—
177C2×#10596831—21no—
178C2×#10596831—21no—
179C2×#10596831—21no—
180C2×#10596831—21no—
181C2×#10596831C22×C411yescharacteristic
182C2×#10596831C22×C411yescharacteristic
183C2×#10596831C2×D411yescharacteristic
184C2×#10596831C2×D411yescharacteristic
185C8×D73—42no—
186C8×D73—42no—
187C4⋊Dic73—21no—
188C4⋊Dic73—21no—
189C4⋊Dic73C22×C411yescharacteristic
190C4⋊Dic73C22×C411yescharacteristic
191#11924007—21no—
192#11924007—21no—
193#11924007—21no—
194#11924007—21no—
195#11924007—21no—
196#11924007—21no—
197C22⋊Dic73—21no—
198#11924005—21no—
199#11924005—21no—
200C4×Dic73—21no—
201C4×Dic73—21no—
202C4×Dic73—21no—
203C4×Dic73—21no—
204C4×Dic73—42no—
205C4×Dic73—42no—
206C4×Dic73C22×C411yescharacteristic
207C4×Dic73C22×C411yescharacteristic
208C4×Dic73C2×D411yescharacteristic
209C4×Dic73C2×D411yescharacteristic
210C4×Dic73C2×Q812yes—
211C22×C4×C73—21no—
212C22×C4×C73C2411yescharacteristic
213M16×C73—21no—
214M16×C73—21no—
215M16×C73—21no—
216M16×C73C2×Q812yes—
217C2×C8×C73—21no—
218C2×C8×C73—21no—
219C2×C8×C73—21no—
220C2×C8×C73—21no—
221C2×C8×C73—21no—
222C2×C8×C73—21no—
223C2×C8×C73C22×C411yescharacteristic
224C2×C8×C73C22×C411yescharacteristic
225C2×C8×C73C2×D411yescharacteristic
226C2×C8×C73C2×D411yescharacteristic
227C4⋊C4×C73C22×C411yescharacteristic
228C4⋊C4×C73C22×C411yescharacteristic
229C4⋊C4×C73C22×C411yescharacteristic
230C4⋊C4×C73C22×C411yescharacteristic
231C22⋊C4×C73—21no—
232C22⋊C4×C73—21no—
233C42×C73—21no—
234C42×C73—21no—
235C42×C73—21no—
236C42×C73—21no—
237C23×C73C2×#4111yescharacteristic
238C22×D73—21no—
239C2×C4×C73—22no—
240C2×C4×C73—21no—
241C2×C4×C73—21no—
242C2×C4×C73—21no—
243C2×C4×C73—21no—
244C2×C4×C73—41no—
245C2×C4×C73—41no—
246C2×C4×C73C23×C411yescharacteristic
247C2×C4×C73C22×D411yescharacteristic
248C2×C4×C73C22×Q811yescharacteristic
249C2×C4×C73C2×#4111yescharacteristic
250C2×C4×C73H(5,𝔽2)11yescharacteristic
251C2×C4×C732−1+411yescharacteristic
252C22⋊D73—22no—
253C22⋊D73—22no—
254C2×Dic73—21no—
255C2×Dic73—22no—
256C2×Dic73—21no—
257C2×Dic73—21no—
258C2×Dic73—22no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73C4×D411yescharacteristic
263C2×Dic73C4×D411yescharacteristic
264C2×Dic73C4×D411yescharacteristic
265C2×Dic73C4×D411yescharacteristic
266D292—21no—
267D292C4×Q812yes—
268C4×D73—22no—
269C4×D73—22no—
270C4×D73—42no—
271C4×D73—81no—
272Dic146—21no—
273Dic146—42no—
274Dic146—81no—
275Dic146C4×Q812yes—
276C8×C73—21no—
277C8×C73—21no—
278C8×C73—21no—
279C8×C73—21no—
280C8×C73—22no—
281C8×C73—41no—
282#10596831—21no—
283#10596831—21no—
284#10596831—21no—
285#10596831—21no—
286#10596831—41no—
287#10596831—41no—
288C22×C73—21no—
289C22×C73—21no—
290C22×C73C2×C4×D411yescharacteristic
291C22×C73#55211yescharacteristic
292C22×C73#55411yescharacteristic
293C2×D73—22no—
294C2×D73—41no—
295C2×D73—41no—
296C4×C73—42no—
297C4×C73—41no—
298C4×C73—41no—
299C4×C73—41no—
300C4×C73—41no—
301C4×C73C2×C4×Q811yescharacteristic
302C4×C73#52011yescharacteristic
303C4×C73#54611yescharacteristic
304C4×C73D4×Q811yescharacteristic
305Dic73—22no—
306Dic73—21no—
307Dic73—21no—
308Dic73—21no—
309Dic73—42no—
310Dic73—41no—
311Dic73—41no—
312Dic73#44412yes—
313#22983—731nomaximal, Sylow
314C2×C73—21no—
315C2×C73—21no—
316C2×C73#236011yescharacteristic
317C2×C73C2×#44411yescharacteristic
318C2×C73#295611yescharacteristic
319D73—42no—
320#3076—731no—
321#2962—731no—
322#2906—731no—
323#1994—731no—
324#1950—731no—
325#1946—731no—
326#1946—731no—
327#1922—731no—
328#1922D7311yescharacteristic, complemented
329#1866—731no—
330#1866—731no—
331#1767—731no—
332#1765—731no—
333#1765—731no—
334C4×#104—731no—
335C73#2298311yescharacteristic, complemented, Sylow
336C2×#131—731no—
337C2×#131—731no—
338#533—731no—
339C4×#41—731no—
340#479—731no—
341#479—731no—
342#478—731no—
343#478—731no—
344#477—731no—
345#476—731no—
346#475—731no—
347#474—731no—
348#446—1461no—
349#446—1461no—
350#446—1461no—
351C8×Q8—1461no—
352C2×#108—731no—
353C2×#108—731no—
354#427—731no—
355#427C2×D7311yescharacteristic
356#424—731no—
357#424—731no—
358#424—731no—
359#424—1461no—
360#424—1461no—
361#424C2×D7311yescharacteristic
362#423—731no—
363#423—1461no—
364C2×C4⋊C8—731no—
365C2×C4⋊C8—731no—
366C2×C4⋊C8—1461no—
367C2×#104—731no—
368C2×#104—731no—
369#416—731no—
370#416—731no—
371#405—731no—
372#405C2×D7311yescharacteristic
373C4×M16—731no—
374C2×#97—731no—
375C2×#97—731no—
376C43—731no—
377#341—731no—
378#341C2×D7311yescharacteristic
379#339—731no—
380#339—731no—
381#339—731no—
382#339—731no—
383#339C2×D7311yescharacteristic
384#339C2×D7311yescharacteristic
385#336—731no—
386#336C2×D7311yescharacteristic
387C2×#41—731no—
388#131—2921no—
389#131—2921no—
390#131—1462no—
391#131—1462no—
392C2×M16—731no—
393C2×M16—731no—
394C2×M16—1461no—
395C2×M16—1461no—
396C2×M16C22×D7311yescharacteristic
397C2×M16C22×D7311yescharacteristic
398C22×C8—731no—
399C22×C8—731no—
400C22×C8—1461no—
401C22×C8—1461no—
402C22×C8C22×D7311yescharacteristic
403C22×C8C22×D7311yescharacteristic
404C4⋊Q8—731no—
405C4⋊Q8—731no—
406#125—731no—
407#125—731no—
408C22⋊Q8—1461no—
409C22⋊Q8—1461no—
410C4×Q8—731no—
411C4×Q8—731no—
412C4×Q8—1461no—
413C4×Q8—1461no—
414C4×D4—731no—
415C4×D4—731no—
416C4×D4—1461no—
417C4×D4—1461no—
418#117—731no—
419#117—1461no—
420#117C22×D7311yescharacteristic
421#117C22×D7311yescharacteristic
422C2×C42—731no—
423C2×C42—731no—
424C2×C42—1461no—
425C2×C42—1461no—
426C2×C42—1461no—
427C2×C42C22×D7311yescharacteristic
428#108—2921no—
429#108—2921no—
430#107—731no—
431#107—731no—
432#107C4×D7311yescharacteristic
433#107C4×D7311yescharacteristic
434#106—731no—
435#106—731no—
436#106C4×D7311yescharacteristic
437#106C4×D7311yescharacteristic
438C4⋊C8—21no—
439C4⋊C8—21no—
440C4⋊C8—731no—
441C4⋊C8—731no—
442C4⋊C8—731no—
443C4⋊C8—731no—
444C4⋊C8—731no—
445C4⋊C8—731no—
446C4⋊C8—731no—
447C4⋊C8—731no—
448C4⋊C8—1461no—
449C4⋊C8—1461no—
450C4⋊C8—1461no—
451C4⋊C8—1461no—
452C4⋊C8—1461no—
453C4⋊C8—1461no—
454C4⋊C8—1461no—
455C4⋊C8—1461no—
456#104—734no—
457#104—1462no—
458#103—731no—
459#103—731no—
460#103—731no—
461#103—731no—
462#102—731no—
463#102—731no—
464#102—731no—
465#102—731no—
466#97—21no—
467#97—2921no—
468#97—2921no—
469#97—1461no—
470#97—1461no—
471C4×C8—21no—
472C4×C8—1461no—
473C4×C8—1461no—
474#41—732no—
475#41—732no—
476#41—1461no—
477#41—1461no—
478C2×Q8—731no—
479C2×Q8—1461no—
480C2×Q8—1461no—
481C2×D4—731no—
482C2×D4—1461no—
483C22×C4—21no—
484C22×C4—731no—
485C22×C4—731no—
486C22×C4—1461no—
487C22×C4—1461no—
488C22×C4—1461no—
489C22×C4—1461no—
490C22×C4C23×D7311yescharacteristic
491M16—21no—
492M16—21no—
493M16—21no—
494M16—2922no—
495M16—2922no—
496M16—1461no—
497M16—1461no—
498M16—1461no—
499M16—1461no—
500M16Q8×D7312yes—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—2922no—
508C2×C8—2922no—
509C2×C8—731no—
510C2×C8—731no—
511C2×C8—731no—
512C2×C8—731no—
513C2×C8—1461no—
514C2×C8—1461no—
515C2×C8—1461no—
516C2×C8—1461no—
517C2×C8—1461no—
518C2×C8—1461no—
519C2×C8C2×C4×D7311yescharacteristic
520C2×C8C2×C4×D7311yescharacteristic
521C2×C8D4×D7311yescharacteristic
522C2×C8D4×D7311yescharacteristic
523C4⋊C4—731no—
524C4⋊C4—731no—
525C4⋊C4—1461no—
526C4⋊C4—1461no—
527C4⋊C4—1461no—
528C4⋊C4—1461no—
529C4⋊C4—1461no—
530C4⋊C4—1461no—
531C4⋊C4—1461no—
532C4⋊C4—1461no—
533C4⋊C4C2×C4×D7311yescharacteristic
534C4⋊C4C2×C4×D7311yescharacteristic
535C4⋊C4C2×C4×D7311yescharacteristic
536C4⋊C4C2×C4×D7311yescharacteristic
537C22⋊C4—21no—
538C22⋊C4—21no—
539C22⋊C4—1461no—
540C22⋊C4—1461no—
541C22⋊C4—1461no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—2922no—
547C42—2922no—
548C42—731no—
549C42—731no—
550C42—731no—
551C42—731no—
552C42—732no—
553C42—1461no—
554C42—1461no—
555C42—1461no—
556C42—1461no—
557C23—1461no—
558C23#41×D7311yescharacteristic
559Q8—2922no—
560Q8—5841no—
561Q8—732no—
562Q8—1461no—
563D4—732no—
564D4—1462no—
565D4—1462no—
566D4—1461no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—2922no—
575C2×C4—2921no—
576C2×C4—2921no—
577C2×C4—2921no—
578C2×C4—5841no—
579C2×C4—731no—
580C2×C4—731no—
581C2×C4—731no—
582C2×C4—731no—
583C2×C4—1462no—
584C2×C4—1461no—
585C2×C4—1461no—
586C2×C4—1462no—
587C2×C4—1461no—
588C2×C4—1462no—
589C2×C4—1462no—
590C2×C4C22×C4×D7311yescharacteristic
591C2×C4C2×D4×D7311yescharacteristic
592C2×C4C2×Q8×D7311yescharacteristic
593C2×C4#41×D7311yescharacteristic
594C2×C4#1455663711yescharacteristic
595C2×C4#1455663811yescharacteristic
596C8—21no—
597C8—21no—
598C8—22no—
599C8—21no—
600C8—21no—
601C8—41no—
602C8—2921no—
603C8—2921no—
604C8—1461no—
605C8—1461no—
606C8—1461no—
607C8—1461no—
608C22—21no—
609C22—21no—
610C22—2921no—
611C22—2921no—
612C22—1462no—
613C22C4×D4×D7311yescharacteristic
614C22#1798927311yescharacteristic
615C22#1798929311yescharacteristic
616C4—42no—
617C4—41no—
618C4—41no—
619C4—41no—
620C4—41no—
621C4—2922no—
622C4—2921no—
623C4—2921no—
624C4—732no—
625C4—1461no—
626C4—1462no—
627C4—1461no—
628C4—1461no—
629C4C4×Q8×D7311yescharacteristic
630C4#1798906711yescharacteristic
631C4#1798922911yescharacteristic
632C4#1798925011yescharacteristic
633C2—21no—
634C2—21no—
635C2—2922no—
636C2#2261293911yescharacteristic
637C2#444×D7311yescharacteristic
638C2#2261883611yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1922C2
socleC22×C73C2×C4×D4
prime radicalC2×#11924028C23
cosocleC22×C4×C73C24
hypercenter#1922D73
hypocenterC73#22983
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1922

Complements

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

NH ≅ G/Ntype
2. #22618875C2semidirect
4. #22610906C2semidirect
5. #22610705C2semidirect
6. #22610685C2semidirect
7. #22610681C2semidirect
8. #22610533C2semidirect
11. #22609783C2semidirect
15. C73×#1922C2semidirect
20. #17988897C4semidirect
21. #17988893C4semidirect
22. #17988890C4semidirect
23. #17988886C4semidirect
24. #17988881C4semidirect
25. #17988876C4semidirect
26. #17988871C4semidirect
27. #17988866C4semidirect
124. #14556517Q8semidirect
328. #1922D73semidirect
335. C73#22983semidirect

Maximal subgroups

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

subgroupindex
#226197172
#226188752
#226184982
#226109062
#226107052
#226106852
#226106812
#226105332
#226102092
#226102072
#226097832
#226097772
#226097742
#226097352
C73×#19222
#2298373