Finite Group Databasebeta

← #32993322

Subgroups — #32993322

Order 18688.

898 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988635C211yesmaximal, characteristic, complemented
2C2×#17988605C211yesmaximal, characteristic
3C22×#14556519C211yesmaximal, characteristic
4#22618375C211yesmaximal, characteristic
5C2×#17988357C211yesmaximal, characteristic, complemented
6#22609625C211yesmaximal, characteristic, complemented
7C2×#339×C73C211yesmaximal, characteristic, complemented
8C22×#11924028C2211yescharacteristic
9C22×#11924012C2211yescharacteristic
10C22×#11924012C2211yescharacteristic
11C22×#11924012C2211yescharacteristic
12#17988635—22no—
13#17988635—22no—
14C2×#14556526—22no—
15#17988605—22no—
16#17988605—22no—
17C2×#14556519—22no—
18C2×#14556519—22no—
19C2×#14556519—22no—
20C2×#14556519C414yescomplemented
21#17988555—22no—
22#17988553—22no—
23#17988550—22no—
24#17988357—22no—
25#17988342—22no—
26C22×M16×C73C2211yescharacteristic
27C2×#117×C73C2211yescharacteristic
28C22×C42×C73C2211yescharacteristic
29#339×C73—22no—
30C2×#11924028—21no—
31C2×#11924028—21no—
32C2×#11924028—21no—
33C2×#11924028—21no—
34C2×#11924028—21no—
35C2×#11924028—21no—
36C2×#11924028—21no—
37C2×#11924028—21no—
38C2×#11924028—21no—
39C2×#11924028—21no—
40C2×#11924028C2×C412yes—
41C2×#11924028D412yes—
42C22×Dic146Q811yescharacteristic
43C22×D292Q811yescharacteristic
44C22×C22⋊D73—21no—
45C22×C4×D73—21no—
46C2×#11924012—21no—
47C2×#11924012—21no—
48C2×#11924012—21no—
49C2×#11924012—21no—
50C2×#11924012—21no—
51C2×#11924012—21no—
52C2×#11924012—21no—
53C2×#11924012—21no—
54C2×#11924012—21no—
55C2×#11924012—21no—
56C2×#11924012—21no—
57C2×#11924012—21no—
58C2×#11924012—21no—
59C2×#11924012—21no—
60C2×#11924012C2×C412yes—
61C2×#11924012D412yes—
62C2×#11924012D412yes—
63C2×#11924012D412yes—
64C22×#10596831—21no—
65C22×#10596831—21no—
66C22×#10596831D412yes—
67#14556526—28no—
68#14556519—28no—
69#14556519—28no—
70#14556503—28no—
71C23×C4×C73—21no—
72C23×C4×C73C2311yescharacteristic
73C2×M16×C73—21no—
74C2×M16×C73—21no—
75C2×M16×C73—21no—
76C2×M16×C73—21no—
77C2×M16×C73—21no—
78C2×M16×C73D412yes—
79C22×C8×C73—21no—
80#117×C73—21no—
81#117×C73—21no—
82#117×C73—21no—
83#117×C73D412yes—
84C2×C4⋊C4×C73—21no—
85C2×C22⋊C4×C73—21no—
86C2×C42×C73—21no—
87C2×C42×C73—21no—
88C2×C42×C73—21no—
89C2×C42×C73—21no—
90C2×C42×C73—21no—
91C2×C42×C73—21no—
92C2×C42×C73C2×C412yes—
93C23×D73—21no—
94#11924028—22no—
95#11924028—22no—
96#11924028—22no—
97#11924028—22no—
98#11924028—22no—
99#11924028—22no—
100#11924028—22no—
101#11924028—22no—
102#11924028—41no—
103#11924028—41no—
104#11924028—41no—
105#11924028—41no—
106#11924028—41no—
107#11924028—41no—
108#11924028—41no—
109#11924028—41no—
110C2×Dic146—21no—
111C2×Dic146—21no—
112C2×Dic146—21no—
113C2×Dic146—21no—
114C2×Dic146—21no—
115C2×Dic146C4⋊C412yes—
116C2×D292—21no—
117C2×D292—21no—
118C2×D292—21no—
119C2×D292—21no—
120C2×D292—21no—
121C2×D292C4⋊C412yes—
122C2×C22⋊D73—22no—
123C2×C22⋊D73—22no—
124C2×C22⋊D73—41no—
125C2×C22⋊D73—41no—
126C2×C22⋊D73—41no—
127C2×C22⋊D73—41no—
128C22×Dic73—21no—
129C2×C4×D73—22no—
130C2×C4×D73—22no—
131C2×C4×D73—41no—
132C2×C4×D73—41no—
133C2×C4×D73—41no—
134C2×C4×D73—41no—
135#11924012—22no—
136#11924012—22no—
137#11924012—22no—
138#11924012—22no—
139#11924012—22no—
140#11924012—41no—
141#11924012—41no—
142#11924012—41no—
143#11924012—41no—
144#11924012—41no—
145#11924012—41no—
146#11924012C22⋊C414yes—
147C2×#10596831—22no—
148C2×#10596831—22no—
149C2×#10596831—22no—
150C2×#10596831—22no—
151C2×#10596831—22no—
152C2×#10596831—22no—
153C2×#10596831—22no—
154C2×#10596831—41no—
155C2×#10596831—41no—
156C24×C73#4111yescharacteristic
157C22×C4×C73—22no—
158C22×C4×C73—44no—
159C22×C4×C73—41no—
160C22×C4×C73—41no—
161C22×C4×C73—41no—
162C22×C4×C73C22×C411yescharacteristic
163C22×C4×C73C2×D411yescharacteristic
164C22×C4×C73C2×D411yescharacteristic
165C22×C4×C73C2×D411yescharacteristic
166C22×C4×C73C2×D411yescharacteristic
167C22×C4×C73C2×D411yescharacteristic
168C22×C4×C73C2×D411yescharacteristic
169C22×C4×C73C2×D411yescharacteristic
170C22×C4×C73C2×Q811yescharacteristic
171C22×C4×C73#4111yescharacteristic
172C22×C4×C73#4111yescharacteristic
173C22×C4×C73#4111yescharacteristic
174C22×C4×C73#4111yescharacteristic
175C22×C4×C73#4111yescharacteristic
176M16×C73—22no—
177M16×C73—22no—
178M16×C73—41no—
179M16×C73—41no—
180C2×C8×C73—22no—
181C2×C8×C73—41no—
182C2×C8×C73—41no—
183C4⋊C4×C73—22no—
184C4⋊C4×C73—41no—
185C22⋊C4×C73—22no—
186C22⋊C4×C73—41no—
187C42×C73—22no—
188C42×C73—22no—
189C42×C73—22no—
190C42×C73—41no—
191C42×C73—41no—
192C42×C73—41no—
193C23×C73—42no—
194C23×C73C4×D411yescharacteristic
195C23×C73#12111yescharacteristic
196C23×C73C22⋊Q811yescharacteristic
197C23×C73#12311yescharacteristic
198C23×C73#12411yescharacteristic
199C23×C73#12411yescharacteristic
200C23×C73#12611yescharacteristic
201C22×D73—22no—
202C22×D73—41no—
203C22×D73—41no—
204C22×D73—82no—
205C2×C4×C73—24no—
206C2×C4×C73—42no—
207C2×C4×C73—44no—
208C2×C4×C73—44no—
209C2×C4×C73—42no—
210C2×C4×C73—42no—
211C2×C4×C73—82no—
212C2×C4×C73C2×C22⋊C411yescharacteristic
213C2×C4×C73C2×C4⋊C411yescharacteristic
214C2×C4×C73#11711yescharacteristic
215C2×C4×C73C4×D411yescharacteristic
216C2×C4×C73C4×D411yescharacteristic
217C2×C4×C73C4×D411yescharacteristic
218C2×C4×C73#12011yescharacteristic
219C2×C4×C73#12011yescharacteristic
220C2×C4×C73#12011yescharacteristic
221C2×C4×C73#12111yescharacteristic
222C2×C4×C73#12111yescharacteristic
223C2×C4×C73#12111yescharacteristic
224C2×C4×C73#12111yescharacteristic
225C2×C4×C73#12111yescharacteristic
226C2×C4×C73#12111yescharacteristic
227C2×C4×C73#12111yescharacteristic
228C2×C4×C73C22⋊Q811yescharacteristic
229C2×C4×C73C22⋊Q811yescharacteristic
230C2×C4×C73C22⋊Q811yescharacteristic
231C2×C4×C73C22⋊Q811yescharacteristic
232C2×C4×C73C22⋊Q811yescharacteristic
233C2×C4×C73#12311yescharacteristic
234C2×C4×C73#12311yescharacteristic
235C2×C4×C73#12311yescharacteristic
236C2×C4×C73#12311yescharacteristic
237C2×C4×C73#12411yescharacteristic
238C2×C4×C73#12611yescharacteristic
239C2×C4×C73#12711yescharacteristic
240C22⋊D73—42no—
241C22⋊D73—42no—
242C22⋊D73—42no—
243C22⋊D73—42no—
244C2×Dic73—22no—
245C2×Dic73—41no—
246C2×Dic73—41no—
247D292—22no—
248D292—22no—
249D292—41no—
250D292—41no—
251C4×D73—42no—
252C4×D73—42no—
253C4×D73—42no—
254C4×D73—42no—
255Dic146—22no—
256Dic146—22no—
257Dic146—41no—
258Dic146—41no—
259C8×C73—42no—
260#10596831—24no—
261#10596831—42no—
262#10596831—42no—
263C22×C73—22no—
264C22×C73—22no—
265C22×C73—22no—
266C22×C73—41no—
267C22×C73—41no—
268C22×C73—41no—
269C22×C73—41no—
270C22×C73#38511yescharacteristic
271C22×C73#38811yescharacteristic
272C22×C73#39011yescharacteristic
273C22×C73#39311yescharacteristic
274C22×C73#39411yescharacteristic
275C22×C73#39611yescharacteristic
276C22×C73#39711yescharacteristic
277C2×D73—44no—
278C2×D73—42no—
279C2×D73—82no—
280C2×D73—82no—
281C4×C73—44no—
282C4×C73—82no—
283C4×C73#38011yescharacteristic
284C4×C73#38511yescharacteristic
285C4×C73#38611yescharacteristic
286C4×C73#38811yescharacteristic
287C4×C73#39211yescharacteristic
288C4×C73#39411yescharacteristic
289C4×C73#39611yescharacteristic
290C4×C73#39911yescharacteristic
291Dic73—42no—
292#11114—731nomaximal, Sylow
293C2×C73—21no—
294C2×C73—21no—
295C2×C73—21no—
296C2×C73—21no—
297C2×C73—21no—
298C2×C73—21no—
299C2×C73#143911yescharacteristic
300C2×C73#186112yes—
301D73—82no—
302C2×#430—731no—
303C2×#421—731no—
304C22×#104—731no—
305#2879—731no—
306C2×#344—731no—
307C2×#339D7311yescharacteristic, complemented
308#1728—731no—
309C73#1111411yescharacteristic, complemented, Sylow
310C22×#41—731no—
311C22×M16—731no—
312C22×M16—731no—
313C22×M16—731no—
314C22×M16C2×D7311yescharacteristic
315C2×#117C2×D7311yescharacteristic
316C22×C42C2×D7311yescharacteristic
317#430—1462no—
318#430—1462no—
319C2×#108—1462no—
320#421—1462no—
321#421—1462no—
322C2×#104—734no—
323C2×#104—1462no—
324C2×#104—1462no—
325C2×#104—1462no—
326#408—1462no—
327#408—1462no—
328#407—1462no—
329#344—1462no—
330#339—22no—
331#336—1462no—
332C2×#41—732no—
333C2×#41—732no—
334C2×#41—1461no—
335C2×#41—1461no—
336C2×#41—1461no—
337C2×#41—1461no—
338C2×#41—1461no—
339C2×#41—1461no—
340C2×#41—1461no—
341C2×#41—1461no—
342C2×#41—1461no—
343C2×#41—1461no—
344C22×Q8—731no—
345C22×D4—731no—
346C22×D4—1461no—
347C23×C4—21no—
348C23×C4—1461no—
349C23×C4C22×D7311yescharacteristic
350C2×M16—21no—
351C2×M16—21no—
352C2×M16—21no—
353C2×M16—21no—
354C2×M16—21no—
355C2×M16—732no—
356C2×M16—732no—
357C2×M16—732no—
358C2×M16—732no—
359C2×M16—1461no—
360C2×M16—1461no—
361C2×M16—1461no—
362C2×M16—1461no—
363C2×M16—1461no—
364C2×M16—1461no—
365C2×M16—1461no—
366C2×M16—1461no—
367C2×M16—1461no—
368C2×M16—1461no—
369C2×M16—1461no—
370C2×M16—1461no—
371C2×M16—1461no—
372C2×M16—1461no—
373C2×M16D29212yes—
374C22×C8—21no—
375C22×C8—732no—
376C22×C8—1461no—
377C22×C8—1461no—
378#117—21no—
379#117—21no—
380#117—21no—
381#117C22⋊D7312yes—
382C2×C4⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C42—21no—
385C2×C42—21no—
386C2×C42—21no—
387C2×C42—21no—
388C2×C42—21no—
389C2×C42—21no—
390C2×C42C4×D7312yes—
391#108—1468no—
392#104—1468no—
393#104—1468no—
394C22⋊C8—1468no—
395C24—1461no—
396C24#1192402711yescharacteristic
397#41—2921no—
398#41—2921no—
399#41—2921no—
400#41—2921no—
401#41—2921no—
402#41—2921no—
403#41—2921no—
404#41—2921no—
405#41—1462no—
406#41—1462no—
407#41—1462no—
408#41—1462no—
409#41—1462no—
410#41—1462no—
411#41—1462no—
412#41—1462no—
413C2×Q8—732no—
414C2×Q8—1461no—
415C2×Q8—1461no—
416C2×Q8—1461no—
417C2×Q8—1461no—
418C2×Q8—1461no—
419C2×D4—2921no—
420C2×D4—2921no—
421C2×D4—2921no—
422C2×D4—2921no—
423C2×D4—732no—
424C2×D4—1462no—
425C2×D4—1462no—
426C2×D4—1461no—
427C2×D4—1461no—
428C2×D4—1461no—
429C2×D4—1461no—
430C2×D4—1461no—
431C22×C4—22no—
432C22×C4—44no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—2921no—
437C22×C4—2921no—
438C22×C4—2921no—
439C22×C4—2921no—
440C22×C4—1461no—
441C22×C4—1462no—
442C22×C4—1462no—
443C22×C4C2×C4×D7311yescharacteristic
444C22×C4D4×D7311yescharacteristic
445C22×C4D4×D7311yescharacteristic
446C22×C4D4×D7311yescharacteristic
447C22×C4D4×D7311yescharacteristic
448C22×C4D4×D7311yescharacteristic
449C22×C4C2×C22⋊D7311yescharacteristic
450C22×C4C2×D29211yescharacteristic
451C22×C4Q8×D7311yescharacteristic
452C22×C4#1192402811yescharacteristic
453C22×C4#1192402811yescharacteristic
454C22×C4#1192402811yescharacteristic
455C22×C4#1192402811yescharacteristic
456C22×C4#1192402911yescharacteristic
457M16—22no—
458M16—22no—
459M16—41no—
460M16—41no—
461M16—2921no—
462M16—2921no—
463M16—2921no—
464M16—2921no—
465M16—2921no—
466M16—2921no—
467M16—734no—
468M16—1462no—
469M16—1462no—
470M16—1462no—
471M16—1462no—
472M16—1462no—
473C2×C8—22no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—2921no—
477C2×C8—2921no—
478C2×C8—1462no—
479C2×C8—1462no—
480C2×C8—1462no—
481C2×C8—1462no—
482C2×C8—1462no—
483C2×C8—1462no—
484C2×C8—1462no—
485C4⋊C4—22no—
486C4⋊C4—41no—
487C22⋊C4—22no—
488C22⋊C4—41no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—41no—
493C42—41no—
494C42—41no—
495C23—42no—
496C23—2921no—
497C23—2921no—
498C23—5842no—
499C23—1462no—
500C23#1455655111yescharacteristic
501C23#1455656311yescharacteristic
502C23#1455657111yescharacteristic
503C23#1455657311yescharacteristic
504C23#1455657711yescharacteristic
505C23#1455657711yescharacteristic
506C23#1455658611yescharacteristic
507Q8—2921no—
508Q8—2921no—
509Q8—1462no—
510Q8—1462no—
511D4—2922no—
512D4—2922no—
513D4—2922no—
514D4—2922no—
515D4—2921no—
516D4—2921no—
517D4—1462no—
518D4—1462no—
519C2×C4—24no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—82no—
526C2×C4—2921no—
527C2×C4—2921no—
528C2×C4—2922no—
529C2×C4—2922no—
530C2×C4—2922no—
531C2×C4—2922no—
532C2×C4—1462no—
533C2×C4C22⋊C4×D7311yescharacteristic
534C2×C4C4⋊C4×D7311yescharacteristic
535C2×C4#1455654811yescharacteristic
536C2×C4C4×C22⋊D7311yescharacteristic
537C2×C4#1455655311yescharacteristic
538C2×C4C4×D29211yescharacteristic
539C2×C4#1455655911yescharacteristic
540C2×C4#1455656111yescharacteristic
541C2×C4#1455656111yescharacteristic
542C2×C4#1455656411yescharacteristic
543C2×C4#1455656411yescharacteristic
544C2×C4#1455656411yescharacteristic
545C2×C4#1455656411yescharacteristic
546C2×C4#1455656411yescharacteristic
547C2×C4#1455656511yescharacteristic
548C2×C4#1455656611yescharacteristic
549C2×C4#1455656711yescharacteristic
550C2×C4#1455656911yescharacteristic
551C2×C4#1455656911yescharacteristic
552C2×C4#1455656911yescharacteristic
553C2×C4#1455656911yescharacteristic
554C2×C4#1455657411yescharacteristic
555C2×C4#1455657611yescharacteristic
556C2×C4#1455657611yescharacteristic
557C2×C4#1455657611yescharacteristic
558C2×C4#1455657911yescharacteristic
559C2×C4#1455658511yescharacteristic
560C2×C4#1455658711yescharacteristic
561C8—42no—
562C8—2922no—
563C8—2922no—
564C8—1464no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—2922no—
573C22—2924no—
574C22—5842no—
575C22—5842no—
576C22#1798846111yescharacteristic
577C22#1798847611yescharacteristic
578C22#1798848711yescharacteristic
579C22#1798850011yescharacteristic
580C22#1798850211yescharacteristic
581C22#1798850911yescharacteristic
582C22#1798851611yescharacteristic
583C4—44no—
584C4—82no—
585C4—2922no—
586C4#1798844611yescharacteristic
587C4#1798845811yescharacteristic
588C4#1798846511yescharacteristic
589C4#1798847811yescharacteristic
590C4#1798849611yescharacteristic
591C4#1798850511yescharacteristic
592C4#1798851111yescharacteristic
593C4#1798852211yescharacteristic
594C2—21no—
595C2—21no—
596C2—21no—
597C2—21no—
598C2—21no—
599C2—21no—
600C2—5842no—
601C2#2260817511yescharacteristic
602C2#2261017212yes—
subgroupG/H
centerC2×C4#14556561
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC2×#339×C73C2
socleC22×C73#385
prime radicalC22×#11924028C22
cosocleC23×C4×C73C23
hypercenterC2×#339D73
hypocenterC73#11114
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#339

Complements

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

NH ≅ G/Ntype
1. C2×#17988635C2semidirect
5. C2×#17988357C2semidirect
6. #22609625C2semidirect
7. C2×#339×C73C2semidirect
20. C2×#14556519C4semidirect
307. C2×#339D73semidirect
309. C73#11114semidirect

Maximal subgroups

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

subgroupindex
C2×#179886352
C2×#179886052
C22×#145565192
#226183752
C2×#179883572
#226096252
C2×#339×C732
#1111473