Finite Group Databasebeta

← #33090244

Subgroups — #33090244

Order 18688.

814 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×Dic73C211yesmaximal, characteristic
2#22612576C211yesmaximal, characteristic, complemented
3#22612568C211yesmaximal, characteristic
4#22610787C211yesmaximal, characteristic
5#22610659C211yesmaximal, characteristic, complemented
6#22610657C211yesmaximal, characteristic, complemented
7#22610631C211yesmaximal, characteristic
8#22610625C211yesmaximal, characteristic
9#22610624C211yesmaximal, characteristic, complemented
10#22610618C211yesmaximal, characteristic, complemented
11#106×Dic73C211yesmaximal, characteristic, complemented
12#22609763C211yesmaximal, characteristic, complemented
13#22609758C211yesmaximal, characteristic
14#22609617C211yesmaximal, characteristic, complemented
15C73×#1968C211yesmaximal, characteristic
16C2×Q8×Dic73C2211yescharacteristic
17C2×D4×Dic73C2211yescharacteristic
18SD16×Dic73C418yescomplemented
19C2×#14556523C2211yescharacteristic
20C2×#14556523C2211yescharacteristic
21C2×#14556515C2211yescharacteristic
22C2×#14556515C2211yescharacteristic
23C2×#14556512C2211yescharacteristic
24C2×#14556512C2211yescharacteristic
25C2×#14556502C2211yescharacteristic
26C2×#14556501C2211yescharacteristic
27C2×#14556501C2211yescharacteristic
28C2×C8×Dic73C2211yescharacteristic
29C2×C8×Dic73C2211yescharacteristic
30#17988494C2211yescharacteristic
31#17988490C2211yescharacteristic
32#17988485C2211yescharacteristic
33#17988485C2211yescharacteristic
34#17988483C2211yescharacteristic
35#17988483C2211yescharacteristic
36#17988481—21no—
37#17988475—21no—
38#17988472—21no—
39#17988470—21no—
40#17988454—21no—
41#17988451—21no—
42#17988448—21no—
43#17988445—21no—
44C4⋊C4×Dic73C2211yescharacteristic
45C4⋊C4×Dic73C2211yescharacteristic
46#17988433C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988349C2211yescharacteristic
52#17988349C2211yescharacteristic
53C22×SD16×C73C2211yescharacteristic
54C2×#106×C73C2211yescharacteristic
55C2×#103×C73C2211yescharacteristic
56C2×#102×C73C2211yescharacteristic
57C2×C4×C8×C73C2211yescharacteristic
58#391×C73C2211yescharacteristic
59#390×C73C2211yescharacteristic
60C23×Dic73—21no—
61C22×#10596831C2311yescharacteristic
62C22×#10596831C2311yescharacteristic
63Q8×Dic73—22no—
64Q8×Dic73C2×C414yes—
65D4×Dic73—22no—
66D4×Dic73C2×C414yes—
67C2×C4⋊Dic73—21no—
68C2×C4⋊Dic73C2311yescharacteristic
69C2×C4⋊Dic73C2311yescharacteristic
70C2×C4⋊Dic73C2311yescharacteristic
71C2×C4⋊Dic73C2311yescharacteristic
72C2×#11924007—21no—
73C2×#11924007—21no—
74C2×#11924007—21no—
75C2×#11924007—21no—
76C2×C22⋊Dic73—21no—
77C2×C22⋊Dic73—21no—
78C2×C22⋊Dic73—21no—
79C2×C4×Dic73—21no—
80C2×C4×Dic73—21no—
81C2×C4×Dic73—21no—
82C2×C4×Dic73C2311yescharacteristic
83C2×C4×Dic73C2311yescharacteristic
84#14556523—22no—
85#14556523C2×C414yes—
86#14556515—22no—
87#14556515C2×C414yes—
88#14556512—22no—
89#14556512C2×C414yes—
90#14556502—22no—
91#14556501—22no—
92#14556501—22no—
93C8×Dic73—22no—
94C8×Dic73C2×C414yes—
95#14556498—21no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103#14556498D411yescharacteristic
104#14556498D411yescharacteristic
105#14556498D411yescharacteristic
106#14556498D411yescharacteristic
107C22×Q8×C73C2311yescharacteristic
108C22×D4×C73C2311yescharacteristic
109C2×SD16×C73—22no—
110C2×SD16×C73—22no—
111C2×SD16×C73C2×C414yes—
112C22×C8×C73C2311yescharacteristic
113C22×C8×C73C2311yescharacteristic
114C2×C4⋊C4×C73C2311yescharacteristic
115C2×C4⋊C4×C73C2311yescharacteristic
116C2×C22⋊C4×C73—21no—
117C2×C22⋊C4×C73—21no—
118C2×C42×C73C2311yescharacteristic
119#106×C73—22no—
120#103×C73—22no—
121#102×C73—22no—
122C4×C8×C73—22no—
123#95×C73—21no—
124#95×C73—21no—
125C22×Dic73—21no—
126C22×Dic73—21no—
127C22×Dic73—24no—
128C22×Dic73—21no—
129C22×Dic73—21no—
130C22×Dic73—81no—
131C22×Dic73—81no—
132C22×Dic73C2×D411yescharacteristic
133C22×Dic73C2×D411yescharacteristic
134C22×Dic73C2×D411yescharacteristic
135C22×Dic73C2×D411yescharacteristic
136C2×#10596831—21no—
137C2×#10596831—21no—
138C2×#10596831—21no—
139C2×#10596831—21no—
140C2×#10596831C22×C412yes—
141C2×#10596831#4112yes—
142C4⋊Dic73—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73—24no—
145C4⋊Dic73—21no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73C22×C412yes—
150C4⋊Dic73C22×C412yes—
151#11924007—81no—
152#11924007—81no—
153#11924007—81no—
154#11924007—81no—
155C22⋊Dic73—24no—
156C22⋊Dic73—81no—
157C22⋊Dic73—81no—
158C4×Dic73—21no—
159C4×Dic73—21no—
160C4×Dic73—24no—
161C4×Dic73—81no—
162C4×Dic73—81no—
163C4×Dic73C22×C412yes—
164C4×Dic73#4112yes—
165C24×C73—21no—
166C2×Q8×C73—21no—
167C2×Q8×C73—21no—
168C2×Q8×C73—21no—
169C2×Q8×C73—21no—
170C2×Q8×C73—21no—
171C2×Q8×C73C22×C412yes—
172C2×D4×C73—21no—
173C2×D4×C73—21no—
174C2×D4×C73—21no—
175C2×D4×C73—21no—
176C2×D4×C73—21no—
177C2×D4×C73C22×C412yes—
178C22×C4×C73—21no—
179C22×C4×C73—21no—
180C22×C4×C73—21no—
181C22×C4×C73C2×D411yescharacteristic
182C22×C4×C73C2×D411yescharacteristic
183C22×C4×C73C2411yescharacteristic
184SD16×C73—28no—
185C2×C8×C73—21no—
186C2×C8×C73—21no—
187C2×C8×C73C22×C412yes—
188C2×C8×C73#4112yes—
189C2×C8×C73#4112yes—
190C2×C8×C73#4112yes—
191C4⋊C4×C73—21no—
192C4⋊C4×C73—21no—
193C4⋊C4×C73—21no—
194C4⋊C4×C73—21no—
195C22⋊C4×C73—81no—
196C22⋊C4×C73—81no—
197C42×C73—21no—
198C42×C73—21no—
199C23×C73—22no—
200C23×C73—41no—
201C23×C73—41no—
202C23×C73—41no—
203C23×C73—41no—
204C23×C73—41no—
205C23×C73—41no—
206C23×C73C22×D411yescharacteristic
207Q8×C73—22no—
208Q8×C73—22no—
209Q8×C73—41no—
210Q8×C73—41no—
211D4×C73—22no—
212D4×C73—22no—
213D4×C73—41no—
214D4×C73—41no—
215C2×C4×C73—22no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—41no—
223C2×C4×C73—41no—
224C2×C4×C73—41no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73—41no—
228C2×C4×C73—41no—
229C2×C4×C73—41no—
230C2×C4×C73C23×C411yescharacteristic
231C2×C4×C73C2×#4111yescharacteristic
232C2×C4×C73C2×#4111yescharacteristic
233C2×C4×C73C2×#4111yescharacteristic
234C2×C4×C73C2×#4111yescharacteristic
235C2×C4×C73C2×#4111yescharacteristic
236C2×Dic73—21no—
237C2×Dic73—21no—
238C2×Dic73—21no—
239C2×Dic73—21no—
240C2×Dic73—22no—
241C2×Dic73—22no—
242C2×Dic73—21no—
243C2×Dic73—21no—
244C2×Dic73—21no—
245C2×Dic73—21no—
246C2×Dic73—21no—
247C2×Dic73—21no—
248C2×Dic73—41no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251C2×Dic73—41no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—84no—
259C2×Dic73C4×D412yes—
260C2×Dic73C4×D412yes—
261C8×C73—22no—
262C8×C73#11714yes—
263#10596831—22no—
264#10596831—22no—
265C22×C73—42no—
266C22×C73—42no—
267C22×C73—42no—
268C22×C73—81no—
269C22×C73—81no—
270C22×C73—81no—
271C22×C73—81no—
272C22×C73C2×C4×D411yescharacteristic
273C22×C73#53211yescharacteristic
274C22×C73#53211yescharacteristic
275C22×C73#57711yescharacteristic
276C22×C73#57711yescharacteristic
277C22×C73#57711yescharacteristic
278C22×C73#57711yescharacteristic
279C4×C73—41no—
280C4×C73—41no—
281C4×C73—42no—
282C4×C73—81no—
283C4×C73—81no—
284C4×C73C2×#11711yescharacteristic
285C4×C73C4×#4111yescharacteristic
286C4×C73#52911yescharacteristic
287C4×C73#52911yescharacteristic
288Dic73—22no—
289Dic73—22no—
290Dic73—41no—
291Dic73—41no—
292Dic73—42no—
293Dic73—42no—
294Dic73—81no—
295Dic73—81no—
296#21280—731nomaximal, Sylow
297C2×C73—82no—
298C2×C73#229311yescharacteristic
299C2×C73#294911yescharacteristic
300C2×C73#295211yescharacteristic
301C2×C73#317311yescharacteristic
302C2×C73#317311yescharacteristic
303C2×C73#317411yescharacteristic
304C2×C73#317411yescharacteristic
305C2×C4×SD16—731no—
306#2320—731no—
307#2319—731no—
308#1968—731no—
309#1968D7311yescharacteristic
310#1941—731no—
311#1941—731no—
312#1936—731no—
313#1936—731no—
314#1935—731no—
315#1935—731no—
316C4×#106—731no—
317#1763—731no—
318#1762—731no—
319#1726—731no—
320C73#2128011yescharacteristic, complemented, Sylow
321C22×SD16C2×D7311yescharacteristic
322C2×C4×Q8—731no—
323C2×C4×D4—731no—
324C4×SD16—738no—
325C2×#106—731no—
326C2×#106—731no—
327C2×#106C2×D7311yescharacteristic
328C2×#103—731no—
329C2×#103—731no—
330C2×#103C2×D7311yescharacteristic
331C2×#102—731no—
332C2×#102—731no—
333C2×#102C2×D7311yescharacteristic
334C2×#97—731no—
335C2×#97—731no—
336C2×#97—731no—
337C2×C4×C8—731no—
338C2×C4×C8—731no—
339C2×C4×C8C2×D7311yescharacteristic
340#391—731no—
341#391C2×D7311yescharacteristic
342#390—731no—
343#390C2×D7311yescharacteristic
344#389—731no—
345#389—731no—
346#389—731no—
347#389—731no—
348#388—1461no—
349#388—1461no—
350#387—1461no—
351#387—1461no—
352#383—1461no—
353#382—1461no—
354#381—1461no—
355#380—1461no—
356C4×C4⋊C4—731no—
357C4×C4⋊C4—731no—
358#376—731no—
359#340—731no—
360#340—731no—
361#340—731no—
362#340—731no—
363#340—731no—
364#340—731no—
365C22×Q8C22×D7311yescharacteristic
366C22×D4C22×D7311yescharacteristic
367C23×C4—1461no—
368C2×SD16—22no—
369C2×SD16—22no—
370C2×SD16C4×D7314yes—
371C22×C8—731no—
372C22×C8—731no—
373C22×C8C22×D7311yescharacteristic
374C22×C8C22×D7311yescharacteristic
375C4×Q8—734no—
376C4×Q8—1462no—
377C4×D4—734no—
378C4×D4—1462no—
379C2×C4⋊C4—731no—
380C2×C4⋊C4—731no—
381C2×C4⋊C4—731no—
382C2×C4⋊C4—731no—
383C2×C4⋊C4—1461no—
384C2×C4⋊C4—1461no—
385C2×C4⋊C4—1461no—
386C2×C4⋊C4—1461no—
387C2×C4⋊C4—1461no—
388C2×C4⋊C4C22×D7311yescharacteristic
389C2×C4⋊C4C22×D7311yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—1461no—
393C2×C22⋊C4—1461no—
394C2×C22⋊C4—1461no—
395C2×C42—731no—
396C2×C42—731no—
397C2×C42—1461no—
398C2×C42—1461no—
399C2×C42—1461no—
400C2×C42C22×D7311yescharacteristic
401#106—22no—
402#106—734no—
403#106—1462no—
404#103—22no—
405#103—734no—
406#103—1462no—
407#102—22no—
408#102—734no—
409#102—1462no—
410#97—1462no—
411#97—1462no—
412#97—1462no—
413C4×C8—22no—
414C4×C8—734no—
415C4×C8—1462no—
416#95—21no—
417#95—21no—
418#95—731no—
419#95—731no—
420#95—731no—
421#95—731no—
422#95—1461no—
423#95—1461no—
424#95—1461no—
425#95—1461no—
426#95—1461no—
427#95—1461no—
428#95—1461no—
429#95—1461no—
430C24—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8C2×C4×D7312yes—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4C2×C4×D7312yes—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—5841no—
447C22×C4—5841no—
448C22×C4—731no—
449C22×C4—731no—
450C22×C4—731no—
451C22×C4—731no—
452C22×C4—1461no—
453C22×C4—1461no—
454C22×C4—1461no—
455C22×C4—1461no—
456C22×C4—1464no—
457C22×C4D4×D7311yescharacteristic
458C22×C4D4×D7311yescharacteristic
459C22×C4C23×D7311yescharacteristic
460SD16—28no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—732no—
464C2×C8—732no—
465C2×C8—1461no—
466C2×C8—1461no—
467C2×C8—1461no—
468C2×C8—1461no—
469C2×C8C2×C4×D7312yes—
470C2×C8#1192402712yes—
471C2×C8#1192402712yes—
472C2×C8#1192402712yes—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—5841no—
478C4⋊C4—5841no—
479C4⋊C4—5841no—
480C4⋊C4—5841no—
481C4⋊C4—732no—
482C4⋊C4—732no—
483C4⋊C4—1464no—
484C4⋊C4—1461no—
485C4⋊C4—1461no—
486C4⋊C4—1461no—
487C4⋊C4—1461no—
488C4⋊C4—1461no—
489C4⋊C4—1461no—
490C22⋊C4—81no—
491C22⋊C4—81no—
492C22⋊C4—5841no—
493C22⋊C4—5841no—
494C22⋊C4—1464no—
495C42—21no—
496C42—21no—
497C42—5841no—
498C42—5841no—
499C42—732no—
500C42—732no—
501C42—1464no—
502C42—1461no—
503C42—1461no—
504C23—22no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23C2×D4×D7311yescharacteristic
512Q8—22no—
513Q8—22no—
514Q8—41no—
515Q8—41no—
516D4—22no—
517D4—22no—
518D4—41no—
519D4—41no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—2921no—
536C2×C4—2921no—
537C2×C4—2921no—
538C2×C4—2921no—
539C2×C4—2921no—
540C2×C4—2921no—
541C2×C4—2921no—
542C2×C4—2921no—
543C2×C4—2921no—
544C2×C4—2921no—
545C2×C4—5844no—
546C2×C4—732no—
547C2×C4—732no—
548C2×C4—1461no—
549C2×C4—1461no—
550C2×C4—1461no—
551C2×C4—1461no—
552C2×C4—1461no—
553C2×C4—1461no—
554C2×C4—1462no—
555C2×C4—1462no—
556C2×C4—1461no—
557C2×C4—1461no—
558C2×C4—1461no—
559C2×C4—1461no—
560C2×C4C22×C4×D7311yescharacteristic
561C2×C4#41×D7311yescharacteristic
562C2×C4#41×D7311yescharacteristic
563C2×C4C2×#1192402711yescharacteristic
564C2×C4C2×#1192402711yescharacteristic
565C2×C4C2×#1192402711yescharacteristic
566C8—22no—
567C8—1462no—
568C8—1462no—
569C8#1455654714yes—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22C4×D4×D7311yescharacteristic
578C22#1798913211yescharacteristic
579C22#1798913511yescharacteristic
580C22#1798939311yescharacteristic
581C22#1798939311yescharacteristic
582C22#1798939411yescharacteristic
583C22#1798939411yescharacteristic
584C4—41no—
585C4—41no—
586C4—42no—
587C4—81no—
588C4—81no—
589C4—2921no—
590C4—2921no—
591C4—2922no—
592C4—2922no—
593C4—5841no—
594C4—5841no—
595C4—1462no—
596C4—1462no—
597C4C2×#1455654711yescharacteristic
598C4C4×#1192402711yescharacteristic
599C4#1798911611yescharacteristic
600C4#1798911811yescharacteristic
601C2—82no—
602C2#2261239311yescharacteristic
603C2#2261878511yescharacteristic
604C2#2261880911yescharacteristic
605C2#2262054111yescharacteristic
606C2#2262054511yescharacteristic
607C2#2262055211yescharacteristic
608C2#2262055611yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1968C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
cosocleC22×C4×C73C24
hypercenter#1968D73
hypocenterC73#21280
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1968

Complements

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

NH ≅ G/Ntype
2. #22612576C2semidirect
5. #22610659C2semidirect
6. #22610657C2semidirect
9. #22610624C2semidirect
10. #22610618C2semidirect
11. #106×Dic73C2semidirect
12. #22609763C2semidirect
14. #22609617C2semidirect
18. SD16×Dic73C4semidirect
320. C73#21280semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×Dic732
#226125762
#226125682
#226107872
#226106592
#226106572
#226106312
#226106252
#226106242
#226106182
#106×Dic732
#226097632
#226097582
#226096172
C73×#19682
#2128073