Finite Group Databasebeta

← #33094106

Subgroups — #33094106

Order 18688.

826 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988946C211yesmaximal, characteristic
2#22612677C211yesmaximal, characteristic, complemented
3#22612644C211yesmaximal, characteristic
4#22610892C211yesmaximal, characteristic, complemented
5#22610625C211yesmaximal, characteristic
6#22610618C211yesmaximal, characteristic, complemented
7#22610611C211yesmaximal, characteristic
8#22610604C211yesmaximal, characteristic, complemented
9#22610064C211yesmaximal, characteristic, complemented
10#22609763C211yesmaximal, characteristic, complemented
11#22609758C211yesmaximal, characteristic
12#103×Dic73C211yesmaximal, characteristic, complemented
13#102×Dic73C211yesmaximal, characteristic
14#22609617C211yesmaximal, characteristic, complemented
15C73×#1958C211yesmaximal, characteristic
16C2×#14556591C2211yescharacteristic
17C2×Q8×Dic73C2211yescharacteristic
18C2×D4×Dic73C2211yescharacteristic
19#17988946C418yescomplemented
20C2×#14556515C2211yescharacteristic
21C2×#14556515C2211yescharacteristic
22C2×#14556512C2211yescharacteristic
23C2×#14556512C2211yescharacteristic
24C2×#14556502C2211yescharacteristic
25C2×#14556501C2211yescharacteristic
26C2×#14556501C2211yescharacteristic
27C2×C8×Dic73C2211yescharacteristic
28C2×C8×Dic73C2211yescharacteristic
29#17988494C2211yescharacteristic
30#17988491—21no—
31#17988491—21no—
32#17988490C2211yescharacteristic
33#17988483C2211yescharacteristic
34#17988483C2211yescharacteristic
35#17988480—21no—
36#17988480—21no—
37#17988472—21no—
38#17988472—21no—
39#17988462—21no—
40#17988462—21no—
41#17988457C2211yescharacteristic
42C4⋊C4×Dic73C2211yescharacteristic
43C4⋊C4×Dic73C2211yescharacteristic
44#17988433C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988349C2211yescharacteristic
52#17988349C2211yescharacteristic
53C2×#124×C73C2211yescharacteristic
54C2×#103×C73C2211yescharacteristic
55C2×#103×C73C2211yescharacteristic
56C2×#102×C73C2211yescharacteristic
57C2×#102×C73C2211yescharacteristic
58C2×C4×C8×C73C2211yescharacteristic
59#382×C73C2211yescharacteristic
60C22×Dic146—21no—
61C23×Dic73—21no—
62C22×#10596831C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64#14556591—22no—
65#14556591C2×C414yescomplemented
66Q8×Dic73—22no—
67Q8×Dic73—22no—
68D4×Dic73—22no—
69D4×Dic73—22no—
70C2×C4⋊Dic73—21no—
71C2×C4⋊Dic73—21no—
72C2×C4⋊Dic73—21no—
73C2×C4⋊Dic73—21no—
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×#11924007—21no—
79C2×#11924007—21no—
80C2×C22⋊Dic73—21no—
81C2×C22⋊Dic73—21no—
82C2×C22⋊Dic73—21no—
83C2×C4×Dic73—21no—
84C2×C4×Dic73—21no—
85C2×C4×Dic73—21no—
86C2×C4×Dic73C2311yescharacteristic
87C2×C4×Dic73C2311yescharacteristic
88#14556515C2×C414yescomplemented
89#14556515C2×C414yescomplemented
90#14556512C2×C414yes—
91#14556512C2×C414yes—
92#14556502C2×C414yescomplemented
93#14556501—22no—
94#14556501—22no—
95C8×Dic73—22no—
96C8×Dic73—22no—
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
109C22×C8×C73C2311yescharacteristic
110C22×C8×C73C2311yescharacteristic
111#124×C73—22no—
112#124×C73C2×C414yes—
113C2×C4⋊C4×C73C2311yescharacteristic
114C2×C4⋊C4×C73C2311yescharacteristic
115C2×C22⋊C4×C73—21no—
116C2×C22⋊C4×C73—21no—
117C2×C42×C73C2311yescharacteristic
118#103×C73—22no—
119#103×C73—22no—
120#102×C73—22no—
121#102×C73—22no—
122C4×C8×C73—22no—
123#95×C73—21no—
124#95×C73—21no—
125C2×Dic146—24no—
126C2×Dic146—81no—
127C2×Dic146—81no—
128C22×Dic73—21no—
129C22×Dic73—21no—
130C22×Dic73—21no—
131C22×Dic73—21no—
132C22×Dic73—42no—
133C22×Dic73—81no—
134C22×Dic73—81no—
135C22×Dic73C2×D411yescharacteristic
136C22×Dic73C2×D411yescharacteristic
137C22×Dic73C2×D411yescharacteristic
138C22×Dic73C2×D411yescharacteristic
139C2×#10596831—21no—
140C2×#10596831—21no—
141C2×#10596831—21no—
142C2×#10596831—21no—
143C2×#10596831C22×C412yes—
144C2×#10596831C22×C412yes—
145C4⋊Dic73—24no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—42no—
153C4⋊Dic73—81no—
154C4⋊Dic73—81no—
155C4⋊Dic73C22×C412yes—
156C4⋊Dic73C22×C412yes—
157#11924007—81no—
158#11924007—81no—
159C22⋊Dic73—42no—
160C22⋊Dic73—81no—
161C22⋊Dic73—81no—
162C4×Dic73—21no—
163C4×Dic73—21no—
164C4×Dic73—42no—
165C4×Dic73—81no—
166C4×Dic73—81no—
167C4×Dic73#4112yes—
168C4×Dic73#4112yes—
169C24×C73—21no—
170C2×Q8×C73—21no—
171C2×Q8×C73—21no—
172C2×Q8×C73—21no—
173C2×Q8×C73—21no—
174C2×Q8×C73—21no—
175C2×Q8×C73C22×C412yes—
176C2×D4×C73—21no—
177C2×D4×C73—21no—
178C2×D4×C73—21no—
179C2×D4×C73—21no—
180C2×D4×C73—21no—
181C2×D4×C73C22×C412yes—
182C22×C4×C73—21no—
183C22×C4×C73—21no—
184C22×C4×C73—21no—
185C22×C4×C73C2×D411yescharacteristic
186C22×C4×C73C2×D411yescharacteristic
187C22×C4×C73C2411yescharacteristic
188C2×C8×C73—21no—
189C2×C8×C73—21no—
190C2×C8×C73—21no—
191C2×C8×C73—21no—
192C2×C8×C73#4112yes—
193C2×C8×C73#4112yes—
194C4⋊C4×C73—21no—
195C4⋊C4×C73—21no—
196C4⋊C4×C73—21no—
197C4⋊C4×C73—21no—
198C22⋊C4×C73—24no—
199C22⋊C4×C73—24no—
200C42×C73C22×C412yes—
201C42×C73#4112yes—
202C23×C73—22no—
203C23×C73—41no—
204C23×C73—41no—
205C23×C73—41no—
206C23×C73—41no—
207C23×C73—41no—
208C23×C73—41no—
209C23×C73C22×D411yescharacteristic
210Q8×C73—22no—
211Q8×C73—22no—
212Q8×C73—41no—
213Q8×C73—41no—
214D4×C73—22no—
215D4×C73—22no—
216D4×C73—41no—
217D4×C73—41no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—22no—
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×C73—41no—
231C2×C4×C73C4×D412yes—
232C2×C4×C73C4×D412yes—
233C2×C4×C73C23×C411yescharacteristic
234C2×C4×C73C2×#4111yescharacteristic
235C2×C4×C73C2×#4111yescharacteristic
236C2×C4×C73C2×#4111yescharacteristic
237C2×C4×C73C2×#4111yescharacteristic
238C2×C4×C73C2×#4111yescharacteristic
239C2×Dic73—21no—
240C2×Dic73—21no—
241C2×Dic73—21no—
242C2×Dic73—21no—
243C2×Dic73—22no—
244C2×Dic73—22no—
245C2×Dic73—21no—
246C2×Dic73—21no—
247C2×Dic73—21no—
248C2×Dic73—21no—
249C2×Dic73—21no—
250C2×Dic73—21no—
251C2×Dic73—41no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73—84no—
262C2×Dic73#12412yes—
263C2×Dic73#12412yes—
264Dic146—84no—
265C8×C73—22no—
266C8×C73—22no—
267#10596831—22no—
268#10596831—22no—
269C22×C73—42no—
270C22×C73—42no—
271C22×C73—42no—
272C22×C73—81no—
273C22×C73—81no—
274C22×C73—81no—
275C22×C73—81no—
276C22×C73C2×C4×D411yescharacteristic
277C22×C73C2×#12411yescharacteristic
278C22×C73#53211yescharacteristic
279C22×C73#57611yescharacteristic
280C22×C73#57711yescharacteristic
281C22×C73#57711yescharacteristic
282C22×C73#57811yescharacteristic
283C4×C73—22no—
284C4×C73—22no—
285C4×C73—42no—
286C4×C73—81no—
287C4×C73—81no—
288C4×C73C4×#4111yescharacteristic
289C4×C73C4×#4111yescharacteristic
290C4×C73#52911yescharacteristic
291C4×C73#52911yescharacteristic
292Dic73—22no—
293Dic73—22no—
294Dic73—41no—
295Dic73—41no—
296Dic73—42no—
297Dic73—42no—
298Dic73—81no—
299Dic73—81no—
300#21654—731nomaximal, Sylow
301C2×C73—82no—
302C2×C73C4×#12411yescharacteristic
303C2×C73#294611yescharacteristic
304C2×C73#294711yescharacteristic
305C2×C73#313611yescharacteristic
306C2×C73#313711yescharacteristic
307C2×C73#317311yescharacteristic
308C2×C73#317511yescharacteristic
309C2×#489—731no—
310#2331—731no—
311#2326—731no—
312#1990—731no—
313#1958D7311yescharacteristic
314#1936—731no—
315#1935—731no—
316#1934—731no—
317#1933—731no—
318#1836—731no—
319#1763—731no—
320#1762—731no—
321C4×#103—731no—
322C4×#102—731no—
323#1726—731no—
324C73#2165411yescharacteristic, complemented, Sylow
325C2×C4⋊Q8—731no—
326C2×#124C2×D7311yescharacteristic
327C2×C4×Q8—731no—
328C2×C4×D4—731no—
329#489—738no—
330C2×#103—731no—
331C2×#103—731no—
332C2×#103C2×D7311yescharacteristic
333C2×#103C2×D7311yescharacteristic
334C2×#102—731no—
335C2×#102—731no—
336C2×#102C2×D7311yescharacteristic
337C2×#102C2×D7311yescharacteristic
338C2×#97—731no—
339C2×#97—731no—
340C2×#97—731no—
341C2×C4×C8—731no—
342C2×C4×C8—731no—
343C2×C4×C8C2×D7311yescharacteristic
344#391—731no—
345#391—1461no—
346#391—1461no—
347#390—731no—
348#389—731no—
349#389—731no—
350#388—1461no—
351#388—1461no—
352#387—1461no—
353#387—1461no—
354#385—1461no—
355#385—1461no—
356#384—731no—
357#382C2×D7311yescharacteristic
358C4×C4⋊C4—731no—
359C4×C4⋊C4—731no—
360#376—731no—
361#340—731no—
362#340—731no—
363#340—731no—
364#340—731no—
365#340—731no—
366#340—731no—
367#340—731no—
368#340—731no—
369C22×Q8—1461no—
370C22×Q8C22×D7311yescharacteristic
371C22×D4C22×D7311yescharacteristic
372C23×C4—1461no—
373C22×C8—731no—
374C22×C8—731no—
375C22×C8C22×D7311yescharacteristic
376C22×C8C22×D7311yescharacteristic
377C4⋊Q8—734no—
378C4⋊Q8—1462no—
379#124—22no—
380#124C4×D7314yes—
381C4×Q8—1462no—
382C4×Q8—1462no—
383C4×D4—1462no—
384C4×D4—1462no—
385C2×C4⋊C4—731no—
386C2×C4⋊C4—731no—
387C2×C4⋊C4—731no—
388C2×C4⋊C4—731no—
389C2×C4⋊C4—1461no—
390C2×C4⋊C4—1461no—
391C2×C4⋊C4—1461no—
392C2×C4⋊C4—1461no—
393C2×C4⋊C4—1461no—
394C2×C4⋊C4—1461no—
395C2×C4⋊C4C22×D7311yescharacteristic
396C2×C4⋊C4C22×D7311yescharacteristic
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—1461no—
400C2×C22⋊C4—1461no—
401C2×C22⋊C4—1461no—
402C2×C42—731no—
403C2×C42—731no—
404C2×C42—1461no—
405C2×C42—1461no—
406C2×C42—1461no—
407C2×C42C22×D7311yescharacteristic
408#103—22no—
409#103—22no—
410#103—734no—
411#103—734no—
412#102—22no—
413#102—22no—
414#102—734no—
415#102—734no—
416#97—734no—
417#97—1462no—
418#97—1462no—
419C4×C8—22no—
420C4×C8—1462no—
421C4×C8—1462no—
422#95—21no—
423#95—21no—
424#95—731no—
425#95—731no—
426#95—731no—
427#95—731no—
428#95—1461no—
429#95—1461no—
430#95—1461no—
431#95—1461no—
432#95—1461no—
433#95—1461no—
434C24—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—5841no—
441C2×Q8—5841no—
442C2×Q8—1464no—
443C2×Q8C2×C4×D7312yes—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4C2×C4×D7312yes—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—2922no—
454C22×C4—5841no—
455C22×C4—5841no—
456C22×C4—731no—
457C22×C4—731no—
458C22×C4—731no—
459C22×C4—731no—
460C22×C4—1461no—
461C22×C4—1461no—
462C22×C4—1461no—
463C22×C4—1461no—
464C22×C4D4×D7311yescharacteristic
465C22×C4D4×D7311yescharacteristic
466C22×C4C23×D7311yescharacteristic
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—732no—
472C2×C8—732no—
473C2×C8—1461no—
474C2×C8—1461no—
475C2×C8—1461no—
476C2×C8—1461no—
477C2×C8#1192402712yes—
478C2×C8#1192402712yes—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—2922no—
484C4⋊C4—5841no—
485C4⋊C4—5841no—
486C4⋊C4—5841no—
487C4⋊C4—5841no—
488C4⋊C4—732no—
489C4⋊C4—732no—
490C4⋊C4—1461no—
491C4⋊C4—1461no—
492C4⋊C4—1461no—
493C4⋊C4—1461no—
494C4⋊C4—1464no—
495C4⋊C4—1461no—
496C4⋊C4—1461no—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—2922no—
500C22⋊C4—5841no—
501C22⋊C4—5841no—
502C42—2922no—
503C42—5841no—
504C42—5841no—
505C42—732no—
506C42—732no—
507C42—1461no—
508C42—1461no—
509C42C2×C4×D7312yes—
510C42#1192402712yes—
511C23—22no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23C2×D4×D7311yescharacteristic
519Q8—22no—
520Q8—22no—
521Q8—41no—
522Q8—41no—
523Q8—5844no—
524D4—22no—
525D4—22no—
526D4—41no—
527D4—41no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—2921no—
542C2×C4—2921no—
543C2×C4—2921no—
544C2×C4—2921no—
545C2×C4—2921no—
546C2×C4—2921no—
547C2×C4—2921no—
548C2×C4—2921no—
549C2×C4—2921no—
550C2×C4—2921no—
551C2×C4—5844no—
552C2×C4—732no—
553C2×C4—732no—
554C2×C4—1461no—
555C2×C4—1461no—
556C2×C4—1461no—
557C2×C4—1461no—
558C2×C4—1461no—
559C2×C4—1461no—
560C2×C4—1462no—
561C2×C4—1462no—
562C2×C4—1461no—
563C2×C4—1461no—
564C2×C4—1461no—
565C2×C4—1461no—
566C2×C4#1455655112yes—
567C2×C4#1455655112yes—
568C2×C4C22×C4×D7311yescharacteristic
569C2×C4#41×D7311yescharacteristic
570C2×C4#41×D7311yescharacteristic
571C2×C4C2×#1192402711yescharacteristic
572C2×C4C2×#1192402711yescharacteristic
573C2×C4C2×#1192402711yescharacteristic
574C8—22no—
575C8—22no—
576C8—1462no—
577C8—1462no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22C2×#1455655111yescharacteristic
586C22#124×D7311yescharacteristic
587C22#1798913511yescharacteristic
588C22#1798938911yescharacteristic
589C22#1798939411yescharacteristic
590C22#1798939411yescharacteristic
591C22#1798940111yescharacteristic
592C4—22no—
593C4—22no—
594C4—42no—
595C4—81no—
596C4—81no—
597C4—2921no—
598C4—2922no—
599C4—2922no—
600C4—2921no—
601C4—5841no—
602C4—5841no—
603C4—1462no—
604C4—1462no—
605C4C4×#1192402711yescharacteristic
606C4C4×#1192402711yescharacteristic
607C4#1798911811yescharacteristic
608C4#1798911811yescharacteristic
609C2—82no—
610C2#2261247811yescharacteristic
611C2#2261876611yescharacteristic
612C2#2261877711yescharacteristic
613C2#2262028411yescharacteristic
614C2#2262029511yescharacteristic
615C2#2262054511yescharacteristic
616C2#2262056711yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1958C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
cosocleC22×C4×C73C24
hypercenter#1958D73
hypocenterC73#21654
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1958

Complements

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

NH ≅ G/Ntype
2. #22612677C2semidirect
4. #22610892C2semidirect
6. #22610618C2semidirect
8. #22610604C2semidirect
9. #22610064C2semidirect
10. #22609763C2semidirect
12. #103×Dic73C2semidirect
14. #22609617C2semidirect
19. #17988946C4semidirect
65. #14556591C2×C4semidirect
88. #14556515C2×C4semidirect
89. #14556515C2×C4semidirect
92. #14556502C2×C4semidirect
324. C73#21654semidirect

Maximal subgroups

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

subgroupindex
C2×#179889462
#226126772
#226126442
#226108922
#226106252
#226106182
#226106112
#226106042
#226100642
#226097632
#226097582
#103×Dic732
#102×Dic732
#226096172
C73×#19582
#2165473