Finite Group Databasebeta

← #33057136

Subgroups — #33057136

Order 18688.

650 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619974C211yesmaximal, characteristic
2#22618965C211yesmaximal, characteristic, complemented
3#22608821C211yesmaximal, characteristic, complemented
4#22608814C211yesmaximal, characteristic, complemented
5#22608809C211yesmaximal, characteristic, complemented
6#22608795C211yesmaximal, characteristic, complemented
7#22608793C211yesmaximal, characteristic, complemented
8#22608786C211yesmaximal, characteristic, complemented
9C8×#11924015C211yesmaximal, characteristic, complemented
10C8×#11924013C211yesmaximal, characteristic, complemented
11#22608714C211yesmaximal, characteristic, complemented
12#22608391C211yesmaximal, characteristic, complemented
13C2×#17988314C211yesmaximal, characteristic, complemented
14#22608215C211yesmaximal, characteristic, complemented
15C73×#2994C211yesmaximal, characteristic, complemented
16C2×C4×D292C2211yescharacteristic
17#17988920C411yescharacteristic
18#17988903C411yescharacteristic
19#17988884C411yescharacteristic
20#17988884C411yescharacteristic
21#17988799C411yescharacteristic
22#17988799C411yescharacteristic
23#17988737C411yescharacteristic
24#17988720C411yescharacteristic
25C4×#11924015C2211yescharacteristic
26C4×#11924015C2211yescharacteristic
27C4×#11924013C2211yescharacteristic
28C4×#11924013C2211yescharacteristic
29#17988666C2211yescharacteristic
30#17988666C2211yescharacteristic
31#17988657—21no—
32#17988657—21no—
33C8×D292C2211yescharacteristic
34C8×D292C2211yescharacteristic
35#17988644C2211yescharacteristic
36#17988640C2211yescharacteristic
37#17988549—21no—
38C22⋊C8×D73—21no—
39C2×C4×#10596831C2211yescharacteristic
40#17988339C2211yescharacteristic, complemented
41#17988339C2211yescharacteristic, complemented
42#17988337C2211yescharacteristic, complemented
43#17988337C2211yescharacteristic, complemented
44Q8⋊#10596831C2211yescharacteristic, complemented
45Q8⋊#10596831C2211yescharacteristic, complemented
46D4⋊#10596831C2211yescharacteristic, complemented
47D4⋊#10596831C2211yescharacteristic, complemented
48#17988314C2211yescharacteristic
49#17988314C2211yescharacteristic
50#17988314C2211yescharacteristic
51#17988314C2211yescharacteristic
52#17988309C2211yescharacteristic, complemented
53#17988309C2211yescharacteristic, complemented
54C8×#10596831C2211yescharacteristic, complemented
55C8×#10596831C2211yescharacteristic, complemented
56#529×C73C2211yescharacteristic
57#446×C73C2211yescharacteristic, complemented
58C8×Q8×C73C2211yescharacteristic, complemented
59#436×C73C2211yescharacteristic, complemented
60#435×C73—21no—
61C8×D4×C73—21no—
62C8×D4×C73C2211yescharacteristic, complemented
63#433×C73—21no—
64#431×C73C2211yescharacteristic
65#424×C73—21no—
66C2×C4⋊C8×C73C2211yescharacteristic
67C22×D292C2×C411yescharacteristic
68C22×C4×D73—21no—
69C2×#11924015C2×C411yescharacteristic
70C2×#11924015C2×C411yescharacteristic
71C2×#11924013C2×C411yescharacteristic
72C2×#11924013C2×C411yescharacteristic
73C2×#11924011—21no—
74C2×#11924011—21no—
75C22×#10596831—21no—
76C2×C8×D73—21no—
77C2×C8×D73—21no—
78C4×D292—21no—
79C4×D292—21no—
80C4×D292C2×C411yescharacteristic
81C4×D292C2×C411yescharacteristic
82C4×D292C2311yescharacteristic
83C4×D292C2311yescharacteristic
84C2×C4⋊Dic73C2×C411yescharacteristic
85C2×#11924005—21no—
86#14556524C2×C411yescharacteristic
87#14556524C2×C411yescharacteristic
88#14556522C2×C411yescharacteristic
89#14556522C2×C411yescharacteristic
90C4⋊#10596831C2×C411yescharacteristic
91C4⋊#10596831C2×C411yescharacteristic
92C4⋊#10596831C2311yescharacteristic
93C4⋊#10596831C2311yescharacteristic
94#14556515C2×C411yescharacteristic
95#14556515C2×C411yescharacteristic
96#14556512C2×C411yescharacteristic
97#14556512C2×C411yescharacteristic
98#14556511C2×C411yescharacteristic
99#14556511C2×C411yescharacteristic
100#14556511C2×C411yescharacteristic
101#14556511C2×C411yescharacteristic
102C22⋊#10596831—21no—
103C22⋊#10596831C2×C411yescharacteristic
104C22⋊#10596831C2×C411yescharacteristic
105#14556503—21no—
106#14556503—21no—
107#14556503—21no—
108#14556503—21no—
109#14556502—21no—
110C4×#10596831—21no—
111C4×#10596831C2311yescharacteristic
112C4×#10596831C2311yescharacteristic
113C4×#10596831C2311yescharacteristic
114C4×#10596831C2311yescharacteristic
115C2×M16×C73—21no—
116C22×C8×C73—21no—
117C22×C8×C73—21no—
118#126×C73—21no—
119#125×C73C2×C411yescharacteristic
120#124×C73C2×C411yescharacteristic
121#123×C73—21no—
122C22⋊Q8×C73C2×C411yescharacteristic
123#121×C73C2×C411yescharacteristic
124C4×Q8×C73C2311yescharacteristic
125C4×D4×C73—21no—
126C4×D4×C73C2311yescharacteristic
127#117×C73—21no—
128C2×C42×C73C2311yescharacteristic
129C4⋊C8×C73—21no—
130C4⋊C8×C73—21no—
131C4⋊C8×C73—21no—
132C4⋊C8×C73D411yescharacteristic
133C4⋊C8×C73D411yescharacteristic
134C4⋊C8×C73C2311yescharacteristic
135C4⋊C8×C73C2311yescharacteristic
136C22⋊C8×C73—21no—
137C22⋊C8×C73—21no—
138C22⋊C8×C73D411yescharacteristic
139C22⋊C8×C73D411yescharacteristic
140#97×C73—21no—
141C4×C8×C73—21no—
142C4×C8×C73C2311yescharacteristic
143C4×C8×C73C2311yescharacteristic
144C23×D73—21no—
145C2×D292—22no—
146C2×D292—21no—
147C2×D292—41no—
148C2×D292C22×C411yescharacteristic
149C2×D292C22×C411yescharacteristic
150C22×Dic73—21no—
151C2×C4×D73—21no—
152C2×C4×D73—21no—
153C2×C4×D73—21no—
154C2×C4×D73—21no—
155C2×C4×D73—81no—
156C2×C4×D73—81no—
157#11924015—22no—
158#11924015—41no—
159#11924013—22no—
160#11924013—41no—
161#11924011—81no—
162#11924011—81no—
163C2×#10596831—24no—
164C2×#10596831—21no—
165C2×#10596831—21no—
166C2×#10596831—21no—
167C2×#10596831—21no—
168C2×#10596831C22×C411yescharacteristic
169C2×#10596831C22×C411yescharacteristic
170C2×#10596831C22×C411yescharacteristic
171C2×#10596831C22×C411yescharacteristic
172C8×D73—81no—
173C8×D73—81no—
174C4⋊Dic73—21no—
175C4⋊Dic73C22×C411yescharacteristic
176C4⋊Dic73C22×C411yescharacteristic
177#11924005—21no—
178#11924005—21no—
179#11924005—21no—
180#11924005—21no—
181C2×Q8×C73C22×C411yescharacteristic
182C2×D4×C73—21no—
183C2×D4×C73C22×C411yescharacteristic
184C22×C4×C73—21no—
185C22×C4×C73C22×C411yescharacteristic
186C22×C4×C73C2×D411yescharacteristic
187C22×C4×C73#4111yescharacteristic
188M16×C73—81no—
189C2×C8×C73—21no—
190C2×C8×C73—21no—
191C2×C8×C73—81no—
192C2×C8×C73—81no—
193C2×C8×C73C2×D411yescharacteristic
194C2×C8×C73C2×D411yescharacteristic
195C2×C8×C73C2×D411yescharacteristic
196C2×C8×C73C2×D411yescharacteristic
197C4⋊C4×C73—21no—
198C4⋊C4×C73—21no—
199C4⋊C4×C73—21no—
200C4⋊C4×C73—21no—
201C4⋊C4×C73C22×C411yescharacteristic
202C4⋊C4×C73C22×C411yescharacteristic
203C22⋊C4×C73—21no—
204C22⋊C4×C73—21no—
205C22⋊C4×C73—21no—
206C22⋊C4×C73—21no—
207C22⋊C4×C73—21no—
208C42×C73—21no—
209C42×C73C22×C411yescharacteristic
210C42×C73C2×D411yescharacteristic
211C42×C73#4111yescharacteristic
212C42×C73C2411yescharacteristic
213C23×C73—21no—
214C23×C73C4×D411yescharacteristic
215C22×D73—21no—
216C22×D73—21no—
217C22×D73—42no—
218C22×D73—42no—
219C22×D73—81no—
220Q8×C73—21no—
221Q8×C73—21no—
222D4×C73—21no—
223D4×C73—21no—
224D4×C73—42no—
225C2×C4×C73—22no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73—41no—
234C2×C4×C73—41no—
235C2×C4×C73C4×D411yescharacteristic
236C2×C4×C73C4×D411yescharacteristic
237C2×C4×C73C4×D411yescharacteristic
238C2×C4×C73C23×C411yescharacteristic
239C2×C4×C73C22×D411yescharacteristic
240C2×C4×C73C2×#4111yescharacteristic
241C2×Dic73—21no—
242C2×Dic73—21no—
243C2×Dic73—81no—
244D292—21no—
245D292—21no—
246D292—21no—
247D292—42no—
248D292#13112yes—
249C4×D73—41no—
250C4×D73—41no—
251C4×D73—41no—
252C4×D73—41no—
253C4×D73—81no—
254C4×D73—81no—
255C8×C73—21no—
256C8×C73—21no—
257C8×C73—21no—
258C8×C73—21no—
259C8×C73—41no—
260C8×C73—41no—
261#10596831—22no—
262#10596831—22no—
263#10596831—21no—
264#10596831—21no—
265#10596831—21no—
266#10596831—21no—
267#10596831—41no—
268#10596831—41no—
269C22×C73—21no—
270C22×C73—21no—
271C22×C73—41no—
272C22×C73—41no—
273C22×C73—41no—
274C22×C73#44312yes—
275C22×C73C2×C4×D411yescharacteristic
276C2×D73—22no—
277C2×D73—41no—
278C2×D73—41no—
279C2×D73—41no—
280C2×D73—41no—
281C2×D73—41no—
282C2×D73—84no—
283C4×C73—21no—
284C4×C73—21no—
285C4×C73—21no—
286C4×C73—21no—
287C4×C73—41no—
288C4×C73—41no—
289C4×C73—41no—
290C4×C73C2×#13111yescharacteristic
291C4×C73#56811yescharacteristic
292C4×C73#57611yescharacteristic
293C4×C73#57711yescharacteristic
294Dic73—41no—
295Dic73—41no—
296#18014—731nomaximal, Sylow
297C2×C73—22no—
298C2×C73—81no—
299C2×C73#293311yescharacteristic
300C2×C73#294611yescharacteristic
301C2×C73C2×#44311yescharacteristic
302D73—42no—
303D73—81no—
304#3105—731no—
305#2994D7311yescharacteristic, complemented
306#2974—731no—
307#1590—731no—
308#1589—731no—
309#1588—731no—
310#1586—731no—
311#1586—731no—
312#1585—731no—
313C8×SD16—731no—
314C8×D8—731no—
315#1573—731no—
316#1494—731no—
317C2×D4⋊C8—731no—
318#1450—731no—
319C73#1801411yescharacteristic, complemented, Sylow
320#529C2×D7311yescharacteristic
321C2×C4×D4—731no—
322#484—731no—
323#481—731no—
324#478—731no—
325#478—731no—
326#463—731no—
327#463—731no—
328#450—731no—
329#447—731no—
330#446C2×D7311yescharacteristic, complemented
331C8×Q8C2×D7311yescharacteristic, complemented
332C4×SD16—731no—
333C4×SD16—731no—
334C4×D8—731no—
335C4×D8—731no—
336#436—731no—
337#436—731no—
338#436C2×D7311yescharacteristic, complemented
339#435—21no—
340#435—1461no—
341#435—1461no—
342C8×D4—21no—
343C8×D4—731no—
344C8×D4—731no—
345C8×D4C2×D7311yescharacteristic, complemented
346#433—21no—
347#432—731no—
348#431—731no—
349#431C2×D7311yescharacteristic
350#424—21no—
351C2×C4⋊C8C2×D7311yescharacteristic
352#407—1461no—
353C2×C22⋊C8—1461no—
354C2×C4×C8—731no—
355#335—731no—
356#335—731no—
357#334—731no—
358#334—731no—
359Q8⋊C8—731no—
360Q8⋊C8—731no—
361D4⋊C8—731no—
362D4⋊C8—731no—
363D4⋊C8—731no—
364D4⋊C8—731no—
365D4⋊C8—731no—
366D4⋊C8—731no—
367#322—731no—
368#322—731no—
369C82—731no—
370C82—731no—
371C22×D4—731no—
372C23×C4—1461no—
373C2×SD16—731no—
374C2×SD16—731no—
375C2×D8—731no—
376C2×D8—731no—
377C2×M16—21no—
378C2×M16—1461no—
379C2×M16—1461no—
380C22×C8—21no—
381C22×C8—21no—
382C22×C8—1461no—
383C22×C8—1461no—
384C22×C8—1461no—
385#126—21no—
386#125C4×D7311yescharacteristic
387#124C4×D7311yescharacteristic
388#123—21no—
389C22⋊Q8C4×D7311yescharacteristic
390#121C4×D7311yescharacteristic
391C4×Q8C22×D7311yescharacteristic
392C4×D4—21no—
393C4×D4—731no—
394C4×D4—731no—
395C4×D4—731no—
396C4×D4—731no—
397C4×D4—1461no—
398C4×D4—1461no—
399C4×D4C22×D7311yescharacteristic
400#117—21no—
401C2×C4⋊C4—731no—
402C2×C22⋊C4—1461no—
403C2×C42C22×D7311yescharacteristic
404#107—731no—
405#107—731no—
406#106—731no—
407#106—731no—
408C4⋊C8—21no—
409C4⋊C8—21no—
410C4⋊C8—21no—
411C4⋊C8—731no—
412C4⋊C8—731no—
413C4⋊C8—731no—
414C4⋊C8—731no—
415C4⋊C8C22⋊D7311yescharacteristic
416C4⋊C8C22⋊D7311yescharacteristic
417C4⋊C8C22×D7311yescharacteristic
418C4⋊C8C22×D7311yescharacteristic
419#103—731no—
420#103—731no—
421#102—731no—
422#102—731no—
423#102—731no—
424#102—731no—
425#102—731no—
426#102—731no—
427C22⋊C8—21no—
428C22⋊C8—21no—
429C22⋊C8—731no—
430C22⋊C8—731no—
431C22⋊C8—1461no—
432C22⋊C8—1461no—
433C22⋊C8—1461no—
434C22⋊C8—1461no—
435C22⋊C8—1461no—
436C22⋊C8C22⋊D7311yescharacteristic
437C22⋊C8C22⋊D7311yescharacteristic
438#97—21no—
439#97—1461no—
440C4×C8—21no—
441C4×C8—731no—
442C4×C8—731no—
443C4×C8—731no—
444C4×C8—731no—
445C4×C8—1461no—
446C4×C8C22×D7311yescharacteristic
447C4×C8C22×D7311yescharacteristic
448C24—1461no—
449C2×Q8C2×C4×D7311yescharacteristic
450C2×D4—21no—
451C2×D4—2921no—
452C2×D4—731no—
453C2×D4—731no—
454C2×D4—1462no—
455C2×D4—1461no—
456C2×D4C2×C4×D7311yescharacteristic
457C22×C4—21no—
458C22×C4—5841no—
459C22×C4—5841no—
460C22×C4—1461no—
461C22×C4—1461no—
462C22×C4—1461no—
463C22×C4—1461no—
464C22×C4—1461no—
465C22×C4C2×C4×D7311yescharacteristic
466C22×C4C2×C22⋊D7311yescharacteristic
467C22×C4#1192402811yescharacteristic
468SD16—2921no—
469SD16—1462no—
470D8—2921no—
471D8—1462no—
472M16—81no—
473M16—5841no—
474M16—5841no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—81no—
478C2×C8—81no—
479C2×C8—5841no—
480C2×C8—5841no—
481C2×C8—731no—
482C2×C8—731no—
483C2×C8—731no—
484C2×C8—731no—
485C2×C8—1461no—
486C2×C8—1461no—
487C2×C8—1464no—
488C2×C8—1461no—
489C2×C8—1461no—
490C2×C8C2×C22⋊D7311yescharacteristic
491C2×C8C2×C22⋊D7311yescharacteristic
492C2×C8C2×C22⋊D7311yescharacteristic
493C2×C8C2×C22⋊D7311yescharacteristic
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—731no—
499C4⋊C4—731no—
500C4⋊C4—1461no—
501C4⋊C4C2×C4×D7311yescharacteristic
502C4⋊C4C2×C4×D7311yescharacteristic
503C22⋊C4—21no—
504C22⋊C4—21no—
505C22⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—1461no—
509C22⋊C4—1461no—
510C22⋊C4—1461no—
511C22⋊C4—1461no—
512C42—21no—
513C42C2×C4×D7311yescharacteristic
514C42C2×C22⋊D7311yescharacteristic
515C42#1192402811yescharacteristic
516C42C23×D7311yescharacteristic
517C23—21no—
518C23—2922no—
519C23—2922no—
520C23—5841no—
521C23—1461no—
522C23—1461no—
523C23C4×C22⋊D7311yescharacteristic
524Q8—21no—
525Q8—21no—
526D4—21no—
527D4—21no—
528D4—42no—
529D4—2922no—
530D4—732no—
531D4—1461no—
532D4—1461no—
533D4—1461no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—2921no—
545C2×C4—2921no—
546C2×C4—2921no—
547C2×C4—2921no—
548C2×C4—5841no—
549C2×C4—5841no—
550C2×C4—5841no—
551C2×C4—1461no—
552C2×C4—1461no—
553C2×C4C4×C22⋊D7311yescharacteristic
554C2×C4C4×C22⋊D7311yescharacteristic
555C2×C4C4×C22⋊D7311yescharacteristic
556C2×C4C22×C4×D7311yescharacteristic
557C2×C4C22×C22⋊D7311yescharacteristic
558C2×C4C2×#1192402811yescharacteristic
559C8—21no—
560C8—21no—
561C8—21no—
562C8—21no—
563C8—41no—
564C8—41no—
565C8—2921no—
566C8—2921no—
567C8—1462no—
568C8—1462no—
569C8—1461no—
570C8—1461no—
571C8—1461no—
572C8—1461no—
573C22—21no—
574C22—21no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—2921no—
579C22—2921no—
580C22—2921no—
581C22—2921no—
582C22—2921no—
583C22—5844no—
584C22—1462no—
585C22#1798870212yes—
586C22C2×C4×C22⋊D7311yescharacteristic
587C4—21no—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—41no—
592C4—41no—
593C4—41no—
594C4—2921no—
595C4—2921no—
596C4#131×D7311yescharacteristic
597C4#1798935211yescharacteristic
598C4#1798939011yescharacteristic
599C4#1798939511yescharacteristic
600C2—22no—
601C2—81no—
602C2—2922no—
603C2—5841no—
604C2#2261865311yescharacteristic
605C2#2261875911yescharacteristic
606C2C2×#1798870211yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC73×#2994C2
socleC22×C73C2×C4×D4
prime radical#17988720C4
cosocleC42×C73C24
hypercenter#2994D73
hypocenterC73#18014
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2994

Complements

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

NH ≅ G/Ntype
2. #22618965C2semidirect
3. #22608821C2semidirect
4. #22608814C2semidirect
5. #22608809C2semidirect
6. #22608795C2semidirect
7. #22608793C2semidirect
8. #22608786C2semidirect
9. C8×#11924015C2semidirect
10. C8×#11924013C2semidirect
11. #22608714C2semidirect
12. #22608391C2semidirect
13. C2×#17988314C2semidirect
14. #22608215C2semidirect
15. C73×#2994C2semidirect
40. #17988339C22semidirect
41. #17988339C22semidirect
42. #17988337C22semidirect
43. #17988337C22semidirect
44. Q8⋊#10596831C22semidirect
45. Q8⋊#10596831C22semidirect
46. D4⋊#10596831C22semidirect
47. D4⋊#10596831C22semidirect
52. #17988309C22semidirect
53. #17988309C22semidirect
54. C8×#10596831C22semidirect
55. C8×#10596831C22semidirect
57. #446×C73C22semidirect
58. C8×Q8×C73C22semidirect
59. #436×C73C22semidirect
62. C8×D4×C73C22semidirect
305. #2994D73semidirect
319. C73#18014semidirect
330. #446C2×D73semidirect
331. C8×Q8C2×D73semidirect
338. #436C2×D73semidirect
345. C8×D4C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226199742
#226189652
#226088212
#226088142
#226088092
#226087952
#226087932
#226087862
C8×#119240152
C8×#119240132
#226087142
#226083912
C2×#179883142
#226082152
C73×#29942
#1801473