Finite Group Databasebeta

← #33159235

Subgroups — #33159235

Order 18688.

740 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22614539C211yesmaximal, characteristic, complemented
2#22611452C211yesmaximal, characteristic
3#22611248C211yesmaximal, characteristic
4#22611167C211yesmaximal, characteristic
5#22611025C211yesmaximal, characteristic
6#22610791C211yesmaximal, characteristic, complemented
7#22610565C211yesmaximal, characteristic, complemented
8#22610562C211yesmaximal, characteristic, complemented
9#22610523C211yesmaximal, characteristic, complemented
10#22610490C211yesmaximal, characteristic, complemented
11#22610272C211yesmaximal, characteristic, complemented
12#22610265C211yesmaximal, characteristic
13#22609946C211yesmaximal, characteristic, complemented
14C4×#14556512C211yesmaximal, characteristic
15C73×#2367C211yesmaximal, characteristic
16C22×#11924018C2211yescharacteristic
17C22×#11924016C2211yescharacteristic
18C2×#14556524C2211yescharacteristic
19C2×#14556522C2211yescharacteristic
20C2×C4⋊#10596831C2211yescharacteristic
21C2×#14556515C2211yescharacteristic
22C2×#14556515C2211yescharacteristic
23C2×#14556515C2211yescharacteristic
24C2×#14556514C2211yescharacteristic
25C2×#14556514C2211yescharacteristic
26C2×#14556512C2211yescharacteristic
27C2×#14556512C2211yescharacteristic
28C2×#14556512C2211yescharacteristic
29C2×C4×#10596831C2211yescharacteristic
30C73⋊#401—21no—
31#17988525—21no—
32#17988493C2211yescharacteristic
33#17988493C2211yescharacteristic
34#17988493C2211yescharacteristic
35#17988485C2211yescharacteristic
36#17988468—21no—
37#17988454—21no—
38#17988453C2211yescharacteristic
39C4×C4⋊Dic73C2211yescharacteristic
40#17988349C2211yescharacteristic
41#17988349C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988342C2211yescharacteristic
45#17988342C2211yescharacteristic
46#17988342C2211yescharacteristic
47#17988342C2211yescharacteristic
48C2×#124×C73C2211yescharacteristic
49C2×C4×Q8×C73C2211yescharacteristic
50C2×C4×D4×C73C2211yescharacteristic
51#391×C73C2211yescharacteristic
52#390×C73C2211yescharacteristic
53#388×C73—21no—
54#388×C73—21no—
55#385×C73—21no—
56C4×C4⋊C4×C73C2211yescharacteristic
57C4×C4⋊C4×C73C2211yescharacteristic
58C4×C22⋊C4×C73—21no—
59C22×Dic146C2311yescharacteristic
60C2×#11924018—22no—
61C2×#11924018—22no—
62C2×#11924018—41no—
63C2×#11924016—22no—
64C2×#11924016—41no—
65C2×#11924016—41no—
66C22×#10596831C2311yescharacteristic
67C22×#10596831C2311yescharacteristic
68C22×#10596831C2311yescharacteristic
69C22×#10596831C2311yescharacteristic
70C2×C4⋊Dic73C2311yescharacteristic
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×#11924007—21no—
74C2×C4×Dic73—21no—
75#14556524—22no—
76#14556522—41no—
77C4⋊#10596831—41no—
78#14556515—22no—
79#14556515—41no—
80#14556515—41no—
81#14556514—22no—
82#14556514—41no—
83#14556512—22no—
84#14556512—22no—
85#14556512—41no—
86C4×#10596831—41no—
87#14556498—21no—
88#14556498—21no—
89#14556498—21no—
90#14556498—21no—
91#14556498—21no—
92#14556498—21no—
93#14556498—21no—
94#14556498—21no—
95#14556498—21no—
96#14556498—21no—
97C22×Q8×C73C2311yescharacteristic
98C22×D4×C73C2311yescharacteristic
99C23×C4×C73—21no—
100#124×C73—22no—
101#124×C73—22no—
102C4×Q8×C73—22no—
103C4×Q8×C73—41no—
104C4×D4×C73—22no—
105C4×D4×C73—41no—
106C2×C4⋊C4×C73—21no—
107C2×C4⋊C4×C73D411yescharacteristic
108C2×C4⋊C4×C73D411yescharacteristic
109C2×C4⋊C4×C73C2311yescharacteristic
110C2×C4⋊C4×C73C2311yescharacteristic
111C2×C22⋊C4×C73—21no—
112C2×C22⋊C4×C73—21no—
113C2×C22⋊C4×C73—21no—
114C2×C22⋊C4×C73—21no—
115C2×C22⋊C4×C73—21no—
116C2×C42×C73—21no—
117C2×C42×C73—21no—
118C2×C42×C73C2311yescharacteristic
119C2×C42×C73C2311yescharacteristic
120C2×C42×C73C2311yescharacteristic
121#95×C73—21no—
122#95×C73—21no—
123#95×C73—21no—
124#95×C73D411yescharacteristic
125#95×C73D411yescharacteristic
126C2×Dic146—21no—
127C2×Dic146—21no—
128C2×Dic146—21no—
129C2×Dic146—21no—
130C2×Dic146—21no—
131C2×Dic146—21no—
132C22×Dic73—21no—
133C22×Dic73—21no—
134C22×Dic73—21no—
135C22×Dic73—21no—
136#11924018—44no—
137#11924016—44no—
138C2×#10596831—21no—
139C2×#10596831—21no—
140C2×#10596831—21no—
141C2×#10596831—21no—
142C2×#10596831—21no—
143C2×#10596831—21no—
144C2×#10596831—21no—
145C2×#10596831—21no—
146C2×#10596831—21no—
147C2×#10596831—21no—
148C2×#10596831—21no—
149C2×#10596831#4112yes—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156#11924007—42no—
157C4×Dic73—42no—
158C24×C73—21no—
159C2×Q8×C73—21no—
160C2×Q8×C73—21no—
161C2×Q8×C73—21no—
162C2×Q8×C73—21no—
163C2×Q8×C73—21no—
164C2×Q8×C73#4112yes—
165C2×D4×C73—21no—
166C2×D4×C73—21no—
167C2×D4×C73—21no—
168C2×D4×C73—21no—
169C2×D4×C73—21no—
170C2×D4×C73#4112yes—
171C22×C4×C73—21no—
172C22×C4×C73—21no—
173C22×C4×C73—21no—
174C22×C4×C73—42no—
175C22×C4×C73—42no—
176C22×C4×C73—81no—
177C22×C4×C73C2×D411yescharacteristic
178C22×C4×C73C2×D411yescharacteristic
179C22×C4×C73C2×D411yescharacteristic
180C22×C4×C73C2×D411yescharacteristic
181C22×C4×C73C2×D411yescharacteristic
182C22×C4×C73C2×D411yescharacteristic
183C22×C4×C73C2411yescharacteristic
184C4⋊C4×C73—22no—
185C4⋊C4×C73—21no—
186C4⋊C4×C73—21no—
187C4⋊C4×C73—21no—
188C4⋊C4×C73—41no—
189C4⋊C4×C73—42no—
190C4⋊C4×C73#4112yes—
191C22⋊C4×C73—24no—
192C22⋊C4×C73—24no—
193C22⋊C4×C73—42no—
194C22⋊C4×C73—42no—
195C22⋊C4×C73—81no—
196C42×C73—21no—
197C42×C73—21no—
198C42×C73—21no—
199C42×C73—21no—
200C42×C73—21no—
201C42×C73—21no—
202C42×C73—42no—
203C42×C73—81no—
204C23×C73—22no—
205C23×C73—41no—
206C23×C73—41no—
207C23×C73—41no—
208C23×C73—41no—
209C23×C73—41no—
210C23×C73—41no—
211C23×C73C22×D411yescharacteristic
212Q8×C73—22no—
213Q8×C73—22no—
214Q8×C73—41no—
215Q8×C73—41no—
216D4×C73—22no—
217D4×C73—22no—
218D4×C73—41no—
219D4×C73—41no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—22no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—22no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—41no—
240C2×C4×C73—41no—
241C2×C4×C73—41no—
242C2×C4×C73—41no—
243C2×C4×C73—41no—
244C2×C4×C73—41no—
245C2×C4×C73—41no—
246C2×C4×C73—84no—
247C2×C4×C73#13512yes—
248C2×C4×C73C2×#4111yescharacteristic
249C2×C4×C73C2×#4111yescharacteristic
250C2×C4×C73C2×#4111yescharacteristic
251C2×C4×C73C2×#4111yescharacteristic
252C2×C4×C73H(5,𝔽2)11yescharacteristic
253C2×C4×C73H(5,𝔽2)11yescharacteristic
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266Dic146—41no—
267Dic146—41no—
268Dic146—41no—
269Dic146—41no—
270#10596831—22no—
271#10596831—41no—
272#10596831—41no—
273#10596831—41no—
274C22×C73—42no—
275C22×C73—42no—
276C22×C73—42no—
277C22×C73—81no—
278C22×C73—81no—
279C22×C73—81no—
280C22×C73—81no—
281C22×C73#52511yescharacteristic
282C22×C73#54611yescharacteristic
283C22×C73#55011yescharacteristic
284C22×C73C2×#13511yescharacteristic
285C22×C73#57511yescharacteristic
286C22×C73#57711yescharacteristic
287C22×C73#57811yescharacteristic
288C4×C73—22no—
289C4×C73—41no—
290C4×C73—41no—
291C4×C73—42no—
292C4×C73—41no—
293C4×C73—42no—
294C4×C73—41no—
295C4×C73—41no—
296C4×C73—81no—
297C4×C73#52911yescharacteristic
298C4×C73#53811yescharacteristic
299C4×C73#55111yescharacteristic
300C4×C73#55311yescharacteristic
301Dic73—81no—
302Dic73—81no—
303Dic73—81no—
304Dic73—81no—
305#27070—731nomaximal, Sylow
306C2×C73—82no—
307C2×C73#246711yescharacteristic
308C2×C73#310211yescharacteristic
309C2×C73#312011yescharacteristic
310C2×C73#329611yescharacteristic
311C2×C73#331311yescharacteristic
312C2×C73#339311yescharacteristic
313C2×C73#340111yescharacteristic
314#2514—731no—
315#2367D7311yescharacteristic
316#2092—731no—
317#2055—731no—
318#2042—731no—
319#2015—731no—
320#1969—731no—
321#1927—731no—
322#1927—731no—
323#1921—731no—
324#1916—731no—
325#1878—731no—
326#1877—731no—
327#1805—731no—
328C4×#102—731no—
329C73#2707011yescharacteristic, complemented, Sylow
330C22×Q16—731no—
331C22×SD16—731no—
332C2×#124C2×D7311yescharacteristic
333C2×C4×Q8C2×D7311yescharacteristic
334C2×C4×D4C2×D7311yescharacteristic
335C2×#107—731no—
336C2×#106—731no—
337C2×C4⋊C8—731no—
338C2×#103—731no—
339C2×#103—731no—
340C2×#103—731no—
341C2×#103—731no—
342C2×#103—731no—
343C2×#102—731no—
344C2×#102—731no—
345C2×#102—731no—
346C2×C4×C8—731no—
347#401—1461no—
348#400—1461no—
349#391—731no—
350#391—731no—
351#391—731no—
352#391C2×D7311yescharacteristic
353#390C2×D7311yescharacteristic
354#389—731no—
355#388—21no—
356#388—21no—
357#387—1461no—
358#385—21no—
359#383—1461no—
360#382—731no—
361C4×C4⋊C4—731no—
362C4×C4⋊C4C2×D7311yescharacteristic
363C4×C4⋊C4C2×D7311yescharacteristic
364C4×C22⋊C4—21no—
365#340—731no—
366#340—731no—
367#340—731no—
368#340—731no—
369#336—731no—
370#336—731no—
371#336—731no—
372#336—731no—
373C22×Q8—731no—
374C22×Q8C22×D7311yescharacteristic
375C22×D4C22×D7311yescharacteristic
376C23×C4—21no—
377C2×Q16—2921no—
378C2×Q16—1462no—
379C2×Q16—1462no—
380C2×SD16—2921no—
381C2×SD16—2921no—
382C2×SD16—1462no—
383C22×C8—731no—
384C22×C8—731no—
385C22×C8—731no—
386C22×C8—731no—
387#124—22no—
388#124—22no—
389C4×Q8—22no—
390C4×Q8—41no—
391C4×D4—22no—
392C4×D4—41no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—731no—
395C2×C4⋊C4—731no—
396C2×C4⋊C4—731no—
397C2×C4⋊C4—1461no—
398C2×C4⋊C4C22⋊D7311yescharacteristic
399C2×C4⋊C4C22⋊D7311yescharacteristic
400C2×C4⋊C4C22×D7311yescharacteristic
401C2×C4⋊C4C22×D7311yescharacteristic
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C42—21no—
408C2×C42—21no—
409C2×C42—1461no—
410C2×C42C22×D7311yescharacteristic
411C2×C42C22×D7311yescharacteristic
412C2×C42C22×D7311yescharacteristic
413#107—1462no—
414#106—2921no—
415C4⋊C8—2921no—
416#103—2921no—
417#103—2921no—
418#103—2921no—
419#103—1462no—
420#103—1462no—
421#102—2921no—
422#102—1462no—
423#102—1462no—
424C4×C8—2921no—
425#95—21no—
426#95—21no—
427#95—21no—
428#95—1461no—
429#95—1461no—
430#95—1461no—
431#95—1461no—
432#95—1461no—
433#95—1461no—
434#95—1461no—
435#95—1461no—
436#95—1461no—
437#95—1461no—
438#95C22⋊D7311yescharacteristic
439#95C22⋊D7311yescharacteristic
440C24—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—1461no—
447C2×Q8—1461no—
448C2×Q8—1461no—
449C2×Q8—1461no—
450C2×Q8—1461no—
451C2×Q8—1461no—
452C2×Q8#1192402812yes—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4#1192402812yes—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—81no—
465C22×C4—1461no—
466C22×C4—1461no—
467C22×C4—1461no—
468C22×C4—1461no—
469C22×C4C2×C22⋊D7311yescharacteristic
470C22×C4C2×C22⋊D7311yescharacteristic
471C22×C4C2×C22⋊D7311yescharacteristic
472C22×C4C2×C22⋊D7311yescharacteristic
473C22×C4C2×C22⋊D7311yescharacteristic
474C22×C4C2×C22⋊D7311yescharacteristic
475C22×C4C23×D7311yescharacteristic
476Q16—2924no—
477SD16—2924no—
478C2×C8—732no—
479C2×C8—1461no—
480C2×C8—1461no—
481C2×C8—1461no—
482C2×C8—1461no—
483C2×C8—1461no—
484C2×C8—1461no—
485C2×C8—1461no—
486C2×C8—1461no—
487C2×C8—1461no—
488C2×C8—1461no—
489C2×C8—1461no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—41no—
495C4⋊C4—42no—
496C4⋊C4—2922no—
497C4⋊C4—1461no—
498C4⋊C4—1461no—
499C4⋊C4—1461no—
500C4⋊C4—1461no—
501C4⋊C4—1461no—
502C4⋊C4—1461no—
503C4⋊C4#1192402712yes—
504C22⋊C4—24no—
505C22⋊C4—24no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—81no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—42no—
516C42—81no—
517C42—2922no—
518C23—22no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23C22×C22⋊D7311yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—41no—
529Q8—41no—
530Q8—2921no—
531Q8—2921no—
532Q8—2921no—
533Q8—2921no—
534D4—22no—
535D4—22no—
536D4—41no—
537D4—41no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—84no—
565C2×C4—2921no—
566C2×C4—2921no—
567C2×C4—2921no—
568C2×C4—2921no—
569C2×C4—2921no—
570C2×C4—2921no—
571C2×C4—2921no—
572C2×C4—2921no—
573C2×C4—2921no—
574C2×C4—2921no—
575C2×C4—2921no—
576C2×C4—2921no—
577C2×C4#1455661112yes—
578C2×C4#41×D7311yescharacteristic
579C2×C4C2×#1192402711yescharacteristic
580C2×C4C2×#1192402811yescharacteristic
581C2×C4C2×#1192402811yescharacteristic
582C2×C4#1455663611yescharacteristic
583C2×C4#1455663711yescharacteristic
584C8—2921no—
585C8—2921no—
586C8—2921no—
587C8—1462no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22#1798909811yescharacteristic
596C22#1798923111yescharacteristic
597C22#1798925611yescharacteristic
598C22C2×#1455661111yescharacteristic
599C22#1798938511yescharacteristic
600C22#1798939811yescharacteristic
601C22#1798940211yescharacteristic
602C4—22no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—41no—
607C4—42no—
608C4—42no—
609C4—41no—
610C4—81no—
611C4—5841no—
612C4—5841no—
613C4—5841no—
614C4—5841no—
615C4#1798911711yescharacteristic
616C4#1798917011yescharacteristic
617C4#1798926611yescharacteristic
618C4#1798928311yescharacteristic
619C2—82no—
620C2#2261397611yescharacteristic
621C2#2261994911yescharacteristic
622C2#2262014511yescharacteristic
623C2#2262176611yescharacteristic
624C2#2262199711yescharacteristic
625C2#2262291511yescharacteristic
626C2#2262299111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2367C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
hypercenter#2367D73
hypocenterC73#27070
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2367

Complements

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

NH ≅ G/Ntype
1. #22614539C2semidirect
6. #22610791C2semidirect
7. #22610565C2semidirect
8. #22610562C2semidirect
9. #22610523C2semidirect
10. #22610490C2semidirect
11. #22610272C2semidirect
13. #22609946C2semidirect
329. C73#27070semidirect

Maximal subgroups

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

subgroupindex
#226145392
#226114522
#226112482
#226111672
#226110252
#226107912
#226105652
#226105622
#226105232
#226104902
#226102722
#226102652
#226099462
C4×#145565122
C73×#23672
#2707073