Finite Group Databasebeta

← #33158667

Subgroups — #33158667

Order 18688.

792 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22612945C211yesmaximal, characteristic
2#22611335C211yesmaximal, characteristic, complemented
3#22611302C211yesmaximal, characteristic, complemented
4#22611120C211yesmaximal, characteristic
5#22611120C211yesmaximal, characteristic
6#22610692C211yesmaximal, characteristic, complemented
7#22610656C211yesmaximal, characteristic, complemented
8#22610539C211yesmaximal, characteristic, complemented
9#22610513C211yesmaximal, characteristic, complemented
10#22610509C211yesmaximal, characteristic, complemented
11#22610191C211yesmaximal, characteristic, complemented
12#22610123C211yesmaximal, characteristic, complemented
13#22610015C211yesmaximal, characteristic, complemented
14C4×#14556511C211yesmaximal, characteristic, complemented
15C73×#2497C211yesmaximal, characteristic, complemented
16C2×#14556565C2211yescharacteristic
17C2×C4×D292C2211yescharacteristic
18C2×#14556551—21no—
19C22×C4⋊Dic73C2211yescharacteristic
20C2×#14556522C2211yescharacteristic
21C2×#14556522C2211yescharacteristic
22C2×#14556522C2211yescharacteristic
23C2×#14556522C2211yescharacteristic
24C2×C4⋊#10596831C2211yescharacteristic
25C2×#14556511C2211yescharacteristic
26C2×#14556511C2211yescharacteristic
27C2×#14556511C2211yescharacteristic
28C2×#14556511C2211yescharacteristic
29C2×C22⋊#10596831C2211yescharacteristic
30C2×C22⋊#10596831C2211yescharacteristic
31C2×C4×#10596831C2211yescharacteristic
32#17988489C2211yescharacteristic
33#17988485C2211yescharacteristic
34#17988479—21no—
35#17988464—21no—
36#17988461—21no—
37C4×C4⋊Dic73C2211yescharacteristic
38#17988349C2211yescharacteristic
39#17988349C2211yescharacteristic
40#17988349C2211yescharacteristic
41#17988349C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988342C2211yescharacteristic
47#17988342C2211yescharacteristic
48#400×C73—21no—
49#396×C73—21no—
50#389×C73C2211yescharacteristic
51#389×C73C2211yescharacteristic
52#388×C73—21no—
53#387×C73—21no—
54#384×C73C2211yescharacteristic
55#380×C73C2211yescharacteristic
56#380×C73C2211yescharacteristic
57C4×C4⋊C4×C73C2211yescharacteristic
58C4×C22⋊C4×C73C2211yescharacteristic
59C22×D292C2311yescharacteristic
60C22×C22⋊D73—21no—
61C23×Dic73—21no—
62C22×C4×D73—21no—
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67#14556565—22no—
68#14556565—22no—
69C4×D292—22no—
70C4×D292—41no—
71#14556551—44no—
72C2×C4⋊Dic73—22no—
73C2×C4⋊Dic73—21no—
74C2×C4⋊Dic73—41no—
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×#11924007—21no—
79C2×#11924005—21no—
80C2×#11924005—21no—
81C2×#11924005—21no—
82C2×#11924005—21no—
83C2×C4×Dic73—21no—
84#14556522—22no—
85#14556522—22no—
86#14556522—22no—
87#14556522—41no—
88C4⋊#10596831—22no—
89#14556511—22no—
90#14556511—22no—
91#14556511—22no—
92#14556511—41no—
93C22⋊#10596831—22no—
94C22⋊#10596831—41no—
95C4×#10596831—41no—
96#14556498—21no—
97#14556498—21no—
98C23×C4×C73C2311yescharacteristic
99C2×C4⋊C4×C73—21no—
100C2×C4⋊C4×C73—21no—
101C2×C4⋊C4×C73—21no—
102C2×C4⋊C4×C73—21no—
103C2×C4⋊C4×C73C2311yescharacteristic
104C2×C4⋊C4×C73C2311yescharacteristic
105C2×C4⋊C4×C73C2311yescharacteristic
106C2×C4⋊C4×C73C2311yescharacteristic
107C2×C22⋊C4×C73—21no—
108C2×C22⋊C4×C73—21no—
109C2×C22⋊C4×C73D411yescharacteristic
110C2×C22⋊C4×C73D411yescharacteristic
111C2×C42×C73—21no—
112C2×C42×C73C2311yescharacteristic
113C2×C42×C73C2311yescharacteristic
114#95×C73—21no—
115#95×C73—21no—
116#95×C73—21no—
117#95×C73—21no—
118#95×C73—21no—
119#95×C73D411yescharacteristic
120#95×C73D411yescharacteristic
121C23×D73—21no—
122C2×D292—21no—
123C2×D292—21no—
124C2×D292—21no—
125C2×D292—21no—
126C2×D292—21no—
127C2×D292#4112yes—
128C2×C22⋊D73—24no—
129C2×C22⋊D73—42no—
130C2×C22⋊D73—42no—
131C22×Dic73—21no—
132C22×Dic73—21no—
133C22×Dic73—21no—
134C22×Dic73—42no—
135C22×Dic73—42no—
136C22×Dic73—81no—
137C2×C4×D73—42no—
138C2×C4×D73—42no—
139C2×C4×D73—81no—
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—21no—
150C2×#10596831—21no—
151C2×#10596831#4112yes—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73—42no—
157C4⋊Dic73#4112yes—
158C4⋊Dic73#4112yes—
159#11924007—42no—
160#11924005—24no—
161#11924005—42no—
162#11924005—42no—
163#11924005—81no—
164C4×Dic73—42no—
165C24×C73C2×D411yescharacteristic
166C22×C4×C73—21no—
167C22×C4×C73—21no—
168C22×C4×C73—21no—
169C22×C4×C73—21no—
170C22×C4×C73—21no—
171C22×C4×C73—21no—
172C22×C4×C73—21no—
173C22×C4×C73—21no—
174C22×C4×C73—21no—
175C22×C4×C73—21no—
176C22×C4×C73C2×D411yescharacteristic
177C22×C4×C73C2×D411yescharacteristic
178C22×C4×C73C2×D411yescharacteristic
179C22×C4×C73C2×D411yescharacteristic
180C22×C4×C73C2×D411yescharacteristic
181C22×C4×C73C2411yescharacteristic
182C4⋊C4×C73—21no—
183C4⋊C4×C73—21no—
184C4⋊C4×C73—21no—
185C4⋊C4×C73—21no—
186C4⋊C4×C73—21no—
187C4⋊C4×C73—21no—
188C4⋊C4×C73—21no—
189C4⋊C4×C73—21no—
190C4⋊C4×C73—42no—
191C4⋊C4×C73—81no—
192C4⋊C4×C73—81no—
193C4⋊C4×C73—81no—
194C22⋊C4×C73—41no—
195C22⋊C4×C73—81no—
196C22⋊C4×C73—81no—
197C22⋊C4×C73SD1614yes—
198C42×C73—21no—
199C42×C73—21no—
200C42×C73—21no—
201C42×C73—21no—
202C42×C73—81no—
203C23×C73—21no—
204C23×C73—21no—
205C23×C73—21no—
206C23×C73—21no—
207C23×C73—21no—
208C23×C73—21no—
209C23×C73C2×SD1612yes—
210C23×C73C22×D411yescharacteristic
211C22×D73—22no—
212C22×D73—41no—
213C22×D73—41no—
214C22×D73—41no—
215C22×D73—41no—
216C22×D73—41no—
217C22×D73—41no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—41no—
232C2×C4×C73—41no—
233C2×C4×C73—41no—
234C2×C4×C73—41no—
235C2×C4×C73—41no—
236C2×C4×C73—41no—
237C2×C4×C73—41no—
238C2×C4×C73—41no—
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—41no—
247C2×C4×C73C2×SD1612yes—
248C2×C4×C73C2×SD1612yes—
249C2×C4×C73C2×#4111yescharacteristic
250C2×C4×C73C2×#4111yescharacteristic
251C2×C4×C73C2×#4111yescharacteristic
252C2×C4×C73C2×#4111yescharacteristic
253C2×C4×C73H(5,𝔽2)11yescharacteristic
254C2×C4×C732−1+411yescharacteristic
255C22⋊D73—48no—
256C2×Dic73—22no—
257C2×Dic73—22no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265C2×Dic73—84no—
266D292—22no—
267D292—22no—
268D292—41no—
269D292—41no—
270C4×D73—84no—
271#10596831—22no—
272#10596831—41no—
273#10596831—41no—
274#10596831—41no—
275C22×C73—22no—
276C22×C73—22no—
277C22×C73—22no—
278C22×C73—41no—
279C22×C73—41no—
280C22×C73—41no—
281C22×C73—41no—
282C22×C73#52511yescharacteristic
283C22×C73#54611yescharacteristic
284C22×C73#54711yescharacteristic
285C22×C73C22×SD1611yescharacteristic
286C22×C73#57511yescharacteristic
287C22×C73#57611yescharacteristic
288C22×C73#57711yescharacteristic
289C2×D73—42no—
290C2×D73—42no—
291C2×D73—42no—
292C2×D73—81no—
293C2×D73—81no—
294C2×D73—81no—
295C2×D73—81no—
296C4×C73—22no—
297C4×C73—22no—
298C4×C73—41no—
299C4×C73—41no—
300C4×C73—41no—
301C4×C73—81no—
302C4×C73—81no—
303C4×C73—81no—
304C4×C73—81no—
305C4×C73#52911yescharacteristic
306C4×C73#53911yescharacteristic
307C4×C73#55311yescharacteristic
308C4×C73#55611yescharacteristic
309Dic73—42no—
310Dic73—42no—
311Dic73—81no—
312#27032—731nomaximal, Sylow
313C2×C73—42no—
314C2×C73#246611yescharacteristic
315C2×C73#310111yescharacteristic
316C2×C73#311911yescharacteristic
317C2×C73#329511yescharacteristic
318C2×C73#332111yescharacteristic
319C2×C73#333511yescharacteristic
320C2×C73#335511yescharacteristic
321D73—82no—
322#2497D7311yescharacteristic, complemented
323#2361—731no—
324#2071—731no—
325#2065—731no—
326#2034—731no—
327#2034—731no—
328#1948—731no—
329#1941—731no—
330#1923—731no—
331#1920—731no—
332#1919—731no—
333#1864—731no—
334#1853—731no—
335#1823—731no—
336C4×#102—731no—
337C73#2703211yescharacteristic, complemented, Sylow
338C2×#121—731no—
339C2×C4×D4—731no—
340C2×C4×D4—1461no—
341C22×C4⋊C4—731no—
342C2×#106—731no—
343C2×#106—731no—
344C2×#106—731no—
345C2×#106—731no—
346C2×C4⋊C8—731no—
347C2×#102—731no—
348C2×#102—731no—
349C2×#102—731no—
350C2×#102—731no—
351C2×C22⋊C8—731no—
352C2×C22⋊C8—731no—
353C2×C4×C8—731no—
354#400—21no—
355#396—21no—
356#390—731no—
357#389—731no—
358#389C2×D7311yescharacteristic
359#389C2×D7311yescharacteristic
360#388—21no—
361#388—1461no—
362#387—21no—
363#386—1461no—
364#385—1461no—
365#384C2×D7311yescharacteristic
366#380C2×D7311yescharacteristic
367#380C2×D7311yescharacteristic
368C4×C4⋊C4—731no—
369C4×C4⋊C4C2×D7311yescharacteristic
370C4×C22⋊C4C2×D7311yescharacteristic
371#340—731no—
372#340—731no—
373#340—731no—
374#340—731no—
375#340—731no—
376#340—731no—
377#340—731no—
378#340—731no—
379#336—731no—
380#336—731no—
381C22×D4—731no—
382C22×D4—1461no—
383C23×C4—1461no—
384C23×C4—1461no—
385C23×C4C22×D7311yescharacteristic
386C22×C8—731no—
387C22×C8—731no—
388C22×C8—731no—
389C22×C8—731no—
390#121—1462no—
391#121—1462no—
392C4×D4—2921no—
393C4×D4—2924no—
394C4×D4—1462no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—2921no—
400C2×C4⋊C4—731no—
401C2×C4⋊C4—731no—
402C2×C4⋊C4—731no—
403C2×C4⋊C4—1461no—
404C2×C4⋊C4—1461no—
405C2×C4⋊C4—1462no—
406C2×C4⋊C4C22×D7311yescharacteristic
407C2×C4⋊C4C22×D7311yescharacteristic
408C2×C4⋊C4C22×D7311yescharacteristic
409C2×C4⋊C4C22×D7311yescharacteristic
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—1461no—
413C2×C22⋊C4—1461no—
414C2×C22⋊C4—1461no—
415C2×C22⋊C4—1461no—
416C2×C22⋊C4C22⋊D7311yescharacteristic
417C2×C22⋊C4C22⋊D7311yescharacteristic
418C2×C42—21no—
419C2×C42—1461no—
420C2×C42C22×D7311yescharacteristic
421C2×C42C22×D7311yescharacteristic
422#106—2921no—
423#106—1462no—
424#106—1462no—
425#106—1462no—
426C4⋊C8—1462no—
427#102—2921no—
428#102—1462no—
429#102—1462no—
430#102—1462no—
431C22⋊C8—2921no—
432C22⋊C8—1462no—
433C4×C8—2921no—
434#95—21no—
435#95—21no—
436#95—21no—
437#95—21no—
438#95—21no—
439#95—1461no—
440#95—1461no—
441#95C22⋊D7311yescharacteristic
442#95C22⋊D7311yescharacteristic
443C24—1461no—
444C24C2×C22⋊D7311yescharacteristic
445C2×D4—2922no—
446C2×D4—2922no—
447C2×D4—732no—
448C2×D4—1464no—
449C2×D4—1461no—
450C2×D4—1461no—
451C2×D4—1461no—
452C2×D4—1461no—
453C2×D4—1461no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—2922no—
465C22×C4—2922no—
466C22×C4—2922no—
467C22×C4—2922no—
468C22×C4—5841no—
469C22×C4—5841no—
470C22×C4—1461no—
471C22×C4—1461no—
472C22×C4—1461no—
473C22×C4C2×C22⋊D7311yescharacteristic
474C22×C4C2×C22⋊D7311yescharacteristic
475C22×C4C2×C22⋊D7311yescharacteristic
476C22×C4C2×C22⋊D7311yescharacteristic
477C22×C4C2×C22⋊D7311yescharacteristic
478C22×C4C23×D7311yescharacteristic
479C2×C8—732no—
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—
490C2×C8—1461no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—42no—
500C4⋊C4—81no—
501C4⋊C4—81no—
502C4⋊C4—81no—
503C4⋊C4—2922no—
504C4⋊C4—2922no—
505C4⋊C4—732no—
506C4⋊C4—732no—
507C4⋊C4—1461no—
508C4⋊C4—1461no—
509C4⋊C4—1461no—
510C4⋊C4—1461no—
511C22⋊C4—41no—
512C22⋊C4—81no—
513C22⋊C4—81no—
514C22⋊C4—2922no—
515C22⋊C4—2922no—
516C22⋊C4—5841no—
517C22⋊C4—1464no—
518C22⋊C4#1192401514yes—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—81no—
524C42—2922no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—2921no—
532C23—2921no—
533C23—2921no—
534C23—2921no—
535C23—2921no—
536C23—2921no—
537C23—1462no—
538C23C2×#1192401512yes—
539C23C22×C22⋊D7311yescharacteristic
540D4—2928no—
541D4—2921no—
542D4—2921no—
543D4—1462no—
544D4—1462no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—2921no—
575C2×C4—2921no—
576C2×C4—2921no—
577C2×C4—2921no—
578C2×C4—2921no—
579C2×C4—2921no—
580C2×C4—2921no—
581C2×C4—5844no—
582C2×C4—5844no—
583C2×C4—1462no—
584C2×C4—1462no—
585C2×C4C2×#1192401512yes—
586C2×C4C2×#1192401512yes—
587C2×C4#41×D7311yescharacteristic
588C2×C4#41×D7311yescharacteristic
589C2×C4#41×D7311yescharacteristic
590C2×C4#41×D7311yescharacteristic
591C2×C4#1455663711yescharacteristic
592C2×C4#1455663811yescharacteristic
593C8—2921no—
594C8—2921no—
595C8—2921no—
596C8—1462no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—2922no—
605C22—2922no—
606C22—2922no—
607C22—5841no—
608C22—5841no—
609C22—5841no—
610C22—5841no—
611C22#1798909411yescharacteristic
612C22#1798922511yescharacteristic
613C22#1798923611yescharacteristic
614C22C22×#1192401511yescharacteristic
615C22#1798938511yescharacteristic
616C22#1798939011yescharacteristic
617C22#1798939811yescharacteristic
618C4—22no—
619C4—22no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—81no—
624C4—81no—
625C4—81no—
626C4—81no—
627C4—2922no—
628C4—2922no—
629C4—5841no—
630C4#1798912011yescharacteristic
631C4#1798917811yescharacteristic
632C4#1798928811yescharacteristic
633C4#1798930411yescharacteristic
634C2—42no—
635C2—5842no—
636C2#2261396311yescharacteristic
637C2#2261993511yescharacteristic
638C2#2262012411yescharacteristic
639C2#2262174911yescharacteristic
640C2#2262211111yescharacteristic
641C2#2262228211yescharacteristic
642C2#2262253111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2497C2
socleC23×C73C22×D4
prime radicalC2×#14556565C22
hypercenter#2497D73
hypocenterC73#27032
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2497

Complements

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

NH ≅ G/Ntype
2. #22611335C2semidirect
3. #22611302C2semidirect
6. #22610692C2semidirect
7. #22610656C2semidirect
8. #22610539C2semidirect
9. #22610513C2semidirect
10. #22610509C2semidirect
11. #22610191C2semidirect
12. #22610123C2semidirect
13. #22610015C2semidirect
14. C4×#14556511C2semidirect
15. C73×#2497C2semidirect
322. #2497D73semidirect
337. C73#27032semidirect

Maximal subgroups

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

subgroupindex
#226129452
#226113352
#226113022
#226111202
#226106922
#226106562
#226105392
#226105132
#226105092
#226101912
#226101232
#226100152
C4×#145565112
C73×#24972
#2703273